An Introduction to Riemann Surfaces

Cornerstones

Series Editors Charles L. Epstein, University of Pennsylvania, Philadelphia, PA, USA Steven G. Krantz, Washington University, St. Louis, MO, USA

Advisory Board Anthony W. Knapp, Emeritus, State University of New York at Stony Brook, Stony Brook, NY, USA

For further volumes: www.springer.com/series/7161 Terrence Napier Mohan Ramachandran

An Introduction to Riemann Surfaces Terrence Napier Mohan Ramachandran Department of Mathematics Department of Mathematics Lehigh University SUNY at Buffalo Bethlehem, PA 18015 Buffalo, NY 14260 USA USA [email protected] [email protected]

ISBN 978-0-8176-4692-9 e-ISBN 978-0-8176-4693-6 DOI 10.1007/978-0-8176-4693-6 Springer New York Dordrecht Heidelberg London

Library of Congress Control Number: 2011936871

Mathematics Subject Classiﬁcation (2010): 14H55, 30Fxx, 32-01

© Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identiﬁed as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.birkhauser-science.com) To Raghavan Narasimhan

Preface

A Riemann surface X is a connected 1-dimensional complex manifold, that is, a connected Hausdorff space that is locally homeomorphic to open subsets of C with complex analytic coordinate transformations (this also makes X a real 2-dimensional smooth manifold). Although a connected second countable real 1-dimensional smooth manifold is simply a line or a circle (up to diffeomorphism), the real 2-dimensional character of Riemann surfaces makes for a much more interesting topological characterization and a still more interesting complex analytic characterization. A noncompact (i.e., an open) Riemann surface satisﬁes analogues of the classical theorems of complex analysis, for example the Mittag-Lefﬂer theorem, the Weier- strass theorem, and the Runge approximation theorem (this development began only in the 1940s with the work of, for example, Behnke and Stein). On the other hand, by the maximum principle, a compact Riemann surface admits only constant holomorphic functions. However, compact Riemann surfaces do admit a great many meromorphic functions. This property leads to powerful theorems, in particular, the crucial RiemannÐRoch theorem. The theory of Riemann surfaces occupies a unique position in modern mathematics, lying at the intersection of analysis, algebra, geometry, and topology. Most earlier books on this subject have tended to focus on its algebraic-geometric and number-theoretic aspects, rather than its analytic aspects. This book takes the point of view that Riemann surface theory lies at the root of much of modern analysis, and it exploits this happy circumstance by using it as a way to introduce some fundamental ideas of analysis, as well as to illustrate some of the interactions of analysis with geometry and topology. The analytic methods applied in this book to the study of one complex variable are also useful in the study of several complex variables. Moreover, they contain the essence of techniques used generally in the study of partial differential equations and in the application of analytic tools to problems in geometry. Thus a careful reader will be rewarded not only with a good command of the classical theory of Riemann surfaces, but also with an introduction to these important modern techniques. While much of the book is intended for students at the second-year graduate level, Chaps. 1 and 2 and Sect. 5.2 (along with the required

vii viii Preface background material) could serve as the basis for the complex analysis component of a year-long first-year graduate-level course on real and complex analysis. A suc- cessful student in such a course would be well prepared for further study in analysis and geometry. The analytic approach in this book is based on the solution of the inhomogeneous CauchyÐRiemann equation with L2 estimates (or the L2 ∂¯-method) in a holomorphic line bundle with positive curvature. This powerful technique from several complex variables (see, for example, AndreottiÐVesentini [AnV], Hörmander [Hö], Skoda [Sk1], [Sk2], [Sk3], [Sk4], [Sk5], and Demailly [De1]) takes an especially nice form on a Riemann surface. Moreover, the 1-variable version serves as a gen- tle introduction to, and demonstration of, this important technique. For example, one may sometimes, with care, check signs in the formulas for higher dimensions by considering the dimension-one case. On the other hand, the higher-dimensional analogues of the main theorems considered in this book do require additional hypotheses. Two central features of this book are a simple construction of a strictly subharmonic exhaustion function (which is a modified version of the construction in [De2]) and a simple construction of a positive-curvature Hermitian metric in the holomorphic line bundle associated to a nontrivial effective divisor. The simplicity of these constructions and the power of the L2 ∂¯-method make this approach to Riemann surfaces very efficient. The recent book of Varolin [V]alsousesL2 methods. How- ever, although there is some overlap in the choice of topics, the proofs themselves are quite different. For a different treatment of this material that uses the solution of the inhomogeneous CauchyÐRiemann equation, but not L2 methods, the reader may refer to, for example, [GueNs]. This book also contains (in Chaps. 5 and 6) proofs of some fundamental facts concerning the holomorphic, smooth, and topological structure of a Riemann surface, such as the Koebe uniformization theorem, the biholomorphic classification of Riemann surfaces, the embedding theorems, the integrability of almost complex structures, Schönflies’ theorem (and the Jordan curve theorem), and the existence of a smooth structure on a second countable surface. The approach in this book to the above facts differs from the usual approaches (see, for example, [Wey], [Sp], or [AhS]) in that it mostly relies on the L2 ∂¯-method (in place of harmonic functions and forms) and on explicit holomorphic attachment of disks and annuli (in place of triangulations). The above facts, along with the facts concerning compact Riemann surfaces considered in Chap. 4, constitute some of the background required for the study of Teichmüller theory (as considered in, for example, [Hu]). Riemann first introduced Riemann surfaces partly as a way of understanding multiple-valued holomorphic functions (see, for example, [Wey]or[Sp]), and Rie- mann surface theory has since grown into a vast area of study. Weyl’s book [Wey] was the first book on the subject, and some elements of the point of view in [Wey] (and in the similar book [Sp]) are present in this book. For different approaches to the study of open Riemann surfaces, the reader may refer to, for example, [AhS]or [For]. For further study of compact Riemann surfaces (the main focus of most books on Riemann surfaces), the reader may refer to, for example, [FarK], [For], or [Ns4]. Preface ix

Much of the background material required for study of this book is provided in detail in Part III (Chaps. 7Ð11). It is strongly recommended that, rather than ﬁrst studying all of Part III, the reader instead focus on Parts I and II, and consult the appropriate sections in Part III only as needed (tables providing the interdependence of the sections appear after table of contents). In fact, the background material has been placed in the last part of the book instead of the ﬁrst in order to make such con- sultation more convenient. The main prerequisite for the book is some knowledge of point-set topology (as in, for example, [Mu]) and elementary measure theory (as in, for example, Chap. 1 of [Rud1]), although parts of these subjects are reviewed in Sects. 7.1 and 9.1. On the other hand, smooth manifolds (as in Chap. 9 and in, for example, [Mat], [Ns3], and [Wa]) and Hilbert spaces (as in Chap. 7 and in, for example, [Fol] and [Rud1]) are essential objects in this book, so it would be to the reader’s advantage to have had some previous experience with these objects. It may surprise the reader to learn that previous experience with complex analysis in the plane (as in, for example, [Ns5]), while helpful, is not necessary. In fact, as indicated earlier, Chaps. 1 and 2 and Sect. 5.2 (along with the required background material and the corresponding exercises) together actually provide the material for a fairly complete course on complex analysis on domains in the plane along with the analogous study of complex analysis on Riemann surfaces (other suggested course outlines appear after the tables listing the interdependence of the sections). Exercises are included for most of the sections, and some of the theory is developed in the exercises.

Acknowledgements The authors would ﬁrst like to especially thank Raghavan Narasimhan for his comments on the ﬁrst draft of this book, and for all he has taught them. Next, for their helpful comments and corrections, the authors would like to thank Charles Epstein and Dror Varolin, as well as the following students: Breeanne Baker, Qiang Chen, Tom Concannon, Patty Garmirian, Spyro Roubos, Kathleen Ryan, Brittany Shelton, Jin Yi, and Yingying Zhang. On the other hand, any mistakes or incoherent statements in the book are entirely the fault of the authors. Finally, the authors would like to thank their families and friends for their patience during the preparation of this book.

Bethlehem, PA, USA T. Napier Buffalo, NY, USA M. Ramachandran

Contents

Interdependence of Sections ...... xv Suggested Course Outlines ...... xvii

Part I Analysis on Riemann Surfaces 1 Complex Analysis in C ...... 3 1.1 Holomorphic Functions ...... 3 1.2 Local Solutions of the CauchyÐRiemann Equation ...... 6 1.3PowerSeriesRepresentation...... 12 1.4 Complex Differentiability ...... 16 1.5 The Holomorphic Inverse Function Theorem in C ...... 18 1.6 Examples of Holomorphic Functions ...... 21 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms .. 25 2.1DefinitionsandExamples...... 26 2.2 Holomorphic Functions and Mappings ...... 30 2.3 Holomorphic Attachment ...... 35 2.4 Holomorphic Tangent Bundle ...... 38 2.5DifferentialFormsonaRiemannSurface...... 45 2.6 L2 Scalar-Valued Differential Forms on a Riemann Surface .... 53 2.7 The Distributional ∂¯ Operator on Scalar-Valued Forms ...... 60 2.8 Curvature and the Fundamental Estimate for Scalar-Valued Forms . 63 2.9 The L2 ∂¯-Method for Scalar-Valued Forms of Type (1, 0) ..... 65 2.10 Existence of Meromorphic 1-Forms and Meromorphic Functions . 67 2.11 Radó’s Theorem on Second Countability ...... 71 2.12 The L2 ∂¯-Method for Scalar-Valued Forms of Type (0, 0) ..... 73 2.13 Topological Hulls and Chains to Infinity ...... 77 2.14 Construction of a Subharmonic Exhaustion Function ...... 82 2.15 The Mittag-Leffler Theorem ...... 88 2.16 The Runge Approximation Theorem ...... 91

xi xii Contents

3TheL2 ∂¯-Method in a Holomorphic Line Bundle ...... 101 3.1 Holomorphic Line Bundles ...... 101 3.2 Sheaves Associated to a Holomorphic Line Bundle ...... 115 3.3Divisors...... 119 3.4 The ∂¯ Operator and Dolbeault Cohomology ...... 124 3.5 Hermitian Holomorphic Line Bundles ...... 128 3.6 L2 Forms with Values in a Hermitian Holomorphic Line Bundle . . 131 3.7 The Connection and Curvature in a Line Bundle ...... 134 3.8 The Distributional ∂¯ Operator in a Holomorphic Line Bundle . . . 139 3.9 The L2 ∂¯-Method for Line-Bundle-Valued Forms of Type (1, 0) . . 140 3.10 The L2 ∂¯-Method for Line-Bundle-Valued Forms of Type (0, 0) . . 142 3.11PositiveCurvatureonanOpenRiemannSurface...... 146 3.12 The Weierstrass Theorem ...... 151

Part II Further Topics 4 Compact Riemann Surfaces ...... 157 4.1 Existence of Holomorphic Sections on a Compact Riemann Surface 157 4.2 Positive Curvature on a Compact Riemann Surface ...... 160 4.3 Equivalence of Positive Curvature and Positive Degree ...... 162 4.4 A Finiteness Theorem ...... 164 4.5 The RiemannÐRoch Formula ...... 165 4.6 Statement of the Serre Duality Theorem ...... 167 4.7 Statement of the ∂¯-Hodge Decomposition Theorem ...... 177 4.8 Proof of Serre Duality and ∂¯-Hodge Decomposition ...... 181 4.9 Hodge Decomposition for Scalar-Valued Forms ...... 185 5 Uniformization and Embedding of Riemann Surfaces ...... 191 5.1 Holomorphic Covering Spaces ...... 192 5.2 The Riemann Mapping Theorem in the Plane ...... 204 5.3 Holomorphic Attachment of Caps ...... 207 5.4 Exhaustion by Domains with Circular Boundary Components . . . 209 5.5 Koebe Uniformization ...... 211 5.6 Automorphisms and Quotients of C ...... 216 5.7 Aut (P1) and Uniqueness of the Quotient ...... 218 5.8AutomorphismsoftheDisk...... 220 5.9 Classiﬁcation of Riemann Surfaces as Quotient Spaces ...... 224 5.10SmoothJordanCurvesandHomology...... 228 5.11 Separating Smooth Jordan Curves ...... 236 5.12 Holomorphic Attachment and Removal of Tubes ...... 241 5.13 Tubes in a Compact Riemann Surface ...... 249 5.14 Tubes in an Arbitrary Riemann Surface ...... 254 5.15 Nonseparating Smooth Jordan Curves ...... 264 5.16 Canonical Homology Bases in a Compact Riemann Surface ....267 5.17 Complements of Connected Closed Subsets of P1 ...... 275 5.18 Embedding of an Open Riemann Surface into C3 ...... 279 Contents xiii

5.19 Embedding of a Compact Riemann Surface into Pn ...... 290 5.20 Finite Holomorphic Branched Coverings ...... 294 5.21 Abel’s Theorem ...... 302 5.22 The AbelÐJacobi Embedding ...... 306 6 Holomorphic Structures on Topological Surfaces ...... 311 6.1 Almost Complex Structures on Smooth Surfaces ...... 311 6.2 Construction of a Special Local Coordinate ...... 320 6.3RegularityofSolutionsonanAlmostComplexSurface...... 322 6.4 The Distributional ∂¯, Connection, and Curvature ...... 326 6.5 L2 SolutionsonanAlmostComplexSurface...... 328 6.6 Proof of Integrability ...... 329 6.7 Statement of Schönﬂies’ Theorem ...... 332 6.8 Harmonic Functions and the Dirichlet Problem ...... 338 6.9 Proof of Schönﬂies’ Theorem ...... 350 6.10 Orientable Topological Surfaces ...... 356 6.11 Smooth Structures on Second Countable Topological Surfaces . . . 364

Part III Background Material 7 Background Material on Analysis in Rn and Hilbert Space Theory . 375 7.1MeasuresandIntegration...... 375 7.2 Differentiation and Integration in Rn ...... 382 7.3 C∞ Approximation...... 390 7.4DifferentialOperatorsandFormalAdjoints...... 392 7.5 Hilbert Spaces ...... 397 7.6 Weak Sequential Compactness ...... 403 8 Background Material on Linear Algebra ...... 407 8.1 Linear Maps, Linear Functionals, and Complexifications ...... 407 8.2 Exterior Products ...... 409 8.3 Tensor Products ...... 412 9 Background Material on Manifolds ...... 415 9.1 Topological Spaces ...... 415 9.2 The Definition of a Manifold ...... 419 9.3 The Topology of Manifolds ...... 423 9.4 The Tangent and Cotangent Bundles ...... 428 9.5 Differential Forms on Smooth Curves and Surfaces ...... 437 9.6 Measurability in a Smooth Manifold ...... 447 9.7 Lebesgue Integration on Curves and Surfaces ...... 449 9.8 Linear Differential Operators on Manifolds ...... 462 9.9 C∞ Embeddings ...... 465 9.10 Classification of Second Countable 1-Dimensional Manifolds . . . 469 9.11 Riemannian Metrics ...... 475 xiv Contents

10 Background Material on Fundamental Groups, Covering Spaces, and (Co)homology ...... 477 10.1 The Fundamental Group ...... 477 10.2 Elementary Properties of Covering Spaces ...... 483 10.3TheUniversalCovering...... 489 10.4DeckTransformations...... 492 10.5 Line Integrals on C∞ Surfaces ...... 498 10.6 Homology and Cohomology of Second Countable C∞ Surfaces . . 503 10.7 Homology and Cohomology of Second Countable C0 Surfaces . . 511 11 Background Material on Sobolev Spaces and Regularity ...... 531 11.1 Sobolev Spaces ...... 532 11.2 Uniform Convergence of Derivatives ...... 534 11.3TheStrongFriedrichsLemma...... 535 11.4 The Sobolev Lemma ...... 541 11.5 Proof of the Regularity Theorem ...... 542 References ...... 545 Notation Index ...... 549 Subject Index ...... 553 Interdependence of Sections

The main prerequisites for each section appear in the tables below. If, for Sect. x, facts proved in Sect. y are needed only for a nonessential result, or only one or two small facts from Sect. y are needed, then Sect. y is omitted from the list of required sections for Sect. x. If Sect. x requires Sect. y and Sect. y requires Sect. z, then, in most cases, Sect. z is omitted from the list of required sections for Sect. x (for example, Sect. 2.3 requires Sects. 2.1 and 2.2, which in turn require Chap. 1 and Sect. 9.1). Italicized section numbers correspond to Part III sections, that is, sections containing background material.

Part I Require(s) Part II Require(s) 1.1Ð1.6 7.1Ð7.4 4.1 3.10 2.1Ð2.2 Chap. 1, 9.1 4.2 2.14 (2.11 is not required), 2.3 2.1, 2.2 4.1 2.4 2.1, 2.2, 8.1, 9.2, 9.4 4.3 4.2 2.5 2.4, 8.2, 9.5Ð9.7 4.4Ð4.9 4.2 2.6 2.5, 7.5 5.1Ð5.5 10.1Ð10.5, 2.3, 2.16 2.7 2.5, 9.8 5.6Ð5.9 5.5 2.8, 2.9 2.6, 2.7 5.10, 5.11 9.9, 10.1Ð10.6 2.10 2.9 or 3.9 5.12Ð5.14 5.5, 5.10, 5.11 2.11 2.10 5.15 5.10, 5.11, 10.7 2.12 2.10 5.16 5.13, 5.15 (Chap. 4 for 2.13 9.1Ð9.3 Corollary 5.16.4) 2.14 2.8, 2.11, 2.13 5.17 5.2Ð5.4, 5.15 2.15 2.12, 2.14 5.18 5.17 2.16 2.12, 2.14 5.19 Chap. 4 3.1Ð3.10 8.3, 2.5, 2.7, 2.8 5.20 5.1 3.11, 3.12 2.14, 3.10 5.21Ð5.22 Chap. 4, 5.20, 10.6, 10.7 6.1Ð6.6 2.9, 9.11, Chap. 11, 7.6 6.7Ð6.9 5.15, 5.17, 9.10 6.10 5.10, 5.11, 10.1Ð10.7 6.11 6.7Ð6.10

Suggested Course Outlines

A typical one-semester course would begin with careful study of Sects. 1.1 and 1.2, a quick reading of Sects. 1.3Ð1.6, careful study of Sects. 2.1 and 2.2, some (possibly brief) consideration of Sect. 2.3, and careful study of Sects. 2.4 and 2.5.The following are some of the natural directions in which the course could then proceed: • Holomorphic and meromorphic functions on an open Riemann surface: Sec- tions 2.6Ð2.16 and, possibly, 5.2. • Holomorphic line bundles on an open Riemann surface: Sections 2.7, 2.8, 3.1Ð3.10, 2.11, 2.13, 2.14, 3.11, and 3.12. • Uniformization: Sections 2.6Ð2.14, 2.16, and 5.1Ð5.9. • Compact Riemann surfaces: Sections 2.7, 2.8, 3.1Ð3.10, 2.13, 2.14 (here, one needs the results of Sect. 2.14 only for an open subset of a compact Riemann surface, which is automatically second countable, so Radó’s theorem from Sect. 2.11 is not required for this case), 4.1Ð4.9, and 5.19. • Integrability of almost complex structures on a surface: Sections 2.6Ð2.9 and 6.1Ð6.6.

Suggested Topics for Further Study Beyond a one-semester course, the reader may wish to consider, for example: • The characterization of a Riemann surface in terms of holomorphic attachment of tubes: Sections 5.10Ð5.14. • Further topics concerning compact Riemann surfaces: Sections 5.10Ð5.13, 5.15, 5.16, 5.19Ð5.22. • The embedding theorem for an open Riemann surface: Sections 5.17 and 5.18. • The existence of smooth structures on second countable topological surfaces: Sections 5.17 and 6.7Ð6.11.

xvii

Part I Analysis on Riemann Surfaces

Chapter 1 Complex Analysis in C

In this chapter, following Hörmander [Hö], we consider elementary deﬁnitions and facts concerning complex analysis in C from the point of view of local solutions of the inhomogeneous Cauchy–Riemann equation ∂u/∂z¯ = v.Theglobal solution of the analogous inhomogeneous Cauchy–Riemann equation on a Riemann surface (see Chaps. 2 and 3) will allow us to obtain analogues of some of the central theorems of complex analysis in the plane (for example, the Riemann mapping theorem, the Mittag-Lefﬂer theorem, and the Weierstrass theorem) for open Riemann surfaces, as well as some of the central theorems of the theory of compact Riemann surfaces (for example, the Riemann–Roch theorem). A reader who is familiar with complex analysis in the plane may wish to read Sects. 1.1 and 1.2 carefully, but only skim Sects. 1.3–1.6.

1.1 Holomorphic Functions

√ 2 Identifying the complex plane C with R under√(x, y) → z = x + iy = x + −1y for (x, y) ∈ R2 (we will sometimes denote i by −1), we may define the first-order constant-coefficient linear differential operators ∂ 1 ∂ ∂ ∂ 1 ∂ ∂ ≡ − i and ≡ + i . ∂z 2 ∂x ∂y ∂z¯ 2 ∂x ∂y

The operators ∂/∂z and ∂/∂z¯ obey the standard sum, product, and quotient rules. Moreover, if γ : (a, b) → C is a (real) differentiable function on an interval (a, b) and u is a C1 function on a neighborhood of the image, then

d ∂u dγ ∂u dγ u(γ (t)) = (γ (t)) + (γ (t)) . dt ∂z dt ∂z¯ dt

T. Napier, M. Ramachandran, An Introduction to Riemann Surfaces, Cornerstones, 3 DOI 10.1007/978-0-8176-4693-6_1, © Springer Science+Business Media, LLC 2011 4 1 Complex Analysis in C

Observe that if f is a C1 function on an open set in C, and u = Re f and v = Im f , then

∂f ∂u ∂v ∂u ∂v = 0 ⇐⇒ = and =− . ∂z¯ ∂x ∂y ∂y ∂x

We refer to the above as the homogeneous Cauchy–Riemann equation(s).Weuse a similar notation if elements of C are represented by other names (for example, ζ = s + it).

Deﬁnition 1.1.1 Let be an open subset of C. A function f ∈ C1() is called holomorphic (or analytic or complex analytic)on if ∂f /∂ z¯ ≡ 0on. We denote the set of all holomorphic functions on by O(), and for each function f ∈ O(), we deﬁne d ∂f ∂f ∂f f = [f(z)]≡ = =−i . dz ∂z ∂x ∂y We also say that f is a primitive for the function f . A holomorphic function on C is also called an entire function.

The proof of the following is left to the reader (see Exercise 1.1.1):

Theorem 1.1.2 Any sum, product, quotient, or composition of holomorphic functions is holomorphic on its domain. Moreover, if f and g are holomorphic functions, then

(f + g) = f + g ,(fg) = f g + fg , f g − fg (f/g) = , and (f ◦ g) = (f ◦ g) · g . g2

Consequently, every rational function of z is holomorphic on its domain, and for every n ∈ Z, d(zn)/dz = nzn−1.

An important tool in this book (and in the general study of analysis on manifolds) is the machinery of differential forms. The required definitions and facts appear in Chap. 9. For now, we take an informal approach to differential forms on open subsets of C, which will suffice for this chapter. We have the real differentials dx and dy on the complex plane C (identified with R2) and the complex differentials

dz = dx + idy and dz¯ = dx − idy.

Any complex 1-form α may be written

α = Pdx+ Qdy= adz+ bdz,¯ 1.1 Holomorphic Functions 5

= 1 − for some (unique) pair of complex-valued functions P and Q and for a 2 (P = 1 + iQ) and b 2 (P iQ).Ifα is continuous (i.e., if the coefﬁcients P and Q are continuous) on a neighborhood of the image of a C1 path γ = (u + iv):[r, s]→C, with u = Re γ and v = Im γ , then s du dv α = P(γ(t)) + Q(γ (t)) dt γ r dt dt s dγ dγ = a(γ(t)) + b(γ (t)) dt. r dt dt The exterior product of dx and dy is denoted by dx ∧ dy =−dy ∧ dx (and dx ∧ dx = dy ∧ dy = 0). Any complex 2-form β may be written

β = fdx∧ dy = f · (i/2)dz∧ dz,¯ for some (unique) function f .Iff is integrable on a Lebesgue measurable set E ⊂ C, then i fdλ= fdx∧ dy = f · dz∧ dz,¯ E E E 2 where λ denotes Lebesgue measure (see Sect. 7.1). If f is a C1 function on an open set in C, then the differential (or exterior derivative)off satisﬁes ∂f ∂f ∂f ∂f df = dx + dy = dz+ dz¯ = ∂f + ∂f,¯ ∂x ∂y ∂z ∂z¯ where ∂f ∂f ∂f ≡ dz and ∂f¯ ≡ dz.¯ ∂z ∂z¯ In particular, f is holomorphic if and only if ∂f¯ = 0 (equivalently, df = hdz for some function h). If α = Pdx+ Qdy is a 1-form of class C1 (i.e., the coefﬁcients P and Q are of class C1), then the exterior derivative of α is given by ∂Q ∂P dα = dP ∧ dx + dQ∧ dy = − dx ∧ dy. ∂x ∂y

= + ¯ = 1 − = 1 + Equivalently, for α adz bdz (with a 2 (P iQ) and b 2 (P iQ)), ∂b ∂a dα = da ∧ dz+ db ∧ dz¯ = − dz∧ dz¯ = ∂α + ∂α,¯ ∂z ∂z¯ where ∂b ∂a ∂α = (∂b) ∧ dz¯ = dz∧ dz¯ and ∂α¯ = (∂a)¯ ∧ dz =− dz∧ dz.¯ ∂z ∂z¯ 6 1 Complex Analysis in C

According to Stokes’ theorem, if C is a smooth domain and α is a C1 1-form on a neighborhood of , then α = dα, ∂ where the left-hand side is given by the sum of the integrals of α over the ﬁnitely many boundary curves of , each parametrized so that lies to the left (i.e., each boundary curve is positively oriented relative to ).

Exercises for Sect. 1.1 1.1.1. Prove Theorem 1.1.2.

1.2 Local Solutions of the CauchyÐRiemann Equation

For 0

(z0; R) ≡{z ∈ C ||z − z0|

(z0; r, R) ≡{z ∈ C | r<|z − z0|

Lemma 1.2.1 Let be a smooth relatively compact domain in C, and let f be a C1 function on a neighborhood of . (a) Cauchy integral formula. For each point z ∈ , we have 1 f(ζ) ∂f /∂ ζ¯ f(z)= dζ + dζ ∧ dζ¯ . 2πi ∂ ζ − z ζ − z (b) Cauchy’s theorem. We have + ∂f ∧ ¯ = f(ζ)dζ ¯ dζ dζ 0. ∂ ∂ζ

Remarks 1. In particular, for f holomorphic on a neighborhood of , we get (the more standard versions) 1 f(ζ) f(z)= dζ ∀ z ∈ and f(ζ)dζ = 0. 2πi ∂ ζ − z ∂ 2. The formula in part (a) is also called the Cauchy–Pompeiu integral formula or the ∂¯-Cauchy integral formula. 1.2 Local Solutions of the Cauchy–Riemann Equation 7

Proof of Lemma 1.2.1 Observe that the function ζ → (ζ − z)−1 is locally integrable in C, so the integrals in (a) are deﬁned. For z ∈ and 0

Lemma 1.2.2 (Local solution of the inhomogeneous Cauchy–Riemann equation) 0 k Let D be a disk in C, let α ∈ C (∂D), let k ∈ Z>0 ∪{∞}, let β be a C function on a neighborhood of D, and, for each point z ∈ D, let 1 α(ζ) β(ζ) f(z)= dζ + dζ ∧ dζ¯ . 2πi ∂D ζ − z D ζ − z

Then f ∈ Ck(D) and ∂f /∂ z¯ = β on D. In particular, f is holomorphic on D \ supp β.

Proof Setting D = (z0; R), we may ﬁx a number r with 0

∞ Thus g is holomorphic and of class C on (z0; r).

Similarly, setting R ≡ R + r +|z0|, we get, for each point z ∈ (z0; r), 1 −1 1 −1 h(z) =− β1(ζ + z) · ζ dλ(ζ) =− β1(ζ + z) · ζ dλ(ζ) π C π (z −z;R) 0 1 −1 =− β1(ζ + z) · ζ dλ(ζ);

π (z0;R )

k and hence, since the function ζ → 1/ζ is integrable on (z0; R ), h is of class C on (z0; r). Moreover, setting γ ≡ ∂β1/∂z¯, we get, for each point z ∈ (z0; r), + ∂h 1 ∂ −1 1 γ(ζ z) (z) =− [β1(ζ + z)]·ζ dλ(ζ) =− dλ(ζ) ¯ ¯ ∂z π (z0;R ) ∂z π C ζ 1 γ(ζ) 1 ∂β /∂ζ¯ =− dλ(ζ) = 1 dζ ∧ dζ.¯ π C ζ − z 2πi C ζ − z

Since supp β1 ⊂ D, the Cauchy integral formula (Lemma 1.2.1)nowgives∂h/∂z¯ = β1 = β on (z0; r), and since the choice of r ∈ (0,R) was arbitrary, the claim follows.

Theorem 1.2.3 If ⊂ C is an open set and f ∈ O(), then f ∈ C∞() and f ∈ O().

Proof For any disk D , the Cauchy integral formula (Lemma 1.2.1) and the local solution of the Cauchy–Riemann equation (Lemma 1.2.2) together imply that f is of class C∞ on D. It follows that f ∈ O(), because ∂f ∂ ∂f ∂ ∂f = = = 0. ∂z¯ ∂z¯ ∂z ∂z ∂z¯

Theorem 1.2.4 For every open set ⊂ C, every compact set K ⊂ , and every ∞ = linear differential operator A with coefﬁcients in Lloc(), there is a constant C C(,K,A) such that

Af L∞(K) ≤ Cf Lp() ∀p ∈[1, ∞],f∈ O().

Proof Since A has bounded coefﬁcients in a neighborhood of K, we may as- | | sume without loss of generality that A = ∂ α /∂xα1 ∂yα2 for some multi-index α = (α1,α2) ∈ Z≥0 × Z≥0, and applying Hölder’s inequality in a relatively compact neighborhood of K in , we see that we may also assume that p = 1. Given a point z0 ∈ K, we may choose r, R1,R2 ∈ R with

The function (z, ζ ) → f(ζ)· (∂η/∂ζ¯ )(ζ ) · (ζ − z)−1, and its derivatives of arbitrary order in x and y (for z = x +iy), are bounded on (z0; r)×[ (z0; R2)\ (z0; R1)]. Differentiation past the integral now gives, for z ∈ (z0; r), [ ] =−1 ∂η [ −· −1] Af (z) f(ζ) ¯ (ζ )A (ζ ) (z) dλ(ζ ). π (z0;R2)\ (z0;R1) ∂ζ Hence

|Af |≤C f 1 ≤ C f 1 sup z0 L ( (z0;R2)\ (z0;R1)) z0 L () (z0;r) for some constant Cz0 independent of the choice of f . Covering K by ﬁnitely many such disks (z0; r), we get the desired constant C.

Corollary 1.2.5 For every open set ⊂ C and every nonempty compact set K ⊂ , there is a constant C = C(,K) such that |f(w)− f(z)|≤Cf Lp()|w − z| for all w,z ∈ K, p ∈[1, ∞], and f ∈ O() (cf. Deﬁnition 7.2.3).

Proof Given a disk D , Theorem 1.2.4 provides a (real) constant B such that | |≤ ∈ O ∈[ ∞] supD ∂f /∂ z B f Lp() for every f () and p 1, . Hence, for every pair of points w,z ∈ D, for every function f ∈ O(), and for every number p ∈[1, ∞],wehave 1 d |f(w)− f(z)|= [f(z+ t(w− z))] dt 0 dt 1 ∂f = (z + t(w− z)) · (w − z) dt 0 ∂z

≤ Bf Lp()|w − z|.

Corollary 1.2.6 If {fn} is a sequence of holomorphic functions converging uniformly on compact subsets of an open set ⊂ C to a function f , then f ∈ O(), (k) and for each k ∈ Z≥0, the sequence of kth derivative functions {fn } converges uniformly to f (k) on compact subsets of . 10 1 Complex Analysis in C

Proof For each disk D and each point z ∈ D, the Cauchy integral formula gives 1 f (ζ ) 1 f(ζ) f(z)= lim f (z) = lim n dζ = dζ. →∞ n →∞ n n 2πi ∂D ζ − z 2πi ∂D ζ − z

(k) (k) Thus Lemma 1.2.2 implies that f is holomorphic on D. That fn → f uniformly on each compact set K ⊂ follows easily from Theorem 1.2.4.

Corollary 1.2.7 (Montel’s theorem) If {fn} is a sequence of holomorphic functions that is uniformly bounded on each compact subset of an open set ⊂ C, then some subsequence fnk converges uniformly on compact subsets of to a function f ∈ O().

Proof For any compact set K ⊂ , Corollary 1.2.5 implies that there is a constant C>0 such that |fn(w) − fn(z)|≤C|w − z| for all w,z ∈ K and for every n ∈ Z>0. Ascoli’s theorem (together with Cantor’s diagonal process) now provides a subsequence converging uniformly on compact subsets of , and Corollary 1.2.6 implies that the limit is holomorphic.

1 ⊂ C According to Deﬁnition 7.4.2,forLloc functions u and v on an open set , we have ∂u ∂ ≡ u = v ¯ ¯ ∂z distr ∂z distr if and only if ∂ϕ u − dλ = vϕdλ¯ ∀ϕ ∈ D(). ∂z

C ∈ 1 Theorem 1.2.8 (Regularity theorem) If is an open subset of , u Lloc(), k k k ∈ Z>0 ∪{∞}, β ∈ C (), and (∂u/∂z)¯ distr = β, then u ∈ C () (i.e., there is a unique Ck function that is equal to u almost everywhere in ). In particular, if (∂u/∂z)¯ distr = 0, then u ∈ O() (i.e., there is a unique holomorphic function that is equal to u almost everywhere in ).

Proof It suffices to show that u is of class Ck on every disk D . Thus, by subtracting the Ck solution v of ∂v/∂z¯ = β in D provided by Lemma 1.2.2,wemay assume without loss of generality that β ≡ 0. To show that u is holomorphic on D,wefixadiskD with D D and a ∞ nonnegative C function κ with compact support in the unit disk (0; 1) in C such that C κdλ= 1. By Lemma 7.3.1 and Lemma 7.4.4, for each sufficiently large n ∈ Z>0, the function un given by 2 un(z) ≡ u(ζ )n κ(n(z− ζ))dλ(ζ) ∀z ∈ D 1.2 Local Solutions of the Cauchy–Riemann Equation 11

C∞ ¯ = ∈ O − → is of class and ∂un/∂z 0; that is, un (D ). Moreover, un u L1(D ) 0 →∞ { | } as n . Hence Theorem 1.2.4 implies that the sequence un D is Cauchy in C0 · ( (D), L∞(D)) and therefore uniformly convergent. Corollary 1.2.6 now implies that u is holomorphic on D and therefore that u ∈ Ck(D).

Theorem 1.2.9 (Mean value property) If f ∈ O( (z0; R)), then for every r ∈ (0,R), 1 2π 1 f(z ) = f z + reiθ dθ = f (ζ ) dλ(ζ ). 0 0 2 2π 0 πr (z0;r)

Proof Lemma 1.2.1 gives the ﬁrst equality. For the second, we integrate to get r2 r 1 r 2π f(z ) = f(z )s ds = f z + seiθ sdθds 2 0 0 2π 0 0 0 0 = 1 f (ζ ) dλ(ζ ). 2π (z0;r)

Theorem 1.2.10 (Riemann’s extension theorem) Let ⊂ C be an open set, let z0 ∈ , and let f be a function that is holomorphic on \{z0}. If

2 lim (z − z0)f (z) = 0 or f ∈ L (), z→z0 ˜ ˜ then there exists a (unique) function f ∈ O() such that f = f on \{z0}.

Proof Let a(z) = z − z0 for each z ∈ C and ﬁx R>0 with D ≡ (z0; R) . → → ∈ 1 ∈ D Assuming ﬁrst that a(z)f (z) 0asz z0, we get f Lloc().Ifϕ (D), then for every r ∈ (0,R), Stokes’ theorem gives ∂ϕ f dz∧ dz¯ =− d[fϕdz]= f (z)ϕ(z) dz. ¯ D\ (z0;r) ∂z D\ (z0;r) ∂ (z0;r)

Letting r → 0+, we get ∂ϕ f dλ = 0. D ∂z¯

Thus (∂f/∂z)¯ distr = 0in, and the regularity theorem (Theorem 1.2.8)impliesthat f ∈ O(). 2 ∗ For f ∈ L () and for z ∈ (z0; R/2), the mean value property (Theo- rem 1.2.9) and the Schwarz (or Hölder) inequality give, for r =|z − z0|/2, 1 |a(z)f (z)|= a(ζ)f(ζ)dλ(ζ) 2 πr (z;r) 1 ≤ a 2 ; ·f 2 ; πr2 L ( (z r)) L ( (z r)) 12 1 Complex Analysis in C

1 ≤ a 2 ; ·f 2 ; πr2 L ( (z0 3r)) L ( (z0 3r)) = √9 → → + f L2( (z ;3r)) 0asr 0 . 2π 0

It follows that a(z)f (z) → 0asz → z0, and hence that f ∈ O().

Exercises for Sect. 1.2 1.2.1. Prove the following general version of Lemma 1.2.2.Letμ be a positive measure in C that is defined on the collection of Lebesgue measurable subsets of C and that satisfies μ(C \ K) = 0 for some compact set K ⊂ C,let f ∈ L1(C,μ), and let 1 f(ζ) u(z) =− dμ(ζ) π C ζ − z for every point z ∈ C at which the above integral is defined. Then we have the following: (i) The function u is defined and holomorphic on C \ K. ∈ ∞ = (ii) If, for some open set ,wehavef L () and dμ dλ in , then u is defined and continuous on and (∂u/∂z)¯ distr = f on . ∈ Ck ∈ Z ∪{∞} (iii) If, for some open set ,wehavef () with k >0 and dμ = dλ in , then u is defined and of class Ck on and ∂u/∂z¯ = f on .

1.3 Power Series Representation In this section, we verify that holomorphic functions are precisely those functions that can be expressed locally as sums of power series, and we consider some important consequences.

∞ − n Theorem 1.3.1 (Abel) Foranypowerseries n=0 cn(z z0) , either there is an R ∈ (0, ∞] such that the series converges uniformly and absolutely on compact subsets of D ≡{z ∈ C ||z − z0| R, or the series converges only when z = z0, in which case we set R = 0. Furthermore, the power series ∞ ∞ n−1 cn n+1 n · cn(z − z ) and (z − z ) 0 n + 1 0 n=1 n=0 have this same radius of convergence R; and if R>0, then on the above set D, the sum of the ﬁrst is equal to f , and the sum of the second has derivative f .

Proof Let R = sup{|z − z0||the series converges at z}. Then, clearly, the series diverges for |z − z0| >R.If0

n with r

Conversely, we have the following:

Theorem 1.3.2 If R ∈ (0, ∞], z0 ∈ C, and f is a holomorphic function on the set D ={z ∈ C ||z − z0|

Proof If 0

− ∞ f(ζ) f(ζ)/(ζ z0) n −n−1 = = f (ζ )(z − z0) (ζ − z0) , ζ − z 1 −[(z − z0)/(ζ − z0)] n=0 and for each n ∈ Z≥0,

n −n−1 −1 n+1 |f (ζ )(z − z0) (ζ − z0) |≤ max |f |·r · (r/s) . ∂ (z0;s)

Thus the Weierstrass M -test gives uniform convergence in ζ and, integrating term = ∞ − n ∈ ; by term, we get f(z) n=0 cn(z z0) for each z (z0 r), where 1 f(ζ) c = dζ ∀n ∈ Z≥ . n − n+1 0 2πi ∂ (z0;s) (ζ z0)

For n ∈ Z≥0, according to Theorem 1.3.1, differentiation of the above power series (n) (n) term by term n times gives f (z). Evaluating at z0, we get f (z0) = cn · n!,as claimed.

Corollary 1.3.3 Let be a domain (i.e., a connected open set) in C, and let f be a nonconstant holomorphic function on . Then: (m) (a) For every point z0 ∈ , f (z0) = 0 for some m ∈ Z≥0. (b) For every point z0 ∈ , there are a unique m ∈ Z≥0 and a unique function m g ∈ O() such that g(z0) = 0 and f(z)= (z − z0) g(z) for every point z ∈ . (c) Identity theorem. The set f −1(0) has no limit points in . 14 1 Complex Analysis in C

(d) For u = Re f and v = Im f , the sets u−1(0) and v−1(0) are nowhere dense in .

Remark Suppose f is a holomorphic function on an open set ⊂ C and z0 ∈ C. It follows from part (b) that if f is not identically zero on some connected neighborhood of z0 in , then there exist a unique nonnegative integer m and a func- m tion g ∈ O() such that g(z0) = 0 and f(z)= (z − z0) g(z) for every point z ∈ .

The nonnegative integer m is called the order of f at z0 and is denoted by ordz0 f . = = For m ordz0 f>0, z0 is called a zero of order m.Form 1, we also say that f has a simple zero at p.Iff vanishes on some neighborhood of z0, then we say that =∞ f has order ordz0 f at z0. ≡ ∞ [ (n)]−1 Proof of Corollary 1.3.3 The set A n=0 f (0) is closed in , and by The- orem 1.3.2, A is also open. Part (a) now follows. (n) For (b), observe that by (a), the set {n ∈ Z≥0 | f (z0) = 0} is nonempty. Taking m to be the minimum of this set and applying Theorem 1.3.2, we get the claim. For the proof of (c), suppose z0 ∈ . By part (b), there are a nonvanishing holomorphic function g on a neighborhood U of z0 in and an integer m ∈ Z≥0 such m −1 that f(z)= (z − z0) g(z) for each z ∈ U. Hence f (0) ∩ U ⊂{z0} and the claim follows. Finally, for the proof of (d), observe that if u ≡ 0onadiskD ⊂ , then on D, we have ∂v/∂z = ∂v/∂z¯ = i∂u/∂z¯ ≡ 0, and hence dv ≡ 0 and v is constant on D. But by the identity theorem (part (c) above), this is impossible. Thus u−1(0) is nowhere dense. Since v is the real part of the holomorphic function −if , it follows that v−1(0) is also nowhere dense.

Theorem 1.3.4 (Maximum principle) If the modulus |f | of a holomorphic function f on a domain ⊂ C attains a local maximum at some point z0 ∈ , then f is constant on .

Proof Let u = Re f and let D = (z0; R) be a disk on which |f |≤|f(z0)|. By Corollary 1.3.3, it sufﬁces to show that u is constant on D. For this, we may assume without loss of generality that f(z0) = 0 (otherwise, we have f ≡ 0ona neighborhood) and hence that f(z0) = 1 (divide by f(z0)). The mean value property (Theorem 1.2.9)gives 1 1 1 = f(ζ)dλ(ζ) = u(ζ ) dλ(ζ ) ≤ dλ(ζ) = . 1 Re 2 2 2 1 πR D πR D πR D Hence the nonnegative function 1 − u integrates to 0, and therefore u ≡ 1onD.

Theorem 1.3.5 (Open mapping theorem) If f is a nonconstant holomorphic function on a domain ⊂ C, then the image U ≡ f()is open.

Proof Suppose w0 = f(z0) ∈ ∂U for some z0 ∈ . By the identity theorem (Corol- lary 1.3.3), there is a disk D containing z0 such that w0 is not in the compact 1.3 Power Series Representation 15 set K ≡ f(∂D). Choosing a point w1 ∈ C\U with |w1 −w0| < dist(w1,K), we get −1 a function g ≡ (f − w1) ∈ O() whose maximum modulus on D is not attained at any point in ∂D. This contradicts the maximum principle (Theorem 1.3.4), so U must be open.

For a holomorphic function on a punctured disk, a punctured plane, or an annulus, we have the Laurent series representation provided by the following theorem, the proof of which is left to the reader (see Exercise 1.3.2):

Theorem 1.3.6 (Laurent series representation) Let z0 ∈ C. { } (a) Let cn n∈Z be a collection of complex constants, and let r and R be the ∞ n ∞ n radii of convergence for the power series n =1 c−nζ and n=0 cnζ , respec- ∞ − n tively. If 1/r < R , then the Laurent series n=−∞ cn(z z0) is absolutely | − n| ∞ summable (i.e., n∈Z cn(z z0) < ) and uniformly convergent (i.e., the ∞ − −n ∞ − n series n=1 c−n(z z0) and n=0 cn(z z0) are uniformly convergent) on compact subsets of the set A ≡{z ∈ C | 1/r < |z − z0|

∞ − n+1 ∞ n−1 (ii) The power series n=1( n)c−nζ and n=1 ncnζ have radii of convergence r and R, respectively, and ∞ n−1 f (z) = ncn(z − z0) ∀z ∈ A; n=−∞

and ∞ n−1 − + ∞ n+1 + (iii) The power series n=2 c−nζ /( n 1) and n=0 cnζ /(n 1) have radii of convergence r and R, respectively, and if c−1 = 0, then the holomorphic function

∞ c z → F(z)≡ n (z − z )n+1 n + 1 0 n=−∞

(the coefﬁcient cn/(n + 1) is taken to be 0 for n =−1) satisﬁes F = f on A. 16 1 Complex Analysis in C

(b) Let s and S be constants with 0 ≤ s

∞ n and let r and R be the radii of convergence of the power series n=1 c−nζ ∞ n ≤ ≤ and n=0 cnζ , respectively. Then 1/r s

~~Exercises for Sect. 1.3 1.3.1 Complete the proof of Theorem 1.3.1. 1.3.2 Prove Theorem 1.3.6.~~

~~1.4 Complex Differentiability~~

~~Complex differentiability provides a third characterization of holomorphic functions.~~

~~Deﬁnition 1.4.1 A function f deﬁned on a neighborhood of a point z0 ∈ C is complex differentiable at z0 if~~

~~f(z)− f(z0) lim→ z z0 z − z0 exists. We say that f is complex differentiable on an open set ⊂ C if f is complex differentiable at each point in .~~

If f is a function that is holomorphic on a neighborhood of a point z0 ∈ C, then applying part (b) of Corollary 1.3.3 to the holomorphic function f − f(z0),we get a holomorphic function h on a neighborhood of z0 such that f(z)− f(z0) = (z − z0)h(z) for all z. It follows that f is complex differentiable at z0. Conversely, if a function f is complex differentiable at a point z0 ∈ C, then f is continuous at z0, and computing the limit in Deﬁnition 1.4.1 separately along the x and y directions, we get

~~∂f ∂f ∂f f(z)− f(z ) ∂f (z ) =−i (z ) = (z ) = lim 0 and (z ) = 0. 0 0 0 → 0 ∂x ∂y ∂z z z0 z − z0 ∂z¯~~

Consequently, if a function f is of class C1 and complex differentiable on an open set, then f is holomorphic with derivative given by the limit appearing in Deﬁni- tion 1.4.1. Thus we have the following: 1.4 Complex Differentiability 17

~~Lemma 1.4.2 Let f be a C1 function on an open set ⊂ C. Then f ∈ O() if and only if f is complex differentiable on . Moreover, if this is the case, then~~

f(z)− f(z ) f (z ) = lim 0 ∀z ∈ . 0 → z z0 z − z0 It is not evident that a complex differentiable function on an open set is of class C1, so we cannot conclude from the above that such a function is holomorphic. Nevertheless, this is the case (see Exercise 1.4.2). In fact, by a theorem of Looman and Menchoff, any continuous function that satisﬁes the Cauchy–Riemann equations on an open set is holomorphic (see [Ns5]). On the other hand, in this book, we will need only the C1 case addressed in Lemma 1.4.2.

Exercises for Sect. 1.4 1.4.1 Prove Goursat’s theorem:IfS ≡ (a, b) × (c, d) ⊂ C is a bounded open rectangular region and f is a function that is complex differentiable on a neighborhood of S, then f(z)dz= 0 (cf. Lemma 1.2.1). ∂S ≡| |= = Outline of a proof. One must show that M0 ∂S f(z)dz 0. Let S0 S. Breaking up S into four congruent rectangular regions, one sees that for one of the rectangles S , M ≡| f(z)dz|≥ M0 . Proceeding inductively, one gets a 1 1 ∂S1 4 decreasing sequence of open rectangular regions {Sν} such that for each ν ∈ Z>0, −ν −ν Sν has side lengths 2 (b − a) and 2 (d − c) and −ν Mν ≡ f(z)dz ≥ 4 M0. ∂Sν

There exists a point z0 in the intersection of the closures, and the function f(z)−f(z0) ∂f − − (z0) if z ∈ S \{z0}, g(z) ≡ z z0 ∂z 0ifz = z0, ≤ | |· →∞ is continuous. For a constant r>0, one gets M0 supSν g r. Letting ν , one gets the claim. 1.4.2 Let f be a complex differentiable function on an open rectangular region S ⊂ C,letz0 = x0 + iy0 ∈ S with x0,y0 ∈ R, and let F : S → C be the function given by x y z = x + iy → f(t+ iy0)dt + i f(x+ it)dt; x0 y0 that is, F(z) is obtained by integration of f(z)dz along the horizontal line segment from z0 to x + iy0 followed by integration along the vertical line segment from x + iy0 to z. (a) Show that ∂F/∂y = if on S. Applying Goursat’s theorem (see Ex- ercise 1.4.1), express F(z) in terms of similar integrals that yield ∂F/∂x = f on S. Conclude from this that F is holomorphic and F = f . 18 1 Complex Analysis in C

(b) Using part (a), prove that a function on an open set is holomorphic if and only if it is complex differentiable. 1.4.3 Show that complex differentiability provides a more direct proof that the sum of a power series is holomorphic than that appearing in the proof of Theo- ∞ − n rem 1.3.1. More precisely, suppose that n=0 cn(z z0) is a power series with radius of convergence R ∈ (0, ∞], and f(z) is the sum of this series for each point z ∈ D ≡{z ∈ C ||z − z0|

~~∞ n−1 ncn(z − z0) n=1~~

also converges for z ∈ D (and therefore that this power series has radius of convergence at least R). (b) Let g(z) be the sum of the series in part (a) for each point z ∈ D. Prove directly from Deﬁnition 1.4.1 (without invoking Corollary 1.2.6) that f is complex differentiable on D with f = g. Using the fact that the limit of a uniformly convergent sequence of continuous functions is continuous, show that f is of class C1, and hence that f is holomorphic.

~~1.5 The Holomorphic Inverse Function Theorem in C~~

A holomorphic function f mapping an open set bijectively onto an open set U is called a biholomorphism (or a biholomorphic mapping) if the inverse function f −1 is holomorphic. We also say that f maps biholomorphically onto U and that and U are biholomorphic. Observe that the chain rule implies that f is nonvanishing and [f −1] = 1/(f ◦ f −1). A function that maps a neighborhood of each point biholomorphically onto an open set is called a local biholomorphism (or a locally biholomorphic mapping). The following fact is fundamental in the study of Riemann surfaces:

~~Theorem 1.5.1 (Holomorphic inverse function theorem in C) Let f be a holomorphic function on an open set ⊂ C.~~

~~(a) If z0 ∈ and f (z0) = 0, then f maps some neighborhood of z0 biholomorphically onto a neighborhood of w0 ≡ f(z0). (b) If f is one-to-one, then f maps biholomorphically onto an open set.~~

~~The main step of the proof is the following inequality: 1.5 The Holomorphic Inverse Function Theorem in C 19~~

~~Lemma 1.5.2 Let D beadiskinC, let f be a holomorphic function on D, and let~~

~~C1 and C2 be positive constants with |f |≥C1 and |f |≤C2 on D. Then C |f(z ) − f(z )|≥C |z − z |− 2 |z − z |2 ∀z ,z ∈ D. 2 1 1 2 1 2 2 1 1 2~~

~~In particular, if diam D<2 C1/C2, then f is injective.~~

~~Proof For every pair of points z1,z2 ∈ D,wehave~~

~~f(z2) − f(z1) − f (z1)(z2 − z1) 1 d = f(z1 + t(z2 − z1)) dt − f (z1)(z2 − z1) 0 dt 1 ~~

~~= (z2 − z1) f (z1 + t(z2 − z1)) − f (z1) dt 0 1 1 d = (z2 − z1) f (z1 + st(z2 − z1)) dsdt 0 0 ds 1 1 2 = (z2 − z1) t · f (z1 + st(z2 − z1))dsdt. 0 0 Therefore~~

~~|f(z2) − f(z1)|≥|f (z1)|·|z2 − z1| 1 1 2 −|z2 − z1| t ·|f (z1 + st(z2 − z1))| dsdt, 0 0 and the claim follows.~~

~~Proof of Theorem 1.5.1 For the proof of part (a), observe that we may choose a constant R>0 so small that D ≡ (z0; R) ⊂ , C1 ≡ infD |f | > 0, and C2 ≡ | | supD f~~

h(w) − h(w1) h(w) − h(w1) = → 1/f (h(w1)) as w → w1; w − w1 f(h(w))− f(h(w1)) so h is a complex differentiable function. Moreover, h has continuous partial derivatives, since ∂h ∂h ∂h 1 =−i = = ; ∂x ∂y ∂z f ◦ h and hence by Lemma 1.4.2, h ∈ O(U). 20 1 Complex Analysis in C

For the proof of (b), observe that by the open mapping theorem (Theorem 1.3.5), f maps homeomorphically onto an open set . Moreover, by the identity theorem (Corollary 1.3.3), the zero set Z ≡ (f )−1(0) of the holomorphic function f has no limit points in , and hence its image Y ≡ f(Z) has no limit points in . There- fore, by part (a), f maps \ Z biholomorphically onto \ f(Z), and Riemann’s extension theorem (Theorem 1.2.10) implies that f −1 is holomorphic on .

Exercises for Sect. 1.5 1.5.1 Give a proof of the holomorphic inverse function theorem (Theorem 1.5.1) that uses the C∞ inverse function theorem (Theorem 9.9.1). 1.5.2 A subset S of Rn is convex if (1 − t)x + ty = x + t(y − x) ∈ S for all points x,y ∈ S and every number t ∈[0, 1] (in other words, S contains every line segment with endpoints in S). A real-valued function ϕ on a convex set S ⊂ Rn is convex if ϕ((1−t)x+ty) ≤ (1−t)·ϕ(x)+t ·ϕ(y) for all points x,y ∈ S and every number t ∈[0, 1] (in other words, the graph of the restriction of the function to a line segment in S lies below the line segment between the endpoints on the graph). (a) Prove that if ϕ is a convex function on a convex set S ⊂ Rn, then for every a ∈ R, {x ∈ S | ϕ(x) < a} is a convex set. (b) For a real-valued C2 function ϕ on an open set ⊂ Rn,theHessian at a n point p ∈ is the bilinear function (Hess(ϕ))p(·, ·) on R given by

~~n ∂2ϕ (Hess(ϕ))p(u, v) = (p)uivj ∂xi∂xj i,j=1~~

n for all u = (u1,...,un), v = (v1,...,vn) ∈ R . Prove that if is convex and n (Hess(ϕ))p(u, u) ≥ 0 ∀u ∈ R ,p∈ , then ϕ is convex. Hint. First consider the case n = 1. (c) Let f be a holomorphic function on a neighborhood of 0 with f(0) = 0 and f (0) = 0. Prove that for every sufﬁciently small >0, f maps the disk (0; ) biholomorphically onto a convex neighborhood of 0. Hint. After applying Theorem 1.5.1 to obtain a local holomorphic inverse g in a neighborhood of 0, show that the function |g|2 is convex near 0. 1.5.3 Let be a convex domain in C (see Exercise 1.5.2), f ∈ O(), and let C1

~~and C2 be positive constants with |f |≥C1 and |f |≤C2 on . (a) Prove that C |f(z ) − f(z )|≥C |z − z |− 2 |z − z |2 ∀z ,z ∈ . 2 1 1 2 1 2 2 1 1 2~~

~~(b) Prove that if diam <2C1/C2, then f maps biholomorphically onto a domain in C. 1.6 Examples of Holomorphic Functions 21~~

~~1.6 Examples of Holomorphic Functions~~

~~In this section, we consider some examples of holomorphic functions.~~

~~Example 1.6.1 (Rational functions) As noted in Theorem 1.1.2, every rational function n +···+ + → anz a1z a0 z m bmz +···+b1z + b0 is holomorphic (on its domain).~~

~~Example 1.6.2 (Exponential, logarithmic, and power functions) The exponential function on C is deﬁned by~~

~~z = x + iy → exp(z) ≡ ex cos y + iex sin y = ex · eiy~~

(see Example 7.2.8). One may verify directly that the exponential function is nonvanishing and holomorphic with derivative function z → exp z (see Exercise 1.6.1) and that exp(z) = exp(z)¯ for each z ∈ C. Fixing w ∈ C, one may also verify directly that exp(w + x) = exp w · exp x for x ∈ R, and the identity theorem (Theo- rem 1.3.3) applied to the holomorphic function z → exp(w + z) then implies that exp(w +z) = exp w ·exp z for all w,z ∈ C. The power series representation centered at 0 is given by ∞ zn exp(z) = ∀z ∈ C. n! n=0 We also denote the exponential function by z → ez. The holomorphic map exp: C → C∗ is surjective, and by the holomorphic inverse function theorem (Theorem 1.5.1), the map is locally biholomorphic. Thus the local inverses determine a multiple-valued holomorphic function, which is called the logarithmic function on C∗. This multiple-valued function is denoted by z → log z, provided there is no danger that it will be confused with the real-valued (and single- valued) logarithmic function x → log x on (0, ∞). The derivative of the logarithmic function is the (single-valued) holomorphic function z → 1/ exp(log z) = 1/z.Alo- cal holomorphic inverse of the exponential function is a single-valued holomorphic branch of the logarithmic function. The difference of any two single-valued holomorphic branches on a domain is a holomorphic function with values in the discrete set 2πiZ, and hence this difference must be constant. We may describe the logarithmic function more explicitly as follows. Recall that polar coordinates in R2 are given by C∞ local inverses of the surjective local diffeomorphism (0, ∞) × R → R2 given by (r, θ) → reiθ (see Examples 7.2.8, 9.2.8, and 9.7.20). Thus the local inverses provide a multiple-valued C∞ function z = reiθ → θ, which is denoted by z → arg z and is called the argument function. For any C∞ local inverse = (ρ, α): → () ⊂ (0, ∞) × R of the map (r, θ) → reiθ on a domain ⊂ C∗ (in particular, ρ : z → |z|), the composition 22 1 Complex Analysis in C of the exponential function with the C∞ function z → L(z) = log |z|+iα(z) is the identity. Thus L is a holomorphic branch of the logarithmic function. Con- versely, given a holomorphic branch L = τ + iα of the logarithmic function on a domain ⊂ C∗ with τ = Re L and α = Im L,wehavez = eτ(z) · eiα(z) for each z ∈ . It follows that τ : z → log |z| and that (eτ ,α) is a C∞ local inverse of the map (r, θ) → reiθ. Thus the logarithmic function is given by

~~z → log |z|+i arg z.~~

In particular, the real-valued logarithmic function on (0, ∞) is the restriction of any branch L = τ + iα of the logarithmic function with α ≡ 0on(0, ∞). For any λ>0 with λ = 1, the exponential function with base λ is the holomorphic function given by z → λz ≡ ez log λ for z ∈ C, and the logarithmic function with → ≡ log z base λ is the multiple-valued holomorphic function given by z logλ z log λ for z ∈ C∗ (i.e., given by the local holomorphic inverses of the exponential function with base λ). For any ζ ∈ C∗, the corresponding power function is the multiple-valued holomorphic function given by z → zζ ≡ exp(ζ log z) for z ∈ C∗.

~~Example 1.6.3 (Trigonometric functions) The trigonometric functions are deﬁned, at each suitable z,by~~

~~i − 1 − sin z ≡− (eiz − e iz), cos z ≡ (eiz + e iz), 2 2 sin z cos z tan z ≡ , cot z ≡ , cos z sin z 1 1 sec z ≡ , csc z ≡ . cos z sin z~~

Exercises for Sect. 1.6 1.6.1 Verify that the exponential function z → exp z is nonvanishing and holomorphic with derivative function z → exp z. 1.6.2 Using the identity exp(w + z) = exp w · exp z, prove that for all w,z ∈ C,

~~sin(w + z) = sin w cos z + cos w sin z, cos(w + z) = cos w cos z − sin w sin z.~~

∞ zn ∈ C 1.6.3 Prove directly that the power series n=0 n! converges for every z , and conclude from Theorem 1.3.1 that the sum determines an entire function E with derivative E = E (thus Theorem 1.3.1 leads to a different approach to the exponential function). 1.6.4 Holomorphic functions of several complex variables. A complex-valued C1 function f on an open subset of Cn is called holomorphic if f is holomorphic in each variable. That is, for each j = 1,...,n, and each choice of 1.6 Examples of Holomorphic Functions 23

z1,...,zj ,...,zn ∈ C, the function ζ → f(z1,...,zj−1,ζ,zj+1,...,zn) is a holomorphic function on the open set {ζ ∈ C | (z1,...,zj−1,ζ,zj+1,...,zn) ∈ } (with the obvious adjustments for j = 1orn). A C1 map F = m (f1,...,fm): → C is called a holomorphic map if fi is a holomorphic function for each i = 1,...,m. For each j = 1,...,n,weset ∂ 1 ∂ ∂ ∂ 1 ∂ ∂ = − i = + i . and ¯ ∂zj 2 ∂x j ∂y j ∂zj 2 ∂x j ∂y j

(a) Prove that a C1 function f on an open set ⊂ Cn is holomorphic if and only if ∂f ≡ 0 ∀j = 1,...,n. ∂z¯j (b) Prove that any composition of holomorphic mappings (of open subsets of complex Euclidean spaces) is holomorphic.

~~Chapter 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms~~

In this chapter, we consider some elementary properties of Riemann surfaces, as well as a fundamental technique called the L2 ∂¯-method, Radó’s theorem on second countability of Riemann surfaces, and analogues of the Mittag-Leffler theorem and the Runge approximation theorem for open Riemann surfaces. Viewing holomorphic functions as solutions of the homogeneous Cauchy–Riemann equation ∂f /∂ z¯ = 0inC allows one to very efficiently obtain their basic properties (see Chap. 1). The intrinsic form of the homogeneous Cauchy–Riemann equation on a Riemann surface is given by ∂f¯ = 0 (see Sect. 2.5). In order to obtain holomorphic functions (and holomorphic 1-forms) on a Riemann surface (even on an open subset of C), it is useful to consider the inhomogeneous Cauchy–Riemann equation ∂α¯ = β. One well-known approach to solving this differential equation (as well as differential equations in many other contexts) is to consider weak solutions in L2. This is the approach taken in this book. In order to do so, we must develop suitable versions of an L2 space of differential forms (see Sect. 2.6) and an (intrinsic) distributional ∂¯ operator (see Sect. 2.7). The relatively simple approaches to the above appearing in this book are, in part, special to Riemann surfaces; but they do contain important elements of the higher-dimensional versions (see, for example, [Hö]or [De3] for the higher-dimensional versions). In this chapter, we consider only scalar-valued differential forms. In Chap. 3,we will consider the analogue for forms with values in a holomorphic line bundle. The solution in line bundles is more efficient in some ways, and it also generalizes more readily to higher-dimensional complex manifolds. In Sects. 2.1–2.9, we consider the definition and basic properties of a Riemann surface, the L2 spaces of differential forms, and the fundamental theorem regarding the solution of the (inhomogeneous) Cauchy–Riemann equation for scalar-valued differential forms. In the remaining sections, we apply the above to obtain some important facts, namely, the existence of meromorphic 1-forms and functions, Radó’s theorem on second countability of Riemann surfaces, the Mittag-Leffler theorem, and the Runge approximation theorem (see [R] for a historical perspective).

T. Napier, M. Ramachandran, An Introduction to Riemann Surfaces, Cornerstones, 25 DOI 10.1007/978-0-8176-4693-6_2, © Springer Science+Business Media, LLC 2011 26 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms

~~2.1 Deﬁnitions and Examples~~

~~In this section, we consider the deﬁnition of a Riemann surface and some examples.~~

Deﬁnition 2.1.1 Let X be a Hausdorff space. (a) A homeomorphism : U → U of an open set U ⊂ X onto an open set U ⊂ C is called a local complex chart of dimension 1(ora1-dimensional local complex chart)inX. We also denote this local complex chart by (U,,U). (b) Two 1-dimensional local complex charts (U1,1,U1) and (U2,2,U2) in X are holomorphically compatible if the coordinate transformations

◦ −1 : ∩ → ∩ 1 2 2(U1 U2) 1(U1 U2) and ◦ −1 −1 = ◦ −1 : ∩ → ∩ 1 2 2 1 1(U1 U2) 2(U1 U2) are holomorphic; that is, each is a biholomorphism (see Fig. 2.1). (c) A family of holomorphically compatible 1-dimensional local complex charts A ={ } = (Ui,i,Ui ) i∈I that covers X (i.e., that satisﬁes X i∈I Ui )iscalled a holomorphic atlas of dimension 1onX. (d) Two 1-dimensional holomorphic atlases A1 and A2 on X are holomorphically equivalent if A1 ∪ A2 is a holomorphic atlas (this is an equivalence relation). (e) An equivalence class R of holomorphic atlases on X is called a holomorphic structure (or a complex analytic structure) of dimension 1 on X, and the pair (X, R) (which is usually denoted simply by X)iscalledacomplex manifold of dimension 1(oracomplex 1-manifold or a complex analytic manifold of dimension 1). If, in addition, X is connected, then the pair (X, R) (which again is usually denoted simply by X) is called a Riemann surface. (f) A 1-dimensional local complex chart (U,,U) in a holomorphic atlas in the holomorphic structure on a complex 1-manifold X is called a local holomorphic chart in X. Setting z = , we call z a local holomorphic coordinate, and for each point p ∈ U, we call (U, z) a local holomorphic coordinate neighborhood of p.

~~Fig. 2.1 Holomorphic coordinate transformations 2.1 Deﬁnitions and Examples 27~~

Remarks 1. Higher-dimensional complex manifolds are also considered in this book, but only in some of the exercises (see, for example, Exercise 2.2.6). For this reason, all local complex charts, all local holomorphic charts, and all holomorphic atlases should be assumed to be of dimension 1 unless otherwise indicated. 2. A holomorphic structure on X determines a unique underlying real 2-dimensional C∞ (in fact, real analytic) structure, with C∞ atlas consisting of local C∞ coordinates given by (x, y) = (Re z, Im z) for any local holomorphic coordinate z (see Chap. 9). A map from (to) an open subset of X from (respectively, to) an open subset of a manifold or complex 1-manifold is said to be of class Ck if the map is of class Ck with respect to the underlying Ck structures. 3. In many treatments, manifolds are assumed to be second countable (often without comment). In fact, according to a theorem of Radó that will be proved in Sect. 2.11, every Riemann surface is automatically second countable. 4. A local holomorphic chart (U,,V) and a local holomorphic coordinate neighborhood (U, z = ) are really the same object, but one generally uses the former terminology when emphasizing the mapping properties, and the latter when emphasizing the coordinate properties. 5. We will call a local holomorphic coordinate neighborhood (U, z = ) in which (U) is a disk a holomorphic coordinate disk (or simply a coordinate disk). We will use the analogous terminology in similar contexts (for example, a holomorphic coordinate annulus will refer to a local holomorphic coordinate neighborhood with image an annulus). Similarly, given a set S ⊂ U and a property of (U),we will often say that S has this property in (or with respect to) the local coordinate neighborhood. For example, if (S) is a closed rectangle in R2, then we will say that S is a closed rectangle in (U, z). We will also use terminology analogous to the above in the context of topological and C∞ manifolds. 6. It turns out that the theory of compact Riemann surfaces and that of noncompact Riemann surfaces differ. A noncompact Riemann surface is also called an open Riemann surface. 7. For convenience, most of the main theorems, as well as some of the elementary facts, in this book are stated as applying to Riemann surfaces. However, analogues, with the appropriate modiﬁcations, also hold for complex 1-manifolds (see Exam- ple 2.1.4 below).

~~We now consider some examples. The veriﬁcations of some details are left to the reader.~~

~~Example 2.1.2 (The complex plane) The complex plane C together with the holomorphic structure determined by the holomorphic atlas {(C,z → z, C)} is a Rie- mann surface.~~

Example 2.1.3 An open set in a complex 1-manifold X has a natural induced ∩ ∩ holomorphic structure in which each triple of the form (U , U∩,(U )), where (U,,U) is a local holomorphic chart in X, is a local holomorphic chart in (see Exercise 2.1.1). Unless otherwise indicated, we take this to be the holomorphic structure on such a given open subset . 28 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms

~~Fig. 2.2 The Riemann sphere~~

≡ Example 2.1.4 Let X α∈A Xα be a disjoint union of complex 1-manifolds {Xα}α∈A, and let ια : Xα → X be the inclusion map for each index α ∈ A. Then X has a unique (natural) holomorphic structure in which each triple of the form ◦ −1 (ια(U), ια ,U ), where (U,,U ) is a local holomorphic chart in Xα for some α ∈ A, is a local holomorphic chart in X (see Exercise 2.1.2). Unless otherwise indicated, we take this to be the holomorphic structure on such a given disjoint union. In particular, every complex 1-manifold is equal to a disjoint union of Riemann surfaces, that is, of its connected components. Consequently, most statements about Riemann surfaces may be easily modiﬁed to give an analogous statement about complex 1-manifolds.

Example 2.1.5 (The Riemann sphere) The Riemann sphere (or extended complex plane) is the one-point compactiﬁcation P1 = C ∪{∞}of C (see Deﬁnition 9.1.11) together with the holomorphic structure determined by the holomorphic atlas

~~∗ {(C,z → z, C), (C ∪{∞},z → 1/z, C)}~~

(where 1/∞=0). The Riemann sphere is diffeomorphic to the unit sphere S2 ⊂ R3 (see Examples 9.2.3) under the stereographic projection S2 → P1 (see Fig. 2.2 and Exercise 2.1.3), i.e., the mapping (x ,x ,x ) → x1 + i x2 ((0, 0, 1) → ∞). 1 2 3 1−x3 1−x3

Example 2.1.6 (Complex tori) A lattice in C is a subgroup of the form = Zξ1 + Zξ2, where ξ1,ξ2 ∈ C are complex numbers that are linearly independent over R. We may associate to an equivalence relation ∼ given by

~~z ∼ w ⇐⇒ z − w ∈~~

(in other words, the equivalence class of each element z ∈ C is z + ). Let us denote the corresponding quotient space and quotient mapping by ϒ : C → X. That is, X is the set of equivalence classes for ∼, ϒ is the mapping z → z + , and X is given the quotient topology (i.e., U ⊂ X is open if and only if its inverse image ϒ−1(U) is open). Observe that ϒ is an open mapping. For if U ⊂ C is open, then ϒ−1(ϒ(U)) is { + } = + ∈ C the union of the open sets U ξ ξ∈ . For each point z u1ξ1 u2ξ2 with 2.1 Deﬁnitions and Examples 29

~~Fig. 2.3 A complex torus u1,u2 ∈ R,theset 1 1 Uz ≡ z + t ξ + t ξ |−~~

2 is the image of the open square (u1 − 1/2,u1 + 1/2) × (u2 − 1/2,u2 + 1/2) ⊂ R 2 under the real linear isomorphism R → C given by (t1,t2) → t1ξ1 +t2ξ2, and hence Uz is a relatively compact neighborhood of z in C (see Fig. 2.3). It is also easy to check that

Uz ∩ (Uz + ( \{0})) =∅ and ϒ(U z) = X. In particular, the openness of ϒ, together with the ﬁrst equality, implies that ϒ maps Uz homeomorphically onto ϒ(Uz); and the second equality implies that X is compact. Furthermore, X is Hausdorff. For it is clear that each of the sets ϒ(Uz) is Hausdorff, while if w ∈ ∂Uz, then the neighborhoods 1 1 V ≡ z + t ξ + t ξ |−

If U ⊂ C is an open set with U ∩ (U + ξ) =∅for each ξ ∈ \{0}, then ϒ maps U homeomorphically onto an open set U ⊂ X. Thus we get a local com- −1 plex chart (U, (ϒ U ) ,U ) in X. The collection of such local complex charts de- −1 termines a holomorphic structure on X.Forif(V, (ϒ V ) ,V ) is another such 30 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms

−1 local complex chart in X and ≡ (ϒ ) ◦ ϒ −1 ∩ is the associ- V (ϒ U ) (U V) ated coordinate transformation, then z → (z) − z is a continuous function on ∩ + = −1 ∩ U (V ) (ϒ U ) (U V) with values in the discrete set . Hence this function must be locally constant, and it follows that is holomorphic. Thus X is a compact Riemann surface. The Riemann surface X is called a complex torus because topologically, X is the torus T2 = S1 × S1. In fact, we have a commutative diagram of C∞ maps α R2 C

ϒ0 ϒ β T2 X where α is the real linear isomorphism (and therefore diffeomorphism) given by ∞ 2 1 1 (t1,t2) → t1ξ1 + t2ξ2, ϒ0 is the C mapping onto the real torus T = S × S 2πit 2πit given by (t1,t2) → (e 1 ,e 2 ), and the induced mapping β is a well-deﬁned diffeomorphism. For it is easy to verify that β is a well-deﬁned bijection. Moreover, 2 given a point t = (t1,t2) ∈ R , ϒ0 maps the neighborhood V ≡ (t1 −1/2,t1 +1/2)× (t2 − 1/2,t2 + 1/2) diffeomorphically onto a neighborhood of ϒ0(t) (with inverse given by the product of 1/2π and a C∞ argument function in each coordinate). Since locally, β is a composition of the C∞ map ϒ,theC∞ map α, and a local C∞ ∞ −1 ∞ inverse of ϒ0, β must be a C map. Similarly, β is also a C map, and therefore β is a diffeomorphism.

Exercises for Sect. 2.1 2.1.1 Verify that in any open subset of a complex 1-manifold, the local complex charts described in Example 2.1.3 form a holomorphic atlas. 2.1.2 Verify that in any disjoint union of complex 1-manifolds, the local complex charts described in Example 2.1.4 form a holomorphic atlas. 2.1.3 Verify that the Riemann sphere (Example 2.1.5) is a Riemann surface, and verify that the stereographic projection (see Fig. 2.2) is a diffeomorphism of the unit sphere S2 onto P1.

~~2.2 Holomorphic Functions and Mappings~~

~~Given a holomorphic structure, one gets the corresponding holomorphic functions and mappings.~~

Deﬁnition 2.2.1 Let X and Y be complex 1-manifolds. (a) A holomorphic function on an open set ⊂ X is a complex-valued function f on such that f ◦ −1 ∈ O(( ∩ U)) for every local holomorphic chart (U,,U) in X. We denote the set of holomorphic functions on by O(). (b) A continuous mapping : X → Y is holomorphic if the function ◦ belongs to O( −1(U)) for every local holomorphic chart (U,,U) in Y . 2.2 Holomorphic Functions and Mappings 31

(c) A bijective holomorphic mapping : X → Y with holomorphic inverse is called a biholomorphism (or a biholomorphic mapping). If such a biholomorphism exists, then we say that X and Y are biholomorphically equivalent (or simply biholomorphic). For Y = X,wealsocall an automorphism of X. (d) A holomorphic mapping : X → Y is a local biholomorphism (or a locally biholomorphic mapping) if maps a neighborhood of each point in X biholomorphically onto an open subset of Y .

Remarks 1. A holomorphic function on an open subset of a complex 1-manifold X is precisely a holomorphic mapping of into C. 2. A function f on an open subset of a complex 1-manifold X is holomorphic if and only if for each point p ∈ , there exists a local holomorphic chart (U,,U) in X such that p ∈ U and f ◦−1 ∈ O((∩U)). Similarly, a continuous mapping of complex 1-manifolds : X → Y is holomorphic if and only if for each point p ∈ X, there exists a local holomorphic chart (U,,U) in Y such that (p) ∈ U and ◦ ∈ O( −1(U)) (see Exercise 2.2.1). Furthermore, a holomorphic mapping of complex 1-manifolds is of class C∞ with respect to the underlying C∞ structures (in fact, the mapping is real analytic with respect to the underlying real analytic structures). 3. Biholomorphic equivalence is an equivalence relation (see Exercise 2.2.2), and we usually identify any two biholomorphic complex 1-manifolds. 4. Any sum or product of holomorphic functions is holomorphic, and any composition of holomorphic mappings is holomorphic (see Exercise 2.2.3). In particular, the set of holomorphic functions on an open set is an algebra. 5. In Example 2.1.6, the quotient mapping ϒ : C → X of C to the complex torus X is locally biholomorphic, because the local holomorphic charts are given by local inverses of ϒ. Although all complex tori are diffeomorphic, they are not all biholomorphic (see Exercise 5.9.1).

~~Many of the elementary theorems for holomorphic functions on domains in the plane immediately give analogues for holomorphic mappings on Riemann surfaces. For example, we have the following:~~

Theorem 2.2.2 Let X and Y be Riemann surfaces. (a) Identity theorem. If : X → Y is a nonconstant holomorphic mapping, then the ﬁber −1(p) over each point p ∈ Y is discrete in X (i.e., −1(p) has no limit points in X). Moreover, if : X → Y is a holomorphic map that is not identically equal to , then {x ∈ X | (x) = (x)} is discrete in X. (b) Riemann’s extension theorem. A continuous mapping : X → Y that is holomorphic on the complement of a discrete subset of X is holomorphic. (c) Maximum principle. If f ∈ O(X) and |f | attains a local maximum at some point p ∈ X, then f is constant. In particular, if X is compact, then every holomorphic function on X is constant. (d) Open mapping theorem. If : X → Y is a nonconstant holomorphic mapping, then the image (X) is open. 32 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms

Proof The proof of the first part of (a) is provided here, and the proofs of the second part of (a) and of (b)–(d) are left to the reader (see Exercise 2.2.4). Suppose : X → Y is a holomorphic mapping for which the fiber F ≡ −1(p) over some point p ∈ Y has a limit point q ∈ X. We may fix local holomorphic charts (U, ,U) in X and (V,,V) in Y such that q ∈ U, U is connected, and (U) ⊂ V .It follows that for the holomorphic function f ≡ ◦ ◦ −1 : U → C, the fiber f −1((p)) = (F ∩ U) has a limit point (q) in U , and hence f ≡ (p) on = −1 ◦ U by the identity theorem in the plane (see Corollary 1.3.3). Thus U ◦ f ◦ ≡ p, and therefore U ⊂ F . Hence the interior F of F is nonempty, and by the ◦ above argument, ∂(F)=∅. It follows that F = X, and hence that is constant.

Lemma 2.2.3 (Local representation of holomorphic mappings) Let p be a point in a Riemann surface X. (a) If f is a nonconstant holomorphic function on X, then there exist a positive integer m and a local holomorphic coordinate neighborhood (U, z) of p such that z(p) = 0 and f = f(p)+ zm on U. Moreover, m = m(f, p) is unique, and for any local holomorphic coordinate neighborhood (W, w) of p, the function (w − w(p))−m(f − f(p))∈ O(W \{p}) extends to a holomorphic function on W that does not vanish at p. (b) If : X → Y is a nonconstant holomorphic mapping of X toaRiemannsur- face Y , then for some m ∈ Z>0 and for every local holomorphic coordinate neighborhood (V, ζ ) of (p) in Y , there is a local holomorphic coordinate neighborhood (U, z) of p in X such that z(p) = 0 and ζ( )= ζ( (p))+ zm on U ∩ −1(V ). Moreover, the integer m = m( , p) is unique, and for any local holomorphic coordinate neighborhood (W, w) of p, the holomorphic function (w − w(p))−m(ζ( ) − ζ( (p))) on W ∩ −1(V ) \{p} extends to a holomorphic function on W ∩ −1(V ) that does not vanish at p.

Proof For the proof of (a), we may fix a local holomorphic chart (U0,= w,V0) with p ∈ U0. Corollary 1.3.3 then provides a unique integer m ∈ Z>0 and a unique m function g ∈ O(U0) such that g(p) = 0 and f − f(p)= (w − w(p)) · g on U0. Choosing U0 to be sufficiently small, we also get a holomorphic branch L of L(ζ ) the logarithmic function on a neighborhood of g(U0) (that is, e = ζ ), and hence we have the holomorphic mth root function eL/m. The holomorphic func- L(g)/m m tion z ≡ (w − w(p)) · e then satisfies z(p) = 0 and f = f(p)+ z on U0. Moreover, − z ◦ 1 (w(p)) = eL(g(p))/m = 0, so the holomorphic inverse function theorem (Theorem 1.5.1) implies that the function z maps some neighborhood U of p in U0 biholomorphically onto a neighborhood of 0 in C. Finally, we have uniqueness of m.Foriff − f(p)= ζ n on Q for some n ∈ Z>0 and some local holomorphic chart (Q, = ζ,Q ) with p ∈ Q, then by Corollary 1.3.3, since (z ◦ −1)(0) = 0, there is a holomorphic function h on 2.2 Holomorphic Functions and Mappings 33 a neighborhood of 0 in C with h(0) = 0 and z( −1(ξ)) = ξh(ξ) for each ξ ∈ Q near 0. Hence, for each ξ near 0, ξ n = f −1(ξ) − f(p)= z −1(ξ) m = ξ m(h(ξ))m.

~~The uniqueness part of Corollary 1.3.3 implies that n = m (and hm ≡ 1). Thus (a) is proved. The proof of (b) is left to the reader (see Exercise 2.2.5).~~

Deﬁnition 2.2.4 Let be an open subset of a complex 1-manifold X, and let p ∈ . (a) If f is a holomorphic function on and m is a positive integer such that f = zmg on a neighborhood p for some local holomorphic coordinate z with z(p) = 0 and some nonvanishing holomorphic function g (i.e., f(p)= 0 and m is the integer provided by part (a) of Lemma 2.2.3), then we say that f has a zero of order m at p.Form = 1, we also say that f has a simple zero at p. (b) If : → Y is a holomorphic mapping into a complex 1-manifold Y and m is a positive integer such that ζ( ) − ζ( (p)) has a zero of order m at p for some local holomorphic coordinate ζ on a neighborhood of (p) (i.e., m is the integer provided by part (b) of Lemma 2.2.3), then we say that has multiplicity m at p, and we write multp = m.

Deﬁnition 2.2.5 A meromorphic function on an open subset of a complex 1-manifold X is a function f : \ P → C on the complement \ P of a discrete subset P of such that for each point p ∈ P , |f(x)|→∞as x → p. Each point p ∈ P is called a pole of f . We denote the set of meromorphic functions on by M().

Proposition 2.2.6 For any holomorphic function f on the complement X \ P of a discrete subset P of a complex 1-manifold X, the following are equivalent: (i) The function f is a meromorphic function on X with pole set P . (ii) There exists a holomorphic mapping h: X → P1 such that h = f on X \ P and P = h−1(∞). (iii) For each point p ∈ P and for every local holomorphic coordinate neighborhood (U, z) of p in X, there exist a nonvanishing holomorphic function g on a neighborhood V of p in U and a positive integer νp such that − f = (z − z(p)) νp g on V \{p}. (iv) For each point p ∈ P , there exist a local holomorphic coordinate neighborhood (U, z) of p in X, a nonvanishing holomorphic function g on a neighbor- −ν hood V of p in U, and a positive integer νp such that f = (z − z(p)) p g on V \{p}. (v) For each point p ∈ P , there exist a local holomorphic coordinate neighborhood (U, z) of p in X and a positive integer νp such that z(p) = 0 and − f = z νp on U \{p}. 34 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms

~~Moreover, in the above, for each point p ∈ P , the integer νp in (iii)Ð(v) is equal to the multiplicity at p of the holomorphic mapping h in (ii).~~

Proof Clearly, any one of the conditions (ii)–(v) implies (i). Conversely, if f is meromorphic with pole set P , then f extends to a unique continuous mapping h: → P1 with h−1(∞) = P , and Riemann’s extension theorem (Theorem 2.2.2) implies that h is a holomorphic mapping; that is, (ii) holds. If p ∈ P and νp = multph, then by the local representation of holomorphic mappings (Lemma 2.2.3), for every local holomorphic coordinate neighborhood (U, z) of p, there is a nonvanishing holomorphic function u on a neighborhood V of p in U such that − − h 1(∞) ∩ V ={p} and u = (z − z(p)) νp · (1/h) on V \{p}. Setting g ≡ 1/u, we get (iii), and (iv) follows. Moreover, we may choose (U, z) so that z(p) = 0 and 1/h= zνp on U, so we get (v) as well.

Deﬁnition 2.2.7 We identify any meromorphic function f on an open subset of a complex 1-manifold X with the associated holomorphic mapping → P1.If p ∈ f −1(∞) is a pole of f at which this mapping has multiplicity m, then we say that f has a pole of order m at p. We say that f has a zero of order m at q ∈ if ∈ −1 q f (0) and the holomorphic function f \f −1(∞) has a zero of order m at q (note that this is consistent with the above identiﬁcation). For any point p ∈ ,the order of f at p is given by ⎧ − ⎪0ifp/∈ f 1({0, ∞}), ⎨⎪ m if f has a zero of order m at p, ordpf ≡ ⎪−m if f has a pole of order m at p, ⎩⎪ ∞ if f ≡ 0 on a neighborhood of p.

~~We say that f has a simple zero (simple pole) at p if ordp f = 1 (respectively, ordpf =−1).~~

Remarks 1. If f and g are meromorphic functions on a connected neighborhood of a point p in a complex 1-manifold X and neither f nor g is identically zero, then in terms of some local holomorphic coordinate z on a neighborhood of p,we m n have f = z f0 and g = z g0 for some pair of integers m and n and some pair of nonvanishing holomorphic functions f0 and g0 on a neighborhood of p. It follows that each of the functions f + g, fg, and f/g is holomorphic on a neighborhood of p or has a removable singularity or a pole at p. Consequently, any sum, product, or quotient (provided the denominator does not vanish identically on any open set) of meromorphic functions is a meromorphic function, provided we holomorphically extend the resulting function over any removable singularities. In other words, for any connected open subset of a complex 1-manifold, M() is a ﬁeld. 2. Suppose X is a complex 1-manifold, (U, z) is a local holomorphic coordinate ∈ ∈ O \{ } = ∞ − n neighborhood of a point p X, f (X p ), and f n=−∞ cn(z z(p)) is the corresponding Laurent series representation of f about p provided by The- orem 1.3.6. Then f has a pole of order m ∈ Z>0 at p if and only if cn = 0 for all n<−m and c−m = 0. 2.3 Holomorphic Attachment 35

Exercises for Sect. 2.2 2.2.1 Prove that a function f on an open subset of a complex 1-manifold X is holomorphic if and only if for each point p ∈ , there exists a local holomorphic chart (U,,U) in X such that p ∈ U and f ◦ −1 ∈ O(( ∩ U)). Prove also that a continuous mapping of complex 1-manifolds : X → Y is holomorphic if and only if for each point p ∈ X, there exists a local holomorphic chart (U,,U) in Y such that (p) ∈ U and ◦ ∈ O( −1(U)). 2.2.2 Prove that biholomorphic equivalence of complex 1-manifolds is an equivalence relation. 2.2.3 Prove that any sum or product of holomorphic functions on a complex 1- manifold is holomorphic, and any composition of holomorphic mappings of complex 1-manifolds is holomorphic. 2.2.4 Prove the second statement in part (a) of Theorem 2.2.2 (i.e., if , : X → Y are distinct holomorphic mappings of Riemann surfaces, then {x ∈ X | (x) = (x)} is discrete in X). Also prove parts (b)–(d) of Theorem 2.2.2. 2.2.5 Prove part (b) Lemma 2.2.3. 2.2.6 Complex manifolds. A holomorphic atlas of dimension n on a Hausdorff space X is an atlas in which each element is a homeomorphism : U → U of an open set U ⊂ X onto an open set U ⊂ Cn and the coordinate transformations are holomorphic mappings (see Exercise 1.6.4 for the deﬁnition). Two holomorphic atlases of dimension n are holomorphically equivalent if their union is a holomorphic atlas. A complex manifold of dimension n is a Hausdorff space X together with an equivalence class of n-dimensional holomorphic atlases (i.e., an n-dimensional holomorphic structure). The elements of any atlas in the holomorphic structure are called local holomorphic charts. A continuous mapping of complex manifolds is holomorphic if its representation in local holomorphic charts is holomorphic. Prove that the Cartesian product of a pair of Riemann surfaces has a natural 2-dimensional holomorphic structure. 2.2.7 Prove that no two of the Riemann surfaces C, P1, and ≡ (0; 1) are biholomorphic. 2.2.8 Prove the following: (a) Liouville’s theorem. Every bounded entire function is constant. Hint. Apply Theorem 1.2.10 near ∞ in P1 ⊃ C. (b) The fundamental theorem of algebra (Gauss). Every nonconstant complex polynomial has a zero. Hint. Given a nonvanishing complex polynomial function g, consider the holomorphic function 1/g.

~~2.3 Holomorphic Attachment~~

One may produce inﬁnitely many examples of Riemann surfaces by holomorphic attachment. In fact, one goal of this book is a proof (appearing in Chap. 5) that every Riemann surface may be obtained by holomorphic attachment of tubes to a domain 36 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms in the Riemann sphere P1. Holomorphic attachment is not required until Chap. 5,so a reader who prefers to skip this section and move on to other topics in this chapter may do so without fear of missing a required concept. It will be convenient to have holomorphic attachment formulated as follows:

Proposition 2.3.1 (Holomorphic attachment) Let X and X be complex 1-manifolds, and let : G → G be a biholomorphism of an open set G ⊂ X onto an open set G ⊂ X such that −1(K ∩ G) is closed in X for every compact set K ⊂ X (i.e., (x) →∞in the one-point compactiﬁcation of X as x → p ∈ ∂G) and (K ∩ G) is closed in X for every compact set K ⊂ X (i.e., −1(x) →∞ in the one-point compactiﬁcation of X as x → p ∈ ∂G). Let ι: X→ X X and ι : X → X X be the inclusion mappings of X and X into the disjoint union X X, let ∼ be the equivalence relation in the disjoint union X X determined by ι(x) ∼ ι( (x)) for every point x ∈ G, and let

: X X → Y = X ∪ X ≡ (X X )/∼ be the associated quotient space. Then there is a unique 1-dimensional holomorphic structure on Y with respect to which ◦ ι and ◦ ι are biholomorphisms of X and X, respectively, onto open subsets of Y (equivalently, there is a unique holomorphic structure on Y with respect to which is a local biholomorphism).

~~The proof is left to the reader (see Exercise 2.3.1).~~

~~Deﬁnition 2.3.2 For X ⊃ G, X ⊃ G, and : G → G as in Proposition 2.3.1,the complex 1-manifold X ∪ X is called the holomorphic attachment of X and X along .~~

Remarks 1. It is customary and convenient to identify X and X with their (disjoint) images in X X. However, although it is customary to leave out any explicit mention of the inclusion maps ι and ι, we will often mention the inclusion maps when considering specific mappings of the above spaces. This will allow us to avoid any danger of confusion (for example, there is danger of confusion whenever X = X). 2. The natural identification of X X with X X gives a natural identification ∪ ∪ of X X with X −1 X.

In applications, usually one ﬁrst removes a set and then holomorphically attaches a new set of a desired type. For example, several key arguments in Chap. 5 will require that we replace sets with caps (i.e., disks) or tubes (i.e., annuli). For now, we consider the following:

Example 2.3.3 (Holomorphic attachment of a tube) Let Y be a complex 1-manifold; { ; } let R0,R1 > 1; let (Dν ,ν,(0 Rν)) ν∈{0,1} be disjoint local holomorphic charts ≡ −1 ; ⊂ = ≡ ; in Y ;letAν ν ((0 1,Rν )) Dν for ν 0, 1; let T (0 1/R0,R1); and 2.3 Holomorphic Attachment 37

Fig. 2.4 Holomorphic attachment of a tube let A0 ≡ (0; 1/R0, 1) ⊂ T and A1 ≡ (0; 1,R1) ⊂ T . Thus we get a biholomor- : ∪ → ∪ phism A0 A1 A0 A1 given by − 1(1/z) ∈ A if z ∈ A , (z) = 0 0 0 −1 ∈ ∈ 1 (z) A1 if z A1. − − For Z ≡ Y \[ 1((0; 1)) ∪ 1((0; 1))]⊃A ∪ A , −1(K ∩ (A ∩ A )) is 0 1 0 1 0 1 closed in T for every compact set K ⊂ Z, and (K ∩ (A0 ∪ A1)) is closed in Z for every compact set K ⊂ T . We may therefore form the holomorphic attachment ≡ ∪ = ∼ ∈ ∈ X Z T Z T/ , where for p A0 and z A0, the images p0 of p and z0 → ∼ · of z in Z T under the inclusions A0,A0 Z T satisfy p0 z0 if and only if z = ∈ ∈ ∼ 0(p) 1, and for p A1 and z A1, the respective images p0 and z0 satisfy p0 z0 if and only if z = 1(p) (see Fig. 2.4). In other words, X is a complex 1-manifold obtained by removing the unit disks in each of the coordinate disks D0 and D1 and gluing in a tube (i.e., an annulus) T (equivalently, the boundaries of the unit disks are glued together). We call X the complex 1-manifold obtained by holomorphic { ; } attachment of the tube T to Y at the coordinate disks (Dν,ν,(0 Rν )) ν∈{0,1} (or simply at {Dν}ν∈{0,1}). This is a slight abuse of language, since actually we ﬁrst −1 ; ∪ −1 ; removed the set 0 ((0 1)) 1 ((0 1)) before performing the attachment. 38 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms

Observe that if Y is connected, then X is connected; and if Y is compact, then X is compact. If X is compact, then Y is compact. However, X may be connected even if Y is not connected (see Exercise 2.3.2). ∗ ∗ ≤ = Fixing Rν with 1

Exercises for Sect. 2.3 2.3.1 Prove Proposition 2.3.1. 2.3.2 Let X be a complex 1-manifold obtained by holomorphically attaching a tube to a complex 1-manifold Y as in Example 2.3.3. Prove that X is connected if Y is connected. Give an example that shows that X may be connected even if Y is not. 2.3.3 Write out a speciﬁc example of holomorphic attachment of a tube to the Rie- mann sphere (observe that the resulting Riemann surface appears to have the topological type of a torus). 2.3.4 In the notation of Example 2.3.3, verify that the natural inclusion of Z T ∗ into Z T descends to a biholomorphism of X∗ onto X. 2.3.5 Let X and X be complex 1-manifolds, let : G → G be a biholomorphism of an open set G ⊂ X onto an open set G ⊂ X,letι: X→ X X and ι : X → X X be the inclusion mappings of X and X into the disjoint union X X,let∼ be the equivalence relation in the disjoint union X X determined by ι(x) ∼ ι( (x)) for every point x ∈ G, and let Y ≡ (X X)/∼ be the associated quotient space. Prove that if there exists a compact set K ⊂ X for which −1(K ∩G) is not closed in X, then Y is not Hausdorff.

~~2.4 Holomorphic Tangent Bundle~~

~~A complex 1-manifold X has an underlying real 2-dimensional C∞ structure, and therefore a tangent bundle and a complexiﬁed tangent bundle~~

TX: TX→ X and (T X)C : (T X)C → X, respectively (see Sect. 9.4). We recall that for p ∈ X, a tangent vector v ∈ (TpX)C is a linear derivation on the germs of C∞ functions at p; that is, for every pair 2.4 Holomorphic Tangent Bundle 39 of C∞ functions f,g on a neighborhood of p,wehavev(fg) = v(f ) · g(p) + f(p) · v(g) ∈ C. Given a local holomorphic coordinate neighborhood (U, z = x + iy), the vector fields ∂/∂x and ∂/∂y form a basis for the tangent space at each point in U. Furthermore, the corresponding complex vector fields ∂/∂z and ∂/∂z¯ as in Chap. 1 form a different complex basis for the complexified tangent space at each point, and their spans yield a natural decomposition into a sum of 1- dimensional vector spaces. Moreover, a C1 function g on U is holomorphic if and only if ∂g/∂z¯ = ∂g/∂z¯ = 0. Similarly, the differentials dz and dz¯ form a dual basis that yields a decomposition of the complexified cotangent space at each point. The precise definitions and verifications appear below.

~~Deﬁnition 2.4.1 Let X be a complex 1-manifold.~~

(a) For each point p ∈ X, a tangent vector v ∈ (TpX)C is of type (1, 0) (of type (0, 1))ifdf(v)¯ = v(f)¯ = 0 (respectively, df (v) = v(f ) = 0) for every 1,0 holomorphic function f on a neighborhood of p. The subspace (TpX) of (TpX)C formed by the tangent vectors at p of type (1, 0) is called the holo- 0,1 morphic tangent space (or (1, 0)-tangent space)atp. The subspace (TpX) ⊂ (TpX)C formed by the tangent vectors at p of type (0, 1) is called the (0, 1)- tangent space at p. The spaces = : 1,0 ≡ 1,0 → (T X)1,0 (T X)C (T X)1,0 (T X) (TpX) X p∈X

and = : 0,1 ≡ 0,1 → (T X)0,1 (T X)C (T X)0,1 (T X) (TpX) X p∈X are called the holomorphic tangent bundle (or (1, 0)-tangent bundle) and (0, 1)- tangent bundle, respectively. ∈ ∈ ∗ (b) For each p X, an element α (Tp X)C is of type (1, 0) (of type (0, 1))if 0,1 1,0 α(v) = 0 for every tangent vector v ∈ (TpX) (respectively, v ∈ (TpX) ). ∗ 1,0 ⊂ ∗ The subspace (Tp X) (Tp X)C formed by the elements of type (1, 0) is called the holomorphic cotangent space (or (1, 0)-cotangent space)atp.The ∗ 0,1 ⊂ ∗ subspace (Tp X) (Tp X)C formed by the elements of type (0, 1) is called the (0, 1)-cotangent space at p. The spaces ∗ ∗ = ∗ : 1,0 ≡ 1,0 → (T ∗X)1,0 (T X)C (T ∗X)1,0 (T X) (Tp X) X p∈X

~~and ∗ ∗ = ∗ : 0,1 ≡ 0,1 → (T ∗X)0,1 (T X)C (T ∗X)0,1 (T X) (Tp X) X p∈X~~

are called the holomorphic cotangent bundle (or (1, 0)-cotangent bundle) and (0, 1)-cotangent bundle, respectively. The holomorphic cotangent bundle 40 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms

(T ∗X)1,0 is also called the canonical line bundle (or canonical bundle) and is : → also denoted by KX KX X. (c) Given a local holomorphic coordinate neighborhood (U, = z = x + iy) in X (with x = Re z and y = Im z), we call the vector ﬁelds ∂ 1 ∂ ∂ ∂ 1 ∂ ∂ ≡ − i and ≡ + i = ∂/∂z ∂z 2 ∂x ∂y ∂z¯ 2 ∂x ∂y

~~the partial derivative operators with respect to z and z¯, respectively. In other words, denoting the standard complex coordinate on C by w,wehave,forany suitable function f ,~~

~~∂f ∂(f ◦ −1) ∂f ∂(f ◦ −1) = ◦ and = ◦ . ∂z ∂w ∂z¯ ∂w¯~~

~~Proposition 2.4.2 Let X be a complex 1-manifold. (a) For each point p ∈ X, we have direct sum decompositions~~

~~= 1,0 ⊕ 0,1 ∗ = ∗ 1,0 ⊕ ∗ 0,1; (TpX)C (TpX) (TpX) and (Tp X)C (Tp X) (Tp X)~~

~~and we have isomorphisms~~

~~∼ ∼ ∗ 1,0 →= 1,0 ∗ ∗ 0,1 →= 0,1 ∗ (Tp X) ((TpX) ) and (Tp X) ((TpX) )~~

→ 1,0 → 0,1 given by α α (TpX) and α α (TpX) , respectively. (b) Conjugation gives (T X)1,0 = (T X)0,1 and (T ∗X)1,0 = (T ∗X)0,1. (c) For every local holomorphic coordinate neighborhood (U, z = x +iy) of a point p ∈ X, we have

~~1,0 0,1 (TpX) = C · (∂/∂z)p,(TpX) = C · (∂/∂z)¯ p, ∗ 1,0 = C · ∗ 0,1 = C · ¯ (Tp X) (dz)p,(Tp X) (dz)p,~~

~~and~~

~~dz((∂/∂z)p) = 1,dz((∂/∂¯ z)¯ p) = 1,~~

~~dz((∂/∂z)¯ p) = 0,dz((∂/∂z)¯ p) = 0. Moreover, for every complex number ζ = a + ib with a,b ∈ R, we have ∂ ∂ ∂ ∂ a + b = ζ + ζ¯ ¯ ∂x p ∂y p ∂z p ∂z p~~

~~and ζ¯ ζ a(dx)p + b(dy)p = (dz)p + (dz)¯ p. 2 2 2.4 Holomorphic Tangent Bundle 41~~

(d) A C1 function f on an open set ⊂ X is holomorphic if and only if for each point p ∈ , ∂f /∂ z¯ ≡ 0 on U ∩ for some (equivalently, for every) local holomorphic coordinate neighborhood (U, z) of p. If : X → Y is a C1 mapping of X into a complex 1-manifold Y and (r, s) ∈{(1, 0), (0, 1)}, then the following are equivalent: (i) is holomorphic. (ii) ∗(T X)r,s ⊂ (T Y )r,s . (iii) ∗(T ∗Y)r,s ⊂ (T ∗X)r,s .

Proof Let (U, = z, U ) be a local holomorphic chart in X and let p ∈ U. A function f ∈ C1(U) is holomorphic if and only if f ◦ −1 ∈ O(U ).Letw denote the standard complex coordinate on C. Since

∂f ∂(f ◦ −1) ◦ −1 = , ∂z¯ ∂w¯ we see that f is holomorphic if and only if ∂f/∂¯ z¯ = ∂f /∂ z¯ ≡ 0, and hence that 1,0 0,1 (∂/∂z)p ∈ (TpX) \{0} and (∂/∂z)¯ p ∈ (TpX) \{0}. Moreover, ∂ ∂ ∂ ∂ ∂ ∂ = + = i − i . ¯ and ¯ ∂x p ∂z p ∂z p ∂y p ∂z p ∂z p

~~Therefore, since dimC(TpX)C = 2, we have the direct sum decomposition ∂ ∂ 1,0 0,1 (T C = C · ⊕ C · = ⊕ pX) ¯ (TpX) (TpX) . ∂z p ∂z p~~

~~We also have ∂ 1 ∂ i ∂ 1 i 1 i dz = (dx + idy) − = − · 0 + i · · 0 − i · · 1 = 1. ∂z 2 ∂x 2 ∂y 2 2 2 2~~

~~A similar computation gives dz(∂/∂z)¯ = 0, and it follows that~~

~~dz(∂/∂¯ z)¯ = dz(∂/∂z) = 1 and dz(∂/∂z)¯ = dz(∂/∂z)¯ = 0.~~

∈ ∗ 1,0 ¯ ∈ ∗ 0,1 ∗ Thus (dz)p (Tp X) and (dz)p (Tp X) form the basis of (Tp X)C which is dual to the basis (∂/∂z)p,(∂/∂z)¯ p. Parts (a)–(c) now follow easily. If is a C1 mapping of X into a complex 1-manifold Y and (V, ζ ) is a local holomorphic coordinate neighborhood of (p) in Y , then

~~∂(ζ ◦ ) ∗ = d(ζ ◦ )(∂/∂z)¯ = dζ( ∗(∂/∂z))¯ = ( dζ )(∂/∂z)¯ ∂z¯ ∗ = d(ζ¯ ◦ )(∂/∂z) = dζ( ¯ ∗(∂/∂z)) = ( dζ¯ )(∂/∂z).~~

~~Part (c) now gives the equivalence of (i)–(iii) in (d). 42 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms~~

Remarks 1. It follows from the above proposition that if (U, z = x + iy) is a local holomorphic coordinate neighborhood in a complex 1-manifold X, p ∈ U, ∈ ∈ ∗ C1 v (TpX)C, α (Tp X)C, and f is a function on a neighborhood of p, then ∂ ∂ ∂ ∂ v = dx(v) + dy(v) = dz(v) + dz(v)¯ , ¯ ∂x p ∂y p ∂z p ∂z p

~~α = α((∂/∂x)p)(dx)p + α((∂/∂y)p)(dy)p~~

= α((∂/∂z)p)(dz)p + α((∂/∂z)¯ p)(dz)¯ p, ∂f ∂f (df ) = (p)(dz) + (p)(dz)¯ . p ∂z p ∂z¯ p In particular, f is holomorphic if and only if df = (∂f/∂z) · dz.If : X → Y is a holomorphic mapping into a Riemann surface Y , (V, w) is a local holomorphic coordinate neighborhood of (p) in Y , and g = w ◦ , then for all a,b ∈ C, ∂ ∂ ∂g ∂ ∂g ∂ ( ∗) + b = (p) + b (p) . p a ¯ a ¯ ∂z p ∂z p ∂z ∂w (p) ∂z ∂w (p) Consequently, if any one of the following linear mappings is nontrivial (i.e., not the zero mapping), then all of the mappings are isomorphisms:

( ∗)p : (TpX)C → (T(p)Y)C,( ∗)p : TpX → T(p)Y, 1,0 1,0 0,1 0,1 ( ∗)p : (TpX) → (T(p)Y) ,( ∗)p : (TpX) → (T(p)Y) , 1,0 (dg)p : (TpX) → C. 2. We express the decomposition of the complexiﬁed tangent and cotangent bun- ∗ ∗ ∗ dles by writing (T X)C = (T X)1,0 ⊕ (T X)0,1 and (T X)C = (T X)1,0 ⊕ (T X)0,1, and for each pair (r, s) ∈{(1, 0), (0, 1)},welet

~~r,s r,s r,s ∗ ∗ r,s P : (T X)C → (T X) and P : (T X)C → (T X) denote the mappings for which the restrictions~~

r,s 1,0 0,1 r,s P : C = ⊕ → (TpX)C (TpX) (TpX) (TpX) (TpX) and r,s ∗ ∗ 1,0 ∗ 0,1 ∗ r,s P ∗ : (T X)C = (T X) ⊕ (T X) → (T X) (Tp X)C p p p p are the corresponding vector space projections for each point p ∈ X. For any ele- ∗ Pr,s ment ξ of (TpX)C or (Tp X)C, we call ξ the (r, s) part of ξ. 3. The holomorphic tangent bundle (T X)1,0 and cotangent bundle (T ∗X)1,0 are examples of holomorphic line bundles (see Example 3.1.4). The (0, 1) tangent bundle (T X)0,1 and cotangent bundle (T ∗X)0,1 are examples of C∞ line bundles. 4. A reader familiar with vector bundles will recognize the decomposition ∞ (T X)C = (T X)1,0 ⊕ (T X)0,1 as a decomposition of the C vector bundle (T X)C ∞ ∗ into a sum of C subbundles (as is the decomposition of (T X)C). 2.4 Holomorphic Tangent Bundle 43

5. For (r, s) ∈{(1, 0), (0, 1)} and for any open subset of a complex 1-manifold X with inclusion mapping ι: → X, we identify (T )r,s and (T ∗)r,s with the −1 ⊂ r,s −1 ⊂ ∗ r,s sets (T X)r,s () (T X) and (T ∗X)r,s () (T X) , respectively, under the : r,s → −1 ∗ : −1 → ∗ r,s bijections ι∗ (T ) (T X)r,s () and ι (T ∗X)r,s () (T ) , respectively.

~~Guided by Deﬁnition 9.4.5, we make the following deﬁnition:~~

Definition 2.4.3 Let X be a complex 1-manifold. (a) The coefficient functions (or simply the coefficients) of a vector field v on a set S ⊂ X with respect to (or in) a local holomorphic coordinate neighborhood (U, z) are the functions dz(v) = v(z) and dz(v)¯ = v(z)¯ on S ∩ U. (b) We call a vector field v of type (1, 0) on an open set ⊂ X (that is, vp is of type (1, 0) for each point p)aholomorphic vector field if the coefficient function f = dz(v) = v(z): ∩ U → C in every local holomorphic coordinate neighborhood (U, z) is holomorphic (we have dz(v)¯ = v(z)¯ = 0, since v is of type (1, 0)); that is, v = f · (∂/∂z) on ∩ U for some function f ∈ O( ∩ U).

Remark A vector ﬁeld on a set S ⊂ X is of class Ck if and only if its coefﬁcient functions with respect to every local holomorphic coordinate neighborhood (or equivalently, for each point p ∈ S, with respect to some local holomorphic coordinate neighborhood of p) are of class Ck (see Exercise 2.4.1).

~~We close this section with the observation that the holomorphic inverse function theorem for domains in the plane (Theorem 1.5.1) gives the following analogue for Riemann surfaces:~~

~~Theorem 2.4.4 (Holomorphic inverse function theorem for Riemann surfaces) Let : X → Y be a holomorphic mapping of Riemann surfaces.~~

(a) If p ∈ X and (∗)p = 0, then maps some neighborhood of p biholomorphically onto a neighborhood of q ≡ (p). In particular, if f is a holomorphic ∈ = function on a neighborhood of a point p X and (df )p 0, then (U, f U ) is a local holomorphic coordinate neighborhood for some neighborhood U of p. (b) If is one-to-one, then maps X biholomorphically onto an open subset (X) of Y .

~~The proof is left to the reader (see Exercise 2.4.2).~~

Corollary 2.4.5 Let f be a holomorphic function on a Riemann surface X, let r be a positive regular value (see Deﬁnition 9.4.6) of the function ρ ≡|f | (|f | is of class C∞ on the complement of its zero set), and let ≡{p ∈ X ||f(p)|

~~Remark In particular, is a C∞ open set (see Deﬁnition 9.7.14).~~

Proof We have 2ρdρ= dρ2 = fdf¯ + fdf¯ on X \ f −1(0), and hence, given a point p ∈ ∂,wehave(df )p = 0. Therefore, by Theorem 2.4.4, there exists a local ∈ holomorphic chart of the form (U, f U ,U ) with p U. Moreover, we may choose U and U so small that there exists a holomorphic branch g of the logarithmic function on U (see Example 1.6.2). Since g is a biholomorphism (with holomorphic inverse function ζ → eζ ), we get a local holomorphic coordinate neighborhood

~~(U, z = x + iy ≡−log r + g ◦ f) ∩ ={ ∈ | } in which x = Re z = log |f/r|U . In particular, U q U x(q) < 0 .~~

Exercises for Sect. 2.4 2.4.1 Let X be a Riemann surface. (a) For a vector field v on X and for k ∈ Z≥0 ∪{∞}, prove that the following are equivalent (cf. Exercise 9.4.2): (i) The vector field v is of class Ck . (ii) The coefficients of v in every local holomorphic coordinate neighborhood are of class Ck. (iii) For every point in X, there exists a local holomorphic coordinate neighborhood with respect to which the coefficients of v are of class Ck. (iv) The (1, 0) and (0, 1) parts of v are of class Ck. (b) For a vector field v of type (1, 0) on X, prove that the following are equivalent: (i) The vector field v is holomorphic. (ii) The coefficient dz(v) of v in some local holomorphic coordinate neighborhood (U, z) of each point in X is holomorphic. (iii) For every holomorphic function f on an open set U ⊂ X, the function df (v) : p → df (vp) = vp(f ) is holomorphic. (c) Prove that any sum of holomorphic vector fields, and any product of a holomorphic function and a holomorphic vector field, are holomorphic (in particular, the set of holomorphic vector fields on X is a complex vector space). 2.4.2 Prove Theorem 2.4.4. 2.4.3 Let X be a Riemann surface. (a) Prove that there is a unique 2-dimensional holomorphic structure on (T X)1,0 such that for each local holomorphic chart (U, = z, U ) in X, the triple −1 (U), ( ◦ 1,0 ,dz),U × C (T X)1,0 (T X)

~~is a local holomorphic chart in (T X)1,0 (see Exercise 2.2.6 for the deﬁni- tion of a complex manifold, and cf. Exercise 9.4.3). 2.5 Differential Forms on a Riemann Surface 45~~

(b) Prove that a vector ﬁeld v of type (1, 0) on X is holomorphic if and only if the mapping v : X → (T X)1,0 is holomorphic as a mapping of complex manifolds (cf. Exercise 9.4.5). (c) Prove that there is a unique 2-dimensional holomorphic structure on (T ∗X)1,0 such that for each local holomorphic chart (U, = z, U ) in X, the triple − 1 (U), , U × C , (T ∗X)1,0 − where the map : 1 (U) → U ×C α → (z(p), α( ∂ )) (T ∗X)1,0 is given by ∂z ∈ ∈ ∗ 1,0 for each point p U and each element α (Tp X) , is a local holomorphic chart in (T ∗X)1,0. (d) Find (natural) C∞ structures in (T X)0,1 and (T ∗X)0,1.

~~2.5 Differential Forms on a Riemann Surface~~

We recall that a differential form α of degree r on a subset S of a C∞ manifold M r ∗ ∈ r ∗ ∈ is a mapping of S into (T M)C with αp (Tp M)C for each point p S (see Sect. 9.5). On a Riemann surface, the decomposition of the complexiﬁed tangent space into (1, 0) and (0, 1) parts induces a decomposition of differential forms.

Deﬁnition 2.5.1 Let X be a complex 1-manifold, and let (r, s) ∈ Z≥0 × Z≥0. (a) We set ⎧ ⎪ 0 ∗ = × C = ⎪ (T X)C X if (r, s) (0, 0), ⎨ ∗ r,s = r,s ∗ ≡ (T X) if (r, s) (1, 0) or (0, 1), T X 2 ∗ ⎪ (T X)C if (r, s) = (1, 1), ⎩⎪ X × {0} if r ≥ 2ors ≥ 2,

r,s ∗ we let r,s T ∗X : T X → X be the corresponding projection, and we let r,s ∗ = −1 ∈ Tp X r,s T ∗X(p) for each point p X. (b) Elements of r,s T ∗X are said to be of type (r, s). A differential form α on a set ⊂ ∈ r,s ∗ ∈ S X is of type (r, s) if αp Tp X for each point p S.Wealsocallα a (differential) (r, s)-form. (c) For each open set ⊂ X, Er,s () denotes the vector space of C∞ differential forms of type (r, s). The vector space of C∞ (r, s)-forms with compact support in is denoted by Dr,s (). (d) Let α be a differential form of degree q on a set S ⊂ X.Thecoefﬁcient function(s) (or simply the coefﬁcient(s)) of α with respect to (or in) a local holomorphic coordinate neighborhood (U, z) is (are) the function(s) on S ∩ U given 46 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms

by ⎧ ⎪α if q = 0, ⎪ ⎨a ≡ α(∂/∂z), b ≡ α(∂/∂z)¯ (i.e., α = adz+ bdz)¯ if q = 1, α ⎪ ≡ ¯ = = ∧ ¯ = ⎪a α(∂/∂z,∂/∂z) (i.e., α adz dz) if q 2, ⎩⎪ dz∧ dz¯ 0ifq>2.

Remarks 1. For each point p ∈ X,wehaveα ∧ β =¯α ∧ β¯ = 0 for all α, β ∈ 1,0 ∗ ∧ + + ∈ r,s ∗ Tp X. Observe also that ξ ζ is of type (r t,s u) if ξ Tp X and ∈ t,u ∗ ζ Tp X. 2. On any local holomorphic coordinate neighborhood (U, z = x + iy),wehave (i/2)dz∧ dz¯ = dx ∧ dy. r,s ∗ s,r ∗ 3. For all r, s ∈ Z≥0, conjugation gives T X = T X; that is, α¯ ∈ s,rT ∗X for each element α ∈ r,s T ∗X (see Exercise 2.5.1). 4. Clearly, a differential form of type (r, s) is of degree r + s. 5. Exercises 9.5.4 and 2.4.3 provide a natural C∞ structure on r,s T ∗X.

~~Guided by Definition 9.5.2 and Definition 2.4.3, we make the following definition:~~

Definition 2.5.2 Let be an open subset of a complex 1-manifold X. (a) A holomorphic 1-form on is a differential form θ of type (1, 0) on such that for every local holomorphic coordinate neighborhood (U, z) in X, the coefficient function θ(∂/∂z) = θ/dz is holomorphic on ∩ U; that is, for some function f ∈ O( ∩ U),wehaveθ = fdzon ∩ U. The vector space of holomorphic 1-forms on is denoted by X() or (). (b) A meromorphic 1-form on is a differential form θ of type (1, 0) defined on the complement in of a discrete subset P such that for every local holomorphic coordinate neighborhood (U, z) in X, the coefficient function θ/dz is meromorphic on ∩ U with pole set P ∩ U; that is, for some function f ∈ M( ∩ U), we have θ = fdzon ∩ U \ f −1(∞) = ∩ U \ P (we normally say simply θ = fdzon ∩ U). (c) A holomorphic (meromorphic) function on is also called a holomorphic (respectively, meromorphic) 0-form. (d) A meromorphic 1-form θ on has a zero (a pole) of order ν at a point p ∈ if for every local holomorphic coordinate neighborhood (U, z) of p,themero- morphic function θ/dz has a zero (respectively, a pole) of order ν at p.Ifν = 1, then we also say that θ has a simple zero (respectively, a simple pole)atp.

Remarks 1. A holomorphic 1-form is of class C∞ (see Proposition 2.5.3 below). 2. For a nontrivial (i.e., not everywhere zero) meromorphic 1-form on a Riemann surface, the set of zeros (i.e., the set of points at which the 1-form is equal to 0) is discrete (see Exercise 2.5.2). 2.5 Differential Forms on a Riemann Surface 47

~~The proof of the following characterization of continuous, Ck, holomorphic, and meromorphic differential forms is left to the reader (see Exercise 2.5.3):~~

Proposition 2.5.3 (Cf. Proposition 9.5.3 and Definition 9.7.12) Let α be a differential form of degree r defined at points of a subset S of a Riemann surface X, let k ∈ Z≥0 ∪{∞}, and let d ∈[1, ∞]. Then: (a) The following are equivalent: Ck d (i) The differential form α is continuous (of class , measurable, in Lloc). (ii) The coefficients of α in every local holomorphic coordinate neighborhood Ck d are continuous (respectively, of class , measurable, in Lloc). (iii) For every point in S, there exists a local holomorphic coordinate neighborhood with respect to which the coefficients of α are continuous (respec- Ck d tively, of class , measurable, in Lloc). = Ck d Moreover, for r 1, α is continuous (of class , measurable, in Lloc) if and only if the (1, 0) and (0, 1) parts of α are continuous (respectively, of class Ck , d measurable, in Lloc). (b) For S open and α of type (1, 0), α is holomorphic if and only if for every point in S, there exists a local holomorphic coordinate neighborhood (U, z) with respect to which the coefficient α(∂/∂z) = α/dz is holomorphic. (c) Suppose α is of type (1, 0) and S = \ P for some discrete set P in an open set ⊂ X. Then α is a meromorphic 1-form on with pole set P if and only if for each point p ∈ , there exists a local holomorphic coordinate neighborhood (U, z) of p such that the coefficient α(∂/∂z) = α/dz is a meromorphic function on ∩ U with pole set ∩ U ∩ P . Moreover, if α is a meromorphic 1-form on , then α has a zero (a pole) of order ν at a point p ∈ if and only if there exists a local holomorphic coordinate neighborhood (U, z) of p such that the coefficient α/dz has a zero (respectively, a pole) of order ν at p.

~~∗ The decomposition of 1(T X)C yields a decomposition of the exterior derivative operator d (see Deﬁnition 9.5.5):~~

Deﬁnition 2.5.4 Let X be a complex 1-manifold, let α be a differential form of ∗ ∗ degree r on an open subset of X, and let Pp,q : 1(T X)C → p,qT X be the associated projection for each pair (p, q) ∈{(1, 0), (0, 1)}. (a) If α is of class C1, then we deﬁne ⎧ ⎧ ⎨⎪P1,0(dα) if r = 0, ⎨⎪P0,1(dα) if r = 0, ∂α = d(P0,1α) if r = 1, and ∂α¯ = d(P1,0α) if r = 1, ⎩⎪ ⎩⎪ 0ifr ≥ 2, 0ifr ≥ 2.

~~(b) If α is of class C1 and we have ∂α¯ = 0(∂α = 0), then we say that α is ∂¯-closed (respectively, ∂-closed). 48 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms~~

(c) If α = ∂β¯ (α = ∂β)forsomeC1 differential form β of degree r − 1on (in particular, r>0), then we say that α is ∂¯-exact (respectively, ∂-exact). If β may k be chosen to be of class C with k ∈ Z>0 ∪{∞}, then we also say that α is Ck ∂¯-exact (respectively, Ck ∂-exact). If for each point in , there exists a C1 − ¯ = differential form β0 of degree r 1 on a neighborhood U0 with ∂β0 α U0 = ¯ (∂β0 α U0 ), then we say that α is locally ∂-exact (respectively, locally ∂- exact). If for some k ∈ Z>0 ∪{∞}, each of the local forms β0 may be chosen to be of class Ck, then we also say that α is locally Ck ∂¯-exact (respectively, locally Ck ∂-exact). It is also convenient to consider the trivial 0-form α ≡ 0tobeC∞ ∂-exact and C∞ ∂¯-exact, and to write 0 = ∂0 = ∂¯0.

Remark If α is a ∂¯-exact (∂-exact) r-form on a Riemann surface X, then we may choose a C1 differential form β with ∂β¯ = α (respectively, ∂β = α). For r = 1, β is of type (0, 0) and α is of type (0, 1) (respectively, α is of type (1, 0)). For r = 2, α is of type (1, 1) and we have ∂¯P1,0β = ∂β¯ = α (respectively, ∂P0,1β = ∂β = α). Thus, in either case (r = 1or2),α is of type (p, q + 1) (respectively, of type (p + 1,q)) with p + q + 1 = r, and we may choose β to be of type (p, q). Moreover, according to part (c) of Proposition 2.5.5 below, if α is of class Ck for k some k ∈ Z>0 ∪{∞}, then any such β of type (p, q) is also of class C .

Proposition 2.5.5 For any C1 differential form α of degree r on an open subset of a Riemann surface X, we have the following: (a) The exterior derivative satisﬁes dα = ∂α + ∂α¯ , and if α is of class C2, then d2α = ∂¯ 2α = ∂2α = ∂∂α¯ + ∂∂α¯ = 0; in other words,

~~d = ∂ + ∂¯ on C1 forms~~

~~and d2 = ∂2 = ∂¯ 2 = ∂∂¯ + ∂∂¯ = 0 on C2 forms.~~

We also have ∂α = ∂¯α¯ and ∂α¯ = ∂α¯ . Consequently, α is ∂¯-closed (∂-closed) if and only if α¯ is ∂-closed (respectively, ∂¯-closed); and if α = ∂β¯ (α = ∂β) for some differential form β of class C2, then α is ∂¯-closed (respectively, ∂-closed). (b) If α is of type (p, q), then ∂α is of type (p + 1,q)and ∂α¯ is of type (p, q + 1). In particular, ∂α = 0 if p ≥ 1 and ∂α¯ = 0 if q ≥ 1. ¯ k (c) If α is ∂-exact (∂-exact) and of class C for some k ∈ Z>0 ∪{∞}, then α is ∂¯-closed (respectively, ∂-closed) and of type (p, q + 1) (respectively, of type (p + 1,q)) for some pair of integers (p, q). Moreover, there exists a C1 differential form β of type (p, q) such that α = ∂β¯ (respectively, α = ∂β), and any such form β is actually of class Ck. (d) Let (U, z) be a local holomorphic coordinate neighborhood in . If r = 0, then on U, ∂α ∂α ∂α ∂α dα = dz+ dz,¯ ∂α = dz, and ∂α¯ = dz.¯ ∂z ∂z¯ ∂z ∂z¯ 2.5 Differential Forms on a Riemann Surface 49

If r = 1 and α = adz+ bdz¯ on U, then on U, we have ∂b ∂a dα = − dz∧ dz,¯ ∂z ∂z¯ and ∂b ∂a ∂a ∂α = dz∧ dz¯ and ∂α¯ = dz¯ ∧ dz =− dz∧ dz.¯ ∂z ∂z¯ ∂z¯ (e) For any C1 differential form β on , we have

~~∂(α ∧ β) = (∂α) ∧ β + (−1)r α ∧ ∂β~~

and ∂(α¯ ∧ β) = (∂α)¯ ∧ β + (−1)r α ∧ ∂β.¯ (f) If α is of type (0, 0) or (1, 0), then α is a holomorphic r-form if and only if ∂α¯ = 0. If α is a holomorphic 0-form (i.e., a holomorphic function), then ∂α is a holomorphic 1-form. (g) If : Y → X is a holomorphic mapping of a Riemann surface Y into X, then ∂∗α = ∗∂α and ∂¯ ∗α = ∗∂α¯ on −1(). In particular, if α is a holomorphic r-form, then ∗α is also a holomorphic r-form. If α is a meromorphic r-form and (Y) is not contained in the pole set of α, then ∗α is a meromorphic r-form.

Proof Let (U, z) be a local holomorphic coordinate neighborhood in .Ifr = 0, then, by (the remarks following) Proposition 2.4.2, ∂α ∂α dα = dz+ dz¯; ∂z ∂z¯ and since the ﬁrst summand on the right-hand side is of type (1, 0) and the second is of type (0, 1), we get the expressions in (d) for ∂α and ∂α¯ .Ifr = 1 and α = adz+ bdz¯ on U, then since d2 = 0 (on class C2 forms), dz ∧ dz = dz¯ ∧ dz¯ = 0, and dz∧ dz¯ =−dz¯ ∧ dz,wehave dα = d(P0,1α + P1,0α) = d(P0,1α) + d(P1,0α) = ∂α + ∂α,¯

∂α = d(P0,1α) = d(bdz)¯ = db ∧ dz¯ ∂b ∂b ∂b = dz+ dz¯ ∧ dz¯ = dz∧ dz,¯ ∂z ∂z¯ ∂z and ∂α¯ = d(P1,0α) = d(adz) = da ∧ dz ∂a ∂a ∂a = dz+ dz¯ ∧ dz =− dz∧ dz.¯ ∂z ∂z¯ ∂z¯ 50 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms

~~In particular, we get (d). The veriﬁcations of the remaining claims are left to the reader (see Exercise 2.5.5).~~

~~We have the following analogue of the Poincaré lemma (Lemma 9.5.7)forthe operators ∂ and ∂¯:~~

Lemma 2.5.6 (Dolbeault lemma) Let p be a point in a Riemann surface X. Then there exists a neighborhood D of p in X such that for every k ∈ Z>0 ∪{∞}, all k r, s ∈ Z≥0, and every C differential form β of type (r, s +1) (respectively, (r +1,s)) on a neighborhood of D, there is a Ck differential form α of type (r, s) on D with ∂α¯ = β (respectively, ∂α = β) on D. Consequently, every Ck differential form of type (r, s + 1) (of type (r + 1,s)) is locally Ck ∂¯-exact (respectively, locally Ck ∂- exact).

Proof We may ﬁx a local holomorphic coordinate neighborhood (U, z) of p in X and a neighborhood D U such that z(D) is a disk in C. Suppose k ∈ Z>0 ∪{∞}, k r, s ∈ Z≥0, and β is a C differential form of type (r, s + 1) on a neighborhood of D, which we may assume to be U. Clearly, β ≡ 0 = ∂¯0ifr>1ors>0, so we may also assume that r = 0 or 1 and that s = 0. Thus we have either β = bdz¯ or β = bdz∧ dz¯ on U for some function b ∈ Ck(U). By Lemma 1.2.2, there exists a function a ∈ Ck(D) with ∂a/∂z¯ = b on D. Thus ∂a¯ = bdz¯ and ∂(¯ −adz)= bdz∧ dz,¯ and it follows that β = ∂α¯ on D, where α = a if r = 0 and α =−adzif r = 1. If instead, we take β to be of type (r + 1,s), then β¯ is of type (s, r + 1). Hence, by the above, there is a Ck differential form α of type (s, r) such that ∂α¯ = β¯ on D. Thus the Ck form α¯ is of type (r, s) and ∂α¯ = ∂α¯ = β¯ = β on D.

We close this section with some remarks concerning integration on a Riemann surface. We ﬁrst observe that line integrals (Deﬁnition 9.7.18) may be expressed in terms of local holomorphic coordinates. For example, if (U, z) is a local holomorphic coordinate neighborhood in a complex 1-manifold X, α = fdz+ gdz¯ is a continuous differential form of degree 1 on U, and γ :[a,b]→U is a piecewise C1 path, then b d d α = f(γ(t)) z(γ (t)) + g(γ(t)) [z(γ (t))] dt. γ a dt dt For integration of 2-forms, observe that there is a natural orientation (see Sect. 9.7) in a complex 1-manifold X determined by the local holomorphic charts. For if (U, = z = x + iy ↔ (x, y), U ) and (V, = w = u + iv ↔ (u, v), V ) are two local holomorphic charts, then on U ∩ V , 2 du∧ dv dw ∧ dw¯ ∂w J ◦ −1 ◦ = = = > 0. dx ∧ dy dz∧ dz¯ ∂z 2.5 Differential Forms on a Riemann Surface 51

In particular, (i/2)dz∧ dz¯ = dx ∧ dy is a positive C∞ (1, 1)-form (i.e., a positive C∞ 2-form) on U. A positive C∞ (1, 1)-form ω on X is also called a Kähler form. By part (g) of Proposition 9.7.9, for any nonnegative measurable (1, 1)-form α de- ﬁned on a measurable set S ⊂ X,wehave

~~m α = sup α, S j=1 Sj where the supremum is taken over all choices of disjoint measurable subsets S1,...,Sm of S each of which is contained in a local holomorphic coordinate neighborhood.~~

Remark By deﬁnition, a positive C∞ (1, 1)-form on a complex manifold of arbitrary dimension is called a Kähler form if it is d-closed. This condition holds automatically in complex dimension 1, since every 2-form on a C∞ manifold of real dimension 2 is closed.

~~We make the following observation:~~

Proposition 2.5.7 (Cf. Exercise 2.5.8 and Exercise 6.7.1) For a nontrivial meromorphic function f (i.e., f is not everywhere zero) on a compact Riemann surface X, counting multiplicities, the number of zeros is equal to the number of poles. More precisely, if Z is the set of zeros, P is the set of poles, μp is the order of the zero at each point p ∈ Z, and νp is the order of the pole at each point p ∈ P , then ordp f = μp − νp = 0. p∈Z∪P p∈Z p∈P

~~In particular, if f has exactly one (simple) pole, then f maps X biholomorphically onto P1.~~

Proof For r>0 sufﬁciently small, the local representation of meromorphic functions provided by Proposition 2.2.6 allows us to ﬁx disjoint local holomorphic charts { = ; } ∈ ∪ (Up,p zp,(0 2r)) p∈Z∪P in X such that for each point p Z P ,wehave zp(p) = 0 and on Up, μp ∈ = zp if p Z, f −νp zp if p ∈ P.

Thus the meromorphic 1-form θ ≡ df/f satisﬁes, on Up, − μ · z 1 · dz if p ∈ Z, θ = p p p − · −1 · ∈ νp zp dzp if p P. 52 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms

~~Stokes’ theorem now gives 2πiμp − 2πiνp = θ −1 ; p∈Z p∈P p∈Z∪P ∂p ((0 r))~~

=− dθ = 0. \ −1 ; X p∈Z∪P p ((0 r)) In particular, if f is holomorphic except for a simple pole at p ∈ X (i.e., P ={p} −1 and νp = 1), then f (∞) ={p}, and for each point q ∈ X \{p}, the meromorphic function f − f(q)is nonvanishing except for a simple zero at q. Thus f : X → P1 is injective, and by the holomorphic inverse function theorem (Theorem 2.4.4), f maps X biholomorphically onto an open subset of P1. On the other hand, f(X) is compact, hence closed, so we must have f(X)= P1.

Exercises for Sect. 2.5 r,s ∗ s,r ∗ 2.5.1 Verify that for all r, s ∈ Z≥0, conjugation gives T X = T X on a Riemann surface X. 2.5.2 Let θ be a nontrivial meromorphic 1-form on a Riemann surface X.Verify that {x ∈ X | θx = 0} is discrete. 2.5.3 Prove Proposition 2.5.3. 2.5.4 Prove that P1 does not admit any nontrivial holomorphic 1-forms. 2.5.5 Prove parts (a)–(c) and (e)–(g) of Proposition 2.5.5. 2.5.6 Let α be a differential form of type (1, 0) on a Riemann surface X. Prove that α is a holomorphic 1-form if and only if α is holomorphic as a mapping of X into the 2-dimensional complex manifold (T ∗X)1,0 (see Exercise 2.4.3 for a description of the natural holomorphic structure on (T ∗X)1,0). 2.5.7 Let X be a complex torus (Example 2.1.6). Prove that the vector space of holomorphic 1-forms on X is 1-dimensional. 2.5.8 Let be a (nonempty) smooth relatively compact domain in a Riemann surface X, and for any given nontrivial meromorphic function h on X that does not have any zeros or poles in ∂,letμh denote the number of zeros of h in , and νh the number of poles of h in , counting multiplicities. (a) Prove the following version of the argument principle (cf. Proposi- tion 2.5.7 and Exercises 5.1.6, 5.1.7, 5.9.5, 6.7.1, and 6.7.6): If f is a nontrivial meromorphic function on X that does not have any zeros or poles in ∂, then 1 df μf − νf = . 2πi ∂ f (b) Prove the following version of Rouché’s theorem:Iff and g are nontrivial meromorphic functions on X that do not have any zeros or poles in ∂, and |g| < |f | on ∂, then μf − νf = μf +g − νf +g. Hint. Using part (a), show that the integer-valued function t → μf +tg − νf +tg is continuous on the interval [0, 1]. 2.6 L2 Scalar-Valued Differential Forms on a Riemann Surface 53

2.5.9 If X is a complex 1-manifold, p ∈ X, θ is a holomorphic 1-form on V \{p} for some neighborhood V of p in X, (U, = z, (z0; R)) is a local holomor- = phic coordinate neighborhood of p with (p) z0 and U ⊂ V , f = θ/dz = \{ } = ∞ − n (i.e., θ fdzon U p ), and f n=−∞ cn(z z0) is the corresponding Laurent series representation of f , then the coefﬁcient c−1 is called the residue of θ at p and is denoted by respθ. Equivalently, if r ∈ (0,R) and D ≡ −1((0; r)), then 1 resp θ = θ. 2πi ∂D

(a) Prove that the residue of a holomorphic 1-form at an isolated singularity is well defined (that is, prove that the above definition is independent of the choice of the local holomorphic coordinate). (b) Prove the following version of the residue theorem (cf. Exercises 5.1.6, 5.1.7, 5.9.5, 6.7.1, and 6.7.6): If is a (nonempty) smooth relatively compact domain in a Riemann surface X, S is a finite subset of , and θ is a holomorphic 1-form on X \ S, then 1 θ = resp θ. 2πi ∂ p∈S

~~In particular, if X is a compact Riemann surface,S is a ﬁnite subset of X, \ = and θ is a holomorphic 1-form on X S, then p∈S resp θ 0.~~

~~2.6 L2 Scalar-Valued Differential Forms on a Riemann Surface~~

Throughout this section, X denotes a complex 1-manifold. The goal of this section is the development of a suitable L2 space of differential forms. Since the objects that we integrate on oriented surfaces are the 2-forms (see Sect. 9.7) and we wish to pair forms of the same degree and integrate in order to get an inner product, it is natural to consider a pointwise pairing that gives a 2-form. Let us first√ consider integration on C. We have the natural√ unit-length√(1, 0)-form θ = (1/ 2)dz and the natural positive (1, 1)-form ω = −1θ ∧ θ¯ = ( −1/2)dz∧ dz¯. In the notation of Sect. 9.7, the Lebesgue measure λ in C is equal to the measure λω associated to ω (see Definition 9.7.10). If α = aθ and β = bθ are differential forms of type (1, 0) with L2 coefficients, then = ¯ = ¯ = ¯ ∧ ¯ = ∧ ¯ a,b L2(C) abdλ abω abiθ θ iα β. C C C C

~~If α and β are L2 differential forms of type (0, 0) (i.e., L2 functions), then = ¯ = ¯ α, β L2(C) αβdλ αβω. C C 54 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms~~

Finally, if α = aω and β = bω are differential forms of type (1, 1) with L2 coefficients, then α β = ¯ = ¯ = · · a,b L2(C) abdλ abω ω. C C C ω ω The right-hand sides in the above point to a reasonable definition for the L2 inner product of differential forms on the complex 1-manifold X. Observe that the expression for the inner product of a pair of forms of type (0, 0) and (1, 1) involves the Kähler form ω, although that for the inner product of a pair of (1, 0)-forms does not.Thus,onX, for forms of type (0, 0) and (1, 1), we must define the inner product with respect to some choice of a positive (1, 1)-form. For functions, it turns out to be useful to weaken the requirement on the (1, 1)-form by allowing it to be only nonnegative. After all, it is not even a priori clear that X admits a global Kähler form; although, as will be shown in Sect. 2.11, it turns out that every Rie- mann surface is second countable and therefore that X does admit a Kähler form (see Corollary 2.11.3). As will be seen in Sect. 2.9, it is also useful to include a weight function e−ϕ (in an abuse of language, we also call ϕ a weight function). For example, a Riemann surface need not admit any L2 holomorphic 1-forms, but given a holomorphic 1-form, one may construct a C∞ function ϕ that grows so large at infinity that the holomorphic 1-form becomes L2 with respect to the weight function e−ϕ. Finally, observe that iα ∧¯α>0 for every nonzero α ∈ 1,0T ∗X. Based on these considerations, we make the following definition:

Definition 2.6.1 Let S be a measurable subset of X,letϕ be a measurable real- valued function that is defined on S, and let α and β be measurable differential forms of type (p, q) that are defined on S. (a) For (p, q) = (1, 0), we define √ 1/2 ≡ − ∧¯· −ϕ ∈[ ∞] α L2 (S,ϕ) 1α α e 0, . 1,0 S √ If −1α ∧ β¯ · e−ϕ is integrable on S, then we define √ ≡ − ∧ ¯ · −ϕ ∈ C α, β L2 (S,ϕ) 1α β e . 1,0 S (b) For (p, q) = (0, 1), we define √ 1/2 ≡ − − ∧¯· −ϕ ∈[ ∞] α L2 (S,ϕ) 1α α e 0, . 0,1 S √ If − −1α ∧ β¯ · e−ϕ is integrable on S, then we define √ ≡− − ∧ ¯ · −ϕ ∈ C α, β L2 (S,ϕ) 1α β e . 0,1 S 2.6 L2 Scalar-Valued Differential Forms on a Riemann Surface 55

~~(c) If (p, q) = (0, 0) and ω is a nonnegative measurable form of type (1, 1) deﬁned on S, then we deﬁne 1/2 ≡ | |2 −ϕ ∈[ ∞] α L2 (S,ω,ϕ) α e ω 0, . 0,0 S~~

If αβ¯ · e−ϕω is integrable on S, then we define ≡ ¯ · −ϕ ∈ C α, β L2 (S,ω,ϕ) αβ e ω . 0,0 S (d) If (p, q) = (1, 1) and ω is a positive measurable form of type (1, 1) defined on S, then we define α 2 1/2 ≡ −ϕ = ∈[ ∞] α L2 (S,ω,ϕ) e ω α/ω L2 (S,ω,ϕ) 0, . 1,1 S ω 0,0 If the form α β · · e−ϕω ω ω is integrable on S, then we define ≡ α · β · −ϕ = α β ∈ C α, β L2 (S,ω,ϕ) e ω , . 1,1 S ω ω ω ω 2 L0,0(S,ω,ϕ) (e) When there is no danger of confusion (for example, when the choice of (p, q), S, ω,orϕ is understood from the context), we will suppress parts of the notation. For example, we will often write α 2 simply as α , Lp,q(S,ω,ϕ) S,ω,ϕ α S,ϕ, α S,ω, α ω,ϕ, α ϕ, α ω, α L2(S,ω,ϕ), α L2(S,ϕ), α L2(ω,ϕ), α L2(ϕ), α L2(ω),or α . We will also use the analogous simplified notation for α, β 2 ,forα 2 , and for α, β 2 . Moreover, when Lp,q(S,ω,ϕ) Lp,q(S,ϕ) Lp,q(S,ϕ) no mention of a weight function appears in the context, then the weight function will be assumed to be ϕ ≡ 0 and we will write α 2 simply as Lp,q(S,ω,0) α 2 and so on. Lp,q(S,ω) (f) Let γ and η be measurable differential 1-forms defined on S with (r, s) parts γ r,s and ηr,s , respectively, for each (r, s) ∈{(1, 0), (0, 1)}. Then we set 2 ≡ 1,02 + 0,12 γ 2 γ ϕ γ ϕ. L1(S,ϕ)

1,0 1,0 0,1 0,1 We also set γ,η 2 ≡γ ,η ϕ +γ ,η ϕ, provided each of the L1(S,ϕ) summands on the right-hand side is deﬁned. We also use the simpliﬁed notation analogous to that appearing in (e).

~~Deﬁnition 2.6.2 Let S be a measurable subset of X, and let ϕ be a measurable real-valued function that is deﬁned on S. 56 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms~~

= 2 (a) For (p, q) (1, 0) or (0, 1), Lp,q(S, ϕ) consists of all equivalence classes of ∞ measurable differential forms α of type (p, q) on S with α L2(S,ϕ) < , where we identify any two elements that are equal almost everywhere. The 2 set L1(S, ϕ) consists of all equivalence classes of measurable 1-forms α with ∞ α L2(S,ϕ) < , where again, we identify any two elements that are equal almost everywhere. (b) For ω a nonnegative measurable differential form of type (1, 1) that is deﬁned 2 on S, L0,0(S,ω,ϕ)consists of all equivalence classes of measurable functions α ∞ on S with α L2(S,ω,ϕ) < , where we identify any two elements that are equal almost everywhere. (c) For ω a positive measurable differential form of type (1, 1) that is deﬁned 2 on S, L1,1(S,ω,ϕ)consists of all equivalence classes of measurable differential ∞ forms α of type (1, 1) on S with α L2(S,ω,ϕ) < , where we identify any two elements that are equal almost everywhere. (d) When no mention of a weight function appears in the context and there is no danger of confusion, then the weight function will be assumed to be ϕ ≡ 0 2 2 2 and we will write Lp,q(S, 0) simply as Lp,q(S), and Lp,q(S, ω, 0) simply as 2 Lp,q(S, ω).

~~Remark For our purposes, we will need only weight functions ϕ and nonnegative (or positive) (1, 1)-forms ω that are deﬁned and of class C∞ on an open subset of X.~~

Proposition 2.6.3 Let S be a measurable subset of X, and let ϕ be a continuous real-valued function that is deﬁned on S. 2 · · 2 · · (a) The pair (L1(S, ϕ), , L2(S,ϕ)), and the pair (Lp,q(S, ϕ), , L2(S,ϕ)) for (p, q) = (1, 0) or (0, 1), are Hilbert spaces (where the inner product of any two equivalence classes is given by the pairing of any representatives). Moreover, we have the Hilbert space orthogonal decomposition

~~2 = 2 ⊕ 2 L1(S, ϕ) L1,0(S, ϕ) L0,1(S, ϕ).~~

(b) For ω a continuous positive differential form of type (1, 1) that is deﬁned on S = 2 · · and for (p, q) (0, 0) or (1, 1), (Lp,q(S,ω,ϕ), , L2(S,ω,ϕ)) is a Hilbert space (where the inner product of any two equivalence classes is given by the pairing of any representatives). Moreover, in each of the above spaces, any sequence converging to an element α admits a subsequence that converges to α pointwise almost everywhere in S.

Proof Let p,q ∈{0, 1}.Weset L2 (S, ϕ) as in (a) if (p, q) = (1, 0) or (0, 1), L2 ≡ p,q p,q 2 = ; Lp,q(S,ω,ϕ)as in (b) if (p, q) (0, 0) or (1, 1) 2.6 L2 Scalar-Valued Differential Forms on a Riemann Surface 57

∈ ∈ p,q ∗ and for each point r S and each pair of elements η,θ Tr X,weset ⎧ ⎪ ¯ · −ϕ(r) · = ⎪ηθ e ωr if (p, q) (0, 0), ⎪ − ⎨⎪iη ∧ θ¯ · e ϕ(r) if (p, q) = (1, 0), H(η,θ)≡ − ∧ ¯ · −ϕ(r) = ⎪ iη θ e if (p, q) (0, 1), ⎪ ⎩⎪ η θ −ϕ(r) · · e ωr if (p, q) = (1, 1). ωr ωr 2 · · We ﬁrst show that (Lp,q, , ) is an inner product space. For this, observe that if ∈ 2 : → α, β Lp,q, then the (1, 1)-form H(α,β) r H(αr ,βr ) is measurable. Suppose (U, = z = x + iy, U ) is a local holomorphic chart and ω ≡ (i/2)dz∧ dz¯ = dx ∧ dy. Then the map H(η,θ) : → ≡ ∈ C ∀ ∈ p,q ∗ h (η, θ) h(η, θ) η,θ Tr X (ω)r p,q ∗ ∈ ∩ is a Hermitian inner product in Tr X for each point r S U.Wealsohavethe 1/2 corresponding pointwise norm η → |η|h =[h(η,η)] . Hence, for each measurable set R ⊂ S ∩ U and each complex number ζ with |ζ|=1, we have (by the Schwarz inequality) [ ]+ = [ ]+ ≤ | | | | Re(ζ H (α, β)) Re(ζ h(α, β)) dλω α h β h dλω R R R 1/2 1/2 ≤ | |2 | |2 α h dλω β h dλω R R 1/2 1/2 = H(α,α) · H(β,β) R R

(where λω is the positive measure associated to ω as in Deﬁnition 9.7.10). Thus, if S1,...,Sm are disjoint measurable subsets of S each of which lies in some local holomorphic coordinate neighborhood, then, by the Schwarz inequality for sums, m m 1/2 1/2 + [Re(ζ H (α, β))] ≤ H(α,α) · H(β,β) j=1 Sj j=1 Sj Sj m 1/2 m 1/2 ≤ H(α,α) · H(β,β) j=1 Sj j=1 Sj 1/2 1/2 ≤ H(α,α) · H(β,β) =α·β. S S Passing to the supremum, we get + [Re(ζ H (α, β))] ≤α·β < ∞. S 58 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms

~~Taking ζ =±1, ±i, we see that H(α,β) is integrable and therefore that α, β= ∈ C | |= S H(α,β)is deﬁned. Furthermore, choosing ζ so that ζ 1 and~~

~~ζ ·α, β=|α, β|, we get + − |α, β| = ζH(α,β)= [Re(ζ H (α, β))] − [Re(ζ H (α, β))] S S S + ≤ [Re(ζ H (α, β))] ≤α·β. S For each constant s ∈ C,wehave~~

H(sα+ β,sα + β) =|s|2H(α,α)+ 2Re(sH (α, β)) + H(β,β), so H(sα + β,sα + β) is integrable. If α = α a.e. (almost everywhere) in S, then = = 2 clearly, H(α ,α ) H(α,α) and H(α ,β) H(α,β) a.e. Thus Lp,q is a well- defined vector space, and as is now easy to verify, ·, · is a well-defined Hermitian inner product. 2 · · It remains to show that (Lp,q, , ) is complete with respect to the norm α=α, α1/2. We again apply (appropriately modified) standard arguments (cf., { } 2 for example, [Rud1]). We must show that a given Cauchy sequence αν in Lp,q converges. For this, it suffices to show that some subsequence converges. Hence, after replacing the sequence with a suitable subsequence, we may assume without loss of generality that αν+1 −αν < 1. For a local holomorphic chart (U, = z, U ), for ω = dx ∧ dy = (i/2)dz∧ dz¯ and for h(·, ·) = H(·, ·)/ω, as before, let us set

~~N ∞ ϕN ≡ |αν+1 − αν|h ∀N ∈ Z>0 and ϕ ≡ |αν+1 − αν |h. ν=1 ν=1~~

~~For each N ∈ Z>0,wehave~~

~~N N ϕN 2 ∩ ≤ |αν+1 − αν|h 2 ∩ ≤ αν+1 − αν < 1. L (S U,λω ) L (S U,λω ) ν=1 ν=1~~

Fatou’s lemma now implies that ϕ 2 ∩ ≤ 1. In particular, ϕ<∞ almost L (S U,λω ) everywhere in S ∩ U. It follows that the series ∞ α1 + (αν+1 − αν ) ν=1 converges pointwise almost everywhere in S to a measurable differential form α of type (p, q), and hence αν → α pointwise a.e. 2.6 L2 Scalar-Valued Differential Forms on a Riemann Surface 59

∈ 2 − → It remains to show that α Lp,q and that α αν 0. Given >0, we may choose N ∈ Z>0 so that αμ − αν < for all μ,ν > N. For any ν>N, Fatou’s lemma (Theorem 9.7.11)gives α − α 2 = H(α− α ,α− α ) ≤ lim inf H(α − α ,α − α ) ≤ 2. ν ν ν →∞ μ ν μ ν S μ S

= − + ∈ 2 − ≤ → 2 Thus α (α αν) αν Lp,q and α αν . Therefore αν α in Lp,q (and 2 pointwise almost everywhere in S) and Lp,q is a Hilbert space. The proposition (including the orthogonal decomposition in (a)) now follows.

~~We have the following useful version of Theorem 1.2.4:~~

Theorem 2.6.4 Let ϕ be a continuous real-valued function and let ω be a continuous positive differential form of type (1, 1) on X. Then, for every compact set K ⊂ X, there is a constant C = C(X,K,ω,ϕ) > 0 such that

~~| |≤ ∀ ∈ O max f C f L2(X,ω,ϕ) f (X). K~~

~~O ∩ 2 2 Consequently, (X) L0,0(X,ω,ϕ)is a closed subspace of L0,0(X,ω,ϕ).~~

Proof Given a point p ∈ K, we may ﬁx a relatively compact local holomorphic coordinate neighborhood (U, z) on which the function (i/2)(dz∧dz)/ω¯ is bounded. Therefore, for some constant R>0, we have, for every holomorphic function f on U, ∧ ¯ | |2 i ∧ ¯ = | |2 ϕ (i/2)dz dz · −ϕ · ≤ 2 f dz dz f e e ω R f L2(U,ω,ϕ). U 2 U ω

~~Fixing a relatively compact neighborhood Vp of p in U and applying Theo- rem 1.2.4, we get a constant Cp > 0 such that~~

~~| |≤ ≤ ∀ ∈ O sup f Cp f L2(U,ω,ϕ) Cp f L2(X,ω,ϕ) f (X). Vp~~

Covering K by ﬁnitely many such neighborhoods Vp and taking C to be the maximum of the associated constants Cp, we get the claim. The proof that O(X) ∩ 2 L0,0(X,ω,ϕ)is a closed subspace is left to the reader (see Exercise 2.6.3).

Exercises for Sect. 2.6 2.6.1 Show that there are no nontrivial L2 holomorphic 1-forms on C (cf. Exer- cise 2.5.4). 2.6.2 For the function ϕ : z → |z|2 on C, show that there exists a nontrivial holo- 2 C morphic 1-form in L1,0( ,ϕ). 60 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms

2.6.3 Let ϕ be a continuous real-valued function on a Riemann surface X. Prove 2 that the vector space of holomorphic 1-forms in L1,0(X, ϕ) is a closed sub- 2 space of L1,0(X, ϕ). Also prove that if ω is a continuous positive (1, 1)-form 2 on X, then the vector space of holomorphic functions in L0,0(X,ω,ϕ) is a closed subspace (i.e., prove the second part of Theorem 2.6.4).

~~2.7 The Distributional ∂¯ Operator on Scalar-Valued Forms~~

Throughout this section, X again denotes a complex 1-manifold. We now develop a suitable distributional version of the ∂¯ operator. In particular, we are led to consider a modified version of the exterior derivative d called the canonical (or Chern) connection. For locally integrable functions f and g on a local holomorphic coordinate neighborhood (U, z),wehave(∂f/∂z)¯ distr = g (see Definition 9.8.2 and Proposi- tion 9.8.3) if and only if for each function u ∈ D(U),wehave ∂u i i f − dz∧ dz¯ = gu¯ dz∧ dz¯ U ∂z 2 U 2 − ∧ ¯ = ¯ ∧ ¯ (equivalently, U f ( ∂u/∂z)dz dz U gudz dz). One natural definition for the distributional ∂¯ operator is the following:

Deﬁnition 2.7.1 Let α and β be locally integrable differential forms on an open set ¯ ⊂ X. We write ∂distrα = β if for every local holomorphic coordinate neighborhood (U, z), one of the following holds:

(i) On U ∩ , α is a 0-form, β = bdz¯, and (∂α/∂z)¯ distr = b; (ii) On U ∩ , α = a1 dz+ a2 dz¯, β = bdz∧ dz¯, and (∂a1/∂z)¯ distr =−b; (iii) The form α is of degree > 1 and β ≡ 0. ¯ Remark Given a locally integrable differential form α, ∂distrα need not exist as a form (see the remarks following Deﬁnition 7.4.2). For example, let u be the characteristic function of the unit disk ≡ (0; 1). Then ∂u¯ ≡ 0 on the complement ¯ C \ ∂ of the measure-zero set ∂. Thus, if ∂distru were to exist on C, then it would be the zero form, and hence by the regularity theorem (Theorem 1.2.8), u would be holomorphic, which it is not. It also follows that for the measurable differential ¯ form α = udz, ∂distrα does not exist as a form.

It is often more convenient to work with an intrinsic form of an operator, so ¯ we look for an intrinsic description of ∂distr. Distributional differential operators in Rn are deﬁned via integration against test functions (as in Sect. 7.4.2). On a surface, the objects that we integrate on open sets (or, more generally, measurable sets) are 2-forms, and for a differential form, its wedge product with a form of complementary degree is a 2-form. Thus, in the present context, it is natural to 2.7 The Distributional ∂¯ Operator on Scalar-Valued Forms 61 integrate against test forms of complementary degree. For example, suppose α and β are C∞ differential forms of type (1, 0) and (1, 1), respectively, and ∂α¯ = β. Since we integrate C∞ forms of type (1, 1) (i.e., 2-forms) on open sets in X, it is natural to integrate the wedge product of α and the conjugate of a (1, 0)-form, and the scalar product (i.e., the wedge product) of β and a function (i.e., a (0, 0)-form). Given a function f ∈ D(X),wehave

~~β · f¯ = (∂α)¯ · f¯ = (dα) · f¯ = d(α · f)¯ + α ∧ df¯ = d(α · f)¯ + α ∧ ∂f¯ + α ∧ ∂¯f¯ = d(α · f)¯ + α ∧ ∂f .~~

Integrating and applying Stokes’ theorem, we get β · f¯ = α ∧ ∂f . X X It turns out to be useful to insert a weight function ϕ (more precisely, e−ϕ), where ϕ is a real-valued C∞ (usually) function on X. Doing so allows one to obtain estimates on L2 norms that lead to solutions of the Cauchy–Riemann equation as well as certain bounds on the L2 norms of the solutions (see Sect. 2.9). Moreover, a Hermitian metric in a holomorphic line bundle (see Chap. 3) is locally represented by such weight functions, and this point of view yields results that generalize readily to that context. For α, β, and f as above, we have − − − − − β · f¯ · e ϕ = (∂α)¯ · f¯ · e ϕ = (dα) · (fe¯ ϕ) = d(α · e ϕf)¯ + α ∧ d(e ϕf)¯ − − = d(α · e ϕf)¯ + α ∧ ∂(e¯ ϕf)¯ = d(α · e−ϕf)¯ + α ∧ eϕ∂(e−ϕf)· e−ϕ.

Thus β · f¯ · e−ϕ = α ∧ eϕ∂(e−ϕf)· e−ϕ. X X ¯ The above suggests a natural intrinsic form of the deﬁnition of ∂distr on forms of type (1, 0). A similar computation applies to forms of type (0, 0). Based on the above, we make the following deﬁnition:

Deﬁnition 2.7.2 Given a real-valued function ϕ ∈ C∞(X), the associated canonical connection (or the Chern connection) is the operator D = Dϕ = D + D , where for each C1 differential form α on an open subset of X, we deﬁne = ≡ ϕ −ϕ = + ϕ −ϕ ∧ = − ∧ D α Dϕα e ∂(e α) ∂α e (∂e ) α ∂α ∂ϕ α and = ≡ ¯ D α Dϕα ∂α. We call D and D, respectively, the (1, 0) part and (0, 1) part of the connection. 62 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms

Remarks 1. Observe that D = ∂¯ does not depend on the choice of ϕ. 2. We have D = d + eϕ(∂e−ϕ) ∧ (·) = d − ∂ϕ ∧ (·). 3. If α is of type (p, q), then Dα is of type (p + 1,q)(in particular, Dα = 0if p ≥ 1) and Dα is of type (p, q + 1) (Dα = 0ifq ≥ 1). 4. If α and β are C1 differential forms and α is of degree p, then (see Exer- cise 2.7.1)

~~D(α ∧ β) = (dα) ∧ β + (−1)pα ∧ Dβ = (Dα) ∧ β + (−1)pα ∧ dβ.~~

~~∈ C1 5. Dϕ, Dϕ, and Dϕ are deﬁned in the same way as above for ϕ (X). ¯ We now get the following equivalent form for the deﬁnition of ∂distr (which we also denote by Ddistr):~~

Proposition 2.7.3 Let ϕ be a real-valued C∞ function on X, and let α and β be locally integrable differential forms on an open set ⊂ X. ¯ (a) If α is of type (1, 0) and β is of type (1, 1), then ∂distrα = β if and only if α ∧ Dfe−ϕ = β · f¯ · e−ϕ ∀f ∈ D(). ¯ (b) If α is of type (0, 0) and β is of type (0, 1), then ∂distrα = β if and only if α · (−Dγ)· e−ϕ = β ∧ γ · e−ϕ ∀γ ∈ D0,1(). ¯ (c) If α is of type (p, q) with p ≥ 2 or q ≥ 1, then ∂distrα = 0.

Remark It will follow from the proof that if the conditions in Deﬁnition 2.7.1 hold for the forms α and β in some local holomorphic coordinate neighborhood of every point, then they hold in every local holomorphic coordinate neighborhood.

Proof of Proposition 2.7.3 Suppose α = adz and β = bdz ∧ dz¯ in some local holomorphic coordinate neighborhood (U, z) with U ⊂ . Then, for every function f ∈ D(U) (which we may view as a C∞ function with compact support in ), we have βfe¯ −ϕ = b(e−ϕf)dz∧ dz¯ U and ∂ α ∧ Df · e−ϕ = α ∧ ∂(e−ϕf)= a · (e−ϕf)dz∧ dz.¯ U U ∂z Given u ∈ D(U), setting f = eϕu, we see that if the above left-hand sides are always ¯ ¯ equal, then we have ∂distrα = β. Conversely, if ∂distrα = β and f ∈ D(), then, C∞ { }n ≡ choosing ﬁnitely many functions ην ν=1 on such that ην 1onsuppf 2.8 Curvature and the Fundamental Estimate for Scalar-Valued Forms 63 and such that for each ν, the support of ην is contained in some local holomorphic coordinate neighborhood Uν , the above gives −ϕ −ϕ −ϕ α ∧ D f · e = α ∧ D (ην · f)· e = α ∧ D (ην · f)· e ν ν Uν −ϕ −ϕ ¯ −ϕ = βηνfe = βηνfe = βf · e . ν Uν ν

~~Thus part (a) is proved. Part (c) is obvious and the proof of part (b) is left to the reader (see Exer- cise 2.7.2).~~

~~The regularity theorem (Theorem 1.2.8) gives the following in this context:~~

Theorem 2.7.4 If p ∈{0, 1}, α is a locally integrable form of type (p, 0) on X, ¯ k and ∂distrα = β for a form β of class C for some k ∈ Z>0 ∪{∞}, then α is also of k ¯ class C . In particular, if ∂distrα = 0, then α is a holomorphic p-form.

~~Exercises for Sect. 2.7 2.7.1 Let ϕ be a C∞ real-valued function on a Riemann surface X. Show that if α and β are C1 differential forms on X and α is of degree p, then~~

~~p p Dϕ(α ∧ β) = (dα) ∧ β + (−1) α ∧ Dϕβ = (Dϕα) ∧ β + (−1) α ∧ dβ.~~

2.7.2 Prove part (b) of Proposition 2.7.3. ∞ 2.7.3 Let X be a Riemann surface, let ϕ be a real-valued C function on X,let{θν } 2 ∈ 2 be a sequence of holomorphic 1-forms in L1,0(X, ϕ), and let θ L1,0(X, ϕ). ∈ 2 Assume that for each element α L1,0(X, ϕ),wehave

~~lim θν,α 2 =θ,α 2 ν→∞ L1,0(X,ϕ) L1,0(X,ϕ)~~

~~{ } 2 (that is, θν converges weakly to θ in L1,0(X, ϕ)). Prove that θ is a holomorphic 1-form.~~

~~2.8 Curvature and the Fundamental Estimate for Scalar-Valued Forms~~

Throughout this section, X again denotes a complex 1-manifold. Suppose ϕ is a real-valued C∞ function on X. We recall that for any C∞ differential form α on an open subset of X,wehaveDϕα = Dα = D α + D α, where

~~− D α = eϕ∂[e ϕα]=∂α − (∂ϕ) ∧ α and D α = ∂α.¯ 64 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms~~

Consequently, (D)2 = (D)2 = 0 and D2 = DD + DD. Since ∂∂¯ + ∂∂¯ = 0, we have D2α = ∂∂α¯ − (∂ϕ) ∧ ∂α¯ + ∂∂α¯ − ∂¯[(∂ϕ) ∧ α] ¯ ¯ ¯ =−(∂ϕ) ∧ ∂α − (∂∂ϕ)∧ α + (∂ϕ) ∧ ∂α = ϕ ∧ α, ¯ where ϕ ≡ ∂∂ϕ is a differential form of type (1, 1) (of course, θϕ ∧ α = 0if deg α>0).

Deﬁnition 2.8.1 The curvature (or curvature form or Levi form) associated to a ∞ ¯ real-valued C function ϕ on X is the differential form = ϕ ≡ ∂∂ϕ. In other words, is deﬁned by D2 = DD + DD = ∧ (·).

The function ϕ is called subharmonic (strictly subharmonic, harmonic, superharmonic, strictly superharmonic) if the real (1, 1)-form iϕ satisﬁes iϕ ≥ 0(respectively, iϕ > 0, iϕ = 0, iϕ ≤ 0, iϕ < 0). In a slight abuse of language, we also say that ϕ has nonnegative (respectively, positive, zero, nonpositive, negative) curvature.

Remarks 1. The above terminology, with the same deﬁnitions, is also applied to C2 functions. There is also a natural and useful notion of a continuous subharmonic function (see, for example, [Ns5]), but continuous subharmonic functions are not considered in this book. 2. For any local holomorphic coordinate neighborhood (U, z = x + iy) in X and for any real-valued C∞ function ϕ on U,wehave ∂2ϕ ∂2ϕ 1 ∂2ϕ ∂2ϕ iϕ = idz∧ dz¯ = 2 dx ∧ dy = + dx ∧ dy. ∂z∂z¯ ∂z∂z¯ 2 ∂x2 ∂y2 Thus ϕ is subharmonic (strictly subharmonic, harmonic) on U if and only if ∂2ϕ ≥ 0 (respectively, > 0, = 0). ∂z∂z¯ 3. It is easy to see that a strictly subharmonic function cannot attain a local maximum (see Exercise 2.8.2). In fact, one can show that subharmonic functions satisfy a strong maximum principle (see, for example, [Ns5]). In particular, every subharmonic function on a compact Riemann surface is constant.

Proposition 2.8.2 (Fundamental estimate for scalar-valued forms) Let ϕ be a real- C∞ = = = ¯ valued function on X, and let D Dϕ, ϕ, and D ∂. Then, for all functions u, v ∈ D(X), we have = + ¯ −ϕ D u, D v L2(X,ϕ) D u, D v L2(X,ϕ) iuve . X In particular, Du2 =Du2 + i|u|2e−ϕ ≥ i|u|2e−ϕ. L2(X,ϕ) L2(X,ϕ) X X 2.9 The L2 ∂¯-Method for Scalar-Valued Forms of Type (1, 0) 65

~~Remark If i ≥ 0, then we get = + D u, D v L2(X,ϕ) D u, D v L2(X,ϕ) u, v L2(X,i,ϕ)~~

~~Du2 =Du2 +u2 ≥u2 and L2(X,ϕ) L2(X,ϕ) L2(X,i,ϕ) L2(X,i,ϕ).~~

Proof of Proposition 2.8.2 For each pair of functions u, v ∈ D(X), Proposition 2.7.3 gives = ∧ −ϕ = ·¯ −ϕ D u, D v L2(X,ϕ) iD u D ve i(D D u) ve X X − − =− i(D D u) ·¯ve ϕ + iuve¯ ϕ X X =−i vDDu · e−ϕ + iuve¯ −ϕ X X = i (Dv) ∧ Du · e−ϕ + iuve¯ −ϕ X X =−i (Du) ∧ Dv · e−ϕ + iuve¯ −ϕ X X = + ¯ −ϕ D u, D v L2(X,ϕ) iuve . X

Exercises for Sect. 2.8 2.8.1 Show that the function z → |z|2 on C is strictly subharmonic. 2.8.2 Show that if ϕ is a C2 strictly subharmonic function (i.e., i∂∂ϕ¯ > 0) on a Riemann surface X, then ϕ cannot attain a local maximum.

~~2.9 The L2 ∂¯-Method for Scalar-Valued Forms of Type (1, 0)~~

~~The following theorem (in various guises) is the main tool in this book:~~

Theorem 2.9.1 Let X be a Riemann surface, let ϕ be a real-valued C∞ func- ¯ tion on X with i = iϕ = i∂∂ϕ ≥ 0, and let Z ={x ∈ X | x = 0}. Then, ∈ 2 \ for every measurable (1, 1)-form β on X with β X\Z L1,1(X Z,i,ϕ) and = 2 ∈ 2 β 0 a.e. in Z (in particular, β is in Lloc on X), there exists a form α L1,0(X, ϕ) such that = ¯ = ≤ Ddistrα ∂distrα β and α L2(X,ϕ) β L2(X\Z,i,ϕ).

~~k k In particular, if β is of class C for some k ∈ Z>0 ∪{∞}, then α is also of class C and ∂α¯ = β. 66 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms~~

~~2 Remark The form β in the above theorem is in Lloc because β vanishes on Z, while for any compact set K in any local holomorphic coordinate neighborhood (U, z), we have~~

~~· 2 \ ∧ ¯ ≤ A· 2 \ , L1,1(K Z,(i/2)dz dz) L1,1(K Z,i,ϕ) where i 1/2 A ≡ max eϕ < ∞. K (i/2)dz∧ dz¯~~

≡ = Proof of Theorem 2.9.1 Let N β L2(X\Z,i,ϕ) β/(i) L2(X\Z,i,ϕ).For each function f ∈ D(X), the Schwarz inequality and the fundamental estimate (Proposition 2.8.2)give β¯ ¯ −ϕ = · · −ϕ · = f βe f e i f,β/(i) L2(X\Z,i,ϕ) X X\Z i ≤ · f L2(X\Z,i,ϕ) β/(i) L2(X\Z,i,ϕ) ≤ · ≤ · N f L2(X,i,ϕ) N D f L2(X,ϕ). :[ ] →− ¯ −ϕ It follows that the mapping ϒ D f i X f βe is a bounded complex linear [D ] 2 functional on the subspace D (X) of L1,0(X, ϕ). For by the above inequality, ϒ ∈ D | [ ]| ≤ · is well deﬁned, and for each f (X),wehave ϒ D f N D f L2(X,ϕ). In particular, ϒ≤N. By the Hahn–Banach theorem (Theorem 7.5.11), there 2 = exists a bounded linear functional ϒ on L1,0(X, ϕ) such that ϒ D [D(X)] ϒ and ϒ=ϒ. Therefore, by Theorem 7.5.10, there exists a (unique) element ∈ 2 =≤ · =· α L1,0(X, ϕ) such that α L2(X,ϕ) ϒ N and ϒ( ) ,α L2(X,ϕ). More- over, for each f ∈ D(X),wehave iα ∧ Dfe−ϕ = ϒ(Df)= (−i)fβe¯ −ϕ = iβfe¯ −ϕ. X X X = Therefore, by Proposition 2.7.3, Ddistrα β, as required. Finally, the regularity statement at the end follows from Theorem 2.7.4.

~~It is often more convenient to apply Theorem 2.9.1 in one of the following forms, the proofs of which are left to the reader (see Exercises 2.9.1 and 2.9.2):~~

Corollary 2.9.2 Suppose that X is a Riemann surface, ω is a Kähler form on X, ϕ is a real-valued C∞ function on X, ρ is a nonnegative measurable function on X with iϕ ≥ ρω, and Z ={x ∈ X | ρ(x) = 0}. Then, for every measurable (1, 1)-form β on X with β = 0 a.e. in Z and

~~∈ 2 \ = 2 \ + β X\Z L1,1(X Z,ρω,ϕ) L1,1(X Z,ω,ϕ log ρ) 2.10 Existence of Meromorphic 1-Forms and Meromorphic Functions 67~~

~~2 ∈ 2 (in particular, β is in Lloc on X), there exists a form α L1,0(X, ϕ) such that = ¯ = ≤ Ddistrα ∂distrα β and α L2(X,ϕ) β L2(X\Z,ρω,ϕ).~~

~~k k In particular, if β is of class C for some k ∈ Z>0 ∪{∞}, then α is also of class C and ∂α¯ = β.~~

~~ 2 ≤ 2 ≥ Remark The main point is that β L (X\Z,iϕ,ϕ) β L (X\Z,ρω,ϕ) if iϕ ρω.~~

Corollary 2.9.3 Suppose that X is a Riemann surface, ω is Kähler form on X, ϕ ∞ 2 is a real-valued C function on X, and C is a positive constant with iϕ ≥ C ω. ∈ 2 ∈ 2 Then, for every form β L1,1(X,ω,ϕ), there exists a form α L1,0(X, ϕ) such that = ¯ = ≤ −1 Ddistrα ∂distrα β and α L2(X,ϕ) C β L2(X,ω,ϕ).

~~k k In particular, if β is of class C for some k ∈ Z>0 ∪{∞}, then α is also of class C and ∂α¯ = β.~~

~~Remark In Sect. 2.11, we will show that every Riemann surface admits a Kähler form.~~

Exercises for Sect. 2.9 2.9.1 Prove Corollary 2.9.2. 2.9.2 Prove Corollary 2.9.3. 2.9.3 Let X be a Riemann surface, let ϕ be a real-valued C∞ function on X with ∞ i = iϕ ≥ 0, and let Z ={x ∈ X | x = 0}. Prove that for every C (1, 1)- = ∈ 2 \ form β on X with β 0a.e.inZ and β X\Z L1,1(X Z,i,ϕ), there C∞ ∈ 2 exists a form α L0,1(X, ϕ) such that

∂α = β and α 2 ≤β 2 \ . L0,1(X,ϕ) L1,1(X Z,i,ϕ) 2.9.4 Let X be a Riemann surface, let ϕ be a real-valued C∞ function on X,letω ∞ be a nonnegative C (1, 1)-form on X,letZ ={x ∈ X | ωx = 0}, and let β C∞ = ∈ 2 \ be a (1, 1)-form on X such that β 0a.e.inZ and β X\Z L1,1(X Z,ω,ϕ). Prove that β ≡ 0onZ (hence in the C∞ case of Theorem 2.9.1 and in Exercise 2.9.3, there is no loss of generality if we assume that β ≡ 0onZ).

~~2.10 Existence of Meromorphic 1-Forms and Meromorphic Functions~~

The power of Theorem 2.9.1 is demonstrated by the following important application, which will play a crucial role in the proof of such central facts as Radó’s theorem on second countability (see Sect. 2.11) and the Riemann mapping theorem (see Chap. 5): 68 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms

Theorem 2.10.1 For every point in a Riemann surface, there exists a meromorphic 1-form that is holomorphic except for a pole of arbitrary prescribed order ≥ 2 at the point. In fact, for each integer m ≥ 2, there exists a universal constant Cm > 0 such that if X is any Riemann surface, p is any point in X, and (D, = z, (0; 1)) is any local holomorphic chart with p ∈ D and (p) = z(p) = 0, then there exists a meromorphic 1-form θ on X with the following properties: (i) The meromorphic 1-form θ is holomorphic on X \{p} and has a pole of order m at p; ≤ (ii) We have θ L2(X\D) Cm; (iii) The meromorphic 1-form θ − z−m dz on D has at worst a simple pole at p; and − −m+1 ≤ (iv) We have zθ z dz L2(D) Cm.

~~Remark The value for the constant Cm that we will obtain is far from optimal.~~

~~Lemma 2.10.2 For any choice of constants a,b,c ∈ R with a~~

Proof For s>0, the function ⎧ 1 ⎪ s exp( − ) ⎨⎪ s = a t if ac− b as s →∞ (for example, by the dominated convergence theorem). Therefore, by the intermediate value theorem, there is a number s0 > 0 such that μ(s0) = c − b. The function → ≡ t t χ(t) a ρs0 (u) du then has the required properties.

Lemma 2.10.3 Let r>0 be a constant and let ψ be a C∞ strictly subharmonic function on (0; r). Then there is a constant b0 = b0(r, ψ) > 0 such that for every constant b>b0, there exist a constant R = R(b,r,ψ) ∈ (0,r) and a nonnegative C∞ subharmonic function ϕ on C∗ with ϕ ≡ 0 on a neighborhood of C \ (0; r) and ∗ ϕ(z) = ψ(z)− log |z|2 − b ∀z ∈ (0; R).

Proof Setting ρ(z) ≡ ψ(z) − log |z|2 for all z ∈ ∗(0; r), we may ﬁx positive constants R0 and R1 with 0 ρ b>b a c sup(0;R0,R1) . Given a constant 0, we may ﬁx constants and with 2.10 Existence of Meromorphic 1-Forms and Meromorphic Functions 69

∞ c>b>a>b0 > 0, and applying Lemma 2.10.2, we get a C function χ : R → R such that χ ≥ 0 and χ ≥ 0onR, χ(t) = 0fort ≤ a, and χ(t) = t − b for t ≥ c. For R ∈ (0,r) sufﬁciently small, we have ρ>con ∗(0; R). This constant R and the function ϕ on C∗ given by ∗ χ(ρ) on (0; R1), ϕ ≡ 0onC \ (0; R1), then have the required properties. For the choice of χ guarantees that ϕ is nonneg- ∞ ∗ ative and of class C , ϕ vanishes on C \ (0; R0), and ϕ = ρ − b on (0; R). ∗ Furthermore, on (0; R1),wehaveiρ = iψ > 0 and hence

~~iϕ = χ (ρ) · iψ + χ (ρ) · i∂ρ ∧ ∂ρ ≥ 0.~~

Proof of Theorem 2.10.1 The idea of the proof (an idea that is applied in various guises throughout this book) is to ﬁrst produce a C∞ solution, that is, a C∞ 1-form τ that has all of the required properties except that it is not meromorphic (away from p). Forming a suitable L2 solution α of the equation ∂α¯ = ∂τ¯ , one gets a meromorphic 1-form θ = τ − α with the required properties. 2 Letting ζ denote the (standard) coordinate function on C, setting ψ0 ≡|ζ| , and applying Lemma 2.10.3, we get positive constants b, R0, and R1 and a nonnegative ∞ ∗ C subharmonic function ϕ0 on C such that R0 0on (0 R0). We may also ﬁx a constant R2 ∈ (0,R0) and a function η0 ∈ D((0; R0)) such that η0 ≡ 1on(0; R2). −m Given an integer m ≥ 2, we have the meromorphic 1-form γm ≡ ζ dζ on C ∞ ∗ and the C form τm ≡ η0γm of type (1, 0) on C . In particular, τm ≡ 0 on a neigh- ∗ ¯ borhood of C \ (0; R0) and τm is holomorphic on (0; R2). Hence βm ≡ ∂τm = ¯ ∞ ∗ ∂η0 ∧ γm is a C 2-form that vanishes on C \ (0; R2,R0). Observe that

~~Am ≡βm 2 ∗ ; =βm 2 ; ∧ ¯ < ∞ L ( (0 R0),iϕ0 ,ϕ0) L ((0 R2,R0),idζ dζ,ϕ0) and~~

~~m−1 B ≡ζτ − ζγ 2 ∗ =(η − 1)dζ/ζ 2 < ∞. m m m L ( (0;1)) 0 L ((0;R2,1))~~

~~2 ∗ By construction, the function ϕ0 +log |ζ | is bounded on (0; 1), so we may deﬁne a positive constant Cm by~~

~~(ϕ +log |ζ |2)/2 Cm ≡ Am + Bm + sup e 0 · Am < ∞. ∗(0;1)~~

Suppose now that p is a point in a Riemann surface X(D,= z, (0; 1)) is a local holomorphic chart with p ∈ D and (p) = z(p) = 0, and Y ≡ X \{p}.The 70 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms

∗ ∞ form τ on Y that is equal to τm on D \{p} and 0 elsewhere is a C differential −1 form of type (1, 0) that vanishes on X \ ((0; R0)) and that is holomorphic −1 ∗ on ( (0; R2)). The function ϕ on Y given by ϕ = ϕ0(z) on D \{p} and 0 elsewhere is nonnegative, subharmonic, and of class C∞. Moreover, ϕ vanishes on −1 −1 ∗ ∞ X \ ((0; R1)) and iϕ = idz∧ dz>¯ 0on ( (0; R0)).TheC (1, 1)- ¯ ∗ form β ≡ ∂τ on Y is equal to βm on D \{p}, vanishes on −1 Y \ ((0; R2,R0)) ⊃ Z ≡ q ∈ Y | i(ϕ)q = 0 , and satisﬁes

~~β 2 \ =βm 2 ∗ ; = Am < ∞. L (Y Z,iϕ,ϕ) L ( (0 R0),iϕ0 ,ϕ0) Therefore, by Theorem 2.9.1, there exists a C∞ (1, 0)-form α on Y such that~~

~~¯ = ≤ ∂α β and α L2(Y,ϕ) Am.~~

Thus C∞ (1, 0)-form θ ≡ τ − α on Y satisﬁes ∂θ¯ = 0 and is therefore holomor- −1 ∗ phic on Y . Since τ and therefore α are holomorphic on ( (0; R2)),wehave ∗ α −1 ∗ = f(z)dz f ∈ O( ( ; R )) ( (0;R2)) for some function 0 2 . In particular, i i 2 |ζf (ζ)|2 dζ ∧ dζ¯ = α ∧¯α · e−ϕ · e|z| −b ∗ −1 ∗ (0;R2) 2 ( (0;R2)) 2

1 R2−b 2 ≤ e 2 ·α < ∞. 2 L2(Y,ϕ) Hence, by Riemann’s extension theorem (Theorem 1.2.10), the function ζ → ζf (ζ) extends to a (unique) holomorphic function on (0; R2); that is, f extends to a meromorphic function with at worst a simple pole at 0. Therefore, since η0 ≡ 1on (0; R2) and γm is a meromorphic 1-form with a pole of order m>1at0,wesee that θ = τ − α determines a meromorphic 1-form on X that is holomorphic on Y and that has a pole of order m at p. For the bounds, we observe that

~~ =− = ≤ ≤ θ L2(X\D) α L2(X\D) α L2(X\D,ϕ) Am Cm and that~~

− −m+1 ≤ − −m+1 + zθ z dz L2(D) zτ z dz L2(D) zα L2(D) + | |2 ≤ + (ϕ0 log ζ )/2 · Bm sup e α L2(D,ϕ) ∗(0;1) + | |2 ≤ + (ϕ0 log ζ )/2 · Bm sup e α L2(Y,ϕ) ∗(0;1)

~~2 (ϕ0+log |ζ | )/2 ≤ Bm + sup e · Am ≤ Cm. ∗(0;1) 2.11 Radó’s Theorem on Second Countability 71~~

~~Corollary 2.10.4 Every Riemann surface X admits a nonconstant meromorphic function.~~

Proof We may choose two distinct points p and q in X. Applying Theorem 2.10.1, we get two meromorphic 1-forms η and θ that are holomorphic except for poles of order 2 at p and q, respectively. The quotient f = η/θ then determines a nonconstant meromorphic function on X.

Exercises for Sect. 2.10 2.10.1 Show that there exists a constant A>0 such that for each integer m ≥ 2, the m constant Cm ≡ A has the properties described in Theorem 2.10.1. 2.10.2 Prove the following generalization of Theorem 2.10.1.LetX0 be a Riemann surface, let θ0 be a meromorphic 1-form on X0 that is holomorphic except for a pole of order m ≥ 2 at some point p0 ∈ X0,let0 be a relatively compact neighborhood of p0 in X0, and let f0 be a holomorphic function on X0 that vanishes at p0. Then there exists a constant C = C(X0,θ0,0,f0)>0 such that if X is any Riemann surface, p is a point in X, and : → 0 is a biholomorphic mapping of a neighborhood of p onto 0 with (p) = p0, then there exists a meromorphic 1-form θ on X with the following properties: (i) The meromorphic 1-form θ is holomorphic X \{p} and has a pole of order m at p; ≤ (ii) We have θ L2(X\) C; ∗ (iii) The meromorphic 1-form θ − θ0 on has at worst a simple pole at p; and · − · ∗ ≤ (iv) We have f0() θ f0() θ0 L2() C.

~~2.11 Radó’s Theorem on Second Countability~~

~~The goal of this section is the following important theorem:~~

~~Theorem 2.11.1 (Radó) Every Riemann surface X is second countable; that is, the topology in X admits a countable basis.~~

~~The proof considered here is similar to that in [Sp].~~

~~Lemma 2.11.2 Every nonempty open subset of a Riemann surface X has only countably many connected components.~~

Proof Fixing a point p ∈ and a connected neighborhood U of p in , and applying Theorem 2.10.1, we get a nontrivial (i.e., not everywhere zero) holomorphic \{ } ∞ { } 1-form θ on X p with θ L2(X\U) < .If i i∈I is the family of (distinct) 72 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms connected components of that do not meet (i.e., which do not contain) U and I = ∅, then for every ﬁnite set J ⊂ I ,wehave 2 ≥ 2 = 2 θ 2 \ θ 2 θ 2 . L (X U) L ( i∈J i ) L (i ) i∈J ∞ 2 Therefore > ∈ θ 2 , and hence, since the right-hand side is an un- i I L (i ) ordered sum of (strictly) positive terms (see Example 7.1.3), the index set I must be countable.

Proof of Radó’s theorem Corollary 2.10.4 provides a nonconstant meromorphic function, that is, a nonconstant holomorphic mapping f : X → P1.LetP be the countable collection of all sets D ⊂ P1, where D is either a disk in C with ratio- 1 nal radius and center in Q + iQ,orD = P \ (0; r) for some r ∈ Q>0.LetB be the collection of all relatively compact open subsets B of X such that B is a connected component of f −1(D) for some set D ∈ P. It follows that B is a basis for the topology in X (see Exercise 2.11.1). Moreover, by Lemma 2.11.2,thesetf −1(D) has only countably many connected components for each D ∈ P, and therefore the basis B must be countable.

~~Radó’s theorem and Proposition 9.7.6 together give the following:~~

Corollary 2.11.3 Every Riemann surface X admits a Kähler form (i.e., a positive C∞ differential form of type (1, 1)). In fact, given a continuous real (1, 1)-form τ on X, there exists a Kähler form ω with ω ≥ τ on X.

Corollary 2.11.4 (Montel’s theorem for a Riemann surface) If {fn} is a sequence of holomorphic functions on a Riemann surface X and {fn} is uniformly bounded on each compact subset of X, then some subsequence of {fn} converges uniformly on compact subsets of X to a holomorphic function on X.

Proof We may choose a countable open covering {Uν} of X such that for each ν, Uν is relatively compact in some local holomorphic coordinate neighborhood. Mon- tel’s theorem in the plane (Corollary 1.2.7) and Cantor’s diagonal process together { } { } yield a subsequence fnk such that the sequence fnk Uν converges uniformly to { } a holomorphic function on Uν for each ν. It follows that fnk converges uniformly on compact subsets of X to some holomorphic function on X.

Remark A C∞ surface need not be second countable. Moreover, Radó’s theorem is false in higher dimensions; that is, there exist connected 2-dimensional complex manifolds that are not second countable, provided, of course, one does not include second countability as part of the deﬁnition (see, for example, [Hu]).

Exercises for Sect. 2.11 In Exercises 2.11.1 and 2.11.2 below, as in the proof of Radó’s theorem (Theorem 2.11.1), X denotes a Riemann surface; f : X → P1 denotes a nonconstant holomorphic mapping; P denotes the collection of all sets 2.12 The L2 ∂¯-Method for Scalar-Valued Forms of Type (0, 0) 73

D ⊂ P1, where D is either a disk in C with rational radius and center in Q + iQ,or 1 D = P \ (0; r) for some r ∈ Q>0; and B denotes the collection of all relatively compact open subsets B of X such that B is a connected component of −1(D) for some set D ∈ P. 2.11.1 Prove that B is a basis for the topology in X (this fact was used in the proof of Radó’s theorem). 2.11.2 Another way to see that B is countable is to observe that each element B ∈ B meets at most countably many connected components of f −1(D) for each set D ∈ P. Prove this observation, and using it in place of Lemma 2.11.2, prove Radó’s theorem. Remark This argument is essentially the proof of a special case of the Poincaré–Volterra theorem (see, for example, [Ns5]).

~~2.12 The L2 ∂¯-Method for Scalar-Valued Forms of Type (0, 0)~~

It is also useful to consider solutions of the inhomogeneous Cauchy–Riemann equation ∂α¯ = β in which α is a function and β is a differential form of type (0, 1).In fact, this problem is essentially equivalent to the problem considered in Sect. 2.9. In order to obtain the L2 solution, we will have to consider the curvature form associated to a Kähler form ω (i.e., a C∞ positive differential form of type (1, 1)) on a complex 1-manifold X. Suppose θ1 and θ2 are two nonvanishing holomorphic ¯ 1-forms on an open set U ⊂ X and Gj ≡ ω/(iθj ∧ θj ) for j = 1, 2. Then the difference 2 (− log G1) − (− log G2) = log |θ1/θ2| ¯ ¯ is a harmonic function; that is, ∂∂[− log G1]=∂∂[− log G2]. Thus we may make the following deﬁnition:

Deﬁnition 2.12.1 For any Kähler form ω on a complex 1-manifold X, the associated curvature is the unique differential form (X,ω) = ω that satisﬁes = = ¯[− ∧ ¯ ] ω U − log(ω/(iθ∧θ))¯ ∂∂ log(ω/(iθ θ)) for every nonvanishing local holomorphic 1-form θ on an open set U ⊂ X.

Remarks 1. Equivalently, if ω = G · (i/2)dz∧ dz¯ (i.e., G =−2iω(∂/∂z,∂/∂z)¯ = ω(∂/∂x,∂/∂y)) in a local holomorphic coordinate neighborhood (U, z = x + iy), ¯ then ω = − log G = ∂∂[− log G] on U. ∞ 2. If ϕ is a real-valued C function on X, then e−ϕ ω = ω + ϕ .

~~Example 2.12.2 For the Euclidean Kähler form ω ≡ (i/2)dz∧ dz¯ on C, ω ≡ 0.~~

Example 2.12.3 The chordal Kähler form on the Riemann sphere P1 is the unique 2 −2 Kähler form ω for which ωC = 2i(1+|z| ) dz∧dz¯. The corresponding curvature form satisﬁes iω = ω (see Exercise 2.12.1). 74 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms

~~The main goal of this section is the following:~~

~~Theorem 2.12.4 Let X be a Riemann surface, let ω be a Kähler form on X, let ϕ be a real-valued C∞ function on X with~~

i ≡ ie−ϕ ω = iω + iϕ ≥ 0, let ρ ≡ i/ω, and let Z ={x ∈ X | x = 0}={x ∈ X | ρ(x) = 0}. Then, for every = ∈ 2 \ + measurable (0, 1)-form β on X with β 0 a.e. in Z and β X\Z L0,1(X Z,ϕ 2 ∈ 2 log ρ) (in particular, β is in Lloc on X), there exists a function α L0,0(X,ω,ϕ) such that

~~= ¯ = ≤ Ddistrα ∂distrα β and α L2(X,ω,ϕ) β L2(X\Z,ϕ+log ρ).~~

~~k k In particular, if β is of class C for some k ∈ Z>0 ∪{∞}, then α is also of class C and ∂α¯ = β.~~

Remarks 1. In order to apply Theorem 2.12.4, one needs a Kähler form ω.For- tunately, by Corollary 2.11.3, a Kähler form always exists. In fact, this version is actually equivalent to Theorem 2.9.1. 2. Although it is possible to obtain the theorem directly, we will instead prove the theorem by ﬁrst forming the exterior product of the given form and a meromorphic 1-form (provided by Theorem 2.10.1), and then applying the solution for forms of type (1, 0) (Theorem 2.9.1). 2 3. The proof that β is in Lloc in the above (as well as in Corollary 2.12.5 below) is similar to that in the remark following the statement of Theorem 2.9.1.

~~It is often more convenient to apply Theorem 2.12.4 in one of the following forms, the proofs of which are left to the reader (see Exercises 2.12.2 and 2.12.3):~~

Corollary 2.12.5 Suppose that X is a Riemann surface, ω is a Kähler form on X, ϕ is a real-valued C∞ function on X, ρ is a nonnegative measurable function on X with iω + iϕ ≥ ρω, and Z ={x ∈ X | ρ(x) = 0}. Then, for every measurable = ∈ 2 \ + (0, 1)-form β on X with β 0 a.e. in Z and β X\Z L0,1(X Z,ϕ log ρ) (in 2 ∈ 2 particular, β is in Lloc on X), there exists a function α L0,0(X,ω,ϕ)such that

~~= ¯ = ≤ Ddistrα ∂distrα β and α L2(X,ω,ϕ) β L2(X\Z,ϕ+log ρ).~~

~~k k In particular, if β is of class C for some k ∈ Z>0 ∪{∞}, then α is also of class C and ∂α¯ = β.~~

Corollary 2.12.6 Suppose that X is a Riemann surface, ω is a Kähler form on X, ϕ is a real-valued C∞ function on X, and C is a positive constant for which + ≥ 2 ∈ 2 iω iϕ C ω on X. Then, for every form β L0,1(X, ϕ), there exists a func- 2.12 The L2 ∂¯-Method for Scalar-Valued Forms of Type (0, 0) 75

~~∈ 2 tion α L0,0(X,ω,ϕ)such that = ¯ = ≤ −1 Ddistrα ∂distrα β and α L2(X,ω,ϕ) C β L2(X,ϕ).~~

~~k k In particular, if β is of class C for some k ∈ Z>0 ∪{∞}, then α is also of class C and ∂α¯ = β.~~

Proof of Theorem 2.12.4 According to Theorem 2.10.1, we may ﬁx a nontrivial meromorphic 1-form θ on X, and we may let Q be the (discrete) set of zeros and poles of θ. On the Riemann surface Y ≡ X \ Q, the function τ ≡ ω/(iθ ∧ θ)¯ is then ∞ positive and of class C , and we have − log τ = ω; that is, iϕ−log τ = i = ρω. The form β0 ≡ θ ∧ β is a measurable differential form of type (1, 1) that vanishes on Z ∩ Y and that satisﬁes (since the set Q is discrete, and therefore of measure 0) 2 β0 2 L (Y \Z,iϕ−log τ ,ϕ−log τ) θ ∧ β 2 = 2 = −(ϕ−log τ) β0 L2(Y \Z,ρω,ϕ−log τ) e ρω Y \Z ρω ∧ θ¯ ∧ β¯ ω = θ β · · · −(ϕ+log ρ) ¯ e ω Y \Z ω ω iθ ∧ θ iθ ∧ θ¯ β¯ = ∧ · · · −(ϕ+log ρ) iθ β ¯ e Y \Z iθ ∧ θ θ = − ∧ ¯ · −(ϕ+log ρ) = 2 ∞ ( i)β β e β L2(X\Z,ϕ+log ρ) < . X\Z

~~2 ∈ Therefore, β0 is in Lloc on Y , and by Theorem 2.9.1, there exists a form α0 2 − = ¯ = L1,0(Y, ϕ log τ) such that Ddistrα0 ∂distrα0 β0 in Y and~~

~~α 2 ≤β 2 =β 2 . 0 L (Y,ϕ−log τ) 0 L (Y \Z,iϕ−log τ ,ϕ−log τ) L (X\Z,ϕ+log ρ)~~

The measurable function α ≡−α0/θ on X (again, Q is a set of measure 0) then satisﬁes ∧¯ 2 = | |2 −ϕ = | |2 −ϕ = iα0 α0 −ϕ α 2 α e ω α0/θ e ω e ω L (X,ω,ϕ) iθ ∧ θ¯ Y Y Y −(ϕ−log τ) = iα0 ∧¯α0 · e Y = 2 ≤ 2 α0 L2(Y,ϕ−log τ) β L2(X\Z,ϕ+log ρ).

2 ¯ = In particular, α is in Lloc on X, and it remains to show that ∂distrα β on X.For each point p ∈ X, we may choose a local holomorphic coordinate neighborhood (U, = z, (0; 2)) such that U ∩ Q ⊂{p} and z(p) = 0. The meromorphic function f ≡ θ/dz on U is then nonvanishing and holomorphic on U \{p}⊂Y , and 76 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms

≡ ≡ ¯ 2 ⊂ 1 the measurable functions a α0/dz and b β/dz on U are in Lloc Lloc. In particular, α =−a/f and β0 = fbdz∧ dz¯ on U. We may also choose a nonnegative C∞ function χ such that χ ≡ 1onC \ (0; 2) and χ ≡ 0on(0; 1). For every C∞ function η with compact support in D ≡ −1((0; 1)), the dominated convergence theorem gives (here, we denote the coordinate on C by w) ∂η i α · · dz∧ dz¯ ∂z¯ 2 D ∂η i = lim χ(z/)α · · dz∧ dz¯ →0+ ∂z¯ 2 D −a ∂η i = lim χ(z/) · · dz∧ dz¯ + →0 D f ∂z¯ 2 ∂ η i = lim − a · χ(z/) · dz∧ dz¯ + →0 D ∂z¯ f 2 1 ∂χ η i + a (z/) · dz∧ dz¯ D ∂w¯ f 2 i = lim − b · χ(z/)η· dz∧ dz¯ + →0 D 2 1 ∂χ i − α (z/)η · dz∧ dz¯ D ∂w¯ 2 i 1 ∂χ i =− bη · dz∧ dz¯ − lim α (z/)η · dz∧ dz¯ . + D 2 →0 D ∂w¯ 2

∈ 2 ¯ ≡ C \ ; On the other hand, since α Lloc(X) and ∂χ/∂w 0on (0 1, 2),wehave, for some constant C>0, 1 ∂χ · i ∧ ¯ ≤ ◦ −1 → → + α (z/)η dz dz C α L2((0;,2)) 0as 0 . D ∂w¯ 2 ¯ It follows that ∂distrα = β.

~~Exercises for Sect. 2.12 2.12.1 Verify that (see Example 2.12.3) there is a unique Kähler form ω on P1 such that 2 −2 ωC = 2i(1 +|z| ) dz∧ dz.¯~~

Verify also that iω = ω. 2.12.2 Prove Corollary 2.12.5. 2.12.3 Prove Corollary 2.12.6. 2.12.4 Let X be a Riemann surface, let ω be a Kähler form on X,letϕ be a ∞ real-valued C function on X with i ≡ ie−ϕ ω = iω + iϕ ≥ 0, let 2.13 Topological Hulls and Chains to Inﬁnity 77

ρ ≡ i/ω, and let Z ={x ∈ X | x = 0}={x ∈ X | ρ(x) = 0}. Prove C∞ = ∈ that for every (1, 0)-form β on X with β 0a.e.inZ and β X\Z 2 \ + C∞ ∈ 2 L1,0(X Z,ϕ log ρ), there exists a function α L0,0(X,ω,ϕ) such that = ≤ ∂α β and α L2(X,ω,ϕ) β L2(X\Z,ϕ+log ρ) (cf. Exercises 2.9.3 and 2.9.4).

~~2.13 Topological Hulls and Chains to Inﬁnity~~

As suggested by Theorem 2.9.1 and its applications (see also Theorem 3.9.1 and its applications), it is useful to have available a large supply of C∞ strictly subharmonic ∞ functions on an (open) Riemann surface, i.e., C functions ϕ for which iϕ > 0 (see Definition 2.8.1). In Sect. 2.14, it will be shown that every open Riemann surface admits a C∞ strictly subharmonic exhaustion function. Recall that a function ρ on a Hausdorff space X is an exhaustion function if {x ∈ X | ρ(x) < a} X for each a ∈ R (see Definition 9.3.10). The first proof of the existence of a C∞ strictly subharmonic exhaustion function on a connected noncompact Riemannian manifold was obtained by Greene and Wu [GreW]. Demailly [De2] provided an elementary proof using a local construction. Demailly’s proof may be modified (as well as simplified) to give an exhausting C∞ strict subsolution for an arbitrary second-order linear elliptic differential operator with continuous coefficients on a second countable noncompact C∞ manifold (for the details see [NR]). The proof of the existence of a C∞ strictly subharmonic exhaustion function on an open Riemann surface appearing in this book is adapted from [NR]. In this section, we first consider some of the topological facts required for the proof.

Deﬁnition 2.13.1 For a subset A of a Hausdorff space X,thetopological hull hX(A) of A in X is the union of A with all of the connected components of X \ A that are relatively compact in X. An open subset of X is called topologically Runge in X if hX() = .

~~Remark Intuitively, one obtains hX(A) by ﬁlling in the holes of A (see Fig. 2.5).~~

~~The proofs of the following properties are left to the reader (see Exercise 2.13.1):~~

~~Lemma 2.13.2 Let X be a Hausdorff space.~~

(a) For every set A ⊂ X, we have hX(hX(A)) = hX(A) ⊃ A. (b) If A ⊂ B ⊂ X, then hX(A) ⊂ hX(B). (c) If X is locally connected and A is a closed subset of X, then hX(A) is closed. 78 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms

~~Fig. 2.5 A subset and its topological hull~~

~~For a compact subset of a suitable topological space (such as a manifold), the topological hull has a nice form (cf. [Mal] and [Ns5]):~~

Lemma 2.13.3 If K is a compact subset of a connected, noncompact, locally connected, locally compact Hausdorff space X, then hX(K) is compact; X \ hX(K) has only ﬁnitely many components; and for each point p ∈ X \ hX(K), there is a connected noncompact closed subset C of X with p ∈ C ⊂ X \ hX(K).

~~Remark One gets a version in which X has more than one, but only ﬁnitely many, connected components by applying the lemma to each connected component (see Exercise 2.13.2).~~

Proof of Lemma 2.13.3 The lemma is trivial for K empty, so we may assume that K = ∅. Since X is locally connected, hX(K) is a closed set whose complement has no relatively compact components (by Lemma 2.13.2). Since X is locally compact Hausdorff, we may choose a relatively compact neighborhood of K in X.The components of X \ K are open and disjoint, so only ﬁnitely many meet the compact set ∂ ⊂ X \ K. By replacing with the union of and all of the relatively compact components of X \ K meeting ∂, we may assume that no relatively compact component of X \ K meets ∂. On the other hand, every component E of X \ K must satisfy E ∩ K = ∂E = ∅. For E is open and closed in X \ K,so∂E ⊂ K, while E = X,so∂E = E \ E = ∅ (E cannot be both open and closed in the connected space X). It follows that if E meets X \ , then E meets ∂, and hence E is not relatively compact in X. Thus

X \ ⊂ E1 ∪···∪Em for ﬁnitely many components E1,...,Em of X \ K, none of which are relatively compact in X. It follows that hX(K) ⊂ and X \ hX(K) = E1 ∪···∪Em. Applying the above argument with K = in place of K, we get a relatively compact neighborhood of K in X such that X \ ⊂ X \ hX(K ) = ∪···∪ \ E1 Ek, where E1,...,Ek are distinct components of X K , none of which = ⊂ are relatively compact in X. For each i 1,...,m, we get Ej Ei for some 2.13 Topological Hulls and Chains to Inﬁnity 79 j ∈{1,...,k}. Thus, if Ai is the set of points in Ei that lie some connected noncompact closed subset of X that is contained in Ei , then Ai = ∅. Given a point p ∈ Ei ∩ Ai , we may choose a relatively compact connected neighborhood V of p in Ei . We may then choose a point q ∈ Ai ∩ V and a connected noncompact closed subset B of X with q ∈ B ⊂ Ei.ThesetC ≡ V ∪ B is then a connected noncompact closed subset of X that lies in Ei and that contains V . Thus V ⊂ Ai , and hence the nonempty set Ai is both open and closed in the connected set Ei . Therefore, Ai = Ei , and the lemma follows.

Lemma 2.13.4 Let X be a second countable, noncompact, connected, locally connected, locally compact Hausdorff space. Then there is a sequence of compact sets ◦ { }∞ = ∞ ⊂ = Kν ν=1 such that X ν=1 Kν , and for each ν, Kν Kν+1 and hX(Kν ) Kν . { Proof By Lemma 9.3.6, we may ﬁx a sequence of compact sets Hν} with X = ∞ ≡ ν=1 Hν . Set K1 hX(H1).GivenKν , we may choose a compact set Kν+1 with ◦ ∪ ⊂ ≡ Hν Kν Kν+1, and we may set Kν+1 hX(Kν+1). This yields the desired sequence.

Lemma 2.13.5 Let X be a connected, locally connected, locally compact Hausdorff space; let B be a countable collection of connected open subsets that is a basis for the topology in X; let K be a compact subset of X; and let U be a connected component of X \ hX(K). Then, for each point p ∈ U, there exists a sequence of basis elements {Bj } that tends to inﬁnity (i.e., {Bj } is a locally ﬁnite family in X) such that p ∈ B1, and for each j, Bj U and Bj ∩ Bj+1 = ∅.

Proof Lemma 2.13.3 provides a connected noncompact closed subset C of X with p ∈ C ⊂ U, and Lemma 9.3.6 provides a countable locally finite (in X) covering A of C by basis elements that are relatively compact in U. For each point q ∈ C, there is a finite sequence of elements B1,...,Bk of A that forms a chain from p to q; that is, p ∈ B1, q ∈ Bk, and Bj ∩Bj+1 = ∅ for j = 1,...,k−1 (we will call k the length of the chain). For the set E of points q in C for which there is a chain from p to q is clearly nonempty and open in C. On the other hand, E is also closed, because if q ∈ E, then q ∈ B for some set B ∈ A and there must be some point r ∈ B ∩ E. A chain B1,...,Bk from p to r yields the chain B1,...,Bk,B from p to q. Thus E = C. Observe that if q ∈ C and B1,...,Bk is a chain of minimal length from p to q, then the sets B1,...,Bk are distinct. Now since C is noncompact and closed, we may choose a locally finite sequence of points {qν} in C; i.e., qν →∞in X (for example, we may fix an increasing sequence {ν} of relatively compact open subsets of X with union X and choose q ∈ C \ for each ν). For each ν, we may choose a chain B(ν),...,B(ν) of ν ν 1 kν minimal length from p to qν . Since the elements of A are relatively compact in U A (ν) and is locally finite in X, there are only finitely many possible choices for Bj for each j (only finitely many elements of A are in some chain of length j from p). Moreover, for each fixed j ∈ Z>0,wehavekν >j for ν 0, because the set of 80 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms points in C joined to p by a chain of length ≤ j is relatively compact in C while qν →∞. Therefore, after applying a diagonal process and passing to the associated subsequence of {qν}, we may assume that for each j, there is an element Bj ∈ A (ν) = { } with Bj Bj for all ν 0. Thus we get an infinite chain of distinct elements Bj from p to infinity as required in (ii) (local finiteness in X is guaranteed since A is locally finite and the elements {Bj } are distinct).

~~Remark The above lemma actually holds even if the basis elements are not necessarily connected, but it is easier to picture chains of connected sets.~~

Exercises for Sect. 2.13 2.13.1 Prove Lemma 2.13.2. 2.13.2 Let K be a compact subset of a noncompact, locally connected, locally compact Hausdorff space X. Prove that if X has only finitely many connected components, then hX(K) is compact, X \ hX(K) has only finitely many components, and for each point p ∈ X \ hX(K), there is a connected noncompact closed subset C of X with p ∈ C ⊂ X \ hX(K). 2.13.3 Let K be a compact subset of a connected, noncompact, locally connected, locally compact Hausdorff space X. (a) Prove that hX(K) is equal to the intersection of all topologically Runge neighborhoods of K. (b) Prove that for every neighborhood U of hX(K) in X, there exists a topologically Runge open set with hX(K) ⊂ ⊂ U. Hint. Show that one may assume without loss of generality that U X and that there exists a finite covering of ∂U by connected noncompact closed subsets of X that lie in X \ hX(K). Then set equal to the complement of these sets in U. 2.13.4 Prove that if X is a locally connected Hausdorff space and A is a connected subset of X, then hX(A) is connected. 2.13.5 Let be a relatively compact open subset of a connected, noncompact, locally connected, locally compact Hausdorff space X. Prove that hX() is a relatively compact open subset of X and that X \ hX() has only finitely many connected components (cf. Exercise 2.13.6 below). 2.13.6 This exercise will be applied in Exercise 2.16.5 (cf. [Ns5]). (a) Prove that if Y is a locally compact Hausdorff space, C is a compact connected component of Y , and U is a neighborhood of C in Y , then there exists a set A such that C ⊂ A ⊂ U and A is open and closed in Y . Hint. First work in a relatively compact neighborhood V of C in Y . Let D be the intersection of all subsets of V that are open and closed in V and that contain C. Show that each neighborhood of D in V contains a set A ⊃ D that is open and closed in V . Using this observation, prove that D is connected and conclude that D = C. Finally, construct the desired open and closed subset of Y for a given neighborhood U. 2.13 Topological Hulls and Chains to Infinity 81

(b) Let be an open subset of a locally compact Hausdorff space X, and let C be a compact connected component of X \. Prove that for every neighborhood U of C in X, there is a neighborhood V of C in U with ∂V ⊂ . (c) Let be an open subset of a locally compact Hausdorff space X. Prove that hX() is open. 2.13.7 Let X be a connected, locally path connected, locally compact Hausdorff space; let B be a countable collection of connected open subsets that is a basis for the topology in X;letK be a compact subset of X; and let U be a connected component of X \ hX(K). Prove that for each point p ∈ U, there is a proper continuous map γ :[0, ∞) → X with γ(0) = p and γ([0, ∞)) ⊂ U (i.e., a path in U from p to ∞). In the remaining exercises in this section, and in some exercises in later sections, generalizations in various contexts are obtained by considering a different hull as follows. For a subset A of a Hausdorff space X,theex- ∗ tended topological hull hX(A) of A in X is the union of A with all of the connected components of X \ A that do not contain any connected noncompact closed subsets of X. 2.13.8 Prove that Lemma 2.13.2 holds with the extended topological hull in place ⊂ ∗ of the topological hull. Prove also that hX(A) hX(A). ∗ = 2.13.9 Give an example of a closed set K in a Hausdorff space X with hX(K) hX(K). ∗ = 2.13.10 Prove that if is an open subset of a Hausdorff space X, then hX() hX(). 2.13.11 Prove that if K is a compact subset of a connected, noncompact, locally ∗ = connected, locally compact Hausdorff space X, then hX(K) hX(K). 2.13.12 Let X be a second countable connected, locally connected, locally compact Hausdorff space, and let K be a closed subset of X. (a) Prove that if B is a countable collection of connected open subsets that is a basis for the topology in X and U is a connected component of \ ∗ ∈ X hX(K), then for each point p U, there exists a sequence of basis elements {Bj } that tends to infinity (i.e., {Bj } is a locally finite family in X) such that p ∈ B1 and for each j, Bj U and Bj ∩ Bj+1 = ∅ (cf. Lemma 2.13.5). (b) Prove that if D ⊂ X \ K is a closed subset of X with no compact connected components, then there exist a countable locally finite (in X) family of disjoint connected noncompact closed sets {Cλ}λ∈ and a locally finite (in X) family of disjoint connected open sets {Uλ}λ∈ such that D ⊂ C ≡ Cλ and Cλ ⊂ Uλ ⊂ U λ ⊂ X \ K ∀λ ∈ . λ∈ Hint. First show that there is a countable locally finite covering A of D by connected open relatively compact subsets of X \ K each of which meets D, that X \ K contains the closure C ≡ V of the union V 82 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms

of the elements of A, that the family of components {Vγ }γ ∈ of V is countable and locally ﬁnite, and that Vγ is noncompact for each γ ∈ . Choosing suitable neighborhoods {Uλ}λ∈ of the connected components {Cλ}λ∈ of C, one gets the claim. 2.13.13 Let K be a subset of a second countable, connected, noncompact, locally connected, locally compact Hausdorff space X. ∗ (a) Prove that, if K is closed, then hX(K) is equal to the intersection of all topologically Runge neighborhoods of K (cf. Exercise 2.13.3). Give an example that shows that this need not be the case if K is not closed. (b) Give an example of such a space X and a closed subset K such that = = ∗ K hX(K) hX(K), but not every neighborhood of K contains a neighborhood of K that is topologically Runge in X.

~~2.14 Construction of a Subharmonic Exhaustion Function~~

~~This section contains the construction, alluded to in the beginning of Sect. 2.13,of a C∞ strictly subharmonic exhaustion function on an open Riemann surface. The main goal is the following:~~

Theorem 2.14.1 Suppose X is an open Riemann surface, K is a compact subset of X satisfying hX(K) = K, τ is a continuous real-valued function on X, θ is a continuous real (1, 1)-form on X, and is a neighborhood of K in X. Then there exists a C∞ exhaustion function ϕ on X such that

~~(i) On X, ϕ ≥ 0 and iϕ ≥ 0; (ii) On X \ , ϕ>τand iϕ ≥ θ; and (iii) We have supp ϕ ⊂ X \ K.~~

Remark It sufﬁces to consider the case τ ≥ 0 and θ ≥ 0, since in general, one may construct ϕ with the nonnegative function |τ|≥τ and the nonnegative form θ + + θ − ≥ θ in place of τ and θ, respectively.

~~Setting K = =∅and choosing θ>0 (as we may by Corollary 2.11.3), we get the following:~~

~~Corollary 2.14.2 Every open Riemann surface admits a C∞ strictly subharmonic exhaustion function.~~

~~After ﬁxing a Kähler form, one may work with a convenient 0-form in place of the curvature form:~~

Deﬁnition 2.14.3 The Laplace operator associated to a Kähler form ω on a com- ∞ plex 1-manifold X is the second-order linear differential operator ω with C co- 2.14 Construction of a Subharmonic Exhaustion Function 83 efﬁcients given by i∂∂ϕ¯ ϕ ≡ ω ω 2 for every C function ϕ. In particular, for ϕ real-valued, we have ωϕ = iϕ/ω.

~~Remarks 1. Writing ω = G · (i/2)dz ∧ dz¯ = Gdx ∧ dy with respect to a local holomorphic coordinate z = x + iy, we get 2 ∂2 1 ∂2 ∂2 ω = = + . G ∂z∂z¯ 2G ∂x2 ∂y2~~

For example, if ω = (i/2)dz∧ dz¯ = dx∧ dy is the standard Euclidean volume form on C, then ∂2 1 ∂2 ∂2 ω = 2 = + , ∂z∂z¯ 2 ∂x2 ∂y2 which is 1/2 the standard Laplace operator. 2. Clearly, a real-valued C∞ (or C2) function ϕ on an open set ⊂ X is subharmonic (strictly subharmonic) if and only if ωϕ ≥ 0 (respectively, ωϕ>0) on .

For the rest of this section, X will denote an open Riemann surface. According to Radó’s theorem, X is second countable, and applying Corollary 2.11.3, we get a Kähler form ω on X. Instead of proving Theorem 2.14.1 directly, we will prove the following equivalent version (in Exercise 2.14.1, the reader is asked to verify that the two versions are equivalent):

Theorem 2.14.4 (Cf. [GreW], [De2], and [NR]) Suppose K is a compact subset of X satisfying hX(K) = K, ρ is a continuous real-valued function on X, and is a neighborhood of K in X. Then there exists a C∞ exhaustion function ϕ on X such that

~~(i) On X, ϕ ≥ 0 and ωϕ ≥ 0; (ii) On X \ , ϕ>ρand ωϕ>ρ; and (iii) We have supp ϕ ⊂ X \ K.~~

~~The main step in the proof of Theorem 2.14.4 is the following:~~

Proposition 2.14.5 Let K be a compact subset of X satisfying hX(K) = K. Then, for each point p ∈ X \ K, there is a nonnegative C∞ function α on X such that ωα ≥ 0 on X, supp α ⊂ X \ K, α(p) > 0, and ωα(p) > 0.

Remark According to the maximum principle for subharmonic functions (see, for example, [Ns5]), a nonconstant subharmonic function on a domain cannot attain a maximum value (in particular, any subharmonic function on a compact Riemann 84 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms surface is constant). Therefore, the converse of the proposition also holds; that is, if such a function α exists for some point p ∈ X \ K, then the component of X \ K containing p is not relatively compact in X.

~~Assuming Proposition 2.14.5 for now, we may prove Theorem 2.14.4.~~

Proof of Theorem 2.14.4 By Proposition 9.3.11, we may assume that ρ is a positive exhaustion function. Let K0 = K. By Lemma 2.13.4 , we may choose a sequence { } = ∞ of nonempty compact sets Kν such that X ν= Kν and such that for each ◦ 1 ν = 1, 2, 3,...,Kν−1 ⊂ Kν and hX(Kν ) = Kν . Given a point p ∈ X \, there is a unique ν = ν(p) > 0 with p ∈ Kν \Kν−1, and ∞ Proposition 2.14.5 provides a nonnegative C function αp and a relatively compact neighborhood Vp of p in X \ Kν−1 such that ωαp ≥ 0onX, supp αp ⊂ X \ Kν−1, and αp >ρand ωαp >ρon Vp (one obtains the last two conditions by multiplying by a sufficiently large positive constant). We may choose a sequence of points { } { } { } pk in X, and corresponding functions αpk and neighborhoods Vpk , so that {Vp } forms a locally finite covering of X \ (for example, we may take {pk} to be k ∞ an enumeration of the countable set ν= Zν , where for each ν, Zν is a finite set ◦ 0 ◦ \[ ∪ ] { } \[ ∪ ] of points in X Kν−1 such that Vp p∈Zν covers Kν Kν−1 ). The col- { } ⊂ \ lection supp αpk is then locally finite in X, since supp αpk X Kν−1 whenever ∈ ∞ pk Kν−1. Hence the sum k=1 αpk is locally finite and therefore convergent to C∞ ≥ ≥ a function ϕ on X satisfying ϕ αpk >ρ and ωϕ ωαpk >ρ on Vpk for { } \ \ each k. Therefore, since Vpk covers X , we get ϕ>ρand ωϕ>ρon X . In particular, ϕ is an exhaustion function. Similarly, we also have supp ϕ ⊂ X \ K, and ϕ ≥ 0 and ωϕ ≥ 0onX.

It remains to prove Proposition 2.14.5. The idea is to form a sequence of functions, each of which is subharmonic outside a small set and strictly subharmonic on the bad set of the previous function. Multiplying each function by a sufﬁciently large positive constant (obtained inductively), one pushes these small bad sets off to inﬁnity.

~~Lemma 2.14.6 There exists a nonnegative C∞ function χ on R such that χ ≡ 0 on (−∞, 0] and χ,χ,χ > 0 on (0, ∞).~~

~~Proof For example, the C∞ function 0ift ≤ 0, χ(t)= exp(t − (1/t)) if t>0, has the required properties. The veriﬁcation is left to the reader (see Exer- cise 2.14.3).~~

~~Lemma 2.14.7 For every r ∈ (0, 1), there exists a nonnegative C∞ function α on P1 such that 2.14 Construction of a Subharmonic Exhaustion Function 85~~

~~(i) We have α>0 on (0; 1) and α ≡ 0 on P1 \ (0; 1); and (ii) The function α is strictly subharmonic on the annulus (0; r, 1) (and therefore subharmonic on P1 \ (0; r)).~~

Proof Letting χ : R →[0, ∞) be the function provided by Lemma 2.14.6, it is easy to see that the function given by α(∞) = 0 and −| |2 2 − 2 α(z) ≡ χ e z /r − e 1/r ∀z ∈ C has the required properties. Again, the veriﬁcation is left to the reader (see Exer- cise 2.14.4).

Proposition 2.14.8 Let c ∈ ≡ (0; 1), and let : P1 → P1 be the map given by z → (z − c)/(1 −¯cz) (with (1/c)¯ =∞and (∞) =−1/c¯). Then we have the following: (i) is an automorphism of P1 (i.e., a biholomorphic mapping of P1 onto itself ); (ii) (c) = 0, (0) = 1 −|c|2, and (c) = 1/(1 −|c|2); and (iii) () = and (∂) = ∂.

The proof is left to the reader (see Exercise 2.14.5). It follows from the above that is an automorphism of . In Chapter 5, we will see that in fact, every ∈ automorphism of is of the form b for some b ∂ and some as above (see Theorem 5.8.2).

Lemma 2.14.9 For every nonempty relatively compact open subset U of the unit disk ≡ (0; 1), there exists a nonnegative C∞ function β on P1 such that (i) We have β>0 on and β ≡ 0 on P1 \ ; and (ii) The function β is strictly subharmonic on the set \ U (and therefore subharmonic on P1 \ U).

Proof Fixing a point c ∈ U, we get the automorphism : z → (z − c)/(1 −¯cz) of P1 mapping c to 0 as in Proposition 2.14.8, and for r ∈ (0, 1) sufﬁciently small, we have −1((0; r))⊂ U. By Lemma 2.14.7, there exists a nonnegative C∞ function α on P1 such that α>0on, α ≡ 0onP1 \ , and α is strictly subharmonic on (0; r, 1). The function β ≡ α() then has the required properties.

Proof of Proposition 2.14.5 By Lemma 2.13.5, given a point p ∈ X \ K, there is a locally ﬁnite (in X) sequence of relatively compact open subsets {Vm} of X \ K such that p ∈ V1 and such that for each m,wehaveVm ∩ Vm+1 = ∅ and there is a biholomorphism of a neighborhood of the closure V m of Vm onto a neighborhood of the closure of the unit disk ≡ (0; 1) that maps Vm onto . Hence we may { }∞ choose a sequence of nonempty open sets Wm m=0 with disjoint closures such that p ∈ W0 V1 and such that for each m ≥ 1, Wm Vm ∩ Vm+1 (see Fig. 2.6). C∞ { }∞ By Lemma 2.14.9, there is a sequence of nonnegative functions βm m=1 such that for each m, βm ≡ 0onX \ Vm, ωβm ≥ 0onX \ Wm, and 86 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms

~~Fig. 2.6 Construction of a subharmonic function that is strictly subharmonic near the point p~~

~~{ }∞ βm > 0 and ωβm > 0onW m−1. We will choose positive constants Rm m=1 inductively so that for each m = 1, 2, 3,..., m ≥ 0onX \ Wm, ω Rj βj > 0onW . j=1 0~~

~~Fix R1 > 0. Given R1,...,Rm−1 > 0 with the above property, using the fact that ωβm > 0onW m−1, we get, for Rm 0, m m−1 ω Rj βj = Rmωβm + ω Rj βj > 0onW m−1. j=1 j=1~~

~~On X \ (Wm−1 ∪ Wm),wehaveωβm ≥ 0 and hence m m−1 ω Rj βj ≥ ω Rj βj ≥ 0. j=1 j=1~~

On W 0 ⊂ X \(Wm−1 ∪Wm), the above middle expression, and hence the expression m ≥ on the left, is positive. Moreover, j=1 Rj βj R1β1 > 0onW 0. Proceeding, we get the sequence {Rm}.Thesum Rmβm is locally ﬁnite in X and the sequence of sets {Wm} is locally ﬁnite in X, so the sum converges to a function α with the required properties.

Remark When second countability of a particular open Riemann surface X is evident (for example, when X is a proper nonempty open subset of a compact Rie- mann surface), one gets Theorem 2.14.4 without any reliance on Radó’s theorem, and hence with almost no reliance on Sects. 2.6–2.12.

~~The existence of strictly subharmonic exhaustion functions allows one to use the fullpoweroftheL2 ∂¯-method. For example, we have the following:~~

Theorem 2.14.10 Let p ∈{0, 1} and let β be a locally square-integrable differential form of type (p, 1) on the open Riemann surface X. Then there exists a locally 2.14 Construction of a Subharmonic Exhaustion Function 87

~~¯ square-integrable differential form α of type (p, 0) with ∂distrα = β. In particular, if k k ¯ β is of class C for some k ∈ Z>0 ∪{∞}, then α is also of class C and ∂α = β.~~

Proof We may ﬁx a Kähler form ω on X. Applying Theorem 2.14.4, with the compact set given by K =∅and the function ρ chosen (with the help of Lemma 9.7.13) ∈ 2 2 +| | so large that β L0,1(X, ρ) or L1,1(X,ω,ρ)and ρ>1 iω/ω , we get a positive C∞ strictly subharmonic (exhaustion) function ϕ on X such that

~~+ ≥ ∞ = iω iϕ ω and β L2(X,ϕ) < if p 0, ≥ ∞ = iϕ ω and β L2(X,ω,ϕ) < if p 1.~~

~~Corollary 2.12.6 and Corollary 2.9.3 now give the desired differential form α.~~

~~In Sects. 2.15 and 2.16, we will consider other applications to open Riemann surfaces.~~

Exercises for Sect. 2.14 2.14.1 Prove that Theorem 2.14.1 and Theorem 2.14.4 are equivalent. 2.14.2 Let ϕ be a real-valued C∞ function on a Riemann surface X. (a) Prove that if χ is a real-valued C∞ function on R, then ¯ χ(ϕ) = χ (ϕ)χ(ϕ) + χ (ϕ)∂ϕ ∧ ∂ϕ.

From this conclude that if ϕ is subharmonic and χ and χ are nonnegative, then χ(ϕ) is subharmonic. Show also that if ϕ is strictly subharmonic and χ > 0 and χ ≥ 0, then χ(ϕ) is strictly subharmonic. (b) Prove that if ϕ is a strictly subharmonic exhaustion function on X, τ is a continuous real-valued function on X, and θ is a continuous real (1, 1)-form on X, then there exists a C∞ function χ : R → R such that χ(ϕ)>τ and iχ(ϕ) ≥ θ. 2.14.3 Verify that the function constructed in the proof of Lemma 2.14.6 has the required properties. 2.14.4 Verify that the function constructed in the proof of Lemma 2.14.7 has the required properties. 2.14.5 Prove Proposition 2.14.8. 2.14.6 Let X be an open Riemann surface. (a) Prove that if q ∈{0, 1} and β is a C∞ differential form of type (1,q) on X, then there exists a C∞ differential form α of type (0,q) with ∂α = β. (b) Prove that if ω is a Kähler form on X, then there exists a C∞ strictly subharmonic function ϕ on X such that iϕ = ω. 2.14.7 This exercise requires Exercises 2.13.8 and 2.13.12. Let X be an open Rie- mann surface, let ω be a Kähler form on X, and let K be a closed subset ∗ = of X satisfying hX(K) K. 88 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms

(a) Prove that for each connected component U of X \ K and each point ∞ p ∈ X\K, there is a nonnegative C function α on X such that ωα ≥ 0 on X, supp α ⊂ U, α(p) > 0, and ωα(p) > 0 (cf. Proposition 2.14.5). (b) Suppose that U is a connected component of X \ K, C is a connected noncompact closed subset of X contained in U, and ρ is a continuous real-valued function on X. Prove that there exists a C∞ function ϕ on X such that (i) We have ϕ ≥ 0 and ωϕ ≥ 0onX; (ii) We have ϕ>ρand ωϕ>ρon C; and (iii) We have supp ϕ ⊂ U. Hint. First show that there exists a connected locally connected closed set D in X with C ⊂ D ⊂ U (for example, by forming the union of the closures of elements of a suitable locally ﬁnite covering of C by coordinate disks each of which meets C). Then show that there is a sequence { } = of compact sets Kν such that X ν Kν and such that for each ν, ◦ Kν ⊂ Kν+1 and hD(Kν ∩ D) = Kν ∩ D. Proceed now as in the proof of Theorem 2.14.4. (c) Suppose that D ⊂ X \ K is a closed subset of X with no compact connected components and ρ is a real-valued continuous function on X. Prove that there exists a C∞ function ϕ on X such that (i) We have ϕ ≥ 0 and ωϕ ≥ 0onX; (ii) We have ϕ>ρand ωϕ>ρon D; and (iii) We have supp ϕ ⊂ X \ K.

~~2.15 The Mittag-Lefﬂer Theorem~~

One of the main applications of the L2 ∂¯-method is the construction of a holomorphic or meromorphic function (or section of a holomorphic line bundle) with prescribed values on a given discrete set, or as in the generalization of the classical Mittag-Lefﬂer theorem below, with some prescribed Laurent series terms (see Theorem 1.3.6). This generalization is due to Behnke–Stein [BehS](seealsoFlo- rack [Fl]):

Theorem 2.15.1 (Mittag-Lefﬂer theorem) Let X be an open Riemann surface, let P be a discrete subset of X (i.e., a closed set with no limit points in X), and for each point p ∈ P , let Up be a neighborhood of p with Up ∩ P ={p}, let fp be a holomorphic function on Up \{p}, and let mp be a positive integer. Then there exists a holomorphic function f on X \ P such that for each point p ∈ P , f − fp extends to a holomorphic function on Up that either vanishes on a neighborhood of p or has a zero of order at least mp at p (in other words , if z is a local holo- = − n morphic coordinate on a neighborhood of p, and f n∈Z an(z z(p)) and = − n fp n∈Z bn(z z(p)) are the corresponding Laurent series expansions cen- = = tered at p, then amp−n bmp−n for n 1, 2, 3,...). 2.15 The Mittag-Lefﬂer Theorem 89

Remark It follows that one may actually choose the above function f ∈ O(X \ P) so that for each point p ∈ P , f − fp extends to a holomorphic function on Up with a zero of order equal to mp at p (see Exercise 2.15.1).

~~The proofs of Corollaries 2.15.2, 2.15.3, and 2.15.4 below are left to the reader (see Exercises 2.15.2, 2.15.3, and 2.15.4).~~

~~Corollary 2.15.2 Let P be a discrete subset of an open Riemann surface X.~~

(a) If fp is a meromorphic function on a neighborhood of p and mp ∈ Z>0 for each point p ∈ P , then there exists a function f ∈ M(X) such that f is holomorphic on X \ P and ordp(f − fp) ≥ mp for every p ∈ P . (b) If fp is a holomorphic function on a neighborhood of p and mp ∈ Z>0 for each point p ∈ P , then there exists a function f ∈ O(X) with ordp(f − fp) ≥ mp for every p ∈ P . (c) If ζp ∈ C for each point p ∈ P , then there exists a function f ∈ O(X) with f(p)= ζp for every p ∈ P .

Corollary 2.15.3 (Behnke–Stein theorem [BehS]) Every open Riemann surface X is Stein; that is, X has the following properties: (i) (Holomorphic convexity) If P is any inﬁnite discrete subset of X, then there exists a holomorphic function on X that is unbounded on P ; (ii) (Separation of points) If p,q ∈ X and p = q, then there exists a holomorphic function f on X such that f(p)= f(q); and (iii) (Global functions give local coordinates) For each point p ∈ X, there exists a holomorphic function f on X such that (df )p = 0.

Corollary 2.15.4 Let X be an open Riemann surface, let P be a discrete subset of X, and for each point p ∈ P , let Up be a neighborhood of p with Up ∩ P ={p}, let θp be a holomorphic 1-form on Up \{p}, and let mp be a positive integer. Then there exists a holomorphic 1-form θ on X \ P such that for each point p ∈ P , θ − θp extends to a holomorphic 1-form on Up that either vanishes on a neighborhood of p or has a zero of order at least mp at p.

Proof of Theorem 2.15.1 Let Y = X \ P . We may choose a Kähler form ω on X, and we may choose a locally ﬁnite family of disjoint local holomorphic coordinate { } { } neighborhoods (Vp,zp) p∈P and a family of open sets Wp p∈P such that for each ∈ ∈ = ≡ pointp P ,wehavep Wp Vp Up and zp(p) 0. Let V p∈P Vp and ≡ W p∈P Wp. ∞ By cutting off, we may construct a real-valued C function ρ0 on Y such that 2 supp ρ0 ⊂ V and ρ0 = mp log |zp| on Wp \{p} for each p ∈ P . We may also ﬁx ∞ a C function γ on Y such that supp γ ⊂ V and γ = fp on Wp \{p} for each p ∈ P . The form β = ∂γ¯ is then a C∞ differential form of type (0, 1) on Y with 2 supp β ⊂ V \ W . Since log |zp| is a harmonic function on Vp \{p} for each p ∈ P , by applying Theorem 2.14.4 (or Theorem 2.14.1), with the compact set equal to the 90 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms empty set and the function ρ chosen (with the help of Lemma 9.7.13) so large that ∈ 2 −| | +| |+| | C∞ β L0,1(Y, ρ ρ0 ) and ρ>1 iω/ω ωρ0 on Y , we get a positive + + ≥ strictly subharmonic (exhaustion) function ρ1 on X such that iω iρ0 iρ1 ω Y β 2 < ∞ on and such that L (Y,ρ0+ρ1) . Corollary 2.12.6 now provides a C∞ function α on Y such that ¯ ∂α = β and α 2 < ∞. L (Y,ω,ρ0+ρ1) In particular, the C∞ function f ≡ γ − α is holomorphic on Y . Moreover, for each p ∈ P ,wehavef − fp =−α on Wp \{p}. On the other hand, for some positive constant C (depending on p), we have

~~−mp 2 zp α L (Wp\{p},(i/2)dzp∧dz¯p)~~

~~=α 2 ≤ Cα 2 L (Wp\{p},(i/2)dzp∧dz¯p,ρ0) L (Wp\{p},ω,ρ0+ρ1)~~

~~≤ Cα 2 < ∞. L (Y,ω,ρ0+ρ1)~~

−mp Hence the holomorphic function zp α on Wp \{p} is square-integrable, and therefore, by Riemann’s extension theorem (Theorem 1.2.10), this function extends to a holomorphic function on Wp. Thus α extends holomorphically past p with order at least mp at p, and therefore f − fp extends to a holomorphic function on Up with order at least mp at p.

Exercises for Sect. 2.15 2.15.1 Prove that in the Mittag-Leffler theorem (Theorem 2.15.1), one may actually choose the function f ∈ O(X \ P) so that for each point p ∈ P , f − fp extends to a holomorphic function on Up with a zero of order equal to mp at p. 2.15.2 Prove Corollary 2.15.2. 2.15.3 Prove the Behnke–Stein theorem (Corollary 2.15.3). 2.15.4 Prove Corollary 2.15.4. 2.15.5 Let f be a meromorphic function on an open Riemann surface X. Prove that there exist holomorphic functions g and h on X such that h is not the zero function and f = g/h. 2.15.6 The goal of this exercise is a generalization of the Mittag-Leffler theorem that, in higher dimensions, is known as the solution of the additive Cousin problem (or the Cousin problem I). Let X be an open Riemann surface, let P be a discrete subset of X,let{mp}p∈P be a collection of positive integers, let {Ui}i∈I be an open covering of X, and for each pair of indices i, j ∈ I , let fij be a holomorphic function on Ui ∩ Uj with ordpfij ≥ mp for each point p ∈ P ∩ Ui ∩ Uj . Assume that the family {fij } satisfies the (additive) cocycle relation

~~fik = fij + fjk on Ui ∩ Uj ∩ Uk ∀i, j, k ∈ I.~~

~~Prove that there exist functions {gi}i∈I such that 2.16 The Runge Approximation Theorem 91~~

(i) For each index i ∈ I , gi ∈ O(Ui) and ordp gi ≥ mp for each point p ∈ P ∩ Ui ; and (ii) For each pair of indices i, j ∈ I ,wehavefij = gj − gi on Ui ∩ Uj . Prove also that the above implies the standard Mittag-Lefﬂer theorem (The- orem 2.15.1). ∞ Hint. Using a C partition of unity {λν} such that each point in P lies in supp λν for exactly one index ν and such that for each ν, supp λν ⊂ Uk ∞ ν for some index kν ∈ I , one may form a C solution of the problem of the = · {¯ } form vi ν λν fkν i . In particular, the forms ∂vi agree on the overlaps and therefore determine a well-deﬁned (0, 1)-form β on X. Suitable weight functions (as in the proof of Theorem 2.15.1) now give a suitable solution of ∂α¯ = β.

~~2.16 The Runge Approximation Theorem~~

According to the Mittag-Lefﬂer theorem (Theorem 2.15.1), on an open Riemann surface X, one may prescribe values for a holomorphic function (to arbitrary order) at the points in a given discrete set. The identity theorem implies that it is not possible to prescribe values on a set that is not discrete. However, for a compact set K with hX(K) = K, one can uniformly approximate a holomorphic function on a neighborhood of K by a global holomorphic function. In other words, we have the following Riemann surface analogue, due to Behnke and Stein [BehS], of the classical Runge approximation theorem [Run] for domains in the plane:

Theorem 2.16.1 (Runge approximation theorem) Suppose K is a compact subset of an open Riemann surface X with hX(K) = K, f0 is a holomorphic function on a neighborhood of K in X, and >0. Then there exists a holomorphic function f on X such that |f − f0| <on K.

The converse is also true (see Exercise 2.16.4). Until now, we have not applied, in an essential way, the L2 estimate in Theorem 2.9.1; but this estimate will play an important role in the proof of this approximation theorem. We will also consider the more general version Theorem 2.16.3, so the reader may wish to skip the proof of Theorem 2.16.1 below and instead, consider only the proof of Theorem 2.16.3.

∞ Proof of Theorem 2.16.1 Multiplying f0 by a C function that has support contained in a small neighborhood of K but that is equal to 1 on some smaller neighborhood, and then extending by 0 to all of X and choosing suitable neighborhoods, ∞ we get a C function τ on X and open sets 0 and 1 such that K ⊂ 0 1 X, τ = f0 on 0, and supp τ ⊂ 1. WemayalsoﬁxaKählerformω on X (by Corollary 2.11.3). Applying The- orem 2.14.1 (with the compact set and its neighborhood given by the empty set, and the nonnegative (1, 1)-form chosen so that its sum with iω is greater than or ∞ equal to ω), we get a positive C strictly subharmonic (exhaustion) function ρ0 on 92 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms

+ ≥ X such that iω iρ0 ω on X. Applying the theorem again (this time with the compact set given by K), we get a nonnegative C∞ subharmonic (exhaustion) function ρ1 on X such that supp ρ1 ⊂ X \ K and ρ1 > 0onX \ 0. In particular, r ≡ inf \ ρ1 > 0. 1 0 ¯ For each positive integer ν,letϕν ≡ ρ0 + νρ1. The form β ≡ ∂τ is a compactly supported C∞ differential form of type (0, 1) on X. Since we have

~~+ = + + ≥ iω iϕν iω iρ0 νiρ1 ω,~~

~~Corollary 2.12.6 provides a C∞ function α on X such that ∂α¯ = β and~~

~~ 2 ≤ 2 =¯ 2 ≤ −νr¯ 2 α 2 β 2 ∂τ 2 e ∂τ 2 . L (X,ω,ϕν ) L (X,ϕν ) L (1\0,ϕν ) L (1\0,ρ0)~~

~~By construction, we have ρ1 ≡ 0 on some relatively compact neighborhood 2 of K in 0, and hence~~

~~ 2 = 2 ≤ 2 α 2 α 2 α 2 . L (2,ω,ρ0) L (2,ω,ϕν ) L (X,ω,ϕν )~~

Therefore, by choosing ν sufﬁciently large, we can make α 2 L (2,ω,ρ0) arbitrarily small. Furthermore, the C∞ function f ≡ τ − α on X is actually holomorphic ¯ ∂f = f − f 2 =α 2 (since 0), and we have 0 L (2,ω,ρ0) L (2,ω,ρ0). Applying The- orem 2.6.4 and choosing ν 0, we get |f − f0| <on K.

~~We have the following consequence (the converse, which also holds, is considered in Exercise 2.16.5):~~

Corollary 2.16.2 Let be a topologically Runge open subset of an open Riemann surface X. Then, for every holomorphic function f0 on , for every compact set K ⊂ , and for every >0, there exists a holomorphic function f on X such that |f − f0| <on K.

~~Proof For every compact set K ⊂ ,wehavehX(K) ⊂ hX() = . Theo- rem 2.16.1 now gives the claim.~~

~~We also have the following more general version of Theorem 2.16.1:~~

Theorem 2.16.3 (Runge approximation with poles at prescribed points) Suppose K is a compact subset of a Riemann surface X, P is a ﬁnite subset of X\K, Y = X\P , hY (K) = K, f0 is a holomorphic function on a neighborhood of K in X, and >0. Then there exists a meromorphic function f on X such that f is holomorphic on X \ P = Y , f has a pole at each point in P , and |f − f0| <on K.

~~For the proof, we will need the following combined version of Lemma 2.10.3 and Theorem 2.14.1: 2.16 The Runge Approximation Theorem 93~~

Lemma 2.16.4 Suppose X is a Riemann surface, K is a compact subset of X, P is a \ = \ = { } finite subset of X K, Y X P , hY (K) K, (Up,zp) p∈P is a collection of disjoint local holomorphic coordinate neighborhoods in X with p ∈ Up and zp(p) = 0 for each p ∈ P , ω is a Kähler form on X, and is a neighborhood of K in X. Then, for every sufficiently large positive constant b, there exist open sets {Vp}p∈P with p ∈ Vp Up for each point p ∈ P such that for every sufficiently large positive constant R (depending on the above choices), there exists a nonnegative C∞ subharmonic exhaustion function ϕ on Y with the following properties:

~~(i) On Y \ , ϕ>0 and iϕ ≥ ω; (ii) We have supp ϕ ⊂ Y \ K; and 2 2 (iii) For each p ∈ P , we have ϕ = R · (|zp| − log |zp| − b) on Vp \{p}.~~

Proof If P =∅and X is compact, then we have K = X and the claim is trivial. Thus we may assume without loss of generality that Y is noncompact. By shrink- { } ing and the sets (Up,zp) p∈P if necessary, we may also assume without loss of generality that Up X \ for each point p ∈ P .WehavehY (K) = K, so The- orem 2.14.1 provides a nonnegative C∞ subharmonic exhaustion function α on Y such that supp α ⊂ Y \ K and such that α>0 and iα ≥ ω on Y \ .Forb 0, Lemma 2.10.3 provides, for each point p ∈ P , a nonnegative C∞ subharmonic func- 2 2 tion βp on X \{p} such that βp ≡ 0onX \ Up and βp =|zp| − log |zp| − b>0 on Wp \{p} for some relatively compact neighborhood Wp of p in Up (the ﬁnite- ness of P allows us to choose a single sufﬁciently large constant b that works for ∈ each of the points p P ). Choosing a relatively compact neighborhood Vp of p ∈ ≡ ≡ in Wp for each p P , setting V p∈P Vp W p∈P Wp, and choosing a ∞ nonnegative C function η on X such that η ≡ 1 on a neighborhood ofX \ W and ⊂ \ = · + · supp η X V , we see that if R 0, then the function ϕ η α p∈P R βp has the required properties.

Proof of Theorem 2.16.3 We have Y = X \ P and hY (K) = K, and as in the proof of Lemma 2.16.4, we may assume without loss of generality that Y is noncompact. ∞ Multiplying f0 by a C function that has support contained in a small neighborhood of K but that is equal to 1 on some smaller neighborhood, and then extending by0toallofX and choosing suitable neighborhoods, we get a C∞ function τ on X and open sets 0 and 1 such that K ⊂ 0 1 X \ P , τ = f0 on 0, and supp τ ⊂ 1. We may also choose disjoint local holomorphic coordinate neighbor- { } ∈ ∈ \ hoods (Up,zp) p∈P in X such that for each p P ,wehavep Up X 1 and zp(p) = 0.WemayalsoﬁxaKählerformω on X (by Corollary 2.11.3). By applying Lemma 2.16.4 (with the compact set and its neighborhood given by the empty set, and the Kähler form chosen so that its sum with iω is greater than or equal to ω), we get a positive constant b0, open sets {Vp}p∈P with p ∈ Vp Up for each point p ∈ P , a positive integer k, and a positive C∞ strictly subharmonic + ≥ (exhaustion) function ρ0 on Y such that iω iρ0 ω on Y and such that for each 2 2 point p ∈ P , ρ0 = k(|zp| − log |zp| − b0) on Vp \{p}. Applying Lemma 2.16.4 again (this time with the compact set given by K) and shrinking each of the neigh- 94 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms

∞ borhoods Vp for p ∈ P , we get a positive constant b1 and a nonnegative C subharmonic (exhaustion) function ρ1 on Y such that supp ρ1 ⊂ Y \ K, ρ1 > 0onX \ 0, 2 2 and ρ1 =|zp| − log |zp| − b1 on Vp \{p} for each p ∈ P (here, we apply the lemma to get constants b and R, and we then divide the resulting function by R and ≡ ≡ C∞ set b1 b/R). In particular, r inf1\0 ρ1 >0.Wemayalsoﬁxa function ≡ η with compact support in the neighborhood p∈P Vp of P such that η 1ona neighborhood of P . Now for each positive integer ν,letϕν ≡ ρ0 + νρ1. Given a positive integer μ ∞ (μ+ν+k) and a constant δ>0, we get a C function γ on Y by setting γ = δη/zp on ¯ Up \{p} for each p ∈ P and γ = τ elsewhere. The form β ≡ ∂γ is then a compactly supported C∞ differential form of type (0, 1) on Y . Since we have

~~+ = + + ≥ iω iϕν iω iρ0 νiρ1 ω,~~

Corollary 2.12.6 provides a C∞ function α on Y such that ∂α¯ = β and 2 ≤ 2 =¯ 2 + 2 −(μ+ν+k) ¯ 2 α 2 β 2 ∂τ 2 δ z ∂η 2 L (Y,ω,ϕν ) L (Y,ϕν ) L (1\0,ϕν ) p L (Vp,ϕν ) p∈P ≤ −νr¯ 2 + 2 −(μ+ν+k) ¯ 2 e ∂τ 2 δ z ∂η 2 . L (1\0,ρ0) p L (Vp,ϕν ) p∈P

By construction, we have ρ1 ≡ 0 on some relatively compact neighborhood 2 of K in 0, and hence 2 = 2 ≤ 2 α 2 α 2 α 2 . L (2,ω,ρ0) L (2,ω,ϕν ) L (Y,ω,ϕν ) Therefore, by choosing ν sufficiently large and then choosing δ>0 sufficiently ν α 2 arbitrarily small. small (depending on ), we can make L (2,ω,ρ0) The C∞ function f ≡ γ − α on Y is actually holomorphic, since ∂f¯ = 0. Near ∈ ν+k = −μ − ν+k each point p P , the function zp α δzp zp f is holomorphic except for an + − − + | |2 | ν k |2 =| |2 ( (ν k)log zp ) isolated singularity at p, and zp α α e is locally integrable near p by the choice of ϕν . Therefore, by Riemann’s extension theorem (Theo- ν+k rem 1.2.10), zp α extends to a holomorphic function in a neighborhood of p, −(μ+ν+k) and hence the function −α = f − γ , which is equal to f − δzp near p, is meromorphic on a neighborhood of p with at worst a pole of order ν + k at p. It follows that f extends to a meromorphic function on X that is holomorphic on X \ P and that has a pole of order μ + ν + k at each point p ∈ P . Finally, since f − f 2 =α 2 ν δ 0 L (2,ω,ρ0) L (2,ω,ρ0), Theorem 2.6.4 implies that for 0 and sufficiently small (depending on ν), we have |f − f0| <on K.

Remarks 1. Note that we may choose the function f in the above proof to have a pole of any order >ν+ k at each point in P (see Exercise 2.16.3). 2. A version in which the discrete set P may be inﬁnite is considered in Exer- cise 2.16.6. 2.16 The Runge Approximation Theorem 95

~~We close this section with a consequence of Theorem 2.16.3 that will play an important role in the proof of the Riemann mapping theorem in Chap. 5 (see Sect. 5.4).~~

Lemma 2.16.5 Given a compact subset K of an open Riemann surface X, there exist a holomorphic function f on a neighborhood Y of K in X, a positive regular value r of the function |f |, and a C∞ domain such that is a connected component of {x ∈ Y ||f(x)|

Proof Clearly, we may assume that K is nonempty and connected. Thus we may choose a compact set K1 X with K ⊂ K1 and hX(K1) = K1 (in fact, we could replace K with hX(K) and set K = K1, but this would require the fact, considered in Exercise 2.13.4, that the topological hull of a connected set in a manifold is connected, and is not really necessary here). Similarly, we may choose a relatively compact neighborhood 0 of K1 in X and a compact set K2 such that ⊂ ⊂ \ = ∂0 K2 X K1 and hX\K1 (K2) K2 (see Exercise 2.13.2). In particular, the set X \ (K1 ∪ K2) = (X \ K1) \ K2 has only finitely many components, and hence we may choose a finite set P ⊂ X \ (K1 ∪ K2) that meets each of these components. The domain Y ≡ X \ P then contains K1 ∪ K2, and furthermore, hY (K1 ∪ K2) = K1 ∪ K2. For each component of Y \ (K1 ∪ K2) is of the form U \ P , where U is a component of X \ (K1 ∪ K2), and by construction, P must meet U. Thus U \ P must contain points arbitrarily close to P = X \ Y , and hence U \ P cannot be relatively compact in Y . Now, by applying Theorem 2.16.3 to a locally constant function on a neighborhood of K1 ∪ K2 that is equal to 0 on K1 and 3 on K2, we get a meromorphic function g on X such that g is holomorphic on X \ P = Y , g has a pole at each point in P , |g| < 1onK1 ⊂ 0, and |g| > 2onK2 ⊃ ∂0. Since g is ≡| | nonconstant, the set of positive critical values of the function ρ g Y is countable (dg = 0 at any critical point of ρ in Y \ g−1(0) = X \ g−1({0, ∞}) since 2ρdρ= dρ2 =¯gdg+ gdg¯ on this set). Thus we may fix a regular value r ∈ (1, 2). The connected component of {x ∈ Y | ρ(x)2on∂0. Setting ≡ f g Y , we get the desired objects.

Exercises for Sect. 2.16 2.16.1 Let X be an open Riemann surface. Using the Runge approximation theorem (not the results of Sect. 2.15), prove the following (cf. Corol- lary 2.15.3): (i) Separation of points.Ifp,q ∈ X and p = q, then there exists a holomorphic function f on X such that f(p)= f(q); and (ii) Global holomorphic functions give local coordinates. For each point p ∈ X, there exists a holomorphic function f on X such that (df )p = 0. 2.16.2 Let P be a ﬁnite subset of a compact Riemann surface X, and let U be a neighborhood of P . Prove that there is a meromorphic function f on X 96 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms

such that f is holomorphic on X \ P , f has a pole at each point in P , and the set of zeros of f is contained in U. 2.16.3 Verify that in the proof of the Theorem 2.16.3, the constructed function f may be chosen to have a pole of any order >ν+ k at each point in P . 2.16.4 Let X be an open Riemann surface and let K be a nonempty compact subset of X. (a) Prove that hX(K) ={p ∈ X ||f(p)|≤maxK |f |∀f ∈ O(X)}. Hint. Show that for each point p ∈ X \hX(K),wehavehX(hX(K)∪ {p}) = hX(K) ∪{p}. Then form a holomorphic approximation to the function that is equal to 0 near hX(K) and 1 near p. Given a point p ∈ hX(K) \ K, apply the maximum principle. (b) Prove that hX(K) = K if and only if for every holomorphic function f0 on a neighborhood of K in X and for every >0, there exists a holomorphic function f on X such that |f − f0| <on K. Hint. Assuming the approximation condition, suppose U is a connected component of X \ K with U X. Fixing a point p ∈ U,there- sults of Sect. 2.16 (or Sect. 2.15) provide a function f ∈ M(X) that is holomorphic except for a pole at p. Show that there is a sequence {gn} in O(X) that converges uniformly to f on K. Applying the maximum { } principle to the sequence gn U (along with the Cauchy criterion), one gets a continuous function on U that is holomorphic on U and that is equal to f on ∂U. This leads to a contradiction. 2.16.5 This exercise requires facts proved in Exercise 2.13.6. Let be a nonempty open subset of an open Riemann surface X. Prove that the following are equivalent: (i) is topologically Runge. (ii) For every compact set K ⊂ ,wehavehX(K) ⊂ . (iii) For every holomorphic function f0 on , every compact set K ⊂ , and every >0, there exists a holomorphic function f on X such that |f − f0| <on K (that is, is holomorphically Runge). Hint. The proof of (iii)⇒(i) is similar to that of part (b) of Exer- cise 2.16.4. 2.16.6 Let X be a Riemann surface, let K be a compact subset of X,letP be a discrete subset of X with P ⊂ X \ K, and let Y = X \ P . Assume that hY (K) = K. { } (a) Suppose that (Up,zp) p∈P is a locally finite collection of disjoint local holomorphic coordinate neighborhoods in X with p ∈ Up and zp(p) = 0 for each p ∈ P , ω is a Kähler form on X, and is a neighborhood of K in X. Prove that for every collection of sufficiently large positive constants {bp}p∈P , there exist open sets {Vp}p∈P with p ∈ Vp Up for each point p ∈ P such that for every collection of sufficiently large positive constants {Rp}p∈P (depending on the above choices), there exists a nonnegative C∞ subharmonic exhaustion function ϕ on Y with the following properties (cf. Lemma 2.16.4): 2.16 The Runge Approximation Theorem 97

(i) On Y \ , ϕ>0 and iϕ ≥ ω; (ii) We have supp ϕ ⊂ Y \ K; and 2 2 (iii) For each p ∈ P ,wehaveϕ = Rp · (|zp| − log |zp| − bp) on Vp \{p}. (b) Suppose that f0 is a holomorphic function on a neighborhood of K in X and >0. Prove that there exists a meromorphic function f on X such that f is holomorphic on X \ P = Y , f has a pole at each point in P , and |f − f0| <on K (cf. Theorem 2.16.3). 2.16.7 Let X be an open Riemann surface, let K ⊂ X be a compact subset with hX(K) = K,letP ⊂ X be a discrete subset with P ⊂ X \ K,letf0 be a holomorphic function on a neighborhood of K in X, and for each point p ∈ P ,letfp be a holomorphic function on Up \{p} for some neighborhood Up of p in X with Up ∩ P ={p}, and let mp be a positive integer. Prove that for every >0, there exists a holomorphic function f on X \ P such that |f − f0| < on K and such that for each point p ∈ P , f − fp extends to a holomorphic function on Up that either vanishes on a neighborhood of p or has a zero of order at least mp at p (note that this is a combined version of the Mittag-Lefﬂer theorem and the Runge approximation theorem). 2.16.8 Suppose K is compact subset of an open Riemann surface X with hX(K) = K and θ0 is a holomorphic 1-form on a neighborhood of K in X. Prove that there exists a sequence of holomorphic 1-forms {θν } on X such that for every local holomorphic coordinate neighborhood (U, z) in X, θν/dz → θ0/dz uniformly on compact subsets of K ∩ U. 2.16.9 This exercise requires Exercises 2.13.8, 2.13.10, 2.13.12, and 2.14.7. Let be a topologically Runge open subset of an open Riemann surface X. (a) Suppose that ω is a Kähler form on X and ϕ is a real-valued C∞ function on X with iω + iϕ ≥ 0onX. Prove that for every holomorphic function f0 on , every closed set K ⊂ , and every >0, there exists a holomorphic function f on X such that f − f0 2 <. L0,0(K,ω,ϕ) (b) Suppose that ϕ is a C∞ subharmonic function on X. Prove that for every holomorphic 1-form θ0 on , every closed set K ⊂ , and every >0, there exists a holomorphic 1-form θ on X such that θ − θ0 2 <. L1,0(K,ϕ) 2.16.10 The goal of this exercise is a special case of the Mergelyan–Bishop theorem. Let λ denote Lebesgue measure on C. (a) Prove that if ρ0 is a continuous complex-valued function on a compact set K ⊂ C and >0, then there exists a C∞ function ρ on C such that |ρ − ρ0| <on K. Hint. Patch together local constant approximations using a C∞ partition of unity. (b) Prove that if S is a measurable subset of C and z ∈ C, then 1 λ(S) dλ(ζ) ≤ 2πR + ∀R>0. S |ζ − z| R √ | − | ≤ Conclude from this that in particular, S(1/ ζ z )dλ(ζ) 8πλ(S). 98 2 Riemann Surfaces and the L2 ∂¯-Method for Scalar-Valued Forms

(c) Prove that if f ∈ C∞(C) and is a smooth relatively compact domain in C, then 1 f(ζ) ∂f f(z)− dζ ≤ · λ()/π. − sup 8 2πi ∂ ζ z ∂ζ¯ (d) HartogsÐRosenthal theorem (see [HarR]). Suppose K is a compact set of measure 0 in C. Prove that if f0 is a continuous function on K and >0, then there exists a holomorphic function f on a neighborhood of K in C such that |f − f0| <on K. Prove also that if in addition, C \ K is connected, then there actually exists an entire function f such that |f − f0| <on K. (e) Prove that given an open set ⊂ C and a compact set K ⊂ , there exists a constant C = C(K,) > 0 such that ∂f | |≤ + ∀ ∈ C∞ max f C f L2() sup f (). K ∂ζ¯

Hint. First consider the case in which K = (z0; R) and = (z0; 4R) for some point z0 ∈ C and some R>0. For this, apply the Cauchy integral formula (Lemma 1.2.1) to the disk (z0; 3R) and to the function ηf , where η ∈ D((z0; 3R)) and η ≡ 1on(z0; 2R).For the general case, cover K by finitely many disks of the form (z0; R) with (z0; 4R) ⊂ . (f) Let X be a Riemann surface. Prove that given an open set ⊂ X, a compact set K ⊂ , a Kähler form ω on X, and a real-valued C∞ function ϕ on X, there exists a constant C = C(K,,ω,ϕ) > 0 such that ∧ 1/2 | |≤ + ∂f ∂f ∀ ∈ C∞ max f C f L2(,ω,ϕ) sup f (). K ω (g) BishopÐKodama localization theorem (see [Bis] and [Kod]). Let X be an open Riemann surface, let K be a compact subset of X, and let f0 be a continuous function on K. Assume that each point p ∈ K admits a neighborhood U in X such that for every >0, there exists a holomorphic function f on a neighborhood of the compact set K ≡ K ∩ U in X with |f − f0| < on K . Prove that for every >0, there exists a holomorphic function f on a neighborhood of K in X such that |f − f0| <on K. Hint. By replacing X with a large domain, one may assume that there are a Kähler metric ω and a C∞ strictly subharmonic function ϕ on X as in Corollary 2.12.6. Fix a finite covering of K by relatively compact open subsets of X with the above approximation property, and fix a suitable partition of unity. Given δ>0, one may form local approximations to within δ on each of these sets. Patching these local 2.16 The Runge Approximation Theorem 99

approximations using the partition of unity, one gets a C∞ function τ ; and the (0, 1)-form ∂τ¯ is controlled by δ at points in K. Multiplying ∂τ¯ by a C∞ (cutoff) function that is equal to 1 on K and that vanishes outside a small neighborhood of K (this function depends on the choice of δ), one gets a (1, 0)-form β that is controlled by δ everywhere in X. Corollary 2.12.6 then provides a solution of the equation ∂α¯ = β along with an L2 estimate. Guided by part (f), one sees that if δ>0 is sufﬁ- ciently small, then the restriction of τ − α to a small neighborhood of K (which depends on the choice of δ) has the required properties. (h) Let X be an open Riemann surface, and let K ⊂ X be a compact set of measure 0. Prove that for every continuous function f0 on K and for every >0, there exists a holomorphic function f on a neighborhood of K in X such that |f − f0| <on K. Prove also that if in addition, hX(K) = K, then there actually exists a function f ∈ O(X) such that |f − f0| <on K.

Remarks According to Mergelyan’s theorem (see [Me] and [Rud1]), given a compact set K ⊂ C with connected complement, a continuous function f0 on K that is holomorphic on the interior of K, and a constant >0, there exists an entire function f with |f − f0| < on K. The Mergelyan–Bishop theorem includes the natural analogue for an open Riemann surface X: Given a compact set K ⊂ X with hX(K) = K, a continuous function f0 on K that is holomorphic on the interior of K, and a constant >0, there exists a holomorphic function f on X with |f − f0| < on K. The proof of the Hartogs– Rosenthal theorem outlined in parts (b)–(d) is due to Hartogs and Rosenthal. The proof of the Bishop–Kodama localization theorem outlined in parts (e)–(g) is due to Jarnicki and Pﬂug (see [JP] and [Ga]).

~~Chapter 3 The L2 ∂¯-Method in a Holomorphic Line Bundle~~

In this chapter, we consider a useful generalization of the notion of a holomorphic function, namely, that of a holomorphic section of a holomorphic line bundle. We ﬁrst consider the basic properties of holomorphic line bundles as well as those of sheaves and divisors. We then proceed with a discussion of the solution of the inhomogeneous CauchyÐRiemann equation with L2 estimates in this more general setting. In this setting, there is a natural generalization of Theorem 2.9.1 for Hermitian holomorphic line bundles (E, h) with positive curvature; that is, ih > 0, where h ¯ is a natural generalization of the curvature form ϕ = ∂∂ϕ considered in Sect. 2.8. In fact, Sects. 3.6Ð3.9 may be read in place of most of the material in Sects. 2.6Ð2.9. We then consider applications, mostly to the study of holomorphic line bundles on open Riemann surfaces (holomorphic line bundles on compact Riemann surfaces are considered in greater depth in Chap. 4). For example, in Sect. 3.11, we prove that every holomorphic line bundle on an open Riemann surface admits a positive- curvature Hermitian metric (this follows easily from the results of Sect. 2.14); and we then obtain a slightly more streamlined proof of (a generalization of) the Mittag- Lefﬂer theorem (Theorem 2.15.1). In Sect. 3.12, we prove the Weierstrass theorem (Theorem 3.12.1), according to which every holomorphic line bundle on an open Riemann surface is actually holomorphically trivial.

~~3.1 Holomorphic Line Bundles~~

Throughout this section, X denotes a complex 1-manifold. In this section, we consider the basic properties of holomorphic line bundles and differential forms with values in a holomorphic line bundle. In order to introduce some of the terminology, we ﬁrst consider a set-theoretic version.

Deﬁnition 3.1.1 Let Y be a topological space. (a) A (set-theoretic) complex line bundle over (or on) Y consists of a set E,asur- jective mapping : E → Y , and a choice of a 1-dimensional complex vector

T. Napier, M. Ramachandran, An Introduction to Riemann Surfaces, Cornerstones, 101 DOI 10.1007/978-0-8176-4693-6_3, © Springer Science+Business Media, LLC 2011 102 3 The L2 ∂¯-Method in a Holomorphic Line Bundle

−1 space structure in the ﬁber Ep ≡ (p) for each point p ∈ Y . We usually let the total space E or the projection map : E → Y represent the line bundle, rather than referring to the triple consisting of the set E,themap, and the choice of vector space structures in the ﬁbers. A local trivialization of E is a bijection of the form = (, ): −1(U) → U × C, where U is an open subset of Y and : −1(U) → C is a surjective mapping for which the restriction ≡ : → C ∈ p Ep Ep is a complex linear isomorphism for each point p U. In other words, we have a commutative diagram

−1(U) U × C pr U U in which is a bijection that is linear on each ﬁber. A local trivialization of the form (X, ) is called a global trivialization. (b) Given two local trivializations (U1,1 = (, 1)) and (U2,2 = (, 2)) of ∗ a complex line bundle E over Y ,themapg : U1 ∩ U2 → C determined by ◦ −1 = · ∈ ∩ × C 1 2 (x, ζ ) (x, g(x) ζ) for all (x, ζ ) (U1 U2) (that is, for each point x ∈ U1 ∩ U2, g(x) = (1)x/(2)x is the nonzero scalar representing the → −1 linear isomorphism ζ 1(2 (x, ζ ))) is called a transition function (from 2 to 1 or from 2 to 1). A ={ } (c) A collection of local trivializations (Ui,i) i∈I of a complex line bun- = dle E on Y with Y i Ui is called a line bundle atlas for E.

Remark A more standard approach is to consider only line bundles for which the total space E is a topological space, the projection is continuous, and the local trivializations are homeomorphisms. The above weaker deﬁnition is more convenient for our purposes.

~~A holomorphic structure allows one to consider a holomorphic version:~~

~~Deﬁnition 3.1.2 Let : E → X be a complex line bundle over X.~~

~~(a) Two local trivializations (U1,1 = (, 1)) and (U2,2 = (, 2)) of E are ∗ holomorphically compatible if the transition functions g12,g21 : U1 ∩U2 → C , given by~~

1 (1)p g12(p) = = ∀p ∈ U1 ∩ U2, g21(p) (2)p are holomorphic. (b) A line bundle atlas for E for which the transition functions are holomorphically compatible is called a holomorphic line bundle atlas. (c) Two holomorphic line bundle atlases A1 and A2 for E are holomorphically equivalent if A1 ∪ A2 is a holomorphic line bundle atlas (this is an equivalence relation). 3.1 Holomorphic Line Bundles 103

(d) An equivalence class S of holomorphic line bundle atlases for E is called a holomorphic line bundle structure in E over X. The pair (E, S) (usually denoted simply by E) is called a holomorphic line bundle over (or on) X. (e) If E is equipped with a holomorphic line bundle structure, then any local trivialization (U, ) in any holomorphic line bundle atlas in the holomorphic line bundle structure is called a local holomorphic trivialization for E.Ifwe have U = X, then (X, ) is a (global) holomorphic trivialization for E and E is holomorphically trivial.Thetrivial line bundle over X is the line bundle 1X ≡ X × C → X.

Remarks 1. A topological line bundle over a topological space and a C∞ line bundle over a C∞ manifold are defined analogously. The natural higher-rank analogues, in which the local trivializations are fiberwise linear bijections of the form −1(U) → U × Cr , are called vector bundles. 2. A holomorphic line bundle on X has a natural underlying C∞ structure, an underlying C∞ line bundle structure, and a 2-dimensional holomorphic structure (see Exercise 2.2.6 for the definition of a complex manifold and Exercise 3.1.9 for the verification).

~~Deﬁnition 3.1.3 Let : E → X be a holomorphic line bundle over X.~~

(a) A section of E onasetB ⊂ X is a mapping s : B → E such that ◦ s = IdB . For each point p ∈ B, we usually denote the value s(p) ∈ Ep by sp.IfB is an open set, then we also call s a local section of E.IfB = X, then we also call s a global section. (b) Let s beasectionofE onasetB. Given a local trivialization (U, (, )), the function (s): B ∩ U → C (i.e., the function p → (sp)) is called the representation of s in the local trivialization. (c) We say that a section s of E on a set B is continuous (of class Ck with ∈ Z ∪{∞} d ∈[ ∞] k ≥0 , holomorphic, measurable, Lloc with d 1, ) if the representation (s) of s in every local holomorphic trivialization (U, (, )) (equivalently, in some local holomorphic trivialization in a neighborhood of each point Ck d in B) is a continuous (respectively, , holomorphic, measurable, Lloc) function. (d) For B ⊂ X an open set, we call a section s of E on the complement in B of a discrete subset P of B a meromorphic section of E on B if the representation (s) of s in every local holomorphic trivialization (U, (, )) (equivalently, in some local holomorphic trivialization in a neighborhood of each point in B) is a meromorphic function on B ∩ U with set of poles P ∩ U. We say that s has a zero (a pole) of order m at a point p ∈ B if the representation (s) of s in every (equivalently, in some) local holomorphic trivialization (U, (, )) over a neighborhood U of p has a zero (respectively, a pole) of order m at p. 104 3 The L2 ∂¯-Method in a Holomorphic Line Bundle

For any point p ∈ B,theorder of s at p is given by ⎧ ⎪0ifp is neither a zero nor a pole of s, ⎨⎪ m if s has a zero of order m at p, ordps = ⎪−m if s has a pole of order m at p, ⎩⎪ ∞ if s ≡ 0 on a neighborhood of p.

We say that s has a simple zero (simple pole) at p if ordps = 1 (respectively, ordps =−1). (e) For any open set U ⊂ X, the set of holomorphic sections of E on U is denoted O O 0 by (E)(U) or (U, (E)) (or HDol(U, E)); the set of meromorphic sections of E on U is denoted by M(E)(U) or (U,M(E)); and the set of C∞ sections of E on U is denoted by E(E)(U) or (U,E(E)).ThesetofC∞ sections with compact support in U is denoted by D(E)(U) or D(U, E).

Remarks 1. It is not hard to verify that, as noted in the above definition, if a section of a holomorphic line bundle on X has a continuous (Ck , holomorphic, meromor- d phic, measurable, Lloc) representation in some local holomorphic trivialization in a neighborhood of each point, then the representation in every local holomorphic trivialization is continuous (respectively, Ck, holomorphic, meromorphic, measurable, d Lloc). The analogous statement holds for zeros and poles of a meromorphic section. The verifications are left to the reader (see Exercise 3.1.1). 2. For a nontrivial (i.e., not everywhere zero) meromorphic section of a holomorphic line bundle over a Riemann surface, the set of zeros (i.e., the set of points at which the section is equal to the zero element of the fiber) is discrete (see Exer- cise 3.1.2). 3. One goal of this chapter is a proof that every holomorphic line bundle on an open Riemann surface admits a great many (nontrivial) holomorphic sections. On the other hand, a holomorphic line bundle on a compact Riemann surface need not admit a nontrivial holomorphic section (see Exercise 3.1.6). Much of Chap. 4, and part of this chapter, are concerned with determining when (and how many) nontrivial holomorphic sections exist. 4. We may identify a section x → (x, f (x)) of the trivial line bundle 1X = X × C → X with the complex-valued function f .

: 1,0 → Example 3.1.4 The holomorphic tangent bundle (T X)1,0 (T X) X and holo- : ∗ 1,0 → morphic cotangent bundle (T ∗X)1,0 (T X) X have natural holomorphic line bundle structures. For given two local holomorphic coordinate neighborhoods (U1,z1) and (U2,z2) in X, we have the local trivialization (Uj ,((T X)1,0 ,dzj )) of 1,0 (T X) over Uj for j = 1, 2, and we have holomorphic transition functions

dz1 ∂z1 dz2 ∂z2 g12 = = and g21 = = = 1/g12. dz2 ∂z2 dz1 ∂z1 = For each j 1, 2, we also have the local trivialization (Uj ,((T ∗X)1,0 ,ρj )) of ∗ 1,0 (T X) over Uj , where ρj (α) = α((∂/∂zj )p) for each point p ∈ Uj and each 3.1 Holomorphic Line Bundles 105

~~∈ ∗ 1,0 element α (Tp X) . The transition functions are the holomorphic functions~~

~~ρ1 ∂/∂z1 ∂z2 ∂/∂z2 ∂z1 g12 = = = and g21 = = = 1/g12. ρ2 ∂/∂z2 ∂z1 ∂/∂z1 ∂z2~~

The local holomorphic sections of (T X)1,0 are precisely the local holomorphic vector ﬁelds, and the local holomorphic sections of (T ∗X)1,0 are precisely the local holomorphic 1-forms (and the analogous statements hold for continuous, Ck, mea- d surable, meromorphic, and Lloc vector ﬁelds and forms). The holomorphic cotangent bundle (T ∗X)1,0 is also called the canonical line bundle (or canonical bundle) : → and is denoted by KX KX X.

~~1 1 Example 3.1.5 For the open subsets U0 ≡ C and U∞ ≡ P \{0} of P ,let~~

~~E ≡[(U0 × C) (U∞ × C)]/∼, where for (z, ζ ) ∈ (U0 \{0}) × C and (w, ξ) ∈ (U∞ \{∞}) × C,~~

~~(z, ζ ) ∼ (w, ξ) ⇐⇒ z = w and ζ = z · ξ;~~

1 let ρ : (U0 × C) (U∞ × C) → E be the quotient map; and let E : E → P be the 1 (well-deﬁned) surjective mapping given by E : ρ(z,ζ) → z. Then E : E → P is a line bundle, which is called the hyperplane bundle. Moreover, for the local trivializations

≡[ ]−1 : −1 = × C → × C (E,0) ρ U0×C E (U0) ρ(U0 ) U0 and ≡[ ]−1 : −1 = × C → × C (E,∞) ρ U∞×C E (U∞) ρ(U∞ ) U∞ , we have the holomorphic transition functions g0∞ from ∞ to 0 and g∞0 from 0 to ∞ determined by

0 1 g0∞ = = : z → z. ∞ g∞0 Thus E, together with these local trivializations, is a holomorphic line bundle. The −1 holomorphic section s of E on U∞ determined by z → ρ(z,1) = (E,∞) (z, 1) for (z, 1) ∈ U∞ × C (i.e., the section that is represented by the constant function ∞(s): z → 1 in the local holomorphic trivialization (U∞,(E,∞))) is represented in the local holomorphic trivialization (U0,(E,0)) by the holomorphic function 0(s): z → z. It follows that s extends to a unique holomorphic section of E on P1 that is nonvanishing except for a simple zero at the point 0.

~~Section 3.3 contains a simple method for producing many (in fact, it turns out, all) examples of holomorphic line bundles on a Riemann surface. 106 3 The L2 ∂¯-Method in a Holomorphic Line Bundle~~

Given a local holomorphic trivialization (U, = (, )) of a holomorphic line bundle : E → X, we get a nonvanishing holomorphic section t on U given by = −1 = −1 ∈ tp p (1) (p, 1) for all p U; that is, t is the section with local representation p → 1. Conversely, according to Proposition 3.1.6 below, a nonvanishing local holomorphic section provides a local holomorphic trivialization. In other words, local holomorphic trivializations are equivalent to nonvanishing local holomorphic sections; and the latter point of view is often more convenient than the former. Fur- ther characterizations of continuity, continuous differentiability, etc., of sections in terms of local holomorphic sections are also contained in the proposition.

Proposition 3.1.6 Suppose that : E → X is a holomorphic line bundle, k ∈ Z≥0 ∪{∞}, and d ∈[1, ∞]. Then we have the following: (a) If t is a nonvanishing holomorphic section of E on an open set U ⊂ X, then the −1 mapping : (U) → C given by ξ → ξ/t(p) for all p ∈ U and ξ ∈ Ep determines a local holomorphic trivialization (U, = (, )) of E (with (t) ≡ 1). (b) The product fs of a continuous (Ck , holomorphic, meromorphic, measurable) function f and a continuous (respectively, Ck , holomorphic, meromorphic, mea- k surable) section s of E and the sum s1 + s2 of two continuous (respectively, C , holomorphic, meromorphic, measurable) sections s1 and s2 of E are continuous (respectively, Ck , holomorphic, meromorphic, measurable) sections. The sum of d d ∈[ ∞] + = two Lloc sections is in Lloc, and for d 1, with (1/d) (1/d ) 1, the d d 1 product of an Lloc function and an Lloc section is in Lloc. (c) A section s : S → E onasetS ⊂ X is continuous (Ck, holomorphic, meromor- d : → phic, measurable, Lloc) if and only if the quotient s/t p s(p)/t(p) is con- Ck d tinuous (respectively, , holomorphic, meromorphic, measurable, Lloc) for every nonvanishing local holomorphic section t of E (equivalently, for some nonvanishing local holomorphic section t on a neighborhood of each point in S). Furthermore, if s is meromorphic, then s has a zero (a pole) of order m at a point p ∈ S if and only if the meromorphic function s/t has a zero (respectively, a pole) of order m at p for every (equivalently, for some) nonvanishing local holomorphic section t of E on a neighborhood of p.

Proof Given a nonvanishing holomorphic section t of E on an open set U ⊂ X,the −1 mapping : (U) → C given by ξ → ξ/t(p) for all p ∈ U and ξ ∈ Ep determines a bijection = (, ): −1(U) → U × C. Moreover, if (V, = (, ϒ)) is a local holomorphic trivialization of E, then (by Deﬁnition 3.1.3) the representing function g = ϒ(t): U ∩ V → C∗ = C \{0} is holomorphic and we have

~~ϒ(ξ)= g(p) · (ξ) ∀p ∈ U ∩ V, ξ ∈ Ep.~~

Thus g is a holomorphic transition function from the local trivialization (, ) to (, ϒ), and hence (, ) is a local holomorphic trivialization. One gets part (b) by considering local representations of the sections. Finally, (c) follows from parts (a) and (b). 3.1 Holomorphic Line Bundles 107

Deﬁnition 3.1.7 Let E : E → X and E : E → X be holomorphic line bundles over complex 1-manifolds X and X,letϒ : X → X be a holomorphic map, and let : E → E be a mapping such that E ◦ = ϒ ◦ E and such that for each ∈ : → point p X, the mapping Ep Ep Eϒ(p) is linear. Then: (a) The map is called a line bundle homomorphism (or line bundle map) along ϒ. Unless otherwise indicated, we will assume that a given line bundle homomorphism is taken along the identity map. (b) We call a continuous (Ck, holomorphic) line bundle homomorphism if the function ◦ (s) is a continuous (respectively, Ck , holomorphic) function for every local holomorphic section s of E and every local holomorphic trivialization (U, (E , )) of E . (c) A bijective holomorphic line bundle homomorphism along the identity with holomorphic inverse line bundle homomorphism is called a holomorphic line bundle isomorphism (and the two bundles are said to be isomorphic).

Example 3.1.8 If : X → Y is a holomorphic mapping (a Ck mapping with k ∈ Z>0 ∪{∞})ofX into a complex 1-manifold Y , then the associated tan- ∗ ∗ gent mapping ∗ : (T X)1,0 → (T Y )1,0 and pullback mapping : (T Y)1,0 → (T ∗X)1,0 are holomorphic (respectively, Ck−1) line bundle homomorphisms (see Exercise 3.1.3).

Remarks 1. A holomorphic line bundle E → X is holomorphically trivial if and ∼ only if E = 1X. 2. As we will see, every holomorphic line bundle on an open Riemann surface is holomorphically trivial (Theorem 3.12.1). However, even in that context, the ab- stract point of view is still useful, since, for example, there is usually no natural choice of a global holomorphic trivialization. 3. A real linear (or conjugate linear) homomorphism of holomorphic line bundles is defined analogously, with the mapping real linear (respectively, conjugate linear) on each fiber. Continuous and Ck real linear and conjugate linear line bundle homomorphisms and isomorphisms are defined analogously. 4. We often identify two isomorphic line bundles without comment, although, occasionally one must proceed with some caution in doing so.

Deﬁnition 3.1.9 For holomorphic line bundles E : E → X and E : E → X: ∗ ≡ ∗ (a) The dual bundle of E is given by E p∈X Ep, and the corresponding pro- ∗ jection E∗ : E → X is determined by α → p for each point p ∈ X and each ∈ ∗ linear functional α Ep. ⊗ ≡ ⊗ (b) The tensor product bundle of E and E is given by E E p∈X Ep : ⊗ → Ep (see Sect. 8.3), and the corresponding projection E⊗E E E X is → ∈ ∈ ⊗ determined by ξ p for each point p X and each element ξ Ep Ep.

~~We will assume that any given dual or tensor product bundle as above has the holomorphic line bundle structure provided by the following: 108 3 The L2 ∂¯-Method in a Holomorphic Line Bundle~~

Proposition 3.1.10 Let E : E → X, E : E → X, and E : E → X be holomorphic line bundles. Then we have the following: ∗ (a) There is a (unique) holomorphic line bundle structure in E∗ : E → X such that for each nonvanishing holomorphic section t of E on an open set U ⊂ X, : −1 → × C → ∈ the map E∗ (U) U given by α (p, α(t(p))) for all p U and ∈ ∗ α Ep is a local holomorphic trivialization. (b) There is a unique holomorphic line bundle structure in E ⊗ E such that for each pair of holomorphic sections t of E and t of E on an open set U ⊂ X, ⊗ ⊗ = ⊗ the tensor product section t t (where (s s )p sp sp for each point p and each pair of sections s,s) is a local holomorphic section of E ⊗ E. (c) For the above holomorphic line bundle structures, we have the (natural) holomorphic line bundle isomorphisms ∗ ∗ (i) E → (E ) given by ξ → λξ , where λξ (α) = α(ξ) for all ξ ∈ Ep and ∈ ∗ ∈ α Ep with p X; ∗ ⊗ → = × C ⊗ → ⊗ ∈ ∗ ⊗ (ii) E E 1X X given by α ξ α(ξ) for all α ξ Ep Ep with p ∈ X; (iii) 1X ⊗ E → E given by (p, ζ ) ⊗ ξ → ζξ for all p ∈ X, ζ ∈ C, and ξ ∈ Ep; ⊗ → ⊗ ⊗ → ⊗ ⊗ ∈ ⊗ (iv) E E E E given by η ξ ξ η for all η ξ Ep Ep with p ∈ X; and (v) (E ⊗ E) ⊗ E → E ⊗ (E ⊗ E) given by (η ⊗ ξ)⊗ τ → η ⊗ (ξ ⊗ τ)for ⊗ ∈ ∈ ∈ ∈ all η ξ Ep, ξ Ep, and τ Ep with p X.

Proof For the proof of (a), we consider two nonvanishing holomorphic sections t and u of E on open sets U and V , respectively. Proposition 3.1.6 then implies that ≡ : ∩ → C∗ ∈ ∗ the function g t/u U V is holomorphic. Thus, for each element α Ep with p ∈ U ∩ V ,wehaveα(t(p)) = g(p)α(u(p)). It follows that the expression in (a) yields a bijection and that the transition functions for any two such bijections are holomorphic. Thus these mappings form a holomorphic line bundle atlas that determines a holomorphic line bundle structure on E∗, and (a) follows. The proofs of (b) and (c) are left to the reader (see Exercise 3.1.4).

~~∗ 1,0 Example 3.1.11 The canonical line bundle KX = (T X) is the dual of the holomorphic tangent bundle (T X)1,0.~~

~~Example 3.1.12 The dual bundle E∗ → P1 of the hyperplane bundle E → P1 (Ex- ample 3.1.5) is called the tautological bundle.~~

Remarks Let E, E, E be holomorphic line bundles on X. 1. Guided by part (c) of Proposition 3.1.10, we identify (E∗)∗ with E, E∗ ⊗ E with 1X,1X ⊗ E with E, E ⊗ E with E ⊗ E, and (E ⊗ E ) ⊗ E with E ⊗ (E ⊗ E). 2. We denote (E ⊗ E) ⊗ E = E ⊗ (E ⊗ E) simply by E ⊗ E ⊗ E;weset 0 E = 1X; and for any positive integer r,weset

~~r factors Er ≡ E ⊗···⊗E. 3.1 Holomorphic Line Bundles 109~~

~~3. One may treat E∗ as a multiplicative inverse of E, and the trivial line bundle 1X as a multiplicative identity.~~

~~The proof of the following is left to the reader (see Exercise 3.1.5):~~

Proposition 3.1.13 Let E : E → X and E : E → X be holomorphic line bundles, let S ⊂ X, let k ∈ Z≥0 ∪{∞}, and let d,d ∈[1, ∞] with (1/d) + (1/d ) = 1. Then we have the following: (a) For any section α of E∗ on S, the following are equivalent: (i) The section α is continuous (of class Ck, holomorphic, measurable). (ii) The function α(s) is continuous (respectively, of class Ck, holomorphic, measurable) for every local continuous (respectively, class Ck, holomorphic, measurable) section s of E. (iii) For each point p ∈ S, the function α(s) is continuous (respectively, of class Ck, holomorphic, measurable) for some nonvanishing continuous (respectively, Ck , holomorphic, measurable) section s of E on a neighborhood of p. (b) For any section α of E∗ on S, the following are equivalent: d (i) The section α is in Lloc. d (ii) The function α(s) is in Lloc for every local continuous section s of E. ∈ d (iii) For each point p S, the function α(s) is in Lloc for some nonvanishing continuous section s of E on a neighborhood of p. d d Moreover, if α is in Lloc and s is a section of E in Lloc on S, then the func- 1 tion α(s) is in Lloc. (c) The tensor product s ⊗ s of continuous (Ck, holomorphic, meromorphic, measurable) sections s of E and s of E on S is continuous (respectively, Ck, holomorphic, meromorphic, measurable). Furthermore, if s is nonvanishing, then the section s−1 of E∗ is continuous (respectively, Ck, holomorphic, meromorphic, measurable). If t is a meromorphic section of E on X that is not identically zero on any connected component of X, then t−1 is a meromorphic section ∗ d d of E . Finally, if u is a section of E in Lloc on S and u is a section of E in Lloc ⊗ 1 on S, then u u is in Lloc. (d) Any section s of E∗ ⊗ E on X determines a line bundle homomorphism : E → E given by ξ → (sp/t)(ξ) · t for each point p ∈ X, each element ∈ \{ } ∈ Ck t Ep 0 , and each element ξ Ep; and is continuous (of class , holomorphic) if and only if the section s is continuous (respectively, of class Ck , holomorphic). Conversely, any line bundle homomorphism : E → E deter- ∗ −1 mines a section s of E ⊗ E on X given by sp ≡ t ⊗ (t) for each point k p ∈ X and each element t ∈ Ep \{0}; and s is continuous (of class C , holomorphic) if and only if is continuous (respectively, of class Ck, holomorphic). Moreover, the above associations are inverse mappings.

~~We now consider differential forms with values in a holomorphic line bundle. 110 3 The L2 ∂¯-Method in a Holomorphic Line Bundle~~

Deﬁnition 3.1.14 Let E : E → X be a holomorphic line bundle. For each pair p,q ∈ Z≥0 and for r = p + q, we deﬁne p,q ∗ ⊗ ≡ p,q ∗ ⊗ ⊂ r ∗ ⊗ ≡ r ∗ ⊗ T X E Tx X Ex (T X)C E (Tx X)C Ex x∈X x∈X ∗ ∗ ( p,qT X ⊗E = X ×{0} for p>1orq>1, r (T X)C ⊗E = X ×{0} for r>2). The corresponding projections are the (surjective) mappings

∗ ∗ ∗ p,q ∗ r p,qT X⊗E : T X ⊗ E → X and r (T X)C⊗E : (T X)C ⊗ E → X → ∈ ∈ p,q ∗ ⊗ r ∗ ⊗ given by α x for each x X and each α Tx X E or (Tx X)C E. The elements of p,qT ∗X ⊗ E are said to be of type (p, q). For each point x ∈ X, we also set

~~[ p,q ∗ ⊗ ] ≡ p,q ∗ ⊗ ⊂[ r ∗ ⊗ ] ≡ r ∗ ⊗ T X E x Tx X Ex (T X)C E x (Tx X)C Ex .~~

~~Given a nonnegative integer s, another holomorphic line bundle F : F → X, and a point x ∈ X, the mapping~~

r ∗ s ∗ r+s ∗ [ (T X)C ⊗ E]x ×[ (T X)C ⊗ F ]x →[ (T X)C ⊗ E ⊗ F ]x given by α β (α, β) → α ∧ β ≡ ∧ ⊗ t ⊗ u, t u for any choice of t ∈ Ex \{0} and u ∈ Fx \{0}, is called the exterior product (or wedge product).

~~Remarks 1. For any holomorphic line bundle E on X, the spaces~~

0,0 ∗ ∼ 1,0 ∗ T X ⊗ E = 1X ⊗ E = E and KX ⊗ E = T X ⊗ E are actually tensor products of holomorphic line bundles, so they have natural holomorphic line bundle structures (provided by Proposition 3.1.10). For p>1orq>1,

∗ p,qT X ⊗ E = X ×{0} =∼ X has the natural holomorphic structure inherited from X. 2. p,qT ∗X ⊗ E and r T ∗X ⊗ E have natural C∞ structures (see Exer- cise 3.1.10). Moreover, p,qT ∗X ⊗ E and r T ∗X ⊗ E are C∞ vector bundles (see, for example, [Wel]). 3. The above deﬁnition of the exterior product is independent of the choice of the nonzero elements of the ﬁbers of the line bundles (see Exercise 3.1.7).

~~Guided by Definition 9.5.2, Definition 9.7.12, Definition 2.5.1, Definition 2.5.2, and Proposition 3.1.6, we make the following definition: 3.1 Holomorphic Line Bundles 111~~

~~Deﬁnition 3.1.15 Let E : E → X be a holomorphic line bundle, let k ∈ Z≥0 ∪ {∞}, and let d ∈[1, ∞].~~

(a) For each r ∈ Z≥0,anE-valued differential form of degree r (or an r-form with values in E or an E-valued r-form)inX onasetS ⊂ X is a mapping α of S into ∗ r ∗ (T X)C ⊗ E such that r (T X)C⊗E ◦ α = IdS (in particular, α/s is a scalar- valued differential form of degree r for every nonvanishing local holomorphic section s of E). We usually denote the value of α at x by αx for each point ∈ ∈ p,q ∗ ⊗ ∈ x S.Ifαx Tx X Ex for each point x S, then we say that α is of type (p, q) or that α is an E-valued (p, q)-form. (b) An E-valued differential form α is continuous (of class Ck, holomorphic, mero- d morphic, measurable, in Lloc) if the scalar-valued differential form α/s is continuous (respectively, of class Ck, holomorphic, meromorphic, measurable, in d Lloc) for every nonvanishing local holomorphic section s of E. A sequence of d { } d Lloc differential forms αν with values in E converges in Lloc to an E-valued { } d differential form α if αν/s converges to α/s in Lloc for every nonvanishing local holomorphic section s of E. (c) For each open set U ⊂ X and each nonnegative integer r,thesetofC∞ E-valued r-forms on U is denoted by Er (U, E) or Er (E)(U).Thesetof C∞ E-valued r-forms with compact support in is denoted by Dr (U, E) r ∞ or D (E)(U). Similarly, for p,q ∈ Z≥0,thesetofC E-valued (p, q)- forms on U is denoted by Ep,q(U, E) or Ep,q(E)(U), and the set of C∞ E- valued (p, q)-forms with compact support in U is denoted by Dp,q(U, E) or Dp,q(E)(U).ThesetofE-valued holomorphic 1-forms on U is denoted by (U,E) or (E)(U).

Remarks 1. Let α be an r-form with values in a holomorphic line bundle E on a subset S of X. Given a nonvanishing holomorphic section s of E on an open set U, ≡ = ⊗ we get the scalar-valued form β α/s with α S∩U β s.Ifα is of type (1, 0), then α is actually a section of the holomorphic line bundle 1,0T ∗X ⊗ E.Ifr = 1, then for each point x ∈ S, we may also identify αx with the linear mapping of the tangent space at x into Ex given by α v → x (v) · t, t for any choice of t ∈ Ex \{0} (the above is independent of the choice of t). For r = 2 and x ∈ S, we may identify αx with the skew-symmetric bilinear pairing of the tangent space at x into Ex given by α (u, v) → x (u, v) · t t for any choice of t ∈ Ex \{0}. The existence of the above identiﬁcations is the reason one calls α a differential form with values in E.Fors a nonvanishing holomorphic section on an open set U and β = α/s on S ∩ U, the above also leads us to denote = ⊗ · ⊗ · α S∩U β s simply by β s or by s β or s β (see Sect. 8.3). 112 3 The L2 ∂¯-Method in a Holomorphic Line Bundle

2. For α and β differential forms with values in holomorphic line bundles E and F , respectively, we may form the exterior product α ∧ β with values in E ⊗ F . Identifying E ⊗ F with F ⊗ E, we see that if α and β are of degree r and m, respectively, then α ∧ β = (−1)rmβ ∧ α. 3. Let α be an E-valued r-form on a set S,let(U, z) be a local holomorphic coordinate neighborhood, and let s be a nonvanishing holomorphic section of E on an open set V . Then, on S ∩ U ∩ V ,wehaveα = as, where a = α/s,ifr = 0; α = adz⊗ s + bdz¯ ⊗ s, where a and b are the coefficients of α/s,ifr = 1; and α = adz∧ dz¯ ⊗ s, where a is the coefficient of α/s,ifr = 2. 4. A holomorphic 1-form θ with values in a holomorphic line bundle E over X may be viewed as a holomorphic section of the holomorphic line bundle KX ⊗ E. 1,1 ∗ 0,1 ∗ We also have an identification of T X ⊗ E with T X ⊗ KX ⊗ E under the mapping α = θ ∧ β · s =−β ∧ θ · s → −β · (θ ⊗ s); so a (1, 1)-form with values in E may be identified with a (0, 1)-form with values in KX ⊗ E (see Sect. 3.10). On the other hand, certain operators and pairings to be considered in this chapter and Chap. 4 are not completely preserved under this mapping, so one must proceed with some caution when making this identification.

~~The proof of the following is left to the reader (see Exercise 3.1.8):~~

Proposition 3.1.16 Let E and F be holomorphic line bundles on X, let k ∈ Z>0 ∪ {∞}, and let d,d ∈[1, ∞]. Then we have the following: (a) A differential form α on a set S ⊂ X with values in E is continuous (of class Ck , d holomorphic, meromorphic, measurable, in Lloc) if and only if for each point in S, the scalar-valued differential form α/s is continuous (respectively, of Ck d class , holomorphic, meromorphic, measurable, in Lloc) for some nonvanishing local holomorphic section s of E on a neighborhood of the point. Ase- d { } d quence of Lloc(S) differential forms αν with values in E converges in Lloc(S) to an E-valued differential form α if and only if for each point in S, {αν/s} con- d verges to α/s in Lloc for some nonvanishing local holomorphic section s of E on a neighborhood of the point. (b) Let α and β be differential forms with values in E and F , respectively. If α and β are continuous (of class Ck , holomorphic, meromorphic, measurable), then α + β and α ∧ β are continuous (respectively, of class Ck, holomorphic, d + d meromorphic, measurable). If α and β are in Lloc, then α β is in Lloc. If α d d + = ∧ and β are in Lloc and Lloc, respectively, and (1/d) (1/d ) 1, then α β is ∧ 1 locally integrable (i.e., α β is in Lloc).

Remark It follows from the above proposition that the set of continuous (Ck, holo- d morphic, meromorphic, measurable, Lloc) differential forms with values in a given holomorphic line bundle over a subset of a complex 1-manifold is a vector space. 3.1 Holomorphic Line Bundles 113

Exercises for Sect. 3.1 3.1.1 Let : E → X be a holomorphic line bundle over a Riemann surface X. (a) Verify that, as noted in Definition 3.1.3, if a section of E has a contin- Ck d uous ( , holomorphic, meromorphic, measurable, Lloc) representation in some local holomorphic trivialization in a neighborhood of each point in X, then the representation in every local holomorphic trivialization is continuous (respectively, Ck, holomorphic, meromorphic, measurable, d Lloc). (b) Verify that if p ∈ X and s is a holomorphic section of E on X such that the representation of s in some local holomorphic trivialization in a neighborhood of p has a zero of order m at p, then s has a zero of order m at p (i.e., the representation in every local holomorphic trivialization in a neighborhood of p has a zero of order m at p). (c) Verify that if p ∈ X and s is a holomorphic section of E on X \{p} such that the representation of s in some local holomorphic trivialization in a neighborhood of p has a pole of order m at p, then s is meromorphic with a pole of order m at p (i.e., the representation in every local holomorphic trivialization in a neighborhood of p is meromorphic with a pole of order m at p). 3.1.2 Let s be a nontrivial meromorphic section of a holomorphic line bundle E on a Riemann surface X. Verify that {x ∈ X | sx = 0} is discrete. 3.1.3 Prove that if : X → Y is a holomorphic mapping (a Ck mapping with k ∈ Z>0 ∪{∞}) of Riemann surfaces X and Y , then the associated tangent ∗ ∗ mapping ∗ : (T X)1,0 → (T Y )1,0 and pullback mapping : (T Y)1,0 → (T ∗X)1,0 are holomorphic (respectively, Ck−1) line bundle homomorphisms. 3.1.4 Prove parts (b) and (c) of Proposition 3.1.10. 3.1.5 Prove Proposition 3.1.13. 3.1.6 Let E be a nontrivial holomorphic line bundle on a compact Riemann surface X. Prove that (X,O(E)) = 0or (X,O(E∗)) = 0. 3.1.7 Verify that the exterior product as given in Definition 3.1.14 is well defined (i.e., independent of the choice of the nonzero elements of the fibers of the line bundles). 3.1.8 Prove Proposition 3.1.16. 3.1.9 Let : E → X be a holomorphic line bundle over a complex 1-manifold X. Prove that there is a unique structure of a 2-dimensional complex manifold on E for which is a holomorphic mapping and : −1(U) → C is a holomorphic function for every local holomorphic trivialization (U, (, )) (see Exercise 2.2.6 for the definition of a complex manifold and a holomorphic mapping). Prove also that a section of E is holomorphic if and only if it is holomorphic as a mapping of complex manifolds. 3.1.10 Let E : E → X be a holomorphic line bundle over a complex 1-manifold X. ∞ ∗ (a) Prove that there is a (unique) structure of a C manifold on 1(T X)C ⊗ E such that for each local holomorphic chart (U, = z, U ) in X and 114 3 The L2 ∂¯-Method in a Holomorphic Line Bundle

∞ each local holomorphic trivialization (U, (E,ϒ)), we get a local C chart −1 2 4 ∗ (U), , U × C = U × R 1(T X)C⊗E ∗ in 1(T X)C ⊗ E, where ∂ ∂ : α ⊗ ξ → z(c), ϒ(ξ) · α ,ϒ(ξ)· α ¯ ∂z c ∂z c

~~for each point c ∈ U, each element ξ ∈ Ec, and each element α ∈ 1 ∗ = −1 (Tc X)C. In other words, setting ξ (E,ϒ) (c, 1),wehave~~

~~2 ((a(dz)c + b(dz)¯ c) ⊗ ξ)= (z(c), a, b) ∀(a, b) ∈ C .~~

(b) Prove that there is a (unique) structure of a C∞ manifold on 0,1T ∗X ⊗ E such that for each local holomorphic chart (U, = z, U ) in X and ∞ each local holomorphic trivialization (U, (E,ϒ)), we get a local C chart − 1 (U), , U × C = U × R2 0,1T ∗X⊗E ∗ in 0,1(T X)C ⊗ E, where ∂ : α ⊗ ξ → z(c), ϒ(ξ) · α ¯ ∂z c

for each point c ∈ U, each element ξ ∈ Ec, and each element α ∈ 0,1 ∗ ⊗ = ∗ 1,0 ⊗ Tc X. Note that the holomorphic line bundle KX E (T X) E has a C∞ structure provided by Exercise 3.1.9. (c) Prove that there is a (unique) structure of a C∞ manifold on 1,1T ∗X ⊗ ∗ E = 2(T X)C ⊗E such that for each local holomorphic chart (U, = z, U ) in X and each local holomorphic trivialization (U, (E,ϒ)),we get a local C∞ chart

~~− 1 (U), , U × C = U × R2 1,1T ∗X⊗E~~

~~in 1,1T ∗X ⊗ E, where ∂ ∂ : α ⊗ ξ → z(c), ϒ(ξ) · α , ¯ ∂z c ∂z c~~

for each point c ∈ U, each element ξ ∈ Ec, and each element α ∈ 1,1 ∗ Tc X. (d) Prove that the projection mappings to X in the above are C∞ mappings. (e) Prove that an E-valued differential form is continuous (of class Ck)if and only if it is continuous (respectively, of class Ck ) as a mapping. 3.2 Sheaves Associated to a Holomorphic Line Bundle 115

~~3.2 Sheaves Associated to a Holomorphic Line Bundle~~

Sheaf theory is a convenient and powerful tool for describing local objects and for passing to global objects. In this book, rather than consider the general theory of sheaves, we instead mostly consider and apply a few specific types of sheaves associated to a holomorphic line bundle. We first consider some examples before considering a formal definition. Throughout this section, : E → X denotes a holomorphic line bundle over a complex 1-manifold X.

Example 3.2.1 The sheaf O(E) of holomorphic sections of E consists of the assignment U → O(E)(U) = (U,O(E)) of the collection of holomorphic sections O(E)(U) to each open set U ⊂ X together U : O → O with the associated restriction maps ρV (U, (E)) (V, (E)) given by → ⊃ O s s V for open sets U V ( (E) is also called a locally free analytic sheaf of rank 1). The sheaf of holomorphic functions O is given by U → O(U), and we identify O with O(1X). Let p ∈ X and let ∼p be the equivalence relation on the set of local holomorphic sections of E that are deﬁned on a neighborhood of p determined by

~~s ∼p t ⇐⇒ s = t on some neighborhood of p.~~

Each of the associated equivalence classes is called a germ of a holomorphic section of E at p.ThesetO(E)p of germs at p is called the stalk of O(E) at p.We denote by germps the germ represented by a local holomorphic section s of E on a neighborhood of p. For each open set U ⊂ X, (U,O(E)) is a module over (U,O) (we view (∅, O(E)) as the trivial module {0}). The operations descend to the level of germs, making O(E)p a module over the ring Op for each point p ∈ X. More precisely, for neighborhoods U, V , and W of p, sections s ∈ (U,O(E)) and t ∈ (V,O(E)), and a function f ∈ O(W), we deﬁne · ≡ · germpf germps germp(f s), + ≡ + germps germpt germp(s t), − ≡ − germps germpt germp(s t).

Example 3.2.2 The definitions of the sheaf M = M(1X) of meromorphic functions, the sheaf M(E) of meromorphic sections of E (a sheaf of modules over M), the germ of a meromorphic section of E, and the stalk M(E)p of M(E) at p ∈ X are analogous to the definitions in Example 3.2.1. ∼ Example 3.2.3 The definitions of the sheaf = (1X) = O(KX) of holomorphic ∼ 1-forms (a sheaf of O-modules), the sheaf (E) = O(KX ⊗ E) of E-valued holomorphic 1-forms (a sheaf of O-modules), the sheaf of meromorphic 1-forms (which 116 3 The L2 ∂¯-Method in a Holomorphic Line Bundle is isomorphic to M(KX) as a sheaf of M-modules), the sheaf of E-valued meromorphic 1-forms (which is isomorphic to M(KX ⊗ E) as a sheaf of M-modules), and the associated germs and stalks, are analogous to the definitions in Exam- ple 3.2.1.

Example 3.2.4 The deﬁnitions of the sheaves E = E0, Er , Ep,q, E(E), Er (E), and Ep,q(E) (which are sheaves of modules over E), and the associated germs and stalks, are analogous to the deﬁnitions in Example 3.2.1.

Example 3.2.5 Given a holomorphic line bundle homomorphism : E → F (along the identity), we get corresponding sheaf morphisms ∗ : O(E) → O(F ) and ∞ ∗ : M(E) → M(F ) given by s → (s). Similarly, a C line bundle homomorphism : E → F induces sheaf morphisms Er (E) → Er (F ) and Ep,q(E) → Ep,q(F ) (locally, the mappings are given by α → (α/t) ⊗ (t) for any nonvanishing local holomorphic section t of E).

Example 3.2.6 Let p ∈ X.Thesetmp of germs at p of holomorphic functions that O r vanish at p is the maximal ideal in the ring p, and for each positive integer r, mp is precisely the set of germs at p of holomorphic functions that vanish at p to order at least r. For each open set U ⊂ X,letF(U) be the O(U)-submodule of O(E)(U) consisting of the holomorphic sections that vanish at p to order at least r (in particular, F(U) = O(E)(U) if p/∈ U). Together with the given restriction mappings, these submodules form a subsheaf F of O(E). For each point x ∈ X,thestalk Fx at x is the collection of germs that are represented by functions in F(U) for some neighborhood U of x. Thus O = F = (E)x if x p, x r · O = mp (E)p if x p. O r · O O The p-submodule mp (E)p of (E)p consists of the germs at p of holomorphic sections s of E with a zero of order at least r at p. The quotient O r · O ={ + r · O | ∈ O } (E)p/mp (E)p ξ mp (E)p ξ (E)p is a complex vector space of dimension r. For in terms of a local holomorphic coordinate z vanishing at r and a nonvanishing holomorphic section t of E in a neigh- 2 r−1 borhood of p, the elements ξ0,...,ξr−1 of V represented by t,zt,z t,...,z t, respectively, form a basis. For each open set U ⊂ X,let O(E) /mr · O(E) if p ∈ U, S(U) ≡ p p p 0ifp/∈ U,

⊂ U : S → S and for any open set V U,letρV (U) (V ) be the obvious restriction map; U ∈ S = S U i.e., ρV is the identity if p V (so that (U) (V )) and ρV is identically 0 if 3.2 Sheaves Associated to a Holomorphic Line Bundle 117 p/∈ V . Then S is a sheaf of modules over O. We may form germs and stalks at a point x by identifying elements that have the same image under some restriction map in some neighborhood of x. Thus Sx = O(E)x Fx at each point x ∈ X. Because there is this single nonzero r-dimensional vector space stalk at p, S is called a skyscraper sheaf. The induced maps O(E)(U) → S(U) given by germ t + mr · O(E) ∈ O(E) /mr · O(E) if p ∈ U, t → p p p p p p 0ifp/∈ U, determine a sheaf morphism, which we denote by O(E) → S. This morphism may also be identiﬁed with the (quotient) module maps O(E)x → Sx = O(E)x Fx at the level of stalks.

Example 3.2.7 Let Y be a topological space, and let R be a ring. Then the associated constant sheaf R is given by R(U) ≡ R for each nonempty open set U ⊂ Y , and R(∅) ≡ 0. We also denote R by R. In particular, we have the zero sheaf U → {0}, which is denoted by 0. Constant sheaves of modules, groups, etc., are deﬁned analogously.

~~We now consider the formal deﬁnitions.~~

Deﬁnition 3.2.8 Let Y be a topological space. (a) A sheaf F on Y is an assignment of a set F(U) to each open set U ⊂ Y and a U : F → F ⊃ restriction mapping ρV (U) (V ) to each pair of open sets U V such that ⊂ U = (i) For each open set U Y ,wehaveρU IdU on U; ⊃ ⊃ U = V ◦ U (ii) For open sets U V W ,wehaveρW ρW ρV ; { } = ∈ F (iii) If Ui i∈I is a family of open subsets of Y , U i∈I Ui , and s,t (U) with ρU (s) = ρU (t) for each i ∈ I , then s = t; and Ui Ui { } = ∈ F (iv) If Ui i∈I is a family of open subsets of Y , U i∈I Ui , si (Ui) for U each i ∈ I , and ρUi (s ) = ρ j (s ) for every pair of indices i, j ∈ I , Ui∩Uj i Ui ∩Uj j then there exists an s ∈ F(U) such that ρU (s) = s for each i ∈ I . Ui i The elements of F(U) are called sections of F over U.ThesetF(U) is also denoted by (U,F) or H 0(U, F). (b) A sheaf R on Y together with the assignment of a ring structure to R(U) for each open set U such that the restriction mappings are homomorphisms with respect to these ring structures is called a sheaf of rings. Given a sheaf of rings R on Y ,asheaf of R-modules on Y is a sheaf F on Y together with the assignment of a module structure to F(U) over the ring R(U) for each open set U ⊂ Y such that the restriction mappings commute with the associated sum and product operations. Sheaves of groups, ideals, vector spaces, and other algebraic structures are deﬁned analogously. 118 3 The L2 ∂¯-Method in a Holomorphic Line Bundle

(c) A morphism (or sheaf mapping) : F → G of sheaves F and G is a choice of a morphism U : F(U) → G(U) (i.e., a map preserving the algebraic structures) for each open set U such that for open sets U ⊃ V ,wehavethecommutative diagram F(U) U G(U) ρU ρU V V F(V ) V G(V )

If each of the above maps U is an inclusion, then we call F a subsheaf of G and we call an inclusion. If each of the above maps is an isomorphism, then we call a sheaf isomorphism. (d) Let F be a sheaf on Y , and for each point p ∈ Y ,let∼p be the equivalence relation on the set of sections of F over neighborhoods of p deﬁned as follows: Given s ∈ F(U) and t ∈ F(V ) with p ∈ U ∩ V ,wehaves ∼p t if and only ⊂ ∩ U = V if there is an open set W U V with ρW (s) ρW (t). For each section s of F over a neighborhood U of p, we call the equivalence class represented by s the germ of s at p and we denote this germ by germps. The collection of germs at p is called the stalk at p and is denoted by Fp.IfF is a sheaf of R-modules on Y for some sheaf of rings R, then for each point p ∈ Y , the stalk Rp is a ring and the stalk Fp is a module over Rp with the natural induced operations. A sheaf morphism : F → G induces a well-deﬁned morphism : F → G ≡ ∈ p p p given by p(germpt) germpU (t) for each point p Y , each neighborhood U, and each section t ∈ F(U). These induced morphisms are inclusions (isomorphisms) if and only if U is injective (respectively, bijective) for each open set U (see Exercise 3.2.2). If U is surjective for each open set U, then p : Fp → Gp is surjective for each point p ∈ M. However, the converse is false (see Exercise 3.2.3).

Remarks 1. Given a sheaf morphism : F → G,thekernel K is the subsheaf with sections over a nonempty open set U given by ker U and the restriction mappings inherited from F. In general, the assignment U → im U need not be a sheaf. It is an example of what is called a presheaf, and it therefore determines a sheaf (called the image), but we will not consider presheaves and their associated sheaves in this book. 2. The skyscraper sheaf is an example of a quotient sheaf. Again, to deﬁne quotient sheaves in general, one must consider presheaves and their associated sheaves. F ∅ 3. For a sheaf , since we may write as a union of the form i∈∅ Ui , and since for s,t ∈ F(∅), the equality ρ∅ (s) = ρ∅ (t) then holds vacuously for i ∈∅, Ui Ui the axiom (iii) implies that F(∅) contains at most one element. Similarly, the condition (iv) implies that F(∅) is nonempty and therefore that F(∅) is a singleton. In particular, for F a sheaf of modules, we have F(∅) = 0.

~~In many contexts, it is often convenient to consider exact sequences of sheaves. However, this is done at the level of stalks, not sections: 3.3 Divisors 119~~

~~Deﬁnition 3.2.9 A sequence of sheaf morphisms~~

~~1 2 3 m F0 −→ F1 −→ F2 −→···−→ Fm on a topological space Y is called exact if for each point p ∈ Y and each index j = 2,...,m,wehave~~

~~ker(j )p = im(j−1)p.~~

~~An exact sequence of sheaves does not always yield an exact sequence at the level of sections on nonempty open sets (see Exercise 3.2.3).~~

Exercises for Sect. 3.2 3.2.1 Verify that the examples given in this section are indeed sheaves. 3.2.2 Verify that the morphisms on stalks induced by a sheaf mapping are well deﬁned and that they preserve the given algebraic structures (group, ring, or module). Also verify that these morphisms are inclusions (isomorphisms) if and only if U is injective (respectively, bijective) for each open set U. 3.2.3 Prove that if 0 → F → G → H is an exact sequence of sheaves on a topological space Y , then for every nonempty open set U ⊂ Y , the sequence

~~0 → F(U) → G(U) → H(U)~~

~~is exact. Give an example to show that the analogous statement for an exact sequence F → G → 0 need not hold.~~

~~3.3 Divisors~~

Divisors are convenient objects for encoding zeros and poles of meromorphic functions and sections. In particular, they allow one to produce many (in fact, it turns out, all) examples of holomorphic line bundles on a Riemann surface. Throughout this section, X denotes a complex 1-manifold.

Definition 3.3.1 For a given complex 1-manifold X: (a) A divisor on X is a mapping D : X → Z with discrete support; that is, the set D−1(Z \{0}) ∩ K is finite for every compact set K ⊂ X. The Abelian group consisting of the set of divisors on X (together with the natural addition) is denoted by Div(X). (b) Given a meromorphic section s of a holomorphic line bundle on X, the function div(s): X → Z ∪{+∞}is given by p → ordps. In particular, if s does not vanish identically on any nonempty open subset of X, then D = div(s) is a divisor that is called the divisor of s and s is called a defining section for D. The divisor D = div(f ) of a meromorphic function f that does not vanish identically on any nonempty open subset of X is called a principal divisor and f is called a solution of D (or a defining function for D). 120 3 The L2 ∂¯-Method in a Holomorphic Line Bundle

(c) If the difference D − D of two divisors D and D on X is a principal divisor, then we write D ∼ D and we say that D and D are linearly equivalent. (d) A divisor D is called effective if D ≥ 0. · Remarks 1. We may identify Div(X) with the group of formal sums p∈S νp p, { } where S is a discrete subset of X and νp p∈S is a collection of integers (we identify · · = two such sums p∈S νp p and p∈T μp p if and only if νp μp whenever ∈ ∩ = ∈ \ = ∈ \ p S T ,νp 0ifp S T , and μp 0ifp T S). More precisely, to each · = such sum p∈S νp p we associate the divisor D given by D(p) νp for each point p ∈ S and D(q)= 0 for each point q ∈ X \ S. Conversely, to each divisor D · we associate the sum p∈supp D D(p) p (to the zero divisor we associate the zero sum). 2. Given meromorphic sections s and t of holomorphic line bundles E and F , respectively, on X that are not identically zero on any nonempty open subset of X, we have

~~div(s ⊗ t)= div(s) + div(t) and div(s/t) = div(s) − div(t)~~

(where s/t = s ⊗ t −1 is the associated meromorphic section of E ⊗ F ∗). Conse- quently, linear equivalence of divisors is an equivalence relation (see Exercise 3.3.1). 3. If : E → F is a holomorphic isomorphism of holomorphic line bundles and s ∈ (X,O(E)), then div((s)) = div(s).

Proposition 3.3.2 Let D be a divisor on X; let F be the set of pairs λ = (U, f ) consisting of an open set U ⊂ X and a meromorphic function f on U with = ≡ × C div(f ) D U (i.e., f is a local deﬁning function for D); let λ U for = ∈ F [ ]≡ ∼ ∼ each element λ (U, f ) ; let D λ∈F λ/ , where is the equivalence relation determined by

f0 (p0,ζ0) ∼ (p1,ζ1) ⇐⇒ p0 = p1 and ζ0 = (p0) · ζ1 f1 = ∈ F ∈ = for elements λj (Uj ,fj ) and (pj ,ζj ) λj for j 0, 1(here, f0/f 1 is the unique extension of this quotient to a nonvanishing holomorphic function on ∩ : →[ ] U0 U1); and let ρ λ∈F λ D be the corresponding quotient map. Then we have the following: (a) The mapping :[D]→X given by ρ(p,ζ)→ p, and for each λ = (U, f ) ∈ F, −1 the mapping λ : (U) = ρ(λ) → C given by ρ(p,ζ)→ ζ for (p, ζ ) ∈ λ, are well deﬁned. Moreover, :[D]→X is a holomorphic line bundle with { } holomorphic line bundle atlas (U, (, λ)) λ=(U,f )∈F , and the section s [ ] \ −1 −∞ = of D on X D (( , 0)) determined by λ(s U ) f for each element λ = (U, f ) ∈ F is a well-deﬁned meromorphic section with div(s) = D. In particular, [D] is (holomorphically) trivial if and only if D is a principal divisor (i.e., D ∼ 0), and s is holomorphic if and only if D is effective. 3.3 Divisors 121

(b) For any holomorphic line bundle E on X with a deﬁning section t for D (i.e., t is a meromorphic section of E with div(t) = D), there is a (natural) isomorphism E →[D] given by multiplication by the nonvanishing holomorphic section s/t of E∗ ⊗[D]; that is, the isomorphism is given by

s ∗ ξ → ξ · (p) ∈ (E ⊗ E ⊗[D]) =[D] ∀p ∈ X, ξ ∈ E . t p p p Moreover, this is the unique isomorphism mapping t to s. (c) For any holomorphic line bundle E on X and any meromorphic section t of E with divisor D = div(t), we have E =∼ [D] if and only if D ∼ D.

Proof It is easy to see that is a well-defined map, and for each λ = (U, f ) ∈ F, −1 (, λ): (U) = ρ(λ) → U × C is a well-defined bijection. Since supp D is discrete, a local defining function for D exists in a neighborhood of each point, −1 and hence is surjective. Moreover, for each point p ∈ X, [D]p = (p) has a natural well-defined 1-dimensional complex vector space structure determined by

ρ(p,ζ)+ η · ρ(p,ξ)= ρ(p,ζ + η · ξ) for each λ = (U, f ) ∈ F with p ∈ U, each pair of elements (p, ζ ), (p, ξ) ∈ λ, and η ∈ C λ = (U, f ) ∈ F p ∈ U :[ ] → C each scalar ; and for each with , λ [D]p D p is a complex linear isomorphism. Thus [D] is a complex line bundle with local { } = ∈ F = trivializations (U, (, λ)) λ=(U,f )∈F .Ifλj (Uj ,fj ) for j 0, 1, then the corresponding transition functions are the nonvanishing holomorphic functions

λ0 f0 λ1 f1 = : U0 ∩ U1 → C and = : U0 ∩ U1 → C, λ1 f1 λ0 f0 so these local trivializations determine a holomorphic line bundle structure in [D]. Moreover, for each point p ∈ U0 ∩ U1 with D(p) ≥ 0, we have

~~−1 = f0 · = λ0 ((, λ1 ) (p, f1(p))) (p) f1(p) f0(p). f1~~

Thus the section s is well deﬁned, and since λ(s) = f ∈ M(U) for each λ = ∈ F = = = (U, f ) , s is a meromorphic section with div(s) D (div(s) U div(f ) D U for each such λ). Thus (a) is proved, and part (b) follows. For the proof of part (c), suppose E is a holomorphic line bundle on X, t is a meromorphic section of E on X that does not vanish identically on any open subset of X, and D = div(t).IfE =∼ [D], then the image of t is a meromorphic section t of [D] with divisor D, and t/s is a meromorphic function with divisor div(t/s) = D − D, and hence D ∼ D. Conversely, if D ∼ D and f is a deﬁning function for D −D, then t/f is a meromorphic section of E with divisor div(t/f ) = D − (D − D) = D, and part (b) implies that E =∼ [D].

Definition 3.3.3 Given a divisor D on X, the holomorphic line bundle [D] and the defining section provided by Proposition 3.3.2 are called the holomorphic line bun- 122 3 The L2 ∂¯-Method in a Holomorphic Line Bundle dle associated to D and the associated defining section, respectively. Given a holomorphic line bundle E over X, OD(E) is the subsheaf of M(E), considered as a sheaf of O-modules, for which for every open set U ⊂ X, OD(E)(U) ⊂ M(E)(U) O ∈ M + ≥ is the (U)-submodule of sections s (E)(U) with div(s) D U 0 (in particular, for D ≥ 0, O−D(E) is a subsheaf of O(E)). For E the trivial line bundle, we let OD ≡ OD(E).

Remarks 1. Given two divisors D and D on X with associated holomorphic line bundles [D] and [D], respectively, and associated defining sections s and s, respectively, s ⊗s and s−1 are meromorphic sections of [D]⊗[D] and [D]∗, respectively, with divisors div(s ⊗ s) = D + D and div(s−1) =−D. Thus Proposition 3.3.2 gives natural isomorphisms [D]⊗[D] =∼ [D + D] and [D]∗ =∼ [−D]. 2. According to Proposition 3.3.2,ifE is a holomorphic line bundle on X and t is a meromorphic section with divisor D = div(t) (note that the product of t and any nonvanishing holomorphic function on X also has divisor D), then we may identify the pair (E, t) with ([D],s), where s is the associated defining section for D. Thus a divisor on X may be viewed as a holomorphic line bundle together with a choice of a meromorphic section. In particular, a holomorphic line bundle on a Riemann surface is isomorphic to the line bundle associated to some divisor if and only if E admits a nontrivial meromorphic section. It turns out that every holomorphic line bundle on a Riemann surface has this property (see Corollary 3.11.7 and Theorem 4.2.3). 3. Let D be a divisor on a Riemann surface X, and let s be the associated defining section of [D]. Then OD and O([D]) are isomorphic as sheaves of O-modules under the sheaf morphism f → f · s.IfE is a holomorphic line bundle on X, then OD(E) and O(E ⊗[D]) are isomorphic as sheaves of O-modules under the sheaf morphism t → t ⊗ s (see Exercise 3.3.4).

Example 3.3.4 The hyperplane bundle E → P1 (see Example 3.1.5) is equal to the line bundle associated to the divisor D with D(0) = 1 and D(q) = 0forq ∈ P1 \{0}. In fact, for any point p ∈ P1 \{0}, the meromorphic function f given by z → (z − p)/z if p ∈ C∗ and z → 1/z if p =∞has a simple pole at 0 and a simple ∼ zero at p. Thus, for Dp = 1 · p,wehaveDp = D + div(f ), and hence [Dp] = E. ∗ The tautological bundle is E =[−Dp].

Deﬁnition 3.3.5 Let D be an effective divisor and let E be a holomorphic line bundle on X.Theskyline sheaf of E along D is the sheaf QD(E) of O-modules with sections over any nonempty open set U given by QD(E)(U) = 0ifU ∩ supp D =∅ and otherwise by QD(E)(U) = O(E)p/O−D(E)p p∈U∩supp D D(p) = O(E)p mp · O(E)p p∈U∩supp D ∼ ∼ (D(p)) = CD(p) = C p∈U∩supp D , p∈U∩supp D 3.3 Divisors 123

~~U : Q → Q ⊃ and with each restriction mapping ρV D(E)(U) D(E)(V ) for U V given by the module projection~~

~~s ={sp}p∈U∩supp D → {sp}p∈V ∩supp D for supp V ∩ D = ∅, and s → 0 otherwise. We also denote QD(E) by O(E)/ O−D(E).~~

Remarks 1. It is easy to verify that QD(E) is a sheaf of O-modules (see Exer- cise 3.3.3). At the level of stalks, for each point p ∈ X, we have the natural isomor- ∼ phism QD(E)p = O(E)p/O−D(E)p. 2. A skyscraper sheaf is a skyline sheaf for a divisor D = p for some point p. 3. The sheaf QD(E) is an example of a quotient sheaf.

Observation 3.3.6 Let E be a holomorphic line bundle on X. Given an effective divisor D on X with associated holomorphic deﬁning section s, we get the holomorphic line bundle map : E → E ⊗[D] given by ξ → (ξ) = ξ ⊗ sp for p ∈ X and ξ ∈ Ep. We also get the corresponding sheaf morphism ∗ : O(E) → O(E ⊗[D]) given by ξ → ξ ⊗ s for any local holomorphic section ξ.Wealso denote D = div(s) by div(). We also get the natural induced sheaf morphism O(E ⊗[D]) → QD(E ⊗[D]). This yields the corresponding exact sequence of sheaves (see Exercise 3.3.5)

~~0 → O(E) → O(E ⊗[D]) → QD(E ⊗[D]) → 0.~~

If G is any holomorphic line bundle on X and t is any holomorphic section of G with div(t) = D, then we may identify (G, t) with ([D],s), and hence we may identify the above exact sequence with the exact sequence

0 → O(E) → O(E ⊗ G) → QD(E ⊗ G) → 0 associated to the holomorphic line bundle map given by ξ → (ξ) = ξ ⊗ t ∼p for p ∈ X and ξ ∈ Ep. In fact, any holomorphic homomorphism : E → F = E ⊗ E∗ ⊗ F of holomorphic line bundles that is not indentically zero over any nonempty open subset of X may be viewed as above. For Proposition 3.1.13 provides a corresponding holomorphic section t of E∗ ⊗ F , and may be identiﬁed with the holomorphic line bundle mapping : ξ → ξ ⊗ t(p)for p ∈ X and ξ ∈ Ep. Setting D = div() = div(t), we get the exact sequence

0 → O(E) → O(F ) → QD(F ) → 0, which we may identify with the exact sequence ∗ ∗ 0 → O(E) → O(E ⊗ E ⊗ F)→ QD(E ⊗ E ⊗ F)→ 0, given by multiplication by t, as well as with the exact sequence

~~0 → O(E) → O(E ⊗[D]) → QD(E ⊗[D]) → 0, given by multiplication by the associated deﬁning section for D. 124 3 The L2 ∂¯-Method in a Holomorphic Line Bundle~~

~~Deﬁnition 3.3.7 For X a compact Riemann surface, the degree of a divisor D on X is the integer deg D ≡ D(p) (deg D = 0ifD = 0). p∈supp D~~

Remarks 1. The mapping D → deg D gives a homomorphism deg: Div(X) → Z for X a compact Riemann surface. 2. By Proposition 2.5.7,forX a compact Riemann surface, any principal divisor has degree 0. Consequently, if D and D are two divisors with isomorphic associated line bundles, then deg D = deg D. Thus, for any holomorphic line bundle E on X that admits a nontrivial meromorphic section s, we may deﬁne the degree of E to be the integer deg E ≡ deg(div(s)), which is independent of the choice of s (and depends only on the isomorphism equivalence class of E). 3. If X is a compact Riemann surface and D is a divisor of negative degree on X, then [D] has no nontrivial global holomorphic sections, since the divisor of such a section must have nonnegative degree. In particular, if D is a nontrivial effective divisor, then D has a nontrivial holomorphic section, while [−D]=[D]∗ has no nontrivial holomorphic sections (of course, for the trivial case D ≡ 0, (X,O([D])) = (X,O([−D])) = (X,O) =∼ C).

Exercises for Sect. 3.3 3.3.1 Verify that linear equivalence of divisors is an equivalence relation. 3.3.2 Let G be the set of isomorphism equivalence classes of holomorphic line bundles on a Riemann surface X. (a) Show that G together with the product ⊗ and the identity element 1X is an Abelian group. (b) Show that the mapping in D → [D] determines an injective homomorphism into G of the quotient group of Div(X) by the subgroup of principal divisors (it will later follow that this mapping is actually an isomorphism). 3.3.3 Verify that for any divisor D and any holomorphic line bundle E, OD(E) (see Definition 3.3.3) is a subsheaf of O-modules of M(E). Verify also that for D ≥ 0, QD(E) (see Definition 3.3.5) is a sheaf of O-modules. 3.3.4 Prove that if D is a divisor on a Riemann surface X, s is a meromorphic section of [D] with div(s) = D, and E is a holomorphic line bundle on X, then O (E) and O(E ⊗[D]) are isomorphic as sheaves of O-modules under D ∼ the sheaf morphism t → t ⊗ s (in particular, OD = O([D]) under f → fs). 3.3.5 Verify that the (first) associated sequence of sheaf mappings in Observa- tion 3.3.6 is exact.

~~3.4 The ∂¯ Operator and Dolbeault Cohomology~~

Throughout this section, E and F denote holomorphic line bundles over a complex 1-manifold X. There is no canonical way to generalize the exterior derivative oper- 3.4 The ∂¯ Operator and Dolbeault Cohomology 125 ator d and the operator ∂ on scalar-valued forms to an operator on E-valued forms, although there is a very useful generalization of the canonical connection that depends on the choice of a Hermitian metric in E (see Sect. 3.7). On the other hand, the holomorphic structure in E provides a natural generalization of the operator ∂¯.

~~Deﬁnition 3.4.1 Let α be an E-valued r-form on an open set ⊂ X. (a) If α is of class C1, then ∂α¯ is the unique E-valued (r + 1)-form that satisﬁes~~

~~∂α¯ = ∂(α/s)¯ · s on U~~

for every nonvanishing holomorphic section s of E on an open subset U of . (b) If α is of class C1 and ∂α¯ = 0, then we say that α is ∂¯-closed. (c) If α = ∂β¯ for some C1 E-valued (r − 1)-form β on , then we say that α is ¯ k ∂-exact.Ifβ may be chosen to be of class C for some k ∈ Z>0 ∪{∞}, then we also say that α is Ck ∂¯-exact. If for each point in , there exists a C1 E-valued − ¯ = (r 1)-form β0 on a neighborhood U0 such that ∂β0 α U , then we say that ¯ 0 α is locally ∂-exact.Ifforsomek ∈ Z>0 ∪{∞}, each of the local forms β0 may be chosen to be of class Ck, then we also say that α is locally Ck ∂¯-exact.Itis also convenient to consider the trivial 0-form α ≡ 0tobeC∞ ∂¯-exact and to write 0 = ∂¯0.

The operator ∂¯ is well deﬁned because if α is a C1 differential form and s and t are nonvanishing holomorphic sections of E on an open set U, then s/t is a holomorphic function and hence s α s α ∂(α/t)¯ · t = ∂¯ · · t = · ∂¯ · t = ∂(α/s)¯ · s. t s t s The Dolbeault lemma (applied to a local representation of the form) implies that for ¯ k ∈ Z>0 ∪{∞}and p,q ∈ Z≥0,everyE-valued ∂-closed (p, q + 1)-form of class Ck is locally Ck ∂¯-exact. Proposition 2.5.5 has the following analogue, the proof of which is left to the reader (see Exercise 3.4.1):

Proposition 3.4.2 For any C1 E-valued r-form α on an open set ⊂ X, we have the following: (a) If α is of class C2, then ∂¯2α = 0. (b) If α is of type (p, q), then ∂α¯ is of type (p, q +1). In particular, ∂α¯ = 0 if q ≥ 1. ¯ k ¯ (c) If α is ∂-exact and of class C for some k ∈ Z>0 ∪{∞}, then α is ∂-closed and of type (p, q + 1) for some pair of integers (p, q). Moreover, there exists a C1 E-valued differential form β of type (p, q) such that α = ∂β¯ , and any such form β is actually of class Ck. (d) If (U, z) is a local holomorphic coordinate neighborhood, s is a nonvanishing holomorphic section of E on U, and a,b ∈ C1(U), then ∂a ∂a ∂(as)¯ = dz¯ · s and ∂(adz¯ + bdz)¯ · s =− dz∧ dz¯ · s. ∂z¯ ∂z¯ 126 3 The L2 ∂¯-Method in a Holomorphic Line Bundle

~~(e) If α is of type (0, 0) or (1, 0), then α is an E-valued holomorphic r-form if and only if ∂α¯ = 0.~~

~~Deﬁnition 3.4.3 For any q ∈ Z≥0,theqth Dolbeault cohomology of E over X is the quotient vector space~~

¯ ¯ q ≡ E0,q →∂ E0,q+1 E0,q−1 →∂ E0,q HDol(X, E) ker (X, E) (X, E) im (X, E) (X, E) ¯ 0,−1 (here, we set ∂ = 0onE (X, E) ≡ 0). For the trivial line bundle 1X (i.e., for q ≡ q ¯ scalar-valued (0,q)-forms), we set HDol(X) HDol(X, 1X). For each ∂-closed ∈ E0,q q form α (X, E), we call the corresponding equivalence class in HDol(X, E) the Dolbeault cohomology class of α, and we denote this class by [α] q ,by HDol(X,E) ¯ ∞ [α]Dol,orsimplyby[α]. We say that two ∂-closed C (0,q)-forms are Dolbeault cohomologous if they represent the same Dolbeault cohomology class (it follows from part (c) of Proposition 3.4.2 that two C∞ ∂¯-closed E-valued (0,q)-forms are cohomologous if and only if their difference is ∂¯-exact).

q ¯ C∞ In other words, HDol(X, E) isgivenbythe∂-closed E-valued (0,q)-forms, where we identify two such forms α and β if and only if α − β is ∂¯-exact. Clearly, 0 = O q = HDol(X, E) (X, (E)) and HDol(X, E) 0forq>1. The Dolbeault Exact Sequence Let : E → F be a holomorphic line bundle mapping that is not identically zero over any nonempty open subset of X. Equiv- alently, for the corresponding holomorphic section s of E∗ ⊗ F and the effective divisor D = div() = div(s) (s is the associated deﬁning section for D under the identiﬁcation of E∗ ⊗ F with [D]), we have F = E ⊗[D] and (ξ) = ξ ⊗ s(p) for each point p ∈ X and each vector ξ ∈ Ep (see Observation 3.3.6). We get the associated exact sequence of sheaves

~~∗ 0 → O(E) → O(F ) → QD(F ) → 0. We will now see that also induces an exact sequence of linear maps of vector spaces~~

0 → (X,O(E)) → (X,O(F )) → (X,QD(F )) → 1 → 1 → HDol(X, E) HDol(X, F ) 0. The second map (X,O(E)) → (X,O(F )) is given by t → (t) = t ⊗ s, which is clearly injective. The third map D(p) (X,O(F )) → (X,QD(F )) = O(F )p/mp O(F )p p∈supp D is given by t → {germ t + mD(p)O(F ) } . p p p p∈supp D 3.4 The ∂¯ Operator and Dolbeault Cohomology 127

The section t maps to 0 if and only if t vanishes to order at least D(p) at each point p ∈ supp D, that is, if and only if t/s is a holomorphic section of E. Thus the kernel of this third map is the image of the second map. Q → 1 For the fourth map (X, D(F )) HDol(X, E), which is also called the connecting homomorphism, suppose ξ is a nonzero element of (X,QD(F )).Wemay choose a C∞ section t of F = E ⊗[D] such that t is holomorphic on a neighborhood U of supp D and ξ ={germ t + mD(p)O(F ) } (for we may form a p p p p∈supp D representing holomorphic section in a neighborhood of supp D and then cut off this section). Thus ∂t¯ is a C∞ form of type (0, 1) with values in E ⊗[D] that vanishes on a neighborhood of supp D ={s = 0}. Thus ∂t/s¯ extends to a unique C∞ form θ of type (0, 1) with values in E on X, and we get the associated Dolbeault cohomology class η ≡[θ]Dol.Themapξ → η is well defined. For if t ∈ E(F )(X) ∈ O with t U (F )(U ) is another such representing section, θ is the extension of ¯ ∞ ∂t /s, and η ≡[θ ]Dol, then (t − t )/s extends to a unique C section u of E on X. Thus θ − θ = ∂u¯ , and hence η = η. Furthermore, if we may choose the representing section t for ξ to be holomorphic on X, then we get θ = ∂t/s¯ ≡ 0, and hence ¯ η =[θ]Dol = 0. Conversely, if ξ → η = 0, then for t and θ as above, θ = ∂v for some C∞ section v of E on X, and hence (t/s) − v is a holomorphic section of E on X \ supp D. Since t is holomorphic near supp D, Riemann’s extension theorem (Theorem 1.2.10) implies that v is holomorphic near supp D, and hence the section t ≡ t − v ⊗ s of F must be holomorphic on X. Moreover, we have t → ξ,sowe get exactness at this step. 1 → 1 =[ ] → [ ⊗ ] The fifth map HDol(X, E) HDol(X, F ) is given by η θ Dol θ s Dol. This is a well-defined linear map because for any C∞ section u of E on X,wehave ¯ ⊗ = ¯ ⊗ ∈ 1 C∞ (∂u) s ∂(u s).Givenaclassη HDol(X, E) with a representing form θ, we may apply the Dolbeault lemma (Lemma 2.5.6) and cut off to get a C∞ section v of E on X such that ∂v¯ = θ on some neighborhood of supp D. Thus, by replacing θ with θ − ∂v¯ , we see that we may choose the representing form θ to vanish near ¯ ∞ supp D.If[θ ⊗ s]Dol = 0, then we have θ ⊗ s = ∂t for some C section t of F on X that is holomorphic near supp D. Setting

ξ ≡{germ t + mD(p)O(F ) } ∈ (X,Q (F )), p p p p∈supp D D we get ξ → η. Conversely, if t is any C∞ section of F that is holomorphic near supp D and that represents an element ξ of (X,QD(F )), θ is the unique extension ¯ ∞ ¯ of ∂t/s to a C form on X, and η =[θ]Dol, then we have θ ⊗ s = ∂t on X, and hence η → [θ ⊗ s]Dol = 0. It remains to show that the map H 1 (X, E) → H 1 (X, F ) is surjective. Sup- ∞ Dol Dol pose τ is a C form of type (0, 1) with values in F on X and λ ≡[τ]Dol. By applying the Dolbeault lemma and cutting off as above, we may assume that τ vanishes on some neighborhood of supp D in X. Hence τ/s extends to a unique C∞ form θ of type (0, 1) with values in E on X, and we have η ≡[θ]Dol → λ. We may summarize the above remarks as follows:

Theorem 3.4.4 Any holomorphic line bundle mapping : E → F over X with divisor D = div() (in particular, is not identically zero over any nonempty open 128 3 The L2 ∂¯-Method in a Holomorphic Line Bundle subset of X) induces an exact sequence of linear maps

~~0 → (X,O(E)) → (X,O(F )) → (X,QD(F )) → 1 → 1 → HDol(X, E) HDol(X, F ) 0.~~

~~Deﬁnition 3.4.5 We call the sequence provided by Theorem 3.4.4 the Dolbeault exact sequence corresponding to the line bundle map.~~

~~Exercises for Sect. 3.4 3.4.1 Prove Proposition 3.4.2 (cf. Proposition 2.5.5 and Exercise 2.5.5).~~

~~3.5 Hermitian Holomorphic Line Bundles~~

Throughout this section, X denotes a complex 1-manifold. In order to consider L2 forms with values in a holomorphic line bundle, we need some notion of a pointwise norm (or length) of the values. This is provided by a Hermitian metric.

Deﬁnition 3.5.1 A Hermitian metric h in a holomorphic line bundle : E → X on X is a choice of a Hermitian inner product hp(·, ·) in the ﬁber Ep of E over each point p ∈ X such that for each pair of local C∞ sections s and t of E,the ∞ function h(s, t): p → hp(sp,tp) is of class C . For each point p ∈ X and each ∈ ≡ | |2 ≡ pair of elements u, v Ep, we also write h(u, v) hp(u, v) and u h h(u, u). If E is equipped with a Hermitian metric h, then we call (E, h) or E a Hermitian holomorphic line bundle.

Remarks 1. According to Proposition 3.11.1 in Sect. 3.11 below, every holomorphic line bundle on a Riemann surface admits a Hermitian metric. The main point is that according to Radó’s theorem (Theorem 2.11.1), every Riemann surface is second countable, after which the construction of a Hermitian metric (using a partition of unity) is standard. For now, we will avoid using Proposition 3.11.1 (until Sect. 3.11), and thereby avoid any reliance on Radó’s theorem. This will also allow us to avoid using (for now) most of the material from Sects. 2.6Ð2.9 (in fact, Sects. 3.6Ð3.9 below may be read in place of most of the material in Sects. 2.6Ð2.9). 2. If (E, h) is a Hermitian holomorphic line bundle on X and s is a nonvanishing ≡− | |2 local holomorphic section of E, then, setting ϕ log s h, we get u v h(u, v) = · · e−ϕ. s s

Thus such a nonvanishing local holomorphic section s allows one to locally represent the Hermitian metric h by the weight function ϕ (i.e., by e−ϕ ). In particular, it k follows that if u and v are sections of E that are continuous (C with k ∈ Z≥0 ∪{∞}, measurable), then the function h(u, v) is continuous (respectively, Ck, measurable). 3.5 Hermitian Holomorphic Line Bundles 129

For the trivial line bundle 1X = X × C, the Hermitian metrics are precisely the Her- −ϕ ∞ mitian metrics of the form e h0, where ϕ is a real-valued C function on X and h0 ¯ is the standard Hermitian metric given by h0((x, ζ ), (x, ξ)) = ζ · ξ for all x ∈ X and ζ,ξ ∈ C; in other words, a Hermitian metric in the trivial line bundle is equivalent to a C∞ weight function ϕ.

Example 3.5.2 Let : E → P1 be the hyperplane bundle, and let s be the holomorphic section of E considered in Example 3.1.5, which is nonvanishing except for a simple zero at 0. Then there is a unique Hermitian metric h in E with | |2 =| |2 +| |2 ∈ P1 | |2 = sz h z /(1 z ) for z ( s∞ h 1). For the section s/z extends to a holomorphic section t on P1 that is nonvanishing except for a simple zero at ∞ (in −1 1 the notation of Example 3.1.5, tz = (E,0) (z, 1) for z ∈ C). For z ∈ P and u, v ∈ Ez, we may then set ⎧ · ⎨ (u/t(z)) (v/t(z)) if z ∈ C, 1+|z|2 h(u, v) = h (u, v) ≡ z ⎩ · (u/s(z)) (v/s(z)) if z ∈ P1 \{0} 1+|1/z|2 (with 1/z = 0forz =∞). One may now verify that h is well deﬁned on the overlap C∗ and that h is a Hermitian metric with the required properties (see Exercise 3.5.1).

Lemma 3.5.3 A Kähler form on X is equivalent to a Hermitian metric in the holomorphic tangent bundle. More precisely: (a) If ω is a Kähler form on X, then there is a (unique) Hermitian metric g in (T X)1,0 such that g(u,v) =−iω(u,v)¯ for each point p ∈ X and each pair of 1,0 tangent vectors u, v ∈ (TpX) . (b) If g is a Hermitian metric in (T X)1,0, then there is a (unique) Kähler form ω on X such that ω(u,v)¯ = ig(u,v) for each point p ∈ X and each pair of tangent 1,0 vectors u, v ∈ (TpX) .

Proof Given a Kähler form ω, we may define g (asin(a))byg(u,v) = gp(u, v) ≡ 1,0 −iω(u,v)¯ for each p ∈ X and all u, v ∈ (TpX) . Given a point p ∈ X and a local holomorphic coordinate neighborhood (U, z) of p,wehaveω = G(i/2)dz∧ dz¯ on U for some positive C∞ function G, and hence for each pair of tangent vectors 1,0 u0,v0 ∈ (TpX) ,wehavegp(u0,v0) = (G(p)/2) · dz(u0)dz(v0). It follows easily ∞ that gp is a Hermitian inner product and that the function g(u,v) is of class C for each choice of local C∞ vector fields u and v of type (1, 0). Thus g is a Hermitian metric, and it is clear that g is uniquely determined. Conversely, if g is a Hermitian metric in (T X)1,0, then we may define a C∞ differential 2-form ω by

1,0 0,1 1,0 0,1 ω(u,v) = ωp(u, v) ≡ ig(u , v ) − ig(v , u ) for each point p ∈ X and each pair of complex tangent vectors u and v at p with (1, 0) parts u1,0 and v1,0, respectively, and (0, 1) parts u0,1 and v0,1, respectively. 130 3 The L2 ∂¯-Method in a Holomorphic Line Bundle

1,0 Clearly, for p ∈ X and u, v ∈ (TpX) ,wehaveω(u,v)¯ = ig(u,v). Moreover, in any local holomorphic coordinate neighborhood (U, z = x + iy),wehave 2 2 ∂ ∂ ω = ω(∂/∂z,∂/∂z)dz¯ ∧ dz¯ = idz ∧ dz¯ = 2 dx ∧ dy, ∂z g ∂z g so ω is a C∞ positive real (1, 1)-form, that is, a Kähler form. Again, uniqueness follows.

~~Deﬁnition 3.5.4 A Hermitian metric g in the holomorphic tangent bundle 1,0 = ∗ (T X) KX is called a Kähler metric on X.~~

According to Lemma 3.5.3, a Kähler metric has a corresponding Kähler form and vice versa. According to the following lemma, the proof of which is left to the reader (see Exercise 3.5.2), Hermitian metrics in holomorphic line bundles induce Hermitian metrics in dual bundles and tensor product bundles:

~~Lemma 3.5.5 Let (E, h), (E,h), and (E,h) be Hermitian holomorphic line bundles on X. Then: (a) There exists a unique Hermitian metric h∗ in the dual bundle E∗ such that~~

α(s)β(s) h∗(α, β) = h∗ (α, β) ≡ , p | |2 s h ∈ ∈ ∗ ∈ \{ } for each point p X, each pair α, β Ep, and each element s Ep 0 . ∈ | |2 = | |2 Equivalently, for each nonzero element s E, 1/s h∗ 1/ s h. (b) There exists a unique Hermitian metric h ⊗ h in the tensor product bundle E ⊗ E such that

~~ (h ⊗ h )(u ⊗ u ,v⊗ v ) = h(u, v) · h (u ,v )~~

∈ ∈ ∈ for each point p X, each pair u, v Ep, and each pair u ,v Ep. (c) Under the identification of (E∗)∗ with E, we have (h∗)∗ = h (where (h∗)∗ is the Hermitian metric obtained by applying part (a) to (E∗,h∗)); under the identification of E ⊗ E with E ⊗ E, we have h ⊗ h = h ⊗ h (where h ⊗ h and h ⊗ h are the Hermitian metrics obtained by applying part (b)); and under the identification of (E ⊗ E) ⊗ E with E ⊗ (E ⊗ E), we have (h ⊗ h) ⊗ h = h ⊗ (h ⊗ h).

Deﬁnition 3.5.6 Let (E, h), (E,h), and (E,h) be Hermitian holomorphic line bundles on X. The Hermitian metric h∗ in E∗ given by Lemma 3.5.5 is called the dual Hermitian metric. The Hermitian metric h ⊗ h in E ⊗ E given by Lemma 3.5.5 is called the tensor product Hermitian metric. Under the natural iden- tiﬁcation of (E ⊗ E ⊗ E,(h⊗ h) ⊗ h) with (E ⊗ E ⊗ E,h⊗ (h ⊗ h)),we 3.6 L2 Forms with Values in a Hermitian Holomorphic Line Bundle 131

~~ set h ⊗ h ⊗ h ≡ (h ⊗ h ) ⊗ h = h ⊗ (h ⊗ h ). For any r ∈ Z>0, we also denote the Hermitian metric h ⊗···⊗h in Er by hr .~~

If (E, h) and (E,h) are Hermitian holomorphic line bundles over a complex 1- manifold X, and s and s are nonvanishing local holomorphic sections of E and E, ≡− | |2 ≡− | |2 respectively, then, setting ϕ log s h and ϕ log s h , we get − | −1|2 = | |2 =− − | ⊗ |2 = + log s h∗ log s h ϕ and log s s h⊗h ϕ ϕ . Thus h∗ and h ⊗ h are locally represented by the weight functions −ϕ and ϕ + ϕ, respectively.

~~Exercises for Sect. 3.5 3.5.1 Verify that the function h constructed in Example 3.5.2 is a Hermitian metric with the required properties. 3.5.2 Prove Lemma 3.5.5.~~

~~3.6 L2 Forms with Values in a Hermitian Holomorphic Line Bundle~~

Throughout this section, (E, h) denotes a Hermitian holomorphic line bundle on a complex 1-manifold X. The goal of this section is the development of a suitable L2 space of differential forms. As discussed in Sect. 2.6, since the objects that we integrate on oriented real 2-dimensional manifolds are the 2-forms and we wish to pair forms of the same degree and integrate in order to get an inner product, it is natural to consider a pointwise pairing that gives a 2-form. For this, the following notation is convenient (see [De3]):

Deﬁnition 3.6.1 Given a point x ∈ X, integers m, n ∈ Z≥0, and elements α ∈ m ∗ n ∗ [ (T X)C ⊗ E]x and β ∈[ (T X)C ⊗ E]x ,weset ( α ) ∧ ( β ) ·|s|2 if m + n ≤ 2, { } ={ } ={ }≡ s s h α, β (E,h) α, β h α, β 0ifm + n>2,

~~m+n ∗ ∈ in (Tx X)C for any nonzero element s Ex .~~

In other words, {α, β} is equal to the product of the exterior product of the form parts (with the second factor conjugated) and the inner product of the line bundle parts. Observe that the above definition does not depend on the choice of s.For 2 m = n = 0, we have {α, β}h = h(α, β). Moreover, if ϕ =−log |s| for a nonvanishing local holomorphic section s of E, and α and β are differential forms with values in E and αs = α/s and βs = β/s, then { } = ∧ ¯ · −ϕ α, β h αs βs e , 132 3 The L2 ∂¯-Method in a Holomorphic Line Bundle which is the pointwise pairing that appears in the definition of the L2 inner product of scalar-valued forms (see Sect. 2.6). In particular, if α and β are continuous (Ck ∈ Z ∪{∞} { } with k ≥0 , measurable), then the scalar-valued differential form α, β h is continuous (respectively, Ck , measurable). Moreover, if d,d ∈[1, ∞], (1/d) + = d d { } 1 (1/d ) 1, and α and β are in Lloc and Lloc, respectively, then α, β h is in Lloc. Based on the above, we make the following definition (cf. Definition 2.6.1):

Definition 3.6.2 Let S be a measurable subset of X, and let α and β be E-valued measurable differential forms of type (p, q) that are defined on S. (a) For (p, q) = (1, 0), we define 1/2 ≡ { } ∈[ ∞] α L2 (S,E,h) i α, α h 0, . 1,0 S

If {α, β}h is integrable on S, then we deﬁne ≡ { } ∈ C α, β L2 (S,E,h) i α, β h . 1,0 S (b) For (p, q) = (0, 1), we deﬁne 1/2 ≡ − { } ∈[ ∞] α L2 (S,E,h) i α, α h 0, . 0,1 S

If {α, β}h is integrable on S, then we define ≡− { } ∈ C α, β L2 (S,E,h) i α, β h . 0,1 S (c) If (p, q) = (0, 0) and ω is a nonnegative measurable form of type (1, 1) defined on S, then we define 1/2 1/2 ≡ | |2 · = { } · ∈[ ∞] α L2 (S,E,ω,h) α h ω α, α h ω 0, . 0,0 S S · ={ } · If h(α, β) ω α, β h ω is integrable on S, then we define ≡ · = { } · ∈ C α, β L2 (S,E,ω,h) h(α, β) ω α, β h ω . 0,0 S S (d) If (p, q) = (1, 1) and ω is a positive measurable form of type (1, 1) defined on S, then we define α 2 1/2 α 1/2 ≡ = α α L2 (S,E,ω,h) ω , ω 1,1 S ω h S ω ω h

~~=α/ω 2 ∈[0, ∞]. L0,0(S,E,ω,h) 3.6 L2 Forms with Values in a Hermitian Holomorphic Line Bundle 133~~

~~If {α/ω,β/ω}hω is integrable on S, then we deﬁne α α ≡ β = β ∈ C α, β L2 (S,E,ω,h) , ω , . 1,1 S ω ω ω ω 2 h L0,0(S,E,ω,h)~~

(e) When there is no danger of confusion (for example, when the choice of (p, q), S, E, ω,orh is understood from the context), we will suppress parts of the notation. For example, we will often write α 2 simply as Lp,q(S,E,ω,h) α S,E,ω,h, α S,ω,h, α ω,h, α h, α E , α ω, α S,h, α L2(S,h), α S,E, α L2(S,E,ω,h), α L2(S,ω,h), α L2(S,ω,h), α L2(ω,h), α L2(S,h), α L2(S,E), α L2(h), α L2(E),or α . The analogous simpliﬁed notation will be used for α, β 2 ,forα 2 , and for α, β 2 . Lp,q(S,E,ω,h) Lp,q(S,E,h) Lp,q(S,E,h) (f) Let γ and η be measurable E-valued 1-forms deﬁned on S with (r, s) parts γ r,s and ηr,s , respectively, for each (r, s) ∈{(1, 0), (0, 1)}. Then we set 2 ≡ 1,02 + 0,12 γ 2 γ h γ h.Wealsoset L1(S,E,h)

~~1,0 1,0 0,1 0,1 γ,η 2 ≡γ ,η h +γ ,η h, L1(S,E,h)~~

~~provided each of the summands on the right-hand side is deﬁned. We also use the simpliﬁed notation analogous to that appearing in (e).~~

Definition 3.6.3 Let S be a measurable subset of X. = 2 (a) For (p, q) (1, 0) or (0, 1),thesetLp,q(S,E,h) consists of all equivalence classes of measurable E-valued differential forms α of type (p, q) on S with ∞ α L2(S,h) < , where we identify any two elements that are equal almost ev- 2 erywhere. The set L1(S,E,h)consists of all equivalence classes of measurable ∞ E-valued 1-forms α on S with α L2(S,h) < , where again, we identify any two elements that are equal almost everywhere. (b) For ω a nonnegative measurable differential form of type (1, 1) that is defined 2 on S, L0,0(S,E,ω,h) consists of all equivalence classes of measurable sec- ∞ tions α of E on S with α L2(S,ω,h) < , where we identify any two elements that are equal almost everywhere. (c) For ω a positive measurable differential form of type (1, 1) that is defined on S, 2 L1,1(S,E,ω,h)consists of all equivalence classes of E-valued measurable dif- ∞ ferential forms α of type (1, 1) on S with α L2(S,ω,h) < , where we identify any two elements that are equal almost everywhere. (d) Suppose E = 1X = X × C → X is the trivial holomorphic line bundle, h0 is the standard Hermitian metric in E, and ϕ is the real-valued C∞ function on X = −ϕ 2 ≡ for which h e h0. Then, for a suitable pair (p, q),wesetLp,q(S, ϕ) 2 2 ≡ 2 Lp,q(S,E,h), and for a suitable ω, Lp,q(S,ω,ϕ) Lp,q(S,E,ω,h)(cf. Defi- 2 · · · nition 2.6.2). The analogous notation is used for L1, and for , L2 and L2 (cf. Definition 2.6.1). 134 3 The L2 ∂¯-Method in a Holomorphic Line Bundle

Remarks 1. In Definitions 3.6.2 and 3.6.3, we could allow the Hermitian metric h to be only measurable (in an appropriate sense) and defined only on the given measurable set S, but this is not necessary for our purposes. 2. If ω is a positive measurable (1, 1)-form defined on a measurable set S ⊂ X, ∈ 2 + = ∈ 2 + = α Lp,q(S,E,ω,h) with p q 0or2orα Lp,q(S,E,h) with p q 1, ∈ 2 + = ∈ 2 + = β Lr,s (S,E,ω,h) with r s 0or2orβ Lr,s (S,E,h) with r s 1, and = = (p, q) (r, s), then we set α, β L2 0.

~~The proof of the following proposition, which is left to the reader (see Exer- cise 3.6.1) is similar to that of Proposition 2.6.3:~~

Proposition 3.6.4 Let S be a measurable subset of X. 2 · · 2 · · = (a) The pair (L1(S,E,h), , h), and the pair (Lp,q(S,E,h), , h) for (p, q) (1, 0) or (0, 1), are Hilbert spaces (where the inner product on any two equivalence classes is given by the pairing of any representatives). Moreover, we have the Hilbert space orthogonal decomposition 2 = 2 ⊕ 2 L1(S,E,h) L1,0(S,E,h) L0,1(S,E,h). (b) For ω a continuous positive differential form of type (1, 1) that is deﬁned on S = 2 · · and for (p, q) (0, 0) or (1, 1), (Lp,q(S,E,ω,h), , ω,h) is a Hilbert space (where the inner product on any two equivalence classes is given by the pairing of any representatives). Moreover, in each of the above spaces, any sequence converging to an element α admits a subsequence that converges to α pointwise almost everywhere in S.

Exercises for Sect. 3.6 3.6.1 Prove Proposition 3.6.4. 3.6.2 Prove a version of Theorem 2.6.4 for sections of a Hermitian holomorphic line bundle. In other words, prove that if X is a Riemann surface, ω is a Kähler form on X, and (E, h) is a Hermitian holomorphic line bundle on X, then for every compact set K ⊂ X, there is a constant C = C(X,K,ω,h) > 0 such that | | ≤ ∀ ∈ O max s h C s L2(X,ω,h) s (X, (E)). K

~~3.7 The Connection and Curvature in a Line Bundle~~

Throughout this section, (E, h) denotes a Hermitian holomorphic line bundle on a complex 1-manifold X.Ifα is a C∞ differential form with values in E, then the form ∂α¯ is well defined by ∂(α/s)¯ · s = ∂α¯ for every nonvanishing local holomorphic section s of E (see Definition 3.4.1). However, in order to define an analogue of the exterior derivative d for line-bundle-valued differential forms, one must add a 3.7 The Connection and Curvature in a Line Bundle 135 suitable correction term to the exterior derivative given by a local trivialization. For a Hermitian holomorphic line bundle, there is a canonical choice.

Theorem 3.7.1 There exists a unique operator D = DE = Dh = D(E,h) with the following properties: (i) If α is a C1 r-form with values in E on an open set ⊂ X, then Dα is a C0 (r + 1)-form with values in E on ; (ii) If α is a C1 E-valued differential form on an open set ⊂ X and s is any nonvanishing holomorphic section of E on an open set U ⊂ , then on U,

= · +| |−2 | |2 ∧ = · + | |2 ∧ Dα d(α/s) s s h ∂( s h) α d(α/s) s (∂ log s h) α ≡− | |2 =[ ]· (that is, for ϕ log s h, Dhα Dϕ(α/s) s); (iii) If ρ is a C1 p -form and α is a C1 E-valued r-form on an open set ⊂ X, then D(ρ ∧ α) = dρ ∧ α + (−1)pρ ∧ Dα; (iv) If α and β are C1 E-valued differential forms on an open set ⊂ X with α of { } ={ } + − r { } degree r, then d α, β h Dα,β h ( 1) α, Dβ h.

Proof Let α and β be C1 E-valued differential forms on an open set ⊂ X,let r ≡ deg α,lets be a nonvanishing holomorphic section of E on an open set U ⊂ , and let αs ≡ α/s and βs ≡ β/s on U. We ﬁrst show that D is well deﬁned by (ii). If t is another nonvanishing holo- ¯ morphic section on U, then setting αt ≡ α/t and f ≡ t/s, we get, since ∂f = 0, · +| |−2 | |2 ∧ (dαt ) t t h ∂( t h) α = −1 · +| |−2| |−2 | |2| |2 ∧ d(f αs) fs f s h ∂( f s h) α =− −2 ∧ · + · +| |−2 ¯ ∧ +| |−2 | |2 ∧ f df (αs ) fs d(αs) s f f∂f α s h ∂( s h) α = · +| |−2 | |2 ∧ (dαs) s s h ∂( s h) α.

~~Thus Dα is well deﬁned and (i) follows easily. If ρ is a C1 p-form on , then~~

∧ = ∧ · +| |−2 | |2 ∧ ∧ D(ρ α) d(ρ αs) s s h ∂( s h) ρ α = ∧ + − p ∧ · + − p ∧| |−2 | |2 ∧ dρ α ( 1) ρ (dαs) s ( 1) ρ s h ∂( s h) α = dρ ∧ α + (−1)pρ ∧ Dα on U. Thus (iii) holds. For the proof of (iv), observe that

~~{ } = [ ∧ ·| |2] d α, β h d αs βs s h = ∧ ·| |2 + − r ∧ ·| |2 + | |2 ∧ ∧ (dαs) βs s h ( 1) αs dβs s h (d s h) αs βs 136 3 The L2 ∂¯-Method in a Holomorphic Line Bundle~~

= ∧ ·| |2 + − r ∧ ·| |2 (dαs) βs s h ( 1) αs dβs s h + | |2 ∧ ∧ + − r ∧ | |2 ∧ (∂ s h) αs βs ( 1) αs (∂ s h) βs =[ +| |−2 | |2 ∧ ]∧ ·| |2 (dαs) s h (∂ s h) αs βs s h + − r ∧ [ +| |−2 | |2 ∧ ]·| |2 ( 1) αs dβs s h (∂ s h) βs s h =[ ]∧ ·| |2 + − r ∧ [ ]·| |2 (Dα)/s βs s h ( 1) αs (Dβ)/s s h ={ } + − r { } Dα,β h ( 1) α, Dβ h on U. Thus (iv) is proved.

Deﬁnition 3.7.2 The operator D provided by Theorem 3.7.1 is called the canonical connection (or the Chern connection)in(E, h). We write D = D + D = D + ∂¯, where for each C1 E-valued differential form α, Dα ≡ ∂α¯ is the (0, 1) part and Dα ≡ Dα − Dα is the (1, 0) part. We also write = = = = = = ¯ = = D DE Dh D(E,h),DDE Dh D(E,h), and ∂ D DE.

~~−ϕ For E = 1X and ϕ the weight function representing h (i.e., h = e h0, where h0 is ≡ ≡ the standard Hermitian metric), we set Dϕ Dh and Dϕ Dh (cf. Deﬁnition 2.7.2).~~

Let α be a C∞ differential form with values in E,lets be a nonvanishing local = =− | |2 holomorphic section of E,letαs α/s, and let ϕ log s h. Then we have = · + ϕ −ϕ ∧ = ϕ [ −ϕ ]· =[ ]· D α ∂αs s e ∂(e ) α e ∂ e αs s Dϕαs s and ¯ ¯ D α = ∂α =[∂αs]·s. Consequently, (D)2 = (D)2 = 0 and D2 = DD + DD. Locally, since ∂∂¯ + ∂∂¯ = 0, we have

2 ¯ ϕ −ϕ ¯ D α = (∂∂αs) · s + e ∂(e ) ∧ ∂αs · s ¯ ¯ ϕ −ϕ + (∂∂αs) · s + ∂(e ∂(e ) ∧ αs) · s ¯ ϕ −ϕ ¯ = ∂(e ∂(e )) ∧ α = ∂∂ϕ ∧ α = h ∧ α, ¯ where h = ∂∂ϕ is a well-deﬁned scalar-valued form of type (1, 1).

Deﬁnition 3.7.3 (Cf. Deﬁnition 2.8.1)Thecurvature (or curvature form)of(E, h) is the scalar-valued (1, 1)-form , which we also denote by E, h,or(E,h), = ¯ − | |2 given locally by ∂∂( log s h) for every nonvanishing local holomorphic section s of E. In other words,

~~D2 = DD + DD = ∧ (·). 3.7 The Connection and Curvature in a Line Bundle 137~~

~~¯ For E = 1X and ϕ the weight function representing h,wesetϕ ≡ h = i∂∂ϕ (cf. Deﬁnition 2.8.1).~~

Remarks 1. If g is a Kähler metric with Kähler form ω, and (U, z) is a local holo- = ∧ ¯ | |2 = morphic coordinate neighborhood, then ω G(i/2)dz dz and ∂/∂z g G/2 ∞ ¯ on U for some positive C function G. Hence g = ∂∂(− log G) = ω (see Defi- nition 2.12.1). 2. In a slight abuse of language, we say that (E, h) has nonnegative (positive, zero, nonpositive, negative) curvature if ih ≥ 0 (respectively, ih > 0, ih = 0, ih ≤ 0, ih < 0). A holomorphic line bundle E that admits a Hermitian metric of positive curvature is called positive. 3. For any nonvanishing local holomorphic section s of E and any Kähler form ω, we have 2 ih = i− | |2 =[ω(− log |s| )]·ω log s h h (see Definitions 2.8.1 and 2.14.3). Combined with the results of Sect. 2.14,this observation will later allow us to construct Hermitian metrics of positive curvature (see Sects. 3.11 and 4.2). ∗ ∗ 4. The dual Hermitian metric h in E satisfies ih∗ =−ih, and if (E ,h) is a second Hermitian holomorphic line bundle on X, then ih⊗h = ih + ih .

Example 3.7.4 Recall that the hyperplane bundle E → P1 is the holomorphic line bundle E =[D] for the divisor D with D(0) = 1 and D(q) = 0forq ∈ P1 \{0} (see Examples 3.1.5, 3.3.4, and 3.5.2). If s is the holomorphic section of E with div(s) = D considered in Example 3.1.5, and h is the Hermitian metric in E with | |2 =| |2 +| |2 | |2 = s h z /(1 z ) ( s∞ h 1) considered in Example 3.5.2, then ∧ ¯ ¯ · ∧ ¯ ∧ ¯ ¯ 2 dz dz zz dz dz dz dz h =−∂∂ log |s| = − = . h 1 +|z|2 (1 +|z|2)2 (1 +|z|2)2

~~In particular, ih > 0. In fact, 2ih is the chordal Kähler form considered in Ex- ample 2.12.3.~~

~~The following will play a role analogous to that of Proposition 2.8.2 in the L2 ∂¯-method:~~

~~ ¯ Proposition 3.7.5 (Fundamental estimate) Let D = Dh, = h, and D = ∂. Then, for every t ∈ D(E)(X), we have 2 = 2 + | |2 ≥ | |2 D t L2(X,h) D t L2(X,h) i t h i t h. X X~~

~~Remarks 1. In particular, if in the above, i ≥ 0, then~~

~~ 2 = 2 + 2 ≥ 2 D t L2(X,h) D t L2(X,h) t L2(X,i,h) t L2(X,i,h). 138 3 The L2 ∂¯-Method in a Holomorphic Line Bundle~~

~~2. The corresponding equality for inner products of C∞ compactly supported sections also holds (see Exercise 3.7.2).~~

Proof of Proposition 3.7.5 Given t ∈ D(E)(X), we may choose a ﬁnite cover- { }m ing Uν ν=1 of supp t by relatively compact open subsets of X such that for each ν, there exists a nonvanishing holomorphic section sν of E on Uν . We may also choose C∞ { }m ⊂ ≡ functions ην ν=1 such that supp ην Uν for each ν and ην 1onsuppt. ≡− | |2 ≡ ∈ C∞ Thus, setting ϕ log sν h and fν t/sν (Uν) for each ν, and applying Proposition 2.8.2, we get 2 = { } = { } Dht L2(X,h) i Dht,Dht h i Dht,Dh(ην t) h X ν X = i{[D f ]·s , [D (η f )]·s } ϕν ν ν ϕν ν ν ν h Uν − = [ ]∧ · ϕν i Dϕν fν Dϕν (ηνfν ) e Uν − − =− ¯ ∧ ¯ · ϕν + ϕν i(∂fν) ∂(ην fν) e iϕν fνηνfνe Uν Uν =− { } + ¯ · ·| |2 i D t,D (ηνt) h ην ih t h U U ν ν =− { } + | |2 i D t,D t h ih t h X X = 2 + | |2 D t L2(X,h) ih t h. X

Exercises for Sect. 3.7 3.7.1 Let (E, h) be a Hermitian holomorphic line bundle on a Riemann surface X. Prove that for every point p ∈ X, there exists a nonvanishing holomorphic | |2 = | |2 = section t of E on a neighborhood of p such that tp h 1 and (d t h)p 0. Conclude from this that for every C1 E-valued differential form α on a neighborhood of p,wehave(D(E,h)α)p = (d(α/t) · t)p. 3.7.2 Prove that in Proposition 3.7.5, the corresponding equality for inner products of C∞ compactly supported sections also holds (cf. Proposition 2.8.2). That is, prove that if (E, h) is a Hermitian holomorphic line bundle on a Riemann = = = ¯ ∈ D surface X, D Dh, h, D ∂, and s,t (E)(X), then = + D s,D t L2(X,h) D s,D t L2(X,h) h(s, t)i. X 3.7.3 Prove that if (E, h) is a Hermitian holomorphic line bundle on a compact Riemann surface X with negative curvature (i.e., −ih > 0), then (X,O(E)) = 0. 3.8 The Distributional ∂¯ Operator in a Holomorphic Line Bundle 139

~~3.8 The Distributional ∂¯ Operator in a Holomorphic Line Bundle~~

Throughout this section, X denotes a complex 1-manifold, (E, h) denotes a Hermi- tian holomorphic line bundle on X, and D = D + D = D + ∂¯ denotes the associated canonical connection. The following is a natural deﬁnition for the distributional ∂¯ operator in E:

Deﬁnition 3.8.1 Let α and β be locally integrable differential forms on X with val- ¯ ues in E. We write D α = ∂distrα = β if for every nonvanishing local holomorphic distr ¯ section s of E,wehave∂distr(α/s) = (β/s) (see Deﬁnition 2.7.1).

~~The following proposition gives an equivalent, and more intrinsic, version (cf. Proposition 2.7.3):~~

Proposition 3.8.2 Let α and β be locally integrable differential forms with values in E on an open set ⊂ X. ¯ (a) If α is of type (1, 0) and β is of type (1, 1), then ∂distrα = β if and only if { } = { } ∀ ∈ D α, D t h β,t h t (E)(). ¯ (b) If α is of type (0, 0) and β is of type (0, 1), then ∂distrα = β if and only if − { } = { } ∀ ∈ D0,1 α, D γ h β,γ h γ (E)(). ¯ (c) If α is of type (p, q) with p ≥ 2 or q ≥ 1, then ∂distrα = 0.

Remark The proof will show that if the condition in Deﬁnition 3.8.1 holds for the forms α/s and β/s for some nonvanishing holomorphic section s on a neighborhood of each point, then it holds for every choice of s.

Proof of Proposition 3.8.2 Suppose α is of type (1, 0) and β is of type (1, 1).If s is a nonvanishing holomorphic section of E on an open set U ⊂ X, αs = α/s, = ≡− | |2 C∞ βs β/s, and ϕ log s h, then given a section t of E with compact support in U, we get, for f ≡ t/s ∈ D(U) (which we may view as a function in D()), { } = · ¯ · −ϕ β,t h βs f e U and {α, D t} = {α, (D f)· s} = α ∧ D f · e−ϕ. h h ϕ h s ϕ U U This observation and Proposition 2.7.3 together imply that if the above left-hand ¯ ¯ sides are always equal, then ∂distrα = β. Conversely, if ∂distrα = β and t ∈ D(E)(), 140 3 The L2 ∂¯-Method in a Holomorphic Line Bundle C∞ { }m ≡ then, choosing ﬁnitely many functions ην ν=1 on such that ην 1on supp t and such that for each ν, the support of ην is contained in some neighborhood Uν on which there exists a nonvanishing holomorphic section of E, we get, by the above, { } = { } = { } = { } α, Dht h α, Dh(ην t) h β,ην t h β,t h. ν Uν ν Uν Thus part (a) is proved. Part (c) is obvious, and the proof of part (b) is left to the reader (see Exercise 3.8.1).

~~Theorem 2.7.4 immediately gives the following in this context:~~

Theorem 3.8.3 If p ∈{0, 1}, α is a locally integrable form of type (p, 0) with values ¯ k in E, and ∂distrα = β for an E-valued form β of class C for some k ∈ Z>0 ∪{∞}, k ¯ then α is also of class C . In particular, if ∂distrα = 0, then α is a holomorphic p-form with values in E.

~~It is sometimes convenient to consider L2 versions of Dolbeault cohomology (cf. Deﬁnition 3.4.3).~~

Z1 = 2 Deﬁnition 3.8.4 Let ω be a Kähler form on X;let L0,1(X,E,h); and let 1 1 ¯ B ⊂ Z be the subspace consisting of elements of the form ∂distrα, where α ∈ 2 2 ¯ 2 L0,0(X,E,ω,h) is an L section for which ∂distrα exists and is in L0,1(X,E,h). The ﬁrst L2 Dolbeault cohomology of (X,E,ω,h)is then the quotient vector space 1 ≡ Z1 B1 HDol,L2 (X,E,ω,h) / .

The ﬁrst C∞ L2 Dolbeault cohomology of (X,E,ω,h)is the quotient vector space 1 ≡ Z1 B1 HDol,L2∩E (X,E,ω,h) ∞/ ∞, Z1 = Z1 ∩ E0,1 B1 = Z1 ∩ ¯[ 2 ∩ E0,0 ] where ∞ (X, E) and ∞ ∂ L0,0(X,E,ω,h) (X, E) .

~~Exercises for Sect. 3.8 3.8.1 Prove part (b) of Proposition 3.8.2.~~

~~3.9 The L2 ∂¯-Method for Line-Bundle-Valued Forms of Type (1, 0)~~

~~The following is a direct generalization of Theorem 2.9.1:~~

Theorem 3.9.1 Let X be a Riemann surface, let (E, h) be a Hermitian holomorphic line bundle on X with i = ih ≥ 0, and let Z ={x ∈ X | x = 0}. 3.9 The L2 ∂¯-Method for Line-Bundle-Valued Forms of Type (1, 0) 141

Then, for every measurable E-valued (1, 1)-form β on X with β = 0 a.e. in Z and ∈ 2 \ 2 β X\Z L1,1(X Z,E,i,h) (in particular, β is in Lloc on X), there exists a form ∈ 2 α L1,0(X,E,h)such that = ¯ = ≤ Ddistrα ∂distrα β and α L2(X,E,h) β L2(X\Z,E,i,h).

~~k k In particular, if β is of class C for some k ∈ Z>0 ∪{∞}, then α is also of class C and ∂α¯ = β.~~

2 Remark The form β is in Lloc because β vanishes on Z, and hence for any compact set K in any local holomorphic coordinate neighborhood (U, z), and for any nonvanishing holomorphic section s of E on U,wehave

~~β/s 2 ∧ ¯ ≤ Aβ 2 \ < ∞, L1,1(K,(i/2)dz dz) L1,1(K Z,i,h) where 1/2 = i | |−2 ∞ A sup s h < . K (i/2)dz∧ dz¯~~

~~It is often more convenient to apply Theorem 3.9.1 in one of the following forms (cf. Corollary 2.9.2 and Corollary 2.9.3), the proofs of which are left to the reader (see Exercises 3.9.1 and 3.9.2):~~

Corollary 3.9.2 Suppose that X is a Riemann surface, (E, h) is a Hermitian holomorphic line bundle on X, ω is a Kähler form on X, ρ is a nonnegative measurable function on X with ih ≥ ρω, and Z ={x ∈ X | ρ(x) = 0}. Then, for every measurable E-valued (1, 1)-form β on X with β = 0 a.e. in Z and ∈ 2 \ = 2 \ − log ρ β X\Z L1,1(X Z,E,ρω,h) L1,1(X Z,E,ω,e h) (in particular, β is 2 ∈ 2 in Lloc on X), there exists a form α L1,0(X,E,h)such that = ¯ = ≤ Ddistrα ∂distrα β and α L2(X,h) β L2(X\Z,ρω,h).

~~k k In particular, if β is of class C for some k ∈ Z>0 ∪{∞}, then α is also of class C and ∂α¯ = β.~~

Corollary 3.9.3 Suppose that X is a Riemann surface, (E, h) is a Hermitian holomorphic line bundle on X, ω is a Kähler form on X, and C is a positive constant ≥ 2 ∈ 2 with ih C ω. Then, for every form β L1,1(X,E,ω,h), there exists a form ∈ 2 α L1,0(X,E,h)such that

~~ = ¯ = ≤ −1 Ddistrα ∂distrα β and α L2(X,h) C β L2(X,ω,h).~~

~~k k In particular, if β is of class C for some k ∈ Z>0 ∪{∞}, then α is also of class C and ∂α¯ = β. 142 3 The L2 ∂¯-Method in a Holomorphic Line Bundle~~

≡ = Proof of Theorem 3.9.1 Let N β L2(X\Z,i,h) β/(i) L2(X\Z,i,h), and let V ≡ D(E)(X). For each section t ∈ V, the Schwarz inequality and the fundamental estimate (Proposition 3.7.5)give { } = β · = t,β h h t, i t,β/(i) L2(X\Z,i,h) X X\Z i ≤ · t L2(X\Z,i,h) β/(i) L2(X\Z,i,h) ≤ · ≤ · N t L2(X,i,h) N D t L2(X,h). :[ ]→− { } It follows that the mapping ϒ D t i X t,β h is a well-deﬁned bounded V 2 complex linear functional on the subspace D of L1,0(X,E,h). For by the above inequality, ϒ is well deﬁned, and for each t ∈ V,wehave|ϒ[Dt]| ≤ · ≤ N D t L2(X,h). In particular, ϒ N. By the HahnÐBanach theorem (The- 2 orem 7.5.11), there exists a bounded linear functional ϒ on L1,0(X,E,h) such = = that ϒ DV ϒ and ϒ ϒ . Therefore, by Theorem 7.5.10, there exists a ∈ 2 =≤ · = (unique) element α L1,0(X,E,h) such that α L2(X,h) ϒ N and ϒ( ) · ∈ V ,α L2(X,h). Moreover, for each t ,wehave { } = = − { } = { } i α, D t h ϒ(D t) ( i) t,β h i β,t h. X X X = Therefore, by Proposition 3.8.2, Ddistrα β, as required. Finally, the regularity statement at the end follows from Theorem 3.8.3.

~~Exercises for Sect. 3.9 3.9.1 Prove Corollary 3.9.2. 3.9.2 Prove Corollary 3.9.3.~~

~~3.10 The L2 ∂¯-Method for Line-Bundle-Valued Forms of Type (0, 0)~~

Throughout this section, X denotes a Riemann surface (although all of the results to be stated also hold for X a complex 1-manifold), g denotes a Kähler metric on X with Kähler form ω, and (E, h) denotes a Hermitian holomorphic line bundle on X. The goal of this section is a proof that with appropriate curvature assumptions, one may solve the inhomogeneous CauchyÐRiemann equation ∂α¯ = β for β adiffer- ential form of type (0, 1) with values in E (cf. Sect. 2.12). We ﬁrst consider the following:

~~(1,q) ∗ (0,q) ∗ Proposition 3.10.1 For q = 0, 1, let q : T X ⊗ E→ T X ⊗ KX ⊗ E be the map given by~~

~~q q q : α = θ ∧ γ · s = (−1) γ ∧ θ · s → (−1) γ · (θ ⊗ s) 3.10 The L2 ∂¯-Method for Line-Bundle-Valued Forms of Type (0, 0) 143~~

~~∈ ∈ (0,q) ∗ ∈ ∈ for each x X, γ Tx X, θ (KX)x , and s Ex . Then we have the following:~~

(a) For each q = 0, 1, q is a well-deﬁned mapping for which the restriction (1,q) ∗ ⊗ → (0,q) ∗ ⊗ ⊗ Tx X Ex Tx X (KX E)x is a linear isomorphism for each point x ∈ X (in fact, 0 is equal to the identity). (b) If α is an E-valued differential form of type (1,q), then α is continuous (Ck with ∈ Z ∪{∞} p ∈[ ∞] ⊗ k ≥0 , measurable, in Lloc with p 1, ) if and only if the KX E- k valued differential form q (α) of type (0,q) is continuous (respectively, C , p = measurable, in Lloc). For q 0, α is a local holomorphic 1-form with values in E if and only if 0(α) is a local holomorphic section of KX ⊗ E. ∈ ∈ (1,q) ∗ ⊗ (c) For each point x X and all α, β Tx X Ex , we have { } · = { } = 0(α), 0(β) g∗⊗h ω i α, β h for q 0, − { } = α β · = i 1(α), 1(β) g∗⊗h , ω for q 1. ω ω h

~~2 2 (d) For each q = 0, 1, q preserves L inner products and L norms. More precisely, suppose α and β are measurable E-valued differential forms of type (1,q)on a measurable set S. Then we have~~

~~ = = 0(α) L2(S,ω,g∗⊗h) α L2(S,h) for q 0, = = 1(α) L2(S,g∗⊗h) α L2(S,ω,h) for q 1.~~

~~Moreover, for q = 0,~~

~~ = 0(α), 0(β) L2(S,ω,g∗⊗h) α, β L2(S,h),~~

~~where both of the above terms are deﬁned if either one is deﬁned. For q = 1,~~

~~ = 1(α), 1(β) L2(S,g∗⊗h) α, β L2(S,ω,h),~~

where again, both of the above terms are deﬁned if either one is deﬁned. (e) For any locally integrable E-valued differential form α of type (1, 0) on X, ¯ ¯ ∂distrα exists if and only if ∂distr[0(α)] exists. Moreover, if these objects do ¯ ¯ exist, then ∂distr[0(α)]=1[∂distrα].

~~Proof In terms of a local holomorphic coordinate z and a nonvanishing local holomorphic section s of E, the maps are given by~~

~~0 : adz· s → adz⊗ s,~~

1 : adz∧ dz¯ · s =−adz¯ ∧ dz· s → −adz¯ · (dz ⊗ s); and parts (a) and (b) follow. Fixing a point r ∈ X and choosing the local holo- | |2 = = ∧ ¯ morphic coordinate z so that dz g∗ 2atr, we get ωr (i/2)(dz dz)r .For 144 3 The L2 ∂¯-Method in a Holomorphic Line Bundle

α = adz· s and β = bdz· s at r,wehave { } · = ¯ ·| ⊗ |2 · = ¯ ·| |2 ·| |2 · 0(α), 0(β) g∗⊗h ω ab dz s g∗⊗h ω ab dz g∗ s h ω = ¯ ·| |2 · ∧ ¯ = { } ab s h idz dz i α, β h. For α = adz∧ dz¯ · s and β = bdz∧ dz¯ · s at r,wehave

− { } =− {− ¯ · ⊗ − ¯ · ⊗ } ∗ i 1(α), 1(β) g∗⊗h i adz (dz s), bdz (dz s) g ⊗h =− ¯ ¯ ∧ ·| ⊗ |2 iab(dz dz) dz s g∗⊗h = ¯ ∧ ¯ · | |2 iab(dz dz) 2 s h α β = (i/2) · (i/2) · 4ω ·|s|2 ω · s ω · s h α β = , · ω. ω ω h Parts (c) and (d) now follow. Finally, for the proof of (e), we may assume without loss of generality that X is an open subset of C.Ifα = adz· s is a locally integrable (1, 0)-form with values in E, ¯ then, according to Deﬁnition 2.7.1 and Deﬁnition 3.8.1, the existence of ∂distr0(α) ¯ and the existence of ∂distrα are each equivalent to the existence of ∂a b ≡ ∈ L1 (X). ¯ loc ∂z distr Moreover, if this function does exist, then ¯ ¯ 1(∂distrα) = 1(−bdz∧ dz¯ · s) = bdz¯ · (dz ⊗ s) = (∂distra) · (dz ⊗ s) ¯ = ∂distr[0(α)].

∞ Remarks 1. 1 is an example of a C line bundle isomorphism. 2. By the above proposition, the operator D = ∂¯ and its distributional version are preserved (in the appropriate sense) under the identiﬁcations given by 0 and 1. However, the operators D, D, and (associated to the Hermitian metrics in the holomorphic line bundles E and KX ⊗ E)arenot preserved under these identi- ﬁcations; in fact, even the resulting types do not correspond. For example, in the ∞ notation of Proposition 3.10.1,foraC (1, 0)-form α ≡ adz∈ E1,0(1C)(C) with values in the trivial line bundle over C,wehave∂α = Dα = 0(asa(2, 0)-form), = ∂a · ∈ E1,0 C but D 0(α) ∂z dz dz (KC)( ). 3. The above proposition also gives the mapping (1,q) ∗ ⊗ ∗ ⊗ → (0,q) ∗ ⊗ ⊗ ∗ ⊗ = (0,q) ∗ ⊗ ( T X) KX E ( T X) KX KX E ( T X) E for q = 0, 1. 3.10 The L2 ∂¯-Method for Line-Bundle-Valued Forms of Type (0, 0) 145

~~We have the following facts, which are equivalent to, respectively, Theo- rem 3.9.1, Corollary 3.9.2, and Corollary 3.9.3:~~

Corollary 3.10.2 Assume that i ≡ ig + ih ≥ 0 on X, and let ρ ≡ i/ω and Z ≡{x ∈ X | ρ(x) = 0}={x ∈ X | x = 0}. Then, for every measurable E-valued = ∈ 2 \ − log ρ (0, 1)-form β on X with β 0 a.e. in Z and β X\Z L0,1(X Z,E,e h), ∈ 2 there exists a section α L0,0(X,E,ω,h)such that = ¯ = ≤ Ddistrα ∂distrα β and α L2(X,E,ω,h) β L2(X\Z,E,e− log ρ h).

~~k k In particular, if β is of class C for some k ∈ Z>0 ∪{∞}, then α is also of class C and ∂α¯ = β.~~

Corollary 3.10.3 Suppose that ρ is a nonnegative measurable function on X with ig + ih ≥ ρω, and Z ={x ∈ X | ρ(x) = 0}. Then, for every mea- = ∈ surable E-valued (0, 1)-form β on X with β 0 a.e. in Z and β X\Z 2 \ − log ρ ∈ 2 L0,1(X Z,E,e h), there exists a section α L0,0(X,E,ω,h)such that = ¯ = ≤ Ddistrα ∂distrα β and α L2(X,E,ω,h) β L2(X\Z,E,e− log ρ h).

~~k k In particular, if β is of class C for some k ∈ Z>0 ∪{∞}, then α is also of class C and ∂α¯ = β.~~

2 Corollary 3.10.4 Suppose that C is a positive constant with ig + ih ≥ C ω ∈ 2 ∈ 2 on X. Then, for every β L0,1(X,E,h), there exists a section α L0,0(X,E,ω,h) such that = ¯ = ≤ −1 Ddistrα ∂distrα β and α L2(X,ω,h) C β L2(X,h).

~~k k In particular, if β is of class C for some k ∈ Z>0 ∪{∞}, then α is also of class C and ∂α¯ = β.~~

~~Corollary 3.10.4 immediately gives the following example of a vanishing theorem (see Deﬁnitions 3.4.3 and 3.8.4):~~

Corollary 3.10.5 Let X be a Riemann surface and let E be a holomorphic line bundle on X. (a) If g is a Kähler metric on X with Kähler form ω and h is a Hermitian metric in 2 E satisfying ih ≥ C ω on X for some positive constant C, then 1 ⊗ ∗ ⊗ = 1 ⊗ ∗ ⊗ = HDol,L2 (X, KX E,ω,g h) HDol,L2∩E (X, KX E,ω,g h) 0.

~~2 Equivalently, if ig + ih ≥ C ω, then 1 = 1 = HDol,L2 (X,E,ω,h) HDol,L2∩E (X,E,ω,h) 0. 146 3 The L2 ∂¯-Method in a Holomorphic Line Bundle~~

(b) Assume that X is compact. If E is positive (i.e., there exists a Hermitian metric h 1 ⊗ = in E with ih > 0 on X), then HDol(X, KX E) 0. Equivalently, if there exist a Kähler metric g on X and a Hermitian metric h in E with ig +ih > 0, 1 = then HDol(X, E) 0. Consequently, if E is positive, then, given a holomorphic 1 ⊗ r = line bundle F on X, we have HDol(X, F E ) 0 for all r 0.

~~The proof of Corollary 3.10.2 appears below, while the proofs of Corol- lary 3.10.3, Corollary 3.10.4, and Corollary 3.10.5 are left to the reader (see Ex- ercises 3.10.1Ð3.10.3).~~

ˆ ∗ ⊗ Proof of Corollary 3.10.2 Let β be the form of type (1, 1) with values in KX E corresponding to β (see the third remark following Proposition 3.10.1); that is, if, in terms of a local holomorphic coordinate z and a nonvanishing local holomorphic section s of E,wehaveβ = bdz¯ · s, then ∂ ∂ βˆ = bdz¯ ∧ dz· ⊗ s =−bdz∧ dz¯ · ⊗ s . ∂z ∂z

~~We have~~

~~ ˆ =ˆ =ˆ β L2(X\Z,ρω,g⊗h) β L2(X\Z,ω,e− log ρ g⊗h) β L2(X\Z,ω,g⊗(e− log ρ h)) = ∞ β L2(X\Z,e− log ρ h) < and~~

i ∗ ⊗ ⊗ = i ∗ + i = i = ρω. (KX E,g h) (KX,g) h Therefore, by Theorem 3.9.1, there exists a form αˆ of type (1, 0) with values in ∗ ⊗ ˆ Ck ˆ Ck ∈ KX E (with α of class if β, and therefore β,isofclass for some k Z ∪{∞} ˆ = ¯ ˆ = ˆ ˆ ≤ˆ >0 ) such that Ddistrα ∂distrα β and α L2(X,g⊗h) β L2(X\Z,ρω,g⊗h). The identiﬁcation given by (the third remark following) Proposition 3.10.1 gives the desired solution α.

Exercises for Sect. 3.10 3.10.1 Prove Corollary 3.10.3. 3.10.2 Prove Corollary 3.10.4. 3.10.3 Prove Corollary 3.10.5 (for the proof of the last statement in part (b), the reader may wish to apply Proposition 3.11.1 from Sect. 3.11).

~~3.11 Positive Curvature on an Open Riemann Surface~~

~~According to Radó’s theorem (Theorem 2.11.1), every Riemann surface is second countable. From second countability, one gets the following: 3.11 Positive Curvature on an Open Riemann Surface 147~~

~~Proposition 3.11.1 Every holomorphic line bundle : E → X on a Riemann surface X admits a Hermitian metric.~~

Proof Since X is second countable, there exist a locally ﬁnite collection of local { } C∞ { } holomorphic trivializations (Uν,(,ν )) ν∈N and a partition of unity λν with supp λν ⊂ Uν (and λν ≥ 0) for each ν ∈ N (see Sect. 9.3). The pairing h(u, v) = hp(u, v) ≡ λν(p)ν(u)ν(v) ∀p ∈ X, u, v ∈ Ep ν∈N is then a Hermitian metric in E. The veriﬁcation is left to the reader (see Exer- cise 3.11.1).

We now (begin to) address the natural question as to when a holomorphic line bundle on a Riemann surface admits a positive-curvature Hermitian metric. For a holomorphic line bundle on an open Riemann surface, the facts proved in Sect. 2.14 provide a Hermitian metric with arbitrarily large positive curvature. In other words, we have the following:

Theorem 3.11.2 Let X be an open Riemann surface, let E be a holomorphic line bundle on X, and let θ be a continuous real (1, 1)-form on X. Then E admits a Hermitian metric h with ih ≥ θ. In particular, if g is a Kähler metric on X with associated Kähler form ω, then E admits a Hermitian metric h with

~~ih ≥ ω and ig + ih ≥ ω.~~

Proof Let us ﬁx a Hermitian metric k in E and a Kähler metric g with Kähler form ω on X (provided by Proposition 3.11.1). Given a continuous real (1, 1)-form θ on X, Theorem 2.14.4 provides a C∞ (exhaustion) function ϕ such that θ ik ig ωϕ ≥ 1 + + + . ω ω ω

~~−ϕ Thus the Hermitian metric h ≡ e k satisﬁes ih =[ωϕ]·ω +ik ≥ θ, ih ≥ ω, and ig + ih ≥ ω.~~

Remarks 1. When second countability of a particular Riemann surface X is evident (for example, when X is an open subset of a compact Riemann surface), one may construct Hermitian metrics and, for X an open Riemann surface, C∞ strictly subharmonic exhaustion functions (as in Sect. 2.14) and positive-curvature Hermitian metrics (as above) without any reliance on Radó’s theorem, and hence with almost no reliance on Sects. 2.6, 2.7, 2.9Ð2.12, 3.6, and 3.8Ð3.10. 2. A holomorphic line bundle E on a compact Riemann surface need not admit a positive-curvature Hermitian metric. In Chap. 4, it will be shown that in fact, a holomorphic line bundle E on a compact Riemann surface admits a Hermitian metric of positive curvature if and only if E is the line bundle associated to a divisor of positive degree (Theorem 4.3.1). 148 3 The L2 ∂¯-Method in a Holomorphic Line Bundle

~~The above considerations, together with the facts proved in Sects. 3.9 and 3.10, give the following (cf. Theorem 2.14.10):~~

Theorem 3.11.3 Let X be an open Riemann surface, let E be a holomorphic line bundle on X, let p ∈{0, 1}, and let β be a locally square-integrable differential form of type (p, 1) with values in E on X. Then there exists a locally square-integrable ¯ differential form α of type (p, 0) with values in E such that ∂distrα = β. If β is of k k ¯ class C for some k ∈ Z>0 ∪{∞}, then α is also of class C and ∂α = β.

Proof We may ﬁx a Kähler metric g with Kähler form ω, and a Hermitian metric k in E. Applying Theorem 2.14.4, with the compact set given by K =∅and the func- ∈ 2 −ρ tion ρ chosen (with the help of Lemma 9.7.13) so large that β L0,1(X,E,e k) or 2 −ρ +| |+| | C∞ L1,1(X,E,ω,e k) and ρ>1 ik/ω ig/ω , we get a positive strictly subharmonic (exhaustion) function ϕ on X such that the Hermitian metric h = e−ϕk satisﬁes + ≥ ∞ = ig ih ω and β L2(X,E,h) < if p 0, ≥ ∞ = ih ω and β L2(X,E,ω,h) < if p 1. Corollary 3.10.4 and Corollary 3.9.3 now provide the desired E-valued differential form α.

~~In particular, we have the following vanishing theorem:~~

~~Corollary 3.11.4 If E is a holomorphic line bundle on an open Riemann surface X, 1 = then HDol(X, E) 0.~~

The construction of a holomorphic section of a holomorphic line bundle with prescribed values on a given discrete set (or prescribed Laurent series about each of the points in the discrete set) is one of the main applications of the L2 ∂¯-method. For an open Riemann surface, the above considerations give the following generalization of the Mittag-Lefﬂer theorem (Theorem 2.15.1):

Theorem 3.11.5 (Mittag-Lefﬂer theorem for a holomorphic line bundle) Let X be an open Riemann surface, let E be a holomorphic line bundle on X, let P be a discrete subset of X, and for each point p ∈ P , let Up be a neighborhood of p with Up ∩ P ={p}, let tp be a holomorphic section of E on Up \{p}, and let mp be a positive integer. Then there exists a holomorphic section s of E on X \ P such that for each point p ∈ P , s − tp extends to a holomorphic section on Up that either vanishes on a neighborhood of p or has a zero of order at least mp at p (in other words, if z is a local holomorphic coordinate and u is a nonvanishing = − n holomorphic section of E on a neighborhood of p, and s/u n∈Z an(z z(p)) = − n and tp/u n∈Z bn(z z(p)) are the corresponding Laurent series expansions = = centered at p, then amp−n bmp−n for n 1, 2, 3,...). 3.11 Positive Curvature on an Open Riemann Surface 149

Remarks 1. Recall that we also denote the value of a section v of a line bundle E at a point p by vp. The section tp in the above theorem should not be confused with such a value. 2. It follows that one may actually choose the above section s ∈ (X \ P,O(E)) so that for each point p ∈ P , s − tp extends to a holomorphic section on Up with a zero of order equal to mp at p (see Exercise 3.11.3). 3. In Exercise 3.11.2, the reader is asked to give a proof that is a direct generalization of the proof of Theorem 2.15.1. The proof provided below, which uses the language of divisors, is more efﬁcient (in particular, the proof is, in part, a demonstration of the quiet power of divisors). It should also be noted, however, that according to the Weierstrass theorem (Theorem 3.12.1), every holomorphic line bundle on an open Riemann surface is holomorphically trivial; so it turns out that the Mittag- Lefﬂer theorem (Theorem 2.15.1) and its generalization for sections of a line bundle (Theorem 3.11.5) are actually equivalent.

~~The proof of the following is left to the reader (see Exercise 3.11.4):~~

~~Corollary 3.11.6 (Cf. Corollary 2.15.2) Let X be an open Riemann surface, let E be a holomorphic line bundle on X, and let P be a discrete subset of X.~~

(a) If tp is a meromorphic section of E on a neighborhood of p and mp ∈ Z>0 for each point p ∈ P , then there exists a section s ∈ (X,M(E)) such that s is holomorphic on X \ P and ordp(s − tp) ≥ mp for every p ∈ P . (b) If tp is a holomorphic section of E on a neighborhood of p and mp ∈ Z>0 for each point p ∈ P , then there exists a holomorphic section s ∈ (X,O(E)) with ordp(s − tp) ≥ mp for every p ∈ P . (c) If ζp ∈ Ep for each point p ∈ P , then there exists a section s ∈ (X,O(E)) with s(p) = ζp for every p ∈ P .

Of course, for a given holomorphic line bundle on a compact Riemann surface X, a holomorphic section with prescribed values on a given discrete set need not exist (for example, (X,O(1X)) = O(X) = C). Much of Chap. 4 is devoted to determining when (and how many) such sections exist.

Proof of Theorem 3.11.5 We may assume without loss of generality that P = ∅; ≡ · we may let D be the (nontrivial) effective divisor given by D p∈P mp p (i.e., D(p) = mp for each point p ∈ P and D vanishes on X \ P ); we may fix a holomorphic section u ∈ (X,O([D])) with div(u) = D; and finally, we may fix a C∞ section γ of the holomorphic line bundle E ⊗[−D] on the open set Y ≡ X \ P such that γ = tp/u near each point p ∈ P . The E ⊗[−D]-valued (0, 1)-form ∂γ¯ extends to a C∞ E ⊗[−D]-valued (0, 1)- form β on X that vanishes on a neighborhood of P . Theorem 3.11.3 provides a C∞ section α of E ⊗[−D] on X with ∂α¯ = β. In particular, α is holomorphic near P , and the section s ≡ (γ − α) ⊗ u = γ ⊗ u − α ⊗ u of E ⊗[−D]⊗[D]=E is holomorphic on Y . Finally, near each point p ∈ P ,wehaves − tp =−α ⊗ u, and 150 3 The L2 ∂¯-Method in a Holomorphic Line Bundle therefore, since α ⊗u is holomorphic near p with ordp(α ⊗u) ≥ ordpu = mp, s has the required properties.

Corollary 3.11.7 (Cf. Theorem 4.2.3) Any holomorphic line bundle E on an open Riemann surface X is equal to the line bundle associated to some nontrivial effective divisor D on X; in other words, E admits a nontrivial (i.e., not everywhere zero) holomorphic section with at least one zero.

~~Further generalizations of the Mittag-Lefﬂer theorem, and also of the Runge approximation theorem, are considered in the exercises.~~

Exercises for Sect. 3.11 3.11.1 Verify that the function h constructed the proof of Proposition 3.11.1 is a Hermitian metric. 3.11.2 Give a proof of Theorem 3.11.5 that is analogous to the proof of Theo- rem 2.15.1 given in Sect. 2.15 (and that does not use divisors). 3.11.3 Prove that in Theorem 3.11.5, one may actually choose the holomorphic section s on X \ P so that for each point p ∈ P , s − tp extends to a holomorphic section on Up with a zero of order equal to mp at p. 3.11.4 Prove Corollary 3.11.6 3.11.5 The goal of this exercise is a generalization of the BehnkeÐStein theorem (Corollary 2.15.3). Let (E, h) be a Hermitian holomorphic line bundle on an open Riemann surface X. Prove that: (i) Holomorphic convexity. If S is any infinite discrete subset of X, then there exists a holomorphic section s of E on X such that |s|h is unbounded on S; (ii) Separation of points.Ifp,q ∈ X and p = q, then there exists a holomorphic section s of E on X such that s(p) = 0 and s(q) = 0; and (iii) Global sections give local coordinates. For each point p ∈ X, there exists a holomorphic section s of E on X such that sp = 0 and (d(s/t))p = 0 for each nonvanishing holomorphic section t of E on a neighborhood of p. 3.11.6 The goal of this exercise is a version of Theorem 3.11.5 that is analogous to the version of the Mittag-Leffler theorem considered in Exercise 2.15.6. Let X be an open Riemann surface, let E be a holomorphic line bundle on X,let P be a discrete subset of X,let{mp}p∈P be a collection of positive integers, let {Ui}i∈I be an open covering of X, and for each pair of indices i, j ∈ I ,let tij ∈ (Ui ∩ Uj , O(E)) with ordptij ≥ mp for each point p ∈ P ∩ Ui ∩ Uj . Assume that the family {tij } satisfies the (additive) cocycle relation

~~tik = tij + tjk on Ui ∩ Uj ∩ Uk ∀i, j, k ∈ I.~~

~~Prove that there exist sections {si}i∈I such that (i) For each index i ∈ I , si ∈ (Ui, O(E)) and ordp si ≥ mp for each point p ∈ P ∩ Ui ; and 3.12 The Weierstrass Theorem 151~~

(ii) For each pair of indices i, j ∈ I ,wehavetij = sj − si on Ui ∩ Uj . Prove also that the above implies Theorem 3.11.5. ∞ Hint. Using a C partition of unity {λν} such that each point in P lies in supp λν for exactly one index ν and such that for each ν, supp λν ⊂ Uk for ∞ ν some index kν ∈ I , one may form a C solution of the problem of the form = · {¯ } vi ν λν tkν i . In particular, the E-valued forms ∂vi agree on the overlaps and therefore determine a well-deﬁned E-valued (0, 1)-form β on X. 3.11.7 The goal of this exercise is a version of the Runge approximation theorem (Theorem 2.16.1) for sections of a holomorphic line bundle. Let X be an open Riemann surface, let (E, h) be a Hermitian holomorphic line bundle on X, and let K ⊂ X be a compact set satisfying hX(K) = K. Prove that for every holomorphic section t of E on a neighborhood of K in X and for every >0, there exists a section s ∈ (X,O(E)) such that |s − t|h < on K. 3.11.8 The goal of this exercise is a version of Theorem 2.16.3 for sections of a holomorphic line bundle. Suppose K is a compact subset of a Riemann surface X, (E, h) is a Hermitian holomorphic line bundle on X, P is a ﬁnite subset of X \ K, Y = X \ P , hY (K) = K, t is a holomorphic section of E on a neighborhood of K in X, and >0. Prove that there exists a meromorphic section s of E on X such that s is holomorphic on Y , s has a pole at each point in P , and |s − t|h <on K. 3.11.9 The goal of this exercise is a generalization of Exercise 2.16.7. Let X be an open Riemann surface, let (E, h) be a Hermitian holomorphic line bundle on X,letK ⊂ X be a compact subset with hX(K) = K,letP ⊂ X be a discrete subset with P ⊂ X \ K,lett0 be a holomorphic section of E on a neighborhood of K in X, and for each point p ∈ P ,lettp be a holomorphic section of E on Up \{p} for some neighborhood Up of p in X with Up ∩ P ={p}, and let mp be a positive integer. Prove that for every >0, there exists a holomorphic section s of E on X \ P such that |s − t0|h < on K and such that for each point p ∈ P , s − tp extends to a holomorphic section on Up that either vanishes on a neighborhood of p or has a zero of order at least mp at p.

~~3.12 The Weierstrass Theorem~~

~~The main goal of this section is the following analogue, due to Florack [Fl], of the classical theorem of Weierstrass:~~

~~Theorem 3.12.1 (Weierstrass theorem) Every holomorphic line bundle E on an open Riemann surface X is (holomorphically) trivial; that is, equivalently, every divisor on X has a solution.~~

Recall that a solution of a divisor D on a complex 1-manifold X is a meromorphic function f such that f does not vanish identically on any open set and 152 3 The L2 ∂¯-Method in a Holomorphic Line Bundle div(f ) = D (see Deﬁnition 3.3.1). To prove the theorem, we must show that E admits a nonvanishing holomorphic section. For this, the main step will be to show that E is C∞ trivial; i.e., E admits a nonvanishing C∞ section. The ∂¯-method will then provide a correction that yields a suitable holomorphic section. For a divisor D with [D]=E and a meromorphic section s of E with div(s) = D, E is C∞ trivial if and only if there exists a C∞ function ρ on X \ D−1((−∞, 0)) such that s is nonvanishing on X \ supp D and s/ρ extends to a nonvanishing C∞ section of E on X (equivalently, ρ is nonvanishing on X \ supp D and for each point p ∈ supp D, there are a local holomorphic coordinate neighborhood (U, z) with z(p) = 0 and a nonvanishing C∞ function ψ on U such that ρ = zD(p) · ψ on U \{p}). Such a function ρ is called a weak solution of the divisor D. The construction of a weak solution very closely parallels that of a strictly subharmonic exhaustion function provided in Sect. 2.14. The ﬁrst step is the following local construction:

Lemma 3.12.2 Let (V,, ≡ (0; 1)) be a local holomorphic chart in a Rie- mann surface X. Then, for each pair of distinct points p,q ∈ V , there exists a weak solution ρ of the divisor D = p − q with ρ ≡ 1 on X \ V .

Proof Given a pair of distinct points p,q ∈ V , setting ζ = (p) and ξ = (q),we may choose a holomorphic branch of the function z − ζ z → log z − ξ on the complement \[ζ,ξ] of the line segment [ζ,ξ] from ζ to ξ (see Exer- cise 3.12.2). In other words, there exists a holomorphic function g on \[ζ,ξ] such that z − ζ eg(z) = ∀z ∈ \[ζ,ξ]. z − ξ We may also choose R with |ζ |, |ξ|

~~Lemma 3.12.3 Let X be an open Riemann surface, let K ⊂ X be a compact set with hX(K) = K, and let p ∈ X \ K. Then there exists a weak solution of the divisor D = 1 · p on X with ρ ≡ 1 on K.~~

Proof By Lemma 2.13.5, there exists a locally ﬁnite (in X) sequence of relatively { }∞ \ ∈ compact open subsets Vm m=1 of X K such that p V1 and such that for each m, Vm is biholomorphic to a disk and Vm ∩ Vm+1 = ∅. Hence we may choose a se- { }∞ = ∈ ∩ ≥ quence of points pm m=0 such that p0 p and pm Vm Vm+1 for each m 1. 3.12 The Weierstrass Theorem 153

By Lemma 3.12.3, for each m = 1, 2, 3,..., there exists a weak solution ρm of the divisor Dm = pm−1 − pm with ρm ≡ 1onX \ Vm. The locally ﬁnite product ρm then yields a weak solution ρ of the divisor D = 1 · p with ρ ≡ 1onK.

~~Lemma 3.12.4 Every divisor D on an open Riemann surface X has a weak solution.~~

{ }∞ Proof By Lemma 2.13.4, there exists a sequence of compact sets Kν ν=0 such =∅ = ∞ = that K0 and X ν=0 Kν , and such that for each ν 0, 1, 2,...,wehave ◦ Kν ⊂ Kν+1 and hX(Kν) = Kν . For each point p ∈ supp D, there is a unique index ν(p) such that p ∈ Kν(p)+1 \ Kν(p). Moreover, by Lemma 3.12.3, there exists a weak solution ρp of the divisor Dp = 1 · p on X such that ρp ≡ 1onKν(p). Since the collection of points in supp D and the collection of sets {X \ Kν} are locally ﬁnite in X, the product D(p) ρp p∈supp D is locally ﬁnite and yields a weak solution of the divisor D.

Proof of Theorem 3.12.1 By Lemma 3.12.4, E admits a nonvanishing C∞ section v. The quotient β = (∂v)/v¯ is then a C∞ scalar-valued form of type (0, 1) (which, locally, may be thought of as ∂¯ log v). By Theorem 3.11.3 (or Theorem 2.14.10 or Corollary 3.11.4), there exists a C∞ function α on X such that ∂α¯ = β.The nonvanishing C∞ section s ≡ e−α · v of E is then holomorphic because ¯ − ∂v − ∂s¯ =−e α · · v + e α · ∂v¯ = 0. v

Remarks 1. Since, according to the Weierstrass theorem, a holomorphic line bundle on an open Riemann surface is holomorphically trivial, one need only really consider scalar-valued differential forms (along with weights) as in Chap. 2 in this case. 2. Suitable higher-dimensional analogues of the Mittag-Lefﬂer and Weierstrass theorems, due to Oka and Cartan, hold on Stein manifolds (see, for example, [Hö] or [KaK]). The problem of obtaining an analogue of the Mittag-Lefﬂer theorem (the additive Cousin problem or the Cousin problem I) and the problem of obtaining an analogue of the Weierstrass theorem (the multiplicative Cousin problem or the Cousin problem II) played central roles in the development of several complex variables; and these analogues are fundamental tools in the subject.

Exercises for Sect. 3.12 3.12.1 Let D be a divisor on a Riemann surface X,letE =[D], and let s be a meromorphic section of E with div(s) = D. Verify that E is C∞ trivial if and only if there exists a C∞ function ρ on X \ D−1((−∞, 0)) such that s 154 3 The L2 ∂¯-Method in a Holomorphic Line Bundle

is nonvanishing on X \ supp D and s/ρ extends to a nonvanishing C∞ section of E on X. Verify also that a C∞ function ρ on X \ D−1((−∞, 0)) has the above property if and only if ρ is nonvanishing on X \ supp D and for each point p ∈ supp D, there are a local holomorphic coordinate neighborhood (U, z) with z(p) = 0 and a nonvanishing C∞ function ψ on U such that ρ = zD(p) · ψ on U \{p}. 3.12.2 Verify that given a pair of distinct points ζ,ξ ∈ C, there exists a holomorphic branch of the function z − ζ z → log z − ξ on the complement C \[ζ,ξ] of the line segment [ζ,ξ] from ζ to ξ. In other words, there exists a holomorphic function g on C \[ζ,ξ] such that z − ζ eg(z) = ∀z ∈ C \[ζ,ξ] z − ξ (this fact was used in the proof of Lemma 3.12.2). 3.12.3 Let D be a divisor on a compact Riemann surface X. Prove that the associated holomorphic line bundle E =[D] is C∞ trivial (i.e., E admits a nonvanishing C∞ section) if and only if deg D = 0. 3.12.4 As is the case for the Mittag-Lefﬂer theorem (see Exercises 2.15.6 and 3.11.6), the Weierstrass theorem may be stated in terms of cocycles (in higher dimensions, this is known as the solution of the multiplicative Cousin problem or the Cousin problem II). Let X be a Riemann surface. (a) Suppose : E → X is a holomorphic line bundle and {fij = i/ j }i,j∈I are the transition functions associated to some holomorphic { = } line bundle atlas (Ui,i (, i)) i∈I for E. Prove that E is (holomorphically) trivial if and only if there exists a collection {fi}i∈I such that fi is a nonvanishing holomorphic function on Ui for each i ∈ I and fij = fj /f i for all i, j ∈ I . (b) Suppose {Ui}i∈I is an open covering of X and fij is a nonvanishing holomorphic function on Ui ∩ Uj for each pair of indices i, j ∈ I .As- sume that the family {fij }i,j∈I satisﬁes the (multiplicative) cocycle relation

fik = fij fjk on Ui ∩ Uj ∩ Uk ∀i, j, k ∈ I. Prove that there exists a unique holomorphic line bundle : E → X for which there exists a holomorphic line bundle atlas {(Ui,i = (, i))}i∈I with transition functions given by i/j = fij for each pair i, j ∈ I . Conclude that in particular, if X is an open Riemann surface, then there exists a collection {fi}i∈I such that fi is a nonvanishing holomorphic function on Ui for each i ∈ I and fij = fj /fi for all i, j ∈ I . Part II Further Topics

~~Chapter 4 Compact Riemann Surfaces~~

In this chapter, we consider some facts concerning holomorphic line bundles (and their holomorphic sections) on compact Riemann surfaces. We first consider conditions that guarantee the existence of holomorphic sections with prescribed values. Unlike the open Riemann surface case (in which one has Theorem 3.11.5), a holomorphic line bundle need not have the positivity required for such a section to exist. For example, the space of holomorphic functions on a compact Rie- mann surface X has dimension 1, and a negative holomorphic line bundle on X has no nontrivial global holomorphic sections. After the above considerations, we consider the fact that a holomorphic line bundle is positive if and only if its degree is positive, and we then consider finiteness of Dolbeault cohomology. The RiemannÐRoch formula is then proved (using finiteness). Finally, we consider the Serre duality theorem and the Hodge decomposition theorem, and some of their consequences. Different approaches to the above facts appear in (for example) [For] and [Ns4].

~~4.1 Existence of Holomorphic Sections on a Compact Riemann Surface~~

~~In this section, we consider consequences of the following:~~

Theorem 4.1.1 Let E be a holomorphic line bundle on a compact Riemann surface X. If there exist a Kähler metric g on X and a Hermitian metric h in E with ig + ih > 0 on X, then for every effective divisor D on X, the linear map

~~(X,O(E ⊗[D])) −→ (X,QD(E ⊗[D]))~~

~~= (X,O(E ⊗[D])/O−D(E ⊗[D]))~~

T. Napier, M. Ramachandran, An Introduction to Riemann Surfaces, Cornerstones, 157 DOI 10.1007/978-0-8176-4693-6_4, © Springer Science+Business Media, LLC 2011 158 4 Compact Riemann Surfaces D(p) = O(E ⊗[D])p (mp · O(E ⊗[D])p) p∈suppD =∼ CD(p) =∼ Cdeg D p∈suppD is surjective. In other words, given a holomorphic section tp of E ⊗[D] on a neighborhood of p in X for each point p ∈ suppD, there exists a holomorphic section s of E ⊗[D] on X such that ordp(s − tp) ≥ D(p) for each point p ∈ suppD. In particular,dim(X,O(E ⊗[D])) ≥ deg D.

~~1 = Proof By Corollary 3.10.5,wehaveHDol(X, E) 0, and therefore the Dolbeault exact sequence (see Theorem 3.4.4 and Deﬁnition 3.4.5)is~~

~~0 → (X,O(E)) → (X,O(E ⊗[D])) → (X,QD(E ⊗[D])) → 0.~~

Furthermore, any choice of a holomorphic section tp of E ⊗[D] on a neighborhood of p in X for each point p ∈ suppD determines a section ξ ∈ (X,QD(E ⊗[D])). Every section s ∈ (X,O(E ⊗[D])) with image ξ in (X,QD(E ⊗[D])) then satisﬁes ordp(s − tp) ≥ D(p) for each point p ∈ suppD.

~~Remark One may also prove a stronger version (see Exercise 4.1.4) that is analogous to the Mittag-Lefﬂer theorem (Theorem 2.15.1 and Theorem 3.11.5).~~

~~The proof of the following equivalent version of Theorem 4.1.1 is left to the reader (see Exercise 4.1.1):~~

~~Corollary 4.1.2 Let E be a positive holomorphic line bundle on a compact Riemann surface X. Then, for every effective divisor D on X, the associated linear map~~

~~(X,O(KX ⊗ E ⊗[D])) −→ (X,QD(KX ⊗ E ⊗[D])) is surjective. In particular,dim(X,O(KX ⊗ E ⊗[D])) ≥ deg D.~~

Corollary 4.1.3 Let X be a compact Riemann surface, let F be a holomorphic line bundle on X, and let E be a positive holomorphic line bundle on X. Then for every integer r 0 and for every effective divisor D on X (r does not depend on the choice of D), the linear map

~~r r (X,O(F ⊗ E ⊗[D])) −→ (X,QD(F ⊗ E ⊗[D])) is surjective. In particular,dim(X,O(F ⊗ Er ⊗[D])) ≥ deg D.~~

~~The proof is left to the reader (see Exercise 4.1.2). 4.1 Holomorphic Sections on a Compact Riemann Surface 159~~

Corollary 4.1.4 Given an effective divisor D on a compact Riemann surface X, there exists a Kähler form ω0 on X such that for every Hermitian holomorphic line bundle (E, h) on X with ih ≥ ω0, the associated linear map

∼ deg D (X,O(E)) −→ (X,QD(E)) = C is surjective (in particular,dim(X,O(E)) ≥ deg D). In fact, we may choose ω0 so that if (E, h) is a Hermitian holomorphic line bundle on X with ih ≥ ω0 and tp is a holomorphic section of E on a neighborhood of p in X for each point p ∈ suppD, then there exists a holomorphic section s of E on X such that ordp(s − tp) = D(p) for each point p ∈ suppD.

~~Proof We may ﬁx a Kähler metric g on X with associated Kähler form ω and a Hermitian metric k in [D] on X. We may now choose a Kähler form ω0 such that~~

~~ω0 + ig − ik > 0~~

(for example, we may take ω0 = Cω for C 0). Hence, if (E, h) is a Hermitian holomorphic line bundle on X with ih ≥ ω0, then the Hermitian holomorphic line bundle (E ⊗[−D],h⊗ k∗) satisﬁes

~~ig + ih⊗k∗ = ig + ih − ik ≥ ig + ω0 − ik > 0.~~

Applying Theorem 4.1.1 to the line bundle E ⊗[−D]⊗[D] =∼ E, we get surjectivity of the mapping (X,O(E)) −→ (X,QD(E)). Now, for each point p ∈ suppD, we may ﬁx a holomorphic section up of E on a neighborhood of p such that ordpup = D(p). Applying the above to the divisor 2D, we get a Kähler metric ω0 such that if (E, h) is a Hermitian holomorphic line bundle on X with ih ≥ ω0 and tp is a holomorphic section of E on a neighborhood of p for each point p ∈ suppD, then there exists a section s ∈ (X,O(E)) with ordp(s − tp − up) ≥ 2D(p) > D(p), and hence ordp(s − tp) = D(p), for each point p ∈ suppD.

Corollary 4.1.5 Let D be an effective divisor on a compact Riemann surface X, let F be a holomorphic line bundle on X, and let E be a positive holomorphic line bundle on X. Then, for r 0(depending on the choice of D, F , and E), the associated linear map

~~r r (X,O(F ⊗ E )) −→ (X,QD(F ⊗ E ))~~

r is surjective (in particular,dim(X,O(F ⊗ E )) ≥ deg D). In fact, if r 0 and tp is a holomorphic section of F ⊗ Er on a neighborhood of p in X for each point p ∈ suppD, then there exists a holomorphic section s of F ⊗ Er on X such that ordp(s − tp) = D(p) for each point p ∈ suppD.

~~The proof is left to the reader (see Exercise 4.1.3). 160 4 Compact Riemann Surfaces~~

Exercises for Sect. 4.1 4.1.1 Prove Corollary 4.1.2. 4.1.2 Prove Corollary 4.1.3. 4.1.3 Prove Corollary 4.1.5. 4.1.4 The goal of this problem is a generalization of Theorem 4.1.1 that is analogous to the Mittag-Lefﬂer theorem (Theorem 2.15.1 and Theorem 3.11.5). Let E be a holomorphic line bundle on a compact Riemann surface X.As- sume that there exist a Kähler metric g on X and a Hermitian metric h in E with ig + ih > 0onX.LetD be an effective divisor on X, and for each point p ∈ suppD,lettp be a holomorphic section of E ⊗[D] on Up \{p} for some neighborhood Up of p with Up ∩ suppD ={p}. Prove that there exists a holomorphic section s of E ⊗[D] on X \ suppD such that for each point p ∈ suppD, s − tp extends to a holomorphic section up of E ⊗[D] on Up with ordpup ≥ D(p).

~~4.2 Positive Curvature on a Compact Riemann Surface~~

According to Theorem 3.11.2, every holomorphic line bundle E on an open Rie- mann surface is positive (i.e., E admits a positive-curvature Hermitian metric). This is not true in general for a holomorphic line bundle E on a compact Riemann surface. It will be shown in Sect. 4.3 (see Theorem 4.3.1) that in fact, a holomorphic line bundle E on a compact Riemann surface is positive if and only if E is the line bundle associated to a divisor of positive degree. For now, we consider the following weaker fact:

Theorem 4.2.1 Let X be a compact Riemann surface. If E =[D] is the holomorphic line bundle associated to a nontrivial effective divisor D on X (i.e., E is a holomorphic line bundle that admits a nontrivial global holomorphic section with at least one zero), then E is positive.

~~For the proof, it will be convenient to have the following construction of a nonnegative-curvature Hermitian metric in the holomorphic line bundle associated to a divisor given by a single point:~~

Lemma 4.2.2 Let X be a Riemann surface, let p ∈ X, let D be the divisor given by D(p) = 1 and D(q) = 0 for each point q ∈ X \{p} (i.e., D = 1 · p), let s be a holomorphic section of the associated holomorphic line bundle E =[D] on X with div(s) = D, and let U be a neighborhood of p. Then there exists a Hermitian metric h in E on X such that | |2 = = \ (i) We have s h 1 and ih 0 at each point in X U, and (ii) We have ih ≥ 0 on X and ih > 0 at p. 4.2 Positive Curvature on a Compact Riemann Surface 161

Proof Clearly, we may assume that there is a holomorphic coordinate z on U with z(p) = 0. Applying Lemma 2.10.3, we get a constant b>0, a nonnegative C∞ subharmonic ϕ on the set Y ≡ X \{p}, and a relatively compact neighborhood V of p in U such that ϕ ≡ 0 on a neighborhood of X \ U and ϕ =|z|2 − log |z|2 − b | |2 = −ϕ on V . We now get a Hermitian metric h in X by requiring that s h e .More ∈ | |2 =| |2 −ϕ precisely, for any ξ E Y ,weset ξ h ξ/s e . We may extend s/z to a ∈ nonvanishing holomorphic section t of E on U, and for any ξ E V ,weset | |2 =| |2 −|z|2+b ξ h ξ/t e . It is now easy to see that h is well defined and that h has the properties (i) and (ii). =[ ] = m Proof of Theorem 4.2.1 Suppose E D , where D j=1 νj pj is a nontrivial effective divisor with νj > 0 for each j. For each j = 1,...,m, Lemma 4.2.2 provides [ ] ≥ a Hermitian metric hj in pj with ihj 0onX and ihj > 0 on some neighbor- ≡ ν1 ⊗···⊗ νm hood Vj of pj in X. The Hermitian metric k h1 hm in E then satisfies ik ≥ 0onX and ik > 0 on the neighborhood V ≡ V1 ∪···∪Vm of suppD.Ap- plying Corollary 2.14.2 in X \ suppD and cutting off, we get a C∞ function α on X such that iα > 0 on a neighborhood of X \ V . Thus, for >0 sufficiently small, − α the Hermitian metric h ≡ e k satisfies ih = iα + ik > 0.

Remark Since the Riemann surface X in the above theorem is compact, one gets second countability without appealing to Radó’s theorem. Therefore, although a fact from Sect. 2.14 (Corollary 2.14.2) was applied in the above proof, there was no real dependence on Radó’s theorem and therefore almost no dependence on Sects. 2.6, 2.7, 2.9Ð2.12, 3.6, and 3.8Ð3.10.

It follows from Theorem 4.2.1 that every compact Riemann surface admits a positive holomorphic line bundle. Moreover, we have the following consequence, which, when combined with Corollary 3.11.7, implies that holomorphic line bundles and divisors on Riemann surfaces are actually equivalent:

Theorem 4.2.3 Any holomorphic line bundle E on a compact Riemann surface X is equal to the line bundle associated to some nontrivial divisor D. In fact, given any ﬁnite set S ⊂ X and any choice of integers {mp}p∈S , there exists a meromorphic section s of E such that ordps = mp for each point p ∈ S.

Proof Suppose S ⊂ X is a nonempty ﬁnite set and mp ∈ Z for each point p ∈ S.Fix- ing a positive holomorphic line bundle (provided by Theorem 4.2.1) and applying Corollary 4.1.5, we get a holomorphic line bundle F on X such that E ⊗ F and F admit global holomorphic sections u and v, respectively, with ordpu = max(mp, 0) and ordpv = max(−mp, 0) for each point p ∈ S. The quotient s ≡ u/v is then a nontrivial meromorphic section of E,ordps = mp for each point p ∈ S, and E =[D] for the nontrivial divisor D ≡ div(s). 162 4 Compact Riemann Surfaces

Exercises for Sect. 4.2 4.2.1 Let X be a compact Riemann surface, let E be a holomorphic line bundle on X,letp ∈ X, and let m ∈ Z≥2. Prove that if E admits a Hermitian metric with ih ≥ 0, then there exists a meromorphic 1-form θ with values in E on X such that θ is holomorphic on X \{p} and θ has a pole of order m at p (cf. Theorem 2.10.1). Hint. Apply Corollary 4.1.2 to the effective divisor D = m · p, the positive holomorphic line bundle E ⊗[p], and a local holomorphic section tp of KX ⊗ E ⊗[(m + 1)p] with a zero of order 1 at p. Divide by a suitable power of a section of [p] with divisor p.

4.3 Equivalence of Positive Curvature and Positive Degree In this section we consider the more complete characterization of positivity of holomorphic line bundles on compact Riemann surfaces alluded to in Sects. 3.11 and 4.2.

~~Theorem 4.3.1 Let D be a divisor on a compact Riemann surface X, and let E =[D] be the associated holomorphic line bundle. Then E is positive if and only if deg D>0.~~

~~Proof If E is positive, then by Corollary 4.1.5,forr 0, Er admits a nontrivial holomorphic section s that has at least one zero. Setting D = div(s), we get~~

1 1 deg D = deg(rD) = deg D > 0. r r Conversely, suppose deg D>0. To show that E is positive, it sufﬁces to show that Er =[rD] admits a nontrivial holomorphic section for some positive integer r. For the divisor of the section will be nontrivial (since deg E = r deg D>0), and hence it will follow from Theorem 4.2.1 that Er admits a positive-curvature Hermi- tian metric h . The Hermitian metric h in E determined by |ξ|2 =|ξ r |2/r for each r h hr ∈ = −1 · ξ E (see Exercise 4.3.1) will then satisfy ih r ihr > 0. If D ≥ 0, then clearly, E admits a nontrivial holomorphic section. Assuming now that D is not effective, we have D = D+ − D−, where D± are (unique) nontrivial (i.e., not everywhere zero) effective divisors with disjoint supports, and E = E+ ⊗ E− with E± ≡[D±]. By Theorem 4.2.1 and Corollary 4.1.3,form a sufﬁciently large positive integer and n an arbitrary positive integer, we have

~~m+n m dim (X,O(E+ )) = dim (X,O(E+ ⊗[nD+])) ≥ deg(nD+) = n deg D+. For any effective divisor G, multiplication by a holomorphic section of [G] with divisor G yields the exact sequence~~

~~0 → (X,O([(m + n)D+ − G])) → (X,O([(m + n)D+])) ∼ deg G → (X,O([(m + n)D+])/O−G([(m + n)D+])) = C . 4.3 Equivalence of Positive Curvature and Positive Degree 163~~

~~Setting G = (m + n)D−, we get~~

~~m+n m+n dim (X,O(E )) ≥ dim (X,O(E+ )) − (m + n) deg D−~~

~~≥ n deg D+ − (m + n) deg D−~~

~~= n deg D − m deg D−.~~

~~Fixing m and choosing n 0, we get the claim.~~

~~Remark One may also prove the theorem by applying the RiemannÐRoch formula, which will be considered in Sect. 4.5 (see Exercise 4.5.1).~~

Corollary 4.3.2 If D is a divisor of positive degree on a compact Riemann surface X and E =[D], then Er admits a nontrivial holomorphic section for r 0 and no nontrivial holomorphic sections for r<0.

~~Proof By Theorem 4.3.1, E>0, and hence Er admits a nontrivial holomorphic section s for r 0 (for example, by Corollary 4.1.5). For r<0, we have deg Er = r deg D<0, so (X,O(Er )) = 0.~~

Observe also that facts considered earlier concerning positive holomorphic line bundles on compact Riemann surfaces have equivalent forms in terms of divisors of positive degree. For example, Theorem 4.3.1 and Corollary 3.10.5 together give the following vanishing theorem:

Corollary 4.3.3 If D is a divisor of positive degree on a compact Riemann sur- 1 ⊗[ ] = face X, then HDol(X, KX D ) 0. Moreover, for any holomorphic line bundle 1 ⊗[ ] = F on X, we have HDol(X, F rD ) 0 for r 0.

~~Some other examples, which follow from the results of Sect. 4.1, are summarized in the following corollary, the proof of which is left to the reader (see Exer- cise 4.3.2):~~

~~Corollary 4.3.4 Let D0 be a divisor of positive degree on a compact Riemann surface X. (a) For every effective divisor D on X, the associated linear map~~

~~(X,O(KX ⊗[D0 + D])) −→ (X,QD(KX ⊗[D0 + D]))~~

is surjective (in particular,dim(X,O(KX ⊗[D0 + D])) ≥ deg D). (b) If F is any holomorphic line bundle on X, then for every sufﬁciently large positive integer r and every effective divisor D on X (r is independent of the choice of D), the linear map

~~(X,O(F ⊗[rD0 + D])) → (X,QD(F ⊗[rD0 + D]))~~

~~is surjective (in particular,dim(X,O(F ⊗[rD0 + D])) ≥ deg D). 164 4 Compact Riemann Surfaces~~

~~(c) If F is any holomorphic line bundle on X and D is any effective divisor on X, then for every sufﬁciently large positive integer r (depending on the choice of D0, F , and D), the linear map~~

~~(X,O(F ⊗[rD0])) −→ (X,QD(F ⊗[rD0]))~~

is surjective (in particular,dim(X,O(F ⊗[rD0])) ≥ deg D). In fact, if r 0 and tp is a holomorphic section of F ⊗[rD0] on a neighborhood of p in X for each point p ∈ suppD, then there exists a holomorphic section s of F ⊗[rD0] on X such that ordp(s − tp) = D(p) for each point p ∈ suppD.

Exercises for Sect. 4.3 4.3.1 As in the proof of Theorem 4.3.1,letE be a holomorphic line bundle, let r be a positive integer, and let hr be a Hermitian metric in the r-fold tensor r ∈ r = ⊗···⊗ ∈ r | |2 = power E . For each element ξ E,letξ ξ ξ E , and let ξ h |ξ r |2/r. Verify that this determines a Hermitian metric h in E with = hr h (1/r)hr . 4.3.2 Prove Corollary 4.3.4. 4.3.3 Prove that if E is a holomorphic line bundle on a compact Riemann surface X − 1 = and deg E deg KX > 0, then HDol(X, E) 0.

~~4.4 A Finiteness Theorem~~

~~The vector space of holomorphic sections of a holomorphic line bundle on a compact Riemann surface is always ﬁnite-dimensional. In fact, we have the following:~~

~~Theorem 4.4.1 (Finiteness theorem) If E is a holomorphic line bundle on a compact Riemann surface X, then~~

~~O = 0 ≤ + ∞ dim (X, (E)) dim HDol(X, E) max(deg E 1, 0)< and 1 ∞ dim HDol(X, E) < .~~

~~Proof Fix a point p ∈ X and let G be the divisor with supp G ={p} and G(p) = 1 (i.e., G = 1 · p). For each positive integer r, we have the linear map~~

~~∼ r−1 (X,O(E)) → (X,Q(r−1)G(E)) = C (={0} if r = 1).~~

~~If dim (X,O(E)) ≥ r, then the kernel contains a nontrivial section. On the other hand, such a section determines an effective divisor D such that E =[D] and~~

~~deg E = deg D ≥ r − 1. 4.5 The RiemannÐRoch Formula 165~~

~~Thus we get the desired bound on dim (X,O(E)). Multiplication by a holomorphic section of [rG] with divisor rG gives the exact sequence of sheaves~~

~~0 → O(E) → O(E ⊗[rG]) → QrG(E ⊗[rG]) → 0 and the corresponding Dolbeault exact sequence~~

0 → (X,O(E)) → (X,O(E ⊗[rG])) → (X,QrG(E ⊗[rG])) → 1 → 1 ⊗[ ] → HDol(X, E) HDol(X, E rG ) 0. By Theorem 4.2.1 and Corollary 3.10.5 (or by Corollary 4.3.3), for r 0, we have 1 ⊗[ ] = HDol(X, E rG ) 0, and hence the map Cr =∼ Q ⊗[ ] → 1 (X, rG(E rG )) HDol(X, E) 1 ∞ is surjective. Thus dim HDol(X, E) < .

~~Deﬁnition 4.4.2 For a compact Riemann surface X,thegenus of X is the nonnegative integer ≡ 1 = 1 genus(X) dim HDol(X, 1X) dim HDol(X).~~

~~Exercises for Sect. 4.4 4.4.1 Prove that if E is a holomorphic line bundle on a compact Riemann surface X, 1 ≤ − + then dim HDol(X, E) max(deg KX deg E 1, 0) (cf. Exercise 4.3.3).~~

~~4.5 The RiemannÐRoch Formula~~

In this section, we consider the following formula for the dimension of the space of holomorphic sections of a holomorphic line bundle on a compact Riemann surface (this formula is much more precise than the estimates for the dimension considered in previous sections):

~~Theorem 4.5.1 (RiemannÐRoch formula) If E is a holomorphic line bundle on a compact Riemann surface X, then O − 1 = − + dim (X, (E)) dim HDol(X, E) 1 genus(X) deg E.~~

Remark For a divisor D on a compact Riemann surface X,weset q ≡ q [ ] = h (D) dim HDol(X, D ) for q 0, 1. Setting g ≡ genus(X) and d ≡ deg D, we get the following equivalent form for the RiemannÐRoch formula, which is easier to remember:

~~h0(D) − h1(D) = 1 − g + d. 166 4 Compact Riemann Surfaces~~

~~The proof given here, which is standard, is similar to that in [For]. The main point is the following induction step:~~

Lemma 4.5.2 Suppose A and B are divisors on a compact Riemann surface X such that B = A + p for some point p ∈ X and such that the Riemann–Roch formula holds for A or for B. Then the Riemann–Roch formula holds both for A and for B.

~~Proof Multiplication by a holomorphic section of the line bundle [p] with divisor p = B − A yields the Dolbeault exact sequence~~

~~0 → (X,O([A])) → (X,O([B])) → (X,QB−A([B])) → 1 [ ] → 1 [ ] → HDol(X, A ) HDol(X, B ) 0.~~

~~Setting~~

~~V ≡ im[(X,O([B])) → (X,QB−A([B]))] = [ Q [ ] → 1 [ ] ]⊂ Q [ ] =∼ C ker (X, B−A( B )) HDol(X, A ) (X, B−A( B )) and~~

~~W ≡ [ Q [ ] → 1 [ ] ] im (X, B−A( B )) HDol(X, A ) = [ 1 [ ] → 1 [ ] ] ker HDol(X, A ) HDol(X, B ) , we get dim V + dim W = dim (X,QB−A(B)) = 1, and therefore~~

~~h0(B) − h1(B) − deg B = (h0(A) + dim V) − (h1(A) − dim W) − deg A − 1 = h0(A) − h1(A) − deg A.~~

~~The lemma now follows.~~

~~Proof of Theorem 4.5.1 For D = 0, we have~~

~~h0(D) − h1(D) = 1 − g = 1 − g + d.~~

~~For an arbitrary divisor D on X,wehaveD = B − A for effective divisors~~

~~A = p1 +···+pm and B = q1 +···+qn.~~

Setting B0 = 0 and Bν = q1 +···+qν for ν = 1,...,n, and applying Lemma 4.5.2 inductively on ν, we get the RiemannÐRoch formula for the divisor B = Bn. Setting D0 = B and Dμ = B − p1 −···−pμ for μ = 1,...,m, and applying Lemma 4.5.2 inductively on μ, we get the RiemannÐRoch formula for the divisor D = Dm. 4.6 Statement of the Serre Duality Theorem 167

Exercises for Sect. 4.5 4.5.1 Using the RiemannÐRoch formula in place of Theorem 4.3.1,giveadiffer- ent proof of Corollary 4.3.2 (i.e., a different proof that for any divisor D of positive degree on a compact Riemann surface X, [rD] admits a nontrivial holomorphic section for r 0). Use this to then give a different proof of Theorem 4.3.1.

~~4.6 Statement of the Serre Duality Theorem~~

Throughout this section, E denotes a holomorphic line bundle on a Riemann sur- ∗ ∼ ∗ ∼ face X. Recall that we have the natural identifications E ⊗ E = E ⊗ E = 1X determined by s ⊗ ξ ↔ ξ ⊗ s → ξ · s = ξ(s) for ξ ∈ E∗ and s ∈ E. Combining this with the wedge product, one gets a natural pointwise bilinear pairing sE(·, ·) of forms with values in E and forms with values in E∗ that is related to the point- {· ·} wise pairing , h of forms with values in E associated to a Hermitian metric h as {· ·} described in Definition 3.6.1 ( , h is not, however, bilinear; it is linear in the first entry and conjugate linear in the second). One then also gets an associated integra- 2 tion pairing SE(·, ·) that is related to the L pairing. The precise definitions are as follows:

~~Deﬁnition 4.6.1 Let m, n ∈ Z≥0. (a) For each point x ∈ X, the bilinear pairing~~

~~m ∗ n ∗ ∗ m+n ∗ sE(·, ·):[ (T X)C ⊗ E]x ×[ (T X)C ⊗ E ]x →[ (T X)C]x~~

~~is given by~~

sE(α, β) ≡ α0 ∧ β0 · s · ξ = α0 ∧ β0 · ξ(s) m ∗ n ∗ ∗ for all α = α0 ⊗ s ∈[ (T X)C ⊗ E]x and β = β0 ⊗ ξ ∈[ (T X)C ⊗ E ]x ∈ ∈ ∗ with s Ex and ξ Ex . (b) If m ∈{0, 1, 2} and α and β are measurable differential forms of degree m and 2 − m, respectively, with values in E and E∗, respectively, on X, then we deﬁne

~~SE(α, β) ≡ sE(α, β), X provided the above integral is deﬁned.~~

Remarks 1. It is easy to see that sE(·, ·) is well defined. In fact, under the identi- ∗ ∼ fication of E ⊗ E = 1X,wehavesE(α, β) = α ∧ β as in Definition 3.1.14 (see Exercise 4.6.1). · · · · {· ·} 2. The pairings sE( , ) and SE( , ) are bilinear, unlike the pairings , h and · · , L2 , which are linear in the first entry, but conjugate linear in the second. 168 4 Compact Riemann Surfaces

mn mn 3. It is easy to see that sE(α, β) = (−1) sE∗ (β, α) and SE(α, β) = (−1) · SE∗ (β, α) for all (suitable) α and β of degree m and n, respectively. Furthermore, if α is of type (p, q) and β is of type (r, s), then sE(α, β) is of type (p + r, q + s) (the value being 0 if p + r>1orq + s>1). 4. Suppose α and β are differential forms with values in E and E∗, respectively. k It is easy to verify that if α and β are continuous (C , measurable), then sE(α, β) is continuous (respectively, Ck, measurable).

Lemma 4.6.2 For C∞ differential forms α and β with values in E and E∗, respectively, on X: ¯ ¯ deg α ¯ (a) We have ∂(sE(α, β)) = sE(∂α,β)+ (−1) sE(α, ∂β). (b) If α is of type (p, 0) and β is of type (1 − p,0) with p ∈{0, 1}, and α or β has ¯ p ¯ compact support, then SE(∂α,β)+ (−1) SE(α, ∂β)= 0. In particular, if α is a C∞ section of E with compact support and β is a holomorphic 1-form with ∗ ∗ values in E (i.e., β ∈ (X,O(KX ⊗ E )) under the usual identiﬁcation given ¯ by Proposition 3.10.1), then SE(∂α,β)= 0.

Proof The local representation of sE(·, ·) gives Part (a) (see Exercise 4.6.2). For the proof of (b), observe that sE(α, β) is a form of type (1, 0) and therefore ¯ d(sE(α, β)) = ∂(sE(α, β)). Thus, integrating the expressions in (a) and applying Stokes’ theorem, we get (b).

~~The above lemma allows us to make the following deﬁnition:~~

~~Deﬁnition 4.6.3 For X a compact Riemann surface, the Serre pairing~~

~~· · : 1 × O ⊗ ∗ → C SE( , ) HDol(X, E) (X, (KX E ))~~

0,1 is given by SE([θ]Dol,s)≡ SE(θ, s) for each form θ ∈ E (E)(X) and each sec- ∗ tion s ∈ (X,O(KX ⊗ E )).TheSerre mapping is the linear mapping E : O ⊗ ∗ →[ 1 ]∗ ιS (X, (KX E )) HDol(X, E) given by s → SE(·,s).

Remarks 1. Lemma 4.6.2 implies that the Serre pairing is well deﬁned. 2. In a slight abuse of notation, we denote both the general integration pairing in Deﬁnition 4.6.1 and the Serre pairing by SE(·, ·). 3. Here, we are identifying each global holomorphic section s ∈ (X, ∗ ∗ O(KX ⊗ E )) of KX ⊗ E with the corresponding holomorphic 1-form with values in E∗. 4. If α = adz¯ · t and β = bdz· ξ for a local holomorphic coordinate z and for local holomorphic sections t and ξ of E and E∗, respectively, then

~~sE(α, β) = ab · dz¯ ∧ dz· ξ(t)=−ab · dz∧ dz¯ · ξ(t). 4.6 Statement of the Serre Duality Theorem 169~~

~~One goal of the remainder of this chapter is the following:~~

Theorem 4.6.4 (Serre duality theorem) If E is a holomorphic line bundle on a E : → · compact Riemann surface X, then the Serre mapping ιS s SE( ,s) gives an isomorphism ∼ E : O ⊗ ∗ −→= [ 1 ]∗ ιS (X, (KX E )) HDol(X, E) .

~~Remark Equivalently, we have an isomorphism~~

~~⊗ ∗ =∼ KX E : O = O ⊗ ⊗ ∗ ∗ −→ [ 1 ⊗ ∗ ]∗ ιS (X, (E)) (X, (KX (KX E ) )) HDol(X, KX E ) .~~

Letting K and D be divisors on X with KX =[K] and E =[D] (i.e., K is the divisor of some meromorphic 1-form η and we identify (KX,η) with ([K],s), where s is the associated deﬁning section for K, and the analogous identiﬁcation holds for E and D), we also get the equivalent forms ∼ [D] : O [ − ] −→= [ 1 [ ] ]∗ ιS (X, ( K D )) HDol(X, D ) , ∼ [K−D] : O [ ] −→= [ 1 [ − ] ]∗ ιS (X, ( D )) HDol(X, K D ) .

1X : →[ 1 ]∗ Corollary 4.6.5 For X compact, the Serre map ιS (X) HDol(X) , which is given by 1X [ ] = ∧ ιS (α)( θ Dol) θ α X ∀ ∈ = O [ ] ∈ 1 α (X) (X, (KX)), θ Dol HDol(X), is an isomorphism. In particular, genus(X) = dim (X).

The Serre duality theorem and the RiemannÐRoch formula have many applications. We consider a few of the immediate applications here. We ﬁrst observe that Serre duality gives the following equivalent version of the RiemannÐRoch formula:

~~Theorem 4.6.6 For any holomorphic line bundle E on a compact Riemann surface X, we have ∗ dim (X,O(E)) − dim (X,O(KX ⊗ E )) = 1 − genus(X) + deg E.~~

~~Equivalently, for any divisor D of degree d on a compact Riemann surface X of genus g and for any divisor K with [K]=KX, we have~~

~~h0(D) − h0(K − D) = 1 − g + d.~~

~~Corollary 4.6.7 For any compact Riemann surface X of genus g, we have~~

~~deg KX = 2g − 2. 170 4 Compact Riemann Surfaces~~

~~Proof The above RiemannÐRoch formula (applied to E = KX) and Corollary 4.6.5 together imply that g = dim (X,O(KX)) = dim (X,O) + 1 − g + deg KX = 1 + 1 − g + deg KX.~~

~~The claim now follows.~~

~~The following application will play a role in the proof of the AbelÐJacobi embedding theorem (Theorem 5.22.2):~~

Theorem 4.6.8 Up to biholomorphism, the only compact Riemann surface of genus 0 is P1. In fact, if X is a compact Riemann surface that is not biholomorphic to P1, then for every point p ∈ X, there is a holomorphic 1-form θ on X with θp = 0.

Proof The veriﬁcation that genus(P1) = 0 (i.e., that P1 has no nontrivial holomorphic 1-forms) is left to the reader (cf. Exercise 2.5.4). Suppose now that X is a compact Riemann surface of genus g, and p ∈ X is a point at which every holomorphic 1-form on X vanishes. Fixing a divisor K with [K]=KX, letting D = 1 · p, and ﬁxing a section s ∈ (X,O([D])) with div(s) = D, we get an isomorphism =∼ (X,O([K])) −→ (X,O([K − D])) given by θ → θ/s. The RiemannÐRoch formula (in the form of Theorem 4.6.6) then gives h0(D) − g = h0(D) − h0(K) = h0(D) − h0(K − D) = 1 − g + deg D = 1 − g + 1, and hence h0(D) = 2. Thus there exists a section t ∈ (X,O([D])) \ C · s, and f ≡ t/s: X → P1 is a nonconstant meromorphic function that is holomorphic except for a simple pole at p (t(p)= 0, since otherwise, f would be a nonconstant holomorphic function on X). Therefore, by Proposition 2.5.7, f is a biholomorphism of X onto P1, and the claim follows.

Remarks 1. The canonical line bundle of a compact Riemann surface X of genus 1 is trivial (see Exercise 4.6.3). 2. It follows from the above theorem that if θ1,...,θg is a basis for (X) for a compact Riemann surface X of genus g>0, then we get a Kähler form g ¯ ω ≡ iθj ∧ θj . j=1

We recall that if a holomorphic line bundle E is the line bundle associated to a nontrivial effective divisor, then E is positive. However, as the following example shows, the converse is false (see [Ns4]):

Observation 4.6.9 Any compact Riemann surface X of genus g>1 admits a positive holomorphic line bundle with no nontrivial holomorphic sections. To see this, 4.6 Statement of the Serre Duality Theorem 171 we ﬁrst observe that there exists an effective divisor D such that deg D = g and h0(D) = 1. For there exists a nontrivial holomorphic 1-form on X and therefore a point p1 ∈ X at which the form does not vanish. Thus the vector space V ≡{ ∈ O | = } 1 θ (X, (KX)) θp1 0 is of dimension g − 1 > 0. Given points p1,...,pm−1 ∈ X and subspaces

(X,O(KX)) = V0 ⊃ V1 ⊃ V2 ⊃···⊃Vm−1, ≤ V ={ ∈ O | = ··· = = } where 1 0, so E ≡[F ] is positive by Theorem 4.3.1. However, letting P be the divisor with P(p) = 1 and P(x) = 0forx = p (i.e., P = 1 · p), and choosing a section t ∈ (X,O([P ])) with div(t) = P , we get the injective linear mapping (X,O(E)) → (X,O([D])) given by multiplication by t. Since s(p) = 0, s cannot be in the image of this mapping. Therefore dim (X,O(E)) < h0(D) = 1, and hence E has no nontrivial holomorphic sections.

~~Remark The above observation is false in genus 1 (see Exercise 4.6.3).~~

We record here for later use the observation that Lemma 4.6.2 leads one to the ¯ following equivalent characterization of the operator ∂distr, which is completely intrinsic, unlike Deﬁnition 3.8.1, which involves the choice of a local holomorphic coordinate and a nonvanishing local holomorphic section, and unlike the characterization in Proposition 3.8.2, which involves the choice of a Hermitian metric:

Lemma 4.6.10 Let p ∈{0, 1} and let α and β be locally integrable differential forms of type (p, 0) and type (p, 1), respectively, with values in E on X. Then ¯ ∂distrα = β if and only if ¯ p ¯ p SE(α, ∂γ)+ (−1) SE(β, γ ) = 0 (i.e., SE∗ (∂γ,α)− (−1) SE∗ (γ, β) = 0) for every form γ ∈ D1−p,0(E∗)(X). In particular, α is holomorphic if and only if

~~¯ ¯ 1−p,0 ∗ SE(α, ∂γ)= 0 (i.e., SE∗ (∂γ,α)= 0) ∀γ ∈ D (E )(X). 172 4 Compact Riemann Surfaces~~

Proof For p = 1, suppose α = adz⊗ s and β = bdz∧ dz¯ ⊗ s on U for some local holomorphic coordinate neighborhood (U, z) and some nonvanishing holomorphic section s of E on U. For every function f ∈ D(U), we may extend f/s by 0 to a C∞ differential form γ of type (0, 0) with values in E∗ on X and compact support in U. Thus

SE(β, γ ) = bf d z ∧ dz¯ U and ¯ ¯ ∂f SE(α, ∂γ)= a · dz∧ ∂f = a · · dz∧ dz.¯ U U ∂z¯ ¯ Deﬁnition 3.8.1 and the above together now imply that if SE(α, ∂τ)−SE(β, τ) = 0 0,0 ∗ ¯ for every form τ ∈ D (E )(X), then ∂distrα = β. ¯ ∞ Conversely, suppose ∂distrα = β and γ is a C compactly supported differential ∗ ∞ form of type (0, 0) with values in E on X. We may form a ﬁnite collection of C { }m ≡ compactly supported functions ην ν=1 on X such that ην 1onsuppγ and such that for each ν, the support of ην is contained in some local holomorphic coordinate neighborhood (Uν,zν ) on which there is a nonvanishing holomorphic section sν ∞ ∞ of E. Setting aν ≡ α/(dzν ⊗ sν) ∈ C (Uν ), bν ≡ β/(dzν ∧ dz¯ν ⊗ sν) ∈ C (Uν ), and fν ≡ ην γ · sν ∈ D(Uν) for each ν, we get ∂f S (α, ∂γ)¯ = S (α, ∂(η¯ γ))= a · ν · dz ∧ dz¯ E E ν ν ∂z¯ ν ν ν ν Uν ν = bνfν · dzν ∧ dz¯ν = SE(β, ηνγ)= SE(β, γ ). ν Uν ν

~~The proof of the case p = 0 is left to the reader (see Exercise 4.6.4).~~

Remark Recall that we have the natural identification of an E-valued (1, 1)-form with a KX ⊗ E-valued (0, 1)-form provided by Proposition 3.10.1. In the present context, this identification can lead to some confusion, since the above pairings are preserved only up to sign. To see this, let (U, z) be local a holomorphic coordinate neighborhood, let s be a nonvanishing holomorphic section of E on U, and let α = dz ⊗ s and β = dz¯ ⊗ s−1.Viewingα as a (1, 0)-form with values in E and ∗ β as a (0, 1)-form with values in E , we get sE(α, β) = dz ∧ dz¯.Viewingα as −1 a (0, 0)-form with values in KX ⊗ E and β =−dz ∧ dz¯ ⊗ ((∂/∂z) ⊗ s ) as a ∗ ⊗ ∗ =− ∧ ¯ =− (1, 1)-form with values in KX E , we get sKX⊗E(α, β) dz dz sE(α, β). Fortunately, confusion can be avoided by specifying the reference line bundle (E or KX ⊗ E) in the subscript for s(·, ·) and S(·, ·). Moreover, for our purposes, the sign is not crucial. The behavior of the pairings with respect to these identifications is summarized in Proposition 4.6.11 below (cf. Proposition 3.10.1), the proof of which is left to the reader (see Exercise 4.6.5). The full proposition is not used in this book, so the reader may wish to omit it. 4.6 Statement of the Serre Duality Theorem 173

~~Proposition 4.6.11 For q ∈{0, 1} and for any holomorphic line bundle F on X, let F : (1,q) ∗ ⊗ → (0,q) ∗ ⊗ ⊗ q T X F T X KX F denote the mapping given by~~

~~F : = ∧ · = − q ∧ · → − q · ⊗ q α θ γ s ( 1) γ θ s ( 1) γ (θ s)~~

∈ ∈ ∈ (0,q) ∗ ∈ for each x X, θ (KX)x , γ Tx X, and s Ex . In particular, we have K∗ ⊗F X : (1,q) ∗ ⊗ ∗ ⊗ → (0,q) ∗ ⊗ ⊗ ∗ ⊗ = q T X (KX F) T X KX KX F (0,q)T ∗X ⊗ F . (a) For each q = 0, 1 and x ∈ X, we have

~~ K∗ ⊗E∗ − = − 1−q E X 1 sE(α, β) ( 1) sKX⊗E q (α), 1−q (β)~~

~~(1,q) ∗ (0,1−q) ∗ ∗ if α ∈ ( T X ⊗ E)x and β ∈ ( T X ⊗ E )x ; and we have~~

~~ ∗ ⊗ − ∗ 1−q KX E 1 E s (α, β) = (−1) s ∗ ⊗ (α), (β) E KX E q 1−q~~

(0,q) ∗ (1,1−q) ∗ ∗ if α ∈ ( T X ⊗ E)x and β ∈ ( T X ⊗ E )x . (b) If q ∈{0, 1} and α is a measurable differential form of type (1,q) with values in E and β is a measurable differential form of type (0, 1 − q) with values in E∗ on X, then

K∗ ⊗E∗ − = − 1−q E X 1 ; SE(α, β) ( 1) SKX⊗E q (α), 1−q (β) here the left-hand side is defined if and only if the right-hand side is defined. If q ∈{0, 1} and α is a measurable differential form of type (0,q) with values in E and β is a measurable differential form of type (1, 1 − q) with values in E∗ on X, then ∗ 1−q K ⊗E −1 E∗ S (α, β) = (−1) S ∗ ⊗ X (α), (β) ; E KX E q 1−q here again, the left-hand side is defined if and only if the right-hand side is defined.

Exercises for Sect. 4.6 Exercises 4.6.3 and 4.6.6Ð4.6.10 require the Serre duality theorem (and its consequences), so the reader may wish to postpone consideration of these exercises until after consideration of the proof in Sect. 4.8. 4.6.1 Verify that in the notation of Definition 4.6.1, under the identification of ∗ ∼ E ⊗ E = 1X,wehavesE(α, β) = α ∧ β as in Definition 3.1.14. 4.6.2 Prove part (a) of Lemma 4.6.2. 4.6.3 Let X be a compact Riemann surface of genus g. (a) Prove that if g = 1, then KX is trivial and dim (X,O(E)) = deg E for every holomorphic line bundle E of positive degree on X. O m = − − (b) Prove that if g>1, then dim (X, (KX )) (2m 1)(g 1) for every m ∈ Z>1. 174 4 Compact Riemann Surfaces

4.6.4 Prove Lemma 4.6.10 for p = 0. 4.6.5 Prove Proposition 4.6.11. Some facts concerning hyperelliptic Riemann surfaces and Weier- strass gaps are developed in Exercises 4.6.6Ð4.6.10 below (see also Exerci- ses 5.20.3Ð5.20.6 and 5.20.8 and, for example, [FarK] for further discussion). 4.6.6 A compact Riemann surface is called hyperelliptic if it admits a divisor D such that deg D = 2 and h0(D) ≥ 2. (a) Prove that a compact Riemann surface X is hyperelliptic if and only if there exists a meromorphic function on X with exactly two poles (counting multiplicity). (b) Prove that every compact Riemann surface of genus g ≤ 2 is hyperelliptic. 4.6.7 Given a point p in a compact Riemann surface X, a positive integer ν is called a gap (or a Weierstrass gap)atp if there does not exist a function f ∈ M(X) that is holomorphic on X \{p} and that has a pole of order ν at p. A positive integer that is not a gap is called a nongap (or a Weierstrass nongap)atp. The goal of this exercise is the following:

Theorem (Weierstrass gap theorem) At any point p in a compact Riemann surface X of genus g, (i) The sum of any two nongaps is a nongap; and (ii) There are exactly g gaps, and for g>0, the gaps are given by positive integers ν1,...,νg (the gap sequence) satisfying 1 = ν1 <ν2 < ···< νg < 2g.

Prove the Weierstrass gap theorem following the outline below. (a) Prove that the sum of any two nongaps at p is a nongap. (b) Prove that for g = 0, there are no gaps at p. (c) Let g>0 and let ν ∈ Z>0. Fixing a holomorphic section s of [p] with div(s) = p, we get the Dolbeault exact sequence (Sect. 3.4)

α 0 → (X,O([(ν − 1)p])) → (X,O([νp])) → (X,Qp([νp])) → 1 [ − ] → 1 [ ] → HDol(X, (ν 1)p ) HDol(X, νp ) 0, where α is the map given by u → u ⊗ s. Prove that ν is a gap if and only if α is surjective, and conclude from this that

~~h0((ν − 1)p) if ν is a gap, h0(νp) = h0((ν − 1)p) + 1ifν is nongap.~~

~~(d) Let g>0 and let K be a divisor with [K]=KX. Use the RiemannÐRoch formula in the form of Theorem 4.6.6 to prove that~~

~~h0(K − νp) + 1ifν is a gap, h0(K − νp + p) = h0(K − νp) if ν is nongap. 4.6 Statement of the Serre Duality Theorem 175~~

(e) Let g>0. Prove that 1 is a gap and that each gap is at most 2g − 1. (f) Let g>0. Prove that if ν is a gap, then there exists a holomorphic section t of [K − (ν − 1)p] with t(p)= 0. Forming such a section and multiplying by sν−1 for each gap ν (for s as in (c)), show that one gets linearly independent holomorphic 1-forms θ1,...,θr , where r is the number of gaps. Conclude that r ≤ g. (g) Let g>0. Given a nontrivial holomorphic 1-form θ with a zero of order ν − 1atp (θp = 0ifν = 1), show that ν must be a gap. Show also that by subtracting a suitable multiple of one of the holomorphic 1-forms θ1,...,θr produced in part (f), one gets a holomorphic 1-form that is either trivial or that has a zero of order ≥ ν at p. Using this observation, complete the proof of the theorem. 4.6.8 Let α1 <α2 < ··· <αg = 2g be the nongaps in (1, 2g] at a point p in a compact Riemann surface of genus g>1 (as in Exercise 4.6.7). (a) Prove that αj + αg−j ≥ 2g for j = 1,...,g− 1. Hint.Wehavej + 1 nongaps αg−j <α1 + αg−j <α2 + αg−j < ···< αj + αg−j for each j. g ≥ + (b) Prove that j=1 αj g(g 1), and prove that equality holds if and only if αj + αg−j= 2g for j = 1,...,g− 1. g = + = (c) Prove that j=1 αj g(g 1) if and only if (α1,α2,...,αg) (2, 4,...,2g). Hint.Ifαj + αg−j = 2g for j = 1,...,g − 1, then in particular, we have α1 + αg−1 = 2g = αg. Hence, if g>2, then the nongap α1 + αg−2 lies in (αg−2,αg). Apply this observation and induction to get α1 + αj−1 = αj for j = 2,...,g. 4.6.9 Let X be a compact Riemann surface of genus g>0. (a) For each pointp ∈ X,theweight (or Weierstrass weight)atp is the in- ≡ g − = ··· teger w(p) j=1(νj j), where (ν1,...,νg) (with 1 ν1 < < νg < 2g) is the gap sequence (as in Exercise 4.6.7). Prove that for each point p ∈ X,wehave0≤ w(p) ≤ g(g−1)/2. Prove also that w(p) = 0if and only if the gap sequence is (1, 2, 3,...,g)(this gap sequence is called the generic gap sequence) and w(p) = g(g − 1)/2 if and only if the gap sequence is (1, 3, 5,...,2g − 1) (this gap sequence is called the hyperelliptic gap sequence, and we then say that p is a hyperelliptic point). Hint. Use Exercise 4.6.8. (b) Given a basis θ = (θ1,...,θg) for (X,O(KX)), the associated Wron- ∈ O g(g+1)/2 skian is the section W(θ) (X, (KX )) given by

~~f1 ... fg~~

~~(∂/∂z)(f1) . . . (∂/∂z)(fg) + W(θ)= · g(g 1)/2 . . . (dz) . . . g−1 g−1 (∂/∂z) (f1)...(∂/∂z) (fg)~~

~~on each local holomorphic coordinate neighborhood (U, z), where fj ≡ θj /dz for j = 1,...,g. Prove that the Wronskian W(θ) is well 176 4 Compact Riemann Surfaces~~

deﬁned by the above (i.e., W(θ) does not depend on the choice of the = local holomorphic coordinate z). Prove also that if θ (θ ,...,θg) is 1 ∗ another basis for (X,O(KX)), then there is a constant ζ ∈ C such that W(θ) = ζ · W(θ) on X. (c) A point p ∈ X is called a Weierstrass point if w(p) > 0. Prove that if θ = (θ1,...,θg) is a basis for (X,O(KX)), then the set of Weierstrass points in X is equal to the set of zeros of the Wronskian W(θ) and the order of the zero of W(θ) at each Weierstrass point p is equal to w(p). Hint.Forg = 1, one may prove this by using the triviality of the canonical line bundle (Exercise 4.6.3). Assume that g>1, let p ∈ X, and let (ν1,...,νg) be the gap sequence at p. Choose the basis θ = (θ1,...,θg) as in part (f) of Exercise 4.6.7. Fix a local holomorphic coordinate neighborhood (U, z) with z(p) = 0, and replace each θj with its product with ν −1 a suitable constant, so that fj ≡ θj /dz = z j + higher-order terms for each j = 1,...,g.Givenak-tuple of holomorphic functions h = (h1,...,hk) on U with k>1, set

~~h1 ... hk~~

~~(∂/∂z)(h1) . . . (∂/∂z)(hk) W(h)≡ ∈ O . . . (U). . . . k−1 k−1 (∂/∂z) (h1)...(∂/∂z) (hk)~~

~~− ··· = μj 1 Show that for arbitrary positive integers μ1 < <μk,forhj z = = k − for j 1,...,k, and for m j=1(μj j),wehave~~

~~m W(h)= (μj − μi)z . 1≤i~~

ν −1 ν −1 Conclude from this that for f = (f1,...,fg) and h = (z 1 ,...,z g ), we have ordp(W(f ) − W(h))>w(p), and then that W(f) has a zero of order w(p)at p. = − + (d) Prove that p∈X w(p) (g 1)g(g 1) (in particular, every compact Riemann surface of genus g>1 has a Weierstrass point). (e) Assume that g>1 and let m be the number of Weierstrass points. Prove the following: (i) We have 2g + 2 ≤ m ≤ (g − 1)g(g + 1). (ii) We have m = 2g + 2 if and only if X has at least 2g + 2 hyperelliptic points. In particular, m = 2g + 2 if and only if every Weierstrass point in X is a hyperelliptic point. (iii) If m = 2g + 2, then X is a hyperelliptic Riemann surface (see Exer- cise 4.6.6 for the deﬁnition of a hyperelliptic Riemann surface, and see Exercise 5.20.4 for the converse). (f) Prove that if g = 2, then every Weierstrass point in X is a hyperelliptic point (in particular, this gives a proof of the fact that a compact Riemann 4.7 Statement of the ∂¯-Hodge Decomposition Theorem 177

surface of genus 2 is hyperelliptic that differs slightly from the proof suggested by the hint in Exercise 4.6.6). 4.6.10 Prove that every nontrivial automorphism of a compact Riemann surface X of genus g>1 has at most 2g + 2 fixed points. Hint.Given ∈ Aut(X) \{IdX}, fix a point p ∈ X that is not a fixed point of . Show that there exist a nongap α at p and a function f ∈ M(X) such that 1 <α≤ g + 1, f is holomorphic on X \{p}, and f has a pole of order α at p. Show that the meromorphic function f − f ◦ has exactly 2α zeros (counting multiplicities).

~~4.7 Statement of the ∂¯-Hodge Decomposition Theorem~~

Throughout this section, E denotes a holomorphic line bundle on a Riemann sur- 2 face X. One may relate the pairing SE(·, ·) and the L inner product associated to a Hermitian metric h in E by considering the unique operator ∗¯ satisfying ∗¯ = 2 SE(α, β) α, β L2 for suitable L E-valued 1-forms α and β. To see this, we ﬁrst observe that a Hermitian metric induces a natural conjugate linear isomorphism of E and E∗.

~~Deﬁnition 4.7.1 For a Hermitian metric h in E with dual metric h∗ in E∗,the associated ﬂat operator is the C∞ conjugate linear isomorphism ∗ = (E,h) = h = E : E → E given by~~

|s|2s−1 if s = 0, s → (s) = s ≡ h(·,s)= h 0ifs = 0. The associated sharp operator is the (C∞ conjugate linear) inverse mapping = = = ≡ −1 : ∗ → # #(E,h) #h #E (E,h) E E, for which we write ξ → ξ #.

Remarks 1. It is easy to verify that and # are C∞ conjugate linear isomorphisms (see Exercise 4.7.1); that is, their restrictions to each ﬁber are conjugate linear isomorphisms, and s and ξ # are C∞ sections for all local holomorphic (or C∞) sections s and ξ of E and E∗, respectively. It follows that if s is a section of E which Ck ∈ Z ∪{∞} s ∈[ ∞] is continuous ( with k ≥0 , measurable, Lloc with s 1, ), then s is Ck s continuous (respectively, , measurable, Lloc). 2. One can form analogous operators for a general ﬁnite-dimensional inner product space (of arbitrary dimension). 3. This notation originated in the classical tensor calculus, in which the operators and # denoted, respectively, the lowering and raising of indices (in analogy with the corresponding musical notation). 178 4 Compact Riemann Surfaces

Proposition 4.7.2 For any Hermitian metric h in E with corresponding dual Her- mitian metric h∗ in E∗: ∗ ∗ ∗ ∗ (a) Under the identiﬁcation ((E ) ,(h ) ) = (E, h), we have E∗ = #E. # = · = ∗ ∈ ∈ ∗ ∈ (b) We have h(s, ξ ) ξ s h (ξ, s ) for all s Ex and ξ Ex with x X. ∗ (c) We have h (s ,t ) = h(t, s) = h(s, t) for all s,t ∈ Ex with x ∈ X. In particular, |s |h∗ =|s|h for all s ∈ E. # # = ∗ = ∗ ∈ ∗ ∈ (d) We have h(ξ ,ζ ) h (ζ, ξ) h (ξ, ζ ) for all ξ,ζ Ex with x X. In par- # ∗ ticular, |ξ |h =|ξ|h∗ for all ξ ∈ E .

∈ ∈ ∈ ∗ = = Proof Suppose x X, s Ex , and ξ Ex with s 0 and ξ 0. Then h(s, ξ #) = (#ξ)· s = ξ · s and ∗ = · ∗ −1 | |2 −1 = · | |2 ·| −1|2 = · h (ξ, s ) (ξ s)h (s , s hs ) (ξ s) s h s h∗ ξ s. Thus (b) is proved. Part (a) follows, since (b) gives

~~h(s, ξ ) = s · ξ = ξ · s = h(s, ξ #).~~

~~For s and t as in (c), we have ∗ h (s,t) = s · t = h(t, #s) = h(t, s).~~

~~Part (d) now follows from parts (a) and (c).~~

∗ ∗ Deﬁnition 4.7.3 The Hodge star operator ∗: (T X)C → (T X)C is the mapping given by ∗: (α + β)¯ → −iα + iβ¯ ∈ (1,0) ∗ ∈ for all α, β Tx X with x X.Theconjugate Hodge star operator is the ∗ ∗ mapping ∗:¯ (T X)C → (T X)C given by

~~∗:¯ γ → ∗γ =∗γ ∗ for all γ ∈ (T X)C.~~

~~In other words, for a local holomorphic coordinate z,wehave∗ dz =−idz and ∗ dz¯ = ∗ dz = idz¯. The proof of the following is left to the reader (see Ex- ercise 4.7.2).~~

Proposition 4.7.4 The Hodge star operator ∗ and conjugate Hodge star operator ∗¯ on X have the following properties: ∈ ∗ ∗¯ ∗ (a) For each point r X, the restrictions of and to (Tr X)C are linear and conjugate linear, respectively, isomorphisms. If α is a continuous (Ck with ∈ Z ∪{∞} s ∈[ ∞] ∗ ∗¯ k ≥0 , measurable, Lloc with s 1, )1-form, then α and α are Ck s continuous (respectively, , measurable, Lloc). 4.7 Statement of the ∂¯-Hodge Decomposition Theorem 179

(b) We have ∗∗ = ∗¯∗=−¯ Id. (c) For (p, q) = (1, 0) or (0, 1), ∗ maps (p,q)T ∗X onto (p,q)T ∗X and ∗¯ maps (p,q)T ∗X onto (q,p)T ∗X. (d) Each of the operators ∗ and ∗¯ maps the real cotangent bundle T ∗X onto itself and ∗=∗¯ on T ∗X. = ∈ ∈ (p,q) ∗ (e) If (p, q) (1, 0) or (0, 1), x X, and α, β Tx X, then

~~α ∧ ∗¯β = α ∧ ∗β = (−1)q iα ∧ β.¯~~

In particular, α ∧ ∗¯α ≥ 0. (f) Let α and β be measurable forms of degree 1 on a measurable set S ⊂ X, and let ϕ be a measurable real-valued function on S. Then (see Deﬁnition 2.6.1) 2 = ∧ ∗¯ · −ϕ =∗ 2 =∗¯ 2 α L2(S,ϕ) α α e α L2(S,ϕ) α L2(S,ϕ). S ∈ 2 ∗ ∗ ∈ 2 Moreover, if α, β L1(S, ϕ) (which is equivalent to α, β L1(S, ϕ) as well ∗¯ ∗¯ ∈ 2 as to α, β L1(S, ϕ)), then =∗ ∗ =∗¯ ∗¯ = ∧ ∗¯ · −ϕ α, β L2(S,ϕ) α, β L2(S,ϕ) β, α L2(S,ϕ) α β e . S

~~For forms with values in a holomorphic line bundle, one may combine the conjugate Hodge star operator with the ﬂat and sharp operators as follows:~~

Definition 4.7.5 Let h be a Hermitian metric in E,let = (E,h) be the associated ∗ flat operator, let # = #(E,h) be the associated sharp operator, and let ∗:¯ (T X)C → ∗ (T X)C be the conjugate Hodge star operator. Then the associated conjugate Hodge star-flat operator

~~∗¯ = ∗¯ = ∗¯ = ∗¯ : 1 ∗ ⊗ → 1 ∗ ⊗ ∗ (E,h) h E (T X)C E (T X)C E and the associated conjugate Hodge star-sharp operator~~

~~∗¯# = ∗¯ # = ∗¯ # = ∗¯# : 1 ∗ ⊗ ∗ → 1 ∗ ⊗ (E,h) h E (T X)C E (T X)C E are the mappings given by~~

~~∗¯ (α ⊗ s) = (∗¯α) ⊗ s and ∗¯#(α ⊗ ξ)= (∗¯α) ⊗ ξ #~~

~~∈ 1 ∗ = ∗ ∈ ∈ ∗ ∈ for α (Tx X)C (Tx X)C, s Ex , and ξ Ex with x X.~~

Remark The conjugate linearity of ∗¯, , and # make the above operators well deﬁned (and conjugate linear on the ﬁbers), since for ζ ∈ C\{0} and for α, s, and ξ as above, we have (∗¯(ζ α)) ⊗ (s/ζ ) = ζ¯ · (1/ζ)¯ · (∗¯α) ⊗ s = (∗¯α) ⊗ s 180 4 Compact Riemann Surfaces and (∗¯(ζ α)) ⊗ (ξ/ζ )# = ζ¯ · (1/ζ)¯ · (∗¯α) ⊗ ξ # = (∗¯α) ⊗ ξ #.

∗¯ ∗¯# · · The basic properties of E and E and the relationship between the pairings SE( , ) · · and , L2(X,E,h) are contained in the following proposition, the proof of which is left to the reader (see Exercise 4.7.3).

Proposition 4.7.6 Given a Hermitian metric h in E with the associated Hermitian metric h∗ in E∗, and given a pair of indices (p, q) = (1, 0) or (0, 1), the corresponding conjugate Hodge star-ﬂat and conjugate Hodge star-sharp operators have the following properties: ∗¯ = ∗¯# ∗¯# ◦ ∗¯ =− ∗¯ ◦ ∗¯# =− ∗¯ (p,q) ∗ ⊗ = (a) We have E∗ E, E E Id, E E Id, E( T X E) (q,p) ∗ ⊗ ∗ ∗¯# (p,q) ∗ ⊗ ∗ = (q,p) ∗ ⊗ T X E , and E( T X E ) T X E. If α is an Ck Z ∪{∞} s E-valued 1-form that is continuous ( with ≥0 , measurable, Lloc with ∈[ ∞] ∗¯ Ck s s 1, ), then Eα is continuous (respectively, , measurable, Lloc). (p,q) ∗ (b) If x ∈ X and α, β ∈ ( T X ⊗ E)x , then ∗¯ = − q { } = − p {∗¯ ∗¯ } sE(α, Eβ) ( 1) i α, β h ( 1) i Eβ, Eα h∗ . ∗¯ ≥ ∈ ∈ (p,q) ∗ ⊗ ∈ In particular, sE(α, Eα) 0. If x X, α ( T X E)x , and β (q,p) ∗ ∗ ( T X ⊗ E )x , then =− − q { ∗¯ # } sE(α, β) ( 1) i α, Eβ h. (c) If α is a measurable differential form of degree 1 with values in E on X, then (see Deﬁnition 3.6.2)

~~ 2 = ∗¯ =∗¯ 2 α L2(X,E,h) SE(α, Eα) Eα L2(X,E∗,h∗). (d) The conjugate Hodge star-ﬂat and conjugate Hodge star-sharp operators determine conjugate linear isomorphisms~~

∗¯ : 2 → 2 ∗ ∗ E Lp,q(X,E,h) Lq,p(X, E ,h ), ∗¯ : 2 → 2 ∗ ∗ E L1(X,E,h) L1(X, E ,h ), ∗¯# =− ∗¯ −1 : 2 ∗ ∗ → 2 E ( E) Lq,p(X, E ,h ) Lp,q(X,E,h), ∗¯# =− ∗¯ −1 : 2 ∗ ∗ → 2 E ( E) L1(X, E ,h ) L1(X,E,h), satisfying ∗¯ ∗¯ = ∈ 2 (i) Eα, Eβ L2 β,α L2 , for all α, β L1(X,E,h); ∗¯# ∗¯# = ∈ 2 ∗ ∗ (ii) Eα, Eβ L2 β,α L2 , for all α, β L1(X, E ,h ); = ∗¯ ∈ 2 (iii) α, β L2 SE(α, Eβ), for all α, β L1(X,E,h) (in particular, the ∗¯ right-hand side SE(α, Eβ) is deﬁned); 4.8 Proof of Serre Duality and ∂¯-Hodge Decomposition 181

~~ ∗¯# =− ∈ 2 ∈ 2 ∗ ∗ (iv) α, Eβ L2 SE(α, β) for all α L1(X,E,h) and β L1(X, E ,h ) (in particular, SE(α, β) is deﬁned).~~

~~Another goal of the remainder of this chapter is the following:~~

Theorem 4.7.7 (∂¯-Hodge decomposition theorem) For every Hermitian holomorphic line bundle (E, h) on a compact Riemann surface X, we have the orthogonal decomposition E0,1 = ∗¯# O ⊗ ∗ ⊕ ¯ E0,0 (E)(X) E((X, (KX E ))) ∂( (E)(X)) 2 · · E0,1 with respect to the restriction of the L inner product , L2 (X,E,h) to (E)(X) ∗ 0,1 ∗ (here, we treat elements of (X,O(KX ⊗ E )) as (1, 0)-forms with values in E ).

∗¯ ∗¯# Remark There is also a natural deﬁnition for E and E on forms of degree 0 and 2, but it depends on the choice of a Kähler metric. The operators map a 0-form to a 2-form and a 2-form to a 0-form. One may also deﬁne such an operator on a complex manifold of higher dimension n. The operator then maps a (p, q)-form to an (n − p,n − q)-form (see, for example, [De3]or[MKo]). In our case, we have n = 1 and (p, q) = (1, 0) or (0, 1),so(n − p,n − q) = (q, p). One should not view the conjugate Hodge star operator as simply switching the indices (p, q) in general, only in the special (1-dimensional) case considered in this book.

Exercises for Sect. 4.7 4.7.1 Verify that for any Hermitian holomorphic line bundle, the associated operators and # are C∞ conjugate linear isomorphisms. 4.7.2 Prove Proposition 4.7.4. 4.7.3 Prove Proposition 4.7.6.

~~4.8 Proof of Serre Duality and ∂¯-Hodge Decomposition~~

The Serre duality theorem and the ∂¯-Hodge decomposition theorem will be proved simultaneously using orthogonal decomposition in a Hilbert space. Throughout this section, E denotes a holomorphic line bundle on a compact Riemann surface X.By Theorem 4.2.3, we may choose divisors D and K in X with [D]=E and [K]=KX. Finally, we may ﬁx a Hermitian metric h in E. According to Proposition 4.7.6,we then have

~~∗¯# O ⊗ ∗ → 2 ∗ ∗ −→E 2 (X, (KX E )) L1,0(X, E ,h ) L0,1(X,E,h), ∗¯ # where E is a (surjective) conjugate linear isomorphism that satisﬁes~~

~~# # 2 ∗ ∗ ∗¯ α,∗¯ β 2 =β,α 2 ∗ ∗ ∀α, β ∈ L (X, E ,h ). E E L0,1(X,E,h) L1,0(X,E ,h ) 1,0 182 4 Compact Riemann Surfaces~~

~~2 Lemma 4.8.1 In L0,1(X,E,h), we have~~

~~∗¯# O ⊗ ∗ = ¯ E0,0 ⊥ E((X, (KX E ))) (∂( (E)(X))) .~~

~~O ⊗ ∗ → 1 Moreover, the conjugate linear map (X, (KX E )) HDol(X, E) given by~~

~~→ [∗¯# ] α Eα Dol~~

~~E : O ⊗ ∗ → 1 ∗ and the Serre mapping ιS (X, (KX E )) (HDol(X, E)) are injective.~~

~~∗ 0,0 Proof If α ∈ (X,O(KX ⊗ E )) and t ∈ E (E)(X), then by Proposition 4.7.6 and Lemma 4.6.2 (or Lemma 4.6.10),~~

~~¯ ∗¯# =− ¯ = ¯ = ∂t, Eα L2 SE(∂t,α) SE(t, ∂α) 0.~~

~~∈ ¯ E0,0 ⊥ = ∗¯# ≡−∗¯ ∈ Conversely, suppose β (∂( (E)(X))) .Wehaveβ Eα, where α Eβ 2 ∗ ∗ ∈ E0,0 L1,0(X, E ,h ), and for every section t (E)(X),wehave~~

~~=¯ ∗¯# =− ¯ 0 ∂t, Eα L2 SE(∂t,α).~~

Lemma 4.6.10 now implies that α is a holomorphic 1-form with values in E∗. That → [∗¯# ] the map α Eα Dol is injective follows easily. ∈ E Finally, if α ker ιS, then by Proposition 4.7.6,

~~= E ·[∗¯# ] = ∗¯ # =−∗¯# 2 0 (ιS(α)) Eα Dol SE( Eα, α) Eα L2 .~~

~~∗¯# = = E Thus Eα 0 and hence α 0. Therefore ιS is injective.~~

Lemma 4.8.2 The following are equivalent: (i) The range ∂(¯ E0,0(E)(X)) is closed in the subspace E0,1(E)(X) of the Hilbert 2 space L0,1(X,E,h). E0,1 = ∗¯# O ⊗ ∗ ⊕ ¯ E0,0 ¯ (ii) (E)(X) E((X, (KX E ))) ∂( (E)(X)) (i.e., the ∂-Hodge decomposition theorem holds for E). O ⊗ ∗ → 1 (iii) The conjugate linear map (X, (KX E )) HDol(X, E) given by

~~→ [∗¯# ] α Eα Dol~~

~~is bijective. (iv) The Serre mapping~~

~~E : O ⊗ ∗ → 1 ∗ ιS (X, (KX E )) (HDol(X, E))~~

~~is bijective (i.e., the Serre duality theorem holds for E). 4.8 Proof of Serre Duality and ∂¯-Hodge Decomposition 183~~

Proof The equivalence of (i)Ð(iii) follows from Lemma 4.8.1 and general Hilbert space theory (see Corollary 7.5.7). For the proof that (i)Ð(iii) and (iv) are equivalent, we observe that given a lin- 1 : → − [∗¯ # ] ear functional τ on HDol(X, E), we get the linear functional ρ t τ( Et Dol) ∗ 2 ∗ ∗ on the ﬁnite-dimensional subspace (X,O(KX ⊗ E )) ⊂ L (X, E ,h ). Thus ∗ 1,0 there exists an element s ∈ (X,O(KX ⊗ E )) such that ρ =·,s 2 ∗ ∗ .If L1,0(X,E ,h ) the orthogonal decomposition (ii) holds, then for each α ∈ E0,1(E)(X),wehave = ∗¯# + ¯ ∈ O ⊗ ∗ C∞ α Et ∂β for some holomorphic section t (X, (KX E )) and some section β ∈ E0,0(E)(X). Therefore, by Proposition 4.7.6,

# τ([α]Dol) = τ([∗¯ t] ) =−ρ(t)=−t,s 2 ∗ ∗ =−s,t 2 ∗ ∗ E Dol L1,0(X,E ,h ) L1,0(X,E ,h ) # # # =−∗¯ t,∗¯ s 2 =−α, ∗¯ s 2 = SE(α, s) E E L0,1(X,E,h) E L0,1(X,E,h) = E ·[ ] ιS(s) α Dol.

~~E E Thus the Serre map ιS is surjective, and therefore, by Lemma 4.8.1, ιS is bijective. Conversely, if there exists an element~~

α ∈ E0,1(E)(X) ∩ cl(∂(¯ E0,0(E)(X))) \ ∂(¯ E0,0(E)(X)) ⊂ ∗¯# O ⊗ ∗ ⊥ ( E((X, (KX E )))) , [ ] ∈ 1 V ∗¯# O ⊗ ∗ then α Dol HDol(X, E) is not in the image of E((X, (KX E ))). Hence 1 there exists a linear functional τ on H (X, E) such that τ([α]Dol) = 1 and τ = 0 Dol ∗ on V. On the other hand, for each s ∈ (X,O(KX ⊗ E )),wehave E ·[ ] = =− ∗¯# = ιS(s) α Dol SE(α, s) α, Es L2(X,E,h) 0, E = E so ιS(s) τ . Thus ιS is not surjective if this is the case.

~~Next, we observe that in order to obtain Serre duality for the divisor D, it sufﬁces to obtain Serre duality for some divisor ≤ D.~~

~~Lemma 4.8.3 If there exists an effective divisor F on X for which the Serre map~~

~~[D−F ] : O [ − + ] → 1 [ − ] ∗ ιS (X, ( K D F )) (HDol(X, D F )) is bijective, then the Serre map~~

~~[D] : O [ − ] → 1 [ ] ∗ ιS (X, ( K D )) (HDol(X, D )) is also bijective.~~

Proof Fix a nontrivial section u ∈ (X,O([F ])). Given a nontrivial linear func- ∈ 1 [ ] ∗ ∈ tional τ (HDol(X, D )) , we may convert τ into a linear functional τu 184 4 Compact Riemann Surfaces

~~1 [ − ] ∗ (HDol(X, D F )) by setting [ ] ≡ [ · ] ∀[ ] ∈ 1 [ − ] τu( θ Dol) τ( θ u Dol) θ Dol HDol(X, D F ).~~

In other words, τu is the composition of τ and the surjective linear mapping 1 [ − ] = 1 [ ]⊗[ ]∗ → 1 [ ] HDol(X, D F ) HDol(X, D F ) HDol(X, D ) induced by multiplication by u (as in the Dolbeault exact sequence). By hypothesis, there exists a section t ∈ (X,O([K − D + F ])) ⊂ E 1,0([−D + F ])(X) such that = [D−F ] = · τu ιS (t) S[D−F ]( ,t). We will now show that the meromorphic section t/u of [K − D] is actually holo- [D] = [ ] ∈ 1 [ ] morphic and that ιS (v) τ . Given an element θ Dol HDol(X, D ),wemay form local solutions of ∂γ¯ = θ near supp div(u) and cut off to get a C∞ section α of [D] on X such that ∂α¯ = θ near supp div(u). Thus, by replacing θ with θ − ∂α¯ , we see that we may choose the representing class θ to vanish on a neighborhood ∞ of supp div(u). Therefore, θ/u extends by 0 to a well-deﬁned C form θu of type (0, 1) with values in [D − F ] (recall that this is how we proved surjectivity of such a map in the Dolbeault exact sequence in Sect. 3.4) and we have

~~τ([θ] ) = τ([θ · u] ) = τ ([θ ] ) = S[ − ](θ ,t) Dol u Dol u u Dol D F u~~

= s[D−F ](θ/u, t) = s[D](θ, t/u). X X In fact, for E ≡[D], the meromorphic 1-form v ≡ t/u with values in E∗ =[−D] is holomorphic. For given a pole p of v, we may choose a local holomorphic coordinate neighborhood (U, = z) of p in X such that z(p) = 0, u and t are nonvanishing on U \{p}, and there exist a nonvanishing holomorphic section s of E and a meromorphic function f with v = fdz/son U. We may also ﬁx a C∞ function λ with compact support in U such that λ ≡ 1onP ≡ −1((0; R)) U for some R>0. Since 1/f has a zero at p, the section λs/(zf) of E on U \{p} extends to a unique C∞ section β of E on X. Moreover, the form γ ≡ ∂β¯ = (∂λ)¯ · s/(zf) vanishes outside a compact subset of U and on P . Thus ¯ = = [ ] = = ∂λ ∧ 0 τ(0) τ( γ Dol) sE(γ, v) dz \ z X U P dz dz = d λ =− =−2πi = 0. U\P z ∂P z We have therefore arrived at a contradiction, and hence v = t/u has no poles. There- fore, for θ as above, we get

~~τ([θ]Dol) = sE(θ, v) = SE([θ]Dol,v). X = · = E Thus τ SE( ,v) ιS(v) as required. 4.9 Hodge Decomposition for Scalar-Valued Forms 185~~

Proof of Theorem 4.6.4 and Theorem 4.7.7 According to the above lemmas, it suf- ﬁces to show that for the divisor D on X, Serre duality holds for the line bundle [D − F ] for some effective divisor F .LetE =[D]. Fixing a point p ∈ X and applying Theorem 4.2.1 (or Theorem 4.3.1), we see that [−D + mp] will admit a positive-curvature Hermitian metric provided m 0. Fixing a Kähler form ω and replacing D with D − mp and E with [D − mp], we may assume without loss of generality that E admits a Hermitian metric k with −ik ≥ ω (after choosing m 0 or replacing ω with the product of ω and a small positive constant). According to Lemma 4.8.2, it remains to show that the range ∂(¯ E0,0(E)(X)) E0,1 2 ∈ is closed in the subspace (E)(X) of L0,1(X,E,k). For this, suppose β E0,1 { } E0,0 ¯ − → (E)(X) and αν is a sequence in (E)(X) with ∂αν β L2(X,E,k) 0. Observe that the fundamental estimate (Proposition 3.7.5)gives ∂α¯ 2 =D α2 − |α|2 · i ≥α2 ∀α ∈ E0,0(E)(X). k k k k k L2 (X,E,ω,k) X 0,0

~~It follows that the sequence {αν} is Cauchy, and hence that the sequence converges ∈ 2 ∈ E0,1 to some α L0,0(X,E,ω,k).Ifγ (E)(X), then by Proposition 3.8.2,~~

Dkγ Dkγ − {α, D γ } =− α, =− lim αν, k k ν→∞ X ω L2(X,E,ω,k) ω L2 =− lim {α ,D γ } = lim {∂α¯ ,γ} →∞ ν k k →∞ ν k ν X ν X ¯ = i lim ∂α ,γ 2 = iβ,γ 2 = {β,γ} . →∞ ν L L k ν X ¯ Applying Proposition 3.8.2 (again), we get ∂distrα = β. The regularity theorem (The- orem 3.8.3) now implies that α is of class C∞ and ∂α¯ = β. Thus the range is closed in E0,1(E)(X), and the theorems follow.

~~4.9 Hodge Decomposition for Scalar-Valued Forms~~

~~Applying Theorem 4.7.7 to the trivial line bundle, we get the following:~~

~~Theorem 4.9.1 (Hodge decomposition for scalar-valued forms) If X is a compact Riemann surface, then we have the orthogonal decompositions~~

~~E0,1(X) = (X) ⊕ ∂(¯ E0,0(X)),~~

~~E1,0(X) = (X)⊕ ∂(E0,0(X)), 186 4 Compact Riemann Surfaces~~

~~E1(X) = (X)⊕ (X) ⊕ d(E0(X)) ⊕ ∗¯d(E0(X)),~~

~~ d ker E1(X) → E2(X) = (X)⊕ (X) ⊕ d(E0(X)), ¯ where (X) = (X,O(KX)) and (X) ={θ | θ ∈ (X)} (not the closure of (X)).~~

~~Remark Recall that we think of a (1, 0)-form and a (0, 1)-form as orthogonal.~~

Proof of Theorem 4.9.1 The first equality follows from Theorem 4.7.7 and the second follows by conjugation of the spaces in the first. For the third and fourth equalities, we first observe that ∗¯d(E0(X)) ⊥ ker d in E1(X), and hence ∗¯d(E0(X)) is orthogonal to (X) ⊕ (X) ⊕ d(E0(X)) ⊂ ker d. For if α ∈ ker d and v ∈ E0(X), then Proposition 4.7.4 and Stokes’ theorem give ∗¯ = ∧ ∗¯∗¯ =− ∧ = = α, dv L2 α dv α dv d(αv) 0. X X X Next, we observe that the first two equalities yield the orthogonal decomposition

~~E1(X) = (X)⊕ (X) ⊕ ∂(E0,0(X)) ⊕ ∂(¯ E0,0(X)).~~

~~Given ρ ∈ E0,0(X),wehavedρ = ∂ρ + ∂ρ¯ and ∗¯dρ = ∗¯∂ρ + ∗¯∂ρ¯ = ∂(i¯ ρ)¯ + ∂(−iρ)¯ . Thus~~

~~d(E0(X)) ⊕ ∗¯d(E0(X)) ⊂ ∂(E0,0(X)) ⊕ ∂(¯ E0,0(X)).~~

Conversely, given α, β ∈ E0,0(X), by setting 1 i ρ ≡ (α + β) and τ ≡− (α¯ − β),¯ 2 2 we get ∂α + ∂β¯ = dρ + ∗¯ dτ. Thus the third equality holds. The fourth equality follows since the right-hand side is contained in ker d and ∗¯d(E0(X)) ⊥ ker d.

~~For a compact Riemann surface X, the space of harmonic 1-forms is given by~~

~~Harm1(X) ≡ (X)⊕ (X).~~

Clearly, the space Harm1(X) ∩ E1(X, R) of real harmonic 1-forms consists of all elements of the form α +¯α with α ∈ (X). Recall that for each r ∈ Z≥0,therth complex de Rham cohomology (see Deﬁnition 10.6.1)ofaC∞ surface M is given by r = r C = Er →d Er+1 Er−1 →d Er HdeR(M) HdeR(M, ) ker (M) (M) /im (M) (M) , and the rth real de Rham cohomology is given by r R = Er R →d Er+1 R Er−1 R →d Er R HdeR(M, ) ker (M, ) (M, ) /im (M, ) (M, ) 4.9 Hodge Decomposition for Scalar-Valued Forms 187

~~(where we set d = 0onE−1(M) = 0). Thus we get the following consequence of Hodge decomposition (Theorem 4.9.1), the proof of which is left to the reader (see Exercise 4.9.1):~~

Corollary 4.9.2 For any compact Riemann surface X, we have the following: [ ] ∈ 1 (a) Every complex de Rham cohomology class θ deR HdeR(X) has a unique 1 → 1 harmonic representative; that is, the map Harm (X) HdeR(X) given by θ → [θ]deR is a (surjective) complex linear isomorphism. [ ] ∈ 1 R (b) Every real de Rham cohomology class θ deR HdeR(X, ) has a unique har- 1 ∩ E1 R → 1 R monic representative; that is, the map Harm (X) (X, ) HdeR(X, ) given by θ → [θ]deR is a (surjective) real linear isomorphism. 1 R = 1 = · Consequently,dimR HdeR(X, ) dimC HdeR(X) 2 genus(X).

~~Remark In particular, for any compact Riemann surface X,wehave~~

~~1 1 1 2 · genus(X) = dimC H (X, C) = dimR H (X, R) = rank H (X, Z)~~

~~= dimC H1(X, C) = dimR H1(X, R) = rank H1(X, Z)~~

(see Sects. 10.6 and 10.7). Hence the (holomorphic) genus of a compact Riemann surface is actually a topological invariant; i.e., it does not depend on the holomorphic structure or even on the C∞ structure. In other words, any two compact Riemann surfaces that are homeomorphic have the same genus. The converse, though true, is not proved in this book (see, for example, [T] for a proof).

~~Theorem 4.9.3 (Poincaré duality theorem) Let X be a compact Riemann surface, and let r ∈{0, 1, 2}. Then the map [ ] [ ] → [ ] [ ] ≡ ∧ ( α deR, β deR) P( α deR, β deR) α β X~~

· · : r × 2−r → C gives a well-deﬁned bilinear pairing P( , ) HdeR(X) HdeR (X) . Fur- [ ] → [ ] ≡ · [ ] thermore, the map β deR ιP( β deR) P( , β deR) gives a (surjective) lin- : 2−r → r ∗ 2 = ear isomorphism ιP HdeR (X) (HdeR(X)) . In particular,dimC HdeR(X) 0 = dimC HdeR(X) 1.

Proof The pairing is well deﬁned. For if α ∈ Er (X) ∩ ker d, β ∈ E2−r (X) ∩ ker d, ρ ∈ Er−1(X) (we set ρ = 0ifr = 0), and τ ∈ E2−r−1(X) (we set τ = 0ifr = 2), then by Stokes’ theorem, (α + dρ) ∧ (β + dτ) X = α ∧ β + α ∧ dτ + dρ ∧ β + dρ ∧ dτ X X X X 188 4 Compact Riemann Surfaces = α ∧ β + (−1)r d(α ∧ τ)+ d(ρ ∧ β)+ d(ρ ∧ dτ) X X X X = α ∧ β. X

[ ] ∈ 1 1,0 ∈ Now, given α deR HdeR(X), we may apply Corollary 4.9.2 to get α (X) 0,1 1,0 0,1 and α ∈ (X) with [α]deR =[α ]deR +[α ]deR, and we may choose the representative α so that α = α1,0 + α0,1.For(p, q) = (1, 0) or (0, 1),wehave

~~(−1)pi αp,q ∧ α = (−1)pi αp,q ∧ αp,q ≥ 0,~~

~~p,q p,q with equality at a point x if and only if αx = 0. Thus, if α is nontrivial, then~~

~~p p,q P([(−1) i α ]deR, [α]deR)>0.~~

It follows that ιP([α]deR) = 0 if and only if [α]deR = 0, and hence that ιP maps 1 1 ∗ HdeR(X) isomorphically onto (HdeR(X)) . 0 = C 2 It is easy to verify that HdeR(X) . To determine HdeR(X), let us ﬁx a Kähler metric g with Kähler form ω on X. To a differential form θ ∈ E1,1(X),wemay ˆ ¯ associate a differential form θ of type (0, 1) with values in KX. Applying the ∂- ∞ Hodge decomposition theorem, we get a C section ρˆ of KX and a holomorphic ⊗ ∗ = ˆ = ∗¯# + ¯ ˆ section ζ of KX KX 1X (i.e., a complex constant) with θ (K ,g∗)ζ ∂ρ. X ¯ ¯ Letting ρ be the scalar-valued (1, 0)-form associated to ρˆ, we get θ =− ζω+ ∂ρ = −¯ + [ ] = ≡ ζω dρ (see Exercise 4.9.2). We also have ω deR 0, since v X ω>0, so 2 = C[ ] =∼ C ∈ C HdeR(X) ω deR . For any ξ,η ,wehave(hereweidentifyξ and η with the corresponding constant functions x → ξ and x → η)

[ ] [ ] = [ ] [ ] = · P( ξ deR, ηω deR) P( ηω deR, ξ deR) ξη v, which vanishes if and only if ξ = 0orη = 0. It now follows that the linear mappings : 0 → 2 ∗ : 2 → 0 ∗ ιP HdeR(X) (HdeR(X)) and ιP HdeR(X) (HdeR(X)) are (surjective) isomorphisms.

Exercises for Sect. 4.9 4.9.1 Prove Corollary 4.9.2. 4.9.2 Let X be a Riemann surface. According to Proposition 3.10.1, we may identify a (1, 1)-form with a corresponding (0, 1)-form with values in KX, and we ∗ may identify a (1, 0)-form with values in KX with a corresponding section of ⊗ ∗ = KX KX 1X, i.e., with a function. Let g be a Kähler metric with associated Kähler form ω on X, and let ζ ∈ C. Viewing the constant function ζ as ∗ a holomorphic 1-form with values in KX, show that the scalar-valued (1, 1)- # ¯ form associated to the K -valued (0, 1)-form ∗¯ ∗ ζ is equal to −ζω(this X (KX,g ) fact was used in the proof of the Poincaré duality theorem). 4.9 Hodge Decomposition for Scalar-Valued Forms 189

4.9.3 Let (E, h) be a Hermitian holomorphic line bundle on a compact Riemann surface X. Prove the Dh-Hodge decomposition theorem: E1,0 = O ⊗ ⊕ E0,0 (E)(X) (X, (KX E)) Dh( (E)(X)). ∗¯# ¯ = ∈ Hint. First prove that E(∂u) Dh(#E(iu)) for each section u E0,0 ∗ ∗¯# ¯ (E )(X). Then apply the operator E to the summands in the ∂-Hodge decomposition for E0,1(E∗)(X).

~~Chapter 5 Uniformization and Embedding of Riemann Surfaces~~

In this chapter, we consider certain complex analytic characterizations of Riemann surfaces (some topological and C∞ characterizations appear in Chap. 6). The ﬁrst goal is the following Riemann surface analogue of the classical Riemann mapping theorem in the plane:

~~Theorem 5.0.1 (Riemann mapping theorem) A simply connected Riemann surface is biholomorphic to the Riemann sphere P1, to the complex plane C, or to the unit disk ={z ∈ C ||z| < 1}.~~

Exactly one of the above outcomes will hold, since no two of the Riemann surfaces P1, C, and are biholomorphic (see Exercise 2.2.7). Theorem 5.0.1 is also called the uniformization theorem. Parts of the theorem were first obtained by Riemann. The first complete proofs were given by Koebe [Koe1], [Koe2] and Poincaré [P]. The proof given in this chapter is essentially due to Simha [Sim] and Demailly [De3], although the idea of forming compactifications of exhausting subdomains is due to Koebe. The theorem allows one to classify Riemann surfaces as quotients of the Riemann sphere, the plane, or the disk (see Sect. 5.9). The second goal of this chapter is the fact that every Riemann surface X may be obtained by holomorphic attachment of tubes at elements of a locally finite sequence of coordinate disks in a domain in P1. In particular, for X compact, this allows one to form a canonical homology basis. Finally, we consider finite holomorphic branched coverings (see Sect. 5.20)as well as the following embedding theorems:

• Every open Riemann surface admits a holomorphic embedding into C3 (see Sect. 5.18). This fact is due to Narasimhan [Ns1] and is the 1-dimensional version of the BishopÐNarasimhanÐRemmert embedding theorem for Stein manifolds. • Every compact Riemann surface admits a holomorphic embedding into some (higher-dimensional) complex projective space (see Sect. 5.19). This fact is the 1-dimensional version of the Kodaira embedding theorem.

T. Napier, M. Ramachandran, An Introduction to Riemann Surfaces, Cornerstones, 191 DOI 10.1007/978-0-8176-4693-6_5, © Springer Science+Business Media, LLC 2011 192 5 Uniformization and Embedding of Riemann Surfaces

~~• Every compact Riemann surface of positive genus admits a holomorphic embedding into some (higher-dimensional) complex torus (see Sect. 5.22). This fact is the AbelÐJacobi embedding theorem.~~

~~5.1 Holomorphic Covering Spaces~~

~~The general theory of covering spaces is reviewed in Chap. 10 (see Sects. 10.1Ð 10.5). In this section, we consider covering spaces of Riemann surfaces.~~

Definition 5.1.1 A holomorphic covering space (or a 1-dimensional complex covering manifold or, for X connected, a covering Riemann surface) of a complex 1-manifold X is a covering space ϒ : X → X in which X is a complex 1-manifold and the mapping ϒ is locally biholomorphic (see Definition 10.2.1). In other words, each point in X has a connected neighborhood U such that ϒ maps each connected component of ϒ−1(U) biholomorphically onto U. The holomorphic covering space ϒ : X → X is holomorphically equivalent to a holomorphic covering space ϒˇ : Xˇ → X if there exists a biholomorphism : Xˇ → X that is a lifting of ϒˇ ; i.e., is a fiber-preserving biholomorphism. A simply connected holomorphic covering space ϒ : X → X of a Riemann surface is also called the holomorphic universal covering space (or universal covering Riemann surface).

~~Remark Holomorphic equivalence of coverings is an equivalence relation (see Ex- ercise 5.1.1).~~

Proposition 5.1.2 Let X be a complex 1-manifold and let ϒ : X → X be a (topological) covering space. (a) If X is a complex 1-manifold and ϒ is holomorphic, then ϒ : X → X is a holomorphic covering space. (b) In general, there is a unique holomorphic structure on X with respect to which ϒ : X → X is a holomorphic covering space.

Proof Part (a) follows from the holomorphic inverse function theorem (Theo- rem 2.4.4). For the proof of (b), ﬁrst observe that X is Hausdorff, since X is Hausdorff (see Proposition 10.2.10). Next observe that we may form a holomor- A ={ } phic atlas (Ui,i,Ui ) i∈I on X such that Ui is connected and evenly cov- −1 ered by ϒ for each i ∈ I . For each i ∈ I , Ui ≡ ϒ (Ui) is the union of a col- { (λ)} lection of disjoint connected open sets Ui λ∈i in X each of which is mapped (λ) = ◦ homeomorphically onto Ui . Setting i i ϒ (λ) , we get a complex at- Ui A≡{ (λ) (λ) } ∈ ∈ ∈ las (Ui ,i ,Ui ) λ∈i ,i∈I in X. Suppose i, j I , λ i , μ j , and (λ) ∩ (μ) = ∅ Ui Uj . Then the coordinate transformation 5.1 Holomorphic Covering Spaces 193

(λ) ◦[ (μ)]−1 = ◦ −1 : (μ) (λ) ∩ (μ) i j i j (μ) (λ)∩ (μ) j (Ui Uj ) j (Ui Uj ) −→ (λ) (λ) ∩ (μ) ⊂ ∩ i (Ui Uj ) i(Ui Uj ) is holomorphic, and hence the atlas determines a holomorphic structure on X. More- over, ϒ is holomorphic with respect to this holomorphic structure, since

~~(λ) −1 i ◦ ϒ ◦[ ] = Id ∈ O(U ) ∀i ∈ I,λ ∈ i. i Ui i~~

Finally, given an arbitrary holomorphic structure on X with respect to which ∈ ∈ (λ) = ϒ is a local biholomorphism, and given i I and λ i , the mapping i i ◦ ϒ (λ) is a composition of a biholomorphism on Ui ⊂ X and a biholomor- Ui (λ) (λ) phism on Ui , and therefore i is biholomorphic. Thus the elements of the atlas A are holomorphically compatible with the local holomorphic charts on X, and uniqueness follows.

~~Remark Given a covering space ϒ : X → X of a complex 1-manifold X, we will assume that X has the induced holomorphic structure unless otherwise indicated.~~

Theorem 5.1.3 (Holomorphic lifting theorem) Let ϒ : X → X be a covering Rie- mann surface of a Riemann surface X, let : Y → X be a holomorphic mapping −1 of a Riemann surface Y to X, let y0 ∈ Y , let x0 = (y0), and let xˆ0 ∈ ϒ (x0). (a) If : Y → X is a lifting of , then is holomorphic. If is a local biholomorphism (a holomorphic covering map), then is a local biholomorphism (respectively, a holomorphic covering map). (b) If ∗π1(Y, y0) ⊂ ϒ∗π1(X,xˆ0), then there is a unique lifting of to a holomorphic map : Y → X with (y0) =ˆx0. Conversely, if such a lifting exists, then ∗π1(Y, y0) ⊂ ϒ∗π1(X,xˆ0).

~~Proof The proof of (a) is left to the reader (see Lemma 10.2.4 and Exercise 5.1.2). Part (b) follows from the (topological) lifting theorem (Theorem 10.2.5) together with part (a).~~

Corollary 5.1.4 Let ϒ : X → X and ϒˇ : Xˇ → X be covering Riemann surfaces −1 ˇ −1 of a Riemann surface X, let x0 ∈ X, let xˆ0 ∈ ϒ (x0), and let xˇ0 ∈ ϒ (x0). If ˇ ˇ ϒ∗π1(X,xˆ0) ⊂ ϒ∗π1(X,xˇ0), then there exists a unique commutative diagram of holomorphic covering maps

~~Xˇ ϒ ϒˇ ϒ X X 194 5 Uniformization and Embedding of Riemann Surfaces~~

ˇ ˇ with ϒ(xˆ0) =ˇx0. Moreover, if ϒ∗π1(X,xˆ0) = ϒ∗π1(X,xˇ0), then ϒ is a biholomorphism; that is, the two coverings are holomorphically equivalent. In particular, up to holomorphic equivalence of holomorphic coverings, every Riemann surface has a unique holomorphic universal covering space.

~~Proof The holomorphic lifting theorem gives the existence and uniqueness of the commutative diagram. Corollary 10.2.6 then implies that if the image groups are equal, then ϒ is a biholomorphism.~~

~~Recall the following (see Deﬁnition 2.2.1):~~

~~Deﬁnition 5.1.5 A biholomorphism of a Riemann surface X onto itself is called an automorphism of X. The group of automorphisms of X (with product given by composition) is denoted by Aut (X).~~

Recall that a deck transformation (or covering transformation) of a covering space ϒ : X → X is a homeomorphism : X → X such that ϒ ◦ = ϒ, and the group of deck transformations is denoted by Deck(ϒ) (see Sect. 10.4). If the above is a Riemann surface covering, then the holomorphic lifting theorem (Theo- rem 5.1.3) implies that Deck(ϒ) is a subgroup of Aut (X) . Moreover, the holomorphic lifting theorem and Theorem 10.4.5 together give the following (the details of the proof are left to reader in Exercise 5.1.3):

Theorem 5.1.6 Let ϒ : X → X be the (holomorphic) universal covering space of a Riemann surface X. Then ≡ Deck(ϒ) is a subgroup of Aut(X) that acts properly discontinuously and freely on X (see Deﬁnition 10.4.3), and there is a unique holomorphic structure on \X giving rise to a commutative diagram X ϒ ϒ =∼ X \X in which ϒ is a holomorphic covering map and the map \X → X is a biholomorphism. In other words, ϒ : X → \X is a holomorphic covering that we may identify with the original holomorphic covering ϒ : X → X. It follows that if ϒ : X → X is the universal covering space of a Riemann surface X and Deck(ϒ) = Deck(ϒ), then X =∼ \X =∼ X and we may identify the holomorphic coverings ϒ : X → X and ϒ : X → X. Conversely, a group of automorphisms acting properly discontinuously and freely gives rise to a holomorphic universal covering space (cf. Theorem 10.4.6):

~~Theorem 5.1.7 Let X be a simply connected Riemann surface, let be a subgroup of Aut (X) that acts properly discontinuously and freely on X, and let 5.1 Holomorphic Covering Spaces 195~~

ϒ = ϒ : X → X ≡ \X be the corresponding quotient space mapping. Then we have the following: (a) There is a unique holomorphic structure on X with respect to which the quotient map ϒ : X → X is a holomorphic universal covering space and = Deck(ϒ). (b) If is a subgroup of , then we have a commutative diagram

X ϒ ϒ ϒ X X ≡ \X of holomorphic covering maps. Moreover, if x˜0 ∈ X, x0 = ϒ(x˜0), and xˆ0 = ˜ ϒ(x0), and : = → : = → ˆ χ Deck(ϒ) π1(X, x0) and χ Deck(ϒ) π1(X,x0) are the corresponding group isomorphisms, then

~~−1 χ ◦ χ = ϒ∗ : π1(X,xˆ0) → π1(X, x0). In other words, the action of [ˆγ ]∈π1(X,xˆ0) on X is the same as that of ϒ∗[ˆγ ]=[ϒ(γ)ˆ ]∈π1(X, x0).~~

Proof For the proof of part (a), we ﬁrst observe that Theorem 10.4.6 implies that ϒ : X → X is a covering map and Deck(ϒ) = .LetA be the collection of local { ◦ −1 } complex charts in X of the form (U, (ϒ U ) ,U ) , where U is an evenly covered connected open subset of X and (U ,,U) is a local holomorphic chart in X for which U is a connected component of ϒ−1(U). We will show that A is a holomorphic atlas in X. Clearly, the neighborhoods in A cover X.If(U, ◦ −1 ◦ −1 A (ϒ U ) ,U ) and (V, (ϒ V ) ,V ) are two local complex charts in and p ∈ U ∩ V , then we may choose a connected evenly covered neighborhood W of p ∩ ≡ −1 ∈ in U V . Setting W (ϒ U ) (W), we get an element γ such that the connected open set γ(W) ⊂ ϒ−1(V ) meets, and hence is contained in, V . Thus = ◦ −1 the coordinate transformation on (W ) (ϒ U ) (W) is given by ◦ −1 ◦[ ◦ −1]−1 = ◦ ◦ −1 ◦ ◦ −1 = ◦ ◦ −1 (ϒ V ) (ϒ U ) γ (ϒ U ) ϒ γ , and is therefore holomorphic as a composition of holomorphic maps. Thus A is a holomorphic atlas. The map ϒ : X → X is holomorphic with respect to this holo- ◦ −1 ∈ A morphic structure, since given a local holomorphic chart (U, (ϒ U ) ,U ) , ◦ −1 ◦ = A we have (ϒ U ) ϒ on U . Finally, if is a holomorphic atlas on X with respect to which the covering map ϒ is holomorphic, (Q,,Q) ∈ A, and ◦ −1 ∈ A (U, (ϒ U ) ,U ) , then the coordinate transformation on ◦ −1 ∩ (ϒ U ) (U Q) is given by ◦[ ◦ −1]−1 = ◦ ◦ −1 (ϒ U ) ϒ , 196 5 Uniformization and Embedding of Riemann Surfaces and is therefore holomorphic as a composition of holomorphic maps. The inverse function theorem then implies that its inverse map is also holomorphic. Thus A and A determine the same holomorphic structure on X, and we have uniqueness. For the proof of (b), suppose is a subgroup of . Then Theorem 10.4.6 gives the required commutative diagram of topological covering maps, and part (a) gives the holomorphic structure on X with respect to which ϒ is a holomorphic covering map. It follows that the covering map ϒ : X → X is also holomorphic, because locally, the map may be written as the composition of the holomorphic map ϒ with a local holomorphic inverse of ϒ.

Corollary 5.1.8 Let X be a Riemann surface, let x0 ∈ X, and let be a subgroup of π1(X, x0). Then, up to holomorphic equivalence of holomorphic covering spaces, −1 there are a unique covering Riemann surface ϒ : X → X and a point xˆ0 ∈ ϒ (x0) such that the homomorphism ϒ∗ : π1(X,xˆ0) → π1(X, x0) maps π1(X,xˆ0) isomorphically onto . In fact, if ϒ : X → X is the universal covering space, then we have the commutative diagram ϒ X = \ X X ϒ ϒ

~~(here, we identify π1(X, x0) ⊃ with Deck(ϒ)).~~

Proof This follows immediately from Theorem 10.3.3 and Theorem 5.1.2 (which give the existence of the holomorphic universal cover), Theorem 5.1.7 (the construction of a holomorphic covering from a properly discontinuous free group action by automorphisms), and Corollary 5.1.4 (holomorphic equivalence of holomorphic coverings with the same image fundamental group).

We now consider some fundamental examples: Example 5.1.9 The holomorphic universal covering map of the punctured plane C∗ is the map ϒ : C → C∗ given by z → exp(2πiz).Forϒ is a surjective locally biholomorphic mapping with local inverses given by holomorphic branches of the multiple-valued holomorphic function ζ → (log ζ)/(2πi) (see Example 1.6.2). Given a point z0 = x0 + iy0 ∈ C with x0,y0 ∈ R, we may form the connected neighborhood 2πit U ≡{ρe | 0 <ρ<∞,x0 − (1/2)

~~= ϒ({x + iy | y ∈ R,x0 − (1/2)~~

~~−1 of ζ0 = ϒ(z0). The inverse image ϒ (U) is then the union of the collection of disjoint domains {Un}n∈Z, where for each n, Un is the inﬁnite strip~~

~~Un ={z = x + iy | y ∈ R,x0 − (1/2) + n~~

~~Fig. 5.1 The universal covering of the punctured plane~~

∗ Fix a point z0 ∈ C,letζ0 ≡ ϒ(z0) ∈ C , and for each t ∈[0, 1],letγ(t)˜ = z0 + t 2πit and γ(t)= ϒ(γ(t))˜ = ζ0e . Clearly, the map 0 : z → z + 1 is a deck transfor- Z → → n mation, so we have an injective homomorphism Deck(ϒ) given by n 0 n −1 (where is the translation z → z + n). Moreover, the fiber ϒ (ζ0) is equal to 0 ∗ ∼ ∼ z0 + Z,sowealsohavesurjectivity.Thusπ1(C ) = Deck(ϒ) = Z. In fact, since the ∗ lifting γ˜ of γ satisfies γ(˜ 0) = z0 and γ(˜ 1) = z0 + 1 = 0(z0), π1(C ,ζ0) is the in- [ ] [ ] → finite cyclic group generated by γ ζ0 . That is, γ ζ0 1 under the composition of the isomorphism Deck(ϒ) → Z (given by → (z0) − z0) and the isomorphism → [ ] → =˜ ˜ π1(X, ζ0) Deck(ϒ) (given by α ζ0 , where (z0) α(1) for α the lifting of α with α(˜ 0) = z0). ∼ ∼ ∗ Any proper subgroup of Z = Deck(ϒ) = π1(C ,ζ0) must be of the form = mZ for some positive integer m; that is, is the infinite cyclic subgroup generated by [γ ]m . We have a holomorphic universal covering map ϒ : C → C∗ given ζ0 2πiz/m by z → e . In fact, ϒ is the composition of the holomorphic universal covering map C → C∗ given by z → e2πiz and the biholomorphism C → C given by z → z/m. The universal covering map ϒ has associated deck transformation group Deck(ϒ) = ⊂ Deck(ϒ), because ⊂ Deck(ϒ), and for any point z ∈ C, · = + Z = −1 C∗ =∼ \C z z m ϒ (ϒ(z)). Therefore, and the holomorphic covering space of C∗ associated to the subgroup as in Corollary 5.1.8 is the finite holomorphic covering C∗ → C∗ given by exp(2πiz/m)→ exp(2πiz); that is, the mapping is given by z → zm.

Example 5.1.10 Let us consider complex tori from the point of view of holomorphic covering spaces (see Example 2.1.6). Recall that a lattice in C is a subgroup of the form = Zξ1 + Zξ2, where ξ1,ξ2 ∈ C are linearly independent over R.Wehave an injective homomorphism → Aut (C) under which each element ξ ∈ maps to the automorphism given by z → z + ξ. Thus we may identify with a subgroup of Aut(C). Clearly, acts freely. Moreover, is the image of Z2 under the real linear 2 isomorphism α : R → C given by (t1,t2) → t1ξ1 + t2ξ2, and hence is a discrete subset of C. Hence, if K is a compact subset of C, then the set

~~K ={ξ ∈ ||ξ|≤diam K} is ﬁnite. Since K ∩ (( \ K ) · K) =∅, it follows that must also act properly discontinuously. Thus we have the holomorphic covering map~~

~~ϒ : C → X = \C, where X is a complex torus. 198 5 Uniformization and Embedding of Riemann Surfaces~~

~~Topologically, X is the torus T2 = S1 × S1. In fact, we have a commutative diagram of C∞ maps α R2 C~~

ϒ0 ϒ β T2 X where α is the real linear isomorphism (and therefore a diffeomorphism) given by 2 (t1,t2) → t1ξ1 + t2ξ2 (we may consider Z , which is mapped onto , as a group of diffeomorphisms of R2 that is given by translations and that acts properly discon- ∞ 2 1 1 tinuously and freely), ϒ0 is the C covering map onto the real torus T = S × S 2πit 2πit given by (t1,t2) → (e 1 ,e 2 ), and β is the induced diffeomorphism.

Example 5.1.11 The holomorphic universal covering of the punctured unit disk ∗ = ∗(0; 1) ={ζ ∈ C | 0 < |ζ | < 1} is the holomorphic mapping ϒ : H → ∗ given by ϒ(z) = exp(2πiz) ∈ ∗(0; 1) for each point z in the upper half-plane H ≡{z ∈ C | Im z>0}. For the inverse image of ∗ under the universal covering map C → C∗ given by z → exp(2πiz) (Example 5.1.9)isH, so the restriction of the covering map to H gives the universal covering of ∗. Given a point −1 z0 = x0 + iy0 ∈ H with x0,y0 ∈ R, the inverse image ϒ (U) of the connected neighborhood 1 1 U ≡ ρ exp(2πit)| 0 <ρ<1,x − 0,x −

~~∗ (an open sector) of ζ0 ≡ ϒ(z0) in (0; 1) is equal to the union of the collection of disjoint domains {Un}n∈Z, where for each n ∈ Z, Un is the inﬁnite half-strip~~

~~Un ≡{z = x + iy | y>0,x0 − (1/2) + n~~

~~∗ ∗ ∗ H → = \H given by z → e2πiz/m and → given by z → zm. 5.1 Holomorphic Covering Spaces 199~~

~~Fig. 5.2 The universal covering of the punctured unit disk~~

~~The mapping z → 1/z gives a biholomorphism ∗ → C\(0; 1) = (0; 1, ∞), so the universal covering of (0; 1, ∞) is the mapping H → (0; 1, ∞) given by − z → 1/e2πiz = e 2πiz.~~

Example 5.1.12 For r ∈ (0, 1), the holomorphic universal covering of the annulus A ≡ (0; r, 1) ={ζ ∈ C | r<|ζ | < 1} is the holomorphic mapping ϒ : H → A given by z → exp(2πiL(z)), where π log r L: H → B ≡ ζ ∈ C | 0 < Im ζ< =− log λ 2π → is the holomorphic branch of the logarithmic function z logλ z with base λ ≡ exp(−2π 2/ log r) >1 (see Example 1.6.2) given by 1 z → · (log |z|+i arccot(Re z/ Im z)) log λ ≡ −1 : R → (and the arccotangent function is given by arccot (cot (0,π)) (0,π)). For B is the inverse image of A under the universal covering map C → C∗ given by ζ → e2πiζ (as in Examples 5.1.9 and 5.1.11), L is a biholomorphism of H onto B (with inverse map given by the exponential function ζ → eζ log λ = λζ with base λ), and ϒ is the composition of the two mappings. iα iθ Fix z0 = R0e 0 ∈ H with R0 > 0 and 0 <α0 <π, and let ζ0 ≡ ϒ(z0) = ρ0e 0 with

~~− 2π log R0 (log R0)(log r) ρ =|ζ |=e 2πα0/ log λ = e(α0/π) log r and θ = =− . 0 0 0 log λ π For each t ∈[0, 1],let~~

~~t −1 2πit γ(t)˜ ≡ z0 · λ = L (L(z0) + t) and γ(t)≡ ϒ(γ(t))˜ = ζ0 · e .~~

As in Examples 5.1.9 and 5.1.11, the deck transformations of the universal covering map B → A are precisely the translations ζ → ζ + n for n ∈ Z. Pulling back to H, we see that the deck transformations of ϒ : H → A are given by − z → L 1(L(z) + n) = exp((L(z) + n) log λ) = λnz 200 5 Uniformization and Embedding of Riemann Surfaces

~~Fig. 5.3 The universal covering of an annulus~~

∼ ∼ for n ∈ Z. Thus π1(A) = Deck(ϒ) = Z with generator 0 : z → λz, and in fact, [ ] π1(A, ζ0) is the inﬁnite cyclic (free) group generated by γ ζ0 . Observe also that the inverse image ϒ−1(U) of the connected neighborhood iθ U ≡{ρe | r<ρ<1,θ0 − π<θ<θ0 + π}⊂A of ζ0 is equal to the union of the disjoint domains {Un}n∈Z, where for each n ∈ Z, n−(1/2) n+(1/2) Un ≡{z ∈ C | Im z>0,λ R0 < |z| <λ R0} and ϒ maps U biholomorphically onto U (see Fig. 5.3). n ∼ ∼ Any proper subgroup of Z = Deck(ϒ) = π1(A, ζ0) must be of the form = mZ for some positive integer m; that is, is the inﬁnite√ cyclic subgroup generated by [γ ]m , that is, by m : z → λmz. Setting Aˇ = (0; m r,1),wehave √ζ0 0 exp(−2π2/ log m r) = λm, and hence we have a holomorphic universal covering map ϒ : H → Aˇ given by 2πi z → exp L(z) . m

The universal covering map ϒ has associated deck transformation group ˇ ∼ Deck(ϒ) = ⊂ Deck(ϒ). Therefore, A = \H and the holomorphic covering space of A associated to the subgroup as in Corollary 5.1.8 is the ﬁnite holomorphic covering Aˇ → A given by exp(2πiL(z)/m) → exp(2πiL(z)); that is, the mapping is given by z → zm. For s,t ∈ R with 0

Exercises for Sect. 5.1 5.1.1 Prove that holomorphic equivalence of holomorphic covering spaces is an equivalence relation. 5.1.2 Prove part (a) of Theorem 5.1.3. 5.1.3 Prove Theorem 5.1.6. 5.1.4 Prove that for r ∈ (0, 1), the universal covering map ϒ : H → (0; r, 1) appearing in Example 5.1.12 extends to a universal covering map H \{0}→ (0; r, 1) (where H and (0; r, 1) are the closures of H and (0; r, 1),respectively). 5.1.5 For each loop γ in C and each point z0 ∈ C \ γ([0, 1]),thewinding number of γ around z0 is given by n(γ ; z0) ≡˜γ(1) −˜γ(0), where γ˜ is a lifting of γ 2πiζ under the universal covering mapping C → C \{z0} given by ζ → z0 + e (see Example 5.1.9). (a) Show that the winding number is an integer that is independent of the choice of the lifting of the loop to the universal cover. ∈ C [ ] → ; (b) Show that for distinct points z0,z1 , the mapping γ z1 n(γ z0) =∼ determines a well-deﬁned group isomorphism π1(C \{z0},z1) −→ Z. (c) Prove that for each loop γ in C and each point z0 ∈ C \ γ([0, 1]), 1 dz n(γ ; z0) = (see Deﬁnition 10.5.3). 2πi γ z − z0

5.1.6 Consideration of winding numbers (see Exercise 5.1.5) yields versions of the Cauchy integral formula and Cauchy’s theorem (cf. Lemma 1.2.1 and Exer- cise 6.7.1), the residue theorem (cf. Exercises 2.5.9, 5.9.5, 6.7.1, and 6.7.6), and the argument principle and Rouché’s theorem (cf. Exercises 2.5.8, 2.9.5, 6.7.1, and 6.7.6) for general loops. Let be a domain in C,letγ be loop in that is path homotopic in to a constant loop, and let C ≡ γ([0, 1]). (a) Cauchy’s theorem. Prove that for each function f ∈ O(), f(z)dz= 0. γ

~~Hint. Apply Theorem 10.5.6. (b) Cauchy integral formula. Prove that for each function f ∈ O() and each point z0 ∈ \ C, 1 f(z) dz = f(z0)n(γ ; z0). 2πi γ z − z0~~

~~(c) Prove that there is a domain 0 such that C ⊂ 0 and γ is path homotopic in 0 to a constant loop. Conclude that in particular,~~

~~n(γ ; z0) = 0 ∀z0 ∈ C \ 0. 202 5 Uniformization and Embedding of Riemann Surfaces~~

~~(d) Residue theorem. Prove that if S is a discrete subset of that is contained in \ C, and θ is a holomorphic 1-form on \ S, then 1 θ = n(γ ; p)respθ 2πi γ p∈S~~

(note that the right-hand side is actually a ﬁnite sum by part (c)). Hint. Using part (c), show that one may assume without loss of generality that S is a ﬁnite set. Writing θ = f(z)dz, and letting gp be the principal part of f about p (i.e., the sum of the negative-exponent terms of the Laurent series about p) for each p ∈ S, one gets a holomorphic −[ ] 1-form θ p∈S gp(z) dz on . (e) Argument principle. Prove that if f is a nontrivial meromorphic function on with zero set Z and pole set P both contained in \ C, then 1 df = ; n(γ z0) ordz0 f. 2πi γ f z0∈Z∪P

(f) Rouché’s theorem. Suppose f and g are nontrivial meromorphic functions on that do not have any zeros or poles in C, and |g| < |f | on C. Let Zf denote the zero set of f , Pf the pole set of f , Zf +g the zero set of f + g, and Pf +g the pole set of f + g. Prove that ; = ; + n(γ z0) ordz0 f n(γ z0) ordz0 (f g). z0∈Zf ∪Pf z0∈Zf +g ∪Pf +g

5.1.7 The goal of this problem to obtain more general (homological) versions of C the facts considered in Exercise 5.1.6. Let be a domain in ,letγ1,...,γm ≡ m [ ] ∈ C be loops in ,letC j=1 γj ( 0, 1 ), and let a1,...,am . Assume that m aj β = 0 j=1 γj ∞ for every C closed 1-form β on (in the language of Sects. 10.6 and 10.7, the 1-cycle ξ = aj γj is homologous to0in). Note that according to Theorem 10.5.6, this is the case if, for example, γj is path homotopic in to a constant loop for each j. (a) Cauchy’s theorem. Prove that for each function f ∈ O(),

~~m aj f(z)dz= 0. j=1 γj 5.1 Holomorphic Covering Spaces 203~~

~~(b) Cauchy integral formula. Prove that for each function f ∈ O() and each point z0 ∈ \ C,~~

~~m m aj f(z) dz = f(z0) aj n(γj ; z0). 2πi z − z0 j=1 γj j=1~~

~~(c) Prove that there is an open set 0 such that C ⊂ 0 and~~

~~m aj n(γj ; z0) = 0 ∀z0 ∈ C \ 0. j=1~~

Hint. Observe that the left-hand side of the above equation is a continuous function of z0. (d) Residue theorem. Prove that if S is a discrete subset of that is contained in \ C, and θ is a holomorphic 1-form on \ S, then

~~m m aj θ = aj n(γj ; p)respθ 2πi j=1 γj p∈S j=1~~

(note that the right-hand side is actually a finite sum by part (c)). Hint. In the proof of part (d) of Exercise 5.1.6, one reduced to the case in which S is finite; but it is not clear that one can do so here (although it is possible to do so using, for example, the homological fact considered in Exercise 5.17.5). Assuming that S is infinite, let {pn} be an enumeration of S, and writing θ = f(z)dz, let gn be the principal part of f about pn for each n (see part (d) of Exercise 5.1.6). Fix a sequence of compact sets {Kν} such that ◦ ⊂ = = ∞ Kν Kν+1 and h(Kν ) Kν for each ν and ν=1 Kν .For each n, apply the Runge approximation theorem to get a holomor- | − | −n phic function hn on such that gn hn < 2 on Kν whenever ∈ −[ ∞ − ] pn Kν . Show that θ n=1(gn hn) dz is a holomorphic 1-form on . (e) Argument principle. Prove that if f is a nontrivial meromorphic function on with zero set Z and pole set P both contained in \ C, then m a df m j = ; aj n(γj z0) ordz0 f. 2πi γj f j=1 z0∈Z∪P j=1 (f) Rouché’s theorem. Suppose f and g are nontrivial meromorphic functions on that do not have any zeros or poles in C, and |g| < |f | on C. Let Zf denote the zero set of f , Pf the pole set of f , Zf +g the zero set of f + g, and Pf +g the pole set of f + g. Prove that 204 5 Uniformization and Embedding of Riemann Surfaces

~~m ; aj n(γj z0) ordz0 f z0∈Zf ∪Pf j=1 m = ; + aj n(γj z0) ordz0 (f g). z0∈Zf +g∪Pf +g j=1~~

~~5.2 The Riemann Mapping Theorem in the Plane~~

~~The goal of this section is the following:~~

~~Theorem 5.2.1 (Riemann mapping theorem in the plane) Any simply connected domain in C with = C is biholomorphic to the unit disk = (0; 1).~~

~~The proof given here is due to Fejér and F. Riesz. We ﬁrst consider the following:~~

Proposition 5.2.2 Let f be a nonvanishing holomorphic function on a simply connected Riemann surface X. Then there exists a holomorphic function h such that eh = f on X (i.e., h is a single-valued holomorphic branch of the function log f ). In particular, for every positive integer q, there is a holomorphic function k = eh/q such that kq = f (i.e., k is a single-valued holomorphic qth root of f ).

Proof By Corollary 10.5.7, the holomorphic 1-form θ = df/f is exact, and hence there is a holomorphic function h on X such that dh = θ (∂h¯ is equal to the (0, 1) part of θ, which is 0). Fixing a point p ∈ X and a single-valued holomorphic branch g of the function log f on a connected neighborhood U of p, we may choose h so that h(p) = g(p). Since dh = θ = dg on U, it follows that the holomorphic functions eh and f agree on U and therefore on X.

~~The biholomorphism will be obtained as a limit of a sequence of holomorphic mappings, so the following observation will be required:~~

Lemma 5.2.3 Let X be a Riemann surface, and let {fν } be a sequence of injective holomorphic functions on X (i.e., a sequence of biholomorphic mappings of X onto open sets in C) that converges uniformly on compact subsets of X to a nonconstant holomorphic function f . Then f is injective.

Proof Given a point a ∈ X, the identity theorem (Theorem 2.2.2) implies that we may choose a relatively compact connected neighborhood D of a with f(∂D)⊂ C \{f(a)}. Hence there exists a δ>0 such that |f − f(a)| >δ on ∂D, and for ν 0, we also get |fν − fν(a)| >δon ∂D.Forν 0, we have δ>|fν(a) − f(a)|, and it follows that f(a)∈ fν(D), because any point ζ in the complement of the domain fν(D) satisﬁes

~~|fν(a) − ζ |≥dist(fν (a), ∂(fν (D))) = dist(fν(a), fν(∂D)) ≥ δ. 5.2 The Riemann Mapping Theorem in the Plane 205~~

Now if a0 and a1 are distinct points in X, then we may choose disjoint neighborhoods D0 and D1 of a0 and a1, respectively, as above with f(aj ) ∈ fν (Dj ) for j = 1, 2 and ν 0. For such an index ν,wealsohavefν(D0) ∩ fν(D1) =∅(since fν is injective), and hence f(a0) = f(a1).

~~The ﬁrst step in the proof of Theorem 5.2.1 is the following:~~

Lemma 5.2.4 Let ⊂ C be a domain such that = C and such that any nonvanishing holomorphic function on has a single-valued holomorphic square root, and let a ∈ . Then there exists an injective holomorphic map f : → ≡ (0; 1) with f(a)= 0.

Proof Fixing a point z0 ∈ C \ , we may form a function h ∈ O() such that 2 [h(z)] = z − z0 for each point z ∈ . In particular, h is injective. Moreover, no two points w,z ∈ can satisfy h(w) =−h(z). For if such a pair exists, then squaring, we get w = z. But then h(w) = h(z) = 0, and squaring again, we get z − z0 = 0, which is impossible, since z0 ∈/ . By the open mapping theorem, h() is a domain, and hence for some δ>0, we have (h(a); δ) ⊂ h(). It follows that h() ∩ (−h(a); δ) =∅.Forifζ ∈ (−h(a); δ), then −ζ ∈ (h(a); δ) ⊂ h(). Since h() ∩ (−h()) =∅,wemusthaveζ/∈ h(). It is now easy to check that for any sufﬁciently large constant R>0, the function f on given by

~~1 h(z) − h(a) f(z)= · ∀z ∈ R h(z) + h(a) has the required properties (see Exercise 5.2.1).~~

~~Theorem 5.2.1 is an immediate consequence of Proposition 5.2.2 and the following version:~~

Theorem 5.2.5 Let ⊂ C be a domain such that = C and such that any nonvanishing holomorphic function on has a single-valued holomorphic square root. Then is biholomorphic to the unit disk ≡ (0; 1).

~~Proof Fixing a point a ∈ , Lemma 5.2.4 implies that~~

~~F ≡{f ∈ O() | f is injective, f()⊂ , and f(a)= 0} =∅.~~

≡ | |∈ ∞] { } F In particular, M supf ∈F f (a) (0, . We may choose a sequence fν in | |→ →∞ such that fν(a) M as ν , and after applying Montel’s theorem (Corol- lary 1.2.7) and passing to a subsequence, we may assume without loss of generality that {fν} converges uniformly on compact subsets of to a holomorphic function f . Corollary 1.2.6 then implies that |f (a)|=M. In particular, M ∈ (0, ∞), f is nonconstant, and therefore by Lemma 5.2.3, f is injective. We also have |f |≤1, and hence the maximum principle implies that |f | < 1on. 206 5 Uniformization and Embedding of Riemann Surfaces

~~It remains to show that f()= . For each point c ∈ ,letc be the automorphism of the disk given by~~

~~z − c c(z) = ∀z ∈ 1 −¯cz~~

(see Proposition 2.14.8). If there exists a point c ∈ \f()⊂ C∗, then the injective holomorphic function c ◦ f : → is nonvanishing, and hence there exists an 2 injective holomorphic function h: → such that h = c ◦ f . Differentiation at a then gives 2|c|1/2|h(a)|=(1 −|c|2)M. The injective holomorphic function g ≡ h(a) ◦ h: → satisﬁes g(a) = 0, and hence g ∈ F. However, we have − |g (a)|=(1 −|c|) 1 · (1 −|c|2)M/(2 |c|) = (1 +|c|)M/(2 |c|)>M.

~~Thus we have arrived at a contradiction, and therefore f is a biholomorphism of onto the disk .~~

~~Corollary 5.2.6 If is a simply connected domain in P1, and P1 \ contains more than one point, then is biholomorphic to (0; 1).~~

Proof Fixing a point ξ ∈ C \ and replacing with its image under the automorphism P1 → P1 given by z → 1/(z − ξ) if necessary, we may assume that is a proper domain in C. The claim now follows from Proposition 5.2.2 and Theo- rem 5.2.5.

~~The proof of the Riemann mapping theorem gives the following:~~

~~Corollary 5.2.7 If is a domain in P1 and every closed C∞ 1-form on is exact, then is biholomorphic to P1, to C, or to (0; 1). In particular, is simply connected.~~

~~The proof is left to the reader (see Exercise 5.2.2).~~

Remark According to the OsgoodÐTaylor–Carathéodory theorem (see [OT] and [C]), if ⊂ P1 is a simply connected region and ∂ is a Jordan curve (see Sect. 5.10), then any biholomorphism of onto the disk may be extended to a homeomorphism of onto . This fact is useful (and appealing), but it is not proved (or applied) in this book. However, a related fact called Schönﬂies’ theorem (Theorem 6.7.1) is considered in Chap. 6.

~~Exercises for Sect. 5.2 5.2.1 Verify that the function f constructed in the proof of Lemma 5.2.4 has the required properties. 5.2.2 Prove Corollary 5.2.7. 5.3 Holomorphic Attachment of Caps 207~~

~~Fig. 5.4 Three disks (or caps) and a domain with three boundary coordinate annuli~~

~~5.3 Holomorphic Attachment of Caps~~

As considered in Sect. 2.3, one may construct examples of Riemann surfaces by holomorphic attachment. In this section, we consider holomorphic attachment of caps (i.e., disks), which in particular, may be used to compactify a suitable domain. Such compactiﬁcations will play an important role in the proof of the Riemann mapping theorem for a Riemann surface. Let X be a complex 1-manifold; let ⊂ X be an open set; let {R0j }j∈J and { } ∞ { ; } R1j j∈J be collections of numbers in (1, );let (Uj ,j ,(0 1/R0j ,R1j )) j∈J ⊂ be a locally ﬁnite family of local holomorphic charts in X with ∂ j∈J Uj ; ∈ ≡ −1 ; ≡ ; ≡ and for each j J ,letAj j ((0 1,R 1j )), Dj (0 R1j ), and Aj ; ⊂ ∩ ⊂ (0 1,R1j ) Dj . Assume that W j∈J Aj for some neighborhood W of ∂ in X, and that for each index j ∈ J , Aj ⊂ , Aj ∩ Ak =∅ for each ∈ \{ } ∩ = −1 ; { } k J j , and Uj ∂ j (∂(0 1)) (note that the sets Uj need not be disjoint; for example, we may let ≡ C\∂(0; 1) ⊂ X ≡ C, U0 = U1 ≡ (0; 1/2, 2), 0 : z → 1/z, 1 : z → z, A0 ≡ (0; 1/2, 1), and A1 ≡ (0; 1, 2)). Then we have the biholomorphism

∼ : ≡ −→= ≡ ⊂ ≡ A Aj A Aj D Dj , j∈J j∈J j∈J where for every index j ∈ J and every point p ∈ Aj , (p) is the image of j (p) → ={ } under the holomorphic inclusion Aj A (see Fig. 5.4, in which J 1, 2, 3 ). The set −1(K ∩ A) is closed in for every compact set K ⊂ D, and the set (K∩A) is closed in D for every compact set K ⊂ . Thus we get the holomorphic 208 5 Uniformization and Embedding of Riemann Surfaces

~~Fig. 5.5 Holomorphic attachment of the caps attachment~~

Y ≡ D ∪ = D /∼, ∈ ∈ ∈ where for j J , p Aj , and z Aj , the images p0 of p and z0 of z under the → ∼ = inclusions Aj ,Aj D satisfy p0 z0 if and only if j (p) z. In particular, : → : → ∈ for the holomorphic inclusions Y and Dj Dj Y for j J ,wehave = \ ; () Y Dj ((0 1)). j∈J In other words, Y is a complex 1-manifold obtained by gluing caps (i.e., disks) at the boundary of (see Fig. 5.5). We say that Y is obtained by holomorphic attachment of caps (or disks) to at the boundary. Observe that Y is compact if and only if X.IfY is connected, then is connected, but the converse is false. Observe also that if in some local holomorphic coordinate neighborhood of some boundary component, is the complement of a concentric circle in an annulus, then we are separating the two pieces and gluing a cap over each piece (as pictured in Fig. 5.5, one cuts a tube into two new tubes, and then places a cap over each of the two new ends). This process of holomorphic tube removal will be considered in more detail in Sect. 5.12. It will be applied in the characterization of a Riemann surface by holomorphic tube attachment (see Sects. 5.13 and 5.14). 5.4 Exhaustion by Domains with Circular Boundary Components 209

Although it is natural to picture D as being glued onto (instead of being glued onto D), is chosen above to map a subset of into the union of disks D (instead of the reverse direction). The mapping is chosen in the above direction in order to make the notation ﬁt with the notation of holomorphic tube attachment and removal (see Sect. 2.3 and Sect. 5.12). The points of view are, of course, equivalent, ∪ = ∪ since we have D −1 D. ∗ ∗ ≤ ∈{ } ∈ Fixing a number Rνj with 1

~~Exercises for Sect. 5.3 5.3.1 In the notation of this section, verify that the natural inclusion of D∗ into D descends to a biholomorphism of Y ∗ onto Y (cf. Exercise 2.3.4).~~

~~5.4 Exhaustion by Domains with Circular Boundary Components~~

In this section, we see that every Riemann surface admits an exhaustion by relatively compact domains for which each boundary component is a concentric circle in a coordinate annulus, a fact that will play an important role in the proof of the Riemann mapping theorem. More precisely, we obtain the following:

Proposition 5.4.1 Given a compact subset K of a Riemann surface X, there exist a compact Riemann surface X, a ﬁnite (possibly empty) collection of disjoint local { ; }n holomorphic charts (Dj , j ,(0 Rj )) j=1 in X with Rj > 1 for each j, and a biholomorphism of a connected relatively compact neighborhood of K in X onto \ n −1 ; X j=1 j ((0 1)). \ n −1 ; ∗ Remark By replacing with the inverse image of X j=1 j ((0 Rj )), ∗ ∈ where Rj (1,Rj ) is sufﬁciently close to 1 for each j, we see that we may { }n ∞ choose so that there exist constants Rj j=1 in (1, ) and disjoint local holo- { ; }n ⊂ ∪···∪ morphic charts (Uj ,j ,(0 1/Rj ,Rj )) j=1 in X such that ∂ U1 Un = ∩ = −1 ; and such that for each j 1,...,n,wehaveUj ∂ j (∂(0 1)) and ≡ −1 ; = ∩ Aj j ((0 1,Rj )) Uj (in particular, is a smooth domain). In fact, in the proof of Proposition 5.4.1, X is constructed by holomorphic attachment of caps to such a domain.

~~We ﬁrst consider the following: 210 5 Uniformization and Embedding of Riemann Surfaces~~

Lemma 5.4.2 Suppose f is a holomorphic function on a Riemann surface X, s0 is a positive regular value of the function ρ ≡|f |, is a connected component of {x ∈ | } X ρ(x) < s0 , and S is a nonempty compact connected√ component of ∂. Then m there exist m ∈ Z>0 and r, t ∈ R with 0

~~Proof At each point x ∈ S,wehave~~

~~2 f(x)(df)x + f(x)(df )x = (dρ )x = 2ρ(x)(dρ)x = 0.~~

Hence (df )x = 0, and by the holomorphic inverse function theorem (Theo- rem 2.4.4), we may form a local holomorphic chart of the form (Ux , x = ∈ ∩ ; ∩ ; f Ux ,Ux) such that x Ux and the sets Ux (0 s0) and Ux ∂(0 s0) are connected. In particular, the connected set −1 ∩ ; = ∩ −1 ; = ∩ −1 −∞ x (Ux (0 s0)) Ux f ((0 s0)) Ux ρ (( ,s0)) −1 must meet, and therefore lie in, the connected component of ρ ((−∞,s0)). ∩ = −1 ∩ ; Thus Ux x (Ux (0 s0)), and similarly, −1 ∩ ; = ∩ −1 ; = ∩ −1 = ∩ = ∩ x (Ux ∂(0 s0)) Ux f (∂(0 s0)) Ux ρ (s0) Ux ∂ Ux S. Fixing a relatively compact connected neighborhood M of S in x∈S Ux ,wesee that f M is a local biholomorphism and −1 −1 M ∩ = M ∩ f ((0; s0)) and S = M ∩ ∂ = M ∩ f (∂(0; s0)). : → ; In particular, the restriction f S S ∂(0 s0) is a local diffeomorphism, and hence by part (a) of Lemma 10.2.11, a ﬁnite C∞ covering map. Therefore f(S)= ∂(0; s0), and by part (b) of Lemma 10.2.11, there exist constants r0 and t0 and a relatively compact connected neighborhood A of S in M such that 0 0, we have a commutative diagram A (0; r, t)

~~f A (0; r0,t0) in which is a biholomorphism and the map (0; r, t) → (0; r0,t0) is the ﬁnite holomorphic covering map given by z → zm. Thus has the required properties.~~

Proof of Proposition 5.4.1 Clearly, we may assume that X is noncompact and that K is nonempty. According to Lemma 2.16.5, there exist a holomorphic function f on a domain Y ⊃ K, a positive regular value s0 of the function |f |, and a 5.5 Koebe Uniformization 211

~~∞ C domain that is a connected component of {x ∈ Y ||f(x)|~~

~~5.5 Koebe Uniformization~~

~~We are now ready to prove the Riemann mapping theorem for a Riemann surface (Theorem 5.0.1). In fact, we will prove something stronger.~~

~~Deﬁnition 5.5.1 An orientable C∞ surface M is planar if every closed C∞ 1-form with compact support in M is exact.~~

Remarks 1. Of course, every exact 1-form θ of class C∞ is C∞-exact; in fact, every potential for θ is of class C∞. 2. Planarity of second countable orientable C∞ surfaces is actually a topological condition (see Exercise 5.5.2).

Lemma 5.5.2 Let M be an orientable C∞ surface. (a) If M is simply connected, then M is planar. (b) If M is planar, then every domain in M is planar. (c) Every domain in the Riemann sphere P1 is planar. 212 5 Uniformization and Embedding of Riemann Surfaces

Proof Part (a) follows immediately from Corollary 10.5.7. For part (b), we simply observe that a closed C∞ 1-form with compact support in a domain in an orientable planar C∞ surface M extends by 0 to a C∞ compactly supported closed differential 1-form on M, and therefore the form is exact. Finally, part (c) follows from (a) and (b).

~~In particular, the Riemann mapping theorem (Theorem 5.0.1) is an immediate consequence of the Riemann mapping theorem in the plane (Theorem 5.2.1) and the following:~~

~~Theorem 5.5.3 (Koebe uniformization theorem [Koe1], [Koe2], [Koe3], [Koe4]) Any planar Riemann surface is biholomorphic to a domain in P1.~~

The proof given here is essentially due to Simha [Sim] and Demailly [De3], although the idea of forming compactiﬁcations of exhausting subdomains is due to Koebe. We ﬁrst consider the compact case.

Lemma 5.5.4 Any compact planar Riemann surface is biholomorphic to P1. In fact, there exists a universal constant C>0 such that if X is any compact planar Riemann surface, p is any point in X, and (D, = z, (0; 1)) is any local holomorphic chart with p ∈ D and (p) = z(p) = 0, then there exists a biholomorphism F : X → P1 for which (i) The meromorphic function F − (1/z) has a removable singularity at p (in particular, F(p)=∞); and ≤ + −1 ≤ (ii) We have dF L2(X\D) C and zdF z dz L2(D) C.

Proof By Theorem 2.10.1, there exists a universal constant C>0 such that if X is any compact planar Riemann surface, p is a point in X, and (D, = z, (0; 1)) is a local holomorphic chart with p ∈ D and (p) = z(p) = 0, then there exists a meromorphic 1-form θ on X with the following properties: (i) The meromorphic 1-form θ is holomorphic on X \{p} and has a pole of order 2 at p; ≤ (ii) We have θ L2(X\D) C; (iii) The meromorphic 1-form θ + z−2 dz on D has at worst a simple pole at p; and + −1 ≤ (iv) We have zθ z dz L2(D) C (note that the meromorphic 1-form provided by Theorem 2.10.1 has been multiplied by −1). For some constant a ∈ C and for some holomorphic function g on (0; 1),wehave 1 a θ = − + + g(z) · dz on D. z2 z In particular, by Stokes’ theorem, 2πia = θ =− dθ = 0. ∂−1((0;1/2)) X\−1((0;1/2)) 5.5 Koebe Uniformization 213

The holomorphic function g has a primitive, i.e., a function G ∈ O((0; 1)) with G = g (one may see this by observing that the closed 1-form g(z)dz must be exact, since (0; 1) is simply connected; or one may apply Theorems 1.3.1 and 1.3.2). −1 The meromorphic function h0 on D given by h0 = z + G(z) then satisﬁes dh0 = ∞ ∂h0 = θ on D \{p}. Fixing a C function η with compact support in D that is equal −1 ∞ to 1 on ((0; 1/2)), we see that θ −d(ηh0) extends from X\{p} to a closed C ∞ 1-form θ1 on X (with θ1 ≡ 0 near p). Since X is planar, there exists a C function h1 ∞ on X with dh1 = θ1. Thus F ≡ ηh0 + h1 is a C function on X \{p} that satisﬁes dF = θ. In particular, ∂F¯ = 0(the(0, 1) part of θ is 0), so F is holomorphic on X \{p}. Furthermore, the holomorphic function F − (1/z) on D \{p} extends to ∞ a C , and therefore holomorphic, function on D (which is equal to G(z) + h1 on −1((0; 1/2))). Hence F is a meromorphic function on X that is holomorphic on X \{p} and that has a simple pole at p, and Proposition 2.5.7 now implies that F is a biholomorphism of X onto P1.TheL2 bounds for dF = θ follow from the construction of θ.

~~Lemma 5.5.5 Let M be an orientable C∞ surface, let r, R ∈ R with 0~~

Proof It is easy to see that N is connected, and Lemma 5.5.2 implies that N is planar if M is planar. For the converse, suppose N is planar and θ is a closed C∞ 1-form with compact support in M. Corollary 10.5.7 provides a C∞ function u on D with = C∞ ≡ du θ D. Fixing a function η with compact support in D such that η 1on −1((0; s)) for some s ∈ (r, R), we see that (after extending ηu by 0 to all of M), ∞ θ0 ≡ θ − d(ηu) is a closed C 1-form with compact support in N. Therefore, since ∞ = N is planar, there exists a C function v on N with dv θ0 N . On the other hand, −1 since dv = θ0 ≡ 0 on the coordinate annulus ((0; r, s)), the function v must be equal to a constant on this set, and we may assume without loss of generality that ∞ this constant is 0. Thus v extends by 0 to a C function v0 on M with dv0 = θ0. Therefore θ = θ0 + d(ηu) = d(v0 + ηu) is exact and M is planar.

Proof of Theorem 5.5.3 By Lemma 5.5.4, it remains to prove that any given noncompact planar Riemann surface X is biholomorphic to a domain in P1.Wemay { }∞ apply Proposition 5.4.1 to get a sequence of nonempty domains ν ν=1 such that = ∞ X ν=1 ν and such that for each ν,wehaveν ν+1 and there exist a compact Riemann surface Xν , a ﬁnite collection of disjoint local holomorphic charts {(D(ν), (ν),(0; R(ν)))}nν in X with {R(ν)}nν in (1, ∞), and a biholomor- j j j j=1 ν j j=1 \ nν (ν) −1 ; phism ν of ν onto Xν j=1( j ) ((0 1)). Let us ﬁx a point p ∈ 1 and a local holomorphic chart (D, = z, (0; 1)) with p ∈ D 1 and (p) = z(p) = 0. For each ν, Lemma 5.5.2 implies that ν is planar and therefore, by Lemma 5.5.5, Xν is planar (by downward induction \ k (ν) −1 ; on k, one sees that the Riemann surface Xν j=1( j ) ((0 1)) is planar for k = 1,...,nν ). Therefore, by Lemma 5.5.4, for some constant C>0 (independent 1 of ν) and for each ν, there exists a biholomorphism Fν : Xν → P such that for 214 5 Uniformization and Embedding of Riemann Surfaces

~~1 fν ≡ Fν ◦ ν : ν → P , the meromorphic function fν − (1/z) on D has a removable singularity at p and the meromorphic 1-form θν ≡ dfν satisﬁes~~

~~−1 2 ≤ + 2 ≤ θν L (ν \D) C and zθν z dz L (D) C.~~

Fixing a point q ∈ 1 \{p}, we may also assume that fν(q) = 0 for each ν ∈ Z>0 (simply by replacing Fν with Fν − Fν(ν(q))). We will show that by passing to a subsequence, we may assume that the sequence of meromorphic functions {fν } converges to the desired biholomorphism of X onto a domain in P1. For this, we observe that for each compact set K ⊂ X \{p} and each ν ∈ Z>0 with ν ⊃ K,we have

~~ = + θν L2(K) θν L2(K\D) θν L2(K∩D) 1 −1 −2 ≤ 2 + + + 2 θν L (ν \D) (zθν z dz) z dz L (K∩D) z L2(K∩D)~~

≤ + C + −2 = C z dz L2(K∩D) AK , minK∩D |z| where clearly, the positive constant AK is independent of the choice of ν.Given a point s ∈ X \{p}, we may ﬁx a local holomorphic chart (U, = ζ,(0; 2)) −1 with U X \{p} and s = (0).Forsomeν0 ∈ Z>0,wehaveU ⊂ ν and 2 ∞ p θν = (∂fν/∂ζ )dζ on U for all ν ≥ ν0. The above L -estimate and the L /L - estimate (Theorem 1.2.4) together imply that the moduli of the holomorphic func- { } tions ∂fν/∂ζ ν≥ν0 are uniformly bounded by some positive constant B (depending on (U, ζ ))onthesetV ≡ −1((0; 1)). Setting s = q, we see that q lies in the set Q of points x ∈ X \{p} for which there exist a neighborhood W and a posi- { } tive integer μ such that fν W ν≥μ is uniformly bounded. Clearly, Q is open, and given a point x ∈ Q, we may choose the above point s and local holomorphic chart (U, = ζ,(0; 2)) so that s ∈ Q and x ∈ V . It follows that Q = X \{p}, and hence {fν} is uniformly bounded on each compact subset of X \{p}. Thus, after applying Montel’s theorem (Corollary 2.11.4) and passing to a suitable subsequence, we may assume without loss of generality that the sequence {fν} converges uniformly on compact subsets of X \{p} to a holomorphic function f on X \{p} with f(q)= 0. Furthermore, for each ν, the holomorphic function gν ≡ fν − (1/z) on D (recall that fν − (1/z) has a removable singularity at p) satisﬁes ∂gν ∂fν 1 z · = z + ∂z L2(D,(i/2)dz∧dz)¯ ∂z z L2(D,(i/2)dz∧dz)¯ 1 1 = √ zθν + dz ≤ C. 2 z L2(D)

~~Thus, after again passing to a subsequence, we may also assume that the sequence of holomorphic functions {z · ∂gν/∂z} converges uniformly on compact subsets of 5.5 Koebe Uniformization 215~~

~~D to a holomorphic function h with h(p) = 0. In particular, on D \{p},~~

~~∂f ∂f ∂g 1 h 1 ← ν = ν − → − as ν →∞. ∂z ∂z ∂z z2 z z2~~

The function h/z (with the singularity at p removed) is holomorphic on D, and therefore this function has a primitive. It follows that f is a meromorphic function on X, f is holomorphic on X \{p}, f has a simple pole at p, and f − (1/z) has a removable singularity at p. In particular, f is nonconstant and Lemma 5.2.3 implies : \{ }→C : → P1 that f X\{p} X p is injective. Since the holomorphic mapping f X satisﬁes f(p)=∞, f is injective on all of X, and therefore f is a biholomorphism of X onto a domain in P1.

~~The Koebe uniformization theorem and Corollary 5.2.7 together give the following:~~

~~Corollary 5.5.6 If X is Riemann surface and every closed C∞ 1-form on X is exact, then X is biholomorphic to P1, to C, or to (0; 1). In particular, X is simply connected.~~

~~We also have the following equivalent version (see Sect. 10.6):~~

~~1 C = Corollary 5.5.7 If X is a Riemann surface and HdeR(X, ) 0, then X is biholomorphic to P1, to C, or to (0; 1). In particular, X is simply connected.~~

Remark As shown in Sects. 10.6 and 10.7, the vanishing of the complex (de Rham) 1 C =∼ 1 C cohomology vector space HdeR(X, ) H (X, ) is equivalent to the vanishing of 1 R =∼ 1 R the real (de Rham) cohomology vector space HdeR(X, ) H (X, ),aswellasto the vanishing of H1(X, R) (in fact, for any choice of a subring A of C containing Z, it is equivalent to the vanishing of H1(X, A)).

Exercises for Sect. 5.5 5.5.1 The goal of this problem is a stronger version of the Koebe uniformization theorem. Prove that there exists a universal constant C>0 such that if X is any planar Riemann surface, p is a point in X, and (D, z = , (0; 1)) is a local holomorphic chart with p ∈ D and (p) = z(p) = 0, then there exists a biholomorphism f of X onto a domain in P1 such that (i) The meromorphic function f − (1/z) has a removable singularity at p (in particular, f(p)=∞); and ≤ + −1 ≤ (ii) We have df L2(X\D) C and zdf z dz L2(D) C. 5.5.2 This problem requires the topological characterization of cohomology in terms of Cechˇ 1-forms appearing in Sect. 10.7.LetM and N be homeomorphic second countable orientable C∞ surfaces. Prove that M is planar if and only if N is planar. 216 5 Uniformization and Embedding of Riemann Surfaces

~~5.6 Automorphisms and Quotients of C~~

The uniformization theorem allows one to obtain a characterization of Riemann surfaces as quotient spaces of P1, C, and = (0; 1). This characterization is considered in Sects. 5.6Ð5.9. We ﬁrst determine, in this section, all of the Riemann surfaces with universal covering space C. For this, we ﬁrst observe the following:

Theorem 5.6.1 Let (a,b) : C → C be the mapping z → az + b for each pair (a, b) ∈ C∗ × C. Then we have the following: ∗ (a) For each pair (a, b) ∈ C × C, the map (a,b) : C → C is an automorphism of C. This automorphism has a ﬁxed point (given by b/(1 − a)) if and only if a = 1. ∗ (b) Aut (C) ={(a,b) | (a, b) ∈ C × C}.

Proof Part (a) is easily veriﬁed. For the proof of (b), suppose ∈ Aut (C). Then (z) →∞as z →∞,so extends to an automorphism of P1 that ﬁxes ∞.In particular, the function z → ((z)−(0))/z is holomorphic on P1 (since the meromorphic function − (0) on P1 has a simple pole at ∞ and a simple zero at 0) and therefore this function is constant.

~~Remark The pair (G ≡ C∗ × C, ·), where~~

~~(a, b) · (c, d) = (ac, ad + b) ∀(a, b), (c, d) ∈ G, is a group, and the map G → Aut (C) given by (a, b) → (a,b) (for (a,b) as above) is a (surjective) group isomorphism (see Exercise 5.6.1).~~

Examples 5.1.9 and 5.1.10, along with the following lemma, imply that any Rie- mann surface with holomorphic universal cover C must be (up to biholomorphism) C, C∗, or a complex torus (this will be examined further in Sect. 5.9).

Lemma 5.6.2 For any subgroup of Aut (C) that acts properly discontinuously and freely, and for ϒ = ϒ : C → X = \C the associated Riemann surface quotient, we have the following: (a) is a group of translations, and either ={0}, or = Zξ for some ξ ∈ C∗, or is a lattice (here, we identify any ζ ∈ C with the translation z → z + ζ ). (b) If ={0}, then X = C and ϒ = IdC. (c) If = Zξ for some ξ ∈ C∗, then ϒ is holomorphically equivalent to the cov- ∗ 2πiz ering ϒ0 : C → C given by z → e (as in Example 5.1.9). More precisely, there is a commutative diagram

~~C α C~~

~~ϒ ϒ 0 β C∗ X 5.6 Automorphisms and Quotients of C 217~~

in which α is the linear isomorphism given by z → ξz and β is a biholomorphism. (d) If is a lattice, then there exists a complex number τ such that Im τ>0 and such that the holomorphic covering space ϒ0 : C → X0 = 0\C corresponding to the lattice 0 ≡ Z + Zτ is holomorphically equivalent to ϒ. More precisely, for some (nonunique) τ ∈ C with Im τ>0, there is a commutative diagram

~~C α C~~

~~ϒ ϒ 0 β X0 X~~

in which α is a linear isomorphism (with α(0) = ) and β is a biholomorphism. Moreover, if X = \C is any complex torus that is biholomorphic to X, then ϒ0 : C → X0 is also equivalent in the above sense to the covering ϒ : C → X.

Proof By Theorem 5.6.1, every automorphism of C must be of the form z → az + b for constants a ∈ C∗ and b ∈ C. Moreover, has a ﬁxed point if and only if a = 1. Therefore, every element of must be a translation, and hence we may identify with a subgroup of C. Now must be a discrete set in C. For if there is a sequence {ξν} in that converges to a point ξ that is not an element of the sequence, then choosing a compact set K containing a neighborhood of {0,ξ}, we see that 0 + ξν ∈ K ∩ (K + ξν) for all large ν, which is impossible since acts properly discontinuously. In particular, if = {0}, then we may choose an element ξ1 ∈ \{0} of minimal modulus. We then have ∩ (Rξ1) = Zξ1.Forifr ∈ R and rξ1 ∈ , then rξ1 − rξ1 (where r denotes the ﬂoor of r) is an element of with modulus (r − r)|ξ1| < |ξ1|, and hence r ∈ Z. The complement of Zξ1 in must be a closed set that does not meet Rξ1. Hence, if this complement is nonempty, then we may choose an element ξ2 ∈ with ξ2 ∈ Zξ1 of minimal distance from [0, 1]·ξ1, and an element uξ1 ∈[0, 1]·ξ1 at which this minimal distance is attained. For any ζ ∈ \ Zξ1 and t ∈ R,wehave

|ζ − tξ1|=|(ζ − tξ1) − (t − t)ξ1|≥|ξ2 − uξ1| > 0, so ξ2 is actually of minimal distance from Rξ1, and this minimal distance is attained at uξ1. In particular, ξ1 and ξ2 are linearly independent over R. It also follows that = Zξ1 + Zξ2. For given an element ξ ∈ , there must exist real numbers t1,t2 ∈ R with ξ = t1ξ1 + t2ξ2. In order to show that ξ ∈ Zξ1 + Zξ2, we may assume without loss of generality that t2 ∈[0, 1), since we may replace ξ with t1ξ1 + (t2 − t2)ξ2. Thus ξ is an element of with distance from Rξ1 at most

~~|ξ − (t1 + t2u)ξ1|=t2|ξ2 − uξ1| < |ξ2 − uξ1|.~~

~~Therefore, by the choice of ξ2,wemusthaveξ ∈ Zξ1. Thus the claim (a) is proved. 218 5 Uniformization and Embedding of Riemann Surfaces~~

The claim (b) is trivial, and the claim (c) is easy to verify (see Exercise 5.6.2). For the proof of (d), let us assume that is a lattice generated by ξ1,ξ2 ∈ C. Since we may replace ξ2 with −ξ2 if necessary, we may choose the generators so that the number τ ≡ ξ2/ξ1 satisﬁes Im τ>0. The linear isomorphism α : C → C given by z → ξ1z then maps the lattice 0 ≡ Z + Zτ onto . It is now easy to verify that we get the desired commutative diagram. Finally, suppose ϒ : C → X is the universal covering of a complex torus X = \C, and γ : X → X is a biholomorphism. Then γ lifts to an automorphism γ˜ of the universal covering C, and hence γ˜ must be of the form z → az + b with a ∈ C∗ and b ∈ C. For each element ξ ∈ ,wehave

aξ =˜γ(ξ)−˜γ(0) ∈ and hence a ⊂ . By symmetry, we then have a−1 ⊂ and hence a = . ≡ ≡ = = Thus ξ1 aξ1,ξ2 aξ2 generate , and ξ2/ξ1 ξ2/ξ1 τ . Thus we get the analogous commutative diagram for C → X.

Exercises for Sect. 5.6 5.6.1 In the notation of Theorem 5.6.1, show that the pair (G ≡ C∗ × C, ·), where (a, b) · (c, d) = (ac, ad + b) for all (a, b), (c, d) ∈ G, is a group, and that the map G → Aut (C) given by (a, b) → (a,b) is a (surjective) group isomorphism. 5.6.2 Prove part (c) of Lemma 5.6.2.

~~5.7 Aut (P1) and Uniqueness of the Quotient~~

Suppose : P1 → P1 is an automorphism of P1. The mapping : P1 → P1 given by (0) = 0, (∞) =∞, and, for z ∈ C∗, ⎧ ⎪ (z)−(0) ∞ ∈ C ⎨ (z)−(∞) if (0), ( ) , (z) = (z) − (0) if (∞) =∞, ⎩⎪ 1 =∞ (z)−(∞) if (0)

(for (0), (∞) ∈ C,weset (−1(∞)) = 1) is then an automorphism of P1. For is a bijective continuous mapping that is holomorphic on C \{−1(∞)},so Riemann’s extension theorem (Theorem 2.2.2) implies that is a bijective holomorphic mapping. In particular, the restriction C : C → C is an automorphism of C with (0) = 0, and hence by Theorem 5.6.1, there exists a constant α ∈ C∗ such that (z) = αz for each z ∈ P1. Solving this equation for (z), we see that there exist constants a,b,c,d ∈ C such that for each point z ∈ C \{−1(∞)},we have cz + d ∈ C∗ and az + b (z) = . cz + d 5.7 Aut (P1) and Uniqueness of the Quotient 219

~~−1 Given two points z1,z2 ∈ C \{ (∞)},wehave~~

az1 + b az2 + b z1 = z2 ⇐⇒ = ⇐⇒ (ad − bc)(z1 − z2) = 0. cz1 + d cz2 + d Choosing the two points to be distinct, we see that ad − bc = 0. Furthermore, for z ∈ C \{−1(∞)},wehave az + b → a/c (=∞ if c = 0) as z →∞ cz + d and (since a(−d/c)+ b = 0forc = 0 and a(−d/c)+ b =∞for c = 0) az + b →∞ as z →−d/c (=∞ if c = 0). cz + d Therefore, (∞) = a/c and (−d/c) =∞. We express this by saying that the equation az + b (z) = cz + d holds for all z ∈ P1.

Deﬁnition 5.7.1 A Möbius transformation is a mapping : P1 → P1 = C ∪{∞} of the form az + b : z → , cz + d where a,b,c,d ∈ C with ad − bc = 0 (here, we set (∞) = a/c and (−d/c) =∞).

Observe that the constants a,b,c,d giving a Möbius transformation as above are not unique, since for any constant λ ∈ C∗, the constants λa, λb, λc, λd give the same Möbius transformation . As shown above, every automorphism of P1 is a Möbius transformation. The converse, as well as other appealing facts, also hold. For the statements, it will be convenient to have the following terminology:

Deﬁnition 5.7.2 Let F = R or C, and let n ∈ Z>0. (a) The general linear group GL(n, F) over F is the group of nonsingular n × n matrices with entries in F (with product given by matrix multiplication). (b) The special linear group SL(n, F) over F is the subgroup of GL(n, F) given by the nonsingular n × n matrices with entries in F and determinant 1. (c) The projectivized special linear group PSL(n, F) over F is the quotient of the group SL(n, F) by the normal subgroup {I,−I}; that is,

~~PSL(n, F) = SL(n, F)/A ∼−A. 220 5 Uniformization and Embedding of Riemann Surfaces~~

~~Remark It is easy to see that GL(n, R),SL(n, R), and PSL(n, R) are subgroups of GL(n, C),SL(n, C), and PSL(n, C), respectively.~~

Theorem 5.7.3 The Möbius transformations have the following properties: (a) The Möbius transformations together with the product given by composition form a group that is equal to the automorphism group of P1. (b) We have a surjective homomorphism GL(2, C) → Aut (P1) given by ab az + b A = → , where (z) = ∀z ∈ P1. cd A A cz + d

(c) The map [A]→A (for A as in (b)) is a well-deﬁned (surjective) isomorphism of PSL(2, C) onto Aut (P1). (d) Every Möbius transformation that is not the identity has exactly one or two ﬁxed points in P1. In particular, up to biholomorphism, the only Riemann surface with universal covering P1 is P1 itself. 1 (e) Given ζ1,ζ2,ζ3,ξ1,ξ2,ξ3 ∈ P with ζi = ζj and ξi = ξj for all i, j ∈{1, 2, 3} with i = j, there exists a unique Möbius transformation with (ζj ) = ξj for j = 1, 2, 3. (f) If C is a circle in P1 (i.e., C is a circle in C or C = L ∪{∞}for some line L in C), then the image of C under any Möbius transformation is also a circle in P1.

~~The proof is left to the reader (see Exercise 5.7.1).~~

~~Exercises for Sect. 5.7 5.7.1 Prove Theorem 5.7.3 (part of it was already proved in the discussion preceding the statement of the theorem).~~

~~5.8 Automorphisms of the Disk~~

~~The following fact will allow us to characterize the automorphism group of the disk:~~

Theorem 5.8.1 (Schwarz’s lemma) Let f : → be a holomorphic mapping of the unit disk ≡ (0; 1) to itself with f(0) = 0. Then we have |f(z)|≤|z| for all z ∈ and |f (0)|≤1. Moreover, if |f(z)|=|z| for some point z ∈ \{0} or |f (0)|=1, then there is a constant c ∈ C such that |c|=1 and f(z)= cz for all z ∈ .

~~Proof Riemann’s extension theorem (Theorem 1.2.10) implies that the (continuous) function h on given by f(z)/z if z ∈ \{0}, h(z) = f (0) if z = 0, 5.8 Automorphisms of the Disk 221~~

| |≤ ∈ is holomorphic. Moreover, we have lim supz→c h(z) 1 for each point c ∂ and therefore, by the maximum principle, |h|≤1on. Furthermore, if |h(z)|=1for some point z ∈ , then h is equal to a constant c of modulus 1.

Theorem 5.8.2 (Cf. Proposition 2.14.8) For each point c ∈ ≡ (0; 1), let 1 1 c : P → P be the map given by − z c 1 c(z) = ∀z ∈ P . 1 −¯cz Then we have the following: ∈ = = (a) For each point c , c is a Möbius transformation, c(c) 0, c(0) −| |2 = −| |2 1 c , and c(c) 1/(1 c ). (b) For each choice of c ∈ and b ∈ ∂, the Möbius transformation ≡ bc −1 ¯ ¯ has inverse z → (z) = −c(bz) = b−bc(z), maps onto , and the : → restriction is an automorphism of . Furthermore, every automorphism of is of this form.

2 Proof For c ∈ ,wehave1· 1 −¯c · c = 1 −|c| > 0, so c is a Möbius transformation. The remaining properties in (a) are left to the reader (see Exercise 2.14.5). We have − 1 −c 1 1 1 c = · , −¯c 1 1 −|c|2 c¯ 1 and hence for each point z ∈ P1, + −1 z c (z) = = −c(z). c 1 +¯cz

Therefore, for b ∈ ∂ and = bc,wehave ¯ + + −1 ¯ bz c ¯ z bc ¯ 1 (z) = − (bz) = = b = b− (z) ∀z ∈ P . c + bc 1 +¯cbz¯ 1 bcz It is easy to check that (∂) ⊂ ∂, and hence by the above, we have (∂) = ∂. In particular, () must be either or P1 \ , and it follows that = = ∈ : → () , since (c) 0 . Therefore, is an automorphism. Finally, suppose is an arbitrary automorphism of . Then 0 ≡ (0) ◦ is an automorphism of that maps 0 to 0. But then

−1 · = −1 ◦ = (0 ) (0) 0(0) (0 0) (0) 1. Since by the Schwarz lemma, each of the factors in the ﬁrst of the above expressions has modulus at most 1, each must have modulus equal to 1. Therefore, again by the Schwarz lemma, there is a constant b ∈ ∂ such that 0(z) = bz for all z ∈ . ∈ = = Therefore, for all z , (z) −(0)(bz) b−b(¯ 0)(z). 222 5 Uniformization and Embedding of Riemann Surfaces

~~It is often more convenient to work with the upper half-plane~~

~~H ={z ∈ C | Im z>0}, which is biholomorphic to the unit disk .~~

Theorem 5.8.3 For H ={z ∈ C | Im z>0} and = (0; 1), we have the following: : P1 → P1 → 1−z = −iz+i (a) The mapping given by z i 1+z z+1 is a Möbius transformation that maps onto H and ∂ onto R ∪{∞}. (b) The set GL+(2, R) ≡{A ∈ GL(2, R) | det A>0} is a subgroup of GL(2, R), and we have a surjective homomorphism GL+(2, R) → Aut (H) given by ab A = → , cd A H

~~where A is the Möbius transformation given by az + b (z) = ∀z ∈ P1. A cz + d~~

~~(c) The map [A]→AH (for A as in (b) and for [A] the equivalence class represented by A) is a well-deﬁned surjective isomorphism of PSL(2, R) onto Aut (H).~~

Proof We have (−i)(1) − (1)(i) =−2i = 0, so is a Möbius transformation. Moreover, (1) = 0, (i) = 1, and (−1) =∞. Therefore, by Theorem 5.7.3, maps ∂, which is the unique circle containing 1,i,−1, onto the unique circle containing 0, 1, ∞, i.e., onto R ∪{∞}. In particular, must map onto exactly one of the two connected components of P1 \ (R ∪{∞}), and therefore, since (0) = i ∈H,wehave () = H. Thus (a) is proved. ab = ∈ + R If A cd GL (2, ), then clearly, the corresponding Möbius transformation az + b : z → A cz + d maps the circle R∪{∞}onto the circle R∪{∞}, and therefore A maps H onto one of the connected components of P1 \ (R ∪{∞}). On the other hand, since ad − bc = det A>0, we have ai + b ac + bd ad − bc A(i) = = + i ∈ H. ci + d c2 + d2 c2 + d2

Thus AH ∈ Aut (H). For the converse, observe that any automorphism of H is of the form ◦ ◦ −1 H for some automorphism of , and hence by Theorem 5.8.2 and Theo- rem 5.7.3, the automorphism must be of the form H for some Möbius transformation with (H) = H and (R ∪{∞}) = R ∪{∞}. Guided by the arguments 5.8 Automorphisms of the Disk 223 in Sect. 5.7, let us set ⎧ − ⎪ μ μ(0) if (0), (∞) ∈ R, ⎪ 1 −(∞) ⎨ ab − A = = μ μ(0) if (∞) =∞, cd ⎪ 01 ⎪ ⎩ 0 μ =∞ 1 −(∞) if (0) , where μ =±1 is chosen so that det A>0, i.e., so that A ∈ GL+(2, R). Then A ◦ is a Möbius transformation that maps 0 to 0, ∞ to ∞, R ∪{∞}to R ∪{∞}, and H to H, and hence A ◦ must be of the form z → rz for some r ∈ (0, ∞) ⊂ R. − r 0 = 1 ∈ + R = Therefore, for B A 01 GL (2, ),wehave B , and applying Theo- rem 5.7.3, we get (b) and (c).

Remark Clearly, Theorem 5.7.3 provided the recipe for the construction of the biholomorphism of onto H; we only needed to choose a Möbius transformation mapping three points arranged in counterclockwise order on ∂ to three corresponding points arranged in the same order on the circle R ∪{∞}oriented with H on the left. On the other hand, one can also verify the claims in the theorem directly without applying Theorem 5.7.3.

~~Next, we see that Example 5.1.11 and Example 5.1.12 are the only examples of Riemann surfaces with universal covering =∼ H and inﬁnite cyclic fundamental group.~~

Lemma 5.8.4 If X is a Riemann surface with universal covering ≡ (0; 1) =∼ ∼ H and fundamental group π1(X) = Z, then X is biholomorphic to the punctured disk ∗ ≡ ∗(0; 1) or to the annulus (0; r, 1) for some r ∈ (0, 1).

Proof We have the universal covering map ϒ : H → X = \H for some infinite cyclic subgroup of Aut (H) that acts properly discontinuously and freely. By The- orem 5.8.3, is generated by H for some Möbius transformation of the form : z → (az + b)/(cz + d) with a,b,c,d ∈ R and ad − bc = 1. In particular, has one or two fixed points in P1, and since has no fixed points in H and (z)¯ = (z) for each point z ∈ P1, the fixed point or points must lie in R ∪{∞}. Suppose has exactly one fixed point λ ∈ R ∪{∞}. The Möbius transformation −1 ∈ R z−λ if λ , 1 : z → z if (∞) =∞, restricts to an automorphism of H and maps λ to ∞. Hence the Möbius transforma- ◦ ◦ −1 H tion 1 1 also restricts to an automorphism of and has the unique fixed ∞ ◦ ◦ −1 → + point . It follows that 1 1 is a translation z z β for some constant β ∈ R∗ = R \{0}. By replacing with −1 (which has the same unique fixed point λ) if necessary, we may assume that β>0. Thus the Möbius transformation 224 5 Uniformization and Embedding of Riemann Surfaces

−1 : z → β 1(z) restricts to an automorphism of H and the Möbius transforma- −1 ∼ tion 0 = ◦ ◦ is given by z → z + 1. Letting 0 = Z be the subgroup of Aut(H) generated by 0H, we get a commutative diagram of holomorphic mappings

~~H H~~

ϒϒ 0 X 0 ∗ 2πiz where ϒ0 is the universal covering map z → e as in Example 5.1.11, and 0 is a biholomorphism. Let us now assume that has exactly two fixed points μ,ν ∈ R ∪{∞}, which we may order so that μ>νif μ,ν ∈ R. The Möbius transformation ⎧ ⎪ z−μ ∈ R ⎨ z−ν if μ,ν , : z → z − μ if ν =∞, ⎩⎪ −1 =∞ z−ν if μ , restricts to an automorphism of H and maps μ to 0 and ν to ∞. Hence the Möbius −1 transformation 0 = ◦ ◦ also restricts to an automorphism of H and has fixed points 0 and ∞. It follows that 0 is a dilation z → λz for some constant λ>0, and by replacing with −1 (which has the same fixed points) if necessary, ∼ we may assume that λ>1. Letting 0 = Z be the subgroup of Aut (H) generated by 2 0H and r ≡ exp(−2π / log λ) ∈ (0, 1), we get a commutative diagram of holomorphic mappings

~~H H~~

ϒϒ 0 X 0 (0; r, 1) where ϒ0 is the universal covering map z → exp(2πiL(z)) (where L is the branch → 1 | |+ = + ∈ H of logλ z given by z log λ (log z i arccot(x/y)) for each z x iy )asin Example 5.1.12, and 0 is a biholomorphism.

~~Exercises for Sect. 5.8 5.8.1 Prove part (a) of Theorem 5.8.2. 5.8.2 Prove that two annuli (0; r, 1) and (0; s,1) with r, s ∈ (0, 1) are biholomorphic if and only if r = s.~~

~~5.9 Classiﬁcation of Riemann Surfaces as Quotient Spaces~~

~~The results of the previous sections may be summarized as follows: 5.9 Classiﬁcation of Riemann Surfaces as Quotient Spaces 225~~

Theorem 5.9.1 Up to biholomorphism, every Riemann surface X is biholomorphic to exactly one of the following: (i) P1; (ii) C; (iii) C∗; (iv) \C, where = Z + τZ and τ ∈ C with Im τ>0; (v) \, where is a subgroup of − 1 z c Aut () = bc | b ∈ S ,c∈ , and c : z → 1 −¯cz

~~that acts properly discontinuously and freely (equivalently, \H, where is a subgroup of az + b Aut (H) = : z → | a,b,c,d ∈ R with ad − bc = 1 =∼ PSL(2, R) cz + d~~

~~that acts properly discontinuously and freely).~~

The Riemann mapping theorem also allows one to characterize Riemann surfaces according to their fundamental groups. For example, the Riemann mapping theorem, Lemma 5.6.2, and Lemma 5.8.4 give the following characterization of Riemann surfaces with inﬁnite cyclic fundamental group:

∼ Theorem 5.9.2 If X is a Riemann surface with fundamental group π1(X) = Z, then X is biholomorphic to exactly one of the following: (i) C∗ = C \{0}; (ii) ∗ = ∗(0; 1) ={z ∈ C | 0 < |z| < 1}; (iii) the annulus (0; r, 1) ={z ∈ C | r<|z| < 1} for some r ∈ (0, 1).

~~Remark According to Exercise 5.8.2, the constant r in (iii) is unique (for a given X).~~

~~Deﬁnition 5.9.3 Let X be a Riemann surface with universal covering X. Then X is called (i) elliptic if X =∼ P1; (ii) parabolic if X =∼ C; (iii) hyperbolic if X =∼ =∼ H.~~

Remarks 1. By the Riemann mapping theorem, every Riemann surface is of exactly one of the above types. 2. A complex torus, that is, a compact Riemann surface with universal covering C, is also called an elliptic curve (which should not be confused with the elliptic Riemann surface P1). 226 5 Uniformization and Embedding of Riemann Surfaces

Exercises for Sect. 5.9 5.9.1 Let = Z + Zτ and = Z + Zτ be lattices in C with Im τ,Im τ > 0(as in Lemma 5.6.2). Prove that the complex tori X ≡ \C and X ≡ \C are = aτ+b biholomorphic if and only if τ cτ+d for some matrix ab ∈ SL(2, Z) ={2 × 2 integral matrices in SL(2, R)}. cd

Find an a example of a pair of complex tori that are not biholomorphic. 5.9.2 Prove that if a Riemann surface X has noncyclic Abelian fundamental group, then X is biholomorphic to a complex torus. 5.9.3 A Kähler form ω is said to be of constant curvature λ ∈ R if iω = λω (see Definition 2.12.1). (a) Show that the Kähler form ω ≡ y−2 dx ∧ dy on H is of constant curvature −1. Show also that if is a group of automorphisms acting properly discontinuously and freely on H, then ∗ω = ω for each element ∈ . (b) Let X be a Riemann surface. Using part (a) along with Examples 2.12.2 and 2.12.3, conclude that: (i) If X is elliptic, then X admits a Kähler form of constant curvature 1. (ii) If X is parabolic, then X admits a Kähler form of constant curvature 0. (iii) If X is hyperbolic, then X admits a Kähler form of constant curvature −1. (c) Let X be a compact Riemann surface. Prove that the converse of each of the statements (i)Ð(iii) in part (b) holds for X; that is, prove that X is elliptic (parabolic, hyperbolic) if and only if X admits a Kähler form ω of constant curvature 1 (respectively, 0, −1). Show that the converses do not hold in general for open Riemann surfaces. (d) Let X be a compact Riemann surface of genus g. Using part (c) above and the results of Chap. 4 (other approaches will be considered in Exer- cise 5.16.1 and Sect. 5.22), prove that (i) X is elliptic if and only if g = 0. (ii) X is parabolic if and only if g = 1. (iii) X is hyperbolic if and only if g>1. Hint. According to Exercise 4.6.3, the canonical line bundle of a compact Riemann surface of genus 1 is trivial. 5.9.4 An element g of a group is called a torsion element if g is of finite order (i.e., gm = e for some positive integer m). A group in which only the identity is a torsion element is said to be torsion-free. (a) Prove that every torsion element of Aut (H) must have fixed point. (b) Prove that the fundamental group of every Riemann surface is torsion- free. 5.9.5 The uniformization theorem leads to a natural generalization of the notion of the winding number of a loop around a point (see Exercises 5.1.5, 5.1.6, and 5.1.7). Let X be an open Riemann surface, let ϒ : X → X be the universal 5.9 Classification of Riemann Surfaces as Quotient Spaces 227

covering of X,letγ be a loop in X that is path homotopic to a constant loop, let C = γ([0, 1]), and let γ˜ be a lifting of γ toaloopinX. In particular, we have X = H or C. Given a point p ∈ X \ C,weset nX(γ ; p) ≡ n(γ˜; z0) ∈ Z −1 z0∈ϒ (p)

(the right-hand side is a ﬁnite sum by Exercise 5.1.6). (a) Prove that if p ∈ X \ C and θ is a holomorphic 1-form on X \{p} with respθ = 1, then 1 nX(γ ; p) = θ. 2πi γ Show that such a 1-form θ always exists (see Corollary 2.15.4 and Exer- cise 2.15.4), and conclude that nX(γ ; p) is independent of the choice of the holomorphic universal covering map ϒ and the lifting γ˜ . (b) Prove that there is a domain such that C ⊂ X and γ is path homotopic in to a constant loop. Conclude that in particular, nX(γ ; p) = 0 for each point p ∈ X \ . (c) Residue theorem. Prove that if S is a discrete subset of X that is contained in X \ C, and θ is a holomorphic 1-form on X \ S, then 1 θ = nX(γ ; p)respθ 2πi γ p∈S

(note that the right-hand side is actually a ﬁnite sum by part (b)). (d) Argument principle. Prove that if f is a nontrivial meromorphic function on X with zero set Z and pole set P both contained in X \ C, then 1 df = nX(γ ; p)ordp f. 2πi f γ p∈Z∪P

(e) Rouché’s theorem. Suppose f and g are nontrivial meromorphic functions on X that do not have any zeros or poles in C, and |g| < |f | on C. Let Zf denote the zero set of f , Pf the pole set of f , Zf +g the zero set of f + g, and Pf +g the pole set of f + g. Prove that nX(γ ; p)ordp f = nX(γ ; p)ordp(f + g). p∈Zf ∪Pf p∈Zf +g∪Pf +g

(f) Suppose X is simply connected, S is a discrete subset of X, and θ is a holomorphic 1-form on X \ S. Prove that there exists a function f ∈ O(X \ S) with df = θ (i.e., θ is exact) if and only if respθ = 0for each point p ∈ S. 228 5 Uniformization and Embedding of Riemann Surfaces

~~5.10 Smooth Jordan Curves and Homology~~

Sections 5.10Ð5.17 rely on the Koebe uniformization theorem (Theorem 5.5.3)as well as elementary facts concerning the first homology and cohomology groups of a second countable C∞ surface (as reviewed in Sects. 10.6 and 10.7), but we will avoid using the classification of Riemann surfaces as quotient spaces considered in Sects. 5.6Ð5.9. The main goal is the fact that every Riemann surface may be obtained by holomorphic attachment of tubes to a planar domain (as in Sects. 2.3 and 5.12). The idea of the proof is to find holomorphic annuli (i.e., tubes) in the Riemann surface, and then to inductively remove them. Reversing the process then gives the desired construction of the given Riemann surface. Each annulus to be removed is found in a small neighborhood of some diffeomorphic image of the circle S1 (that is, a smooth Jordan curve) that represents a nonzero homology class (in fact, one may take the holomorphic annulus to be the small neighborhood, but it will be slightly easier to allow for other possibilities). In this section, we consider the first step, which is to show that the first homology group of a second countable C∞ surface is generated by smooth Jordan curves.

Deﬁnition 5.10.1 Let M be a topological surface. By a Jordan curve (or a Jordan loop or a simple loop or a simple closed curve)inM, we will mean either a loop = [ ] γ :[a,b]→M for which γ [a,b) is injective or the image C γ( a,b ) of such a loop. We will often identify γ with the homeomorphism S1 → C given by e2πit → γ(a+ t(b − a)) for each t ∈[0, 1] (recall that a bijective continuous mapping of compact Hausdorff spaces is a homeomorphism).

Definition 5.10.2 Let M be a C∞ surface and let γ :[a,b]→M be a path. (a) For c ∈ (a, b), we say that γ is smooth at c if on some neighborhood of c, γ is a C∞ map with nonvanishing tangent vector γ˙ . We say that γ is smooth from the right at a if γ extends to a path in M that is defined on some interval [a − ,b] for some >0 and that is smooth at a. Similarly, we say that γ is smooth from the left at b if γ extends to a path that is defined on some interval [a,b + ] for some >0 and that is smooth at b. We call γ a smooth path (or a smooth curve) if γ is smooth at each point in (a, b), smooth from the right at a, and smooth from the left at b. We also call the image of a smooth path a smooth path (or a smooth curve). (b) We say that γ is loop-smooth at a (or, equivalently, at b)ifγ(a)= γ(b) (i.e., γ is a loop) and on some neighborhood of 1 = e2πi0 = e2πi(1) ∈ C in S1,the associated map γˆ : S1 → M given by e2πit → γ(a+ t(b− a)) for t ∈[0, 1] is ∞ C with nonvanishing tangent map γˆ∗. Equivalently, on some neighborhood of +Z − R → → − t−a − a (b a),themap M given by t γ(t b−a (b a)) (where x is the floor of x ∈ R)isC∞ with nonvanishing tangent vector (see Exercise 5.10.1). We call γ a smooth loop (or a smooth closed curve)ifγ is a smooth path that is loop-smooth at a and b (in particular, γ is a loop); that is, the associated map S1 → M is a C∞ immersion (see Definition 9.9.3). We also call the image of a smooth loop a smooth loop (or a smooth closed curve). 5.10 Smooth Jordan Curves and Homology 229

~~(c) We call γ a piecewise smooth path (or piecewise smooth curve)inM if for = ··· = some partition a t0~~

Remarks 1. As usual, we assume that all given loops and paths have domain [0, 1] or S1 unless otherwise noted. 2. The Jordan curves in a surface M are precisely those subsets that are homeomorphic to S1. 3. The smooth Jordan curves in a smooth surface M are precisely the 1-dimensional compact connected C∞ submanifolds (see Theorem 9.10.1). 4. A smooth path γ :[a,b]→M may be a loop and still not be a smooth loop. There is the added requirement that γ meet itself smoothly at the base point γ(a)= γ(b). 5. If a path γ :[a,b]→M is smooth from the right at c, then we may deﬁne the right-hand tangent vector γ˙+(c) to be the tangent vector at c of some smooth extension of γ [c,c+) for some >0 to a smooth path on a neighborhood of c.Sim- ilarly, if γ is smooth from the left at c, then we get a left-hand tangent vector γ˙−(c). Actually, one only needs γ to be C1 from the right at c in the former case, and from the left at c in the latter case. If γ is a loop that is loop-smooth at a (and b), then we may deﬁne the tangent vector at the base point by γ(a)˙ =˙γ(b)≡˙γ+(a) =˙γ−(b). 6. The same terminology is applied to paths in 2-dimensional topological and C∞ manifolds.

We will now see that one may smooth a piecewise smooth injective or Jordan curve. For the proof, it will be convenient to ﬁrst make the pieces of the curve transverse at the nonsmooth points, and to then choose special coordinates in a neighborhood of a corner at which two pieces of the curve meet transversely. Two curves in a C∞ surface are transverse at a point of intersection that is a smooth point for each of the curves if their tangent vectors at the point are linearly independent. The following two lemmas accomplish these steps. 230 5 Uniformization and Embedding of Riemann Surfaces

Lemma 5.10.3 Let α :[a,b]→M and β :[c,d]→M be smooth paths in a C∞ surface M such that 0 ∈ (a, b)∩(c, d), α(0) = β(0), and α(˙ 0) and β(˙ 0) are linearly independent (i.e., α and β are transverse at this point). Then there exists a local C∞ chart (W,,W) in M such that for some r>0, we have [−r, r]⊂α−1(W) ∩ β−1(W) and

~~(α(t)) = (t, 0) and (β(t)) = (0,t) ∀t ∈[−r, r].~~

Proof The claim is local, so we may assume without loss of generality that M is 2 an open subset of R with α(0) = β(0) = (0, 0) ∈ M. Writing α = (α1,α2) and β = (β1,β2), we see that since α (0) = 0(α (0) and β (0) are linearly independent), = we may apply a linear change of coordinates to get α1(0) 0. Hence, by the 1- dimensional C∞ inverse function theorem (the easy case of Theorem 9.9.1), we may assume that α1 extends to a diffeomorphism of an open interval containing [a,b] → −1 − −1 onto an open interval containing 0. The map (x, y) (α1 (x), y α2(α1 (x))) is then a diffeomorphism of neighborhoods of (0, 0) in R2 (with C∞ inverse mapping (u, v) → (α1(u), v + α2(u))) and hence, applying this diffeomorphism to get a change of coordinates, we may assume that α(t) = (t, 0) ∈ R2 for each t ∈[a,b]. = Similarly, since α (0) and β (0) are linearly independent, we have β2(0) 0, and −1 −1 hence the mapping (x, y) → (x − β1(β (y)), β (y)) is a diffeomorphism of ∞ 2 2 neighborhoods of (0, 0) (with C inverse mapping (u, v) → (u + β1(v), β2(v))). − −1 −1 = This diffeomorphism maps the point (t, 0) to (t β1(β2 (0)), β2 (0)) (t, 0). Therefore, after applying this diffeomorphism, we get α(t) = (t, 0) and β(t) = (0,t) for all t near 0.

Lemma 5.10.4 Let M be a C∞ surface, let α : I =[a,b]→M be an injective piecewise smooth path for which 0 ∈ (a, b) and α is smooth at each point in (a, b) \ {0}, let V be a neighborhood of p = α(0) in M, and let U be a neighborhood of 0 in (a, b) ∩ α−1(V ). Then there exists an injective piecewise smooth path β :[a,b]→ M such that for some partition a = t0

~~Remark In fact, as will be clear from the proof, we may choose β so that m ≤ 4 and β = α on [a,0] (see Fig. 5.6).~~

Proof of Lemma 5.10.4 Since α is injective, we may ﬁx a constant R>0 and a ∞ local C chart (W, = (1,2), (−R,R) × (−R,R)) in M with p ∈ W ⊂ V , (p) = (0, 0), and α−1(W) ⊂ U.Forsomer>0 with [−r, r]⊂α−1(W),we :[− ]→ = = have smooth paths α1,α2 r, r W with α1 [−r,0] α [−r,0] and α2 [0,r] ≡ = ≡ = α [0,r]. Setting τ (α1) (τ1,τ2) and ρ (α2) (ρ1,ρ2) and applying Lemma 5.10.3 (to α1 and an arbitrary smooth path transverse to α1 at p), we see 2 that we may choose so that τ(t)= (τ1(t), τ2(t)) = (t, 0) ∈ R for each t ∈[−r, r] 5.10 Smooth Jordan Curves and Homology 231

~~Fig. 5.6 Modifying an injective piecewise smooth path to get transversality~~

~~Fig. 5.7 The two possible local representations of the path and their local modiﬁcations~~

~~(in particular, r~~

~~−1 α([−s,s]) ⊂ α1([−s,s]) ∪ α2([−s,s]) ⊂ ((−S,S) × (−S,S)) ⊂ W \ α([a,−r]∪[r, b]).~~

The lemma is trivial if α˙ −(0) and α˙+(0) are linearly independent, so we may = = assume that ρ (0) (ρ1(0), 0). Since α2 is smooth, we then have ρ1(0) 0, and [− ] hence we may assume that ρ1 is nonvanishing on r, r . We now consider the two possible cases (see Fig. 5.7). If ρ2 = 0 at all points in [0,r] that are near 0, then, after shrinking r, we may assume that ρ2 ≡ 0on[0,r], [− ] and since α is injective, we must then have ρ1 > 0on r, r . In this case, we let γ :[0,s]→[0,ρ1(s)]×[0,s]⊂[−S,S]×[−S,S] be the injective piecewise smooth path from (0, 0) to ρ(s)= (ρ1(s), 0) given by the vertical line segment path in R2 from (0, 0) to (0,s), followed by the line segment path from (0,s) to ρ(s), where each segment is parametrized so as to be smooth. If there exist nonnegative points arbitrarily close to 0 at which ρ2 is nonzero, then we may choose s so that = = ρ2(s) 0 and ρ2(s) 0. We may then let

γ :[0,s]→[−|ρ1(s)|, |ρ1(s)|] × [−|ρ2(s)|, |ρ2(s)|] ⊂ [−S,S]×[−S,S] be the injective piecewise smooth path from (0, 0) to ρ(s) given by the vertical line segment path from (0, 0) to (0,ρ2(s)), followed by the horizontal line segment path from (0,ρ2(s)) to ρ(s), where again, each segment is parametrized so as to be smooth. In particular, in this case, since ρ1 is either strictly increasing on [−r, r] or strictly decreasing on [−r, r], γ will meet ρ([s,r]) only at the point γ(s)= ρ(s). In either case, it is easy to verify that the piecewise smooth path β :[a,b]→M 232 5 Uniformization and Embedding of Riemann Surfaces

~~Fig. 5.8 Smoothing a piecewise smooth Jordan curve given by β ≡ α on [a,0]∪[s,b] and β ≡ −1(γ ) on [0,s] has the required properties.~~

~~Lemma 5.10.5 Let M be a C∞ surface, let α : I =[a,b]→M be an injective piecewise smooth path or a piecewise smooth Jordan curve that is loop-smooth at a and b, let a = t0~~

In other words, an injective piecewise smooth path (or a piecewise smooth Jor- dan curve that is loop-smooth at the base point) is path homotopic to an injective smooth path (respectively, a smooth Jordan curve) that agrees with the original path outside an arbitrarily small neighborhood of the nonsmooth points, as well as at the nonsmooth points (see Fig. 5.8).

Proof of Lemma 5.10.5 Clearly, we may assume that the neighborhoods {Vj } are disjoint. Next we observe that we may assume without loss of generality that α˙ −(tj ) and α˙ +(tj ) are linearly independent (i.e., that we have transversality) for each j = 1,...,m−1. For we may choose a constant u>0 so that 2u

{}k − with Vi i=1, where if si tj is one of the points in Uj at which βj is not smooth, ⊂ + ⊂ then Ui Uj tj and Vi Vj are suitable sufﬁciently small neighborhoods of si and α(sˆ i), respectively, we may assume that α has the desired transversality property. By Lemma 5.10.3,forsomer>0, we may choose disjoint local C∞ charts { − + × − + }m−1 (Wj ,j ,(tj r, tj r) (tj r, tj r)) j=1 such that for each j = 1,...,m− 1, we have Wj Vj ,

~~−1 (tj − r, tj + r)⊂ (tj−1,tj+1) ∩ α (Wj ) ∩ Uj , and (t, tj ) if t ∈ (tj − r, tj ], j (α(t)) = (tj ,t) if t ∈[tj ,tj + r)~~

(one obtains j by applying the lemma to the curve t → α(t + tj ) and then letting j be the sum of the resulting local chart with the point (tj ,tj )). Since either = α [a,b] is injective or α [a,b) is injective with α(a) α(b), we may choose a constant s ∈ (0,r)such that for each j = 1,...,m− 1, −1 [ − + ]×[ − + ] ∩ [ − ]∪[ + ] =∅ j ( tj s,tj s tj s,tj s ) α( a,tj r tj r, b ) . Fixing a nondecreasing C∞ function λ: R →[0, ∞) such that λ ≡ 0 on a neighborhood of (−∞, 0] and λ ≡ 1 on a neighborhood of [s,∞) (for example, for a nonneg- C∞ = s ative function χ with compact support in (0,s) such that q 0 χ(v)dv > 0, : → −1 t the function λ t q 0 χ(v)dv has the required properties), we see that the path β :[a,b]→M given by ⎧ ⎪ ∈[ ]\ m−1[ + ] ⎨α(t) if t a,b j=1 tj ,tj s , − t → 1((1 − λ(t − t )) · (t, t ) ⎩⎪ j j j + λ(t − tj ) · (tj ,t)) if t ∈[tj ,tj + s] for some j, has the required properties. The details of the veriﬁcation are left to the reader (see Exercise 5.10.2).

~~We record here for later use (see Sect. 5.15) the following consequence of Lemma 5.10.5:~~

~~Lemma 5.10.6 If p and q are distinct points in a C∞ surface M, then there exists an injective smooth path in M from p to q.~~

Proof Fixing a point p ∈ M, we must show that the set N of points x ∈ M \{p} for which there is an injective smooth path in M from p to x is equal to the set M \{p}. Clearly, N = ∅. Given a point x0 ∈ M \{p}, we may choose a lo- ∞ cal C chart (U,,(0; 2)) in M with (x0) = 0 and p/∈ U, and we may set D ≡ −1((0; 1)).IfD meets N, then there exists an injective smooth path α 234 5 Uniformization and Embedding of Riemann Surfaces

−1 in M from p to a point in D, and we may set t0 ≡ min α (D) ∈ (0, 1). Hence, ∈ given a point x D, the product of suitable reparametrizations of the paths α [0,t0] −1 and t → ((1 − t)(α(t0)) + t(x)) is an injective piecewise smooth path in M from p to x. Applying Lemma 5.10.5, we get an injective smooth path in M from p to x. Thus D ⊂ N. From this, it follows that N is open with empty boundary in the connected set M \{p}, and hence that N = M \{p}.

~~We now come to the main goal of this section:~~

Proposition 5.10.7 Let M be a C∞ surface. Then, for every (continuous) loop α in M, there exist smooth Jordan curves α1,...,αn in M such that n θ = θ α j=1 αj for every closed C∞ 1-form θ on M. Consequently, if M is second countable and A is a subring of C containing Z, then every 1-cycle in M with coefﬁcients in A is homologous (as a 1-cycle over C) to an A-linear combination of smooth Jordan curves.

Remark The second part of the above proposition follows immediately from the ﬁrst part and Lemma 10.6.6. The only reason for including the second countability condition in this second part is that in this book, homology is considered only for second countable surfaces. Second countability does not play an essential role in the proof. It also follows that for M second countable, H1(M, A) is generated by those homology classes that are represented by smooth Jordan curves, provided M is orientable (or M admits an orientable C∞ surface structure) if A = R and A = C (see Sects. 10.6 and 10.7).

Proof of Proposition 5.10.7 Let α :[0, 1]→M be a loop in M. We may assume without loss of generality that M is second countable, since (for example) we may replace M with a connected relatively compact neighborhood of α([0, 1]) (second countability will allow us to use the convenient language of ho- ∞ mology). We may choose a partition 0 = t0 2 and that the claim holds for partitions of [0, 1] into fewer than m intervals. 5.10 Smooth Jordan Curves and Homology 235

~~If α(r) = α(s) for some r, s, and j with~~

~~t0 ≤ r ≤ tj−1~~

~~−1 r ≡ min([tj−1,tj ]∩α (α([tj ,tj+1]))) ∈ (tj−1,tj ] and −1 s ≡ max([tj ,tj+1]∩α (α(r))) ∈[tj ,tj+1).~~

~~If r~~

By replacing α with β2, we get a path in which the corresponding segment satisﬁes r = s (the partition corresponding to β2 has the m intervals given by [ ] ⊂ −1 ∈{ }\{ + } [ ] ⊂ −1 ϕ( tk−1,tk ) β2 (Dk) for k 1,...,m j,j 1 , ϕ( tj−1,r ) β2 (Dj ), and [ ] ⊂ −1 ϕ( s,tj+1 ) β2 (Dj+1)). Performing this procedure inductively on j, we see that we may assume that α([tj−1,tj )) and α([tj ,tj+1]) are disjoint and that α([tj−1,tj ]) and α((tj ,tj+1]) are disjoint for each j = 1,...,m− 1. A similar argument shows that we may assume without loss of generality that α([tm−1,tm)) and α([t0,t1]) are disjoint and that α([tm−1,tm]) and α((t0,t1]) are disjoint. Thus we may assume that α is a piecewise smooth Jordan curve. Finally, ﬁxing a point a ∈ (0,t1) at which α is smooth, we see that α is homologous to the piecewise smooth Jordan curve γ given by t → α(t + a) on [0, 1 − a] and t → α(t − 1 + a) on [1 − a,1]. Since γ is also loop-smooth at 0, Lemma 5.10.5 implies that γ is homologous to a smooth Jordan curve, and the claim follows.

Exercises for Sect. 5.10 5.10.1 Let γ :[a,b]→M be a loop in a C∞ surface M. Verify that γ is loop- smooth at a (i.e., on some neighborhood of 1 = e2πi·0, the associated map γˆ : S1 → M given by e2πit → γ(a + t(b − a)) for t ∈[0, 1] is C∞ with nonvanishing tangent map) if and only if on some neighborhood of + Z − ⊂ R R → → − t−a − a (b a) ,themap M given by t γ(t b−a (b a)) is C∞ with nonvanishing tangent vector. 236 5 Uniformization and Embedding of Riemann Surfaces

5.10.2 Verify that the path β constructed in the proof of Lemma 5.10.5 has the required properties. 5.10.3 Prove that every Jordan curve α :[0, 1]→M (injective path) in a C∞ surface M is path homotopic to a smooth Jordan curve (respectively, a smooth injective path) β. Prove also that if 0 = t0 < ···

~~5.11 Separating Smooth Jordan Curves~~

In this section, we consider some properties of smooth Jordan curves that allow us to ﬁnd and remove annuli. The main goals are to show that a smooth Jordan curve in an oriented smooth surface admits a neighborhood that is diffeomorphic to an annulus, and to show that the de Rham pairing of a separating smooth Jordan curve and the cohomology class of a compactly supported C∞ closed 1-form is zero.

Lemma 5.11.1 Let γ :[0, 1]→M be an injective smooth path or a smooth Jordan curve in an oriented C∞ surface M, and let C = γ([0, 1]). (a) If γ is a smooth Jordan curve, then for some R>1, there exists an orientation-preserving diffeomorphism of some neighborhood A of C in M onto the annulus (0; 1/R, R) with (γ (t)) = e2πit for each t ∈[0, 1] (see Fig. 5.9). Consequently, M \ C has either one or two connected components, and each of these connected components has boundary equal to C. (b) If γ is an injective smooth path, then for some >0, there exists an orientation- preserving diffeomorphism of some neighborhood R of C in M onto the open rectangle (−,1 + ) × (−,) with (γ (t)) = (t, 0) for each t ∈[0, 1]. Consequently, M \ C is connected.

Remarks 1. For a smooth Jordan curve in a nonorientable surface, one may ﬁnd a neighborhood that is diffeomorphic to either an annulus or a Möbius band. This fact will be proved and applied in the discussion of C∞ structures on surfaces in Chap. 6 (see Sect. 6.10). 2. As indicated in part (a), the existence of annular neighborhoods implies that the complement M \C of a smooth Jordan curve C in an oriented smooth surface M has exactly one or two connected components, and each of these connected components has boundary equal to C. On the other hand, one may also obtain this fact directly (see Exercise 5.11.3). 5.11 Separating Smooth Jordan Curves 237

~~Fig. 5.9 A smooth annular neighborhood of a smooth Jordan curve~~

3. For our purposes, the easier fact that any smooth Jordan curve is separating in some neighborhood of the curve would sufﬁce. However, it will be convenient to have the annular neighborhood provided by Lemma 5.11.1. 4. Given a,b,c,d ∈ R with a<1 1. Similarly, part (b) holds for any >0.

Proof of Lemma 5.11.1 We prove part (a) and leave the proof of part (b), which is similar, to the reader (see Exercise 5.11.1). Assuming that γ is a smooth Jordan 1 2πit curve, we may form the associated mapping γ0 : S → C given by e → γ(t) 2πit and the mapping α : R → C given by t → γ0(e ) = γ(t− t) (which is actually the extension of γ to a periodic mapping with period 1). By Theorem 9.9.2, C is a ∞ compact C submanifold of M, γ0 is a diffeomorphism, and α is a local diffeomorphism (in fact, α is a C∞ covering map). Observe that C∞ local extensions of local inverses of these maps have nonvanishing differentials at points in C. Hence we ∞ may form a ﬁnite covering of C by positively oriented local C charts {(Uj ,j = = × − }m = (xj ,yj ), Rj Ij ( δj ,δj )) j=1 in M such that for each j 1,...,m, Ij is an open interval, δj > 0, C ∩ Uj ={p ∈ Uj | yj (p) = 0}=α(Ij ), and xj (α(t)) = t for each point t ∈ Ij . In particular, since α∗ maps (T R)t bijectively onto (T C)γ(t) for each point t ∈ R, = we have dxj T(C∩Uj ∩Uk) dxk T(C∩Uj ∩Uk) for each pair of indices j,k.Onthe = other hand, we have dyj T(C∩Uj ) 0 for each j,sowehave

~~(dxj ∧ dyj )p = (dxk ∧ dyj )p ∀p ∈ C ∩ Uj ∩ Uk.~~

For we have (dxk)p = a(dxj )p + b(dyj )p for some scalars a,b ∈ R, and restricting to TC, we see that a = 1. C∞ { }m ⊂ We may also form nonnegative functions λj j=1 such that supp λj Uj for ∞ each j and λj ≡ 1 on a neighborhood of C. Thus we may deﬁne a C mapping : M → C = R2 by

m m 2πi(xj +iyj ) −2πyj = λj · e = λj · e · (cos 2πxj , sin 2πxj ) j=1 j=1 238 5 Uniformization and Embedding of Riemann Surfaces m m −2πyj −2πyj = λj · e · cos 2πxj , λj · e · sin 2πxj = ( 1, 2) j=1 j=1

2πi(x +iy ) (as usual, we have extended λj · e j j by 0 to all of M for each j). For − ∈ = 1 = + ↔ = 2πixj (p) = p C and ζ γ0 (p) u iv (u, v),wehaveζ e (i.e., (u, v) (cos(2πxj (p)), sin(2πxj (p))))forany index j with p ∈ Uj . Hence m m − − = · 2πixj (p) = · 1 = 1 (p) λj (p) e λj (p) γ0 (p) γ0 (p) j=1 j=1 and (since dλj = d( λj ) = 0 near C) (d ) = ((d ) ,(d ) ) p 1 p 2 p m = (−2π)λj (p)(v(dxj )p + u(dyj )p), j=1 m 2πλj (p)(u(dxj )p − v(dyj )p) . j=1

Therefore, since u2 + v2 =|ζ |2 = 1, we have m 2 2 (d 1 ∧ d 2)p = 4π λj (p)λk(p)v (dxj ∧ dyk)p j,k=1 m 2 2 + 4π λj (p)λk(p)u (dxk ∧ dyj )p j,k=1 m 2 2 = 4π λj (p)λk(p)v (dxj ∧ dyk)p j,k=1 m 2 2 + 4π λj (p)λk(p)u (dxj ∧ dyk)p j,k=1 m 2 = 4π λj (p)λk(p)(dxj ∧ dyk)p. j,k=1

On the other hand, if P is the set of indices j for which p ∈ Uj , then we have (dxj ∧ dyk)p = (dxk ∧ dyk)p for all j,k ∈ P . Therefore 2 (d 1 ∧ d 2)p = 4π λj (p)λk(p)(dxk ∧ dyk)p j,k∈P 2 = 4π λk(p)(dxk ∧ dyk)p > 0. k∈P 5.11 Separating Smooth Jordan Curves 239

Therefore, by continuity, d 1 ∧ d 2 > 0 on some neighborhood of C, and hence C∞ by the inverse function theorem (Theorem 9.9.1 and Theorem 9.9.2), is an orientation-preserving local diffeomorphism. Furthermore, the intersection of −1(S1) with a sufﬁciently small neighborhood of C must be equal to C. For if this were not the case, then we could choose a sequence {pν} in M \ C that converged to 1 a point p ∈ C and that satisﬁed (pν) ∈ S for each ν. Setting qν ≡ γ0( (pν)) ∈ C for each ν, we would get

~~qν = γ0( (pν)) → γ0( (p)) = p.~~

But then, for large ν, both pν and qν would lie in a neighborhood of p on which = −1 = was injective. Since (qν ) γ0 (qν) (pν ), we have arrived at a contradiction. Lemma 10.2.11 now implies that maps a neighborhood H of C diffeomorphically onto a neighborhood of S1. Choosing R>1 sufﬁciently close to 1, we see that −1 ; the restriction of to ( H ) ((0 1/R, R)) has the required properties. −1 Finally, setting Q1 ≡ (0; 1/R, 1) and Q2 ≡ (0; 1,R), we see that (Qj ) must lie in some connected component Pj of M \ C for each j = 1, 2. Any con- −1 −1 nected component of M \ C containing neither (Q1) nor (Q2) would have to be a connected component of M, since it would have no boundary points in M. Therefore, since M is connected, the set of connected components of M \ C is −1 {P1,P2}. Finally, observe that ∂Pj = (∂Qj ) = C for j = 1, 2.

~~Deﬁnition 5.11.2 A closed subset of a connected topological space X is called separating in X if X \ K is not connected. If X \ K is connected, then K is called nonseparating in X.~~

For example, a circle in C is separating; in fact, as will be proved in Chap. 6, every Jordan curve in C is separating (see Corollary 6.7.2). The smooth Jordan curve C = γ([0, 1]) appearing in Fig. 5.9 is nonseparating.

~~Lemma 5.11.3 If C is a separating smooth Jordan curve in an oriented smooth = C∞ surface M, then C θ 0 for every closed 1-form θ with compact support.~~

Remark The converse, which is also true, will be proved in Sect. 5.15 (see Propo- sition 5.15.2); and the proof of the converse will be applied in the construction of a canonical homology basis for a compact Riemann surface in Sect. 5.16.

Proof of Lemma 5.11.3 By Lemma 5.11.1, M \ C must have exactly two connected components M1 and M2, and we must have ∂M1 = ∂M2 = C. Orienting C positively with respect to M1 and applying Stokes’ theorem (Theorem 9.7.17), we get = = = θ dθ 0 0. C M1 M1 We close this section with the following elementary observation, which will allow us to choose disjoint tubes for removal from a Riemann surface, thereby reduc- ing its homology. 240 5 Uniformization and Embedding of Riemann Surfaces = { } Lemma 5.11.4 Let M be a topological surface; let K α∈A Kα, where Kα is a countable locally ﬁnite collection of disjoint compact sets each of which is mapped onto a closed disk in the plane in some local chart; and let N = M \ K. Then every path γ :[0, 1]→M with γ(0), γ (1) ∈ N is path homotopic (in M) to a path in N. In particular, N is connected.

~~Proof Suppose ﬁrst that for some constants r and R with 0~~

Exercises for Sect. 5.11 5.11.1 Prove part (b) of Lemma 5.11.1. 5.11.2 Prove that in part (a) of Lemma 5.11.1,if0 : V → W is an orientation- preserving diffeomorphism of a neighborhood V of γ([s0,s1]) onto a neigh- 2πit borhood W of the arc E ≡{e | s0 ≤ t ≤ s1} for some pair s0,s1 with 2πit −1 0

5.11.4 Prove that if a<1 1). 5.11.5 Let γ : R → M be a C∞ embedding of R into an oriented C∞ surface M (i.e., a proper C∞ mapping for which the tangent vector γ˙ is nonvanishing), and let C = γ(R). Prove that there exists an orientation-preserving diffeomorphism of some neighborhood of C in M onto the open strip R × (−1, 1) with (γ (t)) = (t, 0) for each t ∈ R. Conclude that in particular, M \ C has either one or two connected components. 5.11.6 Prove that there exists a universal constant C>0 such that if X is a Riemann surface, p is a point in X, and (D, z = , (0; 1)) is a local holomorphic chart with p ∈ D and (p) = z(p) = 0, then there exists a meromorphic 1-form θ on X such that (cf. Exercise 5.5.1) (i) The meromorphic 1-form θ is holomorphic on X \{p}; ≤ (ii) We have θ L2(X\D) C; (iii) The meromorphic 1-form θ + z−2 dz on D has a removable singularity at p; + −1 ≤ (iv) We have zθ z dz L2(D) C; and = (v) We have γ θ 0 for every separating smooth Jordan curve γ in X \{p}.

~~5.12 Holomorphic Attachment and Removal of Tubes~~

~~In this section we develop notation for the holomorphic attachment and removal of a locally ﬁnite family of disjoint tubes (cf. Example 2.3.3).~~

~~5.12.1 Holomorphic Attachment of Tubes~~

~~Let Y be a complex 1-manifold; let {R0j }j∈J and {R1j }j∈J be numbers in the interval (1, ∞);let~~

{(D0j ,0j ,(0; R0j ))}j∈J and {(D1j ,1j ,(0; R1j ))}j∈J be locally ﬁnite families of disjoint local holomorphic charts in Y such that for all ∈ ∩ =∅ ∈ ≡ −1 ; ⊂ j,k J , D0j D1k , and for each j J ,letAνj νj ((0 1,Rνj )) Dνj for ν = 0, 1, let Tj ≡ (0; 1/R0j ,R1j ),letA0j ≡ (0; 1/R0j , 1) ⊂ Tj , and let A1j ≡ (0; 1,R1j ) ⊂ Tj . Setting ≡ ⊂ ≡ ⊂ ≡ ⊂ ≡ Aν Aνj Dν Dνj Y and Aν Aνj T Tj j∈J j∈J j∈J j∈J 242 5 Uniformization and Embedding of Riemann Surfaces

~~Fig. 5.10 An annulus (or tube) and two disjoint disks~~

= : ∪ → ∪ for ν 0, 1, we get a biholomorphism A0 A1 A0 A1, where for each index ν ∈{0, 1}, each index j ∈ J , and each point z ∈ Aνj , the image p of z under the inclusion Aνj → Aν satisﬁes − 1(1/z) ∈ A if ν = 0, (p) ≡ 0j 0j −1 ∈ = 1j (z) A1j if ν 1.

Setting ≡ \ −1 ; ⊃ ∪ Z Y νj ((0 1)) A0 A1, ν∈{0,1},j∈J −1 ∩ ∪ ⊂ we see that (K (A0 A1)) is closed in T for every compact set K Z, and (K ∩ (A0 ∪ A1)) is closed in Z for every compact set K ⊂ T (see Fig. 5.10). We may therefore form the holomorphic attachment X ≡ Z ∪ T = Z T/∼, ∈{ } ∈ ∈ ∈ where for ν 0, 1 , j J , p Aνj , and z Aνj , the images p0 of p and z0 → ∼ of z in Z T under the inclusions Aνj ,Aνj Z T satisfy p0 z0 if and only if z · 0j (p) = 1forν = 0 and z = 1j (p) for ν = 1(seeFig.5.11,in which J ={1, 2, 3}). In other words, X is a complex 1-manifold obtained by removing the unit disks in each of the coordinate disks D0j and D1j , and gluing in a tube (i.e., an annulus) Tj (equivalently, the boundaries of the unit disks are glued together). We call X the complex 1-manifold obtained by holomorphic attachment of tubes at elements of the locally ﬁnite family of disjoint coordinate disks {(Dνj ,νj ,(0; Rνj ))}ν∈{0,1},j∈J (or simply the complex 1-manifold obtained by holomorphic attachment of tubes at {Dνj }ν∈{0,1},j∈J ). Observe that X is compact if and only if Y is compact. If Y is connected, then X is connected. However, X may be connected even if Y is not (see Exercise 2.3.2). 5.12 Holomorphic Attachment and Removal of Tubes 243

~~Fig. 5.11 Holomorphic attachment of tubes~~

~~∗ ∗ ≤ ∈{ } ∈ Fixing a number Rνj with 1~~

The natural inclusion Z T ∗ ⊂ Z T then descends to a natural biholomorphism =∼ X∗ −→ X (see Exercise 5.12.1), so we may identify X∗ with X. In particular, for each j ∈ J , we may always choose R0j and R1j to be equal and arbitrarily close to 1.

~~5.12.2 Holomorphic Removal of Tubes~~

The inverse operation of tube attachment is tube removal.LetX be a complex 1-manifold, let {R0j }j∈J and {R1j }j∈J be collections of numbers in (1, ∞),let {(Tj ,j ,(0; 1/R0j ,R1j ))}j∈J be a locally ﬁnite collection of disjoint local holo- ∈ ≡ −1 ; ⊂ morphic charts in X, and for each j J ,letA0j j ((0 1/R0j , 1)) Tj 244 5 Uniformization and Embedding of Riemann Surfaces

≡ −1 ; ⊂ = ∈ ≡ and A1j j ((0 1,R1j )) Tj . For each ν 0, 1 and each j J ,letDνj ; ≡ ; ⊂ (0 Rνj ) and Aνj (0 1,Rνj ) Dνj . Setting ≡ ⊂ ≡ ≡ ⊂ ≡ Aν Aνj T Tj and Aν Aνj Dν Dνj j∈J j∈J j∈J j∈J

~~Z ≡ X \[T ∩ ∂(A0 ∪ A1)]=X \ (T ∩ ∂A0) = X \ (T ∩ ∂A1) = \ −1 ; ⊃ ∪ X j (∂(0 1)) A0 A1, j∈J~~

−1 ∩ ⊂ we see that (K (A0 A1)) is closed in Z for every compact set K D0 D1, and (K ∩ (A0 ∪ A1)) is closed in D0 D1 for every compact set K ⊂ Z. Thus we get the holomorphic attachment Y ≡ (D0 D1) ∪ Z = D0 D1 Z/∼, where ∈{ } ∈ ∈ ∈ for ν 0, 1 , j J , p Aνj , and z Aνj , the images p0 of p and z0 of z in → ∼ D0 D1 Z under the inclusions Aνj ,Aνj D0 D1 Z satisfy p0 z0 if and only if z · j (p) = 1forν = 0 and z = j (p) for ν = 1. In other words, Y is the complex 1-manifold obtained by removing the unit circles in each of the tubes (i.e., coordinate annuli) Tj and gluing in caps (i.e., disks) D0j and D1j on each end of the remaining two pieces. That is, Y is obtained by attaching caps to the open subset Z of X. We call Y the complex 1-manifold obtained by holomorphic removal of the locally ﬁnite family of disjoint tubes {(Tj ,j ,(0; 1/R0j ,R1j ))}j∈J from X (or simply by holomorphic removal of {Tj }j∈J ). Observe that X is compact if and only if Y is compact. If Y is connected, then X is connected, but connectivity of X does not imply connectivity of Y (see Exercise 5.12.2). As is the case for holomorphic attachment of tubes, shrinking the annuli about the unit circle does not affect the outcome. More precisely, ﬁxing a num- ∗ ∗ ≤ ∈{ } ∈ ∗ ≡ ber Rνj with 1

~~5.12.3 Holomorphic Reattachment of Tubes~~

Tube attachment and tube removal are inverse operations (up to biholomorphism). In other words, if one removes tubes in a complex 1-manifold X as in Sect. 5.12.2, and then (re)attaches tubes with the disks and constants as in Sect. 5.12.1, then the resulting complex 1-manifold is biholomorphic to the original complex 1-manifold X. In fact, if one reattaches only those tubes in some subfamily, then the resulting complex 1-manifold is biholomorphic to the complex 1-manifold obtained by removal from X of those tubes in the complementary subfamily. Similarly, if one attaches tubes to a complex 1-manifold Y as in Sect. 5.12.1, and then removes tubes with the annuli and constants as in Sect. 5.12.2, then the resulting complex 1-manifold is biholomorphic to the original complex 1-manifold Y . The details of holomorphic reattachment of tubes are provided in this section. The details of the inverse operation (holomorphic attachment followed by holomorphic removal of tubes) are left to the reader (see Exercise 5.12.5). In the notation of Sect. 5.12.2,letY = (D0 D1) ∪ Z,letZ : Z→ Y be the ∈{ } ∈ : → holomorphic inclusion of Z, and for each ν 0, 1 and j J ,letDνj Dνj Y = ; = be the holomorphic inclusion of the disk Dνj (0 Rνj ),letDνj Dνj (Dνj ), − =∼ and let ≡ 1 : D → D . Given a set of indices J⊂ J , holomorphic at- νj Dνj νj νj tachment of tubes at elements of the locally ﬁnite family of disjoint coordinate disks { ; } (Dνj , νj ,(0 Rνj )) ν∈{0,1},j∈J in Y yields a complex 1-manifold X.Morepre- cisely, for each j ∈ J,wemayset

~~A0j ≡ (0; 1/R0j , 1), A1j ≡ (0; 1,R1j ) ⊂ Tj ≡ (0; 1/R0j ,R1j ), and~~

~~ ≡ = = −1 ; ⊂ ⊂ = Aνj Dνj (Aνj ) Z(Aνj ) νj ((0 1,Rνj )) Dνj Y for ν 0, 1.~~

~~As in Sect. 5.12.1, setting ≡ ⊂ ≡ ⊂ ≡ ⊂ ≡ Aν Aνj Dν Dνj Y and Aν Aνj T Tj j∈J j∈J j∈J j∈J~~

for ν = 0, 1, we get a biholomorphism : A0 ∪ A1 → A ∪ A , where for each 0 1 index ν ∈{0, 1}, each index j ∈ J , and each point z ∈ Aνj , the image p of z under the inclusion Aνj → Aν satisﬁes −1 (1/z) = D (1/z) ∈ A if ν = 0, (p) ≡ 0j 0j 0j −1 = ∈ = 1j (z) D1j (z) A1j if ν 1. 246 5 Uniformization and Embedding of Riemann Surfaces

Setting ≡ \ −1 ; = \ ; Z Y νj (0 1) Y Dνj (0 1) ν∈{0,1},j∈J ν∈{0,1},j∈J = ∪ ⊃ ∩ ∪ = ∪ Z(Z) Dνj Z(Z) (D0 D1) A0 A1, ν∈{0,1},j∈J \J we may form the holomorphic attachment (of tubes) X ≡ Z ∪ T , with inclusion mappings : Z→ X and : T → X for each j ∈ J . We call X the complex Z Tj j { } 1-manifold obtained by holomorphic reattachment of the tubes Tj j∈J to Y . Setting J≡ J \ J, holomorphic removal of the locally ﬁnite family of disjoint { ; } tubes (Tj ,j ,(0 1/R0j ,R1j )) j∈J from X yields a complex 1-manifold X. More precisely, setting ≡ ⊂ ≡ ⊂ ≡ ⊂ ≡ Aν Aνj T Tj T and Aν Aνj Dν Dνj j∈J j∈J j∈J j∈J

= : ∪ → for ν 0, 1, we get a biholomorphism A0 A1 A0 A1, where for each index ∈{ } ∈ ∈ ν 0, 1 , each index j J , and each point p Aνj , (p) is the image in A0 A1 of the point ∈ = 1/j (p) A0j for ν 0, ∈ = j (p) A1j for ν 1, → ⊂ under the inclusion Aνj A0 A1 D0 D1. Setting Z ≡ X \[T ∩ ∂(A0 ∪ A1)]=X \ (T ∩ ∂A0) = X \ (T ∩ ∂A1) = \ −1 ; = ∪ ⊃ ∩ = ∪ X j (∂(0 1)) Z Tj Z T A0 A1, j∈J j∈J

≡ ∪ we may form the holomorphic attachment Y (D0 D1) Z, with inclusion : → : → ∈{ } ∈ mappings Z Z Y and Dνj Dνj Y for each ν 0, 1 and j J .One may now verify that the natural mapping Y → X given by → ∀ ∈ Z(p) Z(Z(p)) p Z, (p) → ( (p)) ∀j ∈ J,p ∈ T , Z Tj j j → ∀ ∈{ } ∈ ∈ Dνj (z) Z(Dνj (z)) ν 0, 1 ,j J,z Dνj , is a well-deﬁned biholomorphism that maps (Z) onto (Z(Z)), (Tj ) Z Z Z onto (T ) for each index j ∈ J , and (D ) onto (D ) for each in- Tj j Dνj νj Z νj dex ν ∈{0, 1} and each index j ∈ J(see Exercise 5.12.4). Thus we may identify Y { } with X. In other words, holomorphic reattachment of the tubes Tj j∈J to Y yields { } the same complex 1-manifold as holomorphic removal of the tubes Tj j∈J from X. 5.12 Holomorphic Attachment and Removal of Tubes 247

~~5.12.4 Finite Sequential Holomorphic Removal/Attachment of Tubes~~

If a locally ﬁnite family of disjoint tubes in a complex 1-manifold is divided into ﬁnitely many disjoint subfamilies, and the subfamilies are then inductively removed (i.e., if the subfamilies are removed one at a time), then the resulting complex 1-manifold will be naturally biholomorphic to the complex 1-manifold obtained by removing the tubes simultaneously. More precisely, in the notation of Sect. 5.12.2, { }m suppose Jk k=1 are nonempty disjoint subsets of the index set J with union J ; that { }m = is, Jk k=1 is a partition of J . For each k 1,...,m,let (k) ≡ ⊂ (k) ≡ (k) ≡ ⊂ (k) ≡ Aν Aνj T Tj and Aν Aνj Dν Dνj j∈Jk j∈Jk j∈Jk j∈Jk

~~ = : (k) ∪ (k) → (k) (k) for ν 0, 1, and let k A0 A1 A0 A1 be the associated biholomorphism as in Sect. 5.12.2.Nowlet~~

~~≡ \[ (1) ∩ (1) ∪ (1) ]= \ (1) ∩ (1) = \ (1) ∩ (1) Z1 X T ∂(A0 A1 ) X (T ∂A0 ) X (T ∂A1 ) m = \ −1 ; ⊃ (1) ∪ (1) ∪ (k) X j (∂(0 1)) A0 A1 T , j∈J1 k=2~~

≡ (1) (1) ∪ let Y1 (D0 D1 ) 1 Z1 be the complex 1-manifold obtained by holomorphic (1) removal of the tubes {Tj }j∈J , and let Z : Z1 → Y1, and (1) : Dν → Y1 for 1 1 Dν ν = 0, 1, be the corresponding inclusion mappings. We then have the locally ﬁnite family of disjoint tubes

~~−1 {( (T ), ◦ ,(0; 1/R ,R ))} ∈ Z1 j j Z1 0j 1j j J2 in Y1, and holomorphic removal of these tubes yields the complex 1-manifold~~

~~(2) (2) Y2 ≡ (D D ) ∪ −1 Z2 0 1 2◦ Z1~~

(2) and the corresponding inclusion mappings Z : Z2 → Y2 and (2) : Dν → Y2 2 Dν for ν = 0, 1, where m ≡ \ −1 ; ⊃ (2) ∪ (2) ∪ (k) Z2 Y1 Z1 (j (∂(0 1))) Z1 A0 A1 T . j∈J2 k=3

Proceeding inductively, we get complex 1-manifolds X = Y0,Y1,...,Ym and, for = : → each k 1,...,m, holomorphic inclusion mappings Zk Zk Yk of an open (k) set Zk ⊂ Yk−1 and (k) : Dν → Yk for ν = 0, 1. Moreover, setting Z0 ≡ X, Dν 248 5 Uniformization and Embedding of Riemann Surfaces

≡ ≡ ≡ = Z0 IdX, Zm+1 Ym, and Zm+1 IdYm , we see that for each k 1,...,m, ◦···◦ the composition Zk−1 Z0 is defined on the open set \ −1 ; ⊃ (k) X j (∂(0 1)) T , ∈ k−1 j i=1 Ji we have = \ ◦···◦ −1 ; Zk Yk−1 Zk−1 Z0 (j (∂(0 1))), j∈Jk (k) Z + ◦···◦Z + is defined on (k) (Dν ) for ν = 0, 1, and m 1 k 1 Dν (k) (k) (k) (k) Yk = (D D ) ∪ ◦ ◦···◦ −1 Zk = D D Zk/∼ 0 1 k (Zk−1 Z0 ) 0 1 is the complex 1-manifold obtained by holomorphic removal of the tubes { ◦···◦ } Zk−1 Z0 (Tj ) j∈Jk from Yk−1 (in other words, under the appropriate identifications, Yk is the complex { } 1-manifold obtained by holomorphic removal of the tubes Tj j∈Jk from Yk−1). On the other hand, as in Sect. 5.12.2, holomorphic removal of the tubes {Tj }j∈J yields the complex 1-manifold Y ≡ (D0 D1) ∪ Z = D0 D1 Z/∼, where ≡ \ −1 ; Z X j (∂(0 1)), j∈J

∼ = and : A0 ∪ A1 → A A is the biholomorphism for which the restriction 0 1 (k) (k) → (k)∪ (k) is equal to the composition of the inclusion A0 A1 A0 A1 A0 A1 ∼ : (k) ∪ (k) →= (k) (k) = and the biholomorphism k A0 A1 A0 A1 for each k 1,...,m. : → : → = Let Z Z Y and Dν Dν Y for ν 0, 1 be the corresponding inclusion mappings. One may now verify that the natural mapping Y → Ym, which sends the im- ∈ ◦ ··· ◦ ∈ age Z(p) in Y of any point p Z to the point Zm Z0 (p) Ym, and which for ν = 0, 1 and k = 1,...,m, sends the image in Y of any point ∈ (k) (k) → q Dν under the composition of the inclusions Dν and Dν Dν to the point Z + ◦···◦Z + ◦ (k) (q) in Ym, is a well-deﬁned biholomorphism (see Exer- m 1 k 1 Dν cise 5.12.6). Thus we may identify Ym with Y . Similarly, one may consider ﬁnite sequential holomorphic attachment of tubes, and verify that the resulting complex 1-manifold is naturally biholomorphic to the complex 1-manifold obtained by simultaneous holomorphic attachment of the tubes (see Exercise 5.12.7).

Exercises for Sect. 5.12 5.12.1 In the notation of Sect. 5.12.1, verify that the natural inclusion of Z T ∗ into Z T descends to a biholomorphism of X∗ onto X. 5.13 Tubes in a Compact Riemann Surface 249

5.12.2 Let X be a complex 1-manifold obtained by holomorphic attachment of tubes to a complex 1-manifold Y as in Sect. 5.12.1. Prove that X is connected if Y is connected. Give an example that shows that X may be connected even if Y is not (cf. Exercise 2.3.2). ∗ 5.12.3 In the notation of Sect. 5.12.2, verify that the natural inclusion of D0 ∗ ∗ D1 Z into D0 D1 Z descends to a biholomorphism of Y onto Y . 5.12.4 Verify that the mapping Y → X in Sect. 5.12.3 is a well-defined biholomorphism. 5.12.5 Verify that holomorphic tube removal undoes tube attachment. That is, in analogy with Sect. 5.12.3, verify that if one holomorphically attaches tubes to a complex 1-manifold Y as in Sect. 5.12.1, and then holomorphically removes some of the tubes as in Sect. 5.12.2, then the resulting complex 1-manifold is biholomorphic to the complex 1-manifold obtained by holomorphic attachment of the remaining tubes to Y . 5.12.6 Verify that the mapping Y → Ym in Sect. 5.12.4 is a well-defined biholomorphism. 5.12.7 In analogy with the description of finite sequential holomorphic removal of tubes in Sect. 5.12.4, provide details for the process of finite sequential holomorphic attachment of tubes to a complex 1-manifold. Include a verification that the resulting complex 1-manifold is (naturally) biholomorphic to the complex 1-manifold obtained by simultaneous holomorphic attachment of the tubes. 5.12.8 Describe, with proofs, C∞ and C0 versions of attachment, cap attachment, tube attachment, and tube removal in which the biholomorphisms are re- placed with diffeomorphisms in second countable 2-dimensional C∞ manifolds and homeomorphisms in second countable 2-dimensional topological manifolds, respectively. ∼ 5.12.9 Let T = (0; 1/R0,R1) be a holomorphic coordinate annulus (i.e., a tube) in a Riemann surface X as in Sect. 5.12.2 (with R0,R1 > 1). Assume that the complex 1-manifold obtained by holomorphic removal of T is connected, but not simply connected. Prove that there is an infinite covering Riemann surface ϒ : X → X (cf. Exercise 5.9.4) such that (i) T isevenlycoveredbyϒ; and −1 (ii) For any connected component T0 of ϒ (T ), we have Deck(ϒ) · T0 = ϒ−1(T ).

~~5.13 Tubes in a Compact Riemann Surface~~

~~Recall that we are working toward a proof that any Riemann surface may be obtained by holomorphic attachment of tubes to a planar domain. We consider the compact case in this section.~~

Theorem 5.13.1 Up to biholomorphism, every compact Riemann surface may be obtained by holomorphic attachment of tubes at ﬁnitely many disjoint coordinate 250 5 Uniformization and Embedding of Riemann Surfaces

Fig. 5.12 Representation of a compact Riemann surface via holomorphic attachment of finitely many tubes to the Riemann sphere disks in the Riemann sphere P1. More precisely, suppose X is a compact Riemann surface and K ⊂ X is a union of finitely many disjoint compact sets, each of which is a closed disk in some local holomorphic chart (i.e., some local holomorphic chart maps the set onto a closed disk in C). Then, in the notation of Sect. 5.12.1, X is bi- ∪ holomorphic to the compact Riemann surface Z T obtained by holomorphic at- { } = = P1 tachment of tubes Tj j∈J (with T j∈J Tj ) to the Riemann surface Y at elements of a finite family of disjoint coordinate disks {Dνj }ν∈{0,1},j∈J (see Fig. 5.12). Moreover, the biholomorphism may be chosen so that the image of K is disjoint from the image of T in Z ∪ T .

~~According to Sect. 5.12.3, the following theorem, which is in a form that is more convenient for the proof, is equivalent to Theorem 5.13.1:~~

Theorem 5.13.2 For every compact Riemann surface, one may obtain a Rie- mann surface that is biholomorphic to P1 by holomorphic removal of finitely many disjoint tubes. More precisely, suppose X is a compact Riemann surface and K ⊂ X is a union of finitely many disjoint compact sets, each of which is a closed disk in some local holomorphic chart. Then, in the notation of Sect. 5.12.2, one may apply (finite) holomorphic tube removal to get a compact Riemann surface ∼ 1 Y ≡ (D0 D1) ∪ Z = P , and one may choose the tubes to be disjoint from K.

Lemma 5.13.3 Let X be a compact Riemann surface, and let K ⊂ X be a union of ﬁnitely many disjoint compact sets, each of which is a closed disk in some local holomorphic chart. If H1(X, R) = 0, then there exists a local holomorphic chart ∗ ∗ ∗ ∞ (A,,(0; 1/R ,R )) in X \ K with R > 1 andaclosedC 1-form θ on X such : → −1 2πit = that the smooth Jordan curve γ t (e ) satisﬁes γ θ 1. In particular, the image C ≡ γ([0, 1]) is nonseparating in X.

Proof By hypothesis, there exists a nonzero homology class in H1(X, R), and since every class contains a linear combination of loops based at a point in X \ K (Propo- sition 10.6.7), we may assume that this class is represented by a loop in X with 5.13 Tubes in a Compact Riemann Surface 251 base point in X \ K. By Lemma 5.11.4, we may also assume that this representing loop lies in X \ K. According to Proposition 5.10.7, this loop is homologous in X \ K, and therefore, in X, to a sum of smooth Jordan loops. It follows that ∞ there exist a smooth Jordan curve β :[0, 1]→X \ K and a closed C real 1-form τ = on X with β τ 0. Lemma 5.11.1 provides a diffeomorphism of a relatively compact neighborhood U of the smooth Jordan curve B ≡ β([0, 1]) in X \ K onto an annulus that maps B onto a concentric circle. By Proposition 5.4.1, there exists a smooth relatively compact domain in U such that B ⊂ , and such that each of the ﬁnitely many connected components L of ∂ has a neighborhood that admits a biholomorphism onto an annulus that maps L onto a concentric circle. Moreover, by Lemma 5.11.1, \ B = 1 ∪ 2, where 1 and 2 are disjoint smooth relatively compact domains in U, and for each j = 1, 2, B is a boundary component of j and (∂j ) \ B is a union of boundary components of ∂ (B is separating in U and therefore in U). By Stokes’ theorem, τ =± τ = 0 (the sign (∂1)∩(∂) β depends on the choice of orientation). Hence the integral of τ along some boundary component of (which is a boundary component of 1) is nonzero. Choosing a suitable local holomorphic chart (A,,(0; 1/R∗,R∗)) in X \ K mapping this boundary component onto S1, letting γ : t → −1(e2πit), letting θ be a suitable rescaling of τ , and applying Lemma 5.11.3, we get the claim.

Remark In the above proof, we used Proposition 5.4.1 in order to get a holomorphic coordinate annulus in X with nonseparating boundary components. In fact, Theo- rem 5.9.2 implies that a slight shrinking of the neighborhood U is itself biholomorphic to an annulus (U is itself biholomorphic to an annulus, to ∗,ortoC∗). A reader who has worked through the discussion in Sect. 5.9 of the classiﬁcation of Riemann surfaces as quotients by groups of automorphisms may prefer to apply Theorem 5.9.2 in the above proof in place of Proposition 5.4.1 (see Exercise 5.13.1).

Proof of Theorem 5.13.2 We proceed by induction on d ≡ dimR H1(X, R), which, according to Proposition 10.6.7, is ﬁnite, since X is compact. The case d = 0fol- lows from the Koebe uniformization theorem (Theorem 5.5.3) or just Lemma 5.5.4. Assuming that d>0, Lemma 5.13.3 provides a number R∗ > 1, a local holomor- ∗ ∗ ∞ phic chart (A,,(0; 1/R ,R )) in X \ K,andaclosedC 1-form θ on X such : → −1 2πit = that the smooth Jordan curve γ t (e ) satisﬁes γ θ 1, and (hence) the image C ≡ γ([0, 1]) is nonseparating in X. Fixing a coordinate annulus T = −1((0; 1/R, R)) A for some R ∈ (1,R∗), holomorphic removal of the tube T as in Sect. 5.12.2 yields a compact Riemann ≡ ; ⊂ ≡ ; = surface Y . More precisely, setting Aν (0 1,R) Dν (0 R) for ν 0, 1, −1 −1 A0 ≡ ((0; 1/R, 1)) ⊂ T , and A1 ≡ ((0; 1,R))⊂ T , we get a biholo- : ∪ → ∈{ } morphism A0 A1 A0 A1, where for each index ν 0, 1 and each point ∈ p Aν , (p) is the image in A0 A1 of the point ∈ = 1/(p) A0 for ν 0, ∈ = (p) A1 for ν 1, 252 5 Uniformization and Embedding of Riemann Surfaces

~~Fig. 5.13 Reduction of the dimension of the ﬁrst homology via holomorphic removal of a tube~~

~~ → ⊂ ≡ \ under the inclusion Aν A0 A1 D0 D1.ThesetZ X C is connected, since C is nonseparating. Thus we get the compact Riemann surface~~

Y0 ≡ (D0 D1) ∪ Z = (D0 D1) Z/∼, ∈{ } ∈ ∈ where for ν 0, 1 , z Aν , and p Aν , the images z0 of z and p0 of p in → ∼ D0 D1 Z under the inclusions Aν,Aν D0 D1 Z satisfy z0 p0 if and only if z · (p) = 1forν = 0 and z = (p) for ν = 1. In other words, Y0 is a compact Riemann surface obtained by removing the unit circle C in the tube (i.e., annulus) T and gluing in caps (i.e., disks) D0 and D1 on each end of the remaining two pieces (see Fig. 5.13). That is, Y0 is obtained by attaching caps to the open subset Z of X. : → : → = Letting Z Z Y0 and Dν Dν Y0 for ν 0, 1 be the corresponding inclusion mappings, we get \ = ∪ = ∪ Z(T C) Z(A0 A1) D0 (A0) D1 (A1) and ⊂ \ = \ ∪ Z(K) Z(X T) Y0 D0 (D0) D1 (D1) . ∗ ≡ ; ∗ ⊃ = → Setting Dν (0 R ) Dν for ν 0, 1, we also see that the inclusion D0 D1 ∗ ∗ ∪ Y0 extends to a biholomorphism of D0 D1 onto a neighborhood of D0 (D0)

~~D1 (D1) (see the remarks at the end of Sect. 5.12.2). It now sufﬁces to show that dim H1(Y0, R)~~

[ ] ∈ R be the quotient map. Given an element ξ H1 H1(Y0, ), Lemma 5.11.4 implies that we may choose the representative ξ to be a real linear combination of loops in \ ∪ = \ Y0 D0 (D0) D1 (D1) Z(X T), [ ] → [ −1 ] and hence we may form the mapping ξ H1(Y0,R) ρ( (Z )∗ξ H1(X,R)). To see that this mapping is well defined, linear, and injective, let us consider a 1-cycle ξ that is a linear combination of loops in Z(X \ T), and let us set ≡ −1 = −1 ≡ ∈ R ξ (Z )∗ξ (Z)∗ ξ and t θ . ξ C∞ ≡ − = Given a closed 1-form η on X,fors γ η,wehave γ (η sθ) 0, and hence, since the path homotopy class of γ generates the fundamental group of A ⊃ T , − C∞ (η sθ) A is exact. Cutting off a potential, we get a real-valued function λ with compact support in A such that η − sθ = dλ on a neighborhood of T . Thus we get C∞ = −1 ∗ − − a well-defined closed 1-form τ on Y0 by setting τ (Z ) (η sθ dλ) at \ = ∪ each point in Z(X T)and τ 0 at each point in D0 (D0) D1 (D1). We then have τ = η − s θ = η − st = η − t η = η. ξ ξ ξ ξ ξ γ ξ−tγ [ ] = [ ] = [ ] ∈ V In particular, if ξ H1(Y0,R) 0, then ξ H1(X,R) t γ H1(X,R) , and it follows that the mapping is well defined. Linearity is easy to verify. For injectivity, observe that if, for ξ as above, we have [ξ ]H (X,R) = u[γ ]H (X,R) ∈ V for some u ∈ R, 1 ∞ 1 then we must have u = t.GivenaclosedC 1-form τ on Y0, τ is exact on a ∪ C∞ neighborhood of D0 (D0) D1 (D1), and hence for some real-valued function ∪ λ0 with compact support in some small neighborhood of D0 (D0) D1 (D1),we have τ + dλ0 ≡ 0 on some (smaller) neighborhood of this set. Thus we may define C∞ = ∗ + \ aclosed 1-form η0 on X by setting η0 Z(τ dλ0) at each point in X T , and η0 = 0 at each point in T . We then have

~~τ = (τ + dλ0) = η0 = t η0 = 0 ξ ξ ξ γ [ ] = (since γ lies in T ), and hence ξ H1(Y0,R) 0.~~

Remarks 1. The above proof does not really use the full Koebe uniformization theorem (or the Riemann mapping theorem), just the more elementary Proposition 5.4.1 and Lemma 5.5.4. The proof of the general case in Sect. 5.14 will use the full Koebe uniformization theorem (but not the Riemann mapping theorem). R → R R ·[ ] 2. It turns out that the injective linear map H1(Y0, ) H1(X, )/ γ H1 in the above proof is never surjective, because dim H1(X, R) is even for any compact Riemann surface X. This is a consequence of the existence of a canonical homology basis (see Sect. 5.16), as well as of the Hodge decomposition theorem (see Sect. 4.9). 254 5 Uniformization and Embedding of Riemann Surfaces

Exercises for Sect. 5.13 5.13.1 Give a slightly different (and shorter) proof of Lemma 5.13.3 by applying Theorem 5.9.2 in order to show that U may be chosen to be biholomorphic to an annulus (see the remark following the proof of Lemma 5.13.3). 5.13.2 Prove that every compact oriented C∞ surface M may be obtained by C∞ oriented attachment (i.e., the local charts are positively oriented and the attaching maps are orientation-preserving diffeomorphisms) of a ﬁnite family of disjoint tubes (see Exercise 5.12.8) to a compact oriented C∞ surface N with H1(N, R) = 0 (and hence H1(N, A) = 0 for any subring A of C containing Z). Using this fact together with Lemma 5.5.4 and Theorem 5.9.2, give a slightly different proof of Theorem 5.13.1. Note. It will follow from the results of Chap. 6 that in fact, N is diffeomorphic to a sphere.

~~5.14 Tubes in an Arbitrary Riemann Surface~~

~~The goal of this section is the following general tube-attachment theorem (see [Ri] for a more complete classiﬁcation of noncompact second countable topological surfaces):~~

Theorem 5.14.1 Up to biholomorphism, every Riemann surface may be obtained by holomorphic attachment of tubes at elements of a locally finite family of disjoint coordinate disks in a domain in P1. More precisely, suppose X is a Riemann surface, and K ⊂ X is a union of a locally finite family of disjoint compact sets, each of which is a closed disk in some local holomorphic chart. Then, in the notation of Sect. 5.12.1, X is biholomorphic to the Riemann surface Z ∪ T obtained by { } = P1 holomorphic attachment of tubes Tj j∈J (with T j∈J Tj ) to a domain Y in at elements of a locally finite family of disjoint coordinate disks {Dνj }ν∈{0,1},j∈J . Moreover, the biholomorphism may be chosen so that the image of K is disjoint from the image of T in Z ∪ T .

~~As in the compact case, it is more convenient to prove the following version, which, according to Sect. 5.12.3, is equivalent to Theorem 5.14.1:~~

Theorem 5.14.2 For every Riemann surface X, one may obtain a planar Riemann surface by holomorphic removal of a locally ﬁnite family of disjoint tubes. More precisely, suppose X is a Riemann surface, and K ⊂ X is a union of a locally ﬁnite family of disjoint compact sets, each of which is a closed disk in some local holomorphic chart. Then, in the notation of Sect. 5.12.2, one may apply holomorphic tube removal to get a Riemann surface Y ≡ (D0 D1) ∪ Z that is biholomorphic to a domain in P1. Moreover, the tubes may be chosen to be disjoint from K.

The idea of the proof is as follows. We ﬁrst form an exhaustion of X by rela- { }∞ tively compact smooth domains ν ν=1 such that each boundary component is a 5.14 Tubes in an Arbitrary Riemann Surface 255 concentric circle in a coordinate annulus (i.e., in a tube). For each connected component of ν+1 \ ν , by holomorphically removing the tubes about all but one of the connected components of ∂∩ ∂ν , we see that we may assume that ∂∩ ∂ν is connected (see Lemma 5.14.4 and Fig. 5.14). Holomorphic removal of the tubes about the boundary components of the domains {ν} then yields a disjoint union of compact Riemann surfaces (see Fig. 5.16). By Theorem 5.13.2, we may then get a disjoint union of copies of the Riemann sphere by holomorphic removal of tubes in each of these components (see Fig. 5.17). Finally, by holomorphically reattaching the boundary tubes about the boundary components of the domains {ν }, we get a planar Riemann surface (see Fig. 5.18).

Lemma 5.14.3 Suppose X is an open Riemann surface, and K ⊂ X is a union of a locally ﬁnite family of disjoint compact sets, each of which is a closed disk in some { }∞ local holomorphic chart. Then there exists a sequence of smooth domains ν ν=1 in X such that ∈ Z = ∞ (i) ν ν+1 for each ν >0 and X ν=1 ν ; and (ii) For each ν ≥ 1 and for each connected component C of ∂ν , there is a local holomorphic chart (A,,(0; 1/R, R)) for some R = R(C) > 0 with −1 C ⊂ A ⊂ X \ K, (A ∩ ν) = (0; 1,R), and C = (∂(0; 1)). = ∞ { } Proof We may write K j=1 Kj for disjoint connected compact sets Kj , each of which is either the empty set or a closed disk in some local holomorphic chart. We { }∞ = ⊂ may also choose compact sets Lν ν=1 with ν Lν X and Lν Lν+1 for each ν, { }∞ ∩ =∅ and positive integers jν ν=1 such that for each ν, jν

Lemma 5.14.4 Suppose X is an open Riemann surface, and K ⊂ X is a union of a locally ﬁnite family of disjoint compact sets, each of which is a closed disk in some local holomorphic chart. Then, in the notation of Sect. 5.12.2, one may apply holomorphic tube removal to get a Riemann surface Y ≡ (D0 D1) ∪ Z with : → { }∞ holomorphic inclusion Z Z Y and a sequence of smooth domains ν ν=1 in Y such that setting 0 =∅, we have the following (see Fig. 5.14): (i) The closures of the removed tubes are disjoint from K (in particular, K ⊂ Z); ≥ = ∞ (ii) ν ν+1 for each ν 1 and Y ν=1 ν ; (iii) For each ν ≥ 1 and for each connected component C of ∂ν , there is a local holomorphic chart (A,,(0; 1/R, R)) for some R = R(C) > 0 with −1 C ⊂ A Z(Z \ K), (A ∩ ν ) = (0; 1,R), and C = (∂(0; 1)); (iv) For each ν ≥ 0, the (ﬁnitely many) connected components of ν+1 \ ν lie in distinct connected components of Y \ ν , and for ν ≥ 1, if is a connected 256 5 Uniformization and Embedding of Riemann Surfaces

~~Fig. 5.14 Holomorphic removal of certain boundary tubes for an exhaustion yielding an exhaustion with separating boundary components~~

component of ν+1 \ ν and V is the connected component of Y \ ν containing , then one of the boundary components of is a boundary component C of ν satisfying ∂V = ∂ ∩ ∂ν = C; and (v) For each ν ≥ 1, each boundary component C of ∂ν is separating in Y .

{ }∞ Proof By Lemma 5.14.3, there exist a family of smooth domains ν ν=1, disjoint local holomorphic charts ∗ ∗ A(ν),(ν),(0; 1/R ,R ) j j ν ν ν∈Z>0,j∈{1,...,mν } ∈ Z ∗ with mν >0 and Rν > 1 for each ν, disjoint smooth Jordan curves C(ν) , j ν∈Z>0,j∈{1,...,mν } { (ν)} ∈{ } = and disjoint domains j ν∈Z≥0,j∈{1,...,kν } with kν 1,...,mν (setting m0 1) for each ν ≥ 0 such that setting 0 ≡∅,wehave ≥ = ∞ (i) ν ν+1 for each ν 0 and X ν=1 ν ; ≥ ⊂ mν (ν) \ = (ii) For each ν 1, ∂ν j=1 Aj X K, and for each j 1,...,mν , (ν)(A(ν) ∩ ) = (0; 1,R∗) and C(ν) = ((ν))−1(∂(0; 1)) (in particular, j j ν ν j j = mν (ν) ∂ν j=1 Cj ); (ν) (ν) (iii) For each ν ≥ 0, ,..., are the connected components of + \ (in 1 kν ν 1 ν = (0) = particular, k0 1 and 1 1); and ≥ = (ν) (ν) (iv) For each ν 1 and each j 1,...,kν , Cj is a connected component of ∂j (thus we have chosen exactly one of the common connected components of 5.14 Tubes in an Arbitrary Riemann Surface 257

(ν) = ∂ν and j for each j 1,...,kν , and holomorphic attachment of caps at the remaining boundary curves {C(ν)}mν will yield a Riemann surface in j j=kν +1 { (ν)}kν which the images of the boundary curves Cj j=1 will be separating). (ν) \ ≥ = In particular, we have Aj ν+1 ν−1 for each ν 1 and each j 1,...,mν , { (ν)} { } so the family Aj is locally ﬁnite in X. Finally, choosing constants Rν with ∗ ≥ 1

~~∞ mν ∞ mν ≡ (ν) ⊂ ≡ (ν) C Cj T Tj , ν=1 j=kν +1 ν=1 j=kν +1~~

~~Z ≡ X \ C ⊃ X \ T ⊃ K (i.e., the condition (i) in the statement of the lemma holds), and ≡ (ν) = Dμ Dμj for μ 0, 1. ν≥1,kν~~

~~∗ ∗ ≡ (ν) = Dμ Dμj for μ 0, 1, ν≥1,kν~~

(ν) ∪ T (ν),...,(ν) ∪ T (ν), 1 1 kν kν ∩ ∩ each of which meets Z ν . Thus it follows by induction that Z ν is connected = ∞ ∩ \ for each ν, and hence that Z ν=1(Z ν ) is connected. Since Y Z(Z) is covered by a union of a locally ﬁnite family of (disjoint) domains each of which meets Z(Z) (and is biholomorphic to a disk), it follows that Y is connected. 258 5 Uniformization and Embedding of Riemann Surfaces

Next we observe that if 0 ≤ ν<μ, then Z ∩ μ \ ν has exactly kν distinct connected components, each of which contains exactly one of the sets (ν),...,(ν). 1 kν For the proof of this fact, we proceed by induction on μ, the case μ = ν + 1 being trivial. Assuming that the claim holds for Z ∩ μ \ ν , each of the disjoint connected sets (μ) ∪ T (μ),...,(μ) ∪ T (μ) must meet exactly one of the k connected 1 1 kμ kμ ν ∩ \ = (μ) ∩ components of Z μ ν , because for each j 1,...,kμ, Tj μ meets, and therefore lies in, exactly one such connected component. Thus the claim holds for μ + 1, and therefore for all μ>ν. It also follows that for each ν ≥ 0, the set Z \ ν has exactly kν distinct connected components; and we may denote these connected { (ν)}kν (ν) ⊂ (ν) = components by Vj j=1, where j Vj for each j 1,...,kν . Furthermore, ≥ = (ν) for each ν 1 and each j 1,...,kν , Cj is a separating curve in Z. In fact, \ (ν) Z Cj has the two connected components (ν) ∩ ∪ (ν) ∪ (ν) Vj and (Z ν) (Vi Ti ). 1≤i≤kν ,i =j

We now let 0 =∅, and for each ν ≥ 1, we let ν be the union of Z(Z ∩ ν) ∪ with those (ﬁnitely many) connected components of D0 (D0) D1 (D1) that meet this set. For each ν ≥ 1, ν is then a relatively compact domain in Y with distinct boundary components (C(ν)),..., (C(ν)), and since maps Z biholomor- Z 1 Z kν Z phically onto Z(Z), ν satisﬁes the conditions (ii) and (iii) in the statement of the lemma. Furthermore, by construction, for each ν ≥ 0, ν+1 \ ν is the union of the { (ν)}kν (ν) disjoint connected open sets j j=1, where for each ν and j, j is the union (ν) ∪ of Z(j ) with those connected components of D0 (D0) D1 (D1) that meet (ν) \ { (ν)}kν Z(j ), and Y ν is the union of disjoint connected open sets Wj j=1, where (ν) ⊃ (ν) (ν) for each ν and j, Wj j is the union of Z(Vj ) with those connected com- ∪ (ν) ≥ ponents of D0 (D0) D1 (D1) that meet Z(Vj ). Finally, for each ν 1 and ∈{ } (ν) = (ν) ∩ = (ν) each j 1,...,kν , the boundary component Z(Cj ) ∂j ∂ν ∂Wj (ν) \ (ν) of ν (and of j ) is separating in Y , since Y Z(Cj ) has (only) the two connected components (ν) (ν) (ν) W and ν ∪ W ∪ Z(T ) . j i i 1≤i≤kν ,i =j

Proof of Theorem 5.14.2 The compact case is Theorem 5.13.2,sowemayassume that X is noncompact. Step 1. Choice of a suitable exhaustion by domains. We may holomorphically remove relatively compact tubes with closures disjoint from K to get a Riemann surface X with the properties considered in Lemma 5.14.4. Moreover, we may assume that the local holomorphic chart on each tube actually extends to a neighborhood of the closure. Thus, if we let K be the union of the image of K with the closures of the holomorphically attached caps in X, and holomorphic removal of 5.14 Tubes in an Arbitrary Riemann Surface 259

Fig. 5.15 The initial (modiﬁed) Riemann surface X tubes in X with closures disjoint from K yields a planar Riemann surface Y , then, as in Sect. 5.12.4, we may identify Y with a Riemann surface obtained by holomorphic removal of tubes in X that are disjoint from K. Therefore, by replacing X with X and K with K, we may assume without loss of generality that there exist a { }∞ family of (nonempty) smooth domains ν ν=1, disjoint local holomorphic charts ∗ ∗ A(ν),(ν),(0; 1/R ,R ) j j ν ν ν∈Z>0,j∈{1,...,mν } ∈ Z ∗ with mν >0 and Rν > 1 for each ν, disjoint smooth Jordan curves

~~{ (ν)} Cj ν∈Z>0,j∈{1,...,mν },~~

{ (ν)} = ≥ disjoint domains j ν∈Z≥0,j∈{1,...,mν } with m0 1, and for each ν 0, disjoint { (ν)} ≡∅ domains Vj j∈{1,...,mν } such that setting 0 ,wehave(seeFig.5.15) ≥ = ∞ (i) ν ν+1 for each ν 0 and X ν=1 ν ; ≥ ⊂ mν (ν) \ = (ii) For each ν 1, ∂ν j=1 Aj X K, and for each j 1,...,mν ,

(ν) (ν) ∩ = ; ∗ (ν) = (ν) −1 ; j (Aj ν) (0 1,Rν ) and Cj (j ) (∂(0 1)) = mν (ν) (in particular, ∂ν j=1 Cj ); ≥ (ν) (ν) (iii) For each ν 0, 1 ,...,mν are the distinct connected components of (0) (ν) (ν) + \ = ν 1 ν (in particular, 1 1), V1 ,...,Vmν are the distinct connected \ (ν) ⊂ (ν) = components of X ν , and j Vj for each j 1,...,mν ; and ≥ = (ν) (iv) For each ν 1 and each j 1,...,mν , the smooth Jordan curve Cj is sepa- (ν) = (ν) ∩ = (ν) rating in X and Cj ∂j ∂ν ∂Vj is a boundary component of ν (ν) (ν) and of j (and of Vj ). (ν) \ ≥ = In particular, we have Aj ν+1 ν−1 for each ν 1 and each j 1,...,mν , { (ν)} { } so the family Aj is locally ﬁnite in X. Finally, choosing constants Rν with ∗ ≥ 1

~~Fig. 5.16 The complex 1-manifold Y0 obtained by holomorphic removal of the boundary tubes for the exhaustion of X~~

(ν) ≡ (ν) −1 ; (ν) (ν) = (ν) ≡ ; Tj (j ) ((0 1/Rν,Rν )) Aj and D0j D1j (0 Rν ) for each ν ≥ 1 and each j = 1,...,mν , and ≡ (ν) ⊂ ≡ (ν) ⊂ \ C0 Cj T0 Tj X K, ν,j ν,j ≡ \ = (ν) ⊃ Z0 X C0 j K, ν,j ≡ (ν) = Dμ Dμj for μ 0, 1. ν,j

Step 2. Removal of the boundary tubes, removal of the interior tubes, and { (ν)} reattachment of the boundary tubes. Holomorphic removal of the tubes Tj as in Sect. 5.12.2 yields a second countable complex 1-manifold Y0 = (D0 ∪ D1) 0 Z0 (for the appropriate biholomorphism 0). Denoting the associated in- : → : → = \ clusions by Z0 Z0 Y0 and Dμ Dμ Y0 for μ 0, 1, we get Z0 (X ∗ = \ ∪ (ν) ≡ ; ∗ ⊃ (ν) T 0) Y0 (D0 (D0) D1 (D1)). Setting Dμj (0 Rν ) Dμj for each choice of ν, j, and μ, and setting ∗ ∗ ≡ (ν) = Dμ Dμj for μ 0, 1, ν,j we also see that the inclusion D0 D1 → Y0 extends to a biholomorphism of ∗ ∗ ∪ D0 D1 onto a neighborhood of D0 (D0) D1 (D1) (see the remarks at the end of Sect. 5.12.2). { (ν)} The complex 1-manifold Y0 has distinct connected components Yj ν≥0,1≤j≤mν (ν) ⊂ (ν) (ν) with Z0 (j ) Yj for each ν and j; in fact, Yj may be viewed as the compact (ν) Riemann surface obtained by holomorphic attachment of caps to j at the boundary as in Sect. 5.3 (see Fig. 5.16). Therefore, by Theorem 5.13.2, we may apply holomorphic tube removal in each of these connected components to get a Riemann surface that is biholomorphic to P1. Thus simultaneous holomorphic removal of these tubes from Y0 yields a complex 1-manifold Y1 in which each connected component is biholomorphic to P1 (see Fig. 5.17). Moreover, we may choose the tubes 5.14 Tubes in an Arbitrary Riemann Surface 261

~~Fig. 5.17 The complex 1-manifold Y1 obtained by holomorphic removal of tubes in each connected component of Y0~~

Fig. 5.18 The desired Riemann surface Y obtained by holomorphic reattachment of the boundary tubes to Y1 (equivalently, via holomorphic removal of tubes from X) so that their closures do not meet the set

~~∪ ∪ Z0 (K) D0 (D0) D1 (D1).~~

As in Sect. 5.12.4, we may identify Y1 with the Riemann surface obtained by holomorphic removal of all of the above tubes (those removed in the construction of Y0 together with those removed in the construction of Y1) from X. Hence holo- { (ν)} morphic reattachment of the tubes Tj to Y1 as in Sect. 5.12.3 yields a Riemann surface Y that is biholomorphic to the Riemann surface obtained by holomorphic removal of a locally finite collection of disjoint tubes in X whose closures do not meet T 0 ∪ K (see Fig. 5.18). Step 3. Planarity of Y . By the Koebe uniformization theorem (Theorem 5.5.3), it now suffices to show that Y is planar. For this, we denote by T the union of the tubes in X \ (T 0 ∪ K) that were removed in order to get Y , we denote by C ⊂ T the union of the corresponding (unit) circles in each of the tubes in T (which were cut out in the tube-removal process), we set Z = X \ C, and we let Z : Z→ Y denote the corresponding holomorphic inclusion. The complement Y \ Z(Z) is a union of a locally finite family of disjoint compact sets, each of which is a closed disk in some ∼ local holomorphic chart. By construction, Y is connected, and hence Z = Z(Z) is connected (for example, by Lemma 5.11.4). Moreover, for each ν = 0, 1, 2,...,the ∩ = \ ∩ (ν) = (ν) \ ∩ (ν) = (ν) \ set Z ν ν C, and the sets Z j j C and Z Vj Vj C for j = 1,...,mν , are connected (see Exercise 5.14.1). 262 5 Uniformization and Embedding of Riemann Surfaces

Let 0 =∅, and for each ν = 1, 2, 3,...,letν ⊂ Y be the connected component of Y \ Z(∂ν ) containing Z(Z ∩ ν ) = Z(ν \ C) (which we may identify with the Riemann surface obtained by holomorphic removal from ν of the tubes given by those connected components of T that are contained in ν ). Simi- ≥ = (ν) larly, for each ν 0 and each j 1,...,mν ,letj be the connected component of \ ∩ (ν) = (ν) \ Y Z(C0) containing Z(Z j ) Z(j C) (which we may identify with (ν) the Riemann surface obtained by holomorphic removal from j of the tubes given (ν) (ν) by those connected components of T that are contained in j ), and let Wj be the \ ∩ (ν) = (ν) \ connected component of Y Z(∂ν ) containing Z(Z Vj ) Z(Vj C) (which we may identify with the Riemann surface obtained by holomorphic removal from V (ν) of the tubes given by those connected components of T which are j (ν) { } = contained in Vj ). Then ν is a sequence of smooth domains with Y ν ν ≥ ≥ { (ν)} { (ν)} and ν ν+1 for each ν 0. Moreover, for each ν 0, j and Wj are the distinct connected components of ν+1 \ ν and Y \ ν , respectively, and, for = (ν) ⊂ (ν) ≥ each j 1,...,mν ,wehavej Wj . Furthermore, for each ν 1, the distinct (ν) = (ν) ∩ = (ν) boundary components of ν are given by Z(Cj ) ∂j ∂ν ∂Wj for = ≥ = (ν) j 1,...,mν . Moreover, for each ν 1 and j 1,...,mν , Z(Cj ) is separating in Y with complement equal to the union of the disjoint connected open sets (ν) ∪ (ν) ∪ (ν) Wj and ν Wi Z(Ti ). 1≤i≤mν ,i =j To verify that Y is planar, we must show that every closed C∞ 1-form θ with ≥ = (ν) compact support on Y is exact. For each ν 1 and each j 1,...,mν , Z(Cj ) is separating in Y ,so θ = 0 (for either orientation) (ν) Z(Cj ) by Lemma 5.11.3. Hence, since there is a biholomorphism of a neighborhood of (ν) (ν) Z(Tj ) onto an annulus that maps Z(Cj ) onto a concentric circle, θ is exact on this neighborhood. Therefore, by replacing θ with the difference of θ and the differential of a C∞ compactly supported function on Y with differential equal to θ on a neighborhood of Z(T0) (note that we may choose the function to be compactly supported because θ is identically zero on a neighborhood of all but ﬁnitely many connected components of Z(T0)), we may assume that ⊂ \ ⊂ (ν) supp θ Y Z(T0) j . ν,j

Identifying Y1 with the complex 1-manifold obtained by holomorphic removal { (ν) } of the tubes Z(Tj ) from Y (see Sect. 5.12.4), we get the holomorphic inclu- \ : \ → sion Y Z(C0) Y Z(C0) Y1. Thus the restriction θ Y \Z(C0) determines a 5.14 Tubes in an Arbitrary Riemann Surface 263

1-form on the image Y \ (C )(Y \ Z(C0)) in Y1 that extends by 0 to a closed ∞ Z 0 compactly supported C 1-form τ on Y1. On the other hand, each connected com- 1 ponent P of Y1 is a copy of P in which, for some unique choice of ν and j, ∩ \ = (ν) P Y \Z(C0)(Y Z(C0)) Y \Z(C0)(j ),

\ (ν) and the complement P Y \Z(C0)(j ) is a union of ﬁnitely many disjoint connected compact sets, each of which is mapped onto a closed disk by some local = C∞ holomorphic chart. Hence τ P dλ for some function λ on P , and in partic- (ν) ∩ (ν) (ν) ular, λ is constant on the domain Y \Z(C0)(Tj j ). Pulling back to j and extending by the associated constant, we see that for each ν ≥ 1 and each = C∞ (ν) (ν) ∪ (ν) j 1,...,mν , there exists a function ρj on the domain j Z(Tj ) (ν) = C∞ (0) such that dρj θ (ν)∪ (ν) . Similarly, we also have a function ρ1 on j Z(Tj ) (0) = (0) = 1 1 with dρ1 θ (0) . 1 = (0) = (0) ∈ C∞ = Now let η1 ρ1 on 1 1 . Given a function ην (ν) with dην ≥ ∩ θ ν for some ν 1 (in particular, ην is locally constant on T0 ν ), by replacing (ν) (ν) each function ρj with the sum of ρj and some constant, we may assume that (ν) = ρj ην on the domain [ (ν) ∪ (ν) ]∩ = (ν) ∩ j Z(Tj ) ν Z(Tj ) ν.

(ν) + + = + (ν) (ν) = = The function ην 1 given by ην 1 ν ην and ην 1 ∪ ρj for j j Z(Tj ) C∞ = 1,...,mν is then a function on ν+1 with dην θ + . Thus, by induction, ∞ ν 1 ∞ we get a sequence of C functions {ην}, and we get a well-deﬁned C function η = = = on Y with dη θ by setting η ν ην for each ν 1, 2, 3,....

Remark One may instead complete the proof of the above theorem by showing = more directly that γ θ 0 for every loop γ in Y . For any such loop is homologous (ν) to a sum of loops, each of which lies in one of the sets j (here, one uses the fact { (ν)} that the smooth Jordan curves Cj are separating in Y ). One may also choose the loops to avoid C0. Integrating the form τ on Y1 over the images of these loops in Y1, one gets the claim.

Exercises for Sect. 5.14 5.14.1 Verify that in the proof of Theorem 5.14.2, for each ν = 0, 1, 2,...,theset ∩ = \ ∩ (ν) = (ν) \ ∩ (ν) = (ν) \ Z ν ν C, and the sets Z j j C and Z Vj Vj C for j = 1,...,mν , are connected. 5.14.2 According to Sard’s theorem (see, for example, [Mi]), the set of critical values of a C∞ function on a second countable C∞ manifold is a set of measure 0. Using Sard’s theorem, Theorem 9.10.1, Lemma 5.11.1, and arguments similar to those appearing in this section, prove that every second 264 5 Uniformization and Embedding of Riemann Surfaces

countable oriented C∞ surface may be obtained by C∞ oriented attachment of a locally ﬁnite family of disjoint tubes to an oriented planar C∞ surface N (cf. Exercises 5.12.8 and 5.13.2). Using this fact together with Theorem 5.5.3 and Theorem 5.9.2, construct a slightly different proof of Theorem 5.14.1.

~~5.15 Nonseparating Smooth Jordan Curves~~

The goal of this section is a proof of equivalence of the topological (as in Def- inition 5.11.2) and analytic (as in Lemma 5.11.3) characterizations of separating smooth Jordan curves in an oriented smooth surface M; that is, the goal is a proof that for every nonseparating smooth Jordan curve C in M, there exists a compactly supported closed C∞ 1-form with nonzero integral along C. This fact will not be applied directly until Sect. 5.17, but the construction of the associated 1-form will be a key element in the construction in Sect. 5.16 of a canonical homology basis for a compact Riemann surface. The main point of this section is the following:

Lemma 5.15.1 For any smooth Jordan curve β :[0, 1]→M with nonseparating image B = β([0, 1]) in an oriented C∞ surface M, we have the following: (a) There exists a smooth Jordan curve α :[0, 1]→M with image A ≡ α([0, 1]) such that A ∩ B ={α(0)}={β(0)} and such that the pair of tangent vectors ˙ α(˙ 0), β(0) is a positively oriented basis for the tangent space Tα(0)M. (b) For any choice of a smooth Jordan curve α with the properties in (a), the sets A ≡ α([0, 1]) and A ∪ B are nonseparating. Moreover, for any choice of neighborhoods U and V of A and B, respectively, there exist closed com- ∞ pactly supported C real 1-forms θ and τ on M such that supp θ ⊂ V \ B, ⊂ \ = = = = supp τ U A, α θ 1, β θ 0, α τ 0, β τ 1, and for any loop γ ∈ Z ∈ Z in M, we have γ θ and γ τ (hence, for M second countable, we have [ ] [ ] ∈ 1 Z θ H 1 , τ H 1 H (M, )).

Proof According to Lemma 5.11.1, there exists a positively oriented local C∞ chart (T, ,(0; 1/R, R)) with R>1, B ⊂ T , and (β(t)) = e2πit for each t ∈[0, 1]. −1 Fixing R0 ∈ (1,R),wemayletα1 :[−1, 1]→T be the path from (1/R0) to −1 (R0) given by

~~− − − → 1 t = 1 t log R0 = 1 t log R0 + t (R0) (e ) (e 0i).~~

By Lemma 5.10.6, there is an injective smooth path α2 in the domain M \ B from −1 −1 (R0) to (1/R0), and we may set ≡ −1 [ ] ∈[ s0 max α2 (α1( 0, 1 )) 0, 1), ≡ [ ]∩ −1 [− ] ∈ ] s1 min( s0, 1 α2 (α1( 1, 0 ))) (s0, 1 , 5.15 Nonseparating Smooth Jordan Curves 265

~~Fig. 5.19 Transverse nonseparating smooth Jordan curves and a smooth annular neighborhood~~

~~≡ −1 ∈ ] t0 α1 (α2(s0)) (0, 1 , ≡ −1 ∈[− t1 α1 (α2(s1)) 1, 0).~~

We then get a piecewise smooth Jordan curve α3 in M by setting ⎧ ⎨⎪α1(3t) if t ∈[0,t0/3], s1−s0 α3(t) = α2(s0 + + − (3t − t0)) if t ∈[t0/3, 1 + (t1/3)], ⎩⎪ 3 t1 t0 α1(3(t − 1)) if t ∈[1 + (t1/3), 1].

−1 Moreover, α3 is loop-smooth at α3(0) = β(0) = (1), α3 is smooth at each point in (0, 1) \{t0/3, 1 + (t1/3)}, and α3((0, 1)) ⊂ M \ B. Applying Lemma 5.10.5,we get a smooth Jordan curve α that agrees with α3 outside a small neighborhood of the set {t0/3, 1 + (t1/3)}, and that maps that small neighborhood into a small neighborhood of {α3(t0/3), α3(1 + (t1/3))}. In particular, we may choose α so that α([0, 1]) ∩ B ={α(0)}={−1(1)}={β(0)}, and the pair α(˙ 0), β(˙ 0) forms a positively oriented basis. Thus part (a) is proved. For the proof of part (b), suppose that α is an arbitrary smooth Jordan curve for which the image A ≡ α([0, 1]) meets B only in the point α(0) = β(0) and ˙ the pair α(˙ 0), β(0) forms a positively oriented basis for Tα(0)M.Letαˆ : R → M be the 1-periodic C∞ immersion given by t → α(t − t). Given a neighborhood V of B in M, we may choose the annular neighborhood T to be relatively compact in V . Finally, let us denote the two connected components of T \ B by −1 −1 T0 ≡ ((0; 1,R))and T1 ≡ ((0; 1/R, 1)) (see Fig. 5.19). Since polar coordinates (r, θ) (see Example 9.7.20) near (β(0)) = 1forma positively oriented local C∞ chart, we have <(dr∧ dθ) ∗α(˙ ), ∗β(˙ ) 0 0 0 ∂ = (dr ∧ dθ) ∗α(˙ 0), 2π ∂θ 1 d = 2πdr( ∗α(˙ 0)) = 2π r( (α(t)))ˆ . dt t=0 It follows that the function t → | (α(t))ˆ | is strictly increasing on a neighborhood of each integer. Hence we may ﬁx a number s ∈ (0, 1/2) such that α([0,s]) ⊂ T0 and α([1 − s,1]) ⊂ T1. 266 5 Uniformization and Embedding of Riemann Surfaces

We may choose a real-valued C∞ function ϕ on M \ B such that supp ϕ V , ∞ ϕ ≡ 0onT0, and ϕ ≡ 1onT1. Hence dϕ extends to a unique closed C 1-form \ ⊂ \ = θ on M with supp θ V T V B, and in particular, we have β θ 0. Since θ = 0onT ⊃ α([0, 1]\[s,1 − s]) and θ = dϕ on M \ B ⊃ α([s,1 − s]),wehave θ = θ = ϕ(α(1 − s)) − ϕ(α(s)) = 1. α α [s,1−s] It also follows that θ has an integer-valued integral along each loop γ in M.To see this, we may assume that γ(0) = γ(1) ∈ T \ B, and we may choose a partition 0 = t0

Proposition 5.15.2 For any orientable C∞ surface M: = (a) A smooth Jordan curve C in M is separating if and only if C θ 0 for every closed C∞ 1-form θ with compact support in M. (b) M is planar (i.e., every C∞ compactly supported closed 1-form on M is exact) if and only if every smooth Jordan curve C in M is separating.

~~Proof Part (a) follows from Lemma 5.11.3 and Lemma 5.15.1. Part (b) follows from Proposition 5.10.7, Proposition 10.5.5, and part (a).~~

Exercises for Sect. 5.15 5.15.1 Let M be an oriented second countable C∞ surface, let M∗ = M ∪{∞} be the one-point compactiﬁcation of M (see Deﬁnition 9.1.11), and let β :[0, 1]→M be a smooth Jordan curve with image B ≡ β([0, 1]). (a) Assuming that B is separating in M but nonseparating in M∗, prove that: (i) There exists a C∞ embedding α : R → M with image A ≡ α(R) such that A ∩ B ={α(0)}={β(0)} and such that the pair of tangent vectors α(˙ 0), β(˙ 0) is a positively oriented basis for the tangent space Tα(0)M; and (ii) If α : R → M is any C∞ embedding with the properties in (a), then the set A ≡ α(R) is nonseparating in M,thesetA ∪ B is nonseparating in M∗, and for any choice of a neighborhood U of A, there ∞ exists a closed C real 1-form τ on M such that supp τ ⊂ U \ A = ∈ Z and β τ 1, and such that γ τ for every loop γ in M (cf. Ex- ercise 5.11.5). ∗ = (b) Assuming that B is separating in M , prove that β θ 0 for every C∞ [ ] = closed 1-form θ on M (i.e., β H1 0).

~~5.16 Canonical Homology Bases in a Compact Riemann Surface~~

In this section, we construct a canonical homology basis associated to holomorphically attached tubes in a compact Riemann surface. The required facts concerning homology and cohomology appear in Sects. 10.6 and 10.7.

Theorem 5.16.1 Let g ∈ Z≥0, and let X be a compact Riemann surface obtained by holomorphic attachment of g disjoint tubes to P1 (as in Sect. 5.12.1 and Theo- rem 5.13.1). Then we have the following (the case g = 3 is illustrated in Fig. 5.20): (a) For F = R or C, we have

~~1 1 2g = rank H1(X, Z) = rank H (X, Z) = dimF H1(X, F) = dimF H (X, F).~~

~~(b) There exist smooth Jordan curves α1,β1,...,αg,βg in X with images Aj = αj ([0, 1]) and Bj = βj ([0, 1]) for j = 1,...,g such that 268 5 Uniformization and Embedding of Riemann Surfaces~~

~~Fig. 5.20 A canonical homology basis~~

(i) For all i, j = 1,...,g with i = j, Ai ∩ Aj = Ai ∩ Bj = Bi ∩ Bj =∅; (ii) For each j = 1,...,g, Aj ∩ Bj ={αj (0)}={βj (0)}; and ˙ (iii) For each j = 1,...,g, the pair of tangent vectors α˙j (0), βj (0) is a posi- ˙ ˙ tively oriented basis for the tangent space Tαj (0)X (i.e., ω(α(0), β(0)) > 0 for every positive 2-form ω). (c) If α1,β1,...,αg,βg are smooth Jordan curves in X satisfying the conditions [ ] [ ] [ ] [ ] (i)Ð(iii) in (b), then the homology classes α1 H1 , β1 H1 ,..., αg H1 , βg H1 form a basis for the A-module H1(X, A) for any subring A of C containing Z. Given neighborhoods Uj of Aj and Vj of Bj in X for j = 1,...,g, there ex- C∞ { }g { }g ⊂ \ ist closed 1-forms θj j=1 and τj j=1 such that supp θj Vj Bj and supp τj ⊂ Uj \ Aj for j = 1,...,g, and for any subring A of C containing Z, {[ ] }g ∪{[ ] }g 1 A the cohomology classes θj H 1 j=1 τj H 1 j=1 form a basis for H (X, ) {[ ] }g ∪{[ ] }g = that is dual to the basis αj H1 j=1 βj H1 j=1; that is, for i, j 1,...,g, [ ] [ ] = [ ] [ ] = ( θi H 1 , βj H1 )deR ( τi H 1 , αj H1 )deR 0

~~and 1 if i = j, ([θi] 1 , [αj ]H )deR = ([τi] 1 , [βj ]H )deR = H 1 H 1 0 if i = j.~~

(d) Suppose T1,...,Tg are open subsets of X with disjoint closures such that X \ (T 1 ∪···∪T g) is connected and for each j = 1,...,g, Tj is a relatively compact subset of a local holomorphic chart that maps Tj onto an annulus. Then holomorphic removal of the tubes T1,...,Tg yields a compact Riemann surface that is biholomorphic to P1. Moreover, we may choose smooth Jordan 5.16 Canonical Homology Bases in a Compact Riemann Surface 269

curves α1,β1,...,αg,βg as in (b) so that for each j = 1,...,g, Bj is a concentric circle in Tj with respect to the biholomorphism onto an annulus and ˆ Ai ∩ T j =∅for i = 1,...,j,...,g. (e) Suppose T1,...,Tg are open subsets of X with disjoint closures such that for each j = 1,...,g, Tj is a relatively compact subset of a local holomorphic chart that maps Tj onto an annulus. Assume that there exist smooth Jordan curves α1,β1,...,αg,βg as in (b) such that for each j = 1,...,g, ⊂ [ ] Bj Tj . Then βj βj (0) generates the fundamental group π1(Tj ,βj (0)) for each j = 1,...,g, and holomorphic removal of the tubes T1,...,Tg yields a compact Riemann surface that is biholomorphic to P1.

Remarks 1. Part (a) of the theorem (together with the discussion of homology in Sects. 10.6 and 10.7) implies that g, the number of tubes that one attaches to P1 in order to construct a given compact Riemann surface X, depends only on the topology of X, not on the holomorphic structure or even the C∞ structure. 2. It will follow from the results of Chap. 6 that any compact orientable topological surface M (see Sect. 6.10) may be obtained by attaching g tubes to the sphere, where 2g = rank H1(M, Z). One may also obtain this fact directly (see, for example, [T]). 3. Actually, any two compact orientable topological surfaces are homeomorphic if and only if they have isomorphic homology groups (again, see [T]). Of course, two homeomorphic compact Riemann surfaces need not be biholomorphic. For example, two complex tori need not be biholomorphic, although all complex tori are diffeomorphic (see Exercise 5.9.1). 4. Part (e) is a partial converse of part (d), but in (e), one may not be able to choose each coordinate annulus Tj so that the given curve Bj becomes a concentric [ ] circle. However, any choice of such a concentric circle in Tj will represent βj H1 −[ ] or βj H1 , and one may homotope αj slightly to get transversality. 5. For α1,β1,...,αg,βg as in part (b), choosing a path γj from a ﬁxed base point = =[ ∗ ∗ −] =[ ∗ ∗ −]∈ x0 to αj (0) βj (0) and letting uj γj αj γj ,vj γj βj γj π1(X, x0) for each j = 1,...,g, one can show that u1,v1,...,ug,vg generate π1(X, x0) (see Exercise 5.17.3), and that in fact, π1(X, x0) is equal to the free group on these −1 −1 ··· −1 −1 = generators modulo the relation u1v1u1 v1 ugvgug vg e (see, for example, [Hat]). However, we will not use this fact.

~~Before addressing the proof of Theorem 5.16.1, we consider a deﬁnition and some consequences.~~

[ ] Deﬁnition 5.16.2 Let X be a compact Riemann surface. A homology basis α1 H1 , [ ] [ ] [ ] Z A β1 H1 ,..., αg H1 , βg H1 for H1(X, ) (which will also be a basis for H1(X, ) for any subring A of C containing Z) with homology classes represented by smooth Jordan curves α1,β1,...,αg,βg satisfying the conditions in part (b) of Theo- rem 5.16.1 is called a canonical homology basis.

~~Applying Theorem 5.13.1, Lemma 5.11.1, Theorem 5.9.2, and Theorem 5.16.1, we get the following: 270 5 Uniformization and Embedding of Riemann Surfaces~~

Corollary 5.16.3 Let X be a compact Riemann surface, let 2g = rank H1(X, Z), and let α1,β1,...,αg,βg be smooth Jordan curves satisfying the conditions in [ ] [ ] [ ] [ ] part (b) of Theorem 5.16.1 (in particular, α1 H1 , β1 H1 ,..., αg H1 , βg H1 is a canonical homology basis). Then there exist open subsets T1,...,Tg of X with disjoint closures such that X \ (T 1 ∪···∪T g) is connected and such that for each j = 1,...,g, Tj is a relatively compact subset of a local holomorphic chart that ⊂ [ ] maps Tj onto an annulus, Bj Tj , βj βj (0) generates the fundamental group ˆ π1(Tj ,βj (0)), Ai ∩ T j =∅for all i = 1,...,j,...,g, and holomorphic removal 1 of the tubes T1,...,Tg yields a Riemann surface that is biholomorphic to P .

~~Theorem 5.13.1, Theorem 5.16.1, Corollary 4.9.2, and Corollary 4.6.5 together give the following:~~

~~Corollary 5.16.4 Up to biholomorphism, every compact Riemann surface X may be obtained by holomorphic attachment of g disjoint tubes to P1, where = 1 = O = g dimC HDol(X) dim (X, (KX)) genus(X).~~

Proof of Theorem 5.16.1 Throughout this proof, A will denote a subring of C containing Z. Step 1. We construct smooth Jordan curves satisfying (b). By hypothesis, we { }g { }g may choose open subsets Tj j=1 of X with disjoint closures, constants R0j j=1 { }g ∞ and R1j j=1 in (1, ), and a biholomorphism j of a neighborhood of T j onto a domain in C with j (Tj ) = (0; 1/R0j ,R1j ) for each j = 1,...,g such that if = −1 2πit ∈[ ] = [ ] ⊂ = βj (t) j (e ) for each t 0, 1 , Bj βj ( 0, 1 ) Tj for each j 1,...,g, and Z = X \ (B1 ∪ ··· ∪ Bg), then there is a biholomorphism Z of Z onto a 1 1 domain in P with complement P \ Z(Z) equal to the union of a ﬁnite collection of disjoint compact sets, each of which is mapped onto a closed disk by some 1 local holomorphic chart in P . Observe that X1 ≡ X \ (B2 ∪···∪Bg) = Z ∪ T1 is connected, since Z and T1 are connected and Z ∩ T1 = ∅. Moreover, since Z = X1 \ B1 is connected, B1 is nonseparating in X1. Therefore, by Lemma 5.15.1, there exists a smooth Jordan curve α1 with image A1 = α1([0, 1]) in X1 such that ˙ A1 ∩ B1 ={α1(0)}={β1(0)} and the pair of tangent vectors α˙1(0), β1(0) is a positively oriented basis for the tangent space Tα1(0)X. Moreover, the sets A1, B1, and A1 ∪ B1 are nonseparating in X1 (for any choice of α1). Suppose k ∈{2,...,g} and we have constructed smooth Jordan curves α1,...,αk−1 such that for each j = 1,...,k− 1, we have Aj ≡ αj ([0, 1]) ⊂ Xj ≡ X \ (A1 ∪···∪Aj−1 ∪ B1 ∪···∪Bj ∪···∪Bg),

Xj is connected, Aj , Bj , and Aj ∪Bj are nonseparating in Xj , Aj ∩Bj ={αj (0)}= ˙ {βj (0)}, and the pair of tangent vectors α˙j (0), βj (0) is a positively oriented basis for the tangent space Tαj (0)X. Applying Lemma 5.15.1 to the smooth Jordan curve βk in the Riemann surface Xk ≡ X \ (A1 ∪···∪Ak−1 ∪ B1 ∪···∪Bk ∪···∪Bg) 5.16 Canonical Homology Bases in a Compact Riemann Surface 271

~~(Bk is nonseparating in Xk because Ak−1 ∪ Bk−1 is nonseparating in Xk−1 and Xk \ Bk = Xk−1 \ (Ak−1 ∪ Bk−1)), we get a smooth Jordan curve αk such that~~

Ak ≡ αk([0, 1]) ⊂ Xk, ˙ Ak ∩ Bk ={αk(0)}={βk(0)}, the pair of tangent vectors α˙ k(0), βk(0) is a positively oriented basis for the tangent space Tαk(0)X, and (for any choice of such an αk) the sets Ak, Bk, and Ak ∪ Bk are nonseparating in Xk. Thus, by induction, we get smooth Jordan curves αj and βj for j = 1,...,g satisfying the conditions in (b). [ ] [ ] [ ] [ ] Step 2. We show that the homology classes α1 H1 , β1 H1 ,..., αg H1 , βg H1 represented by the Jordan curves α1,β1,...,αg,βg constructed in Step 1 are linearly independent in H1(X, A). We may ﬁx domains U1,V1,...,Ug,Vg in X such that for each j = 1,...,g,wehaveAj ⊂ Uj , Bj ⊂ Vj , and for each i = ˆ 1,...,j,...,g, (Ui ∪Vi)∩(Uj ∪Vj ) =∅. By Lemma 5.15.1, for each j = 1,...,g, ∞ we may choose closed C 1-forms θj and τj on X such that supp θj ⊂ Vj \ Bj , supp τ ⊂ U \ A , θ = τ = 1, θ = τ = 0, θ, τ ∈ Z for every j j j αj j βj j βj j αj j γ γ loop γ in X, and (clearly) ˆ θj = θj = τj = τj = 0fori = 1,...,j,...,g. αi βi αi βi [ ] [ ] [ ] [ ] It follows that the homology classes α1 H1 , β1 H1,..., αg H1 , βg H1 are linearly A = g · + g · ∈ A independent over .Forifξ j=1 rj αj j=1 sj βj Z1(X, ), with rj ,sj ∈ A for j = 1,...,g, then for each j = 1,...,g,wehave

~~rj = θj and sj = τj . ξ ξ~~

So if ξ is homologous to 0, then rj = sj = 0 for each j = 1,...,g. Similarly, the [ ] [ ] [ ] [ ] ∈ 1 A cohomology classes θ1 H 1 , τ1 H 1 ,..., θg H 1 , τg H 1 H (X, ) are linearly independent. [ ] [ ] [ ] [ ] Step 3. We show that the cohomology classes θ1 H 1 , τ1 H 1 ,..., θg H 1 , τg H 1 1 represented by the 1-forms θ1,τ1,...,θg,τg constructed in Step 2 span H (X, A). For this, we must show that each closed C∞ 1-form ρ on X with integrals along loops taking values in A is cohomologous to a linear combination over A of the differential forms θ1,τ1,...,θg,τg. By replacing ρ with the 1-form g g ρ − ρ · θj − ρ · τj , j=1 αj j=1 βj we may assume without loss of generality that ρ = ρ = 0forj = 1,...,g. αj βj In this case, we will show that in fact, ρ is exact. 272 5 Uniformization and Embedding of Riemann Surfaces For each j = 1,...,g,wehave ρ = 0, and hence, since the path homotopy βj class [βj ] generates the fundamental group of a neighborhood of Tj , ρ is exact on this neighborhood. Choosing a function with differential ρ on this neighborhood and subtracting from ρ the differential of a suitable cut-off of the function, we may assume that ρ ≡ 0onTj for each j = 1,...,g. It follows that there is a unique C∞ ˆ P1 ˆ = ≡ g \ ∪ closed 1-form ρ on with ρ 0 on the open set W Z( j=1(Tj Bj )) P1 \ ∗ ˆ = P1 C∞ ( Z(Z)) and Zρ ρ on Z. Since is simply connected, there is a function λˆ on P1 with dλˆ =ˆρ. In particular, λˆ is locally constant on W . On the other hand, for each j = 1,...,g, choosing s ∈ (0, 1/2) with αj ([0,s]∪[1 − s,1]) ⊂ Tj , we get ˆ ˆ 0 = ρ = ρ = ρˆ = λ(Z(αj (1 − s))) − λ(Z(αj (s))). αj αj [s,1−s] Z(αj [s,1−s])

Thus λˆ has the same constant value on the two connected components −1 ; −1 ; Z(j ((0 1/R0j , 1))) and Z(j ((0 1,R1j ))) ˆ of the set Z(Tj \ Bj ) ⊂ W . It follows that the function λ ◦ Z extends to a ∞ unique C function λ on X (which is constant on Tj for each j = 1,...,g ), and = = ∗ ˆ = = = we have dλ ρ, since dλ Zρ ρ on Z and dλ 0 ρ on Tj . Thus [ ] [ ] [ ] [ ] 1 A θ1 H 1 , τ1 H 1 ,..., θg H 1 , τg H 1 is a basis for H (X, ). [ ] [ ] [ ] [ ] Step 4. We show that the classes α1 H1 , β1 H1 ,..., αg H1 , βg H1 form a basis for H1(X, A). Given a 1-cycle ξ ∈ Z1(X, A), the 1-cycle g g ζ ≡ θj · αj + τj · βj ∈ Z1(X, A) j=1 ξ j=1 ξ satisﬁes

θj = τj = 0forj = 1,...,g, ξ−ζ ξ−ζ [ ] [ ] [ ] [ ] and it follows that ξ is homologous to ζ . Thus α1 H1 , β1 H1 ,..., αg H1 , βg H1 is a basis for H1(X, A). In particular, part (a) is proved. Step 5. We prove part (c). Suppose that α1,β1,...,αg,βg are arbitrary smooth Jordan curves in X satisfying the conditions (i)Ð(iii) in (b), and for each j = 1,...,g, Uj is a neighborhood of Aj and Vj is a neighborhood of Bj . Then, for each j = 1,...,g, αj and βj are nonseparating in X (since by Lemma 5.11.1 and Lemma 5.10.3, the connected set βj ((0, 1)) meets each connected component of X\Aj and the connected set αj ((0, 1)) meets each connected component of X\Bj ). ∞ By Lemma 5.15.1, there exist C closed 1-forms θ1,τ1,...,θg,τg on X such that for i, j = 1,...,g, supp θj ⊂ Vj \ Bj , supp τj ⊂ Uj \ Aj , 1ifi = j, θ = τ = j j = αi βi 0ifi j, 5.16 Canonical Homology Bases in a Compact Riemann Surface 273 = = [ ] [ ] ∈ 1 Z α τj β θj 0, and θj H 1 , τj H 1 H (X, ). The argument in Step 2 shows i i [ ] [ ] [ ] [ ] that the homology classes α1 H1 , β1 H1 ,..., αg H1 , βg H1 are linearly indepen- A [ ] [ ] [ ] [ ] dent in H1(X, ), and the cohomology classes θ1 H 1 , τ1 H 1 ,..., θg H 1 , τg H 1 1 A [ ] [ ] [ ] are linearly independent in H (X, ). In particular, α1 H1 , β1 H1 ,..., αg H1 , [ ] R βg H1 is a basis for the 2g-dimensional vector space H1(X, ) (aswellasfor C [ ] [ ] [ ] [ ] 1 R H1(X, )) with dual basis θ1 H 1 , τ1 H 1 ,..., θg H 1 , τg H 1 for H (X, ) (as well 1 C [ ] [ ] [ ] as for H (X, )). The argument in Step 4 shows that α1 H1 , β1 H1 ,..., αg H1 , [ ] A [ ] βg H1 must also span H1(X, ), and hence the dual cohomology classes θ1 H 1 , [ ] [ ] [ ] 1 A τ1 H 1 ,..., θg H 1 , τg H 1 span H (X, ). Thus part (c) is proved. Step 6. We prove part (d). The proof of (d) is similar to the proof of Theo- rem 5.13.1.LetBj be a concentric circle in the holomorphic coordinate annulus Tj = \ g \ g for j 1,...,g. By hypothesis, X j=1 T j and therefore X j=1 Bj are connected. Holomorphic removal of the tube Tg yields a compact Riemann surface with real ﬁrst homology space of dimension < 2g. However, since the ﬁrst homology of a compact Riemann surface is of even dimension, the dimension of the homology must have actually dropped by at least 2. On the other hand, removal of Tg still leaves (the images of) the remaining tubes, each of which contains a nonseparating smooth Jordan curve (given by the image of Bj for some j ∈{1,...,g − 1}). Proceeding inductively, we will not get a planar Riemann surface before we have removed all of the g tubes T1,...,Tg, at which point we get a compact Riemann surface with zero homology, that is, a Riemann surface that is biholomorphic to P1. Observe also that Step 1 provides the desired Jordan curves satisfying (b). Step 7. We prove part (e). Observe that for each j, the path homotopy class [βj ] [ ] = in π1(Tj ) cannot be the identity, because by (c), βj H1(X,Z) 0. Thus we have k [βj ]=[γj ] j for some kj ∈ Z\{0}, where γj is a loop corresponding to a concentric ∞ circle in Tj . By Lemma 5.15.1, there exist C closed 1-forms η1,...,ηg on X such that for all i, j = 1,...,g, 1ifi = j, η = j = βi 0ifi j, ∈ Z and such that γ ηj for each loop γ in X. In particular, for each j,wehave 1 = η = k · η ∈ k Z, and hence we must have k =±1. Thus [β ] gener- βj j j γj j j j j ates π1(Tj ), and Cj ≡ γj ([0, 1]) is nonseparating. Consequently, by subtracting the differential of a suitable C∞ function, we may also assume that supp ηj ⊂ X \ (T 1 ∪···∪T j ∪···∪T g).

It follows by induction that for j = 2,...,g, X \ (T 1 ∪···∪T j−1) is a domain in which Cj and T j are nonseparating, and hence X \ (T 1 ∪···∪T g) is connected. Part (d) now gives the claim (e).

Remark The above proof did not use Theorems 10.7.16 and 10.7.18 in an essential way. In fact, a proof of these theorems for the special case of a compact Riemann surface is actually contained within the above. 274 5 Uniformization and Embedding of Riemann Surfaces

Exercises for Sect. 5.16 5.16.1 Using Exercise 5.9.2 and the results of this section, prove that every compact Riemann surface of genus 1 is biholomorphic to a complex torus (a different approach was considered in Exercise 5.9.3, and yet another approach will be considered in Sect. 5.22). 5.16.2 Let X be a compact Riemann surface of genus 2. Prove that there is a covering Riemann surface ϒ : X → X such that (i) X is obtained by holomorphic attachment of a locally ﬁnite inﬁnite { }∞ C sequence of disjoint tubes Tν ν=1 to ; (ii) We have Deck(ϒ) =∼ Z2; ∩ =∅ ∈ \{ } (iii) We have Tν (Tν) for each Deck(ϒ) IdX and each ∈ Z ν >0; and · = ∞ (iv) We have Deck(ϒ) T1 ν=1 Tν . Hint. Apply Exercise 5.16.1 (cf. Exercise 5.12.9). 5.16.3 The goal of this exercise is a proof of a theorem of Abel (see part (d) below). [ ] [ ] [ ] [ ] Let α1 H1 , β1 H1 ,..., αg H1 , βg H1 be a canonical homology basis for a compact Riemann surface X of genus g>0. ∞ (a) Prove that if η1 and η2 are two closed C 1-forms on X, then

~~g η1 ∧ η2 = η1 · η2 − η1 · η2 . X j=1 αj βj βj αj~~

Hint. Choose the representatives α1,β1,...,αg,βg to be smooth Jor- dan curves as in part (b) of Theorem 5.16.1. For each j = 1,...,g, ∞ construct C annular neighborhoods Jj ⊃ Aj and Tj ⊃ Bj ,asin Lemma 5.11.1. Also choose the neighborhoods so that (J i ∪ T i) ∩ (J j ∪ T j ) =∅for i = j. Form dual closed 1-forms θj and τj with supp θj ⊂ Tj \ Bj and supp τj ⊂ Jj \ Aj for j = 1,...,g. Observe that ∧ = ∧ = = ∧ = for all i, j, X θi θj X τi τj 0, and for i j, X θi τj 0. Show that θj has a potential λj on Tj (in particular, λj is locally constant near Bj ∪ ∂Tj ), and hence θj ∧ τj = d(λj τj ) on Tj . Applying Stokes’ ∧ = theorem, prove that X θj τj 1, and then prove the claim. (b) Prove that for every holomorphic 1-form η on X,

~~g 2 = · η L2 2 Im η η . j=1 βj αj~~

~~(c) Prove that there is a unique basis η1,...,ηg for (X) such that for all i, j = 1,...,g, 1ifi = j, η = i = αj 0ifi j. 5.17 Complements of Connected Closed Subsets of P1 275~~

~~(d) For the basis η1,...,ηg in part (c), let Z = (zij ) be the complex g × g matrix with entries~~

zij ≡ ηi ∀i, j = 1,...,g. βj = = Prove that Z is symmetric (i.e., zij zji for all i and j) and Im Z g (Im zij ) is positive definite (i.e., i,j=1 Im zij uiuj > 0 for all g (u1,...,ug) ∈ R \{0}). Hint. For symmetry, first observe that ηi ∧ ηj ≡ 0 for all i and j. Then apply part (a). In order to prove that Im Z is positive definite, first " # = prove that ηi,ηj L2 2Imzij for all i and j. 5.16.4 Prove a C∞ version of Theorem 5.16.1 for an oriented compact C∞ surface M obtained by C∞ oriented attachment of g tubes to a compact planar oriented C∞ surface N (cf. Exercises 5.12.8 and 5.13.2).

~~5.17 Complements of Connected Closed Subsets of P1~~

~~The following fact will be applied in the construction of embeddings of open Riemann surfaces (in Sect. 5.18) and in the proof of Schönﬂies’ theorem (in Sects. 6.7Ð6.9).~~

~~Lemma 5.17.1 The complement and boundary of a simply connected domain in P1 are connected, and each connected component of the complement of any connected closed subset of P1 is simply connected.~~

Remark Each connected component of the complement of a nonempty proper open subset of a topological surface must meet, and therefore contain, a boundary component of the set. Hence the complement is connected if the boundary is connected (it is easy to see that the converse is false).

Proof of Lemma 5.17.1 Let be a nonempty domain in P1. Suppose ﬁrst that is simply connected, and that A and B are disjoint (and possibly empty) subsets of ∂ that are open in ∂ and that satisfy ∂ = A ∪ B. Then A = (∂) \ B and B = (∂) \ A are also compact, so we may choose disjoint open subsets U and V of P1 with A ⊂ U and B ⊂ V . In particular, ∂U ∩ ∂= ∂V ∩∂=∅. The Riemann mapping theorem implies that is equal to the union of an increasing sequence of relatively compact domains with connected boundary, so the compact set (∂U ∪∂V)∩ = (∂U ∪∂V)∩ is contained in such a domain 0. Consequently, each of the sets ( \ 0) ∩ U = ( \ 0) ∩ U and ( \ 0) ∩ V = ( \ 0) ∩ V is both open and closed in the connected open set \ 0.If 1 A = ∅, then the ﬁrst of the above sets is nonempty, and hence \ 0 ⊂ U ⊂ P \ V ; so B ⊂ (∂) ∩ V =∅. Thus ∂ is connected, and it follows that P1 \ is also connected. 276 5 Uniformization and Embedding of Riemann Surfaces

Assuming now that the domain is a connected component of the complement P1 \ K of a nonempty connected compact set K P1, we will show that must be simply connected. For this, suppose that θ is an arbitrary closed C∞ 1-form on . Given a loop γ in , Proposition 5.4.1 provides a domain 0 such that γ([0, 1]) ⊂ 0 , ∂0 is the union of disjoint smooth Jordan curves C1,...,Cm, and for some r>1 and for each j = 1,...,m, there is a lo- ; = −1 ; cal holomorphic chart (Uj ,j ,(0 1/r,r)) in with Cj j (∂(0 1)) and ∩ = −1 ; P1 \ 0 Uj j ((0 1,r)).LetEj be the connected component of 0 con- 1 taining Cj for each j. Since Cj is a separating smooth Jordan curve in P (for example, by Proposition 5.15.2), it follows that Ej ∩ 0 = Cj (if this were not 1 the case, then there would exist a path in the set (Ej \ Cj ) ∪ 0 ⊂ P \ Cj from −1 ; −1 ; j ((0 1/r,1)) to j ((0 1,r))). In particular, the connected components E1,...,Em are distinct and therefore disjoint. Reordering so that K ⊂ E1,we 1 see that the set 1 ≡ 0 ∪ E2 ∪ ··· ∪ Em is a smooth domain in P that lies 1 in the connected component of P \ K and that satisﬁes ∂1 = C1 ⊂ and ∩ = −1 ; 1 U1 1 ((0 1,r)). Now Stokes’ theorem implies that θ = dθ = 0. Hence, since ∂ = ∂1 1 1 = −1 ; C1 1 (∂(0 1)), the restriction of θ to U1 must be exact. Forming a potential and cutting off, we get a C∞ function ρ on such that dρ = θ on a neighborhood of ∂1. Hence the 1-form τ that is equal to θ − dρ at each point in 1 and 0 at each 1 ∞ point in P \ 1 is closed and of class C , and therefore θ = (θ − dρ) = τ = 0. γ γ γ

~~It follows that θ is exact (on ), and therefore, by Corollary 5.2.7, must be simply connected.~~

Exercises for Sect. 5.17 5.17.1 Two additional approaches to the proof of the second part of Lemma 5.17.1 (each connected component of the complement of any connected closed subset of P1 is simply connected) are outlined in parts (a) and (b) below. Let K be a connected closed subset of P1, and let be nonempty connected component of P1 \ K. (a) According to Sard’s theorem (see, for example, [Mi]), the set of critical values of a C∞ function on a second countable C∞ manifold is a set of measure 0. Prove that is simply connected by modifying the proof of Lemma 5.17.1 so that Sard’s theorem, Theorem 9.10.1, and Lemma 5.11.1 are applied in place of Proposition 5.4.1. (b) Suppose θ is a closed C∞ 1-form on . Applying Lemma 5.15.2,show that θ integrates to 0 along every smooth Jordan curve C in , and applying Proposition 5.10.7, conclude that θ is exact. From this, conclude that is simply connected. 5.17 Complements of Connected Closed Subsets of P1 277

5.17.2 A Riemann surface is said to be of finite topological type if its fundamental group is finitely generated. Prove that a Riemann surface X is of finite topological type if and only if X is biholomorphic to the complement X =∼ Y \ K in some compact Riemann surface Y of some (possibly empty) compact set K ⊂ Y that is equal to a union of finitely many disjoint compact sets, each of which is either a singleton or a closed disk in some local holomorphic chart (X is said to be of finite type if K may be chosen to be a finite set). Hint. Assuming that π1(X) is finitely generated, show that after holomorphic removal of finitely many tubes, one may assume that X P1 (see Sect. 5.13). Suppose X is a topologically Runge smooth domain, and 1 U1,...,Um are the connected components of P \ . Applying Proposi- tion 5.15.2, prove that ∂ has exactly m connected components. Assuming that ∞∈U1 and fixing a point zj ∈ Uj \ X for each j = 2,...,m,show ∞ that the closed C 1-forms θj = dz/(z − zj ) for j = 2,...,m on X represent linearly independent cohomology classes, and from this, obtain a bound on m. Also show that ∪ U1 ∪···∪Uj ∪···∪Um is simply connected, and use this observation to obtain Y and K. 5.17.3 Let X be a compact Riemann surface of genus g;letα1,β1,...,αg,βg be smooth Jordan curves in X with the properties in part (b) of Theorem 5.16.1; ≡ [ ] ≡ [ ] = let Aj αj ( 0, 1 ) and Bj βj( 0, 1 ) for j 1,...,g;letK be a con- \[ g ∪ ] nected compact subset of X j=1(Aj Bj ) with connected comple- \ { }m ment X K in X;let Dν ν=1 be relatively compact smooth domains in \[ ∪ g ∪ ] X K j=1(Aj Bj ) with disjoint closures and connected boundaries; and for each ν = 1,...,m,letKν be a connected compact subset of Dν and ≡ [ ] = γν a smooth Jordan curve with Cν γν ( 0,1 ) ∂Dν (cf. Exercise 5.17.2). ≡ \[ ∪ m ] (a) Prove that the open set X K ν=1 Dν is connected. \[ ∪ m ] (b) Prove that if Y is the connected component of X K ν=1 Kν con- ∈ ˆ = ∗ ∗ − ˆ = ∗ ∗ − taining , p , αj λj αj λj and βj λj βj λj for some path = = ˆ = ∗ ∗ − λj in from p to αj (0) βj (0) for j 1,...,g, and γν ην γν ην for some path ην in \ K from p to γν(0) for ν = 1,...,m, then the ˆ ˆ path homotopy classes of the loops αˆ1,...,αˆg, β1,...,βg, γˆ1,...,γˆm in Y generate π1(Y, p). In particular, the path homotopy classes of the ˆ ˆ loops αˆ 1,...,αˆg, β1,...,βg in X (in X \ K) generate π1(X, p) (respectively, generate π1(X \ K,p)). Hint. Set K0 ≡ K. For the proof of (b), first consider the case g = 0. That 1 is, for X = P , prove that [ˆγ1],...,[ˆγm] generate π1(Y, p). For this, prove first that for each ν = 0,...,m, Kν has a small simply connected neighborhood in which the complement of Kν is biholomorphic to a punctured disk or an annulus (the associated neighborhood of K0 will be denoted by D0). Deduce from this that fixing a point zν ∈ Kν for each ν, there is a continuous mapping of X \{z0,...,zm} into Y that fixes points in an arbitrarily large compact subset of Y . Prove also that on the other hand, every loop inY with \{ } \ base point p is homotopic within X z0,...,zm to a loop in X ν Dν . Using these observations, reduce to the case in which Kν ={zν} for each ν. 278 5 Uniformization and Embedding of Riemann Surfaces

5.17.4 Let X be a Riemann surface. The commutator of elements a and b of a group G is the element [a,b]≡aba−1b−1.Thecommutator subgroup of G is the subgroup generated by the set of commutators and is denoted by [G, G]. (a) Prove that for any group G, [G, G] is a normal subgroup of G and the quotient group G/[G, G] is an Abelian group. Prove also that [G, G] is the smallest normal subgroup of G with this property, that is, if H is any normal subgroup of G for which G/H is Abelian, then H ⊃[G, G]. (b) Prove that for any point p ∈ X,themap π1(X, p) [π1(X, p), π1(X, p)]→H1(X, Z)

~~given by~~

~~[ ] ·[ ]→[ ] γ π1(X,p) π1(X, p), π1(X, p) γ H1(X,Z)~~

is a well-deﬁned group isomorphism. Hint. In exhausting smooth topologically Runge subdomains, construct generating loops as in Exercise 5.17.3 along with dual closed 1-forms. In order to obtain the dual closed 1-forms associated to the boundary Jordan curves, apply Lemma 5.15.1 and Exercise 5.15.1. A C Z A =∼ (c) Prove that for any subring of containing ,wehaveH1 (X, ) H1(X, A) (see the remarks at the end of Sect. 10.7 for the deﬁnition of A the singular homology group H1 (X, ), and the exercises for Sect. 10.7 for some of its properties). (d) Prove that for any subring A of C containing Z, the mapping

~~[ ] → [ ] · ξ H1(X,A) ( ξ H1(X,A), )deR~~

1 gives an injective homomorphism H1(X, A) → Hom(H (X, A), A) (according to Theorem 10.7.18, this homomorphism is also surjective if π1(X) is ﬁnitely generated). 5.17.5 Let K be a compact subset of a Riemann surface X. Prove that there exist domains and such that K ⊂ X and such that for every C∞ closed 1-form ρ on , there exist a C∞ closed 1-form τ on X and a function λ ∈ D() such that ρ − τ = dλ on . Conclude from this that

~~[ 1 → 1 ]= [ 1 → 1 ] im HdeR(X) HdeR() im HdeR( ) HdeR() .~~

Hint. For a large smooth topologically Runge subdomain , construct generating boundary loops along with dual closed 1-forms as in the above exercises. Also form a large subdomain .Forρ as above, one can then form a C∞ closed 1-form θ on X such that ρ − θ is exact near ∂. Forming a potential near ∂ and cutting off, one gets a function λ ∈ D() such that ρ −θ −dλ vanishes near ∂. The restriction of ρ −θ −dλ to then extends ∞ to a C closed 1-form τ0 on X. The 1-form τ ≡ τ0 + θ and the function λ then have the required properties. 5.18 Embedding of an Open Riemann Surface into C3 279

~~5.18 Embedding of an Open Riemann Surface into C3~~

So far in this chapter, we have considered two characterizations of Riemann surfaces: as quotients of the planar domains P1, C, and (0; 1), and as planar domains to which tubes have been holomorphically attached. In the remaining sections of this chapter, we will consider other characterizations. Speciﬁcally, we will see in this section that every open Riemann surface admits a holomorphic embedding into C3, in Sect. 5.19 that every compact Riemann surface admits a holomorphic embedding into a higher-dimensional complex projective space, in Sect. 5.20 that every compact Riemann surface admits a ﬁnite holomorphic branched covering mapping onto P1, and in Sect. 5.22 that every compact Riemann surface of positive genus admits a holomorphic embedding into a higher-dimensional complex torus.

~~n Deﬁnition 5.18.1 Let F = (f1,...,fn): X → C be a mapping of a complex 1-manifold X into some complex Euclidean space Cn.~~

(a) F is holomorphic if fj is a holomorphic function for each j = 1,...,n. (b) F is a holomorphic immersion if for each point p ∈ X,wehave(dfj )p = 0for some j ∈{1,...,n}, that is, the linear map

~~1,0 n (dF )p ≡ ((df1)p,...,(dfn)p): (TpX) → C is nonsingular. (c) F is called a holomorphic embedding if F is a proper injective holomorphic immersion.~~

~~The goal of this section is a proof of the following theorem of Narasimhan [Ns1]:~~

~~Theorem 5.18.2 Every open Riemann surface X admits a (proper) holomorphic embedding into C3.~~

Remarks 1. Given a holomorphic embedding F : X → Cn of a Riemann surface X into Cn, we may identify X with its image Y ≡ F(X). In fact, it is a consequence of the theory of several complex variables that every holomorphic function on X is the restriction of some holomorphic function on Cn (see, for example, [Hö]or[KaK]). 2. The higher-dimensional analogue for Stein manifolds (and Stein spaces) was obtained independently by Bishop, Narasimhan, and Remmert. The proof given here for a Riemann surface is essentially Narasimhan’s original proof (with some modi- ﬁcations), which may be generalized to higher-dimensional Stein manifolds [Ns2]. One may adapt Bishop’s proof (see [Hö]) and Remmert’s proof of the higher- dimensional case to the Riemann surface case. 3. Compact Riemann surfaces do not admit holomorphic embeddings into Eu- clidean spaces, since every holomorphic function on a compact Riemann surface is constant. However, every compact Riemann surface admits a holomorphic embedding into a complex projective space (see Sect. 5.19). Moreover, every compact Riemann surface of positive genus admits a holomorphic embedding into a higher- dimensional complex torus (see Sect. 5.22). 280 5 Uniformization and Embedding of Riemann Surfaces

~~The main tool in the proof of the embedding theorem is the following uniform version of the Runge approximation theorem for a locally ﬁnite family of disjoint coordinate disks (see [Ns1]):~~

Theorem 5.18.3 Let be an open subset of an open Riemann surface X such that is equal to the union of a locally ﬁnite family of disjoint relatively compact open subsets of X, each of which is biholomorphic to a disk. Then, for every holomorphic function g on , every closed subset K of X with K ⊂ , and every positive continuous function ρ on , there is a holomorphic function f on X such that |f − g| <ρ on K.

~~The idea of Narasimhan’s proof of the embedding theorem is as follows. First, one forms a covering of the Riemann surface as in the following:~~

Proposition 5.18.4 In any open Riemann surface X, there exist three nonempty open subsets 0, 1, and 2 such that X = 0 ∪ 1 ∪ 2 and such that for j = 0, 1, 2, j is the union of a locally ﬁnite family of disjoint relatively compact open subsets of X, each of which is biholomorphic to a disk.

By applying Theorem 5.18.3 in order to uniformly approximate a suitable locally constant function on j for each j = 0, 1, 2, one gets a proper holomorphic mapping of X into C3. A Baire category argument then provides an approximation of this mapping that is a holomorphic embedding. Our ﬁrst goal is the proof of Theorem 5.18.3. Recall that the topological hull hY (A) of a subset A of a Hausdorff space Y is the union of A with all of the connected components of Y \ A that are relatively compact in Y (see Deﬁnition 2.13.1).

Lemma 5.18.5 Let X be a connected, locally connected, locally compact Hausdorff { }∞ space; let K0 be a compact subset of X; and let Cν ν=1 be a locally ﬁnite family of disjoint connected compact subsets of X. Then there exists a compact subset K ◦ of X such that hX(K) = K, K0 ⊂ K, and for each ν, Cν ⊂ K or Cν ∩ K =∅.

Proof By local ﬁniteness, there exists a positive integer μ such that ∩ =∅ Cν K0 for each ν>μ. Thus we may choose a relatively compact neigh- ∪ μ \ ∞ borhood U of the compact set K0 ν=1 Cν in the open set X ν=μ+1 Cν .In particular, for each ν, Cν is contained in U, or in a connected component of X \ U that is relatively compact in X, or in a connected component of X \ U that is not relatively compact in X. Lemma 2.13.2 and Lemma 2.13.3 together now imply that the set K ≡ hX(U) has the required properties.

{ }m Lemma 5.18.6 Let Kν ν=0 be a family of disjoint compact subsets of a topological surface M such that hM (K0) = K0 and such that for each index ν = 1,...,m, there is a local chart on a neighborhood of Kν that maps Kν onto a closed disk. Then the ≡ m = compact set K ν=0 Kν satisﬁes hM (K) K. 5.18 Embedding of an Open Riemann Surface into C3 281

Proof If U is a connected component of M \ K0, then U is not relatively compact in M, and hence by Lemma 5.11.4, U \ (K1 ∪···∪Km) is a connected open set that is not relatively compact in X. Thus each of the connected components of M \ K is of this form, and therefore hM (K) = K.

~~For the proof of Theorem 5.18.3, we will apply the L2 ∂¯-method with a weight function provided by the following:~~

Lemma 5.18.7 Let X be an open Riemann surface. Suppose that ⊂ X is the union of a locally ﬁnite family of disjoint relatively compact open subsets of X, each of which is biholomorphic to a disk, ω is a positive continuous (1, 1)-form on X, ρ is a positive continuous function on X, and K is a closed subset of X with K ⊂ . Then there exists a C∞ strictly subharmonic function ϕ on X such that

~~(i) On X, iϕ ≥ ω; (ii) On K, ϕ<−ρ; and (iii) On X \ , ϕ>ρ.~~

Proof We may assume without loss of generality that has infinitely many con- { }∞ nected components ν ν=1. Replacing each of these connected components with a slightly smaller coordinate disk and replacing K with a suitable (larger) subset of , we may also assume that the components of have disjoint closures and that for each ν, the biholomorphism of ν onto a disk extends to a biholomorphism on a neighborhood of ν , and the image of K ∩ ν isacloseddisk.Wemayalsofixa neighborhood of K with ⊂ . By applying Lemma 5.18.5 inductively (together with Lemma 9.3.6 and Radó’s { }∞ theorem), we get a sequence of nonempty compact sets Km m=1 such that ∞ = ⊂ = m=1 Km X and 1 K1, and such that for each m,wehavehX(Km) Km, ◦ Km ⊂ Km+1, each connected component ν of lies either in Km or in X \ Km, and ∩ (Km+1 \ Km) = ∅. Thus by reordering, we may assume that for some strictly increasing sequence of positive integers {νm}, we have, for each m, ∞ ∪···∪ ⊂ \ ⊃ 1 νm Km and X Km ν. ν=νm+1 ν = ≡∅ ≡ ≡ m ⊂ \ − = Set K0 0 , ν0 0, and m ν=ν − +1 ν Km Km 1 for each m ∞ m 1 1, 2, 3,.... Theorem 2.14.1 provides a C strictly subharmonic function ϕ0 on X ∞ such that iϕ ≥ ω and ϕ0 >ρ.GivenC subharmonic functions ϕ0,...,ϕm−1 0 ∞ on X, we may choose a real-valued C function αm on X such that supp αm ⊂ m and on ∩ m, αm + ϕ0 +···+ϕm−1 < −ρ and αm is subharmonic. Lemma 5.18.6 ∞ and Theorem 2.14.1 provide a nonnegative C subharmonic function ψm on X such ≡ ∪ ∩ + ≥ ≥ \ that ψm 0onKm−1 (K m), and ψm αm 0 and iψm+αm 0onm . C∞ ≡ + ≥ \ Thus the function ϕm ψm αm is subharmonic on X, ϕm 0onX , ≡ m − ∩ νm ϕm 0onKm−1, and by induction, j=0 ϕj < ρ on K ν=1 ν . Thus we get 282 5 Uniformization and Embedding of Riemann Surfaces { } ∞ a sequence of functions ϕm , and it follows that the locally finite sum m=0 ϕm determines a C∞ strictly subharmonic function ϕ with the required properties.

Proof of Theorem 5.18.3 We may assume without loss of generality that has { }∞ infinitely many connected components ν ν=1. Hence we have local holomorphic { ; }∞ charts (ν,ν ,(0 1)) ν=1, and we may fix nonempty neighborhoods and D of K in such that ⊂ D and, for each ν, ν(D ∩ ν ) is a relatively compact disk in (0; 1). We may also fix a C∞ function τ on X such that supp τ ⊂ and τ ≡ g on D. In particular, the support of the C∞ (0, 1)-form γ ≡ ∂τ¯ lies in \ D. Fixing a Kähler form ω on X and applying Theorem 2.14.1, we get a C∞ strictly + ≥ subharmonic function ϕ0 on X such that iω iϕ0 ω. By Theorem 2.6.4,for each ν, there exists a constant δν ∈ (0, 1) such that |h| <ρ on K ∩ ν for every ∈ O ∩ function h ( ν) with h L2(∩ ,ω,ϕ ) <δν . By applying Lemma 5.18.7, ∞ ν 0 we may form a C subharmonic function ϕ1 on X such that for each ν, ϕ1 < 2logδν on ∩ ν , and for ϕ ≡ ϕ0 + ϕ1,

2 = 2 −ν γ 2 γ 2 < 2 . L (ν ,ϕ) L (ν \D,ϕ) 2 = ∞ 2 In particular, γ 2 = γ 2 < 1, and by Corollary 2.12.6, there L (X,ϕ) ν 1 L (ν ,ϕ) C∞ ¯ = exists a function α on X such that ∂α γ and α L2(X,ω,ϕ) < 1. The function f ≡ τ − α is then a holomorphic function on X. Furthermore, since τ = g on D, α must be holomorphic on D ⊃ , and for each ν,

~~α 2 ≤ δ ·α 2 ≤ δ ·α 2 <δ . L (∩ν ,ω,ϕ0) ν L (∩ν ,ω,ϕ) ν L (X,ω,ϕ) ν~~

~~It follows that |f − g|=|α| <ρon K.~~

We now turn to the construction of the three subsets in Proposition 5.18.4.For this, we form a graph that divides X into small simply connected sets (actually, as the reader is asked to prove in Exercise 6.11.4, X admits a triangulation; but we will not use this more general fact). The three sets then consist of a union of small neighborhoods of the vertices in the graph, a union of small neighborhoods of the open edges, and ﬁnally, the complement of the graph. For the construction of the graph, we will need some elementary facts, the ﬁrst of which is the following analogue of the identity theorem (Corollary 1.3.3) for real analytic functions:

~~Lemma 5.18.8 If the zero set Z = f −1(0) of a real-valued real analytic function f on an open interval I ⊂ R has a limit point in I , then f ≡ 0.~~

Proof Given a point a ∈ Z,forsome>0, the function f may be written as a → = ∞ − k − + ⊂ power series x f(x) k=0 ck(x a) on the interval (a ,a ) I .This series converges absolutely, so we may deﬁne a holomorphic function g on the disk 5.18 Embedding of an Open Riemann Surface into C3 283

(a; ) in C by ∞ k g(z) = ck(z − a) ∀z ∈ (a; ). k=0 If a is a limit point of Z, then the identity theorem implies that g is constant. ◦ Thus the interior Z is nonempty. The above also implies that the boundary of the ◦ ◦ nonempty open set Z does not meet I , and hence that Z = Z = I .

~~The desired graph will be constructed as the union of coordinate circles and line segments, and the following lemma will give local ﬁniteness of intersections:~~

Lemma 5.18.9 Suppose f : 0 → 1 is a biholomorphism of domains 0 and 1 in C, and for each j = 0, 1, zj ∈ C, rj > 0, and (zj ; rj ) j . Then the intersection f(∂(z0; r0)) ∩ ∂(z1; r1) is inﬁnite if and only if f(∂(z0; r0)) = ∂(z1; r1).

2πit Proof Let ρ : R → C be the periodic function t → |f(z0 + r0e ) − z1|.Ifthe −1 above intersection is inﬁnite, then the set ρ (r1) has a limit point t0 ∈ R.Onthe other hand, on some neighborhood of t0, the function log ρ is the restriction to R of the real part of the composition of a holomorphic branch of the logarithmic function 2πiζ with the holomorphic function ζ → f(z0 + r0e ) − z1, and hence log ρ is real analytic (in fact, the composition of real analytic functions is real analytic, so one can show that ρ is itself real analytic near t0). Lemma 5.18.8 then implies that ρ −1 is constant near t0, and hence the interior of the set ρ (r1) in R is nonempty; and, moreover, this interior has no boundary points. Thus ρ ≡ r1 on R, and hence f(∂(z0; r0)) ⊂ ∂(z1; r1). The reverse containment follows by symmetry.

~~In order to get simply connected neighborhoods of the edges in a graph, we will apply the following elementary fact:~~

Lemma 5.18.10 If I is an open interval in R and W is a neighborhood of the set I ×{0} in R2, then there exists a simply connected neighborhood V of I ×{0} in W ∩ (I × R) such that V ∩ (R ×{0}) = I ×{0} and V \ R = V \ (I ×{0}) has exactly two connected components:

~~V− ≡{(x, y) ∈ V | y<0} and V+ ≡{(x, y) ∈ V | y>0}, both of which are nonempty. In particular, I ×{0}=(I × R) ∩ ∂V− ∩ ∂V+.~~

~~Proof The sets~~

2 V− ≡{(x, y) ∈ R | x ∈ I,y < 0, and {x}×[y,0]⊂W} and 2 V+ ≡{(x, y) ∈ R | x ∈ I, y > 0, and {x}×[0,y]⊂W} 284 5 Uniformization and Embedding of Riemann Surfaces are open connected sets. For if (x1,y1), (x2,y2) ∈ V− with x1 0 sufﬁciently small, we have −>yj for j = 1, 2, and the connected set

({x1}×[y1, 0)) ∪ ([x1,x2]×{−}) ∪ ({x2}×[y2, 0)) lies in V−. Similar arguments show that V+ is also connected, and that V− and V+ are open. The set V ≡ V− ∪ V+ ∪ (I ×{0}) ⊂ W ∩ (I × R) is a connected neighborhood of I ×{0} with V ∩ R = I ×{0}, because each point in I ×{0} has an open rectangular neighborhood of the form J × (−δ,δ) ⊂ W for some open interval J ⊂ I and some δ>0, and hence this neighborhood is contained in V and meets both V− and V+. Finally, the set V is simply connected because if α = (u, v) is a loop in V with base point (t0, 0) ∈ I ×{0}, then the map

~~(t, s) → (u(t), (1 − s)v(t)) is a path homotopy in V from α toaloopβ in I ×{0}, and the map~~

~~(t, s) → (u(t) + s(t0 − u(t)), 0) is a path homotopy in I ×{0} from β to the trivial loop.~~

Proof of Proposition 5.18.4 We first construct a suitable graph. Since X is second countable (by Radó’s theorem), Lemma 9.3.6 provides a locally finite sequence of { } local holomorphic charts (Uν ,ν ,Uν) such that ∈ Z ≡ ; (i) For each ν >0,wehave (0 1) Uν ; { }≡{ −1 } (ii) The family Dν ν () is a (locally finite) covering of X; and (iii) If μ,ν ∈ Z>0 with Dμ ∩ Dν = ∅, then Dμ Uν . ≡ ∞ In particular, the subset M0 ν=1 ∂Dν of X is closed, and by Lemma 5.18.9, the set ∂Dμ ∩ ∂Dν is finite for all μ,ν ∈ Z>0 for which ∂Dμ = ∂Dν . We will now construct additional sets so that the union becomes a connected set that has connected intersection with each of the sets {Dν}. For all μ,ν ∈ Z>0,thesetDν ∩ ∂Dμ has only finitely many connected com- −1 ponents, because each nonempty connected component is the image under μ of an open arc of the circle S1 = ∂, and any endpoint of the arc must map into the finite set ∂Dμ ∩ ∂Dν . It follows that for each ν,thesetM0 ∩ Dν has only finitely many connected components (and each of these connected components is =∅ { }∞ compact). Setting L0 , we now construct sequences of closed sets Mk k=0 and { }∞ compact sets Lk k=0 inductively as follows. Suppose we have constructed closed sets M0,...,Mk−1 and compact sets L1,...,Lk−1 so that for each j = 1,...,k−1, Mj = Mj−1 ∪ Lj and Mj ∩ Dν has only finitely many connected components for each ν.IfMk−1 ∩ Dν is connected for each ν, then we set Lk =∅and Mk = Mk−1. Otherwise, letting ν be the minimal index in Z>0 for which the set Mk−1 ∩ Dν is not connected, we may choose a set Lk ⊂ Dν such that ν(Lk) is the line segment from some point in a connected component A of ν(Mk−1 ∩ Dν) to a point 5.18 Embedding of an Open Riemann Surface into C3 285

~~Fig. 5.21 Local construction of the graph~~

in ν(Mk−1 ∩ Dν) \ A with length (ν (Lk)) = dist(A, ν(Mk−1 ∩ Dν) \ A) (see Fig. 5.21). Thus the set Mk ≡ Mk−1 ∪ Lk is closed, and the number of connected components of Mk ∩ Dν is strictly less than that of Mk−1 ∩ Dν . Observe also that for any μ ∈ Z>0 with Lk ∩ Dμ = ∅, either Lk ⊂ Dμ (and hence the endpoints of Lk lie in Mk−1 ∩ Dμ) or every connected component of Lk ∩ Dμ meets ∂Dμ (and hence meets M0 ∩ Dμ). Thus, in either case, the number of connected components of Mk ∩ Dμ is not more than that of Mk−1 ∩ Dμ. Thus we get sequences of sets {Mk} and {Lk}, and it follows from the construction that for each ν,thesetMk ∩ Dν is connected for each k 0. Moreover, by local finiteness of the family {Dν}, the family of compact sets {Lk} is locally finite, and hence for every compact S, Mk+1 ∩ S = Mk ∩ S for k 0. Thus the set ∞ ∞ X1 ≡ Mk = M0 ∪ Lk k=1 k=1 is a closed subset of X, and X1 ∩ Dν is connected for each ν. We may also choose a discrete subset X0 of X1 such that X0 meets ∂Dν for each ν, X0 contains ∂Dμ ∩ ∂Dν whenever μ,ν ∈ Z>0 with ∂Dμ = ∂Dν , and X0 contains the endpoints of Lk for each k. In particular, the collection of connected components of X1 \ X0 is locally finite in X, and each connected component is relatively compact in X. Observe also that for each k, in some local holomorphic chart, Lk is a line segment whose interior points do not meet Mk−1 ⊃ M0. Thus the sets Lk ∩ ∂Dν for each ν, and Lk ∩ Lj for each j

Fig. 5.22 The three desired open sets compact connected open subsets of X, and such that if V is one of these subsets (i.e., one of the connected components of 1), then V is biholomorphic to a disk, V contains exactly one connected component L of X1 \ X0, and V \ L has exactly two connected components (see Fig. 5.22). We will now show that the third set may be taken to be the open set 2 ≡ X \ X1 (see Fig. 5.22). Given a connected component of 2,wehave ∩ Dν = ∅ for some ν. Since ∂Dν ⊂ X1,wemusthave ⊂ Dν \ X1 X. Thus ν () ⊂ is a connected component of the complement of the connected compact set 1 ν(X1 ∩Dν) in P . Therefore, Lemma 5.17.1 and the Riemann mapping theorem in the plane imply that is biholomorphic to a disk. Furthermore, ∂ is a connected 1 subset of X1 ∩Dν that cannot be a single point (for if p ∈ ∂, then P \{ν(p)} is a 1 connected open set that meets, but is not contained in, ν() ⊂ ,soP \{ν(p)} must meet ∂ν () = ν(∂)). Hence ∂ meets some connected component L of X1 \ X0 with L ⊂ Dν . On the other hand, if V is the connected component of 1 containing L, then V \ L = V− ∪ V+, where V− and V+ are nonempty disjoint connected open subsets of X \ X1, and hence must contain V− or V+.It follows that the collection of connected components of 2 must be locally ﬁnite in X.

Remarks 1. As is clear from the above proof, the closure of each connected component of X1 \ X0 is the image of a smooth injective path with endpoints in X0. Thus the set X1 maybeviewedasagraph embedded in X with vertices equal to the points of X0. 2. The above decomposition of X into manifolds of increasing dimension X0, X1 \ X0, X \ X1 (i.e., X2 \ X1 for X2 = X) is an example of a stratiﬁcation of X. 3. One may also push the construction further to get a triangulation of X (see Exercise 6.11.4). 4. The set X1 \ X0 is a real analytic submanifold of X \ X0. On the other hand, for our purposes, it is not really necessary that the local charts, in which the connected components of X1 \ X0 become intervals, be holomorphic. Continuous local charts would sufﬁce, and in fact, the existence of suitable local C∞ charts would follow from smoothness of the components, Theorem 9.10, and a natural analogue of Lemma 5.11.1. However, that one may choose the local charts to be holomorphic follows directly from the construction.

The above facts will allow us to produce a proper holomorphic mapping into C3, and the following lemma will allow us to approximate the mapping by an embedding: 5.18 Embedding of an Open Riemann Surface into C3 287

Lemma 5.18.11 Let X be an open Riemann surface, and let ω be a Kähler form on X. Then there exists a Kähler form ω on X such that every C∞ function ϕ on X with iϕ ≥ ω has the following properties: (i) For every pair of distinct points p,q ∈ X and every constant ζ ∈ C, there exists a function f ∈ O(X) ∩ L2(X,ω,ϕ)with f(p)− f(q)= ζ . ∈ ∈ 1,0 ∗ (ii) For every point p X and every cotangent vector η Tp X, there exists a 2 function f ∈ O(X) ∩ L (X,ω,ϕ)with (df )p = η.

Proof We may fix a locally finite collection of local holomorphic charts { = ; }∞ (Dν,zν ν ,(0 3)) ν=1 = −1 ; ∈ D ; ≡ with X ν ν ((0 1)), and a nonnegative function λ ((0 3)) with λ 1 ; ∈ −1 ; on (0 2). For each point p ν ((0 1)),wehave ¯ 2 i∂∂[λ(zν) log |zν − zν(p)| ] 2 ∂ λ 2 (∂λ/∂z)(zν ) = (zν ) log |zν − zν(p)| + 2Re idzν ∧ dz¯ν ≥−Cνω ∂z∂z¯ z¯ν −¯zν(p) on Dν \{p}, where the constant 2 ∧ ¯ ∂ λ ∂λ idzν dzν Cν ≡ (log 16) max + 2max · max ¯ −1 ∂z∂z ∂z ν (supp λ) ω is independent of the choice of the point p. Moreover, a standard partition of unity ∞ argument yields a C function ρ on X such that ρ>1 + 2Cν +|ω/ω| on Dν for each ν, and we may set ω ≡ ρ · ω. ∞ Suppose now that ϕ is a C function on X with iϕ ≥ ω . Given two (not necessarily distinct) points p,q ∈ X and a constant ζ ∈ C, we may fix indices μ and ∈ −1 ; ∈ −1 ; = = ν with p μ ((0 1)) and q ν ((0 1)) (and with μ ν if p q) and a function τ ∈ D(X) such that τ ≡ ζ on a neighborhood of p and τ ≡ 0ona neighborhood of q if p = q, and τ ≡ ζ · (zμ − zμ(p)) on a neighborhood of p if p = q.TheC∞ (0, 1)-form γ ≡ ∂τ¯ then has compact support in the Riemann surface Y ≡ X \{p,q}, and the C∞ function

~~2 2 ψ ≡ ϕ + λ(zμ) log |zμ − zμ(p)| + λ(zν) log |zν − zν (q)| on Y satisﬁes~~

iω + iψ ≥ ω. Therefore, by Corollary 2.12.6, there exists a C∞ function α on Y such that ∂α¯ = γ ∞ and α L2(Y,ω,ψ) < . In particular, α is holomorphic near p and q. Moreover, if p = q, then for some constant C>0, we have | − |2 ∧ ¯ ≤ · 2 ∞ α/(zμ zμ(p)) idzμ dzμ C α 2 −1 < , −1 L (μ ((0;1)),ω,ψ) μ ((0;1)) 288 5 Uniformization and Embedding of Riemann Surfaces and the analogous inequality holds near q. Therefore, by Riemann’s extension theorem (Theorem 1.2.10), α extends to a C∞ function on X (which we will also denote by α) that vanishes at p and q. Thus f ≡ τ −α is then a holomorphic function on X, − = = \ −1 ∪ −1 ⊂ f(p) f(q) ζ , and since ϕ ψ on X (μ (supp λ) ν (supp λ)) Y ,we 2 have f ∈ L (X,ω,ϕ). Thus the property (i) holds. If p = q and η = ζ ·(dzμ)p, then | − 2|2 ∧ ¯ ≤ · 2 ∞ α/(zμ zμ(p)) idzμ dzμ C α 2 −1 < −1 L (μ ((0;1)),ω,ψ) μ ((0;1)) for some constant C>0, and hence α extends to a C∞ function on X (which we will also denote by α) that is holomorphic near p with a zero of order at least 2 at p. Hence f ≡ τ − α ∈ L2(X,ω,ϕ) is then a holomorphic function on X satisfying (df )p = η, and the property (ii) also holds.

Lemma 5.18.12 Let X be an open Riemann surface, let be an open subset that is equal to the union of a locally ﬁnite family of disjoint relatively compact open subsets of X, each of which is biholomorphic to a disk, let K be a closed subset ⊂ ∈ 1,0 ∗ of X with K , let C be a countable subset of X, let ηp Tp X for each point p ∈ C, let D be a countable set of pairs of distinct points (p, q) in X, let ζp,q ∈ C for each pair (p, q) ∈ D, and let ρ be a positive continuous function on X. Then there exists a holomorphic function f on X such that |f | <ρon K, (df )p = ηp for each point p ∈ C, and f(p)− f(q) = ζp,q for each pair (p, q) ∈ D.

Proof We may assume without loss of generality that has infinitely many con- { }∞ ⊂ nected components ν ν=1, we may fix a neighborhood of K with , and we may fix Kähler forms ω and ω on X with the properties described in Lemma 5.18.11. Theorem 2.14.1 then provides a C∞ strictly subharmonic function α on X with iα ≥ ω . By Theorem 2.6.4, for each ν, there exists a constant δν ∈ (0, 1) such that |h| <ρ on K ∩ ν for every function h ∈ O( ∩ ν ) with ∞ 2 C h L (∩ν ,ω,α) <δν . Moreover, Lemma 5.18.7 provides a subharmonic function β on X such that β<2logδν on ∩ ν for each ν = 1, 2, 3,.... Setting ≡ + ∈ O ϕ α β, we see that if h (X) and h L2(X,ω,ϕ) < 1, then for each ν,

2 ≤ · 2 h L (∩ν ,ω,α) δν h L (∩ν ,ω,ϕ) <δν, and hence |h| <ρon K. According to Theorem 2.6.4, O(X) ∩ L2(X,ω,ϕ) is a closed subspace of L2(X,ω,ϕ), and is therefore itself a Hilbert space. By construction, for each point ∈ ∈ 1,0 ∗ ∈ \{ } p X, each cotangent vector η Tp X, each point q X p , and each constant ζ ∈ C,thesets

2 Ap,η ≡{f ∈ O(X) ∩ L (X,ω,ϕ)| (df )p = η}, 2 Bp,q,ζ ≡{f ∈ O(X) ∩ L (X,ω,ϕ)| f(p)− f(q) = ζ}, are nonempty, and it follows from Theorem 2.6.4 and Theorem 1.2.4 that the above sets are also open in O(X) ∩ L2(X,ω,ϕ). Moreover, these sets are dense. For if 5.18 Embedding of an Open Riemann Surface into C3 289

2 f ∈ O(X) ∩ L (X,ω,ϕ) with (df )p = η (respectively, f(p)− f(q)= ζ ), then ﬁxing h ∈ Ap,0 (respectively, h ∈ Bp,q,0), we get f + h ∈ Ap,η (respectively, p ∗ f + h ∈ Bp,q,ζ ) for every ∈ C , and (f + h) − f →0as → 0. According to the Baire category theorem (see for example, [Mu]or[Rud1]), the intersection of any countable collection of dense open subsets of a Hilbert space (or any complete metric space) is dense. Hence there must exist an element f of ∩ Ap,ηp Bp,q,ζp,q p∈C (p,q)∈D with f L2(X,ω,ϕ) < 1, and the claim follows.

Proof of Theorem 5.18.2 By Proposition 5.18.4, there exists a covering of X by three open sets 0, 1, and 2 such that for each j = 0, 1, 2, the connected com- { (ν)}∞ ponents j ν=1 of j form a locally finite collection of relatively compact open subsets of X and each connected component is biholomorphic to a disk. By shrinking these sets slightly and applying Theorem 5.18.3, we may also assume that | − | (ν) there exists a holomorphic function gj on X such that gj ν < 1onj for 3 ν = 1, 2, 3,.... In particular, the holomorphic mapping g = (g0,g1,g2): X → C is proper. We will approximate g by a (proper) holomorphic embedding. WemayfixclosedsubsetsK1 and K2 of X such that K1 ⊂ 1, K2 ⊂ 2, and X = 0 ∪ K1 ∪ K2, and we may set f0 ≡ g0. By the identity theorem, the set C ≡{p ∈ X | (df0)p = 0} is discrete, and fixing a countable dense subset C of X containing C, we see that the set D of pairs of distinct points (p, q) in X with p ∈ C and f0(p) = f0(q) is countable. Applying Lemma 5.18.12, we get a holomorphic function h1 on X such that (dh1)p = −(dg1)p for each point p ∈ C, h1(p) − h1(q) = g1(q) − g1(p) for each pair (p, q) ∈ D, and |h1| < 1onK1. Thus (ν) the holomorphic function f1 ≡ g1 + h1 satisfies |f1 − ν| < 2onK1 ∩ for 1 each ν, (df1)p = 0 for each point p ∈ C ⊃ C (in particular, (f0,f1) is a holo- 2 morphic immersion into C ), and f1(p) = f1(q) for each pair (p, q) ∈ D.Itnow suffices to show that the set E ⊂ X × X of pairs of distinct points (p, q) with (f0(p), f1(p)) = (f0(q), f1(q)) is countable. For the above argument will then pro- | − | ∩ (ν) vide a holomorphic function f2 such that f2 ν < 2onK2 2 for each ν and 3 f2(p) = f2(q) for each pair (p, q) ∈ E. The mapping f = (f0,f1,f2): X → C will then be a holomorphic embedding. Since the set of points in X at which df0 = 0ordf1 = 0 is countable, and E ∩ ({p}×X) is countable for each point p ∈ X, it suffices to show that the set of points (p, q) ∈ E at which (df0)p = 0 and (df1)p = 0 is countable. Therefore, since X ×X is second countable, is suffices to show that each such point (p, q) is an isolated point in E. By the open mapping theorem, the holomorphic inverse function theorem, and the identity theorem, there exist disjoint connected neighborhoods U and V of p and q, respectively, such that for j = 0, 1, fj maps U biholomorphi- ⊂ C ⊂ −1 ∩ ={ } cally onto a domain Uj , fj (V ) Uj , and fj (fj (p)) V q . For each point (r, s) ∈ E ∩ (U × V),wehave −1 = = −1 (f0 U ) (f0(s)) r (f1 U ) (f1(s)). 290 5 Uniformization and Embedding of Riemann Surfaces

∈ ∩ ∩ −1 Since C is dense in X, we may ﬁx a point r C U f0 (f0(V )) and a point ∈ ∩ −1 ∈ = s V f0 (f0(r)). We then get (r, s) D, and hence f1(r) f1(s); that is, −1 ◦ − −1 ◦ (f0 U ) f0 (f1 U ) f1 is not identically zero on the domain V .Itfol- lows that we may choose V so small that q is the only zero of the above holomorphic function, that is, so that E ∩ (U × V)⊂ U ×{q}. Hence E ∩ (U × V)= { −1 }={ } ((f0 U ) (f0(q)), q) (p, q) , and the claim follows.

Exercises for Sect. 5.18 5.18.1 Let X be an open Riemann surface. Using only Theorem 5.18.3 and Proposi- tion 5.18.4, prove directly (without appealing to the Baire category theorem) that there exists a proper holomorphic immersion of X into C3. 5.18.2 Prove Theorem 5.18.3 directly using the Runge approximation theorem (Theorem 2.16.1) in place of the L2 ∂¯-method. 5.18.3 The proof of Theorem 5.18.2 shows that every open Riemann surface admits a holomorphic immersion into C2. Prove this fact directly by applying the Mittag-Lefﬂer theorem (Theorem 2.15.1).

~~5.19 Embedding of a Compact Riemann Surface into Pn~~

In this section, we consider an embedding theorem for a compact Riemann surface X. As mentioned in Sect. 5.18, X cannot admit a holomorphic embedding into a Euclidean space, because O(X) = C. However, X does admit a holomorphic embedding into a complex projective space.

n Deﬁnition 5.19.1 Let n ∈ Z>0.Then-dimensional complex projective space P is the quotient space + Pn ≡ Cn 1 \{0}/∼, where for all points w,z ∈ Cn+1 \{0},wehavew ∼ z if and only if w = λz for ∗ n+1 some scalar λ ∈ C . We call any representative (ζ0,...,ζn) ∈ C \{0} for a point n ζ ∈ P homogeneous coordinates for ζ , and we write ζ =[ζ0,...,ζn].

Remarks 1. In other words, Pn is the space of complex lines in Cn+1 that pass through the origin. n 2. For each j = 0,...,n,thesetUj ≡{[ζ0,...,ζn]∈P | ζj = 0} is well defined n and open (in the quotient topology), and the mapping Uj → C given by [ζ0,...,ζn]→(ζ0/ζj ,...,ζj /ζj ,...,ζn/ζj ) Pn = n is a well-defined homeomorphism (see Exercise 5.19.1). Moreover, j=0 Uj . 3. For n = 1, P1 (as defined above) is a connected compact Riemann surface with local holomorphic charts (U0, [ζ0,ζ1]→ζ1/ζ0, C) and (U1, [ζ0,ζ1]→ζ0/ζ1, C) 5.19 Embedding of a Compact Riemann Surface into Pn 291 given by the mappings in Remark 2 above, and the quotient mapping C2 \{0}→P1 is holomorphic. Moreover, although we also denote the Riemann sphere C ∪{∞} by P1, there is no ambiguity because we have a biholomorphism C2 \{0}/∼→ C ∪{∞}given by [ζ0,ζ1]→ζ0/ζ1 ([ζ0, 0]→∞). 4. For an arbitrary n ≥ 1, Pn is a connected compact complex manifold of dimension n. The local holomorphic charts are provided by the mappings in Remark 2 above, and the quotient map Cn+1 \{0}→Pn is holomorphic (see Exercise 5.19.1). We will not use this fact directly, but it justifies the use of the terminology appearing in Definition 5.19.2 below (cf. Definition 5.18.1).

Deﬁnition 5.19.2 Let X be a complex 1-manifold, and let F : X → Pn be a continuous mapping into a complex projective space Pn. For each j = 0,...,n,let n n j : Uj → C be the homeomorphism of the open set Uj ≡{[ζ0,...,ζn]∈P | n ζj = 0} onto C given by [ζ0,...,ζn]→(ζ0/ζj ,...,ζj /ζj ,...,ζn/ζj ). −1 n (a) F is called holomorphic if the mapping j ◦F : F (Uj ) → C is holomorphic for each j = 0,...,n. −1 n (b) F is called a holomorphic immersion if the mapping j ◦ F : F (Uj ) → C is a holomorphic immersion for each j = 0,...,n. (c) F is called a holomorphic embedding if X is compact and F is an injective holomorphic immersion (i.e., F is a proper injective holomorphic immersion).

~~One may obtain holomorphic mappings into complex projective spaces from holomorphic sections of a holomorphic line bundle as follows:~~

Lemma 5.19.3 Let E be a holomorphic line bundle on a Riemann surface X, and let s = (s0,...,sn) be an (n + 1)-tuple of holomorphic sections of E with no com- n mon zeros. Then the mapping κs : X → P given by s (x) s (x) κ (x) ≡ 0 ,..., n ∀x ∈ X and ξ ∈ E \{0 } s ξ ξ x x is well deﬁned and holomorphic.

Proof For x ∈ X and ξ,η ∈ Ex \{0x},wehave s (x) s (x) η s (x) s (x) 0 ,..., n = · 0 ,..., n , ξ ξ ξ η η so the mapping is well deﬁned. Moreover, given a point x0 ∈ X and an index j with sj (x0) = 0, then sj is nonvanishing on some neighborhood U of x0 in X and the composition of the mapping [ζ0,...,ζn]→(ζ0/ζj ,...,ζj /ζj ,...,ζn/ζj ) with the mapping κs U is the mapping x → s0(x)/sj (x),...,sj (x)/sj (x),...,sn(x)/sj (x) , which is holomorphic. 292 5 Uniformization and Embedding of Riemann Surfaces

Deﬁnition 5.19.4 Let E be a holomorphic line bundle on a Riemann surface X, and let s = (s0,...,sn) be an (n + 1)-tuple of holomorphic sections of E with no common zeros. Then the associated holomorphic mapping X → Pn given by s (x) s (x) x → 0 ,..., n ∀x ∈ X and ξ ∈ E \{0 } ξ ξ x x is denoted by κs or by [s0,...,sn].

Example 5.19.5 Let E → P1 be the hyperplane bundle (see Examples 3.1.5 and 3.3.4); that is, E =[D], where D is the divisor with D(0) = 1 and D(x) = 0 for each 1 point x ∈ P \{0}.Lets0 be a holomorphic section with div(s0) = D.Wealsohave the holomorphic section s1 ≡ s0/z (s1(∞) = 0), and the corresponding mapping =[ ] C2 \{ } ∼→C∪{∞} κ(s0,s1) s0,s1 is equal to inverse of the biholomorphism ( 0 )/ described in the remarks following Deﬁnition 5.19.1 (see Exercise 5.19.2).

~~Recall that the vector space of holomorphic sections of a holomorphic line bundle on a compact Riemann surface is ﬁnite-dimensional (Theorem 4.4.1).~~

Lemma 5.19.6 Let E be holomorphic line bundle on a compact Riemann surface X, and let s = (s0,...,sn) for a basis s0,...,sn for (X,O(E)). (a) A point p ∈ X satisﬁes s(p) = (0,...,0) if and only if every holomorphic section of E on X vanishes at p. (b) Suppose that the sections s0,...,sn have no common zeros. Then the associated map κs =[s0,...,sn] is injective if and only if for each pair of distinct points p,q ∈ X, there exists a holomorphic section t of E with t(p)= 0 and t(q) = 0. (c) Suppose (again) that the sections s0,...,sn have no common zeros. Then the associated map κs =[s0,...,sn] is a holomorphic immersion if and only if for each point p ∈ X, there is a holomorphic section t withasimplezeroatp.

Remark In the standard terminology (see, for example, [GriH]), the point p in part (a) is a base point for the holomorphic sections (or, more precisely, for the linear system of divisors associated to the nontrivial holomorphic sections of E). In (b), the holomorphic sections are said to separate points in X. In (c), the holomorphic sections are said to give local coordinates in X.

Proof of Lemma 5.19.6 If p ∈ X with s(p) = (0,...,0), then every section t ∈ (X,O(E)) must vanish at p (since t is a linear combination of the sections s0,...,sn). Part (a) now follows. Let us assume for the rest of this proof that the sections s0,...,sn have no common zeros. Given distinct points p,q ∈ X, there exists an index j ∈{0,...,n} with sj (q) = 0. If sj (p) = 0, then the section sj separates the points p and q as in (b), and we have κs(p) = κs(q). Assuming that sj (p) = 0 and that κs is injective, the corresponding points s0(p)/sj (p),...,sj (p)/sj (p),...,sn(p)/sj (p) 5.19 Embedding of a Compact Riemann Surface into Pn 293 and s0(q)/sj (q),...,sj (q)/sj (q),...,sn(q)/sj (q) in Cn must be distinct, and therefore s (p) s (q) i = i for some i ∈{0,...,j,...,nˆ }. sj (p) sj (q) Thus the holomorphic section

si(p) t ≡ si − · sj sj (p) satisﬁes t(p) = 0 and t(q) = 0. For the converse, suppose that sj (p) = 0 and sj (q) = 0, and that there exists a holomorphic section t with t(p)= 0 and t(q) = 0. ∈ C = n We then have constants ζ0,...,ζn with t i=0 ζisi , and hence n n si(p) si(q) ζi = 0 and ζi = 0. sj (p) sj (q) i=0 i=0 It follows that the points s0(p)/sj (p),...,sj (p)/sj (p),...,sn(p)/sj (p) and s0(q)/sj (q),...,sj (q)/sj (q),...,sn(q)/sj (q) n in C must be distinct and therefore that κs(p) = κs(q). The claim (b) now follows. For the proof of (c), let p ∈ X and let j ∈{0,...,n} with sj (p) = 0. If ˆ κs is a holomorphic immersion, then for some i ∈{0,...,j,...,n},wehave (d(si/sj ))p = 0, and hence the holomorphic section

si(p) t ≡ si − · sj sj (p) has a simple zero at p. Conversely, given a section t∈ (X,O(E)) with a simple ∈ C = n zero at p, we have constants ζ0,...,ζn with t i=0 ζisi , and hence n 0 = (d(t/sj ))p = ζi · (d(si/sj ))p. i=0 ˆ Thus (d(si/sj ))p = 0forsomei ∈{0,...,j,...,n}, and (c) follows.

Deﬁnition 5.19.7 A holomorphic line bundle E on a compact Riemann surface X is called ample if E is positive; that is, E admits a positive-curvature Hermitian metric or, equivalently (by Theorem 4.3.1), deg E>0. The line bundle E is called 294 5 Uniformization and Embedding of Riemann Surfaces

~~n very ample if the map κs : X → P associated to any basis s for (X,O(E)) exists and is a holomorphic embedding.~~

~~Remark Since the line bundle associated to any nontrivial effective divisor has positive degree, if E is very ample, then E is ample.~~

~~The results of Chap. 4 and the above observations now give us the following:~~

Theorem 5.19.8 Let E be a holomorphic line bundle on a compact Riemann surface X. Then we have the following: (a) If E is ample, then Ed is very ample for every d 0. (b) If deg E>2, then KX ⊗ E is very ample. (c) If deg E>2 · genus(X), then E is very ample.

Proof Clearly, part (a) follows from part (c). Moreover, part (c) follows from part (b) ∗ ⊗ applied to the holomorphic line bundle KX E, since this line bundle has degree −2 genus(X) + 2 + deg E by Corollary 4.6.7. Thus it sufﬁces to prove part (b). Assuming that deg E>2, suppose p,q ∈ X are two (possibly equal) points and D is the divisor given by D = p + q. Then, applying Corollary 4.1.2 to the positive holomorphic line bundle E ⊗[−D] (we have deg E ⊗[−D] > 2 − 2 = 0) and the effective divisor D, we get a holomorphic section t of KX ⊗ E ⊗[−D]⊗[D]= KX ⊗ E such that t(p)= 0 and t(q) = 0ifp = q, and t has a simple zero at p if p = q. Thus, by Lemma 5.19.6,themapκs is deﬁned for any basis s of (X,O(KX ⊗ E)), and κs is a holomorphic embedding.

Remark According to the higher-dimensional analogue of Theorem 5.19.8, which is called the Kodaira embedding theorem,ifE is an ample holomorphic line bundle on a connected compact complex manifold X, then Ed is very ample for d 0 (see, for example, [GriH], [MKo], or [Wel]).

Exercises for Sect. 5.19 5.19.1 Prove that Pn is a complex manifold of dimension n with local holomorphic charts given by the mappings in Definition 5.19.2, and the quotient map Cn+1 \{0}→Pn is holomorphic (see Exercise 2.2.6 for the required definitions). =[ ] 5.19.2 Verify that the mapping κ(s0,s1) s0,s1 in Example 5.19.5 is equal to the inverse of the biholomorphism (C2 \{0})/∼→C ∪{∞}described in the remarks following Definition 5.19.1.

~~5.20 Finite Holomorphic Branched Coverings~~

As we will see in this section, any nonconstant proper holomorphic mapping of Rie- mann surfaces is actually a finite holomorphic covering map away from a discrete 5.20 Finite Holomorphic Branched Coverings 295 set of branch points (see Proposition 5.20.1 and Definition 5.20.2 below). We first recall some terminology and facts concerning the topology of manifolds (Sect. 9.3). A continuous mapping of Hausdorff spaces is called proper if the inverse image of every compact subset of the target is compact (see Definition 9.3.9). A sequence in a manifold converges to a point p if for each neighborhood U of p, all but finitely many terms of the sequence lie in U. According to Theorem 9.3.2, a compact subset K of a manifold has the BolzanoÐWeierstrass property; that is, every sequence in K admits a convergent subsequence.

Proposition 5.20.1 Let : X → Y be a nonconstant proper holomorphic mapping of Riemann surfaces X and Y , and let mq = multq ∈ Z>0 denote the multiplicity of at q for each point q ∈ X. Then we have the following: (a) is surjective, and counting multiplicities, the cardinalities of the ﬁbers of are all equal to some ﬁnite constant n ∈ Z>0; that is, mq = n ∀p ∈ Y. q∈−1(p)

−1 (b) If p ∈ Y and Z ≡ (p), then there exist a connected neighborhood V of p { } −1 = and disjoint connected open sets Uq q∈Z such that (V ) q∈Z Uq and ∈ ∈ : → such that, for each q Z, we have q Uq , Uq Uq V is a surjective proper holomorphic mapping, and, counting multiplicities, the number of points in each ﬁber of Uq is equal to mq . (c) The set of critical values

~~C ≡ ({q ∈ X | (∗)q = 0}) = ({q ∈ X | multq >1})~~

: \ −1 → \ is discrete, and the mapping X\−1(C) X (C) Y C is a ﬁnite (proper) holomorphic covering map. (d) If −1(p) is a singleton for some point p ∈ Y \ C, then is a biholomorphism of X onto Y .

Proof By the open mapping theorem (Theorem 2.2.2), ≡ (X) is an open subset of Y .Ifp ∈ , then there is a sequence {qν} in X such that (qν) → p, and by replacing this sequence with a suitable subsequence, we may assume that {qν} converges to some point q ∈ X. But we then have (qν ) → (q), and hence p = (q) ∈ . Thus is both open and closed in the Riemann surface Y , and hence is surjective. Once again, let us fix a point p ∈ Y and let Z ≡ −1(p). Since Z is both compact (because is proper) and discrete (by the identity theorem), Z must be finite. The local representation of holomorphic mappings (Lemma 2.2.3) provides a local holomorphic coordinate neighborhood (V, = ζ,V ) with p ∈ V and ζ(p)= 0, { = } disjoint local holomorphic coordinate neighborhoods (Uq ,q zq ,Uq ) q∈Z in X, and positive integers {mq }q∈Z such that for each q ∈ Z,wehaveq ∈ Uq , zq (q) = 0, mq (Uq ) ⊂ V , and ζ() = zq on Uq .For>0 sufficiently small, we have 296 5 Uniformization and Embedding of Riemann Surfaces ; ⊂ −1 −1 ; ⊂ ≡ (0 ) V and ( ((0 ))) U q∈Z Uq . For if this were not the case, then there would exist a sequence {qν } in X \ U such that (qν ) → p and qν → q ∈ X \ U with (q) = p. But we would then have q ∈ Z \ U, which is ; 1/mq ⊂ ∈ impossible. We may also choose so small that (0 ) Uq for each q Z. Therefore, by replacing V and V with (0; ) and −1((0; )), respectively, − ; 1/mq 1 ; 1/mq and replacing Uq and Uq with (0 ) and q ((0 )), respectively, for each q ∈ Z, we may assume that the local holomorphic charts are given by

~~ = = ; { = ; 1/mq } (V, ζ,V (0 )) and (Uq ,zq ,Uq (0 )) q∈Z.~~

In particular, for each point q ∈ Z, is represented on Uq by the holomorphic map (0; 1/mq ) → (0; ) given by z → zmq , and hence, counting multiplicities, the ∈ cardinality of the fiber of Uq over y is equal to mq for each point y V , and part (b) follows. Furthermore, counting multiplicities, the number of points in each = fiber over each point in V is equal to the constant n q∈Z mq . It follows that the function on Y given by the cardinality of each fiber, again counting multiplicities, is locally constant and therefore constant. The claim (c) now follows from Lemma 10.2.11. The claim (d) follows from the above (or from part (a) and Theorem 2.4.4).

Definition 5.20.2 A proper holomorphic mapping : X → Y of a complex 1- manifold X onto a Riemann surface Y for which the restriction to each connected component is surjective and for which almost every fiber consists of n points (i.e., every fiber has n points counting multiplicities) is called a finite holomorphic branched covering map (or a finite holomorphic ramified covering map) with n sheets. Each critical point p of is called a branch point (or ramification point) of order m − 1, where m = multp>1. The sum of all of the orders of the branch points of is called the branching order of .

~~According to Corollary 2.10.4, every Riemann surface admits a nonconstant meromorphic function. Thus we have the following:~~

~~Theorem 5.20.3 Every compact Riemann surface admits a ﬁnite holomorphic branched covering map onto P1.~~

~~We now record the following fact for use in the proof of Abel’s theorem (see Sect. 5.21):~~

Proposition 5.20.4 Let X be a complex 1-manifold, let Y be a Riemann surface, let : X → Y be a ﬁnite holomorphic branched covering map with n sheets, and let C ⊂ Y be the set of critical values. (a) If γ :[0, 1]→Y is an injective path with γ((0, 1)) ⊂ Y \ C, then there exist exactly n distinct liftings γ1,...,γn of γ to paths in X. Moreover, the sets { }n ∈ −1 = γj ((0, 1)) j=1 are disjoint, and if p (γ (0)) and m multp, then exactly m of the liftings have initial point p. 5.20 Finite Holomorphic Branched Coverings 297

~~(b) If f is a holomorphic function on X, then the function Y \ C → C given by y → f(x) x∈−1(y)~~

has a unique extension to a holomorphic function on Y . (c) Given a holomorphic 1-form α on X, there is a unique holomorphic 1-form β on Y such that if γ is any injective path in Y with γ((0, 1)) ⊂ Y \C, and γ1,...,γn are the distinct liftings of γ to paths in X (as given by part (a)), then

~~n β = α. γ j=1 γj~~

Proof If γ is an injective path in Y with γ((0, 1)) ⊂ Y \C, then, since the restriction X \ −1(C) → Y \ C is an (unbranched) covering map with exactly n points in each ﬁber, there exist exactly n distinct liftings δ1,...,δn of γ (0,1) to paths in −1 X\ (C). Moreover, the images of these liftings are disjoint. For if δi(t0) = δj (t1) for some pair of indices i, j ∈{1,...,n} and some pair of numbers t0,t1 ∈ (0, 1), then

γ(t0) = (δi (t0)) = (δj (t1)) = γ(t1), and the injectivity of γ implies that t0 = t1. Uniqueness of liftings in a covering space then implies that δi = δj , and hence i = j.Nowletp1,...,pk be the distinct −1(γ ( )) m ≡ i = ,...,k points in 0 , and let i multpi for each 1 (in particular, = k n i=1 mi ). Then, by Proposition 5.20.1, there exist a connected neighborhood −1 = V of γ(0) in Y and disjoint connected open sets U1,...,Uk such that (V ) k = ∈ ∩ −1 ⊂ i=1 Ui and such that for each i 1,...,k,wehavepi Ui , Ui (C) { } : → pi , and Ui Ui V is a finite branched holomorphic covering map with mi sheets. Let J be the connected component of γ −1(V ) containing 0. Then, for each j = 1,...,n, δj (J ∩ (0, 1)) lies in Ui for some unique index i, and properness + implies that δj (t) → pi as t → 0 . A similar argument applies at t = 1. Thus δj extends to a unique lifting γj :[0, 1]→X of γ , and this lifting satisfies γj (0) = pi . Furthermore, fixing a point t0 ∈ J ∩ (0, 1), we see that for each i = 1,...,k, each −1 of the mi points in (γ (t0)) ∩ Ui will be equal to the value at t = t0 for exactly one of the paths δ1,...,δn. Thus exactly mi of the paths γ1,...,γn will have initial value pi , and part (a) is proved. For the proof of (b), suppose f ∈ O(X) and h: Y \ C → C is the function given → ⊂ \ by y x∈−1(y) f(x).IfU Y C is a domain that is evenly covered by the \ −1 → \ holomorphic covering map X (C) Y C, and U1,...,Un are the distinct −1 = n connected components of the inverse image (U), then we have h U j=1 fj , = ≡ ◦ −1 : → C where for each j 1,...,n, fj f ( Uj ) U . It follows that h is a holomorphic function on Y \ C. Moreover, by properness, h must also be bounded on K \ C for each compact set K ⊂ Y , and hence by Riemann’s extension theorem (Theorem 1.2.10), h extends to a unique holomorphic function on Y . 298 5 Uniformization and Embedding of Riemann Surfaces

Finally, for the proof of (c), suppose α is a holomorphic 1-form on X. In analogy with part (b), the corresponding holomorphic 1-form on Y is obtained by averaging over the fibers. More precisely, if U ⊂ Y \ C is a domain that is evenly covered by −1 the holomorphic covering map X\ (C) → Y \C, and U1,...,Un are the distinct −1 connected components of (U), then we let ηU be the holomorphic 1-form on U ≡ n = given by ηU j=1 αj , where for each j 1,...,n, αj is the unique holomor- ∗ = phic 1-form on U satisfying ( Uj ) αj α Uj . These holomorphic 1-forms agree on the overlaps, and therefore they determine a well-defined holomorphic 1-form η on Y \ C. Given a point q ∈ C, we may form neighborhoods U and V of −1(q) : → and q, respectively, such that U U V is a finite holomorphic branched cov- ∈ O = ering map and such that for some function f (U),wehaveα U df (for we −1 may choose V so that the connected components of (V ) lie in a union of dis- → joint simply connected open sets). By part (b), the function y x∈−1(y) f(y) on V \ C extends to a unique holomorphic function g on V , and as is clear from the construction of η,wehavedg = η on V \ C. It follows that η extends to a holomorphic 1-form β on Y . Suppose γ is an injective path in Y with γ((0, 1)) ⊂ Y \ C, and γ1,...,γn are the distinct liftings of γ . For each ∈ (0, 1/2), there is a partition = t0 < ···< tk = 1 − such that for each ν = 1,...,k, γ([tν−1,tν]) is contained in an evenly \ { }n covered domain in Y C. The paths γj [tν−1,tν ] j=1 are then the distinct liftings of γ [tν−1,tν ], and hence

~~k k n n β = β = α = α. γ [ − ] γ [ ] γ [ ] γ [ − ] ,1 ν=1 tν−1,tν ν=1 j=1 j tν−1,tν j=1 j ,1 ~~

Letting → 0+, we get n β = α. γ j=1 γj Finally, suppose β is another holomorphic 1-form with the above property. Each point p ∈ Y \C has a connected neighborhood U in Y \C on which β = df for some function f ∈ O(U). Since f may be obtained by integration of β along injective paths in U with initial point p, and each of these integrals must agree with the corresponding integral of β,wehaveβ = df = β on U. Thus β = β on Y \ C and therefore on Y .

Exercises for Sect. 5.20 5.20.1 Suppose X is a complex 1-manifold, Y is a Riemann surface, : X → Y is a ﬁnite holomorphic branched covering map with n sheets, C ⊂ Y is the set of critical values, f ∈ O(X), and P(z1,...,zn) is a complex polynomial in n variables that is symmetric (that is, P(zσ(1),...,zσ(n)) = P(z1,...,zn) n for every point (z1,...,zn) ∈ C and every permutation σ of {1,...,n}). 5.20 Finite Holomorphic Branched Coverings 299

For each point y ∈ Y \ C,letg(y) = P(x1,...,xn), where x1,...,xn are the distinct points in −1(y). Prove that there is a unique holomorphic extension of g to a holomorphic function on Y . Exercises 5.20.2Ð5.20.9 below require facts stated in Sect. 4.6 (cf. [FarK]). 5.20.2 RiemannÐHurwitz formula.Let: X → Y be a ﬁnite holomorphic branched covering mapping of compact Riemann surfaces X and Y ,letb be the branching order of , and let n be the number of sheets. Prove that b genus(X) = + n · (genus(Y ) − 1) + 1. 2 Hint. We may ﬁx a nontrivial meromorphic 1-form β on Y (Theo- rem 2.10.1), and we may let α be the nontrivial meromorphic 1-form on X given by α ≡ ∗β. Set D = div(α) and E = div(β). Given a point q ∈ Y and a point p ∈ −1(q), apply the local representation of holomorphic maps to see that D(p) = m − 1 + mE(q),

where m ≡ multp. Now sum over all of the critical values of and nonzero points of E, and apply Corollary 4.6.7. 5.20.3 Prove that a compact Riemann surface X is hyperelliptic (see Exercise 4.6.6) if and only if X admits a 2-sheeted holomorphic branched covering mapping onto P1. 5.20.4 Let X be a hyperelliptic Riemann surface of genus g, and let : X → P1 be a 2-sheeted holomorphic branched covering map (see Exercises 4.6.6 and 5.20.3). (a) Prove that has exactly 2g + 2 branch points, and that each branch point is of order 1. Hint. Apply the RiemannÐHurwitz formula (Exercise 5.20.2). (b) Prove that if g>1, then every Weierstrass point in X is a hyperelliptic point (see Exercise 4.6.9) and the set of Weierstrass points is precisely the set of branch points of (in particular, X has exactly 2g + 2 Weier- strass points, and combining this with Exercise 4.6.9, we see that a compact Riemann surface of genus g>1 is hyperelliptic if and only if it has exactly 2g + 2 Weierstrass points). Hint. Show that the branch points are hyperelliptic (Weierstrass) points and apply Exercise 4.6.9. (c) Prove that if g>1 and : X → P1 is any 2-sheeted holomorphic branched covering map, then = ◦ for some automorphism ∈ Aut (P1). Hint. Show that after composing with automorphisms of P1 (i.e., with Möbius transformations), we may assume that there are Weier- strass points p,q,r ∈ X with (p) = (p) = 0, (q) = (q) = 1, and (r) = (r) =∞. Then consider the meromorphic function /. 5.20.5 Let X be a hyperelliptic Riemann surface of genus g>1, and let : X → P1 be a 2-sheeted holomorphic branched covering map (see Exer- cise 5.20.3). 300 5 Uniformization and Embedding of Riemann Surfaces

(a) Prove that there exists a unique nontrivial automorphism (i.e., an automorphism that is not the identity) J of X such that ◦ J = . Prove also that J 2 = Id, that the fixed points of J are precisely the 2g + 2 branch points of (i.e., the 2g + 2 Weierstrass points in X), and that J is independent of the choice of (see Exercise 5.20.4). The associated automorphism J is called the hyperelliptic involution or the sheet interchange. (b) Let J be the hyperelliptic involution on X. Prove that if ∈ Aut (X) \ {Id,J} (i.e., is not in the subgroup generated by J ), then has at most four fixed points (in particular, J is the unique nontrivial automorphism of X with at least 2g + 2 fixed points). Hint. Applying Exercise 5.20.4, one sees that ◦ = ◦ for some Möbius transformation . Show that if m is the number of fixed points of , then the number of fixed points of is at least m/2. (c) Prove that the hyperelliptic involution J commutes with every automorphism of X (i.e., J is in the center of Aut (X)). 5.20.6 Let X be a compact Riemann surface of genus g>1. In this exercise, the theorem of Schwarz on finiteness of the automorphism group Aut (X) (with a bound on the order of the group) is obtained as an application of Exer- cises 4.6.6Ð4.6.10 and Exercises 5.20.2Ð5.20.5 (a more precise bound on the order of Aut (X) is obtained in Exercise 5.20.9 below). (a) Let S be the set of Weierstrass points in X. Prove that (S) = S for ∈ → each Aut (X). Conclude from this that the map S is a group homomorphism of Aut (X) into the permutation group of S. (b) Prove that if X is not hyperelliptic, then the homomorphism considered in (a) is injective, and the order of Aut (X) is at most [(g − 1)g(g + 1)]!. Hint. Apply Exercises 4.6.9 and 4.6.10. (c) Prove that if X is hyperelliptic with hyperelliptic involution J (see Ex- ercise 5.20.5), then the kernel of the homomorphism considered in (a) is equal to {1,J}, and the order of Aut (X) is at most 2 ·[(2g + 2)!]. 5.20.7 One way to obtain examples of finite holomorphic branched coverings is to form quotients of Riemann surfaces by finite groups of automorphisms. Throughout this exercise, X will denote a Riemann surface, and will denote a (finite) subgroup of order n in Aut (X). The quotient space by (see Definition 10.4.3),

~~Y ≡ \X = X/[p∼q ⇐⇒ (p) = q for some ∈ ],~~

with quotient map ϒ : X → Y , will be assumed to be equipped with the quotient topology. For each point p ∈ X, p will denote the subgroup of consisting of those elements that ﬁx p (i.e., p is the isotropy subgroup at p). (a) Let p ∈ X and let m be the order of p. Prove that there exist a local holomorphic chart (D,,(0; 1)) and a primitive mth root of unity ω (i.e., ωm = 1butωk = 1if1≤ k

the (rotation) group of automorphisms of (0; 1) given by z → ωkz for k = 0, 1, 2,...,m− 1. Prove also that p is a cyclic group. Hint. Fix a local holomorphic chart (U,,U) with p = −1(0). Applying Exercise 1.5.2, show that for every sufficiently small >0, ◦ ◦ −1 maps the disk (0; ) U biholomorphically onto a relatively compact convex neighborhood of 0 in U for each ∈ p, and ◦ ◦ −1((0; )) is a convex neighborhood of 0 on which ∈p −1 the restriction of ◦ ◦ is an automorphism for each ∈ p. Show that any convex domain is simply connected, apply the Riemann mapping theorem (Theorem 5.2.1) in order to obtain the desired local holomorphic chart, and finally, apply Theorem 5.8.2 in order to obtain the desired primitive mth root. (b) Prove that ϒ : X → Y is an open mapping and that Y is a Hausdorff space. (c) Prove that there is a unique holomorphic structure on Y (i.e., a unique structure of a Riemann surface) for which ϒ : X → Y is a finite holomorphic branched covering map with n sheets. Prove also that a point p ∈ X is a branch point of order m − 1 if and only if p is of order m>1. Hint.Givenp ∈ X, fix a local holomorphic chart (D, = z, (0; 1)) as in (a) in which p = −1(0) and the restrictions of the distinct ele- k ments of p are represented by the rotations z → ω z for k = 0,..., m − 1. Show that D may be chosen so that D ∩[( \ p) · D]=∅. Then show that this yields a local complex chart (D = ϒ(D), = ζ,(0; 1)) in Y determined by (ϒ(q)) = ((q))m for q ∈ D (in other words, the restriction of ϒ to D is represented by the finite holomorphic branched covering map (0; 1) → (0; 1) given by z → ζ = zm). Show that the corresponding atlas is a holomorphic atlas in Y and that ϒ is an n- sheeted finite holomorphic branched covering map with the stated properties. 5.20.8 Let X be a compact Riemann surface of genus g>1. Prove that X is hyperelliptic (see Exercises 4.6.6 and 5.20.3Ð5.20.5) if and only if X admits a holomorphic involution (i.e., ∈ Aut(X) and 2 = Id) with exactly 2g + 2 fixed points. Hint. Given a nontrivial holomorphic involution of X, Exercise 5.20.7 provides a 2-sheeted branched holomorphic covering map X → Y ≡ \X, where ={1, } is the group generated by . Now apply Exer- cises 5.20.2 and 5.20.3 and Theorem 4.6.8. 5.20.9 According to Exercise 5.20.6, every compact Riemann surface of genus g>1 has finite automorphism group. In this exercise, the following bound on the order of the automorphism group is obtained as an application of Exercises 5.20.2Ð5.20.8:

Theorem (Hurwitz) Let X be a compact Riemann surface of genus g>1, and let n be the order of the automorphism group ≡ Aut (X). Then n ≤ 84(g − 1). 302 5 Uniformization and Embedding of Riemann Surfaces

For the proof, let ϒ : X → Y ≡ \X be the n-sheeted ﬁnite holomorphic branched covering provided by Exercise 5.20.7, let h ≡ genus(Y ), and let q1,...,qk be the distinct critical values of ϒ. (a) Prove that for each j = 1,...,k, the groups p ≡{ ∈ | (p) = p} −1 for p ∈ ϒ (qj ) have the same order mj > 1 and that mj divides n. (b) Prove that

k 1 2g − 2 = n · (2h − 2) + n · 1 − mj j=1 (which is interpreted as 2g − 2 = n · (2h− 2) if ϒ has no critical values). (c) Prove that if h ≥ 2, then n ≤ g − 1. (d) Prove that if h = 1, then n ≤ 4(g − 1). (e) Prove that if h = 0 and k ≥ 5, then n ≤ 4(g − 1). (f) Prove that if h = 0 and k = 4, then n ≤ 12(g − 1). (g) Prove that if h = 0 and k = 3, then n ≤ 84(g − 1). (h) Prove that if h = 0, then k>2.

~~5.21 Abel’s Theorem~~

According to the Weierstrass theorem (Theorem 3.12.1), every holomorphic line bundle on an open Riemann surface is holomorphically trivial. Equivalently, every divisor on an open Riemann surface has a solution; that is, there exists a meromorphic function with arbitrary prescribed discrete zeros and poles. This is not the case for a compact Riemann surface. For example, any divisor that is linearly equivalent to the zero divisor must have degree 0. In fact, most divisors of degree zero on a compact Riemann surface of positive genus do not have a solution. The goal of this section is Abel’s theorem (Theorem 5.21.2), which provides a necessary and sufficient condition for a divisor of degree 0 on a compact Riemann surface to have a solution. Abel’s theorem will also allow us to form a holomorphic embedding into a higher-dimensional complex torus (see Sect. 5.22). The approach taken here is similar to the approach in, for example, [For]. Observe that for a Riemann surface X, we may identify the group of divisors with finite support in X with the group of integral 0-chains C0(X, Z) (see Defini- tion 10.6.3) under the isomorphism D → D(p) · p. p∈supp D

Thus we have the homomorphism ∂ : C1(X, Z) → Div(X) (the boundary operator) and the group of integral 1-cycles Z1(X, Z) = ker ∂. Moreover, we have the following fact, the proof of which is left to the reader (see Exercise 5.21.1).

~~Lemma 5.21.1 For any divisor D on a compact Riemann surface X, we have deg D = 0 if and only if D ∈ im(∂ : C1(X, Z) → Div(X)). 5.21 Abel’s Theorem 303~~

Recall that every holomorphic line bundle on a Riemann surface is equal to the line bundle associated to some nontrivial divisor (Corollary 3.11.7 and Theo- rem 4.2.3). Moreover, a divisor D has a solution (that is, D is the divisor of some meromorphic function) if and only if the associated holomorphic line bundle is holomorphically trivial (Proposition 3.3.2). The goal of this section is the following:

Theorem 5.21.2 (Abel’s theorem) Suppose D is a divisor of degree 0 on a compact Riemann surface X. Then D has a solution (i.e., the associated holomorphic line bundle E =[D] is holomorphically trivial) if and only if there exists an integral ∈ Z = = ∈ 1-chain ξ C1(X, ) in X such that ∂ξ D and ξ θ 0 for every θ (X).

We ﬁrst recall the notion of a weak solution of a divisor D on a Riemann surface X (see Sect. 3.12). Let E =[D] and ﬁx a meromorphic section s of E with div(s) = D (in particular, s is nontrivial). Then E is C∞ trivial if and only if there is a C∞ function ρ on X \ D−1((−∞, 0)) such that ρ is nonvanishing on X \ supp D and s/ρ extends to a nonvanishing C∞ section v of E on X. Such a function ρ is called a weak solution of the divisor D. Equivalently, a C∞ function ρ on X \D−1((−∞, 0)) is a weak solution if ρ is nonvanishing on X \supp D and for each point p ∈ supp D, there are a local holomorphic coordinate neighborhood (U, z) with z(p) = 0 and a nonvanishing C∞ function ψ on U such that ρ = zD(p) · ψ on U \{p}.Ob- serve that the logarithmic differential dρ/ρ is locally integrable on X, since writing ρ = zD(p) · ψ on a coordinate neighborhood (U, z) with z(p) = 0 as above, we get, near p, dρ dz dψ = D(p) · + . ρ z ψ Observe also that for v = s/ρ as above, we have

~~∂v¯ ∂ρ¯ =− on X \ supp D, v ρ and for (U, z) as above, we have~~

~~∂v¯ ∂ψ¯ =− on U. v ψ~~

Lemma 5.21.3 If U is a disk in C, z0,z1 ∈ U, τ is a weak solution of the divisor F = z1 − z0 with τ ≡ 1 on C \ U, and f is a holomorphic function on a neighborhood of U, then 1 dτ ∧ df = f(z1) − f(z0). 2πi U τ

∞ Proof The function z → τ(z) · (z − z0)/(z − z1) extends to a nonvanishing C function ψ on C.For>0 sufﬁciently small, Stokes’ theorem (Theorem 9.7.17) 304 5 Uniformization and Embedding of Riemann Surfaces and the Cauchy integral formula and Cauchy’s theorem (Lemma 1.2.1) together give 1 dτ ∧ df 2πi U\((z ;)∪(z ;)) τ 0 1 1 dτ 1 dτ = f · + f · 2πi ∂(z ;) τ 2πi ∂(z ;) τ 1 0 1 f(z) 1 f(z) = dz− dz 2πi ∂(z ;) z − z1 2πi ∂(z ;) z − z0 1 0 1 dψ + f · 2πi ∂(z ;)∪∂(z ;) ψ 0 1 1 dψ = f(z1) − f(z0) + df ∧ . 2πi (z0;)∪(z1;) ψ Letting → 0+, we get the claim.

Lemma 5.21.4 Let W be a relatively compact neighborhood of the image C = γ([0, 1]) of a path γ in a Riemann surface X. Then there exists a weak solution ρ of the divisor D = ∂γ = γ(1) − γ(0) such that ρ ≡ 1 on X \ W and 1 dρ θ = ∧ θ ∀θ ∈ (X). γ 2πi X ρ

~~Proof By Lemma 3.12.2, there exist a partition 0 = t0~~

Proof of Theorem 5.21.2 For the proof, we let D be a divisor of degree 0, which we may assume to be nontrivial. Suppose D has a solution, that is, there exists a nonconstant meromorphic function f with div(f ) = D. Thus f : X → P1 is an n-sheeted ﬁnite holomorphic branched covering map with ﬁnite set of critical values C, and we may choose an injective path γ :[0, 1]→P1 with γ(0) =∞, γ(1) = 0, and γ((0, 1)) ⊂ P1 \ (C ∪{0, ∞}). According to Proposition 5.20.4, γ has exactly n 5.21 Abel’s Theorem 305

−1 distinct liftings γ1,...,γn to paths in X. Moreover, for each point p ∈ f (∞), exactly −D(p) of these liftings have initial point p, and for each point q ∈ f −1(0), exactly D(q) of these liftings have terminal point q. It follows that the integral 1- ≡ n = chain ξ j=1 γj satisﬁes ∂ξ D. Furthermore, if θ is a holomorphic 1-form on X, then Proposition 5.20.4 provides a holomorphic 1-form β on P1 such that θ = β. ξ γ

1 However, P has no nontrivial holomorphic 1-forms (see, for example, Exer- = cise 2.5.4 and Theorem 4.6.8), so ξ θ 0. Conversely, suppose we have an integral 1-chain ξ in X such that ∂ξ = D and such that every holomorphic 1-form on X integrates to 0 along ξ. Applying Lemma 5.21.4, we get a weak solution ρ of D such that ∂ρ¯ dρ ∧ θ = ∧ θ = 0 ∀θ ∈ (X) X ρ X ρ

(see Exercise 5.21.2). Fixing a meromorphic section s of E =[D] with div(s) = D, the section s/ρ extends to a nonvanishing C∞ section v of E. Hence the C∞ differential form η ≡ ∂v/v¯ of type (0, 1), which agrees with −∂ρ/ρ¯ on X \ supp D, satisﬁes " # = − ∧¯= ∀ ∈ η,τ L2 (X) ( i)η τ 0 τ (X). 0,1 X

Applying the Hodge decomposition theorem for scalar-valued forms (Theo- rem 4.9.1), we get a C∞ function ϕ with η = ∂ϕ¯ on X. The nonvanishing C∞ section t ≡ e−ϕv of E then satisﬁes

~~− − ∂t¯ =−e ϕη · v + e ϕ∂v¯ = 0, and therefore t is holomorphic. Thus E =[D] is holomorphically trivial and D has a solution.~~

Exercises for Sect. 5.21 5.21.1 Prove Lemma 5.21.1. 5.21.2 Verify the claim in the proof of Abel’s theorem that for any integral 1-chain ξ in a compact Riemann surface X such that ∂ξ = D and such that every holomorphic 1-form on X integrates to 0 along ξ, there exists a weak solution ρ of D such that ∂ρ¯ dρ ∧ θ = ∧ θ = 0 ∀θ ∈ (X). X ρ X ρ 306 5 Uniformization and Embedding of Riemann Surfaces

~~5.22 The AbelÐJacobi Embedding~~

The goal of this section is the fact that every compact Riemann surface of positive genus admits a holomorphic embedding into a higher-dimensional complex torus. We first consider the basic facts and definitions concerning such higher-dimensional complex tori, which are analogous to those corresponding to 1-dimensional complex tori considered in Example 2.1.6 (see also Example 5.1.10). The verifications of the basic facts are left to the reader. n For n ∈ Z>0,alattice in C is a subgroup of the form = Zλ1 +···+Zλ2n, n where λ1,...,λ2n ∈ C are elements that are linearly independent over R.Wehave an isomorphism of Cn onto the group of translations in Cn (which is, of course, a subgroup of the group of self-diffeomorphisms of Cn) under which each element ζ ∈ Cn is identified with the translation given by z → z+ζ . Thus we may identify with a subgroup of the group of translations in Cn. Clearly, acts freely. Moreover, is the image of Z2n under the real linear isomorphism α : R2n → Cn given by n (t1,...,t2n) → t1ξ1 +···+t2nξ2n, and hence is a discrete subset of C .Wehave the quotient map ϒ : Cn → T ≡ \Cn, and the quotient space (which is a quotient group) T is called a complex torus of (complex) dimension n. As in Example 5.1.10, Theorem 10.4.6 implies that T admits a unique structure of a 2n-dimensional compact smooth manifold for which ϒ is the C∞ universal covering map. In fact, one may show that the local charts given by local inverses of ϒ form a holomorphic atlas, and hence that T is a (fundamental) example of a complex manifold of complex dimension n and that ϒ is a holomorphic covering map (see Exercise 5.22.1). We will not use this fact directly, but it justifies the use of the terminology appearing in Definition 5.22.1 below. As in the 1-dimensional case, we also have a commutative diagram of C∞ maps

~~α R2n Cn~~

ϒ ϒ 0 β T2n T where α is the real linear isomorphism given by (t1,...,t2n) → t1λ1 +···+t2nλ2n, ∞ 2n 1 1 ϒ0 is the C covering map onto the real torus T = S × ··· × S given by 2πit 2πit (t1,...,t2n) → (e 1 ,...,e 2n ), and β is the induced diffeomorphism.

n n Deﬁnition 5.22.1 Let n ∈ Z>0,let be a lattice in C ,letϒ : C → T be the associated quotient map onto the complex torus T ≡ \Cn, and let : X → T be a continuous mapping of a complex 1-manifold X into T. (a) is called holomorphic if the mapping ◦ is holomorphic for each local inverse of ϒ. 5.22 The AbelÐJacobi Embedding 307

(b) is called a holomorphic immersion if the mapping ◦ is a holomorphic immersion for each local inverse of ϒ. (c) is called a holomorphic embedding if X is compact and is an injective holomorphic immersion (i.e., is a proper injective holomorphic immersion).

~~The main goal of this section is the following:~~

Theorem 5.22.2 (AbelÐJacobi embedding theorem) Let X be a compact Riemann surface of genus g>0, and let θ = (θ1,...,θg) be a (complex) basis for (X). Then we have the following: (a) The mapping [ ] → ≡ ξ H1(X,Z) θ θ1,..., θg ξ ξ ξ g determines a well-deﬁned injective group homomorphism H1(X, Z) → C , and g the image θ of this homomorphism is a lattice in C . In fact, the image of any g integral basis for H1(X, Z) is a real basis for C . g (b) Let ϒθ : C → Tθ be the covering map associated to the complex torus g Tθ ≡ θ \C . Given a point p0 ∈ X, the mapping

~~p → ϒθ θ = ϒθ θ1,..., θg , γ γ γ~~

~~where γ is an arbitrary path in X from p0 to p, determines a well-deﬁned : → holomorphic embedding Jθ,p0 X Tθ .~~

Deﬁnition 5.22.3 Given a compact Riemann surface X of genus g>0 and a ba- g sis θ = (θ1,...,θg) for (X), the lattice θ in C provided by the AbelÐJacobi embedding theorem is called the period lattice associated to θ. The corresponding g complex torus θ \C is called the Jacobi variety of X and is denoted by Jac(X) or ∈ : → Jac(X, θ). Given a point p0 X, the holomorphic embedding Jθ,p0 X Jac(X) provided by the AbelÐJacobi embedding theorem is called the AbelÐJacobi embedding associated to the basis θ and the point p0.

Remarks 1. As deﬁned above, the period lattice, Jacobi variety, and AbelÐJacobi embedding depend on the choice of a basis for (X) (and the AbelÐJacobi embedding depends on the choice of an initial point p0 ∈ X). A more intrinsic, but equivalent, point of view also exists (see, for example, [GriH]). 2. For a higher-dimensional smooth complex projective variety (or even for a connected compact complex manifold) with nontrivial holomorphic 1-forms, one may form the analogous holomorphic mapping, which is called the Albanese map. The target complex torus is called the Albanese variety. This mapping is important in algebraic geometry (see, for example, [GriH]). In higher dimensions, the Albanese map need not be an embedding (for example, for Y a compact Riemann surface of positive genus, X ≡ Y × P1 is a smooth projective variety that admits 308 5 Uniformization and Embedding of Riemann Surfaces nontrivial holomorphic 1-forms, but the corresponding Albanese map is constant on {y}×P1 for each point y ∈ Y ).

~~For genus 1, the AbelÐJacobi embedding theorem immediately gives the following (cf. Exercises 5.9.3 and 5.16.1):~~

~~Corollary 5.22.4 Every compact Riemann surface of genus 1 is biholomorphic to a complex torus (of dimension 1).~~

Proof of Theorem 5.22.2 It follows from the definition of integral homology (Def- g inition 10.7.9) that the mapping H1(X, Z) → C in (a) is a well-defined group homomorphism. Moreover, since ¯ ¯ ¯ (θ, θ)≡ (θ1,...,θg, θ1,...,θg) ⊕ =∼ 1 C is a basis for (X) (X) HdeR(X, ) (Corollary 4.9.2), the (well-defined) 2g linear map H1(X, C) → C given by [ ] → ¯ = ¯ ¯ = ξ H1 θ, θ θ1,..., θg, θ1,..., θg θ, θ ξ ξ ξ ξ ξ ξ ξ ξ is injective, and by Theorem 10.7.18, the images under this map of the elements of any integral basis ξ1,...,ξ2g for H1(X, Z) ⊂ H1(X, C) form a complex basis 2g g for C . Setting λj ≡ θ ∈ C for each j = 1,...,2g, it follows that λ1,...,λ2g ξj Cg ∈ R = 2g = is a real basis for .Forifaj for j 1,...,2g and j=1 aj λj 0, then we have 2g 2g 2g aj θ, θ = aj λj , aj λj = (0, 0), j=1 ξj ξj j=1 j=1 and therefore a1 =···=a2g = 0. Part (a) now follows. For the proof of (b), let us fix a point p0 ∈ X. We will let p and q denote points in X, and we will let α and β denote two paths in X with α(0) = β(0) = p0, α(1) = p, and β(1) = q. = − = ∈ = If p q, then α θ β θ α∗β− θ θ , and hence ϒθ ( α θ) ϒθ ( β θ). Thus : → the associated map Jθ,p0 X Tθ is well defined. Conversely, if p and q satisfy = Jθ,p0 (p) Jθ,p0 (q), then

θ − θ = λ = θ ∈ θ α β ξ for some integral 1-cycle ξ ∈ Z1(X, Z). Hence the 1-chain ζ ≡ α − β − ξ ∈ C1(X, Z) has boundary divisor D = ∂ζ = p − q and satisﬁes η = 0 ∀η ∈ (X). ζ 5.22 The AbelÐJacobi Embedding 309

Abel’s theorem (Theorem 5.21.2) now implies that p = q.ForifD were nontrivial (i.e., if p and q were distinct points), then Abel’s theorem would provide a nontrivial meromorphic function f : X → P1 with div(f ) = D, and by Proposition 2.5.7 (or

Proposition 5.20.1), f would be a biholomorphism. Thus Jθ,p0 is injective. g We may ﬁx a holomorphic mapping F = (f1,...,fg): U → C on a connected = = = neighborhood U of p in X such that dF (df1,...,dfg) θ on U, F(p) α θ, and F(U)is contained in a connected neighborhood V of F(p)that ϒθ maps homeomorphically onto its image ϒθ (V ). Therefore, if W is a connected neighborhood of = : → ⊂ Cg Jθ,p0 (p) ϒθ (F (p)) in ϒθ (V ), W (W) is a local inverse of ϒθ , and ≡ −1 ∩ ◦ : → Q ϒθ (W) V , then the homeomorphism ϒθ Q Q (W) is given by −1 z → z + λ for some (constant) element λ ∈ θ . Thus, on the neighborhood F (Q) of p,wehave ≡ ◦ = ◦ ◦ = + G Jθ,p0 ϒθ F F λ.

Thus G is holomorphic and (dG)p = ((θ1)p,...,(θg)p). In particular, since the nontrivial holomorphic 1-forms θ1,...,θg have no common zeros (by Theo- = : → rem 4.6.8), we have (dG)p 0. Thus the mapping Jθ,p0 X Tθ is a holomorphic embedding.

Exercises for Sect. 5.22 5.22.1 Verify that an n-dimensional complex torus X = \Cn admits the structure of an n-dimensional complex manifold (see Exercise 2.2.6) for which the quotient mapping Cn → X is holomorphic with local holomorphic inverses.

~~Chapter 6 Holomorphic Structures on Topological Surfaces~~

In this chapter, we address the natural problem of determining conditions for a topological surface to admit a holomorphic structure. One necessary condition is, of course, that the surface be orientable. According to Radó’s theorem (Theo- rem 2.11.1), another necessary condition is that the surface be second countable. It turns out that these two conditions are also sufficient. Sections 6.1–6.6 consist of a proof of the theorem of Korn and Lichtenstein, according to which every almost complex structure on a smooth surface is integrable; that is, it is induced by a holomorphic structure. It then follows that every orientable second countable smooth surface admits a holomorphic structure. The proof of integrability appearing in this chapter is based on the proofs of the higher-dimensional analogue appearing in [Hö] and [De3]. Sections 6.7–6.11 consist of a proof that every second countable topological surface admits a smooth structure. The first part of the chapter then implies that any second countable orientable topological surface admits a holomorphic structure. One of the main tools in the proof of the existence of smooth structures is Schönflies’ theorem, and a proof (due to Kneser and Radó) of this fact appears in Sects. 6.7–6.9. The proof of integrability and the proof of the existence of smooth structures may be read independently.

~~6.1 Almost Complex Structures on Smooth Surfaces~~

The main goal of Sects. 6.1–6.6 is the existence of holomorphic structures on second countable oriented smooth surfaces. One obtains a holomorphic structure by ﬁrst producing an almost complex structure.

~~Deﬁnition 6.1.1 Let X be a 2-dimensional smooth manifold. (a) An almost complex structure on X is a choice of subsets 1,0 1,0 0,1 0,1 (T X) = (TpX) and (T X) = (TpX) p∈X p∈X~~

T. Napier, M. Ramachandran, An Introduction to Riemann Surfaces, Cornerstones, 311 DOI 10.1007/978-0-8176-4693-6_6, © Springer Science+Business Media, LLC 2011 312 6 Holomorphic Structures on Topological Surfaces

of (T X)C such that 1,0 0,1 (i) For each point p ∈ X, (TpX) and (TpX) are complex subspaces of 1,0 0,1 1,0 0,1 (TpX)C with (TpX)C = (TpX) ⊕ (TpX) and (TpX) = (TpX) ; and (ii) For each point in X, there is a nonvanishing C∞ vector ﬁeld Z on a neigh- 1,0 0,1 borhood U such that (TpX) = C·Zp (hence (TpX) = C·Zp) for each point p ∈ U. We write (T X)C = (T X)1,0 ⊕(T X)0,1. The manifold X together with an almost complex structure is called an almost complex manifold of dimension 2 (or an almost complex surface if X is connected). (b) For any almost complex structure decomposition (T X)C = (T X)1,0 ⊕(T X)0,1, ∗ ∗ ∗ we have the associated decomposition (T X)C = (T X)1,0 ⊕ (T X)0,1 of the cotangent bundle with summands ∗ 1,0 = ∗ 1,0 ∗ 0,1 = ∗ 0,1 (T X) (Tp X) and (T X) (Tp X) , p∈X p∈X where for each point p ∈ X, ∗ 1,0 ≡{ ∈ ∗ | = ∀ ∈ 0,1} ∼ [ 1,0]∗ (Tp X) α (Tp X)C α(v) 0 v (TpX) = (TpX) and

~~∗ 0,1 ≡ ∗ 1,0 ={ ∈ ∗ | = ∀ ∈ 1,0} (Tp X) (Tp X) α (Tp X)C α(v) 0 v (TpX) ∼ 0,1 ∗ = [(TpX) ]~~

∗ = ∗ 1,0 ⊕ ∗ 0,1 (in particular, (Tp X)C (Tp X) (Tp X) ). (c) Let Z be a nonvanishing C∞ vector field with values in (T X)1,0 on an open set U ⊂ X, and let θ be the 1-form on U determined by θ(Z) ≡ 1 and θ(Z) ≡ 0. For any vector field v, we call the functions a ≡ θ(v) and b ≡ θ(v)¯ (i.e., v = aZ + bZ)thecoefficients of v with respect to Z and Z. For any 1-form α,we call the functions A ≡ α(Z) and B ≡ α(Z) (i.e., α = Aθ + Bθ¯)thecoefficients of α with respect to θ and θ¯. (d) An almost complex structure (T X)C = (T X)1,0 ⊕ (T X)0,1 on X is integrable if this decomposition is equal to the decomposition of (T X)C into (1, 0) and (0, 1) parts associated to some holomorphic structure on X; that is, the almost complex structure is induced by the holomorphic structure.

Remark For an almost complex structure (T X)C = (T X)1,0 ⊕ (T X)0,1,weusethe notation and terminology analogous to that used in the integrable case. For example, an element of (T X)1,0 is of type (1, 0).

Recall that by definition, a vector field or differential form on a smooth manifold Ck ∈ Z ∪{∞} d ∈[ ∞] is continuous ( with k ≥0 , measurable, in Lloc with d 1, ) if its C∞ Ck d coefficients in local charts are continuous (respectively, , measurable, in Lloc). We have the following analogous characterization on an almost complex surface: 6.1 Almost Complex Structures on Smooth Surfaces 313

~~Proposition 6.1.2 Let (T X)C = (T X)1,0 ⊕(T X)0,1 be an almost complex structure on a smooth surface X, let k ∈ Z≥0 ∪{∞}, and let d ∈[1, ∞].~~

(a) If v is a vector field on a set S ⊂ X, then the following are equivalent: Ck d (i) The vector field v is continuous ( , measurable, in Lloc). (ii) The coefficients of v with respect to Z and Z are continuous (respectively, Ck d C∞ Z , measurable, in Lloc) for every nonvanishing local vector field of type (1, 0). (iii) For every point p ∈ S, the coefficients of v with respect to Z and Z are Ck d continuous (respectively, , measurable, in Lloc) for some nonvanishing C∞ vector field Z of type (1, 0) on a neighborhood of p. (iv) The (1, 0) and (0, 1) parts of v are continuous (respectively, Ck, measur- d able, in Lloc). (b) If α is a 1-form on a set S ⊂ X, then the following are equivalent: Ck d (i) The 1-form α is continuous ( , measurable, in Lloc). (ii) For every nonvanishing C∞ local vector field Z of type (1, 0), the coefficients of α with respect to the corresponding dual 1-forms θ and θ¯ (determined by θ(Z) ≡ 1 and θ(Z) ≡ 0) are continuous (respectively, Ck , d measurable, in Lloc). (iii) For every point p ∈ S, there exists a nonvanishing C∞ local vector field Z of type (1, 0) on a neighborhood of p such that the coefficients of α with respect to the corresponding dual 1-forms θ and θ¯ are continuous (respec- Ck d tively, , measurable, in Lloc). (iv) The (1, 0) and (0, 1) parts of α are continuous (respectively, Ck, measur- d able, in Lloc). (c) A nonvanishing local vector field Z of type (1, 0) is continuous (Ck, measurable) if and only if its dual 1-form θ (of type (1, 0)) is continuous (respectively, Ck , measurable). In particular, for each point in p ∈ X, there exists a nonvanishing C∞ differential form of type (1, 0) on a neighborhood of p.

Proof Let Z be a nonvanishing C∞ vector ﬁeld of type (1, 0) on an open set U ⊂ X, and let θ and θ¯ be the corresponding dual 1-forms. Suppose also that we have a local C∞ chart (U, = (x, y), (U)). Then the corresponding coefﬁcients a = dx(Z) and b = dy(Z) of Z are of class C∞, and we have

∂ ∂ ∂ ∂ Z = a + b and Z =¯a + b¯ . ∂x ∂y ∂x ∂y Hence the matrix C ≡ ab is nonsingular at each point and a¯ b¯ ¯ − 1 b −b C 1 = ab¯ −¯ab −¯aa 314 6 Holomorphic Structures on Topological Surfaces has C∞ entries. Setting A ≡ θ(∂/∂x) and B ≡ θ(∂/∂y), we get θ = Adx + Bdy and 1 θ(Z) A = = C . 0 θ(Z) B Hence the column vector A − 1 = C 1 · B 0 has C∞ entries, and therefore θ is a C∞ form of type (1, 0). Armed with this fact, it is now easy to verify (a)–(c) (see Exercise 6.1.1).

~~Proposition 6.1.3 Let X be a smooth surface.~~

(a) If X admits an almost complex structure (T X)C = (T X)1,0 ⊕ (T X)0,1, then X is orientable. In fact, there is a unique orientation for which iα∧¯α>0 for each α ∈ (T ∗X)1,0 \{0}. (b) If X is oriented and second countable, then X admits an almost complex structure that induces (as in (a)) the given orientation.

Proof For the proof of (a), suppose (T X)C = (T X)1,0 ⊕ (T X)0,1 is an almost complex structure. If (U,(x,y)) is a connected local C∞ coordinate neighborhood, then either iα ∧¯α ∀ ∈ ∈ ∗ 1,0 \{ } > 0 p U,α (Tp X) 0 , (dx ∧ dy)p or ∧¯ iα α ∀ ∈ ∈ ∗ 1,0 \{ } < 0 p U,α (Tp X) 0 . (dx ∧ dy)p For if θ is a nonvanishing C∞ differential form of type (1, 0) on an open set V ⊂ U, then

∧¯ i(θ ∧ θ)¯ iα α =| |2 p ∀ ∈ ∈ ∗ 1,0 \{ } α/θp p V,α (Tp X) 0 , (dx ∧ dy)p (dx ∧ dy)p and the real-valued C∞ function i(θ ∧ θ)/(dx¯ ∧ dy) is nonvanishing. So if, for some choice of θ and V , i(θ ∧ θ)/(dx¯ ∧ dy) > 0 at some point in V , then the set of points p ∈ U at which the inequality i(α ∧¯α)/(dx ∧ dy)p > 0 holds for every ∈ ∗ 1,0 \{ } α (Tp X) 0 is nonempty with empty boundary in U, and is therefore equal to U. Similarly, if (x, y) and (x ,y ) are two choices of local C∞ coordinates for which the above quotients are positive, then we have (dx ∧ dy)/(dx ∧ dy )>0, and hence they have compatible orientations. The claim (a) now follows. As motivation for the proof of (b), we ﬁrst examine how the orientation in C = R2 relates to the standard (integrable) almost complex structure. For the standard coor- 6.1 Almost Complex Structures on Smooth Surfaces 315 dinate z = x + iy ↔ (x, y), we have, at each point p, 2 1,0 1,0 ∂ 1 ∂ ∂ (TpR ) = (TpC) = C · = C · − i . ∂z p 2 ∂x p ∂y p

2 2 Under the identiﬁcation of TpR with {p}×R , we see that (∂/∂y)p is obtained by multiplication of (∂/∂x)p by i, that is, by rotation of the vector counterclockwise by 90°. On a general surface, one may deﬁne a counterclockwise 90° rotation if one has a Riemannian metric (to determine the rotational angle measure) and an orientation (to determine the counterclockwise sense). Suppose now that X is a second countable oriented smooth surface. By Propo- sition 9.11.2, X admits a Riemannian metric g. For each point p ∈ X,wemay 1,0 = 1 − let (TpX) be the collection of all vectors of the form w 2 (u iv), where u, v ∈ TpX have the following properties:

~~(i) We have g(u,v) = 0(i.e.,u ⊥ v) and |u|g =|v|g; and ≥ ∈ 2 ∗ (ii) We have α(u,v) 0 for every positive α Tp X.~~

0,1 1,0 Setting (TpX) ≡ (TpX) for each point p ∈ X, we now show that the spaces 1,0 1,0 0,1 0,1 (T X) ≡ (TpX) and (T X) ≡ (TpX) p∈X p∈X determine an almost complex structure (T X)C = (T X)1,0 ⊕(T X)0,1 on X. For this, let us fix a positively oriented local C∞ coordinate neighborhood (U,(x,y)) in X. Then we have C∞ real vector fields ∂/∂x (∂/∂y) − g(∂/∂y,u)u u ≡ and v ≡ |∂/∂x|g |(∂/∂y) − g(∂/∂y,u)u|g that satisfy g(u,v) ≡ 0 and |u|g =|v|g ≡ 1onU. Moreover, for every point p ∈ U ∈ 2 ∗ and every positive α Tp X,wehave α α(up,vp) = · (dx ∧ dy)(up,vp) (dx ∧ dy)p α 1 1 = · · > 0. (dx ∧ dy)p |(∂/∂x)p|g |(∂/∂y)p − g((∂/∂y)p,up)up|g Thus the values of the nonvanishing C∞ vector field Z ≡ (1/2)(u − iv) on U lie in (T X)1,0. For each point p ∈ U and each complex number ζ = a + ib with a,b ∈ R, we have 1 ζ · Zp = ((aup + bvp) − i(−bup + avp)), 2 + − + =− + = | + |2 =| |2 =|− + g(aup bvp, bup avp) ab ba 0, aup bvp g ζ bup |2 ∈ 2 ∗ avp g, and for each positive α (Tp X),

~~2 α(aup + bvp, −bup + avp) =|ζ | α(up,vp) ≥ 0. 316 6 Holomorphic Structures on Topological Surfaces~~

1,0 Thus C · Zp ⊂ (TpX) . Conversely, given an element w = (1/2)(r − is) ∈ 1,0 = + ∈ R ⊥ | |2 =| |2 = 2 + 2 (TpX) ,wehaver aup bvp with a,b , s r, and s g r g a b . Thus s =±(−bup + avp). On the other hand, we also have (dx ∧ dy)(r, s) ≥ 0 and

~~2 2 (dx ∧ dy)(r, −bup + avp) = (a + b )(dx ∧ dy)(up,vp) ≥ 0~~

(with equality if and only if r = s = 0). Thus s =−bup + avp, and hence for ζ = 1,0 1,0 a + ib,wehavew = ζ · Zp. Therefore, (Tp X) = C · Zp. It also follows that Zp and Zp are linearly independent over C, since (dx ∧ dy)(up, −vp)<0. Thus we have produced an almost complex structure (T X)C = (T X)1,0 ⊕ (T X)0,1.The veriﬁcation that this almost complex structure induces the given orientation is left to the reader (see Exercise 6.1.2).

~~A holomorphic structure induces an almost complex structure. The main goal of Sects. 6.1–6.6 is the following:~~

Theorem 6.1.4 (Integrability theorem) Every almost complex structure on a smooth surface is integrable. In fact, every almost complex structure is induced by a unique (1-dimensional) holomorphic structure.

~~Radó’s theorem (Theorem 2.11.1) and the above integrability theorem together give the following:~~

~~Corollary 6.1.5 Any smooth surface that admits an almost complex structure must be second countable.~~

~~Proposition 6.1.3 and the integrability theorem together give the following:~~

~~Corollary 6.1.6 Every oriented second countable smooth surface admits a holomorphic structure that induces the given orientation.~~

Deﬁnition 6.1.7 Let (T X)C = (T X)1,0 ⊕ (T X)0,1 be an almost complex structure on a 2-dimensional smooth manifold X,letα be a C1 differential form of degree r ∗ ∗ on an open set ⊂ X, and let Pp,q : 1(T X)C → p,qT X be the associated projection for each pair (p, q) ∈{(1, 0), (0, 1)}. Then we deﬁne ⎧ ⎧ ⎨⎪P1,0(dα) if r = 0, ⎨⎪P0,1(dα) if r = 0, ∂α ≡ d(P0,1α) if r = 1, and ∂α¯ ≡ d(P1,0α) if r = 1, ⎩⎪ ⎩⎪ 0ifr ≥ 2, 0ifr ≥ 2.

Remarks Proposition 6.1.2 allows us to make the above deﬁnition because according to this proposition, the (r, s) part of a Ck differential form is of class Ck.Wealso recall that we use the notation and terminology analogous to that used in the integrable case. For example, a differential form α is ∂¯-closed if ∂α¯ = 0. According to Proposition 6.1.8 below, the operators ∂ and ∂¯ retain at least some of the properties that these operators have in the integrable case (cf. Proposition 2.5.5). 6.1 Almost Complex Structures on Smooth Surfaces 317

Proposition 6.1.8 Let (T X)C = (T X)1,0 ⊕(T X)0,1 be an almost complex structure on a smooth surface X. Then (a) The operators d, ∂, and ∂¯ satisfy d = ∂ + ∂¯ on C1 forms and d2 = ∂2 = ∂¯ 2 = ∂∂¯ + ∂∂¯ = 0 on C2 forms. (b) If α is a C1 differential form of type (p, q), then ∂α is of type (p + 1,q)and ∂α¯ is of type (p, q + 1). In particular, ∂α = 0 if p ≥ 1 and ∂α¯ = 0 if q ≥ 1. (c) If α and β are C1 differential forms, then

~~∂(α ∧ β) = (∂α) ∧ β + (−1)deg αα ∧ ∂β~~

~~and ∂(α¯ ∧ β) = (∂α)¯ ∧ β + (−1)deg αα ∧ ∂β.¯~~

The proof is left to the reader (see Exercise 6.1.3). Note that we have not included an analogue of the property (d) of Proposition 2.5.5 (as well as (c), (f), and (g)). The formation of such an analogue is at the heart of the proof of the integrability theorem. As a ﬁrst step toward the proof of the integrability theorem, we show that the integrability condition is completely local. For this, we will need the following:

Lemma 6.1.9 Let X be an almost complex surface, let f = u + iv: X → C be a C1 function with u = Re f and v = Im f (which we may also view as a C1 map- 2 ¯ ping f = (u, v): X → R ), and let p ∈ X. Assume that (∂f)p = 0. Then the complex linear extension (f∗)p : (TpX)C → (Tf(p)C)C of the real linear tangent map 2 (f∗)p : TpX → Tf(p)C = Tf(p)R preserves the almost complex structures; that is,

~~r,s r,s (f∗)p(TpX) ⊂ (Tf(p)C) ∀(r, s) ∈{(1, 0), (0, 1)}.~~

~~Moreover, if, in addition, (df )p = 0(i.e., (∂f )p = 0), then the real linear maps~~

~~2 2 ((du)p,(dv)p): TpX → R and (f∗)p : TpX → TpR = TpC, and the complex linear maps~~

~~(f∗)p : (TpX)C → (TpC)C, 1,0 1,0 = ◦ ∗ 1,0 : → C (df )p (TpX) dz (f )p (TpX) (TpX) , ¯ 0,1 0,1 = ¯ ◦ ∗ 0,1 : → C (df)p (TpX) dz (f )p (TpX) (TpX) , are bijective.~~

Proof To avoid confusion, when thinking of the complex-valued function f as a mapping of X into the Riemann surface C, let us denote this mapping by F . Letting z = x + iy denote the complex coordinate in C = R2 with Re z = x and Im z = y, ¯ we then have f = z ◦ F and f =¯z ◦ F . Note also that for each point a ∈ C, (dz)a 318 6 Holomorphic Structures on Topological Surfaces

1,0 0,1 maps (TaC) isomorphically onto C and vanishes on (TaC) , while (dz)¯ a maps 0,1 1,0 (TaC) isomorphically onto C and vanishes on (TaC) . Given a tangent vector 1,0 v ∈ (TpX) ,wehave ¯ ¯ 0 = ∂f(v)¯ = df (v)¯ = dz(F∗v)¯ and 0 = df (v)¯ = df(v)= dz(F¯ ∗v).

~~r,s r,s It follows that (F∗)p(TpX) ⊂ (TF(p)C) for (r, s) ∈{(1, 0), (0, 1)}. Further- more, if (df )p = 0, then we may choose a tangent vector w ∈ (TpX)C with~~

~~0 = df (w) = ∂f (w) = df (P1,0w).~~

1,0 Thus (df )p : (TpX) → C is a nontrivial linear mapping of 1-dimensional complex vector spaces, and is therefore an isomorphism. It follows that the complex linear maps −1 1,0 1,0 (F∗) 1,0 = dz 1,0 ◦ (df ) 1,0 : (T X) → (T C) p (TpX) (TF(p)C) p (TpX) p F(p) and −1 ¯ 0,1 0,1 (F∗) 0,1 = dz¯ 0,1 ◦ (df) 0,1 : (T X) → (T C) p (TpX) (TF(p)C) p (TpX) p F(p) are bijective. In particular, the complexiﬁed tangent map (F∗)p : (TpX)C → (TF(p)C)C is bijective, and hence, by Proposition 8.1.3, the corresponding real tangent map (F∗)p : TpX → TF(p)C must also be bijective, as must be the map

~~2 ((du)p,(dv)p) = ((dx)p,(dy)p) ◦ F∗ : TpX → R .~~

~~We also have the following uniqueness property:~~

Lemma 6.1.10 Let X be a Riemann surface with induced almost complex structure (T X)C = (T X)1,0 ⊕ (T X)0,1 and corresponding (0, 1) part of the d operator denoted by ∂¯. (a) Let (T X)C = (T X)1,0 ⊕ (T X)0,1 be a second almost complex structure on X with corresponding (0, 1) part of the d operator denoted by ∂¯. If p ∈ X and there exists a local holomorphic chart (U,,U ) in X with p ∈ U and ¯ (∂)p = 0, then the two almost complex structures agree at p; that is,

1,0 1,0 0,1 0,1 (TpX) = (TpX) and (TpX) = (TpX) . (b) Any holomorphic structure on the smooth surface underlying X that induces the same given almost complex structure is equal to the given holomorphic structure on X.

Proof In the situation of part (a), by Lemma 6.1.9, (∗)p maps (TpX)C isomorphi- 1,0 1,0 cally onto (T(p)C)C, and both of the subspaces (TpX) and (TpX) isomorphi- 1,0 C ⊂ C cally onto T(p) (T(p) )C. Part (a) now follows. 6.1 Almost Complex Structures on Smooth Surfaces 319

For proof of (b), we observe that on a smooth surface, any two 1-dimensional holomorphic structures with the same underlying smooth structures and induced (almost) complex structures have the same ∂¯ operator. Hence, if (U, z) is a local holomorphic coordinate neighborhood with respect to one of the holomorphic structures, then ∂z¯ = 0, and hence z is a local holomorphic coordinate in the other holomorphic structure. Thus, holomorphic atlases for the two holomorphic structures are holomorphically equivalent; i.e., the holomorphic structures are the same.

Lemma 6.1.11 Let X be an almost complex surface. If for each point p ∈ X, there exists a nonvanishing C∞ ∂¯-closed differential form of type (1, 0) on a neighborhood of p, then the almost complex structure is integrable.

Proof By hypothesis, there exists a covering {Uj } of X by open sets such that ∞ for each j, there is a nonvanishing C form αj of type (1, 0) on Uj with ¯ dαj = ∂αj = 0. By the Poincaré lemma (Lemma 9.5.7), we may also assume that ∞ for each j, there exists a C function j on Uj with dj = αj , and therefore ¯ ∂j = 0. According to Lemma 6.1.9,foruj ≡ Re j and vj ≡ Im j , duj ∧ dvj is nonvanishing, and hence by the C∞ inverse function theorem (Theorem 9.9.1 and ∞ Theorem 9.9.2), we may also assume that j is a C diffeomorphism of Uj onto an open subset Uj of the plane. For each pair of indices j,k, applying Lemma 6.1.9, C∞ ≡ ◦ −1 ∩ we see that for the coordinate transformation fjk j k on k(Uj Uk), we have ¯ = ◦ −1 ◦ P0,1 = ◦ −1 ◦ P0,1 ∂fjk (dj ) (k )∗ (∂j ) ((k)∗) 0,1 −1 = (∂j ) ◦ P ◦ ((k)∗) ≡ 0, where Pr,s denotes the projection of tangent vectors onto their (r, s) parts for (r, s) ∈ { } { } (1, 0), (0, 1) ; that is, fjk is holomorphic. Thus the atlas (Uj ,j ,Uj ) determines a holomorphic structure on X, and by Lemma 6.1.10, the induced complex structure is equal to the given almost complex structure.

Remark An almost complex structure on a higher-dimensional C∞ manifold is induced by a holomorphic structure if and only if for every C∞ differential form β of type (0, 1), the 2-form dβ has no (2, 0) part (see, for example, [Hö]). Of course, on a Riemann surface, this condition holds automatically.

Exercises for Sect. 6.1 6.1.1 Complete the proof of Proposition 6.1.2. 6.1.2 Verify that the almost complex structure constructed in the proof of part (b) of Proposition 6.1.3 induces the given orientation. 6.1.3 Prove Proposition 6.1.8. 6.1.4 Let g1 and g2 be two Riemannian metrics on an oriented smooth surface X. Prove that g1 and g2 induce the same almost complex structure (as in the proof of Proposition 6.1.3) if and only if there exists a C∞ real-valued function ϕ ϕ on X such that g2 = e g1. 320 6 Holomorphic Structures on Topological Surfaces

~~6.2 Construction of a Special Local Coordinate~~

According to Lemma 6.1.11, to obtain integrability in an almost complex surface, it suffices to obtain a local ∂¯-closed C∞ differential form of type (1, 0) that is nonzero at a given point p. On a Riemann surface, the L2 ∂¯-method provides a technique for producing nonconstant holomorphic (or meromorphic) functions and forms (and sections of holomorphic line bundles). Most of the L2 ∂¯-method on a Riemann surface as considered in Chap. 2 works without change on an almost complex surface, but there are three difficulties. The first is that on a Riemann surface, the technique relied on the (obvious) existence of nonconstant local holomorphic functions and forms. But on an almost complex surface, it is the existence of such local functions and forms that is the problem to be solved. The second difficulty is that the construction of a positive-curvature weight function (or Hermitian metric) in Sect. 2.14 used a local holomorphic coordinate. Again, in the present context, the existence of a local holomorphic coordinate is the problem to be solved. The third difficulty is that the proof of regularity for ∂¯ on a Riemann surface (Theorem 1.2.8 and Theo- rem 2.7.4) relied on normal families of holomorphic functions, which again, are not a priori available on an almost complex surface. The proof of regularity provided in this chapter, which is similar to the proofs in [De3] and [Hö], requires elements of Sobolev space theory (see Chap. 11). For the first two difficulties, following De- mailly [De3], the idea is to work with a local C∞ complex coordinate z such that although ∂z¯ does not vanish identically, ∂z¯ does vanish to high order at the point p. The existence of such a coordinate is given by the following lemma, which is the main goal of this section:

Lemma 6.2.1 Let X be an almost complex surface, let p ∈ X, and let m ∈ Z≥0. Then there exists a C∞ function f on a neighborhood of p in X such that ¯ (∂f )p = (df )p = 0 and ∂f has vanishing derivatives of order ≤ m at p (see Deﬁ- nition 9.5.8).

Proof Since the claim is local, we may assume that X is an open subset of C to which an almost complex structure (T X)C = (T X)1,0 ⊕(T X)0,1 has been assigned. This almost complex structure may be different from that induced by C. In particular, ∂ and ∂¯ will be the operators associated to the almost complex structure on X, not the standard holomorphic structure on C. We may also assume that p = 0 ∈ C. By replacing z with z¯ and X with a small neighborhood of p if necessary, we may assume without loss of generality that the coordinate function z in C satisﬁes ∂z = 0 on X. Thus we may form the C∞ function

~~(∂z)¯ (∂z)¯ μ ≡ = on X. (∂z) (∂¯z)¯~~

Note that the above denotes a quotient of (0, 1)-forms, not a partial derivative. In other words, μ is the C∞ function deﬁned by ∂z¯ = μ · ∂z = μ · ∂¯z¯. 6.2 Construction of a Special Local Coordinate 321

~~We proceed by induction on m. For the case m = 0, we simply set f(z)= z − μ(0)z¯. We then get ¯ ¯ ¯ (∂f)0 = (∂z)0 − μ(0)(∂z)¯ 0 = 0, and hence~~

(∂f )0 = (df )0 = (dz)0 − μ(0)(dz)¯ 0 = 0 (since dz and dz¯ are pointwise linearly independent). Thus the claim is proved for m = 0. Assume now that m>0 and that the desired function exists for lower orders. In other words, there exists a C∞ function h on a neighborhood of 0 such that ¯ ∂h has vanishing derivatives of order ≤ m − 1 at 0 and (dh)0 = 0. In particular, Lemma 6.1.9 and the C∞ inverse function theorem imply that (Re h, Im h) gives local C∞ coordinates in a neighborhood of 0. Therefore, after replacing the coordinate z with the local coordinate h − h(0), we may assume that the C∞ function μ has vanishing derivatives of order ≤ m−1 at 0 (see Proposition 9.5.9). Equivalently, the degree-m Taylor polynomial P for μ centered at 0 is either homogeneous of degree m or trivial. Thus, expressing P as a polynomial in z and z¯ (which we may do, since x = Re z = 2−1(z +¯z) and y = Im z = (2i)−1(z −¯z)), we get m m−1 m P(z)= A0z + A1z z¯ +···+Amz¯ , for some complex constants A0,...,Am. Setting

m m−1 1 2 1 m+1 Q(z) ≡ A z z¯ + A z z¯ +···+Am z¯ , 0 1 2 m + 1 we get ∂Q ∂Q ∂Q = P and (0) = (0) = 0. ∂z¯ ∂z ∂z¯ We will show that the function f(z)≡ z − Q(z) has the required properties. We have ∂Q ∂Q ∂f¯ = df ◦ P0,1 = dz− dz− dz¯ ◦ P0,1 ∂z ∂z¯ ∂Q = ∂z¯ − · ∂z¯ − P · ∂¯z¯ ∂z ∂Q = μ − P − μ · ∂z. ∂z Since μ − P has vanishing derivatives of order ≤ m at 0, ∂Q/∂z = 0 at 0, and μ has vanishing derivatives of order ≤ m−1 at 0, we see that ∂f¯ has vanishing derivatives of order ≤ m at 0. Similarly, we have ∂Q ∂Q (df ) = (dz) − (0) · (dz) − (0) · (dz)¯ = (dz) = 0. 0 0 ∂z 0 ∂z¯ 0 0 Thus f has the required properties and the lemma follows. 322 6 Holomorphic Structures on Topological Surfaces

~~6.3 Regularity of Solutions on an Almost Complex Surface~~

For regularity, we will use Theorem 11.0.1, which is a general regularity theorem for first-order differential operators satisfying a certain estimate. In order to obtain the required (1, 0)-form in Lemma 6.1.11 on a neighborhood of a given point in an almost complex surface, it suffices to work locally. Thus it suffices to consider a nonempty relatively compact domain X in R2 (obtained by replacing the given surface with a connected local C∞ coordinate neighborhood) on which we have a given almost complex structure TXC = (T X)1,0 ⊕(T X)0,1 and in which p = (0, 0). Letting (x, y) denote the real coordinates in R2, we get a second almost complex structure under the identification R2 = C given by (x, y) ↔ z = x + iy. To avoid confusion, we will use the notation Pr,s , ∂, ∂¯, r,s TX, and r,s T ∗X only for the operators and spaces associated to the given almost complex structure on X, not the standard holomorphic structure on C. According to Lemma 6.2.1, Lemma 6.1.9, and the C∞ inverse function theorem (Theorem 9.9.1 and Theorem 9.9.2), we may also assume that the (1, 0)-form ∂z¯ has vanishing derivatives of order ≤ 2atp = 0 (actually, for the results considered in this section, order 0 would suffice, but we will need vanishing derivatives of order ≤ 2 in later sections). In particular, we have (∂z)0 = (dz)0 = 0, so we may also assume that ∂z = 0 at every point in X (replacing X with a small neighborhood of 0 if necessary). We now derive local formulas for ∂ and ∂¯. Given a local C1 function f ,wehave ∂f ∂f ∂f¯ = df ◦ P0,1 = dz+ dz¯ ◦ P0,1 ∂z ∂z¯ ∂f ∂f ∂f ∂f = ∂z¯ + ∂¯z¯ = ∂z¯ + ∂z ∂z ∂z¯ ∂z ∂z¯ ∂f ∂f = + μ · · ∂z = Zf · ∂z, ∂z¯ ∂z where μ is the C∞ function on X given by

(∂z)¯ (∂z)¯ μ ≡ = (i.e., ∂z¯ = μ · ∂¯z),¯ (∂z) (∂¯z)¯ and Z is the C∞ vector ﬁeld (and therefore linear differential operator of order 1 with C∞ coefﬁcients) on X given by ∂ ∂ Z ≡ + μ · . ∂z¯ ∂z Furthermore, by the choice of the local representation of X, the function μ has vanishing derivatives of order ≤ 2atp = 0. In particular, we may assume that |μ|2 < 3/16 on X.Wealsohave ∂ ∂ ∂f = (∂¯f)¯ = Zf¯ · ∂z = Zf · ∂z, where Z ≡ Z = +¯μ · . ∂z ∂z¯ 6.3 Regularity of Solutions on an Almost Complex Surface 323

Given a local C1 differential form α of type (1, 0),wehaveα = f∂z for the C1 function f ≡ α/(∂z). Thus ∂f ∂f ∂α¯ = (∂f)¯ ∧ ∂z+ f ∂∂z¯ = + μ · · ∂z∧ ∂z+ f ∂∂z¯ ∂z¯ ∂z √ √ √ ∂f ∂f −1 √ −1 = 2 −1 + μ · · ∂z∧ ∂z+ 2 −1f ∂∂z¯ ∂z¯ ∂z 2 2 √ ∂f ∂f = 2 −1 + μ · + τ · f ω = Af¯ · ω, ∂z¯ ∂z where ω is the nonvanishing real C∞ differential form of type (1, 1) given by √ −1 ω ≡ ∂z∧ ∂z, 2 τ is the C∞ function on X given by √ −1 ∂∂z¯ ∂∂z¯ τ ≡ · = 2 ω ∂z∧ ∂z

(see Exercise 6.3.1 for a more explicit expression for τ in terms of μ), and A and A¯ are the linear differential operators of order 1 with C∞ coefﬁcients on X given by √ ∂ ∂ √ A ≡−2 −1 +¯μ · +¯τ =−2 −1(Z +¯τ) ∂z ∂z¯ and √ ∂ ∂ √ A¯ ≡ 2 −1 + μ · + τ = 2 −1(Z + τ). ∂z¯ ∂z In particular, since ∂z¯ has vanishing derivatives of order ≤ 2at0,τ has vanishing derivatives of order ≤ 1at0.Forβ = f ∂z a C1 differential form of type (0, 1),we have ∂β = ∂¯β¯ = ∂(¯ f∂z)¯ = (A¯f¯ · ω) = Af · ω. We may summarize the above as follows:

Lemma 6.3.1 For any given point p in an almost complex surface X, in order to show that there exists a nonvanishing ∂¯-closed C∞ differential form of type (1, 0) on a neighborhood of p, it sufﬁces to consider the case in which: (i) X is a relatively compact domain in R2 together with a given almost complex structure TXC = (T X)1,0 ⊕ (T X)0,1, as well as the holomorphic structure associated to the identiﬁcation R2 = C given by (x, y) ↔ z = x + iy, and p = 0; (ii) The (1, 0)-form ∂z is nonvanishing on X; 324 6 Holomorphic Structures on Topological Surfaces

(iii) We have C∞ functions μ and τ and a nonvanishing C∞ real differential form ω of type (1, 1) on X such that μ and τ have vanishing derivatives of order ≤ 2 and of order ≤ 1, respectively, at 0, |μ|2 < 3/16 on X, and √ √ −1 −1 ∂z¯ = μ · ∂z, ω = ∂z∧ ∂z, and ∂∂z¯ = τ · ω; 2 2 and (iv) For every local C1 function f , we have

~~∂f¯ = Zf · ∂z and ∂f = Zf · ∂z,~~

~~where ∂ ∂ ∂ ∂ Z = + μ · and Z ≡ +¯μ · , ∂z¯ ∂z ∂z ∂z¯ and we have~~

~~∂(f∂z)¯ = Af¯ · ω and ∂(f∂z) = Af · ω,~~

where A¯ and A are the linear differential operators of order 1 with C∞ coefﬁ- cients on X given by √ ∂ ∂ √ A¯ ≡ 2 −1 + μ · + τ = 2 −1(Z + τ) ∂z¯ ∂z and √ ∂ ∂ √ A ≡−2 −1 +¯μ · +¯τ =−2 −1(Z +¯τ). ∂z ∂z¯

~~Fixing the choices and notation in the above lemma for the rest of this section, the main goal is the following regularity property for the operator A¯:~~

~~Proposition 6.3.2 (Regularity) If is a relatively compact open subset of X, and ∈ 2 ¯ ∈ C∞ ∈ C∞ f Lloc() with Adistrf (), then f ().~~

Observe that for a local C1 function f on X,wehave 2 2 2 2 ∂f ∂f ∂f ∂f + = 2 + 2 . ∂x ∂y ∂z ∂z¯ Furthermore, if f ∈ D(X), then (denoting Lebesgue measure by λ) we get 2 2 ∂f ∂f ∂f ∂f = dλ = dλ ∂z L2(X) X ∂z X ∂z ∂z ∂f ∂f¯ ∂2f¯ = · dλ =− f · dλ X ∂z ∂z¯ X ∂z∂z¯ 6.3 Regularity of Solutions on an Almost Complex Surface 325 ∂f ∂f¯ ∂f ∂f ∂f 2 = · dλ = dλ = ¯ ¯ ¯ ¯ . X ∂z ∂z X ∂z ∂z ∂z L2(X) Thus Proposition 6.3.2 will follow immediately from the general ﬁrst-order regularity theorem (Theorem 11.0.1) together with the following:

~~Lemma 6.3.3 For every open set X, there is a constant C>0 such that ∂f 2 ≤ C · f 2 +Af¯ 2 ∀f ∈ D(). ¯ L2() L2() ∂z L2()~~

Proof Given an open set X and a function f ∈ D(),wehave ∂f 2 ∂f 2 = Af¯ − i · μ · − i · τ · f 4 ¯ 2 2 ∂z L2() ∂z L2() 2 ≤ ¯ 2 + · | |2 · ∂f 2 Af L2() 16 sup μ ∂z L2() + · | |2 · 2 16 sup τ f L2().

Thus, since |μ|2 < 3/16 on X (by construction) and τ is bounded on , there is a constant C>0 independent of the choice of f such that ∂f 2 ∂f 2 4 ≤ C · f 2 +Af¯ 2 + 3 · ¯ L2() L2() ∂z L2() ∂z L2() ∂f 2 = C f 2 +Af¯ 2 + 3 . L2() L2() ¯ ∂z L2() The desired inequality now follows.

Finally, we note that the real volume forms ω and dx∧dy = (i/2)dz∧dz¯ induce the same orientation in X, since X is connected and the quotient ω η ≡ dx ∧ dy is a nonvanishing C∞ function that is equal to 1 at 0 (in fact, as the reader is asked to verify in Exercise 6.3.1, η = 1/(1 −|μ|2). We will use this orientation when integrating 2-forms.

Exercises for Sect. 6.3 6.3.1 Show that η = 1/(1 −|μ|2) and 1 ∂|μ|2 ∂μ ∂μ ∂|μ|2 τ = + − ·|μ|2 + μ · . 1 −|μ|2 ∂z¯ ∂z ∂z ∂z 326 6 Holomorphic Structures on Topological Surfaces

~~6.4 The Distributional ∂¯, Connection, and Curvature~~

We continue in this section with the notation of Sect. 6.3. In particular, we have the positive (1, 1)-form ω = (i/2)∂z ∧ ∂z on X. We also fix a real-valued C∞ function ϕ on X. For every measurable set S ⊂ X and every continuous real (1, 1)- · · · · · form θ defined at points of S, we may define , L2(S,ϕ), , L2(S,θ,ϕ), L2(S,ϕ), · 2 and L2(S,θ,ϕ) on scalar-valued differential forms and the associated L (Hilbert) 2 2 ≥ spaces Lp,q(S, ϕ) and Lp,q(S,θ,ϕ) (with θ 0orθ>0 wherever appropriate) with respect to the almost complex structure (T X)C = (T X)1,0 ⊕ (T X)0,1 exactly as in Sect. 2.6. We may also define the corresponding canonical connection (or Chern connection) = = + = + ≡ ϕ [ −ϕ · ]+¯ D Dϕ D D Dϕ Dϕ e ∂ e ( ) ∂ on scalar-valued differential forms exactly as in Sect. 2.7, and the corresponding curvature ¯ = ϕ ≡ ∂∂ϕ as in Sect. 2.8. In particular, we have D2 = D D + D D = ∧ (·).

~~Deﬁnition 6.4.1 Let α and β be locally integrable differential forms on an open ¯ subset U of X. We write ∂distrα = β if one of the following holds:~~

~~(i) On U, α is a 0-form, β = b∂z, and Zdistrα = b; ¯ (ii) On U, α = a1∂z+ a2∂z, β = bω, and Adistr(a1) = b; (iii) The form α is of degree > 1 and β ≡ 0.~~

~~In analogy with Proposition 2.7.3, we have the following equivalent form for the ¯ deﬁnition of ∂distr:~~

Proposition 6.4.2 Let α and β be locally integrable differential forms on an open set U ⊂ X. ¯ (a) If α is of type (1, 0) and β is of type (1, 1), then ∂distrα = β if and only if α ∧ D fe−ϕ = β · f¯ · e−ϕ ∀f ∈ D(U). U U ¯ (b) If α is of type (0, 0) and β is of type (0, 1), then ∂distrα = β if and only if α · (−D γ)· e−ϕ = β ∧ γ · e−ϕ ∀γ ∈ D0,1(U). U U ¯ (c) If α is of type (p, q) with p ≥ 2 or q ≥ 1, then ∂distrα = 0.

Proof Part (c) is trivial and the proof of (b) is left to the reader (see Exercise 6.4.1). For the proof of (a), suppose α = a∂z and β = bω are locally integrable forms of 6.4 The Distributional ∂¯, Connection, and Curvature 327 type (1, 0) and (1, 1), respectively, on an open set U ⊂ X. By the (weak) Friedrichs ∞ lemma (Lemma 7.3.1), we may choose a sequence of C functions {aν} converging 1 = → 1 to a in Lloc. Setting αν aν∂z for each ν, we get αν α in Lloc. Recall also that we have ω = ηdx∧ dy. ¯ ¯ If ∂distrα = β, then Adistra = b. Hence, for every function f ∈ D(U),wehave −ϕ −ϕ −ϕ α ∧ D f · e = lim αν ∧ D f · e = lim αν ∧ ∂(e f) ν→∞ ν→∞ U U U ¯ −ϕ ¯ −ϕ = lim αν ∧ ∂(e f)= lim (∂αν) · (e f) ν→∞ ν→∞ U U ¯ −ϕ ¯ −ϕ = lim A(aν) · e f · ω = lim A(aν ) · e fη· dx ∧ dy ν→∞ ν→∞ U U ¯∗ −ϕ = lim aν · A (e fη)· dx ∧ dy ν→∞ U = a · A¯∗(e−ϕfη)· dx ∧ dy U = b · e−ϕfη· dx ∧ dy = βfe¯ −ϕ. U U Conversely, suppose α ∧ D f · e−ϕ = βfe¯ −ϕ ∀f ∈ D(U). U U Given u ∈ D(U), setting f = eϕu/η, we get a · A¯∗u · dx ∧ dy = a · A¯∗(e−ϕfη)· dx ∧ dy U U ¯∗ −ϕ = lim aν · A (e fη)· dx ∧ dy ν→∞ U −ϕ −ϕ ¯ −ϕ = lim αν ∧ D f · e = α ∧ D f · e = βfe ν→∞ U U U = b · e−ϕfη· dx ∧ dy = b ·¯u · dx ∧ dy. U U ¯ ¯ Hence Adistra = b and therefore ∂distrα = β. Thus (a) is proved.

Remark The main point of the above proof is that the characterization holds for C∞ C∞ 1 forms, and the forms are dense in Lloc. One may also prove Proposition 6.4.2 more directly without density of the C∞ forms (see Exercise 6.4.2).

~~The proof of Proposition 2.8.2 (with Proposition 6.4.2 in place of Proposi- tion 2.7.3) gives the following identical fundamental estimate: 328 6 Holomorphic Structures on Topological Surfaces~~

Proposition 6.4.3 (Fundamental estimate) For all functions u, v ∈ D(X), we have = + ¯ −ϕ D u, D v L2(X,ϕ) D u, D v L2(X,ϕ) iϕuve . X In particular, 2 = 2 + | |2 −ϕ ≥ | |2 −ϕ D u L2(X,ϕ) D u L2(X,ϕ) iϕ u e iϕ u e . X X

~~Remark If iϕ ≥ 0 in the above, then we get~~

~~ 2 = 2 + 2 D u, D v L (X,ϕ) D u, D v L (X,ϕ) u, v L (X,iϕ ,ϕ) and~~

~~ 2 = 2 + 2 ≥ 2 D u 2 D u 2 u 2 u 2 . L (X,ϕ) L (X,ϕ) L (X,iϕ,ϕ) L (X,iϕ ,ϕ)~~

Exercises for Sect. 6.4 6.4.1 Prove part (b) of Proposition 6.4.2. 6.4.2 Prove part (a) of Proposition 6.4.2 by direct computation without using the density of the space of C∞ forms. Hint. Apply the formulas for η and τ appearing in Exercise 6.3.1.

~~6.5 L2 Solutions on an Almost Complex Surface~~

Carrying over the notation of Sects. 6.3 and 6.4, we get the following analogue of Theorem 2.9.1, which gives local existence of solutions of the Cauchy–Riemann equation on an almost complex surface along with L2 estimates. The proof, which is left to the reader (see Exercise 6.5.1), is identical to the proof of Theorem 2.9.1, except that one must use Proposition 6.4.2 in place of Proposition 2.7.3, Propo- sition 6.4.3 in place of Proposition 2.8.2, and Proposition 6.3.2 in place of Theo- rem 2.7.4.

Theorem 6.5.1 Assume that i = iϕ ≥ 0, and let Z ={x ∈ X | x = 0}. Then, = ∈ for every measurable (1, 1)-form β on X with β 0 a.e. in Z and β X\Z 2 \ ∈ 2 L1,1(X Z,i,ϕ), there exists a form α L1,0(X, ϕ) such that = ¯ = ≤ Ddistrα ∂distrα β and α L2(X,ϕ) β L2(X\Z,i,ϕ).

~~In particular, if β is of class C∞, then α is also of class C∞ and ∂α¯ = β.~~

~~The proof of the following corollary is also left to the reader (see Exercise 6.5.2): 6.6 Proof of Integrability 329~~

Corollary 6.5.2 (Cf. Corollary 2.9.3) Suppose that C is a positive constant with ≥ 2 ∈ 2 ∈ iϕ C ω on X. Then, for every β L1,1(X,ω,ϕ), there exists a form α 2 L1,0(X, ϕ) such that = ¯ = ≤ −1 Ddistrα ∂distrα β and α L2(X,ϕ) C β L2(X,ω,ϕ).

~~In particular, if β is of class C∞, then α is also of class C∞ and ∂α¯ = β.~~

~~Exercises for Sect. 6.5 6.5.1 Prove Theorem 6.5.1. 6.5.2 Prove Corollary 6.5.2.~~

~~6.6 Proof of Integrability~~

~~The integrability theorem (Theorem 6.1.4) follows from Lemma 6.1.11 together with the following:~~

~~Lemma 6.6.1 In the notation of Sects. 6.3Ð6.5, there exists a nonvanishing C∞ ∂¯-closed differential form of type (1, 0) on a neighborhood of p = 0 in X ⊂ C.~~

Proof Let ϕ(z) =|z|2 + log |z|2 for each point z ∈ C∗, and for each >0, let 2 2 ϕ(z) =|z| + log(|z| + ) ∀z ∈ C. ∗ Then ϕ ≥ ϕ on C for each >0. On X (with the choices and notation given in Lemma 6.3.1), we have ∂¯|z|2 = z∂z+¯z∂z¯ = (z +¯zμ) · ∂z, and hence i i ∂|z|2 ∧ ∂¯|z|2 = |z +¯zμ|2∂z∧ ∂z =|z +¯zμ|2 · ω 2 2 and i i i i i ∂∂¯|z|2 = ∂z∧ ∂z+ z∂∂z+ ∂z¯ ∧ ∂z¯ + z∂¯ ∂z¯ 2 2 2 2 2 = (1 + zτ¯ +|μ|2 +¯zτ) · ω.

Thus, for each >0, we have ∂∂¯|z|2 ∂|z|2 ∧ ∂¯|z|2 i = i∂∂ϕ¯ = i∂∂¯|z|2 + i − i ϕ (|z|2 + ) (|z|2 + )2 1 + 2Re(zτ)¯ +|μ|2 |z +¯zμ|2 = 2 · (1 + 2Re(zτ)¯ +|μ|2) + − · ω |z|2 + (|z|2 + )2

~~= 2 · (1 + ρ ) · ω, 330 6 Holomorphic Structures on Topological Surfaces where (using |z +¯zμ|2 =|z|2(1 +|μ|2) + 2Re(z2μ)¯ )~~

(|z|2 + )(1 + 2Re(zτ)¯ +|μ|2) −|z +¯zμ|2 ρ ≡ 2Re(zτ)¯ +|μ|2 + (|z|2 + )2 (1 +|μ|2) + (|z|2 + )2Re(zτ)¯ − 2Re(z2μ)¯ = 2Re(zτ)¯ +|μ|2 + . (|z|2 + )2 Since μ and τ have vanishing derivatives of orders ≤ 2 and ≤ 1, respectively, at 0, after replacing X with a relatively compact neighborhood of 0, we may assume that there exists a constant C>0 such that |μ|≤C|z|3 and |τ|≤C|z|2 on X (see Exercise 9.5.10). Therefore

|z|3 |z|5 ρ ≥−2C|z|3 − 2C − 2C ≥−2C(|z|3 + 2|z|). |z|2 + (|z|2 + )2 Replacing X again with a sufﬁciently small neighborhood of 0, we see that we may ≥− ≥ assume without loss of generality that ρ 1/2, and hence iϕ ω on X,for every >0. Finally, recall that we have the positive C∞ function η = ω/(dx ∧ dy), which we may assume to be bounded above and below by positive constants. Now let β be the C∞ form of type (1, 1) given by

~~β ≡ ∂∂z¯ = 2iτ · ω = 2iτ · η · dx ∧ dy.~~

For every >0, we have − − −| |2 2 = | |2 ϕ ≤ | |2 ϕ ≤ 2| |2 z β L2(X,ω,ϕ ) 4 τ e ω 4 τ e ηdλ 4C z e ηdλ. X X X Replacing X again with a sufﬁciently small neighborhood of 0, we may assume ≤ without loss of generality that β L2(X,ω,ϕ ) 1 for every >0. ∞ According to Corollary 6.5.2, for every >0, there exists a C form α of type (1, 0) on X such that ¯ = 2 ≤ 2 ≤ ∀ ∂α β and α L (X,ϕδ ) α L (X,ϕ ) 1 δ>.

Fix a sequence of positive numbers {δν} with δν 0. Applying weak sequential compactness in a Hilbert space (see Theorem 7.6.1) and Cantor’s diagonal process, we get a sequence of positive numbers {ν} such that ν 0 and such that for each 2 μ ∈ Z , there is a form θ ∈ L (X, ϕ ) with θ 2 ≤ 1 and >0 μ 1,0 δμ μ L (X,ϕδμ )

~~2 γ,α 2 →γ,θ 2 as ν →∞ ∀γ ∈ L (X, ϕ ) ν L (X,ϕδμ ) μ L (X,ϕδμ ) 1,0 δμ~~

(for each μ,wehave,forν 0, <δ and hence α 2 ≤ 1). Since ν μ ν L (X,ϕδμ ) R2 2 X , the set of (equivalence classes of) forms in L1,0(X, ϕδ) is actually equal 2 to L1,0(X) for each δ>0 (although the inner products differ). In particular, given ∈ 2 = ϕδμ · ∈ Z aformγ0 L1,0(X), the above (with γ e γ0) shows that for each μ >0, 6.6 Proof of Integrability 331

→ we have γ0,αν L2(X) γ0,θμ L2(X). Thus in fact, we have a single form α such = ¯ = = that α θμ for every μ. In particular, we have ∂distrα Ddistrθμ β (for example, by Proposition 6.4.2), and by Fatou’s lemma, we have 2 1 iα ∧¯αe−|z| = iα ∧¯αe−ϕ ≤ . 2 1 X |z| X

The regularity property (Proposition 6.3.2) implies that α is of class C∞, and ﬁnite- ness of the above integral implies that α = 0 at the point p = 0. Thus γ ≡ ∂z − α is a ∂¯-closed C∞ form of type (1, 0) on X, and at 0, we have γ = dz− 0 = 0. Thus the lemma is proved (and integrability follows).

Exercises for Sect. 6.6 6.6.1 The integrability theorem (together with Sard’s theorem) may be used to construct a proof of the Koebe uniformization theorem (Theorem 5.5.3) that differs slightly from the proof appearing in Chap. 5. (a) Let = ∅ be a C∞ relatively compact domain in an oriented smooth surface M. By applying Theorem 9.10.1 and Lemma 5.11.1, prove (directly, without using Proposition 5.4.1) that there are an oriented compact C∞ surface M , a ﬁnite (possibly empty) collection of disjoint positively ori- C∞ { ; }n ented local charts (Dj ,j ,(0 Rj )) j=1 in M with Rj > 1for each j, and an orientation-preserving diffeomorphism : → ( )

of a connected relatively compact neighborhood of in M onto a = \ n −1 ; domain ( ) in M such that () M j=1 j ((0 1)). Prove also that M is planar if and only if is planar. (b) For the objects in part (a), prove that for any almost complex structure (T M)C = (T M)1,0 ⊕ (T M)0,1 inducing the given orientation in M,

~~there exists an almost complex structure (T M )C = (T M )1,0 ⊕(T M )0,1 in M such that~~

~~1,0 1,0 0,1 0,1 ∗(TpM) = (T(p)M ) and ∗(TpM) = (T(p)M )~~

for every point p in some neighborhood of in . (c) According to Sard’s theorem (see, for example, [Mi]), the set of critical values of a C∞ function on a second countable smooth manifold is a set of measure zero. Using this fact together with Proposition 9.3.11 and The- orem 9.9.2, prove that each compact subset of a second countable C∞ surface M lies in some C∞ relatively compact domain in M. (d) Let K be a compact subset of a Riemann surface X. By applying parts (a)–(c) above together with Theorem 6.1.4, prove that there exist a C∞ domain with K ⊂ X and a biholomorphism : → ( ) of a connected relatively compact neighborhood of in X onto a domain ( ) in a compact Riemann surface X . Prove also that if X is planar, then one may choose X to be planar. 332 6 Holomorphic Structures on Topological Surfaces

(e) Give a proof of the Koebe uniformization theorem that is analogous to the proof appearing in Sect. 5.5 but that uses parts (a)–(d) above in place of Proposition 5.4.1. Exercises 6.6.2–6.6.4 below require the discussion of homology and cohomology appearing in Sects. 10.6 and 10.7, as well as some of the facts and exercises from Chap. 5. ∞ 6.6.2 Prove that if M is a compact oriented C surface with H1(M, R) = 0, then M is diffeomorphic to a sphere. 6.6.3 Prove that if M is a second countable noncompact oriented C∞ surface with 2 H1(M, R) = 0, then M is diffeomorphic to R . 6.6.4 Let M be a second countable C∞ surface. (a) Prove that the universal cover of M is diffeomorphic to the plane R2 or the sphere S2. Hint. Apply integrability, uniformization, and Exercise 6.6.8. (b) Prove that if M is orientable, then π1(M) is torsion-free and ∼ H1(M, Z) = π1(M) [π1(M), π1(M)], while if M is nonorientable, then every torsion element of π (M) has ∼ 1 order 1 or 2 and π1(M)/ = Z/2Z for some torsion-free normal subgroup of π1(M). Hint. Apply Exercise 5.9.4 and Exercise 5.17.4. (c) Let A be a subﬁeld of C containing Z, and let F = R or C. Prove that if M A =∼ A is orientable, then H1 (M, ) H1(M, ), while if M is nonorientable, F =∼ F then H1 (M, ) H1(M, ). (d) Prove that if M is orientable, then for any subring A of C containing Z, the mapping [ ] A → [ ] A · ξ H1(M, ) ξ H1(M, ), deR 1 gives an injective homomorphism H1(M, A) → Hom(H (M, A), A) (according to Theorem 10.7.18, this homomorphism is also surjective if π1(X) is ﬁnitely generated). (e) Assume that M is orientable, and let K ⊂ M be a compact set. Prove that there exist domains and such that K ⊂ M and such that for every C∞ closed 1-form ρ on , there exist a C∞ closed 1-form τ on M and a function λ ∈ D( ) such that ρ − τ = dλ on . Conclude from this that [ 1 → 1 ]= [ 1 → 1 ] im HdeR(M) HdeR() im HdeR( ) HdeR() . Hint. See Exercise 5.17.5.

~~6.7 Statement of Schönﬂies’ Theorem~~

The main goal of the rest of this chapter is the fact that every second countable topological surface admits a C∞ structure. For the proof, one forms a C0 atlas and 6.7 Statement of Schönﬂies’ Theorem 333

Fig. 6.1 The homeomorphism provided by Schönflies’ theorem then inductively modifies the local charts on the overlaps to get C∞ compatibility. For the modification, one applies the following (see Fig. 6.1):

Theorem 6.7.1 (Schönﬂies’ theorem) Let γ : S1 → P1 be a Jordan curve with image C ≡ γ(S1) in the Riemann sphere P1. Then there exists a homeomorphism 1 1 1 1 1 : P → P such that S1 = γ and such that P1\S1 : P \ S → P \ C is a diffeomorphism.

~~In particular, we have the following:~~

Corollary 6.7.2 (Jordan curve theorem) If γ : S1 → P1 is a Jordan curve with image C ≡ γ(S1), then P1 \ C has exactly two connected components. Moreover, the boundary of each of these connected components is equal to C.

Remark One may choose the homeomorphism in Schönﬂies’ theorem to map the unit disk (0; 1) onto either of the connected components of P1 \ C by replacing with the homeomorphism z → (z/|z|2) = (1/z)¯ (0 → (∞), ∞→(0))if necessary.

A proof of Schönﬂies’ theorem, which is due to Kneser and Radó [KnR], appears in Sect. 6.9 (a different approach, as well as a more complete treatment of triangulations and the classiﬁcation of surfaces, can be found in [T]). In fact, their proof allows one to choose the diffeomorphism so that the restriction of −1 to one of the connected components of P1 \ C, and the restriction of 1/−1 to the other connected component, are harmonic. For parts of the proof, we will apply the Riemann mapping theorem in the plane (Theorem 5.2.1), although it is not really necessary. For now, we consider a preliminary observation, namely, that a Jordan curve in P1 that is piecewise smooth away from some point on the curve is separating.

Lemma 6.7.3 Let α :[a,b]→P1 be a Jordan curve in P1 such that for some point ∈ ∈ c (a, b), α [a,r] and α [s,b] are piecewise smooth paths for all r (a, c) and s ∈ (c, b). Then the image C ≡ α([a,b]) is separating in P1.

Proof We have C = P1 (for example, C is not simply connected), so by applying a suitable Möbius transformation, we may assume without loss of generality that ∞ ∈/ C. By choosing a smooth point u ∈ (a, b) and replacing α with the path given by t → α(t −a +u) for t ∈[a,b+a −u] and t → α(t −b +u) for t ∈[b +a −u, b] 334 6 Holomorphic Structures on Topological Surfaces

(and modifying c accordingly), we may assume that α is loop-smooth at a (see Deﬁnition 5.10.2). We may also ﬁx r0,r1 ∈ R such that a0, β((0,))⊂ U0 and β((1 − ,1)) ⊂ U1. Consequently, we may choose R>0 so small that for D ≡ (α(c); R),wehaveD P1 \ (U ∪ β([0, 1]) ∪ α([a,r1])), and setting

~~r ≡ min{t ∈[a,c]|α(t) ∈ D} and s ≡ max{t ∈[c,b]|α(t) ∈ D},~~

~~1 we get r1~~

Exercises for Sect. 6.7 The exercises for this section require Schönflies’ theorem (and its consequences), so the reader may wish to postpone consideration of these exercises until after consideration of the proof in Sect. 6.9. 6.7.1 Let γ :[0, 1]→C be a (continuous) Jordan curve in C with interior (i.e., is the unique bounded connected component of C \ C provided by Corol- lary 6.7.2), and let C = γ([0, 1]). Assume that γ is oriented so that is on 1 the left; that is, for the corresponding homeomorphism γ0 : S → C given by e2πit → γ(t), after choosing the homeomorphism : P1 → P1 with S1 = γ0 provided by Schönflies’ theorem (Theorem 6.7.1) to map the unit disk ≡ (0; 1) diffeomorphically onto (see the remark following the statement of Corollary 6.7.2), γ and γ0 are directed so that this diffeomorphism is orientation-preserving (to ensure this, one simply replaces γ with the reverse curve and with the mapping z → (z)¯ if necessary). (a) Prove that γ is homotopic in to a trivial loop. (b) Prove the following version of Cauchy’s theorem (cf. Lemma 1.2.1 and Exercises 5.1.6 and 5.1.7): If f is a holomorphic function on a neighbor- = hood of , then γ f(z)dz 0. (c) Prove the following version of the Cauchy integral formula (cf. Lem- ma 1.2.1 and Exercises 5.1.6 and 5.1.7): If f is a holomorphic function 6.7 Statement of Schönflies’ Theorem 335

~~on a neighborhood of and z0 ∈ , then 1 f(z) f(z0) = dz. 2πi γ z − z0~~

(d) Prove the following version of the residue theorem (cf. Exercises 2.5.9, 5.1.6, 5.1.7, 5.9.5, and 6.7.6): If S is a ﬁnite subset of and θ is a holomorphic 1-form on \ S for some neighborhood of , then 1 θ = respθ. 2πi γ p∈S

(e) Prove the following version of the argument principle (cf. Exercises 2.5.8, 5.1.6, 5.1.7, 5.9.5, and 6.7.6): If f is a nontrivial meromorphic function on a neighborhood of with no zeros or poles in ∂, and counting multiplicities, μ is the number of zeros of f in and ν is the number of poles in , then 1 df μ − ν = . 2πi γ f (f) Prove the following version of Rouché’s theorem (cf. Exercises 2.5.8, 5.1.6, 5.1.7, 5.9.5, and 6.7.6): If f and g are nontrivial meromorphic functions on a neighborhood of that do not have any zeros or poles in ∂, and |g| < |f | on ∂, then μf − νf = μf +g − νf +g, where counting multiplicities, μf is the number of zeros of f in , νf is the number of poles of f in , μf +g is the number of zeros of f + g in , and νf +g is the number of poles of f + g in . 6.7.2 Let M be a topological surface. (a) Let α :[0, 1]→M be an injective path with image A ≡ α([0, 1]). Prove 0 that for every t0 ∈ (0, 1), there exist numbers a,b and a local C chart

~~(U,,U = (−1, 1)×(a, b)) in M such that 0 ≤ a~~

6.7.4 Prove that for any nonempty relatively compact domain M in a topological surface M, the following are equivalent: (i) Each point in ∂ admits a local coordinate neighborhood (U, = (x, y)) in M such that ∩ U ={q ∈ U | x(q) < 0}. (ii) There exist disjoint Jordan curves C1,...,Cn such that ∂ = C1 ∪··· ∪ Cn and such that for each j = 1,...,n, Cj is separating in the connected component of M \ (C1 ∪···∪Cj ∪···Cn) containing Cj . (iii) There exist Jordan curves γj :[0, 1]→M, with corresponding continuous 1-periodic extensions γ˜j : R → M,forj = 1,...,n, such that the images Cj ≡ γj ([0, 1]) for j = 1,...,nare disjoint; ∂ = C1 ∪···∪Cn; and for each index j = 1,...,n and for each point t0 ∈ R, there is a local coordinate neighborhood (U, = (x, y), U = (−1, 1) × (a, b)) ∩ ={ ∈ | } ∈ ⊂˜−1 in M with U q U x(q) < 0 , t0 (a, b) γj (U), and (γ˜j (t)) = (0,t)for every t ∈ (a, b). Prove also that if M is a smooth surface, then each of the local charts (U, ) in (i) and (iii) may be chosen so that the restriction U\∂ is a diffeomorphism of U \ ∂ onto its image. Hint. Apply Exercise 6.7.2 and Theorem 9.10.1. 6.7.5 In this exercise, we consider some facts concerning orientation preservation. (a) Let M be a C∞ surface, let ⊂ M be a C∞ open set, let p ∈ ∂, and let (U, = (x, y), U ) and (V, = (u, v), V ) be local C∞ charts in M with p ∈ U ∩ V ∩ ∂ and ∩ U ={p ∈ U | x(p) < 0} and ∩ V ={p ∈ V | u(p) < 0}. Assume that and induce the same orientation in ∂ at p; that is, for the inclusion mapping ι: U ∩ V ∩ ∂ → U ∩ V ,wehave ∗ ∗ (ι dv)p/(ι dy)p > 0. Prove that and induce the same orientation in some neighborhood of p in U ∩ V (and hence in the connected component of U ∩ V containing p). (b) Let = (u, v): U → V be a homeomorphism of neighborhoods U and V of (0, 0) in R2 such that (i) (0, 0) = (0, 0); (ii) U ∩ ((−∞, 0) × R) ={(x, y) ∈ U | u(x,y) < 0}; \ { }×R \ (iii) U\({0}×R) maps U ( 0 ) diffeomorphically onto V ({0}×R); and (iv) The function t → v(0,t)is increasing. Prove that there exists a neighborhood W of (0, 0) in U for which the restriction W\({0}×R) is orientation-preserving. Hint. Produce a smooth Jordan curve C bounding a smooth domain U ∩ ((−∞, 0) × R) such that part of C is a directed line segment with lying to the left, and another part is the inverse image under of a directed line segment with () lying to the left. Then apply part (a). (c) Let M be an oriented smooth surface, and let (U, = (x, y), U = (−1, 1) × (a, b)) be a local C0 chart in M for which the restriction ≡ : −1 \ { }× → \ { }× 0 −1(U \({0}×(a,b))) (U ( 0 (a, b))) U ( 0 (a, b)) 6.7 Statement of Schönﬂies’ Theorem 337

is a diffeomorphism. Prove that 0 must be either orientation-preserving or orientation-reversing. Hint. Show that it suffices to consider the case M C ⊂ P1.Fix a suitable disk D U that meets the y-axis. Applying Schönflies’ theorem, one may form a homeomorphism : P1 → P1 that agrees with −1 on ∂D and that maps D diffeomorphically onto its image = −1 (D) (D) U. In particular, D is orientation-preserving or -reversing. Now apply part (b) to the homeomorphism 0 ◦ near points in (∂D) \ ({0}×R). (d) Prove that for any Jordan curve γ : S1 → P1 with image C, the restriction 1 1 1 P1\S1 : P \ S → P \ C of the homeomorphism provided by Schön- flies’ theorem must be either an orientation-preserving or an orientation- reversing diffeomorphism (in particular, by replacing with the homeomorphism z → (z/|z|2) = (1/z)¯ , one may always choose the homeomorphism so that this restriction is orientation-preserving). (e) Let M be an oriented smooth surface, and let α :[0, 1]→M be an injective path with image A ≡ α([0, 1]). Prove that for every t0 ∈ (0, 1), there exist numbers a,b and a local C0 chart (U,,U = (−1, 1) × (a, b)) in M such that 0 ≤ a

~~(a) Residue theorem. Prove that if S is a ﬁnite subset of and θ is a holomorphic 1-form on X, then~~

~~n 1 θ = res θ. 2πi p j=1 γj p∈S~~

(b) Argument principle. Prove that if f is a nontrivial meromorphic function on X with no zeros or poles in ∂, and counting multiplicities, μ is the number of zeros of f in and ν is the number of poles in , then

~~n 1 df μ − ν = . 2πi f j=1 γj~~

(c) Rouché’s theorem. Prove that if f and g are nontrivial meromorphic functions on X that do not have any zeros or poles in ∂, and |g| < |f | on ∂, then μf − νf = μf +g − νf +g, where counting multiplicities, μf is the number of zeros of f in , νf is the number of poles of f in , μf +g is the number of zeros of f + g in , and νf +g is the number of poles of f + g in . 6.7.7 Prove that if M is an orientable C∞ surface and γ :[0, 1]→M is a Jordan curve with image C = γ([0, 1]), then for some R>1, there exists a neighborhood of C in M and a homeomorphism : → (0; 1/R, R) with (γ (t)) = e2πit for each t ∈[0, 1]. Hint. Apply Exercises 6.7.2 and 6.7.5. 6.7.8 State and prove an analogue of Lemma 5.15.1 for a continuous Jordan curve with nonseparating image in an orientable smooth surface M (see Exer- cise 6.7.7). Using this fact (and Exercise 6.7.5), prove that the image C of a = C∞ Jordan curve γ in M is separating if and only if γ θ 0 for every closed 1-form θ with compact support in M (a different proof of this analogue of Proposition 5.15.2 was suggested in Exercise 6.7.2).

~~6.8 Harmonic Functions and the Dirichlet Problem~~

Kneser and Radó’s proof of Schönﬂies’ theorem relies on the solution of the Dirich- let problem, but only for harmonic functions on simply connected domains in P1 with complement containing at least two points. In this section, we consider an approach to the Dirichlet problem that is an ad hoc version of the important Perron method. In particular, by restricting our attention to simply connected domains, we are able to avoid some work by applying the Riemann mapping theorem in the plane. However, it should be noted that the general Perron method can be developed entirely from elementary results, and it leads to a more complete solution. Moreover, most of the results concerning open Riemann surfaces appearing in this book can be obtained by using the Perron method in place of the L2 ∂¯-method (see, for example, [AhS]or[For]). 6.8 Harmonic Functions and the Dirichlet Problem 339

A complex-valued C2 function u on a complex 1-manifold X is harmonic if ∂∂u¯ = 0 (cf. Deﬁnition 2.8.1). A real-valued C2 function ρ on X is subharmonic (strictly subharmonic)ifi∂∂ρ¯ ≥ 0 (respectively, i∂∂ρ¯ >0).

Lemma 6.8.1 Let X be a Riemann surface. (a) If u is a real-valued harmonic function on X and X is simply connected, then u = Re f for some holomorphic function f on X. In particular, the real-valued harmonic functions on a Riemann surface are precisely the functions that are locally equal to the real part of a local holomorphic function. (b) Strong maximum principle for harmonic functions. If u is a real-valued harmonic function on X that attains a local maximum or local minimum value, then u is constant. (c) If {un} is a sequence of real-valued harmonic functions on X that is uniformly { } bounded on each compact subset of X, then some subsequence unk converges uniformly on compact subsets of X to a harmonic function on X. (d) Weak maximum principle for subharmonic functions. If X is a relatively compact domain in X and ϕ is a continuous real-valued function on that is C2 = and subharmonic on , then max ϕ max∂ ϕ.

Proof If u is a real-valued harmonic function on X, then θ ≡ ∂u is a (closed) holomorphic 1-form, since ∂θ¯ =−∂∂u¯ = 0. Therefore, if X is simply connected, then by Corollary 10.5.7,2θ = df for some C∞ function f on X. Since θ is of type (1, 0), we have ∂f¯ = 0, so f is holomorphic. On the other hand, 1 ∂(u− Re f)= θ − df = 0 and ∂(u¯ − Re f)= ∂(u− Re f)= 0, 2 so u − Re f is constant. Choosing f to agree with u at some point, we get u = Re f on X in this case, and (a) follows. For a real-valued harmonic function u on X, part (a) implies that eu is locally the modulus of a holomorphic function (for u = Re f ,wehaveeu =|ef |). Therefore, if u attains a local maximum value of c at some point, then the maximum principle for holomorphic functions (Theorem 1.3.4) implies that u ≡ c in a neighborhood of the point. Hence the interior of u−1(c) is nonempty. Corollary 1.3.3 implies that u ≡ c near any point in . Hence is both open and closed in X, and therefore = X. Suppose {un} is a sequence of real-valued harmonic functions on X that is uniformly bounded on each compact subset of X. Each point has a neighborhood D X such that D is biholomorphic to a disk and for some constant R = R(D) > 0, |un|≤R on D for each n ∈ Z>0. In particular, by part (a), for each n,wehave = ∈ O ≡ fn un D Re fn for some function fn (D). The holomorphic function Fn e −R u R then satisﬁes 0

Fig. 6.2 Local representation of the zero set of a harmonic function monic function log |F |. Covering X by a locally finite collection of such coordinate disks D and applying Cantor’s diagonal process, we get the claim (c). Finally, for the proof of (d), assuming that max ϕ>max∂ ϕ, we reason to a contradiction. By replacing with a sufficiently large relatively compact open subset of , we may assume without loss of generality that = X. Fixing a point p ∈ X \ and applying Corollary 2.14.2, we get a positive C∞ strictly subharmonic function ρ on X \{p}.For>0 sufficiently small and for ψ ≡ ϕ + ρ,we have max ψ>max∂ ψ; and hence ψ attains its maximum value at some interior point q ∈ . On the other hand, for any local holomorphic coordinate neighborhood (U, z = x + iy) of q in ,wehave ∂2ψ ∂2ψ (q) + (q) > 0; ∂x2 ∂y2 so we have arrived at a contradiction.

The following local description of the zero set of a harmonic function is an easy consequence of Lemma 6.8.1 and the local description of holomorphic mappings given by Lemma 2.2.3. The proof is left to the reader (see Exercise 6.8.2).

Lemma 6.8.2 Let Z ={p ∈ X | u(p) = 0} be the zero set of a nonconstant real- valued harmonic function u on a Riemann surface X. Then, for each point p ∈ Z, there exist a constant Rp > 0, a positive integer mp, and a local holomorphic chart = ; = −1 = mp (Up,p zp,(0 Rp)) such that p p (0) and u Im(zp ) on Up. In particular, ∩ = ∪···∪ ∩ ; (Z Up) (L0 Lmp−1) (0 Rp), ij π/m where Lj ≡{re p | r ∈ R} for each j = 0,...,mp − 1. Moreover, mp = 1 if and only if (du)p = 0, and the set of points p ∈ Z at which mp > 1 is discrete in X.

~~Remark The case mp = 6 is pictured in Fig. 6.2. Clearly, the zero set Z in the above lemma is closed. The above characterization implies that in particular, Z is a locally ﬁnite graph.~~

For a nonempty open subset of a Riemann surface X and a continuous function ρ : ∂ → C, the associated (classical) Dirichlet problem is that of ﬁnding a 6.8 Harmonic Functions and the Dirichlet Problem 341

: → C = continuous function u that is harmonic on and that satisfies u ∂ ρ. This problem is not always solvable. For example, let ≡ (0; 1),let ≡ \{0}, and suppose u is a continuous function on = that is harmonic on and that satisfies u(0) = 1 and u ≡ 0on∂. By replacing u with Re u, we may assume that u is real-valued. The maximum principle (Lemma 6.8.1) implies that 0 0 sufficiently small, the harmonic function v : z → u(z)+ log |z|2 on , which approaches −∞ at 0 and which vanishes at ∂, will have some positive values and therefore will attain a local maximum at some point in . This contradicts the maximum principle, so no such solution u of the given Dirichlet problem can exist. For a nonempty relatively compact domain in a Riemann surface and a continuous function on ∂, the maximum principle implies that the associated Dirichlet problem has at most one solution. For if u and v are solutions, then w ≡ u − v is a continuous function on that vanishes on ∂ and that is harmonic on . Hence, by the maximum principle, the maximum and minimum values of Re w and Im w must be 0, and therefore w ≡ 0. By applying the Perron method, one can prove that the Dirichlet problem is solvable on any relatively compact domain in a Riemann surface for which each boundary component contains more than one point (see, for example, [AhS]). For our purposes, the following will suffice:

~~Proposition 6.8.3 The Dirichlet problem is solvable on each connected component of the complement P1 \ K of any connected closed set K P1.~~

~~We now work toward the proof. We ﬁrst consider the (Poisson)formulaforthe solution on a disk. As motivation, we consider the following consequence of the Cauchy integral formula:~~

Lemma 6.8.4 (Mean value property) If R>0 and u: (0; R) → R is a continuous function that is harmonic on (0; R), then 1 2π R2 −|z|2 1 ζ + z dζ = u(Reiθ) · dθ = u(ζ ) · u(z) iθ 2 Re 2π 0 |Re − z| 2πi ∂(0;R) ζ − z ζ for each point z ∈ (0; R). In particular, we have the mean value property 1 2π 1 u(ζ ) u(0) = u(Reiθ)dθ = dζ. 2π 0 2πi ∂(0;R) ζ = ∈ O ; Proof By Lemma 6.8.1, u (0;R) Re f for some function f ((0 R)).Let v ≡ Im f . For each r ∈ (0,R), the Cauchy integral formula (Lemma 1.2.1)gives 1 f(ζ) 1 u(ζ ) 1 v(ζ) f(0) = dζ = dζ + dζ 2πi ∂(0;r) ζ 2πi ∂(0;r) ζ 2π ∂(0;r) ζ 1 2π i 2π = u(reiθ)dθ + v(reiθ)dθ. 2π 0 2π 0 342 6 Holomorphic Structures on Topological Surfaces

Comparing real parts, we see that 1 u(ζ ) 1 2π u(0) = dζ = u(reiθ)dθ, 2πi ∂(0;r) ζ 2π 0 and letting r → R− (and observing that u(reiθ) → u(Reiθ) uniformly), we get 1 u(ζ ) 1 2π u(0) = dζ = u(Reiθ)dθ. 2πi ∂(0;R) ζ 2π 0 Given a point z ∈ (0; R), Proposition 2.14.8 provides the automorphism (i.e., the Möbius transformation) ζ − (z/R) : ζ → 0 1 − (z/R)ζ¯ of P1, which maps (0; 1) onto itself. Setting

R2ζ − R2z (ζ) ≡ R (ζ/R) = ∀ζ ∈ P1, 0 R2 −¯zζ we get an automorphism ∈ Aut (P1) that maps (0; R) onto itself. Applying the above mean value property at 0 to the harmonic function u(−1), we get −1 − 1 u( (ξ)) u(z) = u ◦ 1(0) = dξ 2πi ∂(0;R) ξ 1 u(ζ ) = · (ζ ) dζ 2πi ∂(0;R) (ζ) 1 R2 −|z|2 = u(ζ ) · dζ 2 2πi ∂(0;R) (ζ − z)(R −¯zζ) 1 2π R2 −|z|2 = u(Reiθ) · Reiθ dθ iθ 2 iθ 2π 0 (Re − z)(R −¯zRe ) 2π 2 −| |2 = 1 iθ · R z u(Re ) iθ −iθ dθ 2π 0 (Re − z)(Re −¯z) 1 2π R2 −|z|2 = u(Reiθ) · dθ. iθ 2 2π 0 |Re − z| Combining the above with the equality |ζ |2 −|z|2 ζ + z = Re , |ζ − z|2 ζ − z we get the desired formula. 6.8 Harmonic Functions and the Dirichlet Problem 343

Lemma 6.8.5 (Poisson formula) Given a continuous function ρ : ∂(0; R) → C for some R>0, the function u: (0; R) → C given by ⎧ ⎨ 1 2π R2 −|z|2 ρ(Reiθ) · dθ if z ∈ (0; R), ≡ | iθ − |2 u(z) ⎩2π 0 Re z ρ(z) if z ∈ ∂(0; R), is the unique solution of the associated Dirichlet problem (that is, u is continuous ; ; = on (0 R), u is harmonic on (0 R), and u ∂(0;R) ρ).

~~Remark The function K: (ζ, z) → (R2 −|z|2)/(2πR|ζ − z|2) is called the Poisson = ∈ ; kernel.Wehaveu(z) ∂(0;R) ρ(ζ)K(ζ,z)ds(ζ) for z (0 R).~~

Proof of Lemma 6.8.5 Clearly, we may assume without loss of generality that ρ is |ζ |2−|z|2 = [ ζ +z ] real-valued. As in the proof of Lemma 6.8.4, the equality |ζ −z|2 Re ζ −z then implies that u (0;R) is equal to the real part of the function 1 ζ + z dζ z → ρ(ζ)· 2πi ∂(0;R) ζ − z ζ 1 ρ(ζ) z ρ(ζ)/ζ = dζ + dζ. 2πi ∂(0;R) ζ − z 2πi ∂(0;R) ζ − z

Moreover, by Lemma 1.2.2 (or by differentiation past the integral), the above function is holomorphic, and therefore u is harmonic on (0; R). It remains to show that u is continuous at each boundary point z0 ∈ ∂(0; R). Let λ denote the Lebesgue measure on R, and for each δ>0, let

~~iθ Nδ ≡{θ ∈[0, 2π]||Re − z0| < 2δ}.~~

For each δ>0 and each z ∈ (0; R) ∩ (z0; δ), by applying Lemma 6.8.4 to the constant (and therefore harmonic) function ζ → 1, we get 1 2π R2 −|z|2 |u(z) − u(z )|= ρ(Reiθ) · dθ 0 iθ 2 2π 0 |Re − z| 1 2π R2 −|z|2 − ρ(z ) · dθ 0 iθ 2 2π 0 |Re − z| 1 R2 −|z|2 ≤ |ρ(Reiθ) − ρ(z )|· dλ(θ) 0 | iθ − |2 2π Nδ Re z 1 R2 −|z|2 + |ρ(Reiθ) − ρ(z )|· dλ(θ) 0 | iθ − |2 2π [0,2π]\Nδ Re z 344 6 Holomorphic Structures on Topological Surfaces 2π 2 −| |2 iτ 1 R z ≤ sup |ρ(Re ) − ρ(z0)|· dθ ∈ 2π |Reiθ − z|2 τ Nδ 0 2π − 1 + δ 2 · 2sup|ρ|· (R2 −|z|2)dθ ∂(0;R) 2π 0 −2 2 2 = sup |ρ − ρ(z0)|+δ · 2sup|ρ|·(R −|z| ). (z0;2δ)∩∂(0;R) ∂(0;R) Now, given >0, the continuity of ρ implies that we may choose δ>0sosmall that the ﬁrst term on the right-hand side is less than /2. For z ∈ (0; R)∩ (z0; δ) sufﬁciently close to z0, the second term will then also be less than /2. Thus u is continuous at z0.

~~The solution of the Dirichlet problem on more general domains requires the following fundamental tool from the Perron method:~~

~~Deﬁnition 6.8.6 Let be an open subset of a complex 1-manifold X, and let p ∈ ∂. A real-valued function β on is called a C∞ barrier at p on if β is a C∞ subharmonic function and~~

~~lim β(z) = 0 but lim sup β(z) < 0 ∀q ∈ (∂) \{p}. z→p z→q~~

~~Remark One may also deﬁne a notion of a continuous subharmonic function and consider continuous barriers.~~

For the construction of a barrier in our case, we will need some elementary geometric facts. We ﬁrst recall that any holomorphic function f on a neighborhood of a point z0 ∈ C with f (z0) = 0isconformal at z0.Thatis,ifα and β are two smooth paths in C with α(s0) = β(t0) = z0, then f preserves the angle measure

~~Re(α (s0)β (t0)) arccos |α (s0)|·|β (t0)| between the (tangent vectors of) the paths, because~~

~~2 2 (f ◦ α) (s0)(f ◦ β) (t0) = α (s0)β (t0)|f (z0)| and |f (z0)| > 0. We also have the following fact, the proof of which is left to the reader (see Exer- cise 6.8.3):~~

Lemma 6.8.7 Let C be a circle of radius R in the plane, let M bealineinthe plane that meets C in two points, let l be the length of the line segment given by the intersection of M with the closed disk bounded by C, and let d be the distance from the center of the circle to the line M. Then, at each of the two points of intersection, the two angles between M and C are given by l/2 l/2 θ = 2 · arctan ∈ (0,π/2] and θ = π − θ = 2 · arctan . 0 R + d 1 0 R − d 6.8 Harmonic Functions and the Dirichlet Problem 345

~~Fig. 6.3 The image of the domain under a branch of the logarithmic function~~

~~Lemma 6.8.8 If K is a connected closed subset of P1 containing more than one point, is a connected component of P1 \ K, and p ∈ ∂, then there exists a C∞ barrier β at p on .~~

Proof This proof is based on the arguments in [AhS](seealso[BerG]). By applying an automorphism of P1, we may assume that p =∞and that 0 ∈ K.By Lemma 5.17.1, is simply connected, and hence by Proposition 5.2.2, there exists a (single-valued) holomorphic function L on with eL(z) = z for each point z ∈ ⊂ C∗ (i.e., L is a single-valued branch of the logarithmic function). Setting u ≡ Re L and v ≡ Im L,wehaveu(z) = log |z| and v(z) is an argument of z for each point z ∈ . The function L maps biholomorphically onto a domain ⊂ C. Since u →−∞at 0 and +∞ at ∞, the intermediate value theorem implies that for each r ∈ R,wehave ∩ (r + iR) = r + iAr , where Ar is a nonempty open subset of R in which no two distinct points differ by a multiple of 2π. In particular, Ar is equal to the union of a (nonempty) countable collection of disjoint open intervals { } ={ } Irj j∈Jr (arj ,brj ) j∈Jr

(the connected components) of total length (i.e., Lebesgue measure) λ(Ar ) = (brj − arj ) ≤ 2π, j∈Jr with Jr = Z>0 or Jr ={1,...,mr } for some mr ∈ Z>0 (see Fig. 6.3). We will construct a subharmonic function γ on such that γ(ζ) is bounded above by a negative constant for ζ in the intersection with any half-plane {Re ξr} for which the value at any point ζ will be the product of −2/π and a sum of angle measures. Each of these angle measures will be for an angle formed by the vertical line Re ξ = r and, for some j ∈ Jr , the circle passing through ζ and the endpoints of the line segment r +iI rj .Asζ approaches any point in r +iIrj , the associated circle will approach the line r + iR and the angle measure will approach π. Thus the limit of the function at points in r + iAr will be at most −2. We will get the function γr on as an extension of the composition of a suitable nondecreasing convex function 346 6 Holomorphic Structures on Topological Surfaces and the above function. As Re ζ →∞, the above circles will grow large and the above angle measures will shrink to 0; and it will follow that γr (ζ ) → 0. To get a function γ that is bounded away from 0 on the intersection of with any left = −ν →∞ half-plane, we will then let γ 2 γrν for some sequence rν . We now consider the details of the construction. For each number r ∈ R and each 1 1 index j ∈ Jr , the Möbius transformation rj : P → P given by r + ia − ζ ζ → rj r + ibrj − ζ

1 (r + iarj → 0, r + ibrj → ∞, ∞→1) is an automorphism of P = C ∪{∞}that maps the segment r + iI rj onto {∞} ∪ (−∞, 0], the segment {∞} ∪ (r + i(R \ Irj )) onto [0, ∞) ∪{∞}(here, of course, for a ∈ R, (−∞,a]={x ∈ R | x ≤ a} and [a,∞) ={x ∈ R | x ≥ a}, while {∞} denotes the singleton consisting of the point 1 at inﬁnity in P ), and the half-plane Hr ={ζ ∈ C | Re ζ>r} onto the upper half- plane H ={ζ ∈ C | Im ζ>0}. We may also form a single-valued argument function that is the unique harmonic function α : C \ i(−∞, 0]→(−π/2, 3π/2) satisfying eiα(ξ) = ξ/|ξ| for each ξ ∈ C \ i(−∞, 0] (i.e., α is the imaginary part of a branch of the logarithmic function on the simply connected domain C \ i(−∞, 0]). Thus

2 − α ≡− α ◦ : 1(C \ i(−∞, 0]) → (−3, 1) rj π rj rj is a harmonic function, −2 <αrj < 0onHr , αrj ≡−2onr + iIrj , and αrj ≡ 0 on r + i(R \ I rj ). Moreover, the fact that a Möbius transformation maps circles in P1 to circles (Theorem 5.7.3), the fact that (local) biholomorphisms are conformal, Lemma 6.8.7, and the inequality 0 < arctan x0 together give the following estimate: 2 brj − arj 4 brj − arj − · < − arctan ≤ αrj (ζ ) < 0 ∀ζ ∈ Hr π Re ζ − r π 2(Re ζ − r)

(as pictured in Fig. 6.4,forζ ∈ Hr , the inverse image under rj of the ray M ≡ [0, ∞) · rj (ζ ) is an arc of a circle that meets the ray r + i(−∞,arj ] at the vertex + = −1 r iarj rj (0), thus determining an angle of the same measure as the angle formed by M and the nonnegative x-axis). The Weierstrass M-test and Lemma 6.8.1 together now imply that the (possibly ﬁnite) series α converges uniformly on compact subsets of H to a har- j∈Jr rj r monic function αr satisfying 2 brj − arj 2 λ(Ar ) 4 0 >αr (ζ ) > − · =− · ≥− ∀ζ ∈ Hr . π Re ζ − r π Re ζ − r Re ζ − r j∈Jr

Now, according to Lemma 2.10.2, we may ﬁx a C∞ function χ : R →[−1, ∞) such that χ ,χ ≥ 0onR, χ ≡−1on(−∞, −3/2], and χ(t)= t for each t ≥−1/2 (simply form the function as in the lemma with a =−3/2, b =−1, and c =−1/2, 6.8 Harmonic Functions and the Dirichlet Problem 347

Fig. 6.4 Harmonic functions associated to two components of ∪ (r + iR) and then subtract 1 to get χ). On the other hand, for each r ∈ R, each index j ∈ Jr , and each point ζ ∈ r + iIrj ⊂ r + iAr = ∩ ∂Hr ,wehaveαrj (ζ ) =−2 and hence

~~αr ≤ αrj < −3/2 at all points in Hr near ζ . It follows that the function γr : →[−1, 0) given by −1ifζ ∈ \ Hr , γr (ζ ) = χ(αr (ζ )) if ζ ∈ ∩ Hr ,~~

∞ is a C subharmonic function. Choosing a strictly increasing sequence of real num- { } →∞ ≡ ∞ −ν : →[− bers rν with rν , we see that the function γ ν=1 2 γrν 1, 0) is equal to the ﬁnite sum μ−1 − −μ+1 + −ν ≤− −μ+1 2 2 γrν 2 ν=1 \ on the open set H rμ for each μ. Since these open sets form an increasing sequence with union , it follows that γ is a C∞ subharmonic function on . Fur- thermore, γ is bounded above by a negative constant on the intersection of with any left half-plane {ζ ∈ C | Re ζ0, we may choose μ >0 so large that ν=μ+1 2 −μ 2 </2. Hence, if ζ ∈ with Re ζ>rμ, then μ μ − + −ν ≥− + −ν − 4 0 >γ(ζ)> 2 γrν (ζ ) 2 χ . 2 2 Re ζ − rν ν=1 ν=1

Since χ(0) = 0, we see that for each ζ ∈ with Re ζ rμ,wehave 0 >γ(ζ)>−. It follows that the function β ≡ γ(L): →[−1, 0) is a C∞ barrier at p =∞on (note that the function z → log |z|=Re L(z) is bounded above on each compact subset of C).

~~The last fact we will need for the proof of Proposition 6.8.3 is the following (for a much more general version, see, for example, [Mu]): 348 6 Holomorphic Structures on Topological Surfaces~~

Theorem 6.8.9 (Tietze extension theorem in C) If ρ : K → R is a continuous function on a nonempty compact subset K of C, then there exists a continuous function : C → R = = = ρ0 such that ρ0 K ρ,infC ρ0 minK ρ, and supC ρ0 maxK ρ.

Proof We may fix a number R>0 so that K ⊂ (0; R); a sequence {Dν}, where Dν = (ζν; rν) (0; 2R)\ K for each ν, the collection {Dν} is locally finite in C \ ; \ ⊂ K, and (0 R) K Dν ; a sequence of nonnegative continuous functions {λν} with supp λν ⊂ Dν for each ν, λν ≤ 1onC, and λν ≡ 1on(0; R) \ K; and for each ν, a point zν ∈ K with dist(zν,ζν) = dist(K, ζν)>rν . Clearly, the function α : C → R given by ρ on K, α ≡ ρ(zν ) · λν on C \ K, = is continuous on C \ K and satisfies αK ρ. We now show that α is continuous at each point z0 ∈ K.Given>0, we may choose δ1 > 0 so that

~~|α(z) − α(z0)|=|ρ(z)− ρ(z0)| < ∀z ∈ (z0; 3δ1) ∩ K.~~

~~By local ﬁniteness, there is an N ∈ Z>0 such that Dν lies in the δ1-neighborhood of K each ν>N. In particular, for each ν>N,wehave~~

~~rν < dist(K, ζν ) =|zν − ζν| <δ1.~~

We may also choose a number δ such that 0 <δ<δ1, (z0; δ) ⊂ (0; R), and (z0; δ)∩ Dν =∅for each ν = 1,...,N. Therefore, if z ∈ C \ K with |z − z0| <δ, then we have z ∈ (0; R), z/∈ Dν for each ν = 1,...,N, and for each ν>Nwith z ∈ Dν ,wehave

~~|zν − z0|≤|zν − ζν|+|ζν − z|+|z − z0| < 3δ1.~~

~~Therefore |α(z) − α(z0)|= λν (z) · (ρ(zν) − ρ(z0)) ≤ λν (z) ·|ρ(zν) − ρ(z0)| <.~~

Thus the extension α is continuous at z0. Hence the function ρ0 : C → R given by ⎧ ⎨⎪α(z) if minK ρ ≤ α(z) ≤ maxK ρ, z → min ρ if α(z) < min ρ, ⎩⎪ K K maxK ρ if α(z) > maxK ρ, has the required properties.

Proof of Proposition 6.8.3 Given a connected closed set K P1, a connected component of P1 \ K, and a continuous function ρ : ∂ → C, we show that the associated Dirichlet problem is solvable. By considering Re ρ and Im ρ and applying 6.8 Harmonic Functions and the Dirichlet Problem 349 a suitable automorphism of P1, we may assume without loss of generality that ρ is real-valued and ∞∈. Any Dirichlet problem on the complement of a singleton has the constant solution, so we may also assume that K contains more than one point. By the Tietze extension theorem (Theorem 6.8.9), there exists a continuous function ρ0 : C → R with |ρ0|≤M ≡ max |ρ| on C ⊃ ∂ and ρ0 = ρ on ∂.We will produce solutions on elements of an exhausting sequence of coordinate disks for (with boundary values given by ρ0). We will then pass to a convergent subsequence; and ﬁnally, we will use barriers to show that the limit is a solution of the Dirichlet problem. By Lemma 5.17.1 and the Riemann mapping theorem in the plane (see Corol- lary 5.2.6), there exists a biholomorphism : → of onto the unit disk ≡ (0; 1). We may choose a sequence of numbers {Rν} in (0, 1) converging to 1 such that

~~−1 ∞∈ν ≡ ((0; Rν)) ∀ν = 1, 2, 3,....~~

For each ν, Lemma 6.8.5 provides a (unique) real-valued continuous function uν = | |≤ on ν such that uν ν is harmonic and uν ∂ν ρ0 ∂ν . In particular, uν M on ν by the maximum principle (part (b) of Lemma 6.8.1). Applying part (c) of Lemma 6.8.1 and passing to a subsequence, we may assume that the sequence {uν } converges uniformly on compact subsets of to a real-valued harmonic function on (more precisely, setting uˆν = uν on ν and uˆν = 0on \ ν , we see that for each μ, a subsequence of {ûν } will converge uniformly on μ, and hence Cantor’s diagonal process provides a subsequence converging uniformly on compact subsets of ). It now suffices to show that the function u: →[−M,M]⊂R given by lim →∞ u on , u ≡ ν ν ρ on ∂, is continuous at each point z0 ∈ ∂.Given>0, there is a δ1 > 0 such that |ρ0(z) − ρ0(z0)| < for each z ∈ C with |z − z0| <δ1. By Lemma 6.8.8, there exists a barrier β at z0 on . Since β<0on and lim supz→ζ β(z) < 0 at each ζ ∈ (∂) \{z } β< β point 0 ,wehavesup\(z0;δ1) 0; and hence, by replacing with the product of β and a sufficiently large positive constant, we may assume that β<−2M on \ (z0; δ1). For each ν,wehave

~~uν + β ≤−M ≤−|ρ(z0)| <ρ(z0) + on (∂ν ) \ (z0; δ1) and~~

uν + β ≤ ρ0 <ρ(z0) + on (∂ν) ∩ (z0; δ1). Therefore, by the weak maximum principle for subharmonic functions (part (d) of Lemma 6.8.1), we have uν + β<ρ(z0) + on ν for each ν, and passing to the 350 6 Holomorphic Structures on Topological Surfaces limit, we get u + β ≤ ρ(z0) + on . Therefore, since β → 0atz0 and ρ is continuous, we must have

~~lim sup u(z) ≤ ρ(z0) + ∀>0; z→z0 u(z) ≤ ρ(z ) = u(z ) −ρ and hence lim supz→z0 0 0 . The same argument applied to 0, −ρ, and −uν →−u gives~~

~~lim sup(−u(z)) ≤−u(z0), z→z0 ≥ = and hence lim infz→z0 u(z) u(z0). Thus limz→z0 u(z) u(z0), and therefore u is continuous.~~

~~Exercises for Sect. 6.8 6.8.1 State and prove a (Poisson) formula for the solution of the Dirichlet problem onadisk(z0; R) with center z0. 6.8.2 Prove Lemma 6.8.2. 6.8.3 Prove Lemma 6.8.7.~~

~~6.9 Proof of Schönﬂies’ Theorem~~

~~We are now ready to prove Schönﬂies’ theorem (following [KnR]). Throughout this section, denotes the unit disk (0; 1). We begin with some preliminary observations.~~

Lemma 6.9.1 Let γ : S1 → P1 be a Jordan curve with image C ≡ γ(S1), and let be a connected component of P1 \ C. Then ∂ is an inﬁnite connected subset of C (that is, γ −1(∂) is either S1 or a closed arc in S1), the solution : → C of the = −1 : → S1 Dirichlet problem in with boundary values given by ∂ (γ ) ∂ ∂ C∞ exists, and is a proper mapping of into .

Proof Observe that C = P1, since (for example) C is not simply connected. Ac- cording to Lemma 5.17.1, ∂ is a nonempty connected compact subset of C. More- over, if ∂ were a singleton {q}, then we would have = P1 \{q} (the connected set P1 \{q} would meet , but not ∂), which is clearly impossible because C ⊂ P1 \ . Thus γ −1(∂) is either S1 oraclosedarcinS1. According to Proposition 6.8.3, the solution : → C of the Dirichlet problem in with = −1 : → S1 ≡ boundary values given by ∂ (γ ) ∂ ∂ exists. Let u Re and ≡ ∈ | |= ≡ | | v Im . We may choose a point p with (p) R max , a line L in C with L ∩ (0; R) ={(p)}, and constants a,b,c ∈ R such that the function λ: (x + iy) → ax + by + c for x,y ∈ R satisﬁes L = λ−1(0) and λ<0on(0; R). Thus the continuous function λ() ≡ au + bv + c : → R, which is harmonic 6.9 Proof of Schönﬂies’ Theorem 351 on , attains its maximum at p. On the other hand, λ() is nonconstant because if λ() were constant, then we would have (∂) ⊂ () ={(p)}, which is = −1 : → S1 impossible since ∂ (γ ) ∂ ∂ is injective and ∂ is not a singleton. Therefore, by the maximum principle (Lemma 6.8.1), p ∈ ∂, and it follows that R = 1 (one may also see that R = 1 by applying Lemma 6.8.1 to the C∞ subharmonic function ||2 on ) and () ⊂ . Finally, since (∂) ⊂ ∂,the : → C∞ restriction must be a proper mapping; that is, the inverse image of every compact subset of is compact.

Lemma 6.9.2 Suppose that γ : S1 → P1 is a Jordan curve, C ≡ γ(S1), and for each 1 connected component of P \ C, there exists a continuous mapping : → = −1 : → S1 ⊂ : → such that () ∂ (γ ) ∂ ∂ , () , and () is a 1 proper local diffeomorphism. Then P \C has exactly two connected components 0 and 1 (in particular, the Jordan curve theorem holds for γ ), ∂0 = ∂1 = C, and = : → for each ν 0, 1, ν ν is a homeomorphism for which the restriction : → : P1 → P1 (ν ) ν ν is a diffeomorphism. Consequently, the mapping given by − 1(z) if z ∈ , 0 (z) ≡ − 1(1/z)¯ if z ∈ P1 \ , 1 1 1 1 1 is a homeomorphism for which S1 = γ and P1\S1 : P \ S → P \ S is a diffeomorphism (in particular, Schönﬂies’ theorem holds for γ ).

1 Proof Given a connected component 0 of P \ C, Lemma 10.2.11 implies that the : → C∞ proper local diffeomorphism (0 ) 0 0 is a (finite) covering map and therefore a diffeomorphism (since is simply connected). Thus = ⊂ ⊂ 0 (0) 0 (0) , : → and hence 0 0 is a continuous bijection, and therefore by compactness −1 = = = S1 a homeomorphism. In particular, γ (∂0) 0 (∂0) ∂ , and hence 1 1 ∂0 = C. Furthermore, P \ C is not connected. For if 0 = P \ C, then would be a homeomorphism of P1 onto . But these two spaces are not homeomorphic. For example, if we remove two distinct points from P1, then we get a domain that is biholomorphic to C∗ and therefore is not simply connected. However, if we remove any two distinct points in ∂ from , the resulting space is still simply connected. 1 Now, letting 1 be a connected component of P \ C with 1 = 0, we get a : ∪ = ∪ ∪ → P1 = homeomorphism 0 1 0 C 1 by setting 0 on 0 = → P1 \ and 1/1 on 1 (observe that the homeomorphism given by z → 1/z¯ is equal to the identity on ∂); and maps 0 ∪ 1 diffeomorphically onto P1 \ C. Suppose there exists a third connected component of P1 \ C.By applying a Möbius transformation, we may assume that 0 ∈ 0 and ∞∈1, and 2πit we may fix r>0 with (0; r) 0. Letting γ0(t) = γ(e ) for each t ∈[0, 1] and fixing a path α in 0 \{0}≈ \{ (0)} from γ0(0) = γ0(1) to r,wesee − 0 ∼ that the loop β = α ∗ γ0 ∗ α generates the fundamental group π1(0 \{0},r)= 352 6 Holomorphic Structures on Topological Surfaces

2πit π1(0 \{0},r) (the veriﬁcation is left to the reader). Thus the loop σ : t → re [ ]m ∈ \{ } ∈ Z represents the element β r π1(0 0 ,r)for some m , and hence 1 1 1 2πi = dz = m dz = m dz. σ z β z γ0 z

~~However, γ0is path homotopic to the constant loop in ≈ and therefore in C∗ ⊃ ,so (1/z) dz = 0. Thus we have arrived at a contradiction, and hence γ0 1 0 ∪ C ∪ 1 = P .~~

Proof of Theorem 6.7.1 Let γ : S1 → P1 be a Jordan curve, let C ≡ γ(S1), and let be a connected component of P1 \C. According to Lemma 6.9.1, ∂ is an inﬁnite connected subset of C, the solution : → C of the Dirichlet problem in with = −1 : → S1 boundary values given by ∂ (γ ) ∂ ∂ exists, and the restriction : → C∞ ≡ ≡ is a proper mapping. Let u Re and v Im . According to C∞ Lemma 6.9.2, it sufﬁces to show that is a local diffeomorphism. By the inverse function theorem (Theorem 9.9.1 and Theorem 9.9.2), this is the case if and only if du ∧ dv is nowhere 0 in . So, for the rest of this proof, we will assume that there exists a point p ∈ at which (du ∧ dv)p = 0, and we will reason to a contradiction. This condition implies that there exist constants a,b ∈ R that are not both 0, but that satisfy a(du)p + b(dv)p = 0. The continuous function

ρ ≡ au + bv − au(p) − bv(p): → R is then harmonic on and satisﬁes ρ(p) = 0 and (dρ)p = 0. The argument will proceed (in several steps) as follows (see Fig. 6.5). We will ﬁrst show that ρ is nonconstant. The local description of zero sets of nonconstant harmonic functions in Lemma 6.8.2 will then imply that for Z ≡ ρ−1(0), Z ∩ is a graph with at least four edges emanating from p. These edges cannot be rejoined by a sequence of edges in Z ∩ \{p}, since if this were possible, then we would get a Jordan curve in Z enclosing a region in . The maximum principle would then imply that ρ vanishes on this region. Similar arguments will then imply that the (at least four) connected components of Z ∩ \{p} approach at least four distinct points in ∂. But these four boundary points must then map into the intersection of the line L ≡{ax + by − au(p) − bv(p) = 0} with ∂, which consists of only two points. Since ∂ is injective, this will yield the desired contradiction. The steps below contain the details of the above argument. Step 1. Proof that ρ is nonconstant. The line L in C given by

L ≡{z ∈ C | a · Re z + b · Im z − au(p) − bv(p) = 0} contains the point (p) ∈ , and hence L ∩ ∂ contains exactly two points, which we will denote by ξ1 and ξ2.Ifρ is constant, then () ⊂ L, and hence (∂) ⊂{ξ1,ξ2}, which, according to Lemma 6.9.1, is impossible. Thus ρ is nonconstant, and in particular, the set S ≡{q ∈ | ρ(q) = 0 and (dρ)q = 0} is a discrete subset of . 6.9 Proof of Schönﬂies’ Theorem 353

~~Fig. 6.5 Three possibilities for the zero set Z, the ﬁrst two of which are ruled out by the maximum principle, the third by the injectivity of on the boundary~~

~~Step 2. Proof of the nonexistence of a separating Jordan curve that lies in the zero set of ρ and that does not meet ∂ in more than one point.Let − Z ≡ 1(L ∩ ) ={z ∈ | ρ(z)= 0}.~~

Suppose there exists a separating Jordan curve A in P1 such that A ⊂ Z and A ∩ ∂ contains at most one point. Then P1 \ A has a connected component that does not meet the connected subset C \ (A ∩ ∂) of P1 \ A. In particular, ∩ ∂ =∅ (otherwise, since ∂ is connected and not a singleton, would contain points in (∂) \ (A ∩ ∂) ⊂ C \ (A ∩ ∂)). On the other hand, ⊂ P1 \ (otherwise, the set ∂, which, according to Lemma 6.9.1, is inﬁnite, would be contained in the set A \ = A ∩ ∂, which contains at most one point), so we must have ⊂ .But then, since ∂ ⊂ A ⊂ Z, the maximum principle implies that ρ is constant on the open set and therefore on . This contradicts Step 1, so no such Jordan curve A exists. Step 3. Local description of certain injective paths near a point of Z ∩.Givena point q ∈ Z ∩, Lemma 6.8.2 implies that for some R>0 and some m ∈ Z>0, there is a local holomorphic chart (D, = ζ,(0; R)) such that D ⊂ , q = −1(0), and (Z ∩ D) = (L0 ∪···∪Lm−1) ∩ (0; R) for m distinct lines L0,...,Lm−1 through 0. Suppose α :[a,b]→Z \{q} is an injective path, α([a,b]) ∩ ∂ contains ∈ at most one point, and for some point c (a, b), α [a,r] and α [s,b] are piecewise smooth paths (in P1) for all r ∈ (a, c) and s ∈ (c, b). Then α([a,b]) meets at most one of the 2m connected components of Z ∩ D \{q}.ForifA and B are two distinct connected components of Z ∩D \{q} meeting α([a,b]), then since α is injective and | | | | the functions ζ A and ζ B are homeomorphisms onto the interval (0,R), there ∈[ ] ∈ −1 | |= | | are unique numbers r, s a,b such that r α (A), α(r) minα−1(A) ζ(α) , ∈ −1 | |= | | s α (B), and α(s) minα−1(B) ζ(α) . By exchanging A with B and r with s if necessary, we may assume that r

Step 4. Proof that the closure of each connected component of Z ∩ \ S is the image of an injective path that is smooth at each point in . The (closed) zero set Z −1 of ρ satisﬁes Z ∩ ∂ ⊂ ({ξ1,ξ2}) (a set with at most two points), Z ∩ is the zero set of the nonconstant harmonic function ρ , and p is a point in the discrete set S ={q ∈ Z ∩ | (dρ)q = 0}. By Lemma 6.8.2 (or Theorem 9.9.2), Z ∩ \ S is a (properly embedded) 1-dimensional smooth submanifold of the open set \ S. Thus,byStep2,Lemma6.7.3 (or Proposition 5.15.2), and Theorem 9.10.1, each connected component A of Z ∩ \ S is a noncompact connected 1-dimensional smooth submanifold of \ S that is the image of a diffeomorphism α : (0, 1) → A. We may choose a sequence {sν } in (0, 1) such that sν → 0 and such that the sequence {α(sν)} converges to a point q0 ∈ A ⊂ . Since α : (0, 1) → \ S is a proper mapping, we have q0 ∈/ A and therefore q0 ∈ S ∪ (Z ∩ ∂). Similarly, we may choose a sequence {tν} in (0, 1) such that tν → 1 and such that the sequence {α(tν)} converges to a point q1 ∈ S ∪ (Z ∩ ∂). If q0 ∈ S, then by Step 3 and Lemma 6.8.2, there is a local holomorphic chart = ; ∩ ={ }= −1 (D0,0 ζ0,(0 R0)) such that D0 , D0 S q0 0 (0),

0(Z ∩ D0) = (L0 ∪···∪Lm−1) ∩ (0; R0) for m>1 distinct lines L0,...,Lm−1 through 0, and A meets (and hence contains) exactly one of the 2m connected components of Z ∩ D0 \{q0}=Z ∩ D0 \ S. ∩ = −1 After applying a rotation, we may assume that A D0 0 ((0,R0)).Itfol- ≡ −1 − ≡ ∪ −1 − lows that for w0 0 ( R0/2),thesetA0 A 0 (( R0/2,R0)) is a connected noncompact 1-dimensional smooth submanifold of the open subset 0 ≡ 1 \[(S \{q0}) ∪{w0}] of P , and that the connected set A is open in A0. More- −1 over, since 0(α(sν )) → 0, it follows that the connected open set α (D0) = −1 −1 α ( ((0,R0))) must be equal to an interval of the form (0,r0) for some point 0 + r0 ∈ (0, 1). In particular, α(t) → q0 as t → 0 (0(α) is a strictly increasing real- ∞ valued C function on (0,r0)) and q1 ∈ α([r0, 1)) \ D0. As above, if q1 ∈ S, then there exist a connected neighborhood D1 with D1 ∩ S ={q1}, a point w1 ∈ D1 \{q1}, and a connected noncompact 1-dimensional 1 smooth submanifold A1 of the open subset 1 ≡ \[(S \{q1}) ∪{w1}] of P such that q1 ∈ A1 ⊂ A ∪ D1 and A is a connected open set in A1. We then have −1 − α (D1) = (r1, 1) for some point r1 ∈ (0, 1) and α(t) → q1 as t → 1 . Further- more, we may choose the neighborhood D1 so that D1 ∩ D0 =∅if q0 ∈ S (and hence 0

~~The above proof also gives the following:~~

Theorem 6.9.3 (Kneser–Radó [KnR]) If γ : S1 → P1 is a Jordan curve, C ≡ γ(S1), and is a connected component of P1 \C, then the solution of the Dirichlet prob- = −1 : → S1 lem in with boundary values given by ∂ γ C is a homeomorphism of onto that maps diffeomorphically onto .

Exercises for Sect. 6.9 6.9.1 Show that in the proof of Schönﬂies’ theorem, we could have formed a smooth path to the boundary. In other words, prove that if ρ is a nonconstant real-valued harmonic function on a simply connected domain C and p ∈ Z ≡ ρ−1(0), then there exists a proper C∞ embedding α : R → such that α(0) = p and α(R) ⊂ Z.

~~6.10 Orientable Topological Surfaces~~

In this section we consider a natural notion of orientability for a second countable topological surface. The simplest example of a nonorientable C∞ surface is the Möbius band (Example 9.7.1). So one natural way in which to define orientability of a (second countable) topological surface is to require that the surface not contain a Möbius band. In order to verify that the definition is consistent with the definition of orientability of a C∞ surface (Definition 9.7.2), we will need some preliminary facts. The first is a version of Lemma 5.11.1 for a nonorientable C∞ surface: 6.10 Orientable Topological Surfaces 357

Lemma 6.10.1 Let γ :[0, 1]→M be a smooth Jordan curve in a C∞ surface M, and let C = γ([0, 1]). If C does not admit an orientable neighborhood in M, then there exists a diffeomorphism : → B of some neighborhood of C in M onto the Möbius band

B ≡[0, 1]×(0, 1)/(0,t)∼ (1, 1 − t) ∀t ∈ (0, 1), with the quotient map :[0, 1]×(0, 1) → B, such that (γ (t)) = (t, 1/2) for each point t ∈[0, 1] (here, (t, 1/2) denotes an ordered pair, not an interval).

Proof We proceed as in the proof of Lemma 5.11.1, but here, we must put a twist 1 in the constructed band. We have the associated diffeomorphism γ0 : S → C given by e2πit → γ(t), and the C∞ covering map α : R → C given by

~~2πit t → γ0(e ) = γ(t− t) (where t denotes the ﬂoor of t). We may form local C∞ charts { = = × − }m (Uj ,j (xj ,yj ), Rj Ij ( δj ,δj )) j=1 in M and a partition 0 = t0~~

~~(Um,m = (xm,ym), Rm) = (U1,(1 + x1, −y1), (1 + I1) × (−δ1,δ1)) and such that for each j = 1,...,m, Ij is an open interval containing [tj−1,tj ], 0 <δj < 1/2,~~

C ∩ Uj ={p ∈ Uj | yj (p) = 0}=α(Ij ), and xj (α(t)) = t for each point t ∈ Ij . We may also choose the local charts so that !(I1) = !(Im)<1/2 and Ij ⊂ (0, 1) for j = 2,...,m− 1. We may ﬁx disjoint { }m { }m connected open sets Wj j=1 and connected open sets Vj j=1 such that

(i) We have γ([tj−1,tj ]) ⊂ Vj ⊂ Uj for each j = 1,...,m; (ii) We have Vm ∩ V1 ⊂ Wm ⊂ Um = U1, and for each j = 1,...,m− 1, we have Vj ∩ Vj+1 ⊂ Wj ⊂ Uj ∩ Uj+1; (iii) We have Vi ∩ Vj =∅whenever γ([ti−1,ti]) ∩ γ([tj−1,tj ]) =∅(that is, whenever i and j are indices with 1 ≤ i

Let V ≡ V1 ∪···∪Vm. We now alter the induced orientations inductively as follows. Suppose that 1 ≤ k ≤ m − 2 and that for j = 2,...,k, the local coordinates (xj−1,yj−1) and (xj ,yj ) induce the same orientation on the connected open subset Wj−1 of Uj−1 ∩ Uj ; that is,

~~dxj ∧ dyj > 0onWj−1 dxj−1 ∧ dyj−1 358 6 Holomorphic Structures on Topological Surfaces~~

(we assume that this holds vacuously for k = 1). Then, by replacing yk+1 with −yk+1 if necessary, we may assume that the above holds for j = k + 1 as well. Proceeding inductively, we see that we may assume that

~~dxj ∧ dyj > 0onWj−1 ⊃ Vj−1 ∩ Vj ∀j = 2,...,m− 1. dxj−1 ∧ dyj−1~~

In fact, the above condition holds for j = m as well. For if this were not the case, then the local coordinates (xm, −ym) = (1+x1,y1) (on Um = U1) would induce the same orientation as the local coordinates (xm−1,ym−1) on the set Wm−1 ⊃ Vm−1 ∩ Vm and the same orientation as the local coordinates (x1,y1) on Wm ⊃ Vm ∩ V1.But then V would be a connected orientable neighborhood of C, so we would arrive at a contradiction. Thus the condition holds for j = m. C∞ { }m−1 + m−1 ≡ Let us now ﬁx functions λ and λj j=2 such that λ j=2 λj 1ona neighborhood 0 of C,0≤ λ ≤ 1, supp λ ⊂ V1 ∪Vm, and for each j = 2,...,m−1, 0 ≤ λj ≤ 1 and supp λj ⊂ Vj . We may deﬁne the characteristic functions − 1onV ∩ 1([0, 1/2) × (−δ ,δ )), χ ≡ 1 1 1 0 \ −1 [ × − 0onV 1 ( 0, 1/2) ( δ1,δ1)), and 1onV ∩ −1((1/2, 1) × (−δ ,δ )), χ ≡ m m m 1 \ −1 × − 0onV m ((1/2, 1) ( δm,δm)) −1 − × − = −1 × − (observe that 1 (( 1/2, 0) ( δ1,δ1)) m ((1/2, 1) ( δm,δm)),so χ0 · χ1 ≡ 0butχ0 + χ1 ≡ 1onV ∩ U1 = V ∩ Um), and the mapping : 0 → B given by

~~ ≡ χ0 · λ · (x1,(1/2) + y1) + χ1 · λ · (xm,(1/2) + ym) m−1 + λj · (xj ,(1/2) + yj ) . j=2~~

C∞ { = × }2 The Möbius band B has the atlas (Qj ,j ,R (0, 1) (0, 1)) j=1, where ≡ ≡ −1 ≡ [ ]\{ } × Q1 (R), 1 ( R) , Q2 (( 0, 1 1/2 ) (0, 1)), and (s + (1/2), t) if (s, t) ∈[0, 1/2) × (0, 1), 2( (s, t)) ≡ (s − (1/2), 1 − t) if (s, t) ∈ (1/2, 1]×(0, 1).

~~If is a sufﬁciently small connected neighborhood of C in 0, then~~

~~∩ ⊃ ∩ −1 { }× − = ∩ −1 { }× − Wm 1 ( 0 ( δ1,δ1)) m ( 1 ( δm,δm)). 6.10 Orientable Topological Surfaces 359~~

Moreover, we have λ ≡ 1 and λj ≡ 0onWm ∩ for j = 2,...,m − 1, ∩ ⊂ ∩ ◦ = 1 + 1 + (Wm ) Q2, and on Wm ,wehave2 ( 2 x1, 2 y1).Itfol- C∞ lows that is a mapping on , and in fact, Wm∩ is a diffeomorphism. We also have (\ Wm) ⊂ Q1, and on a neighborhood of the set \ Wm,wehave

(ρ1,ρ2) ≡ 1 ◦ m−1 = λ · (χ0x1 + χ1xm) + λj · xj , j=2 m−1 −1 −1 −1 λ · (χ0 · (2 + y1) + χ1 · (2 + ym)) + λj · (2 + yj ) , j=2 ∞ which is of class C . Given a point p ∈ C \ Wm,wehave 2 (dρ1 ∧ dρ2)p = λ (p) · ((χ0(p) dx1 + χ1(p) dxm) ∧ (χ0(p) dy1 + χ1(p) dym))p m−1 + λ(p)λj (p)((χ0(p) dx1 + χ1(p) dxm) ∧ dyj )p j=2 m−1 + λ(p)λj (p)(dxj ∧ (χ0(p) dy1 + χ1(p) dym))p j=2 m−1 + λi(p)λj (p)(dxi ∧ dyj )p i,j=2

(here we have used the fact that for p = γ(t),wehave(xj (p), yj (p)) = (t, 0) and + = ∧ = ∧ d(λ j dλj )p 0). Therefore, since (dxj dyj )p (dxk dyj )p for all j and k with p ∈ Uj ∩ Uk,wehave

2 2 (dρ1 ∧ dρ2)p = λ (p)χ0(p)(dx1 ∧ dy1)p + λ (p)χ1(p)(dxm ∧ dym)p m−1 + λ(p)λj (p)(dxj ∧ dyj )p j=2 m−1 + λ(p)λj (p)χ0(p)(dx1 ∧ dy1)p j=2 m−1 + λ(p)λj (p)χ1(p)(dxm ∧ dym)p j=2 m−1 + λi(p)λj (p)(dxj ∧ dyj )p. i,j=2 360 6 Holomorphic Structures on Topological Surfaces

Since the local charts induce the same orientations on the sets Vi ∩ Vj for all i = 1,...,m and j = 2,...,m − 1, the above 1-form is nonzero. Thus, by continuity, we may choose so that dρ1 ∧ dρ2 is nonvanishing on a neighborhood ∞ of \ Wm, and therefore, by the C inverse function theorem (Theorem 9.9.1 and Theorem 9.9.2), is a local diffeomorphism of onto a neighborhood of ([0, 1]×{1/2}) in B with (γ(t)) = ((t, 1/2)) for each point t ∈[0, 1]. As in the proof of Lemma 5.11.1, we may choose so small that is a diffeomorphism onto (). Moreover, we may assume that for some ∈ (0, 1/2), we have () = ([0, 1]×((1/2)−,(1/2)+)). On the other hand, the mapping : B → () given by

(s, (1/2) + t)→ (s, (1/2) + 2t) ∀t ∈ (−1/2, 1/2) is a well-deﬁned diffeomorphism that is equal to the identity on ([0, 1]×{1/2}). Thus the mapping ≡ −1 ◦ : → B is a diffeomorphism with the required properties.

~~Lemma 6.10.2 Let M be a C∞ surface. If every smooth Jordan curve in M admits an orientable neighborhood, then M is orientable.~~

Proof We ﬁrst show that if α :[a,b]→M is a path for which the inverse image α−1(x) of each point x ∈ M contains at most two points, then the image α([0, 1]) admits an orientable neighborhood in M. For this, we let

~~d ≡ sup{t ∈ (a, b]|α([a,t]) admits an orientable neighborhood}∈(a, b], c ≡ inf{t ∈[a,d) | α([t,d]) admits an orientable neighborhood}∈[a,d).~~

If α(c) = α(d), then we have α([c,c + ]) ∩ α([d − ,d]) =∅for some (sufﬁciently small) ∈ (0,(d − c)/2); and we may choose connected open sets 0, 1, and such that 0 ∩ 1 =∅, α([c,c + ]) ⊂ 0, α([d − ,d]) ⊂ 1, α([c + , d − ]) ⊂ , and the connected open sets 0 ∪ and ∪ 1 are orientable. We may choose orientations on each of these sets that agree on their (connected) intersection , and hence 0 ∪ ∪ 1 is an orientable neighborhood of α([c,d]), and in particular, [c,d]=[a,b]. Thus we may assume without loss of generality that α(c) = α(d), and we will denote this point by p. Next, we observe that the connected set α((c, d)) admits an orientable neighbor- −1 hood in M.Forα(p) ={c,d}, so there exists a sequence of open intervals {Iν } = = such that (c, d) ν Iν and such that for each ν 1, 2, 3,...,wehave

−1 Iν Iν+1 (c, d) and α (α(I ν)) ∩[c,d]⊂Iν+1. ∞ By the choice of c and d, for each ν there exists a nonvanishing C real 2-form ων on a relatively compact connected neighborhood Uν of α(Iν+1) in M. Moreover, by inductively replacing ων with −ων whenever necessary, we may assume that

~~(ων+1/ων )q > 0 ∀q ∈ α(Iν+1) (a connected set) ∀ν = 1, 2, 3,.... 6.10 Orientable Topological Surfaces 361~~

∞ For each ν = 1, 2, 3,..., we may also choose a nonnegative C function λν on M such that supp λν ⊂ Uν , λν ≡ 1onα(Iν ), and λν ≡ 0onα([c,d]\Iν+1).If{ν} is a sequence of positive numbers converging sufﬁciently fast to 0, then the series ∞ νλν ων ν=1 ∞ converges uniformly on compact subsets of M(that is, for every nonvanishing C real 2-form ω on an open set U, the series ν λν ων/ω converges uniformly on compact subsets of U) to a continuous real 2-form ω. Moreover, for each index μ = 1, 2, 3,... and each point q ∈ α(Iμ),wehave ∞ (ω/ωμ)q = ν λν(q)(ων/ωμ)q ≥ μ > 0. ν=1

Thus ω is nonzero at each point in α((c, d)) and therefore at each point in some connected neighborhood of α((c, d)) in M \{p}. Hence this neighborhood is orientable by Proposition 9.7.4. We may now ﬁx a local C∞ chart (U,,(0; 1)) with p = −1(0) and α([c,d]) ⊂ U, a constant δ ∈ (0,(d− c)/2) with α([c,c + δ]∪[d − δ,d]) ⊂ U, and a constant r ∈ (0, 1) so small that for D ≡ −1((0; r)), D ∩ α([c + δ,d − δ]) =∅. Let 0 and 1 be the connected components of ∩ U containing α((c, c + δ]) and α([d − δ,d)), respectively. By replacing the neighborhood with the connected component of the neighborhood ( \ D) ∪ 0 ∪ 1 containing α((c, d)),wemay assume that ∩ D = (0 ∪ 1) ∩ D and ∩ D = (0 ∪ 1) ∩ D. We may also choose the orientation in so that it agrees with the orientation induced by in the connected open set 0 ⊂ U ∩ . For the proof of the existence of an orientable neighborhood of α([a,b]), it sufﬁces to show that the two orientations also agree in 1. For if they do agree, then ∪ D is an orientable neighborhood of α([c,d]), and the claim follows. In particular, we may assume without loss of generality that 0 = 1 (i.e., that 0 ∩ 1 =∅). Setting P ≡ −1((0; r/2)) and applying Lemma 5.10.6, we get an injective smooth path β :[0, 1]→ with β(0) ∈ α((c, d)) ∩ 0 ∩ P and β(1) ∈ α((c, d)) ∩ 1 ∩ P . Setting

~~s ≡ max{t ∈[0, 1]|β(t) ∈ 0 ∩ P }∈(0, 1),~~

u ≡ min{t ∈[s,1]|β(t) ∈ 1 ∩ P }∈(s, 1), we get the piecewise smooth Jordan curve γ obtained by joining the path β [s,u] (with β((s, u)) ⊂ \ P ) and the path in P ⊂ D with image under equal to the line segment from (β(u)) to (β(s)). Applying Lemma 5.10.5, we get a smooth Jordan curve C in ∪ D that is the union of two arcs A and B (i.e., A and B are each the homeomorphic image of a compact interval) such that A ⊂ , B ⊂ D, and A ∩ B ={p0,p1} with pi ∈ i ∩ D for i = 0, 1(seeFig.6.6). In particular, since 362 6 Holomorphic Structures on Topological Surfaces

Fig. 6.6 Construction of an orientable neighborhood of the curve by hypothesis, every smooth Jordan curve admits an orientable neighborhood, there exist connected open sets V and W such that A ⊂ V ⊂ , B ⊂ W ⊂ D, V ∩ W is equal to the union of two disjoint connected open sets Q0 and Q1 with pi ∈ Qi ⊂ i ∩ D for i = 0, 1, and V ∪ W is orientable. Fixing the orientation in V ∪ W so that the orientation induced in V agrees with that induced by , we see that it induces the same orientation in Q0 ⊂ V ∩0 ∩D as . But then V ∪W and must induce the same orientation in the connected open set W ⊂ D and therefore in Q1 ⊂ V ∩ 1 ∩ D. Similarly, since V ∪ W and induce the same orientation in V ⊃ Q1, and must induce the same orientation in Q1 ⊂ 1 ∩ D and therefore in 1. Therefore, since ∩ D = (0 ∪ 1) ∩ D, the neighborhood ∪ D of α([c,d]) is orientable, and the claim concerning α follows. We now get an orientation in M as follows. Let us ﬁx a point p0 ∈ M and C∞ = ; = −1 a local chart (U0,0 (x0,y0), (0 1)) with p0 0 (0). Given a point p1 ∈ M \{p0}, we may choose an injective smooth path α1 in M from p0 to p1 and, ∞ by the above, a nonvanishing C real 2-form ω1 on a neighborhood of α1([0, 1]) ∧ such that (ω1)p0 /(dx0 dy0)p0 > 0. By Theorem 9.9.2, we may now ﬁx a lo- C∞ = ; = −1 ∧ cal chart (U1,1 (x1,y1), (0 1)) such that p1 1 (0), (dx1 dy1)p1 / (ω1)p > 0, and α1([0, 1]) ∩ U1 is connected. Consequently, (dx1 ∧ dy1)/ω1 > 0at 1 ∞ each point in α1([0, 1]) ∩ U1. We may cover M by such local C charts (note that even p0 ∈ U1 for a suitable choice of p1 ∈ M \{p0}). In order to check compatibility, let us suppose that p2 ∈ M \{p0} with p2 = p1 (the proof for p2 = p1 is similar) and that we have chosen an associated path α2, form ω2, and local chart (U2,2 = (x2,y2), (0; 1)).Ifp2 ∈ U1 \{p0}, then we may form an injective smooth path β :[0, 1]→U1 \{p0} from p1 to p2, and we may set ≡ −1 [ ] ∈[ ] ≡ −1 r max β (α1( 0, 1 )) 0, 1 and s α1 (β(r)). − The product of suitable reparametrizations of α1 [0,s], β [r,1], and α2 (omitting β if r = 1) is then a piecewise smooth loop γ . Moreover, γ takes any given value at most 6.10 Orientable Topological Surfaces 363 twice, so by the above, there is a nonvanishing real C∞ 2-form ω on a neighborhood [ ] ∧ = of γ( 0, 1 ) with ωp0 /(dx0 dy0)p0 > 0. Consequently, for q α1(s),wehave

(dx ∧ dy )q (dx ∧ dy )q (ω )q 1 1 = 1 1 · 1 > 0. ωq (ω1)q ωq ∧ Similarly, we have (dx2 dy2)p2 /ωp2 > 0, and hence (dx ∧ dy ) (dx ∧ dy ) ω 1 1 p2 = 1 1 p2 · p2 > . ∧ ∧ 0 (dx2 dy2)p2 ωp2 (dx2 dy2)p2

So the orientations agree in a neighborhood of p2 ∈ U1 ∩ U2.Ifp2 ∈ M \{p0} is ∞ arbitrary and p ∈ U1 ∩ U2, then we may form a local C coordinate neighborhood (U, = (x, y), (0; 1)) of p as above but with U ⊂ U1 ∩ U2. By the above, each of the local charts 1 and 2 induces the same orientation in U as and therefore as the other. Thus the orientations are compatible on the overlap, and hence M is orientable.

~~Proposition 6.10.3 Let M be a C∞ surface. Then M is nonorientable if and only if some open subset of M is homeomorphic to the Möbius band.~~

Proof If M is nonorientable, then by Lemma 6.10.2, there exists a smooth Jordan curve C in M that does not admit an orientable neighborhood. Lemma 6.10.1 then provides a homeomorphism (in fact, a diffeomorphism) of some neighborhood onto the Möbius band. Conversely, suppose M is orientable and : → B is a homeomorphism of some open subset of M onto the Möbius band

B ≡[0, 1]×(0, 1)/(0,t)∼ (1, 1 − t) ∀t ∈ (0, 1) with quotient map :[0, 1]×(0, 1) → B (in particular, is second countable). Let γ :[0, 1]→B be the Jordan curve given by s → (s, 1/2), and let C ≡ γ([0, 1]). For the continuous mapping R : B ×[0, 1]→B given by

R( (s, t), u) = (s, (1/2)+(1−u)(t −1/2)) ∀(s,t,u)∈[0, 1]×(0, 1)×[0, 1], we have R(p,u) = p for each point (p, u) ∈ (C ×[0, 1])∪(B ×{0}), and R(p,1) ∈ C for each point p ∈ B; that is, R is a strong deformation retraction of B onto C. Hence, since [γ ] generates π1(C), [γ ] must also generate π1(B) (given a loop τ in B with base point τ(0) = γ(0), the mapping (t, u) → R(τ(t),u) is a path homotopy from τ toaloopinC). Moreover, the differential of the C∞ function given by ∞ (s, t) → s for all (s, t) ∈ (0, 1) × (0, 1) extends to a unique closed C 1-form θ ∼ = R = R ·[ ] R = R on B with γ θ 1. It follows that H1(B, ) γ H1(B, ) , and hence the ≡ −1 R = R ·[ ] =∼ R Jordan curve β (γ ) satisﬁes H1(, ) β H1(,R) (see Sect. 10.7). Now β is homologous in to a sum of smooth Jordan curves (by Proposi- tion 5.10.7), so there exists a smooth Jordan curve α :[0, 1]→ with nonzero 364 6 Holomorphic Structures on Topological Surfaces

[ ] ≡ [ ] homology class α H1(,R). On the other hand, the image A α( 0, 1 ) is nonseparating in . For Stokes’ theorem implies that any separating smooth Jordan curve for which the complement has a relatively compact connected component in must be homologous to 0, while the complement of any compact subset of B has exactly one connected component that is not relatively compact in B (any connected component that is not relatively compact must contain the connected open set ([0, 1]×((0,)∪ (1 − ,1))) for some sufﬁciently small ∈ (0, 1/2)). There- fore, by Lemma 5.15.1, there exists a second smooth Jordan curve σ in for which [ ] [ ] the homology classes α H1(,R) and σ H1(,R) are linearly independent, which is [ ] R impossible because α H1 spans H1(, ). Thus we have arrived at a contradiction, and the proposition follows.

~~The above proposition implies that the following definition is not in conflict with the definitions of orientability and nonorientability for a smooth surface given in Sect. 9.7:~~

Deﬁnition 6.10.4 A second countable topological surface M is called nonorientable if some open subset of M is homeomorphic to the Möbius band. The surface M is called orientable if no open subset of M is homeomorphic to the Möbius band.

Exercises for Sect. 6.10 6.10.1 Let γ :[0, 1]→M be a (continuous) Jordan curve in a C∞ surface M, and let C ≡ γ([0, 1]). Assume that C does not admit an orientable neighborhood in M. Prove that there exists a homeomorphism : → B of some neighborhood of C in M onto the Möbius band

~~B ≡[0, 1]×(0, 1)/(0,t)∼ (1, 1 − t) ∀t ∈ (0, 1)~~

with quotient map :[0, 1]×(0, 1) → B such that (γ (t)) = (t, 1/2) for each point t ∈[0, 1] (cf. Exercise 6.7.7). 6.10.2 Let B be the Möbius band with associated C∞ quotient map :[0, 1]× (0, 1) → B (see Exercise 6.10.1), let γ(t) ≡ (t, 1/2) for each point t ∈[0, 1], and let C ≡ γ([0, 1]). Prove that C is nonseparating in B, = C∞ but γ θ 0 for every closed 1-form θ with compact support in B (cf. Proposition 5.15.2).

~~6.11 Smooth Structures on Second Countable Topological Surfaces~~

~~The main goal of this section is the following:~~

~~Theorem 6.11.1 Every second countable topological surface admits a smooth (surface) structure. 6.11 Smooth Structures on Second Countable Topological Surfaces 365~~

Note that it is redundant to refer to a smooth surface structure, since no topological surface is homeomorphic to a manifold of dimension n = 2 (see Exercise 6.11.1). It follows from the above theorem and the results of Sect. 6.10 that the deﬁnition of the homology group H1(M, A) provided in Sect. 10.7 (Deﬁnition 10.7.9) applies to any orientable second countable topological surface M and any subring A of C containing Z. That this group is isomorphic to the singular homology group is considered in Exercise 6.11.9. Theorem 6.11.1, Proposition 6.10.3, and Corollary 6.1.6 together give the following:

~~Corollary 6.11.2 Every second countable orientable topological surface admits a (1-dimensional) holomorphic structure.~~

We now address the proof of the theorem. The idea of the proof is to inductively form smooth structures on successively larger open sets. In the inductive step, one attaches a topological disk D ≈ (0; r) to an open set with a smooth structure S, after first using Schönflies’ theorem to modify the homeomorphism on the overlap so that it becomes smooth with respect to S. We first consider the following case, in which one attaches an inner disk to a topological annulus .

Lemma 6.11.3 Let K be a compact subset of ≡ (0; 1), and let S0 be a (2-dimensional) smooth structure on \ K. Then there exists a smooth structure S on such that S = S0 near ∂; that is, S and S0 induce the same restricted smooth structure on the complement \ K of some compact subset K ⊃ K of .

Proof We may assume without loss of generality that K = (0; r) for some r ∈ (0, 1). A generator for the fundamental group of the annulus \ K = (0; r, 1) is homologous to a sum of Jordan loops that are smooth with respect to the smooth structure S0 (by Proposition 5.10.7). Hence there is a Jordan curve γ :[0, 1]→ \ K that is not homologous to 0 and that is smooth with respect to S0.Theim- age C ≡ γ([0, 1]) is separating in P1 (by the Jordan curve theorem) and therefore in \ K. Moreover, \ K is orientable (as shown in Sect. 6.10, orientability of surfaces is a topological property), so Stokes’ theorem implies that no connected component of ( \ K)\ C is relatively compact in \ K (one may also see this directly). Therefore, choosing ∈ (0,(1 − r)/2) so small that C ⊂ (0; r + ,1 − ), we see that the set ( \ K) \ C has exactly two connected components U0 and U1, where U0 ⊃ (0; r, r + ) and U1 ⊃ (0; 1 − ,1). Hence the domain 0 ≡ (0; r + ) ∪ U0 satisﬁes ∂0 = C. By Lemma 5.11.1,forsomeR>1, there are a relatively compact connected neighborhood A of C in \ K and a homeomorphism 0 : A → (0; 1/R, R) 1 such that 0(C) = S , 0 is a diffeomorphism with respect to the smooth structure on the domain A induced by S0 and the standard smooth structure on the target ; ∩ = ∩ = −1 ; (0 1/R, R), and A 0 A U0 0 ((0 1/R, 1)). By Schönﬂies’ theorem, : P1 → P1 = : → S1 there exists a homeomorphism 1 such that 1 C 0 C C and 1 0 maps 0 diffeomorphically (with respect to the standard smooth structure 366 6 Holomorphic Structures on Topological Surfaces

~~Fig. 6.7 Construction of the local chart (,,(0; R))~~

~~Fig. 6.8 Extension of (a modiﬁcation of) the smooth structure S0 on to a smooth structure S on a neighborhood of ∪ K~~

1 on both the domain and the target) onto (and 1 maps P \ diffeomor- 1 P \0 0 1 phically onto P \ ). Setting ≡ 0 ∪ A 0 and (z) for z ∈ , (z) ≡ 1 0 ∈ \ = \ = −1 { ∈ C | ≤| | } 0(z) for z A 0 0 0 ( ζ 1 ζ

= : \ → (0; 1,R) \0 0 \0 0 is a diffeomorphism with respect to the smooth structure induced by S0 on the domain and the standard smooth structure on the target (see Fig. 6.7). Now let A be the collection of local charts that are equal either to (,,(0; R)) ∞ or to (U \ 0, \ ,(U \ 0)) for some local C chart (U, , (U)) in the U 0 ∞ smooth surface ( \ K,S0). Then A is a C atlas on , and the correspond- ∞ ∞ ing C structure S induces the same C structure as S0 on the open subset \ 0 ⊂ \ K.

~~The following lemma is the main step in the proof of Theorem 6.11.1:~~

Lemma 6.11.4 Let M be a second countable topological surface, let and be open subsets with ⊂ , let S0 be a (2-dimensional) smooth structure on , let (D,,≡ (0; 1)) be a local C0 chart in M, and let K be a compact subset of D. Then there exists a smooth structure S on a neighborhood of ∪ K such that S and S0 induce the same smooth structure on \ D (see Fig. 6.8). 6.11 Smooth Structures on Second Countable Topological Surfaces 367

Proof We may fix an open set 0 with ⊂ 0 ⊂ 0 ⊂ (by Lemma 9.3.6) −1 and a constant r ∈ (0, 1) with K ⊂ D0 ≡ ((0; r)) D. Given distinct points ζ1,...,ζm ∈ ∂(0; r) arranged in counterclockwise order, we let ζ0 ≡ ζm, and for each j = 1,...,m,weletαj :[0, 1]→(0; r) be an injective parametrization of the line segment from ζj−1 to ζj ,weletβj :[0, 1]→∂(0; r) be an injective parametrization of the counterclockwise arc of the circle ∂(0; r) from ζj−1 to ≡ − ∗ ≡ [ ] : S1 → ζj ,weletγj αj βj ,weletCj γj ( 0, 1 ), and we let ηj Cj be the 2πit associated homeomorphism given by e → γj (t). Exactly one of the two con- 1 nected components of P \ Cj , which we will denote by Vj , is contained in (0; r) and satisfies ∂Vj = Cj (this follows from Schönflies’ theorem, but it is also easy to −1 verify directly in this case). We also set Uj ≡ (Vj ) ⊂ D0 for j = 1,...,m and −1 pj ≡ (ζj ) for j = 0,...,m. We may choose such boundary points ζ0,ζ1,...,ζm = ζ0 so close together that ∞ if j ∈{1,...,m} with U j ∩ 0 = ∅, then U j is contained in some local C ∞ chart (Wj ,j ,(0; 1)) in the C surface (, S0). Schönflies’ theorem (Theo- rem 6.7.1) then allows us to modify into a diffeomorphism with respect to S0 on each such set Uj , while leaving the values elsewhere unchanged. More precisely, if j ∈{1,...,m} with U j ∩ 0 = ∅, then Schönflies’ theorem provides a homeomorphism → V j that maps diffeomorphically onto Vj and that is equal 1 1 to the homeomorphism ηj : S → Cj on S = ∂, as well as a homeomorphism j (U j ) = j (Uj ) → that maps j (Uj ) diffeomorphically onto and that −1 ◦ ◦ −1 : → S1 = is equal to the homeomorphism ηj j j (∂Uj ) on j (∂Uj ) ∂j (Uj ). The associated composition U j → j (U j ) → → V j then yields : → = : → a homeomorphism j U j V j such that j ∂Uj ∂Uj ∂Uj ∂Vj and : → j Uj Uj Vj is a diffeomorphism with respect to the smooth structure on Uj induced by S0 and the standard smooth structure on Vj ⊂ C (see Fig. 6.9). Thus the mapping : D0 → (0; r) given by ⎧ ⎨⎪(x) if x ∈ D0 \ (U1 ∪···∪Um), (x) ≡ (x) if x ∈ U with j ∈{1,...,m} and U ∩ =∅, ⎩⎪ j j 0 j (x) if x ∈ U j with j ∈{1,...,m} and U j ∩ 0 = ∅, is a well-defined homeomorphism that maps each of the sets Uj ⊂ 0 with closure meeting 0 diffeomorphically, with respect to S0, onto its image Vj ⊂ C (which has the standard C∞ structure). Setting m m m −1 −1 Q ≡ (0; r) V j = (0; r) V j = D0 U j , j=1 j=1 j=1

~~0 we may now let A1 be the collection of local C charts in the open set~~

~~1 ≡ (0 \{p1,...,pm}) ∪ D0 = (0 \ Q) ∪ D0, 368 6 Holomorphic Structures on Topological Surfaces~~

~~Fig. 6.9 Construction of a smooth structure outside the vertices of the polygonal region Q~~

; which are equal either to (D0, D0 ,(0 r)) or to W ∩ \ Q, ,(W ∩ \ Q) 0 (W∩0\Q) 0 ∞ for some local C chart (W, , (W)) in (, S0) with W ∩0 = ∅. This collection ∞ ∞ A1 is then a C atlas, because for any local C chart (W, , (W)) in (, S0),we have m −1 (W ∩ 0 \ Q) ∩ D0 = W ∩ 0 ∩ Vj j=1 = W ∩ 0 ∩ Uj ,

1≤j≤m,U j ∩0=∅ ∞ and with respect to the C structure S0, maps this set diffeomorphically onto its ∞ image. Clearly, the C structure S1 in 1 determined by A1 agrees with S0 in the open subset 0 \ Q ⊃ 1 \ D ⊃ \ D. So far, we have obtained a smooth structure on the set 1 ⊃ ( \{p1,..., pm}) ∪ K. Letting J ⊂{1,...,m} be the set of indices j for which pj ∈ 0,we will expand the set of deﬁnition for our smooth structure so as to include these 0 points. We may choose a collection of disjoint local C charts {(Ej ,j ,)}j∈J with −1 pj = (0) and Ej ⊂ 0 ∩ D for each index j ∈ J . In particular, for each j ∈ J , j ∗ we have Ej \{pj }⊂1. Hence the local charts in = \{0} given by the compo- ∞ − C S 1 ∗ sitions of local charts in (1, 1) meeting Ej with j determine a smooth ∗ − ∗ R 1 ∗ : R → \{ } S structure j on for which the mapping j ( , j ) (Ej pj , 1) is a diffeomorphism. Therefore, by Lemma 6.11.3, there is a smooth structure Qj on that induces the same smooth structure as Rj on the annulus (0; uj , 1) for ∈ some uj (0, 1). ≡ −1 ; A Setting N j∈J j ((0 uj )), we may now let 2 be the collection of local C0 charts in the open set

~~2 ≡ 0 ∪ D0 = 1 ∪{pj | j ∈ J }⊃ ∪ K 6.11 Smooth Structures on Second Countable Topological Surfaces 369~~

−1 ◦ C∞ that are equal either to (j (W), j ,P)for some local chart (W,,P)in ∞ (, Qj ),orto(W \ N, \ ,(W\ N)) for some local C chart (W, , (W)) W N ∞ in (1, S1) with W ⊂ N. The collection A2 is then a C atlas on 2 that induces ∞ the same C structure as S1 on the open set

~~2 \ N = 1 \ N ⊃ 1 \ D ⊃ \ D. ∞ Since S1 induces the same C structure as S0 on \ D, the claim follows.~~

~~We are now ready for the proof of Theorem 6.11.1. In fact, we will prove the following slightly stronger version:~~

Theorem 6.11.5 Let M be a second countable topological surface, let S0 be a (2-dimensional) smooth structure on some (nonempty) open subset of M, and let K be a closed subset of M with K ⊂ . Then there exists a smooth structure S on M that induces the same smooth structure as S0 on some neighborhood of K.

Proof We will assume that M \ is noncompact, the proof for M compact being similar. We may ﬁx an open set 0 with K ⊂ 0 ⊂ 0 ⊂ , a locally ﬁnite collec- C0 { ≡ ; }∞ { }∞ tion of local charts (Uν ,ν, (0 1)) ν=1, and a sequence rν ν=1 in (0, 1) ≡ −1 ; \ such that Dν ν ((0 rν )) Uν M K for each ν and ∞ M \ 0 ⊂ Dν. ν=1

We now set 0 ≡ , 0 ≡ 0, and ν ≡ 0 ∪ D1 ∪···∪Dν for each ν = 1, 2, 3,..., and we proceed inductively. Suppose that for each μ = 0,...,ν− 1, we ∞ have constructed a (2-dimensional) C structure Sμ on an open set μ ⊃ μ.As- ∞ sume also that for each μ = 1,...,ν−1, Sμ and Sμ−1 induce the same C structure on the set μ−1 \ U μ. Then, by Lemma 6.11.4, there exists a smooth structure Sν on an open set ν ⊃ ν−1 ∪ Dν = ν that induces the same smooth structure as Sν−1 on the set ν−1 \ U ν . Thus we get a sequence of sets with smooth structures {(ν, Sν)}. These smooth structures eventually stabilize on any given compact set to give a well-deﬁned global smooth structure. More precisely, by local ﬁniteness, given a point p ∈ M, there are an index ν0 and a neighborhood W of p such that W ∩ Uν =∅for all ν>ν0. In particular, we have W ⊂ ν , and we may choose ∞ 0 W so that there is a local C chart (W,,V) for (ν , Sν ). By construction, this ∞ 0 0 local chart is a local C chart with respect to Sν for every ν ≥ ν0. It is now easy to see that the collection A of all such local charts is a smooth surface atlas on M, and that the associated smooth structure S induces the same smooth structure on the \ S neighborhood M ν U ν of K as 0.

~~Exercises for Sect. 6.11 6.11.1 Let be a nonempty open subset of R2. Prove that is not homeomorphic to an open subset of Rn for n = 2. 370 6 Holomorphic Structures on Topological Surfaces~~

Hint. The complement of a point in any connected neighborhood in R2 is not simply connected. 6.11.2 Prove that if M is a compact orientable topological surface with H1(M, R) = 0, then M is homeomorphic to a sphere. 6.11.3 Prove that if M is a second countable noncompact orientable topological 2 surface with H1(M, R) = 0, then M is homeomorphic to R . 6.11.4 Let M be a second countable topological surface. The goal of this exercise is a proof of a theorem of Radó, according to which M admits a triangulation. A triangle in M is a homeomorphism : T → T of a geometric closed triangle T ⊂ R2 onto a compact set T ⊂ M (in a slight abuse of notation, we also denote by T ). The image of each vertex of T is called a vertex of (or a vertex of T ), and the image of each edge of T is called an edge of (or an edge of T ). A triangulation of M is a locally ﬁnite covering {Ti}i∈I of M by triangles such that for each pair of distinct indices i, j ∈ I , Ti ∩ Tj is either empty, a common vertex of Ti and Tj , or a common edge of Ti and Tj . ◦ ◦ (a) Prove that if : T → T is a triangle in M, then (T ) = T , ◦ (∂T ) = ∂T, ∂T is a separating Jordan curve in M, and T is a connected component of M \ ∂T. (b) Suppose {Ti}i∈I is a triangulation of M, i ∈ I , and E is an edge of Ti . Prove that there is a unique index j ∈ I \{i} such that E is an edge of Tj . Prove also that Ti and Tj have exactly two vertices in common, namely, the endpoints of E. (c) Prove, in the following way, that if M is orientable, then M admits a triangulation (this is a special case of the above theorem of Radó). By the results of this chapter, M admits the structure of a Riemann surface. The proof of Proposition 5.18.4 gives a locally ﬁnite covering of M by triangles with disjoint interiors. Add more edges so that no vertex will lie in the interior of an edge, and no two distinct edges will meet in more than one point. (d) Prove, in the following way, that M admits a triangulation even if M is nonorientable (that is, prove the above theorem of Radó). By Theorem 6.11.1, M admits the structure of a C∞ surface. Apply the 1-dimensional C∞ version of Sard’s theorem (see Exercise 9.6.5) in order to construct a suitable graph in M as in the proof of Proposi- tion 5.18.4, and then proceed as in the proof of part (c). 6.11.5 Let γ :[0, 1]→M be a Jordan curve with image C in a topological surface M. (a) Prove that if C admits a (topologically) orientable neighborhood, then for some R>1, there exist a neighborhood of C in M and a homeomorphism : → (0; 1/R, R) with (γ (t)) = e2πit for each t ∈[0, 1] (cf. Exercise 6.7.7). (b) Prove that if C does not admit a (topologically) orientable neighborhood, then there exists a homeomorphism : → B of some neigh- 6.11 Smooth Structures on Second Countable Topological Surfaces 371

~~borhood of C in M onto the Möbius band~~

~~B ≡[0, 1]×(0, 1)/(0,t)∼ (1, 1 − t) ∀t ∈ (0, 1),~~

with quotient map :[0, 1]×(0, 1) → B, such that (γ (t)) = (t, 1/2) for each point t ∈[0, 1] (cf. Exercise 6.10.1). 6.11.6 Let M be a second countable orientable topological surface, let γ :[0, 1]→ M be a Jordan curve, and let C ≡ γ([0, 1]). Prove that C is separating in M = ˇ if and only if γ θ 0 for every Cech 1-form θ with compact support in M. Prove also that every Jordan curve in M is separating if and only if every Cechˇ 1-form θ with compact support in M is exact. Hint. Apply Exercise 6.7.2. 6.11.7 Prove that every compact orientable topological surface M may be obtained by C0 attachment of a locally ﬁnite family of disjoint tubes (see Exer- cise 5.12.8) to the unit sphere S2. 6.11.8 Let M be a nonorientable second countable topological surface. Prove that there exists a connected orientable covering space ϒ : M → M for which each ﬁber contains exactly two elements (as in Exercise 9.7.8, ϒ : M → M is called the orientable double cover of M). 6.11.9 This exercise requires facts considered in Exercises 6.6.2–6.6.4. Let M be a second countable topological surface. (a) Prove that the universal cover of M is homeomorphic to the plane R2 or the sphere S2. (b) Prove that if M is orientable, then π1(M) is torsion-free and ∼ H1(M, Z) = π1(M)/[π1(M), π1(M)],

while if M is nonorientable, then every torsion element of π1(M) has ∼ order 1 or 2 and π1(M)/ = Z/2Z for some torsion-free normal subgroup of π1(M). (c) Let A be a subﬁeld of C containing Z, and let F = R or C. Prove that A =∼ A if M is orientable, then H1 (M, ) H1(M, ), while if M is nonori- F =∼ F entable, then H1 (M, ) H1(M, ). (d) Prove that if M is orientable, then for any subring A of C containing Z, the mapping [ ] → [ ] · ξ H1(M,A) ( ξ H1(M,A), )deR 1 gives an injective homomorphism H1(M, A) → Hom(H (M, A), A) (according to Theorem 10.7.18, this homomorphism is also surjective if π1(X) is ﬁnitely generated). (e) Assume that M is orientable, and let K ⊂ M be a compact set. Prove that there exist domains and such that K ⊂ M and such that

~~im[H 1(M, C) → H 1(, C)]=im[H 1( , C) → H 1(, C)].~~

~~Part III Background Material~~

~~Chapter 7 Background Material on Analysis in Rn and Hilbert Space Theory~~

~~In this chapter, we recall some basic deﬁnitions and facts concerning integration and Hilbert spaces.~~

~~7.1 Measures and Integration~~

~~In this section, we recall some basic facts from measure theory. The proofs, which are omitted, can be found in, for example, [Fol], [Rud1], or [WZ].~~

Deﬁnition 7.1.1 Let X beaset. (a) A σ -algebra in X is a collection M of subsets of X such that (i) We have ∅∈M; (ii) X \ S ∈ M for each set S ∈ M; and ∈ { } (iii) j∈J Sj M for each countable family of sets Sj j∈J in M. The pair (X, M), which is often denoted simply by X, is then called a measurable space, and any element of M is called a measurable set. A mapping f : S → Y of a measurable set S ⊂ X into a topological space Y (see Sect. 9.1) is called a measurable mapping if f −1(U) is measurable for each open set U ⊂ Y .ForY =[−∞, ∞] or C,wealsocallf a measurable function. (b) A positive measure on X (or on (X, M))isafunction μ: M→[0, ∞] on ∅ = ∞ = ∞ a σ -algebra M in X such that μ( ) 0 and μ( j=1 Sj ) j=1 μ(Sj ) for every sequence of disjoint measurable sets {Sj }. For each measurable set S, μ(S) is called the measure of S. The pair (X, μ) (or triple (X, M,μ))iscalled a measure space. The measure μ is complete if every subset of a set of measure 0 is measurable (and hence of measure 0). We say that a property of points holds almost everywhere (abbreviated a.e.) in a set S ⊂ X if the property holds for all points in the complement S \ A of some set A ∈ M of measure 0.

Remarks 1. If f is a sum or product of measurable functions, a composition of a continuous function with a measurable mapping, or a supremum, inﬁmum, or limit of a sequence of measurable functions, then f is measurable.

T. Napier, M. Ramachandran, An Introduction to Riemann Surfaces, Cornerstones, 375 DOI 10.1007/978-0-8176-4693-6_7, © Springer Science+Business Media, LLC 2011 376 7 Background Material on Analysis in Rn and Hilbert Space Theory

2. We will occasionally consider measure spaces (X, M,μ) that are not complete (see Sect. 9.7). However, letting M denote the collection of sets of the form A = E ∪ B, where E ∈ M and B is a subset of some set of measure 0, and letting μ(A)ˆ = μ(E) for each such set A, one gets a well-deﬁned complete measure space (X, M , μ)ˆ called the completion of (X, M,μ)(see, for example, [Rud1]).

Deﬁnition 7.1.2 Let (X, μ) be a measure space, and let f be a measurable function on a measurable set S ⊂ X. (a) If f ≥ 0a.e.inS, then the integral of f over S is given by m f(x)dμ(x)≡ fdμ≡ sup rj μ(Sj ) ∈[0, ∞], S S j=1

where the supremum is taken over all choices of disjoint measurable subsets S1,...,Sm of S and constants r1,...,rm ∈[0, ∞] with f ≥ rj a.e. in Sj for each j = 1,...,m. (b) We say that f is integrable on S if the nonnegative extended real numbers + − R+ ≡ [Re f ] dμ, R− ≡ [Re f ] dμ, S S + − I+ ≡ [Im f ] dμ, I− ≡ [Im f ] dμ, S S are ﬁnite. If this is the case, then the integral of f over S is given by fdμ≡ R+ − R− + iI+ − iI−. S

Remarks 1. Every measurable subset Y of a measure space (X, μ) inherits a positive measure given by S → μ(S) for every measurable set S with S ⊂ Y . Integrals of functions over measurable subsets of S with respect to this induced measure then agree with the corresponding integrals with respect to μ. In an abuse of notation, we also denote the induced measure by μ. 2. A simple function ϕ on a measurable space X is a measurable complex-valued function with ﬁnite range. Letting r1,...,rn be the distinct values of ϕ,wehave = n = −1 = ⊂ ϕ j=1 rj χSj , where Sj ϕ (rj ) for j 1,...,n, and for any set E X, χE denotes the corresponding characteristic function (i.e., χE ≡ 1onE and χE ≡ 0 on X \ E). If μ is a positive measure on X, S ⊂ X is measurable, and ϕ ≥ 0, then = n ∩ we have S ϕdμ j=1 rj μ(Ej S). Consequently, if f is a measurable function on S and f ≥ 0 a.e., then fdμ= sup ϕdμ| ϕ is a nonnegative simple function and ϕ ≤ f a.e. in S . S S

Example 7.1.3 (Counting measure) Let X be a set, let M be the collection of all subsets of X (which is clearly a σ -algebra), and let μ(S) ∈ Z≥0 ∪{∞}⊂[0, ∞] be 7.1 Measures and Integration 377 the number of elements of S for each set S ⊂ X. Then μ is a positive measure that is called the counting measure on X. For any function f : X →[0, ∞] and any set ⊂ = S X,wehave S fdμ x∈S f(x), the (unordered) sum of f over S (recall that ≡ F x∈S f(x) supF ∈F x∈F f(x), where is the set of all ﬁnite subsets of S). In −1 particular, this sum is inﬁnite if f ((0, ∞]) is uncountable. A function f : X → C is integrable if and only if x∈S f(x) is absolutely summable, and if this is the case, then the integral of f over S is equal to the sum.

Example 7.1.4 (Lebesgue measure) For n ∈ Z>0,ann-cell is a Cartesian product = ×···× ⊂ Rn R Q I1 In of n intervals in . For any n-cell Q as above, the volume ≡ n ∈ ∞] ⊂ of Q is given by vol(Q) i=1 (Ii) (0, . For any set S X,theLebesgue outer measure of S is given by

∞ ∗ λ (S) ≡ inf vol(Qj ) |{Qj } is a sequence of open n-cells that covers S . j=1 ∗ ∞ ≤ ∞ ∗ This function is countably subadditive; that is, λ ( j=1 Sj ) j=1 λ (Sj ) for any n n sequence of subsets {Sj } of R .AsetS ⊂ R is Lebesgue measurable if it satisﬁes the Carathéodory criterion:

~~λ∗(A) = λ∗(A ∩ S) + λ∗(A \ S) ∀A ⊂ Rn.~~

~~The collection M of Lebesgue measurable sets is a σ -algebra, and the function~~

≡ ∗ : →[ ∞] λ λ M M 0, , which is called Lebesgue measure, is a positive measure with the following properties: (i) Completeness. Every subset of a set of Lebesgue measure 0 is measurable (in fact, any set of Lebesgue outer measure 0 is Lebesgue measurable). (ii) Regularity. Every open set is Lebesgue measurable (hence M contains the Borel σ -algebra, i.e., the smallest σ -algebra containing the open sets). More- over, if a set S ⊂ Rn is Lebesgue measurable, then for every >0, there exist aclosedsetA and an open set B such that A ⊂ S ⊂ B and λ(B \ A) < . Con- sequently, there exist sets F and G such that F is a countable union of closed sets (i.e., F is an Fσ ), G is a countable intersection of open sets (i.e., G is a Gδ), F ⊂ S ⊂ G, and λ(G \ F)= 0 (the completeness property (i) and the measurability of Borel sets imply that the converse is also true). (iii) Normalization. For each n-cell Q, λ(Q) = vol(Q). (iv) Translation invariance. For each Lebesgue measurable set S ⊂ Rn and each point x ∈ Rn,thesetS + x is Lebesgue measurable and λ(S + x) = λ(S). = For n 1 and for I an interval with endpoints a

~~Theorem 7.1.5 (Convergence theorems) Let (X, μ) be a measure space, and let {fν} be a sequence of measurable functions on a measurable set S ⊂ X.~~

~~(a) Monotone convergence theorem. If 0 ≤ f1 ≤ f2 ≤···, then f = f . lim→∞ ν lim→∞ ν S ν ν S~~

~~(b) Fatou’s lemma. If fν ≥ 0 for each ν, then lim inf f ≤ lim inf f . →∞ ν →∞ ν S ν ν S~~

(c) Lebesgue’s dominated convergence theorem. If fν → f and there is a nonnegative integrable function g such that |fν|≤g on S for each ν ∈ Z>0, then f and fν for each ν are integrable and f = lim f . →∞ ν S ν S

Deﬁnition 7.1.6 Let V be a vector space over the ﬁeld F = R or C. (a) A function ·: V →[0, ∞) is called a norm if (i) For each vector v ∈ V \{0},wehavev > 0; (ii) For each vector v ∈ V and each scalar a ∈ F,wehaveav=|a|v; and (iii) Triangle inequality.Wehaveu + v≤u+v for all u, v ∈ V. The pair (V, ·) (or simply V)iscalledanormed vector space. (b) A sequence {vj } in V converges to v in (V, ·) if vj − v→0asj →∞. We also call v the limit of {vj }, and we write

~~v = lim vj or vj → v as j →∞. j→∞~~

A sequence that is not convergent is said to be divergent. (c) A sequence {vj } in (V, ·) is a Cauchy sequence if for every >0, there exists an N ∈ Z>0 such that vi − vj <for all i, j > N. If every Cauchy sequence in V converges, then (V, ·) is called a complete normed vector space or a Banach space. (d) A set U ⊂ V is open in (V, ·) if for each vector v ∈ U,theball of radius r centered at v, B(v; r) ≡{u ∈ V |u − v 0. A set D ⊂ V is closed if its complement V \ D is open. Equivalently, D is closed if the limit of every convergent sequence of vectors {vj } in V with ◦ vj ∈ D for each j lies in D.Theinterior S of a set S ⊂ V is the union of all open sets that are contained in S.Theclosure S (or cl(S))ofS is the intersection of all closed sets containing S; that is, S is the set of limits of sequences in S that converge in V.

Remarks 1. If {uj } and {vj } are sequences in a normed vector space (V, ·), and uj → u and vj → v, then uj + vj → u + v, and for each scalar c, cvj → cv. 7.1 Measures and Integration 379

2. For vectors u and v in (V, ·),wealsohave|u−v| ≤ u + v.In particular, if vj → v, then vj →v. 3. The closure W of any (vector) subspace W of (V, ·) is a subspace. 4. Any closed subspace of a Banach space is a Banach space with respect to the restriction of the given norm. 5. If V is a ﬁnite-dimensional vector space, then V is complete with respect to any norm. Moreover, any two norms in V are equivalent, so convergence of sequences, openness of sets, etc., are independent of the choice of norm. Moreover, a sequence {vj } in V converges to a vector v if and only if α(vj ) → α(v) for every linear ∗ functional α on V (equivalently, for some basis α1,...,αn of the dual space V , αk(vj ) → αk(v) for k = 1,...,n).

~~Example 7.1.7 For F = R or C, Fn together with the Euclidean norm n 1/2 2 n ζ = |ζj | ∀ζ = (ζ1,...,ζn) ∈ F j=1~~

~~(in fact, with any norm) is a Banach space (we set F0 ={0}).~~

Deﬁnition 7.1.8 Let (X, μ) be a measure space, and let p ∈[1, ∞]. (a) Let f be a measurable function on X.Ifp<∞, then for each measurable function f ,theLp norm of F is given by 1/p p f Lp(X,μ) ≡ |f | dμ . X The L∞ norm of f is given by

~~f L∞(X,μ) ≡ inf{R ∈[0, ∞] | |f |≤R a.e.}.~~

~~p We also denote the L norm by ·Lp(μ) or ·Lp(X) or ·Lp when the measure or space is clear from the context. (b) The Lp space is given by~~

~~p L (X, μ) ≡{f | f is a measurable function with f Lp < ∞}/∼,~~

where f ∼ g if and only if f = g a.e. We usually denote each equivalence class by a representative f . We also denote the Lp space by Lp(μ) or Lp(X) or Lp when the measure or the space are clear from the context. (c) For Lebesgue measure λ on Rn, we often denote Lp(Y, λ) simply by Lp(Y ) ⊂ Rn p for any Lebesgue measurable set Y .WeletLloc(Y ) denote the set of measurable functions f on Y such that each point in Y admits a neighborhood U Rn ∈ p ∩ p in such that f U∩Y L (U Y) (in particular, for Y an open set, Lloc(Y ) is the set of functions for which the restriction to each compact subset of Y is in Lp), where as in (b), we identify any two measurable functions that are equal 1 almost everywhere. Elements of Lloc are called locally integrable. 380 7 Background Material on Analysis in Rn and Hilbert Space Theory

Theorem 7.1.9 If (X, μ) is a complete measure space and p ∈[1, ∞], then p (L (X, μ), ·Lp ) is a Banach space. Moreover, if {fν } is a sequence that con- p p verges in L (X, μ) to a function f ∈ L (X, μ), then some subsequence of {fν } converges pointwise almost everywhere to f .

Remarks 1. In this book, the fact that ·Lp is a norm is used, in an essential way, only for p ∈{1, 2, ∞}.Forp = 1 this follows easily from the inequality | f |≤ |f |;forp =∞, this follows easily from the triangle inequality for |·|; and for = · p 2, this follows from the observation that L2 is the norm associated to an inner product (see Sect. 7.5). Completeness of Lp is used only for p = 2or∞. For the measure spaces of interest in this book, completeness of L2 follows from Proposition 2.6.3. The proof of completeness of Lp for p ∈[1, ∞) is similar. The proof of completeness of L∞ is relatively easy (see, for example, [Rud1]). The claim concerning pointwise convergence of a subsequence may be veriﬁed directly { } as follows. We may choose a subsequence fνk such that ∞ ∞ ∞ − p = | − |p > fνk f Lp fνk f k=1 k=1 (for example, by the monotone convergence theorem). Hence the integrand is ﬁnite almost everywhere, and the claim follows. 2. For any compact set K ⊂ Rn, the vector space C0(K) of continuous functions on K is a closed vector subspace of L∞(K), since uniform convergence preserves continuity. ∈[ ∞] 1 + 1 = 3. We also recall that for p,q 1, with p q 1 and for measurable ≤ · functions f,g on (X, μ),wehavetheHölder inequality fg L1(X,μ) f Lp(X,μ) gLq (X,μ). In this book, this inequality is used, in an essential way, only for the (relatively easy) cases p = 1 and p = 2. 4. For a measurable set Y ⊂ Rn, we say that a sequence of measurable functions { } p fν on Y converges in Lloc(Y ) to a function f on Y if each point in Y admits a p ∩ neighborhood U such that f U∩Y and fν U∩Y for each ν are in L (U Y), and fν − f Lp(U∩Y) → 0.

Theorem 7.1.10 (Fubini’s theorem) Let m, n ∈ Z>0, and let f be a (Lebesgue) measurable function on Rm+n = Rm × Rn. Then (a) For almost every point x ∈ Rm, the function f(x,·) is measurable on Rn. | | Rm (b) The function x → Rn f(x,y) dλ(y) is measurable on . Moreover, f is integrable on Rm+n if and only if this function is integrable on Rm. (c) If f is nonnegative or integrable on Rm+n, then fdλ= f(x,y)dλ(y) dλ(x) Rm+n Rm Rn = f(x,y)dλ(x) dλ(y). Rn Rm 7.1 Measures and Integration 381

~~A proof is outlined in Exercise 7.1.3 (a proof for a general σ -ﬁnite product measure space can be found in, for example, [Fol]or[Rud1]).~~

Exercises for Sect. 7.1 7.1.1 Let λ∗ and λ denote Lebesgue outer measure and Lebesgue measure, respectively, on Rn (see Example 7.1.4). (a) Verify that λ∗ is countably subadditive. (b) Verify that λ is a positive measure and that λ has the properties (i)Ð(iv) in Example 7.1.4. 7.1.2 Let f be a nonnegative measurable function on a measure space (X, μ). Prove { }∞ ≤ ≤ that there is a sequence of simple functions ϕν ν=1 on X such that 0 ϕν ∈ Z → →∞ → ϕν+1 for each ν >0, ϕν f as ν , and ϕν A f A uniformly on any set A ⊂ X on which f is bounded. Hint. For example, consider

~~ν ν 2 j − 1 ϕ ≡ · χ −1 [ − ν ν + ν · χ −1 [ ∞] . ν 2ν f ( (j 1)/2 ,j/2 )) f ( ν, ) j=1~~

7.1.3 The goal of this exercise is a proof of Fubini’s theorem (Theorem 7.1.10). Let m n m+n m m, n ∈ Z>0, and for every set S ⊂ R × R = R and every point x ∈ R , n let Sx ≡{y ∈ R | (x, y) ∈ S}. Also ﬁx an open (m + n)-cell R = A × B with m n A R and B R ;letS0 be the collection of all measurable sets S ⊂ R n for which the set Sx ⊂ B ⊂ R is measurable for almost every point x ∈ A; and let S ⊂ S0 be the subcollection consisting of all sets S ∈ S0 for which the function on A given (almost everywhere) by x → λ(Sx ) is measurable and

~~λ(S) = λ(Sx )dλ(x) A~~

m+n n (here, λ(S) and λ(Sx ) denote the Lebesgue measure in R and R , respectively). (a) Prove that S0 is a σ -algebra in R, and that S0 contains the collection of Borel subsets of R. (b) Prove that ∅∈S and that R \ S ∈ S for each set S ∈ S. { } S ⊂ (c) Let Sν be a sequence of sets in . Prove that if Sν Sν+1 for each ν { } ∈ S ⊃ or the sets Sν are disjoint, then ν Sν . Prove also that if Sν Sν+1 ∈ S for each ν, then ν Sν . (d) Prove that every Gδ and every Fσ in R are in S. Hint. First show that every open subset of Rd is a countable union of disjoint d-cells. (e) Prove that every set S ⊂ R of measure 0 is in S. Hint. S is contained in a Gδ of measure 0, and Lebesgue measure is complete. (f) Prove that S is precisely the collection of Lebesgue measurable subsets of R. 382 7 Background Material on Analysis in Rn and Hilbert Space Theory

~~m+n n (g) Let E be a measurable subset of R . Prove that the set Ex ⊂ R is m measurable for almost every point x ∈ R , the function x → λ(Ex ) is measurable, and ~~

λ(E) = λ(Ex)dλ(x). Rm (h) Prove Fubini’s theorem (Theorem 7.1.10). Hint. First prove that the theorem holds for the characteristic function of a measurable set, and then that the theorem holds for a nonnegative simple function. Obtain the theorem for a nonnegative measurable function by applying Exercise 7.1.2. Finally, prove that the theorem holds for a measurable complex-valued function. 7.1.4 Let be an open subset of Rn and let p ∈[1, ∞). The goal of this exercise is a proof that the vector space of continuous functions with compact support p in is dense in Lloc( ). This fact will be applied in the proof of the Friedrichs lemma (Lemma 7.3.1). Let K be compact subset of . (a) Prove that if f is a nonnegative locally integrable function on , then for every >0, there exists a nonnegative simple function ϕ on Rn such that ϕ − f Lp(K) <. Hint.ApplyExercise7.1.2. (b) Prove that if C is a compact subset of an open set U ⊂ Rn, then there exists a nonnegative continuous function η : Rn →[0, 1] such that η ≡ 1 on C and supp η ⊂ U. (c) Prove that if E is a measurable subset of Rn with characteristic function χE, then for every >0, there exists a nonnegative continuous function n η : R →[0, 1] such that η − χELp(K) <. Hint.Forδ>0 sufﬁciently small, ﬁx an open set U and a compact set C such that C ⊂ E ∩ K ⊂ U and λ(U \ C) < δ (see Example 7.1.4). Now apply part (b) above. ∈ p (d) Prove that if f Lloc( ), then for every >0, there exists a continuous function g with compact support in such that g − f Lp(K) <. 7.1.5 Prove that if is an open subset of Rn and p ∈[1, ∞), then the vector space of continuous functions with compact support in is dense in Lp( ). Hint. Apply Exercise 7.1.4.

~~7.2 Differentiation and Integration in Rn~~

~~In this section, we recall some facts concerning differentiation and integration on Rn that will allow us to consider differentiation and integration on manifolds (see Chap. 9).~~

Deﬁnition 7.2.1 Let k ∈ Z≥0 ∪{∞,ω},letF = R or C, and let be an open subset of Rn. (a) A function f : → F is of class Ck if one of the following holds: 7.2 Differentiation and Integration in Rn 383

(i) We have k = 0 and f is continuous; (ii) We have k ∈ Z>0 and all of the partial derivatives of f of order ≤ k exist and are continuous on ; (iii) We have k =∞and f has continuous partial derivatives of all orders; (iv) We have k = ω and f is real analytic (that is, in a neighborhood of n every point a = (a1,...,an) ∈ R , we may write f as a power se- ∞ = − m1 ··· − mn ries f(x1,...,xn) m ,...,m =0 cm1,...,mn (x1 a1) (xn an) for 1{ n } F some choice of constants cm1,...,mn in ). It is convenient to consider any function on R0 ={0} to be of class Ck. (b) The vector space of Ck functions on with values in F is denoted by Ck( , F), or simply by Ck( ). We also denote C∞( , F) by E( , F) or by E( ).The vector space of C∞ F-valued functions with compact support in is denoted by D( , F) or by D( ). For an arbitrary set S ⊂ Rn, the vector space of continuous F-valued functions on S is denoted by C0(S, F) or by C0(S). (c) For f ∈ C1( , F) and p ∈ ,thedifferential of f at p is the F-linear map n (df )p : F → F given by

~~n ∂f n (df )p(v) = vj · (p) ∀v = (v1,...,vn) ∈ F . ∂xj j=1~~

~~We also denote by df the mappings p → (df )p and (p, v) → (df )p(v) (the latter mapping df : × Fn → F being of class Ck−1 if f ∈ Ck( ) with k ∈ Z>0 ∪{∞}).~~

Deﬁnition 7.2.2 Let k ∈ Z≥0 ∪{∞,ω},letF = R or C,let be an open subset n m of R , and let F = (f1,...,fm): → F be a mapping. k k (a) F is of class C if fj ∈ C ( ) for each j = 1,...,m. (b) If F is of class C1, then for each point p ∈ ,thedifferential of F at p is the n m F-linear map (dF )p = ((df1)p,...,(dfm)p): F → F given by

~~n ∂F n (dF )p(v) = vj · (p) ∀v = (v1,...,vn) ∈ R . ∂xj j=1~~

n m We also denote by dF the mapping p → (dF )p ∈ Hom(F , F ) and the map- n m ping (p, v) → (dF )p(v) (the latter mapping dF : × F → F being of class k−1 k C if F is of class C with k ∈ Z>0 ∪{∞}). (c) For F of class C1 and m = n,theJacobian determinant is the function ∂fi JF = det(dF ) = det : → F. ∂xj

(d) If F maps bijectively onto an open set ⊂ Rm and both F and F −1 are of class C∞, then we call F a diffeomorphism of onto (in particular, m = n). 384 7 Background Material on Analysis in Rn and Hilbert Space Theory

Remarks 1. If F : → Rm, then under the inclusion Rm ⊂ Cm, we may consider, n m for each point p ∈ , the differential (dF )p : C → C , which is the complex linear n m extension of the real linear map (dF )p : R → R . 2. Although we often identify C with R2, one must proceed with caution when considering differentials. For if F is a C1 mapping of an open set ⊂ Rn into R2m n 2m and p ∈ , then the complex linear extension (dF )p : C → C of the real lin- n 2m ear differential (dF )p : R → R is clearly not the same as the differential mapping Cn → Cm that one obtains by considering F as a mapping of into Cm.In this book, the context will determine which differential mapping is being considered. 3. According to the chain rule,ifF : → Rm and G: → Fl are Ck map- n m k pings with ⊂ R , ⊂ R , and k ∈ Z>0 ∪{∞}, then G ◦ F is of class C and

~~n l d(G◦ F)p = (dG)F(p) ◦ (dF )p : C → C~~

−1 for each point p ∈ F ( ) (where for F = C, the above mapping (dF )p is the complex linear extension). In particular, if F : → is a diffeomorphism, ∈ −1 = −1 = then for each point p , (dF )p d(F )F(p), and hence m n in this case. 4. By the intermediate value theorem, a continuous injective real-valued function f on an interval I ⊂ R is either strictly increasing or strictly decreasing, and f maps I homeomorphically onto an interval J ; that is, f −1 : J → I is continuous (in particular, the image of any endpoint of I is an endpoint of J ). It follows easily that if I is open and f is of class C∞ with nonvanishing derivative, then f : I → J is a diffeomorphism. The higher-dimensional analogue is called the C∞ inverse function theorem (Theorem 9.9.1). This (nontrivial) theorem and its consequences are applied in the study of the topological and holomorphic structure of Riemann surfaces in Sects. 5.10Ð5.17, and in the study of holomorphic structures on smooth surfaces in Chap. 6.

Deﬁnition 7.2.3 A function f onasetS ⊂ Rn is said to be locally Lipschitz on S if for each point in S, there are a neighborhood U and a constant C = C(U) > 0 such that |f(y)− f(x)|≤Cy − x for all x,y ∈ S ∩ U. We say that f is uniformly Lipschitz on S if there is a constant C>0 such that |f(y)− f(x)|≤Cy − x for all x,y ∈ S.

~~Lemma 7.2.4 If f is a C1 function on an open set ⊂ Rn, then f is uniformly Lipschitz on every compact subset of .~~

~~The proof is left to the reader (see Exercise 7.2.1).~~

Proposition 7.2.5 (Differentiation past the integral) Let ⊂ Rn, let (X, μ) be a measure space, and let u be a complex-valued function on . Assume that the func- 7.2 Differentiation and Integration in Rn 385 tion u(t, ·) is integrable on X for each ﬁxed t ∈ , and let v(t) = u(t, x) dμ(x) ∀t ∈ . X

(a) Suppose that t0 ∈ , u(·,x) is continuous at t0 for each fixed x ∈ X, and there exists a nonnegative integrable function g on X with |u(t, x)|≤g(x) for each point (t, x) ∈ × X. Then v is continuous at t0. (b) Suppose that is open, j ∈{1,...,n}, ∂u/∂tj is defined on × X, and there exists a nonnegative integrable function g on X with |(∂u/∂tj )(t, x)|≤g(x) for each point (t, x) ∈ × X. Then ∂v/∂tj is defined on and ∂v ∂u (t) = (t, x) dμ(x) ∀t = (t1,...,tn) ∈ . ∂tj X ∂tj

Proof Part (a) follows from the dominated convergence theorem. Clearly, for the proof of (b), we may assume that n = 1, that is an open interval, and that u is real- valued. As a limit of a sequence of measurable functions, ∂u/∂t is measurable in the variable x ∈ X. By hypothesis, |(∂u/∂t)(t, x)|≤g(x) for each point (t, x) ∈ ×X, and hence ∂u/∂t is integrable in x ∈ X. For distinct points t,t0 ∈ and for x ∈ X, we have, by the mean value theorem, − u(t, x) u(t0,x) ∂u − (t0,x) ≤ 2g(x). t − t0 ∂t

~~Thus, once again, the dominated convergence theorem gives the claim.~~

n Lemma 7.2.6 (Taylor’s formula to order one) If a = (a1,...,an) ∈ R and f is a C2 function on a neighborhood of a with a + t(x − a) ∈ for each point x ∈ and each t ∈[0, 1] (that is, is starlike about a), then for each point x = (x1,..., xn) ∈ , we have

n n f(x)= f(a)+ bj · (xj − aj ) + cij (x)(xi − ai)(xj − aj ), j=1 i,j=1 where for all indices i, j = 1,...,n(and for coordinate functions u = (u1,...,un)), ∂ bj ≡ [f(u)] ∂uj u=a and 1 ∂2 cij (x) ≡ (1 − t) [f(u)] dt. 0 ∂ui∂uj u=a+t(x−a) 386 7 Background Material on Analysis in Rn and Hilbert Space Theory

Proof For each point x = (x1,...,xn) ∈ , the fundamental theorem of calculus and integration by parts give 1 d f(x)− f(a)= [f(a+ t(x − a))] dt 0 dt d =−(1 − t) [f(a+ t(x − a))] dt t=1 d + (1 − t) [f(a+ t(x − a))] dt t=0 1 d2 + ( − t) [f(a+ t(x − a))] dt. 1 2 0 dt The chain rule now gives the desired expression for f(x).

~~Remark Differentiation past the integral (Proposition 7.2.5) implies that if f ∈ Ck( ) for some k ∈ Z≥2 ∪{∞}, then each of the functions x → cij (x) in the lemma is of class Ck−2.~~

We close this section with a consideration of the behavior of integrals under diffeomorphisms. We ﬁrst recall that the image P = A(Q) of the compact n-cell n Q =[0,s1]×···×[0,sn]⊂R (with si > 0fori = 1,...,n) under a linear map A: Rn → Rn is a compact parallelepiped of measure

~~λ(P ) =|det A|·vol(Q) =|det A|·s1 ···sn.~~

This formula is trivial for n = 1, and relatively easy to verify for n = 2(seeEx- ercise 7.2.2), and ultimately, these are the only cases required in this book. For a general diffeomorphism, we have the following:

Theorem 7.2.7 (Change of variables formula) Let F : → be a diffeomorphism of an open set ⊂ Rn onto an open set ⊂ Rn. Then the composition u ◦ F of any measurable function u on with F is a measurable function on . Moreover, if u is a nonnegative measurable function on , then

udλ= (u ◦ F)·|JF | dλ, where JF = det(dF ) is the Jacobian determinant of F . If u is a (complex-valued) integrable function on , then the function (u ◦ F)|JF | is integrable on and the above equality holds. In particular, the image F(E) ⊂ of a set E ⊂ is measurable if E is measurable, and F(E)is of measure 0 if E is of measure 0.

A proof is outlined in Exercise 7.2.3. The proof may be modiﬁed to give the change of variables formula for a C1 diffeomorphism (i.e., a bijective C1 mapping with C1 inverse), but only the C∞ case is applied in this book. 7.2 Differentiation and Integration in Rn 387

Example 7.2.8 Polar coordinates. Polar coordinates play an important role in the study of complex analysis in the plane. We first recall one of the many equivalent constructions of the real trigonometric functions.Thereal arcsine function is → [− ] [− ] the strictly increasing homeomorphism x arcsin x of 1, 1 onto π/2,π/2 → x − 2 −1/2 ≡ 1 − 2 −1/2 given by x 0 (1 t ) dt, where π −1(1 t ) dt. The restriction to (−1, 1) is a diffeomorphism onto (−π/2,π/2); that is, the function and its inverse function (−π/2,π/2) → (−1, 1) are of class C∞.Thereal sine function is the unique extension θ → sin θ of the inverse function of the arcsine function to a 2π-periodic function that satisfies sin(θ + π) =−sin θ for each θ ∈ R.Thereal cosine function is the unique 2π-periodic function θ → cos θ determined by cos θ = 1 − sin2 θ ∀θ ∈[−π/2,π/2] and cos(θ + π)=−cos θ ∀θ ∈ R. The real tangent function,thereal cotangent function,thereal secant function, and the real cosecant function are then given by sin θ π tan θ ≡ ∀θ ∈ R + Z · π , cos θ 2 cos θ cot θ ≡ ∀θ ∈ R \ (Z · π), sin θ 1 π sec θ ≡ ∀θ ∈ R + Z · π , cos θ 2 1 csc θ ≡ ∀θ ∈ R \ (Z · π). sin θ The verification that these real trigonometric functions are of class C∞ with the familiar derivative functions is straightforward (see Exercise 7.2.4). We also get ≡ −1 ≡ the remaining real inverse trigonometric functions arccos (cos [0,π]) ,arctan −1 ±∞ ≡± (tan [−π/2,π/2]) (arctan( ) π/2), etc., and their derivative functions. The familiar formulas for the sine and cosine of a sum will follow from consideration of the complex exponential function in Chap. 1 (see Example 1.6.2 and Exercise 1.6.2). 2 For any fixed θ0 ∈ R, the mapping [0, ∞) ×[θ0,θ0 + 2π)→ R given by

(r, θ) → (r cos θ,r sin θ) is a continuous surjection. Moreover, the corresponding restrictions yield a bijection 2 of the domain (0, ∞) ×[θ0,θ0 + 2π) onto the punctured plane R \{(0, 0)}, and a 2 diffeomorphism of the domain (0, ∞) × (θ0,θ0 + 2π) onto the complement R \ P 2 of the ray P ≡{(r cos θ0,rsin θ0) | r ≥ 0} in R (see Exercise 7.2.5). For any point (x, y) ∈ R2 and any pair (r, θ) ∈[0, ∞) × R with (r cos θ,r sin θ)= (x, y), we call (r, θ) polar coordinates for (x, y). Under the identiﬁcation of R2 with C = R + iR, 388 7 Background Material on Analysis in Rn and Hilbert Space Theory we also write eiθ ≡ cos θ + i sin θ = (cos θ,sin θ) ∀θ ∈ R. The Jacobian determinant of the diffeomorphism (r, θ) → (r cos θ,r sin θ) described above is given by cos θ −r sin θ = r. sin θrcos θ Hence, if f is an integrable function or a nonnegative measurable function on R2, then for any choice of θ0 ∈ R, Fubini’s theorem (Theorem 7.1.10) and the change of variables formula (Theorem 7.2.7)give ∞ ∞ f(x,y)dx dy = f (x, y) dλ((x, y)) −∞ −∞ R2 = f(rcos θ,r sin θ)r dλ((r, θ)) (0,∞)×(θ0,θ0+2π) ∞ θ0+2π = f(rcos θ,r sin θ)rdθ dr. 0 θ0 In particular, the function (x, y) → rp = (x2 + y2)p/2 is locally integrable on R2 for p>−2 (and integrable on the complement of any neighborhood of (0, 0) for p<−2).

Remark Recall that, similarly, one may define the real logarithmic function to be ∞ → R → ≡ x −1 the diffeomorphism (0, ) given by x log x 1 t dt, and one may define the real exponential function x → exp(x) = ex to be the corresponding C∞ inverse function. Clearly, these functions have derivative functions x → 1/x and x → 1/(1/ exp(x)) = exp(x), respectively. Moreover, the verifications of the basic algebraic properties of these functions are straightforward (see Exercise 7.2.6).

Exercises for Sect. 7.2 7.2.1 Prove Lemma 7.2.4. 7.2.2 Prove that the image P = A(Q) of the compact 2-cell Q =[0,s1]×[0,s2]⊂ 2 2 2 R (with si > 0fori = 1, 2) under a linear map A: R → R satisﬁes

λ(P ) =|det A|·vol(Q) =|det A|·s1 · s2. 7.2.3 The goal of this exercise is a proof of the change of variables formula (The- orem 7.2.7). Let F = (f1,...,fn): → be a diffeomorphism of open ⊂ Rn ∈ = 1 n : → sets , , and for each point p ,letGp (gp,...,gp) n Gp( ) ⊂ R be the diffeomorphism given by → −1 − = −1 − x (dF )p (F (x) F(p)) (dF )F(p)(F (x) F (p)). Also, ﬁx a relatively compact open subset U of . 7.2 Differentiation and Integration in Rn 389

(a) Prove that there exists a positive constant C such that for each point a = (a1,...,an) ∈ U, each positive number s for which the compact cu- bic n-cell R =[a1,a1 + s]×···×[an,an + s] is contained in U, and each point x = (x1,...,xn) ∈ R,wehave

~~− 2 ≤ i ≤ + 2 ∀ = Cs ga(x) s Cs i 1,...,n.~~

~~Hint. Apply Lemma 7.2.6. (b) Prove that for C>0 and R =[a1,a1 + s]×···×[an,an + s]⊂U as in (a), we have~~

~~2 2 F(R)⊂F(a)− (dF )a(Cs ,...,Cs ) 2 2 + (dF )a([0,s+ 2Cs ]×···×[0,s+ 2Cs ]),~~

~~and conclude from this that the compact set F(R)satisﬁes~~

n λ(F (R)) ≤|JF (a)|·vol(R) · (1 + 2Cs) . ⊂ ≤ |J | (c) Prove that the open set F(U) satisﬁes λ(F (U)) U F dλ. Hint. First prove that for each ν0 ∈ Z>0, U is equal to a countable union of disjoint n-cells of the form

~~−ν −ν −ν −ν [i12 ,(i1 + 1)2 ) ×···×[in2 ,(in + 1)2 )~~

∈ Z ∈ Z with i1,...,in and ν >ν0 . ∗ ≤ |J | ⊂ (d) Prove that λ (F (E)) E F dλ for every measurable set E . Con- clude that in particular, the image of every set of measure 0 under F is a set of measure 0. Finally, prove that the image of every measurable set under F is measurable. (e) Prove the change of variables formula (Theorem 7.2.7). Hint.Foru a nonnegative measurable function on , show that u ◦ F is measurable. Given disjoint measurable sets E1,...,Em ⊂ and constants r1,...,rm ∈[0, ∞] with u ≥ rj on Ej for each j = 1,...,m, prove that m rj λ(Ej ) ≤ (u ◦ F)|JF | dλ. j=1 ≤ ◦ |J | Pass to the supremum to get udλ (u F) F dλ. Apply this to −1 the diffeomorphism F and the function (u ◦ F)·|JF | on to get the reverse inequality. 7.2.4 Verify that the trigonometric functions (as deﬁned in Example 7.2.8)areof class C∞ with derivative functions 390 7 Background Material on Analysis in Rn and Hilbert Space Theory

d d sin θ = cos θ, cos θ =−sin θ, dθ dθ d d tan θ = sec2 θ, cot θ =−csc2 θ, dθ dθ d d sec θ = sec θ tan θ, csc θ =−csc θ cot θ. dθ dθ 2 7.2.5 For a ﬁxed number θ0 ∈ R,let:[0, ∞) ×[θ0,θ0 + 2π)→ R be the continuous mapping given by (r, θ) → (r cos θ,r sin θ). (a) Prove that maps the set (0, ∞) ×[θ0,θ0 + 2π) bijectively onto the punctured plane R2 \{(0, 0)} (as claimed in Example 7.2.8). (b) Prove that maps the domain (0, ∞) × (θ0,θ0 + 2π) diffeomorphically 2 onto the complement R \ P of the ray P ={(r cos θ0,rsin θ0) | r ≥ 0} in R2. 7.2.6 Verify that the real logarithmic and exponential functions (as deﬁned in the remark following Example 7.2.8) satisfy

~~x+y x y log(xy) = log x + log y ∀x,y ∈ R>0 and e = e · e ∀x,y ∈ R.~~

~~7.3 C∞ Approximation~~

~~A convenient way in which to obtain a C∞ approximation of a locally integrable function is to form a molliﬁer as follows:~~

∞ Lemma 7.3.1 (Friedrichs) Let κ be a nonnegative C function with compact sup- n = port in the unit ball B(0; 1) in R such that Rn κdλ 1, and let be an open sub- Rn ∈ 1 δ ≡ −n set of . For each function u Lloc( ) and for each δ>0, let κ (x) δ κ(x/δ) n n n for each point x ∈ R , let δ ≡{x ∈ R | dist(x, R \ ) > δ}, and let δ uδ(x) ≡ u(y)κ (x − y)dλ(y) = u(x − y)κδ(y) dλ(y) B(0;δ) = u(x − δy)κ(y) dλ(y) B(0;1)

∈ ∈ 1 for every x δ. Then, for every function u Lloc( ), we have the following: ∞ (a) The function uδ belongs to C ( δ) for every δ>0. ⊂ − → → + → (b) For every compact set K , uδ u L1(K) 0 as δ 0 (i.e., uδ u in 1 ∈ C0 → Lloc( )). Moreover, if u ( ), then uδ u uniformly on K. 7.3 C∞ Approximation 391

~~Remarks 1. A stronger version of the lemma (which is applied in Chap. 6) appears in Sect. 11.3 (see Lemma 11.3.1). 2. An example of such a function κ is given by~~

C−1 exp(−1/(1 − 4x2)) if x < 1/2, κ(x) = 0ifx≥1/2, where − − 2 C = e 1/(1 4 x ) dλ(x). B(0;1/2) 3. The family of functions {κδ} is called a (positive) molliﬁer (or an approximation of the identity), and {uδ} is called a molliﬁcation (or a regularization)ofu.The convolution of two (suitable) functions f and g on Rn is the function f ∗ g : x → f(x− y)g(y)dλ(y). Rn

~~δ n Thus, in the above lemma, uδ is the convolution of κ and an extension of u to R .~~

~~Proof of Lemma 7.3.1 Part (a) follows from differentiation past the integral (Propo- sition 7.2.5). For the proof of (b), we ﬁx a compact set K ⊂ and we set~~

~~Kδ ≡{x ∈ Rn | dist(x, K) ≤ δ}~~

~~n δ for each δ>0. Fixing δ0 with 0 <δ0 < dist(K, R \ ), we get K 0 ⊂ . Suppose ﬁrst that u is continuous. If for 0 <δ≤ δ0,weset~~

~~δ Mδ ≡ sup{|u(x) − u(y)||x,y ∈ K 0 , |x − y| <δ},~~

~~+ then by uniform continuity, we have Mδ → 0asδ → 0 . For each x ∈ K,wehave |uδ(x) − u(x)|= u(x − δy)κ(y) dλ(y) − u(x)κ(y)dλ(y) B(0;1) B(0;1) ~~

≤ |u(x − δy) − u(x)|·κ(y)dλ(y) ≤ Mδ. B(0;1) → → + − → Thus uδ u uniformly on K as δ 0 , and hence in particular, uδ u L1(K) 0asδ → 0+. ∈ 1 For a general u Lloc( ),given>0, we may choose a continuous function − ∈ v on such that v u L1(Kδ0 ) </3 (see Exercise 7.1.4). For each δ (0,δ0), Fubini’s theorem (see Theorem 7.1.10 and Exercise 7.3.1) then implies that − = − − − vδ uδ L1(K) (v(x δy) u(x δy))κ(y) dλ(y) dλ(x) K B(0;1) 392 7 Background Material on Analysis in Rn and Hilbert Space Theory ≤ |(v(x − δy) − u(x − δy))κ(y)| dλ(y)dλ(x) K B(0;1) = |v(x − δy) − u(x − δy)|dλ(x) κ(y)dλ(y) B(0;1) K = |v(x) − u(x)| dλ(x) κ(y)dλ(y) B(0;1) K+(−δy) ≤ |v(x) − u(x)| dλ(x) κ(y)dλ(y) B(0;1) Kδ0 = − v u L1(Kδ0 ) </3. − On the other hand, for δ>0 sufﬁciently small, we have vδ v L1(K) </3. Hence − ≤ − + − + − uδ u L1(K) uδ vδ L1(K) vδ v L1(K) v u L1(K) <. Thus (b) is proved.

~~One consequence of the Friedrichs lemma is the following uniqueness property:~~

~~Rn ∈ 1 Lemma 7.3.2 If is an open subset of , u, v Lloc( ), and uϕ dλ = vϕ dλ ∀ϕ ∈ D( ), then u = v almost everywhere in .~~

Proof We may assume without loss of generality that v ≡ 0. In the notation of Lemma 7.3.1,wehaveuδ ≡ 0on δ for every δ>0. On the other hand, for ev- ⊂ − → ery compact set K ,wehave uδ u L1(K) 0, and therefore, for some se- { } = → = quence δν ,0 uδν u pointwise almost everywhere in K. Thus u 0almost everywhere in .

Exercises for Sect. 7.3 7.3.1 Verify that the function (x, y) → |v(x − δy) − u(x − δy)| for (x, y) ∈ K × B(0; 1) in the proof of Lemma 7.3.1 is measurable (hence Fubini’s theorem applies).

~~7.4 Differential Operators and Formal Adjoints~~

n Deﬁnition 7.4.1 For each multi-index α = (α1,...,αn) ∈ (Z≥0) ,weset|α|= α1 +···+αn and ∂ α α1 αn |α| = ∂ ··· ∂ = ∂ α1 ··· αn ∂x ∂x1 ∂xn ∂x1 ∂xn 7.4 Differential Operators and Formal Adjoints 393

(in particular, for α = (0,...,0), (∂/∂x)αu = u for every function u). A linear dif- n ferential operator of order k ∈ Z≥0 with coefﬁcients {aα} on an open set ⊂ R is a linear map A from Ck( ) to the space of complex-valued functions on given by α ∂ k Au = aα u ∀u ∈ C ( ), n ∂x α∈(Z≥0) ,|α|≤k

n where aα is a complex-valued function on for each α ∈ (Z≥0) with |α|≤k.For any open set U ⊂ , we also denote the linear differential operator α ∂ : Ck → C0 aα U (U) (U) n ∂x α∈(Z≥0) ,|α|≤k by A, unless there is danger of confusion. We deﬁne the conjugate operator A¯ by α ¯ ∂ k Au = Au¯ = a¯α u ∀u ∈ C ( ). n ∂x α∈(Z≥0) , |α|≤k

Remark In practice, we usually work with simpler expressions obtained by allowing l the multi-indices to be in (Z≥0) for 1 ≤ l ≤ n. For example, if A is a second-order linear differential operator, then we may write

n ∂2 n ∂ A = aij + bi + c, ∂xi∂xj ∂xi i,j=1 i=1 and refer to {aij }, {bi}, and c as the coefﬁcients of A. Observe also that by replacing { } { 1 } = aij with 2 aij , we may assume that aij aji for all i and j (i.e., that the matrix (aij ) is symmetric).

~~One may consider an extension of a differential operator to a space of functions that are not necessarily differentiable in the following way:~~

~~Deﬁnition 7.4.2 Let A be a linear differential operator of order k with (complex) n |α| coefﬁcients {aα} on an open set ⊂ R such that aα ∈ C ( ) for each multi- index α.~~

(a) The formal transpose of A is the linear differential operator t A of order k given by α t |α| ∂ k Au ≡ (−1) [aα · u]∀u ∈ C ( ). n ∂x α∈(Z≥0) , |α|≤k 394 7 Background Material on Analysis in Rn and Hilbert Space Theory

~~(b) The formal adjoint of A is the linear differential operator A∗ of order k given by α ∗ t ¯ |α| ∂ k A u ≡ Au = (−1) [¯aα · u]∀u ∈ C ( ). n ∂x α∈(Z≥0) , |α|≤k~~

~~∈ 1 (c) For u, v Lloc( ), we say that Au is equal to v in the distributional (or weak) sense, and we write Adistru = v,if u · A∗ϕdλ= v ·¯ϕdλ ∀ϕ ∈ D( ).~~

~~Equivalently, u · t Aϕ dλ = v · ϕdλ ∀ϕ ∈ D( ).~~

Remarks 1. The function Adistru is unique (if it exists) by Lemma 7.3.2. ∈ 1 2. For any function u Lloc( ), Adistr u always exists in a more general sense, → · t D namely, as the linear functional ϕ u Aϕ dλ on ( ) (see, for example, ∈ 1 [Rud2]). For this reason, to say Adistru Lloc( ) will mean that Adistru exists as a 1 function in Lloc( ).

Lemma 7.4.3 Let A and B be linear differential operators with C∞ coefﬁcients on an open set ⊂ Rn, and let k be the order of A. ∈ Ck = ∈ 1 = (a) If u ( ) (or k 0 and u Lloc( )), then Adistru Au. ∈ 1 = ∈ (b) For u, v Lloc( ), we have Adistru v in if and only if each point p ⊂ [ ]= admits a neighborhood U with Adistr u U v U . (c) We have t (t A) = A, (A∗)∗ = A, t (ζ A + B) = ζ t A + t B, and (ζ A + B)∗ = ζA¯ ∗ + B∗ for each ζ ∈ C, t (AB) = t Bt A, and (AB)∗ = B∗A∗. ∈ 1 ∈ C + = + (d) Let u, v Lloc( ) and let ζ . Then (ζ A B)distru ζAdistru Bdistru, pro- ∈ 1 + = + vided Adistru, Bdistru Lloc( ); and Adistr(ζ u v) ζAdistru Adistrv pro- ∈ 1 vided Adistru, Adistrv Lloc( ). ∈ 1 = (e) Suppose u, Bdistru, v Lloc( ). Then we have (AB)distru v if and only if Adistr[Bdistru]=v. 1 (f) Suppose u, Adistru ∈ L ( ), k = 1, the 0th-order term of A vanishes, and ∞ loc ρ ∈ C ( ). Then Adistr[ρu]=ρAdistru + uAρ.

Proof Part (a) is trivial for k = 0, and it follows easily from integration by parts in each variable for k>0. The details are left to the reader (see Exer- cise 7.4.1). ∈ 1 For the proof of (b), suppose that u, v Lloc( ) and each point admits a neighborhood in which Au is equal to v in the distributional sense. Then, given a function ∈ D C∞ { }m ≡ ϕ ( ), we may choose ﬁnitely many functions ην ν=1 such that ην 1 on supp ϕ and such that for each ν, supp ην lies in an open subset Uν of on 7.4 Differential Operators and Formal Adjoints 395 which Au is equal to v in the distributional sense (see, for example, Sect. 9.3). Hence ∗ ∗ u · A ϕdλ= u · A [ην ϕ] dλ = v · ηνϕdλ= vϕdλ.¯ ν Uν ν Uν

The converse is trivial. For (c), we observe that if ϕ,ψ ∈ D( ), then ψ ·[Aϕ] dλ = [t Aψ]·ϕdλ= ψ · t (t A)ϕ dλ, and ψ ·[t (AB)ϕ] dλ = [(AB)ψ]·ϕdλ = [Bψ]·[t Aϕ] dλ = ψ ·[t Bt Aϕ] dλ.

Lemma 7.3.2 now implies that t (t A)ϕ = Aϕ and t (AB)ϕ = t Bt Aϕ. For any function u ∈ C∞( ) and any point p ∈ , there exists a function ϕ ∈ D( ) such that ϕ ≡ u near p, so it follows that t (t A) = A and t (AB) = t Bt A. The remaining claims in (c) are easy to verify directly. The proof of (d) is left to the reader (see Exercise 7.4.1). ∈ 1 ∈ 1 For the proof of part (e), let u Lloc( ) with Bdistru Lloc( ), and let ϕ ∈ D( ). Then t t t t u ·[ (AB)ϕ] dλ = u ·[ B Aϕ] dλ = [Bdistru]· Aϕ dλ.

~~∈ 1 If (AB)distru Lloc( ), then the ﬁrst expression is equal to ~~

~~[(AB)distru]·ϕdλ,~~

~~and comparison with the last expression gives AdistrBdistru = (AB)distru. Similarly, ∈ 1 if AdistrBdistru Lloc( ), then the last expression is equal to ~~

~~[AdistrBdistru]·ϕdλ,~~

and comparison with the ﬁrst expression gives (AB)distru = AdistrBdistru. = = ∈ 1 For the proof of (f), we assume that k 1, Adistru v Lloc( ), the 0th-order term of A vanishes, and ρ ∈ C∞( ). An easy computation shows that for ϕ ∈ D( ), 396 7 Background Material on Analysis in Rn and Hilbert Space Theory we have t A(ρϕ) = ρ ·(t Aϕ)−(Aρ)·ϕ (thus A∗(ρϕ) = ρ ·(A∗ϕ)−(Aρ)¯ ·ϕ). Hence ρu· (t Aϕ) dλ = u · t A(ρϕ) dλ + u · (Aρ) · ϕdλ= [ρv + u · Aρ]·ϕdλ, and (f) follows.

Remarks 1. The terminology in Deﬁnition 7.4.2 is motivated by the notion of the adjoint of a linear operator on a Hilbert space (see, for example, [Rud2]) and the equality ∗ = ∀ ∈ D ϕ,A ψ L2( ) Aϕ, ψ L2( ) ϕ,ψ ( ) (see Sect. 7.5) provided by Lemma 7.4.3. 2. Parts (a) and (b) of Lemma 9.8.3 hold even if A has Ck coefﬁcients.

∞ Lemma 7.4.4 Let κ be a nonnegative C function with compact support in the n n unit ball B(0; 1) in R such that Rn κdλ= 1, and let be an open subset of R . ∈ 1 δ ≡ −n ∈ Rn For each u Lloc( ) and δ>0, let κ (x) δ κ(x/δ) for each point x , let n n δ ≡{x ∈ R | dist(x, R \ ) > δ}, and let δ δ uδ(x) ≡ u(y)κ (x − y)dλ(y) = u(x − y)κ (y) dλ(y) B(0;δ) = u(x − δy)κ(y) dλ(y) B(0;1) n for every x ∈ δ. If A is a constant-coefﬁcient linear differential operator on R , ∈ 1 ∈ 1 [ ]=[ ] and u Lloc( ) with Adistru Lloc( ), then A uδ Adistru δ in δ for every δ>0. In particular, for every compact set K ⊂ , we have − → → + A(uδ) Adistru L1(K) 0 as δ 0 .

~~Proof Given δ>0 and a function ϕ ∈ D( δ), we have, by Fubini’s theorem (see Theorem 7.1.10 and Exercise 7.4.2), t uδ(x)[ Aϕ](x) dλ(x)~~

δ = u(x − y)κδ(y) dλ(y) ·[t Aϕ](x) dλ(x) B(0;δ) δ = u(x − y) ·[t Aϕ](x) dλ(x) κδ(y) dλ(y) B(0;δ) δ = u(x) ·[t Aϕ](x + y)dλ(x) κδ(y) dλ(y) B(0;δ) δ+(−y) = u(x) ·[t A(ϕ(·+y))](x) dλ(x) κδ(y) dλ(y) B(0;δ) δ+(−y) 7.5 Hilbert Spaces 397 δ = [Adistru](x)ϕ(x + y)dλ(x) κ (y) dλ(y) B(0;δ) +(−y) δ

~~= [Adistru]δ(x)ϕ(x) dλ(x) δ~~

~~(that A has constant coefﬁcients gives the fourth equality). Therefore, A(uδ) = Adistr(uδ) =[Adistru]δ (by the uniqueness property provided by Lemma 7.3.2).~~

Exercises for Sect. 7.4 7.4.1 Prove parts (a) and (d) of Lemma 7.4.3. 7.4.2 Verify that the function (x, y) → u(x − y)κδ(y)[t Aϕ](x) for (x, y) ∈ δ × B(0; δ) in the proof of Lemma 7.4.4 is integrable (hence Fubini’s theorem applies).

~~7.5 Hilbert Spaces~~

Hilbert spaces (or complete Hermitian inner product spaces) are (possibly inﬁnite- dimensional) vector spaces in which one has many of the useful geometric tools (for example, orthogonal projection) that one has in Rn. They play a crucial role in this book. In this section, we recall some of the basic elements of Hilbert space theory. Proofs of most of the relevant facts required in this book are either provided or outlined in the exercises. Further facts concerning weak sequential compactness, which are applied only in Chap. 6, are considered in Sect. 7.6. We ﬁrst recall the following:

Deﬁnition 7.5.1 Let V be a complex vector space. (a) A Hermitian inner product (or simply an inner product)onV is a function ·, ·: V × V → C such that for all u, v, w ∈ V and ζ ∈ C,wehave (i) ζu+ v,w=ζ u, w+v,w, (ii) u, v=v,u, and (iii) v,v > 0ifv = 0. The pair (V, ·, ·) is called a Hermitian inner product space (or simply an inner product space). √ (b) The norm associated to ·, · is the function v → v≡ v,v. (c) Two vectors u and v are orthogonal with respect to ·, · if u, v=0 (equivalently, v,u=0). We write u ⊥ v.Ifu ⊥ v and u=v=1, then u and v are orthonormal. A nonempty set S ⊂ V is an orthonormal set if each pair of distinct vectors in S are orthonormal. (d) For every nonempty set S ⊂ V, we deﬁne S⊥ ≡{v ∈ V | v ⊥ u ∀u ∈ S}.

Remark Similarly, a real inner product (or simply an inner product) on a real vector space V is a real symmetric bilinear form g on V such that g(v,v) > 0 for each v ∈ V \{0}. 398 7 Background Material on Analysis in Rn and Hilbert Space Theory

Theorem 7.5.2 For any Hermitian inner product space (V, ·, ·), we have the following: (a) Law of cosines. We have u − v2 =u2 − 2Reu, v+v2 for all u, v ∈ V. (b) Pythagorean theorem. We have u − v2 =u + v2 =u2 +v2 for every pair of orthogonal vectors u, v ∈ V. (c) Parallelogram law. We have u + v2 +u − v2 = 2u2 + 2v2 for all u, v ∈ V. (d) Schwarz inequality. We have |u, v| ≤ u·v for all u, v ∈ V. (e) ·is a norm on V. (f) The function ·, · is continuous; that is, if {uj } and {vj } are sequences converging to u and v, respectively, in (V, ·), then uj ,vj →u, v. ⊥ (g) For any set S ⊂ V, S⊥ is a closed subspace of (V, ·), and S = S⊥, where S = cl(S) is the closure of S. Moreover, if S is a subspace of V, then S ∩ S⊥ ={0}.

Proof The proofs of (a)Ð(c) are left to the reader (see Exercise 7.5.1). It is easy to see that 0=0, that ζv=|ζ |·v for all v ∈ V and ζ ∈ C, and that v > 0 if v ∈ V \{0}. For the proof of (d), suppose u, v ∈ V \{0}. Setting u = u/u and v = v/v, and ﬁxing ζ ∈ C with |ζ |=1 and ζ u,v=|u,v|, we get, for each t ∈ R, 0 ≤ζu − tv 2 = 1 − 2t|u ,v | + t2. The right-hand side is a quadratic polynomial in t that attains its minimum value at t =|u,v|. Substituting this value for t, we get 0 ≤ 1 − 2|u,v|2 +|u,v|2, and hence

~~|u, v| =|u ,v | ≤ 1. u·v Part (d) now follows. For the proof of (e), it remains to verify the triangle inequality. Given u, v ∈ V, we have~~

~~u + v2 =u2 + 2Reu, v+v2 ≤u2 + 2|u, v| + v2 ≤u2 + 2uv+v2 = (u+v)2, and the triangle inequality follows. For the proof of (f), suppose uj → u and vj → v in V. Then~~

~~|uj ,vj −u, v| ≤ |uj ,vj −uj ,v| + |uj ,v−u, v|~~

~~=|uj ,vj − v| + |uj − u, v|~~

~~≤uj ·vj − v+uj − u·v→u·0 + 0 ·v=0, and hence uj ,vj →u, v. 7.5 Hilbert Spaces 399~~

⊥ For the proof of (g), suppose {uj } is a sequence in S converging to u ∈ V. Then, for every vector v ∈ S,wehave0=uj ,v→u, v. Thus u, v=0, and hence ⊥ u ∈ S⊥. Therefore, S⊥ is closed. Since S ⊂ S,wealsohaveS⊥ ⊃ S . Conversely, ⊥ if u ∈ S and v ∈ S, then we may choose a sequence {vj } in S converging to v. ⊥ ⊥ ⊥ Hence 0 =u, vj →u, v, and it follows that u ∈ S . Thus S = S . It is easy to verify that S⊥ is a subspace. Finally, if S is a (vector) subspace of V and v ∈ S ∩ S⊥, then v,v=0, so v = 0. Thus (g) is proved.

~~Deﬁnition 7.5.3 A Hilbert space is a Hermitian inner product space (H, ·, ·) that is complete with respect to the corresponding norm ·.~~

Example 7.5.4 Given a measure space (X, μ), we get the Hilbert space L2(X, μ) with inner product and norm ≡ ¯ = ∀ ∈ 2 u, v L2(X,μ) uvdμ and u L2(X,μ) u, u L2(X,μ) u, v L (X, μ). X

For the inequality |uv|≤(1/2)(|u|2 +|v|2) implies that a product of two L2 functions is integrable (alternatively, one may use Hölder’s inequality). It is then easy to check that the above is an inner product. Completeness is the case p = 2 of Theo- rem 7.1.9. On the other hand, for the related Hilbert spaces considered in this book, completeness will be proved directly (see Proposition 2.6.3 and Proposition 3.6.4).

Example 7.5.5 For any set X, 2(X) denotes the L2 space associated to X together with the counting measure. The inner product and norm are then given by = = | |2 ∀ ∈ 2 u, v 2(X) u(x)v(x) and u 2(X) u(x) u, v (X). x∈X x∈X

~~In particular, taking X = Z>0, we get ∞ ∞ 2 2 u, v 2 = u(n)v(n) u 2 = |u(n)| ∀u, v ∈ (Z ). (Z>0) and (n) >0 n=1 n=1~~

Theorem 7.5.6 If V is a closed subspace of a Hilbert space (H, ·, ·), then we have the direct sum decomposition H = V ⊕ V⊥. That is, for each vector u ∈ H, there are unique vectors v ∈ V and w ∈ V⊥ such that u = v + w. Moreover, we have u − s > w for all s ∈ V \{v}.

The above decomposition H = V ⊕ V⊥ is called the associated orthogonal decomposition. The surjective linear mapping u → v ∈ V (with w = u − v ∈ V⊥)is called the orthogonal projection of H onto V. 400 7 Background Material on Analysis in Rn and Hilbert Space Theory

~~Proof of Theorem 7.5.6 Given a vector u ∈ H, we may choose a sequence {vj } in V such that~~

~~u − vj →r ≡ inf u − v∈[0, ∞). v∈V According to the parallelogram law (see Theorem 7.5.2),~~

2 2 vi − vj =(u − vi) − (u − vj ) 2 2 2 = 2u − vi + 2u − vj −(u − vi) + (u − vj ) 2 2 2 = 2u − vi + 2u − vj − 4u − (1/2)(vi + vj ) 2 2 2 ≤ 2u − vi + 2u − vj − 4r .

Since we can make the last expression arbitrarily small by requiring that i and j be sufﬁciently large, {vj } is a Cauchy sequence, and therefore, {vj } converges to some vector v ∈ V. Setting w ≡ u − v, we see that for each vector s ∈ V, w=r ≤u − s. Fixing ζ ∈ C with |ζ|=1 and ζ w,s=|w,s|, we get, for each t ∈ R,

~~− u − (v + tζ 1s)2 =ζw− ts2 =w2 − 2t|w,s| + t2s2.~~

The quadratic polynomial on the right-hand side attains its minimum value at t = 0, so we must have w,s=0. Thus w ∈ V⊥. The decomposition u = v + w is unique, since V ∩ V⊥ ={0}. In particular, if s ∈ V with u − s=r, then the minimizing = = sequence given by vj s for all j must converge to v; that is, s v.

Corollary 7.5.7 For any subspace W of a Hilbert space (H, ·, ·), we have the following: (a) (W⊥)⊥ = W. (b) If V is a subspace of H with W ∪W⊥ ⊂ V, then W is closed in V (i.e., W ∩V = W) if and only if V = W ⊕ W⊥.

⊥ ⊥ Proof We have W ⊥ W and W = W⊥,soW ⊂ (W⊥)⊥. Conversely, if u ∈ (W⊥)⊥, then we have u = w + v with w ∈ W and v ∈ W⊥. Subtracting, we get v = u − w ∈ (W⊥)⊥ ∩ W⊥ ={0},sou ∈ W. Thus (a) is proved. Suppose W, W⊥ ⊂ V as in (b). Given u ∈ V,wehaveu = w + v, where w ∈ W ⊥ and v ∈ W = W⊥ ⊂ V. In particular, w = u − v ∈ W ∩ V.Thus,ifW is closed in V, then w ∈ W. Conversely, if V = W ⊕ W⊥ and u ∈ W ∩ V, then writing ⊥ ⊥ u = w + v with w ∈ W and v ∈ W⊥ = W , we get v = u − w ∈ W ∩ W ={0}. Thus u = w ∈ W and hence W ∩ V = W.

~~Remark Theorem 7.5.6 and Corollary 7.5.7 do not hold in general for an inner product space (see Exercise 7.5.2). 7.5 Hilbert Spaces 401~~

~~Deﬁnition 7.5.8 Let (V, ·) be a normed vector space over F = R or C. A bounded linear functional α on V is a linear functional α : V → F (i.e., an F-valued linear map) for which the norm~~

~~α≡inf{R ∈[0, ∞] | |α(v)|≤Rv∀v ∈ V} is ﬁnite. The set of all bounded linear functionals on V is called the (norm) dual space of V and is denoted by V∗.~~

Remark On a ﬁnite-dimensional normed vector space V, every linear functional is bounded (see, for example, [Fol]or[Rud1]). Thus the algebraic dual space (see Sect. 8.1) is equal to the norm dual space of V if dim V < ∞.

Proposition 7.5.9 For any normed vector space (V, ·) with V = {0}, we have the following: ∈ V ∗ = | | = | | (a) For each α , α supv∈V\{0} α(v) / v supv∈V;v=1 α(v) . (b) The dual space (V∗, ·) is a Banach space.

~~The proof is left to the reader (see Exercise 7.5.3). The following important theorem allows one to identify a Hilbert space with its dual space up to conjugate linear isomorphism.~~

Theorem 7.5.10 Let (H, ·, ·) be a Hilbert space. Then, for each vector v ∈ H, the mapping H → C given by u → u, v is a bounded linear functional with norm equal to v. Conversely, given a bounded linear functional α on H, there is a unique vector v ∈ H such that α(u) =u, v for all u ∈ H (in particular, α=v).

Proof It follows immediately from the deﬁnition of an inner product that for each v ∈ H, the mapping α : H → C given by u → u, v is a linear functional. The Schwarz inequality implies that |α(u)|≤vu for each vector u ∈ H,soα is a bounded linear functional with α≤v. Setting u = v, we get α≥v, and hence we have equality. Observe also that if w ∈ H with ·,w=α =·,v, then setting u = v − w, we get

u2 =u, u=u, v−u, w=0, and hence v = w. In particular, this observation gives the uniqueness claim in the theorem. Conversely, given an element α ∈ H∗, the kernel V ≡ ker α = α−1(0) is a closed subspace of H (see Exercise 7.5.4). If α ≡ 0, then α =·, 0.Ifα = 0, then we may ﬁx an element w ∈ V⊥ \{0} and we may set ζ ≡ α(w)/w2 ∈ C \{0}. The vector v ≡ ζw∈ V⊥ then satisﬁes

~~|α(w)|2 α(v) = ζα(w)= =v,v. w2 402 7 Background Material on Analysis in Rn and Hilbert Space Theory~~

~~Given a vector u ∈ H = V ⊕ V⊥,wehaveu = r + s with r ∈ V and s ∈ V⊥.In particular,~~

α(s) ⊥ s − v ∈ V ∩ V ={0}, v2 and hence v,v α(u) = α(s) = α(s) =s,v=u, v. v2 Theorem 7.5.11 (HahnÐBanach theorem) If α is a bounded linear functional on a subspace V of a Hilbert space (H, ·, ·) (that is, α is bounded with respect to the norm obtained by restriction to V of the norm on H), then there exists a bounded linear functional β on H such that βV = α and β=α.

Proof We ﬁrst observe that there exists a unique extension of α to a bounded linear functional on the closure V.Forif{vn} is a sequence in V that converges to a vector v, then for all m, n ∈ Z>0,

~~|α(vm) − α(vn)|=|α(vm − vn)|≤αvm − vn.~~

~~It follows that the sequence {α(vn)} in C is Cauchy and therefore convergent. More- over, if {un} is another sequence in V converging to v, then~~

~~|α(un) − α(vn)|≤αun − vn→0.~~

~~Thus we get a well-deﬁned mapping γ : V → C by setting~~

γ(v)= lim α(vn), n→∞ for any v ∈ V and any sequence {vn} in V converging to v. In particular, γ V = α. The linearity properties of limits of sequences imply that γ is a linear functional, and since γ is an extension of α,wehaveγ ≥α. On the other hand, for v ∈ V and {vn} as above, we have

|γ(v)|= lim |α(vn)|≤ lim αvn=αv, n→∞ n→∞ and hence γ ≤α. Thus γ is a bounded linear functional on V and γ =α. Letting P be the orthogonal projection of H onto V, we get a linear functional β ≡ γ ◦ P : H → C with βV = γ . In particular, β≥γ =α. Conversely, the Pythagorean theorem (see Theorem 7.5.2)gives

~~|β(v)|=|γ ◦ P(v)|≤αPv≤αv∀v ∈ H, and hence β≤α. Thus β ∈ H∗ and β=α.~~

~~Remark More general versions of the HahnÐBanach theorem can be found in, for example, [Rud1] and [Rud2]. 7.6 Weak Sequential Compactness 403~~

Exercises for Sect. 7.5 7.5.1 Prove parts (a)Ð(c) of Theorem 7.5.2. 7.5.2 Give examples that show that Theorem 7.5.6 and Corollary 7.5.7 do not hold in general for an inner product space. 7.5.3 Prove Proposition 7.5.9. 7.5.4 Show that the kernel of any element of the dual space of an inner product space is a closed subspace. 7.5.5 Suppose {vn} is a sequence in an inner product space (V, ·, ·), v ∈ V, and u, vn→u, v for every u ∈ V (i.e., {vn} converges weakly to v). Prove that {vn} converges (strongly) to v (i.e., vn − v→0) if and only vn→v.

~~7.6 Weak Sequential Compactness~~

This section is required for the discussion of integrability of almost complex structures in Chap. 6. Throughout this section, (H, ·, ·) denotes a Hilbert space. The main goal is the following:

Theorem 7.6.1 (Weak sequential compactness) The closed unit ball in a Hilbert space (H, ·, ·) is weakly sequentially compact; that is, for every bounded se- { } H { } ∈ H quence uν in , there is a subsequence uνk and a vector u such that ≤ u supν uν and → →∞ ∀ ∈ H uνk ,v u, v as k v .

Remark Conversely, by the uniform boundedness principle (see, for example, [Fol], [Rud1], or [Rud2]), a weakly convergent sequence in H (see Exercise 7.5.5) is bounded (we will not need this converse in this book).

~~For the proof, we use complete orthonormal sets, a fundamental object in Hilbert space theory.~~

~~Deﬁnition 7.6.2 An orthonormal set with dense span in H is called a complete orthonormal set (or a complete orthonormal basis).~~

Theorem 7.6.3 If {uν } is a sequence in H \{0}, then V ≡ Span{uν} has a countable orthonormal basis. Consequently, if H has a countable dense subset (i.e., H is separable), then H has a countable complete orthonormal set.

Proof Let us assume that dim V =∞, since the ﬁnite-dimensional case is similar, but easier. By passing to a subsequence (constructed inductively), we may assume that the vectors {uν } are linearly independent. The Gram–Schmidt orthonormalization process then yields an orthonormal basis {eν } for V determined inductively 404 7 Background Material on Analysis in Rn and Hilbert Space Theory by ν−1 u uν − = uν,eμeμ e ≡ 1 and e ≡ μ 1 for ν>1. 1 u ν − ν−1 1 uν μ=1 uν,eμ eμ Remark A Zorn’s lemma argument shows that in fact, every Hilbert space admits a complete orthonormal set.

~~Lemma 7.6.4 (Bessel’s inequality) Let {ei}i∈I be an orthonormal set in H, and for each vector v ∈ H, let v(i)ˆ =v,ei for each index i ∈ I . Then v2 ≥ |ˆv(i)|2. i∈I~~

~~In particular, at most countably many of the complex numbers {ˆv(i)}i∈I are nonzero.~~

Remarks 1. {ˆv(i)}i∈I is the collection of Fourier coefﬁcients of v with respect to the orthonormal set. 2. One can show that in fact, equality in Bessel’s inequality holds if {ei}i∈I is a complete orthonormal set. In fact, v →v ˆ is an isometric isomorphism onto 2(I) in this case (see, for example, [Rud1]).

Proof of Lemma 7.6.4 Given a vector v ∈ H and a ﬁnite set of indices J ⊂ I ,the Pythagorean theorem (Theorem 7.5.2)impliesthat 2 2 2 2 v = v,ej ·ej + v − v,ej ·ej ≥ |ˆv(j)| . j∈J j∈J j∈J The claim now follows.

Proof of Theorem 7.6.1 Given a sequence {uν} in H with uν≤1 for each ν,we = may assume without loss of generality that supν uν 1. We may also assume V ≡ { } H { } that Span uν is dense in . For if there exists a subsequence uνk converging weakly to a vector u with u≤1 in the Hilbert space V, then letting P : H → V be the orthogonal projection, we get = P → P = ∀∈ H uνk ,v uνk , v u, v u, v v .

Theorem 7.6.3 provides a countable orthonormal basis {ei}i∈I for V, which is then a complete orthonormal set in H, and we may let {ˆv(i)}i∈I ={v,ei}i∈I be the Fourier coefficients of each v ∈ H. Applying Bessel’s inequality and Cantor’s {ˆ }∞ diagonal process, we see that we may assume that the sequence uν(i) ν=1 converges to some complex number ζi with |ζi|≤1 for each i ∈ I . For each finite set J ⊂ I , Bessel’s inequality gives 2 2 2 1 ≥uν ≥ |ûν (j)| → |ζj | as ν →∞. j∈J j∈J | |2 ≤ Thus i∈I ζi 1. 7.6 Weak Sequential Compactness 405

¯ We may define a unique linear functional τ on V by setting τ(ei) = ζi for each i ∈ I . For each vector v ∈ V, we then have 2 2 ¯ 2 2 2 |τ(v)| = v(i)ˆ ζi ≤ |ˆv(i)| · |ζi| ≤v i∈I i∈I i∈I ˆ {ˆ } (note that the sum i∈I v(i)ζi is actually a finite sum, since v(i) has at most finitely many nonzero terms). Thus τ is a bounded linear functional of norm at most 1 on V. Applying the HahnÐBanach theorem (or the density of V) and Theo- rem 7.5.10, we get a (unique) vector u ∈ H such that u≤1 and v,u=τ(v) for each v ∈ V. Now, given a vector v ∈ H and a number >0, we may choose a vector w ∈ V with v − w </3. For some finite set J ⊂ I ,wehavewˆ ≡ 0onI \ J ,so ¯ w,uν= w(j)ˆ · uˆν(j) → w(j)ˆ · ζj = τ(w)=w,u as ν →∞. j∈J j∈J

~~Thus, for ν 0, we have |w,uν−w,u| </3, and hence~~

~~|v,uν−v,u| ≤ |v − w,uν | + |w,uν − u| + |w − v,u| <.~~

~~The claim now follows.~~

~~Chapter 8 Background Material on Linear Algebra~~

In this chapter, we recall some basic deﬁnitions and facts concerning exterior products (which are essential in the discussion of differential forms in Sect. 9.5) and tensor products (which are essential in the discussion of holomorphic line bundles in Chap. 3). In this book, we mostly consider exterior and tensor products in vector spaces of dimension 1 or 2.

~~8.1 Linear Maps, Linear Functionals, and Complexiﬁcations~~

The vector space of linear mappings V → W of vector spaces V and W over F = R or C is denoted by Hom(V, W), which is itself a vector space over F (in general, for modules M and N over a ring R, the set of module homomorphisms from M to N is an R-module that is denoted by Hom(M, N )). The (algebraic) dual space of V is ∗ the vector space V ≡ Hom(V, F).IfdimV = n<∞ and e1,...,en is a basis, then there exist unique linear functionals λ1,...,λn with λi(ej ) = δij for i, j = 1,...,n. These linear functionals form a basis for V∗ that is called the associated dual basis. We also have the canonical isomorphism V =∼ (V ∗)∗ given by v(α) = α(v) for all v ∈ V and α ∈ V∗.IfdimV = 1, then for every pair of vectors u, v ∈ V with v = 0, we denote the unique scalar c ∈ F with u = cv by u/v. We also denote the linear functional u → u/v by v−1.

Remarks 1. If V is inﬁnite-dimensional and {eα}α∈A is a basis for V, then we get a collection of linearly independent linear functionals {λα}α∈A determined by λα(eβ ) = 1ifα = β,0ifα = β. However, these linear functionals will not span the dual space, since the unique linear functional λ determined by λ(eα) = 1 for all α ∈ A will not be in the span. Consequently, V (V∗)∗, since there exists a nontriv- ∗ ial linear functional τ on V with τ(λα) = 0 for all α ∈ A (here, we have used the fact that every linearly independent subset of a vector space is contained in a basis). 2. For a normed vector space (V, · ), the associated norm dual space (see Deﬁnition 7.5.8) is equal to the algebraic dual space if and only if dim V < ∞.

T. Napier, M. Ramachandran, An Introduction to Riemann Surfaces, Cornerstones, 407 DOI 10.1007/978-0-8176-4693-6_8, © Springer Science+Business Media, LLC 2011 408 8 Background Material on Linear Algebra

~~For if {eν} is a sequence of linearly independent vectors, then there exists a linear functional λ with λ(eν) = ν eν for each ν ∈ Z>0.~~

~~One obtains a real vector space from a complex vector space simply by restricting scalar multiplication to real scalars. More precisely, we have the following:~~

Proposition 8.1.1 Let V be a complex vector space with vector addition + and scalar multiplication ·. Then the set V together with vector addition given by + and scalar multiplication given by ·R×V determines a real vector space VR of dimension dimR VR√= 2dimC V. Moreover, for any complex basis {eν} for VC, the collection {eν}∪{ −1eν} is a real basis for VR.

~~The proof is left to the reader (see Exercise 8.1.1).~~

Deﬁnition 8.1.2 For a complex vector space V, the associated real vector space given by Proposition 8.1.1 is called the realiﬁcation (or the underlying real vector space)ofV and is denoted by VR or (in a slight abuse of notation) simply by V.

~~One obtains a complex√ vector space VC from a real vector space V by forming all of the formal sums u + −1v for u, v ∈ V. More precisely, we have the following:~~

Proposition 8.1.3 Let V and W be real vector spaces. (a) The set V ⊕ V, together with the standard direct sum vector addition and with scalar multiplication given by z · (u, v) = (xu − yv,yu + xv) for all (u, v) ∈ V ⊕ V and z = x + iy with x,y ∈ R, is a complex vector space VC of dimension dimC V = dimR V. Denoting the element (u, v) ∈ VC by u + iv for all u, v ∈ V, we have z · (u + iv) = (xu − yv) + i(yu + xv) for all z = x + iy with x,y ∈ R. We also have a real linear inclusion V → VC given by v → v + i0 ↔ (v, 0). (b) The map VC → VC given by u + iv → u + iv ≡ u − iv (i.e., (u, v) → (u, −v)) is a conjugate linear isomorphism. (c) Any real basis for V is a complex basis for VC. (d) If α, β ∈ Hom (V, W) are two real linear maps, then the complex linear extension of the map λ = α + iβ, i.e., the map λ: VC → WC (which, in an abuse of notation, we give the same name) given by λ(u + iv) = α(u)− β(v)+ i(β(u)+ ¯ α(v)) ∈ WC for all u, v ∈ V, is a complex linear mapping. Moreover, setting λ ≡ ¯ α − iβ = α + i(−β), we get λ(w) = λ(w)¯ for each w ∈ VC. The above corre- ∼ spondence gives a (canonical) isomorphism [Hom(V, W)]C = Hom(VC, WC). ∗ ∼ ∗ In particular, we have (V )C = (VC) . Furthermore, α ∈ Hom(V, W) is injective (surjective) if and only if the associated complex linear extension is injective (respectively, surjective).

~~The proof is left to the reader (see Exercise 8.1.2). 8.2 Exterior Products 409~~

Definition 8.1.4 For a real vector space V, the associated complex vector space VC given by Proposition 8.1.3 is called the complexification√ of V. We denote each element (u, v) ∈ VC by w = u + iv = u + −1v, we call Re w = u its real part, we call Im w = v its imaginary part, and we call w¯ = u − iv its conjugate. We identify V with V + i0 ⊂ VC. Given a real vector space W, we identify each element λ ∈[Hom (V, W)]C ⊃ Hom (V, W) with its complex linear extension λ ∈ Hom (VC, WC) (for each λ ∈ Hom (VC, WC),wehaveλ = Re λ + i Im λ, where (Re λ)(v) = Re(λ(v)) and (Im λ)(v) = Im(λ(v)) for each v ∈ V). We also use the ∗ ∗ ∗ identification and notation VC ≡ (V )C = (VC) .

~~Example 8.1.5 (Rn)C = Cn under the identiﬁcation~~

~~n x + iy ↔ (x1 + iy1,...,xn + iyn) ∀x = (x1,...,xn), y = (y1,...,yn) ∈ R .~~

~~Similarly, we have (Cn)R = R2n.~~

Exercises for Sect. 8.1 8.1.1 Prove Proposition 8.1.1. 8.1.2 Prove Proposition 8.1.3. 8.1.3 Prove that the dual space of any inﬁnite-dimensional vector space over F = R or C cannot have a countable basis.

~~8.2 Exterior Products~~

~~Throughout this section, V denotes a vector space over F = R or C. In this book, we require exterior products only of degree ≤ 2, so we will restrict our attention to this case.~~

Deﬁnition 8.2.1 For the vector space V: (a) A bilinear function on V (or V × V)isamapα : V × V → F that is linear in each entry; that is, for all (u1,v1), (u2,v2) ∈ V × V and for each ζ ∈ F,wehave

~~α(ζu1,v1) = α(u1,ζv1) = ζα(u1,v1),~~

~~α(u1 + u2,v1) = α(u1,v1) + α(u2,v1),~~

α(u1,v1 + v2) = α(u1,v1) + α(u1,v2). The vector space of bilinear functions on V × V (a vector subspace of the space of F-valued functions) is denoted by V∗ ⊗ V∗ and is called the tensor product of V∗ with itself. (b) An element θ ∈ V∗ ⊗ V∗ is skew-symmetric (symmetric)ifθ(u,v)=−θ(v,u) (respectively, θ(u,v) = θ(v,u)) for all u, v ∈ V.Forα, β ∈ V∗,theexterior (or wedge) product is the skew-symmetric bilinear function α ∧ β given by (α ∧ β)(u, v) = α(u)β(v)− β(u)α(v) for all u, v ∈ V. The vector (sub)space of skew-symmetric bilinear functions is denoted by 2V∗. 410 8 Background Material on Linear Algebra

Remarks 1. For vector spaces (or modules) U, V, W,amapα : U × V → W that is linear in each factor is called a bilinear pairing. 2. Another standard deﬁnition for ∧ includes a factor of 1/2.

~~Proposition 8.2.2 For the vector space V, we have the following: (a) For all α, β, γ ∈ V∗ and ζ ∈ F,~~

~~(ζ α) ∧ β = α ∧ (ζβ) = ζ · (α ∧ β), (α + β)∧ γ = α ∧ γ + β ∧ γ, α ∧ β =−β ∧ α.~~

2 ∗ ∗ (b) If dim V = 1, then V ={0}. If dim V = 2 and λ1,λ2 is a basis for V , then 2 ∗ 2 ∗ λ1 ∧ λ2 is a basis for V (in particular,dim V = 1). In fact, if e1,e2 is a basis for V with dual basis λ1,λ2, then α = α(e1,e2)λ1 ∧ λ2 for every α ∈ 2V∗. (c) If F = R and α, β ∈ V∗ ⊗ V∗ are two (real) bilinear functions, then the complex bilinear extension of the function λ = α + iβ, i.e., the map λ: VC × VC → C (which we give the same name) given by

~~λ(ζ, ξ) =α(u,r) − α(v,s) − β(u,s)− β(v,r) + i(α(u,s) + α(v,r) + β(u,r)− β(v,s))~~

for ζ = u + iv and ξ = r + is with u, v, r, s ∈ V, is a (complex) bilinear func- ¯ ¯ ¯ ¯ tion on VC. Setting λ = α − iβ = α + i(−β), we get λ(ζ, ξ) = λ(ζ,ξ) for all ζ,ξ ∈ VC. Moreover, if α and β are skew-symmetric, then λ is skew-symmetric. The above correspondence gives (canonical) isomorphisms

~~∗ ∗ ∼ ∗ ∗ 2 ∗ ∼ 2 ∗ [V ⊗ V ]C = VC ⊗ VC and [ V ]C = (VC).~~

Proof The proofs of parts (a) and (c) are left to the reader (see Exercise 8.2.1). For ∗ the proof of (b), suppose dim V = 2, let λ1,λ2 be a basis for V , and let e1,e2 be the dual basis for V. For each τ ∈ 2V∗,wehave,forallu, v ∈ V,

~~τ(u,v)= τ(λ1(u)e1 + λ2(u)e2,λ1(v)e1 + λ2(v)e2)~~

~~= (λ1(u)λ2(v) − λ2(u)λ1(v)) · τ(e1,e2)~~

~~= τ(e1,e2) · (λ1 ∧ λ2)(u, v).~~

~~2 ∗ Thus V = F · λ1 ∧ λ2. On the other hand, (λ1 ∧ λ2)(e1,e2) = 1, so λ1 ∧ λ2 is a basis. It is clear that if dim V = 1, then 2V∗ ={0}.~~

Remarks 1. We set 0V∗ = F, 1V∗ = V∗, and ζ ∧ α = ζ · α ∈ pV∗ if ζ ∈ 0V∗ and α ∈ pV∗ with p ∈{0, 1, 2}.IfdimV ≤ 2, then we set pV∗ ={0} for all p>2, and we set α ∧ β = 0 for all α ∈ pV∗, β ∈ q V∗ with p + q>2. It 8.2 Exterior Products 411

p ∗ q ∗ r ∗ follows that if dim V ≤ 2, p,q,r ∈ Z≥0, α ∈ V , β ∈ V , and γ ∈ V , then α ∧ β = (−1)pqβ ∧ α and (α ∧ β) ∧ γ = α ∧ (β ∧ γ)(we denote the latter simply by α ∧ β ∧ γ ). If dim V > 2, then we leave pV∗ undeﬁned for p>2, since such spaces are not required in this book. 2. For F = R and p ∈{0, 1, 2} or p>2 ≥ dim V, Proposition 8.2.2 allows us to p ∗ p ∗ p ∗ use the identiﬁcation and notation VC ≡[ V ]C = (VC).

Deﬁnition 8.2.3 Assume that F = R and that n ≡ dim V = 1or2. n ∗ (a) We say that two nonzero elements λ1,λ2 ∈ V have equivalent orientations if λ1/λ2 > 0 (this is an equivalence relation with exactly two equivalence classes). (b) An equivalence class as in (a) is called an orientation in V∗ (or in V) and the other equivalence class is called the opposite orientation. If an element α ∈ 2V∗ is in the equivalence class determined by a given orientation, then α is said to be positive (or positively oriented) with respect to the orientation, we write α>0, and we say α induces the orientation.Ifα induces the opposite orientation, then α is said to be negative (or negatively oriented) with respect to the given orientation, and we write α<0. If α>0(< 0) or α = 0, then we say that α is nonnegative (respectively, nonpositive) and we write α ≥ 0 (respectively, α ≤ 0). An ordered basis (v1,...,vn) (= v1 or (v1,v2))forV is positively oriented if each positive element α of nV∗ is positive on the ordered basis (i.e., α(v1)>0ifn = 1, α(v1,v2)>0ifn = 2). (c) Given an orientation in V∗ and given two elements α, β ∈ nV∗, we write α>β (α ≥ β)ifα − β>0 (respectively, α − β ≥ 0). In this context, for a sequence n ∗ {αν} in V , we may deﬁne αν αν inf αν ≡ inf · θ, sup αν ≡ sup · θ, ν ν θ ν ν θ αν αν lim inf αν ≡ lim inf · θ, lim sup αν ≡ lim sup · θ, ν→∞ ν→∞ θ ν→∞ ν→∞ θ

~~for an arbitrary positive element θ ∈ 2V∗, provided the above deﬁned coefﬁ- cients exist in R.~~

~~n ∗ Remark It is easy see that for a sequence {αν} in V as in (c) above, we have lim αν = α if and only if lim inf αν = lim sup αν = α.~~

2 ∗ Example 8.2.4 For the standard dual basis λ1,λ2 in (R ) , the orientation induced by λ1 ∧ λ2 is the right-handed orientation, while that represented by −λ1 ∧ λ2 = λ2 ∧ λ1 is the left-handed orientation.

~~Exercises for Sect. 8.2 8.2.1 Prove parts (a) and (c) of Proposition 8.2.2. 412 8 Background Material on Linear Algebra~~

~~8.3 Tensor Products~~

Tensor products are required for the study of holomorphic line bundles in Chaps. 3 and 4 (as well as in the last part of Chap. 5). Throughout this section, F denotes the ﬁeld R or C. We consider tensor products from the point of view of bilinear functions (cf. Deﬁnition 8.2.1).

Deﬁnition 8.3.1 Let U and V be vector spaces over F. (a) A bilinear function on U × V is a map α : U × V → F that is linear in each entry; that is, for all (u1,v1), (u2,v2) ∈ U × V and for each ζ ∈ F,wehave

~~α(ζu1,v1) = α(u1,ζv1) = ζα(u1,v1),~~

~~α(u1 + u2,v1) = α(u1,v1) + α(u2,v1),~~

~~α(u1,v1 + v2) = α(u1,v1) + α(u1,v2).~~

We call the vector space of bilinear functions on U × V (a vector subspace of the space of F-valued functions) the tensor product of U ∗ and V∗ and we denote it by U∗ ⊗ V∗. For dim U, dim V < ∞,thetensor product of U and V is given by U ⊗ V ≡ (U∗)∗ ⊗ (V∗)∗, under the identiﬁcation of U with (U∗)∗ and V with (V∗)∗. (b) The tensor product of α ∈ U ∗ and β ∈ V∗ is the element α ⊗ β ∈ U∗ ⊗ V∗ deﬁned by [α ⊗ β](u, v) ≡ α(u) · β(v) for each pair (u, v) ∈ U × V.

~~Proposition 8.3.2 Let U, V, and W be ﬁnite-dimensional vector spaces over F. Then we have the following: (a) For all α, β ∈ U ∗, γ,δ ∈ V∗, and ζ ∈ F, we have~~

(ζ α) ⊗ γ = α ⊗ (ζ γ ) = ζ · (α ⊗ γ), (α + β)⊗ γ = α ⊗ γ + β ⊗ γ, α ⊗ (γ + δ) = α ⊗ γ + α ⊗ δ. { }m { }n U∗ V∗ (b) If θi i=1 and λj j=1 are bases for and , respectively, then the ten- ∗ ∗ sor products {θi ⊗ λj }1≤i≤m, 1≤j≤n form a basis for U ⊗ V . In particular, dim U ∗ ⊗ V∗ = mn. (c) There exist unique (surjective) isomorphisms =∼ (i) U ∗ ⊗ V∗ → V∗ ⊗ U∗ satisfying α ⊗ β → β ⊗ α for all α ∈ U∗ and β ∈ V∗; =∼ (ii) (U∗ ⊗V∗)⊗W∗ → U∗ ⊗(V∗ ⊗W∗) satisfying (α ⊗β)⊗γ → α ⊗(β ⊗γ) for all α ∈ U ∗, β ∈ V∗, and γ ∈ W∗; =∼ (iii) V∗ ⊗ V → F satisfying α ⊗ v → α(v) for all α ∈ V∗ and v ∈ V; =∼ (iv) F ⊗ V∗ → V∗ satisfying ζ ⊗ α → ζ · v for all ζ ∈ F and α ∈ V∗; =∼ (v) : U ∗ ⊗ V → Hom(U, V) satisfying (α ⊗ v)(u) = α(u) · v for all α ∈ U ∗, v ∈ V, and u ∈ U; and 8.3 Tensor Products 413

~~=∼ (vi) : U∗ ⊗ V∗ → (U ⊗ V)∗ satisfying (α⊗ β)(u ⊗ v) = α(u)· β(v) for all α ∈ U∗, β ∈ V∗, u ∈ U, and v ∈ V.~~

Proof The proof of (a) is left to the reader (see Exercise 8.3.1). For the proof of (b), let {ei} be the dual basis of U associated to {θi}, and let {fj } be the dual basis of } ∈ U∗ ⊗ V∗ ∈ U × V V associated to {λj . For each τ and each pair (u, v) ,wehave = m = n u i=1 θi(u)ei and v j=1 λj (v)fj , and hence

m n m n τ(u,v)= θi(u)λj (v)τ(ei,fj ) = τ(ei,fj ) · (θi ⊗ λj )(u, v). i=1 j=1 i=1 j=1 = m n · ⊗ Thus τ i=1 j=1 τ(ei,fj ) θi λj , and it follows that the tensor products { ⊗ } U ∗ ⊗ V∗ { } θi λj 1≤i≤m,1≤j≤nspan . Moreover, if ζij 1≤i≤m, 1≤j≤n are elements F = m n · ⊗ of and τ i=1 j=1 ζij θi λj , then for each choice of indices i and j, evaluation of both of the above expressions for τ on (ei,fj ) gives ζij = τ(ei,fj ). Hence, if τ = 0, then we get ζij = 0 for all i and j. Thus we have linear indepen- dence. The proof of (c), which relies on (a) and (b), is left to the reader (see Exer- cise 8.3.1).

Remarks 1. The above proposition is stated mostly with respect to dual spaces, since this is a more convenient setup for the proofs. However, for finite-dimensional vector spaces, the analogous statements also hold with U, V, and W in place of U ∗, V∗, and W∗, respectively, since U =∼ (U ∗)∗,etc. 2. According to part (c-ii), the tensor product operation on finite-dimensional vector spaces is associative up to a canonical isomorphism. For this reason, we denote the k-fold tensor product of finite-dimensional vector spaces V1,...,Vk by V1 ⊗···⊗Vk . We also identify the pairs of spaces that are canonically isomorphic as in (c). 3. For any finite-dimensional vector space V over F, we denote the k-fold tensor product of V by k V.Wealsoset 0 V = F, and we set ζ ⊗ λ = ζ · λ for all ζ ∈ F and v ∈ V. 4. If U and V are finite-dimensional vector spaces over F,dimV = 1, and ∼ v ∈ V \{0}, then for each ξ ∈ U ⊗ V = V ⊗ U, there is a unique vector u ∈ U = ⊗ = k ⊗ ∈ U ∈ V = with ξ u v.Ifξ j=1 uj vj with uj and vj for each j 1,...,k, then k v u = j · u . v j j=1 Equivalently, u = ξ ⊗ v−1 ∈ U ⊗ V ⊗ V∗ under the identification U =∼ U ⊗ V ⊗ V∗ provided by Proposition 8.3.2. We also denote ξ by u · v or v · u, and u by ξ/v. The verification that ξ/v is well defined by the above is left to the reader (see Exercise 8.3.3). Given a vector w ∈ U, we may form the quotient w/v = w ⊗ v−1 ∈ 414 8 Background Material on Linear Algebra

U ⊗ V∗. We may also view this as the quotient of w ∈ U ⊗ V∗ ⊗ V =∼ U and v. Finally, for any integer m,weset ⎧ ⎪ m factors ⎨⎪ V ⊗···⊗V if m>0, Vm ≡ ⎪F = ⎪ if m 0, ⎩ ∗ − − ∗ [V ] m =∼ [V m] if m<0; and for any t ∈ V,weset ⎧ ⎪ m factors ⎨⎪ t ⊗···⊗t ∈ Vm if m>0, tm ≡ ⎪1 ∈ V0 = F if m = 0 and t = 0, ⎩⎪ − − − − (t 1) m = (t m) 1 ∈ Vm if m<0and t = 0.

Exercises for Sect. 8.3 8.3.1 Prove parts (a) and (c) of Proposition 8.3.2. 8.3.2 Let U and V be two ﬁnite-dimensional real vector spaces. Prove that if α, β ∈ U ∗ ⊗ V∗ are two (real) bilinear functions, then the complex bilinear extension of the function λ = α + iβ, i.e., the map λ: UC × VC → C (which we give the same name) given by

~~λ(ζ, ξ) =α(u,r) − α(v,s) − β(u,s)− β(v,r) + i(α(u,s) + α(v,r) + β(u,r)− β(v,s))~~

for ζ = u + iv and ξ = r + is with u, v ∈ U and r, s ∈ V, is a complex bi- ¯ linear function on UC × VC. Setting λ = α − iβ = α + i(−β), prove that ¯ ¯ ¯ λ(ζ, ξ) = λ(ζ,ξ) for all ζ ∈ UC and ξ ∈ VC. Finally, prove that the above ∗ ∗ ∼ ∗ ∗ correspondence gives a (canonical) isomorphism [U ⊗ V ]C = UC ⊗ VC. 8.3.3 Verify that quotients ξ/v, as deﬁned in Remark 4 following the proof of Proposition 8.3.2, are well-deﬁned. Chapter 9 Background Material on Manifolds

~~In this chapter, we recall some basic deﬁnitions and facts concerning analysis on manifolds (mainly of dimension 1 or 2).~~

~~9.1 Topological Spaces~~

~~In this section, we recall some terminology and facts from point-set topology (see, for example, [Mu]).~~

~~Deﬁnition 9.1.1 A topology onasetX is a collection of subsets T such that~~

(i) X ∈ T and ∅∈T ; U T ∈ T (ii) For every subcollection of ,wehave U∈U U ; and (iii) We have U ∩ V ∈ T for all U,V ∈ T . The pair (X, T ) (usually denoted simply by X)iscalledatopological space. Each element of T is called an open subset of X. For each point x ∈ X, any open set U ⊂ X containing x is called a neighborhood of x.

Deﬁnition 9.1.2 Let (X, T ) be a topological space, and let A ⊂ X. (a) A is a closed subset of X if X \ A is open. ◦ (b) The interior A of A is the union of all open subsets of X contained in A,theclosure A (also denoted by cl(A)) is the intersection of all closed sets containing A, ◦ and the boundary ∂A is the set A¯ \ A. (c) A limit point (or accumulation point)ofA in X is a point p ∈ X such that p ∈ A \{p}. A point in A that is not a limit point is called an isolated point of A.IfA has no limit points in X, then A is called a discrete subset. (d) The collection TA ≡{U ∩ A | U ∈ T } is called the subspace topology on A. (e) A mapping : X → Y of X to a topological space Y is continuous if −1(U) ⊂ X is open for each open set U ⊂ Y .If is bijective with continuous

~~T. Napier, M. Ramachandran, An Introduction to Riemann Surfaces, Cornerstones, 415 DOI 10.1007/978-0-8176-4693-6_9, © Springer Science+Business Media, LLC 2011 416 9 Manifolds~~

inverse, then is called a homeomorphism and we say that X and Y are homeomorphic.If maps a neighborhood of each point in X homeomorphically onto an open subset of Y , then is called a local homeomorphism. Depending on the context, C0(X) will denote either the vector space of real-valued continuous functions on X or the vector space of complex-valued continuous functions on X. (f) The set A is compact if every open covering of A admits a finite subcovering; that is, for every familyU ={Ui}i∈I of open subsets of X that satisfies ⊂ ⊂ ⊂ A i∈I Ui ,wehaveA i∈J Ui for some finite set J I .ThesetA is relatively compact in X if the closure A is compact, and if this is the case, then we write A X. If every open covering of A admits a countable subcovering, then we say that A is σ -compact. (g) The set A is connected if for every pair of open sets U and V with A ⊂ U ∪ V , we have A∩U ∩V = ∅ or A∩U =∅or A∩V =∅. For the equivalence relation ∼ on A determined by x ∼ y if and only if x and y lie in a connected subset of A, each equivalence class is (a connected set) called a connected component (or simply a component)ofA. The space X is locally connected if for every point p ∈ X and every neighborhood U of p, there is a connected neighborhood V of p that is contained in U (equivalently, every connected component of every open subset of X is open in X). (h) X is called a Hausdorff space if for each pair of distinct points x,y ∈ X, there exist open sets U and V with x ∈ U, y ∈ V , and U ∩ V =∅.

Remarks 1. An arbitrary intersection of closed sets is closed. 2. The interior of a set is the largest open set contained in the set, and the closure of a set is the smallest closed set containing the set. 3. The subspace topology on a subset is a topology on the subset. 4. Let A ⊂ X. Then A is compact (connected) if and only if A is compact (respectively, connected) with respect to the subspace topology on A. The restriction of any continuous mapping : X → Y to A is continuous. Moreover, if A is compact (connected), then (A) is also compact (respectively, connected). 5. A compact subset of a Hausdorff space is closed. Consequently, a bijective continuous mapping of compact Hausdorff spaces is a homeomorphism. 6. For any set A ⊂ X, A ={x ∈ X | x is an isolated point or limit point of A}.In particular, A is discrete if and only if A is closed and each point in A is an isolated point of A (see Exercise 9.1.1).

Deﬁnition 9.1.3 A collection B of subsets of a set X is a basis for a topology on X if B covers X and for each pair of sets B1,B2 ∈ B and each point x ∈ B1 ∩ B2, there is an element B ∈ B with x ∈ B ⊂ B1 ∩ B2. The collection T = B U ⊂ B B∈U is called the topology generated by B in X, and B is called a basis for T . A topological space with a countable basis is called second countable. 9.1 Topological Spaces 417

~~Remark The topology generated by a basis B as above is a topology on X. More- over, this topology is equal to the intersection of all topologies on X containing the collection.~~

~~Example 9.1.4 A normed vector space (V, · ) together with the associated open subsets is a topological space (see Deﬁnition 7.1.6).~~

Example 9.1.5 Given topological spaces (Xi, Ti) for i = 1,...,n, the collection B ={U1 ×···×Un | Ui ∈ Ti for i = 1,...,n} is a basis for a topology in the Carte- sian product X = X1 ×···×Xn. The topology T in X generated by B is called the product topology. A mapping = (1,...,n): Y → X of a topological space Y into X is continuous if and only if i is continuous for each i = 1,...,n.The details are left to the reader (see Exercise 9.1.2).

Example 9.1.6 The disjoint union of a family of sets {Aλ}λ∈ is the set A ≡ Aλ ≡{(x, λ) | λ ∈ , x ∈ Aλ}. λ∈ ∈ In other words, each element x λ∈ Aλ determines distinct copies { } ∈ (x, λ) λ∈,x∈Aλ of this element in λ∈ Aλ. For each λ , we identify Aλ with its image in A under the natural inclusion map ιλ : x → (x, λ), provided there is no danger of confusion. In fact, given a set of indices ⊂ and a subset Bλ ⊂ Aλ for ∈ each λ , we identify λ∈ Bλ with its image in A under the natural inclusion. For a family of topological spaces {(Xλ, Tλ)}λ∈, the disjoint union X ≡ T ≡{ | ∈ T ∀ ∈ } λ∈ Xλ inherits the (natural) topology λ∈ Uλ Uλ λ λ .In other words, T is the unique topology in X for which for each λ ∈ , the inclusion ιλ : Aλ → X maps Aλ homeomorphically onto an open set ιλ(Aλ).

Example 9.1.7 Let ∼ be an equivalence relation in a topological space X. Then the quotient topology in the corresponding quotient space Y ≡ X/∼ with quotient map : X → Y ≡ X/∼ is the collection of all sets U ⊂ Y for which −1(U) is open in X (see Exercise 9.1.3). Observe that is a surjective continuous mapping with respect to these topologies. Moreover, for an arbitrary surjective mapping : X → Z of X onto a set Z, the ﬁbers are equivalence classes for the equivalence relation given by x ∼ y ⇐⇒ (x) = (y). Hence we get an induced quotient topology in Z.IfZ has a given topology TZ, then the quotient topology may differ from TZ. However, if is continuous with respect to TZ, then TZ is contained in the quotient topology.

Deﬁnition 9.1.8 Let X be a topological space. (a) For p,q ∈ X,apath (or parametrized path or curve or parametrized curve) from p to q in X is a continuous mapping γ :[a,b]→X for some numbers a,b ∈ R with a

(b) X is path connected if there is a path from p to q for each pair of points p,q ∈ X. The space X is locally path connected if for every point p ∈ X and every neighborhood U of p, there is a path connected neighborhood V of p that is contained in U.

Remarks 1. We take the domain of a path to be the interval [0, 1] unless otherwise speciﬁed. 2. In some contexts, we also call the image of a parametrized path (with certain properties) a path (or curve). A 1-dimensional manifold (see Sect. 9.2)isalso sometimes called a curve. 3. A path connected space is connected, and a connected locally path connected space is path connected (see Exercise 9.1.4).

~~Example 9.1.9 The set ({0}×R) ∪{(x, sin(1/x)) | x>0}⊂R2 is connected, but not path connected.~~

~~Deﬁnition 9.1.10 A Hausdorff space X is locally compact if for every point p ∈ X and every neighborhood U of p, there is a relatively compact neighborhood V of p in U.~~

Remarks 1. A general topological space is called locally compact if each point has a neighborhood that lies in a compact set. For Hausdorff spaces, this condition is equivalent to the above (see, for example, [Mu]). 2. It is easy to see that if U is a neighborhood of a compact subset K in a locally compact Hausdorff space, then there exists a relatively compact neighborhood V of K in U (that is, V is compact and V ⊂ U, which we indicate by writing V U).

Deﬁnition 9.1.11 The one-point compactiﬁcation of a locally compact Hausdorff space (X, TX) is the topological space with underlying set X ∪{∞}obtained by adjoining a point ∞ (not in X)toX and with topology given by

~~TX ∪{(X \ K)∪{∞}|K is a compact subset of X}.~~

~~The element ∞ is called the point at inﬁnity.~~

The one-point compactiﬁcation X ∪{∞}of a locally compact Hausdorff space X is a compact Hausdorff space, and if X is noncompact and connected, then X ∪{∞} is connected (see Exercise 9.1.5).

~~Exercises for Sect. 9.1 9.1.1 Prove that for any subset A of a topological space X,wehave~~

~~A ={x ∈ X | x is an isolated point or limit point of A}.~~

~~Also show that A is discrete if and only if A is closed and each point in A is an isolated point of A. 9.2 The Deﬁnition of a Manifold 419~~

9.1.2 Verify that the product topology on the Cartesian product X = X1 ×···×Xn of (ﬁnitely many) topological spaces X1,...,Xn is a topological space (see Example 9.1.5). Also verify that a mapping = (1,...,n): Y → X of a topological space Y into X is continuous if and only if i is continuous for each i = 1,...,n. 9.1.3 Verify that the quotient topology in a quotient space of a topological space is a topology, and that the corresponding quotient map is continuous (see Example 9.1.7). 9.1.4 Prove that any path connected topological space is connected, and that any connected locally path connected topological space is path connected. 9.1.5 Verify that the one-point compactiﬁcation X ∪{∞} of a locally compact Hausdorff space X is a topological space and that X ∪{∞}is compact Haus- dorff. Prove also that if X is noncompact and connected, then X ∪{∞}is connected.

~~9.2 The Deﬁnition of a Manifold~~

Deﬁnition 9.2.1 Let M be a Hausdorff space, and let n ∈ Z≥0. (a) A homeomorphism : U → U of an open set U ⊂ M onto an open set U ⊂ Rn is called a local chart of dimension n (or an n-dimensional local chart) in M. We also denote this local chart by (U,,U). (b) For any two n-dimensional local charts (U1,1,U1) and (U2,2,U2) in M, the mapping ◦ −1 : ∩ → ∩ 1 2 2(U1 U2) 1(U1 U2) is called a coordinate transformation (see Fig. 9.1). A ={ } (c) A collection of n-dimensional local charts (Ui,i,Ui ) i∈I that covers M = (i.e., for which M i∈I Ui )iscalledanatlas of dimension n on M. (d) If M admits an n-dimensional atlas, then M is called a manifold (or a topological manifold or a C0 manifold) of dimension n. M is also called a (topological or C0) n-manifold. (e) A connected 1-dimensional manifold is also called a curve (or a topological curve). A connected 2-dimensional manifold is also called a surface (or a topological surface or a C0 surface).

Remarks 1. In this book, although we do not assume that a given manifold is second countable, we mostly consider second countable manifolds. The reader should also note that it is common to take second countability as a part of the deﬁnition of a manifold (without any speciﬁc comment). 2. Because we also call a path in a topological space a curve, we will mostly avoid referring to a connected 1-dimensional manifold as a curve unless there is no danger of confusion. 420 9 Manifolds

~~Fig. 9.1 Coordinate transformations~~

Deﬁnition 9.2.2 Let M be Hausdorff space, let k ∈ Z≥0 ∪{∞}, and let n ∈ Z≥0. Ck (a) Two n-dimensional local charts (U1,1,U1) and (U2,2,U2) in M are compatible if the coordinate transformations ◦ −1 : ∩ → ∩ 1 2 2(U1 U2) 1(U1 U2) and [ ◦ −1]−1 = ◦ −1 : ∩ → ∩ 1 2 2 1 1(U1 U2) 2(U1 U2) are of class Ck . (b) An atlas consisting of Ck compatible local charts in M is called a Ck atlas on M. k k (c) Two n-dimensional C atlases A1 and A2 on M are C equivalent if A1 ∪ A2 is a Ck atlas (this is an equivalence relation). (d) An equivalence class S of n-dimensional Ck atlases on M is called a Ck structure of dimension n on M. The pair (M, S) (usually denoted simply by M)is called a Ck manifold of dimension n (or a Ck n-manifold). (e) Any local chart (U,,U) in any Ck atlas in the Ck structure on a Ck k n-manifold M is called a local C chart in M. Setting x = (x1,...,xn) = , we call x local Ck coordinates, and for each point p ∈ U, we call (U, x) a local Ck coordinate neighborhood of p. (f) A C∞ manifold is also called a smooth manifold. (g) A connected 2-dimensional Ck manifold (smooth manifold) is also called a Ck surface (respectively, smooth surface). A connected 1-dimensional Ck manifold (smooth manifold) is also called a Ck curve (respectively, smooth curve).

~~Remark A real analytic manifold is deﬁned analogously (i.e., the coordinate transformations are required to be real analytic).~~

n Examples 9.2.3 For n ∈ Z>0, R is a smooth (in fact, real analytic) manifold of dimension n with C∞ atlas {(Rn,x → x,Rn)}. The unit sphere Sn is a C∞ (in fact, C∞ A ={ + }n+1 ∪ real analytic) manifold of dimension n with atlas (Ui ,i,Vi) i=1 { − }n+1 = + (Ui ,i,Vi) i=1 , where for each i 1,...,n 1, : ± ≡{ = ∈ Sn |± }→ ≡{ ∈ Rn | } i Ui x (x1,...,xn+1) xi > 0 Vi t t < 1 9.2 The Deﬁnition of a Manifold 421 is the mapping given by (x1,...,xn+1) → (x1,...,xi,...,xn+1). Any open subset of a Ck manifold M is a Ck manifold with Ck atlas consisting of the restrictions to of local Ck coordinates in M. k k The Cartesian product M1 × M2 of two C manifolds M1 and M2 is a C manifold, called the product manifold, with Ck atlas consisting of local Ck charts given by

(U1 × U2,(x,y) → (1(x), 2(y)), V1 × V2) k for local C charts (U1,1,V1) and (U2,2,V2) in M1 and M2, respectively. k { } For any family of C manifolds Mλ λ∈of dimension n with correspond- {A } ≡ ing atlases λ λ∈, the disjoint union M λ∈ Mλ, with inclusion mappings k ιλ : Mλ → M for λ ∈ , inherits the structure of a C manifold of dimension n with atlas given by A ≡{ ◦ −1 | ∈ A ∈ } (ιλ(U), ιλ ,U ) (U,,U ) λ for some λ . The veriﬁcations of the above are left to the reader (see Exercise 9.2.1).

k Deﬁnition 9.2.4 For k ∈ Z≥0 ∪{∞}, a continuous mapping : M → N of C man- k k ifolds M and N is of class C if for every choice of local C charts (U1,1,V1) and (U2,2,V2) in M and N, respectively, the mapping ◦ ◦ −1 : ∩ −1 → 2 1 1(U1 (U2)) V2 is of class Ck. The vector space of Ck functions on M with values in F = R or C is denoted by Ck(M, F). Depending on the context, Ck(M) will denote either Ck(M, R) or Ck(M, C). We also denote C∞(M, F) by E(M, F) or by E(M).The vector space of C∞ F-valued functions with compact support in M is denoted by D(M, F) or by D(M). A C∞ bijective mapping with C∞ inverse is called a diffeomorphism.AC∞ map for which the restriction U to some neighborhood U of each point maps U diffeomorphically onto an open set (U) is called a local diffeomorphism.

Remarks 1. The chain rule implies that if : M → N is a continuous mapping k k of C manifolds with k ∈ Z≥0 ∪{∞}, then is of class C if and only if for each k point p ∈ M, there exist local C charts (U1,1,V1) and (U2,2,V2) on M and N, −1 respectively, such that p ∈ U1 ∩ (U2) and the mapping ◦ ◦ −1 : ∩ −1 → 2 1 1(U1 (U2)) V2 is of class Ck (see Exercise 9.2.2). In other words, one need only check that the condition holds for elements of some Ck atlas in the Ck structure (not for every local Ck chart). 2. For real analytic manifolds, one uses the analogous terminology. Moreover, the natural analogue of the above property, as well as analogues of many other properties of Ck manifolds, also apply to real analytic manifolds. However, since 422 9 Manifolds a proof that compositions of real analytic mappings are real analytic requires some work (for example, one may form local extensions to complex manifolds and use the fact that compositions of holomorphic maps are holomorphic), and since general facts concerning real analytic manifolds are not required in this book, we will avoid consideration of such facts. 3. It is easy to verify that sums and products of Ck functions are of class Ck,so Ck(M) is an algebra.

k k Deﬁnition 9.2.5 For k ∈ Z≥0 ∪{∞}, a path γ :[a,b]→M is a C path (or a C curve)ifγ extends to a Ck mapping of a neighborhood of [a,b] into M. The path k is piecewise C if there is a partition a = t0

~~We close this section with a consideration of subsets that inherit a C∞ structure.~~

Deﬁnition 9.2.6 Let M be a smooth manifold of dimension n, and let r ∈ {0,...,n}. A nonempty closed set N ⊂ M is a smooth (or C∞) submanifold of dimension r in M if for each point a ∈ N, there exists a local C∞ coordinate neighborhood (U, (x1,...,xn)) of a in M such that

~~N ∩ U ={p ∈ U | xr+1(p) = xr+2(p) =···=xn(p) = 0}.~~

Lemma 9.2.7 A smooth submanifold N of dimension r in a smooth n-manifold M is itself a smooth manifold of dimension r with C∞ atlas consisting of all local charts ∞ of the form (N ∩ U,(x1,...,xr )), where (U, (x1,...,xn)) is a local C chart in M with N ∩ U ={p ∈ U | xr+1(p) = xr+2(p) =···=xn(p) = 0}.

~~Proof Clearly, N is Hausdorff. If (U, = (x1,...,xn)) and (V, = (y1,...,yn)) are two local C∞ charts in M with~~

N ∩ U ={xr+1 =···=xn = 0} and N ∩ V ={yr+1 =···=yn = 0}, ∞ then each of the C maps 0 ≡ (x1,...,xr ) and 0 ≡ (y1,...,yr ) maps N ∩ U and N ∩ V , respectively, homeomorphically onto an open subset of Rr . Moreover, ◦ −1 = −1 0 0 (t1,...,tr ) 0( (t1,...,tr , 0,...,0)) for every point (t1,...,tr ) ∈ 0(N ∩ U ∩ V), and hence the coordinate transforma- ◦ −1 : ∩ ∩ → ∩ ∩ C∞ tion 0 0 0(N U V) 0(N U V)is of class .

~~Remark We assume that a given smooth submanifold has the C∞ structure given by the above lemma unless otherwise indicated.~~

Example 9.2.8 For each R>0, the circle ∂ (0; R) in R2 is a smooth submanifold of R2. For given a point p ∈ ∂ (0; R), polar coordinates (r, θ) (see Example 7.2.8) provide local C∞ coordinates in some neighborhood U of p in R2. Setting ρ = r − R, we get the local C∞ coordinate neighborhood (U, (θ, ρ)) in which U ∩ ∂ (0; R) ={q ∈ U | ρ(q)= 0}. 9.3 The Topology of Manifolds 423

Exercises for Sect. 9.2 9.2.1 Verify the claims in Examples 9.2.3. 9.2.2 Verify that if : M → N is a continuous mapping of Ck manifolds with k k ∈ Z≥0 ∪{∞}, then is of class C if and only if for each point p ∈ M, k there exist local C charts (U1,1,V1) and (U2,2,V2) on M and N, respec- ∈ ∩ −1 ◦ ◦ −1 : ∩ tively, such that p U1 (U2) and the mapping 2 1 1(U1 −1 k (U2)) → V2 is of class C .

~~9.3 The Topology of Manifolds~~

Manifolds have many nice topological properties, especially second countable manifolds. When working in Rn, it is often convenient to phrase topological arguments in terms of sequences. Fortunately, in a manifold, it also often sufﬁces to consider sequences (in a general topological space, the analogous arguments may be phrased in terms of more general objects called nets, as in, for example, [Ke]).

Deﬁnition 9.3.1 Let {aν} be a sequence in a topological space X. We say that {aν } converges (in X) to a point a ∈ X if for every neighborhood U of a in X, there exists an integer μ>0 such that aν ∈ U for all ν ≥ μ. We also say that {aν} is convergent, we call a the limit of {aν}, and we write

~~lim aν = a or aν → a as ν →∞. ν→∞~~

~~If {aν} does not have a limit, then we say that the sequence diverges (or the sequence is divergent).~~

~~The following theorem, the proof of which is left to the reader (see Exer- cise 9.3.1), contains analogues of standard facts concerning topology in Rn:~~

Theorem 9.3.2 Let D be a subset of a manifold M. Then (a) Sequentially closed sets. D is closed if and only if the limit of every convergent sequence {aν} in M with aν ∈ D for all ν lies in D. A point p ∈ M is a limit point of D if and only if p is the limit of a sequence of points in D \{p}. The closure of D is equal to the limits of all sequences in D that converge in M. (b) BolzanoÐWeierstrass property. If D is compact, then every sequence in D admits a subsequence that converges in M to a point in D. (c) Sequential continuity. A mapping : D → N of D into a manifold N is continuous at a point a ∈ D if and only if (aν ) → (a) for every sequence {aν } in D with aν → a.

~~We recall that a topological space is second countable if it has a countable basis.~~

~~Lemma 9.3.3 If (X, T ) is a second countable topological space and B is any basis for T , then there is a countable basis B ⊂ B. 424 9 Manifolds~~

Proof Fix a set B0 ∈ B. By hypothesis, there exists a countable basis A for T .For each pair a = (A1,A2) ∈ A × A, we may ﬁx a set Ba ∈ B with A1 ⊂ Ba ⊂ A2 if such a set exists; and we may set Ba = B0 if no such set exists. Then the collection B ={Ba | a ∈ A × A}⊂B is countable. Given an open set U ⊂ X and a point p ∈ U, there exist an element A2 ∈ A with p ∈ A2 ⊂ U (since A is a basis for T ), an element B ∈ B with p ∈ B ⊂ A2 (since B is a basis for T ), and an element A1 ∈ A with p ∈ A1 ⊂ B ⊂ A2. In particular, for a ≡ (A1,A2) ∈ A × A,wehave p ∈ A1 ⊂ Ba ⊂ A2 ⊂ U. It follows that B is a countable basis for T .

~~Deﬁnition 9.3.4 Let X be a topological space.~~

(a) A family of subsets A ={Ai}i∈I of X is locally finite if each point in X has a neighborhood that meets Ai for at most finitely many indices i ∈ I . (b) We call X paracompact if every open cover of X has a locally finite refinement. { } = That is, for every family of open sets Ui i∈I with X i∈I Ui, there is a { } = locally finite family of open sets Vj j∈J such that X j∈J Vj and such that for each j ∈ J ,wehaveVj ⊂ Ui for some i ∈ I .

~~The proof of the following useful observation is left to the reader (see Exer- cise 9.3.2):~~

~~Lemma 9.3.5 If {Ai}i∈I is a locally ﬁnite family of sets in a topological space X, then Ai = Ai. i∈I i∈I~~

~~The following lemma contains some useful facts concerning the topology of second countable locally compact Hausdorff spaces (for example, second countable manifolds):~~

Lemma 9.3.6 For any second countable locally compact Hausdorff space X, we have the following: { }∞ =∅ = ∞ (a) There exists a sequence of open sets ν ν=0 such that 0 , X ν=1 ν , and ν−1 ν for each ν = 1, 2, 3,.... In particular, X is σ -compact. (b) If X is locally connected and connected, then we may choose the sequence of open sets { ν} in (a) so that ν is connected for each ν. (c) If {Uα}α∈A is a covering of a closed set K ⊂ X by open subsets of X and B is a basis for the topology in X, then there exists a countable locally ﬁnite (in X) covering {Bi}i∈I of K by elements of the basis B such that for each i ∈ I , we have Bi Uα for some α ∈ A. In particular, X is paracompact. (d) If {Uα}α∈A is a covering of a closed set K ⊂ X by open subsets of X and ∈ B B for each α A, α and α are bases for the topology in Uα, then there exist { } { } countable locally ﬁnite (in X) coverings Bi i∈I and Bi i∈I of K such that ∈ ∈ B ∈ B (i) For each i I , we have Bi Bi Uα and Bi α, Bi α, for some α ∈ A; and 9.3 The Topology of Manifolds 425

∈ ∩ = ∅ (ii) If i, j I and Bi Bj , then Bi Bj . (e) If {Vi}i∈I is a locally finite (in X) covering of a closed set K ⊂ X by relatively compact open subsets of X, then there exists a covering {Ki}i∈I of K by compact subsets of X with Ki ⊂ Vi for each i ∈ I . (f) If {Ki}i∈I is a locally finite family of closed subsets of X, then there exists a locally finite family of open sets {Vi}i∈I such that Ki ⊂ Vi for each index i ∈ I . Moreover, if the sets {Ki}i∈I are disjoint, then the sets {Vi}i∈I mayalsobe chosen to be disjoint.

Proof For the proof of (a), let us fix a countable basis B0 for the topology in X. We may assume that X = ∅, ∅ ∈/ B0. Since X is locally compact, the collection of relatively compact elements of B0 is itself a basis (as one may easily check), so we may also assume that each element of B0 is relatively compact in X. Hence we { }∞ may choose a covering of X by relatively compact basis elements Vj j=1, and for ≡ ∪···∪ = ∞ = ∞ each j,wemayletj V1 Vj X. In particular, X j=1 Vj j=1 j . Let j0 = 0 and 0 =∅. Given indices j0 jν−1 so that the compact set jν−1 is contained in jν . Thus we get a { } ≡ = sequence of indices jν , and setting ν jν for ν 0, 1, 2,..., we get a sequence of open sets { ν} with the properties required in (a). If X is also locally connected and connected, then the collection of connected components of elements of B0 is countable, and hence by replacing B0 with this collection, we may assume that each element of the basis B0 is connected. For {Vj } as above, we may instead let j be the connected component of V1 ∪···∪Vj containing V1 for each j = 1, 2, 3,.... The union ≡ j is then equal to X.For if p ∈ , then p ∈ Vj for some j and Vj must meet k for some k. Therefore, for l ≡ max(j, k), Vj ∪ k is a connected subset of V1 ∪···∪Vl containing V1, and hence p ∈ Vj ⊂ l ⊂ . Thus is both open and closed in X and is therefore equal to X. A suitable subsequence of {j } (chosen inductively) will then have each term relatively compact in the next term, as required in (b). For the proof of (c), suppose B is an arbitrary basis for the topology in X.We { }∞ may choose a sequence of open sets ν ν=0 with the properties in (a). For each point p ∈ K, there is a basis element Cp ∈ B such that p ∈ Cp Uα for some α ∈ A and such that for each index ν = 1, 2, 3,...,wehaveCp ν if and only if p ∈ ν , and we have Cp ∩ ν = ∅ if and only if p ∈ ν . For each ν = 1, 2, 3,..., { } there exists a (possibly empty) finite collection of points pi i∈Iν in the compact set K ∩ \ − such that K ∩ \ − ⊂ C . Letting I be the disjoint ν ν 1 ν ν 1 i∈Iν pi union I and letting B = C for each i ∈ I (here, we identify I with its ν∈Z>0 ν i pi ν image in I for each ν), we get a countable covering {Bi}i∈I of K by basis elements each of which is relatively compact in Uα for some α ∈ A. Moreover, for each ν,we have Bi ∩ ν =∅if i ∈ Iμ with μ>ν+ 1, so the family {Bi} is locally finite. B B For the proof of (d), suppose we have two bases α and α for the topology in Uα for each α ∈ A. Again, we may assume that the empty set is not an element of any of these bases. Clearly, we may also assume that K = ∅. By applying part (c) to the covering {Uα} and the basis for the topology in X consisting of all open sets ∩ =∅ ∈ B ∈ B such that either B K or B α and B Uα for some α A, we get a 426 9 Manifolds countable locally finite (in X) covering {Cm}m∈M of K such that for each m ∈ M, ∈ B ∈ we have Cm Uα(m) and Cm α(m) for some α(m) A. Applying part (c) to the covering {Cm}, we get a countable locally finite covering {Vl}l∈L of K by nonempty open sets such that for each l ∈ L,wehaveVl Cm(l) for some m(l) ∈ M. Finally, we apply part (c) (one last time) to the covering {Vl} and the basis for the topology in X consisting of all open sets B such that either B ∩ K =∅or for some l ∈ L, we have B ⊂ Vl , B ∈ Bα(m(l)), and B Cm(k) whenever k ∈ L with B ∩ V k = ∅. Thus we get a countable locally finite (in X) open covering {Bi}i∈I of K such that for each i ∈ I , there is an index l(i) ∈ L with Bi Vl(i) and Bi ∈ Bα(m(l(i))), ∈ ∩ = ∅ ≡ and Bi Cm(k) whenever k L with Bi V k . Setting Bi Cm(l(i)) for each ∈ { } ∈ i I , we get a second covering Bi i∈I of K such that for each i I ,wehave ∈ B = ∅ = Bi α(m(l(i))), Bi , Bi Bi Uα(m(l(i))), and Bi Cm(l(j)) Bj whenever j ∈ I with Bi ∩ V l(j) ⊃ Bi ∩ Bj = ∅. Finally, since {Bi} is a locally finite family of nonempty sets, each of the relatively compact subsets Cm of X will be equal to Cm(l(i)) (and therefore contain Bi ) for only finitely many indices i ∈ I . Therefore, { } { }={ } since the family Cm is locally finite, it follows that the family Bi Cm(l(i)) is also locally finite. The proofs of (e) and (f) are left to the reader (see Exercise 9.3.3).

~~Theorem 9.3.7 Let M be a second countable topological manifold.~~

(a) If K is a closed subset of M and U ={Ui}i∈I is a covering of K by open subsets of M, then there exists a countable family of continuous functions {λj }j∈J on M } with values in [0, 1] such that the family {supp λj j∈J is locally ﬁnite in M, ≡ ∈ ⊂ j∈J λj 1 on a neighborhood of K, and for each j J , we have supp λj ∞ Ui for some i ∈ I . Moreover, if M is a C manifold, then we may choose the ∞ functions {λj } to be of class C . If U is locally ﬁnite and countable, then we may choose the family of functions so that J = I and supp λi ⊂ Ui for each index i. (b) If K0 and K1 are disjoint closed subsets of M, then there exists a continuous function λ: M →[0, 1] such that λ ≡ 0 on a neighborhood of K0 and λ ≡ 1 on ∞ a neighborhood of K1. Moreover, if M is a C manifold, then we may choose λ to be of class C∞.

Remark Recall that the support of a function is the closure of the complement of its zero set. The theorem holds for M a second countable locally compact Hausdorff space (see, for example, [Mu]), but the proof for M a manifold is easier.

Proof of Theorem 9.3.7 Let n = dim M. For the proof of (a), we fix a closed set K ⊂ M and a covering U ={Ui}i∈I of K by open subsets of M. Applying Lemma 9.3.6, we see that we may assume without loss of generality that U is countable and locally finite. Applying Lemma 9.3.6 again, we get a countable locally { } { } finite family of local charts (Bν,ν,ν (Bν )) ν∈N in M and a covering Bν ν∈N ∈ = ; of M such that for each ν N, Bν Bν , ν(Bν ) BRn (0 1) ν (Bν), and either ⊂ \ ⊂ ∈ C∞ Bν M K or Bν Ui(ν) for some index i(ν) I .IfM is a manifold, then 9.3 The Topology of Manifolds 427 we may also assume that ν is a diffeomorphism for each ν. It is easy to verify that the function ρ : R →[0, ∞) given by − e1/(t 1) if t<1, t → 0ift ≥ 1,

∞ 2 is of class C on R. Hence, for each ν ∈ N, ην ≡ ρ(|ν| ) is a nonnegative continuous function that has compact support contained in Bν and that is positive on Bν . ∞ ∞ Moreover, ην is of class C if M is a C manifold. The functions {λi}i∈I given by ν∈N,i=i(ν) ην λi ≡ ∀i ∈ I ν∈N ην (note that in each of the above sums, locally, all but ﬁnitely many terms vanish) then have the properties required in (a). The veriﬁcation of this and the proof of (b) are left to the reader (see Exercise 9.3.4).

Deﬁnition 9.3.8 If U ={Ui}i∈I is an open covering of a topological space M, then any family of continuous functions {λj }j∈J onM with values in [0, 1] such that { } ≡ the family supp λj j∈J is locally ﬁnite in M, j∈J λj 1onM, and for each j ∈ J , supp λj ⊂ Ui for some i ∈ I , is called a partition of unity subordinate to ∞ ∞ the covering U.IfM is a C manifold and λj is of class C for each j ∈ J , then ∞ {λj }j∈J is called a C partition of unity.

~~Deﬁnition 9.3.9 A continuous mapping : X → Y of Hausdorff spaces is called proper if the inverse image of every compact subset of Y is compact.~~

Deﬁnition 9.3.10 A real-valued function ρ on a Hausdorff space X is an exhaustion function if {x ∈ X | ρ(x) < a} X for every a ∈ R. A sequence of sets {Dν} in X ◦ = such that Dν Dν+1 for each ν and ν Dν X is called an exhaustion of X by the sets {Dν}.

Remarks 1. If ρ is an exhaustion function, then any function τ ≥ ρ is also an exhaustion function. 2. A continuous function ρ : X → R is an exhaustion function if and only if ρ is bounded below and ρ is a proper mapping.

Proposition 9.3.11 Every second countable C∞ manifold M admits a positive C∞ exhaustion function. In fact, for every continuous real-valued function τ on M, there exists a C∞ exhaustion function ρ such that ρ>τon M.

Proof We may assume that M is noncompact, and we may choose a countable lo- ∞ cally ﬁnite covering {Uν} by relatively compact open subsets and a C partition of unity {λν} with supp λν⊂ Uν for each ν. Given a continuous function τ : M → R, ∞ + | | · C∞ the locally ﬁnite sum ν=1(ν maxsupp λν τ ) λν then gives a positive exhaustion function that is greater than τ . 428 9 Manifolds

Exercises for Sect. 9.3 9.3.1 Prove Theorem 9.3.2. 9.3.2 Prove Lemma 9.3.5. 9.3.3 Prove parts (e) and (f) of Lemma 9.3.6. 9.3.4 Verify that the functions {λi}i∈I constructed in the proof of part (a) of Theo- rem 9.3.7 have the required properties. Also prove part (b) of the theorem. 9.3.5 Prove that if D is a subset of a second countable manifold M, and every sequence in D admits a subsequence that converges to a point in D, then D is compact (this is a partial converse of part (b) of Theorem 9.3.2).

~~9.4 The Tangent and Cotangent Bundles~~

Recall that if S ={g = 0} for a real-valued C∞ function g with nonvanishing gra- dient on an open subset of R3, then a tangent vector to S at a point p ∈ S is a vector v along which the total differential (dg)p vanishes. The tangent plane to S at p is the plane through p that is parallel to each such tangent vector v. For any such tangent vector v and any real-valued C∞ function f on a neighborhood of p 3 in R ,thevalueof(df )p(v) depends only on the values of the restriction of f to a neighborhood of p in S. One may turn this around and view v as a linear functional ∞ f → (df )p(v) ∈ R that is actually defined on the germs of C functions at p.This point of view provides an efficient approach for defining tangent vectors (complex as well as real) for a smooth manifold.

Deﬁnition 9.4.1 Let M be a smooth manifold and let F = R or C. ∞ (a) Let p ∈ M, and let ∼p be the equivalence relation on the set of F-valued C functions that are deﬁned on a neighborhood of p determined by

~~f ∼p g ⇐⇒ f = g on some neighborhood of p.~~

Each of the associated equivalence classes is called a germ of an (F-valued) C∞ function at p. The set of germs of C∞ functions at p is called the stalk of C∞ C∞ E at p and is denoted by p or p. We denote by germpf the germ represented C∞ by a local function f on a neighborhood of p; i.e., germpf is the germ of f at p. We also consider Ep as an algebra over F with the natural operations. More precisely, for neighborhoods U and V of p and functions f ∈ C∞(U) and g ∈ C∞(V ), we deﬁne · ≡ ∀ ∈ F c germpf germp(cf ) c , · ≡ · germpf germpg germp(f U∩V g U∩V ), + ≡ + germpf germpg germp(f U∩V g U∩V ), − ≡ − germpf germpg germp(f U∩V g U∩V ). 9.4 The Tangent and Cotangent Bundles 429

(b) For each point p ∈ M,atangent vector (over F) at p is a linear functional v : Ep → F that satisfies = · v(germp(fg)) v(germpf germpg) = · + · v(germpf) g(p) f(p) v(germpg) ∈ E for all germs germpf,germpg p (that is, v is a linear derivation on the algebra Ep). For F = R (F = C), v is also called a real tangent vector (respectively, a complex tangent vector). Given a C∞ F-valued function f on a neighborhood ≡ of p, we also write v(f ) v(germpf). (c) For each point p ∈ M, the (real) vector space of real tangent vectors at p is called the tangent space (or real tangent space) to M at p and is denoted by TpM. The complex vector space of complex tangent vectors at p is called the complexified tangent space to M at p and is denoted by (TpM)C. (d) For each point p ∈ M and each F-valued C∞ function f on a neighborhood of p,thedifferential of f at p is the linear functional (df )p on the tangent space at p given by (df )p(v) ≡ v(f ) for each tangent vector v at p.If m ∞ F = (f1,...,fm): U → F is a C mapping (equivalently, each of the func- ∞ m tions f1,...,fm is of class C ) of a neighborhood U into F , then we define (dF )p ≡ ((df1)p,...,(dfm)p), and for each tangent vector v to M at p,weset v(F) ≡ (dF )p(v). ∞ (e) If (U, = (x1,...,xn), U ) is a local C chart in M, f is a function defined on a neighborhood of a point p ∈ U, and j ∈{1,...,n}, then we define ∂f ≡ ∂ [ −1 ] (p) f( (t1,...,tn)) (t ,...,t )=(p), ∂xj ∂tj 1 n provided this partial derivative exists. The corresponding tangent vector ∂f ∂ f → (p) is denoted by . ∂xj ∂xj p (f) For each p ∈ M,thecotangent space and the complexified cotangent space to M at p are, respectively, the dual spaces ∗ ≡ ∗ ∗ ≡[ ]∗ Tp M (TpM) and (Tp M)C (TpM)C . (g) If : M → N is a C∞ mapping of M into a C∞ manifold N, then the induced tangent maps (or pushforward maps) at p ∈ M are the real linear map TpM → T(p)N and the complex linear map (TpM)C → (T(p)N)C, which are both denoted by (∗)p and are given by

~~(∗)p(v)(f ) ≡ v(f ◦ )~~

for every C∞ function f on a neighborhood of (p) and for every tangent vector v at p (with f R-valued and v ∈ TpM if we are considering the tangent mapping of the real tangent spaces). The associated pullback mappings (or 430 9 Manifolds

∗ → ∗ cotangent mappings) are the real linear map T(p)N Tp M and the complex ∗ → ∗ ∗ linear map (T(p)N)C (Tp M)C, which are both denoted by ( )p and are given by the adjoints associated to the tangent mappings. That is,

~~∗ ( )p(α)(v) ≡ α((∗)p(v))~~

for every linear functional α on the tangent space to N at (p) and every tangent vector v to M at p (with α and v real when we are considering the real (co)tangent spaces, and complex when we are considering the complexiﬁed (co)tangent spaces).

Remarks 1. If v is a tangent vector to a smooth manifold M at a point p ∈ M and f is a C∞ function that is equal to a constant c on a neighborhood of p, then v(f ) = 0. For v(1) = v(1 · 1) = v(1) · 1 + 1 · v(1) = 2v(1), and hence v(c) = v(c · 1) = cv(1) = 0. 2. Product rule. It follows from the deﬁnition that if f and g are C∞ functions on a neighborhood of a point p in a smooth manifold M, then

~~(d(fg))p = g(p) · (df )p + f(p)· (dg)p.~~

~~3. Chain rule.Let: M → N and : N → P be C∞ mappings of C∞ manifolds M, N, and P , and let p ∈ M. Then~~

~~∗ ∗ ∗ ([ ◦ ]∗)p = ( ∗)(p) ◦ (∗)p and ([ ◦ ] )p = ( )p ◦ ( )(p).~~

In particular, if is a diffeomorphism (i.e., is bijective with C∞ inverse), ∗ −1 = then (∗)p and ( )p are linear isomorphisms with inverse mappings (∗)p −1 ∗ −1 = −1 ∗ : → = R C ◦ (( )∗)(p) and ( )p (( ) )(p).If N P or , then (d( ))p = (d )(p) ◦ (∗)p (on the real tangent space for P = R and on the com- plexiﬁed tangent space for P = C). For given a tangent vector v,wehave

~~(d( ◦ ))p(v) = v( ◦ ) = (∗)p(v)( ) = (d )(p) ◦ (∗)p(v).~~

A similar argument shows that the above equality also holds if : N → P = Rm m m m or C . Furthermore, taking N = P = R or C and = IdP , we see that the linear maps (∗)p and (d)p have the same kernel and the same rank.

~~Proposition 9.4.2 For any C∞ manifold M of dimension n and any point p ∈ M, we have the following:~~

(a) The map TpM → (TpM)C that associates to each real tangent vector v the complex tangent vector given by f → v(Re f)+ iv(Im f) for every C∞ function f on a neighborhood of p is an injective linear map. Identifying TpM with a real vector subspace of (TpM)C under the above injection, we get the real vector space direct sum decomposition (TpM)C = TpM ⊕ iTpM; and hence we may identify the complexified tangent space (TpM)C with the complexification of the real tangent space TpM (see Definition 8.1.4) with the real and 9.4 The Tangent and Cotangent Bundles 431

imaginary projections given by Re: u + iv → u and Im: u + iv → v for each ∈ ∗ → ∗ pair u, v TpM. The map Tp M (Tp M)C that associates to each real lin- ∈ ∗ ∗ ear functional α Tp M the complex linear functional in (Tp M)C given by → + ∗ v α(Re v) iα(Im v) is an injective real linear map. Identifying Tp M with ∗ a real vector subspace of (Tp M)C under the above injection, we get the real ∗ = ∗ ⊕ ∗ vector space direct sum decomposition (Tp M)C Tp M iTp M; and we may ∗ identify the complexiﬁed cotangent space (Tp M)C with the complexiﬁcation of ∗ the real cotangent space Tp M with the real and imaginary projections given by : + → : + → ∈ ∗ Re α iβ α and Im α iβ β for each pair α, β Tp M. (b) The tangent space TpM is of real dimension n, and (TpM)C is of complex di- ∞ mension n. In fact, if (U, = (x1,...,xn)) is any local C coordinate neigh- { }n borhood of p, then the partial derivative operators (∂/∂xj )p j=1 form a real { }n basis for TpM and a complex basis for (TpM)C. The differentials (dxj )p j=1 ∗ ∗ form the real dual basis for Tp M and the complex dual basis for (Tp M)C.

Proof That the mapping f → v(Re f)+ iv(Im f)is a complex tangent vector for each v ∈ TpM, and that the corresponding mapping TpM → (TpM)C is an injective real linear map, are left to the reader. Moreover, if v ∈ TpM \{0}, then v(f) ∈ R\{0} ∞ for some real-valued C function f on a neighborhood of p. Hence, if w ∈ TpM, then iw(f) ∈ iR,soiw = v. Thus TpM ∩ iTpM ={0}. On the other hand, given a ∈ C∞ nonzero tangent vector w (TpM)C, for each real germ germpf ,wemayset u(f ) ≡ Re[w(f )] and v(f ) ≡ Im[w(f )]. The mappings u and v are real tangent vectors at p. For they are obviously real linear, and for each pair of real C∞ germs ∈ E germpf,germpg p,wehave = [ · + ] u(germp(fg)) Re w(f ) g(p) f(p)w(g) = · + · u(germpf) g(p) f(p) u(germpg), and the analogous equalities hold for v. Identifying u and v with their images in (TpM)C, we also get w = u + iv. Thus we have the direct sum decomposition (TpM)C = TpM ⊕iTpM of (TpM)C into real subspaces. Letting V be the complex- iﬁcation of the real vector space TpM, we have the real linear map V → (TpM)C given by w → Re w + i Im w. It is now easy to verify that this map is a complex linear isomorphism. The veriﬁcation that the analogous decomposition of the cotangent spaces holds is left to the reader. ∞ Suppose now that (U, = (x1,...,xn)) is a local C coordinate neighborhood of p. For each pair i, j = 1,...,n,wehave ∂xi 1ifi = j, (p) = δij = ∂xj 0ifi = j;

{ }n so the real tangent vectors (∂/∂xj )p j=1 are linearly independent. Conversely, ∈ ≡ n · ∈ given a real tangent vector v TpM,wemaysetu j=1 vj (∂/∂xj )p TpM, ∞ where vj ≡ v(xj ) ∈ R for each j = 1,...,n.GivenaC function f on a neighbor- 432 9 Manifolds

~~{ }n C∞ { }n hood of p, Lemma 7.2.6 provides constants bj j=1 and functions cij i,j=1 on a neighborhood of p such that~~

n n f = f(p)+ bj (xj − xj (p)) + cij (xi − xi(p))(xj − xj (p)) j=1 i,j=1 on a neighborhood of p. Thus, using the fact that u and v are derivations, one may now easily check that n v(f) = bj vj + 0 = u(f ). j=1 = { }n Thus v u, and hence (∂/∂xj )p j=1 is a basis for TpM. Clearly, = = { }n (dxj )p((∂/∂xi )p) δij for all i, j 1,...,n,so (dxj )p j=1 is the associated real dual basis for TpM. The corresponding claims regarding the complexiﬁcations also follow.

Remarks 1. It follows from the above proposition that if (U, (x1,...,xn)) is a local C∞ coordinate neighborhood in M, v is a tangent vector to M at p ∈ U, α is an element of the cotangent space at p, and f is a C∞ function a neighborhood of a point p, then n ∂ n ∂ v = (dxj )p(v) · = v(xj ) · , ∂xj ∂xj j=1 p j=1 p n ∂ α = α · (dxj )p, ∂xj j=1 p and n ∂ n ∂f (df )p = df (dxj )p = (p)(dxj )p. ∂xj ∂xj j=1 p j=1

2. Given a complex tangent vector w ∈ (TpM)C,wehavew = u + iv with u, v ∈ TpM. Under the identiﬁcation with the complexiﬁcation of TpM,wehaveRew ≡ u = C∞ and Im w v. It should be noted that for a complex function germ germpf , u(f ) = u(Re f)+ iu(Im f)need not be equal to Re(w(f )) = u(Re f)− v(Im f).

Deﬁnition 9.4.3 Let M be a smooth manifold. (a) The (real) tangent bundle and the complexiﬁed tangent bundle of M are given by TM≡ TpM and (T M)C ≡ (TpM)C, p∈M p∈M 9.4 The Tangent and Cotangent Bundles 433

~~respectively. The associated tangent bundle projections are the (surjective) mappings : → : → TM TM M and (T M)C (T M)C M~~

given by v → p for each point p ∈ M and each tangent vector v in TpM or (TpM)C. (b) The (real) cotangent bundle and the complexiﬁed cotangent bundle of M are given by ∗ ≡ ∗ ∗ ≡ ∗ T M Tp M and (T M)C (Tp M)C, p∈M p∈M

~~respectively. The associated cotangent bundle projections are the (surjective) mappings~~

∗ ∗ ∗ ∗ T M : T M → M and (T M)C : (T M)C → M → ∈ ∗ given by α p for each point p M and each linear functional α in Tp M or ∗ (Tp M)C. (d) Depending on the context, for each C∞ function f on an open set U ⊂ M,the : −1 → R differential (or exterior derivative)off is the function df TM(U) or : −1 → C df (T M)C (U) with restriction to each tangent space equal to (df )p; that is, df (v) = v(f ) for each tangent vector v at each point p ∈ U.ForF = ∞ (f1,...,fm) with f1,...,fm ∈ C (U), we deﬁne dF ≡ (df1,...,dfm). (e) If : M → N is a C∞ mapping into a C∞ manifold N, then the associated tangent maps (or pushforward maps) are the maps ∗ : TM → TN and ∗ : (T M)C → (T N)C given by ∗(v) = (∗)p(v) for every tangent vector v to M at a point p ∈ M. The associated pullback mappings (or cotangent map- ∗ ∗ ∗ ∗ ∗ ∗ pings) are the maps : T N → T M and : (T N)C → (T M)C given by ∗ = ∗ ∗ ∗ ∈ α ( )pα for every element α of T(p)N or (T(p)N)C with p M.

Remarks 1. The tangent and cotangent bundles of a smooth manifold M admit natural C∞ (manifold) structures (see Exercise 9.4.3). They are also examples of C∞ vector bundles (see, for example [Ns3]or[Wel]). 2. For any open subset of M with inclusion mapping ι: → M, we identify ∗ −1 ⊂ −1 ⊂ ∗ T and T with the sets TM( ) TM and T ∗M ( ) T M, respectively, : → −1 ∗ : −1 → ∗ under the bijections ι∗ T TM( ) and ι T ∗M ( ) T .Wemakethe analogous identifications for the complexified tangent and complexified cotangent bundles. 3. For any open set ⊂ Rn, we identify the tangent bundles T and (T )C with the products × Rn and × Cn, respectively, under the bijections

~~n n (T ,d(Id )): T → × R and ((T )C ,d(Id )): (T )C → × C .~~

~~For any C∞ map F : → Rm, we get the corresponding identiﬁcation of the differential mapping dF on T with the differential mapping (p, v) → (dF )p(v) on 434 9 Manifolds~~

× Rm (see Definition 7.2.2). We make the analogous identification for a C∞ mapping F : → Cm. 4. For any C1 function f on a neighborhood of a point p in a smooth manifold M, and any tangent vector v to M at p, we define

n ∂f df (v) = v(f ) ≡ vj (p), ∂xj j=1 ∞ where (x1,...,xn) are local C coordinates in a neighborhood of p and vj = dxj (v) for j = 1,...,n. The verification that v(f ) is independent of the choice of local coordinates, and hence that we have an induced differential df : v → df (v), is left to the reader. Note also that this definition is consistent with the earlier definitions for f of class C∞. Similarly, if : M → N is a C1 mapping of smooth manifolds M and N, then we define ∗ : TM→ TN and ∗ : (T M)C → (T N)C ∞ by (∗v)(f ) ≡ v(f ◦ ) for each tangent vector v to M and each local C function f in N. It is easy to verify that ∗v is a derivation and that we also have (∗v)(f ) ≡ v(f ◦ ) for every C1 function f . The pullback mappings are ∗ ∗ given by α → α ≡ α ◦ ∗. The mappings ∗ and are well-defined mappings of the corresponding tangent and cotangent bundles, respectively. Moreover, ∞ if x = (x1,...,xn) and y = (y1,...,ys) are local C coordinates in a neighborhood of p∈ M and a neighborhood of (p), respectively, then for every tangent vector = · = = v j=1 vj (∂/∂xj )p (i.e., vj dxj (v) for each j 1,...,n), we have s n ∂(yk ◦ ) ∂ ∗v = vj · (p) · , ∂xj ∂yk k=1 j=1 p = s and for every element α k=1 ak(dyk)(p) of the cotangent space to N at (p) (i.e., ak = α((∂/∂yk)p) for each k = 1,...,s), we have n s ∗ ∂(yk ◦ ) α = ak · (p) · (dxj )p. ∂xj j=1 k=1

For any C1 mapping : N → P of N into a C∞ manifold P ,wehave ∗ ∗ ∗ ( ◦ )∗ = ∗ ◦ ∗ and ( ◦ ) = ◦ . ∗ For any C1 function g on N,wehaved(g ◦ ) = (dg) ◦ ∗ = dg. 5. If : M → N is a C1 mapping of smooth manifolds M and N and ∗ ≡ 0, then is locally constant (see Exercise 9.4.1).

Deﬁnition 9.4.4 If γ : I → M is a C1 mapping of an interval I into a smooth manifold M (i.e., γ is C1 on the interior and γ admits local C1 extensions at any endpoint contained in I ), then the tangent vector γ(t˙ 0) to γ at each t0 ∈ I is given by d γ(t˙ ) ≡ γ∗ , 0 dt t0 9.4 The Tangent and Cotangent Bundles 435 where t denotes the standard coordinate function on R (at an endpoint in I ,the tangent map is that of a local C1 extension).

~~∞ By the remarks preceding this deﬁnition, if ◦ γ = (γ1,...,γn) in a local C coordinate neighborhood (U, = (x1,...,xn)) of γ(t0), then n ∂ γ(t˙ 0) = γj (t0) . ∂xj j=1 γ(t0)~~

Definition 9.4.5 Let S be a subset of a smooth manifold M of dimension n. (a) A (real) vector field (complex vector field)onS in M is a map v : S → TM : → ◦ = ◦ = (respectively, v S (T M)C) with TM v IdS (respectively, (T M)C v IdS ). We usually denote the value v(p) by vp ∈ TpM (respectively, (TpM)C)for each point p ∈ S. For any C1 function f on an open set U ⊂ M, df (v) = v(f ) denotes the function on S ∩ U given by p → df (vp) = vp(f ) (that is, df (v) = df ◦ v). If : U → N is a C1 mapping of U into a smooth manifold N, then ∗v denotes the mapping S ∩ U → (T N)C (or TN) given by p → ∗vp (that is, ∗v = ∗ ◦ v). (b) The coefficient functions (or simply the coefficients) of a vector field v on S ∞ with respect to (or in) a local C coordinate neighborhood (U, (x1,...,xn)) ∩ ≡ = = are the functions on S U given by vj dxj (v) v(xj ) for j 1,...,n(that = ∂ is, v vj ∂x ). We say that v is continuous if v has continuous coefficients in ∞j every local C coordinate neighborhood. For S an open set and k ∈ Z≥0 ∪{∞}, we say that v is of class Ck if v has Ck coefficients in every local C∞ coordinate neighborhood.

Remarks 1. For the purposes of this book, although it is occasionally convenient to consider vector fields, it is usually more convenient to consider differential forms instead (see Sect. 9.5). 2. A vector field v is of class Ck if and only if each point admits a local C∞ coordinate neighborhood in which the coefficients of v are of class Ck (see Exer- cise 9.4.2). 3. For a function f and vector fields u and v, we define vector fields fv and u + v by p → f(p)vp and p → up + vp, respectively.

~~We close this section with some terminology:~~

Deﬁnition 9.4.6 Let : M → N be a C∞ mapping of smooth manifolds M and N. A point p ∈ M is a critical point of if the linear mapping (∗)p : TpM → T(p)N is not surjective. The image of a critical point is called a critical value of .Every point in N that is not a critical value is called a regular value of (in particular, each point in N \ (M) is a regular value). 436 9 Manifolds

Exercises for Sect. 9.4 9.4.1 Prove that if : M → N is a C1 mapping of smooth manifolds M and N and ∗ ≡ 0, then is locally constant. 9.4.2 Let M be a smooth manifold, let S ⊂ M, and let k ∈ Z≥0 ∪{∞}. (a) For a vector field v on S, prove that the following are equivalent: (i) The vector field v is of class Ck . (ii) For every point in S, there exists a local C∞ coordinate neighborhood with respect to which the coefficients of v are of class Ck. k (iii) The function df (v) = v(f ): p → vp(f ) is of class C for each local C∞ function f . (b) Prove that any sum of Ck vector fields, and any product of a Ck function and a Ck vector field, is of class Ck. 9.4.3 Let M be a smooth manifold of dimension n. (a) Prove that there are unique C∞ (manifold) structures in TM and in ∞ (T M)C such that for each local C chart (U, = (x1,...,xn), U ) in M, the triples −1 n (U), ( ◦ TM,d −1 ), U × R TM TM(U) and −1 n 2n ◦ −1 × C = × R (T M)C (U), ( (T M)C ,d (U)), U U (T M)C ∞ are local C charts in TM and (T M)C, respectively. That is, the associated local C∞ chart in the tangent bundle is given by

~~v → (x1(p),...,xn(p), dx1(v),...,dxn(v))~~

for each point p ∈ U and each tangent vector v to M at p. (b) Prove that there are unique C∞ (manifold) structures in T ∗M and in ∗ ∞ (T M)C such that for each local C chart (U, = (x1,...,xn), U ) in M, we get local C∞ charts −1 × Rn T ∗M (U), , U and −1 n 2n ∗ C × C = × R (T M)C (U), ,U U ∗ ∗ in T M and (T M)C, respectively, where the mappings and C are given by ∂ ∂ α → x1(p),...,xn(p), α ,...,α ∂x1 p ∂xn p ∈ ∗ ∗ for each point p U and each element α of Tp M and (Tp M)C, respectively. 9.5 Differential Forms on Smooth Curves and Surfaces 437

~~(c) Prove that for each C∞ function f on an open set ⊂ M, the differentials~~

~~: −1 → R : −1 → C df TM( ) and df (T M)C ( )~~

~~are of class C∞ with respect to the smooth structures provided by (a). (d) Prove that if : M → N is a C∞ mapping of M into a C∞ manifold N, then the tangent maps~~

~~∗ : TM→ TN and ∗ : (T M)C → (T N)C~~

~~and the pullback maps~~

~~∗ ∗ ∗ ∗ ∗ ∗ : T N → T M and : (T N)C → (T M)C~~

are of class C∞ with respect to the smooth structures provided by (a) and (b). ∗ ∗ (e) Prove that the inclusion mappings TM → (T M)C and T M → (T M)C ∗ ∗ and the real and imaginary projections TMC → TMand T MC → T M are C∞ mappings. k 9.4.4 Let M be a smooth manifold and let k ∈ Z>0 ∪{∞}. Show that if f ∈ C (M), − then the mapping df : (T M)C → C is of class Ck 1, where (T M)C has the C∞ structure provided by Exercise 9.4.3 above. Show also that if : M → N is a Ck mapping of M into a smooth manifold N, then the tangent mapping ∗ ∗ ∗ ∗ : (T M)C → (T N)C and the pullback mapping : (T N)C → (T M)C are of class Ck−1. 9.4.5 Let v be a vector ﬁeld on a smooth manifold M, and let k ∈ Z≥0 ∪{∞}. Prove that v is a Ck vector ﬁeld if and only if v is of class Ck as a mapping of M ∞ into (T M)C, where (T M)C has the C structure provided by Exercise 9.4.3 above.

~~9.5 Differential Forms on Smooth Curves and Surfaces~~

In this section, we consider differential forms (of degree ≤ 2), that is, objects that are locally exterior products in the cotangent bundle (see Sect. 8.2). For simplicity, we restrict our attention to smooth curves and surfaces (which are the only cases we will need).

Deﬁnition 9.5.1 Let M be a smooth manifold of dimension n = 1or2. (a) For each r = 1, 2, we deﬁne r ∗ ≡ r ∗ ⊂ r ∗ ≡ r ∗ T M Tp M (T M)C (Tp M)C. p∈M p∈M 438 9 Manifolds

~~We also set 0 ∗ ≡ × R = { }×R = 0 ∗ T M M p Tp M p∈M p∈M ⊂ 0 ∗ ≡ × C = { }×C = 0 ∗ (T M)C M p (Tp M)C, p∈M p∈M~~

~~and for r ∈ Z>2,weset r ∗ = r ∗ ≡ ×{ }= { }×{ }= r ∗ T M (T M)C M 0 p 0 ( Tp M)C. p∈M p∈M~~

For each r ∈ Z≥0, the corresponding projections are the (surjective) mappings ∗ ∗ ∗ r ∗ r r T M : T M → M and r (T M)C : (T M)C → M → ∈ r ∗ given by α p for each point p M and each element α of Tp M or r ∗ (Tp M)C. (b) Given a C1 mapping : M → N into a C∞ manifold N of dimension ≤ 2 and a nonnegative integer r, the associated pullback mappings at each point p ∈ M r ∗ → r ∗ are the real linear map T(p)N Tp M and the complex linear map r ∗ → r ∗ ∗ (T(p)N)C (Tp M)C, which are both denoted by ( )p and are defined = ∗ ≡ as follows. For r 1, as in Definition 9.4.1,wesetpα(v) α(∗v) for every linear functional α on the tangent space to N at (p) and every tangent vector v to M at p (with α and v real when we are considering the real (co)tangent spaces, and complex when we are considering the complexified (co)tangent = ∗ ≡ spaces). For r 2, we set pα(u,v) α(∗u, ∗v) for every skew-symmetric bilinear function α on the tangent space to N at (p) and every pair of tangent vectors u, v to M at p (again, with α, u, and v real when we are considering the real (co)tangent spaces, and complex when we are considering the complexified = ∗ ≡ (co)tangent spaces). For r 0, we set p((p), ζ ) (p, ζ ) for each scalar ζ , ∗ ≡ ∗ : r ∗ → r ∗ and for r>2, we set p 0. The pullback mappings T N T M ∗ ∗ ∗ and : r (T N)C → r (T M)C are then the mappings with restriction to r ∗ r ∗ ∗ = T(p)N or (T(p)N)C equal to p for r 0, 1, 2,....

Remarks 1. If : M → N is a C1 mapping of C∞ manifolds of dimension ≤ 2, ∈ ∈ Z ∈ r ∗ ∈ s ∗ ∗ ∧ = p M, r, s ≥0, α (T(p)N)C, and β (T(p)N)C, then (α β) ∗α ∧ ∗β (see Exercise 9.5.1). 2. For a given smooth manifold M of dimension n = 1 or 2 and a given nonnega- ∗ ∗ ∞ tive integer r, r T M and r (T M)C have natural C (manifold) structures (see Exercise 9.5.4). They are also examples of C∞ vector bundles (see, for example, [Ns3]or[Wel]).

~~Deﬁnition 9.5.2 Let M be a smooth manifold of dimension n = 1or2,letS ⊂ M, and let r ∈ Z≥0. 9.5 Differential Forms on Smooth Curves and Surfaces 439~~

(a) A real differential form (complex differential form) of degree r in M on S is a r ∗ r ∗ map α : S → T M (respectively, α : S → (T M)C) with r T ∗M ◦ α = ∗ IdS (respectively, r (T M)C ◦ α = IdS ). We usually denote the value of α at p ∈ r ∗ r ∗ ∈ = by αp Tp M (respectively, (Tp M)C) for each point p S.Forr 0, we identify α with the function p → ζ for αp = (p, ζ ). We also call a differential form of degree r an r-form. (b) Let α be a differential form of degree r on S.Thecoefﬁcient functions (or simply the coefﬁcients)ofα with respect to (or in) a local C∞ coordinate neighborhood (U, (x1,...,xn)) are the functions on S ∩ U given by ⎧ ⎪α r = , ⎪ if 0 ⎪ ⎪ = = ⎪aj α(∂/∂xj ) for j 1,...,n ⎪ ⎨ = n = (i.e., α j=1 aj dxj ) if r 1, ⎪ = = α ⎪a α(∂/∂x1,∂/∂x2) dx ∧dx ⎪ 1 2 ⎪ = ∧ = = ⎪ (i.e., α adx1 dx2) if n r 2, ⎩⎪ 0ifr>n.

(c) We say that a differential r-form α on S is continuous if its coefﬁcients in ev- ∞ ery local C coordinate neighborhood are continuous. For k ∈ Z≥0 ∪{∞},we say that α is of class Ck if the coefﬁcients of α in every local C∞ coordinate neighborhood are of class Ck. (d) For S open, Er (S, F) denotes the set of C∞ differential forms of degree r with values in F = R or C. Depending on the context, Er (S) will denote either Er (S, R) or Er (S, C).ThesetofC∞ F-valued r-forms with compact support (where the support of a differential form is the closure of the set of points in S at which the form is nonzero) in S is denoted by Dr (S, F) or by Dr (S).

Remarks 1. Observe that if α and β are differential forms of degree r and s, respectively, on a subset S of a smooth manifold M of dimension 1 or 2, then the exterior product α ∧ β : p → (α ∧ β)p ≡ αp ∧ βp is an (r + s)-form, with α ∧ β ≡ 0if r + s>dim M (see Sect. 8.2). For r = s,thesumα + β is the r-form given by p → αp + βp. 2. If α is a 1-form and v is a tangent vector at a point in a smooth manifold, then α(v) =¯α(v)¯ .Ifα is a 2-form and u and v are tangent vectors, then α(u,v) =¯α(u,¯ v)¯ (see Sects. 8.1 and 8.2). 3. If : M → N is a C∞ mapping of C∞ manifolds of dimension ≤ 2 and α is an r-form on a set S ⊂ N, then ∗α is an r-form on −1(S) in M.Forr = 0, viewing α as a function on S,wehave∗α = α ◦ .

~~We have the following description of Ck differential forms and their behavior.~~

~~Proposition 9.5.3 Suppose M is a smooth manifold of dimension n = 1 or 2 and k ∈ Z≥0 ∪{∞}. Then we have the following: 440 9 Manifolds~~

(a) Let α be a differential form of degree r on a set S ⊂ M. Then α is a continuous (class Ck ) differential form if and only if for each point in S, there exists a local C∞ coordinate neighborhood with respect to which the corresponding coefﬁcient functions of α are continuous (respectively, of class Ck ). (b) If : N → M is a C∞ mapping of a smooth manifold N into M, α is a differential form on a set in M, and α is continuous (of class Ck), then ∗α is continuous (respectively, of class Ck). (c) The sum and exterior product of any two continuous (class Ck) differential forms is continuous (respectively, of class Ck). In particular, for any open set ⊂ M and for any pair of nonnegative integers r, s, the exterior product (α, β) → α ∧ β yields a mapping E r ( ) × Es( ) → Er+s( ).

Proof Suppose α is a differential r-form defined at points in a set S ⊂ M and ∞ (U, x = (x1,...,xn)) and (V, y = (y1,...,yn)) are local C coordinate neighborhoods in M.Forr = 1, we have, on S ∩ U ∩ V , n n n n ∂yi α = ai dyi = ai dxj = bj dxj , ∂xj i=1 j=1 i=1 j=1 where {ai} and {bj } are the coefficients of α in (V, y) and (U, x), respectively. ∞ Since ∂yi/∂xj ∈ C (U ∩ V) for each pair of indices i and j, it follows that if k the coefficients {ai} are continuous (of class C ), then each of the coefficients n ∂yi k b = = a is also continuous (respectively, of class C ) for each j = 1,...,n. j i 1 i ∂xj Similarly, for n = r = 2, we have

α = ady1 ∧ dy2 = aJ · dx1 ∧ dx2, where J is the C∞ function given by the (Jacobian) determinant ∧ ∂y1 ∂y1 dy1 dy2 ∂x1 ∂x2 J ≡ = . dx1 ∧ dx2 ∂y2 ∂y2 ∂x1 ∂x2 Hence, if the coefﬁcient a is continuous (of class Ck), then the coefﬁcient aJ is continuous (respectively, of class Ck). Part (a) now follows. The proofs of (b) and (c) are left to the reader (see Exercise 9.5.2).

~~Remark In particular, the sets Er ( ) and Dr ( ) associated to an open subset of a smooth manifold are vector spaces.~~

It is easy to see that if f is a Ck function on a smooth manifold M of dimension n = 1 or 2 with k ∈ Z>0 ∪{∞}, then the differential, or exterior derivative, df is a Ck−1 differential form of degree 1 on M. In particular, for every open set ⊂ M,wehave d : E0( ) → E1( ) and d : D0( ) → D1( ). The following lemma provides an exterior derivative operator d : E1( ) → E2( ). 9.5 Differential Forms on Smooth Curves and Surfaces 441

Lemma 9.5.4 Let M be a 2-dimensional smooth manifold. Then there is a unique operator d on local C1 differential forms of degree 1 such that: (i) For every C1 differential form α of degree 1 on an open set ⊂ M, dα is a C0 differential form of degree 2 on ; (ii) For every pair of C1 differential forms α and β of degree 1 on an open set ⊂ M and every constant ζ ∈ C, we have d(α + ζβ) = dα + ζdβ; (iii) For every local C2 function f , we have d2f = d(df) = 0; (iv) For every C1 function f and every C1 differential form α of degree 1 on an open set ⊂ M, we have d(fα) = df ∧ α + fdα; and (v) For every C1 differential form α of degree 1 on an open set ⊂ M and every ⊂ = open set U , we have d(α U ) (dα) U . Moreover, if α is a C1 differential form of degree 1 on an open set ⊂ M, (U,(x,y)) is a local C∞ coordinate neighborhood, and α = Pdx+ Qdy on U ∩ (i.e., P and Q are the coefﬁcients of α in the coordinate neighborhood), then ∂Q ∂P dα = − dx ∧ dy on U ∩ . ∂x ∂y

~~k k−1 In particular, if α is of class C with k ∈ Z>0 ∪{∞}, then dα is of class C .~~

k Proof Suppose α is a C 1-form on an open set ⊂ M with k ∈ Z>0 ∪{∞}. Let us assume for the moment that an operator d satisfying the conditions (i)Ð(v) exists. If (U,(x,y)) is a local C∞ coordinate neighborhood in M, then we have α = Pdx+ Qdy on U ∩ for unique functions P,Q∈ Ck(U ∩ ) (P ≡ α(∂/∂x), Q ≡ α(∂/∂y)). The conditions (i)Ð(v) then imply that on U ∩ , dα = d(P dx)+ d(Qdy) = (dP ) ∧ dx + (dQ) ∧ dy ∂P ∂Q = dy ∧ dx + dx ∧ dy ∂y ∂x ∂Q ∂P = − dx ∧ dy. ∂x ∂y

In particular, dα is of class Ck−1 and uniqueness follows. Motivated by the above, given a C1 1-form α on an open set ⊂ M,wenow deﬁne the C0 2-form dα on by setting ∂Q ∂P (dα) ∩ = − dx ∧ dy U ∂x ∂y whenever (U,(x,y)) is a local C∞ coordinate neighborhood in M, and P and Q are the (unique) C1 functions with α = Pdx+ Qdy on U ∩ . The 2-form dα is well deﬁned because, if (V, (u, v)) is another local C∞ coordinate neighborhood in M and α = Rdu+ Sdvon V ∩ , then on U ∩ V ∩ ,wehave ∂Q ∂ ∂ ∂ ∂u ∂v = α = R + S ∂x ∂x ∂y ∂x ∂y ∂y 442 9 Manifolds

∂2u ∂2v ∂u ∂R ∂v ∂S = R + S + + ∂x∂y ∂x∂y ∂y ∂x ∂y ∂x ∂2u ∂2v ∂u ∂u ∂R ∂u ∂v ∂R ∂v ∂u ∂S ∂v ∂v ∂S = R + S + + + + . ∂x∂y ∂x∂y ∂y ∂x ∂u ∂y ∂x ∂v ∂y ∂x ∂u ∂y ∂x ∂v Applying the analogous calculation to obtain ∂P/∂y and subtracting, we get ∂Q ∂P ∂S ∂R ∂u ∂v ∂v ∂u ∂S ∂R − = − · − = − · det J, ∂x ∂y ∂u ∂v ∂x ∂y ∂x ∂y ∂u ∂v where J is the (Jacobian) matrix ∂u ∂u ≡ ∂x ∂y J ∂v ∂v . ∂x ∂y We also have ∂x ∂x ∂y ∂y dx ∧ dy = du+ dv ∧ du+ dv ∂u ∂v ∂u ∂v ∂x ∂y ∂y ∂x = − · du∧ dv ∂u ∂v ∂u ∂v − = det(J 1) · du∧ dv.

It follows that ∂Q ∂P ∂S ∂R − · dx ∧ dy = − · du∧ dv, ∂x ∂y ∂u ∂v and hence dα is well deﬁned. As one may verify, the local representation of dα also gives the conditions (ii)Ð(v) (see Exercise 9.5.5).

~~Remark Actually, the conditions (i)Ð(iv) in the lemma sufﬁce to uniquely determine the operator d (see Exercise 9.5.5).~~

~~We now collect together the deﬁnition of the exterior derivative for a 0-form provided in Sect. 9.4 and the deﬁnition for a 1-form suggested by Lemma 9.5.4 above.~~

Deﬁnition 9.5.5 Let α be a differential form of degree r ∈ Z≥0 on an open subset of a smooth manifold M of dimension n = 1or2. (a) If α is of class C1, then the exterior derivative of α is the continuous differential form dα of degree r + 1on that is equal to (i) The differential of α if r = 0 (see Sect. 9.4); (ii) The 2-form provided by Lemma 9.5.4 if r = 1 and n = 2; (iii) The trivial (i.e., zero) differential form if r ≥ 2orn = r = 1. 9.5 Differential Forms on Smooth Curves and Surfaces 443

(b) If α is of class C1 and we have dα = 0, then we say that α is closed (or d-closed). (c) If α = dβ for some C1 differential form β of degree r − 1on (in particular, r>0), then we say that α is exact (or d-exact). If β may be chosen to be of k k class C for some k ∈ Z>0 ∪{∞}, then we also say that α is C -exact.For r = 1, we call the function β a potential for α. If for each point in , there 1 exists a C differential form β0 of degree r − 1 on a neighborhood U0 such that = dβ0 α U0 , then we say that α is locally exact (or locally d-exact) and we call β0 a local potential for α.Ifforsomek ∈ Z>0 ∪{∞}, each of the local forms k k β0 may be chosen to be of class C , then we also say that α is locally C -exact. It is also convenient to consider the trivial 0-form α ≡ 0tobeC∞ d-exact and to write 0 = d0.

Remarks Let M be a smooth manifold of dimension n = 1or2. 1. For n = 2, in terms of local C∞ coordinates (x, y) on an open set in M,the exterior derivative of a C1 differential form u of degree 0 (i.e., a C1 function) is ∂u ∂u du = dx + dy, ∂x ∂y and the exterior derivative of a C1 differential 1-form α = Pdx+ Qdy is ∂Q ∂P dα = dP ∧ dx + dQ∧ dy = − dx ∧ dy. ∂x ∂y

1 k 2. For k ∈ Z>0 ∪{∞},aC function u on M is of class C if and only if du is of class Ck−1. Consequently, any exact 1-form α of class Ck−1 is automatically Ck-exact, and in fact, any potential for α is of class Ck. According to the Poincaré lemma (Lemma 9.5.7 below), any closed Ck 2-form is locally Ck -exact. For the purposes of this book, characterizations of global Ck-exactness of a 2-form are not necessary (although a partial solution is considered in Exercise 9.5.9). 3. Any exact differential form α of class C1 on M is closed. For dα = 0 automatically if deg α = 2. If degα = 1, then any potential β for α must be of class C2, and hence dα = d2β = 0. 4. If α and β are C1 differential forms on M, then (see Exercise 9.5.6)

~~d(α ∧ β) = (dα) ∧ β + (−1)deg αα ∧ dβ.~~

~~According to the following fact, the proof of which is left to the reader (see Exercise 9.5.7), exterior differentiation commutes with pullbacks:~~

Proposition 9.5.6 If : M → N is a C2 mapping of smooth manifolds M and N of dimension 1 or 2 and β is a C1 differential form on N, then d∗β = ∗dβ. If deg β = 0, then this also holds for a C1 mapping.

Lemma 9.5.7 (Poincaré lemma) Let p be a point in a smooth surface M. Then there exists a neighborhood U of p in M such that for each k ∈ Z>0 ∪{∞}and 444 9 Manifolds each closed Ck differential form θ of degree r>0 on U, there exists a Ck differential form α on U with dα = θ (in particular, every closed Ck form is locally Ck-exact).

Proof The statement is local, so we may assume without loss of generality that 2 k M = (−1, 1) × (−1, 1) ⊂ R . Suppose k ∈ Z>0 ∪{∞}and θ is a C differential form of degree r>0onM. If r = 1, then there are functions P,Q∈ Ck(M) such that ∂Q ∂P θ = Pdx+ Qdy and 0 = dθ = − dx ∧ dy. ∂x ∂y The function α on M given by x y α(x,y) = P(t,0)dt + Q(x, t) dt ∀(x, y) ∈ M 0 0 then satisﬁes ∂α y ∂ (x, y) = P(x,0) + [Q(x, t)] dt ∂x ∂x 0 y ∂ = P(x,0) + [P(x,t)] dt = P(x,y) 0 ∂t (by Proposition 7.2.5) and ∂α (x, y) = Q(x,y) ∂y for (x, y) ∈ M. Thus dα = θ on M, and in particular, α ∈ Ck(M) (in fact, α ∈ Ck+1(M)). If r = 2, then there is a function f ∈ Ck(M) such that θ = fdx∧ dy. The differential 1-form α = Pdx+ Qdy on M, where 1 y 1 x P(x,y) ≡− f(x,t)dt and Q(x,y) ≡ f(t,y)dt ∀(x, y) ∈ M, 2 0 2 0 is of class Ck (by Proposition 7.2.5) and satisﬁes ∂P/∂y =−f/2 and ∂Q/∂x = f/2. Thus dα = θ.

Remarks 1. In the last part of the above proof, the 1-forms 2Pdxand 2Qdy also have exterior derivative θ. 2. The Poincaré lemma holds for a closed form of positive degree on a manifold of arbitrary dimension.

~~The following terminology is used mainly in Chap. 6:~~

Definition 9.5.8 Let M be a C∞ manifold of dimension n,letp ∈ M, and let k k ∈ Z≥0.AC function f on a neighborhood of p has vanishing derivatives of 9.5 Differential Forms on Smooth Curves and Surfaces 445 order less than or equal to k at p if ∂ α ∂|α|f f(p)= (p) = 0 α1 ··· αn ∂x ∂x1 ∂xn ∞ for every local C coordinate neighborhood (U, x = (x1,...,xn)) of p and every n k multi-index α = (α1,...,αn) ∈ (Z≥0) with |α|≤k.AC vector field or differential form on a neighborhood of p has vanishing derivatives of order ≤ k at p if its coefficients in every local C∞ coordinate neighborhood of p have vanishing derivatives of order ≤ k at p.AC∞ vector field or differential form with vanishing derivatives of order ≤ m at p for every positive integer m is said to have vanishing derivatives of all orders at p.

~~In fact, the above condition is independent of the choice of local coordinates.~~

∞ Proposition 9.5.9 Let M be an n-dimensional C manifold, let p ∈ M, let k ∈ Z≥0, and let f be a Ck function on a neighborhood of p. Then f has vanishing derivatives of order ≤ k at p if (and only if) there exists a local C∞ coordinate neighbor- α hood (V, y = (y1,...,yn)) of p such that (∂/∂y) f(p)= 0 for each multi-index n k α = (α1,...,αn) ∈ (Z≥0) with |α|≤k. Similarly, a C vector field or differential form on a neighborhood of p has vanishing derivatives of order ≤ k at p if (and only if) its coefficients in some local C∞ coordinate neighborhood of p have vanishing derivatives of order ≤ k at p. If θ = fβ, where θ is a Ck vector field or differential form, f is a Ck function, and β is a nonvanishing Ck vector field or differential form, then θ has vanishing derivatives of order ≤ k at p if and only if f has vanishing derivatives of order ≤ k at p.

Proof The claim concerning functions is clear for k = 0. We proceed by induction on k, assuming that k>0 and that the claim holds for nonnegative integers less ∞ than k. Assume that we have a local C coordinate neighborhood (V, y = (y1,..., ∂ α = | |≤ yn)) of p such that ( ∂y ) f(p) 0 for each multi-index α with α k. In particular, by the induction hypothesis, for each j = 1,...,n, the function ∂f /∂yj has vanishing derivatives of order ≤ k − 1atp. Given a local C∞ coordinate neigh- n borhood (U, x = (x1,...,xn)) of p, a multi-index α ∈ (Z≥0) with |α|=k, and an index m with αm > 0, setting n β ≡ (α1,...,αm−1,αm − 1,αm+1,...,αn) ∈ (Z≥0) , we get α β n β ∂ ∂ ∂f ∂ ∂yj ∂f f(p)= (p) = (p). ∂x ∂x ∂xm ∂x ∂xm ∂yj j=1 The product rule yields a sum in which each of the terms contains, for some j, a derivative of ∂f /∂ yj at p of order at most k − 1. Therefore, the above must vanish. The proofs of the claims concerning vector ﬁelds and differential forms are left to the reader (see Exercise 9.5.8). 446 9 Manifolds

Exercises for Sect. 9.5 9.5.1 Prove that if : M → N is a C1 mapping of C∞ manifolds of dimension ≤ 2, ∈ ∈ Z ∈ r ∗ ∈ s ∗ ∗ ∧ p M, r, s ≥0, α (T(p)N)C, and β (T(p)N)C, then (α β) = ∗α ∧ ∗β. 9.5.2 Prove parts (b) and (c) of Proposition 9.5.3. 9.5.3 Let M be a C∞ manifold of dimension n = 1or2,letα be a differential form of degree r on an open set ⊂ M, and let k ∈ Z≥0 ∪{∞}. Prove that for k r = 1, α is of class C if and only if the function α(v): p → αp(vp) is of class Ck for each local C∞ vector ﬁeld v. Prove also that for r = 2, α is of k k class C if and only if the function α(u,v): p → αp(up,vp) is of class C for each pair of local C∞ vector ﬁelds u and v. ∞ 9.5.4 Let M be a C manifold of dimension n = 1 or 2, and let r ∈ Z≥0.Forr = 0 ∗ ∗ ∞ or r>2, r T M and r (T M)C have obvious C (manifold) structures. For r = 1, they have natural C∞ structures provided by Exercise 9.4.3. We now consider the case r = 2. Assume that n = 2. (a) Prove that there are unique C∞ (manifold) structures in 2T ∗M and in 2 ∗ ∞ (T M)C such that for each local C chart (U, = (x1,x2), U ) in M, we get local C∞ charts −1 −1 ∗ (U), , U × R and ∗ (U), C,U × C , 2T M 2(T M)C ∗ ∗ in 2T M and 2(T M)C, respectively, where the mappings and C are given by α → x1(p), x2(p), α (∂/∂x1)p,(∂/∂x2)p α = x1(p), x2(p), (dx1)p ∧ (dx2)p

∈ 2 ∗ 2 ∗ for each point p U and each element α of Tp M and (Tp M)C, respectively. (b) Prove that if : M → N is a Ck mapping of M into a C∞ manifold N of dimension ≤ 2 with k ∈ Z>0 ∪{∞}, then the pullback mappings

~~∗ 2 ∗ 2 ∗ ∗ 2 ∗ 2 ∗ : T N → T M and : (T N)C → (T M)C~~

are of class Ck−1 with respect to the smooth structures provided by (a). ∗ ∗ (c) Prove that the inclusion mapping 2T M → 2(T M)C and the real ∗ ∗ ∞ and imaginary projections 2(T M)C → 2T M are C mappings. 9.5.5 Let M be a 2-dimensional smooth manifold. (a) Verify that the conditions (ii)Ð(v) in Lemma 9.5.4 hold for the operator d constructed in the proof. (b) Prove that in fact, the conditions (i)Ð(iv) in Lemma 9.5.4 uniquely determine the operator d. 9.5.6 Prove that if α and β are C1 differential forms on a smooth manifold M of dimension n = 1 or 2 and r = deg α, then d(α∧β) = (dα)∧β +(−1)r α∧dβ. 9.6 Measurability in a Smooth Manifold 447

9.5.7 Prove Proposition 9.5.6. 9.5.8 Verify the claims concerning vector ﬁelds and differential forms appearing in Proposition 9.5.9. 9.5.9 Let M be a second countable smooth surface, let k be a positive integer, and let θ be an exact differential form of degree 2 and class Ck. Prove that there exists a differential form α of class Ck such that dα = θ. Hint. By deﬁnition, there is a C1 1-form β with dβ = θ. Applying the k 2 Poincaré lemma, one gets local C 1-forms {βi} with dβi = α, and local C { } − = C∞ functions fi with βi β dfi . Forming a suitable partition of unity 2 {λi}, one gets a C 1-form γ ≡ (λiβi + fidλi) with dγ = θ. Now proceed by induction. k+1 9.5.10 Let M be a smooth manifold, let p ∈ M,letk ∈ Z≥0, and let f be a C function on M with vanishing derivatives of order ≤ k at p. Prove that for every local C∞ coordinate neighborhood (U, x) of p, and every compact set K ⊂ U, there exists a constant C>0 such that |f(q)|≤C x(q)− x(p) k+1 for all points q ∈ K.

~~9.6 Measurability in a Smooth Manifold~~

~~In order to consider Lebesgue integration on manifolds, we must ﬁrst extend the deﬁnition of Lebesgue measurability.~~

Deﬁnition 9.6.1 Let M be a smooth manifold of dimension n, and let S ⊂ M. (a) S is called (Lebesgue) measurable if the set (S ∩ U)⊂ Rn is Lebesgue measurable for every local C∞ chart (U,,U) in M. (b) S is of measure 0if(S ∩ U) is a set of Lebesgue measure 0 for every local C∞ chart (U,,U) in M. (c) A statement (P) about points in S is said to hold almost everywhere if the set of points in S at which the statement does not hold is a set of measure 0. (d) A mapping : S → X of S into a topological space X is measurable if the inverse image −1(U) ⊂ S is measurable for every open set U ⊂ X (in particular, S is measurable). For X =[−∞, ∞] or C, is also called a measurable function.

~~We have the following characterizations of measurable sets and mappings:~~

Proposition 9.6.2 In any smooth manifold M, we have the following: (a) AsetS ⊂ M is measurable (of measure 0) if and only if for each point p ∈ S, there exists a local C∞ chart (U,,U) such that p ∈ U and (S ∩ U)⊂ Rn is measurable (respectively, of measure 0). The composition ρ ◦ of a continuous mapping ρ and a measurable mapping is measurable. The image of a measurable subset of M under a diffeomorphism is measurable. 448 9 Manifolds

(b) The collection of measurable subsets of M is a σ -algebra that contains the Borel subsets of M. Moreover, any subset of a set of measure 0 is a set of measure 0. (c) Let S ⊂ M be a measurable set. Then any product or sum of measurable functions (with values in [−∞, ∞] or C) on S is measurable. A mapping = (ψ1,...,ψm) of S into a product X1 × ··· × Xm of topological spaces X1,...,Xm is measurable if and only if ψj is measurable for each j = 1,...,m.

~~Proof Part (a) follows from the deﬁnitions and the change of variables formula (Theorem 7.2.7), and parts (b) and (c) follow from standard arguments.~~

~~In analogy with Deﬁnition 9.5.2, we make the following:~~

Deﬁnition 9.6.3 Let S be a measurable subset of a smooth manifold M of dimension n = 1 or 2. A differential form on S is measurable if its coefﬁcients in every local C∞ coordinate neighborhood are measurable.

~~The proofs of the following facts concerning measurable differential forms are left to the reader (see Exercise 9.6.1):~~

Proposition 9.6.4 (Cf. Proposition 9.5.3) Let M be a smooth manifold of dimension n = 1 or 2, and let α be a differential form on a measurable set S ⊂ M. Then we have the following: (a) The differential form α is measurable if and only if for every point in S, there exists a local C∞ coordinate neighborhood with respect to which the coefﬁcient functions of α are measurable. (b) If : N → M is a diffeomorphism, then α is measurable if and only if the pullback differential form ∗α on −1(S) ⊂ N is measurable. (c) The sum and exterior product of any two measurable differential forms on S is measurable.

Remark Similarly, a vector ﬁeld on a measurable subset of a smooth manifold is measurable if its coefﬁcients in any local C∞ coordinate neighborhood (or in some local C∞ coordinate neighborhood of each point) are measurable.

Exercises for Sect. 9.6 9.6.1 Prove Proposition 9.6.4. 9.6.2 Let α be a differential form of degree r on a measurable subset S of a smooth manifold M of dimension n = 1 or 2. Prove that for r = 1, α is measurable if ∞ and only if the function α(v): p → αp(vp) is measurable for each local C vector ﬁeld v. Also prove that for r = 2, α is measurable if and only if the ∞ function α(u,v): p → αp(up,vp) is measurable for every pair of local C vector ﬁelds u and v. 9.7 Lebesgue Integration on Curves and Surfaces 449

9.6.3 Let v be a vector field on a measurable set S in a smooth manifold M. Prove that the following are equivalent: (i) The vector field v is measurable (i.e., its coefficients in any local C∞ coordinate neighborhood are measurable). ∞ (ii) For any local C coordinate neighborhood (U, (x1,...,xn)), the func- ∩ = = tion dxj (v) is measurable on S U for each j 1,...,n (i.e., v n · ∩ j=1 vj (∂/∂xj ) on S U with measurable coefficients v1,...,vn on S ∩ U). (iii) For every point p ∈ S, there exists a local C∞ coordinate neighborhood (U, (x1,...,xn)) in M such that the function dxj (v) is measurable on S ∩ U for each j = 1,...,n. (iv) The function df (v) = v(f ) is measurable for each local C∞ function f . 9.6.4 Let v be a vector field on a measurable set S in a smooth manifold M. Prove that v is a measurable vector field (i.e., its coefficients in any local C∞ coordinate neighborhood are measurable) if and only if v is measurable as a ∞ mapping of S into (T M)C, where (T M)C has the C (and hence, topological space) structure provided by Exercise 9.4.3. State and prove the analogous claim for a differential form. 9.6.5 Let f : M → N be a C∞ mapping of second countable C∞ manifolds M and N of dimension m and n, respectively. According to Sard’s theorem (see, for example, [Mi]), the set of critical values of f is a set of measure 0. Prove this theorem for the case m = n = 1. Hint. One may assume without loss of generality that M is a bounded open interval in R and N = R. Prove that for every interval I ⊂ M, f(I) ≤ | |· is measurable and λ(f (I)) supI f (I). Show that one may cover any compact subset of the zero set of f by a suitable finite collection of disjoint open intervals on which |f | is small.

~~9.7 Lebesgue Integration on Curves and Surfaces~~

A connected 1-dimensional smooth manifold M is orientable; that is, one may cover M by coordinate intervals with the same sense of left and right (more precisely, the coordinate transformations have positive derivative). The choice of such an atlas is then one of the two orientations in M. One gets the opposite orientation by reversing the direction of each of the intervals in the atlas. For M second countable, the existence of an orientation follows from the fact that every connected second countable 1-dimensional smooth manifold is diffeomorphic to either the circle or the real line (see Theorem 9.10.1). For the general case, one may apply a Zorn’s lemma argument. However, in two dimensions (or higher), no orientation need exist.

~~Example 9.7.1 The Möbius band is the C∞ surface~~

~~M =[0, 1]×(0, 1)/(0,t)∼ (1, 1 − t) ∀t ∈ (0, 1). 450 9 Manifolds~~

The topology is the quotient topology for the quotient map :[0, 1]×(0, 1) → M, C∞ { }2 = × = and a atlas is given by (Uj ,j ,V) j=1, where U1 ((0, 1) (0, 1)), V × = −1 = [ ]\{ } × (0, 1) (0, 1), 1 ( (0,1)×(0,1)) , U2 (( 0, 1 1/2 ) (0, 1)), and (s + (1/2), t) if (s, t) ∈[0, 1/2) × (0, 1), 2((s, t)) = (s − (1/2), 1 − t) if (s, t) ∈ (1/2, 1]×(0, 1).

We cannot choose an atlas so that on overlaps, the local charts agree on what a right- handed frame should be, because of the twist at the end. The details appear after the precise deﬁnitions below.

Throughout this section, M denotes a smooth manifold of dimension n = 1or2 (for a discussion of orientation and integration in higher-dimensional manifolds, the reader may refer to, for example, [Mat], [Ns3], or [Wa]). We recall that the Jacobian 1 m determinant of a C mapping F = (f1,...,fm): U → R of an open subset U of m m R into R is the continuous function JF : U → R given by ∂f1 ··· ∂f1 ∂x1 ∂xm . . . JF ≡ det(dF ) = . .. . . . . ∂fm ··· ∂fm ∂x1 ∂xm In particular, if F is a diffeomorphism, then JF = 0. For m = 1, we get JF = F , and for m = 2, we get

~~∂f1 ∂f2 ∂f2 ∂f1 JF = − . ∂x1 ∂x2 ∂x1 ∂x2~~

Deﬁnition 9.7.2 In the manifold M, we have the following: C∞ (a) Two local charts (U1,1,U1) and (U2,2,U2) in M have compatible orientations if each of the associated coordinate transformations has positive Jaco- bian determinant. That is,

~~J ◦ −1 > 0on2(U1 ∩ U2) 1 2~~

−1 (equivalently, J ◦ −1 = (1/J ◦ −1 ) ◦ (2 ◦ )>0on1(U1 ∩ U2)). 2 1 1 2 1 (b) An atlas in M in which the local charts have compatible orientations is said to be oriented. (c) Two oriented atlases A1 and A2 in M have equivalent orientations if the atlas A1 ∪ A2 is oriented. The veriﬁcation that this gives an equivalence relation is left to the reader (see Exercise 9.7.1). (d) An equivalence class of oriented atlases in M is called an orientation in M.If an orientation exists, then M is said to be orientable. Otherwise, M is called nonorientable. A manifold M together with an orientation is said to be oriented. A local chart in a representing atlas for the orientation is then said to be positively oriented. 9.7 Lebesgue Integration on Curves and Surfaces 451

(e) A diffeomorphism : M → N of M onto an oriented smooth manifold N of dimension n is orientation-preserving if for each positively oriented local C∞ chart (V, ,V) in N, the local C∞ chart (−1(V ), ◦ , V ) is positively oriented in M (hence, −1 is also orientation-preserving).

~~Lemma 9.7.3 The Möbius band is nonorientable.~~

Proof In the notation of Example 9.7.1, the open subsets U1 and U2 are con- 2 nected and orientable, since we have the diffeomorphism j : Uj → V ⊂ R for j = 1, 2. If M is orientable, then we may choose an oriented C∞ atlas A. Since equivalence of orientations is an equivalence relation among oriented atlases, we may choose the orientation on M so that for some map ρ : R2 → R2 of the ∞ form (x, y) → (x, σy) with σ =±1, the local C charts (U1,ρ ◦ 1, σ (V )) and (U2,2,V)induce orientations in U1 and U2, respectively, that are compatible with the orientation on M.ButU1 ∩ U2 = (((0, 1) \{1/2}) × (0, 1)), and for each point (s, t) ∈ 2(((0, 1/2) × (0, 1))) = (1/2, 1) × (0, 1),wehave ◦ ◦ −1 = − (ρ 1) 2 (s, t) (s (1/2), σ t), while for each point (s, t) ∈ 2(((1/2, 1) × (0, 1))) = (0, 1/2) × (0, 1),wehave ◦ ◦ −1 = + − (ρ 1) 2 (s, t) (s (1/2), σ (1 t)).

Thus the Jacobian determinant J ◦ ◦ −1 is equal to σ on ((0, 1/2) × (0, 1)) ρ 1 2 and −σ on ((1/2, 1) × (0, 1)), and hence the Jacobian determinant cannot be everywhere positive. Thus we have arrived at a contradiction, and therefore M is nonorientable.

Remarks 1. In the discussion of smooth structures on surfaces in Chap. 6, we will see that a (topological) Möbius band M is nonorientable with respect to any C∞ structure on M (see Sect. 6.10). 2. If M is an oriented C∞ manifold, then any open subset of M has a natural induced orientation. 3. If M is a connected orientable C∞ manifold, then there are exactly two (opposite) orientations in M (see Exercise 9.7.2).

~~∞ Proposition 9.7.4 For each local C chart (U, = (x1,...,xn), U ) in M, let ω be the nonvanishing C∞ differential form of degree n on U given by dx1 if n = 1, ω ≡ dx1 ∧ dx2 if n = 2.~~

~~∞ (a) Given local C charts (U, = (x1,...,xn), U ) and (V, = (y1,...,yn), V ) in M, we have ω = J ◦ ( ◦−1 ) . ω 452 9 Manifolds~~

In particular, the two local C∞ charts have compatible orientations if and only if ω /ω > 0 on U ∩ V . (b) If M is second countable and oriented, then there exists a nonvanishing C∞ differential form ω of degree n on M such that for every positively oriented ∞ local C chart (U,,U ) in M, we have ω/ω > 0 on U. (c) Given a nonvanishing continuous differential form ω of degree n on M, the ∞ collection A of all local C charts (U,,U ) in M with ω/ω > 0 on U is an oriented atlas in M (and hence A determines a unique orientation in M).

Proof The proof of part (a) and the proof of part (c) are left to the reader (see Exer- cise 9.7.3). For the proof of part (b), we observe that if M is second countable and oriented, then we may choose a countable locally ﬁnite covering of M by positively ∞ ∞ oriented local C charts {(Ui,i,U )}i∈I and a C partition of unity {λi} with ∞ i supp λi ⊂ Ui for each i ∈ I .TheC differential form ≡ · ω λi ωi i∈I · is then of degree n (following standard conventions, here we extend λi ωi by0to a C∞ n-form on M for each i ∈ I ), and given a positively oriented local C∞ chart (U,,U) in M, part (a) implies that ω ωi = λi · > 0 ω ω i∈I on U (in particular, ω is nonvanishing). Thus part (b) is proved.

Definition 9.7.5 (Cf. Definition 8.2.3) Assume that M is oriented, and let S ⊂ M. (a) A real differential form ω of degree n defined at points in S is said to be positive (nonnegative) if for every positively oriented local C∞ coordinate neighborhood (U, = (x1,...,xn)) in M,wehave ω > 0 (respectively, ≥ 0) on S ∩ U, ω

where ω = dx1 if n = 1 and ω = dx1 ∧ dx2 if n = 2 (equivalently, each point in S has some coordinate neighborhood (U, = (x1,...,xn)) for which the above holds). We write ω>0or0<ω (respectively, ω ≥ 0or0≤ ω). Similarly, ω is said to be negative (nonpositive) and we write ω<0or0>ω (respectively, ω ≤ 0or0≥ ω)if−ω>0 (respectively, −ω ≥ 0). For p ∈ M, an ordered basis (v1,...,vn) (= v1 or (v1,v2))forTpM is positively oriented n if each positive α ∈ TpM is positive on the ordered basis (i.e., α(v1)>0if n = 1, α(v1,v2)>0ifn = 2). (b) For two real differential forms α and β of degree n deﬁned at points in S,we write α ≥ β or β ≤ α if α − β ≥ 0. 9.7 Lebesgue Integration on Curves and Surfaces 453

(c) Let α be a real differential form of degree n deﬁned at points in a set S ⊂ M. Given a point p ∈ S, we deﬁne the positive part and negative part of α at p by ≥ − ≤ + ≡ αp if αp 0, − ≡ αp if αp 0, αp and αp 0ifαp < 0, 0ifαp > 0,

~~respectively. In other words,~~

~~ + − + αp − αp (α ) = · ω and (α ) = · ω p ω p ω~~

∈ n ∗ for an (arbitrary) positive element ω Tp M. (d) Given a sequence {αν} of real differential forms of degree n defined at points in S, and given a point p ∈ S, we define (αν)p (αν )p inf(αν)p ≡ inf · ω, lim inf(αν)p ≡ lim inf · ω, ν ν ω ν→∞ ν→∞ ω (αν)p (αν)p sup(αν)p ≡ sup · ω, lim sup(αν)p ≡ lim sup · ω, ν ν ω ν→∞ ν→∞ ω ∈ n ∗ for an arbitrary positive element ω Tp M, provided the above defined coefficients exist in R.

Remarks 1. If α is a measurable (continuous) real differential form of degree n = dim M as in (c) above, then α+ and α− are measurable (respectively, continuous). The analogous statement concerning measurability holds for the inﬁmum, supremum, limit inferior, limit superior, and limit of a sequence of real measurable differential forms of degree n as in (d) above. 2. It is easy see that for a sequence {αν} as in (d) above and for α as in (c), we have lim αν = α if and only if lim inf αν = lim sup αν = α. 3. A nonvanishing C∞ real differential form of degree n on M is also called a volume form.

Proposition 9.7.6 Assume that M is second countable and oriented. Then, for every continuous real differential n-form τ on M, there exists a positive C∞ differential n-form ω such that ω ≥ τ on M.

Proof Proposition 9.7.4 provides a positive C∞ n-form ω on M, and Proposi- tion 9.3.11 provides a C∞ function ρ such that ρ>|τ/ω| on M. The positive C∞ n-form ω ≡ ρω then satisﬁes ω ≥ τ on M.

Lemma 9.7.7 Suppose M is oriented, ω is a measurable nonnegative differential form of degree n deﬁned at points in a measurable set S ⊂ M, (t1,...,tn) n are the standard coordinates in R , ωR1 = dt1, ωR2 = dt1 ∧ dt2, and (U, = 454 9 Manifolds

~~ (x1,...,xn), U ) and (V, = (y1,...,yn), V ) are two positively oriented local C∞ charts in M. Then [−1]∗ω [ −1]∗ω dλ = dλ. (S∩U∩V) ωRn (S∩U∩V) ωRn~~

~~Proof This follows from the change of variables formula (Theorem 7.2.7). For if n = 2 and ω = fdx1 ∧ dx2 = gdy1 ∧ dy2 on U ∩ V , then = · J ◦ = ·|J ◦ | f g ◦−1 g ◦−1 and~~

~~[ −1]∗ = ◦ −1 ∧ = ◦ −1 ·|J | ∧ ω (f )dt1 dt2 (g ) ◦−1 dt1 dt2.~~

Therefore [−1]∗ω dλ = (f ◦ −1)dλ (S∩U∩V) ωR2 (S∩U∩V) = ◦ −1 ◦ ◦ −1 ·|J | (g ) ◦−1 dλ ◦ −1( (S∩U∩V)) [ −1]∗ω = (g ◦ −1)dλ= dλ. (S∩U∩V) (S∩U∩V) ωR2

~~The proof for n = 1 is similar.~~

Definition 9.7.8 (Cf. [AhS], [Sp], [Wey]) Suppose M is oriented, ω is a measurable complex differential form of degree n defined at points in a measurable set S ⊂ M, n (t1,...,tn) are the standard coordinates in R , ωR1 = dt1, and ωR2 = dt1 ∧ dt2. (a) If ω ≥ 0onS and S ⊂ U for some positively oriented local C∞ chart (U,,U) in M, then we define [−1]∗ω ω ≡ dλ ∈[0, ∞] S (S) ωRn

~~(which is independent of the choice of the local chart by Lemma 9.7.7). (b) If ω ≥ 0onS, then we deﬁne the integral of ω over S by~~

~~ m ω ≡ sup ω ∈[0, ∞], S j=1 Sj~~

where the supremum is taken over all choices of disjoint measurable subsets ∞ S1,...,Sm of S, each of which is contained in a (possibly different) local C chart. 9.7 Lebesgue Integration on Curves and Surfaces 455

(c) We say that ω is integrable on S if the nonnegative extended real numbers + − R+ ≡ [Re ω] ,R− ≡ [Re ω] , S S + − I+ ≡ [Im ω] ,I− ≡ [Im ω] , S S are ﬁnite. If this is the case, then we deﬁne the integral of ω over S by ω ≡ R+ − R− + iI+ − iI−. S (d) We say that ω is locally integrable on S if each point in S admits a neighborhood U in M such that ω U∩S is integrable.

Remarks 1. The definitions in (a) and (b) are consistent; that is, for S contained in a local C∞ coordinate neighborhood, the integral of ω ≥ 0 defined in (a) is equal to its integral as defined in (b). 2. On a second countable oriented manifold, one may simplify the above by using a partition of unity (see, for example, [Mat], [Ns3], or [Wa]).

Proposition 9.7.9 Assume that M is oriented, let S be a measurable subset of M, let ω be a nonnegative measurable differential form of degree n deﬁned on S, and let α and β be measurable differential forms of degree n deﬁned on S.

~~(a) The function λω on the σ -algebra of measurable subsets of S deﬁned by ~~

~~λω : R → ω R is a positive measure. (b) If f is a measurable function that satisﬁes 0 ≤ f<∞ a.e. in S, then ~~

~~fω= fdλω. S S~~

(c) If α = 0 a.e. in Z ≡{x ∈ S | ωx = 0} and α ≥ 0 a.e. in S, then α α α = · ω = · dλω. S S\Z ω S\Z ω (d) If α ≥ 0 and β ≥ 0 a.e. in S, then (rα + β) = r α + β ∀r ∈[0, ∞). S S S (e) If α and β are integrable on S and r ∈ C, then rα + β is integrable on S and (rα + β) = r α + β. S S S 456 9 Manifolds

~~(f) If α ≥ β a.e. in S and each of the forms is either nonnegative or integrable on S, then α ≥ β. S S (g) For any open covering B of M, we have~~

~~ m ω = sup ω, S j=1 Sj~~

~~where the supremum is taken over all choices of disjoint measurable subsets S1,...,Sm of S each of which is contained in an element of B.~~

Proof Clearly, we have λω(∅) = 0. Suppose R is the union of a sequence {Sν} of disjoint measurable subsets of S. Given ﬁnitely many disjoint measurable subsets ∞ R1,...,Rm of S, each of which is contained in a local C coordinate neighborhood in M,wehave

m m ∞ ∞ m ∞ ∞ ω = ω = ω ≤ ω = λω(Sν ). ∩ ∩ j=1 Rj j=1 ν=1 Sν Rj ν=1 j=1 Sν Rj ν=1 Sν ν=1 ≤ ∞ ∈[ ∞ ∞ Thus λω(R) ν=1 λω(Sν). Conversely, given t 0, ) with ν=1 λω(Sν)>t, we may choose N ∈ Z>0 and, for each ν = 1,...,N, disjoint measurable subsets { (ν)}mν C∞ Rj j=1 of Sν , each contained in some local coordinate neighborhood, such that N mν t< ω ≤ λω(R). (ν) ν=1 j=1 Rj Part (a) now follows. For (b), observe that by changing f on a set of measure 0, we may assume without loss of generality that 0 ≤ f<∞ on S. The equality is clear if S is contained in a local C∞ coordinate neighborhood (see the remarks following Deﬁni- tion 9.7.8). Applying (a), one gets the equality for a ﬁnite union of disjoint measurable subsets, each of which has the above property. Passing to suprema, one gets ≤ S fω S fdλω. For the reverse inequality, observe that if S1,...,Sm are disjoint measurable subsets of S and r1,...,rm are nonnegative constants with f ≥ rj on Sj for each j, then

~~m m m rj λω(Sj ) = rj ω ≤ fω≤ fω j=1 j=1 Sj j=1 Sj S~~

~~(one may verify the above inequalities directly). The proofs of (c)Ð(g) are left to the reader (see Exercise 9.7.4). 9.7 Lebesgue Integration on Curves and Surfaces 457~~

Deﬁnition 9.7.10 Assume that M is oriented, and let ω be a nonnegative measurable n-form on a measurable set S ⊂ M. Then we denote by λω the positive measure on S provided by Proposition 9.7.9, and we call λω the measure associated to ω.

~~Remark The measure λω is complete if ω>0 almost everywhere (see Exer- cise 9.7.9).~~

~~Applying the standard convergence theorems for measure spaces, we get the following versions for integrals of differential forms:~~

~~Theorem 9.7.11 (Convergence theorems) Assume that M is oriented. Let ω and { }∞ ⊂ ων ν=1 be measurable n-forms on a measurable set S M.~~

~~(a) Monotone convergence theorem. If 0 ≤ ω1 ≤ ω2 ≤ ω3 ≤··· and ων → ω, then ω = lim ω . →∞ ν S ν S~~

~~(b) Fatou’s lemma. If ων ≥ 0 for each ν and ω = lim infν→∞ ων , then ω ≤ lim inf ω . →∞ ν S ν S~~

(c) Dominated convergence theorem. Suppose ων → ω and there exists a nonnegative integrable differential form θ of degree n on S such that for each ν ∈ Z>0 and each point x ∈ S, we have either (ων )x = 0 or θx > 0 and |(ων)x /θx|≤1. Then ω is integrable and ω = lim ω . →∞ ν S ν S

~~Remark It is not necessary to specify that the form ω be measurable, since the limit or limit inferior of a sequence of measurable differential forms of degree n is measurable.~~

Proof of Theorem 9.7.11 For the proof of (a), we let P ={x ∈ S | ωx > 0}.Byhy- pothesis, we have ων = 0 at each point in S \ P . Therefore, by the standard monotone convergence theorem (i.e., the monotone convergence theorem for integration of functions with respect to measures), we have ων ων ω ων = · ω = · dλω → · dλω = ω = ω as ν →∞. S P ω P ω P ω P S

~~The proofs of (b) and (c) are left to the reader (see Exercise 9.7.5).~~

~~Deﬁnition 9.7.12 Let S be a measurable subset of M, and let p ∈[1, ∞]. 458 9 Manifolds~~

(a) Let α be a measurable differential form of degree r in M on S.Forr = 0, we p C∞ ◦ −1 say that α is in Lloc(S) if for every local chart (U,,U ), α is in p ∩ ∩ Lloc((S U)) (that is, each point in (S U)admits a neighborhood V such that the function |α ◦ −1|p is integrable on V ∩ (S ∩ U)). For r>0, we say p C∞ that α is in Lloc(S) if the coefficients of α in every local chart (U,,U ) p ∩ = are in Lloc(S U).Forp 1, we also say that α is locally integrable, and for p = 2, we also say that α is locally square integrable. { }∞ p = { } (b) Let αν ν=1 and α be differential r-forms in Lloc(S).Forr 0, we say that αν p C∞ converges in Lloc(S) to α if for every local chart (U,,U ), the sequence { ◦ −1} ◦ −1 p ∩ αν converges to α in Lloc((S U)) (that is, for each point in ∩ | ◦ −1 − (S U), there exists a neighborhood V in U such that V ∩(S∩U) αν −1 p p α ◦ | dλ → 0). For r>0, we say that {αν} converges in L (S) to α if in ∞ loc every local C chart (U,,U ), the coefficients of the terms of {αν} converge p ∩ in Lloc(S U) to the corresponding coefficients of α.

Remarks 1. It is easy to check that the conditions in (a) and (b) hold precisely when they hold in some local C∞ coordinate neighborhood of each point in S. 2. For M orientable, a measurable n-form α on S ⊂ M is locally integrable (in the above sense) if and only if each point in S admits a neighborhood U such that α is integrable on U ∩ S (as an n-form). The analogous statement for local L1- convergence also holds. p q ∈ 3. If α and β are differential forms in Lloc and Lloc, respectively, with p,q [1, ∞] and p−1 + q−1 = 1, then the exterior product α ∧ β is locally integrable; ∧ 1 that is, α β is in Lloc. p 4. A measurable vector ﬁeld is said to be in Lloc(S) if its coefﬁcients in every C∞ p ∩ local chart (U,,U ) are in Lloc(S U).

Lemma 9.7.13 Assume that M is second countable and oriented, and let ω be a nonnegative locally integrable measurable differential form of degree n on M. Then there exists a positive C∞ function ρ on M so large that e−ρω is integrable.

Proof We may assume that M is noncompact (otherwise take ρ ≡ 1) and we may ∞ choose a locally ﬁnite covering {Uν } = by relatively compact open subsets and a ∞ ν 1 C partition of unity {λν} with supp λν ⊂ Uν and 0 ≤ λν ≤ 1 for each ν. For each ν, we may choose a constant Rν such that −ν 0 0. We then get ∞ ∞ −ρ −ν e ω = Rν λνω< 2 = 1 < ∞. M ν=1 M ν=1 9.7 Lebesgue Integration on Curves and Surfaces 459

Deﬁnition 9.7.14 An open subset of a 2-dimensional C∞ manifold M is called C∞ (or smooth) if for each point p ∈ M, there is a local C∞ coordinate neighborhood (U,(x,y)) such that ∩ U ={q ∈ U | x(q) < 0}.

~~Remark The boundary ∂ of a C∞ open subset of M is either the empty set or a 1-dimensional C∞ submanifold of M.~~

Lemma 9.7.15 Let M be an oriented C∞ manifold of dimension 2, and let be a C∞ open subset of M with nonempty boundary ∂ . Then ∂ has a (natural) induced orientation with an oriented C∞ atlas A consisting of all local C∞ charts ∩ ∩ = of the form (U ∂ ,2 U∩∂ ,2(U ∂ )), where (U, (1,2), U ) is a ∞ positively oriented local C chart in M with ∩ U ={q ∈ U | 1(q) < 0} (see Fig. 9.2). Furthermore, if : M → N is an orientation-preserving diffeomorphism of M onto an oriented C∞ manifold N and ≡ ( ), then the diffeomorphism : → ∂ ∂ ∂ is orientation-preserving with respect to the induced orientations.

∞ Proof The collection A covers ∂ , since if (U, (1,2), U ) is a local C chart that is not positively oriented, U is connected, and ∩ U ={1 < 0}, then the ∞ local C chart (U, (1, −2), U ), where U ≡{(x, −y) | (x, y) ∈ U }, is posi- ∞ tively oriented. Given two positively oriented local C charts (U, = (1,2) = (x, y), U ) and (V, = ( 1, 2) = (u, v), V ) with ∩ U ={x<0} and ∩ V = { } ≡ ◦ −1 u<0 , the coordinate transformation F 2 (2 U∩∂ ) of the associated local C∞ charts in ∂ satisﬁes −1 d ∂v −1 F (t) = d ( ∩ )∗ = ( (0, t)). 2 2 U ∂ dt ∂y On the other hand, we have u<0onU ∩ V ∩ ={p ∈ U ∩ V | x(p) < 0}, u ≡ 0 on U ∩ V ∩ ∂ ={p ∈ U ∩ V | x(p) = 0}, and u>0onU ∩ V \ ={p ∈ U ∩ V | x(p) > 0}. So, at each point in U ∩ V ∩ ∂ ,wehave ∂u ∂u ≥ 0 and = 0, ∂x ∂y and hence ∂u/∂x 0 ∂u ∂v 0 < J ◦ −1 ◦ = = · . ∂v/∂x ∂v/∂y ∂x ∂y It follows that F (t) > 0 and therefore that the atlas is oriented. The proof of the last statement concerning the diffeomorphism : M → N is left to the reader (see Exercise 9.7.6).

Deﬁnition 9.7.16 Let be a C∞ open subset of an oriented 2-dimensional C∞ manifold M with nonempty boundary ∂ . The orientation on ∂ given by Lemma 9.7.15 is called the induced orientation.

~~Remark We assume that the boundary of a C∞ open set in an oriented 2-dimensional C∞ manifold has the induced orientation unless otherwise indicated. 460 9 Manifolds~~

~~Fig. 9.2 The induced orientation on ∂~~

Theorem 9.7.17 (Stokes’ theorem) Let M be an oriented C∞ surface, let be a nonempty smooth domain in M, and let α be a C1 differential form of degree 1 on M such that ∩ supp α is compact. Then α = dα. ∂ ∩ =∅ = In particular, if supp α ∂ , then dα 0. ∗ = Remark Stokes’ theorem should actually be written ∂ ι α dα, where ι: ∂ → M is the inclusion map; but in an abuse of notation, we will usually leave out the pullback map ι∗.

Proof of Stokes’ theorem We may choose a ﬁnite collection { = = × }m (Ui,i (xi,yi), Ui (ai,bi) (0, 1)) i=1 ∞ of positively oriented local C charts such that ∩supp α ⊂ U1 ∪···∪Um M and ∞ for each i = 1,...,m, ∩ Ui ={p ∈ Ui | xi(p) < 0} =∅. We may also choose C { }m m ≡ ∩ ⊂ functions ηi i=1 such that i=1 ηi 1on supp α and supp ηi Ui for each i = 1,...,m. By reordering if necessary, we may assume that there is an index k such that ai < 0

~~We close this section by brieﬂy recalling the related notion of a line integral.~~

Deﬁnition 9.7.18 Let α be a continuous differential form of degree 1 on a C∞ manifold M of dimension 1 or 2, and let γ :[a,b]→M be a piecewise C1 path in M (see Deﬁnition 9.2.5). Then the (line) integral of α along γ is given by b α ≡ α(γ(t))dt˙ = γ ∗α. γ a (a,b)

Lemma 9.7.19 Let α be a continuous 1-form on a C∞ manifold M of dimension n = 1 or 2, and let γ :[a,b]→M be a piecewise C1 path in M. ∞ (a) If (U, (x1,...,xn)) is a local C coordinate neighborhood in M, γ([a,b]) ⊂ U, cj = xj ◦ γ for each j = 1,...,n, and α = a1dx1 +···+andxn on U, then n b = α aj (γ (t))cj (t) dt. γ j=1 a

(b) If N ≡ γ ((a, b)) is itself a 1-dimensional C∞ manifold for which the restriction : → : → γ (a,b) (a, b) N is a diffeomorphism and the inclusion map ι N M is C∞ = ∗ of class , then γ α N ι α, provided N is given the orientation induced by γ .

~~The proof is left to the reader (see Exercise 9.7.7). Line integrals are considered in greater depth in Chap. 10.~~

Example 9.7.20 For polar coordinates (r, θ) in a suitable open subset of R2, and for rectangular coordinates (x, y),wehavedx ∧ dy = rdr∧ dθ. Thus (r, θ) gives positively oriented local C∞ coordinates. Consequently, for each R>0 and each iθ iθ θ0 ∈ R, the diffeomorphism θ → Re of (θ0,θ0 + 2π) onto (∂ (0; R)) \{Re 0 } is orientation-preserving (where the boundary ∂ (0; R) of the smooth domain (0; R) is given the induced orientation). Hence, for any continuous 1-form α = 462 9 Manifolds adr+ bdθ deﬁned at points of ∂ (0; R),wehave θ0+2π α = b(Reiθ)dθ. ∂ (0;R) θ0

Exercises for Sect. 9.7 9.7.1 Prove that equivalence of oriented C∞ atlases on a C∞ manifold M of dimension n = 1 or 2 is an equivalence relation. 9.7.2 Prove that any connected orientable C∞ manifold M of dimension n = 1or2 admits exactly two orientations. 9.7.3 Prove parts (a) and (c) of Proposition 9.7.4. 9.7.4 Prove parts (c)Ð(g) of Proposition 9.7.9. 9.7.5 Prove parts (b) and (c) of Theorem 9.7.11. 9.7.6 Prove the last part of Lemma 9.7.15. That is, prove that if M and N are oriented 2-dimensional C∞ manifolds, is a C∞ open subset of M, : M → N is an orientation-preserving diffeomorphism, and ≡ ( ), then the dif- : → feomorphism ∂ ∂ ∂ is orientation-preserving with respect to the induced orientations. 9.7.7 Prove Lemma 9.7.19. 9.7.8 Let M be a nonorientable C∞ surface. Prove that there exists a surjective C∞ mapping ϒ : M → M of an orientable C∞ surface M onto M such that each point in M admits a connected neighborhood U for which ϒ−1(U) has exactly two connected components, each of which is mapped diffeomorphically onto U by ϒ (ϒ : M → M is called the orientable double cover of M). 9.7.9 Let ω be a nonnegative measurable n-form on an orientable C∞ manifold M of dimension n = 1 or 2. Prove that if ω>0 a.e., then the measure λω is complete.

~~9.8 Linear Differential Operators on Manifolds~~

Throughout this section, M denotes a smooth manifold of dimension n. Linear differential operators on open sets in Rn were considered in Sect. 7.4. A linear differential operator on M is an operator that may be expressed in local C∞ coordinates as a linear differential operator on an open subset of Rn.Morepre- cisely:

Deﬁnition 9.8.1 Let be an open subset of M.Alinear differential operator A of order k ∈ Z≥0 on in M is an operator that associates to each open set V ⊂ and each Ck function u on V , a function Au ∈ C0(V ), and which has the following property. For each point in , there exist a local C∞ coordinate neighborhood (U, = (x1,...,xn), U ) and a linear differential operator A of order k on ( ∩ U) such that for each open set V ⊂ and each function u ∈ Ck(V ), 9.8 Linear Differential Operators on Manifolds 463

~~[ ] = ◦ −1 ◦ we have Au V ∩U A(u ) V ∩U . In other words, there exist functions { } n ∩ aα α∈(Z≥0) ;|α|≤k on U such that~~

~~∂|α| [ ] = · u Au V ∩U aα α n ∂x α∈(Z≥0) ,|α|≤k~~

∩ ⊂ ∈ Ck = on V U for every open set V and every function u (V ) (we have A −1 α n (aα ◦ ) · (∂/∂t) , where (t1,...,tn) are the standard coordinates on R ). We call this a local representation of A with coefﬁcients {aα}, and we write ∂ α A = aα on ∩ U. n ∂x α∈(Z≥0) ,|α|≤k

~~We say that A has Cd coefﬁcients if it has local representations with Cd coefﬁcients in a neighborhood of each point.~~

Remarks 1. It is easy to verify that if A is a linear differential operator of order k with Cd coefficients on an open subset of M, then A has a local representation with Cd coefficients in every local C∞ coordinate neighborhood in M. 2. It follows from the definition that if A is a linear differential operator of order k ⊂ ∈ Ck [ ] = on an open set M, u ( ), and V is an open subset of , then Au V [ ] A u V . 3. Linear differential operators on differential forms are defined similarly (in terms of differential operators on the coefficients in local C∞ coordinate neighborhoods).

~~For distributional solutions of the differential equation Au = v, one consid- ers distributional solutions of the corresponding equations in local C∞ coordinates (cf. Deﬁnition 7.4.2).~~

Deﬁnition 9.8.2 Let k ∈ Z≥0, and let A be a linear differential operator of order k Ck ⊂ ∈ 1 with coefﬁcients on an open set M.Foru, v Lloc( ), we say that Au is equal to v in the distributional (or weak) sense, and we write Adistru = v (and ∈ 1 C∞ Adistru Lloc( )), if for every local coordinate neighborhood (U,,U ) in M −1 with corresponding linear differential operator A (i.e., Af = A(f ◦ ) ◦ for k −1 −1 f ∈ C ), we have [A]distr(u ◦ ) = v ◦ on ( ∩ U).

This property is local; that is, Adistru = v on if and only if Adistru = v on a neighborhood of each point in . Moreover, according to the following, this property is independent of the choice of local C∞ coordinates:

Proposition 9.8.3 Let A be a linear differential operator of order k with Ck coefﬁ- ⊂ ∈ 1 = cients on an open set M, and let u, v Lloc( ). Then Adistru v if and only if for every point in , there exists a local C∞ chart (V, ,V) in M for which the 464 9 Manifolds

−1 −1 associated linear differential operator A satisﬁes [A ]distr(u ◦ ) = v ◦ on ( ∩ V). In particular, Adistru = v if and only if for each point in , there is a = neighborhood U in such that Adistr(u U ) v U .

∞ Proof Let (U,,U ) be a local C coordinate neighborhood, and let A be the associated linear differential operator. Suppose (V, ,V) is a local C∞ coordinate neighborhood in M for which the associated linear differential operator A −1 −1 satisﬁes [A ]distr(u ◦ ) = v ◦ on ( ∩ V). We then have the coordinate transformation F ≡ ◦ −1 : (U ∩ V)→ (U ∩ V), and for each function ϕ ∈ D(( ∩ U ∩ V)), the change of variables formula (Theorem 7.2.7) gives, for each f ∈ C∞(( ∩ U ∩ V)), · ∗ = · f A(ϕ) dλ A(f ) ϕdλ ( ∩U∩V) ( ∩U∩V) = (A(f ◦ ) ◦ −1) · ϕdλ ( ∩U∩V) −1 = (A(f ◦ ) ◦ ) · ϕ ◦ F |JF | dλ ( ∩U∩V)

~~= (A (f ◦ F))· ϕ ◦ F |JF | dλ ( ∩U∩V) = ◦ · ∗ [|J | ◦ ] (f F) A F (ϕ F) dλ ( ∩U∩V) = · |J |· ∗ [|J | ◦ ]◦ −1 f F −1 (A F (ϕ F) F )dλ. ( ∩U∩V)~~

∗ =|J |· ∗ [|J | ◦ ]◦ −1 Therefore, by Lemma 7.3.2, A(ϕ) F −1 (A F (ϕ F) F ), and hence ◦ −1 · ∗ = ◦ −1 · ∗ [|J | ◦ ] (u ) A(ϕ) dλ (u ) A F (ϕ F) dλ ( ∩U∩V) ( ∩U∩V) −1 = (v ◦ ) · ϕ ◦ F |JF | dλ ( ∩U∩V) = (v ◦ −1) · ϕdλ. ( ∩U∩V)

−1 −1 Thus [A]distr(u ◦ ) = v ◦ on ( ∩ U ∩ V). If each point in U admits such a local C∞ coordinate neighborhood (V, ,V), ∈ D ∩ ∈ then given a function ϕ (( U)), there exist functions η1,...,ηm −1 −1 D(( ∩ U)) such that ηj ≡ 1onsuppϕ and [A]distr(u ◦ ) = v ◦ 9.9 C∞ Embeddings 465 on a neighborhood Vj of supp ηj in ( ∩ U) for each j = 1,...,m. Thus

~~ m ◦ −1 · ∗ = ◦ −1 · ∗ (u ) A(ϕ) dλ (u ) A(ηj ϕ)dλ ∩ ( U) j=1 Vj m −1 = (v ◦ ) · ηj ϕdλ j=1 Vj = (v ◦ −1) · ϕdλ; ( ∩U)~~

~~−1 −1 and hence [A]distr(u ◦ ) = v ◦ on ( ∩ U).~~

~~9.9 C∞ Embeddings~~

In this section, we consider some criteria that guarantee that a C∞ mapping is a (local) diffeomorphism, as well as criteria that guarantee that the zero set of a C∞ function, or the image of a C∞ mapping, is a smooth submanifold. We will apply these criteria in the study of the topological and holomorphic structure of Riemann surfaces in Sects. 5.10Ð5.17, and in the study of holomorphic structures on surfaces in Chap. 6. For our purposes, it will sufﬁce to consider curves and surfaces. We ﬁrst recall the following fact from analysis in Rn:

Theorem 9.9.1 (C∞ inverse function theorem) Suppose ⊂ Rn is an open set, n ∞ F : → R is a C mapping with Jacobian determinant JF : → R, and a ∈ is a point at which JF (a) = 0. Then F maps some neighborhood U of a in diffeomorphically onto some open subset F(U)of Rn.

Note that conversely, the Jacobian determinant of a diffeomorphism F : U → V of open subsets U and V of Rn is nonvanishing, because for each point a ∈ U, n n n JF (a) is the determinant of the linear operator (dF )a : R = R → R ,anisomor- −1 phism with inverse linear map (dF )F(a). The proof of the theorem for n = 1is straightforward. A proof for arbitrary n is outlined in Exercise 9.9.1. The proof can be modiﬁed to give a Ck version, but only the C∞ case is applied in this book.

Theorem 9.9.2 Let M be a smooth manifold of dimension n = 1 or 2. (a) If u is a real-valued C∞ function on an open set U ⊂ M and n = 1, then the restriction of u to some neighborhood of a point p ∈ U is a local C∞ coordinate ∞ if and only if (du)p = 0. If u and v are two real-valued C functions on an open set U ⊂ M and n = 2, then the restriction of (u, v) to some neighborhood of a ∞ point p ∈ U gives local C coordinates if and only if (du ∧ dv)p = 0(i.e., ∗ (du)p and (dv)p form a basis for Tp M). 466 9 Manifolds

∞ (b) Suppose n = 2 and v ∈ C (M, R). If (dv)p = 0 at some point p ∈ M, then there exists a real-valued C∞ function u on a neighborhood U of p such that (U, (u, v)) is a local C∞ coordinate neighborhood in M. If dv = 0 at each point in the zero set Z ≡{p ∈ M | v(p) = 0}, then Z is a smooth submanifold of dimension 1 in M. (c) Suppose r ∈{1, 2} and : N → M is a C∞ mapping of an r-dimensional C∞ manifold N into M. If q ∈ N and the tangent map (∗)q : Tq N → T(p)M is injective, then maps a neighborhood V of q in N diffeomorphically onto an r-dimensional submanifold (V ) of some neighborhood U of (q) in M. If is injective and proper and (∗)q : Tq N → T(q)M is injective for every point q ∈ N, then maps N diffeomorphically onto an r-dimensional smooth submanifold (N) of M. Conversely, if maps N diffeomorphically onto a smooth submanifold of M, then ∗ : TN → TM is injective. If p ∈ M and −1 (∗)q : Tq N → TpM is surjective for every point q ∈ L ≡ (p) (i.e., p is a regular value), then L is a C∞ submanifold of dimension r − n in N.

Proof Let p ∈ M.Forn = dim M = 1, we may fix a local C∞ coordinate neighborhood (U, = x). For any real-valued C∞ function u on a neighborhood of p J ≡ J ◦ = in U, the Jacobian determinant u ∂u/∂x (more precisely, u◦−1 ∂u/∂x) ∞ satisfies du = Ju · dx. Similarly, for n = 2 and for (u, v) a pair of real-valued C functions on a neighborhood of p, the Jacobian determinant ∂v ∂v ∂u J ≡ ∂u − (u,v) ∂x ∂y ∂x ∂y in a local C∞ coordinate neighborhood (U,(x,y)) of p satisfies du ∧ dv = J(u,v)dx ∧ dy. Part (a) now follows from the inverse function theorem. ∞ If n = 2, p ∈ M, and v is a real-valued C function on M with (dv)p = 0, then fixing a local C∞ coordinate neighborhood (U,(x,y)) of p, one of the pairs ∗ = (dx)p,(dv)p and (dy)p,(dv)p must be a basis for Tp M. Setting u x in the former case and u = y in the latter case, we see that by (a), (u, v) gives local C∞ coordinates in a neighborhood of p. In particular, if dv = 0 at every point in the zero set Z of v, then Z is a C∞ submanifold of dimension 1. Thus (b) is proved. Suppose r ∈{1, 2} and : N → M is a C∞ mapping of an r-dimensional C∞ manifold N into M.Ifr = n, then in local C∞ charts, the Jacobian determinant of is precisely the determinant of the tangent map ∗ (with respect to the bases provided by the partial derivative operators). So the inverse function theorem implies that if (∗)q is an isomorphism for some point q ∈ N, then maps a neighborhood of q diffeomorphically onto an open subset of M. Suppose r = 1 < 2 = n and q is a point at which (∗)q is injective. Fixing a local C∞ chart (U, = (x, y), V ⊂ R2) in a neighborhood of p ≡ (q) with (p) = (0, 0) ∈ V , we see that by exchanging x and y if necessary, we may assume that the function ≡ x() satisfies (d)q = 0. Therefore, by part (a), we may assume that determines a local C∞ chart (W,,I ⊂ R) in N such that q ∈ W ⊂ −1(U) and I is an open interval containing 0 = (q). In fact, by fixing an open interval J with (0, 0) ∈ I × J ⊂ V , replacing I with an open subinterval I containing 0 that 9.9 C∞ Embeddings 467 is so small that W ≡ −1(I ) ⊂ −1(U ) for U = −1(I × J), and replacing W with W , U with U , and V with I × J , we may assume that V = I × J .TheC∞ function s → v(s) ≡ y(s) − y((−1(x(s)))) on U satisfies ∂ ∂ dv = dy − y((−1(x))) · dx − y((−1(x))) · dy ∂x ∂y ∂ = dy − y((−1(x))) · dx, ∂x ∞ and hence dx ∧ dv = dx ∧ dy = 0. Thus we get a local C chart (U0, 0 = (x, v), I0 × J0) in M, where p ∈ U0 ⊂ U, and I0 and J0 are open intervals with 0 ∈ I0 ⊂ I and 0 ∈ J0 ⊂ J . By shrinking I0 if necessary, we may assume that the −1 neighborhood W0 ≡ (I0) of q in W satisfies (W0) ⊂ U0.Nowift ∈ W0, then −1 (t) ∈ U0 and v((t)) = y((t)) − y(( ((t)))) = 0. Conversely, if s ∈ U0 −1 with v(s) = 0, then setting t ≡ (x(s)) ∈ W0, we get

x((t)) = (t) = x(s) and v((t)) = 0 = v(s), and hence (t) = s. Therefore, (W0) is the 1-dimensional submanifold N0 ≡ −1 ×{ } = −1 ◦ → 0 (I0 0 ) of U0. Furthermore, the map W0 (x N0 ) W0 N0 is a composition of diffeomorphisms, and is therefore a diffeomorphism. The veriﬁca- tions of the remaining claims in (c) are left to the reader.

~~Deﬁnition 9.9.3 A C∞ mapping : N → M of an r-dimensional C∞ manifold N into an n-dimensional C∞ manifold M with r, n ∈{1, 2} is~~

(i) An immersion if the tangent map (∗)p : TpN → TpM is injective for each point p ∈ N; (ii) An embedding (or imbedding)if is a proper injective immersion; (iii) A submersion if the tangent map (∗)p : TpN → TpM is surjective for each point p ∈ N.

Remarks 1. In this book, we require embeddings to be proper, but it is also common practice not to require this as part of the definition. 2. If N is a smooth submanifold of a smooth manifold M, and ι: N → M is the inclusion map, then we identify TNwith ι∗TN⊂ TM.IfM and N are oriented and α is a differential form of degree dim N on M, then, in a slight abuse of notation, we ∗ often write N ι α simply as N α. Note that this does not conflict with the earlier definition of the integral over a measurable subset of M because there, we assumed α to be of degree dim M. 3. Theorem 9.9.2 implies that a C∞ mapping : N → M of C∞ manifolds M and N of dimension n = 1 or 2 is a local diffeomorphism if and only if the tangent map is an isomorphism at each point. 4. It follows from Theorem 9.9.2 that an open subset of a C∞ surface M is smooth if and only if for each point p ∈ ∂ , there exists a real-valued C∞ function ϕ on a neighborhood U of p such that (dϕ)p = 0 and ∩ U ={x ∈ U | ϕ(x) < 0}. 468 9 Manifolds

Exercises for Sect. 9.9 9.9.1 The goal of this exercise is a proof of the C∞ inverse function theorem (The- n n orem 9.9.1). Let be an open subset of R ,letF = (f1,...,fn): → R be a C∞ mapping with Jacobian matrix ⎛ ⎞ ∂f1 ··· ∂f1 ⎜ ∂x1 ∂xn ⎟ ≡ ⎜ . . . ⎟ : → Rn×n ∼ Rn2 JF ⎝ . .. . ⎠ = ∂fn ··· ∂fn ∂x1 ∂xn

and Jacobian determinant JF = det(JF ): → R (clearly, both are of class ∞ C ), and let a = (a1,...,an) ∈ be a point at which JF (a) = 0. For each ∈ = 1 n : → Rn C∞ point b ,letRb (rb ,...,rb ) be the map given by

~~x → F(x)− F(b)− (dF )b(x − b).~~

(a) Show that there is a constant C0 > 0 such that x ≤C0 (dF )a(x) for every vector x ∈ Rn. (b) Prove that for every >0, there is a neighborhood U of a in such that Rb(y) − Rb(x) ≤ y − x for all points b,x,y ∈ U. (c) Prove that there are a constant C1 > 0 and a neighborhood U of a in such that F(y)− F(x) ≥C1 y − x for all points x,y ∈ U. In particular, F U is injective. (d) Prove that if U is a neighborhood of a in on which JF is nonvanishing, then for every point b ∈ Rn,theC∞ function F − b 2 cannot attain a positive local minimum value at a point in U. J (e) Prove that if U is a neighborhood of a in such that F U is nonvanishing and F U is injective, then F maps U homeomorphically onto an open set F(U)in Rn. Hint. Assuming V U is open, but W ≡ F(V) is not open, there must exist a point p ∈ V with q = F(p)∈ ∂W. Fix an open ball B centered at p with B V . Then, by injectivity, q is not in the compact set K ≡ F(∂B). Now ﬁx a point b ∈ Rn \ W with b − q < dist(q, K)/2 and apply part (d). (f) Prove that there exists a relatively compact neighborhood U of a in J such that F U is nonvanishing, F maps U homeomorphically onto an Rn = ≡ −1 : → Rn open set F(U)in , and, for G (g1,...,gn) (F U ) F(U) , we have, for some constant C2 > 0, G(y) − G(x) ≤C2 y − x for all x,y ∈ F(U). (g) Prove that for U as in part (f), we have

~~R (G(x)) lim b = 0 ∀b ∈ U. x→F(b) x − F(b) 9.10 Classiﬁcation of Second Countable 1-Dimensional Manifolds 469~~

(h) Prove that for U as in part (f) and for each point b ∈ U, the ﬁrst-order partial derivatives of G at F(b)exist and ⎛ ⎞ ∂g1 (F (b)) ··· ∂g1 (F (b)) ⎜ ∂x1 ∂xn ⎟ ≡ ⎜ . . . ⎟ = −1 JG(F (b)) ⎝ . .. . ⎠ (JF (b)) . ∂gn (F (b)) ··· ∂gn (F (b)) ∂x1 ∂xn

Hint.Wehavex = F(b) + (dF )b(G(x) − b) + Rb(G(x)) for each point x ∈ F(U). Solve for G(x) (in terms of x and Rb(G(x))), and apply part (g). (i) Prove by induction on k that G is of class Ck for each k = 0, 1, 2,.... 9.9.2 Let be an open subset of a second countable smooth manifold M. Prove that is a smooth open set if and only if there exists a C∞ function ϕ : M → R such that ={x ∈ M | ϕ(x) < 0} and (dϕ)p = 0 for each point p ∈ ∂ .

~~9.10 Classiﬁcation of Second Countable 1-Dimensional Manifolds~~

~~The second countable curves are completely determined by the following:~~

Theorem 9.10.1 Every second countable connected 1-dimensional manifold is homeomorphic to either the circle S1 or the line R. Any second countable connected 1-dimensional smooth manifold is diffeomorphic to either S1 or R.

~~Remarks 1. It is easy to see that every open interval in R is diffeomorphic to R. For given a,b ∈ R with a~~

Throughout this section, M denotes a second countable connected 1-dimensional topological or smooth manifold. The intermediate value theorem implies that a continuous injective real-valued function on an interval I is either strictly increasing or strictly decreasing, and that the function maps the interval homeomorphically onto an interval J (in particular, the image of any endpoint of I is an endpoint of J ). Fur- thermore, if the interval is open and the function is of class C∞ with nonvanishing derivative, then the function is a diffeomorphism.

~~Lemma 9.10.2 Let α : I → M be an injective continuous mapping of an open interval I ⊂ R into M with image A = α(I). Then 470 9 Manifolds~~

(a) The image A is a connected open subset of M onto which α maps I homeomorphically. (b) For M a smooth manifold and α of class C∞ with nonvanishing tangent vector α˙ , α maps I diffeomorphically onto A.

Proof Given a point c ∈ I , there is a local chart (J, , R) in M with α(c) = −1(0). The connected component H of α−1(J ) containing c is then an open subinterval ◦ of I and α H maps H homeomorphically onto an open interval. The claims (a) and (b) now follow.

Lemma 9.10.3 Let α :[a,b]→M and β :[c,d]→M be injective paths, and let A ≡ α([a,b]), A0 ≡ α((a, b)), B ≡ β([c,d]), and B0 ≡ β((c, d)). Assume that M −1 is noncompact, A0 ∩ B0 = ∅, and A ⊂ B0. Then β (A) =[c,r] or [r, d] for some point r ∈[c,d]. ¯ Proof Clearly, A = A0 and B = B0, and by Lemma 9.10.2, A0 and B0 are connected open subsets of M, α maps (a, b) homeomorphically onto A0, and β maps (c, d) homeomorphically onto B0. Of course, α maps [a,b] homeomorphically onto A and β maps [c,d] homeomorphically onto B, since these are compact Hausdorff spaces. Each connected component D of β−1(A) is either a singleton or a closed interval, and hence α−1(β(D)) is either a singleton or a closed interval contained in [a,b].If D ⊂ (c, d), then β(D) is a compact subset of A ∩ B0. Since A ⊂ B0, it follows that α−1(β(D)) is a connected compact subset of [a,b] that is not equal to the entire interval. Hence there is an open interval J such that

~~−1 −1 −1 J ⊂ (a, b) ∩ α (B0) \ α (β(D)) and ∂J ∩ α (β(D)) = ∅.~~

Thus β−1(α(J )) ∪ D D is a connected subset of β−1(A), and we have arrived at a contradiction. By hypothesis, β−1(A) ⊂[c,d] has a connected component with nonempty interior, and by the above, such a connected component must be a closed interval with at least one endpoint equal to c or d. Suppose that we have such a connected component of the form [r, d] with c

~~B \ B0 ={β(c),β(d)}⊂A \{β(u),β(r)}=A \{α(a),α(b)}=A0,~~

−1 and hence A ∪ B = A0 ∪ B0 is open. Suppose instead that u = c. Since α ◦ β is a strictly increasing continuous mapping of [r, d] onto a closed subinterval of [a,b) 9.10 Classiﬁcation of Second Countable 1-Dimensional Manifolds 471 and we have α−1(β(c)) = b, we get a continuous injective mapping

γ : (α−1(β(d)), r − c + b) → A ∪ B by setting − α(t) if t ∈ (α 1(β(d)), b], γ(t)= β(t − b + c) if t ∈[b,r − c + b). Hence, by Lemma 9.10.2, the image of γ is a neighborhood of α(b) = β(c) in M, and hence this point is an interior point of A ∪ B.Wealsohaveα(a) = β(r) ∈ B0 −1 and β(d) = α(α (β(d)) ∈ A0. Thus A ∪ B is again open in this case. Therefore, in either case, since M is connected and noncompact, we have arrived at a contradiction. A similar argument shows that any connected component of β−1(A) of the form [c,r] with r>cmust be equal to β−1(A).

~~Lemma 9.10.4 Let −∞ 0 and g > 0 on [a,b], then h may also be chosen to be of class C∞ with h > 0.~~

Proof We may fix points c,d ∈ (a, b) with c 0 and g > 0, then fixing points u and v with ∞ c 0, we see that the function h given by t −1 h(t) = f(a)+ [λ1(s)f (s) + r (g(d) − q)λ2(s) + λ3(s)g (s)] ds ∀t ∈[a,b] a satisfies h > 0on(a, b), h = f on [a,c], and h = g on [d,b]. 472 9 Manifolds

Lemma 9.10.5 Suppose 0 and U are open subsets of M; I0 is an open interval; α0 : I0 → 0 and β : R → U are homeomorphisms; K0 is a compact subset of 0; and K is a compact subset of U that meets K0. Assume that M is noncompact. Then there exists a homeomorphism α of an open interval I onto an open subset ∪ ⊂ ⊂ ∪ −1 = −1 of M such that K0 K 0 U and α K0 α0 K0 . Moreover, if M is smooth and α0 and β are diffeomorphisms, then α may be chosen to be a diffeomorphism.

Proof If K ⊂ 0, then we may set α ≡ α0 : I ≡ I0 → ≡ 0; so we may assume without loss of generality that K ⊂ 0. We may also assume that K is connected, and we may ﬁx an interval I1 =[a,b] I0 with K0 ⊂ α0((a, b)) and an interval J =[c,d] R with K ⊂ β((c, d)). Let us ﬁrst suppose that 0 ⊂ K. Then we may choose the intervals I1 and J so that α0(I1) ⊂ β(J); and by applying Lemma 9.10.3 (and reparametrizing β if nec- −1 essary), we may assume that for some r ∈ (c, d),wehaveβ (α0(I1)) ∩ J =[c,r] −1 −1 = = −1 −1 with either a<α0 (β(c)) < α0 (β(r)) b or a α0 (β(r)) < α0 (β(c)) < b. ∈ −1 In the former case, we may choose a number s (α0 (β(c)), b) so close to b that K0 ⊂ α0((a, s)). By reparametrizing β again, we may assume that = −1 = −1 = −1 s α0 (β(s)) and r α0 (β(r)) b (in particular, a<α0 (β(c)) < s < b < d and c 0. The set

~~≡ α0((a, b]) ∪ β([b,d)) = α0((a, b)) ∪ β((c, d)) is then a domain in M, and~~

~~K0 ∪ K α0((a, b)) ∪ β((c, d)) = ⊂ 0 ∪ U.~~

Moreover, we may deﬁne a homeomorphism α : I ≡ (a, d) → (which is a diffeomorphism if M is smooth) by setting ⎧ ⎨⎪α0(t) if t ∈ (a, s], α(t) = α (f (t)) if t ∈[s,b], ⎩⎪ 0 β(t) if t ∈[b,d).

−1 = −1 ⊃ In particular, we have α α0 on the set α0((a, s)) K0. In the case in which = −1 −1 a α0 (β(r)) < α0 (β(c)) < b, we may apply the above arguments to β and the homeomorphism αˆ 0 : t → α0(b − t + a) to get a domain and a homeomorphism αˆ : (a, d) → . The homeomorphism α : I ≡ (b − d + a,b) → given by t → α(bˆ − t + a) then has the required properties. Finally, suppose 0 ⊂ K ⊂ U. Then we may choose the homeomorphism (or diffeomorphism) β : R → U and the constants c and d so that c

~~β([a,b]) = α0([a,b]), β(a) = α0(a), and β(b) = α0(b). After choosing numbers s0 and s1 with a~~

~~Therefore, Lemma 9.10.4 provides constants u0, v0, u1, and v1 with~~

~~a 0 and ≡ f1 > 0. We then have the domain β((c, d)) that satisﬁes~~

K0 ∪ K ⊂ α0((s1,s2)) ∪ β((c, d)) = β((c, d)) = , and we have the homeomorphism (diffeomorphism for M smooth) α : I ≡ (c, d) → given by ⎧ ⎪β(t) if t ∈ (c, a]∪[b,d), ⎨⎪ α (f (t)) if t ∈[a,s ], α(t) = 0 0 0 ⎪α (t) if t ∈[s ,s ], ⎩⎪ 0 0 1 α0(f1(t)) if t ∈[s1,b].

Proof of Theorem 9.10.1 We ﬁrst consider the case in which the manifold M is { }∞ noncompact. By Lemma 9.3.6, there exist locally ﬁnite open coverings Uν ν=1 and { }∞ = Vν ν=1 of X such that for each ν 1, 2, 3,..., Vν Uν M, Uν is homeomorphic to R, and for M smooth, Uν is also diffeomorphic to R. We may also choose the coverings so that for each ν,thesetV1 ∪···∪Vν meets Vν+1. For we may let ν1 be the smallest index ν for which Vν = ∅, and given ν1,...,νk,wemayletνk+1 ∈{ } ∪···∪ be the smallest index ν/ ν1,...,νk for which Vν meets Vν1 Vνk (such an index exists because the union of the boundaries of the relatively compact open sets

Vν1 ,...,Vνk cannot be contained in the union of these sets). By induction, we obtain { } a sequence Vνk that covers M (otherwise, the union would have a boundary point that lies in Vν for some ν and we would have ν<νk for k 0). Thus we may { } { } { } { } replace the original coverings Vν and Uν with the coverings Vνk and Uνk , respectively. There exist domains { ν} in M and mappings {αν} such that for each ν = 1, 2, 3,...,

ν ≡ V1 ∪···∪Vν ν ⊂ ν ≡ U1 ∪···∪Uν, −1 = −1 and αν is a homeomorphism of an open interval Iν onto ν with αν ν αν+1 ν . Moreover, for M smooth, each mapping αν is a diffeomorphism. For the existence of 1 and α1 : I1 → 1 is clear, and one may then proceed to apply Lemma 9.10.5 474 9 Manifolds

{ −1 }∞ inductively. Since the sequence of functions αμ Vν μ=ν is constant for each ν, we may then define a homeomorphism (a diffeomorphism for M smooth) γ of = = −1 = M Vν onto an open interval by setting γ Vν αν Vν for each ν 1, 2, 3,.... Fixing a diffeomorphism λ: R → γ(M), we get the desired homeomorphism (diffeomorphism for M smooth) γ −1 ◦ λ: R → M. Finally, suppose M is compact. We may fix a homeomorphism β : R → U onto a neighborhood U of a point p ∈ M, with β a diffeomorphism for M smooth. The set U \{p} has exactly two connected components U0 and U1, and each of the connected components of M \{p} must meet, and therefore contain, at least one of these connected components. It follows that M \{p} has at most two connected components. However, if N ≈ R is a connected component that contains only one of these connected components, say, U0, then, fixing a point q ∈ U0, we see that of the two connected components of N \{q}, one is relatively compact in N, which is clearly impossible. Thus M \{p} is connected, and we may choose a homeomorphism α : (0, 1) → M \{p}, with α a diffeomorphism for M smooth. Since U0 and U1 are disjoint open connected subsets of M \{p} that are not relatively compact in M \{p}, we may choose U and α so that for some r ∈ (0, 1/4), −1 −1 we have α (U0) = (0, 2r) and α (U1) = (1 − 2r, 1). We may also choose the homeomorphism (or diffeomorphism) β so that β(−r) = α(1 − r) and β(r) = α(r). In particular, β([−r, r]) is a connected relatively compact subset of U that meets −1 both U0 and U1, and therefore p ∈ β((−r, r)), and we may set t0 ≡ β (p). −∞ = ∞ = −1 → → − We have β(( ,t0)) U1, β((t0, )) U0, α (β(t)) 1ast t0 , and −1 → → + −1 ◦ −1 ◦ α (β(t)) 0ast t0 ;soα β (−∞,t0) and α β (t0,∞) must be strictly −1 −1 increasing functions with α ◦β([−r, t0)) =[1−r, 1) and α ◦β((t0,r]) = (0,r]. Fixing b0 ∈ (−r, t0) ∩ (−r, 0) and b1 ∈ (t0,r)∩ (0,r) and applying Lemma 9.10.4, −1 we get continuous strictly increasing functions f0 :[−r, b0]→[1 − r, α (β(b0))] −1 and f1 :[b1,r]→[1 + α (β(b1)), 1 + r] such that f0(t) = 1 + t for t near −r, −1 −1 f0(t) = α (β(t)) for t near b0, f1(t) = 1 + α (β(t)) for t near b1, and f1(t) = + C∞ 1 t for t near r. Moreover, f0 and f1 are of class with f0 > 0 and f1 > 0for M smooth. The mapping γ : (−r, 1 + b0) → M given by ⎧ ⎪ ⎪α(t) if t ∈[r, 1 − r], ⎪ ⎨⎪α(f0(t)) if t ∈ (−r, b0], = ∈[ ] γ(t) ⎪β(t) if t b0,b1 , ⎪ ⎪α(−1 + f1(t)) if t ∈[b1,r], ⎩⎪ α(f0(t − 1)) if t ∈[1 − r, 1 + b0), is a surjective continuous mapping that is a local homeomorphism and that satisfies γ(t+1) = γ(t)for each t ∈ (−r, b0). Moreover, γ is a local diffeomorphism if M is smooth. It follows that the mapping S1 → M given by e2πit → γ(t)for t ∈ (−r, 1+ b0) is a well-defined homeomorphism, and this mapping is a diffeomorphism if M is smooth. 9.11 Riemannian Metrics 475

~~9.11 Riemannian Metrics~~

In this section we consider Riemannian metrics (on smooth surfaces), which in this book are applied directly (only) in the construction of an almost complex structure on an orientable surface (Proposition 6.1.3) in Chap. 6.

Deﬁnition 9.11.1 A Riemannian metric g on a 2-dimensional smooth manifold M is a choice of a real inner product gp(·, ·) on the real tangent space TpM at each point p ∈ M such that for each pair of local C∞ vector ﬁelds u and v, the function ∞ g(u,v): p → gp(up,vp) is of class C . For each point p ∈ M and each pair of real ∈ = | |2 = tangent vectors u, v TpM, we also write g(u,v) gp(u, v) and u g g(u, u).

Remarks Riemannian metrics are fundamental objects in the study of the (differential) geometry of manifolds. A Riemannian metric provides a notion of distance, curvature, and volume in a manifold. In this book, we mostly consider related objects, such as Kähler metrics on Riemann surfaces and Hermitian metrics in holomorphic line bundles. Corresponding notions of volume and curvature do play important roles in the analytic techniques applied throughout this book, but the geometric meaning will not be the main consideration (for much more on Riemannian metrics, see, for example, [KobN1] and [KobN2]).

~~Observe that if g is a Riemannian metric on a smooth surface M and (U,(x,y)) is a local C∞ coordinate neighborhood in M, then the functions~~

~~A ≡ g(∂/∂x,∂/∂x), B ≡ g(∂/∂x,∂/∂y) = g(∂/∂y,∂/∂x), C ≡ g(∂/∂y,∂/∂y), are of class C∞ and~~

g(u,v) = Adx(u)dx(v)+ Bdx(u)dy(v)+ Bdy(u)dx(v)+ Cdy(u)dy(v) for every pair of tangent vectors u, v at a point in U. In particular, it follows that if k u and v are vector ﬁelds that are continuous (C with k ∈ Z≥0 ∪{∞}, measurable), then the function g(u,v) is continuous (respectively, Ck, measurable).

~~Proposition 9.11.2 Every second countable smooth surface M admits a Rieman- nian metric.~~

~~Proof There exist a locally ﬁnite collection of local C∞ charts~~

~~{ = } (Uν,ν (xν,yν), Uν ) ν∈N 476 9 Manifolds~~

∞ and a C partition of unity {λν} with supp λν ⊂ Uν (and λν ≥ 0) for each ν ∈ N. The pairing given by g(u,v) = gp(u, v) ≡ λν(p)(dxν(u) dxν (v) + dyν(u) dyν(v)), ν∈N for all p ∈ M and u, v ∈ TpM, is then a Riemannian metric on M. The veriﬁcation is left to the reader (see Exercise 9.11.1).

Exercises for Sect. 9.11 9.11.1 Verify that the pairing deﬁned in the proof of Proposition 9.11.2 is a Rie- mannian metric. Chapter 10 Background Material on Fundamental Groups, Covering Spaces, and (Co)homology

We recall that a path (or parametrized path or curve or parametrized curve)in a topological space X from a point x to a point y is a continuous mapping γ :[a,b]→X with γ(a)= x and γ(b)= y (see Sect. 9.1). We take the domain of a path to be [0, 1], unless otherwise indicated. A loop (or closed curve) with base point p ∈ X is a path in X from p to p. In this chapter, we consider the equivalence relation given by path homotopies. This leads to the fundamental group, which is the group given by the path homotopy equivalence classes of loops at a point, and to covering spaces, both of which are important objects in complex analysis and Riemann surface theory. We also consider homology groups, which are essentially Abelian versions of the fundamental group, and cohomology groups, which are groups that are dual to the homology groups.

~~10.1 The Fundamental Group~~

~~Throughout this section, X denotes a nonempty topological space.~~

~~Deﬁnition 10.1.1 A path homotopy in X is a continuous mapping~~

~~H :[a,b]×[0, 1]→X, where −∞~~

~~Remarks 1. We take the domain of a path homotopy to be [0, 1]×[0, 1], unless otherwise indicated.~~

T. Napier, M. Ramachandran, An Introduction to Riemann Surfaces, Cornerstones, 477 DOI 10.1007/978-0-8176-4693-6_10, © Springer Science+Business Media, LLC 2011 478 10 Fundamental Groups, Covering Spaces, and (Co)homology

~~Fig. 10.1 A path homotopy from γ0 to γ1~~

~~2. Change of parameter.Ifγ :[a,b]→X is a path and ϕ :[a,b]→[a,b] is a continuous mapping with ϕ(a) = a and ϕ(b) = b, then~~

~~H : γ ∼ γ ◦ ϕ, where H(t,s)= γ(t+ s(ϕ(t) − t)) ∀(t, s) ∈[a,b]×[0, 1].~~

~~If δ :[a,b]→X is a path, H : δ ∼ γ , and ψ :[c,d]→[a,b] is a continuous mapping with ψ(c)= a and ψ(d)= b, then~~

~~G : δ ◦ ψ ∼ γ ◦ ψ, where G(t, s) = H(ψ(t),s)∀(t, s) ∈[c,d]×[0, 1].~~

For this reason, we lose nothing by restricting most of our attention to paths with domain [0, 1], since we may always reparametrize so that the domain becomes [0, 1]. 3. The path homotopy relation ∼ is an equivalence relation. To see this, suppose γj :[a,b]→X is a path in X with γj (a) = x and γj (b) = y for j = 0, 1, 2, and Hj : γj−1 ∼ γj for j = 1, 2. Then the mapping

~~(t, s) → γ0(t) ∀(t, s) ∈[a,b]×[0, 1] is a path homotopy from γ0 to itself;~~

(t, s) → H1(t, 1 − s) ∀(t, s) ∈[a,b]×[0, 1] is a path homotopy from γ1 to γ0; and H (t, 2s) ∀(t, s) ∈[a,b]×[0, 1/2], (t, s) → 1 H2(t, 2s − 1) ∀(t, s) ∈[a,b]×[1/2, 1], is a path homotopy from γ0 to γ2.

~~Deﬁnition 10.1.2 Let γ0 :[0, 1]→X and γ1 :[0, 1]→X be paths in X with γ0(1) = γ1(0).~~

~~(a) The product path of γ0 and γ1 is the path γ0 ∗ γ1 :[0, 1]→X from γ0(0) to γ1(1) given by γ0(2t) if t ∈[0, 1/2], γ0 ∗ γ1(t) ≡ γ1(2t − 1) if t ∈[1/2, 1].~~

~~− :[ ]→ (b) The reverse path of γ0 is the path γ0 0, 1 X from γ0(1) to γ0(0) given by − ≡ − ∀ ∈[ ] γ0 (t) γ0(1 t) t 0, 1 . 10.1 The Fundamental Group 479~~

Remarks 1. The product path is obtained by tracing the ﬁrst path at twice the speed, and then tracing the second path at twice the speed. The paths are traced at twice the speed so that the resulting domain is again [0, 1]. 2. The reverse path is obtained by retracing the path in the reverse direction. Clearly, for any path γ (with domain [0, 1]), we have (γ −)− = γ . 3. Occasionally, it is convenient to consider product and reverse paths with domains other than [0, 1].Ifγj :[aj ,bj ]→X is a path in X for j = 0, 1 and γ0(b0) = γ1(a1), then by γ0 ∗ γ1 :[a,b]→X we will mean a path obtained by tracing γ0 and then γ1, after some (nonunique, noncanonical) linear reparametrizations. That is, after choosing an interval [a,b] and a point c ∈ (a, b),wehave − γ (a + t a (b − a )) if t ∈[a,c], γ ∗ γ (t) ≡ 0 0 c−a 0 0 0 1 + t−c − ∈[ ] γ1(a1 b−c (b1 a1))) if t c,b . − :[ ]→ Similarly, by γ0 a,b X we will mean the path given by − t − a γ (t) ≡ γ b + (a − b ) ∀t ∈[a,b]. 0 0 0 b − a 0 0

4. For paths γ0 and γ1 in X and a continuous mapping : X → Y into a topo- ∗ = ∗ − = − logical space Y ,wehave(γ0 γ1) (γ0) (γ1) and (γ0 ) ((γ0)) . 5. The product operation on paths is not associative. Given n paths γ1,...,γn, we will set

γ1 ∗···∗γn ≡ (···((γ1 ∗ γ2) ∗ γ3) ···) ∗ γn. On the other hand, the product is associative up to path homotopy. In fact, we have the following lemma, the proof of which is left to the reader (see Exercise 10.1.1).

Lemma 10.1.3 Let γj :[0, 1]→X be a path with γj (0) = xj and γj (1) = yj for j = 0, 1, 2. For each point p ∈ X, let ep :[0, 1]→X denote the constant loop t → p. ∗ − ∼ − ∗ ∼ ∗ ∼ ∗ ∼ (a) We have γ0 γ0 ex0 , γ0 γ0 ey0 , ex0 γ0 γ0, and γ0 ey0 γ0. Fur- ∼ − ∼ − thermore, if γ0 γ1, then γ0 γ1 . (b) If γ0(1) = γ1(0) and γ1(1) = γ2(0), then (γ0 ∗ γ1) ∗ γ2 ∼ γ0 ∗ (γ1 ∗ γ2). (c) If γ0(1) = γ1(1) = γ2(0) and γ0 ∼ γ1, then γ0 ∗ γ2 ∼ γ1 ∗ γ2. (d) If γ0(1) = γ1(0) = γ2(0) and γ1 ∼ γ2, then γ0 ∗ γ1 ∼ γ0 ∗ γ2. (e) If : X → Y is a continuous mapping of topological spaces and H : γ0 ∼ γ1, then (H): (γ0) ∼ (γ1).

~~Lemma 10.1.3 allows us to associate a group to each point in X as follows:~~

~~Deﬁnition 10.1.4 Let x0 ∈ X.~~

(a) We denote the set of loops in X based at x0 with domain [0, 1] by L(X, x0).For each loop γ ∈ L(X, x0), we denote the path homotopy equivalence class of γ [ ] [ ] [ ] by γ or γ x0 or γ π1(X,x0). 480 10 Fundamental Groups, Covering Spaces, and (Co)homology

(b) The fundamental group of X with base point x0 is the group π1(X, x0) with elements given by the set of equivalence classes of loops in L(X, x0), with the [ ]·[ ]≡[ ∗ ] [ ]−1 ≡[ −] product and inverse given by γ0 γ1 γ0 γ1 and γ0 γ0 , respec- ∈ L [ ] tively, for all γ0,γ1 (X, x0), and the identity element given by ex0 , where ∈ L ex0 (X, x0) is the constant loop based at x0. (c) If : X → Y is a continuous mapping of topological spaces and y0 = (x0), : → [ ] → [ ] then the map ∗ π1(X, x0) π1(Y, y0) given by γ x0 (γ ) y0 is the induced pushforward homomorphism of fundamental groups.

Remarks If : X → Y and : Y → Z are continuous mappings of topological spaces, x0 ∈ X, y0 = (x0), and z0 = (y0), then the induced homomorphisms ∗ : π1(X, x0) → π1(Y, y0) and ∗ : π1(Y, y0) → π1(Z, z0) satisfy

~~∗ ◦ ∗ = ( ◦ )∗ : π1(X, x0) → π1(Z, z0).~~

~~In particular, if is a homeomorphism, then ∗ : π1(X, x0) → π1(Y, y0) is an iso- − morphism with inverse mapping ( 1)∗.~~

Lemma 10.1.5 Let η :[0, 1]→X be a path from x0 to x1 in X. Then the map → [ ] → [ − ∗ ∗ ] π1(X, x0) π1(X, x1) given by γ x0 η γ η x1 is a well-deﬁned surjec- [ ] → [ ∗ ∗ −] tive isomorphism with inverse map given by γ x1 η γ η x0 . Moreover, if : X → Y is a continuous mapping into a topological space Y , yj = (xj ) for j = 0, 1, and λ = (η), then the above isomorphism and the isomorphism [ ] → [ − ∗ ∗ ] γ y0 λ γ λ y1 together give a commutative diagram ∗ π1(X, x0) ∗π1(X, x0) ⊂ π1(Y, y0) =∼ =∼ =∼ ∗ π1(X, x1) ∗π1(X, x1) ⊂ π1(Y, y1)

~~Proof This follows from Lemma 10.1.3.~~

Remark Because all fundamental groups of a path connected topological space X are isomorphic (although not canonically isomorphic), the associated groups are (or more precisely, the associated isomorphism class of groups is) often denoted by π1(X).

Deﬁnition 10.1.6 We say that X is simply connected if X is path connected and =[ ] π1(X) is trivial (that is, π1(X, x0) ex0 x0 for some, hence for every, point x0 ∈ X). We say that X is locally simply connected if for every point p ∈ X and every neighborhood U of p, there exists a simply connected neighborhood V of p in U.

For example, Rn is simply connected, and for n>1, the unit sphere Sn is simply connected (see Exercise 10.1.3). We postpone consideration of examples of nontriv- 10.1 The Fundamental Group 481 ial fundamental groups until Sect. 10.4, since they are most easily considered within the context of covering spaces and deck transformations. The proof of the following consequence of Lemma 10.1.3 is left to the reader (see Exercise 10.1.2):

~~Lemma 10.1.7 If X is simply connected, then any two paths in X with the same initial and terminal points are path homotopic.~~

~~We will also need the following easy observation:~~

~~Lemma 10.1.8 Let α be a path in X, let 0 = t0~~

Proof We prove by induction that βi ≡ α1 ∗ ··· ∗ αi is a reparametrization of = = − α [0,ti ] for each i 1,...,n, the case i 1 being trivial. Assuming that βi 1 is a :[ ]→[ − ] reparametrization of α [0,ti−1], we have continuous mappings ϕ 0, 1 0,ti 1 and ψ :[0, 1]→[ti−1,ti] such that ϕ(0) = 0, ϕ(1) = ti−1, βi−1 = α ◦ ϕ, ψ(0) = ti−1, ψ(1) = ti , and αi = α ◦ ψ. Thus

~~βi = βi−1 ∗ αi = (α ◦ ϕ) ∗ (α ◦ ψ)= α ◦ ρ, where ρ :[0, 1]→[0,ti] is the continuous mapping given by ϕ(2t) if t ∈[0, 1/2], ρ(t)≡ ψ(2t − 1) if t ∈[1/2, 1].~~

~~The claim now follows by induction, and the lemma is then the case i = n.~~

Lemma 10.1.9 Let X be a connected locally simply connected locally compact Hausdorff space. (a) For every compact set K ⊂ X, there is a path connected relatively compact neighborhood of K in X such that im(π1() → π1(X)) is ﬁnitely generated. (b) If X is compact, then π1(X) is ﬁnitely generated. (c) If X is second countable, then π1(X) is countable.

Proof For the proof of (a), suppose K is a compact subset of X and x0 is a point in K. By applying Lemma 9.3.6 and replacing K with the closure of a relatively compact connected neighborhood in X, we may assume that K is connected. The collection of relatively compact simply connected open subsets of X forms a basis for the topology. Therefore, by Lemma 9.3.6, there exist ﬁnite coverings of K by { } { } ∈ nonempty connected open sets Bi i∈I and Bi i∈I in X such that for each i I , ∩ = ∅ Bi K , Bi is simply connected, and Bi Bi , and for every pair of indices ∈ ∩ = ∅ i, j I with Bi Bj ,wehaveBi Bj . In particular, K is contained in the 482 10 Fundamental Groups, Covering Spaces, and (Co)homology ≡ ∈ ∩ = ∅ domain i∈I Bi X. For each pair of indices i, j I with Bi Bj ,we may choose a point pij = pji ∈ Bi ∩ Bj and a path αij = αji in from x0 to pij . Let us also choose these paths so that αii lies in Bi whenever i ∈ I with x0 ∈ Bi . Finally, for each triple of indices i, j, k ∈ I with Bi ∩ Bj = ∅ and Bj ∩ Bk = ∅,we may choose a path βij k in Bj from pij to pjk. Given a loop λ in with base point x0, we may form a partition

~~0 = t0~~

It follows that the image im(π1(, x0) → π1(X, x0)) is (finitely) generated by the finite set of elements {[ ∗ ∗ − ] | ∈ ∩ = ∅ ∩ = ∅} αij βij k αjk x0 i, j, k I, Bi Bj , and Bj Bk . Thus part (a) is proved. Part (b) follows immediately from (a). Finally, for the proof of part (c), we observe that part (a) and σ -compactness (Lemma 9.3.6 ) provide a sequence of do- { } ∈ = mains ν and a point x0 1 such that X ν ν and such that for each ν, ν ν+1 and the image ν of π1(ν,x0) in π1(X, x0) is finitely generated and ∞ therefore countable. Furthermore, π1(X, x0) is equal to the countable set ν=1 ν . For given an element [γ ]∈π1(X, x0),wehaveγ([0, 1]) ⊂ ν for some ν 0, and hence [γ ]∈ν .

~~Exercises for Sect. 10.1 10.1.1 Prove Lemma 10.1.3. 10.1.2 Prove Lemma 10.1.7. 10.1.3 Prove that for n>1, the unit sphere Sn is simply connected. 10.2 Elementary Properties of Covering Spaces 483~~

~~Fig. 10.2 A covering space and an evenly covered set~~

10.2 Elementary Properties of Covering Spaces Deﬁnition 10.2.1 Let ϒ : X → X be a surjective continuous mapping of topological spaces. −1 (a) An open set U ⊂ X is evenly covered by ϒ if U ≡ ϒ (U) = ∈ Uλ, where λ {Uλ}λ∈ is a collection of disjoint open subsets of X and ϒ maps Uλ homeomorphically onto U for each index λ ∈ (see Fig. 10.2). (b) ϒ is called a covering map and X is called a covering space of X if every point in X has a neighborhood that is evenly covered by ϒ. ∞ (c) If X and X are C manifolds and ϒ : X →X is a covering map that is a local ≡ −1 = diffeomorphism (i.e., for U ϒ (U) λ∈ Uλ as in (a), ϒ maps each of ∞ the sets Uλ diffeomorphically onto U), then ϒ : X → X is called a C covering space (or a C∞ covering manifold)ofX.

Remarks 1. The map R → R given by x → x3 is a covering map (in fact, a homeomorphism) that is of class C∞, but this is not a C∞ covering space because the map is not a local diffeomorphism. 2. If X is a locally connected topological space, ϒ : X → X is a surjective continuous mapping of X onto a topological space X, and U is an evenly covered domain in X, then ϒ maps each connected component of ϒ−1(U) homeomorphically onto U.

Example 10.2.2 The map ϒ : R → S1 given by t → e2πit = (cos(2πt),sin(2πt)) is a C∞ covering map. For as considered in Examples 7.2.8, 9.2.8, and 9.7.20,for 1 2πit each point t0 ∈ R, we get an open arc U ≡ S \{e 0 } and we have −1 ϒ (U) = Un, n∈Z where {Un} is the collection of disjoint open intervals given by

~~Un ≡ (t0 + (n − 1)2π,t0 + n2π) ∀n ∈ Z, and ϒ maps Un diffeomorphically onto U for each n.~~

Deﬁnition 10.2.3 Let : X → Z and : Y → Z be continuous mappings of topological spaces. A lifting of (relative to ) is a continuous map : X → Y such that ◦ = . In other words, the following diagram commutes: 484 10 Fundamental Groups, Covering Spaces, and (Co)homology

~~Y X Z~~

Lemma 10.2.4 Let X, Y , and Y be connected topological spaces, let : X → Y be a continuous map, let ϒ : Y → Y be a covering space, and let : X → Y be a lifting of . (a) If is a lifting of that agrees with at some point in X, then = . (b) If is a local homeomorphism, then is a local homeomorphism. (c) If is a covering map and Y is locally connected, then is a covering map. (d) Suppose X and Y are C∞ manifolds, is of class C∞, and Y is a C∞ covering space. Then the lifting is of class C∞. Moreover, if is a local diffeomorphism, then is a local diffeomorphism. If : X → Y is a C∞ covering space, then : X → Y is a C∞ covering space.

~~Proof For the proof of (a), let be a lifting of and assume that the set~~

A ≡ x ∈ X | (x) = (x) is nonempty. Since X is connected, it sufﬁces to show that A is both closed and open. Given a point x0 ∈ X, we may choose a neighborhood U of (x0) such that −1 U ≡ ϒ (U) = Uλ, λ∈ where {Uλ}λ∈ is a collection of disjoint open subsets of Y , and ϒ maps Uλ homeomorphically onto U for each index λ ∈ . Hence there exist unique indices λ and λ with (x0) ∈ Uλ and (x0) ∈ Uλ , that is,

− − (x ) = (ϒ ) 1((x )) and (x ) = (ϒ ) 1((x )). 0 Uλ 0 0 Uλ 0 −1 −1 The set V ≡ (Uλ) ∩ ) (Uλ ) is then a neighborhood of x0.Ifx0 ∈/ A, then λ = λ ,so(V ) ∩ (V ) =∅, and hence V ⊂ X \ A.Ifx0 ∈ A, then λ = λ , and = −1 ◦ = ⊂ on V ,wehave (ϒ Uλ ) , and hence V A. Thus A is both closed and open, and therefore A = X. The proofs of (b)Ð(d) are left to the reader (see Exercise 10.2.1).

~~Theorem 10.2.5 (Lifting theorem) Let ϒ : X → X be a connected covering space −1 of a connected topological space X, let x0 ∈ X, and let x0 ∈ ϒ (x0).~~

(a) Path lifting property. If γ :[a,b]→X is a path in X with γ(a)= x0, then there is a unique lifting γˆ :[a,b]→X of γ to a path in X with γ(a)ˆ =ˆx0. 10.2 Elementary Properties of Covering Spaces 485

~~(b) Path homotopy lifting property. If H :[a,b]×[0, 1]→X is a path homotopy in X with H(a,·) ≡ x0, then there is a unique lifting~~

H:[a,b]×[0, 1]→X of H to a path homotopy in X with H(a,·) ≡ˆx0 (in particular, H(b,·) is constant). (c) The induced homomorphism ϒ∗ : π1(X,xˆ0) → π1(X, x0) is injective. (d) General lifting property. If : Y → X is a continuous mapping of a con- −1 nected locally path connected topological space Y to X, y0 ∈ (x0), and ∗π1(Y, y0) ⊂ ϒ∗π1(X,xˆ0), then there is a unique lifting : Y → X of with (y0) =ˆx0. In fact, for every point y ∈ Y and every path γ :[a,b]→Y ˆ ˆ from y0 to y, we have (y) = λ(b), where λ:[a,b]→X is the unique lifting ˆ of the path λ ≡ (γ ) with λ(a) =ˆx0. Conversely, if such a lifting exists, then ∗π1(Y, y0) ⊂ ϒ∗π1(X,xˆ0).

Proof We ﬁrst prove uniqueness in part (a). Suppose γ :[a,b]→X is a path and ϒ(xˆ0) = x0 = γ(a) as in (a). For any evenly covered neighborhood U of x0,itis clear that if γ lies in U, then there is a unique lifting of γ with initial point x0.In general, if α and β are two liftings with initial point xˆ0, then letting

~~c ≡ sup{t ∈[a,b]|α = β on [a,t]} ∈ (a, b] and forming an evenly covered neighborhood U of γ(c), we may choose >0so that a0 is sufﬁciently small and~~

D ≡ ([c − ,c + ]∩[a,b]) × ([r − ,r + ]∩[0, 1]), then H(D)is contained in some evenly covered open set U.Ifc = a, then ϒ maps a neighborhood U0 of xˆ0 homeomorphically onto U, and we get a (continuous) lifting −1 ◦ : → ˆ { }× [ − + ]∩[ ] (ϒ U0 ) H D D U0 that is equal to x0 along a ( r ,r 0, 1 ), and therefore must be equal to H on D. However, the point (c, r) = (a, r) lies in the set ((c − ,c + )∩[a,b]) × ((r − ,r + )∩[0, 1]), which is open in [a,b]×[0, 1] and is contained in D. Since (c, r) ∈ A, this is not possible. Assuming that c>a, 486 10 Fundamental Groups, Covering Spaces, and (Co)homology we may then choose the number to be in the interval (0,c− a). But then H is continuous on the connected subset [c − ,c) × ([r − ,r + ]∩[0, 1]) of D, and hence the image of this set must be contained in some open set U1 that ϒ maps homeomorphically onto U. Again, applying the local inverse, we get a continuous { − }× [ − + ]∩[ ] lifting of H D that agrees with H along the set c ( r ,r 0, 1 ) and therefore by construction, must agree with H on all of D. Thus we have again arrived at a contradiction, and we see that H must be continuous. Moreover, the connected set H( {b}×[0, 1]) lies in the ﬁber ϒ−1(H (b, 0)), and is therefore a singleton. Thus H is a path homotopy. Uniqueness follows from (a). Part (c) now follows easily, and part (d) is left to the reader (see Exer- cise 10.2.2).

Corollary 10.2.6 Let ϒ : X → X and ϒˇ : Xˇ → X be connected covering spaces of a connected locally path connected topological space X, let x0 ∈ X, let xˆ0 ∈ −1 ˇ −1 ˇ ˇ ϒ (x0), and let xˇ0 ∈ ϒ (x0). If ϒ∗π1(X,xˆ0) ⊂ ϒ∗π1(X,xˇ0), then there exists a unique commutative diagram of covering maps Xˇ ϒ ϒˇ ϒ X X ˇ ˇ with ϒ(xˆ0) =ˇx0. Moreover, if ϒ∗π1(X,xˆ0) = ϒ∗π1(X,xˇ0), then ϒ is a homeomorphism. If X, X, and Xˇ are C∞ manifolds and ϒ and ϒˇ are C∞ covering maps, ∞ then ϒ is a C covering map (and hence ϒ is a diffeomorphism if ϒ∗π1(X,xˆ0) = ˇ ˇ ϒ∗π1(X,xˇ0)).

Proof The lifting theorem (Theorem 10.2.5) and Lemma 10.2.4 give the existence and uniqueness of the commutative diagram. If the image groups are equal, then we also have a corresponding covering map Xˇ → X. The composition with ϒ is then a lifting of ϒ toacoveringmapX → X that ﬁxes x0, and therefore by uniqueness, the composition must be the identity. Thus ϒ is a homeomorphism in this case. The claims in the C∞ case follow from the above and Lemma 10.2.4.

Corollary 10.2.7 Let ϒ : X → X be a connected covering space of a connected locally path connected topological space X, let U be a connected open subset of X, −1 : → and let V be a connected component of ϒ (U). Then the restriction ϒ V V U is a covering space.

Proof Fix a point xˆ0 ∈ V , and let x0 ≡ ϒ(xˆ0). Then, given a point x ∈ U, there is a path γ in U from x0 to x. Hence the unique lifting γˆ to a path in X with γ(ˆ 0) =ˆx0 must lie in V , and we have ϒ(γ(ˆ 1)) = x. Thus ϒ(V) = U, and it follows that : → ϒ V V U is a covering space. 10.2 Elementary Properties of Covering Spaces 487

Deﬁnition 10.2.8 Two covering spaces ϒ : X → X and ϒˇ : Xˇ → X of a topological space X are equivalent if there exists a homeomorphism ϒ : X → Xˇ that is a lifting of ϒ; i.e., ϒ is a ﬁber-preserving homeomorphism.

Remarks 1. By Lemma 10.2.4, if the above are C∞ covering manifolds, then the homeomorphism ϒ is actually a diffeomorphism, and we say that the coverings are C∞ equivalent. 2. Equivalence of coverings is an equivalence relation (see Exercise 10.2.3). 3. Corollary 10.2.6 immediately gives the following:

Proposition 10.2.9 Let ϒ : X → X and ϒˇ : Xˇ → X be connected coverings of a connected locally path connected topological space X, and let x0 ∈ X and −1 xˆ0 ∈ ϒ (x0). Then the two coverings are equivalent if and only if there exists a ˇ −1 ˇ ˇ point xˇ0 ∈ ϒ (x0) such that ϒ∗π1(X,x0) = ϒ∗π1(X,xˇ0).

~~Any covering space of a C∞ manifold has a natural induced C∞ structure.~~

Proposition 10.2.10 Let ϒ : M → M be a covering space. (a) If M is a Hausdorff space, then M is a Hausdorff space. (b) If M is a topological manifold, then M is a topological manifold of the same dimension. (c) If M is a C∞ manifold, then there is a unique C∞ structure on M with respect to which ϒ is a local diffeomorphism (i.e., with respect to which ϒ : M → M is a C∞ covering manifold).

~~Remark Given a covering space ϒ : M → M of a C∞ manifold M, we assume that M is a C∞ manifold with the associated induced C∞ structure unless otherwise indicated.~~

Proof of Proposition 10.2.10 The proofs of parts (a) and (b) are left to the reader (see Exercise 10.2.4). For the proof of (c), observe that if M is a C∞ manifold of dimension n, then M is Hausdorff (by (a)), and we may form a C∞ at- A ={ } las (Ui,i,Ui ) i∈I on M such that Ui is connected and evenly covered −1 for each i ∈ I . For each i ∈ I , Ui ≡ ϒ (Ui) is the union of a collection of { (λ)} disjoint connected open sets Ui λ∈i in M, each of which is mapped homeo- (λ) ≡ ◦ morphically onto Ui . Setting i i ϒ (λ) , we get an n-dimensional atlas Ui A≡{ (λ) (λ) } ∈ ∈ ∈ ∈ (Ui ,i ,Ui ) λ∈i ,i∈I on M. Suppose i I , j I , λ i , μ j , and (λ) ∩ (μ) = ∅ Ui Uj . Then the coordinate transformation

(λ) ◦[ (μ)]−1 = ◦ −1 : (μ) (λ) ∩ (μ) i j i j (μ) (λ)∩ (μ) j (Ui Uj ) j (Ui Uj ) −→ (λ) (λ) ∩ (μ) ⊂ ∩ i (Ui Uj ) i(Ui Uj ) 488 10 Fundamental Groups, Covering Spaces, and (Co)homology is of class C∞, and hence the atlas determines a C∞ structure on M with respect to which ϒ is a local diffeomorphism. Moreover, given an arbitrary C∞ structure on M with respect to which ϒ is a (λ) local diffeomorphism, and given indices i ∈ I and λ ∈ i , the mapping = ∞ ∞i i ◦ ϒ (λ) is a composition of a C diffeomorphism on Ui ⊂ M and a C dif- Ui (λ) C∞ feomorphism on Ui , and is therefore a diffeomorphism. Thus the elements of the atlas A are C∞ compatible with the local C∞ charts on M, and uniqueness follows.

Recall that a continuous mapping of Hausdorff spaces is proper if the inverse image of every compact subset of the target is compact (see Definition 9.3.9). A sequence in a manifold converges to a point p if for every neighborhood U of p,all but finitely many terms of the sequence lie in U. Recall also that a compact subset of a manifold has the BolzanoÐWeierstrass property (see Theorem 9.3.2). These properties allow one to prove the following useful fact concerning finite covering spaces (i.e., covering spaces with finite fibers) of manifolds:

Lemma 10.2.11 For any local homeomorphism : M → N of connected topological manifolds M and N, we have the following: (a) The mapping is proper if and only if is a finite covering map. (b) If K is a nonempty connected compact subset of (M) with connected compact inverse image K ≡ −1(K), is a sufficiently small connected neighborhood of K in N, and is the connected component of −1() containing K, then ≡ : → the restriction ϒ is a finite covering map.

Proof For the proof of (a), let us first assume that is a proper mapping. The map is then surjective. For if y ∈ (M), then we may choose a sequence of points {xν} in M such that yν = (xν) → y. Since is proper, {xν} must lie in a compact subset of M, and hence some subsequence must converge to some point x ∈ M. Continuity then gives (x) = y. Thus (M) is a nonempty subset of N that is closed (by the above) and open (since is a local homeomorphism); and therefore, since N is connected, we have (M) = N. Now given a point y0 ∈ N, we may choose a finite covering of the compact fiber −1 F ≡ (y0) by relatively compact connected open subsets V1,...,Vn of M, each of which is mapped homeomorphically onto a connected neighborhood of y0 by . Each of these sets contains exactly one element of F , and since M is Hausdorff, we may choose the sets to be disjoint. We may also choose a connected neighbor- −1 hood U of y0 in (V1) ∩···∩(Vn) such that (U) ⊂ V ≡ V1 ∪···∪Vn.For if this were not the case, then as in the above proof of surjectivity, there would exist a sequence of points {xν} in M \ V that converges to a point x ∈ M \ V with (x) = y0, which is impossible, since F ⊂ V . Thus the desired neighbor- −1 hood U must exist. Hence (U) = U1 ∪···∪Un, where for each i = 1,...,n, ≡ −1 ∩ = −1 Ui (U) Vi ( Vi ) (U) is a connected neighborhood that is mapped 10.3 The Universal Covering 489 homeomorphically onto U by . Since the sets U1,...,Un are disjoint, it follows that U is evenly covered, and hence that : M → N is a finite covering map. For the proof of the converse, suppose : M → N is a finite covering map. Then, for each point p ∈ N, the inverse image −1(U) of some (in fact, any) evenly covered neighborhood U of p has only finitely many connected components. Fixing a relatively compact neighborhood V of p in U, we see that −1(V) is compact. Since any compact set D ⊂ N admits a finite covering by such sets V , −1(D) must be compact. For the proof of (b), it suffices to show that for any sufficiently small connected neighborhood of K and for the connected component of −1() contain- ≡ −1 ≡ : → ing the connected compact set K (K), the restriction ϒ is proper; for we may then apply part (a). Clearly, for this, we may assume that K = N. Since K is connected and compact, we may choose a relatively compact connected neighborhood V of K in M. Since (M) is open, we may require that (M). Moreover, if is sufficiently small, then will not meet the compact set (∂V ), and hence the connected component of −1() containing K will be a connected open subset of M that meets V , but not ∂V. Thus we will have ⊂ V . Moreover, if C is a compact subset of , then −1(C) is a closed subset of M that does not meet ∂ (since any point in −1(C)∩cl() has a connected neighborhood in −1() that meets, and hence is contained in, ). Thus −1(C) ∩ is compact, ≡ : → the restriction ϒ is a proper mapping, and the claim follows.

Exercises for Sect. 10.2 10.2.1 Prove parts (b)Ð(d) of Lemma 10.2.4. 10.2.2 Prove part (d) of Theorem 10.2.5. 10.2.3 Prove that equivalence of coverings is an equivalence relation. 10.2.4 Prove parts (a) and (b) of Proposition 10.2.10.

~~10.3 The Universal Covering~~

~~Deﬁnition 10.3.1 A simply connected covering space ϒ : X → X of a connected topological space X is called a universal covering space of X.~~

~~Corollary 10.2.6 immediately gives the following:~~

Lemma 10.3.2 (Uniqueness of universal coverings) A connected locally path connected topological space X has at most one universal covering space up to equivalence of coverings. In fact, if x0 ∈ X and for j = 1, 2, ϒj : Xj → X is a univer- ∈ −1 sal covering space of X and xj ϒj (x0), then there is unique homeomorphism : X1 → X2 such that ϒ2 ◦ = ϒ1 and (x1) = x2.

~~The main goal of this section is the following: 490 10 Fundamental Groups, Covering Spaces, and (Co)homology~~

~~Theorem 10.3.3 (Existence of universal coverings) Every connected locally simply connected topological space X has a universal covering space ϒ : X → X.~~

Proof The idea of the construction is to separate the terminal points of paths from a fixed initial point whenever the paths are not homotopic. Fix a point x0 ∈ X, and let P be the set of paths α :[0, 1]→X with initial point α(0) = x0. Then path homotopy equivalence determines an equivalence relation ∼ on P, and we may set X ≡ P/∼. Denoting the path homotopy equivalence class of each path α by [α],we may define a surjective map ϒ : X → X by [α]→α(1). We must define a topology on X with respect to which this is a universal covering. Let A be the collection of all pairs ([α],U), where [α]∈X and U is a simply connected neighborhood of ϒ([α]) = α(1) in X. For each element ξ = ([α],U)∈ A, we get a well-defined map ξ : U → X with ξ (α(1)) =[α] by setting, for each point x ∈ U,

ξ (x) ≡[α ∗ β], where β :[0, 1]→U is an arbitrary path in U from α(1) to x.Forifα ∼ α and β is a path in U from α(1) = α (1) to x ∈ U, then since U is simply connected, β and β , and therefore α ∗ β and α ∗ β , are path homotopic. Clearly, ϒ ◦ ξ = IdU , and hence ξ maps U bijectively onto ξ (U) with inverse mapping ϒ ξ (U). Observe = [ ] = [ ] ∈ A ⊂ = also that if ξ ( α ,U),ζ ( α ,V) with U V , then ξ ζ U . T ⊂ −1 One may verify that the collection of subsets V X such that ξ (V ) is an open set in X for every element ξ ∈ A is a topology on X (see Exer- cise 10.3.1). Moreover, ξ (U) ∈ T for each ξ = ([α],U)∈ A. For given an element ζ = ([β],V)∈ A and a point ∈ −1 = ∩ ⊂ ∩ x1 ζ (ξ (U)) ϒ(ζ (V ) ξ (U)) U V, we may choose a simply connected neighborhood W of x contained in U ∩V , a path γ :[0, 1]→U from α(1) to x1, and a path δ :[0, 1]→V from β(1) to x1. We then get [ ∗ ]= =[ ]−1 = =[ ∗ ] α γ ξ (x1) ϒ ζ (V )∩ξ (U) (x1) ζ (x1) β δ .

~~Hence if x ∈ W and η :[0, 1]→W is a path from x1 to x, then~~

ζ (x) =[β ∗ δ ∗ η]=[α ∗ γ ∗ η]=ξ (x) ∈ ξ (U). ⊂ −1 −1 Thus W ζ (ξ (U)), and hence ζ (ξ (U)) is open. Clearly, with respect to the topology T , ξ is a continuous bijection of U onto the open set ξ (U) for each ξ = ([α],U)∈ A. Moreover, if V ⊂ X is open, then = ∩ = −1 ϒ(V) ϒ(V ξ (U)) ξ (V ) ξ=([α],U)∈A ξ∈A is open; hence ϒ is an open mapping. Thus ϒ : X → X is a surjective local homeomorphism (with local continuous inverses given by the mappings {ξ }ξ∈A). In fact, 10.3 The Universal Covering 491

~~−1 given a point x1 ∈ X and an element ([α],U)∈ A with x1 = α(1), ϒ (U) is equal to the union of disjoint connected open sets~~

{ [ ∗ ] (U)}[ ] ∈ , ( β α ,U) β x0 π1(X,x0) each of which is mapped homeomorphically onto U by ϒ.Forif[γ ]∈ϒ−1(U) and − − δ :[0, 1]→U is a path from x1 to γ(1), then β ≡ γ ∗ δ ∗ α is a loop based at x0 and [γ ]=[β ∗ α ∗ δ]=([β∗α],U)(γ (1)). If, on the other hand, β1 and β2 are loops ∈ = based at x0 and there exist points y1,y2 U with ([β1∗α],U)(y1) ([β2∗α],U)(y2), then, applying ϒ, we see that y1 = y2. Fixing a path δ :[0, 1]→U from x1 to y1 = y2, we get [ ∗ ∗ ]= = =[ ∗ ∗ ] β1 α δ ([β1∗α],U)(y1) ([β2∗α],U)(y2) β2 α δ , and hence [ ] =[ ∗ ∗ ∗ − ∗ −]=[ ∗ ∗ ∗ − ∗ −]=[ ] β1 x0 β1 α δ δ α β2 α δ δ α β2 x0 .

Thus the sets are disjoint, and ϒ : X → X is a covering space. It remains to show that X is simply connected. Let ex0 be the constant loop at x0. Given a path α ∈ P, we may choose a partition 0 = t0

Thus α is path homotopic to the constant loop ex0 , and therefore, again by the path homotopy lifting property, the lifting γ is path homotopic in X to the constant loop [ ] at ex0 . 492 10 Fundamental Groups, Covering Spaces, and (Co)homology

~~Exercises for Sect. 10.3 10.3.1 Verify that the collection of sets T constructed in the proof of Theo- rem 10.3.3 is a topology.~~

~~10.4 Deck Transformations~~

Deﬁnition 10.4.1 A deck transformation (or covering transformation) of a covering space ϒ : X → X is a homeomorphism : X → X such that ϒ ◦ = ϒ (i.e., preserves the ﬁbers). The group determined by the set of deck transformations for the covering with multiplication given by composition is denoted by Deck(ϒ).

Theorem 10.4.2 Let ϒ : X → X be a universal covering of a connected locally −1 path connected topological space X, let x0 ∈ X, let F ≡ ϒ (x0), and let x˜0 ∈ F . We then have the following:

(a) The map Deck(ϒ) → F given by → (x˜0) is a bijection. (b) The map χ : Deck(ϒ) → π1(X, x0) given by → [ϒ(α)˜ ]x , where α˜ is an 0 arbitrary path in X from x˜0 to (x˜0), is a well-deﬁned group isomorphism. −1 (c) Let σ ≡ χ : π1(X, x0) → Deck(ϒ) be the inverse isomorphism. If x ∈ X, γ˜ ˜ [ ] ∈ ˜ is a path from x0 to x, α x0 π1(X, x0), α is the unique lifting of α to a path ˜ in X with α(˜ 0) =˜x0, and γ˜1 is the unique lifting of γ ≡ ϒ(γ)˜ to a path with ˜ =˜ [ ] =˜ [ ] γ1(0) α(1), then σ( α x0 )(x) γ1(1) (that is, σ( α x0 )(x) is equal to the terminal point of the lifting of α ∗ γ with initial point x˜0). (d) The composition π1(X, x0) → Deck(ϒ) → F is the bijection given by [ ] → [ ] ˜ α x0 σ( α x0 )(x0).

(e) Each element ∈ Deck(ϒ) \{Id} has no ﬁxed points. In fact, if U is an evenly −1 covered domain in X and U0 is a component of U ≡ ϒ (U), then the set U1 ≡ (U0) is a component of U that is distinct (hence disjoint) from U0.

Proof By uniqueness of universal coverings up to equivalence (Lemma 10.3.2), for each point y ∈ F , there is a unique homeomorphism : X → X such that ϒ ◦ = ϒ and (x˜0) = y; that is, the map Deck(ϒ) → F is bijective. Thus (a) is proved. Suppose that for j = 1, 2, j ∈ Deck(ϒ), yj = j (x˜0), and α˜j :[0, 1]→X is a path with initial point α˜j (0) =˜x0 and terminal point α˜j (1) = yj . In particular, αj ≡ ϒ(α˜ j ) is a loop based at x0 for j = 1, 2. If y1 = y2 (i.e., 1 = 2), then since X is simply connected, α˜ 1 and α˜ 2, and therefore α1 and α2, are path homotopic. Hence χ is well deﬁned. In general, 1(α˜ 2) is a path in X from y1 to 1 ◦ 2(x˜0), and hence

~~◦ =[ ˜ ∗ ˜ ] =[ ∗ ] =[ ] ·[ ] = · χ(1 2) ϒ(α1 1(α2)) x0 α1 α2 x0 α1 x0 α2 x0 χ(1) χ(2). 10.4 Deck Transformations 493~~

Thus χ is a homomorphism. If α1 is path homotopic to the trivial loop, then the lifting α˜ 1 is also a loop (by the path homotopy lifting property in Theorem 10.2.5) and hence 1(x˜0) = y1 =˜x0. Thus 1 = Id and it follows that χ is injective. Given a loop α based at x0, there are a unique lifting to a path α˜ in X with α(˜ 0) =˜x0 and ˜ =˜ =[ ] a unique deck transformation with (x0) α(1). Clearly, χ() α x0 ,soχ is surjective and (b) is proved. The proofs of (c)Ð(e) are left to the reader (see Exercise 10.4.1).

~~Deﬁnition 10.4.3 Let be a subgroup of the group of self-homeomorphisms of a locally compact Hausdorff space X.~~

~~(a) We denote by ∼ the equivalence relation on X given by~~

~~x ∼ y ⇐⇒ γ(x)= y for some γ ∈ ;~~

by \X the quotient space X/∼, which is given the quotient topology; and by ϒ : X → \X the corresponding quotient map. (b) The group acts properly discontinuously on X if for every compact set K ⊂ X, γ(K)∩ K =∅for all but ﬁnitely many elements γ of . (c) The group acts freely if for every element γ ∈ with γ = Id and every point x ∈ X,wehaveγ(x)= x.

Lemma 10.4.4 Let be a subgroup of the group of self-homeomorphisms of a locally compact Hausdorff space X. Then acts properly discontinuously on X if and only if for each pair of points x,y ∈ X, there exist neighborhoods U of x and V of y such that γ(U)∩ V =∅for all but ﬁnitely many elements γ of .

~~Proof If acts properly discontinuously and x,y ∈ X, then ﬁxing relatively compact neighborhoods U of x and V of y, we see that~~

γ(U)∩ V ⊂ γ(U ∪ V)∩ (U ∪ V)=∅ for all but ﬁnitely many elements γ of . For the proof of the converse, suppose that K ⊂ X is compact and that for each pair of points x,y ∈ K, there exist neighborhoods Uxy of x and Vxy of y such that

Gxy ≡{γ ∈ | γ(Uxy) ∩ Vxy = ∅} is finite. For each point x ∈ K, we may choose a finite set Yx⊂ K such that K ⊂ V , and we may set W equal to the neighborhood U of x. y∈Yx xy x y∈Yx xy ∈ ⊂ n ∈ Thus there exist points x1,...,xn K such that K j=1 Wxj .Ifγ and = ∈ ∈ ⊂ ∈ γ(x) y for some pair of points x,y K, then x Wxj Uxj z, y Vxj z, and ∈ ∈{ } ∈ γ Gxj z for some index j 1,...,n and some point z Yxj . It follows that n { ∈ | ∩ = ∅} ⊂ γ γ(K) K Gxj z = ∈ j 1 z Yxj is a finite set. 494 10 Fundamental Groups, Covering Spaces, and (Co)homology

Theorem 10.4.5 Let ϒ : X → X be the universal covering space of a connected locally simply connected locally compact Hausdorff space X. Then ≡ Deck(ϒ) acts properly discontinuously and freely on X, the quotient map ϒ : X → \X is a covering map, and we have a commutative diagram X ϒ ϒ =∼ X \X in which the mapping \X → X is a homeomorphism. Moreover, if X is a C∞ manifold, then is a group of self-diffeomorphisms of X (with respect to the induced ∞ ∞ C structure) and \X has a unique C structure with respect to which ϒ is a C∞ covering map and the map \X → X is a diffeomorphism. In other words, ϒ : X → \X is a covering that we may identify with the original covering ϒ : X → X. It follows that if ϒ : X → X is the universal covering space of a connected locally simply connected locally compact Hausdorff space (or a connected C∞ manifold) X and Deck(ϒ) = Deck(ϒ), then X =∼ \X =∼ X and we may identify the coverings ϒ : X → X and ϒ : X → X.

Proof of Theorem 10.4.5 Part (e) of Theorem 10.4.2 and Lemma 10.4.4 together imply that acts properly discontinuously and freely. Part (a) of Theorem 10.4.2 implies that ϒ descends to a continuous bijection \X → X. The inverse mapping X → \X is also continuous because ϒ is a surjective open mapping. ∞ Lemma 10.2.4 implies that ϒ is a covering map. Finally, if X is a C manifold and ϒ : X → X is the C∞ universal covering, then Lemma 10.2.4 implies that every deck transformation is a diffeomorphism. Moreover, we get a unique C∞ structure on \X with respect to which the map \X → X is a diffeomorphism (for example, ∞ by Proposition 10.2.10) and Lemma 10.2.4 then implies that ϒ is a C covering map.

~~We also have the following partial converse:~~

Theorem 10.4.6 Let X be a simply connected locally simply connected locally compact Hausdorff space, let be a subgroup of the group of self-homeomorphisms of X that acts properly discontinuously and freely on X, and let ϒ = ϒ : X → X = \X be the corresponding quotient space mapping. We then have the following: (a) The quotient space X is connected, locally simply connected, and locally compact Hausdorff; ϒ : X → X is the universal covering space; and = Deck(ϒ). (b) If is a subgroup of , then we have a commutative diagram X ϒ ϒ X ϒ X ≡ \X 10.4 Deck Transformations 495

~~˜ ∈ = ˜ ˆ = ˜ of covering maps. Moreover, if x0 X, x0 ϒ(x0), and x0 ϒ(x0), and : = → : = → ˆ χ Deck(ϒ) π1(X, x0) and χ Deck(ϒ) π1(X,x0)~~

~~are the corresponding group isomorphisms, then~~

−1 χ ◦ χ = ϒ∗ : π1(X,xˆ0) → π1(X, x0). In other words, the action of [ˆγ ]∈π1(X,xˆ0) on X is the same as that of ϒ∗[ˆγ ]=[ϒ(γ)ˆ ]∈π1(X, x0). (c) If X is a topological manifold, then X is a topological manifold of the same dimension. (d) If X is a C∞ manifold and each element of is a diffeomorphism, then X has a unique C∞ structure with respect to which ϒ : X → X is a C∞ (universal) ⊂ C∞ covering space. Moreover, for as in part (b), ϒ and ϒ are covering maps (with respect to a unique C∞ structure on X).

Proof ϒ is an open mapping because given an open set W ⊂ X, ϒ−1(ϒ(W)) = · W is the union of the open sets {γ(W)}γ ∈. Since X is locally compact Haus- dorff and acts properly discontinously, given a point x0 ∈ X, we may ﬁx a point −1 y0 ∈ ϒ (x0), a relatively compact neighborhood V of y0 in X, and a ﬁnite set 1 ⊂ such that γ(V)∩ V =∅ for each element γ ∈ with γ/∈ 1. Since X is Hausdorff, for each γ ∈ 1 with γ = Id, we may choose disjoint neighborhoods Vγ and Wγ of y0 and γ(y0), respectively. For γ = Id, we set Vγ = Wγ = V . Finally, we may choose a connected neighborhood U0 of y0 with −1 U0 ⊂ V ∩ Vγ ∩ γ (Wγ ). γ ∈1

Setting U ≡ ϒ(U0), we get −1 ϒ (U) = · U0 = γ(U0); γ ∈ and hence, in order to show that U isevenlycoveredbyϒ, it sufﬁces to show that the sets {γ(U0)}γ ∈ are disjoint and that ϒ maps each of these sets homeomorphically onto U. For this, we ﬁrst observe that if α ∈ with U0 ∩ α(U0) = ∅, then α = Id. For we have V ∩ α(V ) = ∅, and hence α ∈ 1 and

~~−1 U0 ∩ α(U0) ⊂ Vα ∩ α (Wα) ∩ α(Vα) ∩ Wα ⊂ Vα ∩ Wα.~~

Thus α = Id, and it follows that the open sets {γ(U0)}γ ∈ are disjoint and ϒ maps each of these sets bijectively and therefore homeomorphically (since ϒ is continuous and open) onto U. To see that X is Hausdorff, suppose x1 and x2 are distinct points in X. Then, for each j = 1, 2, we may choose an evenly covered connected neighborhood Uj of xj 496 10 Fundamental Groups, Covering Spaces, and (Co)homology

−1 −1 and a point yj ∈ ϒ (xj ) such that the component Vj of ϒ (Uj ) containing yj is relatively compact in X. Thus the set K ≡ V 1 ∪ V 2 is compact, and since acts properly discontinuously, we get a ﬁnite set 1 ⊂ such that γ(K)∩ K =∅for each element γ ∈ \ 1. Since X is Hausdorff and y2 ∈/ · y1, for each element γ ∈ 1 we may choose neighborhoods Pγ and Qγ of γ(y1) and y2, respectively, that are disjoint. Thus we get neighborhoods −1 R1 ≡ V1 ∩ γ (Pγ ) and R2 ≡ V2 ∩ Qγ γ ∈1 γ ∈1 of y1 and y2, respectively, and neighborhoods W1 ≡ ϒ(R1) ⊂ U1 and W2 ≡ ϒ(R2) ⊂ U2 of x1 and x2, respectively. The neighborhoods W1 and W2 are disjoint because

~~−1 −1 ϒ (W1) ∩ ϒ (W2) ∩ V2 = ( · R1) ∩ R2 = (1 · R1) ∩ R2 ⊂ Pγ ∩ Qγ =∅. γ ∈1 γ ∈1~~

Thus X is Hausdorff. Moreover, since ϒ is a surjective local homeomorphism, X is also connected, locally simply connected, and locally compact. For the proof that = Deck(ϒ) (and for the proof of (b)), let us ﬁx a point x˜0 ∈ X and let x0 = ϒ(x˜0). Given an element ∈ ,wehaveϒ ◦ = ϒ and therefore ∈ Deck(ϒ). Conversely, if ∈ Deck(ϒ), then ϒ((x˜0)) = ϒ(x˜0), and hence there is an element ∈ ⊂ Deck(ϒ) with (x˜0) = (x˜0). Therefore, by Theorem 10.4.2, = ∈ . Thus = Deck(ϒ). : → = \ For the proof of (b), suppose is a subgroup of , ϒ X X X is ˆ = ˜ the corresponding quotient covering space, and x0 ϒ(x0). The surjective map- : → → ping ϒ X X given by ϒ(x) ϒ(x) is well deﬁned and therefore continuous with respect to the quotient topology. If U is a nonempty domain in X that is −1 evenlycoveredbyϒ, and U0 is a connected component of ϒ (U), then each con- − nected component V of ϒ 1(U ) ⊂ ϒ−1(U) is equal to a connected component 0 −1 −1 of ϒ (U).ForifW is the component of ϒ (U) containing V , then ϒ(W) is a −1 = = connected subset of ϒ (U) (ϒ(ϒ(W)) ϒ(W) U) that meets, and is therefore contained in, U0.SoW = V . Since ϒ maps V homeomorphically onto U and ϒ maps V locally homeomorphically onto U0 (by Corollary 10.2.7), ϒ must map U0 homeomorphically onto U. Thus ϒ is a covering map. To complete the proof of (b), we observe that given an element [ˆγ ]∈π1(X,xˆ0), we may set γ = ϒ( γ)ˆ and we may form the common lifting γ˜ of γ and γˆ to a path in X with γ(˜ 0) =˜x0. We then have

~~−1 −1 [γ ]·x ˜0 = χ ([γ ])(x˜0) =˜γ(1) = χ ([ˆγ ])(x˜0) =[ˆγ ]·x ˜0,~~

~~− so [γ ]=ϒ∗[ˆγ ]=χ(χ 1([ˆγ ])). The proofs of (c) and (d) are left to the reader (see Exercise 10.4.2). 10.4 Deck Transformations 497~~

Corollary 10.4.7 Let X be a connected locally simply connected locally compact Hausdorff space, let x0 ∈ X, and let be a subgroup of π1(X, x0). Then, up to equivalence of covering spaces, there is a unique connected covering space −1 ϒ : X → X such that for some point xˆ0 ∈ ϒ (x0), ϒ∗ : π1(X,xˆ0) → π1(X, x0) maps π1(X,xˆ0) isomorphically onto . In fact, if ϒ : X → X is the universal covering space, then we have the commutative diagram

~~ϒ X X = \X~~

~~ϒ ϒ~~

X ∞ (here, we identify π1(X, x0) ⊃ with Deck(ϒ)). Moreover, if X is a C manifold, and X has the induced C∞ structure, then there is a unique C∞ structure on X with ∞ respect to which ϒ and ϒ are C covering maps.

Proof This follows immediately from Theorem 10.3.3 (existence of the universal cover), Theorem 10.4.6 (the construction of a covering from a properly discontinuous free group action), and Proposition 10.2.9 (equivalence of coverings with the same image fundamental group).

Example 10.4.8 For S1 ={(x, y) ∈ R2 | x2 + y2 = 1},theC∞ universal covering space is given by ϒ : R → S1, where ϒ(t)= (cos(2πt),sin(2πt))= e2πit for each t ∈ R (see Example 10.2.2). If ∈ Deck(ϒ), then e2πi(t) = e2πit for all t ∈ R. Thus t → ((t) − t)/(2π) is a C∞ function with values in Z, and is therefore a constant function. Conversely, any translation of the form t → t + n for n ∈ Z is 1 ∼ ∼ clearly a deck transformation. Thus π1(S ) = Deck(ϒ) = Z. In fact, for x0 = (1, 0), S1 [ ] = 2πit π1( ,x0) is the infinite cyclic (free) group generated by γ x0 , where γ(t) e for each t ∈[0, 1] (since γ has the lifting γ˜ : t → t with γ(˜ 0) = 0 and γ(˜ 1) = 0 + 1 = 1). ∼ 1 Any subgroup of Z = π1(S ,x0) must be of the form = mZ for some nonnegative integer m; that is, is the infinite cyclic subgroup generated by [γ ]m .We x0 also have a universal covering map R → S1 given by t → e2πit/m (the proof is similar to the above), and this covering map has deck transformation group mZ = ⊂ Z. Therefore, S1 =∼ \R and the covering space of S1 associated to the subgroup as in Corollary 10.4.7 is the finite mapping S1 → S1 given by e2πit/m → e2πit, that is, by z → zm (see Example 1.6.2).

~~Exercises for Sect. 10.4~~

~~10.4.1 Prove parts (c)Ð(e) of Theorem 10.4.2. 10.4.2 Prove parts (c) and (d) of Theorem 10.4.6. 498 10 Fundamental Groups, Covering Spaces, and (Co)homology~~

~~10.5 Line Integrals on C∞ Surfaces~~

Throughout this section, M denotes a 2-dimensional C∞ manifold. We first recall C1 that for a continuous differential form θ of degree 1 on M and a piecewise path :[ ]→ = b ˙ γ a,b M,the(line) integral of θ along γ is given by γ θ a θ(γ(t))dt (see Definition 9.7.18). For any point a ∈ R and any map γ :{a}→M, we define ≡ γ θ 0. The chain rule and the fundamental theorem of calculus give the following (see Exercise 10.5.1):

~~Lemma 10.5.1 For any continuous 1-form θ on M and any piecewise C1 path γ :[a,b]→M: =− (a) We have γ − θ γ θ. (b) If a~~

Part (c) of the above lemma suggests that one may deﬁne integration of a locally exact 1-form (for example, a closed C1 form) along a continuous path as a sum of changes of local potentials. The following lemma will imply that the integral is well deﬁned.

~~Lemma 10.5.2 Let θ be a locally exact 1-form on M (in particular, θ is continuous), let γ :[a,b]→M be a (continuous) path in M, let S : a = s0 < ···~~

~~m n αi(γ (si)) − αi(γ (si−1)) = βj (γ (tj )) − βj (γ (tj−1)) . i=1 j=1~~

~~Furthermore, if γ is a piecewise C1 path, then the above number is equal to the integral of θ along γ .~~

Proof By considering a common reﬁnement of the two partitions, we see that we may assume without loss of generality that T is a reﬁnement of S; that is, {s0,...,sm}⊂{t0,...,tn}. Thus, for each j = 1,...,n, we have, for some unique index i, [tj−1,tj ]⊂[si−1,si] and dβj = dαi = θ on some connected neighborhood W of γ([tj−1,tj ]) in Ui ∩ Vj . In particular, αi and βj differ by a constant on W , and hence αi(γ (tj )) − αi(γ (tj−1)) = βj (γ (tj )) − βj (γ (tj−1)). Therefore, 10.5 Line Integrals on C∞ Surfaces 499

m αi(γ (si)) − αi(γ (si−1)) i=1 m = αi(γ (tj )) − αi(γ (tj−1)) i=1 1≤j≤n [tj−1,tj ]⊂[si−1,si ] n = βj (γ (tj )) − βj (γ (tj−1)) . j=1 Finally, Lemma 10.5.1 implies that the above gives the integral when γ is a piecewise C1 path.

Deﬁnition 10.5.3 Let θ be a locally exact 1-form on M. For any (continuous) path γ :[a,b]→M,theintegral of θ along γ is given by n θ ≡ αi(γ (ti)) − αi(γ (ti−1)) , γ i=1 where a = t0

~~Lemma 10.5.4 For any locally exact 1-form θ on M and any path γ :[a,b]→M: =− (a) We have γ − θ γ θ. (b) If a~~

Proof The proofs of parts (a), (b), and (e) are left to the reader (see Exercise 10.5.2). We prove part (c) for any choice of points s0,s1,s2,...,sm and functions α1,...,αm by induction on m, the case m = 1 being trivial. Assume now that m>1 and that the claim holds for positive integers less than m. If there is an index k ∈{1,...,m− 1} with sk

~~m = αi(γ (si)) − αi(γ (si−1)) i=k+1 k − αk−i+1(γ (sk−i )) − αk−i+1(γ (sk−i+1)) i=1 m = αi(γ (si)) − αi(γ (si−1)) . i=1~~

If there is an index k ∈{1,...,m− 1} with c = s0 ≤ sk ≤ sm = d, then for J ≡{t ∈ R | c ≤ t ≤ sk} and K ≡{t ∈ R | sk ≤ t ≤ d}, the induction hypothesis gives θ = θ + θ γ I γ J γ K k m = αi(γ (si)) − αi(γ (si−1)) + αi(γ (si)) − αi(γ (si−1)) i=1 i=k+1 m = αi(γ (si)) − αi(γ (si−1)) . i=1

~~Finally, if sm~~

Proposition 10.5.5 For any locally exact 1-form θ on M, the following are equivalent: (i) The form θ is exact. = (ii) γ θ 0 for every loop γ in M. (iii) The line integrals are independent of path; that is, θ = θ for every pair γ0 γ1 of paths γ0 and γ1 with the same endpoints. 10.5 Line Integrals on C∞ Surfaces 501

Proof We may assume without loss of generality that M is connected. That (i) implies (ii) follows immediately from the definition of the line integral. That (ii) implies (iii) follows immediately from Lemma 10.5.4. Assuming now that the condition (iii) holds, and fixing a point p ∈ M, for each ∈ ≡ point x M we may define f(x) γ θ, where γ is an arbitrary path from p to x in M.Givenq ∈ M, we may fix a C1 function α with dα = θ on some connected neighborhood U of q, and a path η from p to q. Given a point x ∈ U,wemay choose a path γ from q to x in U, and we get f(x)= θ + θ = α(x) − α(q) + θ. γ η γ

~~Hence f and α differ by a constant on U, and therefore f ∈ C1(M) and df = θ.~~

~~One may greatly reduce the collection of loops along which one must check that the integral vanishes in order to get exactness.~~

~~Theorem 10.5.6 Let θ be a locally exact1-form on M, and let γ0 and γ1 be two path homotopic paths in M. Then θ = θ. γ0 γ1~~

~~Proof Let H :[a,b]×[0, 1]→M be a path homotopy from γ0 to γ1, and for each s ∈[0, 1],letγs ≡ H(·,s):[a,b]→M. We may choose partitions~~

~~a = t0~~

1 such that for each pair (i, j) ∈{1,...,m}×{1,...,n}, there is a C function αij with dαij = θ on a connected neighborhood Uij of H([ti−1,ti]×[sj−1,sj ]) in M (see Fig. 10.3). In particular, if i ≥ 2 and j ≥ 1, then αij and αi−1,j differ by a constant on some neighborhood of H({ti−1}×[sj−1,sj ]).Forj = 1,...,n, m = − θ αij (γsj−1 (ti)) αij (γsj−1 (ti−1)) γ sj−1 i=1 m = − + − αij (γsj−1 (ti)) αij (γsj (ti)) αij (γsj (ti)) αij (γsj (ti−1)) i=1 + − αij (γsj (ti−1)) αij (γsj−1 (ti−1)) m−1 = − + αij (γsj−1 (ti)) αij (γsj (ti)) θ γ i=1 sj m + − αi−1,j (γsj (ti−1)) αi−1,j (γsj−1 (ti−1)) i=2 = θ γsj 502 10 Fundamental Groups, Covering Spaces, and (Co)homology

~~Fig. 10.3 Invariance of line integrals under path homotopy~~

~~(here we have used the fact that γs(a) and γs(b) are each constant in s). The claim now follows.~~

Remark One may view the equality − = − αij (γsj−1 (ti)) αij (γsj−1 (ti−1)) αij (γsj−1 (ti)) αij (γsj (ti)) + − αij (γsj (ti)) αij (γsj (ti−1)) + − αij (γsj (ti−1)) αij (γsj−1 (ti−1)) as the replacement of the integral of θ along the segment γsj−1 [ti−1,ti ] with the − − ∗ ∗ → integral along the path λi 1 γsj [ti−1,ti ] λi , where λi is the path s H(ti,s) = = from γsj−1 (ti) to γsj (ti) for i 0,...,m and j 1,...,m (see Fig. 10.3). The integrals are equal because θ is exact on the neighborhood Uij . The integrals along the λi ’s cancel in the telescoping sum.

~~Corollary 10.5.7 If M is simply connected, then any locally exact 1-form on M is exact.~~

~~Corollary 10.5.8 If M is connected, x0 ∈ M, and θis a locally exact 1-form on M, → C + [ ]→ then the map π1(X, x0) ( , ) given by γ γ θ is a well-deﬁned group homomorphism.~~

~~Exercises for Sect. 10.5 10.5.1 Prove Lemma 10.5.1. 10.5.2 Prove parts (a), (b), and (e) of Lemma 10.5.4. 10.6 Homology and Cohomology of Second Countable C∞ Surfaces 503~~

~~10.5.3 Let θ = Pdx+ Qdy for continuous functions P,Qon an open set ⊂ R2. Show that if ∂Q = ∂P ∈ 1 Lloc(), ∂x distr ∂y distr then θ is locally exact.~~

~~10.6 Homology and Cohomology of Second Countable C∞ Surfaces~~

Differential forms and their appropriate generalizations allow one to associate certain Abelian groups and vector spaces to a C∞ surface. In a sense, these groups measure the number of loops (and forms) in the surface when we identify loops (and forms) that are identical up to integration. We will need these groups only for second countable surfaces, so we restrict our attention to that case for simplicity. The approach taken here is similar to the approach in the book of Weyl [Wey]. Throughout this section, F denotes the ﬁeld R or C.

Deﬁnition 10.6.1 For a second countable C∞ surface M and an integer r ∈{0, 1, 2}, the rth de Rham cohomology group with coefﬁcients in F is the quotient group (and quotient vector space) r F ≡ Er F →d Er+1 F HdeR(M, ) ker (M, ) (M, ) − d /im Er 1(M, F) → Er (M, F)

≡ E−1 ≡ r F (here, we set d 0on (M) 0). We also denote HdeR(M, ) simply r ∈ Er F by HdeR(M). For each closed differential form α (M, ), we call the corre- r F sponding equivalence class in HdeR(M, ) the de Rham cohomology class of α, and we denote this class by [α] r F ,by[α] ,orsimplyby[α].Wealsosay HdeR(M, ) deR that two differential forms are cohomologous if they represent the same de Rham cohomology class (i.e., if their difference is C∞-exact). We also let 1 F ≡ E1 F →d E2 F ZdeR(M, ) ker (M, ) (M, ) , 1 F ≡ E0 F →d E1 F BdeR(M, ) im (M, ) (M, ) . r F C∞ F In other words, HdeR(M, ) is given by the closed -valued differential forms of degree r on M, where we identify two closed C∞ differential forms α − C∞ r F and β if and only if α β is -exact. Clearly, HdeR(M, ) is an Abelian group [ ] +[ ] =[ + ] with respect to the induced sum operation α deR β deR α β deR. In fact, r F F HdeR(M, ) is a vector space over with respect to the induced sum and the in- ·[ ] =[ ] duced scalar multiplication ζ α deR ζα deR. For our purposes, we will mainly consider the case r = 1. 504 10 Fundamental Groups, Covering Spaces, and (Co)homology

Although we often use the same notation [α]deR to denote the de Rham coho- r R r C mology class of a closed form α in HdeR(M, ) or HdeR(M, ), there is no real danger of confusion, because according to the following proposition, we may view r R r C HdeR(M, ) as a real vector subspace (and as a subgroup) of HdeR(M, ).

Proposition 10.6.2 For a second countable C∞ surface M and an integer r r r ∈{0, 1, 2}, the map H (M, R) → H (M, C) given by [α] r R → deR deR HdeR(M, ) [α] r C is a well-deﬁned injective real linear map. In particular, identifying HdeR(M, ) r R r C HdeR(M, ) with its image in HdeR(M, ), we get the real direct sum decomposition r C = r R ⊕ r R HdeR(M, ) HdeR(M, ) iHdeR(M, ) given by [ ] =[ ] + [ ] ∀[ ] ∈ r C α deR Re α deR i Im α deR α deR HdeR(M, ) r C r R (thus we may identify HdeR(M, ) with the complexiﬁcation (HdeR(M, ))C).

Proof Let α and β be closed C∞ real differential forms of degree r.Ifα − β = dγ C∞ − [ ] =[ ] for some real differential form γ of degree r 1, then clearly, α deR β deR r C in HdeR(M, ). Thus the map is well deﬁned. It is also easy to check that the map is linear. Finally, if α = dγ for some C∞ complex differential form γ , then we have α = d Re γ (and d Im γ = 0). It follows that the map is injective. [ ] ∈ r C For the direct sum decomposition, observe that if α deR HdeR(M, ), then d Re α = d Im α = 0 and [α]deR =[Re α]deR + i[Im α]deR.Ifρ and τ are closed C∞ [ ] = [ ] ∈ r R ∩ r R real r-forms with ρ deR i τ deR HdeR(M, ) iHdeR(M, ), then we have ρ − iτ = dβ for some C∞ complex (r − 1)-form β (ρ − iτ = 0forr = 0). But then = = − [ ] = [ ] = ρ d(Re β) and τ d( Im β), and hence ρ deR i τ deR 0.

Remark In Sect. 10.7, we will see that for any second countable C∞ surface M, 1 F 1 F HdeR(M, ) is canonically isomorphic to a vector space H (M, ) (the ﬁrst cohomology group) that depends only on the topology of M.Infact,wewillseethat 1 F =∼ 1 F =∼ F HdeR(M, ) H (M, ) Hom(π1(M), )

(where Hom(π1(M), F) is the group of homomorphisms of π1(M) into F, which 1 F has a natural vector space structure). For this reason, we identify HdeR(M, ) with H 1(M, F). Although for the sake of convenience, we will define H 1(M, F) as the space of classes of Cechˇ 1-forms (which are natural topological analogues of closed 1-forms), the resulting space is actually canonically isomorphic to the singular cohomology group with coefficients in F (see the remarks at the end of Sect. 10.7 and Exercise 10.7.13). We will also consider, in Sect. 10.7, the cohomology group H 1(M, A) with coefficients in an arbitrary subring A of C with 1 ∈ A, which is an A-submodule of H 1(M, C). It turns out that under the above 1 isomorphisms, H (M, A) may be identified with the submodule Hom(π1(M), A) of C 1 C Hom(π1(M), ) and with the submodule of HdeR(M, ) consisting of those classes [ ] ∈ 1 C ∈ A θ deR HdeR(M, ) for which γ θ for every loop γ in M. In this section, we focus mainly on cohomology (and homology) over F, and set aside consideration of coefficients in other rings until Sect. 10.7. 10.6 Homology and Cohomology of Second Countable C∞ Surfaces 505

We now turn to homology. For this, we first consider the vector space consisting of all formal finite linear combinations of paths. It will later be convenient (in Sect. 10.7) to have considered formal finite linear combinations over more general rings.

Definition 10.6.3 Let M be a second countable topological surface, and let A be a subring of C that contains 1 (i.e., that contains Z). (a) A 1-chain in M with coefficients in A is a formal finite linear combination

~~m ξ = ai · γi, i=1~~

where ai ∈ A and γi :[0, 1]→M is a (continuous) path for i = 1,...,m.Fol- lowing standard conventions, we identify the above 1-chain ξ with another = n = 1-chain ζ j=1 bj βj in M if for each i 1,...,m, ⎧ ⎨ 1≤j≤n bj if γi ∈{β1,...,βn}, = = al ⎩ βj γi 1≤l≤m 0ifγi ∈{/ β1,...,βn}, γl =γi

~~and for each j = 1,...,n, the analogous statement, with the roles of ξ and ζ switched, holds. A0-chain in M with coefﬁcients in A is a formal ﬁnite linear combination~~

~~m ξ = ai · pi, i=1~~

where ai ∈ A and pi ∈ X for i = 1,...,m (and we have the identifications analogous to those described above for 1-chains). For q = 0, 1, the q-chains in M, together with the obvious addition and scalar multiplication by elements of A,formanA-module (a vector space when A is a field), which we denote by Cq (M, A). (b) The boundary operator ∂ : C1(M, A) → C0(M, A) is the homomorphism (a linear map when A is a field) given by

~~m m ξ = ai · γi → ai · (γi(1) − γi(0)) i=1 i=1~~

~~(which, as one may easily check, is well deﬁned). (c) A 1-cycle in M is a 1-chain ξ for which ∂ξ = 0. The set of 1-cycles in C1(M, A) is a submodule (or subspace), which we denote by~~

~~Z1(M, A) ≡ ker(∂ : C1(M, A) → C0(M, A)). 506 10 Fundamental Groups, Covering Spaces, and (Co)homology~~

Remarks 1. Let P be the set of paths in M, and let G be the A-module consisting of the A-valued functions on P that vanish everywhere except at ﬁnitely many points. Then we may identify G with C1(M, A) under the correspondence f ↔ f(γ)· γ γ ∈P \f −1(0)

(which we set equal to 0 if f ≡ 0). Similarly, we may identify the A-module D consisting of the A-valued functions on M that vanish everywhere except at ﬁnitely many points with C0(M, A) under the correspondence f ↔ f(p)· p p∈M\f −1(0)

(which we set equal to 0 if f ≡ 0). This point of view together with linearity makes it clear that the boundary operator ∂ is well deﬁned by (b). 2. For any pair of subrings A1 and A2 of C with 1 ∈ A1 ⊂ A2, Cr (M, A1) is a submodule of Cr (M, A1) for r = 0, 1 and Z1(M, A1) is a submodule of Z1(M, A2).

Definition 10.6.4 Let M be a second countable C∞ surface. = m · ∈ C C∞ (a) For every 1-chain ξ i=1 ai γi C1(M, ) and every closed 1-form (in fact, for every locally exact 1-form) θ on M,theintegral (or line integral)ofθ along ξ is given by m θ ≡ ai θ ξ i=1 γi (which, as one may easily check, is well defined). (b) We denote by B1(M, F) the subspace of Z1(M, F) consisting of all 1-cycles ξ = C∞ for which ξ θ 0 for every closed 1-form θ on M (note that the definition yields the same space regardless of whether we require the forms θ to be real or complex). (c) The first homology group of M with coefficients in F is the quotient vector space

~~H1(M, F) ≡ Z1(M, F)/B1(M, F).~~

We call the equivalence class in H1(M, F) represented by ξ ∈ Z1(M, F) the [ ] [ ] homology class of ξ, and we denote this class by ξ H1(M,F),by ξ H1 ,orsimply by [ξ]. Two 1-cycles ξ,ζ ∈ Z1(M, F) are said to be homologous if they = C∞ represent the same homology class, that is, if ξ θ ζ θ for every closed 1-form θ on M. ∈ F = C∞ Remarks 1. If ξ C1(M, ) and ξ θ 0 for every closed 1-form θ on M, then ξ is a 1-cycle and hence ξ ∈ B1(M, F) (see Exercise 10.6.1). In other words, the assumption in part (b) that the 1-chain ξ is a 1-cycle is superﬂuous. 10.6 Homology and Cohomology of Second Countable C∞ Surfaces 507

2. It is more standard to define B1(M, F) as the image of a certain boundary operator ∂ on a space of 2-chains (see the remarks at the end of Sect. 10.7),butitturns out that the above definition, which is in a more convenient form for our purposes, is equivalent to the standard definition (see the exercises for Chaps. 5 and 6). Con- sequently, H1(M, F) is equal to the first singular homology group with coefficients in F. 3. We will consider homology and cohomology groups of topological surfaces in Sect. 10.7. In particular, we will see that the first homology group and the first de Rham cohomology group actually depend only on the topology of the surface. 4. Given a subring A of C with 1 ∈ A, the natural analogue of the above definition gives a submodule of Z1(M, A) and a corresponding quotient; but without further hypotheses, the quotient need not coincide with the singular homology group. How- ever, it turns out that the two do coincide for an orientable second countable C∞ surface. With this in mind, for any orientable second countable C∞ surface M and any subring A of C with 1 ∈ A, we define the A-modules A ≡ ∈ A = ∀ ∈ 1 C B1(M, ) ξ Z1(M, ) θ 0 θ ZdeR(M, ) ξ and

H1(M, A) ≡ Z1(M, A)/B1(M, A). On the other hand, in this section, we focus mostly on homology (and cohomology) over F, and set aside consideration of coefﬁcients in other rings until Sect. 10.7.

~~For a second countable C∞ surface M, we have the real direct sum decomposition~~

Z1(M, C) = Z1(M, R) ⊕ iZ1(M, R) = (Z1(M, R))C = + ∈ C = given by ξ Re ξ i Im ξ for each ξ Z1(M, ), where for any q-chain η m ≡ ≡ j=1 aj λj ,Reη j (Re aj )λj and Im η j (Im aj )λj . According to the following proposition, the proof of which is left to the reader (see Exercise 10.6.2), this also holds at the level of homology:

Proposition 10.6.5 For a second countable C∞ surface M, the real linear map R → C [ ] → [ ] H1(M, ) H1(M, ) given by ξ H1(M,R) ξ H1(M,C) is well deﬁned and injective. In particular, we may identify H1(M, R) with its image in H1(M, C), and we have the real direct sum decomposition H1(M, C) = H1(M, R) ⊕ iH1(M, R) given by

~~[ ] =[ ] + [ ] ∀[ ] ∈ C ξ H1(M,C) Re ξ H1(M,R) i Im ξ H1(M,R) ξ H1 H1(M, )~~

~~(thus we may identify H1(M, C) with the complexiﬁcation (H1(M, R))C).~~

~~Lemma 10.6.6 and Proposition 10.6.7 below often allow one to work with loops in place of general paths when considering homology: 508 10 Fundamental Groups, Covering Spaces, and (Co)homology~~

Lemma 10.6.6 Let M be a second countable C∞ surface, let A beasubringofC with 1 ∈ A, and let x0 ∈ M. Then, for every 1-cycle ξ ∈ Z1(M, A), there exists a 1-cycle η ∈ Z1(M, A) such that η is a linear combination over A of loops based = ∈ 1 C at x0 and ξ θ η θ for every θ ZdeR(M, ) (that is, ξ and η are homologous as 1-cycles over C). Moreover, for ξ an integral 1-cycle (that is, for A = Z), one may choose η itself to be a loop based at x0. = m ∈ A Proof For ξ i=1 aiγi Z1(M, ), we prove by induction on m that ξ is homologous to a linear combination of loops. For m = 1, we have 0 = ∂ξ = a1 · (γ1(1) − γ1(0)), and hence a1 = 0orγ1 is a loop. Assume now that the claim holds for any linear combination of strictly fewer than m paths. If aj = 0orγj − = is a loop for some j, then ξ aj γj i=j aiγi is a 1-cycle, which, by the induction hypothesis, is homologous over C to a linear combination of loops. Thus we may assume that aj = 0 and γj is not a loop for each j = 1,...,m. Hence − we may ﬁx distinct indices i ,...,i ∈{1,...,m} and a path λ = γ or γ for 1 k ν iν iν each ν = 1,...,k such that for all μ,ν = 1,...,k,wehaveλμ(0) = λν (0) and λμ(1) = λν(1) if μ = ν, and λμ(1) = λν (0) if and only if ν = μ + 1. Furthermore, we may choose k to be the largest integer for which it is possible to make such a choice. In particular, the point λk(1), which is equal to γik (0) or γik (1), is not an = endpoint for any of the paths γi1 ,...,γik−1 ; and hence, since ∂ξ 0, λk(1) must be ∈{ }\{ } an endpoint for the path γik+1 for some index ik+1 1,...,m i1,...,ik . Setting − λ + equal to γ + if λ (1) = γ + (0), γ if λ (1) = γ + (1), we see that by the k 1 ik 1 k ik 1 ik+1 k ik 1 maximality of k, λk+1(1) = λμ(0) for some unique μ ∈{1,...,k}. Letting λ be the loop λμ ∗ λμ+1 ∗···∗λk+1, and for each ν = μ,...,k+ 1, letting σν be equal to 1 − if λ = γ , −1ifλ = γ , we see that ξ is homologous over C to the 1-cycle ν iν ν iν

~~k+1 ≡ + − + η aiγi (aiν σμaiμ σν )γiν σμaiμ λ. i∈{1,...,m}\{iμ,...,ik+1} ν=μ+1~~

Applying the induction hypothesis to the 1-cycle η − σμaμλ, we get the claim. Moreover, given a loop τ in M, we may fix a path ρ from x0 to τ(0), and τ will − be homologous over C to the loop ρ ∗ τ ∗ ρ with base point at x0. It follows C ≡ n that ξ is actually homologous over to a linear combination η j=1 bj ρj of { }n A = Z = loops ρj j=1 based at x0. Moreover, if , then may assume that bj 1for − each j = 1,...,n(replacing bj with −bj and ρj with ρ whenever bj < 0, we get j = n bj C ∗···∗ η j=1 k=1 ρj ). Hence ξ is then homologous over to the loop ρ1 ρn. ∞ Proposition 10.6.7 Let M be a second countable C surface, and let x0 ∈ M. Then: [ ] =[ ] → [ ] (a) The mapping α x0 α π1(M,x0) α H1(M,F) gives a well-defined group homomorphism π1(M, x0) → H1(M, F). (b) The image of the fundamental group satisfies SpanF im(π1(M, x0) → H1(M, F)) = H1(M, F). 10.6 Homology and Cohomology of Second Countable C∞ Surfaces 509

(c) The image of any set of generators of the group π1(M, x0) spans H1(M, F) as a vector space, and consequently, H1(M, F) has a countable basis. In particular, if π1(M) is ﬁnitely generated (for example, if M is compact), then H1(M, F) is ﬁnite-dimensional . = C∞ ∈ C (d) ξ θ 0 for every exact 1-form θ on M and every 1-cycle ξ Z1(M, ).

Proof Theorem 10.5.6 implies that the mapping in (a) is well deﬁned. Moreover, [ ] [ ] ∈ C∞ given two elements α x0 , β x0 π1(M, x0) and a closed 1-form θ on M, Lemma 10.5.4 implies that θ = θ + θ = θ. α∗β α β 1·α+1·β [ ∗ ] =[ ] +[ ] Thus α β H1(M,F) α H1(M,F) β H1(M,F), and it follows that the mapping is a group homomorphism. Part (b) follows from Lemma 10.6.6, part (c) follows from part (b) and Lemma 10.1.9, and part (d) follows from part (b) and Proposi- tion 10.5.5.

Definition 10.6.8 Let M be a second countable C∞ surface. We call the bilinear 1 1 pairing H (M, F) × H1(M, F) → F (or H1(M, F) × H (M, F) → F) given by deR deR [ ] [ ] = [ ] [ ] ≡ ( θ deR, ξ H1 )deR ( ξ H1 , θ deR)deR θ ξ [ ] ∈ 1 F [ ] ∈ F for every θ deR HdeR(M, ) and ξ H1 H1(M, ) the de Rham pairing on M (Definition 10.6.4 and Proposition 10.6.7 imply that this pairing is well defined).

~~Remark In Sect. 10.7, we will see that we may identify H 1 (M, F) with the dual ∗ deR space (H1(M, F)) under the linear isomorphism [θ]deR → ([θ]deR, ·)deR (see The- orem 10.7.16).~~

For a C∞ mapping : M → N of second countable C∞ surfaces M and N, recall that we have the pullback (linear) mapping ∗ : Er (N, F) → Er (M, F) for each r (see Deﬁnition 9.5.1), and because d ◦ ∗ = ∗ ◦ d (by Proposition 9.5.6), the pullback of a closed (exact, C∞-exact) differential form is also closed (respec- ∞ tively, exact, C -exact). For r = 0, 1, we denote by ∗ : Cr (M, F) → Cr (N, F) the induced pushforward linear map given by m m ξ = aiγi → ai(γi), i=1 i=1 where a1,...,am ∈ F and γ1,...,γm are paths in M if q = 1, points in M if q = 0. As is easy to check, we have ∂ ◦∗ = ∗ ◦∂, and hence ∗(Z1(M, F)) ⊂ Z1(N, F). ∞ It is also easy to check that if θ is a closed C 1-form (or even a locally exact ∗ = ∈ F continuous 1-form) on N, then ξ θ ∗(ξ) θ for every 1-chain ξ C1(M, ). ∗ In particular, ∗B1(M, F) ⊂ B1(N, F).If is a diffeomorphism, then ∗ and − − ∗ − − ∗ are isomorphisms with inverse mappings (∗) 1 = ( 1)∗ and ( ) 1 = ( 1) . 510 10 Fundamental Groups, Covering Spaces, and (Co)homology

~~The above mappings induce linear mappings of the corresponding de Rham cohomology and homology spaces, which have the expected functoriality properties.~~

Proposition 10.6.9 For any C∞ mapping : M → N of second countable C∞ surfaces M and N, we have the following: F → F [ ] → [ ] (a) The mapping H1(M, ) H1(N, ) given by ξ H1(M,F) ∗ξ H1(N,F) is a well-deﬁned linear map, and the diagram H1(M, R) H1(M, C)

~~H1(N, R) H1(N, C)~~

commutes. If is a diffeomorphism, then the map H1(M, F) → H1(N, F) is an [ ] → [ −1 ] isomorphism with inverse mapping ζ H1(N,F) ∗ ζ H1(M,F). (b) If y0 = (x0) for some point x0 ∈ M, then the diagram of induced mappings π1(M, x0) H1(M, F)

~~π1(N, y0) H1(N, F)~~

commutes. Moreover, the image in H1(N, F) of (any generating set for) π1(M, x0) spans the image (vector space) of H1(M, F). 1 F → 1 F [ ] → [ ∗ ] (c) The mapping HdeR(N, ) HdeR(M, ) given by θ deR θ deR is well deﬁned and linear, and the diagram

~~1 R 1 C HdeR(N, ) HdeR(N, )~~

1 R 1 C HdeR(M, ) HdeR(M, ) 1 F → 1 F commutes. Moreover, the map HdeR(N, ) HdeR(M, ) is an isomorphism if is a diffeomorphism. [ ] ∈ 1 F [ ] ∈ F (d) For each θ 1 F H (N, ) and each ξ H1(M,F) H1(M, ), we have HdeR(N, ) deR [ ∗ ] [ ] = [ ] [ ] θ 1 F , ξ H1(M,F) θ 1 F , ∗ξ H1(N,F) . HdeR(M, ) deR HdeR(N, ) deR

~~The proof is left to the reader (see Exercise 10.6.4). Note that the linear map in (c) may be deﬁned in exactly the same way for de Rham cohomology in degrees 0 and 2.~~

Deﬁnition 10.6.10 For : M → N as in Proposition 10.6.9, we call the corresponding mappings of homology and de Rham cohomology the induced pushfor- 10.7 Homology and Cohomology of Second Countable C0 Surfaces 511 ward and pullback mappings, respectively, and we denote these mappings by : F → F ∗ : 1 F → 1 F ∗ H1(M, ) H1(N, ) and HdeR(N, ) HdeR(M, ).

Remark If : M → N and : N → P are C∞ mappings of second countable C∞ ∗ ∗ ∗ surfaces, then ( ◦ )∗ = ∗ ◦ ∗ and ( ◦ ) = ◦ both at the level of q-chains and 1-forms, and at the level of homology and de Rham cohomology (see Exercise 10.6.5).

Exercises for Sect. 10.6 C∞ = 10.6.1 Prove that if ξ is a 1-chain on a surface M and ξ θ 0 for every closed C∞ 1-form θ on M, then ξ is a 1-cycle. 10.6.2 Prove Proposition 10.6.5. ∞ 10.6.3 Let θ be an exact 1-form of class C1 on a second countable C surface M. = ∈ C Prove that ξ θ 0 for every 1-cycle ξ Z1(M, ). 10.6.4 Prove Proposition 10.6.9. 10.6.5 Prove that if : M → N and : N → P are C∞ mappings of second count- ∞ ∗ ∗ ∗ able C surfaces, then ( ◦ )∗ = ∗ ◦ ∗ and ( ◦ ) = ◦ both at the level of q-chains and 1-forms and at the level of homology and de Rham cohomology. [ ] [ ] → [ ∧ ] 10.6.6 Prove that the map ( α deR, β deR) α β deR determines a well-deﬁned 1 C × 1 C → 2 C bilinear pairing HdeR(M, ) HdeR(M, ) HdeR(M, ) for any second countable C∞ manifold M. F = R C R2 \{ } F =∼ F 1 R2 \{ } F =∼ F 10.6.7 Prove that for or , H1( 0 , ) and HdeR( 0 , ) .

~~10.7 Homology and Cohomology of Second Countable C0 Surfaces~~

In Sect. 10.6, homology and cohomology groups were defined using a C∞ structure on a surface. As we will see in this section, these groups are actually, up to isomorphism, topological; that is, one may define analogues on a second countable topological surface that are isomorphic to the corresponding C∞ versions when M is given a C∞ structure. This is particularly appealing because, as is shown in Chap. 6, every second countable topological surface admits a C∞ structure. The fact that these groups do not depend on the choice of the C∞ structure is not crucial for most of our purposes, but it does allow one to see that certain complex analytic objects associated to a Riemann surface are actually topological in nature. For this reason, a proof is outlined in this section. We also consider homology and cohomology with coefficients in an arbitrary subring A of C containing Z. When considering homology with coefficients in A, in order to guarantee that the natural generalization of Definition 10.6.4 (with A in place of F) yields a group that coincides with the first singular homology group (see the remarks at the end of this section and the exercises for Chaps. 5 and 6), we will 512 10 Fundamental Groups, Covering Spaces, and (Co)homology restrict our attention to second countable topological surfaces that admit orientable smooth structures (in fact, as is shown in Chap. 6, this is actually a topological condition). There are many different standard ways of defining homology and cohomology on topological spaces, and with enough restrictions on the spaces, the resulting groups are isomorphic. Here, we consider a variant of Cechˇ cohomology (see, for example, [Wa]) as considered in, for example, [Fa] (for an equivalent approach using potentials on covering spaces, see [For]). The idea is to produce a generalization of the notion of a locally exact 1-form, and to then proceed as before. The discussion in Sect. 10.5 of line integrals along continuous paths suggests that one may do this by considering the local potentials themselves, without mentioning the associated differential 1-form one gets by applying d. Throughout this section, F denotes the field R or C, and A denotes a subring of C that contains 1 (i.e., that contains Z).

Deﬁnition 10.7.1 Let M be a second countable topological surface. (a) We denote by P 1(M, A) (P stands for “potentials”) the collection of families of pairs {(fi,Ui)}i∈I for which {Ui}i∈I is an open covering of M; for each i ∈ I , fi : Ui → C is a continuous function on Ui , with fi real-valued if A ⊂ R; and the function fi − fj is locally constant on the overlap Ui ∩ Uj with values in A for each pair of indices i, j ∈ I . Given two elements F ={(fi,Ui)}i∈I , 1 G ={(gj ,Vj )}j∈J ∈ P (M, A) and a ring element ζ ∈ A, we denote by ζF and F G the elements of P 1(M, A) given by

~~ζF ≡{(ζfi,Ui)}i∈I~~

~~and F G ≡ (f + g ,U ∩ V ) . i Ui ∩Vj j Ui ∩Vj i j (i,j)∈I×J (b) Let ∼ be the equivalence relation in P 1(M, C) given by~~

~~F ={(fi,Ui)}i∈I ∼ G ={(gj ,Vj )}j∈J~~

if and only if fi − gj is locally constant on Ui ∩ Vj for all indices i ∈ I and j ∈ J ; that is, the union of the two families is an element of P 1(M, C) (note that for F,G ∈ P 1(M, A), we do not require that each of the complex-valued functions fi − gj take values in A, just that they be locally constant). (c) We denote the corresponding quotient mapping and quotient space by : 1 C → 1 C ≡ 1 C ∼ Z1(M,C) P (M, ) Z (M, ) P (M, )/ , we let 1 A ≡ 1 A ∼= 1 A ⊂ 1 C Z (M, ) P (M, )/ Z1(M,C)(P (M, )) Z (M, ) and ≡ : 1 A → 1 A Z1(M,A) Z1(M,C) P 1(M,A) P (M, ) Z (M, ), 10.7 Homology and Cohomology of Second Countable C0 Surfaces 513

and we call any element of Z1(M, A) a Cechˇ 1-form (or a continuous closed 1-form)onM over A (or an integral Cechˇ 1-form for A = Z or a real Cechˇ 1-form for A = R or a complex Cechˇ 1-form for A = C). We also equip 1 A A { } Z (M, ) with the -module structure with zero element Z1(M,A)( (0,M) ) and multiplication (by elements of A) and addition given by

~~· ≡ ζ Z1(M,A)(F ) Z1(M,A)(ζ F )~~

and + ≡ Z1(M,A)(F ) Z1(M,A)(G) Z1(M,A)(F G) for all F,G ∈ P 1(M, A) and ζ ∈ A (the veriﬁcation that the above give a well-deﬁned module structure is left to the reader in Exercise 10.7.1). For ={ } ∈ 1 C = F (fi,Ui) i∈I P (M, ) and θ Z1(M,C)(F ),thesupport of θ is the closed subset supp θ of M with complement equal to the union of all open ⊂ ∈ sets U M for which fi U∩Ui is locally constant for each i I (it is easy to see that this set is independent of the choice of the representative F ). Equiv- alently, M \ supp θ is the largest open set U0 for which there is a representative 1 1 1 F ={(fi,Ui)}i∈I ∈ P (M, C) (we may take F ∈ P (M, R) if θ ∈ Z (M, R)) with 0 ∈ I and f0 ≡ 0. (d) For any Cechˇ 1-form θ on M and any path γ :[a,b]→M,theintegral of θ along γ (or the line integral of θ along γ )isgivenby m ≡ − θ fiν (γ (tν)) fiν (γ (tν−1)) γ ν=1

~~1 for any representative F ={(fi,Ui)}i∈I ∈ P (M, C) of θ and any partition P : = ··· = [ ] ⊂ ∈ a t0 <~~

1 Remarks 1. One may view the representing functions {(fi,Ui)}i∈I ∈ P (M, A) for a Cechˇ 1-form θ as generalized local potentials. The requirement that fi be real- valued if A ⊂ R is convenient for our purposes, but not absolutely necessary. Of course, it would be too restrictive to require that fi be A-valued (for example, any integer or rational-valued continuous function is locally constant). 514 10 Fundamental Groups, Covering Spaces, and (Co)homology

2. For a Cechˇ 1-form θ over A, one may not be able to choose a representa- 1 tive {(fi,Ui )}i∈I ∈ P (M, A) with fi ≡ 0 and Ui = M \ supp θ for some i ∈ I . For example, choosing a real-valued continuous function f on R2 with f ≡ 0 on (0; 1) and f ≡ 1/2onR2 \ (0; 2), we get the Cechˇ 1-form θ represented 2 1 2 1 by {(f, R )}∈P (R , Z). If there is a representative {(fi,Ui)}i∈I ∈ P (M, Z) as above with fi ≡ 0 and Ui = M \ supp θ for some i ∈ I , then according to Lemma 10.7.2 below, there is a constant ζ ∈ C such that f − (fi − ζ) is integer- valued. But then, evaluating on R2 \ (0; 2) and (0; 1), we see that both (1/2) − ζ and ζ must be integers, which is impossible. 1 3. By deﬁnition, two elements {(fi,Ui)}i∈I , {(gj ,Vj )}j∈J ∈ P (M, A) represent the same Cechˇ 1-form θ over A if fi − gj is locally constant as a C-valued function for all i ∈ I and j ∈ J . Since fi and gj roughly correspond to local potentials for θ (in a generalized sense), it is natural that the addition of constants not affect θ,even constants not in A. Thus, although it may seem more natural to require that fi − gj take values in A, such a requirement, which would introduce some complications (for example, the exact 1-forms represented by f ≡ 1/2 and g ≡ 0 would differ as integral Cechˇ 1-forms, but not as real Cechˇ 1-forms), is not necessary. Moreover, we have the following fact:

Lemma 10.7.2 Given a second countable topological surface M and two repre- 1 sentatives F ={(fi,Ui)}i∈I ,G ={(gj ,Vj )}j∈J ∈ P (M, A) for a Cechˇ 1-form = = A ∈ C ∈ R θ Z1(M,A)(F ) Z1(M,A)(G) over , there exists a constant ζ (with ζ 1 if A ⊂ R) such that the representative H ≡{(gj − ζ,Vj )}j∈J of θ is in P (M, A) and the function fi − (gj − ζ) is locally constant with values in A on Ui ∩ Vj for all i ∈ I and j ∈ J .

∈ ∈ ∈ ∈ ∩ Proof Fix a point x0 M and indices i0 I and j0 J with x0 Ui0 Vj0 , and set ≡ − ∈ − − ∈ A ζ gj0 (x0) fi0 (x0).ThesetQ of points x M for which fi(x) (gj (x) ζ) for some (and therefore for every) choice of i ∈ I and j ∈ J with x ∈ Ui ∩ Vj is then nonempty. Moreover, given a point x ∈ M and indices i ∈ I and j ∈ J with x ∈ Ui ∩ Vj , the function fi − (gj − ζ) will be constant on some neighborhood W of x in Ui ∩ Vj . It follows that Q is both open and closed in M, and hence Q = M.

~~Analogues of the arguments in Sect. 10.5 give the following:~~

~~Theorem 10.7.3 For any Cechˇ 1-form θ on a second countable topological surface M, we have the following:~~

~~(a) For any path γ :[a,b]→M: =− (i) We have γ − θ γ θ. (ii) If a~~

~~representative of θ such that for each ν = 1,...,m,~~

{ | ≤ ≤ ≥ ≥ }⊂ γ(t) sν−1 t sν or sν−1 t sν Uiν m for some iν ∈ I , then θ = = [fi (γ (sν)) − fi (γ (sν−1))]. γ J ν 1 ν ν (iv) Invariance under continuous reparametrization. For any continuous mapping ϕ :[r, s]→[ a,b] with c ≡ ϕ(r) ≤ d ≡ ϕ(s) and for J ≡{t ∈ R | c ≤ t ≤ d}, we have θ = θ. γ ◦ϕ γ J (b) The following are equivalent: (i) The Cechˇ 1-form θ is exact. = (ii) γ θ 0 for every loop γ in M. (iii) The line integrals are independent of path; that is, θ = θ for every γ0 γ1 pair of paths γ0 and γ1 with the same endpoints. (c) If γ and γ are two path homotopic paths in M, then θ = θ. 0 1 γ0 γ1 (d) If M is simply connected, then θ is exact. 1 (e) Assuming that θ ∈ Z (M,R) if A ⊂ R, given a point x0 ∈ M, we have θ ∈ Z1(M, A) if and only if θ ∈ A for every loop γ based at x Moreover if γ 0. , ∈ 1 A [ ]→ θ Z (M, ), then the mapping γ γ θ determines a well-deﬁned group homomorphism π1(X, x0) → (A, +). (f) If θ ∈ Z1(M, A) and B is a basis for the topology on M, then θ has a represen- 1 tative F ={(fi,Ui)}i∈I ∈ P (M, A) such that {Ui}i∈I is a countable locally ﬁnite family of elements of B (which covers M).

Proof We prove part (e) and leave the proofs of the remaining parts to the reader (see Exercise 10.7.3). For this, let us ﬁx a point x0 ∈ M and an element F = 1 1 {(fi,Ui)}i∈I ∈ P (M, C) representing θ, with F ∈ P (M, R) if A ⊂ R.Ifwe 1 may choose F to be in P (M, A), then given a loop γ based at x0 and a parti- P : = ··· = [ ] ⊂ ∈ tion a t0 <

For the proof of the converse, for each i ∈ I , let us assume that Ui is nonempty and connected (as we may, for example, by part (f)), let us ﬁx a point pi ∈ Ui and a path γ from x to p , and let us set g ≡ f −f (p )+ θ. Then F ∼ G ≡{(g ,U )} ∈ . i 0 i i i i i γi i i i I Moreover, given indices i, j ∈ I and a point x ∈ Ui ∩ Uj , we may choose a path α in Ui from pi to p and β in Uj from pj to p. We then have

~~gi(x) − gj (x) = fi(x) − fi(pi) + θ − fj (x) − fj (pj ) + θ = θ, γi γj η 516 10 Fundamental Groups, Covering Spaces, and (Co)homology~~

≡ ∗ ∗ − ∗ − where η γi α β γj . Hence, if the integral of θ along each loop based 1 1 at x0 lies in A, then G ∈ P (M, A) and therefore θ ∈ Z (M, A). That the mapping [ ]→ → A + γ γ θ determines a well-deﬁned group homomorphism π1(X, x0) ( , ) then follows from parts (a) and (c).

~~On a second countable C∞ surface, every locally exact 1-form is a Cechˇ 1-form. In fact, we have the following:~~

Lemma 10.7.4 Let M be a second countable C∞ surface. To any locally exact 1-form θ over F (i.e., θ is a real 1-form if F = R) on M, we may associate a unique ˇ ˆ = ∈ 1 F ≡{ } ∈ 1 F Cech 1-form θ Z1(M,F)(F ) Z (M, ), where F (fi,Ui) i∈I P (M, ) for an arbitrary choice of an open covering {Ui}i∈I of M and functions {fi}i∈I with ∈ C1 F = ∈ fi (Ui, ) and dfi θ Ui for each i I (i.e., θ is independent of the choice of the collection F of local potentials for θ). Moreover, we have the following: → ˆ 1 F (i) The map θ θ determines a linear isomorphism of ZdeR(M, ) onto the vector subspace of Z1(M, F) consisting of all Cechˇ 1-forms represented by a fam- ∞ ily of F-valued local C functions . = ˆ (ii) For every path γ in M, we have γ θ γ θ for every choice of θ. (iii) The given 1-form θ is exact (as a 1-form) if and only if θˆ is exact (as a Cechˇ 1-form), and if this is the case, then for some F-valued function f ∈ C1(M), we = ˆ = { } have θ df and θ Z1(M,F)( (f, M) ). (iv) The given 1-form θ has compact support if and only if θˆ has compact support.

The proof is left to the reader (see Exercise 10.7.4). On a second countable C∞ surface, we will identify any locally exact 1-form θ with its corresponding Cechˇ 1-form, and we will give the Cechˇ 1-form the same name θ (instead of θˆ).

~~Definition 10.7.5 For a second countable topological surface M,thefirst cohomology group with coefficients in A is the quotient A-module~~

~~H 1(M, A) ≡ Z1(M, A)/B1(M, A) = Z1(M, A)/(B1(M, C) ∩ Z1(M, A)).~~

For each Cechˇ 1-form θ ∈ Z1(M, A), we call the corresponding equivalence class 1 A [ ] in H (M, ) the cohomology class of θ, and we denote this class by θ H 1(M,A), [ ] [ ] ˇ by θ H 1 ,orsimplyby θ . We also say that two Cech 1-forms are cohomologous if they represent the same cohomology class (i.e., if their difference is exact).

ˇ = ={ } ∈ 1 C Given a Cech 1-form θ Z1(M,C)(F ) with F (fi,Ui) i∈I P (M, ) on a second countable topological surface M, we get well-deﬁned Cechˇ 1-forms Re θ and 1 1 Im θ in Z (M, R) represented by Re F ≡{(Re fi,Ui)}i∈I ∈ P (M, R) and Im F ≡ 1 {(Im fi,Ui)}i∈I ∈ P (M, R), respectively, and we have θ = Re θ +i Im θ. It follows that we have a real direct sum decomposition Z1(M, C) = Z1(M, R) ⊕ iZ1(M, R), and that we may identify Z1(M, C) with the complexiﬁcation (Z1(M, R))C (see Exercise 10.7.5). This also holds at the level of cohomology: 10.7 Homology and Cohomology of Second Countable C0 Surfaces 517

Proposition 10.7.6 For a second countable topological surface M, the mapping 1 A → 1 C [ ] → [ ] H (M, ) H (M, ) given by θ H 1(M,A) θ H 1(M,C) is a well-deﬁned injective A-module homomorphism, and identifying H 1(M, A) with its image in H 1(M, C), we have 1 A = [ ] ∈ 1 C ∈ A H (M, ) θ H 1(M,C) H (M, ) θ for every loop γ in M . γ Moreover, we have the real direct sum decomposition

H 1(M, C) = H 1(M, R) ⊕ iH1(M, R) [ ] =[ ] + [ ] [ ] ∈ given by θ H 1(M,C) Re θ H 1(M,R) i Im θ H 1(M,R) for each θ H 1(M,C) H 1(M, C) (thus we may identify H 1(M, C) with the complexiﬁcation (H 1(M, R))C).

Proof The claims regarding the map H 1(M, A)→ H 1(M, C) follow from the deﬁnitions and Theorem 10.7.3 (observe that if A ⊂ R, [θ]∈H 1(M, A), and ∈ A = γ θ for every loop γ in M, then γ Im θ 0 for every loop γ , and hence [θ] has a real representative), so it remains to show that each cohomol- 1 R ∩ 1 R =[ ] = ogy class ξ in H (M, ) iH (M, ) is trivial. We have ξ θ H 1(M,R) [ ] = = i η H 1(M,R), where θ Z1(M,R)(F ) and η Z1(M,R)(G) for some choice of 1 F ={(fj ,Uj )}j∈J ,G ={(gn,Vn)}n∈N ∈ P (M, R) (in particular, the functions {fj } and {gn} are real-valued). Consequently, there exists a continuous function u on M such that for each j ∈ J and n ∈ N, the function fj − ign − u is locally constant on Uj ∩Vn. It follows that the functions fj −Re u and gn −Im u are locally constant on Uj and Vn, respectively, and hence that ξ = 0.

For a second countable surface, although the first cohomology with coefficients in F is clearly a topological object, for any C∞ structure on the surface, it is canonically isomorphic to the first de Rham cohomology.

Theorem 10.7.7 (de Rham) Let M be a second countable C∞ surface. Then, for every Cechˇ 1-form α over F on M, there exists a closed C∞ 1-form β over F on M such that α − β is exact. Consequently, we have a well-deﬁned (surjective) linear ∼ 1 F →= 1 F [ ] → [ ] isomorphism HdeR(M, ) H (M, ) given by β deR β H 1 , and under this isomorphism, assuming that A ⊂ F, we get an A-module isomorphism =∼ [ ] ∈ 1 F ∈ A −→ 1 A θ deR HdeR(M, ) θ for every loop γ in M H (M, ). γ

~~Moreover, if α is a Cechˇ 1-form with compact support, then α − β is exact for some closed C∞ 1-form β over F with compact support.~~

Proof Suppose α is a Cechˇ 1-form over F on M. Applying part (f) of Theo- rem 10.7.3, we get a representing family F ={(fi,Ui)}i∈I for α such that Ui M 518 10 Fundamental Groups, Covering Spaces, and (Co)homology for each i ∈ I and {Ui}i∈I is a countable locally ﬁnite open covering of M.Wemay also choose F so that fi ≡ 0 whenever i ∈ I and Ui ∩ (supp α) =∅. Applying The- C∞ { } ⊂ orem 9.3.7, we get a partition of unity λi i∈I onM such that supp λi Ui for ∈ ≡ : → F each i I . Thus we get a continuous function f i∈I λifi M . For each i ∈ I , we have, on Ui , ∞ fi − f = λj (fi − fj ) ∈ C (Ui), j∈I since fi − fj is locally constant on Ui ∩ Uj for each j ∈ J . Therefore, by ˇ ≡ − { } Lemma 10.7.4,theCech 1-form β α Z1(M,F)( (f, X) ), which is cohomologous to α, is actually a closed C∞ 1-form on M (with compact support if α has compact support). Lemma 10.7.4 and Theorem 10.7.3 now give the remaining claims.

For a group G and an Abelian group A, the set of group homomorphisms from G to A, together with the natural zero element and addition, is an Abelian group denoted by Hom(G, A) (similarly, for modules M and N over a ring R,theset of module homomorphisms from M to N is an R-module, which is denoted by Hom(M, N )). In fact, Hom(G, A) is an A-module. The point of view provided by the Cechˇ 1-forms leads to the following characterization of the ﬁrst cohomology group:

Theorem 10.7.8 (de Rham) Let M be a second countable topological surface, 1 and let x0 ∈ M. Then the mapping H (M, A) → Hom(π1(M, x0), A) given by [ ] → :[ ] → θ H 1 ρ, where ρ γ x0 γ θ, is a well-deﬁned module isomorphism. Con- sequently, for M a second countable smooth surface, the analogous mapping [ ] → 1 F → F θ deR ρ yields a vector space isomorphism HdeR(M, ) Hom(π1(M, x0), ).

Proof Theorem 10.7.3 implies that the mapping is a well-defined injective module homomorphism, so it remains to verify surjectivity. For this, let us fix a locally finite covering {Ui}i∈I of M by simply connected open sets and a continuous partition of unity {ηi}i∈I with supp ηi ⊂ Ui for each i ∈ I , and for each index i ∈ I , let us fix a point pi ∈ Ui and a path γi from x0 to pi . Given a group homomorphism ρ : π1(M, x0) → A, let us set, for each pair of indices i, j ∈ I and each point p ∈ Ui ∩ Uj , ≡ [ ∗ ∗ − ∗ −] ∈ A fij (p) ρ( γj β α γi x0 ) , where α is an arbitrary path in Ui from pi to p, and β is an arbitrary path in Uj from pj to p. The functions {fij } are well defined because Ui is simply connected for each i ∈ I , and since ρ is a group homomorphism, they satisfy the cocycle relation

~~fij + fjk = fik on Ui ∩ Uj ∩ Uk ∀i, j, k ∈ I.~~

Moreover, as is easy to verify, the functions are locally constant. For each j ∈ I , ≡ we get a continuous function fj i∈I ηifij on Uj (here, we have extended the 10.7 Homology and Cohomology of Second Countable C0 Surfaces 519 function ηifij by 0 from a function on Ui ∩ Uj to a continuous function on Uj for each i ∈ I ), and we have fj − fi = ηk(fkj − fki) = ηkfij = fij : Ui ∩ Uj → A ∀i, j ∈ I. k∈I k∈I

Moreover, fi is real-valued for each i ∈ I if A ⊂ R. Thus the family F ≡ 1 {(fi,Ui)}i∈I is an element of P (M, A), and hence F determines a Cechˇ 1-form ≡ θ Z1(M,A)(F ). Givenaloopλ based at x0, we may ﬁx a partition 0 = t0

− − − − ρ([λ]) = ρ([β ∗ γ ]) + ρ([γ ∗ α ∗ β ∗ γ ]) 1 i1 i1 1 2 i2 − − +···+ρ([γ ∗ α − ∗ β ∗ γ ]) + ρ([γ ∗ α ]) im−1 m 1 m im im m − − − − = ρ([γ ∗ α ∗ β ∗ γ ]) + ρ([γ ∗ α ∗ β ∗ γ ]) im m 1 i1 i1 1 2 i2 − − +···+ρ([γ ∗ α − ∗ β ∗ γ ]) im−1 m 1 m im = − + − (fim (λ(tm)) fi1 (λ(t0))) (fi1 (λ(t1)) fi2 (λ(t1))) +···+ − (fim−1 (λ(tm−1)) fim (λ(tm−1))) m = − = (fiν (λ(tν)) fiν (λ(tν−1))) θ. ν=1 λ [ ] Thus ρ is the image of θ H 1 , and we have surjectivity. The claim concerning a second countable C∞ surface follows from the above together with Theorem 10.7.7.

The definitions of the homology groups with coefficients in F for a second countable topological surface are analogous to the definitions in the C∞ case provided in Sect. 10.6. However, in this section, we will also consider homology over the ring A, and when doing so, we will add the condition that the surface admit an orientable smooth surface structure.

Definition 10.7.9 (Cf. Definition 10.6.4)LetM be a second countable topological surface. 520 10 Fundamental Groups, Covering Spaces, and (Co)homology = m · ∈ C (a) For every 1-chain ξ i=1 ai γi C1(M, ) (see Definition 10.6.3) and every Cechˇ 1-form θ on M,theintegral (or line integral)ofθ along ξ is given by m θ ≡ ai θ ξ i=1 γi (which, as one may easily check, is well defined). (b) We denote by B1(M, F) the subspace of Z1(M, F) consisting of all 1-cycles = ˇ ξ for which ξ θ 0 for every Cech 1-form θ on M (note that the definition yields the same space regardless of whether we require the Cechˇ 1-forms θ to be real or complex). The first homology group of M with coefficients in F is the quotient vector space

~~H1(M, F) ≡ Z1(M, F)/B1(M, F).~~

We call the equivalence class in H1(M, F) represented by ξ ∈ Z1(M, F) the ho- [ ] [ ] mology class of ξ, and we denote this class by ξ H1(M,F),by ξ H1 ,orsimply by [ξ]. Two 1-cycles ξ,ζ ∈ Z1(M, F) are said to be homologous if they repre- = ˇ sent the same homology class; that is, if ξ θ ζ θ for every Cech 1-form θ on M. (c) Assuming that M admits an orientable 2-dimensional C∞ structure, we denote by B1(M, A) the A-submodule of Z1(M, A) consisting of all 1-cycles ξ for = ˇ which ξ θ 0 for every Cech 1-form θ on M (again, the definition yields the same module regardless of whether we require the Cechˇ 1-forms θ to be real or complex). The first homology group of M with coefficients in A is the quotient A-module

~~H1(M, A) ≡ Z1(M, A)/B1(M, A).~~

The equivalence class in H1(M, A) represented by ξ ∈ Z1(M, A) is the homol- [ ] [ ] [ ] ogy class of ξ, denoted by ξ H1(M,A),by ξ H1 ,orsimplyby ξ . Two 1-cycles ξ,ζ ∈ Z1(M, A) are homologous if they represent the same homology class.

∞ Remark Although we deﬁned H1(M, A) only for M admitting an orientable C surface structure, this module is clearly a topological object (in particular, for most purposes, we may as well consider M to be an orientable C∞ surface in this case). The extra condition is included in order to make H1(M, A) coincide with the ﬁrst singular homology group (see the remarks at the end of this section and the exercises for Chaps. 5 and 6). In Chap. 6, it is proved that every second countable topological surface admits a smooth structure, and that orientability is actually a topological condition.

~~For a second countable topological surface M and for subrings A1 and A2 of C with Z ⊂ A1 ⊂ A2,wehaveZ1(M, A1) ⊂ Z1(M, A2). We also have the real direct sum decomposition~~

~~Z1(M, C) = Z1(M, R) ⊕ iZ1(M, R) = (Z1(M, R))C 10.7 Homology and Cohomology of Second Countable C0 Surfaces 521~~

= + ∈ C = given by ξ Re ξ i Im ξ for each ξ Z1(M, ), where for any q-chain η m ≡ ≡ j=1 aj λj ,Reη j (Re aj )λj and Im η j (Im aj )λj . According to the following fact, the proof of which is left to the reader (see Exercise 10.7.6), these inclusions also hold at the level of homology:

Proposition 10.7.10 (Cf. Proposition 10.6.5) For a second countable topological R → C [ ] → surface M, the real linear map H1(M, ) H1(M, ) given by ξ H1(M,R) [ ] R ξ H1(M,C) is well deﬁned and injective; and identifying H1(M, ) with its image in H1(M, C), we have the real direct sum decomposition H1(M, C) = H1(M, R) ⊕ iH1(M, R) given by [ ] =[ ] + [ ] ∀[ ] ∈ C ξ H1(M,C) Re ξ H1(M,R) i Im ξ H1(M,R) ξ H1 H1(M, )

(thus we may identify H1(M, C) with the complexiﬁcation (H1(M, R))C). Further- more, for M an orientable second countable smooth surface, the A-module homo- A → C [ ] → [ ] morphism H1(M, ) H1(M, ) given by ξ H1(M,A) ξ H1(M,C) is well de- ﬁned and injective. In particular, H1(M, A) is torsion-free, and we may identify H1(M, A) with its image in H1(M, C).

Recall that an element g of a group is called a torsion element if g is of ﬁnite order (i.e., gm = e for some positive integer m). A group in which only the identity is a torsion element is said to be torsion-free. Also as in the C∞ case, one may often reduce questions regarding homology to questions regarding integration along loops by applying Lemma 10.7.11 and Proposition 10.7.12 below, the proofs of which are left to the reader (see Exer- cises 10.7.7 and 10.7.8):

Lemma 10.7.11 (Cf. Lemma 10.6.6) Let M be a second countable topological surface, and let x0 ∈ M. Then, for every 1-cycle ξ ∈ Z1(M, A), there exists a 1-cycle η ∈ Z1(M, A) such that η is a linear combination over A of loops based at x0 and = ∈ 1 C ξ θ η θ for every θ Z (M, ) (that is, ξ and η are homologous as 1-cycles C ∈ A ∈ 1 A over ). In particular, ξ θ for every θ Z (M, ). Moreover, for ξ an integral 1-cycle (that is, for A = Z), one may choose η itself to be a loop based at x0.

Proposition 10.7.12 (Cf. Proposition 10.6.7) Let M be a second countable topological surface, and let x0 ∈ M. Then: [ ] =[ ] → [ ] (a) The mapping α x0 α π1(M,x0) α H1(M,F) gives a well-defined group homomorphism π1(M, x0) → H1(M, F). For M an orientable second countable C∞ [ ] → [ ] surface, the mapping α x0 α H1(M,Z) gives a well-defined surjective group homomorphism π1(M, x0) → H1(M, Z), and in particular, the image of any set of generators of the group π1(M, x0) generates H1(M, Z) as a group (equivalently, as a Z-module) and H1(M, Z) is countable. (b) The image of the fundamental group satisfies SpanF im(π1(M, x0) → H1(M, F)) = H1(M, F), 522 10 Fundamental Groups, Covering Spaces, and (Co)homology

and consequently, the image of any set of generators of the group π1(M, x0) spans H1(M, F) as a vector space and H1(M, F) has a countable basis. In particular, if π1(M) is ﬁnitely generated (for example, if M is compact), then ∞ H1(M, F) is ﬁnite-dimensional. For M an orientable second countable C surface, we have SpanA im(π1(M, x0) → H1(M, A)) = H1(M, A),

and consequently, the image of any set of generators of the group π1(M, x0) generates H1(M, A) as an A-module and H1(M, A) is countably generated. In particular , if π1(M) is ﬁnitely generated, then H1(M, A) is ﬁnitely generated. = ˇ ∈ C (c) ξ θ 0 for every exact Cech 1-form θ on M and every 1-cycle ξ Z1(M, ).

Definition 10.7.13 Let M be a second countable topological surface. We call the 1 1 bilinear pairing H (M, F)×H1(M, F) → F (or H1(M, F)×H (M, F) → F)given by [ ] [ ] = [ ] [ ] ≡ ( θ H 1 , ξ H1 )deR ( ξ H1 , θ H 1 )deR θ ξ [ ] ∈ 1 F [ ] ∈ F F for every θ H 1 H (M, ) and ξ H1 H1(M, ) the de Rham pairing over on M (Definition 10.7.9 and Proposition 10.7.12 imply that this pairing is well defined). If M also admits an orientable 2-dimensional C∞ structure, then we call the 1 bilinear pairing of A-modules H1(M, A) × H (M, A) → A given by the restriction of the de Rham pairing over C the de Rham pairing over A.

Remark Theorem 10.7.7 and Proposition 10.7.12 imply that although, in the C∞ ∞ case, H1(M, F) was defined (in Definition 10.6.4)usingtheC structure, the topological definition (Definition 10.7.9) actually yields the same vector space (the spaces are literally the same, not just canonically isomorphic). Moreover, under the 1 F =∼ 1 F identification given by the isomorphism HdeR(M, ) H (M, ), the versions of the de Rham pairing (·, ·)deR in Definitions 10.6.8 and 10.7.13 coincide.

~~A continuous mapping : M → N of second countable topological surfaces M and N induces a pullback map ∗ : P 1(N, A) → P 1(M, A) given by~~

~~−1 {(fi,Ui)}i∈I → {(fi ◦ , (Ui))}i∈I .~~

This map determines a well-deﬁned pullback map ∗ : Z1(N, A) → Z1(M, A) → ∗ (which we give the same name) given by Z1(N,A)(F ) Z1(M,A)( F), and this map is an A-module homomorphism. Moreover, ∗B1(N, A) ⊂ B1(M, A). For r = 0, 1, we denote by ∗ : Cr (M, A) → Cr (N, A) the induced pushforward module homomorphism given by

m m ξ = aiγi → ai(γi), i=1 i=1 10.7 Homology and Cohomology of Second Countable C0 Surfaces 523 where a1,...,am ∈ A and γ1,...,γm are paths in M if q = 1, points in M if q = 0. As is easy to check, we have ∂ ◦ ∗ = ∗ ◦ ∂, and hence ∗(Z1(M, A)) ⊂ A ˇ ∗ = Z 1(N, ). It is also easy to check that if θ is a Cech 1-form on N, then ξ θ ∈ A ∗ A ⊂ A ∗(ξ) θ for every 1-chain ξ C1(M, ). In particular, B1(M, ) B1(N, ), assuming that M and N admit orientable C∞ surface structures unless A = R or C. ∗ − − If is a homeomorphism, then ∗ and are isomorphisms and (∗) 1 = ( 1)∗ and (∗)−1 = (−1)∗. The above mappings induce module homomorphisms of the corresponding cohomology and homology spaces that have the expected functoriality properties.

Proposition 10.7.14 For any continuous mapping : M → N of second countable topological surfaces M and N, we have the following: A → A [ ] → [ ] (a) The mapping H1(M, ) H1(N, ) given by ξ H1(M,A) ∗ξ H1(N,A) is a well-deﬁned module homomorphism, and the diagram H1(M, A) H1(M, C)

H1(N, A) H1(N, C) commutes, provided M and N are orientable C∞ surfaces if A is neither R nor C. If is a homeomorphism, then the map H1(M, A) → H1(N, A) is [ ] → [ −1 ] an isomorphism with inverse mapping ζ H1(N,A) ∗ ζ H1(M,A). (b) If y0 = (x0) for some point x0 ∈ M, then the diagram of induced mappings π1(M, x0) H1(M, A)

π1(N, y0) H1(N, A) commutes, provided M and N are orientable C∞ surfaces if A is neither R nor C. Moreover, the image in H1(N, A) of (any generating set for) π1(M, x0) generates the image of H1(M, A) as an A-module. 1 A → 1 A [ ] → [ ∗ ] (c) The mapping H (N, ) H (M, ) given by θ H 1 θ H 1 is a well- deﬁned module homomorphism, and the diagram H 1(N, A) H 1(N, C)

~~H 1(M, A) H 1(M, C) commutes. Moreover, the map H 1(N, A) → H 1(M, A) is an isomorphism if is a homeomorphism. 524 10 Fundamental Groups, Covering Spaces, and (Co)homology~~

[ ] ∈ 1 F [ ] ∈ F (d) For each θ H 1(N,F) H (N, ) and each ξ H1(M,F) H1(M, ), we have ∗ [ ] [ ] F = [ ] [ ∗ ] F θ H 1(M,F), ξ H1(M, ) deR θ H 1(N,F), ξ H1(N, ) deR. The proof is left to the reader (see Exercise 10.7.9).

Deﬁnition 10.7.15 For : M → N as in Proposition 10.7.14, we call the corresponding mappings of homology and cohomology the induced pushforward and pullback mappings, respectively, and we denote these module homomorphisms by

~~∗ 1 1 ∗ : H1(M, A) → H1(N, A) and : H (N, A) → H (M, A).~~

Remark If : M → N and : N → P are continuous mappings of second count- ∗ ∗ ∗ able topological surfaces, then ( ◦ )∗ = ∗ ◦ ∗ and ( ◦ ) = ◦ both at the level of q-chains and Cechˇ 1-forms and at the level of homology and cohomology (see Exercise 10.7.10).

~~We may identify the cohomology space with the dual space of the homology space in a natural way:~~

Theorem 10.7.16 (de Rham) For any second countable topological surface M: [ ] → [ ] · (a) The mapping θ H 1(M,F) ( θ H 1(M,F), )deR gives a linear isomorphism ∼ 1 = ∗ H (M, F) −→ (H1(M, F)) = Hom(H1(M, F), F). [ ] → [ ] · (b) The mapping ξ H1(M,F) ( ξ H1(M,F), )deR gives an injective linear map 1 ∗ 1 H1(M, F)→ (H (M, F)) = Hom(H (M, F), F),

~~and this map is surjective if and only if H1(M, F) is ﬁnite-dimensional. If H1(M, F) is inﬁnite-dimensional, then H1(M, F) has a countable basis, but H 1(M, F) does not.~~

Proof It is easy to see from the deﬁnitions that the mappings in (a) and (b) are linear maps. The injectivity of the mapping in (a) follows from Theorem 10.7.3, and the injectivity of the mapping in (b) follows from the deﬁnition of H1(M, F). For the proof of surjectivity of the mapping in (a), observe that any linear func- ∗ tional τ ∈ (H1(M, F)) determines a group homomorphism τˆ : π1(M) → F given [ ]→ [ ] ˇ ∈ 1 F by γ τ( γ H1 ). Thus Theorem 10.7.8 provides a Cech 1-form θ Z (M, ) such that [ ] =ˆ[ ] = ∀[ ]∈ τ( γ H1 ) τ( γ ) θ γ π1(M), γ and Proposition 10.7.12 implies that [ ] = [ ] [ ] ∀[ ] ∈ F τ( ξ H1 ) ( θ H 1 , ξ H1 )deR ξ H1 H1(M, ). Thus part (a) is proved. 10.7 Homology and Cohomology of Second Countable C0 Surfaces 525

For the proof of (b), observe that if H1(M, F) is finite-dimensional, then by (a), 1 H (M, F) and H1(M, F) must have the same dimension, and therefore the injective linear map in (b) must be surjective. Conversely, if H1(M, F) is infinite- dimensional, then by Proposition 10.7.12, H1(M, F) has a countably infinite basis. But the dual space of any infinite-dimensional vector space cannot have a countable basis (see Exercise 8.1.3), so H 1(M, F) does not have a countable basis. In particular, (H 1(M, F))∗ also does not have a countable basis, so the linear map in (b) cannot be surjective.

~~Theorem 10.7.16 and Proposition 10.7.12 give the following:~~

~~Corollary 10.7.17 If M is a compact topological surface, then 1 dim H (M, F) = dim H1(M, F)<∞.~~

Remark According to Theorem 10.7.7,forM a second countable C∞ surface, 1 F 1 F we may identify H (M, ) with HdeR(M, ); so Theorem 10.7.16 and Corol- 1 F 1 F lary 10.7.17 hold in this case with HdeR(M, ) in place of H (M, ). According to the fundamental theorem of finitely generated Abelian groups, every finitely generated torsion-free Abelian group (G, +) is isomorphic to Zn for some unique integer n called the rank of G; that is, as a Z-module, G has a basis with n elements. Consequently, a modification of the proof of Theorem 10.7.16 gives the following (weaker) version for coefficients in the subring A, the proof of which is left to the reader (see Exercise 10.7.12):

Theorem 10.7.18 (de Rham) For any orientable second countable C∞ surface M: [ ] → [ ] · (a) The mapping θ H 1(M,A) ( θ H 1(M,A), )deR gives a module isomorphism ∼ 1 = H (M, A) −→ Hom(H1(M, A), A).

(b) If π1(M) is ﬁnitely generated (which is the case if, for example, M is compact), then we have the following: [ ] → [ ] · (i) The mapping ξ H1(M,A) ( ξ H1(M,A), )deR gives a module isomorphism ∼ = 1 H1(M, A) −→ Hom(H (M, A), A). {[ ] }m Z Z (ii) If ζi H1 i=1 are linearly independent (over ) elements of H1(M, ), {[ ] }m A A then ζi H1 i=1 are linearly independent (over ) in H1(M, ). {[ ] }n Z A (iii) Every basis ξj H1 j=1 for H1(M, ) is also a basis for H1(M, ), and {[ ] }n 1 Z =∼ Z Z the (dual) basis θj H1 j=1 for H (M, ) Hom(H1(M, ), ) determined by 1 if i = j, ([θ ] 1 , [ξ ] ) = ∀i, j = 1,...,n i H j H1 deR 0 if i = j,

~~is also a (dual) basis for H 1(M, A). 526 10 Fundamental Groups, Covering Spaces, and (Co)homology~~

Remark In the exercises for Chaps. 5 and 6, the reader is asked to prove that the 1 above homomorphism H1(M, A) → Hom(H (M, A), A) is injective for any ori- ∞ entable second countable C surface M (even if π1(M) is not ﬁnitely generated).

~~Theorem 10.7.18 and Proposition 10.7.12 give the following:~~

~~Corollary 10.7.19 If M is a compact orientable C∞ surface, then~~

~~1 1 rank H1(M, Z) = rank H (M, Z) = dim H1(M, F) = dim H (M, F)<∞.~~

Remarks We close this section with some instructive, but nonessential, remarks concerning other (equivalent) versions of homology and cohomology on a second countable topological surface M. 1. Cechˇ cohomology. A reader familiar with Cechˇ (sheaf) cohomology (see, for example, [For]) will recognize the space Z1(M, C) of Cechˇ 1-forms on M as the space of global sections Hˇ 0(M, C0/C), where C0 is the sheaf of germs of continuous functions and C is the sheaf of germs of locally constant functions. The space B1(M, C) of exact Cechˇ 1-forms is the image of Hˇ 0(M, C0). The exact sequence of sheaves 0 → C → C0 → C0/C → 0 yields the exact sequence of Cechˇ cohomology spaces

~~0 → Hˇ 0(M, C) → Hˇ 0(M, C0) → Hˇ 0(M, C0/C)~~

~~→ Hˇ 1(M, C) → Hˇ 1(M, C0) → Hˇ 1(M, C0/C) →··· .~~

~~Since Hˇ 1(M, C0) = 0, we get the isomorphism~~

~~Z1(M, C)/B1(M, C) = Hˇ 0(M, C0/C)/im(Hˇ 0(M, C0) → Hˇ 0(M, C0/C)) =∼ −→ Hˇ 1(M, C).~~

2. Singular homology and cohomology.Thestandard 2-simplex is the closed 2 2 2 triangular region ≡{(x, y) ∈ (R≥0) | x + y ≤ 1}⊂R ;asingular 2-simplex in M is a continuous mapping σ : 2 → M; and a (singular)2-chain in M with coefﬁcients in A is any formal ﬁnite linear combination m ξ = ai · σi, i=1 where ai ∈ A and σi :[0, 1]→M is a singular 2-simplex for i = 1,...,m.We = n ˆ identify the above 2-chain ξ with another 2-chain ζ j=1 bj σj in M if for each i = 1,...,m, ⎧ ⎨ 1≤j≤n bj if σi ∈{ˆσ1,...,σˆn}, = ˆ = al ⎩ σj σi 1≤l≤m 0ifσi ∈{ˆ/ σ1,...,σˆn}, σl =σi 10.7 Homology and Cohomology of Second Countable C0 Surfaces 527 and for each j = 1,...,n, the analogous statement, with the roles of ξ and ζ switched, holds. The module formed by the 2-chains is denoted by C2(M, A). For ν = 0, 1, 2, the νth edge of a singular 2-simplex σ in M is the path σ (ν) in M given by ⎧ ⎨⎪σ(1 − t,t) if ν = 0, t → σ(0,t) if ν = 1, ∀t ∈[0, 1]. ⎩⎪ σ(t,0) if ν = 2,

The boundary operator ∂ : C2(M, A) → C1(M, A) is the homomorphism given by m m = · → (0) − (1) + (2) ξ ai σi ai σi σi σi , i=1 i=1 the image is denoted by A ≡ : A → A B1 (M, ) im(∂ C2(M, ) C1(M, )), A 2 = and each element of B1 (M, ) is called a (singular) 1-boundary.Wehave∂ 0 A A (as is easy to verify), so B1 (M, ) is a submodule of Z1(M, ), and we may define the first singular homology group of M with coefficients in A to be the quotient module A ≡ A A H1 (M, ) Z1(M, ) B1 (M, ). For q = 0, 1, 2, we set q A ≡ A A C(M, ) Hom(Cq (M, ), ), and we call any element of the above module a q-cochain in M with coefficients A = : q A → q+1 A in .Forq 0, 1, the homomorphism δ C(M, ) C (M, ) given by ≡ ∀ ∈ q A ∈ A (δϕ)(ξ) ϕ(∂ξ) ϕ C(M, ), ξ Cq+1(M, ) is called the coboundary operator.Weset 1 A ≡ : 1 A → 2 A Z(M, ) ker(δ C(M, ) C(M, )), 1 A ≡ : 0 A → 1 A B(M, ) im(δ C(M, ) C(M, )), 1 A we call each element of Z(M, ) a(singular)1-cocycle, and we call each element 1 A 2 = 1 A of B(M, ) a(singular)1-coboundary. Clearly, δ 0, so B(M, ) is a sub- 1 A module of Z(M, ), and we may define the first singular cohomology group of M with coefficients in A to be the quotient module 1 A ≡ 1 A 1 A H(M, ) Z(M, ) B(M, ). A 1 A A Further properties of H1 (M, ) and H(M, ), and how they relate to H1(M, ) and H 1(M, A), are considered in Exercises 10.7.13Ð10.7.16 and in the exercises for Chaps. 5 and 6. 528 10 Fundamental Groups, Covering Spaces, and (Co)homology

Exercises for Sect. 10.7 10.7.1 Verify that the operations assigned to Z1(M, A) (as in Definition 10.7.1) determine a well-defined module structure. 10.7.2 Verify that line integrals of closed Cechˇ 1-forms, as defined in Defini- tion 10.7.1, are well defined. 10.7.3 Prove parts (a)Ð(d) and part (f) of Theorem 10.7.3. 10.7.4 Prove Lemma 10.7.4. 10.7.5 Verify that for any second countable topological surface M, we have a real direct sum decomposition Z1(M, C) = Z1(M, R) ⊕ iZ1(M, R). 10.7.6 Prove Proposition 10.7.10. 10.7.7 Prove Lemma 10.7.11. 10.7.8 Prove Proposition 10.7.12. 10.7.9 Prove Proposition 10.7.14. 10.7.10 Prove that if : M → N and : N → P are continuous mappings of second countable topological surfaces, then ( ◦ )∗ = ∗ ◦ ∗ and ( ◦ )∗ = ∗ ◦ ∗ both at the level of q-chains and Cechˇ 1-forms and at the level of homology and cohomology. 10.7.11 In the situation of Theorem 10.7.16, assuming that the homology vector space H1(M, F) is infinite-dimensional, give a direct proof that the associ- [ ] → [ ] · ated injection ξ H1 ( ξ H1 , )deR is not surjective using the fact that, for any infinite-dimensional vector space V over F,wehaveV (V∗)∗ (see Sect. 8.1). 10.7.12 Prove Theorem 10.7.18. A 1 A Some properties of H1 (M, ) and H(M, ), and how they relate to 1 H1(M, A) and H (M, A), are considered in Exercises 10.7.13Ð10.7.16 below. 10.7.13 Let M be a second countable topological surface, and let A be a subring of C containing Z. (a) Suppose α and β are paths in M. (i) Prove that if α(1) = β(0), then α + β − α ∗ β = ∂σ for some sin- + − ∗ ∈ Z gular 2-simplex σ in M (in particular, α β α β B1 (M, )). (ii) Prove that if α(1) = β(0), then −α + β− + α ∗ β = ∂σ for some singular 2-simplex σ in M (in particular, −α + β− + α ∗ β ∈ Z B1 (M, )). : → ∈ (iii) Prove that the constant loop ep0 t p0 at any point p0 M = satisfies ep0 ∂σ for the constant singular 2-simplex σ : (x, y) → p0. (iv) Prove that if α and β are path homotopic and p1 ≡ α(1) = β(1), (0) = then there is a singular 2-simplex σ in M such that σ ep1 (the (1) (2) constant loop at p1), σ = β, and σ = α; and hence α − β = ∂(σ −ˆσ), where σˆ is the constant singular 2-simplex given by ˆ : → − ∈ Z σ (x, y) p1 (in particular, α β B1 (M, )). + − ∈ Z (v) Prove that α α B1 (M, ). (vi) Prove that if σ is a singular 2-simplex with α = σ (2) and β = σ (0), then σ (1) is path homotopic to α ∗ β. 10.7 Homology and Cohomology of Second Countable C0 Surfaces 529

∈ [ ] → [ ] (b) Prove that for each point p0 M, the mapping γ p0 γ A H1 (M, ) → A gives a well-defined homomorphism π1(M, p0) H1 (M, ). Prove A also that the image of this homomorphism generates H1 (M, ) as an A-module and that the map is surjective if A = Z. 1 A 1 A (c) Prove that H(M, ) is (naturally) isomorphic to H (M, ). Hint. Fix a point p0 ∈ M, and applying part (a) above, show 1 A → A that the mapping H(M, ) Hom(π1(M, p0), ) determined by 1 [ϕ] 1 ·[γ ]=ϕ(γ) for all [ϕ] 1 ∈ H (M, A) and [γ ]∈π1(M, p0) H H is a well-defined injective homomorphism. For surjectivity, fix a path ∈ = λx from p0 to x for each point x M, with λp0 ep0 . Show that given a homomorphism F ∈ Hom(π1(M, p0), A), there is a unique singu- ≡ [ ∗ ∗ − ] lar 1-cocycle ϕ determined by ϕ(γ) F( λγ(0) γ λγ(1) ) for each path γ . A ⊂ A (d) Prove that B1 (M, ) B1(M, ), provided M admits an orientable C∞ surface structure if A = R and A = C. Conclude that there is a (nat- A → A ural) surjective homomorphism H1 (M, ) H1(M, ) (the reader is asked to prove bijectivity in the exercises for Chaps. 5 and 6). 10.7.14 Let M be a second countable topological surface, let σ be a singular 2-simplex in M, and let p0 ∈ M. Prove that: (a) If λ0 is a path in M from p0 to σ(0, 0), then there exists a singular (0) = (0) (1) = ∗ (1) (2) = 2-simplex σ0 in M such that σ0 σ , σ0 λ0 σ , and σ0 (2) λ0 ∗ σ . (b) If λ1 is a path in M from p0 to σ(1, 0), then there exists a singular (0) = ∗ (0) (1) = (1) (2) = 2-simplex σ1 in M such that σ1 λ1 σ , σ1 σ , and σ1 (2) ∗ − σ λ1 . (c) If λ2 is a path in M from p0 to σ(0, 1), then there exists a singular (0) = (0) ∗ − (1) = (1) ∗ − 2-simplex σ2 in M such that σ2 σ λ2 , σ2 σ λ2 , and (2) = (2) σ2 σ . 10.7.15 Suppose α and β are path homotopic paths in a second countable topological surface M, σ is a singular 2-simplex in M, ν ∈{0, 1, 2}, and α = σ (ν). Prove that there exists a singular 2-simplex σˆ in M such that σˆ (ν) = β and σˆ (μ) = σ (μ) for μ ∈{0, 1, 2}\{ν}. Conclude that in particular,

~~∂σ = (−1)να − (−1)νβ + ∂σ.ˆ~~

10.7.16 The real projective plane is the quotient space RP2 ≡ S2/[x∼−x ∀x ∈ S2]. Let ϒ : S2 → RP2 be the corresponding quotient map. (a) Prove that RP2 admits a C∞ surface structure for which ϒ is a C∞ covering map. (b) Prove that RP2 is nonorientable (thus ϒ : S2 → RP2 is an orientable double cover as considered in Exercise 9.7.8). (c) Letting F = R or C, prove the following equalities (which demonstrate that the version of integral homology H1(M, Z) considered in this section for orientable C∞ surfaces would be inadequate for nonorientable 530 10 Fundamental Groups, Covering Spaces, and (Co)homology

~~surfaces):~~

RP2 Z =∼ RP2 =∼ Z = Z Z H1 ( , ) π1( ) 2 /2 , RP2 F = RP2 F ={ } H1( , ) H1 ( , ) 0 , 1 RP2 Z = 1 RP2 Z = 1 RP2 F = 1 RP2 F ={ } H ( , ) H( , ) H ( , ) H( , ) 0 . Hint. For the claim concerning the homology groups, let γ be a gen- 2 erating loop for π1(RP ).Ifγ = ∂ξ for some integral 2-cochain ξ, then by applying Exercises 10.7.14 and 10.7.15, reduce to the case in which each of the boundary edges of the singular 2-simplices in ξ is either γ or the constant loop. Applying Exercise 10.7.13, obtain a contradiction (one may also prove this fact by observing that the ﬁrst singular homology group is equal to the Abelianization of the fundamental group, as proved in, for example, [Hat]). Chapter 11 Background Material on Sobolev Spaces and Regularity

~~This chapter is required for the proof of integrability of almost complex structures in Chap. 6. Throughout this chapter, we use the operator notation:~~

∂ Dj = for j = 1,...,n ∂xj and ∂ α ∂|α| Dα = = (D )α1 ···(D )αn = 1 n α1 ··· αn ∂x ∂x1 ∂xn n for each multi-index α = (α1,...,αn) ∈ (Z≥0) (see Sect. 7.4). We also use the following notation. Let us denote the coordinates in Rn × Rn = R2n by

(x, y) = ((x1,...,xn), (y1,...,yn)) = (x1,...,xn,y1,...,yn). β n Given a linear differential operator L = bβ D on an open set ⊂ R , we denote by Lx and Ly the linear differential operators on × given by ∂ β L [h(x,y)]= b (x) [h(x,y)] x β ∂x and ∂ β L [h(x,y)]= b (y) [h(x,y)]. y β ∂y For example, for = R2,wehave

~~(1,2)[ 2 3 4 5]= 4 5 Dx x1 x2 y1 y2 (2x1)(6x2)y1 y2 and (1,2)[ 2 3 4 5]= 2 3 3 3 Dy x1 x2 y1 y2 x1 x2 (4y1 )(20y2 ).~~

T. Napier, M. Ramachandran, An Introduction to Riemann Surfaces, Cornerstones, 531 DOI 10.1007/978-0-8176-4693-6_11, © Springer Science+Business Media, LLC 2011 532 11 Sobolev Spaces and Regularity

(1,2) For L = x1D ,wehave 2 3 = L(x1 x2 ) x1(2x1)(6x2), [ 2 3 4 5]= 4 5 Lx x1 x2 y1 y2 x1(2x1)(6x2)y1 y2 , [ 2 3 4 5]= 2 3 3 3 Ly x1 x2 y1 y2 y1x1 x2 (4y1 )(20y2 ). The main goal of this chapter is the following regularity theorem:

Theorem 11.0.1 (General ﬁrst-order regularity) Let A be a linear differential operator of order 1 with C∞ coefﬁcients on an open set ⊂ Rn. Assume that there exists a constant C>0 such that n 2 + 2 ≤ · 2 + 2 ∀ ∈ u L2() Dj u L2() C u L2() Au L2() u D(). (1) j=1 ∈ 2 ∈ C∞ ∈ C∞ Then, for every function u Lloc() with Adistru (), we have u ().

Remarks 1. The analogous version holds in higher dimensions (see, for example, [Hö]) and for higher-order operators (see, for example, [GiT]or[Ns3]). 2. The term u2 on the left-hand side is redundant. However, it will be L2() convenient to have the theorem stated in this form, since the left-hand side represents the square of the ﬁrst-order Sobolev norm of u (see Deﬁnition 11.1.1), and Sobolev spaces are applied in the proof.

A natural approach to the regularity theorem (and to the study of differential equations in general) is to complete the space of C∞ functions with respect to the norm obtained from Lp norms of derivatives. One obtains the so-called Sobolev spaces. One proves the regularity theorem by applying the Sobolev lemma, which gives an embedding of Sobolev spaces for a large number of derivatives into the Banach space Ck . For our purposes, p = 2 sufﬁces, but Sobolev spaces for arbitrary p ≥ 1 are, in general, very useful. The reader may refer to, for example, [Ad] and [GiT] for more about this important class of spaces.

~~11.1 Sobolev Spaces~~

n Deﬁnition 11.1.1 Let be an open subset of R , and let k ∈ Z≥0. (a) The associated Sobolev space of order k is given by k ≡{ ∈ 2 | α ∈ 2 ∀ ∈ Zn | |≤ } W () u L () Ddistru L () α ≥0 with α k . (b) The Sobolev inner product and Sobolev norm are given by ≡ α α u, v W k() Ddistru, Ddistrv L2() |α|≤k

~~1/2 k u k ≡ u, u u, v ∈ W () and W () W k(), respectively, for all . 11.1 Sobolev Spaces 533~~

(c) The local Sobolev space of order k is given by k ≡{ ∈ 2 | α ∈ 2 ∀ | |≤ } Wloc() u Lloc() Ddistru Lloc() α with α k . ∈ k For all u, v Wloc() and any measurable subset E of ,weset ≡ α α u, v W k(E) Ddistru, Ddistrv L2(E) |α|≤k

~~1/2 u k ≡ u, u C [ , ∞] and W (E) W k(E), provided these numbers exist in (or 0 ). Remark Clearly, u, v W k(E) and u W k(E) are independent of the choice of the neighborhood of E.~~

~~Completeness of L2() gives the following fact, the proof of which is left to the reader (see Exercise 11.1.1).~~

~~k · · Proposition 11.1.2 (W (), , W k()) is a Hilbert space for every open set ⊂ Rn and every nonnegative integer k.~~

The following lemma allows one to manipulate distributional values of differential operators in ways similar to the ways in which one treats standard values. In particular, it allows one to cut off in order to get compact support.

Lemma 11.1.3 Let A and B be linear differential operators of order k ≥ 1 and l, respectively, with C∞ coefﬁcients on an open set ⊂ Rn, and let m ≡ k + l − 1. ∈ l (a) If u Wloc(), then = · β ∈ 2 Bdistru bβ Ddistru Lloc(), |β|≤l = β ⊂ where B |β|≤l bβ D . In particular, for every compact set K , there is ∈ l a constant C>0 such that for every function u Wloc(), we have ≤ Bdistru L2(K) C u W l(K). (b) The commutator [A,B]≡AB −BA is a linear differential operator of order m. ∈ m ∈ l ∈ k−1 (c) If u Wloc() and Adistru Wloc(), then we have Bdistru Wloc (), ∈ 2 ∈ 2 [ ] ∈ 2 (AB)distru Lloc(), (BA)distru Lloc(), A,B distru Lloc(), and

(AB)distr(u) = AdistrBdistru = (BA)distru +[A,B]distru = +[ ] ∈ 2 BdistrAdistru A,B distru Lloc(). In particular, for any nonnegative integer s, the product of a function in C∞() s s and a function in Wloc() is in Wloc(). 534 11 Sobolev Spaces and Regularity

~~(d) We have t B = (−1)lB + L for some linear differential operator L of order max(l − 1, 0) with C∞ coefﬁcients on (t B = B if l = 0).~~

~~Remark As noted above, the commutator of two linear differential operators A and B with C∞ coefﬁcients is given by [A,B]=AB − BA.IfA and B are of order 0, then [A,B]=0.~~

Proof of Lemma 11.1.3 Part (a) follows immediately from Lemma 7.4.3 (note that 2 ⊂ 1 Lloc Lloc). The product rule gives part (b) (see Exercise 11.1.2). For the proof ∈ m [ ] ∈ 2 of part (c), let u Wloc(). Parts (a) and (b) imply that A,B distru Lloc(). Furthermore, for each multi-index α with |α|≤k − 1, we have, by Lemma 7.4.3, α [ ]= α ∈ 2 α − + = Ddistr Bdistru (D B)distru Lloc() (D B is of order k 1 l m), and hence ∈ k−1 ∈ l Bdistru Wloc ().IfAdistru Wloc(), then (a) and Lemma 7.4.3 together give = ∈ 2 (BA)distru BdistrAdistru Lloc(), and hence = = +[ ] ∈ 2 AdistrBdistru (AB)distru (BA)distru A,B distru Lloc(). The proof of (d) is left to the reader (see Exercise 11.1.2).

~~Exercises for Sect. 11.1 11.1.1 Prove Proposition 11.1.2. 11.1.2 Prove parts (b) and (d) of Lemma 11.1.3.~~

~~11.2 Uniform Convergence of Derivatives~~

In the context of differential equations, it is also natural to consider uniform convergence of derivatives in place of Lp convergence. For this, it is useful to have the following well-known fact:

k Proposition 11.2.1 Let k be a nonnegative integer, let {uν} be a sequence of C n functions on an open set ⊂ R , and let {vα}α∈(Z≥0)n, |α|≤k be a collection of func- α tions such that D uν → vα uniformly on compact subsets of for each multi-index k α α with |α|≤k. Then, for v ≡ v(0,...,0), we have v ∈ C () and D v = vα for each α with |α|≤k.

~~The proof is left to the reader (see Exercise 11.2.1).~~

n Corollary 11.2.2 If k ∈ Z≥0, is an open subset of R , and {uν} is a sequence of k ⊂ { α } functions in C () such that for every compact set K , D uν K is a Cauchy C0 ·≡ |·| | |≤ sequence in ( (K), supK ) for each multi-index α with α k, then there k α α exists a function u ∈ C () such that D uν → D u uniformly on compact subsets of for each α with |α|≤k. 11.3 The Strong Friedrichs Lemma 535

~~The proof is again left to the reader (see Exercise 11.2.2).~~

~~Exercises for Sect. 11.2 11.2.1 Prove Proposition 11.2.1. 11.2.2 Prove Corollary 11.2.2.~~

~~11.3 The Strong Friedrichs Lemma~~

~~The goal of this section is the following strong version of the Friedrichs regularization lemma (Lemma 7.3.1):~~

∞ Lemma 11.3.1 (Friedrichs) Let κ be a nonnegative C function with compact sup- n port in the unit ball B(0; 1) in R such that Rn κdλ= 1, let be an open subset of Rn, let k be a nonnegative integer, and let A be a linear differential opera- C∞ ∈ 1 tor of order k with coefﬁcients on . For each function u Lloc() and for n δ −n each δ>0, let δ ≡{x ∈ | dist(x, R \ ) > δ}, let κ (x) ≡ δ κ(x/δ) for each point x ∈ Rn, and let

~~δ δ uδ(x) ≡ u(y)κ (x − y)dλ(y) = u(x − y)κ (y) dλ(y) B(0;δ)~~

= u(x − δy)κ(y) dλ(y) B(0;1) for every x ∈ δ. Finally, let K be a compact subset of , let δ0 be a constant with Rn \ ⊂ δ ≡{ ∈ Rn | ≤ } 0 <δ0 < dist(K, ) (hence K δ0 ), and let K x dist(x, K) δ for each δ>0. Then we have the following: ∈ 1 ∈ C∞ (a) For every function u Lloc() and every δ>0, we have uδ (δ). k + (b) If u ∈ C (), then A(uδ) → Au uniformly on K as δ → 0 . ≤ ∈ k ∈ (c) We have uδ W k(K) u W k(Kδ0 ) for every u Wloc() and δ (0,δ0). ∈ k − → → + → k (d) If u Wloc(), then uδ u W k(K) 0 as δ 0 (i.e., uδ u in Wloc()). (e) There is a constant C>0 such that for every δ ∈ (0,δ0) and every function ∈ max(k−1,0) ∈ 2 u W () with Adistru Lloc(), we have − ≤ A(uδ) (Adistru)δ L2(K) C u W max(k−1,0)().

~~∈ max(k−1,0) ∈ 2 − → (f) If u Wloc () and Adistru Lloc(), then A(uδ) Adistru L2(K) 0 as δ → 0+.~~

Proof The proof given here is similar to the proof of Lemma 5.2.2 in [Hö]. Part (a) follows from Lemma 7.3.1 (or differentiation past the integral). For the proof of (b), we observe that if u ∈ Ck(), α is a multi-index with |α|≤k, and a is a C∞ function on , then by Lemma 7.3.1 and Lemma 7.4.4,wehave 536 11 Sobolev Spaces and Regularity

~~The claim (b) now follows. ∈ 2 ∈ For the proof of (c), suppose u Lloc() and δ (0,δ0). Then, for each point x ∈ K, the Schwarz inequality gives~~

~~2 2 |uδ(x)| = u(x − δy)κ(y) dλ(y) B(0;1) ≤ |u(x − δy)|2κ(y)dλ(y) · κ(y)dλ(y) B(0;1) B(0;1)~~

~~= |u(x − δy)|2κ(y) dλ(y). B(0;1)~~

Applying Fubini’s theorem (Theorem 7.1.10), we get 2 ≤ | − |2 uδ L2(K) u(x δy) dλ(x) κ(y)dλ(y) B(0;1) K = |u(x)|2 dλ(x) κ(y)dλ(y) B(0;1) K+(−δy) ≤ |u(x)|2 dλ(x) κ(y)dλ(y) =u2 . L2(Kδ0 ) B(0;1) Kδ0

∈ k If u Wloc(), then applying Lemma 7.4.4, we get 2 = α 2 = [ α ] 2 uδ W k(K) D (uδ) L2(K) (D )distru δ L2(K) |α|≤k |α|≤k ≤ (Dα) u2 =u2 . distr L2(Kδ0 ) W k(Kδ0 ) |α|≤k

~~Thus (c) is proved. For the proof of (d), we ﬁrst observe that if v ∈ C0(), then~~

~~ − ≤ 1/2 | − |→ → + vδ v L2(K) λ(K) max vδ v 0asδ 0 K~~

~~∈ 2 by part (b) (or Lemma 7.3.1). Given u Lloc() and >0, we may choose a − continuous function v on such that v u L2(Kδ0 ) < /3 (see Exercise 7.1.4). Hence, by part (c), we have~~

− = − vδ uδ L2(K) (v u)δ L2(K) < /3 11.3 The Strong Friedrichs Lemma 537 for each δ ∈ (0,δ0). On the other hand, for δ>0 sufﬁciently small, we also have − vδ v L2(K) < /3. Therefore − ≤ − + − + − uδ u L2(K) uδ vδ L2(K) vδ v L2(K) v u L2(K) < .

− → → + ∈ k Thus uδ u L2(K) 0asδ 0 .Ifu Wloc(), then applying Lemma 7.4.4 (as above), we get − 2 = α − α 2 uδ u W k(K) D (uδ) (D )distru L2(K) |α|≤k = [ α ] −[ α ]2 → → + (D )distru δ (D )distru L2(K) 0asδ 0 . |α|≤k

Thus (d) is proved. We now prove (e) by induction on k.Ifk = 0, then the differential operator A is given by v → a · v for some C∞ function a on . Therefore, if u ∈ L2(), then for each point x ∈ K and each δ ∈ (0,δ0), the Schwarz inequality gives

~~2 |A(uδ)(x) − (Adistru)δ(x)|~~

~~ C > 0 such that for each δ ∈ (0,δ0),wehave − ≤ ≤ A (uδ) (Adistru)δ L2(K) C u W max(k−2,0)() C u W max(k−1,0)()~~

= − ∈ 2 for every such function u. Furthermore, Adistru Adistru Adistru Lloc() and − ≤ − A(uδ) (Adistru)δ L2(K) A (uδ) (Adistru)δ L2(K) + − A (uδ) (Adistru)δ L2(K). = = α Thus we may assume without loss of generality that A A ; i.e., A |α|=k aαD ∞ with aα ∈ C () for each α. For each δ ∈ (0,δ0) and each point x ∈ K, differentiation past the integral ∈ k−1 ∈ 2 (Proposition 7.2.5) gives, for u W () with Adistru Lloc(),

~~A(uδ)(x) − (Adistru)δ(x)~~

~~δ δ = u(y)Ax [κ (x − y)] dλ(y)− [Adistru(y)]κ (x − y)dλ(y) B(x;δ) B(x;δ)~~

~~δ t δ = u(y)Ax [κ (x − y)] dλ(y)− u(y) · ( A)y [κ (x − y)] dλ(y). B(x;δ) B(x;δ)~~

According to Lemma 11.1.3,wehavet A = (−1)kA + L for some linear differential operator L of order k − 1 with C∞ coefﬁcients on .Wealsohave [ δ − ]= α[ δ − ] Ax κ (x y) aα(x)Dx κ (x y) = − k α[ δ − ] ( 1) aα(x)Dy κ (x y) k δ δ = (−1) Ay [κ (x − y)]+M[κ (x − y)], where M is the linear differential operator on × ⊂ Rn × Rn given by [ ]= − k − α[ ] M h(x,y) ( 1) (aα(x) aα(y))Dy h(x,y) . n α∈(Z≥0) , |α|=k

~~Thus A(uδ)(x) − (Adistru)δ(x) = F(x,δ)− G(x, δ), where~~

~~F(x,δ)≡ u(y)M[κδ(x − y)] dλ(y) B(x;δ) and~~

δ G(x, δ) ≡ u(y)Ly [κ (x − y)] dλ(y). B(x;δ) t k−1 t Since L is of order k − 1 and u ∈ W (), Lemma 11.1.3 implies that Ldistru ∈ 2 t ≤ − Lloc() and Ldistru L2(Kδ0 ) C0 u W k 1() for some constant C0 > 0 that is 11.3 The Strong Friedrichs Lemma 539

~~t independent of the choice of u. In particular, we have G(x, δ) = ( Ldistru)δ(x) for each x ∈ K, and, applying part (c), we get~~

~~ · ≤t ≤ − G( ,δ) L2(K) Ldistru L2(Kδ0 ) C0 u W k 1().~~

It remains to bound the L2 norm of F(·,δ). For each multi-index α with |α|=k, α α we have D = D Dj for some multi-index α with |α |=k − 1 and some index α ∞ jα ∈{1,...,n}. Thus, if h(x,y) is a C function on an open subset of × , then [ ]= − k − · α M h(x,y) ( 1) (aα(x) aα(y)) Dy (Djα )y (h(x, y)) |α|=k = − k α − · ( 1) Dy (aα(x) aα(y)) (Djα )y (h(x, y)) |α|=k + − k − α ( 1) (aα(x) aα(y)), Dy (Djα )y(h(x, y)) |α|=k = − k α − · ( 1) Dy (aα(x) aα(y)) (Djα )y (h(x, y)) |α|=k − − k α ( 1) aα(y), Dy (Djα )y (h(x, y)) |α|=k = − k α − · + ( 1) Dy (aα(x) aα(y)) (Djα )y (h(x, y)) Ny(h(x, y)), |α|=k where [·, ·] denotes the commutator (see Lemma 11.1.3) and N is a linear differential operator of order k −1 with C∞ coefﬁcients on (N = 0ifk = 1). Note that the third equality holds because the function aα(x) is constant with respect to the vari- t t k−1 ables y = (y1,...,yn) for each α. Therefore, since N = ( N) and u ∈ W (), Lemma 11.1.3 implies that for each point x ∈ K and each δ ∈ (0,δ0),wehave t F(x,δ)=− Hα(x, δ) + ( Ndistru)δ(x), |α|=k where for each α,

~~≡ α · − · [ δ − ] Hα(x, δ) (Ddistru(y)) (aα(x) aα(y)) (Djα )y κ (x y) dλ(y). B(x;δ)~~

A construction similar to that of the bound on the L2 norm of G(·,δ) yields a constant C1 > 0, which is independent of the choice of u and δ, such that t ≤ ( N distru)δ L2(K) C1 u W k−1().

2 To bound the L norm of each Hα(·,δ), we ﬁrst observe that Lemma 7.2.4 implies that there is a constant C2 > 0 independent of the choice of u and δ such that for each 540 11 Sobolev Spaces and Regularity multi-index α with |α|=k, for each point x ∈ K, and each point y ∈ B(x; δ) ⊂ Kδ0 , we have |aα(x) − aα(y)|≤C2δ and 1 x − y 1 x − y (D ) [κδ(x − y)]=(D ) κ =− (D κ) . jα y jα y δn δ δn+1 jα δ

≡ ; ≡ n | | Setting v vol(B(0 1)) and C3 C2 maxx∈R , |α|=k Djα κ(x) , and applying the Schwarz inequality, we get, for each point x ∈ K and each multi-index α with |α|=k, 2 | |2 ≤ 2 | α | −n Hα(x, δ) C3 Ddistru(y) δ dλ(y) B(x;δ) 2 = 2 | α − | C3 Ddistru(x δy) dλ(y) B(0;1)

~~≤ 2 · · | α − |2 C3 v Ddistru(x δy) dλ(y). B(0;1)~~

Applying Fubini’s theorem (Theorem 7.1.10), we get · 2 ≤ 2 · · | α − |2 Hα( ,δ) L2(K) C3 v Ddistru(x δy) dλ(x) dλ(y) B(0;1) K = 2 · · | α |2 C3 v Ddistru(x) dλ(x) dλ(y) B(0;1) K+(−δy) ≤ 2 · 2 · 2 C3 v u W k−1().

The claim (e) now follows. For the proof of (f), we ﬁx an open set with K ⊂ and a constant Rn \ δ0 with 0 <δ0 < dist(K, ). According to part (e), there is a constant C>0 ∈ ∈ max(k−1,0) ∈ such that for every δ (0,δ0) and every function u Wloc () with Adistru 2 Lloc(),wehave − ≤ A(uδ) (Adistru)δ L2(K) C u W max(k−1,0)().

~~Given such a function u and a number >0, part (d) implies that we may choose a ∈ C∞ − ∈ function v () such that u v W max(k−1,0)() < /(4C). For each δ (0,δ0), we get~~

− A(uδ) Adistru L2(K) ≤ − + − A(uδ) (Adistru)δ L2(K) (Adistru)δ Adistru L2(K) ≤ − − − + − A((u v)δ) (Adistr(u v))δ L2(K) A(vδ) Av L2(K) + − + − (Av)δ Av L2(K) (Adistru)δ Adistru L2(K) 11.4 The Sobolev Lemma 541

< +A(vδ) − Av 2 +(Av)δ − Av 2 4 L (K) L (K) + − (Adistru)δ Adistru L2(K). By part (b), for δ>0 sufﬁciently small, we have − A(vδ) Av L2(K) < /4, and by part (d), for δ>0 sufﬁciently small, we also have − − (Av)δ Av L2(K) < /4 and (Adistru)δ Adistru L2(K) < /4. − Thus A(uδ) Adistru L2(K) < , and the claim (f) is proved.

~~11.4 The Sobolev Lemma~~

~~The goal of this section is the following fact, which allows one to obtain differentiability of functions possessing a high number of L2 distributional derivatives:~~

n Lemma 11.4.1 (Sobolev lemma) For every open set ⊂ R and every k ∈ Z≥0, k+n ⊂ Ck ∈ k+n we have Wloc () (); that is, each function u Wloc () is equal almost everywhere to a function of class Ck.

Proof Multiplying the given function by a C∞ cutoff function and applying part (c) of Lemma 11.1.3, we see that we may assume without loss of generality that ◦ supp u ⊂ K ⊂ K ⊂ for some compact set K. Lemma 11.3.1 then provides a sequence of functions {uν } in D() such that supp uν ⊂ K for each ν and − = − → →∞ uν u W k+n() uν u W k+n(K) 0asν . Applying the inequality

n ∂ ϕ n sup |ϕ|≤ ∀ϕ ∈ D(R ) Rn ∂x1 ···∂xn (see Exercise 11.4.1), we see that for every multi-index α with |α|≤k and for all positive integers μ and ν,wehave,forβ = α + (1,...,1), | α − α |≤ β − β ≤ 1/2 · β − β sup D uμ D uν D uμ D uν L1(K) (λ(K)) D uμ D uν L2(K) ≤ 1/2 · − → →∞ (λ(K)) uμ uν W k+n() 0asμ,ν .

~~Proposition 11.2.1 now implies that u ∈ Ck(). 542 11 Sobolev Spaces and Regularity~~

~~Exercises for Sect. 11.4 11.4.1 Prove the inequality~~

~~n ∂ ϕ n sup |ϕ|≤ ∀ϕ ∈ D(R ). Rn ∂x1 ···∂xn~~

11.5 Proof of the Regularity Theorem We now come to the proof of Theorem 11.0.1. Throughout this section, A denotes a linear differential operator of order 1 with C∞ coefﬁcients on an open set ⊂ Rn. The ﬁrst step is the following observation:

Lemma 11.5.1 If the inequality (1) in Theorem 11.0.1 holds for some constant C>0 and for every function u ∈ D(), then for every positive integer k and every compact set K ⊂ , there is a constant C(K,k) > 0 such that 2 ≤ · 2 + 2 ∀ ∈ u W k(K) C(K,k) u L2() Au W k−1() u D(). (2)

Proof We proceed by induction on k = 1, 2, 3,.... The case k = 1 follows from the (stronger) inequality (1). Thus we may assume that k>1 and that the inequality (2) holds for every compact subset of (for some choice of a constant) for strictly lower orders. We may ﬁx a compact set K with K ⊂ K ⊂ and a function ρ ∈ D() with ρ ≡ 1onK and supp ρ ⊂ K.LetL denote the linear differential operator of order 0 given by multiplication by ρ. Given a multi-index α with |α|=k,wehave α = α ∈{ } | |= − D D Djα for some jα 1,...,n and some multi-index α with α k 1. For each function u ∈ D(), we get = +[ ] A(Djα (ρu)) Djα L(Au) A,Djα L u, [ ] where Djα L and A,Djα L are of order 1. Applying part (a) of Lemma 11.1.3,we get a constant C1(K, k) > 0 independent of the choice of u and α such that α 2 = α 2 D u 2 D Djα (ρu) 2 L (K) L (K) ≤ − · 2 + 2 C(K,k 1) Djα (ρu) 2 ADjα (ρu) k−2 L () W () = − · 2 + 2 C(K,k 1) Djα (ρu) 2 ADjα (ρu) k−2 L (K ) W (K ) ≤ · 2 + 2 C1(K, k) u k−1 Au k−1 W (K ) W (K ) ≤ · − 2 + 2 C1(K, k) C(K ,k 1) u 2 Au k−2 L () W () + 2 Au k−1 W (K ) ≤ · − + · 2 + 2 C1(K, k) (C(K ,k 1) 1) u L2() Au W k−1() . The claim now follows. 11.5 Proof of the Regularity Theorem 543

~~Theorem 11.0.1 now follows immediately from the Sobolev lemma (Lem- ma 11.4.1) together with the following fact:~~

Lemma 11.5.2 If the inequality (1) in Theorem 11.0.1 holds for some constant C>0 and for every function u ∈ D(), then for every nonnegative integer k, we have ∈ 2 | ∈ max(k−1,0) ⊂ k u Lloc() Adistru Wloc () Wloc().

∈ 2 ∈ Proof Given a nonnegative integer k and a function u Lloc() with Adistru max(k−1,0) ∈ k Wloc (), we must show that u Wloc(). We proceed by induction on k. The case k = 0 is trivial, so let us assume that k>0 and the claim holds for the ∈ k−1 Sobolev spaces of order

k−1 Adistr(ρu) = ρ · Adistru +[A,ρ]u ∈ W (), → [ ] ∈ k−1 since v ρv and A,ρ are operators of order 0 and Adistru, u Wloc (). Thus ◦ we may assume that supp u ⊂ K ⊂ K ⊂ for some compact set K. Applying the (strong) Friedrichs lemma (Lemma 11.3.1), we get a sequence of functions in {uν} in D() such that supp uν ⊂ K for each ν and, as ν →∞, − = − → − → uν u W k−1() uν u W k−1(K) 0 and Auν Adistru W k−1() 0. Note that for the convergence of the ﬁrst sequence, we have applied part (d) of the lemma, while for the second sequence, we have applied part (f) to the operator DαA for each multi-index α with |α|≤k − 1, and we have also used the fact that α = α (D A)distru DdistrAdistru (Lemma 7.4.3). By Lemma 11.5.1, for some constant C > 0 and for all positive integers μ and ν, − 2 = − 2 ≤ · − 2 + − 2 uμ uν W k() uμ uν W k(K) C uμ uν L2() Auμ Auν W k−1() .

~~Therefore, since the right-hand side converges to 0 as μ,ν →∞and W k() is a Hilbert space, we have u ∈ W k(), and the lemma follows.~~

Exercises for Sect. 11.5 11.5.1 Let A be a linear differential operator of order 1 with C∞ coefﬁcients on an open set ⊂ Rn. Assume that the inequality (1) in Theorem 11.0.1 holds for some constant C>0 and for every function u ∈ D(). Assume also that the coefﬁcients of A and their partial derivatives of all orders are bounded on . Prove that for every positive integer k, there is a constant Ck > 0 such that 2 ≤ · 2 + 2 ∀ ∈ u W k() Ck u L2() Au W k−1() u D().

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~~Notation Index~~

A Dr , 439 r Adistr, 394, 463 D (E), 111 α ∗ β, 478 ∗ d, 429, 433, 442 A , 394 ∂,47 Aut(X), 194 ∂¯, 47, 125 ¯ B ∂distr, 60, 139 D, D,D, 61, 136 B1(M, A), 506, 520 1 B (M, A), 513 Ddistr, 62, 139 1 F BdeR(M, ), 503 Deck(ϒ), 492 A D B1 (M, ), 527 deg , 124 1 A B(M, ), 527 deg E, 124 ∂S, 415 2, 526 {· ·} , (E,h), 131 ω,82 [D], 121 (z0; R),6 [·] r F , [·]deR, 503 HdeR(M, ) (z ; r, R),6 [·] [·] 0 H q (X,E), Dol, 126 ∗ Dol (z0; R),6 [·, ·], 278, 534 div( ), 123 [·] , [·] , 479 x0 π1(X,x0) div(s), 119 B(x; r), 378 Div(X), 119 C C0 , 383, 416 E Ck , 382, 421 E, 116, 421 cl(S), 378, 415 E cos z,22 (E), 104, 116 Ep,q cot z,22 , 45, 116 Ep,q Cq (M, A), 505, 527 (E), 111, 116 q A Er , 116, 439 C(M, ), 527 csc z,22 Er (E), 111, 116 C∗,6 exp(z),21 ez,21 D D, 421 D(E), 104 F Dp,q ,45 , 177 p,q D (E), 111 Fσ , 377

~~T. Napier, M. Ramachandran, An Introduction to Riemann Surfaces, Cornerstones, 549 DOI 10.1007/978-0-8176-4693-6, © Springer Science+Business Media, LLC 2011 550 Notation Index~~

· · · · · · G , L2 (E,h), , L2 (E,ω,h), , L2(E,h), 132 ˙ p,q p,q 1 γ , 434 · 2 , · 2 , · 2 ,54 − Lp,q (ϕ) Lp,q (ω,ϕ) L (ϕ) γ , 478 · · · 1 L2 (E,h), L2 (E,ω,h), L2(E,h), 132 E p,q p,q 1 (U, (E)), 104 2 2 2 (U,F), 117 Lp,q(ϕ), Lp,q(ω, ϕ), L1(ϕ),55 2 2 2 (U,M(E)), 104 Lp,q(E, h), Lp,q(E,ω,h),L1(E, h), 133 (U,O(E)), 104 Gδ , 377 M genus(X), 165 M, 33, 115 GL(n, F), 219 M(E), 104, 115 multp,33 H H, 198 N ; H1(M, A), 506, 520 n(γ z0), 201 H 1(M, A), 516 H : α ∼ β, 477 O 1 A O H(M, ), 527 , 4, 30, 115 A OD(E), 122 H1 (M, ), 527 r O(E), 104, 115 HdeR(M), 503 H 1 ,H1 , 140 (E), 111, 115 Dol,L2 Dol,L2∩E X, 46, 115 H q (X), H q (X, E), 126 Dol Dol ordp f , 14, 34 Hom(·, ·), 407, 518 ordp s, 104 hq , 165 h∗, 130 P h ⊗ h, 130 P1,28 h ,77 X P 1(M, A), 512 h∗ ,81 X ⊥, 397 ∗, 429, 433, 438, 509, 511, 522, 524 I ∗, 126, 429, 433, 480, 509, 510, 522, 524 ιE , 168 S π1(X, x0), 479 TM,(T M)C , 433 ∗ ∗ J T M ,(T M)C , 433 Jac(X), 307 PSL(n, F), 219 JF , 383, 450 Jθ,p, 307 Q QD(E), 122 K KX,40 R κs , 292 rank G, 525 RP2, 529 L λω, 457 S ∗ p,qT X,45 S, 378, 415 p,q ∗ ⊗ T X E, 110 sE (·, ·), SE (·, ·), 167, 168 ∗ ∗ r T M,r (T M)C, 437 sec z,22 r V, 409 #, 177 (ν) θ , 307 σ , 527 ◦ log z,21 S, 378, 415 Lp, Lp(X, μ), 379 p sin z,22 Lloc, 379 SL(n, F), 219 · · · · · · , L2 (ϕ), , L2 (ω,ϕ), , 2 ,54 p,q p,q L1(ϕ) ∗, ∗¯, 178 Notation Index 551

∗¯ , ∗¯#, 179 V , 416 VC, 409 V∗, 401, 407 T t A, 393 W tan z,22 ∧, 409 h, 136 W k, 532 ω,73 ϕ ,64 ⊗, 409, 412 Z Z (M, A), 505 TM,(TM)C, 429, 432 1 ∗ ∗ 1 A T M,(T M)C, 429, 433 Z (M, ), 513 Z1 (M, F), 503 Tθ , 307 deR p,q 1 A (T X) ,39 Z(M, ), 527 (T ∗X)p,q ,39 zζ ,22

~~Subject Index~~

A Biholomorphic, 18, 31 Abel–Jacobi embedding theorem, 192, 307 Biholomorphism, 18, 31 Abel’s theorem, 303 local, 18, 31 Accumulation point, 415 Bilinear function, 409, 412 Almost complex skew-symmetric, 409 structure, 311 symmetric, 409 integrable, 311 Bishop–Kodama localization theorem, 98 surface, 311 Bishop–Narasimhan–Remmert embedding Almost everywhere, 447 theorem, 191, 279 Ample, 293 Bolzano–Weierstrass property, 423 very, 293 Borel set, 377 Approximation of the identity, 391 Boundary Argument of a set, 415 function, 21 operator ∂, 505, 527 principle, 52, 202, 203, 227, 335, 337, 338 Atlas, 419 C Ck , 420 Canonical homology basis, 269 holomorphic, 26 Canonical line bundle, 40, 105 holomorphically equivalent, 26 degree of, 169 line bundle, 102 Cauchy integral formula, 6, 201, 202, 334 holomorphically equivalent, 102 Cauchy–Pompeiu integral formula, 6 Automorphism, 31, 194 Cauchy–Riemann equation group homogeneous, 4 of , 221 inhomogeneous, 7, 25 2 of C, 216 L solution of H, 222 for line-bundle-valued forms of type of P1, 218–220 (0, 0), 145 for line-bundle-valued forms of type (1, 0), 140 B for scalar-valued forms of type (0, 0), Banach space, 378 74 Barrier, 344 for scalar-valued forms of type (1, 0), Basis for a topology, 416 65 Behnke–Stein theorem, 89 local solution, 7 Bessel’s inequality, 404 Cauchy’s theorem, 6, 201, 202, 334

~~T. Napier, M. Ramachandran, An Introduction to Riemann Surfaces, Cornerstones, 553 DOI 10.1007/978-0-8176-4693-6, © Springer Science+Business Media, LLC 2011 554 Subject Index~~

Cechˇ Complexification, 409 1-form, 513 Connected, 416 exact, 513 component, 416 line integral of, 513 locally, 416 cohomology, 512, 526 Connecting homomorphism, 127 Chain rule, 384, 430 Connection canonical Change of variables formula, 386 for scalar-valued forms, 61 Characteristic function, 376 in a line bundle, 135, 136 Chordal Kähler form, 73, 76, 137 Continuous, 415 C∞ approximation, 390, 535 differential form k C line-bundle-valued, 111 atlas, 420 section, 103 differential form, 439 sequentially, 423 line-bundle-valued, 111 Continuous closed 1-form, 513 function, 382, 421 Convolution, 391 map, 383, 421 Coordinate transformation, 419 Ck section, 103 , 420 vector field, 435 holomorphic, 26 Coordinates Closed homogeneous, 290 differential form local Ck , 420 d, 443 local holomorphic, 26 ∂,47 Cotangent ¯ ∂, 47, 125 bundle, 433 set, 378, 415 (0, 1),39 sequentially, 423 (1, 0),39 Closure, 378, 415 complexified, 433 Coboundary operator δ, 527 holomorphic, 39 Cocycle relation, 90, 150, 154, 518 projections, 433 Coefficients real, 433 of a differential form, 45, 439 map, 430, 433 space, 429 of a vector field, 43, 435 (0, 1),39 Cohomology, 516 (1, 0),39 ˇ Cech, 512, 526 holomorphic, 39 de Rham, 503 Covering Dolbeault, 126 branched, 294–298 Dolbeault L2, 140 map, 483 singular, 504, 527 space, 483 Commutator, 534 C∞, 483 ∞ subgroup, 278 C equivalent, 487 Compact, 416 equivalent, 487 locally, 418 holomorphic, 192 relatively, 416 holomorphically equivalent, 192 universal, 489–491 Complete orthonormal Critical basis, 403 point, 435 set, 403 value, 435 Complex differentiability, 16, 17 Curvature, 64, 136 Complex manifold form, 64, 136 1-dimensional, 26 negative, 64, 137 n-dimensional, 35 nonnegative, 64, 137 Complex torus, 28, 197 nonpositive, 64, 137 Subject Index 555

Curvature (cont.) Dolbeault of a Kähler form, 73 cohomology, 126 positive, 64, 137, 147, 160, 162 class, 126 zero, 64, 137 exact sequence, 128 2 Cycle, 505 L cohomology, 140 lemma, 50 D Dominated convergence theorem, 378, 457 ∂¯-Cauchy integral formula, 6 Dual De Rham basis, 407 cohomology, 503 line bundle, 107 pairing, 509, 522 space theorem, 524, 525 algebraic, 407 Deck transformation, 492 norm, 401 Defining E function, 119 Edge of a singular 2-simplex, 527 local, 120 Embedding section, 119 Abel–Jacobi, 307 Degree C∞, 467 of a divisor, 124 holomorphic of a line bundle, 124 into Cn, 279–290 Diffeomorphism, 383, 421 into Pn, 291 Differential, 383, 429, 433 Entire function, 4 Differential form, 439 Evenly covered, 483 line-bundle-valued, 111 Exact differential form negative, 452 d, 443 negative part, 453 Ck, 48, 125, 443 on a Riemann surface, 45–52 ∂,48 positive, 452 ∂¯, 48, 125 positive part, 453 locally, 48, 125, 443 sequence Exact sequence infimum, 453 Dolbeault, 128 limit inferior, 453 of sheaves, 119 limit superior, 453 Exhaustion supremum, 453 by sets, 427 Differential operator, 392, 462 function, 77, 427 Dirichlet problem, 338–350 strictly subharmonic, 82 Discrete, 415 Exponential function, 21, 22 Disjoint union, 28, 417, 421 real, 388 Distributional Exterior derivative, 442 ¯ ∂, 60, 139 Exterior product, 409 D, 62, 139 differential operator, 394, 463 F solution, 394, 463 Fatou’s lemma, 378, 457 Divisor, 119–124 Finite type, 277 defining section associated to, 121 topological, 277 effective, 120 Finiteness theorem, 164 line bundle associated to, 121 Flat operator, 177 linearly equivalent, 120 Formal of a line bundle map, 123 adjoint, 394 of a section, 119 transpose, 393 principal, 119 Fourier coefficients, 404 solution of, 119, 152, 303 Friedrichs lemma, 390 weak solution of, 152, 303 strong version, 535 556 Subject Index

Fubini’s theorem, 380 removal of tubes, 243–248 Fundamental estimate section, 103 for scalar-valued forms, 64 vector ﬁeld, 43 in a line bundle, 137 Holomorphically compatible on an almost complex surface, 328 local charts, 26 Fundamental group, 479 local trivializations, 102 Fundamental theorem of algebra, 35 Homeomorphism, 416 Homology, 506, 519 G canonical basis for, 267–273 Gap, 174 singular, 278, 507, 511, 520, 527 sequence, 174 Homotopy, 477 generic, 175 Hurwitz’s theorem, 301 hyperelliptic, 175 Hyperelliptic General linear group, 219 gap sequence, 175 Genus, 165, 169 involution, 300 Germ, 115, 117, 428 point, 175 Goursat’s theorem, 17 Riemann surface, 174, 299 Gram–Schmidt orthonormalization process, Hyperplane bundle, 105 403 curvature, 137 Group action free, 493 Hermitian metric in, 129 properly discontinuous, 493 quotient by, 493 I Identity theorem, 13, 31 H Immersion ∞ Hahn–Banach theorem, 402 C , 467 Harmonic, 64, 339 holomorphic 1-form, 186 into Cn, 279 Hartogs–Rosenthal theorem, 98 into Pn, 291 Hausdorff, 416 into a complex torus, 306 Hilbert space, 399 Inner product Hodge Hermitian, 397 conjugate star, 178 real, 397 conjugate star ﬂat operator, 179 Integral, 376 conjugate star sharp operator, 179 differentiation past, 384 decomposition of a differential form, 454 ¯ for ∂, 181 Interior, 378, 415 for scalar-valued forms, 185 Inverse function theorem star, 178 C∞, 384, 465 Holomorphic holomorphic, 18, 43 atlas, 26 attachment, 35–38 J of caps, 207–209 Jacobi variety, 307 of tubes, 36, 241–263 Jacobian determinant, 383, 450 function in C,4 Jordan curve, 228 on a Riemann surface, 30 theorem, 333 map, 30 into Cn, 279 K into Pn, 291 Kähler into a complex torus, 306 form, 51 local representation, 32 metric, 130 1-form, 46 Kodaira embedding theorem, 191, 294 line-bundle-valued, 111 Koebe uniformization, 212 Subject Index 557

L M Lattice, 28, 306 Manifold, 419 period, 307 Ck, 420 Laurent series, 15 complex Law of cosines, 398 of dimension 1, 26 Lifting, 483 product, 420 holomorphic, 193 real analytic, 420 theorem, 484 smooth, 420 Limit point, 415 Maximum principle, 14, 31 Line bundle for harmonic functions, 339 associated to a divisor, 121 for subharmonic functions, 339 dual, 107 Mean value property, 11, 341 Hermitian, 128 Measurable holomorphic, 103 differential form, 448 homomorphism, 107 line-bundle-valued, 111 isomorphism, 107 function, 375, 447 map, 107 map, 375, 447 set-theoretic, 101 section, 103 tensor product, 107 set, 375, 447 Line integral, 461 vector field, 448 along a 1-chain, 506, 520 Measure along a continuous path, 499 complete, 375 Linear functional, 407 completion of, 376 bounded, 401 counting, 376 Liouville’s theorem, 35 for a nonnegative form, 457 Lipschitz, 384 Lebesgue, 377 Local chart, 419 outer, 377 Ck , 420 positive, 375 holomorphic, 26 space, 375 Logarithmic function, 21, 22 Mergelyan–Bishop theorem, 97 real, 388 Meromorphic Loop, 417 function, 33 Lp 1-form, 46 norm, 379 space, 379 existence of, 67 p line-bundle-valued, 111 Lloc differential form, 458 section, 103 line-bundle-valued, 111 Metric function, 379 Hermitian, 128 section, 103 dual, 130 L2 tensor product, 130 inner product Kähler, 130 of functions, 399 Riemannian, 475 of line-bundle-valued forms, 132 Mittag-Leffler theorem, 88, 97, 160 of scalar-valued forms, 54 for a line bundle, 148, 160 norm Möbius of a function, 399 band, 449 of a line-bundle-valued form, 132 transformation, 219 of a scalar-valued form, 54 Mollification, 391 space Mollifier, 391 of functions, 399 Monotone convergence theorem, 378, 457 of line-bundle-valued forms, 132, 133 Montel’s theorem, 10, 72 of scalar-valued forms, 55 Multiplicity, 33 558 Subject Index

N Pole, 33 Neighborhood, 415 of a meromorphic 1-form, 46 Net, 423 of a meromorphic section, 103 Nongap, 174 simple, 34, 46, 104 Nonorientable, 450 Potential, 443 Nonseparating, 239 local, 443 Norm, 378 Power function, 22 complete, 378 Power series, 12, 13 for an inner product, 397 Primitive, 4 Principal part, 202 O Product One-point compactification, 418 path, 478 Open rule, 430 mapping theorem, 14, 31 topology, 417 Riemann surface, 27 Projection map, 102 set, 378, 415 Projective space, 290 Order, 14, 34, 46, 104 Projectivized special linear group, 219 of a pole, 34, 46 Proper map, 427 of a zero, 14, 34, 46 Pullback, 429, 433, 438, 509, 511, 522, 524 Orientable, 450 Pushforward, 429, 433, 480, 509, 510, 522, double cover, 371, 462 524 topological surface, 364 Pythagorean theorem, 398 Orientation, 450 Q compatible, 450 q-boundary, 527 equivalent, 450 q-chain, 505, 526 in a vector space, 411 q-coboundary, 527 induced on a boundary, 459 q-cochain, 527 preserving, 451 q-cocycle, 527 Oriented, 450 Quotient positively, 450 map, 417 Orthogonal, 397 space, 417 decomposition, 399 topology, 417 projection, 399 Orthonormal, 397 R Osgood–Taylor–Carathéodory theorem, 206 Rank, 525 Real projective plane, 529 P Realification, 408 Parallelogram law, 398 Regular value, 435 Partition of unity, 427 Regularity Path, 417 first-order, 532 k C , 422 for ∂¯, 63, 324 connected, 418 for ∂/∂z¯,10 locally, 418 Regularization, 391 piecewise Ck, 422 Representation of a section, 103 Path homotopy, 477 Residue, 53 Perron method, 338 theorem, 53, 201, 203, 227, 335, 338 Planar, 211 Reverse path, 478 Poincaré Riemann mapping theorem, 191 duality, 187 in the plane, 204 lemma, 443 Riemann sphere, 28 Poisson Riemann surface, 26 formula, 343 open, 27 kernel, 343 Riemann–Hurwitz formula, 299 Polar coordinates, 387 Riemann–Roch formula, 165, 169 Subject Index 559

Riemann’s extension theorem, 11, 31 Singular 2-simplex, 526 Rouché’s theorem, 52, 202, 203, 227, 335, 338 Sobolev lemma, 541 Runge Sobolev space, 532 approximation theorem, 91 Special linear group, 219 for a line bundle, 151 Stalk, 115, 117, 428 approximation theorem with poles, 92, 97 Standard 2-simplex, 526 for a line bundle, 151 Stereographic projection, 28 open set Stokes’ theorem, 460 holomorphically, 96 Stratification, 286 topologically, 77 Strictly subharmonic, 64 Strictly superharmonic, 64 S Strong deformation retraction, 363 Sard’s theorem, 263, 276, 331, 449 Structure Schönflies’ theorem, 333 Ck, 420 proof of, 350–356 complex analytic, 26 Schwarz inequality, 398 holomorphic, 26 Second countable, 416 Subharmonic, 64, 339 Section, 103 Submanifold, 422 Ck , 103 Submersion, 467 continuous, 103 Superharmonic, 64 defining Support, 426 associated to a divisor, 121 Surface, 419 holomorphic, 103 Ck, 420 p Lloc, 103 Riemann, 26 measurable, 103 smooth, 420 meromorphic, 103 of a sheaf, 117 T Separable, 403 Tangent Separating, 239 bundle, 432 Sequence (0, 1),39 Cauchy, 378 (1, 0),39 convergent, 378, 423 complexified, 432 divergent, 378, 423 holomorphic, 39 limit of, 378, 423 projections, 433 Serre real, 432 duality, 169 map, 429, 433 mapping, 168 space, 429 pairing, 168 (0, 1),39 Sharp operator, 177 (1, 0),39 Sheaf, 115–119 complexified, 429 constant, 117 holomorphic, 39 definition, 117 real, 429 isomorphism, 117 vector, 429 mapping, 117 (0, 1),39 of holomorphic 1-forms, 115 (1, 0),39 of holomorphic sections, 115 complex, 429 of meromorphic 1-forms, 115 holomorphic, 39 of meromorphic sections, 115 real, 429 skyline, 122 to a path, 434 skyscraper, 116 Tautological bundle, 108 Sheet interchange, 300 Taylor’s formula, 385 σ -algebra, 375 Tensor product, 409, 412 Simple function, 376 line bundle, 107 Simply connected, 480 Tietze extension theorem, 348 560 Subject Index

Topological hull, 77 Vector ﬁeld, 435 extended, 81 Volume form, 453 Topological space, 415 Topology, 415 W product, 417 Weak subspace, 415 compactness, 403 Torsion, 226, 332, 371, 521 convergence, 403 free, 226, 332, 371, 521 solution, 394, 463 Torus Wedge product, 409 complex, 28, 306 Weierstrass real, 30 gap, 174 Total space, 102 gap theorem, 174 Triangulation, 282, 286, 370 nongap, 174 Trigonometric functions, 22 point, 176 real, 387 theorem, 151 weight, 175 Trivialization Weight function, 54 global, 102 Winding number, 201, 226 local, 102 Wronskian, 175 Type (p, q), 43, 110 Z U Zero Upper half-plane, 198 of a holomorphic function, 14, 33 of a meromorphic 1-form, 46 V of a meromorphic function, 34 Vanishing derivatives, 444 of a meromorphic section, 103 Vanishing theorem, 145, 148, 163 simple, 14, 33, 34, 46, 104