Riemann Surfaces by Way of Complex Analytic Geometry Dror Varolin Graduate Studies in Mathematics Volume 125 American Mathematical Society http://dx.doi.org/10.1090/gsm/125 Riemann Surfaces by Way of Complex Analytic Geometry Riemann Surfaces by Way of Complex Analytic Geometry Dror Varolin Graduate Studies in Mathematics Volume 125 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE David Cox (Chair) Rafe Mazzeo Martin Scharlemann Gigliola Staffilani 2010 Mathematics Subject Classification. Primary 30F10, 30F15, 30F30, 30F45, 30F99, 30G99, 31A05, 31A99, 32W05. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-125 Library of Congress Cataloging-in-Publication Data Varolin, Dror, 1970– Riemann surfaces by way of complex analytic geometry / Dror Varolin. p. cm. — (Graduate studies in mathematics ; v. 125) Includes bibliographical references and index. ISBN 978-0-8218-5369-6 (alk. paper) 1. Riemann surfaces. 2. Functions of complex variables. 3. Geometry, Analytic. I. Title. II. Series. QA333.V37 2011 515.93—dc23 2011014621 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2011 by Dror Varolin. All rights reserved. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 161514131211 Dedicated to Erin Contents Preface xi Chapter 1. Complex Analysis 1 §1.1. Green’s Theorem and the Cauchy-Green Formula 1 §1.2. Holomorphic functions and Cauchy Formulas 2 §1.3. Power series 3 §1.4. Isolated singularities of holomorphic functions 4 §1.5. The Maximum Principle 8 §1.6. Compactness theorems 9 §1.7. Harmonic functions 11 §1.8. Subharmonic functions 14 §1.9. Exercises 19 Chapter 2. Riemann Surfaces 21 §2.1. Definition of a Riemann surface 21 §2.2. Riemann surfaces as smooth 2-manifolds 23 §2.3. Examples of Riemann surfaces 25 §2.4. Exercises 36 Chapter 3. Functions on Riemann Surfaces 37 §3.1. Holomorphic and meromorphic functions 37 §3.2. Global aspects of meromorphic functions 42 §3.3. Holomorphic maps between Riemann surfaces 45 §3.4. An example: Hyperelliptic surfaces 54 vii viii Contents §3.5. Harmonic and subharmonic functions 57 §3.6. Exercises 59 Chapter 4. Complex Line Bundles 61 §4.1. Complex line bundles 61 §4.2. Holomorphic line bundles 65 §4.3. Two canonically defined holomorphic line bundles 66 §4.4. Holomorphic vector fields on a Riemann surface 70 §4.5. Divisors and line bundles 74 §4.6. Line bundles over Pn 79 §4.7. Holomorphic sections and projective maps 81 §4.8. A finiteness theorem 84 §4.9. Exercises 85 Chapter 5. Complex Differential Forms 87 §5.1. Differential (1, 0)-forms 87 § ∗0,1 5.2. TX and (0, 1)-forms 89 § ∗ 5.3. TX and 1-forms 89 § 1,1 5.4. ΛX and (1, 1)-forms 90 §5.5. Exterior algebra and calculus 90 §5.6. Integration of forms 92 §5.7. Residues 95 §5.8. Homotopy and homology 96 §5.9. Poincare´ and Dolbeault Lemmas 98 §5.10. Dolbeault cohomology 99 §5.11. Exercises 100 Chapter 6. Calculus on Line Bundles 101 §6.1. Connections on line bundles 101 §6.2. Hermitian metrics and connections 104 §6.3. (1, 0)-connections on holomorphic line bundles 105 §6.4. The Chern connection 106 §6.5. Curvature of the Chern connection 107 §6.6. Chern numbers 109 § 1,0 6.7. Example: The holomorphic line bundle TX 111 §6.8. Exercises 112 Chapter 7. Potential Theory 115 Contents ix §7.1. The Dirichlet Problem and Perron’s Method 115 §7.2. Approximation on open Riemann surfaces 126 §7.3. Exercises 130 Chapter 8. Solving ∂¯ for Smooth Data 133 §8.1. The basic result 133 §8.2. Triviality of holomorphic line bundles 134 §8.3. The Weierstrass Product Theorem 135 §8.4. Meromorphic functions as quotients 135 §8.5. The Mittag-Leffler Problem 136 §8.6. The Poisson Equation on open Riemann surfaces 140 §8.7. Exercises 143 Chapter 9. Harmonic Forms 145 §9.1. The definition and basic properties of harmonic forms 145 §9.2. Harmonic forms and cohomology 149 §9.3. The Hodge decomposition of E (X) 151 §9.4. Existence of positive line bundles 157 §9.5. Proof of the Dolbeault-Serre isomorphism 161 §9.6. Exercises 161 Chapter 10. Uniformization 165 §10.1. Automorphisms of the complex plane, projective line, and unit disk 165 §10.2. A review of covering spaces 166 §10.3. The Uniformization Theorem 168 §10.4. Proof of the Uniformization Theorem 174 §10.5. Exercises 175 Chapter 11. Hormander’s¨ Theorem 177 §11.1. Hilbert spaces of sections 177 §11.2. The Basic Identity 180 §11.3. Hormander’s¨ Theorem 183 §11.4. Proof of the Korn-Lichtenstein Theorem 184 §11.5. Exercises 195 Chapter 12. Embedding Riemann Surfaces 197 §12.1. Controlling the derivatives of sections 198 §12.2. Meromorphic sections of line bundles 201 x Contents §12.3. Plenitude of meromorphic functions 202 §12.4. Kodaira’s Embedding Theorem 202 §12.5. Narasimhan’s Embedding Theorem 204 §12.6. Exercises 210 Chapter 13. The Riemann-Roch Theorem 211 §13.1. The Riemann-Roch Theorem 211 §13.2. Some corollaries 217 Chapter 14. Abel’s Theorem 223 §14.1. Indefinite integration of holomorphic forms 223 §14.2. Riemann’s Bilinear Relations 225 §14.3. The Reciprocity Theorem 228 §14.4. Proof of Abel’s Theorem 229 §14.5. A discussion of Jacobi’s Inversion Theorem 231 Bibliography 233 Index 235 Preface What This book presents certain parts of the basic theory of Riemann surfaces through methods of complex analytic geometry, many of which were developed at one time or another in the past 50 years or so. The first chapter of the book presents a rapid review of the elementary part of basic complex analysis, introducing holomorphic, meromorphic, harmonic, and subharmonic functions and establishing some of the basic local properties of these functions. In the second chapter we define Riemann surfaces and give some exam- ples, both of a concrete and abstract nature. The most abstract example is that any oriented 2-dimensional real manifold has a complex structure. In fact, we state the classical theorem of Korn-Lichtenstein: given any metric on an oriented surface, there is a complex structure that makes that metric locally conformal to the Eu- clidean metric in any holomorphic chart. The proof of this theorem is given only in Chapter 11. Riemann surfaces are non-linear spaces on which, roughly speaking, local complex analysis makes sense, and so we can extend the inherently local notions of holomorphic, meromorphic, harmonic, and subharmonic functions to such sur- faces. This point of view starts the third chapter, but by the end we prove our first main result, namely the constancy of the degree of a holomorphic map between two compact Riemann surfaces. With the recollection of some basic topology of surfaces, we are then able to establish the Riemann-Hurwitz Formula, which puts constraints on holomorphic maps between compact Riemann surfaces. The chap- ter ends with a proof of the Harnack Principle. Up to this point, the presentation is mostly classical, in the sense that it does not differ markedly from other presen- tations of the same material (though perhaps the particular blend of the ideas is unusual). xi xii Preface The novelty of the book begins in the fourth chapter, where we first define and give examples of complex and holomorphic line bundles. There are a number of important, albeit elementary, ideas that are presented in this chapter, and among these the correspondence between divisors and holomorphic line bundles with sec- tions is first established. Another key idea is the correspondence between finite- dimensional vector spaces of sections of a holomorphic line bundle and projective maps. The chapter ends with an elementary proof of the finite-dimensionality of the space of all global holomorphic sections of a holomorphic line bundle over a compact Riemann surface. The fifth chapter begins by defining complex differential forms. These forms play a central role in the rest of the book, but in this particular chapter we focus on their formal aspects—how to integrate them and how to describe the topology of a Riemann surface with them. In the sixth chapter we develop the theory of differentiation for sections of complex line bundles (which is known as defining a connection). A complex line bundle admits many connections and, unlike the trivial bundle, there is no canon- ical choice of a connection for a general complex line bundle. The first goal of the chapter is to describe geometric structures that isolate a subset of the possible ways to differentiate sections. If a geometry is symmetric enough, there will be a canonical way to define differentiation of sections. The first instance of such a scenario is the classical Levi-Civitaˇ Theorem for vector fields on a Riemannian manifold. The main fact we establish in Chapter 6, due to Chern, is that for each holomorphic line bundle with Hermitian metric there is a unique connection that is compatible with these structures in the appropriate sense.