A First Look at K¨Ahler Manifolds
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Hodge Decomposition
Hodge Decomposition Daniel Lowengrub April 27, 2014 1 Introduction All of the content in these notes in contained in the book Differential Analysis on Complex Man- ifolds by Raymond Wells. The primary objective here is to highlight the steps needed to prove the Hodge decomposition theorems for real and complex manifolds, in addition to providing intuition as to how everything fits together. 1.1 The Decomposition Theorem On a given complex manifold X, there are two natural cohomologies to consider. One is the de Rham Cohomology which can be defined on a general, possibly non complex, manifold. The second one is the Dolbeault cohomology which uses the complex structure. We’ll quickly go over the definitions of these cohomologies in order to set notation but for a precise discussion I recommend Huybrechts or Wells. If X is a manifold, we can define the de Rahm Complex to be the chain complex 0 E(Ω0)(X) −d E(Ω1)(X) −d ::: Where Ω = T∗ denotes the cotangent vector bundle and Ωk is the alternating product Ωk = ΛkΩ. In general, we’ll use the notation! E(E) to! denote the sheaf! of sections associated to a vector bundle E. The boundary operator is the usual differentiation operator. The de Rham cohomology of the manifold X is defined to be the cohomology of the de Rham chain complex: d Ker(E(Ωn)(X) − E(Ωn+1)(X)) n (X R) = HdR , d Im(E(Ωn-1)(X) − E(Ωn)(X)) ! We can also consider the de Rham cohomology with complex coefficients by tensoring the de Rham complex with C in order to obtain the de Rham complex! with complex coefficients 0 d 1 d 0 E(ΩC)(X) − E(ΩC)(X) − ::: where ΩC = Ω ⊗ C and the differential d is linearly extended. -
Riemann Surfaces
RIEMANN SURFACES AARON LANDESMAN CONTENTS 1. Introduction 2 2. Maps of Riemann Surfaces 4 2.1. Defining the maps 4 2.2. The multiplicity of a map 4 2.3. Ramification Loci of maps 6 2.4. Applications 6 3. Properness 9 3.1. Definition of properness 9 3.2. Basic properties of proper morphisms 9 3.3. Constancy of degree of a map 10 4. Examples of Proper Maps of Riemann Surfaces 13 5. Riemann-Hurwitz 15 5.1. Statement of Riemann-Hurwitz 15 5.2. Applications 15 6. Automorphisms of Riemann Surfaces of genus ≥ 2 18 6.1. Statement of the bound 18 6.2. Proving the bound 18 6.3. We rule out g(Y) > 1 20 6.4. We rule out g(Y) = 1 20 6.5. We rule out g(Y) = 0, n ≥ 5 20 6.6. We rule out g(Y) = 0, n = 4 20 6.7. We rule out g(C0) = 0, n = 3 20 6.8. 21 7. Automorphisms in low genus 0 and 1 22 7.1. Genus 0 22 7.2. Genus 1 22 7.3. Example in Genus 3 23 Appendix A. Proof of Riemann Hurwitz 25 Appendix B. Quotients of Riemann surfaces by automorphisms 29 References 31 1 2 AARON LANDESMAN 1. INTRODUCTION In this course, we’ll discuss the theory of Riemann surfaces. Rie- mann surfaces are a beautiful breeding ground for ideas from many areas of math. In this way they connect seemingly disjoint fields, and also allow one to use tools from different areas of math to study them. -
Holomorphic Embedding of Complex Curves in Spaces of Constant Holomorphic Curvature (Wirtinger's Theorem/Kaehler Manifold/Riemann Surfaces) ISSAC CHAVEL and HARRY E
Proc. Nat. Acad. Sci. USA Vol. 69, No. 3, pp. 633-635, March 1972 Holomorphic Embedding of Complex Curves in Spaces of Constant Holomorphic Curvature (Wirtinger's theorem/Kaehler manifold/Riemann surfaces) ISSAC CHAVEL AND HARRY E. RAUCH* The City College of The City University of New York and * The Graduate Center of The City University of New York, 33 West 42nd St., New York, N.Y. 10036 Communicated by D. C. Spencer, December 21, 1971 ABSTRACT A special case of Wirtinger's theorem ever, the converse is not true in general: the complex curve asserts that a complex curve (two-dimensional) hob-o embedded as a real minimal surface is not necessarily holo- morphically embedded in a Kaehler manifold is a minimal are obstructions. With the by morphically embedded-there surface. The converse is not necessarily true. Guided that we view the considerations from the theory of moduli of Riemann moduli problem in mind, this fact suggests surfaces, we discover (among other results) sufficient embedding of our complex curve as a real minimal surface topological aind differential-geometric conditions for a in a Kaehler manifold as the solution to the differential-geo- minimal (Riemannian) immersion of a 2-manifold in for our present mapping problem metric metric extremal problem complex projective space with the Fubini-Study as our infinitesimal moduli, differential- to be holomorphic. and that we then seek, geometric invariants on the curve whose vanishing forms neces- sary and sufficient conditions for the minimal embedding to be INTRODUCTION AND MOTIVATION holomorphic. Going further, we observe that minimal surface It is known [1-5] how Riemann's conception of moduli for con- evokes the notion of second fundamental forms; while the formal mapping of homeomorphic, multiply connected Rie- moduli problem, again, suggests quadratic differentials. -
Spectral Data for L^2 Cohomology
University of Pennsylvania ScholarlyCommons Publicly Accessible Penn Dissertations 2018 Spectral Data For L^2 Cohomology Xiaofei Jin University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/edissertations Part of the Mathematics Commons Recommended Citation Jin, Xiaofei, "Spectral Data For L^2 Cohomology" (2018). Publicly Accessible Penn Dissertations. 2815. https://repository.upenn.edu/edissertations/2815 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/edissertations/2815 For more information, please contact [email protected]. Spectral Data For L^2 Cohomology Abstract We study the spectral data for the higher direct images of a parabolic Higgs bundle along a map between a surface and a curve with both vertical and horizontal parabolic divisors. We describe the cohomology of a parabolic Higgs bundle on a curve in terms of its spectral data. We also calculate the integral kernel that reproduces the spectral data for the higher direct images of a parabolic Higgs bundle on the surface. This research is inspired by and extends the works of Simpson [21] and Donagi-Pantev-Simpson [7]. Degree Type Dissertation Degree Name Doctor of Philosophy (PhD) Graduate Group Mathematics First Advisor Tony Pantev Keywords Higgs bundle, Monodromy weight filtration, Nonabelian Hodge theory, Parabolic L^2 Dolbeault complex, Root stack, Spectral correspondence Subject Categories Mathematics This dissertation is available at ScholarlyCommons: https://repository.upenn.edu/edissertations/2815 SPECTRAL DATA FOR L2 COHOMOLOGY Xiaofei Jin A DISSERTATION in Mathematics Presented to the Faculties of the University of Pennsylvania in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy 2018 Supervisor of Dissertation Tony Pantev, Class of 1939 Professor in Mathematics Graduate Group Chairperson Wolfgang Ziller, Francis J. -
Complex Manifolds
Complex Manifolds Lecture notes based on the course by Lambertus van Geemen A.A. 2012/2013 Author: Michele Ferrari. For any improvement suggestion, please email me at: [email protected] Contents n 1 Some preliminaries about C 3 2 Basic theory of complex manifolds 6 2.1 Complex charts and atlases . 6 2.2 Holomorphic functions . 8 2.3 The complex tangent space and cotangent space . 10 2.4 Differential forms . 12 2.5 Complex submanifolds . 14 n 2.6 Submanifolds of P ............................... 16 2.6.1 Complete intersections . 18 2 3 The Weierstrass }-function; complex tori and cubics in P 21 3.1 Complex tori . 21 3.2 Elliptic functions . 22 3.3 The Weierstrass }-function . 24 3.4 Tori and cubic curves . 26 3.4.1 Addition law on cubic curves . 28 3.4.2 Isomorphisms between tori . 30 2 Chapter 1 n Some preliminaries about C We assume that the reader has some familiarity with the notion of a holomorphic function in one complex variable. We extend that notion with the following n n Definition 1.1. Let f : C ! C, U ⊆ C open with a 2 U, and let z = (z1; : : : ; zn) be n the coordinates in C . f is holomorphic in a = (a1; : : : ; an) 2 U if f has a convergent power series expansion: +1 X k1 kn f(z) = ak1;:::;kn (z1 − a1) ··· (zn − an) k1;:::;kn=0 This means, in particular, that f is holomorphic in each variable. Moreover, we define OCn (U) := ff : U ! C j f is holomorphicg m A map F = (F1;:::;Fm): U ! C is holomorphic if each Fj is holomorphic. -
Kähler Manifolds and Holonomy
K¨ahlermanifolds, lecture 3 M. Verbitsky K¨ahler manifolds and holonomy lecture 3 Misha Verbitsky Tel-Aviv University December 21, 2010, 1 K¨ahlermanifolds, lecture 3 M. Verbitsky K¨ahler manifolds DEFINITION: An Riemannian metric g on an almost complex manifiold M is called Hermitian if g(Ix; Iy) = g(x; y). In this case, g(x; Iy) = g(Ix; I2y) = −g(y; Ix), hence !(x; y) := g(x; Iy) is skew-symmetric. DEFINITION: The differential form ! 2 Λ1;1(M) is called the Hermitian form of (M; I; g). REMARK: It is U(1)-invariant, hence of Hodge type (1,1). DEFINITION: A complex Hermitian manifold (M; I; !) is called K¨ahler if d! = 0. The cohomology class [!] 2 H2(M) of a form ! is called the K¨ahler class of M, and ! the K¨ahlerform. 2 K¨ahlermanifolds, lecture 3 M. Verbitsky Levi-Civita connection and K¨ahlergeometry DEFINITION: Let (M; g) be a Riemannian manifold. A connection r is called orthogonal if r(g) = 0. It is called Levi-Civita if it is torsion-free. THEOREM: (\the main theorem of differential geometry") For any Riemannian manifold, the Levi-Civita connection exists, and it is unique. THEOREM: Let (M; I; g) be an almost complex Hermitian manifold. Then the following conditions are equivalent. (i)( M; I; g) is K¨ahler (ii) One has r(I) = 0, where r is the Levi-Civita connection. 3 K¨ahlermanifolds, lecture 3 M. Verbitsky Holonomy group DEFINITION: (Cartan, 1923) Let (B; r) be a vector bundle with connec- tion over M. For each loop γ based in x 2 M, let Vγ;r : Bjx −! Bjx be the corresponding parallel transport along the connection. -
Complex Cobordism and Almost Complex Fillings of Contact Manifolds
MSci Project in Mathematics COMPLEX COBORDISM AND ALMOST COMPLEX FILLINGS OF CONTACT MANIFOLDS November 2, 2016 Naomi L. Kraushar supervised by Dr C Wendl University College London Abstract An important problem in contact and symplectic topology is the question of which contact manifolds are symplectically fillable, in other words, which contact manifolds are the boundaries of symplectic manifolds, such that the symplectic structure is consistent, in some sense, with the given contact struc- ture on the boundary. The homotopy data on the tangent bundles involved in this question is finding an almost complex filling of almost contact manifolds. It is known that such fillings exist, so that there are no obstructions on the tangent bundles to the existence of symplectic fillings of contact manifolds; however, so far a formal proof of this fact has not been written down. In this paper, we prove this statement. We use cobordism theory to deal with the stable part of the homotopy obstruction, and then use obstruction theory, and a variant on surgery theory known as contact surgery, to deal with the unstable part of the obstruction. Contents 1 Introduction 2 2 Vector spaces and vector bundles 4 2.1 Complex vector spaces . .4 2.2 Symplectic vector spaces . .7 2.3 Vector bundles . 13 3 Contact manifolds 19 3.1 Contact manifolds . 19 3.2 Submanifolds of contact manifolds . 23 3.3 Complex, almost complex, and stably complex manifolds . 25 4 Universal bundles and classifying spaces 30 4.1 Universal bundles . 30 4.2 Universal bundles for O(n) and U(n).............. 33 4.3 Stable vector bundles . -
Cohomology of the Complex Grassmannian
COHOMOLOGY OF THE COMPLEX GRASSMANNIAN JONAH BLASIAK Abstract. The Grassmannian is a generalization of projective spaces–instead of looking at the set of lines of some vector space, we look at the set of all n-planes. It can be given a manifold structure, and we study the cohomology ring of the Grassmannian manifold in the case that the vector space is complex. The mul- tiplicative structure of the ring is rather complicated and can be computed using the fact that for smooth oriented manifolds, cup product is Poincar´edual to intersection. There is some nice com- binatorial machinery for describing the intersection numbers. This includes the symmetric Schur polynomials, Young tableaux, and the Littlewood-Richardson rule. Sections 1, 2, and 3 introduce no- tation and the necessary topological tools. Section 4 uses linear algebra to prove Pieri’s formula, which describes the cup product of the cohomology ring in a special case. Section 5 describes the combinatorics and algebra that allow us to deduce all the multi- plicative structure of the cohomology ring from Pieri’s formula. 1. Basic properties of the Grassmannian The Grassmannian can be defined for a vector space over any field; the cohomology of the Grassmannian is the best understood for the complex case, and this is our focus. Following [MS], the complex Grass- m+n mannian Gn(C ) is the set of n-dimensional complex linear spaces, or n-planes for brevity, in the complex vector space Cm+n topologized n+m as follows: Let Vn(C ) denote the subspace of the n-fold direct sum Cm+n ⊕ .. -
Hermitian Geometry
1 Hermitian Geometry Simon M. Salamon Introduction These notes are based on graduate courses given by the author in 1998 and 1999. The main idea was to introduce a number of aspects of the theory of complex and symplectic structures that depend on the existence of a compatible Riemannian metric. The present notes deal mostly with the complex case, and are designed to introduce a selection of topics and examples in differential geometry, accessible to anyone with an acquaintance of the definitions of smooth manifolds and vector bundles. One basic problem is, given a Riemannian manifold (M; g), to determine whether there exists an orthogonal complex structure J on M , and to classify all such J . A second problem is, given (M; g), to determine whether there exists an orthogonal almost-complex structure for which the corresponding 2-form ! is closed. The first problem is more tractible in the sense that there is an easily identifiable curvature obstruction to the existence of J , and this obstruction can be interpreted in terms of an auxiliary almost-complex structure on the twistor space of M . This leads to a reasonably complete resolution of the problem in the first non-trivial case, that of four real dimensions. The theory has both local and global aspects that are illustrated in Pontecorvo's classification [89] of bihermitian anti-self-dual 4-manifolds. Much less in known in higher dimensions, and some of the basic classification questions concerning orthogonal complex structures on Riemannian 6-manifolds remain unanswered. The second problem encompasses the so-called Goldberg conjecture. -
Notes on Riemann Surfaces
NOTES ON RIEMANN SURFACES DRAGAN MILICIˇ C´ 1. Riemann surfaces 1.1. Riemann surfaces as complex manifolds. Let M be a topological space. A chart on M is a triple c = (U, ϕ) consisting of an open subset U M and a homeomorphism ϕ of U onto an open set in the complex plane C. The⊂ open set U is called the domain of the chart c. The charts c = (U, ϕ) and c′ = (U ′, ϕ′) on M are compatible if either U U ′ = or U U ′ = and ϕ′ ϕ−1 : ϕ(U U ′) ϕ′(U U ′) is a bijective holomorphic∩ ∅ function∩ (hence ∅ the inverse◦ map is∩ also holomorphic).−→ ∩ A family of charts on M is an atlas of M if the domains of charts form a covering of MA and any two charts in are compatible. Atlases and of M are compatibleA if their union is an atlas on M. This is obviously anA equivalenceB relation on the set of all atlases on M. Each equivalence class of atlases contains the largest element which is equal to the union of all atlases in this class. Such atlas is called saturated. An one-dimensional complex manifold M is a hausdorff topological space with a saturated atlas. If M is connected we also call it a Riemann surface. Let M be an Riemann surface. A chart c = (U, ϕ) is a chart around z M if z U. We say that it is centered at z if ϕ(z) = 0. ∈ ∈Let M and N be two Riemann surfaces. A continuous map F : M N is a holomorphic map if for any two pairs of charts c = (U, ϕ) on M and d =−→ (V, ψ) on N such that F (U) V , the mapping ⊂ ψ F ϕ−1 : ϕ(U) ϕ(V ) ◦ ◦ −→ is a holomorphic function. -
Arxiv:1208.0648V1 [Math.DG] 3 Aug 2012 4.6
CALCULUS AND INVARIANTS ON ALMOST COMPLEX MANIFOLDS, INCLUDING PROJECTIVE AND CONFORMAL GEOMETRY A. ROD GOVER AND PAWEŁ NUROWSKI Abstract. We construct a family of canonical connections and surrounding basic theory for almost complex manifolds that are equipped with an affine connection. This framework provides a uniform approach to treating a range of geometries. In particular we are able to construct an invariant and effi- cient calculus for conformal almost Hermitian geometries, and also for almost complex structures that are equipped with a projective structure. In the latter case we find a projectively invariant tensor the vanishing of which is necessary and sufficient for the existence of an almost complex connection compatible with the path structure. In both the conformal and projective setting we give torsion characterisations of the canonical connections and introduce certain interesting higher order invariants. Contents 1. Introduction 2 1.1. Conventions 4 2. Calculus on an almost complex affine manifold 5 2.1. Almost complex affine connections 5 2.2. Torsion and Integrability 6 2.3. Compatible affine connections 7 3. Almost Hermitian geometry 8 4. Conformal almost Hermitian manifolds 12 4.1. A canonical torsion free connection 12 4.2. Canonical conformal almost complex connections 14 4.3. Compatibility in the conformal setting 15 4.4. Characterising the distinguished connection 16 4.5. The Nearly Kähler Weyl condition 18 arXiv:1208.0648v1 [math.DG] 3 Aug 2012 4.6. The locally conformally almost Kähler condition 18 4.7. The Gray-Hervella types 20 4.8. Higher conformal invariants 20 4.9. Summary 25 5. Projective Geometry 25 5.1. -
Daniel Huybrechts
Universitext Daniel Huybrechts Complex Geometry An Introduction 4u Springer Daniel Huybrechts Universite Paris VII Denis Diderot Institut de Mathematiques 2, place Jussieu 75251 Paris Cedex 05 France e-mail: [email protected] Mathematics Subject Classification (2000): 14J32,14J60,14J81,32Q15,32Q20,32Q25 Cover figure is taken from page 120. Library of Congress Control Number: 2004108312 ISBN 3-540-21290-6 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2005 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Erich Kirchner, Heidelberg Typesetting by the author using a Springer KTjiX macro package Production: LE-TgX Jelonek, Schmidt & Vockler GbR, Leipzig Printed on acid-free paper 46/3142YL -543210 Preface Complex geometry is a highly attractive branch of modern mathematics that has witnessed many years of active and successful research and that has re- cently obtained new impetus from physicists' interest in questions related to mirror symmetry.