Riemann Surfaces By Way of Analytic Geometry

Riemann Surfaces By Way of Analytic Geometry Dror Varolin Preface The present book arose from the need to bridge what ...

Author: Dror Varolin

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Riemann Surfaces By Way of Analytic Geometry Dror Varolin

Preface The present book arose from the need to bridge what I perceived as a rather substantial gap between what graduate students at Stony Brook know after they have passed their qualifying exams, and higher dimensional complex analytic geometry in its present state. At present, the generic post-qual student at Stony Brook is relatively well-prepared in algebraic topology and differential geometry, but far from so in real and complex analysis, or partial differential equations. Fortunately, the amount of real analysis needed in the approach to Riemann surfaces presented in this book is rather minimal. The deepest results needed are the Hahn-Banach Theorem and the Spectral Theorem for compact, self-adjoint operators (and the latter is not used in a fundamental way). Courses in partial differential equations may often point in directions that typically do not lead to complex geometry, even insofar as the L2 -methods originated by Bochner and Kodaira in the compact setting and by Andreotti and Vesentinni, Hörmander, Kohn, and Morrey in general, and later developed by Bombieri, Catlin, Demailly, Siu, Skoda, and many others. Amazingly, even courses in complex analysis do not typically emphasize the points most important in the study of Riemann surfaces, focusing instead on issues of minimal regularity for solving the CauchyRiemann equations. One exception is the Riemann Mapping Theorem, one proof of which is rather similar to the proof of the Uniformization Theorem we give in Chapter 10. In this book, the Riemann Mapping Theorem, and any other form of classification, take a back seat while the drivers are results based on technique, and especially the applications of solving the ∂¯ (and sometimes ¯ equation. We present as many methods as possible for solving these equations, introducing and dis∂ ∂) cussing Green’s Functions and Runge-type approximation theorems for this purpose, and giving a proof of the Hodge Theorem using basic Hilbert and Sobolev space theory. Perhaps the centerpiece is Hörmander’s Theorem on solution of ∂¯ with L2 estimates. The Proof of Hörmander’s Theorem in one complex dimension simplifies greatly, because a certain boundary condition that arises in the functional analytic formulation of ¯ the ∂-problem on Hilbert spaces is a Dirichlet boundary condition, as opposed to its higher-dimensional ¯ and more temperamental relative, the ∂-Neumann boundary condition of Spencer-Kohn, which requires the introduction of the notion of pseudoconvexity of the boundary of a domain. Unlike many other books on Riemann surfaces, this book tries not to distnguish between compact and non-compact Riemann surfaces unless there is a natural reason to do so. Perhaps it is also best to confess as soon as possible that we have stuck to the classical, confusing terminology, calling a non-compact Riemann surface an open Riemann surface. (Classically the compact Riemann surfaces were called closed, but we did not adapt that part of the classical terminology.) Line bundles play a major role in the book, providing the backdrop and geometric motivation for much of what is done. The 1-dimensional aspect makes the Hermitian geometry rather easy to deal with, and gives the novice a gentle introduction to the higher dimensional differential geometry of Hermitian line bundles. For example the Kähler condition, which plays such a magical role in the higher-dimensional theory, is automatic. Moreover, the meaning of curvature of the line bundle, which is beautifully demonstrated in establishing what has come to be known as the Bochner-Kodaira Identity, is greatly simplified in complex dimension 1. (The Bochner-Kodaira Identity is used to estimate from below the smallest eigen-value of the Laplace-Beltrami operator on sections of a Hermitian holomorphic line bundle.) The identity is obtained through integration-by-parts. At a certain point one must interchange the order of some exterior differential operators. When a non-flat geometry is present, the commutator of these operators is non-trivial, and is the usual definition of curvature. If this commutator, which is a multiplier, is positive, then we obtain a positive lower bound for the smallest eigen-value of a certain geometric Laplacian, and this lower bound is precisely

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what is needed to apply the functional analytic method to prove Hörmander’s Theorem. The main application of Hörmander’s Theorem is to the existence of holomorphic sections of sufficiently positive line bundles. Using these sections, I prove the existence of non-trivial meromorphic functions on Riemann surfaces, non-trivial meromorphic sections of any holomorphic line bundle, and conclude with a proofs of the Kodaira Embedding Theorem and, almost simultaneously, the embedding theorem for open Riemann surfaces. By contrast with line bundles, I made the choice of avoiding both vector bundles and sheaves, two natural extensions of line bundles. The idea was to show simultaneously that (i) these tools are not needed in the basic theory of Riemann surfaces, and (ii) at times the absence of these tools makes the presentation cumbersome. A good example of (i) is the proof of Kodaira embedding without the artifice of sheaf cohomology, but rather by a simple-minded direct construction of certain sections using a beautiful idea first introduced by Bombieri. A nice example of (ii) is provided by a number of the results proved in Chapter 8, such as the Mittag-Leffler Theorem. A second example of (ii) is seen in our proof of the theorem of Riemann-Roch. In the latter we avoided the use of sheaves by “Serre Duality through residues”, i.e., the identification of (H 1 (X, OX (E))∗ and H 0 (X, O(KX ⊗E ∗ ) by use of the Residue Theorem. This approach leads to a simple but not particularly illuminating proof of the Riemann-Roch Theorem. The book ends with two classical results, namely Abel’s Theorem and the Riemann-Roch Theorem. These results constitute an anti-climax of sorts, since they require a number of techniques not particularly relevant to the rest of the book. In this regard, we do not add to what is already in the literature. The inclusion of the two theorems is motivated by seeing them as concluding remarks: Abel’s Theorem and its complement, Jacobi’s Inversion Theorem, are included because they provide a kind of classification for the most central figures in the book, holomorphic line bundles, while the Riemann-Roch Theorem allows one to sharpen the embedding theorem (or in the language of modern analytic geometry, give an effective embedding result). Of course, there are many glaring omissions that would appear in a standard treatise on Riemann surfaces. We do not discuss Weierstrass points and the finiteness of the automorphism group of a compact Riemann surface of genus at least two. Riemann’s Theta Functions are not studied in any great detail. We bring them up only on the torus as a demonstrative tool. As a consequence, we do not discuss Torelli’s Theorem. We also omit discussion of basic algebraic geometry of curves, and of monodromy. There are probably other omissions that I am not even aware of. Psychologically, the most difficult omission for me was that of a discussion of interpolation and sampling on so-called finite open Riemann surfaces. The theory of interpolation and sampling provides a natural setting (in fact, the only non-trivial natural setting I know in one complex dimension) in which to introduce ¯ the twisted ∂-technique of Ohsawa-Takegoshi. This technique has had incredibly powerful applications in both several complex variables and algebraic geometry, and there remain many avenues of research to pursue. I chose to omit this topic because a rigorous treatment of it on Riemann surfaces other than the unit disk and the complex plane requires rather delicate analysis of Green’s Functions on finite Riemann surfaces which, by contrast with the rest of the book, would be disproportionately technical in nature. I have decided to postpone such a presentation to another occasion. ACKNOWLEDGMENT. In my days as a pizza delivery guy for Pizza Pizza in Toronto, I had a colleague named Vlad who used to say: “No money, no funny!” I am grateful to the NSF for its generous financial support. Much of what is presented in this book is motivated by the work of Jean-Pierre Demailly and YumTong Siu, and I am grateful to both of them for all that they have taught me, both in their writings and in person. John D’Angelo and Jeff McNeal were very encouraging in the early parts of the project, and

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gave me the inspiration I needed to start the project. Anna Benini, Ying Chi, Trevor Clark, Caner Koca, Ki Song, Traves Waddington and Yi Zhu attended the course in which I tested out the material in the book, and provided useful questions and remarks that improved the exposition and content. Andy Raich and Colleen Robles read a preliminary version of these notes at Texas A&M, and Colleen communicated corrections and suggestions that were extremely useful. I am grateful to both of them. Most of all, I am indebted to Mohan Ramachandran. Our frequent conversations, beyond giving me great pleasure, led to the inclusion of many omissions and the correction of many errors, and Mohan’s passion for the old literature taught me an enormous amount about the history and development of the subject. Needless to say, the responsibility for the remaining errors and choices of topics rests with me alone.

Dror Varolin Brooklyn, NY 2008

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Contents 1

Complex Analysis I 1.1 Green’s Theorem and the Cauchy Green Formula . . . . . . 1.1.1 Green’s Theorem . . . . . . . . . . . . . . . . . . . 1.1.2 The Cauchy-Green formula . . . . . . . . . . . . . 1.2 Holomorphic functions and Cauchy Formulas . . . . . . . . 1.2.1 The Homogeneous Cauchy-Riemann Equations . . . 1.2.2 Cauchy’s Theorem and Integral Formula . . . . . . . 1.3 Power series . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Series representation . . . . . . . . . . . . . . . . . 1.3.2 Corollaries . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Cauchy Estimates . . . . . . . . . . . . . . . . . . . 1.4 Isolated Singularities of Holomorphic Functions . . . . . . . 1.4.1 Laurent Series . . . . . . . . . . . . . . . . . . . . 1.4.2 Singularities of Holomorphic Functions at a puncture 1.5 The Maximum Principle . . . . . . . . . . . . . . . . . . . 1.6 Compactness Theorems . . . . . . . . . . . . . . . . . . . . 1.6.1 Montel’s Theorem . . . . . . . . . . . . . . . . . . 1.6.2 Köbe’s Compactness Theorem . . . . . . . . . . . . 1.7 Harmonic functions . . . . . . . . . . . . . . . . . . . . . . 1.7.1 Laplacian and harmonic . . . . . . . . . . . . . . . 1.7.2 Harmonic conjugate . . . . . . . . . . . . . . . . . 1.7.3 Mean Value Theorem . . . . . . . . . . . . . . . . . 1.7.4 Maximum Principle . . . . . . . . . . . . . . . . . 1.7.5 Poisson Formula . . . . . . . . . . . . . . . . . . . 1.7.6 Regularity of harmonic functions . . . . . . . . . . 1.8 Subharmonic functions . . . . . . . . . . . . . . . . . . . . 1.8.1 Subharmonic . . . . . . . . . . . . . . . . . . . . . 1.8.2 Basic properties . . . . . . . . . . . . . . . . . . . . 1.8.3 Subharmonic again . . . . . . . . . . . . . . . . . . 1.8.4 Local integrability . . . . . . . . . . . . . . . . . . 1.8.5 Regularity . . . . . . . . . . . . . . . . . . . . . . . 1.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

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Riemann Surfaces 2.1 1-Dimensional Complex Manifolds . . . . . . . 2.1.1 Complex charts . . . . . . . . . . . . . . 2.1.2 Riemann surfaces . . . . . . . . . . . . . 2.1.3 Riemann surfaces as smooth 2-manifolds 2.2 Examples of Riemann surfaces . . . . . . . . . . 2.2.1 Complex manifolds . . . . . . . . . . . . 2.2.2 Examples of quotient Riemann surfaces . 2.2.3 Implicitly defined Riemann surfaces . . . 2.2.4 Projective curves . . . . . . . . . . . . . 2.3 Exercises . . . . . . . . . . . . . . . . . . . . .

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Functions and Maps 3.1 Functions on a Riemann surface . . . . . . . . . . . . . . . . 3.1.1 Holomorphic functions . . . . . . . . . . . . . . . . . 3.1.2 Meromorphic functions . . . . . . . . . . . . . . . . . 3.1.3 The argument principle . . . . . . . . . . . . . . . . . 3.2 Global aspects of meromorphic functions . . . . . . . . . . . 3.2.1 Meromorphic functions on compact Riemann surfaces 3.2.2 Meromorphic functions on P1 . . . . . . . . . . . . . 3.2.3 Meromorphic functions on complex tori . . . . . . . . 3.3 Holomorphic maps between Riemann surfaces . . . . . . . . . 3.3.1 Basic definitions and simple theorems . . . . . . . . . 3.3.2 Meromorphic functions as maps to the Riemann sphere 3.3.3 Multiplicity . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Degree of a holomorphic map . . . . . . . . . . . . . 3.3.5 The Riemann-Hurwitz Formula . . . . . . . . . . . . 3.3.6 An Example: maps between complex tori . . . . . . . 3.4 An example: hyperelliptic surfaces . . . . . . . . . . . . . . . 3.4.1 Gluing surfaces . . . . . . . . . . . . . . . . . . . . . 3.4.2 Hyperelliptic surfaces . . . . . . . . . . . . . . . . . 3.4.3 Meromorphic functions on hyperelliptic surfaces . . . 3.5 Harmonic and Subharmonic Functions . . . . . . . . . . . . . 3.5.1 Definition of Harmonic Functions . . . . . . . . . . . 3.5.2 Harnack’s Principle . . . . . . . . . . . . . . . . . . . 3.5.3 Subharmonic functions . . . . . . . . . . . . . . . . . 3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Complex Line Bundles 4.1 Complex line bundles . . . . . . . . . . . 4.1.1 Basic definitions . . . . . . . . . 4.1.2 Description by transition functions 4.1.3 Description by local sections . . . 6

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4.1.4 Remark: Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . Holomorphic line bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Definition of Holomorphic line bundle . . . . . . . . . . . . . . . . 4.2.2 Picard group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Holomorphic sections and meromorphic functions . . . . . . . . . Two canonically defined holomorphic line bundles . . . . . . . . . . . . . 4.3.1 The canonical bundle of a Riemann surface . . . . . . . . . . . . . 4.3.2 The tangent bundle of a Riemann surface . . . . . . . . . . . . . . 4.3.3 Duality of KX and TX1,0 . . . . . . . . . . . . . . . . . . . . . . . . Holomorphic vector fields on a Riemann surface . . . . . . . . . . . . . . . 4.4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Divisors and Line Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 The line bundle of a divisor . . . . . . . . . . . . . . . . . . . . . 4.5.3 The divisor of a line bundle with meromorphic section . . . . . . . 4.5.4 Summary of the divisor-line bundle correspondence . . . . . . . . . Line bundles over Pn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 The tautological bundle . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 The hyperplane bundle and its global sections . . . . . . . . . . . . 4.6.3 Line bundles over P1 . . . . . . . . . . . . . . . . . . . . . . . . . Holomorphic sections and projective maps . . . . . . . . . . . . . . . . . . 4.7.1 Definition of projective map and morphism . . . . . . . . . . . . . 4.7.2 Description of φW in terms of a basis of W . . . . . . . . . . . . . 4.7.3 All holomorphic maps to projective space are projective morphisms 4.7.4 Resolving the base locus . . . . . . . . . . . . . . . . . . . . . . . A Finiteness Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Complex Differential Forms 5.1 Differential (1, 0)-forms . . . . 5.2 TX∗0,1 and (0,1)-forms . . . . . . 5.3 TX∗ and 1-forms . . . . . . . . . 5.4 Λ1,1 X and (1,1)-forms . . . . . . . 5.5 Exterior algebra and calculus . . 5.6 Integration of 1-forms . . . . . . 5.7 Integration of (1,1)-forms . . . . 5.8 Residues . . . . . . . . . . . . . 5.9 Homotopy and homology . . . . 5.9.1 Homotopy of curves . . 5.9.2 Fundamental Group . . 5.9.3 Homology . . . . . . . 5.10 Poincaré and Dolbeault Lemmas 5.11 Dolbeault Cohomology . . . . .

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Calculus on Line Bundles 6.1 Connections on holomorphic line bundles . . . . . . 6.1.1 General connection on a complex line bundle 6.1.2 (1,0)-connection . . . . . . . . . . . . . . . 6.1.3 Hermitian metrics and connections . . . . . . 6.1.4 The Chern connection . . . . . . . . . . . . 6.1.5 Curvature of the Chern connection . . . . . . 6.1.6 Chern numbers . . . . . . . . . . . . . . . . 6.1.7 Example: the holomorphic line bundle TX1,0 .

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Potential Theory 7.1 The Dirichlet Problem and Perron’s Method . . . . . . . . . . . . . 7.1.1 Definition of the Dirichlet Problem . . . . . . . . . . . . . 7.1.2 Perron Families . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Perron’s Method . . . . . . . . . . . . . . . . . . . . . . . 7.1.4 Aside: Countable topology of Riemann surfaces . . . . . . 7.1.5 A helpful note . . . . . . . . . . . . . . . . . . . . . . . . 7.1.6 Green’s Functions . . . . . . . . . . . . . . . . . . . . . . 7.1.7 Symmetry of the Green’s Function . . . . . . . . . . . . . . 7.1.8 Reproducing formulas . . . . . . . . . . . . . . . . . . . . 7.2 Approximation on open Riemann surfaces . . . . . . . . . . . . . . 7.2.1 Holomorphic hulls, Runge domains and Regular Exhaustion 7.2.2 The Runge Theorem of Behnke-Stein . . . . . . . . . . . . 7.2.3 Function-Theoretic Description of hulls . . . . . . . . . . .

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Solving ∂¯ with smooth data 8.1 The basic result . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Triviality of holomorphic line bundles . . . . . . . . . . . . . . 8.3 The Weierstrass Product Theorem . . . . . . . . . . . . . . . . 8.4 Meromorphic functions as quotients . . . . . . . . . . . . . . . 8.5 The Mittag-Leffler Problem . . . . . . . . . . . . . . . . . . . . 8.5.1 Principal parts and the Mittag-Leffler Problem . . . . . 8.5.2 Solution on open Riemann surfaces . . . . . . . . . . . 8.5.3 Solution on general Riemann surfaces . . . . . . . . . . 8.5.4 Principal parts of meromorphic 1-forms . . . . . . . . . 8.6 Poisson’s Equation on Open Riemann Surfaces . . . . . . . . . 8.6.1 Pfluger’s Harmonic Runge Theorem . . . . . . . . . . . 8.6.2 Solving ∂ ∂¯ with smooth data on open Riemann surfaces

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Harmonic Forms 149 9.1 Harmonic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 9.1.1 The Hodge Star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 9.1.2 Inner products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 8

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9.3 9.4

9.5

9.1.3 The formal adjoint of d . . . . . . . . . . . . . . . . . . . 9.1.4 Laplace-Beltrami operator and harmonic forms . . . . . . 9.1.5 Regularity for the Laplace-Beltrami Operator . . . . . . . The Hodge decomposition of E (X) . . . . . . . . . . . . . . . . 9.2.1 The main theorem . . . . . . . . . . . . . . . . . . . . . 9.2.2 Obstructions from the kernel of ∆ . . . . . . . . . . . . . 9.2.3 A compact subspace of L2 (X) . . . . . . . . . . . . . . . 9.2.4 The Spectral Theorem for Compact Self-adjoint Operators 9.2.5 Completion of the proof of Theorem 9.2.1 . . . . . . . . . Arithmetic and geometric genus . . . . . . . . . . . . . . . . . . Existence of positive line bundles . . . . . . . . . . . . . . . . . . 9.4.1 Compact Riemann surfaces I: Using Hodge Theory . . . . 9.4.2 Compact Riemann surfaces II: An ad hoc technique . . . . 9.4.3 Open Riemann surfaces . . . . . . . . . . . . . . . . . . Proof of the Dolbeault-Serre Isomorphism . . . . . . . . . . . . .

10 Uniformization 10.1 Automorphisms of the plane and the disk . . . . . . . . . . 10.1.1 Aut(C) . . . . . . . . . . . . . . . . . . . . . . . . 10.1.2 Aut(D) . . . . . . . . . . . . . . . . . . . . . . . . 10.2 A review of covering spaces . . . . . . . . . . . . . . . . . 10.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Curve lifting property . . . . . . . . . . . . . . . . 10.2.3 Universal cover . . . . . . . . . . . . . . . . . . . . 10.2.4 Intermediate covers . . . . . . . . . . . . . . . . . . 10.3 The Uniformization Theorem . . . . . . . . . . . . . . . . . 10.3.1 Statement . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Potential-theoretic classification of Riemann surfaces 10.3.3 Riemann surfaces covered by P1 . . . . . . . . . . . 10.3.4 Riemann surfaces covered by C . . . . . . . . . . . 10.3.5 Poincaré classification . . . . . . . . . . . . . . . . 10.3.6 Existence of Positive line bundles: another approach 10.4 Proof of The Uniformization Theorem . . . . . . . . . . . . 11 Hörmander’s Theorem 11.1 Hilbert Spaces of Sections . . . . . . . . . . . . . . . . . . 11.1.1 The Hilbert space of sections . . . . . . . . . . . . . 11.1.2 The Hilbert space of line bundle-valued (0, 1)-forms 11.1.3 The ∂¯ operator on L2 . . . . . . . . . . . . . . . . . 11.1.4 The formal adjoint of ∂¯ . . . . . . . . . . . . . . . . 11.2 The Basic Identity in the L2 method . . . . . . . . . . . . . 11.2.1 An integration-by-parts identity . . . . . . . . . . . 11.2.2 Geometric interpretation of the basic identity . . . . 9

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167 167 167 167 168 168 168 168 169 169 169 170 170 171 171 173 175

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177 177 177 177 178 179 180 180 181

11.3 Hörmander’s Theorem . . . . . . . . . . . . . . . 11.4 Proof of the Korn-Lichtenstein Theorem . . . . . . 11.4.1 Derivation of the PDE . . . . . . . . . . . 11.4.2 The operator ∂¯ω . . . . . . . . . . . . . . . 11.4.3 Hilbert space structures . . . . . . . . . . . 11.4.4 The formal adjoint of ∂¯ω . . . . . . . . . . 11.4.5 The basic estimate . . . . . . . . . . . . . 11.4.6 Positively curved weights . . . . . . . . . 11.4.7 Conclusion of the proof of Theorem 2.1.11 12 Embedding Riemann Surfaces 12.1 Controlling the derivatives of sections . . . . . . . 12.2 Meromorphic sections of line bundles . . . . . . . 12.3 Plenitude of meromorphic functions . . . . . . . . 12.4 Kodaira’s Embedding Theorem . . . . . . . . . . . 12.4.1 Embedding Riemann surfaces in PN . . . . 12.4.2 Embedding Riemann surfaces in P3 . . . . 12.5 Embedding open Riemann surfaces . . . . . . . . . 12.5.1 Statement and outline of the proof . . . . . 12.5.2 The injective immersion of X in C3 . . . . 12.5.3 Analytic polyhedra . . . . . . . . . . . . . 12.5.4 Conclusion of the proof of Theorem 12.5.1

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13 The Riemann-Roch Theorem 13.1 The Riemann-Roch Theorem . . . . . . . . . . . . . . . . . . . 13.1.1 Statement of the theorem . . . . . . . . . . . . . . . . . 13.1.2 Mittag-Leffler Problems . . . . . . . . . . . . . . . . . 13.1.3 Exact sequences of vector spaces . . . . . . . . . . . . 13.1.4 Proof of Theorem 13.1.2 . . . . . . . . . . . . . . . . . 13.2 Some corollaries . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 Necessity of meromorphic plenitude . . . . . . . . . . . 13.2.2 Amplitude . . . . . . . . . . . . . . . . . . . . . . . . 13.2.3 Curves of genus 0,1 and 2 . . . . . . . . . . . . . . . . 13.2.4 The canonical bundle of a curve of positive genus is free 14 Abel’s Theorem 14.1 Indefinite integration of holomorphic forms . . . . . . . 14.1.1 The Jacobian of a curve . . . . . . . . . . . . . 14.1.2 Statement of Abel’s Theorem . . . . . . . . . . 14.2 Bilinear Relations . . . . . . . . . . . . . . . . . . . . . 14.3 Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . 14.3.1 Abelian differentials of the second and third kind 14.3.2 The Reciprocity Theorem . . . . . . . . . . . . 10

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14.4 Proof of Abel’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 14.5 A discussion of Jacobi’s Inversion Theorem . . . . . . . . . . . . . . . . . . . . . 218

11

12

Chapter 1 Complex Analysis I 1.1 1.1.1

Green’s Theorem and the Cauchy Green Formula Green’s Theorem

For the sake of a complete presentation, we begin with Green’s Theorem. Again, we do not give the most general form. The proof can be found in almost any calculus text. T HEOREM 1.1.1. Let D ⊂ R2 be an open connected set whose boundary ∂D is piecewise smooth and positively oriented. (We remind the reader that the boundary ∂D is oriented positively if, when one is moving forward along ∂D, one finds D to one’s left.) Let P and Q be smooth, complex valued functions on some neighborhood of D. Then ZZ I ∂Q ∂P P dx + Qdy = − dA. ∂x ∂y D ∂D R EMARK . Usually Green’s Theorem is stated for real-valued functions P and Q, but by linearity can be complexified.

1.1.2

The Cauchy-Green formula

Green’s Theorem can be used to prove the Cauchy-Green Integral formula. To state the latter, we recall that the complex partial derivatives are defined by ∂f 1 ∂f √ ∂f ∂f 1 ∂f √ ∂f = − −1 and = + −1 . ∂z 2 ∂x ∂y ∂ z¯ 2 ∂x ∂y T HEOREM 1.1.2. Let D ⊂ C be an open connected set with piecewise smooth boundary, and let f : D → C be a C 1 -function. Then for all z ∈ D, Z ZZ 1 f (ζ) ∂f dA(ζ) 1 f (z) = √ dζ − . ¯ π 2π −1 ∂D ζ − z D ∂ζ ζ − z 13

Proof. Fix z ∈ D. Let ε > 0 be such that D(z, ε) ⊂⊂ D. (The Notation A ⊂⊂ B, means ‘A is relatively compact in B’, i.e., the closure of A is compact in B, while D(z, ε) = {x ∈ C ; |x−z| < ε}.) If we apply Green’s Theorem to the functions √ f (ζ) −1 f (ζ) 1 P = √ and Q = √ , 2π −1 ζ − z 2π −1 ζ − z and the domain Dε := D − D(z, ε), we obtain, after collecting terms and using the definition of the complex partial derivatives, the following formula. ZZ Z Z 1 1 ∂f dA(ζ) 1 f (ζ) f (ζ) √ dζ − √ dζ = . (1.1) ¯ π 2π −1 ∂D ζ − z 2π −1 ∂D(z,ε) ζ − z Dε ∂ ζ ζ − z In polar coordinates centered at z, the measure dA is rdrdθ, and since ζ −z = reiθ , the integrand of the right hand integral is locally bounded. It follows that, as ε → 0, the right hand side converges to ZZ ∂f dA(ζ) 1 . ¯ π D ∂ζ ζ − z On the other hand, since f is differentiable, we have Z Z √ √ f (ζ) f (ζ) − f (z) dζ = dζ + 2π −1f (z) → 2π −1f (z) ζ −z ∂D(z,ε) ζ − z ∂D(z,ε) as ε → 0.

1.2 1.2.1

Holomorphic functions and Cauchy Formulas The Homogeneous Cauchy-Riemann Equations

Recall the definition of a holomorphic function. D EFINITION 1.2.1. A C 1 function f is holomorphic in a domain D if and only if it satisfies the homogeneous Cauchy-Riemann equations ∂f ≡0 ∂ z¯

on D.

R EMARK . The Cauchy-Riemann equations, and especially the inhomogeneous equations ∂u = g, ∂ z¯ are certainly the main characters in the present text. We will study several methods for solving the Cauchy-Riemann equations, and through the solutions we will uncover much of the structure underlying a Riemann surface. 14

1.2.2

Cauchy’s Theorem and Integral Formula

As a corollary of the Cauchy-Green formula, we obtain the Cauchy Theorem and the Cauchy Integral Formula for C 1 -functions. T HEOREM 1.2.2. Consider an open set D ⊂ C with piecewise smooth boundary. If f is holomorphic on a neighborhood of D, then 1. (Cauchy’s Theorem) Z f (z)dz = 0. ∂D

2. (Cauchy Integral Formula) f (z) =

Z

1 √

2π −1

∂D

f (ζ) dζ. ζ −z

Proof. The second result is obvious from the Cauchy-Green Formula. For the first result, apply the Cauchy-Green formula to the function z 7→ (z − ζ)f (z).

1.3 1.3.1

Power series Series representation

Suppose f is holomorphic in a neighborhood of the closure of the disk D(0, r). For z ∈ D(0, r) and |ζ| = r we have the series ∞ X zj 1 1 = = . j+1 ζ −z ζ(1 − zζ ) ζ j=0 Applying the Cauchy Integral Formula, we find that ∞ X f (z) = aj z j , j=0

where

Z 1 f (ζ)dζ aj = aj (f, 0) := √ . 2π −1 |ζ|=r ζ n+1 In particular, f is given locally by a convergent power series. A similar result, properly scaled and shifted, holds at any point zo in place of the origin. In that setting, Z ∞ X 1 f (ζ)dζ j f (z) = aj (z − zo ) with aj = aj (f, zo ) = √ . n+1 (ζ − z ) 2π −1 o |ζ−z |=r o j=1 It follows that f is infinitely complex-differentiable, and that Z ∂ n f n! f (ζ)dζ = √ . n ∂z (ζ − zo )n+1 2π −1 |ζ−zo |=r

z=zo

15

(1.2)

1.3.2

Corollaries

The fact that f is given by a convergent power series implies the following theorems. T HEOREM 1.3.1 (Closure in the compact open topology). If a sequence of holomorphic functions converges uniformly, then the limit function is holomorphic. T HEOREM 1.3.2 (Identity Theorem). If two holomorphic functions are defined on an open connected set D ⊂ C and they agree on an open subset of D, then they agree on all of D. D EFINITION 1.3.3 (Order). Let f be holomorphic in a neighborhood of a point x ∈ C. We say that f has a zero of order n at x if n is the smallest integer such that ∂ n f 6= 0. ∂z n z=zo We write n = Ordx f .

1.3.3

Cauchy Estimates

Estimating (1.2) gives us the following result. P ROPOSITION 1.3.4 (Cauchy Estimates). A holomorphic function f : D0 (R) → C on the closed disk of radius R centered at the origin satisfies the estimates n ∂ f n! ≤ n sup |f |. ∂z n R D0 (R) z=0 As a corollary, we have Liouville’s Theorem. C OROLLARY 1.3.5 (Liouville). Any bounded holomorphic function f : C → C is constant. Proof. Let C := supC |f |. For any point p ∈ C and any R > 0, |f 0 (p)| ≤ CR−1 . Thus we have f 0 (p) = 0.

1.4 1.4.1

Isolated Singularities of Holomorphic Functions Laurent Series

Using the Cauchy Integral Formula, we can develop the notion of Laurent series. To this end, let A be an annulus centered at 0, with radii R > r ≥ 0, and let f be holomorphic on A. Choose R0 and r0 such that R > R0 > r0 > r. Then, according to the Cauchy Integral Formula, if r0 < |z| < R0 , Z Z 1 f (ζ) 1 f (ζ) f (z) = √ dζ − √ dζ. 2π −1 |ζ|=R0 ζ − z 2π −1 |ζ|=r0 ζ − z 16

In the first integral, |z| < |ζ| = R0 , while in the second r0 = |ζ| < |z|. Thus in the first integral we write z , ζ −z =ζ 1− ζ and in the second, ζ ζ − z = −z 1 − . z Using that (1 − r)−1 = 1 + r + r2 + ..., we see that X f (z) = an z n ,

(1.3)

n∈Z

where an =

1 √

2π −1

Z |z|=ρ

f (z) dz, z n+1

(1.4)

for some ρ ∈ (r, R). Since f (z)/z n is analytic in A, the latter integral is independent of ρ. D EFINITION 1.4.1. The series (1.3) is called the Laurent series of f . If the annulus has inner radius zero (i.e., it is a punctured neighborhood), then the term a−1 is called the residue of f at 0, and is denoted Res0 f . More generally, if c ∈ C, and g is holomorphic in a punctured neighborhood of c, we define Resc (g) := Res0 (f ), where f (z) = g(z − c). By considering the Laurent series of f at each singularity of f , one obtains the following theorem. T HEOREM 1.4.2 (Residue Theorem). If z1 , ..., zk are distinct points of D and f is holomorphic on D − {z1 , ..., zk }, then Z f (z)dz = Resz1 (f ) + ... + Reszk (f ). ∂D

1.4.2

Singularities of Holomorphic Functions at a puncture

Next we study the local form of a holomorphic function in the neighborhood of a pole or a zero. D EFINITION 1.4.3. Let f be holomorphic in a punctured neighborhood of a point z ∈ C. P 1. We say that f has a pole of order n ≥ 0 if the Laurent series aj (ζ − z)j of f at z has the property that a−n 6= 0 but a−k = 0 for k > n. 2. If f has a pole of order 0 at z, i.e., the Laurent series of f at z is holomorphic across z, we say that z is a removable singularity of f . 3. If the Laurent series of f has infinitely many negative coefficients, we say that f has an essential singularity at z. 17

4. If f does not have an essential singularity at z, we define Ordz (f ) := k where k is the largest integer such that a` = 0 for each ` < k. R EMARK . Note that if f has a pole of order k > 0 then Ordx (f ) = −k. Thus the order of a meromorphic function is different from the pole order of that function. D EFINITION 1.4.4 (Meromorphic function). Let D ⊂ C. 1. We say that f is meromorphic at a point p ∈ D if there is a punctured neighborhood U of p in D such that f is holomorphic on U and p is not an essential singularity for f . 2. We say that f is meromorphic on D if f is meromorphic at every point of D. We have the following characterization of singularities. T HEOREM 1.4.5. Let p ∈ C and U a neighborhood of p. Suppose that f : U − {p} → C is a holomorphic function. 1. If f is bounded in a neighborhood of p, then f extends to a holomorphic function on U . In particular, limz→p f (z) exists. 2. If limz→p |f (z)| = +∞, then f has a pole at z. Proof. We mayP assume p = 0. (1) Let f (z) = n∈Z an z be the Laurent series expansion of f in a small punctured disk centered at the origin. Estimating formula (1.4), we see that if |f | ≤ M then for all sufficiently small ε one has |an | ≤ 2πM ε−n . For n < 0 this means that an = 0. (2) Choose a disk D centered at the origin such that f does not vanish on D. Then the function g(z) = 1/f (z) is holomorphic and bounded on D − {0}. By part (1) it extends to a holomorphic function on D, and since it has limit zero at 0, a look at the power series expansion on g shows the existence of an integer k ≥ 1 such that g(z) = z k h(z) for some holomorphic function h satisfying h(0) 6= 0. It follows that near the origin, f (z) = z −k ϕ(z) for some holomorphic function ϕ satisfying ϕ(0) 6= 0. R EMARK . As a corollary, if f is unbounded on U − {p} and |f (q)| is not arbitrarily large for all points q sufficiently near p, then f has an essential singularity at p. In fact, even more is true. If f has an essential singularity at p, then the image of U is dense in C. Indeed, if not then there is a disk D ⊂ C ∪ {∞} such that f (U − {p}) ⊂ D. We can then choose constants a, b, c, e ∈ C such that a + bf (z) g(z) = ∈ D(0, 1) for all z ∈ U. c + ef (z) Then g extends to p, and thus so does f (as a meromorphic function), contradicting Theorem 1.4.5. 18

P ROPOSITION 1.4.6. If g is a nowhere zero holomorphic function on a disk D, then there exists a holomorphic function h on D such that g = eh on D. Proof. Fix a point p ∈ D. The function g 0 (z)/g(z) is holomorphic on D, and so we can define the function Z z 0 g (ζ)dζ , H(z) = g(ζ) p where the integral is over some path connecting p to z. By Cauchy’s Theorem, the integral is independent of the path connecting p to z, and thus well-defined. Since the integral is path independent, we can calculate H 0 (z) by considering a path parallel to the x-axis moving from z to z + h, h ∈ R. By the Fundamental Theorem of Calculus we find that H 0 (z) = g 0 (z)/g(z). It follows that g 0 (z) g(z)H 0 (z) d g(z) = − = 0. dz eH(z) eH(z) eH(z) Thus g = ceH = eh . C OROLLARY 1.4.7 (Normal Form Theorem). If f is holomorphic in a neighborhood of p and k = Ordp (f ) then there exists a neighborhood U of p and an injective holomorphic function w = g(z) such that g(p) = 0 and f (g −1 (w)) = wk . Proof. We may assume, after translation, that p = 0. By looking at the power series expansion of f , we can write f (z) = z k g(z) with g(0) 6= 0. According to Proposition 1.4.6, we can write −1 f (z) = z k eh(z) for some holomorphic function h. Let g(z) = zek h(z) . Then g 0 (0) 6= 0 and thus, by the implicit function theorem, g is injective. R EMARK . If |a| is sufficiently small but non-zero, then there the equation f (z) = a has exactly k distinct solutions p1 , ..., pk in U , namely pj = g −1 (|a|1/k ωkj ), where ωk is a k th root of unity. For this reason, we sometimes call the number k the multiplicity of f at p. The normal form theorem has three strong consequences. T HEOREM 1.4.8. If U , V are open sets and f : U → V is holomorphic and injective, then the inverse map f −1 : f (U ) → U is holomorphic. Proof. If f is injective then k = 1 since, in the normal coordinate w every point has k preimages. But if k = 1 then f 0 (0) 6= 0. T HEOREM 1.4.9. The zero set of a non-constant holomorphic function is discrete. Proof. This is clear from the normal form theorem. T HEOREM 1.4.10 (Open Mapping Theorem). If U ⊂ C is open and f : U → C is a non-constant holomorphic function, then f (U ) is open. 19

Let f be holomorphic in a punctured neighborhood of 0 and assume 0 is not an essential singularity. In this case, there is an integer k and a holomorphic function g such that f (z) = z k g(z). We then have

k g 0 (z) d log f = + . dz z g(z)

(Observe that, while log is not a well-defined function, its derivative is a well-defined meromorphic function.) Applying the obvious modification of this formula at all points on the domain of a function and using Cauchy’s Theorem, we obtain the following theorem. T HEOREM 1.4.11 (Argument Principle). Let D ⊂ C be an open connected set with piecewise smooth boundary, and let f be a meromorphic function on a neighborhood of D having no zeros or poles on ∂D. Then Z X 1 d √ Ordz (f ). (log f )dz = 2π −1 ∂D dz z∈D

1.5

The Maximum Principle

The next result is important and useful in complex analysis. T HEOREM 1.5.1 (Maximum Principle). If f : {|z| < r} → C is holomorphic and there is a maximum for |f | at some z0 with |z0 | < r, then f is constant. Proof. Clearly we can assume f (z0 ) 6= 0. Reparameterizing the Cauchy formula, we obtain, for any ρ < dist(z0 , {|z| ≥ r}), the formula 1 f (z0 ) = 2π

Z

2π

√

f (z0 + ρe

−1θ

)dθ.

0

Rearranging and taking real parts, we have Z

√

2π

Re 0

1−

f (z0 + ρe f (z0 )

−1θ

)

! dθ = 0.

Since, by hypothesis, the latter integrand is continuous and non-negative, we see that the integrand is constant. By the Cauchy-Riemann equations, a holomorphic function whose real part is constant is itself constant. R EMARK . The maximum principle also trivially follows from the open mapping theorem. We have chosen this proof because it also works for harmonic functions without change. As a corollary of the Maximum Principle, we obtain the well-known Schwarz Lemma. 20

T HEOREM 1.5.2 (Schwarz Lemma). Let f : D → D be a holomorphic self map of the unit disk such that f (0) = 0. Then for all z ∈ D, |f (z)| ≤ |z|. In particular, |f 0 (0)| ≤ 1. Equality holds in √ −1θ the first estimate for some z ∈ D or in the second estimate if and only if f (z) = e z for some constant θ ∈ R. . Then gr is holomorphic on {|z| < 1/r}, and thus we Proof. Fix r ∈ (0, 1). Let gr (z) = f (rz) rz have gr (zo ) ≤ max gr (z) = max |f (rz)|/r ≤ 1/r. |z|=1

|z|=1

Thus |f (rz)| ≤ r|z|. Letting r → 1 gives the estimates. If equality holds at any point zo ∈ D, i.e. g1 (zo ) = 1, then for some r ∈ (|zo |, 1), gr (zo /r) = 1. By the maximum principle gr is constant. Thus |f (rz)| = |rz| holds for |z| < 1/r. It follows that √ −1θ z for some constant θ ∈ R. The proof is complete. f (z) = e

1.6 1.6.1

Compactness Theorems Montel’s Theorem

We begin with our first compactness theorem. T HEOREM 1.6.1 (Montel). Let U ⊂ C be an open set. Let K1 ⊂⊂ K2 ⊂⊂ ... ⊂ U be a sequence of compact sets such that any compact subset K ⊂⊂ U is contained in some Kj . Let M1 ≤ M2 ≤ ..... Then the set of holomorphic functions BK,M := {f : U → C ; sup |f | ≤ Mj , j = 1, 2, ...} Kj

is compact in O(U ) in the compact-open topology. Proof. By the Cauchy Estimates, the family BK,M is uniformly bounded and equicontinuous on each relatively compact open subset V of U . By the Arzela-Ascoli Theorem, every sequence in BK,M |Kj has a uniformly convergent subsequence. By a diagonal argument, every sequence in BK,M has a uniformly convergent subsequence. An application of Theorem 1.3.1 completes the proof. C OROLLARY 1.6.2. Let U ⊂ C be an open set such that C − U has interior points. Then the set F := {f ∈ O(D) ; f (D) ⊂ U } is compact in O(D) in the compact-open topology. Proof. Let p ∈ C − U be an interior point. The transformation f 7→ (f − p)−1 maps F to a family of uniformly bounded holomorphic functions. An application of Montel’s Theorem completes the proof. 21

1.6.2

Köbe’s Compactness Theorem

Let S denote the set of all injective holomorphic functions f : D → C such that f (0) = 0 and f 0 (0) = 1. A fundamental result in complex analysis is the following theorem of P. Köbe. T HEOREM 1.6.3 (Köbe). The set S of all injective holomorphic functions f : D → C such that f (0) = 0 and f 0 (0) = 1 is compact in O(D) in the compact-open topology. Proof. Fix a sequence {fn } ⊂ S . Let Rn := sup{R > 0 ; D0 (R) ⊂ fn (D)}. Since fn−1 |D0 (R) : D0 (R) → D is holomorphic for any D0 (R) ⊂ f (D), the Schwarz Lemma implies that Rn ≤ 1. Choose a point xn ∈ ∂D0 (Rn ) − fn (D) and let gn := fn /xn . Then√ D ⊂ gn (D) 63 1. Now,√gn (D) is simply connected, so there is a holomorphic branch ψ of z − 1 such that ψ(0) = −1. Then hn := ψ ◦ gn satisfies h2n = gn − 1. Now, hn (D) ∩ (−hn (D)) = ∅. Indeed, if w = hn (z) and −w = hn (z 0 ) then gn (−w) = gn (w) and by injectivity w = −w = 0. But then gn (z) = 1, which is impossible. Since D ⊂ gn (D), we have U := ψ(D) ⊂ hn (D), and thus (−U ) ∩ hn (D) = ∅. By Corollary 1.6.2 hn has a convergent subsequence. Thus, since |xn | = Rn ≤ 1, fn = xn (1 + h2n ) has a convergent subsequence. Let f be the limit of this subsequence. Let a ∈ f (D). By the argument principle, for any z with f (z) 6= a, Z 1 f 0 (z)dz ` := √ 2π −1 |z|=r f (z) − a is an integer, which evidently is ≥ 1 for r sufficiently close to 1. Since fnk is arbitrarily close to f on |z| ≤ r, the injectivity of fnk implies that Z Z fn0 k (z)dz 1 1 f 0 (z)dz √ √ = = `. 1= 2π −1 |z|=r fnk (z) − a 2π −1 |z|=r f (z) − a Thus f is injective, and the proof is complete.

1.7 1.7.1

Harmonic functions Laplacian and harmonic

The Laplace operator (or Laplacian) ∆ is defined in a plane domain by ∆u = uxx + uyy = 4∂z ∂z¯u. D EFINITION 1.7.1. A harmonic function u on Ω is a C 2 function for which ∆u ≡ 0. We write u ∈ H(Ω). 22

R EMARK . Later on we will see that the regularity requirement in the definition of harmonic functions can be substantially reduced. Since ∆ is a real operator, a complex valued function is harmonic precisely if its real and imaginary parts are harmonic. ∂2 , one sees that every holomorphic function is harmonic, as is From the formula ∆ = 4 ∂z∂ z¯ the conjugate of a holomorphic function. On a simply connected domain, the converse holds, i.e., every harmonic function is the real part of a holomorphic function. This may be seen as follows. Let 1 1 ∂ − ∂¯ = (−dx ⊗ ∂y + dy ⊗ ∂x ). dc := √ 2 2 −1 Then for a function h one has ddc h =

√

¯ = 2 (∆h) dx ∧ dy. −1∂ ∂h

(A review of exterior calculus, in the setting of Riemann surfaces, is presented in Chapter 5.)

1.7.2

Harmonic conjugate

Suppose u is a real valued harmonic function on a simply connected domain Ω ⊂ C. Then dc u is a closed 1-form. It essentially follows from Stokes’ Theorem and the simple connectivity of Ω that, given zo ∈ Ω and a curve γzo ,z : [0, 1] → Ω satisfying γzo ,z (0) = zo and γzo ,z (1) = z the function Z v(z) := 2 dc u γzo ,z

is well-defined independent of the choice of γzo ,z . (If we change zo , the function v(z) changes by an additive constant.) √ We claim that the function f = u + −1v is holomorphic. Indeed, since v is independent of the curve γzo ,z , by choosing the z-end of γzo ,z to be horizontal or vertical, one obtains from the Fundamental Theorem of Calculus that vx = −uy

and vy = ux .

¯ = 0, and our claim is proved. These equations mean precisely that ∂f R EMARK . In a domain in C it is possible to construct a harmonic conjugate from a harmonic function without recourse to differentiation and integration. In fact, if u = u(x, y) is a real-valued harmonic function on R2 , then the function z z f (z) := 2u , √ 2 2 −1 defines a holomorphic function. The proof of this interesting fact may be found in D’Angelo’s book [D’Angelo-1993]. We have opted for our presentation here because the construction generalizes to manifolds more directly. 23

1.7.3

Mean Value Theorem

P ROPOSITION 1.7.2 (Mean Value Property). If u : Ω → R is harmonic, p ∈ Ω and r is so small that {|z − p| ≤ r} ⊂ Ω, then 1 u(p) = 2π

Z

2π

√

u(p + re

−1θ

)dθ.

0

Proof. Let 1 Ar (u) := 2π

2π

Z

u(p + re

√ −1θ

)dθ.

0

By continuity we have lim Ar (u) = u(p).

r→0

By the divergence form of Green’s Theorem, we have 1 dAr (u) = dr 2πr

Z ∆udxdy = 0. |z−p| 0 there exists a continuous (and even smooth) 26

function ϕ on K such that u|K ≤ ϕ ≤ u|K + ε. This fact is easily established by applying the usual method of smoothing by convolution with an approximate identity, i.e., the function hε (x) = ε−2 h(ε−1 x), where h is a smooth, compactly supported function on the unit disk with values in [0, 1] and total integral 1. Given any function f : X → R on a Hausdorff space, we can define its upper regularization ∗ f by f ∗ (x) := lim sup f (y). y→x

By definition of lim sup, one easily sees that f ∗ is upper semi-continuous, f ∗ ≥ f , and that if g is upper semi-continuous and g ≥ f then g ≥ f ∗ . Thus, in particular, f is upper semi-continuous if and only if f ∗ = f . D EFINITION 1.8.2. A function u : Ω → [−∞, +∞) is called subharmonic (written u ∈ SH(Ω)) if (i) u is upper semi-continuous, and (ii) for each K ⊂⊂ Ω and h ∈ H(interior(K)) ∩ C 0 (K) such that u|∂K ≤ h|∂K , it holds that u ≤ h.

1.8.2

Basic properties

P ROPOSITION 1.8.3. The following basic properties of subharmonic functions hold. 1. If u ∈ SH(Ω) and c is a positive constant, then cu ∈ SH(Ω). 2. If {uα | α ∈ A} ⊂ SH(Ω) and u, defined by u(x) = supα uα (x) is finite and upper semicontinuous, then u ∈ SH(Ω). 3. If u1 ≥ u2 ≥ ... is a sequence of subharmonic functions, then u = lim uj is subharmonic. Proof. 1 and 2 are trivial. To show 3, note first that since {uj } is a decreasing sequence of functions, u−1 [−∞, s) = ∪j u−1 j [−∞, s) and so u is upper semi-continuous. Let K ⊂⊂ Ω and h a continuous function on K which is harmonic on the interior of K and majorizes u on the boundary of K. Fix ε > 0 and let Aj := {z ∈ bdry K | uj (z) ≥ h(z) + ε}. Then Aj is closed (hence compact) and Aj+1 ⊂ Aj . Since ∩Aj = ∅, it follows that for j0 large enough, Aj = ∅ for all j ≥ j0 . Thus uj ≤ h + ε in K for all sufficiently large j, and hence u ≤ h in K. R EMARK . We note that in statement 2 of Theorem 1.8.3, one can drop the requirement that u is upper semi-continuous, but then the conclusion is that u∗ is subharmonic. 27

1.8.3

Subharmonic again

Although our definition of subharmonic functions justifies the name and gives us some useful properties, it is not well adjusted to proving various of the elementary properties of subharmonicity, such as the fact that subharmonicity is preserved under addition and that it is a local property. The next theorem gives three alternative definitions of subharmonicity. T HEOREM 1.8.4. Let u : Ω → [−∞, ∞) be upper semi-continuous. Then the following are equivalent. 1. u ∈ SH(Ω). 2. If D ⊂ Ω is a disc and f is a holomorphic polynomial on D such that u ≤ Re f on ∂D, then u ≤ Re f on D. 3. If δ > 0, z ∈ Ω is of distance more than δ from the boundary of Ω, and dµ is a positive measure on [0, δ], then u(z) ≤ 2π

Rδ 0

2π

Z

1

Z

δ

√

u(z + re

dµ(r)

0

−1θ

)dµ(r)dθ.

(µSM V )

0

4. For each δ > 0 and z ∈ Ω of distance more than δ from the boundary of Ω, there exists a positive measure dµ on [0, δ] that is not supported on {0}, such that (µSM V ) holds. Proof. Clearly 2 follows from 1 and 4 follows from 3. Assuming that 2 holds, let z be a distance at least δ from the boundary of Ω, let 0 < r < δ, and let ϕ be a continuous function on ∂Dz (r) majorizing u there. By approximation, we √ may assume −1θ by z and ϕ is a trigonometric polynomial, which we can extend into D (r) (by replacing e z √ − −1θ e by z¯) as a harmonic polynomial, hence the real part h of a holomorphic polynomial. Thus u ≤ h on Dz (r), and we have, 1 u(z) ≤ h(z) = 2π

Z

2π

√

ϕ(z + re

−1θ

)dθ.

0

Since we can approximate u from above, in L1òc , by continuous functions, we see that 1 u(z) ≤ 2π

Z

2π

√

u(z + re

−1θ

)dθ,

0

and integration over [0, δ] with respect to dµ shows that 3 holds. Assuming that 4 holds, let K ⊂⊂ Ω and suppose h ∈ H(interiorK) ∩ C 0 (K) is such that h ≥ u on bdryK. Let v = u − h and set M := supK v. Being an upper semi-continuous function, v achieves its maximum on K. Assuming, for the sake of contradiction, that M > 0, we find that there exists a nonempty compact F ⊂ K such that v = M on F . Let z0 be the point in F which 28

is of minimal distance > δ to the boundary of K. Since v is upper semi-continuous, v < M on an open subset of Dz0 (δ). Thus Z 2π Z δ √ 1 v(z0 + re −1θ )dµ(r)dθ < M, M = v(z0 ) ≤ Rδ 2π 0 dµ(r) 0 0 and this is the desired contradiction. C OROLLARY 1.8.5.

1. The sum of two subharmonic functions is subharmonic.

2. Subharmonicity is a local property: u ∈ SH(Ω) if and only if every point p ∈ Ω has a neighborhood U such that u|U is subharmonic. 3. If f ∈ O(Ω), then log |f | is subharmonic (and in fact, harmonic away from the zeros of f ). 4. If ϕ : R → R is a convex increasing function then with ϕ(−∞) := lim ϕ(x), x→−∞

ϕ ◦ u is subharmonic whenever u is. 5. If u1 and u2 are subharmonic functions then log(eu1 + eu2 ) is subharmonic. Proof. 1 and 2 follow from Theorem 1.8.4 parts 4 and 2 respectively, while 3 follows from part 2 of Theorem 1.8.4 and the Maximum Principle. 4 follows from Jensen’s inequality. To see 5, note that eui are subharmonic by 4. Let D ⊂ Ω be a closed disc, and let f be a holomorphic polynomial such that eu1 + eu2 ≤ Ref on ∂D. We may assume, after adding an arbitrarily small constant, that f is nowhere zero in D; say, f = eg for some holomorphic function g. Then eu1 + eu2 ≤ |eg |. Now, since uj − Reg is subharmonic, 4 with ϕ the exponential function implies that euj |e−g | is subharmonic for j = 1, 2. Hence by 1, so is (eu1 + eu2 )|e−f |, and since the latter is ≤ 1 on ∂D, we have that log(eu1 + eu2 ) ≤ Ref on D. Then 5 follow from 2.

1.8.4

Local integrability

Another important corollary of Theorem 1.8.4 is the following result. T HEOREM 1.8.6. If Ω ⊂ C is connected and u ∈ SH(Ω), then either u ≡ −∞ or u ∈ L1òc (Ω). R EMARK . In particular, if u is subharmonic and u 6≡ −∞ then u > −∞ a.e. Proof. Let X := {z ∈ Ω | there exists r > 0 with D(z, r) ⊂⊂ Ω and u ∈ L1 (D(z, r)) }. Then X is clearly open, and by the upper semi-continuity (hence boundedness from above) of u and part 3 of Theorem 1.8.4, each z with u(z) > −∞ belongs to X. We claim also that X is closed. Indeed, if p ∈ X then there exists points z arbitrarily close to p such that u(z) > −∞. One of these points is the center of a disc that contains p and is contained in Ω, and again by part 3 of theorem 1.8.4, u is integrable on a neighborhood of p. Thus, since X is both open and closed and since Ω is connected, the theorem is proved. 29

1.8.5

Regularity

Finally, we come to the most important characterization of subharmonicity. To state it, we will use the theory of distributions. T HEOREM 1.8.7. Let u be a distribution. Then u ∈ SH(Ω) if and only if, in the sense of distributions, ∆u ≥ 0. To prove this result, we need the following two lemmas. L EMMA 1.8.8. Let u ∈ C ∞ (Ω). Then u is subharmonic if and only if ∆u ≥ 0. Proof. By Theorem 1.8.4, it suffices to show that the sub-mean value property holds precisely when ∆u ≥ 0. Let us compute the Taylor series of u to order 2 near a given point p. With (x, y) as real coordinates in C, we have √

u(p + re

−1θ

) = u(p) + r (ux (p) cos θ + uy (p) sin θ) r2 + uxx (p)cos2 θ + uyy (p) sin2 θ + uxy sin 2θ + O(r3 ), 2

so that Z

2π

√

u(p + re

−1θ

)

0

dθ r2 − u(p) = ∆u + O(r3 ). 2π 4

Thus, if the mean value property holds, then ∆u ≥ 0. Conversely, if ∆u ≥ 0, then with Z

2π

√

u(p + re

f (r) =

−1θ

0

)

dθ 2π

we have, by Green’s Theorem, 1 f (r) = 2πr 0

Z ∆u dA ≥ 0. Dr (0)

Since u is smooth, lim f (r) = u(p).

r→0

Thus the sub-mean value property is established, and the proof is complete. ∞ L EMMA 1.8.9. √ Let u ∈ C (Ω) be subharmonic, and let ψ ∈ D(D1 (0)) such that ψ ≥ 0 and −1θ ψ(x) = ψ(e x) for all real θ. Then uε := u ∗ ψε is a subharmonic family of functions that decreases with ε.

30

Proof. First, note that ∆(u ∗ ψε ) = (∆u) ∗ ψε and the latter is non-negative because both ∆u and ψ are. Next, Z

Z

1

u(z − εζ)ψ(ζ)dA(ζ) =

uε (z) =

Z

0

D1 (0)

2π

√

u(z − εre

rψ(r)dr

−1θ

)dθ,

0

and since, as was shown at the end of the proof of lemma 1.8.8, the mean value Z

2π

u(z − εreiθ )dθ

0

is an increasing function of ε, the lemma is proved. Proof of Theorem 1.8.7. The idea here is to smooth the distribution u to a family uε of subharmonic functions such that for all x, uε1 (x) ≤ uε2 (x) whenever ε1 ≤ ε2 . Once this has been done, the theorem follows from Lemma 1.8.8 and the fact that a decreasing sequence of subharmonic functions is subharmonic. √ To this end, we let ψ ∈ D(D1 (0)) such that ψ ≥ 0 and ψ(x) = ψ(e −1θ x). Let ψε (x) := ε−2 ψ(x/ε) and set uε = u ∗ ψε . We claim that uε is a smooth family of subharmonic functions decreasing with ε. Seeing that uε is subharmonic is easy: ∆uε = (∆u) ∗ ψε ≥ 0 since ∆u is a positive distribution, and ψ ≥ 0. To see the decreasing property, we use a “double smoothing trick”: u ∗ ψδ is smooth and subharmonic (because ∆(u ∗ ψδ ) = (∆u) ∗ ψδ ≥ 0 ), and hence (u ∗ ψε ) ∗ ψδ = (u ∗ ψδ ) ∗ ψε is a subharmonic family decreasing with ε. By Lemma 1.8.9 the right hand side implies that for each δ, and all ε1 < ε2 u ∗ ψδ ∗ ψε1 ≤ u ∗ ψδ ∗ ψε2 . while the left hand side allows us to let δ → 0. It follows that u ∗ ψε is a decreasing sequence of subharmonic functions, and thus converges to a subharmonic function or else to −∞ identically. But since u ∗ ψε → u, the latter does not happen. The proof is complete.

1.9

Exercises

1.1 Prove the formula (1.1) from Green’s Theorem. 1.2 Prove that if f ∈ O(C) and |f (z)| ≤ C(1 + |z|2 )N then f is a polynomial in z, of degree at most N . 1.3 Prove that if f ∈ O(C) and

R C

|f |2 dA < +∞ then f ≡ 0.

1.4 Prove that if f is holomorphic on the punctured unit disk D − {0} and then f ∈ O(D). 31

R D−{0}

|f |2 dA < +∞

1.5 Let F ⊂ O(C) be a family of entire functions such for each R > 0 there is a constant CR such that Z |fj |2 dA ≤ CR . sup f ∈F

D(0,R)

Show that every sequence in F has a convergent subsequence. 1.6 Find a harmonic function in the punctured unit disk that is not the real part of a holomorphic function. 1.7 Let u be a subharmonic function in the unit disk. Show that the function ψ : [0, ε) → R ∪ [−∞) defined by Z 2π √ 1 ψ(r) := u(er e −1θ )dθ 2π 0 is convex and increasing.

32

Chapter 2 Riemann Surfaces 2.1 2.1.1

1-Dimensional Complex Manifolds Complex charts

D EFINITION 2.1.1. Let X be a topological space. A complex chart is a homeomorphism from an open set U ⊂ X onto an open set V ⊂ C: X C ∪ ∪ ϕ U −→ V ∼ = E XAMPLE 2.1.2. Take U = X = R2 , V = C and ϕ(x, y) = x +

√

−1y.

E XAMPLE 2.1.3. Let X = S 2 , realized as X = {(x, y, w) ∈ R3 ; u2 + v 2 + w2 = 1}. Define the maps ϕN : UN → C and ϕS : US → C by the formulae ϕN (u, v, w) :=

√ u v + −1 1−w 1−w

and

ϕS (u, v, w) :=

√ u v − −1 . 1+w 1+w

Here UN = S√2 − {(0, 0, 1)} and US = S 2 − {(0, 0, −1)}. Let x + −1y = ϕN (u, v, w). We have u = x(1 − w) and v = y(1 − w), and thus from 2 u + v 2 + w2 = 1 we obtain w2 = 1 − (1 − w)2 (x2 + y 2 ). This quadratic equation in w has the two solutions x2 + y 2 − 1 w = 1 or w = 2 . x + y2 + 1 But on UN we have omitted the first solution, so we have the formula √ 2y x2 +y 2 −1 −1 2x ϕN (x + −1y) = x2 +y2 +1 , x2 +y2 +1 , x2 +y2 +1 . 33

We then compute that with z = x +

√

−1y, 1 ϕS ◦ ϕ−1 N (z) = . z

2.1.2

Riemann surfaces

An atlas is a collection of compatible charts for a topological space. The word “compatible” refers to the regularity of the glue used in putting together the coordinate charts, and thus often one says C k -atlas where k is an integer, ∞ or ω. The regularity of the glue then allows us to do calculus (a local procedure) of regularity up to k on pieces of our manifold. In the case of Riemann surfaces, we use an even more regular glue, allowing us to do complex analysis locally. D EFINITION 2.1.4. Let X be a topological Hausdorff space. (a) Two complex charts ϕ : U → C and ψ : V → C are said to be compatible if the map ϕ ◦ ψ −1 : ψ(U ∩ V ) → ϕ(U ∩ V ) is (bi)holomorphic. The map ϕ ◦ ψ −1 is called a transition map, or a change of coordinates. (b) A complex atlas for X is a collection of charts {ϕj : Uj → C ; j ∈ J} any two of which are compatible, and whose domains form an open cover of X: [ Uj . X= j∈J

(c) Two complex atlases are equivalent if their union is also a complex atlas. An atlas is maximal if it contains every equivalent atlas. (Thus every atlas is contained in a unique maximal atlas.) (d) A Riemann surface is a topological Hausdorff space together with a maximal complex atlas. Specifying a maximal complex atlas is often also referred to as specifying a complex structure. (e) (Classical terminology) If a Riemann surface is non-compact, it is called an open Riemann surface. R EMARK . We emphasize the parenthetical remark in (c): one atlas determines a Riemann surface uniquely. E XAMPLE 2.1.5. Let C := (R2 , {ϕ : (x, y) 7→ x +

√

−1y}).

The Riemann surface C is called the affine line or the complex plane. 34

E XAMPLE 2.1.6. Let ∆ := ({(x, y) ∈ R2 ; x2 + y 2 < 1}, {ϕ : (x, y) 7→ x +

√

−1y}).

The Riemann surface ∆ is called the unit disc. E XAMPLE 2.1.7. Let P1 := (S 2 , {ϕN , ϕS }), where ϕN and ϕS are the maps defined in Example 2.1.3. As we showed in that example, the pair {ϕN , ϕS } form an atlas. The Riemann surface P1 thus obtained is called the projective line or the Riemann sphere. In complex analysis, other common notations are C and C ∪ {∞}. R EMARK . We have ∆ ⊂ C ⊂ P1 . The second inclusion is not as clear, but follows from what was done in Example 2.1.3. In general, any open subset of a Riemann surface is a Riemann surface.

2.1.3

Riemann surfaces as smooth 2-manifolds

In view of the Cauchy formula, every holomorphic function on a domain in C is real analytic. Thus every Riemann surface is a real analytic manifold of dimension 2. But in fact, Riemann surfaces have a little more structure. T HEOREM 2.1.8. Every Riemann surface is oriented. Proof. Let f = fαβ := ϕα ◦ ϕ−1 β be a typical transition map. Write f = u + Jacobian determinant of f is det(Df ) = ux vy − uy vx .

√ −1v. Then the

On the other hand, since f is holomorphic, we have vx = −uy and vy = ux . Thus 2 ∂f 2 2 det(Df ) = ux + uy = > 0. ∂z This completes the proof. When a topological 2-manifold X is compact and oriented, we can completely characterize X; it is just a sphere with g handles attached. If g = 1, X is called a torus. Moreover, any two such manifolds with the same number of handles are homeomorphic. D EFINITION 2.1.9. Let X be a compact oriented topological 2-manifold. The number g of handles of X is called the genus of X. One could wonder if orientation is the only obstruction to defining a complex structure on a topological 2-manifold. As it turns out, this is indeed the case. Here, we will state and partially prove a theorem that implies such a complex structure. T HEOREM 2.1.10. Let k ≥ 2. Every oriented 2-manifold of class C k has a complex structure. In particular, every compact oriented 2-manifold has a complex structure. 35

The key idea behind this proof is the following theorem. T HEOREM 2.1.11. Let g be a C 2 -smooth Riemannian metric in√ a neighborhood of 0 in R2 . There exists a positively oriented change of coordinates z = u(x, y) + −1v(x, y) fixing the origin, such that g = a(z) dzd¯ z. Such coordinates are called “isothermal coordinates”. Theorem 2.1.11, which is a classical result in partial differential equations, is called the KornLichtenstein Theorem. Though it is not yet clear here, the saught-after function z can be obtained by solving a partial differential equation that is closely related to the Cauchy-Riemann equations— a system of partial differential equations that figures more prominently in this book than any other circle of ideas. We shall give a proof of Theorem 2.1.11 in Chapter 11, after we learn how to solve the Cauchy-Riemann equations using Hilbert space methods. Proof of Theorem 2.1.10 given Theorem 2.1.11. Fix a Riemannian metric g on X and let {zα := ϕα ; α ∈ A} be an oriented atlas of isothermal coordinates for g . If z and w are two such coordinates, then they are related by a smooth map w = F (z). Now z) g = a(w) dwdw¯ = a(w) (F (z))(Fz dz + Fz¯d¯ z )(Fz¯dz + Fz d¯ (w) 2 2 = a (F (z))(|Fz | + |Fz¯| )dzd¯ z + Fz Fz¯dz 2 + Fz¯Fz d¯ z2. On the other hand, g = a(z) dzd¯ z . It follows that either Fz = 0 or Fz¯ = 0. But if the former holds, then F is not positively oriented. Thus F is a holomorphic function, which completes the proof. R EMARK . Geometrically, the idea behind the proof of Theorem 2.1.10 is the following. First, we put a Riemannian metric on an oriented surface X. This Riemannian metric gives us a notion of rotation through an angle of π2 in each tangent space. Infinitesimally, this means we have a √ notion of multiplying by −1 in the tangent space. Theorem 2.1.11 implies that this infinitesimal √ multiplication by −1 is actually locally constant. Thus we obtain the structure of a Riemann surface.

2.2 2.2.1

Examples of Riemann surfaces Complex manifolds

Riemann surfaces are examples of more general spaces called complex manifolds. (More precisely, Riemann surfaces are 1 complex dimensional complex manifolds.) Since we are going to use some basic complex manifolds, we give their definition here, mostly for the sake of completeness; we shall not use any deep facts about complex manifolds in general. 36

D EFINITION 2.2.1. Let X be a topological Hausdorff space. 1. A complex chart of dimension n on X is a homeomorphism ϕ : U → V where U ⊂ X and V ⊂ Cn . 2. Two complex charts, ϕ1 : U1 → V1 and ϕ2 : U2 → V2 are said to be compatible if the −1 restrictions of ϕ2 ◦ ϕ−1 1 and of ϕ2 ◦ ϕ1 to U1 ∩ U2 are biholomorphic maps onto their images. 3. An atlas is a collection of charts that cover X. Two atlases are said to be equivalent if their union is also an atlas. A maximal atlas is an equivalence class of atlases. A maximal atlas is also called a complex structure. 4. The space X together with a maximal atlas is called a complex manifold. D EFINITION 2.2.2. 1. Let B ⊂ Cn be an open set. A function f : B → C is said to be holomorphic if it is continuous1 and holomorphic in each variable separately. 2. Let M be a complex manifold. A function f : M → C is said to be holomorphic if for any chart ϕ : U → V ⊂ Cn , the function f ◦ ϕ−1 : V → C is holomorphic. 3. Let B ⊂ Cn be an open set. A function f = (f1 , ..., fm ) : B → Cm is said to be holomorphic if each component function fj is holomorphic. 4. Let M1 , M2 be complex manifolds. A function f : M1 → M2 is said to be holomorphic if for every pair of charts ϕ1 : U1 ⊂ M1 → V1 ⊂ Cn1 and ϕ2 : U2 ⊂ M2 → V1 ⊂ Cn2 the function ϕ2 ◦ f ◦ ϕ−1 1 is holomorphic. 5. A holomorphic function f : M1 → M2 between two complex manifolds is said to be an embedding if it is one-to-one and its derivative Df has maximal rank at each point. 6. Two complex manifolds M1 and M2 are said to be biholomorphic if there exists a holomorphic function f : M1 → M2 that is both one-to-one and onto, and whose derivative Df is invertible at each point. 2 T HEOREM 2.2.3. A holomorphic function on a connected compact complex manifold is constant. Proof. Suppose f : M → C is holomorphic. Then |f | is continuous, and thus there is a point p ∈ M at which |f | takes on its maximum. Let ϕ : U → V ⊂ Cn be a complex chart containing p in its interior. Then |f ◦ ϕ−1 | attains its maximum in the interior of V . Applying the maximum principle to each variable separately, we see that f is constant on U . Since U is an open set and M is connected, the principle of analytic continuation implies that f is constant on M . C OROLLARY 2.2.4. If M is a compact complex submanifold of Cn , then M is zero-dimensional. Proof. One applies Theorem 2.2.3 to the restriction of the coordinate functions zj to M . 1 2

It is a fact that a function which is holomorphic in each variable separately is automatically continuous. It turns out that if the map f is one-to-one and onto, then its derivative is automatically invertible.

37

2.2.2

Examples of quotient Riemann surfaces

The examples of Riemann surfaces we considered until now were either trivially subsets of C or were constructed by “bare hands”. But there are other methods of obtaining examples of Riemann surfaces. A canonical idea is to obtain them as quotients of a space by an equivalence relation. We shall now present two examples. E XAMPLE 2.2.5 (The projective line again). Previously we put a complex structure directly on S 2 using the stereographic projection ϕN and its inversion ϕS . But there is another way to obtain P1 . Consider the set P1 of lines through the origin in C2 : P1 := {1−dimensional subspaces of C2 }. There is a map π : C2 − {0} → P1 sending each non-zero vector to the unique 1-dimensional subspace containing it. We denote the image of π using square brackets: π(z0 , z1 ) := [z0 , z1 ]. The map π is the projection associated to the equivalence relation (z0 , z1 ) ∼ (w0 , w1 ) ⇐⇒ ∃ λ ∈ C∗ such that (z0 , z1 ) = λ(w0 , w1 ). Specifying the point in projective space corresponding to the line passing through (z0 , z1 ) by the notation [z0 , z1 ] of points is abusively referred to as giving the “homogeneous coordinates” of the point. Next we define the topology on P1 as follows: U ⊂ P1 is open if and only if π −1 (U ) is open. Note that any topology on P1 for which π is continuous must contain this topology. Observe that, since π(S 3 ) = P1 , where S 3 = {|z| = 1} is the unit sphere in C2 , P1 is compact. We now define charts for P1 : Let (z0 , z1 ) be Euclidean coordinates in C2 . We let U0 := {[1, z1 ] ; z1 ∈ C}

and U1 := {[z0 , 1] ; z0 ∈ C},

i.e., U0 = P1 − {[1, 0]} and U1 = P1 − {[0, 1]}. Define the maps ϕ0 : U0 → C and ϕ1 : U1 → C by z0 z1 and ϕ1 [z0 , z1 ] := . ϕ0 [z0 , z1 ] := z0 z1 Observe that ϕ0 (U0 ∩ U1 ) = C − {0} and that ϕ1 ◦ ϕ−1 0 (ζ) =

1 ζ

1 and ϕ0 ◦ ϕ−1 1 (η) = . η

Thus we have defined a complex atlas, and in so doing, a complex structure for P1 . 38

R EMARK . We leave it as an exercise for the reader to show that P1 , as defined here, is diffeomorphic to S 2 , the unit sphere in R3 . Later we shall see that the Riemann surface so obtained is the same as the one defined previously in Example 2.1.7. In fact, there is only one way to define a complex structure on S 2 . R EMARK (Blowup and Tautological line bundle). We take this opportunity to describe an important construction. Let U := {(z, `) ∈ C2 × P1 ; z ∈ `}. Let B`0 : U → C2 and π : U → P1 denote the restriction to U of the projections from C2 × P1 to the first and second factor respectively. D EFINITION 2.2.6.

(1) B`0 : U → C2 is called the blowup of 0 in C2 .

(2) π : U → P1 is called the Tautological line bundle.3 The reader should ponder the following facts: Statement (1) corresponds to obtaining U from C2 by replacing the origin with a copy of P1 ; and P1 here should be interpreted as the set of possible slopes of the straight lines along which one can approach the origin. Statement (2) corresponds to defining U as the set of all points on the various lines that make up the projective line P1 . E XAMPLE 2.2.7 (Complex Tori). Let ω1 , ω2 ∈ C be R-independent. Consider the lattice L := Zω1 + Zω2 = {nω1 + mω2 ; n, m ∈ Z}. The set L is a discrete subgroup of the additive group C, and thus the quotient space X = XL is well-defined: X = XL := C/L. We denote the projection by π : C → X. We now have a set (actually, a group) X, but in order to endow it with the structure of a Riemann surface, we must first give it the structure of a topological space. We define a set U ⊂ X to be open if its inverse image π −1 (U ) ⊂ C is open. Then π is continuous in this topology, and in fact any topology that makes π continuous must be contained in this topology. The map π is an open mapping. Indeed, if U ⊂ C is open, then π(U ) is open, because [ π −1 (π(U )) = U + ω. ω∈L 3

Line bundles will be defined in Chapter 4.

39

Observe that X is compact. Indeed, the image of the set Pz := {z + tω1 + sω2 ; s, t ∈ [0, 1]} under π covers X: π(Pz ) = X. Next we will build charts on X. To this end, observe that since L ⊂ C is discrete, there is a positive number ε such that the only point of L that is within 2ε of the origin is the origin itself: L ∩ {z ∈ C ; |z| < 2ε} = {0}. By definition, for any such ε, any ω ∈ L − {0} and any z ∈ C, we have the property (D(z, ε) + ω) ∩ D(z, ε) = ∅, where D(z, ε) = {ζ ∈ C ; |ζ − z| < ε}. We define for every z ∈ C the map πz := π|D(z,ε) : D(z, ε) → π(D(z, ε)). We leave it as an exercise for the reader to show that the maps πz are homeomorphisms, and the collection A := {ϕz := πz−1 ; z ∈ C} is an atlas. The interested reader may also wish to prove that XL is diffeomorphic to S 1 × S 1 , the compact oriented manifold of genus 1 also known as the torus. R EMARK . Of course, for every choice of vectors ω1 and ω2 we obtain a different construction of a torus XL . While all of these constructions lead to a single smooth manifold, it is not clear whether all of these constructions lead to the same Riemann surface. In fact, they do not. R EMARK . In fact, all Riemann surfaces can be constructed as quotients of either P1 , C or ∆. This deep and important fact, known as the Uniformization Theorem, will be proved in Chapter 10.

2.2.3

Implicitly defined Riemann surfaces

In this paragraph, we discuss the possibility of defining a Riemann surface as a subset of C2 defined implicitly by a single equation F (x, y) = 0. We begin by describing the local version of this process. 40

G RAPHS OF HOLOMORPHIC FUNCTIONS Let D ⊂ C be an open set and f : D → Cn a holomorphic mapping. That is to say, f = (f1 , ..., fn ) and each fj is holomorphic on D. We define the surface Sf := {(z, w) ∈ D × Cn ; w = f (z)}. We endow Sf with the relative topology: U ⊂ Sf is open if and only if there is an open subset V ⊂ D × Cn such that U = V ∩ Sf . Denote by p1 : C × Cn → C the projection to the first factor: p1 (z, w) = z. We define the following complex charts of Sf : if V ⊂ D × Cn is open and U = V ∩ Sf , then ϕ : U → C is defined to be the restriction of p1 to U : ϕ := p1 |U . We define A to be the set of all such maps ϕ. It is not hard to show that A is a complex atlas for Sf (and that Sf is biholomorphic to D). D EFINITION 2.2.8. Let f : D → Cn be a holomorphic function. The Riemann surface (Sf , A) is called the graph of f . T HE I MPLICIT F UNCTION T HEOREM It is not hard to use the real implicit function theorem to prove a holomorphic version. However, elementary Complex Analysis can be used to give a direct and short proof of the holomorphic implicit function theorem. T HEOREM 2.2.9 (Implicit Function Theorem). Let U be an open subset of C2 (with coordinates (x, y)) containing the origin, and let F : U → C be a continuous function that is holomorphic in each variable separately. Suppose that F (0, 0) = 0 and Fy (0, 0) 6= 0. Then there exists a neighborhood V of the origin in C2 such that for each x0 ∈ proj1 (V ) the equation F (x0 , y) = 0 has a unique solution y0 = ϕ(x0 ) satisfying (x0 , y0 ) ∈ V . Moreover, x 7→ ϕ(x) is holomorphic. Proof. Since F is continuous, there exists ε > 0 such that Fy (x, y) 6= 0 whenever |x|, |y| ≤ ε. Let Z Fy (x, y) 1 dy. n(x) := √ 2π −1 |y|=ε F (x, y) By the Argument Principle n(x) is an integer. Our hypotheses imply that F (0, y) = yg(y) for some holomorphic function g which does not vanish at the origin. Thus n(0) = 1 and by continuity there exists ε0 < ε such that n(x) = 1 for all |x| ≤ ε0 . Thus, by the argument principle, for each such x there is a unique solution y = f (x) of the equation F (x, y) = 0. Moreover, by the Residue Theorem we have the formula Z 1 Fy (x, y) f (x) = y dy, 2πi |y|=ε F (x, y) which shows that f is holomorphic. 41

P LANE CURVES Our next goal is to give a complex structure to the zero set of a holomorphic function in two variables. The idea here is that this zero set is locally a graph over its tangent space whenever the latter is defined. This follows from the implicit function theorem, as we shall see. D EFINITION 2.2.10. An affine plane curve X is the locus of zeros in C2 of a holomorphic function f ∈ O(C2 ): X := {(z, w) ∈ C2 ; f (z, w) = 0}. (p) 6= 0 or A holomorphic function f ∈ O(C2 ) is non-singular at a point p ∈ C2 if either ∂f ∂z ∂f (p) 6= 0. The plane curve X = {f = 0} is non-singular at p ∈ X if f is non-singular at p, and ∂w X is non-singular if it is non-singular at each of its points. We shall now show that a non-singular plane curve X is a Riemann surface. To do so, we must specify an atlas for X. To this end, let p ∈ X. Since X is non-singular at p, one of the partial ∂f derivatives ∂f , ∂f does not vanish at p. Without loss of generality, we may assume ∂w (p) 6= 0. ∂z ∂w Let p = (z0 , w0 ). By the implicit function theorem, there is a neighborhood U of z0 in C and a holomorphic function gp : U → C such that X is the graph of gp over U . Thus the projection πp1 : (z, w) 7→ z onto the first variable gives a homeomorphism (πp1 )−1 (U ) → U . We take this to be our complex chart. We must now check the compatibility of these charts. Suppose p1 and p2 are two points of X whose associated neighborhoods U1 and U2 intersect. If the same partial derivative doesn’t vanish at both points, then the composition πp11 ◦ (πp12 )−1 is just the identity. On the other hand, if different ∂f (p1 ) 6= 0 6= ∂f (p2 ), then we have partials do not vanish, say ∂w ∂z πp22 ◦ (πp11 )−1 (z) = g(z), which is holomorphic. This completes the proof of the following theorem. T HEOREM 2.2.11. A non-singular affine plane curve X = (f = 0) is a Riemann surface. R EMARK . As we shall prove in Chapter 12, every open Riemann surface can be embedded in C3 . At the time of writing of this book, it was unknown whether every open Riemann surface can be embedded in C2 . E MBEDDED CURVES IN COMPLEX MANIFOLDS We note that the construction of the previous paragraph is not limited to working in the affine plane. Let M be a complex manifold of dimension n. Given a holomorphic function f : M → Cn−1 , we can consider its zero set X := {p ∈ M ; f (p) = 0}. We leave it to the reader to state and prove a higher dimensional version of the implicit function theorem in Cn for n > 2, and then use it to prove that if, for any point p ∈ X, df (p) has full rank n − 1, then X is a Riemann surface. One could have other complex manifolds constructed in this way, by considering maps f : M → Ck , 1 ≤ k ≤ dimC (M ). In general, the codimension of the resulting manifold would be k. 42

R EMARK . Of course, if M is a compact complex manifold, then M has no holomorphic functions on it other than constants. Thus the above construction cannot begin. In fact, there are complex manifolds that have no closed complex submanifolds at all. But there are other manifolds, like the complex projective space, that have many complex submanifolds, none of which could be described as the common zero locus of some holomorphic functions. We will soon modify this construction so that we can indeed describe complex submanifolds of projective spaces.

2.2.4

Projective curves

There are many instances in which a subset of a manifold enjoys a certain global symmetry. In this situation, it is often easier to analyze the subset by passing to the quotient space. Perhaps the simplest example of this instance is a subset of Cn that remains invariant under scaling. (That is to say, it is a collection of lines through the origin.) Passing to the quotient is known as projectivization. We begin in C3 . We first study some properties of the quotient space, known as the projective plane and denoted P2 . We then proceed to study basic properties of zero sets of homogeneous polynomials in C3 as per their image in P2 . T HE PROJECTIVE PLANE We define the projective plane to be the set P2 := {1 − dimensional subspaces of C3 }. If (x, y, z) ∈ C3 − {0} then there is a unique line in C3 passing through (x, y, z) and (0, 0, 0). We denote this point of P2 by [x, y, z]. We define the topology on P2 to be the most coarse topology such that the map π : C3 − {0} → P2 sending (x, y, z) to [x, y, z] is continuous. That is to say, a set U ⊂ P2 is open if and only if π −1 (U ) is open in C3 . R EMARK . When referring to points in P2 , x,y and z are called homogeneous coordinates. Of course, homogeneous coordinates of a point are not unique. However, saying that a homogeneous coordinate (or more generally, a homogeneous function) is zero or non-zero is meaningful. Later on, we will discuss a geometric way to interpret homogeneous functions: every such function can be identified with a section of some line bundle on projective space. The projective plane is covered by three rather large open sets U0 := {[x, y, z] ; x 6= 0}, U1 := {[x, y, z] ; y 6= 0} and U2 := {[x, y, z] ; z 6= 0}. 43

Each set Ui is homeomorphic to C2 . The homeomorphisms ϕj : Uj → C2 are given by ϕ0 ([x, y, z]) = xy , xz x z ϕ1 ([x, y, z]) = , y y x y ϕ2 ([x, y, z]) = z , z It is easy to see that ϕ0 ◦

ϕ−1 1 (ζ, η)

= ϕ0 ([ζ, 1, η]) =

1 η , ζ ζ

is well-defined and holomorphic on ϕ1 (U0 ∩ U1 ) = {(ζ, η) ; ζ 6= 0}. Similar calculations for the other possibilities of ϕi ◦ ϕ−1 j show that P2 is a complex manifold of complex dimension 2. Finally, we note that every line through the origin in C3 passes through the unit sphere S 5 := {|x|2 + |y|2 + |z|2 = 1} ⊂ C3 (in fact, it intersects the sphere in a circle) and thus P2 is the image under the continuous map π of the (compact) unit sphere S 5 ⊂ C3 . It follows that P2 is compact. H OMOGENEOUS POLYNOMIALS D EFINITION 2.2.12. A holomorphic function F : C3 → C is said to be homogeneous of degree j if for any p ∈ C3 and any λ ∈ C one has F (λp) = λj F (p).

(2.1)

It is easy to see, by expanding F is a power series, that any homogeneous holomorphic function on C3 of degree j can be written as X A`mn x` y m z n F (x, y, z) = `+m+n=j

for some complex numbers A`,m,n . L EMMA 2.2.13. If F is homogeneous of degree j, then jF (x, y, z) = x

∂F ∂F ∂F +y +z . ∂x ∂y ∂z

Proof. Differentiate equation (2.1) with respect to λ and then set λ = 1. D EFINITION 2.2.14. A homogeneous polynomial F (x, y, z) is said to be non-singular if the system of equations F =

∂F ∂F ∂F = = =0 ∂x ∂y ∂z

has no non-zero solutions. 44

(2.2)

Z ERO SETS OF HOMOGENEOUS POLYNOMIALS As already mentioned, there are no non-constant holomorphic functions on P2 and thus one cannot imitate the construction of affine curves in projective space directly. Instead, one can define subsets ˜ of P2 using homogeneous polynomials in C3 . Fix such a homogeneous polynomial F and let X 3 3 denote its zero set in C . Equation (2.1) shows that, with π : C − {0} → P2 the natural projection, ˜ = X. ˜ Thus no information is lost if we study the image X = π(X). ˜ Henceforth we π −1 ◦ π(X) shall say that X is cut out by F . Let Xj := X ∩ Uj and Yj := ϕj (Xj ), j = 0, 1, 2 Then each Yj ⊂ C2 is an affine plane curve given by the polynomial fj (ζ, η) = 0, where f0 (ζ, η) = F (1, ζ, η),

f1 (ζ, η) = F (ζ, 1, η)

and f2 (ζ, η) = F (ζ, η, 1).

L EMMA 2.2.15. If F is a non-singular homogeneous polynomial in C3 , then each Yj is a smooth affine plane curve. Proof. By symmetry it suffices to consider Y0 . Suppose Y0 is not smooth. Then by the implicit function theorem there exists a solution (ζ0 , η0 ) to the system of equations f0 (ζ, η) =

∂f0 ∂f0 = = 0. ∂ζ ∂η

But then (1, ζ0 , η0 ) is a solution to the system of equations (2.2). Indeed, F (1, ζ0 , η0 ) ∂F (1, ζ0 , η0 ) ∂y ∂F (1, ζ0 , η0 ) ∂z ∂F (1, ζ0 , η0 ) ∂x

= f0 (ζ0 , η0 ) = 0 ∂f0 = (ζ0 , η0 ) = 0 ∂ζ ∂f0 = (ζ0 , η0 ) = 0 ∂η = jF (1, ζ0 , η0 ) − ζ0

∂F ∂F (1, ζ0 , η0 ) − η0 (1, ζ0 , η0 ) = 0, ∂y ∂z

where the last equality follows from Lemma 2.2.13. This completes the proof. Lemma 2.2.15 shows that if X is the projectivization of the zero set of a non-singular homogeneous polynomial, then X can be written as a union of three open sets, each of which is a Riemann surface. It is not hard to convince oneself that each Xj is dense in X, and that X is a smooth manifold. But we want to know more, namely, that X is itself a Riemann surface. We now endow X with the unique maximal atlas A containing the following charts. If p ∈ X, then p ∈ Xj for some j. It follows that yj := ϕj (p) is in Yj , and since the latter is a smooth affine plane curve, there is a chart χj : U → V containing yj . We define a chart ψ : W → V at p by ψ := χj ◦ ϕj

−1 W := ϕ−1 j ◦ χj (U ).

and

In fact, we have the following theorem. 45

T HEOREM 2.2.16. The set X with the maximal atlas A is a Riemann surface. The proof of Theorem 2.2.16 is a simple combination of the holomorphicity of the overlap maps defining an affine curve, and the the holomorphicity of the transition functions defining P2 . We leave it as an exercise to the interested reader. P ROJECTIVE n- SPACE We define Pn to be the set of all 1-dimensional subspaces of Cn+1 . Let (z0 , ..., zn ) be coordinates in Cn+1 . Again, to each point z ∈ Cn+1 − {0} we can associate the unique line [z] ∈ Pn passing through z and 0. We define a map π : Cn−1 − {0} → Pn ; z 7→ [z]. The representation of an arbitrary point as [z0 , ..., zn ] is referred to as homogeneous coordinates. We define the topology of Pn to be the coarsest topology for which π is continuous. We observe again that, with S 2n+1 = {z ∈ Cn+1 ; |z|2 = 1}, π(S 2n+1 ) = Pn . Thus Pn is compact. We define open sets Uj := {[z] ∈ Pn ; zj 6= 0},

j = 0, ..., n,

and homeomorphisms n

ϕj : Uj → C ; [z] 7→

z0 zj−1 zj+1 zn , ..., , , ..., zj zj zj zj

,

j = 0, ..., n.

It is easily checked that ϕj ◦ ϕ−1 i (ζ1 , ..., ζn ) = ϕj [ζ1 : ... : ζi−1 : 1 : ζi+1 : ... : ζn ] z0 zi−1 1 zi+1 zn , ..., , , , ..., = zj zj ζj zj zj is well-defined and holomorphic on ϕi (Ui ∩ Uj ) = {ζ ∈ Cn ; ζj 6= 0}. C OMPLETE INTERSECTIONS To define curves in a higher dimensional projective space Pn , one can again consider zero sets of homogeneous polynomials. However, a single polynomial will only cut out a set whose dimension at a smooth point is n − 1. In order to cut out a curve, we must have n − 1 homogeneous polynomials. 46

D EFINITION 2.2.17. Let F1 , ..., Fn−1 be homogeneous polynomials (possibly of different degrees) in n + 1 variables z0 , ..., zn , and let X be the subset of Pn cut out by F1 , ..., Fn−1 . We call X a complete intersection curve. We further say that X is smooth if for each [x] ∈ X the (n−1)×(n+1) matrix of partial derivatives ∂Fi (x) ∂zj has maximal rank n − 1. Using the higher dimensional Implicit Function Theorem, one has the following proposition. P ROPOSITION 2.2.18. A smooth complete intersection in Pn is a Riemann surface. L OCAL COMPLETE INTERSECTIONS If n ≥ 3, not all 1-dimensional complex submanifolds of Pn are smooth complete intersections. The simplest example is the so-called twisted cubic in P3 , defined to be the image of the map ιV : P1 → P3 ; [z0 : z1 ] 7→ [z03 : z02 z1 : z0 z12 : z13 ]. (Note that in the chart U0 ⊂ P1 we have 2 3 ιV ◦ ϕ−1 0 (t) = ιV ([1, t]) = [1 : t : t : t ],

hence the name “twisted cubic”.) We observe that ιV (P1 ) is cut out by the three equations Z0 Z3 = Z1 Z2 ,

Z0 Z2 = Z12

Z1 Z3 = Z22 .

It turns out that ιV (P1 ) cannot be cut out by just two homogeneous polynomials. Note that if [Z0 , Z1 , Z2 , Z3 ] ∈ ιV (P1 ) is a point at which Z0 6= 0, then in a neighborhood of this point ιV (P1 ) can be cut out by the first two equations, since near such a point the third equation follows from the first two. D EFINITION 2.2.19. A local complete intersection curve X ⊂ Pn is a subset cut out by a family F = {Fα }α∈A of homogeneous polynomials, such that at each point [x] ∈ X, there is a neighborhood U of p in Pn and n − 1 polynomials Fα1 , ..., Fαn−1 ∈ F so that X ∩ U = {Fα1 = ... = Fαn−1 = 0}. We say that X is smooth if at each [x] ∈ X there exist Fα1 , ..., Fαn−1 ∈ F such that the (n − 1) × (n + 1) matrix ∂Fαi (x) ∂xj has full rank n − 1. As in the case of complete intersections, one can use a higher dimensional version of the Implicit Function Theorem to prove the following proposition. 47

P ROPOSITION 2.2.20. Every smooth local complete intersection curve X in Pn is a Riemann surface. It is a fact that every Riemann surface that can be holomorphically embedded in Pn (and must thus be compact) is a local complete intersection. It is also the case that every compact Riemann surface can be holomorphically embedded in Pn for some n. (In fact, n = 3 will do.) The latter will be proved in Chapter 12.

2.3

Exercises

2.1 Consider the singular curve Γ := {(z, w) ; z 2 = w3 } in C2 . Find (i) a singular curve C that is a family of four straight lines meeting pairwise transversely, in such a way that each point of C2 is contained in at most two of these lines, and (ii) a holomorphic map F : C2 → C2 , such that F (C) = Γ. 2.2 Give the structure of a complex manifold to the set of all k-dimensional complex subspaces of the complex vector space Cn . (The resulting manifold is called the Grassmann Manifold G(k, n).) 2.3 Show that the set {[x, y, z] ∈ P2 ; zx2 = y(y − z)(y − 2z)} is a Riemann surface. 2.4 On S 2 = {x ∈ R3 ; ||x|| = 1} consider the map A : x 7→ −x. Let X be the quotient of S 2 by the action of the group {I, A} ⊂ Diffeo(S 2 ). Define the norm of a tangent vector ξ to X to be the Euclidean norm of either of its lifts to S 2 via the quotient map, where these lifts are seen as vectors in R3 . Show that X is not orientable, and thus does not admit the structure of a Riemann surface. 2.5 Show that D and C are not biholomorphic.

48

Chapter 3 Functions and Maps We have already defined holomorphic functions and maps on complex manifolds, thus on Riemann surfaces. In this chapter we dig a little deeper into the notion of functions, and obtain more detailed local and global information.

3.1

Functions on a Riemann surface

In this section we extend to Riemann surfaces some of the ideas in Chapter 1.

3.1.1

Holomorphic functions

Let X be a Riemann surface, p ∈ X a point and f a function defined in a neighborhood W of p. Recall that we defined f to be holomorphic at p if there is a chart ϕ : U → V with p ∈ U such that f ◦ ϕ−1 is holomorphic at ϕ(p), and holomorphic on W if it is holomorphic at each point of W . We leave the proof of the following elementary lemma as an exercise to the interested reader. L EMMA 3.1.1. With X, p, W and f as above, (a) f is holomorphic at p if and only if for every chart ϕ : U → V with p ∈ U , f ◦ ϕ−1 is holomorphic at ϕ(p). (b) f is holomorphic on W if and only if for every atlas {ϕj : Uj → ϕj }j∈J on W , we have that for any p ∈ Uj , f ◦ ϕ−1 j is holomorphic at ϕj (p). E XAMPLE 3.1.2.

(a) Any complex chart is holomorphic on its domain.

(b) If X = C then the notion of holomorphic function agrees with the classical notion. (c) If f and g are holomorphic (at a point p or on W ) then so are f ± g and f g. Moreover, f /g is holomorphic if g is non-zero (at p or at any point of W ). 49

(d) Let X = P1 , [z0 , w0 ] ∈ P1 and let p, q ∈ C[z, w] be homogeneous polynomials of the same degree, such that q(z0 , w0 ) 6= 0. Then f ([z, w]) := p(z, w)/q(z, w) is holomorphic in a neighborhood of [z0 , w0 ]. In fact, f is holomorphic on the (rather large) open set Aq := {[z, w] ; q(z, w) 6= 0}. (e) Consider a torus X = C/L, and let W ⊂ X be an open subset. Then f is holomorphic at p ∈ W if and only if there is a point z ∈ π −1 (p) such that f ◦ π is holomorphic at z. In fact, f is holomorphic on W if and only if f ◦ π is holomorphic on π −1 (W ). (Here π : C → X is the projection defining the quotient.) We can, for instance, take W = X. In this case, a function f on X determines f˜ on C by pullback: f˜ = f ◦ π. To go in the other direction one needs a function on C that is invariant under the action of the lattice, i.e., if f˜(z + ω) = f˜(z) for all ω ∈ L, then there exists f : X → C such that f˜ = f ◦ π. (Traditionally, one says that f˜ is doubly periodic.) Since we know that every holomorphic function on a compact complex manifold is constant, we see that every doubly periodic function on C is constant. (f) Let X be an affine plane curve. The restriction of the coordinate functions to X defines two holomorphic functions on X. In fact, the restriction of any holomorphic function on C2 to X is holomorphic. More generally, if X is a curve in a complex manifold M , then the restriction to X of every holomorphic function on M is holomorphic. It turns out that if M = Cn then the converse is also true: If f is a holomorphic function on X then there exists a holomorphic function F on all of Cn such that F |X = f . (g) Let X ⊂ Pn be a projective algebraic curve that is a local complete intersection. Suppose F, G ∈ C[x0 , ..., xn ] are homogeneous polynomials of the same degree. Then f [x] = F (x)/G(x) is holomorphic on W := {[x] ∈ X ; G(x) 6= 0}. D EFINITION 3.1.3. Let X be a Riemann surface and W ⊂ X an open set. We define OX (W ) := {holomorphic f : W → C } . We might write O(W ) when X is clear from the context.

3.1.2

Meromorphic functions

Recall that, in Chapter 1, we defined the notions of pole and essential singularity for a holomorphic function on a punctured neighborhood. These notions are invariant under local biholomorphic functions, and thus can be extended to Riemann surfaces. D EFINITION 3.1.4. Let f be holomorphic in a punctured neighborhood of a point p ∈ X (a) We say that f has a removable singularity at p if there exists a complex chart ϕ : U → V containing p such that f ◦ ϕ−1 has a removable singularity at ϕ(p). 50

(b) We say that f has a pole of order n at p if there exists a complex chart ϕ : U → V containing p such that f ◦ ϕ−1 has a pole of order n at ϕ(p). (c) We say that f has an essential singularity at p if there exists a complex chart ϕ : U → V containing p such that f ◦ ϕ−1 has an essential singularity at ϕ(p). (d) We say that f has a zero of order n at p if there exists a complex chart ϕ : U → V containing p such that f ◦ ϕ−1 has a zero of order n at ϕ(p). As in analysis on domains in C, we have the following properties of singularities. T HEOREM 3.1.5. Let f be holomorphic in a punctured neighborhood of a point p ∈ X. (a) If f is bounded in a neighborhood of p then f has a removable singularity at p. (b) If limz→p |f (z)| = +∞ then f has a pole at p. (c) If limz→p |f (z)| does not exist, then f has an essential singularity at p. D EFINITION 3.1.6. Let X be a Riemann surface and let W ⊂ X be an open subset. We say that a function f on W is meromorphic at p ∈ W if f is holomorphic in a punctured neighborhood of p and has either a pole at p or a removable singularity at p. We say that f is meromorphic on W if it is meromorphic at every point of W . In other words, if we take a chart ϕ : U → V containing p and expand f ◦ ϕ−1 in a Laurent series in a neighborhood of ϕ(p), then there is a largest integer n such that the Laurent series of f has no coefficients of order less than n. We now have a well-defined notion of meromorphic functions on a Riemann surfaces, and therefore a well-defined notion of the order of a holomorphic function at a point. We write Ordp (f ) for the order of the meromorphic function f at the point p. E XAMPLE 3.1.7. Here are several examples of meromorphic functions. (a) Any holomorphic function is also a meromorphic function. Indeed, every point in the domain of definition is a removable singularity. (b) If X = C then the notion of meromorphic function agrees with the classical notion. (c) If f and g are meromorphic (at a point p or on W ) then so are f ± g, f g and f /g, the latter provided that g 6≡ 0. (d) Let X = P1 . If p, q ∈ C[z, w] are homogeneous polynomials of the same degree such that q 6≡ 0, then f ([z, w]) := p(z, w)/q(z, w) is a meromorphic function on P1 . 51

(e) Consider a torus X = C/L. We define a meromorphic function ℘ : C → C as follows: X 1 1 1 . ℘(z) := 2 + − z (z + ω)2 ω 2 06=ω∈L Ignoring issues of convergence, observe that ℘(z + ω) = ℘(z) for any ω ∈ L, and thus ℘ determines a unique meromorphic function f on the quotient X. Both f and ℘ are called the Weierstrass ℘-function. (f) Let X be an affine plane curve. If g and h are holomorphic functions on C2 then, after restricting f and g to X, the function f = g/h is meromorphic on X. In fact, Ordp (f ) = Ordp (g) − Ordp (h). More generally, let X be a curve in a complex manifold M , then the restriction to X of the quotient of any two holomorphic functions on M is a meromorphic function on X. It turns out that if M = Cn then the converse is also true: If f is a meromorphic function on X then there exist holomorphic functions F and G on Cn such that (F/G)|X = f . (g) Let X ⊂ Pn be a projective algebraic curve that is a local complete intersection. Suppose F, G ∈ C[x0 , ..., xn ] are homogeneous polynomials of the same degree such that G 6≡ 0 on π −1 (X) ⊂ Cn+1 . Then f [x] = F (x)/G(x) is meromorphic on X. D EFINITION 3.1.8. Let X be a Riemann surface and W ⊂ X an open set. We define MX (W ) := {meromorphic f : W → C } . We may write M (W ) when X is clear from the context.

3.1.3

The argument principle

Let Γ be a smooth embedded curve (i.e., an embedding of [0, 1]) in a S Riemann surface X. Let ϕj : Uj → Vj , j = 1, ..., N be N coordinate charts such that Γ ⊂ N j=1 Uj . For each j, let SN Wj ⊂⊂ Uj such that Γ ⊂ j=1 Wj , and let ψj be a smooth function that is ≡ 1 on Wj and whose P −1 N support is in Uj . Put χj := ψ ψj . We define k k=1 Z Γ

N

X df := f j=1

Z

χj (ϕ−1 j (z))

ϕj (Γ∩Uj )

0 (f ◦ ϕ−1 j ) (z)

f ◦ ϕ−1 j (z)

dz.

(3.1)

It is a standard exercise in advanced calculus to show that the definition is independent of choice of functions χj . Moreover, if we choose different coordinate charts, say ϕ˜j : U˜ → V˜ , and let z˜ be 52

the corresponding variable in V˜ then on V ∩ V˜ we have (f ◦ ϕ˜−1 )0 (˜ z) d˜ z −1 f ◦ ϕ˜ (˜ z) (f ◦ ϕ−1 ◦ (ϕ ◦ ϕ˜−1 ))0 (ϕ˜ ◦ ϕ−1 (z)) d(ϕ˜ ◦ ϕ−1 (z)) = f ◦ ϕ˜−1 (ϕ˜ ◦ ϕ−1 (z)) (f ◦ ϕ−1 )0 (ϕ ◦ ϕ˜−1 ◦ ϕ˜ ◦ ϕ−1 (z)) = d(ϕ ◦ ϕ˜−1 )(ϕ˜ ◦ ϕ−1 (z)) · d(ϕ˜ ◦ ϕ−1 (z)) f ◦ ϕ−1 (z) (f ◦ ϕ−1 )0 (z) = dz, f ◦ ϕ−1 (z) where the last two equalities follow from the chain rule. Thus the definition (3.1) is well posed, and we can state the following theorem, which is just Argument Principle on Riemann surfaces. T HEOREM 3.1.9. Let X be a Riemann surface and D ⊂ X an open subset whose closure is compact and whose boundary ∂D is piecewise smooth. If f is a meromorphic function on X with no zeroes or poles on ∂D, then I X df 1 √ = Ordx (f ). 2π −1 ∂D f x∈D Proof. The set A := {x ∈ D ; Ordx (f ) 6= 0} is locally finite, and thus finite. Let x1 , ..., xk be the points of A, and let D1 , ..., Dk be relatively compactScoordinate disks in D with centers at x1 , ..., xk respectively. Consider the domain E := D − kj=1 Dj . The boundary ∂E of E is piecewise smooth, and f 0 /f is holomorphic in E. Thus Z Z k Z df df df X − = = 0. f ∂E f ∂D f j=1 ∂Dj (The last equality is a consequence of Cauchy’s Theorem; the simplest adaptation of Cauchy’s Theorem to Riemann surfaces can be achieved using Stokes’ Theorem, which will be recalled in Chapter 5.) On the other hand, by the argument principle Z √ df = 2π −1Ordxj (f ). ∂Dj f This completes the proof.

3.2 3.2.1

Global aspects of meromorphic functions Meromorphic functions on compact Riemann surfaces

C OROLLARY 3.2.1. Let X be a compact Riemann surface. If f ∈ M (X), then X Ordp (f ) = 0. p∈X

53

Proof. Let p ∈ X such that f (p) 6= 0 or ∞. Let D be a small neighborhood of p such that (i) f has no zeros or poles in D, and (ii) D is the image, under the inverse of some complex chart ϕ, of a disk in the plane with center ϕ(p). Since f has no zeros or poles in D, Theorem 3.1.9 shows that I I X X df df = 0 and Ordp (f ) = Ordp (f ) = . ∂(X−D) f ∂D f p∈X p∈X−D But

I ∂(X−D)

df =− f

I ∂D

df . f

This completes the proof.

3.2.2

Meromorphic functions on P1

Example (d) above says that the quotient of any two homogeneous polynomials of the same degree in C2 descends to a meromorphic function on P1 . In fact, we have the following theorem. T HEOREM 3.2.2. Any meromorphic function on P1 is a quotient of two homogeneous polynomials of the same degree. Proof. Let f be a meromorphic function on P1 . It follows from the compactness of P1 that f has only a finite number of zeroes and poles. Let [zj , wj ], j = 1, ..., N be the points of P1 where the order nj :=Ord[zj ,wj ] (f ) of f is non-zero. Consider the rational function R(z, w) :=

N Y (zwj − wzj )nj . i=1

P Since f ∈ M (P1 ), Corollary 3.2.1 implies that N j=1 nj = 0. Hence R(tz, tw) = R(z, w) for all 2 t ∈ C and (z, w) ∈ C . Thus R descends to a Meromorphic function on P1 , and by construction has the same zeroes and poles as f , counting multiplicity. In other words, f /R is a meromorphic function on P1 with no zeroes or poles. Thus f /R is holomorphic, and so constant. It follows that f is a rational function, as desired.

3.2.3

Meromorphic functions on complex tori

We saw a construction of a function on the torus in Example 3.1.7 (e) above. We shall now construct all functions on the torus. We specialize to a lattice of the form L = Z ⊕ τ Z for some τ with positive imaginary part. Later we will show that every complex torus is biholomorphic to such a torus. Fixing τ with Im(τ ) > 0, we define θ(z) :=

∞ X

√

e

−1π(n2 τ +2nz)

n=−∞

54

.

This series converges absolutely and uniformly on any compact subset of C, and moreover θ(z + 1) = θ(z). On the other hand, θ(z + τ ) = e−

√

−1π(τ +2z)

θ(z).

It follows that the zeroes of θ remain invariant after translation by any element of L, counting multiplicity. Moreover, √ √ √ θ0 (z + 1) = θ0 (z) and θ0 (z + τ ) = e− −1π(τ +2z) θ0 (z) − 2π −1e− −1π(τ +2z) θ(z). P ROPOSITION 3.2.3. Let Π1,τ be the closed parallelogram in C spanned by 1 and τ . Then θ(z) has exactly one zero, counting multiplicity, in Π1,τ . Proof. Observe first that for x ∈ [0, 1], θ(x) =

X

e2π

√

√ −1nx π −1n2 τ

e

n∈Z

and θ(xτ ) = e−π

√

−1x2 τ

X

2

e(n+x)

√

−1πτ

.

n∈Z

A careful analysis shows that these two sums do not vanish. Thus by periodicity, θ does not vanish on ∂Π1,τ . Next, note that θ0 (z + 1) θ0 (z) = θ(z + 1) θ(z)

and

√ θ0 (z + τ ) θ0 (z) = − 2π −1. θ(z + τ ) θ(z)

The first claim is thus a direct application of the argument principle together with the fact that I Z Z Z Z = + − − . ∂Π1,τ

[0,1]

1+τ ·[0,1]

τ +[0,1]

τ ·[0,1]

The proof is complete Using the so-called Jacobi triple product, one can give the following product representation for the function θ: ∞ Y θ(z) = (1 − wm )(1 + q 2m−1 w2 )(1 + q 2m−1 w−2 ), √

m=1 √ −1πτ

. Looking at the third factor when m = 1, we find a zero of θ where w = e −1πz and q = e 2 when w = −q, i.e., when 2z = (1 + τ ). Thus, modulo proof of the product formula, which we omit, one has the following fact. P ROPOSITION 3.2.4. The zero of θ in Π1,τ is at

1+τ . 2

55

Now let θ(x) (z) := θ(z − 12 −

τ 2

− x).

Observe that θ(x) is still periodic with period 1, but that θ(x) (z + τ ) = −e2π

√

−1(z−x) (x)

θ

(z)

and

θ(x) (x) = 0.

Thus, for any finite collection x1 , ..., xN , y1 , ..., yM ∈ C, perhaps with repetitions, the ratio Q (xi ) θ (z) R(z) := Q i (yj ) (z) jθ is meromorphic in C, has period 1, and satisfies X X √ R(z + τ ) = (−1)N −M exp −2π −1 (N − M )z + yj − xi j

!! R(z).

i

In particular, we see that if N =M

and

X

yj −

X

j

xi ∈ Z,

i

then R descends to a meromorphic function on C/L. We can now imitate the proof of Theorem 3.2.2 to deduce the following result. T HEOREM 3.2.5. Any meromorphic function on a P complex torus, whose zeros [x1 ],..., [xk ] and poles [y1 ], ..., [yk ] (possibly with repetitions) satisfy (xj − yj ) ∈ Z is a quotient of two products of θ functions. In Chapter 14 we show that the zeros and poles of every meromorphic function on C/L satisfy the hypotheses of Theorem 3.2.5, and thus every meromorphic function on C/L is a quotient of products of theta functions.

3.3

Holomorphic maps between Riemann surfaces

In this section, we will study the basic properties of holomorphic maps between Riemann surfaces.

3.3.1

Basic definitions and simple theorems

Let X and Y be Riemann surfaces. Recall that if p ∈ X and W is a neighborhood of p, we say that a map F : W → Y is holomorphic at p if there are coordinate charts ϕ1 : U1 → V1 and ϕ2 : U2 → V2 such that p ∈ U1 ⊂ W , F (p) ∈ U2 and ϕ2 ◦ F ◦ ϕ−1 1 : V1 → V2 is holomorphic at ϕ1 (p). As in the definition of holomorphic functions, there are some immediate consequences stemming from the local biholomorphic invariance of Riemann surfaces. We state these in the following lemma, whose proof is omitted. 56

L EMMA 3.3.1. Let F : X → Y be a mapping between Riemann surfaces. 1. F is holomorphic at p if and only if for any pair of charts ϕ1 : U1 → V1 and ϕ2 : U2 → V2 with p ∈ U1 and F (p) ∈ U2 , ϕ2 ◦ F ◦ ϕ−1 1 is holomorphic at ϕ1 (p). 2. F is holomorphic on W if and only if there is a collection of complex S charts {ϕ1,j : U1,j → V } on X and {ϕ : U → V } on Y such that W ⊂ 2,j 2,j 2,j j∈J2 j∈J1 U1,j , F (W ) ⊂ S1,j j∈J1 j∈J2 U2,j and ϕ2,i ◦ F ◦ ϕ1,j is holomorphic for every i ∈ J1 and j ∈ J2 such that it well defined. Of course, every holomorphic function is a holomorphic map into C. The proofs of the following facts are left to the interested reader. 1. Every holomorphic mapping is C ∞ -smooth. 2. The composition of holomorphic maps is holomorphic. 3. If F : X → Y is holomorphic and W ⊂ Y is open, then (a) for any f ∈ OY (W ), F ∗ f := f ◦ F ∈ OX (F −1 (W )). (b) for any f ∈ MY (W ), F ∗ f := f ◦ F ∈ MX (F −1 (W )). D EFINITION 3.3.2. An isomorphism between two Riemann surfaces X and Y is a bijective holomorphic map F : X → Y whose inverse is also holomorphic. R EMARK . We know from the Normal Form Theorem that a bijective holomorphic map between Riemann surfaces has a holomorphic inverse. E XAMPLE 3.3.3. We leave it to the reader to check that the map 2Im (z w) ¯ |z|2 − |w|2 2Re (z w) ¯ , , [z, w] 7→ |z|2 + |w|2 |z|2 + |w|2 |z|2 + |w|2 is an isomorphism from P1 to S 2 with the atlas {ϕN , ϕS }, defined in Chapter 1. As an immediate consequence of the Open Mapping Theorem, we have the following corollary. P ROPOSITION 3.3.4. A non-constant holomorphic mapping between Riemann surfaces maps open sets to open sets. A corollary of the Identity Theorem is the following result. P ROPOSITION 3.3.5. Let X and Y be two Riemann surfaces such that X is connected, and let F, G : X → Y be holomorphic maps. If F = G on some open subset of X, then F ≡ G. The next proposition is a topological fact. 57

P ROPOSITION 3.3.6. If X and Y are two open Riemann surfaces such that X is compact and Y is connected, and if F : X → Y is a non-constant holomorphic map, then Y is compact and F is surjective. Proof. Since F is an open mapping, F (X) is open. Since X is compact, F (X) is closed. It follows that F (X) = Y is compact. P ROPOSITION 3.3.7. Let F : X → Y be a non-constant holomorphic map. Then for each y ∈ Y , F −1 (y) is a discrete set. In particular, if X is compact then F −1 (y) is finite for each y. Proof. Let y be a point in Y and choose a chart ϕ such that ϕ(y) = 0. Then ϕ ◦ F is a holomorphic function, and thus its zero set is discrete. But the zero set of this function is precisely F −1 (y).

3.3.2

Meromorphic functions as maps to the Riemann sphere

In this section we discuss the correspondence between meromorphic functions and maps to P1 . T HEOREM 3.3.8. Let f ∈ M (X). Then there exists a unique holomorphic map Ff : X → P1 such that the following holds: at every p ∈ X there are holomorphic functions gp and hp at p such that, near p, hp Ff = [gp , hp ] and f= . gp Proof. Let f ∈ M (X). If Ordp (f ) ≥ 0,then we take gp = 1 and hp = f . If Ordp (f ) ≤ 0, we take gp = 1/f and hp = 1. Observe that if Ordp (f ) = 0, then [f, 1] = [1, 1/f ]. Since this set of points is open, the identity theorem implies that F is well-defined. To see uniqueness, suppose Ff = [gp0 , h0p ]. Then, h0p /gp0 = hp /gp . If f ≡ 0, then gp ≡ gp0 ≡ 0 and we are done. Otherwise, we see that gp0 /h0p = gp /hp . It follows that at the points where hp 6= 0 6= h0p , [gp0 , h0p ] = [gp , hp ]. Since this is an open set, the uniqueness follows from the Identity Theorem. T HEOREM 3.3.9. Let F : X → P1 be a holomorphic map of Riemann surfaces. Then there exists a unique f ∈ M (X) such that such that F = Ff . Proof. Suppose p ∈ X and F (p) ∈ U0 . Then there is a neighborhood U of p such that F (q) ∈ U0 for all q ∈ U . It follows that ϕ0 ◦ F is holomorphic on U , and in fact F = [1, ϕ0 ◦ F ]. By a similar argument when F (p) ∈ U1 , we see that F = [ϕ1 ◦ F, 1] near p. Since ϕ01◦F = ϕ1 ◦ F on their common domain of definition, we set ϕ1 ◦ F (p) F (p) ∈ U1 f (p) := . 1 F (p) ∈ U0 ϕ0 ◦F (p) It is clear from the definition of P1 that f is a well-defined meromorphic function, and F = Ff . R EMARK . In view of the results of this section, there is no need to distinguish between meromorphic functions and holomorphic functions to P1 . In the text, we will often abuse notation by using phrases like “... a meromorphic function f : X → P1 ...”. 58

3.3.3

Multiplicity

T HEOREM 3.3.10 (Normal Form Theorem). Let F : X → Y be a holomorphic map between Riemann surfaces, and let x ∈ X. Then there exist two coordinate charts ϕ1 : U1 → V1 and ϕ2 : U2 → V2 at x and F (x) respectively, and a unique integer m = mx , such that ϕ1 (x) = ϕ2 (F (x)) = 0 and m ϕ2 ◦ F ◦ ϕ−1 1 (z) = z . m m ˜ Proof. To see uniqueness, suppose ϕ2 ◦ F ◦ ϕ−1 ˜2 ◦ F ◦ ϕ˜−1 1 (z) = z and ϕ 1 (z) = z . Then , with h1 = ϕ1 ◦ ϕ˜−1 ˜−1 1 and h2 = ϕ2 ◦ ϕ 2 , we see that m ˜ m h2 ((h−1 1 (z)) ) = z .

Since h1 and h2 are biholomorphic, a power series expansion shows that m ˜ = m. For existence, choose any pair of coordinate charts. After translations, we can assume that m h(ζ) ϕ˜1 (x) = ϕ2 (F (x)) = 0. It follows that for some integer m, ϕ2 ◦ F ◦ ϕ˜−1 . Let 1 (ζ) = ζ e m−1 h(ζ) 0 ψ(ζ) := ζe . Then ψ (0) 6= 0, and thus ψ has an inverse in some neighborhood of the origin. We let ϕ1 := ψ ◦ ϕ˜1 . D EFINITION 3.3.11. Let X be a Riemann surface, x ∈ X and F : X → Y a holomorphic map to another Riemann surface Y . 1. We call the integer m the multiplicity of F at x. We write m := Multx (F ). 2. If p ∈ X and Multp (f ) ≥ 2, we say that F is ramified at p, and that p is a ramification point for F . 3. If p ∈ X is a ramification point of F , we call F (p) a branch point of F . Finally, we have the following lemma. L EMMA 3.3.12. 1. Let X be a smooth affine plane curve cut out by a non-singular polynomial f (x, y), and let π : X → C be the restriction to X of the map (x, y) 7→ x. Then π is ramified at p ∈ X if and only if ∂f (x, y) = 0. ∂y (x,y)=p 2. Let X be a smooth projective plane curve cut out by a non-singular homogeneous polynomial F (x, y, z), and let G : X → P1 be the restriction to X of the map [x, y, z] 7→ [x, z]. Then G is a holomorphic map, and G is ramified at [x0 , y0 , z0 ] if and only if ∂F (x, y, z) = 0. ∂y (x,y,z)=(x0 ,y0 ,z0 ) 59

Proof. (1) If

∂f ∂y

6= 0, then by the implicit function theorem, π is a chart. Thus, since a chart is a p

local isomorphism, and thus not ramified, p is not a ramification point of π. Conversely, suppose ∂f = 0. Since f is non-singular, we must have ∂y p

∂f ∂x p

6= 0. It follows

from the implicit function theorem that X is cut out, near p, by the equation x − g(y) = 0 for some function g. Now, along X we have f (g(y), y) ≡ 0. Differentiation with respect to y at p = (x0 , y0 ) gives us that fx (p)g 0 (y0 ) + fy (p) = 0. Since fy (p) = 0 6= fx (p), we must have g 0 (y0 ) = 0. But g is precisely the local expression for π, and thus (1) holds. (2) The proof of this part is similar, but slightly more messy. It is left as an exercise to the interested reader. Finally, we leave it to the reader to check the following assertions. P ROPOSITION 3.3.13. Let f ∈ M (X) and F = Ff : X → P1 its associated holomorphic map. 1. If p in not a pole of f then Multp (F ) =Ordp (f − f (p)). 2. If p is a pole of f , then Multp (F ) = −Ordp (f ).

3.3.4

Degree of a holomorphic map

We begin with our first main theorem of this section. T HEOREM 3.3.14. Let F : X → Y be a holomorphic mapping between compact connected Riemann surfaces. Then the function Deg(F ) : Y → Z defined by X Deg(F )(y) = Multx (F ) x∈F −1 (y)

is constant. Proof. Let Yn := {y ∈ Y ; Deg(F )(y) ≥ n}. We claim that Yn is open in Y . Indeed, by compactness, the number of preimages of y is finite. Then, using the normal form of a map, we see that if Multx (F ) = m for x ∈ F −1 (y), then for all points y 0 6= y sufficiently near y have m preimages near x. Since Mult is a positive integer, we see that Deg(F )(y 0 ) ≥ Deg(F )(y), and Yn is indeed open. To show that Yn is closed, let (yk ) be a sequence in Yn , converging to a point y ∈ Y . Since there are only finitely many ramification points in Y , we may assume that for any yk , none of the preimages f −1 (yk ) contain any ramification points. Thus f −1 (yk ) contains n distinct points xk,1 , ..., xk,n . Since X is compact, we may extract from (xk,n )k≥1 a convergent subsequence, and thus obtain n points x1 , ..., xn ∈ X. Now, the points x1 , ..., xn may not be distinct. But if, say, x = xi1 = xi2 = ... = xij , then we claim that the multiplicity of x is at least j. Indeed, by the normal form theorem, the restriction of F to a neighborhood of x is an m-to-one map for some integer m. Moreover, if k is sufficiently large, this map takes the j distinct points xk,i1 , ..., xk,ij to the single point yk . Thus m ≥ j. It follows from this argument that the sum of the multiplicities at the points of the set (of ≤ n) points {x1 , ..., xn } is ≥ n, and thus y ∈ Yn , i.e., Yn is closed. 60

We conclude that, for each Yn , either Yn = Y or Yn = ∅. Now, choose any y ∈ Y . If n = Deg(F )(y), then y ∈ Yn but y 6∈ Yn+1 . It follows that Yn = Y and Yn+k ⊂ Yn+1 = ∅ for all k ≥ 1. Thus Deg(F ) ≡ n on Y , as desired. R EMARK . Recall that a map is proper if the inverse image of any compact set is compact. Theorem 3.3.14 holds even if X and Y are non-compact, so long as the map F : X → Y is proper. The proof holds essentially verbatim. D EFINITION 3.3.15. We denote the constant value of the map Deg(F ) by Deg(F ), and call it the degree of F . We have some corollaries of the theorem on the degree. C OROLLARY 3.3.16. A holomorphic map between compact connected Riemann surfaces is an isomorphism if and only if it has degree 1. C OROLLARY 3.3.17. If f ∈ M (X) is a meromorphic function with a single, simple pole, then X is isomorphic to P1 . Proof. By hypothesis, Deg(F ) = #(F −1 (∞)) = 1, and thus by the previous corollary, F is an isomorphism. R EMARK . We can also give another proof that the sum of the orders of a meromorphic function on a compact Riemann surface is 0. Indeed, according to the theorem on degree, X X Deg(f ) = Multx (f ) = Multy (f ). f (x)=0

f (y)=∞

But then, X

Ordp (f )

p∈X

=

X f (x)=0

=

X

X

Ordx (f ) +

Ordy (f )

f (y)=∞

Multx (f ) −

f (x)=0

X

Multy (f ) = 0,

f (y)=∞

where the second to last equality follows from Proposition 3.3.13.

3.3.5

The Riemann-Hurwitz Formula

D EFINITION 3.3.18. Let S be a compact 2-manifold, possibly with (smooth) boundary. 1. A 0-simplex, or vertex, is a point, a 1-simplex, or edge, is a set homeomorphic to a closed interval, and a 2-simplex, or face, is a set homeomorphic to the triangle {(x, y) ∈ [0, 1] × [0, 1] ; x + y ≤ 1}. 61

2. A triangulation of S is a decomposition of X into faces, edges and vertices, such that the intersection of any two faces is a union of edges and the intersection of any two edges is a union of vertices. a collection of subsets called faces, edges and vertices. A face is an open set homeomorphic to a disk, an edge is a set homeomorphic to an open interval, and a vertex is a point. The closure of a face, called a triangle, consists of the face, which is the interior of the closure, and three edges and three vertices, which are the boundary. each pair of these three boundary vertices bounds an edge. Finally, any non-empty intersection of the closure of any two faces must consist of entire edges and vertices. 3. Let S have a triangulation with total number of faces equal to F , total number of edges equal to E, and total number of vertices equal to V . Then the Euler characteristic of S is the number χ(S) := F − E + V. The main theorem about the Euler characteristic is the following result, stated here without proof. T HEOREM 3.3.19. Every compact 2-manifold has a triangulation, and the Euler characteristic χ(S) is independent of the triangulation. In particular, if S is an orientable 2-manifold without boundary and of genus g, then χ(S) = 2 − 2g. R EMARK . If one believes the first two statements, it is easy to prove the third. Indeed, observe that the Euler characteristic of a sphere is 2, that of a cylinder is 0 and that of a disk is 1. To increase the genus of a given surface, remove two disks and attach a cylinder to the boundary of the resulting surface. The proof then follows by induction. For the reader versed in Topology, the existence of a triangulation of a compact Riemann surface can be established by using the representation of the surface as a quotient of a 4g-gon. The independence of the Euler characteristic of a triangulation is proved by showing that two triangulations can be refined, by adding vertices and edges, to form a common triangulation. The process of refinement is then checked to see that it leaves the Euler characteristic unchanged. If X is a compact Riemann surface, we denote by g(X) the genus of X. We are now ready to discuss the second main theorem of this section. T HEOREM 3.3.20. (R IEMANN -H URWITZ FORMULA ) Let F : X → Y be a non-constant holomorphic map of compact connected Riemann surfaces. Then X 2g(X) − 2 = Deg(F )(2g(Y ) − 2) + (Multx (F ) − 1). x∈X

Proof. Let d = Deg(F ). Take a triangulation of Y such that every branch point is a vertex. (There may, of course, be other vertices.) Suppose this triangulation has F faces, E edges, Vu unbranched vertices and Vb branched vertices. 62

Since the preimage of every unbranched point has d points, we obtain a triangulation of X with dF faces, dE edges, and W vertices, where X

X

W = dVu +

1

y∈Vb x∈F −1 (y)

= dV − dVb +

X

X

1

y∈Vb x∈F −1 (y)

= dV + = dV +

X

X

y∈Vb

x∈F −1 (y)

X

(Multx (F ) − 1)

(Multx (F ) − 1)

x∈X

where the last equality follows because, by definition, Multx (F ) = 1 for all points x ∈ X at which F is unramified. The theorem follows from the definition of the Euler characteristic. R EMARK . Observe that the number B(F ) :=

X

(Multx (F ) − 1),

x∈X

called the branching degree of F , is even. C OROLLARY 3.3.21. Let F : X → Y be a non-constant holomorphic map of compact connected Riemann surfaces. 1. g(Y ) ≤ g(X). 2. If F is unramified and g(X) = 1 then g(Y ) = 1. 3. If F is unramified, g(X) > 1 and Deg(F ) > 1, then g(X) > g(Y ) > 1 and Deg (F )| (g(X)− 1). 4. If g(X) = 0 then g(Y ) = 0. 5. If g(X) = g(Y ) ≥ 1, then either Deg(F ) = 1 or g(X) = 1. In either case, F is unramified. Proof. Use either the Riemann-Hurwitz formula or its equivalent g(X) Deg(F ) − 1 g(Y ) = + − Deg(F ) Deg(F )

63

P

x∈X (Multx (F )

2Deg(F )

− 1)

.

3.3.6

An Example: maps between complex tori

Let L be a lattice in C and X = C/L the associated complex torus. For each a ∈ C, the translation map τa : C → C defined by τa (z) = z + a descends to a holomorphic self-map of the torus X. This map is an automorphism, with τa−1 = τ−a . Now let L0 be another lattice and X 0 the associated torus. Suppose F : X → X 0 is a holomorphic map. We claim there exists a map F˜ : C → C such that the diagram C π↓ X

F˜

−→ F

−→

C ↓π X0

where the vertical arrows are the quotient maps. Indeed, the map π ◦ F : C → X lifts to the universal cover of X, since C is simply connected.1 Moreover, the map F˜ is holomorphic because π and F are, and π is a covering map, hence a local isomorphism. After composing F and F˜ with translations, we may assume that F ([0]) = [0] and that F˜ (0) = 0. In this case, it must hold that F˜ (z +ω) = F˜ (z)+ω 0 (z, ω) where ω 0 : C×L → C is holomorphic and takes values in L0 . By continuity of ω 0 and the discreteness of L0 , it follows that ω 0 does not depend on z. Thus, for each ω ∈ L, the function F˜ (z + ω) − F (z) is constant. In particular, F˜ 0 (z) is doubly periodic, and thus constant. It follows that F˜ (z) = αz for some α ∈ C. Thus ω 0 = αω, and this restricts α somewhat. We summarize everything in the following proposition. P ROPOSITION 3.3.22. Let X = C/L and X 0 = C/L0 be two complex tori, and F : X → X 0 a holomorphic map. Then there exists an affine linear holomorphic function F˜ : C → C sending z to αz + a for some α, a ∈ C. The number a is determined modulo L0 , and the number α must satisfy αL ⊂ L0 . Moreover, F is an isomorphism if and only if αL = L0 . The last statement is easy to verify: the inverse of F is induced from the map G(z) = α1 (z − a). R EMARK . It is now obvious that every complex torus is biholomorphic to a torus whose lattice is generated by the complex numbers 1 and τ , where Im τ > 0. R EMARK . Let F : X → X 0 be induced by the map F˜ (z) = αz, and let d := [L0 : αL] be the index of αL in L0 . Then deg(F ) = d. Let us now determine the automorphisms of a complex torus X = C/L. To this end, let F : X → X be an automorphism. After composing with translation maps, we may assume that F ([0]) = [0]. Let F˜ (z) = αz be the associated lift. Since F is an automorphism, αL = L. We claim that any such α has unit modulus. Indeed, if ω ∈ L − {0} is the closest non-zero lattice element to the origin, then αω must also be the closest element of L − {0} to the origin. It follows that |α| = 1. Clearly we can have α = ±1. But there may be other possibilities. Indeed, suppose α ∈ C−R. Let ω ∈ L − {0} be a number of minimal length. As we saw, αω must also be of minimal length, 1

The basic ideas of covering spaces will be reviewed in Chapter 10

64

and thus if α 6= ±1, ω and αω must generate L. It follows that α2 ω = mαω +nω for some integers m and n. But then α must satisfy a quadratic equation α2 − mα − n = 0 for some integers m and n. Solving this equation gives √ m ± m2 + 4n α= . 2 Since α is a (non-real) root of unity, we must have m2 + 4n < 0, and thus m2 m2 2 1 = |α| = − n+ = −n. 4 4 √ We find that m2 < 4, so the only possibilities for m are 0 or ±1. If m = 0 we obtain α = ± −1 √ √ and if m = ±1 we obtain α = ± 12 ± 2−3 . (This is six possibilities in total.) In the cases α = ± −1 we see that L must be a square lattice, and in the other cases a hexagonal lattice. Thus we have the following proposition. P ROPOSITION 3.3.23. Let X = C/L be a complex torus, and let F ∈ Aut(X) be an automorphism fixing the origin. Then F is induced by the linear map z 7→ αz where 1. if L is a square lattice then α is a 4th root of unity, 2. if L is a hexagonal lattice then α is a 6th root of unity, and 3. otherwise α = ±1. We can now classify all complex tori. As we have already remarked, every torus X is isomorphic to a torus Xτ , with lattice generated by 1 and τ , for some Im τ > 0. Now suppose Xτ ∼ = Xτ 0 . Then we have a + bτ = τ 0 and c + dτ = 1 for some integers a, b, c, d. Thus τ 0 = (a +bτ )/(c + dτ ). a b Observe that, conversely, τ 0 determines τ from such a relation, and thus the matrix lies c d in the group SL(2, Z). We thus obtain the following proposition. P ROPOSITION 3.3.24. Two complex tori Xτ and Xτ 0 are isomorphic if and only if there is a matrix a b ∈ SL(2, Z) c d such that τ 0 = (a + bτ )/(c + dτ ). R EMARK . It is a fact that {τ ∈ C ; Im τ > 0}/SL(2, Z) is a Riemann surface isomorphic to the unit disk. 65

3.4 3.4.1

An example: hyperelliptic surfaces Gluing surfaces

Let X and Y be be two Riemann surfaces, U ⊂ X and V ⊂ Y two open subsets, and ϕ : U `→ V an holomorphic isomorphism. We define an equivalence relation on the disjoint union ` X Y of X and Y By identifying u ∈ U with ϕ(u) ∈ V . The resulting space is denoted X Y /ϕ. ` P ROPOSITION 3.4.1. There is a unique complex structure on X Y /ϕ such that the`natural in` ` clusions X ,→ X Y /ϕ and Y ,→ X Y /ϕ are holomorphic. Moreover, if X Y /ϕ is a Hausdorff space, then it is a Riemann surface. Proof. Simply take the union of the atlases for X and Y . These atlases are compatible on U and V because ϕ is an isomorphism. ` E XAMPLE 3.4.2. The space X Y /ϕ need not be Hausdorff. Indeed, the map obtained by gluing together C and C along C − {0} and C − {0} via the map ϕ(z) = z is not Hausdorff, as the origins of the two copies of C cannot be separated by open sets in the quotient.

3.4.2

Hyperelliptic surfaces

Let h be a polynomial in one complex variable, of degree 2g + 1 + , where = 1 or 0. Suppose all the roots of h have multiplicity 1. We define a smooth affine2 plane curve Xh := {(x, y) ; y 2 = h(x)}, and consider also the open subset Uh = {(x, y) ∈ Xh ; x 6= 0}. Observe that Xh is a smooth curve, since all the roots of h are distinct. Next, let ~(x) = x2g+2 h(1/x). Since ~ also has distinct roots, the affine plane curve X~ := {(x, y) ; y 2 = ~(x)} is smooth as well. We define the open subset U~ := {(x, y) ∈ X~ ; x 6= 0}. Finally, we define the isomorphism ϕ : Uh → U~ by 1 y ϕ(x, y) = , . x xg+1 (Observe that ϕ ◦ ϕ = id.) Let Z := Xh

a

X~ /ϕ.

L EMMA 3.4.3. The set Z is a compact Riemann surface of genus g, and the meromorphic function (x, y) → x on Xh extends to a holomorphic map π : Z → P1 whose degree is 2. The branch points of π are the roots of h, and ∞ if = 0. 2

The adjective affine refers to the fact that y 2 − h(x) is a polynomial– a fact we don’t directly exploit in this book.

66

Proof. To see compactness, observe that Z is the union of (the image in Z of) the compact sets {(x, y) ∈ Xh ; |x| ≤ 1} and {(z, w) ∈ X~ ; |z| ≤ 1}. We verify Hausdorff as follows. Any two points of Xh that lie in Uh can be separated by open sets in Z, since they can be separated by open sets in Uh . The image in Z of a point p in {x = 0} ⊂ Xh can be separated from any other point q of Z: if q ∈ Xh there is again nothing to prove. On the other hand, suppose q ∈ X~ . Take a neighborhood of p in Xh so small that its image under ϕ has very large first coordinates. Then the image of this neighborhood will miss q. Similar arguments applied to X~ show that Z is Hausdorff, and thus Z is a Riemann surface. Observe that the complement in Z of Xh is a finite set. (In fact, it has either one or two points.) We claim that πh : (x, y) 7→ x extends to a meromorphic function on Z. Indeed, on X~ we can define the meromorphic function π~ : (x, y) 7→ 1/x, and this map agrees with the former on ϕ(Uh ): π~ (ϕ(x, y)) = 1/(1/x) = x = πh (x, y). Alternatively, we can look at the associated map π : Z → P1 . In fact, let [1, x] (x, y) ∈ Xh π(x, y) := [1/x, 1] (x, y) ∈ X~ Clearly this map is consistent with the topological quotient, and corresponds to the meromorphic function (x, y) 7→ x on Xh . Finally we turn to the genus. First, it is clear that π has p degree 2, since at any non-root x of h, the preimage of any point x is the pair of points (x, ± h(x)). Moreover, ramification points occur either at the roots of h or, if ε = 0, at ∞ (which is a root of ~ if and only if the degree of h is odd). And in this case, the multiplicity of the ramification points is precisely 2. Thus there are precisely 2g + 2 ramification points, so by the Riemann-Hurwitz formula, we have 2g(Z) − 2 = 2(−2) + 2g + 2, and thus g(Z) = g. We note that the surface Z supports an involution, namely the map σ that sends (x, y) → (x, −y) on either Xh or X~ . Moreover, σ ∗ π = π. D EFINITION 3.4.4. The surface Z thus constructed is called a hyperelliptic Riemann surface of genus g. The involution σ is called the hyperelliptic involution, and the map π : Z → P1 is called the hyperelliptic projection.

3.4.3

Meromorphic functions on hyperelliptic surfaces

We now describe all of the meromorphic functions on a hyperelliptic Riemann surface (Z, σ, π). The essential point is the following lemma. L EMMA 3.4.5. Let f : Z → P1 be a σ-invariant meromorphic function: σ ∗ f = f . Then there is a meromorphic function g : P1 → P1 such that f = g ◦ π. 67

Proof. Simply define g(p) = f (q) where q ∈ π −1 (p). It is clear that g is well-defined. We now show that g is a holomorphic map. First, observe that Z is cut out locally by a holomorphic function of the form (x − p)eψ(x) + c = y 2 . In particular, x is a holomorphic function of y near y = 0. But then g(x(y)) = f (x(y), y) = f (x(−y), −y) = g(x(−y)). Thus the function F = g ◦ x is holomorphic in a neighborhood of y = 0, and satisfies F (−y) = We claim that the power P F (y). n series expansion of F has only even powers. Indeed, if F (z) = an z , then Z F (z)dz 1 an = 2πi |z|= z n+1 Z F (−z)d(−z) 1 = 2πi |z|= (−z)n+1 Z F (z)dz n 1 = (−1) = (−1)n an . n+1 2πi |z|= z (We have also used the fact that z 7→ −z does not change the orientation of the circle.) It follows that X X g(x) = F (y) = a2n y 2n = a2n ((x − p)eψ(x) + c)n is holomorphic, as desired. We can immediately understand the restriction to Xh of meromorphic functions that are σantisymmetric. Indeed, if f : Z → P1 is a meromorphic function with σ ∗ f = −f , then function defined on Xh by f (x, y) (x, y) 7→ y is σ-symmetric and, being a product of meromorphic functions, meromorphic. It follows that on Xh every σ-antisymmetric meromorphic function may be written as a product f (x, y) = yg(x), where g is a meromorphic function of x. It is now easy to describe all meromorphic functions on a hyperelliptic surface. Indeed, let f : Z → P1 be any meromorphic function. Then the functions 1 1 f + := (f + σ ∗ f ) and f − := (f − σ ∗ f ) 2 2 + − + are also meromorphic, and f = f + f . Moreover, f is symmetric and f − is antisymmetric. We thus have the following proposition. P ROPOSITION 3.4.6. Let X ⊂ Z be the affine part of a hyperelliptic surface, i.e., the surface cut out by the equation y 2 = h(x) where h is a polynomial with only simple roots. Then every f ∈ M(Z) may be written, on X, as f (x, y) = g1 (x) + g2 (x)y for some rational functions g1 and g2 . R EMARK . We emphasize that, while the function g1 extends to a meromorphic function on Z, the function g2 does not, since the function y does not transform well with respect to the identifying isomorphism ϕ. 68

3.5 3.5.1

Harmonic and Subharmonic Functions Definition of Harmonic Functions

Of course, a Riemann surface is also a C ∞ manifold of real dimension 2. As such, we can consider on it smooth functions. Recall that a function f is smooth if for any (and thus every) coordinate chart ϕ : U → V , f ◦ ϕ−1 is smooth on V . Suppose given a smooth function f and a chart z : U → V ⊂ C. Let z = x + iy. We can calculate 2 ∂2 ∂ 2 (f ◦ ϕ−1 ) ∂ −1 −1 . + f ◦ ϕ = 4 ∆(f ◦ ϕ ) := ∂x2 ∂y 2 ∂z∂ z¯ Observe that if f and g are complex valued functions, then ! ∂f (g) ∂g ∂f (g) ∂g + ∂ζ ∂ z¯ ∂z ∂ ζ¯ ! 2 ∂g ∂ 2 f (g) ∂g ∂g ∂ 2 f (g) ∂g ∂g + + (∆f )(g). 2 2 ¯ ∂ζ ∂z ∂ z¯ ∂z ∂ z¯ ∂z ∂ζ

∂ ∆(f ◦ g) = 4 ∂z = 4

In particular, if g is holomorphic, we have 2 ∂g ∆(f ◦ g) = (∆f )(g). ∂z

(3.2)

Thus we see that the vanishing of the “local function” ∆f is a property that is invariant under holomorphic changes of coordinates. R EMARK . In fact, the transformation rule for the local functions ∆(f ◦ ϕ) shows that ∆f is not a function at all, but rather the coefficient of a differential 2-form. The invariance of the kernel of the differential operator ∆ leads us to the following definition. D EFINITION 3.5.1. Let X be a Riemann surface. We say that a function f defined in a neighborhood of p is harmonic at p if there is a complex chart ϕ : U → V such that p ∈ U and ∆(f ◦ ϕ−1 ) ≡ 0

on V.

If W ⊂ X is an open subset, we say that f is harmonic on W if it is harmonic at every point of W .

3.5.2

Harnack’s Principle

We conclude this paragraph with a discussion of Harnack’s Principle. First, we note that since harmonic functions are by definition locally harmonic, the following analog of Corollary 1.7.5 holds by the same proof. P ROPOSITION 3.5.2. If a sequence of harmonic functions uj on a Riemann surface X converges locally uniformly to a function u, then u is harmonic on X. 69

Next we turn to Harnack’s Inequality. For its proof, it is convenient to note that, by replacing z with z/r, we can translate the Poisson formula to the disk of radius r: 1 Pr f (z) := 2π

Z |ζ|=r

|ζ|2 − |z|2 dζ . f (ζ) √ 2 |ζ − z| −1ζ

We can also translate the origin to another point a by replacing z and ζ with z − a and ζ − a respectively. Next, one has the following trivial estimate of the Poisson kernel: |z| < r

⇒

r − |z| r2 − |z|2 r + |z| √ ≤ . ≤ r + |z| r − |z| |re −1θ − z|2

We can then easily prove the following result. L EMMA 3.5.3 (Harnack’s Inequality). Let X be a Riemann surface and open

K ⊂⊂ D ⊂ X. Then there exists a constant c = c(K, D) such that for any positive harmonic function u on D and all z1 , z2 ∈ K, 1 u(z1 ) ≤ ≤ c. c u(z2 ) Proof. Fix a small number r with the property that for any z ∈ K, {ζ ∈ X ; |ζ| < r} ⊂ D. (We fix here finitely many coordinate charts covering K.) In the disk of radius r centered at z1 , for any z2 with |z2 − z1 | ≤ r/2 and any positive harmonic function u, we have 1 u(z1 ) 3

r − |z2 − z1 | u(z1 ) r + |z2 − z1 | Z 1 |ζ − z1 |2 − |z2 − z1 |2 dζ √ ≤ u(z2 ) = u(ζ) 2π |ζ−z1 |=r |ζ − z2 |2 −1ζ r + |z2 − z1 | ≤ u(z1 ) ≤ 3u(z1 ). r − |z2 − z1 | ≤

By compactness of K, there is a positive integer N such that for any two points z1 , z2 ∈ K there are n ≤ N distinct points x1 = z1 , x2 , ..., xn−1 , xn = z2 such that |xi − xi+1 | ≤ r/2. Take c = 3N . The proof is complete. As a corollary of Harnack’s Inequality, we have the following result. T HEOREM 3.5.4 (Harnack Principle). Let u1 ≤ u2 ≤ ... be an increasing sequence of harmonic functions on an open subset D of a compact Riemann surface X. Then uj → u uniformly on compact sets, where u is either harmonic on X or else ≡ +∞ on X. 70

Proof. By considering vj := uj − u1 , we may assume that uj ≥ 0 for all j. Let u := lim uj . Since the sequence uj is increasing, u is well-defined pointwise, provided we allow the value +∞. Moreover, by Harnack’s Inequality, the convergence is locally uniform. Consider the decomposition X := {x ; u(x) = +∞} ∪ {x ; u(x) < +∞}. By Harnack’s Inequality, both sets are open, and thus both are closed. Since X is connected, one of these two sets is empty. The result now follows from Proposition 3.5.2.

3.5.3

Subharmonic functions

D EFINITION 3.5.5. We say that an L1 -loc function u is subharmonic if ∆u, computed in the sense of distributions, defines a positive measure on the coordinate chart in question. The formula (3.2) shows that the definition of subharmonic makes sense. Moreover, by the local nature of the definition of subharmonic, the results about subharmonic functions proved in Chapter 1 carry over to their obvious global analogs. We shall return to the study of harmonic and subharmonic functions later on, when we consider the Dirichlet Problem on Riemann surfaces.

3.6

Exercises

3.1 Consider the metric g = (1 − |z|2 )−2 |dz|2 on the unit disk D. Show that if F : D → D is holomorphic and an isomotery of g at one point zo ∈ D, i.e., (1 − |F (zo )|2 )−2 |F 0 (zo )|2 = (1 − |zo |2 )−2 , then F is invertible, and is furthermore an isometry of g at all points of D. √ 3.2 Find all subharmonic functions u on C such that u(z) = u(z + 1) = u(z + −1) for all z ∈ C. 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10

71

72

Chapter 4 Complex Line Bundles 4.1 4.1.1

Complex line bundles Basic definitions

Recall that a complex manifold is by definition equipped with charts taking values in complex euclidean space, and the transition functions are complex differentiable. For the sake of completeness of exposition, we point out only that removing all occurrences of the word “complex” in the previous sentence gives the definition of manifold. D EFINITION 4.1.1. Let M be a manifold, assumed connected for simplicity. A complex line bundle on M is a manifold L together with a map π : L → M having the following properties. 1. (Local triviality) For every p ∈ M there is a neighborhood U of p and a map fU : π −1 (U ) → C such that the map FU : π −1 (U ) 3 v 7→ (π(v), fU (v)) ∈ U × C is a diffeomorphism. 2. (Global linear structure) For each pair of such neighborhoods Uα and Uβ there is a map gαβ : Uα ∩ Uβ → C∗ such that fUα ◦ fU−1 (x, v) = (x, gαβ (x)v). β D EFINITION 4.1.2.

1. The map FU is also called a local trivialization of the line bundle.

2. The maps gαβ are called transition functions. 3. The sets Lx := π −1 ({x}), x ∈ M are called the fibers of the line bundle. E XAMPLE 4.1.3. An important example of a line bundle is the trivial bundle π : M × C → M;

π(x, v) = x.

By definition, every line bundle is locally the same as the trivial bundle. 73

R EMARK . We point out that gαα = 1

and, on Uα ∩ Uβ ∩ Uγ ,

gαβ · gβγ · gγα = 1.

(4.1)

D EFINITION 4.1.4. Let π : L → M be a complex line bundle. A section of π : L → M is an injective map s : M → L such that π ◦ s = idM . The set of all sections of L is denoted Γ(M, L). Observe that, in particular, a section gives us a copy of M inside L. A section bears its name because it cuts L along directions transverse to the fibers of L. Every line bundle has at least one section. D EFINITION 4.1.5. The section oL assigning to x ∈ M the origin in the fiber π −1 ({x}) is called the zero section of L. Every line bundle admits an action of C∗ defined as follows. If π(v) ∈ U , then λ · v := FU−1 (π(v), λfU (v)). We leave it as an exercise to the reader to show that the C∗ action and the zero section are welldefined. D EFINITION 4.1.6. Let L and L0 be two complex line bundles over a manifold M . 1. A homomorphism, or line bundle map, from L to L0 is a map F : L → L0 such that (a) The diagram F

L −→ L0 ↓ ↓ id M −→ M commutes, and (b) Fx := F |Lx : Lx → L0x is a linear map of vector spaces. 2. The line bundles are said to be isomorphic if there are line bundle maps F : L → L0 and G : L0 → L such that F ◦ G = idL0 and G ◦ F = idL . R EMARK . Let L → M and L0 → M be complex line bundles. We leave it to the reader to show that a necessary and sufficient conditions for the line bundles L and L0 are isomorphic is that there 0 0 exist transition functions gαβ for L and gαβ for L0 so that gαβ = hα gαβ h−1 β for some collection {hα } of zero-free local functions. R EMARK . We note that, by our definition of line bundle, L ∼ = L0 if and only if L = L0 . We add this remark because in the community at large, it seems that most people view line bundles as collections of local data (c.f. the previous remark), whereas we view line bundles as concrete geometric objects, which turn out to be equivalence classes of local data. 74

4.1.2

Description by transition functions

A line bundle is completely determined by the following data: A set of neighborhoods Uα , α ∈ A, that cover M , and a set of transition functions gαβ : Uα ∩Uβ → C∗ satisfying the cocycle conditions (4.1). Indeed, given a manifold M and such data gαβ we define ! a L({gαβ ; α, β ∈ A}) := Uα × C / ∼ α

where

`

means disjoint union, and (x, v) ∼ (x, gαβ (x)v) whenever x ∈ Uα ∩ Uβ .

The projection map π is then defined in the obvious way, namely π[x, v] = x. The interested reader is invited to show that the relation ∼ defining the line bundle L({gαβ }) is an equivalence relation, and that if L is a line bundle and {gαβ } is a complete set of transition functions, then L({gαβ }) is isomorphic to L.

4.1.3

Description by local sections

Let L → M be a complex line bundle and let U ⊂ M be an open set. The set π −1 (U ) → U is itself a line bundle, called the restriction of L to U and denoted L|U . We abuse notation and denote the space of sections of this line bundle Γ(U, L). D EFINITION 4.1.7. A frame for L over U is a nowhere vanishing section E ∈ Γ(U, L). If L has a frame over U then the map v 7→ (π(v), v/E(π(v))) gives an isomorphism L|U → U × C. (Note that since E(π(v)) 6= 0, the vector v is a multiple of E(π(v)), and we denote the factor v/E(π(v)).) It follows that L|U is trivial. Conversely, suppose U is an open set in M such that L|U is trivial. Let E : U → V be the section associated to the section x → (x, 1) of U × C via the map FU . Then E is a nowhere vanishing section. The discussion can be summarized as follows. P ROPOSITION 4.1.8. A line bundle is trivial is and only if it has a nowhere vanishing section. The following proposition is obvious. P ROPOSITION 4.1.9. Let EU be a frame for L over U . For each s ∈ Γ(U, L) there exists a function sU such that s = sU EU . 75

Let Uα and Uβ be two open sets in M such that L|Uα and L|Uβ are both trivial. Let Eα and Eβ be frames for L|Uα and L|Uβ respectively. Then these frames are related by Eα = gβα Eβ for some function gαβ : Uα ∩ Uβ → C∗ . It is easy to see that gαβ satisfy the cocycle condition (4.1). If x ∈ Uα ∩ Uβ , then for each v ∈ Lx one has v = vα Eα = vβ Eβ . It follows that gαβ (x)vβ = vα . From proposition 4.1.9 and the definition of the transition functions associated to a frame it follows that a section is determined locally by a function, and the fact that the section is globally defined means that these local functions satisfy linear relations with a non-linear parameter. We can wrap this discussion up in one convenient proposition. P ROPOSITION 4.1.10. . Let L → M be a complex line bundle and Uα , α ∈ A, a collection of locally trivializing charts with frames Eα and transition functions gαβ . Suppose there is a family of functions {sα }α∈A such that, for all α, β ∈ A, gαβ sβ = sα . Then there is a section s ∈ Γ(M, L) such that s|Uα = sα Eα . Proof. Clearly sα Eα = sβ Eβ , so just set s := sα Eα on Uα . We have introduced three realizations of the notion of line bundle: • global realization, • realization by transition functions, and • realization by the spaces of local sections Γ(U, L), U ⊂ M open. It is often important to know all three realizations intimately.

4.1.4

Remark: Vector Bundles

In this book we will not use vector bundles in a serious way. However, in certain definitions it is useful to have the notion of vector bundle available; for example when defining differential forms. The definition of complex vector bundle of rank r mimics that of complex line bundle. The fibers are vector spaces and the transition functions take values in GL(r, C). One important difference between vector bundles and line bundles is that non-commutativity of the group GL(r, C) for r ≥ 2. However, in the present book this issue will not arise. 76

4.2 4.2.1

Holomorphic line bundles Definition of Holomorphic line bundle

D EFINITION 4.2.1 (Holomorphic line bundle). Let X be a complex manifold. A complex line bundle π : L → X is said to be a holomorphic line bundle if the maps fU are holomorphic. R EMARK . Note that if π : L → X is a holomorphic line bundle, then automatically L is a complex manifold and the map π is holomorphic. Moreover, it is easy to see that a complex line bundle L → X is holomorphic if and only if one can find holomorphic transition functions for L.

4.2.2

Picard group

If L → M and L0 → M are complex line bundles, then we can define new complex line bundles 0 are the transition functions for L and L0 , L∗ → M and L ⊗ L0 → M as follows. If gαβ and gαβ then we take transition functions 1 for L∗ gβα = gαβ and 0 gαβ gαβ

for L ⊗ L0 .

We claim that L ⊗ L∗ is always the trivial bundle. The latter can be seen in several ways. For example, the transition functions are the identity. Or one could observe that x 7→ idLx is a nowherezero section of Hom(L, L) ∼ = L ⊗ L∗ . It follows that the set of complex line bundles on a manifold form an Abelian group, with ⊗ denoting the group addition and L 7→ L∗ the group negation. Clearly the holomorphic line bundles form a subgroup. D EFINITION 4.2.2. The group of holomorphic line bundles on a complex manifold X is called the Picard group Pic(X) of X.

4.2.3

Holomorphic sections and meromorphic functions

D EFINITION 4.2.3. A section of a holomorphic line bundle L → X is called holomorphic if it is holomorphic as a map X → L. The set of holomorphic sections of L is denoted ΓO (X, L). D EFINITION 4.2.4. A meromorphic section s of a holomorphic line bundle L → X is a locally finite set P := {x1 , x2 , ...} ⊂ X and a holomorphic section s : X − P → L such that if E is a frame near some xi ∈ P and s = f E, then f does not have an essential singularity at xi . The set of meromorphic sections of L is denoted ΓM (X, L). Let s be a global, non-identically zero meromorphic section of L → X. Then any other global meromorphic section s˜ of L → X is related to s in that there exists a meromorphic function f ∈ M (X) such that s˜ = f · s. 77

It follows that the space of global holomorphic sections can be identified with a subspace of the space of all meromorphic functions. This subspace is defined as follows. If there is no holomorphic section, there is nothing to do. On the other hand, if s is a global holomorphic section, then for any meromorphic function f , the section f s will be holomorphic if and only if for all x ∈ X, Ordx (f ) ≥ −Ordx (s). Note that the multiplicative structure of M (X) does not preserve this subspace. We end this paragraph by observing that when X is compact, the ratio of two global holomorphic sections is either constant or else has non-trivial poles. Indeed, if there are no poles then the ratio is a holomorphic function on a compact Riemann surface.

4.3 4.3.1

Two canonically defined holomorphic line bundles The canonical bundle of a Riemann surface

Let X be a Riemann surface. There are several ways to define the canonical bundle of X. We shall begin with a geometric definition, and then produce local sections and transition functions. We begin by defining germs of holomorphic functions. Consider the set SX,x of all pairs (f, U ) where U is a neighborhood of x in X and f : U → C is a holomorphic function. D EFINITION 4.3.1 (Germs of holomorphic functions). If (f1 , U1 ) and (f2 , U2 ) are two elements of SX,x , we say (f1 , U1 ) and (f2 , U2 ) define the same germ if there is an open subset U ⊂ U1 ∩U2 such that f1 |U = f2 |U . The equivalence class of (f, U ) in SX,x that define the same germ is denoted f x, and is called a germ of a holomorphic function. The set of germs of holomorphic functions is denoted OX,x . The set of germs of holomorphic functions has a vector space structure: if (f, U ) and (g, V ) are two representatives of germs at x, then for any W ⊂ U ∩ V , (f + g, W ) is an element of OX,x , and thus we define f x + g x := f + g x . Scalar multiplication is even easier, since the domain of the constant function can be taken to be all of X. We define c · f x := c · f x . R EMARK . More general than scalar multiplication, one can define products of germs in the obvious way: replace + by · in the definition of addition of germs. In this way OX,x can be given the structure of a ring. Since this ring naturally contains C, we can also see it as an algebra over C. As a complex vector space, OX,x is infinite dimensional. Thus it is better to treat OX,x as an algebra. The collection of these algebras over all x ∈ X is best treated as a sheaf, but we have decided to avoid this perspective. 78

Since the vector space OX,x is too large, we cut it down to something more manageable by defining an equivalence relation on germs. D EFINITION 4.3.2 ((1, 0)-forms). Two germs f x = [(f, U )] and g x = [(g, V )] are said to be tangent, written f x ∼1 g x if there is a coordinate chart ϕ : W → C such that x ∈ W ⊂ U ∩ V and (f ◦ ϕ−1 )0 (ϕ(x)) = (g ◦ ϕ−1 )0 (ϕ(x)). The chain rule implies that the equivalence relation is well-defined. The set of equivalence classes of mutually tangent germs is denoted KX,x . The elements of KX,x are called (1, 0)-forms. The equivalence class of germs tangent to f x is denoted df (x). E XAMPLE 4.3.3. If z : U → C is a complex chart, then dz(x) is a 1-form. The relation ∼1 is clearly linear, in the sense that if f x ∼1 f˜x and g x ∼1 g˜x then f x + ag x ∼1 f˜x + a˜ g x . If we define a · df (x) := d(af )(x)

and df (x) + dg(x) := d(f + g)(x)

then the map d : OX,x → KX,x is linear. P ROPOSITION 4.3.4. The vector space KX,x is 1-dimensional. Proof. This is an easy consequence of the chain rule: in fact, if we fix a coordinate chart z : U → C at x, then any holomorphic function f whose domain lies in U is of the form f˜ ◦ z for some holomorphic function f˜ of one complex variable. Thus df (x) = f˜0 (z(x))dz(x) is a multiple of the (1, 0)-form dz(x). Let now KX :=

[

KX,x .

x∈X

We define the map π : KX → X by π −1 (x) = KX,x . Next we give KX the structure of a line bundle as follows. Let z : U → C be a complex chart for X. Since z is a homeomorphism, the map dz : U 3 x 7→ dz(x) ∈ KX |U is injective. This map is by definition our non-vanishing section over U , and gives the description of KX by local sections: the map U × C 3 (x, c) 7→ c · dz(x) 79

is a complex chart for KX . To see that the the charts constructed are compatible, we must compute the transition functions. To this end, let (zα , Uα ) and (zβ , Uβ ) be two charts on X. If we define hαβ (x) := (zα ◦ zβ−1 )0 (zβ (x)). then by the chain rule we have dzα (x) = d (zα ◦ zβ−1 ) ◦ zβ (x) = hαβ (x)dzβ (x). It is also clear that hαα ≡ 1 and the cocycle condition (4.1) for transition functions of a line bundle is satisfied. R EMARK . Since the transition functions for KX are holomorphic, KX is a holomorphic line bundle. D EFINITION 4.3.5. Let X be a Riemann surface. The line bundle KX with transition functions hαβ defined above is called the canonical bundle of X. R EMARK . Another name for the line bundle KX is the holomorphic cotangent bundle. If this name is used, one usually denotes KX by the symbol TX1,0∗ . Both of these notations have generalizations to higher dimensions. However, while the canonical bundle generalizes to a line bundle on higher dimensional complex manifolds, the holomorphic cotangent bundle generalizes to a vector bundle.

4.3.2

The tangent bundle of a Riemann surface

Let X be a Riemann surface. We denote by Tx (X) the space of holomorphic functions f : D → X where D is a neighborhood of to in C and f (to ) = x. E XAMPLE 4.3.6. Let U be a subset of X such that z : U → z(U ) = D ⊂ C is a complex chart. Then the inverse z −1 : D → X is an element of Tx (X). As in the case of the cotangent bundle, one could consider germs of elements of Tx (X). In this presentation we will not do so, instead going directly to tangency equivalence classes. D EFINITION 4.3.7. We say that f1 ∼1 f2 in Tx (X) if there exists a complex chart zα : Uα → C such that x ∈ Uα , f1 (to,1 ) = f2 (to,2 ) = x and (zα ◦ f1 )0 (to,1 ) = (zα ◦ f2 )0 (to,2 ). An equivalence class is denoted [f ]x , and we set 1,0 TX,x := Tx (X)/ ∼1 . 1,0 The set TX,x is called the tangent space to X at x, and its elements are called tangent vectors.

80

E XAMPLE 4.3.8. The equivalence class [z −1 ]x of the inverse of a complex chart is also denoted ∂ . ∂z x 1,0 Next we give TX,x the structure of a vector space. We define scalar multiplication by

a · [f ]x := [fa ]x ,

where fa (t) := f (a(t − to ) + to ),

and addition by [f ]x + [g]x := [h]x , where, if f (to ) = g(so ) = x then h(t) := zα−1 (zα ◦ f (t + to ) + zα ◦ g(t + so ) − zα (x)). Observe that h(0) = x and that h is well-defined on a sufficiently small neighborhood of 0. An easy exercise using the chain rule shows that addition and scalar multiplication are well-defined. P ROPOSITION 4.3.9. Let U be a coordinate chart on a Riemann surface X, with complex coordinate z : U → C, and let f : X → Y be a holomorphic mapping into another Riemann surface Y . Then ∂ −1 0 −1 . [f ◦ z ]f (x) = f (z (x)) · ∂z x Proof. By the chain rule that the maps Df (z −1 (x))z −1 and f ◦ z −1 are tangent at x. 1,0 Having defined the 1-dimensional complex vector spaces TX,x , our next task is to glue them together. To this end, let [ 1,0 TX1,0 := TX,x x∈X 1,0 We define the map π : TX1,0 → X by π −1 (x) = TX,x . 1,0 We give TX a complex structure as follows. Let x ∈ X and let z : U → C be a complex chart for X. The map ∂ : U → TX1,0 |U ∂z defined by ∂ ∂ (x) := ∂z ∂z x

is clearly injective. This map is by definition our non-vanishing holomorphic section over U . Equivalently, we have a complex chart U × C → TX1,0 |U ; (x, v) 7→ v 81

∂ (x). ∂z

To see that that our complex charts are compatible, we must compute the transition functions. To this end, take an atlas {(zα , Uα )} on X. (Here Uα ⊂ X.) Then zα ◦ zβ−1 : zβ (Uα ∩ Uβ ) → C has a nowhere vanishing derivative. We define gαβ (x) : Uα ∩ Uβ → C∗ by gαβ (x) := (zα−1 ◦ zβ )0 (x). Now

∂ ∂ (x) = [zα−1 ]x = [(zα−1 ◦ zβ ) ◦ zβ−1 ]x = gαβ (x) (x), ∂zα ∂zβ

where the second to last equality follows from Proposition 4.3.9 R EMARK . It is clear that gαα (x) ≡ 1, and the chain rule implies the cocycle condition (4.1). D EFINITION 4.3.10. Let X be a Riemann surface. The line bundle TX1,0 with transition functions gαβ defined above is called the holomorphic tangent bundle of X. R EMARK . Since the transition functions of TX1,0 are holomorphic, TX1,0 is a holomorphic line bundle.

4.3.3

Duality of KX and TX1,0 .

In this short section, we make explicit the observation that ∗ KX = TX1,0 .

(4.2)

There are several ways to see this fact. One such way is to observe that the transition functions are reciprocals. (To see this relation between the transitions functions, one uses the chain rule.) A second way to prove (4.2) is to note that KX ⊗ TX1,0 → X has a global section with no zeros. To construct this section, we may use the local sections dzα and ∂z∂α . A calculation using the chain rule and the bilinearity of the tensor product shows that the section s defined by s := dzα ⊗

∂ ∂zα

on

Uα

is globally defined. A third way to construct the desired isomorphism stems from the possibility of composing local holomorphic maps from C into X with holomorphic functions on X, provided their domains are 1,0 compatible. To be more precise, let ω = df (x) ∈ KX,x and v := [g]x ∈ TX,x , where g(to ) = x. define hω, vi := (f ◦ g)0 (to ). 82

It is easy to show that the definition is independent of the representatives of the equivalence classes, and that this pairing is bilinear and non-degenerate. ∗ The pairing defines an endomorphism TX1,0 → KX by 1,0 ∗ , TX,x 3 v 7→ h·, vi ∈ KX,x

and this endomorphism is non-degenerate by the non-degeneracy of the bilinear form. The three methods of establishing the aforementioned duality (4.2) are closely related. The relation between the first two methods is obvious. The third method of establishing (4.2) is related to the second through the observation that the fiber of the line bundle KX ⊗ TX1,0 → X over x can 1,0 ∗ be thought of as the collection of all endomorphisms from TX,x to KX,x . Thus the non-zero section s gives rise to a non-degenerate endomorphism.

4.4 4.4.1

Holomorphic vector fields on a Riemann surface Definition

D EFINITION 4.4.1. A global holomorphic section of TX1,0 is called a holomorphic vector field. ∂ is a holomorphic E XAMPLE 4.4.2. Let X = C with the global coordinate function z. Then ∂z ∂ ∂ vector field. Since ∂z has no zeros, any holomorphic vector field on C is of the form f ∂z for some entire holomorphic function.

E XAMPLE 4.4.3. Let X = P1 with local coordinates z0 on U0 and z1 on U1 . As we saw, z1 = on U1 ∩ U0 . Consider the holomorphic vector fields ξ0 :=

∂ ∂z0

and η0 = z0

1 z0

∂ ∂z0

on U0 . Then on U0 ∩ U1 , the chain rule tells us that ξ0 =

∂z1 ∂ ∂ = −(z1 )2 ∂z0 ∂z1 ∂z1

η0 =

∂z1 1 ∂ ∂ = −z1 . ∂z0 z1 ∂z1 ∂z1

while

If we define

∂ ∂ and η1 = −z1 , ∂z1 ∂z1 then ξ0 = ξ1 and η0 = η1 on U0 ∩ U1 . By setting ξ1 := −(z1 )2

ξ = ξi and η = ηi on Ui , we obtain two independent holomorphic vector fields on P1 . (Independence here means linear independence in the vector space of sections of TP1,0 .) 1 Later in the book we will see that any holomorphic vector field on P1 is a linear combination of ξ and η. 83

E XAMPLE 4.4.4. Let X be the complex torus C/L. Then on C we have [z]x = [z + a]x for any ∂ on C descends to a well-defined vector field on X, constant a, which means that the vector field ∂z ∂ ∂ often still denoted ∂z . Observe that since the vector field ∂z has no zeros, TX1,0 is trivial. R EMARK . In fact, we will see later that complex tori are the only compact Riemann surfaces with trivial tangent bundle. On the other hand, we will also see that every line bundle on an open Riemann surface is trivial. R EMARK . We will see later on that among compact Riemann surfaces, only P1 and tori have non-zero holomorphic vector fields.

4.4.2

Flow

In this section we will assume the existence and uniqueness theorem for solutions of complex differential equations. The form of the theorem we will use is the following. T HEOREM 4.4.5. (Existence, uniqueness and regularity for ODE) Let U ⊂ C be an open set, U 0 ⊂ U an open subset whose closure is compact, and v a holomorphic function on U . There exists a neighborhood W ⊂ C of 0, and a holomorphic function γ : U 0 × W → U such that, for all (x, t) ∈ U 0 × W , ∂γ(x, t) = v (γ(x, t)) . γ(x, 0) = x and ∂t ˜ is another such function, then γ and γ˜ agree on U 0 × (W ∩ W ˜ ). Moreover, if γ˜ : U 0 × W R EMARK . A proof of Theorem 4.4.5 can be obtained either by using Picard iterates, or by appealing to the Picard Theorem for Lipschitz differential equations using a regularization argument. R EMARK . In the notation of Theorem 4.4.5, if s, t ∈ W and γ(x, s) ∈ U 0 , then by uniqueness s + t ∈ W and γ(γ(x, s), t) = γ(x, s + t). (4.3) Indeed, one only has to differentiate both sides to see that they satisfy the same ODE and initial condition. One often writes γx (t) = γ t (x) = γ(x, t) when one wants to fix x or t respectively. Then the differential equation becomes γx0 (t) = v(γx (t)), while (4.3) reads γ t ◦ γ s = γ s+t . Let X be a Riemann surface and let ξ ∈ H 0 (X, TX1,0 ) be a holomorphic vector field. If z : U → C is a complex chart on X, we define the holomorphic function v = v (z) : z(U ) → C by ξ=v

∂ . ∂z

84

∂ In view of the transformation rules for the sections ∂z , we see that the functions vα := v (zα ) transform as vα (zα (x)) = (zα ◦ zβ−1 )0 (x)vβ (zβ (x)).

If x ∈ Uα , we define the family of curves γx,α : Wα → C with holomorphic parameter x to be the family of curves satisfying the initial value problem 0 γx,α (t) = vα (γx,α (t)),

γx,α (0) = zα (x).

We calculate, using the chain rule, that d zα ◦ zβ−1 ◦ γx,β = vα (zα ◦ zβ−1 ◦ γx,β ), dt and also observe that zα ◦ zβ−1 ◦ γx,β (0) = zα ◦ zβ−1 (zβ (x)) = zα (x). It follows from the Uniqueness Theorem for ODE that γx,α = zα ◦ zβ−1 ◦ γx,β . We define a subset Dξ ⊂ X × C as follows. A pair (x, t) is in Dξ if there exist pairs (x, 0) = (x0 , t0 ), (x1 , t1 ), ..., (xN , tN ) = (xt , t) with xj ∈ Uαj ∩ Uαj+1

and zαj (xj+1 ) = γxj ,αj (tj ),

j = 1, 2, ..., N.

We define the map Fξ : Dξ → X by Fξ (x, t) = xN = xt . Using Theorem 4.4.5, one can verify that Fξ is well-defined. D EFINITION 4.4.6. The space Dξ is called the fundamental domain of ξ, and the map Fξ : Dξ → X is called the flow of ξ. One also writes Fξt (x) := Fξ (x, t). The map Fξt is called the time-t map of ξ. R EMARK . If (x, s) ∈ Dξ and (Fξs (x), t) ∈ Dξ then (x, s + t) ∈ Dξ . D EFINITION 4.4.7. A vector field ξ is said to be complete if Dξ = X ×C. In this case, the previous remark implies that Fξ : X × C → X is a holomorphic C-action, and that {Fξt } ⊂ Aut(X) is a one-parameter subgroup. Of course, the set X × {0} is always in Dξ , and moreover Fξ0 (x) = x for all x ∈ X. 85

L EMMA 4.4.8. If there exists > 0 such that X × {t ∈ C ; |t| ≤ ε} ⊂ Dξ , then ξ is complete. In particular, the time-1 map of a vector field exists if only if that vector field is complete. Proof. If t ∈ C, then let N be so large that δN := |t|/N < ε. Then put xj = Fξ (xj−1 , tδN /|t|) for j = 1, 2, ..., N , where x0 = x. Then inductively applying the above remark about Fξ , we see that (xj , jtδN /t) ∈ Dξ for all j = 1, ..., N . T HEOREM 4.4.9. If X is a compact Riemann surface then each ξ ∈ Γ(X, TX1,0 ) is complete. Proof. By the existence and uniqueness theorem for ODE, to each x ∈ X there is associated a number εx > 0 and a neighborhood Ux such that Ux × {|t| < εx } ⊂ DV . By compactness, there S exist x1 , ..., xk such that ki=1 Uxi = X. Let ε := min{εx1 , ..., εxk }, and apply Lemma 4.4.8. E XAMPLE 4.4.10. Let L ⊂ C be a lattice and X = C/L the complex torus associated to L. The ∂ flow of the vector field c ∂z is Fct ∂ ([z]) = [z + ct]. ∂z

E XAMPLE 4.4.11. Consider the vector fields ξ and η on P1 defined in Example 4.4.3. If we use coordinates z0 on U0 and z1 on U1 , then for x ∈ U0 , z0 ◦ Fξt (x) = z0 (x) + t

and

z0 ◦ Fηt (x) = z0 (x)et ,

z1 (x) 1 + z1 (x)t

and

z1 ◦ Fηt (x) = z1 (x)e−t .

while on U1 , z1 ◦ Fξt (x) =

If we look only at real times, the closures of the integral curves of the vector field ξ are horocycles touching at the fixed point ∞ while the integral curves of η are lines of latitude of the sphere whose poles are 0 and ∞. Perhaps it is also interesting to note that while the flow of ξ restricts to a complete flow on U0 , it does not restrict to a complete flow on U1 .

4.5

Divisors and Line Bundles

In the study of compact Riemann surfaces, one of the most important concepts is the notion of divisor. The concept of divisor is also useful in the study of open Riemann surfaces, but it is not as important as in the compact case. The geometry of divisors is nicely captured using line bundles. In this section we discuss the relationship between divisors and line bundles.

4.5.1

Divisors

D EFINITION 4.5.1. Let X be a Riemann surface. 86

1. A divisor D on a Riemann surface X is a locally finite1 subset p1 , p2 , ... of distinct points of X, together with a collection of integers m1 , m2 , ... with mi associated to pi . The notation is X D= m j pj . j

2. The divisor all of whose points are assigned the integer 0 is written 0. 3. The set of points p1 , p2 , ... is called the support of D. 4. When the support of D is finite, the number deg(D) :=

X

mj ∈ Z

j

is called the degree of D. (The notion of degree is most useful on compact Riemann surfaces.) 5. The collection of divisors on X is denoted Div(X). 6. Equivalently, a divisor is a map D : X → Z whose support is a locally finite subset of X. We sometimes write mj = D(pj ) or X D= D(p)p. p∈X

R EMARK . Perhaps it is useful to keep in mind that divisors are meant to capture the notion of zeros and poles of a meromorphic function locally. Globally a divisor is in general not the set of zeros and poles of a meromorphic function. For example, on a compact Riemann surface we will see that the degree of the divisor of a meromorphic function must be 0, but this necessary condition is still not sufficient. In Chapter 14 we will prove a theorem of Abel that identifies geometrically all those divisors that are the collections of zeros and poles of some meromorphic function on a compact Riemann surface. On the other hand, for open Riemann surfaces there is no obstruction: in Chapter 8 we will prove a theorem, due to Weierstrass, stating that, on an open Riemann surface, every divisor is the divisor of some meromorphic function. The set Div(X) has the structure of an Abelian group, if we define X D ± D0 := (D(p) ± D0 (p)) · p. p∈X

(If we view D as a function, this group law is consistent with the usual addition of functions.) If the Riemann surface X is compact, then any locally finite subset must be finite, and thus in this case Div(X) is just the free abelian group generated by the points of X. 1

It is useful to note that locally finite is not the same as isolated. For example, the set E = {1/n ; n ≥ 1} in the 1 complex plane is not locally finite, but E ∩ D( n1 , ε) = { n1 } for any ε < n(n+1) .

87

E XAMPLE 4.5.2.

1. Let f be a meromorphic function on X. Then we have a divisor X Ord(f ) := Ordp (f )p.

We may sometimes write Ord(f ) = (f )0 − (f )∞ , where (f )0 =

X

Ordp (f ) · p

f (p)=0

is the divisor of zeros of f and X

(f )∞ =

−Ordp (f ) · p

f (p)=∞

the divisor of poles of f . Note that deg(Ord(f )) = 0. 2. Let L → X be a holomorphic line bundle and s a meromorphic section of L over X. Then one has the order divisor X Ord(s) := Ordp (s) · p = (s)0 − (s)∞ . 3. On a Riemann surface the residue of a meromorphic function makes no sense, since to compute the residue of a function we need to make a choice of a local coordinate. On the other hand, the residue of a meromorphic 1-form at a point x ∈ X does make sense. A meromorphic section of the canonical bundle is called a meromorphic 1-form. Let ω be a meromorphic 1-form. The residue of a meromorphic 1-form at a point x ∈ X is defined as follows: if z is a local coordinate with z(x) = 0 and ω = f (z)dz locally, then Z 1 f (z)dz. Resx (ω) = √ 2π −1 |z|=ε Since ω is holomorphic away from 0, the integral is independent of the path of integration, and under a change of coordinates w = w(z), if ω = g(w)dw then by definition g(w) dw = dz f (z), so that Resx (ω) is independent of the choice of coordinates. (Later we will define integration of 1-forms in general using exactly the same principle.) We thus obtain the residue divisor X Res(ω) := Resp (ω) · p. p∈X

Note that since for a locally defined meromorphic function g the derivative g 0 has no residue, deg(Res(df )) = 0 for a meromorphic function f ∈ M (X). 88

4. Let F : X → Y be a holomorphic map. Then we have two divisors associated to F : (a) The branching divisor of F Branch(F ) :=

X

(Multx (F ) − 1) · x

x∈X

in Div(X), whose degree B(F ) is called the branching number of F , and whose support is called the Branching locus. (b) The ramification divisor of F X Ram(F ) := (#{x ∈ X ; f (x) = y and Df (x) = 0} · y y∈Y

in Div(Y ), whose support is called the ramification locus of F , or the set of critical values of F . By the theorem on normal forms, the support of Ram(F ) is locally finite. D EFINITION 4.5.3. Let X be a Riemann surface and D, ∆ ∈ Div(X). 1. D is said to be effective if D(p) ≥ 0 for all p ∈ X. In this case, we write D ≥ 0. If D − ∆ is effective, we may write D ≥ ∆. (In particular, any divisor can be written as a difference of two effective divisors.) 2. D is said to be linearly trivial if D = Ord(f ) for some meromorphic function f on X. In this case, we write D ≡ 0. Classically, linearly trivial divisors are called principal divisors. 3. D and ∆ are said to be are called linearly equivalent if D − ∆ is linearly trivial. In this case we write D ≡ ∆. 4. D is said to be numerically trivial if deg(D) = 0. We write D ≡num 0 5. D and ∆ are said to be numerically equivalent if D − ∆ is numerically trivial. In this case we write D ≡num ∆. As we have shown, every linearly trivial divisor is numerically trivial. Indeed, this assertion is by definition equivalent to the statement that the number of zeros of a meromorphic function, counted with multiplicity, is equal to the number of poles of that function, also counted with multiplicity. On the Riemann sphere P1 , the converse assertion is true: every numerically trivial divisor is principal. This is seen as follows: by choosing projective coordinates wisely, we may guarantee that our divisor does not contain the point at infinity. That is to say, it lies entirely in U0 . Consider the polynomials Y Y p(z0 ) = (z0 − z0 (x))D(x) and q(z0 ) = (z0 − z0 (x))D(x) . D(x)>0

D(x) Finally, consider the map

log C , log 2

any such section s must be identically zero. ΓO (X, L) →

N M

P

j=1

Sending the section s to the N -tuple (P1 , ..., PN ) , where P ∼ = Ck+1 is the space of polynomials th of degree at most k, and Pj is the k Taylor polynomial of the function s/ei at the origin, in the C coordinate zi . The above calculation shows that this map is injective as soon as k > log . It log 2 log C follows that ΓO (X, L) is finite dimensional. (The dimension is bounded above by (b log 2 c + 2)N .) The proof is complete. D EFINITION 4.8.2. On a compact Riemann surface X, the number gO (X) := dimC ΓO (X, KX ) is called the arithmetic genus. As we will see in Chapter 9, the arithmetic genus of a compact Riemann surface X is equal to the geometric genus, i.e., the number of handles of X. 97

98

Chapter 5 Complex Differential Forms There are several kinds of differential forms that will be important to us.

5.1

Differential (1, 0)-forms

D EFINITION 5.1.1. Let X be a Riemann surface. A section ω ∈ Γ(X, KX ) of the canonical bundle is called a (1, 0)-form on X. We have already seen that if (zα , Uα ) is a complex chart on X, then dzα is a section of KX over Uα . It follows that in terms of this chart, ω = fα (zα )dzα , and for different charts the functions fα are related by fα (z α ) = fβ (z β )

dz β . dz α

D EFINITION 5.1.2. If fα is holomorphic (resp. meromorphic) for all α, we say that ω is a holomorphic (resp. meromorphic) 1-form. That is to say, ω is a holomorphic section of the canonical bundle. E XAMPLE 5.1.3. If f : X → C is a complex valued function, then ω=

∂f dzα ∂zα

on Uα

is a (1,0)-form. Moreover, if f is holomorphic (resp. meromorphic), then ω is a holomorphic (resp. meromorphic) 1-form. Indeed, if zα and zβ are two charts, then ∂f ∂zβ ∂f ∂zα ∂f dzα = dzβ = dzβ , ∂zα ∂zα ∂zβ ∂zβ ∂zβ where the last equality follows from the chain rule. 99

D EFINITION 5.1.4. The form ω of the previous example is denoted ∂f . R EMARK . Not every differential (1, 0)-form is of the form ∂f for some function f . For example, a non-trivial holomorphic differential form on a compact Riemann surface cannot be of the form ∂f , for then ∂ ∂f = 0, ∂ z¯α ∂zα and thus f would be a harmonic function, which must be constant. Thus, for instance, the differential form dz on a torus C/L (where z is the coordinate on C, which is defined on C/L only modulo L, but whose differential is consequently well-defined) is not the differential of any function. On the other hand, it is a consequence of the complex conjugate of Theorem 8.1.1 in Chapter 8 that every (1, 0)-form on an open Riemann surface is of the form ∂f for some function f . A local version of this result can be established by taking complex conjugates of the Dolbeault Lemma, proved at the end of the present chapter. E XAMPLE 5.1.5. The (1, 0)-forms ω and η on P1 given by

dz0 −z1−2 dz1

on on

U0 U1

z0−1 dz0 −z1−1 dz1

on on

U0 U1

η= and ω= are meromorphic.

E XAMPLE 5.1.6 (Non-example). There are no non-trivial holomorphic differential (1, 0)-forms on P1 . Indeed, suppose ω is a differential (1, 0)-form on P1 . Then on Ui we have ω = fi (zi )dzi where f is an entire function. Moreover, f0 (z0 ) =

dz1 f1 (z0−1 ) = −z0−2 f1 (z0−1 ). dz0

Thus as z0 → ∞, f (z0 ) has a limit, namely 0. Thus f is holomorphic on the Riemann sphere, hence constant, and thus 0. D EFINITION 5.1.7. Let p ∈ X and let ω be a meromorphic 1-form on X, written locally near p as ω = f dz. We define Ordp (ω) := Ordp (f ). R EMARK . Since dz defines a frame for KX , this definition of Ord is just a special case of the definition already given for meromorphic sections of general line bundles. 100

5.2

TX∗0,1 and (0,1)-forms

D EFINITION 5.2.1. Let X be a Riemann surface. 1. The complex line bundle TX∗0,1 is the line bundle whose transition functions are the complex conjugates of the transition functions for the canonical bundle. 2. A section s ∈ Γ(X, TX∗0,1 ) of TX∗0,1 is called a (0,1)-form. z is a non-vanishing local section of It follows that if z is a local coordinate on X, then dz = d¯ ∗0,1 TX , and thus every (0,1)-form is of the form f d¯ z on the domain of the chart z. R EMARK . In fact, the operation of complex conjugation sends sections of KX to sections of TX∗1,0 . E XAMPLE 5.2.2. If f is a function on X and {(zα , Uα )} is an atlas, then ω=

∂f d¯ zα ∂ z¯α

is a well-defined (0,1)-form on X. ¯ . D EFINITION 5.2.3. The differential (0, 1)-form of the previous example is denoted ∂f R EMARK . Unlike KX , TX∗0,1 is not a holomorphic line bundle. Thus we have no notion of holomorphic (0, 1)-forms. Nevertheless, we will make significant use of (0, 1)-forms in this book.

5.3

TX∗ and 1-forms

D EFINITION 5.3.1. We call an element of Γ(X, KX ) ⊕ Γ(X, TX∗0,1 ) a 1-form. In other words, a 1-form ω on a Riemann surface can be written in terms in a chart (zα , Uα ) as ω = fα dzα + gα d¯ zα . D EFINITION 5.3.2. If f : X → C is a function, we define ¯ df := ∂f + ∂f. R EMARK . If we write z = x +

√

−1y, we find that df =

∂f ∂f dx + dy. ∂x ∂y

R EMARK . Clearly 1-forms can be identified with sections of the vector bundle TX∗ = KX ⊕ TX∗0,1 . We chose to present the definition as we have in order to avoid a thorough treatment of vector bundles in this book. R EMARK . Note that a holomorphic 1-form automatically has no (0, 1)-component. From here on, the expression holomorphic 1-form will be synonymous with the expression holomorphic (1, 0)form. 101

5.4

1,1 ΛX and (1,1)-forms

D EFINITION 5.4.1. Let X be a Riemann surface. 1. The complex line bundle Λ1,1 X is by definition 1,1 ΛX := KX ⊗ TX∗0,1 .

Therefore the transition functions are the squared moduli of the transition functions of KX . 2. A section s ∈ Γ(X, Λ1,1 X ) is called a (1,1)-form. R EMARK . Like TX∗0,1 , Λ1,1 X is not a holomorphic line bundle, and thus there is no notion of a holomorphic (1, 1)-form. Nevertheless, we will make substantial use of (1, 1)-forms in the book. Given a local coordinate z on X, we denote a local section of Λ1,1 X by f dz ∧ d¯ z. √ R EMARK . A formal calculation shows that if the coordinate z = x + −1y is written in terms of real and imaginary parts, then √ −1 dz ∧ d¯ z = dx ∧ dy. 2 Our choice of notation for the local sections of Γ(X, Λ1,1 X ) stems from the fact that we are planning to integrate (1,1)-forms.

5.5

Exterior algebra and calculus

D EFINITION 5.5.1. (wedge product) If ω1 and ω2 are 1-forms, given locally by ω1 = f1 dz + g1 d¯ z and ω2 = f2 dz + g2 d¯ z, Then we set ω1 ∧ ω2 := (f1 g2 − f2 g1 )dz ∧ d¯ z. It is easily seen that ω2 ∧ ω1 = −ω1 ∧ ω2 . R EMARK . A simple calculation shows that if ω1 and ω2 are global 1-forms on X, then the expression ω1 ∧ ω2 , a priori defined only locally, is actually a global (1,1)-form on X. D EFINITION 5.5.2. (pullback) Let F : X → Y be a holomorphic mapping and {zα } an atlas for Y . Let f : Y → C be a function, ω = gα dzα a (1,0)-form, θ = hα d¯ zα a (0,1)-form and η = kα dzα ∧ d¯ zα a (1,1)-form. We define F ∗f F ∗ω F ∗θ F ∗η

:= = = =

f ◦F (F ∗ gα )d(F ∗ zα ) (F ∗ hα )d(F ∗ zα ) (F ∗ kα )d(F ∗ zα ) ∧ d(F ∗ z α ). 102

R EMARK . Observe that F ∗ takes functions to functions, 1-forms to 1-forms, and (1,1)-forms to (1,1)-forms. However, the chain rule shows that F ∗ sends (1, 0)-forms to (1, 0)-forms if and only if it sends (0, 1)-forms to (0, 1)-forms if and only if F is holomorphic. ¯ and df as Recall that for a function f , we defined ∂f , ∂f ∂f =

∂f dz, ∂z

¯ = ∂f d¯ ¯ ∂f z and df = ∂f + ∂f ∂ z¯

We now extend the definition of exterior derivative to forms. D EFINITION 5.5.3. (Exterior derivative) Let X be a Riemann surface, ω = gdz + hd¯ z a 1-form on X and θ a (1, 1)-form on X. Then ∂h dz ∧ d¯ z ∂z ¯ := ∂g ¯ ∧ dz = − ∂g dz ∧ d¯ ∂ω z ∂ z¯ ∂h ∂g ¯ − dω := ∂ω + ∂ω = dz ∧ d¯ z. ∂z ∂ z¯ ¯ := dθ := 0. ∂θ := ∂θ ∂ω := ∂h ∧ d¯ z=

¯ = 0. In fact, Observe that a 1-form ω is holomorphic if and only if ω is a (1, 0)-forms and ∂ω even more is true. L EMMA 5.5.4. If ω is a holomorphic 1-form on a Riemann surface X, then dω = 0. dz ∧ d¯ z = 0. Proof. Since ω = gdz + hd¯ z is holomorphic, h = 0, and then dω = − ∂g ∂ z¯ We leave it to the reader to show that if F : X → Y is a holomorphic map, then ∂F ∗ = F ∗ ∂,

¯ ∗ = F ∗ ∂¯ and dF ∗ = F ∗ d. ∂F

(Of course, the third relation holds for all smooth maps.) L EMMA 5.5.5. Let F : X → Y be a holomorphic map and ω a meromorphic 1-form on Y . For each p ∈ X, Ordp (F ∗ ω) = (1 + OrdF (p) (ω))Multp (F ) − 1. Proof. Choose local coordinates z near p and w near F (p) so that w = F (z) = z n . If the coordinate patches are simply connected, then locally ω = wk ef (w) dw, we we have n

F ∗ ω = nz nk+n−1 ef (z ) dz, and thus the desired formula holds. 103

5.6

Integration of 1-forms

D EFINITION 5.6.1. Let X be a Riemann surface. A path on X is a continuous, piecewise smooth function γ : [a, b] → X. The endpoints of γ are γ(a) and γ(b). We say that γ connects γ(a) and γ(b), and that γ is 1. closed if γ(a) = γ(b), and 2. simple if γ is injective on [a, b], except possibly if γ(a) = γ(b). We shall often abuse notation and refer to the image γ([a, b]) as a path, or to either γ or its image as a curve or an arc. The latter is mostly used when γ is not closed. An injective piecewise smooth continuous map α : [a, b] → [c, d] is called a reparameterization. We shall also say that γ ◦ α−1 : [c, d] → X is a reparameterization of γ. Note that a reparameterization can reverse the orientation of the curve in the surface. The orientation remains the same if α is an increasing function, and reverses if α is decreasing. Let γ : [a, b] → X be a path. Using the compactness of [a, b] it is easily seen that there exist a = a0 < a1 < ... < aN −1 < aN = b such that, with γj = γ|[aj−1 ,aj ] : [aj−1 , aj ] → X,

1 ≤ j ≤ N,

(i) the final point of one path γj is the initial point of the next path γj+1 : γj (aj ) = γj+1 (aj ),

1 ≤ j ≤ N,

and

(ii) each path γj has image contained in a coordinate chart. We shall refer to this as a partition of γ as a charted partition. We are now ready to define integration of a 1-form. D EFINITION 5.6.2. Let X be a Riemann surface, ω a 1-form on X, and γ : [a, b] → X a path with charted partition γ1 , ..., γN and coordinates zj : Uj → C so that γj ([aj−1 , aj ]) ⊂ Uj . Write ω = fj dzj + gj d¯ zj on Uj . We define ( ) Z N Z aj X d(zj ◦ γ) d(zj ◦ γ) + gj (zj ◦ γ(t)) dt. fj (zj ◦ γ(t)) ω := dt dt γ j=1 aj−1 By now it is an easy exercise to see that this definition is independent of the coordinate charts chosen. Moreover, the following properties are easily checked. 1. If γ is a union of paths γ1 , ..., γN , then Z ω= γ

N Z X j=1

104

γj

ω.

2. The integral is C-linear in ω: Z

Z

∀c1 , c2 ∈ C,

c1 ω1 + c2 ω2 = c1 γ

Z ω1 + c2

γ

ω2 γ

3. If F : X → Y is a holomorphic map of Riemann surfaces, then Z Z ω = F ∗ ω. F ◦γ

γ

4. If f is a smooth function defined in a neighborhood of γ([a, b]) then Z df = f (γ(b)) − f (γ(a)). γ

5. If α is a reparameterization that preserves the orientation of γ, then Z Z ω = ω. γ

γ◦α

6. If γ is a path, let −γ denote the same path traversed backwards, i.e., with the opposite orientation. Then Z Z ω = − ω. −γ

5.7

γ

Integration of (1,1)-forms

Let X be a Riemann surface, T ⊂ X a triangle and z : U → C a chart with T ⊂ U . Given a √ −1 (1,1)-form η on a neighborhood of U in X, the restriction of η to U is η = f 2 dz ∧ d¯ z for some function f on U . We then define Z Z η := f ◦ z −1 dxdy, T

z(T )

where the integral on the right is the usual Riemann integral in the plane. The change of variables formula and the fact that the real JacobianRdeterminant of the holomorphic transformation w = ϕ(z) is |ϕ0 (z)|2 show that the definition of T η is independent of the local chart z. Now let D ⊂ X be an open subset with smooth boundary and compact closure. Triangulate D with a finite number of triangles T1 , ..., TN such that for some complex charts zi : Ui → C, Ti ⊂ Ui for all i = 1, ...N . D EFINITION 5.7.1. Z η := D

N Z X j=1

105

Tj

η.

It is easy to verify the following facts. 1. If D = D1 ∪ D2 with D1 ∩ D2 = ∅, then Z Z η= D

Z η+

η.

D1

D2

2. For all c1 , c2 ∈ C and all (1,1)-forms η1 , η2 , Z Z Z 1 2 1 2 c η1 + c η2 = c η1 + c η2 . D

D

D

3. If F : X → Y is an isomorphism of Riemann surfaces, D ⊂ X is finitely triangulated, and η is a 1-form on F (D), then Z Z F ∗ η.

η= F (D)

D

Observe that properties 1,2 and 3 of integration of (1,1)-forms are analogs of properties 1,2 and 3 of integration of 1-forms. Property 5 of integration of 1-forms corresponds to a “change of variables formula”, and is accounted for by property 3 of integration of (1,1)-forms. The analog of property 4, i.e., the fundamental theorem of calculus, is the well known Stokes Theorem. In the plane, it is just Green’s Theorem, and thus on a surface it can be derived from Green’s theorem by a decomposition of the domain into a union of plane domains. We state the theorem in the invariant language of exterior calculus. T HEOREM 5.7.2. (Stokes’ Theorem) Let D be a finitely triangulated open subset of a Riemann surface X with smooth co-oriented boundary ∂D, and let ω be a smooth 1-form on a neighborhood of D. Then Z Z ω= ∂D

dω. D

R EMARK (Remark on regularity). Since our definitions are just patchings of local definitions, developing measure theory from here on out is done in just the same way as in Euclidean spaces. We shall assume that the reader is familiar enough with measure theory to carry out the details. When more subtle ideas of functional analysis are employed in the proof of Hörmander’s Theorem, we provide the full details.

5.8

Residues

The notion of the residue of a function is not well-defined on a Riemann surface. The point is that to compute the residue, one needs to integrate along a path, and on a general Riemann surface integration can only be done for forms (or their more singular analogs). Because plane domains have a global frame for KX , namely dz, the problem in defining the residue of a function disappears in a plane domain. We have already presented the following definition, which we now restate. 106

D EFINITION 5.8.1. Let X be a Riemann surface, p a point of X, and ω a meromorphic 1-form on X. Let U be an open subset of X containing p and having smooth boundary ∂U , such that there is no pole of ω in U − {p}. We define Resp (ω) :=

1 √

2π −1

Z ω, ∂U

where the boundary ∂U is co-oriented: if one walks on the surface forward along ∂U , one finds U on the left. The definition of residue is independent of the domain U chosen as above. To see this independence, we argue as follows. First, let V ⊂⊂ U be a domain containing p. By Stokes’ Theorem, Z

Z

Z

ω−

ω=

∂U

dω = 0, U −V

∂V

where the last equality holds because holomorphic 1-forms on Riemann surfaces are closed. Now if U1 and U2 are domains, we can find V ⊂⊂ U1 ∩ U2 , and thus we conclude that Z Z ω= ω. ∂U1

∂U2

Later on, when we look at the Riemann-Roch Theorem, it will be useful to have the following proposition. P ROPOSITION 5.8.2. Let ω be a meromorphic 1-form on a compact Riemann surface X. Then X

Resp (ω) = 0.

p∈X

Proof. Let x1 , ..., xN be the poles of ω, and take disjoint coordinate disks D1 , ..., DN with coordinates z1 , ..., zN such that zi (xi ) = 0. Let U = X − (D1 ∪ ... ∪ DN ). Then ω is holomorphic on U , so that Z N Z X √ X 0 = − dω = ω = 2π −1 Resp (ω). U

5.9

i=1

∂Di

p∈X

Homotopy and homology

We shall present, in a heuristic manner, some basic ideas in algebraic topology. For the rest of this section we assume the Riemann surface X is connected. 107

5.9.1

Homotopy of curves

Let γ0 , γ1 : [a, b] → X be two paths such that γ0 (a) = γ1 (a) and γ0 (b) = γ1 (b). Recall that a homotopy of the paths γ0 and γ1 is a continuous map Γ : [a, b] × [0, 1] → X such that for all t ∈ [a, b] Γ(t, 0) = γ0 (t), Γ(t, 1) = γ1 (t) and for all s ∈ [0, 1] Γ(a, s) = γ0 (a) = γ1 (a)

and

Γ(b, s) = γ0 (b) = γ1 (b).

In this case, we say that γ0 and γ1 are homotopic. It is not hard to construct a homotopy of two paths whose image is the same, but whose oriented parameterizations are different. Of course, there are many homotopic curves whose images are different.

5.9.2

Fundamental Group

Fix a point p ∈ X and consider the set of all parameterized closed loops in X starting and ending at p. Two such loops are said to be equivalent if they are homotopic. The set of equivalence classes is denoted π1 (X, p). The set π1 (X, p) carries the structure of a group, defined as follows. 1. The product [γ1 ][γ2 ] of two loops γ1 and γ2 is the homotopy class of the loop obtained by first following γ1 and then following γ2 . 2. The inverse of [γ] is the homotopy class of the loop obtained by reversing the loop γ. It is not hard to show that the product and inverse operations are well-defined. D EFINITION 5.9.1. The set π1 (X, p) together with the multiplication and inverse operations outlined above is called the fundamental group. R EMARK . Let p, q ∈ X and fix a path σ whose initial point is p and whose final point is q. Then we can define a map from Fσ π1 (X, q) → π1 (X, p) as follows. If α is a loop starting at q, then we have a loop starting at p, obtained by following σ, then α, then −σ. The homotopy class of this loop is denoted Fσ [α]. It is easy to see that F−σ = Fσ−1 . From now on, we will denote π1 (X, p) by π1 (X).

5.9.3

Homology

We shall pretend that the image D = Γ([a, b] × [0, 1]) of Γ is regular enough to integrate over. (In fact, the image of D need not be so regular; it can be approximated by a domain that is sufficiently regular.) A clever application of Stokes’ Theorem gives us the following proposition. P ROPOSITION 5.9.2. Let ω be a 1-form such that dω = 0. If γ0 and γ1 are homotopic paths, then Z Z ω= ω. γ0

γ1

108

R EMARK . In particular, the hypothesis of the proposition holds for a holomorphic 1-form. It follows that a 1-form determines a map Z [γ] ∈ π1 (X) → ω ∈ C. γ

It is clear from the properties of integrals that this map is a homomorphism, and thus, since C is an Abelian group, the kernel of this map must contain the commutator subgroup of π1 (X). It can be shown that the intersections of a the kernels of the maps obtained by using all closed 1-forms ω is exactly the commutator subgroup of π1 (X). D EFINITION 5.9.3. The quotient group H1 (X) := π1 (X)/[π1 (X), π1 (X)] is called the first deRham homology group of the surface X. (This group can also be realized as a quotient of Abelian group of closed 1-chains modulo boundaries of 2-chains.) Let us denote an equivalence class of a closed loop γ in H1 (X) by {γ}. Let X be a compact Riemann surface, Let Z 1 (X) denoted the set of all smooth, closed 1-forms on X. We say that ω1 and ω2 are cohomologous, and write ω1 ∼ ω2 , if there is a smooth function f : X → C such that ω1 − ω2 = df. Since d2 = 0, Stokes’ Theorem shows us that for any closed loop γ, Z Z ω1 = ω2 . γ

γ

(In fact a little more work is needed; we have to perturb γ a little to make it a union of mutually 1 disjoint, simple closed curves.) The quotient space HdR (X) = Z 1 (X)/ ∼ is an Abelian group, called the first deRham cohomology group. 1 L EMMA 5.9.4. The bilinear pairing h , i : H1 (X) × HdR (X) → C given by Z h{γ}, [ω]i := ω γ 1 is non-degenerate, and thus HdR (X) ∼ = (H1 (X))∗ . R Proof. Fix a smooth 1-form ω on X and suppose γ ω = 0 for all closed loops γ in X. Then the function Z x F (x) := ω xo 1 is well-defined, and dF = ω. It follows that [ω] = 0 in HdR (X), and thus the pairing is nondegenerate.

R EMARK . Some of the results on deRham cohomology can be extended to open Riemann surfaces if we consider differential forms with compact support. We shall not be concerned with this generalization. 109

5.10

Poincaré and Dolbeault Lemmas

T HEOREM 5.10.1 (Poincaré Lemma). Let X be a simply connected Riemann surface and let ω be a 1-form such that dω = 0. Then there exists a function f such that ω = df . Proof. Fix p ∈ X. If x ∈ X, we define Z

x

ω,

f (x) := p

where the integral is over any simple arc connecting p to x. This integral is well-defined because dω = 0 and X is simply connected, and it is an easy exercise in vector calculus to show that df = ω. ¯ = 0. R EMARK . Observe that if α is a (0,1)-form, then ∂α T HEOREM 5.10.2 (Dolbeault Lemma). Let X be a Riemann surface, p ∈ X a point and α a (0,1)form in a simply connected coordinate neighborhood U of p. Then for any relatively compact ¯ . U 0 ⊂⊂ U there is a function f : U 0 → C such that α = ∂f Proof. We can assume that U is a simply connected subset of the plane, and thus that α is given globally by α = g(z)d¯ z. We stress that g need not be holomorphic. Recall the Cauchy-Green formula: If h : U → C is a function, then Z ZZ 1 h(ζ) 1 ∂h dA(ζ) h(z) = dζ + . ¯ 2πi ∂U ζ − z π U ∂ζ ζ − z Let χ : X → [0, 1] be a smooth function that is identically 1 on U 0 and supported in U . Defining ZZ 1 χ(ζ)g(ζ)dA(ζ) , f1 (z) := π ζ −z U we have ZZ ∂f1 1 ∂ 1 = χ(ζ)g(ζ) dA(ζ) ∂ z¯ π ∂ z¯ ζ − z U ZZ ∂ 1 1 χ(ζ)g(ζ) ¯ dA(ζ) = − π ∂ζ ζ − z U ZZ 1 ∂(χ(ζ)g(ζ)) dA(ζ) = π ζ −z ∂ ζ¯ ZU ZZ 1 χ(ζ)g(ζ) 1 ∂(χ(ζ)g(ζ)) dA(ζ) = dζ + 2πi ∂U ζ − z π ζ −z ∂ ζ¯ U = χ(z)g(z). Taking f = f1 |U 0 completes the proof. ¯ = α for a function f R EMARK . As we shall see later on, it is possible to solve the equation ∂f defined on all of U . 110

5.11

Dolbeault Cohomology

We say that two smooth (0, 1)-forms ω and ω 0 are equivalent, writing ω ∼ ω 0 , if there is a smooth function f such that ¯ ω − ω 0 = ∂f. The (first) Dolbeault cohomology of a compact Riemann surface is the group H∂0,1 ¯ (X) := {smooth (0, 1)-forms}/ ∼ . As we will see, if X is an open Riemann surface then H∂0,1 ¯ (X) = 0. By contrast, on a compact 0,1 Riemann surface X the group H∂¯ (X) gives us very interesting information, both topological and complex analytic. T HEOREM 5.11.1 (Dolbeault-Serre Isomorphism). Let X be a compact Riemann surface. Then ∼ H∂0,1 ¯ (X) = ΓO (KX ). 0,1 In particular, H∂0,1 ¯ (X) is finite dimensional, and the dimension of H∂¯ (X) is gO (X), the arithmetic genus of X.

Once the isomorphism stated in Theorem 5.11.1 is established, the finite dimensionality statement is a consequence of Theorem 4.8.1. Theorem 5.11.1 is proved by considering a partial differential operator on (0, 1)-forms whose kernel contains exactly one member of any class in H∂1,0 ¯ (X). This partial differential operator is a natural generalization of the Laplacian to the setting of (0, 1)forms on Riemann surfaces with metrics. Theorem 5.11.1 will be proved at the end of Chapter 9.

111

112

Chapter 6 Calculus on Line Bundles The theory of connections of line bundles can only be made natural if it is embedded inside the theory of connections for vector bundles. Since we want to avoid vector bundles, this could pose a problem for our presentation. Fortunately, when one passes to holomorphic vector bundles on Riemann surfaces, the need for vector bundles can be circumvented with some ad hoc definitions.

6.1 6.1.1

Connections on holomorphic line bundles General connection on a complex line bundle

We begin with the general definition of a connection. D EFINITION 6.1.1. Let X be a Riemann surface and H → X a complex line bundle. A connection on X is a linear map ∇ : Γ(X, H) → Γ(X, TX∗ ⊗ H) that satisfies the Leibniz condition ∇(f s) = df ⊗ s + f ∇s for all s ∈ Γ(X, H) and f ∈ C 1 (X). Let ξ be a frame of H over an open subset U ⊂ X, i.e., a section of H over U such that ξ(x) 6= 0 for all x ∈ U . Then any section of H over U is of the form s = f · ξ for some differentiable function f . By the Leibniz Rule, we have ∇(f · ξ) = df ⊗ ξ + f ω ⊗ ξ where ω is a differential 1-form on U . D EFINITION 6.1.2. The form ω is called the connection form of ∇ in the frame ξ. R EMARK . In the literature one often finds the expression ∇ = d + ω, or ∇s = ds + ωs. These expressions depend on the choice of frame, but often the frame is not explicitly mentioned. Confusion does arise, but after some thought one finds that there is only one possible interpretation. 113

If we change the frame ξ to another frame ξ 0 , then ξ0 = f ξ for some function f . Thus ω 0 ⊗ ξ = (f ω + df ) ⊗ ξ. This calculation shows that in fact the 1-form ω is not globally defined. E XAMPLE 6.1.3.

1. The exterior derivative d is a connection for the trivial bundle O → X.

2. The operator ∂¯ defines a non-trivial connection for TX∗1,0 : if we choose a local coordinate z then ¯ dz) = − ∂f dz ∧ d¯ z. ∂(f ∂ z¯ (We leave it to the reader to check that this definition is independent of the coordinate z.) ∗1,0 ¯ ∗1,0 ∗ ∗ Since Λ1,1 X is a sub-bundle of TX ⊗ TX , ∂α ∈ Γ(X, TX ⊗ TX ), as claimed. Connections are very natural objects associated to line bundles. When we perform multilinear operations on line bundles with connections, we can expect the resulting line bundles to have naturally induced connections. D EFINITION 6.1.4. Let Hi → X be complex line bundles with connection ∇i , i = 1, 2. 1. The connection ∇1 + ∇2 for H1 ⊗ H2 is defined by (∇1 + ∇2 )(s ⊗ σ) = ∇1 s ⊗ σ + s ⊗ ∇2 σ. 2. The connection −∇1 for H1∗ is defined as follows. If ξ is a local frame for H1 , denote by the associated frame for H1∗ . Suppose ∇1 ξ = α ⊗ ξ. Then

1 ξ

−∇1 ( 1ξ ) = −α ⊗ 1ξ . These definitions, whose well-posedness is left to the reader, are natural for several reasons, some of which we cannot discuss because we have chosen a streamlined presentation. The following proposition is one exception. P ROPOSITION 6.1.5. Let ∇ be a connection for H, and denote by d the connection for the trivial bundle given in Example 6.1.3.1. Then (1) ∇ + d = ∇,

and 2) ∇ + (−∇) = d.

Proof. Let ξ be a frame for H, s = f ξ a section of H and g a section of the trivial bundle (i.e., a function). Then (∇ + d)(f ξ ⊗ g) = = = = ∼ =

(∇ + d)((f ξ) ⊗ (g ⊗ 1)) ∇(f ξ) ⊗ (g ⊗ 1) + (f ξ) ⊗ (dg ⊗ 1) (df ⊗ ξ + f ∇(ξ)) ⊗ (g ⊗ 1) + (f ξ) ⊗ (dg ⊗ 1) (d(f g) ⊗ ξ + f g∇ξ) ⊗ 1 ∇(f gξ). 114

This proves 1. Moving on to 2, let us write ∇ξ = α ⊗ ξ for a local section ξ of H. We have (∇ + (−∇))gξ ⊗

1 ξ

= ∇(gξ) ⊗ 1ξ + gξ ⊗ ∇( 1ξ ) = dg ⊗ ξ ⊗ ( 1ξ ) + gα ⊗ ξ ⊗ ( 1ξ ) − gα ⊗ ξ ⊗ ( 1ξ ) = dg ⊗ ξ ⊗ ( 1 ) ∼ = dg. ξ

The proof is complete. R EMARK . If we take into account the group structure of line bundles, then the previous proposition shows that this group structure is respected by connections, provided we take the exterior derivative, seen as a connection for the trivial bundle, to be the identity element.

6.1.2

(1,0)-connection

The line bundle TX∗1,0 of Example 6.1.3.2 is actually a holomorphic line bundle. The connection ∂¯ can be naturally defined for all holomorphic line bundles, as follows. D EFINITION 6.1.6. Let H → X be a holomorphic line bundle. We define the connection ∂¯ as follows. Choose a holomorphic local section ξ. Then ¯ · ξ) = ∂f ¯ ⊗ ξ. ∂(f Note that if ξ 0 is another holomorphic section, then there is a holomorphic nowhere zero function g such that ξ 0 = gξ, and thus if s = f 0 ξ 0 = f ξ, we have f = f 0 g. We obtain ¯ 0 ξ 0 ) = (∂f ¯ 0 )ξ 0 = (∂f ¯ 0 )gξ = (∂(f ¯ 0 g))ξ = (∂f ¯ )ξ = ∂(f ¯ ξ), ∂(f which shows that ∂¯ is well-defined. D EFINITION 6.1.7. A connection ∇ for a holomorphic vector bundle H is said to be a (1, 0)connection if, in some (and hence any) holomorphic frame, the connection form ω is a (1, 0)-form. ¯ Equivalently, the (0, 1)-part of ∇ is the connection ∂.

6.1.3

Hermitian metrics and connections

To a complex line bundle H → X, one can associate its complex conjugate bundle H → X. In terms of transition functions, if gαβ are the transition functions of H, then gαβ are the transition functions for H. Moreover, if s is a section of H, then there is an associated section of H which we denote s¯. The assignment s 7→ s is conjugate-linear. R EMARK . Note that since H → H is an R-linear operation, we have a naturally induced connection on H which we write as ∇¯ s := ∇s. It follows that if ω is the connection form for ∇ with respect to some frame ξ, then the connection form for ∇ with respect to the frame ξ¯ is ω ¯. 115

D EFINITION 6.1.8. A Hermitian metric for a line bundle H → X is a smooth section h of the line bundle H ∗ ⊗ H ∗ → X such that for each v, w ∈ Hx h(v, w) ¯ = h(w, v¯) and h(v, v¯) = 0 ⇐⇒ v = 0. (It follows that h(v, v¯) ∈ R.) E XAMPLE 6.1.9. Let H → X be a holomorphic line bundle and s1 , ..., sk be global sections of H with no common zeros. If we choose a frame ξ, then there are functions f1 , ..., fk such that si = fi ξ. We define h(v, w) ¯ :=

a¯b , |f1 |2 + ... + |fk |2

where v = aξ and w = bξ.

It is clear that if we change frame, the functions fi and the numbers a, b change in such a way that the value of h(v, w) ¯ remains unchanged. A standard abuse of notation is to write h(v, w) ¯ =

v w¯ . |s1 |2 + ... + |sk |2

For example, if X = P1 and H = H → P1 is the hyperplane line bundle, we can take the sections z0 , z1 , where [z0 , z1 ] are homogeneous coordinates. (See Section 4.6). The metric hF S (v, w) ¯

v w¯ |z0 |2 + |z1 |2

is called the Fubini-Study metric. R EMARK . If we take a frame ξ for H over U , then the function ¯ h(ξ, ξ) is nowhere vanishing on U . We can then define ¯ ϕ(ξ) := − log h(ξ, ξ). (ξ)

Thus for and section s = f ξ, we have h(s, s¯) = |f |2 e−ϕ . A standard abuse of notation is to omit reference to the frame and write h(s, s¯) = |s|2 e−ϕ ,

or simply

h = e−ϕ .

D EFINITION 6.1.10. We say that a line bundle is Hermitian if it has a Hermitian metric. 116

R EMARK . A standard partition of unity argument shows that any complex line bundle has a Hermitian metric. Indeed, if we take a locally finite open cover {Uα } of coordinate charts on each of which H is trivial, then we have Hermitian metrics hα (aξα , ¯bξ¯α ) = a¯b. Taking a partition of unity {χα } subordinate to this cover, the object X h := χα hα α

is easily seen to be a Hermitian metric. R EMARK . Locally, if we write h(s, t¯) = st¯e−ϕ , then d(h(s, t¯) = d(st¯e−ϕ ) = dst¯e−ϕ + sdte−ϕ + st¯d(e−ϕ ) = ∇st¯e−ϕ + s∇te−ϕ + d(e−ϕ ) − ωe−ϕ − ω ¯ e−ϕ = h(∇s, t¯) + h(s, ∇t) + ∇h(s, t). The ∇ in the third term on the right hand side is the connection for the line bundle H ∗ ⊗ H ∗ associated to the connection for H. D EFINITION 6.1.11. A connection ∇ for a complex line bundle H is said to be compatible with a Hermitian metric h for H if for any sections s, t and any tangent vector v, d(h(s, t))v = h((∇s)(v), t¯) + h(s, (∇s)v). Equivalently, the connection for H ∗ ⊗ H ∗ associated to the connection for H annihilates h, i.e., ∇h = 0.

6.1.4

The Chern connection

T HEOREM 6.1.12. Let H → X be a holomorphic line bundle with Hermitian metric h. Then there is a unique (1, 0)-connection for H that is compatible with h. Proof. We have d(h(s, t¯) = d(st¯e−ϕ ) = dst¯e−ϕ + sdte−ϕ + st¯d(e−ϕ ) ¯ t¯e−ϕ + s(∂t − (∂ϕ)t + ∂t)e ¯ −ϕ , = (∂s − (∂ϕ)s + ∂s) and thus the formula ∇(f ξ) = (df + f (−∂ϕ)) ⊗ ξ shows the existence of the desired connection. Now suppose we have two such connections, say ∇1 and ∇2 . Then we have an H-valued (1,0)-form θ such that ∇1 s − ∇2 s = θ ⊗ s. 117

Then 0 = d(h(s, t)) − d(h(s, t)) = h(∇1 s, t¯) − h(∇2 s, t¯) − (h(∇1 s, t¯) − h(s, ∇2 t)) = h(θs, t) − h(s, θt). It follows that θ = θ. But since complex conjugation maps (1, 0)-forms to (0, 1)-forms, θ = 0. D EFINITION 6.1.13. On a Hermitian line bundle, the unique (1, 0)-connection compatible with the Hermitian metric is called the Chern connection. If the metric is e−ϕ , then the connection (1,0)-form is −∂ϕ.

6.1.5

Curvature of the Chern connection

Let h be a Hermitian metric for a holomorphic line bundle H. Choose a holomorphic frame ξ and consider the function ¯ ϕ(ξ) := − log h(ξ, ξ). If we choose another holomorphic frame ξ 0 , then there is a nowhere zero holomorphic function g such that ξ 0 = gξ. It follows that 0 ϕ(ξ ) = ϕ(ξ) − log |g|2 . Since log |g|2 is pluriharmonic, we have ¯ (ξ) = ∂ ∂ϕ ¯ (ξ0 ) . ∂ ∂ϕ ¯ (ξ) locally in terms of a frame ξ. Thus there is a global (1, 1)-form Θh defined as ∂ ∂ϕ D EFINITION 6.1.14. The form Θh is called the curvature of the Chern connection associated to h. We have chosen an ad hoc definition, but to give it some meaning, we present the following discussion. The Chern connection, being a (1, 0)-form, can be written as ∇ = ∇1,0 + ∂¯ where locally ∇1,0 s = ∂s − ∂ϕs. Consider next the line bundle TX∗1,0 ⊗ H. We can give this line bundle the natural connection ∇, which is defined by ¯ ⊗ s + α ∧ ∂s. ¯ ∇(α ⊗ s) := dα ⊗ s + α ∧ ∇s = ∂α Thus we have ¯ = (−∂ϕ)∂s ¯ ¯ ¯ ∇∇s = ∇(∂s + (−∂ϕ)s + ∂s) + ∂((−∂ϕ))s = (∂ ∂ϕ)s, ¯ = 0. Thus the curvature is just the second where several times we have used ∂ 2 = ∂¯2 = ∂ ∂¯ + ∂∂ derivative, with respect to the connection. The reader can see that the connection is an analog of the exterior derivative, whose square is 1,1 not quite zero, but rather a 0th -order differential operator with values in ΛX , otherwise known as a multiplier. 118

R EMARK . The definition of curvature as the second covariant derivative can be given in general if one develops the theory of connections for vector bundles rather than just line bundles. The reader who is familiar with this construction of curvature will recognize the formula dω − ω ∧ ω for the curvature 2-form. It is an easy exercise to check that for any complex line bundle, the curvature form is a globally defined (1, 1)-form. In the case of a vector bundle V , the curvature is not gloablly defined as a (1, 1)-form (or 2-form if the manifolds have higher dimension), but it can be made globally defined if we let it take values in the vector bundle End(V ) of endomorphisms of V . E XAMPLE 6.1.15. The trivial bundle with the constant metric has zero connection form and zero curvature. However, there are non-trivial connections whose curvature is zero, both on open and compact Riemann surfaces, in the latter case as long as the genus is positive. We shall return to this point later in the text. D EFINITION 6.1.16. We say that a metric e−ϕ has non-negative (resp. positive) curvature if the (1, 1)-form √ −1 Θh 2π is a non-negative (resp. positive) multiple of an area form on X. R EMARK . If a metric h = e−ϕ has nonnegative curvature, then the local functions ϕ obtained by using holomorphic frames are subharmonic. E XAMPLE 6.1.17. If H → X has global holomorphic sections s0 , ..., sk then the metric defined by ϕ = log(|s0 |2 + ... + |sk |2 ) (6.1) as in Example 6.1.9 has non-negative curvature. For example, the Fubini-Study metric has nonnegative curvature. In fact, the latter metric has positive curvature: suppose we take the local coordinate ζ = z1 /z0 on U0 . Then the Fubini-Study metric is given by the function ϕ = log(1 + |ζ|2 ), and so

√ −1 ¯ −1dz ∧ d¯ z ∂ ∂ϕ = 2 2π 2π(1 + |ζ| )2

√

is strictly positive on U0 = {ζ ∈ Cn }. We note that √ Z Z ∞ −1 2rdr Θh = = 1. 2π P1 (1 + r2 )2 0 In general, the metric (6.1) has positive curvature if and only if, with V := Span{s0 , ..., sk } ⊂ ΓO (X, H), 119

the induced map φ|V | : X → P(V ∗ ) is an immersion. Indeed, in local coordinates where s0 6= 0, writing fj = sj /s0 , 1 ≤ j ≤ k and f = (f1 , ..., fk ), we find that √

X √ −1∂ ∂¯ log( |si |2 ) = −1∂ ∂¯ log(1 + ||f ||2 ) ˙ − (f¯ · df ) ∧ (f · df ) (1 + ||f ||2 )df ∧df (1 + ||f ||2 )2 ˙ df ∧df ≥ , (1 + ||f ||2 )2 =

˙ means which vanishes if and only if the map df annihilates some tangent vector. (Here df ∧df df1 ∧ df1 + ... + dfk ∧ dfk ).

6.1.6

Chern numbers

We begin with the following theorem. T HEOREM 6.1.18. Let X be a compact Riemann surface and h a Hermitian metric for a holomorphic line bundle H → X, and let Θh be the curvature of its Chern connection. Then the number √ Z −1 Θh , c(H) := 2π X is independent of the metric h. Proof. If h and h0 are two metrics for H, then h/h0 is a metric for the trivial bundle, and thus is a smooth function with no zeros. Denote this function by e−f . It follows that √ √ ¯ = d( −1∂f ¯ ). Θh − Θh0 = −1∂ ∂f Thus by Stokes’ Theorem, we see that c(H) is independent of h. As we have previously stated, and shall prove in due course, every holomorphic line bundle H on a Riemann surface has a meromorphic section s that is not identically zero. Using such a section, we can compute the number c(H), as we now do. Consider the function x 7→ f (x) = h(s(x), s(x)) on the set Xs := {x ∈ X ; s(x) 6= 0 or ∞}. Let Xs,ε be the subset of X obtained by removing coordinate disks |zj | < ε about the points xj of X − Xs from X. Let √ −1 ¯ c d := (∂ − ∂). 2 120

By Stokes’ Theorem, we have √ √ Z Z −1 −1 ¯ ∂ ∂ log h(s, s¯) = ddc log h(s, s) 2π Xs,ε 2π Xs,ε Z k X 1 = − dc (log |zj |2mj − ϕ), 2π |zj |=ε j=1 where mj = Ordxj (s). (Note that the orientation of the inner boundary of Xs,ε is opposite to the orientation of the last integral, and hence the minus sign.) A simple calculation shows that Z dc log |z|2 = 2π, |z|=ε

and thus we have √

−1 2π

Z

Z 1 ¯ ∂ ∂ log h(s, s¯) = − deg(Ord(s)) + dc ϕ 2π Xs,ε |zj |=ε √ Z −1 = − deg(Ord(s)) + Θh , 2π X−Xs,ε

(6.2)

where the last equality is again by Stokes’ Theorem. On the other hand, on Xs,ε we have √ √ √ √ ¯ − −1∂ ∂¯ log |s|2 = −1∂ ∂ϕ, ¯ −1∂ ∂¯ log h(s, s¯) = −1∂ ∂ϕ since the log-modulus of a nowhere zero holomorphic function is harmonic, and hence is annihi¯ Thus lated by ∂ ∂. √ √ Z Z −1 −1 ¯ ∂ ∂ log h(s, s¯) = Θh . 2π Xs,ε 2π Xs,ε √ R −1 Adding 2π Θh to both sides of (6.2), we have the following theorem. X−Xs,ε T HEOREM 6.1.19. Let X be a compact Riemann surface, H → X a holomorphic line bundle with Hermitian metric h, and s ∈ ΓM (X, H). Then the number c(H) is an integer given by the formula √ Z −1 Θh = deg(Ord(s)). 2π X In particular, the number Ord(s) is independent of the meromorphic section s. R EMARK . The constancy of the function ΓM (X, H) 3 s 7→ deg(Ord(s)) was already known to us by the argument principle: the quotient of two sections is a meromorphic function, wh ose order divisor has degree zero. D EFINITION 6.1.20. The number c(H) is called the Chern number of the holomorphic line bundle H. We will also define this number to be the degree of H. R EMARK . Chern numbers can be defined for more general connections of complex line bundles, again as the integral of the curvature of any connection of those line bundles. The Chern numbers in this context are also integers, and are independent of the connection, thus agreeing with our definition of Chern numbers in the case of a holomorphic line bundle. 121

6.1.7

Example: the holomorphic line bundle TX1,0

Let X be a real oriented surface, with a Riemannian metric g. In view of Theorem 2.1.11, we can choose a complex √ atlas for X that makes it into a Riemann surface, and such that in local coordinates z = x + −1y, g=

e−ϕ 1 (dx ⊗ dx + dy ⊗ dy) = e−ϕ dz ⊗ d¯ z. 2 2

Notice that this Riemannian metric for X is now a Hermitian metric for TX1,0 , in terms of the complex structure provided by Theorem 2.1.11. In the literature, a Hermitian metric for TX1,0 is often referred to as a Hermitian metric for X. Equivalently, a Hermitian metric for X is a metric that is invariant under√the linear transformation J : TX → TX defined in terms of the holomorphic coordinates z = x + −1y by ∂ J ∂x =

∂ ∂y

and

∂ ∂ J ∂y = − ∂x .

As the reader can easily verify, √ the linear transformation J satisfies J 2 = −Id, and thus has the purely imaginary eigenvalues ± −1. In fact, TX ⊗ C decomposes into the two eigenspaces of J as TX ⊗ C = TX1,0 ⊕ TX0,1 . Notice that the function e−ϕ depends on z in the following way: if z 0 is an overlapping local coordinate such that 01 z0, g = e−ϕ dz 0 ⊗ d¯ 2 then on the intersection of the coordinate neighborhoods, we must have 0 2 −ϕ0 dz e = e−ϕ . dz It follows that the differential (1, 1)-form √ ωg :=

−1 −ϕ e dz ∧ d¯ z 2

is globally defined. D EFINITION 6.1.21. The form ωg is called the metric form, or the area form, associated to g. R EMARK . It turns out that the Chern connection of TX1,0 with the Hermitian metric g agrees with the Levi-Civita connection of the Riemannian metric g on X, after we identify TX1,0 with TX be sending a (1, 0)-vector to its real part. Complex Hermitian manifolds (in arbitrary dimension) whose Levi-Civita connections agree with their Chern connections are called Kähler manifolds. It turns out that being Kähler is equivalent to the property that dωg = 0, which trivially holds on Riemann surfaces. The fact that a Hermitian metric on a Riemann surfaces is automatically Käher is one of relatively few low-dimensional accidents that account for the extraordinarily rich geometric structure of Riemann surfaces. 122

Chapter 7 Potential Theory Recall that we are employing the classical terminology that an open Riemann surface is noncompact.

7.1 7.1.1

The Dirichlet Problem and Perron’s Method Definition of the Dirichlet Problem

D EFINITION 7.1.1 (Dirichlet Problem). Let X be a Riemann surface. The Dirichlet Problem for an open subset Y ⊂ X with non-empty boundary ∂Y is the following boundary value problem. For any continuous function f : ∂Y → R, find a continuous function u on Y such that ∆u = 0 on Y

7.1.2

and u = f

on ∂Y.

Perron Families

To solve the Dirichlet problem, we shall employ a method introduced by Perron. D EFINITION 7.1.2. Let X be a Riemann surface. A non-empty family F of subharmonic functions on X is said to be a Perron family if (P1) For every coordinate disk D ⊂ X and every u ∈ F there exists v ∈ F such that v|D is harmonic and v ≥ u. (P2) For every u1 , u2 ∈ F there exists v ∈ F such that v ≥ max{u1 , u2 }. The main result about Perron families is the following theorem. T HEOREM 7.1.3. Let X be a connected Riemann surface. If F is a Perron family on X, then the upper envelope U = UF := sup v v∈F

is either ≡ +∞ or is harmonic on X. 123

Proof. It suffices to prove the result for disks in X, for by connectivity, it is not possible for U to be harmonic on some disk and ≡ +∞ on another disk. Thus we may assume X = ∆ is the unit disk. Let {zj } ⊂ ∆ be a dense subset of the unit disk. For each j, choose a sequence {vjk ; k ≥ 1} ⊂ F such that U (zj ) = lim vjk (zj ). k→∞

Using (P1 ), choose v1 ∈ F such that v1 is harmonic on D and v1 ≥ v11 . If {v1 , ..., vn } ⊂ F have been chosen, let vn+1 ∈ F harmonic on D and such that vn+1 ≥ max{vn , vm` ; m ≤ n + 1, ` ≤ n + 1}. It follows that lim vn (zj ) = sup vn (zj ) = U (zj ).

n→∞

n

Suppose now that U 6≡ +∞. Without loss of generality, we can assume that U (z1 ) < +∞. By Harnack’s Principle, v := lim vn is harmonic on D. We will show that v = U . Note that even though we already know this on a dense subset, we cannot deduce it everywhere because we do not know that U is continuous. First, v ≤ U since U is the upper envelope of F , so that U ≥ vn . Next, v ≥ w on a dense subset for all w ∈ F . Since each w ∈ F is continuous, v ≥ w for all w ∈ F . It follows that v ≥ U . Thus v = U , and the proof is finished.

7.1.3

Perron’s Method

D EFINITION 7.1.4. Let X be a Riemann surface and Y ⊂⊂ X an open subset whose boundary is non-empty. A point x ∈ ∂Y is said to be regular if there is a neighborhood U of x in X and a continuous function β : Y ∩ U → R such that 1. β|Y ∩U is subharmonic, and 2. 0 = β(x) > β(y) for all y ∈ Y ∩ U − {x}. A function β with these two properties is called a barrier at x. Once we have a regular point, we can refine our choice of barrier. We have the following lemma. L EMMA 7.1.5. Let X be a Riemann surface and Y ⊂⊂ X an open subset whose boundary is non-empty. Let x ∈ ∂Y be a regular point. Fix real numbers m ≤ c. Then there is a neighborhood V of x in X and a continuous function v : Y → R such that 1. v|Y is subharmonic, 2. c = v(x) ≥ v(y) for all y ∈ Y ∩ V , and 3. v|Y −V = m. 124

Proof. Clearly we may assume c = 0. Suppose U is a neighborhood of x and β is a barrier at x with respect to this neighborhood U . Let V be a neighborhood of x such that V ⊂⊂ U . Then sup∂V ∩Y β < 0, and thus there is a constant a > 0 such that aβ|∂V ∩Y < m. Then the function max{m, aβ} on V ∩ Y v := m on Y − V satisfies the required conditions. The proof is complete. D EFINITION 7.1.6. Let X be a Riemann surface and Y ⊂⊂ X an open subset. Fix a continuous function f : ∂Y → R. The Perron class Pf ⊂ C (Y ) consists of all functions u that are subharmonic in Y and satisfy u ≤ f on ∂Y . The properties of subharmonic functions show that Pf is a Perron family. It follows from Theorem 7.1.3 that the function Uf defined by Uf (x) := sup v(x) v∈Pf

is harmonic in Y . We have the following theorem. T HEOREM 7.1.7. If x ∈ ∂Y is a regular point, then lim Uf (y) = f (x).

Y 3y→x

Proof. Fix ε > 0. By continuity of f there exists a relatively compact neighborhood V of x in X with |f (y)−f (x)| < ε for all y ∈ ∂Y ∩V . Let a := min∂Y f and A := max∂Y f . By Lemma 7.1.5 there is a continuous function v : Y → R that is subharmonic on Y , such that v(x) = f (x) − ε, vY ∩V ≤ f (x) − ε and v|Y −V = a − ε. Then v ∈ Pf and thus v ≤ Uf . It follows that lim inf Uf (y) ≥ v(x) = f (x) − ε. Y 3y→x

Next, Lemma 7.1.5 provides a continuous function w : Y → R that is subharmonic on Y , such that w(x) = −f (x), w|Y ∩V ≤ −f (x) and w|Y −V = −A. Now, for every u ∈ Pf and y ∈ ∂Y ∩ V one has u(y) ≤ f (x) + ε. Then u(y) + w(y) ≤ ε for all y ∈ ∂Y ∩ V . Furthermore u(z) + w(z) ≤ A − A = 0 for all z ∈ Y ∩ ∂V . By the maximum principle applied to the subharmonic function u + w on Y ∩ V , we deduce that u + w ≤ ε on Y ∩ V . Thus u|Y ∩V ≤ ε − w|Y ∩V for all u ∈ Pf . Hence lim sup Uf (y) ≤ f (x) + ε. Y 3y→x

Since ε was arbitrary, the proof is complete. 125

As a corollary, we obtain the following result. C OROLLARY 7.1.8. Let X be a Riemann surface and Y ⊂⊂ X an open subset each of whose boundary points is regular. Then the Dirichlet Problem has a solution on Y . We end with a useful sufficient condition for regularity of a boundary point which implies, in particular, that any point at which the boundary is smooth is a regular point. P ROPOSITION 7.1.9. Let X be a Riemann surface, Y ⊂⊂ X an open set and x ∈ ∂Y . Suppose there is a coordinate neighborhood z : U → C of x such that for some p ∈ z(U − Y ) the disk {ζ ∈ z(U ) ; |ζ − p| < |z(x) − p|} lies in z(U ) and does not meet z(Y ∩ U ). Then x is a regular point. Proof. The function β(ζ) := log

|z(x)−p| 2

− log ζ −

p−z(x) 2

is a barrier at x.

7.1.4

Aside: Countable topology of Riemann surfaces

In the definition of manifold, one has to assume a countable topology. Though we would happily make such an assumption in this book, it turns out that for Riemann surfaces, it is not necessary to do so, as was proved by Radó. The key is the ability to solve the Dirichlet Problem. T HEOREM 7.1.10. Every Riemann surface has a countable topology. Proof. Let U ⊂ X be a coordinate neighborhood. Choose disjoint smoothly bounded open disks D1 , D2 ⊂⊂ U whose closures are disjoint, and set Y := X − D1 ∪ D2 . Then ∂Y consists of regular points, and thus there is a continuous function u : Y → R that is harmonic on Y and satisfies u|∂D0 ≡ 0 and u|∂D1 ≡ 1. The (1, 0)-form ω := ∂u is then holomorphic and non-trivial. Consider the universal cover π : Y˜ → Y . Let f be a holomorphic function on Y˜ such that ∂f = π ∗ ω. Since the mapping f : Y˜ → C is continuous and has discrete fibers, it follows from elementary point set topology that Y˜ has a countable topology, and since the mapping π : Y˜ → Y is continuous and surjective, Y has countable topology, as desired.

7.1.5

A helpful note

In the next paragraph we consider Green’s Functions. On a Riemann surface Y with boundary, a Green’s Function is a family of solutions of the distributional boundary value problems √ ¯ x = δx , Gx |∂Y = 0 −1∂ ∂G as x varies over the points of (the interior of) Y . Since for a fixed x such a function is harmonic away from x, and since the difference of any two solutions (ignoring the boundary condition) is 126

also Harmonic, we can expect that the kind of singularity that a Green’s Function Gx possesses at x is in some sense uniform. To see what this singularity is, we work locally. The following lemma, whose prove was already indirectly hinted at on a number of occasions, characterizes the singularity of a Green’s √ √ −1 ¯ c function locally. Recall that with d = 2 (∂ − ∂), −1∂ ∂¯ = ddc . L EMMA 7.1.11. Let ϕ ∈ C0∞ (C). Then Z log |z|2 ddc ϕ = 2πϕ(0). C

That is to say, ddc log |z|2 = 2πδ0 in the sense of currents. Proof. Let R > 0 be such that Support(ϕ) ⊂ D(0, R), and let ε ∈ (0, R). Note that since log |z|2 is locally integrable, Z Z 2 c log |z| dd ϕ = lim log |z|2 ddc ϕ. ε→0

C

D(0,R)−D(0,ε)

Note that dc f ∧ dg = −df ∧ dc g. By integration-by-parts, we have Z Z Z 2 c 2 c 2 log |z| dd ϕ = − d log |z| ∧ d ϕ + log ε dc ϕ D(0,R)−D(0,ε) D(0,R)−D(0,ε) |z|=ε Z = dc log |z|2 ∧ dϕ + O(ε log ε2 ) D(0,R)−D(0,ε) Z Z c 2 = − ϕdd log |z| + ϕdc log |z|2 + O(ε log ε2 ) Z

D(0,R)−D(0,ε) 2π √ −1θ

ϕ(εe

=

|z|=ε

)dθ + O(ε log ε2 ).

0 √

√ d¯ z −1θ (Note that dc log |z|2 = 2−1 dz − , so that when z = εe , dc log |z|2 = dθ.) Thus z z¯ Z Z 2 c log |z|2 ddc ϕ = 2πϕ(0). log |z| dd ϕ = lim C

ε→0

D(0,R)−D(0,ε)

The proof is complete.

7.1.6

Green’s Functions

D EFINITION 7.1.12. Let X be a Riemann surface. A Green’s Function on X with singularity at x ∈ X is a subharmonic function Gx : X → [−∞, 0) such that (G1) Gx is harmonic on X − {x}, (G2) if z is any local coordinate in a neighborhood U of x with z(x) = 0 then Gx − log |z|2 is harmonic in U , and 127

(G3) if H is any other subharmonic function satisfying (G1) and (G2) then Gx ≥ H. Observe that by the very definition of Green’s Function, any Riemann surface that admits a Green’s function must admit a bounded subharmonic function. (In this text we call such surfaces potential-theoretically hyperbolic, a notion we will discuss again in Chapter 10.) The amazing fact is that the converse is true. T HEOREM 7.1.13. If a Riemann surface X admits a bounded non-constant subharmonic function then for any point x ∈ X, X admits a Green’s Function with singularity x. R EMARK . Note that no compact Riemann surface admits a Green’s Function. Some readers may find this confusing, since there are other notions of Green’s Function in the literature such that every compact Hermitian manifold admits such a Green’s Function. These other notions arise because Poisson’s Equation can only be solved for data that is orthogonal to the space of Harmonic Functions; the Green’s Function can then be defined on this orthogonal complement. We shall solve such a PDE on compact Riemann surfaces In Chapter 9, but we will not use the term Green’s Function for the solution operator. Before proving Theorem 7.1.13, we shall need the following lemma. L EMMA 7.1.14. Let X be a Riemann surface admitting a bounded subharmonic function and let K be a compact subset of X such that X − K is connected and has regular boundary. Then there exists a continuous function ϕ : X − K → (0, 1] that is harmonic and non-constant on X − K, and ≡ 1 on ∂K. Proof. Let ψo be a non-constant negative subharmonic function on X and set mo := min(−ψo |K ) and ψ1 := ψo /mo . Then ψ1 is negative, subharmonic, non-constant, and ≤ −1 on K. Note that by the strong maximum principle, ψ1 assumes its maximum −1 at some point p ∈ ∂K. Also, there is a point q ∈ X −K such that ψ(q) > −1, for otherwise p is an interior maximum, and the maximum principle would imply that ψ1 is constant. Let ψ := max{ψ1 , −1}. Then ψ is subharmonic on X, −1 ≤ ψ < 0, ψ(q) < −1 and ψ|K ≡ −1. Let H denote the set of all continuous functions on X that are subharmonic on X − K and satisfy the inequality v + ψ ≤ 0 on X − K. The maximum of two functions in H is clearly in H . Take a function u ∈ H and a disk D in X − K. Solve the Dirichlet Problem on D with boundary value u on ∂D. Let w be the solution, and set v = w in D and v = u in X − (K ∪ D). We have to verify that v is subharmonic. It suffices to verify the sub-mean value property on ∂D. But since w ≥ u in D, we have v ≥ u, with equality on the boundary of D. The desired sub-mean value inequality now follows. Since the function 0 lies in H , H is a Perron family (on X − K). It follows that the function ϕ := sup v v∈H

is harmonic on X − K, and satisfies 0 ≤ ϕ ≤ 1 on X − K. But we need a little more. To get the desired information we produce a good member of H . Toward this end, choose an open set U ⊂⊂ X containing K such that ∂U consists of a finite number of smooth curves. (This is easy to do and we shall give a proof in the next section.) Solve 128

the Dirichlet Problem on U − K for the boundary values 1 on ∂K and 0 and ∂U . By the maximum principle the solution vo satisfies 0 ≤ vo ≤ 1 on U − K. Extend vo to X − U by zero. Then as before, vo is subharmonic on X − K. We claim that vo + ψ ≤ 0 on X − K. Indeed, the latter is subharmonic on X − K and ≤ 0 on ∂U and on X − U . Since ψ ≡ −1 on ∂K, the maximum principle implies that vo + ψ ≤ 0 on U − K. Thus vo ∈ H . By the definition of ϕ, we have vo ≤ ϕ ≤ −ψ. It follows that ϕ is continuous on X − K and ≡ 1 on ∂K, and since −ψ(q) < 1, ϕ is non-constant. Moreover, 0 < ϕ < 1 on X − K. The proof is complete. R EMARK . The function ϕ satisfying the conclusions of Lemma 7.1.14 is a classical object called the harmonic measure of the set K. The harmonic measure has many important interpretations and applications that we will not get into in this book. Proof of Theorem 7.1.13. We are going to use Perron’s Method. To this end, denote by G the family of all non-negative functions u on X that have compact support, are subharmonic on X − {x}, and such that for any local coordinate z near x with z(x) = 0, u + log |z| is subharmonic in |z| < 1. The function uo (z) := −χ{|z| 0. Let Y = Y0 ⊂⊂ Y1 ⊂⊂ Y2 ⊂⊂ ... ⊂ X be a normal exhaustion. By what we have just proved, O(Yi )|Yi−1 is dense in O(Yi−1 ) for all i = 1, 2, ... . Let f0 := f and define inductively fi ∈ O(Yi ) such that sup |fi − fi−1 | < 2−i ε. Yi−1

136

Define g := f +

∞ X

fj − fj−1 .

j=1

Then g converges uniformly on Y and sup |f − g| ≤ Y

∞ X j=1

sup |fj − fj−1 | < ε. Y


7.2.3

Function-Theoretic Description of hulls

P ROPOSITION 7.2.10. Let X be an open Riemann surface and K ⊂ X a compact set. Then b O(X) = x ∈ X ; |f (x)| ≤ sup |f | for all f ∈ O(X) . K K

Proof. Let x be a point in some relatively compact component U ⊂⊂ X − K. Then ∂U ⊂ K. By the Maximum Principle, for any f ∈ O(X), |f (x)| ≤ sup∂U |f | ≤ supK |f |. b O(X) . Let V be an unbounded component of X − K containing x. Conversely, suppose x 6∈ K Take a small disk ∆ ⊂ V centered at x. Then b O(X) ∪ ∆ and \ (K ∪ ∆)O(X) = K

b O(X) ∩ ∆ = ∅. K

By Theorem 7.2.9 the function f (z) :=

b O(X) 0 z ∈K 1 z ∈∆

can be approximated by global holomorphic functions on X. It follows that there is a function F ∈ O(X) such that |F (x)| > 1/2 > sup |F |. K

Thus x 6∈ {y ∈ X ; |f (y)| ≤ supK |f | for all f ∈ O(X)}. The proof is complete.

137

138

Chapter 8 Solving ∂¯ with smooth data In this chapter we use Runge’s Approximation Theorem to prove that on an open Riemann surface the ∂¯ equation can always be solved for smooth closed (0, 1)-forms. We then obtain several fundamental corollaries of this fact. R EMARK . The reader may be wondering what happens for compact Riemann surfaces. The problem will be fully addressed in the next chapter, but for the sake of avoiding confusion, we state ¯ =α the outcome now. For a (0, 1)-form α on a compact Riemann surface, the solvability of ∂u is determined by the Dolbeault cohomology class of α, as stated in Theorem 5.11.1, which will be ¯ = α has a solution if and proved in the next chapter. In fact, we will show that the equation ∂u only if for all holomorphic 1-forms h, Z 1 √ α ∧ h = 0. 2 −1 X ¯ = α has a Note that on P1 there are no non-zero holomorphic 1-forms, so that the equation ∂u solution for all (0, 1)-forms α. For a (1, 1)-form µ on any compact Riemann surface X, we will see in the next chapter that ¯ = µ has a solution β if and only if the equation ∂β Z µ = 0. X

8.1

The basic result

Using the technique of the Runge Approximation Theorem, we can now extend Theorem 7.1.18 to the following result. T HEOREM 8.1.1. Let X be an open Riemann surface. Given any smooth (0, 1)-form θ on X there ¯ = θ. is a smooth function f : X → C such that ∂f Proof. Let Y0 ⊂⊂ Y1 ⊂⊂ ... ⊂ X be a normal exhaustion of X. We shall construct a sequence of functions fn : Yn → C for n = 0, 1, 2, ... such that ¯ n = θ on Yn−1 ∂f and sup |fn+1 − fn | < 1n , n ≥ 0. Yn−1

139

2

¯ 1 = θ on Y0 . Such f1 exists by Theorem 7.1.18. Now suppose Take f1 to be any solution of ∂f ¯ k+1 = θ on Yk . f1 , ..., fk have been chosen. Let uk+1 : Yk+1 → C be a smooth function solving ∂u Then uk+1 − fk is holomorphic on Yk−1 and thus there is a holomorphic function gk ∈ O(X) that is within 2−k of uk+1 − fk on Yk−1 . Let fk+1 := uk+1 − gk . Then ¯ k+1 = θ on Yk ∂f

and

sup |fk+1 − fk | = sup |uk+1 − fk − gk | < 2−k . Yk−1

Yk−1

Pn

Now fn = f0 + j=1 fj − fj−1 . Note that on Yk−1 , k < n, fn = fk + hk,n for some holomorphic function hk,n whose sup norm on Yk−1 is at most 21−k . It follows that fn → f for some smooth function f in C 1 -norm, and thus ¯ = lim ∂f ¯ n = lim θ = θ. ∂f The proof is complete.

8.2

Triviality of holomorphic line bundles

Let L → X be a holomorphic line bundle. Choose a locally trivial open cover {Ui } such that each Uj and each Uk ∩ U` is connected and simply connected, and such that L|Uj is trivial. For each i, let ei ∈ ΓO (Ui , L) be a nowhere zero section, and define the holomorphic functions hij := ei /ej

on

Ui ∩ Uj .

Then hij is a nowhere zero holomorphic function, and thus there are holomorphic functions gij such that hij = egij . Let {χj } be a partition of unity subordinate to the open cover {Uj }. Consider the smooth functions X g˜i := χj gij on Ui . j

Then g˜i − g˜j =

X

χk (gik − gjk ) = gij .

k

¯ = ∂˜ ¯gj on Uj . Let h be It follows that there is a globally defined smooth (0, 1)-form γ such that ∂γ ¯ = γ, and set a smooth function satisfying ∂h gi := g˜i − h. Then gi − gj = gij . Define si := e−gi ei . Then on Ui ∩ Uj one has si = e−gi ei = e−gij e−gj ei = e−gj ej = sj . Hence s = si on Ui is a globally defined section, and evidently s has no zeros. Thus we have proved the following theorem. T HEOREM 8.2.1. Every holomorphic line bundle on an open Riemann surface is trivial. 140

8.3

The Weierstrass Product Theorem

Let X be a Riemann surface and D a divisor on X. We seek to find out when D = Ord(f ) for some f ∈ M (X). If X is a compact Riemann surface, a necessary condition is that deg(D) = 0, but we already know that this condition is not sufficient unless X = P1 . The sufficient condition for surfaces of positive genus will be addressed in Chapter 14; it is the content of Abel’s Theorem. In this paragraph we solve the problem for open Riemann surfaces. Unlike compact Riemann surfaces, there is no obstruction in the open case. In fact, as a corollary of Theorem 8.2.1, we obtain the following generalization of the famous Weierstrass Product Theorem. T HEOREM 8.3.1. Let D be a divisor on an open Riemann surface. Then there is a function f ∈ M (X) such that Ord(f ) = D. Proof. Let D be a divisor, and let L be the line bundle associated to the divisor D. The line bundle L has a canonical meromorphic section sD such that Ord(sD ) = D. But by Theorem 8.2.1, there is a nowhere zero section s ∈ ΓO (X, L). Thus the meromorphic function f := sD /s satisfies Ord(f ) = Ord(sD ) − Ord(s) = Ord(sD ) = D. This completes the proof.

8.4

Meromorphic functions as quotients

Let X be a Riemann surface. If f is a meromorphic function on X, then locally f = g/h for holomorphic functions g and h. However, in general this is not the case globally. Indeed, on a compact Riemann surface there are no non-constant holomorphic functions. But as we shall see in Chapter 12, there are always many meromorphic functions. Amazingly, the only obstruction to writing a meromorphic function as a quotient of holomorphic functions globally is compactness. In fact, we have the following result. T HEOREM 8.4.1. Let X be an open Riemann surface and let f ∈ M (X). Then there are holomorphic functions g, h : X → C such that f = g/h. Proof. Let D := (f )∞ be the divisor of poles of f . Then D defines a holomorphic line bundle L and a canonical section s such that Ord(s) = D. Evidently s is holomorphic. Since X is open, Theorem 8.2.1 provides a nowhere zero section t ∈ ΓO (X, L). Let h = s/t. Then h is a holomorphic function, and Ord(f h) = Ord(f ) + Ord(s) = (f )0 − (f )∞ + (f )∞ = (f )0 . is effective. It follows that g = f h is a holomorphic function. The proof is complete. 141

8.5

The Mittag-Leffler Problem

In this section we consider the problem of specifying the singular parts of a meromorphic function. On a compact Riemann surface, the singular structure of a meromorphic function is not arbitrary. By contrast, on an open Riemann surface we have total freedom to prescribe the singular structure of a meromorphic function.

8.5.1

Principal parts and the Mittag-Leffler Problem

We begin by defining the notion of singular structure, which we call principal part. D EFINITION 8.5.1. Let X be a Riemann surface. A principal part is a collection {(Uj , fj ) ; j ∈ J}, such that 1. U = {Uj ; j ∈ J} is a locally finite open cover, 2. fj ∈ M (Uj ) for all j ∈ J, and 3. fi − fj ∈ O(Ui ∩ Uj ) for all i, j ∈ J. E XAMPLE 8.5.2. Let f be a meromorphic function on X. Choose a locally finite open cover U := {Uj ; j ∈ J} and holomorphic functions gj ∈ O(Uj ). Then the functions fj := f |Uj − gj , together with U form a principal part. Any such principal part is said to be trivial, or may also be called the principal part of a meromorphic function. The Mittag-Leffler Problem is to classify which principal parts are the principal parts of a meromorphic function.

8.5.2

Solution on open Riemann surfaces

We begin by treating open Riemann surfaces. T HEOREM 8.5.3. On an open Riemann surface X, every principal part is trivial. Proof. Let gij := fi − fj ∈ O(Ui ∩ Uj ). Choose a partition of unity {χj } subordinate to the open cover U , and define the (0, 1)-form αj on Uj by ¯ αj := ∂(

X

χi gij ).

i

Observe that X X ¯ ¯ kj ¯ kj = 0. αj − αk = ∂( χi (gij − gik )) = ∂(g χi ) = ∂g i

i

It follows that the (0, 1)-form α defined to be αj in Uj is smooth. By Theorem 8.1.1 there is a ¯ = α. smooth function h such that ∂h 142

Consider the function gj := −h +

X

χi gij .

i

¯ j = αj − α = 0, so that gj ∈ O(Uj ). We also have Then ∂g gj − gk =

X

χi (gij − gik ) = gkj = fk − fj .

i

It follows that fj + gj = fk + gk , and thus we have a global meromorphic function f ∈ M (X) defined to be fj + gj in Uj . Evidently f shows that {(Uj , fj )} is trivial.

8.5.3

Solution on general Riemann surfaces

We turn now to the case of compact Riemann surfaces. E XAMPLE 8.5.4. On a compact Riemann surface there are principal parts that are not the principal parts of meromorphic functions. For example, on a complex torus one can take a single open coordinate chart U and one point p ∈ U , and consider the function fU (z) = (z − p)−1 . There is no meromorphic function with a single, simple pole on a torus, and thus {(U, fU ), (V, 1)}, where V is any open subset of the torus containing the complement of U , is not the principal part of a meromorphic function. Up to a certain point, the method of proof for open Riemann surfaces can be carried over to compact Riemann surfaces. The proof breaks down exactly at the point where we try to solve the ¯ = α; this equation cannot be solved on a general compact Riemann surface. equation ∂h Let us look again at the form α constructed in the proof of Theorem 8.5.3. To emphasize the dependence of the choice of partition of unity, we write α = αχ . Suppose we choose another partition of unity χ. ˜ Then we have X X (χi − χ˜i )(gij − gik ) = gkj (χi − χ˜i ) = gkj (1 − 1) = 0. i

i

Thus we can define the global function ψχ,χ˜ by ψχ,χ˜ =

X

(χi − χ˜i )gij

on Uj ,

i

and we find that ¯ χ,χ˜ . αχ − αχ˜ = ∂ψ ¯ = α is independent of the It follows that the existence of a function h satisfying the equation ∂h choice of partition of unity. The proof of Theorem 8.5.3 is easily modified to obtain one part of the following result. (Recall the definition of Dolbeault cohomology discussed in Section 5.11.) 143

T HEOREM 8.5.5. Let P := {(Uj , fj )} be a principal part on a Riemann surface X and let aP be ¯ the ∂-cohomology class of the form α associated to P by the construction in the proof of Theorem 8.5.3. Then P is the principal part of a meromorphic function if and only if aP = 0. Proof. Since the if direction has already been proved, we prove only the converse. Suppose, then, that we have a meromorphic function f and holomorphic functions gj such that f = fj + gj on Uj . Then X X X ¯ i − gj ∂(1) ¯ =− ¯ i = ∂h, ¯ α = ∂¯ χi (gj − gi ) = − gi ∂χ gi ∂χ i

where h = −

P

i

i

χi gi . Thus aP = 0 by definition. The proof is complete.

R EMARK . We can take one more formal step. Two principal parts are said to be equivalent if their difference is the principal part of a meromorphic function. Our results show that the map sending ¯ the equivalence class [P ] of a principal part P to the ∂-cohomology class αP is 1 − 1. It turns out 1 that [P ] → aP is also surjective onto H∂¯ (X). In other words, every element of H∂1¯ (X) is of the form aP for some principal part P . ¯ One way to treat cohomology while avoiding the ∂-equation is through the study of principal parts of meromorphic functions, also called Laurent Tails in the literature. We shall not take this perspective in the present text.

8.5.4

Principal parts of meromorphic 1-forms

Suppose {Ui }i∈I is an open cover of X and ηi is a meromorphic 1-form on Ui . (We will say {ηi } is subordinate to {Ui }.) Assume that ηi − ηj is holomorphic on Ui ∩ Uj . Then the residue of ηi and that of ηj at some point p ∈ Ui ∩ Uj agree. Thus we can define Resp ({ηi }i∈I ) = Resp (ηi ),

p ∈ Ui .

We can then define a divisor Res({ηi }) :=

X

Resp ({ηi }) · p.

p∈X

With this notation, we have the following theorem. T HEOREM 8.5.6 (Mittag-Leffler Theorem for meromorphic 1-forms). Let X be a compact Riemann surface, {Uj } an open cover of X, and {ηj } a collection of meromorphic 1-forms subordinate to {Uj } such that ηi − ηj is holomorphic on Ui ∩ Uj . Then deg(Res({ηj })) :=

X

Resx ({ηi }) = 0.

x∈X

if and only if there is a meromorphic 1-form η on X such that η − ηj is holomorphic on Uj . 144

Proof. One direction is easy. We already know that for a global meromorphic 1-form η, deg Res(η) = 0. Since η − ηj is holomorphic on Uj for all j, Res(η) = Res({ηj }). We now establish the converse. Let x1 , ..., xN be the set of poles of {ηi }. Choose a partition of unity {χi } subordinate to {Ui }, and let X η˜i := χj (ηi − ηj ) on Uj . j

Then each η˜j is smooth. Moreover, η˜i − η˜j = ηi − ηj is a holomorphic 1-form on Ui ∩ Uj , and thus ∂¯η˜i = ∂¯η˜j

on Ui ∩ Uj .

It follows that the (1, 1)-form α defined by α := ∂¯η˜i

on Ui

is globally defined. We claim that

Z 1 √ α = deg(Res({ηi })). 2π −1 X To see this, let Dj,ε be small coordinate disks of radius ε, each containing exactly one pole xj , j = 1, ..., N , of {ηi }. Observe that each ηj is holomorphic on X − {x1 , ..., xN }, and thus since ηi − ηj = η˜i − η˜j , the form β := ηj − η˜j is smooth on X − {x1 , ..., xN } where it satisfies the equation dβ = −α. Since α is smooth, we have Z Z α = lim α S X

ε→0

X−

j

Dj,ε

Z = − lim

ε→0

= lim

ε→0

dβ S X− j Dj,ε

N Z X j=1

β

∂Dj,ε

X √ X = 2π −1 Resxj ({ηi(j) }) − lim j

√ X = 2π −1 Resxj ({ηi(j) }), j

145

j

ε→0

Z η˜i(j) ∂Dj,ε

where i(j) is any integer such that Dj,ε ⊂ Ui(j) . The last equality holds since η˜i are smooth forms. Thus we have the stated integral identity. R In particular, if deg(Res({η })) = 0, then α = 0. As we will show in Chapter 9 (Corollary i X R ¯ , and thus α = ∂µ ¯ 9.2.2), the vanishing of X α implies that there is a function f such that α = ∂ ∂f ¯ with µ = −∂f . Let ξi := η˜i − µ. Then ∂ξi = 0, so ξi are holomorphic, and ξi − ξj = η˜i − η˜j = ηij = ηi − ηj . It follows that η := ξi − ηi on Ui defines a global meromorphic 1-form, with the same polar structure as {ηi }. This completes the proof.

8.6

Poisson’s Equation on Open Riemann Surfaces

In the final section of this chapter, we solve Poisson’s equation with smooth forcing term on an open Riemann surface. Although there are many applications of the solvability of this equation, we shall not pursue them here.

8.6.1

Pfluger’s Harmonic Runge Theorem

The method we used to prove Runge’s Approximation Theorem for holomorphic functions can be applied to harmonic functions. Local solvability The starting point is the following lemma. L EMMA 8.6.1. Let X be a Riemann surface, Y ⊂⊂ X an open subset such that X − Y 6= ∅ and ω a (1, 1)-form with compact support in Y . Then there is a function f on a neighborhood of Y such that √ ¯ = ω. −1∂ ∂f Proof. Let Y 0 ⊂⊂ X be a smoothly bounded open subset such that Y ⊂ Y 0 and X − Y 0 6= ∅. Let GY 0 be the Green’s function of Y 0 . One simply takes Z 1 f (x) := GY 0 (x, y)ω(y). 2π Y 0

Pfluger’s Theorem Recall that H(X) denotes the set of harmonic functions on a Riemann surface X. T HEOREM 8.6.2. Let X be an open Riemann surface and Y ⊂⊂ X a Runge open set. Then H(X)|Y is dense in H(Y ) in the topology of uniform convergence on compact sets. 146

Proof. The approach is the same as in the proof of Runge’s Theorem 7.2.9. Fix a relatively compact open set Z ⊂⊂ X with Y ⊂⊂ Z. We begin by showing that every harmonic function on Y can be approximated uniformly by harmonic functions in Z. As before, the Hahn-Banach Theorem will achieve our aim if we show that every linear functional that annihilates H(Z) also annihilates H(Y ). By the Riesz Representation Theorem, every such linear functional is represented by a compactly supported real-valued measure dµ on Y . Let K := Supp(dµ). We associate to the measure dµ the following linear functional on (0, 1)-forms θ compactly supported on Z: Z √ ¯ = θ in Z. −1∂ ∂f S(θ) := f dµ, where Y

Note that S is well-defined since if by hypothesis,

√

¯ = −1∂ ∂f

√

¯ then f − g is harmonic on Z and thus −1∂ ∂g,

Z (f − g)dµ = 0. Y

Now, by the proof of Lemma 8.6.1, the function Z 1 GZ (x, y)θ(y) f (x) := 2π Z √ ¯ = θ on Z. Therefore, with solves the equation −1∂ ∂f Z 1 GZ (x, y)dµ(x), σ(y) := 2π Y we have

1 S(θ) = 2π

Z σθ. Z

Observe that σ is harmonic on Z − K and vanishes on ∂Z. By the maximum principle, σ must be b O(Z) . zero on every unbounded component of Z − K in Z. It follows that σ is zero on Z − K b O(Z) ⊂⊂ Y . Thus there is a function g ∈ Co∞ (Z) Now let f ∈ H(Y ). Since Y is Runge, K b O(Z) . Then such that g = f on K Z Z Z √ ¯ = 0, f dµ = gdµ = σ −1∂ ∂g Y

Y

Z

√ ¯ ⊂ Z −K b O(Z) , and hence σ|Supp(√−1∂ ∂g) where the last equality holds since Supp( −1∂ ∂g) ¯ ≡ 0. Thus we have proved that H(Z)|Y is dense in H(Y ). Finally, fix f ∈ H(Y ) and ε > 0. Let Y = Y0 ⊂⊂ Y1 ⊂⊂ Y2 ⊂⊂ ... ⊂ X be a normal exhaustion. By what we have just proved, H(Yi )|Yi−1 is dense in H(Yi−1 ) for all i = 1, 2, ... . Let f0 := f and define inductively fi ∈ H(Yi ) such that sup |fi − fi−1 | < 2−i ε. Yi−1

147

Define g := f +

∞ X

fj − fj−1 .

j=1

Then g converges uniformly on Y and sup |f − g| ≤ Y

∞ X j=1

sup |fj − fj−1 | < ε. Y


8.6.2

Solving ∂ ∂¯ with smooth data on open Riemann surfaces

We can now use the natural analog of the proof of theorem 8.1.1 to solve Poisson’s equation with smooth data. T HEOREM 8.6.3. Let ω be a smooth (1, 1)-form on an open Riemann surface X. Then there is a smooth function ψ : X → C such that √ ¯ = ω. −1∂ ∂ψ Proof. Let Y0 ⊂⊂ Y1 ⊂⊂ ... ⊂ X be a normal exhaustion of X. We shall construct a sequence of functions fn : Yn → C for n = 1, 2, ... such that √ 1 ¯ n = ω on Yn−1 , n ≥ 1. −1∂ ∂f and sup |fn+1 − fn | < 2n−1 Yn−1

√ ¯ 1 = ω on Y0 . Such f0 exists by Lemma 8.6.1. Now suppose Take f1 to be any solution of −1∂ ∂f √ ¯ k+1 = ω f1 , ..., fk have been chosen. Let uk+1 : Yk+1 → C be a smooth function solving −1∂ ∂u on Yk+1 . Then uk+1 − fk is harmonic on Yk−1 and thus there is a harmonic function gk ∈ H(X) that is within 2−k of uk+1 − fk on Yk−1 . Let fk+1 := uk+1 − gk . Then √ ¯ k+1 = ω on Yk and sup |fk+1 − fk | = sup |uk+1 − fk − gk | < 2−k . −1∂ ∂f Yk−1

Yk−1

P Note that fn = f0 + nj=1 fj − fj−1 and that, on Yk−1 , k < n, fn = fk + hk,n for some harmonic function hk,n on Yk−1 whose sup norm on Yk−1 is at most 2−k . It follows that fn → f for some smooth function f in C 1 -norm, and thus √ √ ¯ = lim −1∂ ∂f ¯ n = lim ω = ω. −1∂ ∂f The proof is complete. In the next chapter we will establish an analog of Theorem 8.6.3 for compact Riemann surfaces.

148

Chapter 9 Harmonic Forms 9.1

Harmonic Forms

Let X be a compact Riemann surface and Ω a strictly positive (1, 1)-form on X. We will consider the algebra differential forms on X. The goal is to find canonical representatives in cohomology classes of differential forms, specifically by solving a certain extremal problem defined with regard to Ω. (The reader may wish to recall Paragraph 6.1.7, in which we explained how to identify strictly positive (1, 1)-forms with Hermitian metrics.)

9.1.1

The Hodge Star

We begin by defining the Hodge ? operator on each member of such a triple. For a function f we set ?f := f¯Ω. For a 1-form α = α1,0 + α0,1 we set 1 0,1 α − α1,0 . ?α := √ −1 For a (1, 1)-form ω = f Ω we set ?ω := f¯. Observe that ? maps functions to (1, 1)-forms and vice versa, and fixes 1-forms. Moreover, on functions and (1, 1)-forms ?? = Id while on 1-forms, ?? = −Id.

9.1.2

Inner products

Given two functions f, g : X → C, we define Z (f, g) :=

f ? g. X

149

Given two 1-forms α and β on X, we define Z α ∧ ?β.

(α, β) := X

And given two (1, 1)-forms ω and θ, we define Z (ω, θ) :=

ω ? θ. X

We declare ((f, α, ω), (g, β, θ)) := (f, g) + (α, β) + (ω, θ). D EFINITION 9.1.1. E (X, Ω) denotes the inner product space of all differential forms (which can be graded, so thought of as triples of functions, 1-forms and (1, 1)-forms), together with the above inner product. When we work with the various components, we will write E0 (X, Ω),

E1 (X, Ω) and

E1,1 (X, Ω).

Thus E (X, Ω) = E0 (X, Ω) ⊕ E1 (X, Ω) ⊕ E1,1 (X, Ω). The Hilbert space closures of these spaces of smooth objects are respectively denoted L20 (X, Ω),

L21 (X, Ω),

L21,1 (X, Ω)

and L2 (X, Ω) = L20 (X, Ω) ⊕ L21 (X, Ω) ⊕ L21,1 (X, Ω).

9.1.3

The formal adjoint of d

The formal adjoint of the exterior derivative operator d is the operator d∗ defined to be zero on functions, and defined on 1-forms (resp. (1, 1)-forms) α by the relations (d∗ α, f ) = (α, df )

for all f ∈ E0 (X) (resp. ∈ E1 (X)).

Let us calculate the operators d∗ . (1) (1-forms) We have ∗

Z

Z

Z

df ∧ ?α = − f d(?α) X Z Z = − f ? (?d ? α) = − (?d ? α) ? f = −(?d ? α, f ). α ∧ ?df =

(d α, f ) =

X

X

X

X

150

(2) ((1, 1)-forms) ∗

Z

Z

Z

dβ ? ω = β ∧ d(?ω) X Z Z = − β ∧ ?(?d ? ω) = − (?d ? ω) ∧ ?β = −(?d ? ω, β).

(d ω, β) =

ω ? dβ =

X

X

X

X

Thus in all cases, we have d∗ = − ? d ? .

9.1.4

Laplace-Beltrami operator and harmonic forms

D EFINITION 9.1.2. The Laplace-Beltrami operator associated to (X, Ω) is ∆ := dd∗ + d∗ d. An element in the kernel of ∆ is called a Harmonic form. The term harmonic may be a little confusing, since we already have a notion of harmonic functions. At the very least, these two notions should agree if we are going to use the same name for them. To do the local calculations, it is convenient to introduce the local (real valued) function h associated to the local coordinate function z by the relation √ −1 dz ∧ d¯ z. h(z)Ω = 2 Now, for a function f we have √ √ √ ¯ ) = 2 ? −1∂ ∂¯f¯ = 2 ? fzz¯ −1dz ∧ d¯ ∆f = ?d ? df = ?d −1(∂f − ∂f z = 4h(z)fzz¯. Therefore harmonic functions are harmonic in the previous sense of the word. Next, for a (1, 1)-form ω = f Ω, we have ¯ )= ∆ω = −d(?df¯) = d ? (∂f + ∂f

√ √ ¯ ) = 2 −1∂ ∂f ¯ = h(∆f )Ω. −1d(−∂f + ∂f

so that a form ω = f Ω is harmonic if and only if the function f is harmonic. Thus the map f 7→ f Ω gives a 1 − 1 correspondence between harmonic (1, 1)-forms and harmonic functions. In addition, we also have the following proposition. P ROPOSITION 9.1.3. Global harmonic (1, 1)-forms on a compact Riemann surface X are constant multiples of Ω and conversely. 151

Finally, we turn to 1-forms. If α = f dz + gd¯ z , then √ √ −(dd∗ + d∗ d)α = d(?d −1(f¯d¯ z − g¯dz)) − ?d ? −1(gz − fz¯)2hΩ √ = 2d ? ((fz¯ + gz )hΩ) + 2 ? d( −1h(gz − fz¯)) √ ¯ = 2d(h(fz¯ + gz )) + 2 ? −1(∂(h(gz − fz¯)) + 2∂(h(g z − fz¯)) ¯ ¯ = 2(∂ + ∂)(h(fz¯ + gz )) + 2(∂(h(gz − fz¯)) − ∂(h(gz − fz¯)) ¯ z ) + ∂(hfz¯)) = 4(∂(hg = (hfzz¯ + hz fz¯)dz + (hgzz¯ + hz¯gz )d¯ z. R EMARK . Given a coordinate system z, let us denote by H (z) the real-valued function satisfying √ H (z) −1dz ∧ d¯ z = Ω. (ζ)

One can always choose a coordinate system ζ about a point o such that H (ζ) (o) = 1 and Hζ (o) = 0. To see this, start with any coordinate system z, and write H(z) = a + bz + bz + O(|z|2 ). Note that a ∈ R. By scaling z, we can assume a = 1. Now take z = ζ − 2b ζ 2 . Then dz = 1 − bζ, so √ √ ¯ H (z) −1dz ∧ d¯ z = (1 + bζ + bζ + ...)(1 − bζ)(1 − bζ)dζ ∧ dζ¯ = (1 + O(|ζ|2 )) −1dζ ∧ dζ. √ A coordinate system in which Ω = (1 + O(|ζ|2 )) −1dζ ∧ dζ¯ is called a normal coordinate system. In such a coordinate system, we find that a 1-form is harmonic if and only if its local component functions are harmonic to second order.

9.1.5

Regularity for the Laplace-Beltrami Operator

With the formulas for the Laplacian in hand, we can now easily prove the following result. P ROPOSITION 9.1.4 (Regularity for ∆). Let ξ ∈ E (X) be a smooth element. Then any weak solution η of the equation ∆η = ξ is also smooth. Proof. Evidently the result is local. From the formulas for the Laplacian on 0-forms, 1-forms and (1, 1)-forms, it suffices to prove the following results: given any smooth function h, any solutions f and g for the equations fz = h and gz¯ = h respectively must be smooth. Since f¯z = fz¯, it suffices to prove the regularity of solutions to the first of these two equations. Now, any two solutions differ by a holomorphic, therefore smooth, function, and thus it suffices to prove that there is one smooth solution. But such a smooth solution is easily obtained from our Cauchy-Green formula; see for example (the proof of) Proposition 5.10.2.

9.2

The Hodge decomposition of E (X)

For ease of reading, we omit the notational dependence on Ω. Let us write H (X) := {(f, α, ω) ; ∆f = 0, ∆α = 0 and ∆ω = 0}. 152

We have the decomposition H (X) = H0 (X) ⊕ H1 (X) ⊕ H(1,1) (X), with the obvious meaning for the factors.

9.2.1

The main theorem

Our goal in this section is to establish the following theorem. T HEOREM 9.2.1. There is an orthogonal decomposition E (X) = ∆(E (X)) ⊕ H (X). This decomposition respects the decomposition by degrees of forms. The proof of Theorem 9.2.1 has two parts. The first, rather trivial part, is to show that H (X) is the orthogonal complement of the closure of ∆(E (X)) in L2 (X). This is done in Lemma 9.2.3 below. The second and most complicated part of the proof is to show that ∆(E (X)) is closed in E (X). In view of Proposition 9.1.4, it suffices to find weak solutions of ∆θ = u when u is smooth. The idea for solving the latter equation is to show that the operator I + ∆ has a compact inverse, and then use the Spectral Theorem for compact self-adjoint operators to solve ∆ on the orthogonal complement of its kernel. In establishing the compactness, we need a result of Rellich from measure theory, which we prove below. Before proving Theorem 9.2.1, we state one important corollary. C OROLLARY 9.2.2. Let f : X → C be a smooth function on a compact Riemann surface. Then the equation ∆u = f has a solution if and only if Z f Ω = 0. X

Equivalently, the equation

√

¯ = ω has a solution if and only if −1∂ ∂f

R X

ω = 0.

Proof. Since every harmonic function on X is constant and every harmonic (1, 1)-form is a constant multiple of Ω, a function fR(resp. (1, 1)-form R ω) is orthogonal to the harmonic functions (resp. (1, 1)-forms) if and only if X f Ω = 0 (resp. X ω = 0). The proof is complete. R EMARK . Note the √ contrast between Corollary 9.2.2 and Theorem 8.6.3, the latter of which states ¯ = ω always has a solution on an open Riemann surface. that the equation −1∂ ∂u

9.2.2

Obstructions from the kernel of ∆

L EMMA 9.2.3. Let H(X) := E (X) ∆(E (X)). Then H(X) = H (X). 153

Proof. Consider the equation (η, ∆x) = (∆η, x). If η ∈ H (X), then the right hand side is zero, and thus by the left hand side, η ⊥ ∆(E (X)), i.e., η ∈ H(X). On the other hand, if η ⊥ ∆(E (X)) then the left hand side is zero, and thus ∆η ⊥ E (X). Since smooth objects are dense in our Hilbert space, ∆η = 0. By Lemma 9.2.3, the image ∆(E (X)) is densely contained in the subspace V of E (X) given by the closure in E (X) of the orthogonal complement of the Harmonic forms on X. Our next goal is to show that for every u ∈ V there exists θ in L2 (X) such that ∆θ = u in the weak sense, i.e., for all smooth ξ, (θ, ∆ξ) = (u, ξ). Unfortunately, although we are working in the complement of the Kernel of ∆, and we know that (∆ξ, ξ) = ||dξ||2 + ||d∗ ξ||2 ≥ 0, at this point we do not have any positive lower bounds on the smallest non-zero eigenvalue of ∆. (If we had such estimates, we could solve the equation ∆θ = u with estimates, and we shall take that approach in Chapter 11, where we solve a related equation under additional assumptions.) The trick to obtaining such a positive lower bound is to look at the operator I + ∆. As we will show, (I + ∆)−1 is compact, and thus we can get some crucial information on its eigenvalues from the Spectral Theorem. Of course, this method will not give us an estimate for the smallest positive eigenvalue, but that is not a problem for us now.

9.2.3

A compact subspace of L2 (X)

To prove the compactness of (I + ∆)−1 , we will need to bring into the picture another Hilbert space H, which we define as the closure of the set of smooth forms ξ with respect to the norm q ||ξ||H := ||ξ||2L2 + ||dξ||2L2 + ||d∗ ξ||2L2 . Since || · ||L2 ≤ || · ||H , we see that H is contained in L2 (X). But in fact more is true. We have the following well-known lemma of Rellich. L EMMA 9.2.4 (Rellich’s Compactness Lemma). Let {ξj } ⊂ L2 (X) be a sequence such that sup ||ξj ||H < +∞. j

Then there is a subsequence {ξjk } converging in L2 (X). That is to say, the inclusion H ,→ L2 (X) is a compact operator. 154

Proof. Since the smooth forms are dense, we may assume that the sequence {ξj } consists of smooth forms. Let {Ua } be a finite cover of X by coordinate charts, with subordinate partition of unity {ϕa }. It suffices to extract a convergent subsequence from {ϕa ξj } for each a. We fix a and sometimes do not refer to it in the argument. Since we are working locally, there are vector valued functions Fj (with compact support) such that Z |Fj |2 dA = ||ϕa ξj ||2L2 . R2

(Note that Fj also carries the information of the metric Ω.) A simple calculation in local coordinates shows that Z 2 ||ϕa ξj ||H ∼ (|Fj |2 + |DFj |2 )dA. R2

R EMARK . For x, y ∈ R2 , |x| ≤ |x − y| + |y| and thus |x|2 ≤ 2(|x − y|2 + |y|2 ). It follows that 1 + |x|2 ≤ 1 + 2|x − y|2 + 2|y|2 ≤ 2(1 + |y|2 )(1 + |x − y|2 ), and thus we have the inequality 1 + |x|2 ≤ 2(1 + |x − y|2 ). 1 + |y|2

(9.1)

We work with the Fourier transform Fˆj (ξ) :=

Z

√

e−

−1hξ,xi

Fj (x)dA(x).

R2

Letting ϕ ∈ C ∞ (X) be a smooth function supported in the coordinate neighborhood in question and identically 1 on the support of Fj , we have Fj = ϕFj , and by thus by (9.1) and the product formula for Fourier transforms we have Z p p p 1 + |ξ|2 |Fˆj (ξ)| . 1 + |η|2 |Fˆj (η)| 1 + |ξ − η|2 |ϕ(ξ ˆ − η)|dA(η) R2

and Z p p 2 ˆ |Fˆj (η)| 1 + |ξ|2 |Dϕ(ξ 1 + |ξ| |DFj (ξ)| ≤ ˆ − η)|dA(η) 2 R Z p p ˆ . 1 + |η|2 |Fˆj (η)| 1 + |ξ − η|2 |Dϕ(ξ − η)|dA(η) R2

It follows that (1 + |ξ| )(|Fˆj (ξ)|2 + |DFˆj (ξ)|2 ) . 2

Z

(1 + |η|2 )|Fˆj (η)|2 dA(η)

R2

Z =

(|Fj |2 + |DFj |2 )dA

R2

∼ ||ϕa ξj ||2H , 155

where the last equality follows from the Plancherel Theorem and standard properties of Fourier transforms. These uniform C 1 -bounds on the Fourier transforms of Fj combine with the AscoliArzela Compactness Theorem to produce a locally uniformly convergent subsequence {Fˆjk }. Moreover, by Plancherel’s Theorem we have Z 2 |Fˆjk − Fˆj` |2 dA(ξ) ||Fjk − Fj` || = 2 ZR ≤ |Fˆjk − Fˆj` |2 dA(ξ) |ξ|R Z ||ϕa ξjk ||2H + ||ϕa ξj` ||2H |Fˆjk − Fˆj` |2 dA(ξ) + ≤ . 1 + R2 |ξ|> 0 by the locally uniform convergence of {Fˆjk }, while the second term can be made as small as we like by choosing R sufficiently large. Thus {Fjk } is a Cauchy sequence, and the proof is complete.

9.2.4

The Spectral Theorem for Compact Self-adjoint Operators

In the proof of the Hodge Decomposition Theorem we will make use of the following result from functional analysis. T HEOREM 9.2.5 (Spectral Theorem for Compact Self-adjoint Operators). Let T : H → H be an injective, compact, self-adjoint operator on a separable Hilbert space H. Then there is a discrete sequence of positive numbers λm & 0 such that the vector spaces Hm (T ) := {θ ∈ H ; T θ = λm θ} are finite-dimensional, and we have the orthogonal Hilbert space decomposition H=

∞ M

Hm (T ).

m=1

A proof of the Spectral Theorem can be found in [Rudin-1991].

9.2.5

Completion of the proof of Theorem 9.2.1

By Lemma 9.2.3 it suffices to show that given u ⊥ H (X) there exists θ ∈ E (X) such that ∆θ = u. By Proposition 9.1.4 it suffices to find a weak solution, which is what we now do. L EMMA 9.2.6. The operator I + ∆ has a self-adjoint extension to the dense subset H of L2 (X). 156

Proof. Consider the quadratic form Q(θ, θ) := ||θ||2H defined on H, which we take to be its domain. (Note that on smooth forms θ, Q(θ, θ) = ((I + ∆)θ, θ)L2 .) Clearly Q(θ, θ) ≥ ||θ||2L2 for all θ ∈ H. Now, the inequality |(ξ, η)L2 |2 ≤ ||ξ||2L2 ||η||2L2 ≤ ||ξ||2L2 Q(η, η) shows that, for ξ ∈ L2 (X), the linear functional η 7→ (ξ, η)L2 is continuous on H. By the Riesz Representation Theorem there is an element T ξ ∈ H such that Q(T ξ, η) = (ξ, η)L2 . The operator T : L2 (X) → H satisfies the estimate ||T ξ||2L2 ≤ Q(T ξ, T ξ) = (ξ, T ξ)L2 ≤ ||ξ||L2 ||T ξ||L2 , and thus it is bounded. Moreover, if T ξ = 0 then 0 = Q(T ξ, ξ) = ||ξ||2L2 , so that T is injective. We define F := T −1 , and clearly F |E (X) = I + ∆. Moreover, (F ξ, η)L2 = (T −1 ξ, η)L2 = Q(ξ, η) = Q(η, ξ) = (T −1 η, ξ)L2 = (F η, ξ)L2 = (ξ, F η)L2 , so that F is self-adjoint on its domain H. L EMMA 9.2.7. Let ι : H ,→ L2 (X) be the inclusion. The operator G := ιT : L2 (X) → L2 (X) is compact. Proof. We first observe that for ξ ∈ L2 (X), ||T ξ||2H = Q(T ξ, T ξ) = (ξ, T ξ)L2 ≤ ||ξ||L2 ||T ξ||L2 ≤ ||ξ||H ||T ξ||H ,

(9.2)

which shows that T defines a bounded linear operator from L2 (X) to H. Composing with the natural inclusion ι : H ,→ L2 (X), the result follows from Rellich’s Compactness Lemma and the fact that a composition of a bounded and a compact operator is compact. 157

End of the proof of Theorem 9.2.1. The estimate (9.2) shows that the eigenvalues of G are no more than 1. Moreover, since F is positive semi-definite and self-adjoint, so is G, and thus all its eigenvalues are positive. By the Spectral Theorem there is a discrete set of numbers 1 = λ1 > λ2 > .... } such that λm → 0 and the eigenspaces Km (X) := {θ ∈ E (X) ; Gθ = λm θ} are finite dimensional and provide a decomposition of L2 (X). Observe that for any θ ∈ Km (X), ∆θ = (λ−1 m − 1)θ =

1 − λm θ. λm

In particular, K1 (X) = H (X). R EMARK . It follows immediately that the elements of K1 (X) are smooth. It turns out that all of the vectors in each Km (X) are smooth, a fact we will not use. Let us fix orthonormal bases (m)

{θj

; 1 ≤ j ≤ Nm } ⊂ Km (X),

m = 1, 2, ... .

Let u ∈ V . Decompose u orthogonally in L2 (X) as u=

∞ X m X

(m)

cjm θj

m=2 j=1

(There is no m = 1 term because u ⊥ H (X).) Let ∞ X Nm X λm cjm (m) θ := θ . 1 − λm j m=2 j=1

Then the inequality |λm | 1 ≤ , |1 − λm | 1 − λ2

m≥2

show that θ is convergent and ||θ||L2 ≤

1 ||u||L2 . 1 − λ2

Next, ∞ X Nm ∞ X Nm X X λm cjm (m) (m) ∆θ = ∆θj = cjm θj = u, 1 − λm m=2 j=1 m=2 j=1

and thus we have found our weak solution θ. The proof of Theorem 9.2.1 is complete. 158

9.3

Arithmetic and geometric genus

Observe that a consequence of the Hodge Theorem is the finite-dimensionality of H (X). In fact, we can establish the finite-dimensionality of H (X) using a result we proved in Chapter 4, namely, Theorem 4.8.1. Precisely, we have the following result. T HEOREM 9.3.1. Let X be a compact Riemann surface. Then we have the orthogonal decomposition (9.3) H1 (X) = ΓO (X, KX ) ⊕ ΓO (X, KX ). In particular, H (X) is finite dimensional. Proof. We know that H0 (X) and H(1,1) (X) are 1-dimensional. Since ΓO (X, KX ) is finite dimensional by Theorem 4.8.1, the right hand side of (9.3) is finite dimensional, and in fact has dimension 2gO(X) ,where gO(X) is the arithmetic genus of X. We now turn to the proof of (9.3). It is clear that if α and β are holomorphic 1-forms, then α + β¯ is harmonic. Moreover, the orthogonality of the two summands in the right hand side of the decomposition is clear from the definition of the ? operator. Conversely, suppose σ ∈ H1 (X). First, (∆σ, σ) = |dσ|2 + |d∗ σ|2 , and thus harmonic forms are precisely those forms that are closed and annihilated by d∗ . Next, let σ := σ1,0 + σ0,1 be the unique decomposition into (1, 0) and (0, 1) forms. The equation dσ = 0 means ¯ 1,0 + ∂σ0,1 = 0, ∂σ while the equation d∗ σ = 0 means ¯ 0,1 . ∂σ1,0 = ∂σ But together these two equations imply that ¯ 1,0 = ∂σ0,1 = 0, ∂σ which simply says that σ1,0 and σ0,1 are holomorphic. The proof is complete. Recall that the deRham cohomology group is 1 HdR (X) :=

1,1 Kernel(d : Γ(X, Λ1X ) → Γ(X, ΛX )) . ∞ dC (X)

1 T HEOREM 9.3.2. In each class [α] ∈ HdR (X) there is exactly one harmonic 1-form.

Proof. Since a form α is harmonic if and only if dα = 0 and 159

d∗ α = 0,

every harmonic 1-form is closed. Next, we define the functional N (β) := ||β||2 . on smooth 1-forms β. Among all smooth β ∈ [α], we shall try to minimize N (β). Note that if we do find a minimizer β of N , then for all ε > 0 0 ≤ N (β + εdf ) − N (β) = 2Re (β, df )ε + O(ε2 ) and thus applying this estimate to real and imaginary valued f , we find that d∗ β = 0. Since dβ = 0 by hypothesis, the form β is harmonic, and thus smooth. Next, suppose a minimizer exists. Then it is unique. Indeed, if β1 and β2 are minimizers, then β2 = β1 + df and we have ||β2 ||2 = ||β1 ||2 + ||df ||2 + 2Re (β1 , df ) = ||β1 ||2 + ||df ||2 ≥ ||β1 ||2 , so equality holds if and only if df = 0. The above calculations show that the minimizer of N is orthogonal to the collection of exact 1-forms. However, we cannot yet guarantee that this minimizer exists, since the inner product space ([α], || · ||) is not necessarily closed. This problem is easily surmounted by simply seeking the minimizer β in the Hilbert space closure H of ([α], || · ||). Such a minimizer must be orthogonal to the subspace of all exact 1-forms, and thus orthogonal to its closure. Thus the minimizer is harmonic in the sense of distributions. By Proposition 9.1.4 we can conclude that harmonic in the sense of currents are automatically smooth. Thus our minimizer is smooth, and the proof is complete. C OROLLARY 9.3.3. The arithmetic genus gO (X) of a compact Riemann surface X agrees with the geometric genus of X, i.e., the number of handles of X. Proof. Every handle on X yields exactly two homology classes of closed loops in X. Thus we have 1 dimR H1 (X) = 2g. By Lemma 5.9.4, dimR HdR (X) = 2g. By Theorem 9.3.2, dimR H1 (X) = 2g. But by Theorem 9.3.1, dimR H1 (X) = 2gO (X). It follows that g = gO (X), and the proof is complete.

9.4

Existence of positive line bundles

In this section we prove, in several ways, that every Riemann surface admits a line bundle that has a metric of positive curvature. In the next chapter we will give yet another proof using the uniformization theorem. The existence of positive line bundles is fundamental in Chapter 12, where we prove that every compact Riemann surface embeds in P3 and that every open Riemann surface embeds in C3 . 160

9.4.1

Compact Riemann surfaces I: Using Hodge Theory

P ROPOSITION 9.4.1. Let X be a compact Riemann surface. Then√there is a holomorphic line ¯ is a strictly positive bundle L → X and a smooth Hermitian metric e−ϕ for L such that −1∂ ∂ϕ (1, 1)-form. In fact, we will prove the following, more general result. P ROPOSITION 9.4.2. Let L be a holomorphic line bundle whose chern number c(L) is positive. ¯ is strictly positive. Then there is a smooth metric e−ϕ for L whose curvature ∂ ∂ϕ Proof. Let Ω be an area form, i.e., an everywhere strictly positive (1, 1)-form, and normalize Ω so that Z Ω = 1. X

Fix any smooth metric e−ϕo for L. By Theorem 6.1.18, √

−1 2π

Z

¯ o = c(L). ∂ ∂ϕ

X

√

It follows that the (1, 1)-form η :=

−1 ¯ ∂ ∂ϕo 2π

− c(L)Ω satisfies

Z η = 0. X

By Corollary 9.2.2 there is a smooth function u : X → R such that √ −1 ¯ ∂ ∂u = η. 2π ¯ satisfying The metric eϕ := eu−ϕo therefore has curvature ∂ ∂ϕ √

−1 ¯ ∂ ∂ϕ = c(L)Ω. 2π

The proof is complete. In fact, the Hodge Decomposition Theorem is far too powerful a tool if all one wants is a positive line bundle. In the next two paragraphs we present two constructions of metrics of positive curvature. The first construction, mostly analytic and rather local in its nature, was communicated to us by M. Ramachandran. The second construction is based more on basic calculus, but gives a somewhat less precise result than Proposition 9.4.1 161

9.4.2

Compact Riemann surfaces II: An ad hoc technique

Our construction of positive line bundles on compact Riemann surfaces uses the very powerful Hodge Theorem. If one wants to avoid using the Hodge Theorem, there are a number of other ways to proceed. In this section, we take a particular ad hoc approach that provides us with a less precise conclusion. One might try to construct a metric of positive curvature as follows. Let ω be a strictly positive (1, 1)-form on X such that Z ω = 2π. X

Take a cover {Ui } of X by connected, simply connected coordinate neighborhoods (with coordinate functions zj ) such that the intersections Uij := Ui ∩ Uj and Uijk = Ui ∩ Uj ∩ Uk are also simply connected. On each Ui , we have √ zi . ω = e−ψi (zi ) −1dzi ∧ d¯ We define the function 1 ϕi (zi ) := 2π Note that

Z

√ ¯ log |zi − ζ|2 e−ψi (ζ) −1dζ ∧ dζ.

Ui

√

¯ i=ω −1∂ ∂ϕ

is independent of i, and in particular, that ϕi −ϕj is harmonic. It follows by the simple connectivity of Uij that there is a holomorphic function gij such that ϕi − ϕj = log |gij |2 . In fact, while the squared magnitude of gij is |gij |2 = eϕj −ϕj , the argument of gij can be defined by Z

z

arg gij = 2

dc (ϕi − ϕj ),

vij

where vij ∈ Uij is a chosen point. (By convention we choose vji = vij .) The choice of the curve connecting vij to z is not important, except to declare that the curve remain in Uij which is simply connected (hence the independence on the curve). So let us fix one such curve, Γij (z). We remark only that if we change the point vij , then the function arg gij changes by an additive constant that depends only on i and j. Now, since we have taken ( ) Z z √ gij := exp ϕi − ϕj + 2 −1 dc ϕi − dc ϕj , vij

162

we have gij gjk gki √ = exp 2 −1

(Z

z

dc ϕi − dc ϕj +

Z

vij

z

dc ϕj − dc ϕk +

vjk

Z

)!

z

dc ϕk − dc ϕi

.

vki

We note that in fact gij gjk gki is constant. Indeed, it is holomorphic and of constant magnitude 1. It follows that √ gij gjk gki = e −1θijk for some collection {θijk }i,j,k∈A ⊂ R that is skew symmetric in the indices ijk. R EMARK . Our next goal would be to show that there is are sets {ηij = −ηji }i,j∈A ⊂ R and {mijk }i,j,k∈A ⊂ Z skew symmetric such that Z θijk = ηij + ηjk + ηki + mijk ω. (9.4) X

While (9.4) is true, I could not come up with a direct elementary proof. Nevertheless, we can still get somewhere. After a slight perturbation of the points vij , vik and vjk , we may assume that the numbers θijk are a rational multiple of 2π. (This assertion can be proved rigorously by employing the Implicit Function Theorem.) We take an integer m so that mθijk is an integer multiple of 2π for all i, j, k. Setting g˜ij = gijm , we find that

√

g˜ij g˜jk g˜ki = e

−1mθijk

= 1.

Thus {˜ gij } satisfies the cocycle condition, and hence defines a line bundle H → X. And since e−ϕi |gij |2 = e−ϕj , we see that e−ϕ := {e−mϕi } defines a smooth Hermitian metric for H. Moreover, by construction the curvature current of e−ϕ is √ ¯ = mω, −1∂ ∂ϕ which is strictly positive. Of course, our Hodge-theoretic method tells us that we can find a holomorphic line bundle with a metric whose curvature is ω.

9.4.3

Open Riemann surfaces

In this paragraph we prove that the trivial bundle of an open Riemann surface admits a metric of strictly positive curvature. Of course, since we are dealing with the trivial bundle, our claim is that on an open Riemann surface there is a strictly subharmonic function. In fact, we will prove more. 163

D EFINITION 9.4.3. 1. A continuous map f : A → B of topological spaces is said to be proper if for each compact subset K ⊂ B, f −1 (K) is compact in A. 2. For an interval (−∞, a) ⊂ R, a proper map f : A → (−∞, a) is called an exhaustion of A. T HEOREM 9.4.4 (Behnke-Stein). An open Riemann surface has a strictly subharmonic exhaustion. Proof. Let Y1 ⊂⊂ Y2 ⊂⊂ ... ⊂ X be a normal exhaustion. Choose open sets U1 ⊂ U2 ⊂ ... such that Yj ⊂ Uj ⊂⊂ Yj+1 for all j ≥ 1. For each j, choose holomorphic functions fj1 , ..., fjNj such that 1. supYj |fjk | < 1, 2. for each z ∈ Yj+2 − Uj , maxk |fjk (z)| > 1, and 3. for each z ∈ Yj there exists k such that fjk (z) 6= 0 and dfjk (z) 6= 0. Properties 1, 2 and 3 are easily achieved using the Runge Approximation Theorem. By taking powers of the fjk if necessary, we can guarantee that Nj X

2

−j

|fjk (z)| < 2 ,

z ∈ Yj

and

Nj X

|fjk (z)|2 > j,

z ∈ Yj+2 − Uj .

k=1

k=1

It follows that the function ϕ :=

Nj ∞ X X

|fjk |2

j=1 k=1

converges to a smooth function (in fact, real analytic) on X and satisfies ϕ>j

on X − Uj .

Moreover, for any tangent vector ξ ∈ TX,z , ¯ ¯ = ∂ ∂ϕ(z)(ξ, ξ)

X

|∂fjk (z)(ξ)|2 > 0

j,k

by condition 3 above. Thus ϕ is strictly subharmonic. The proof is finished.

9.5

Proof of the Dolbeault-Serre Isomorphism

In this brief section we prove Theorem 5.11.1. Let α be a d-closed (0, 1)-form on a compact Riemann surface. Then √ √ d∗ α = ?d( −1 α) = ?( −1 dα) = 0, and thus α is harmonic. 164

Next we claim that in every Dolbeault class there is a d-closed form. To this end, let α be a smooth (0, 1)-form. Suppose α is not d-closed. We seek a d-closed (0, 1)-form β and a smooth function f such that ¯ β − α = ∂f. Now, consider the (1, 1)-form dα = ∂α. By Stokes’ Theorem, Z dα = 0. X

It follows from Corollary 9.2.2 that there is a function f such that ¯ = ∂α. −∂ ∂f Letting ¯ β := α + ∂f, we have ¯ = 0. dβ = ∂β = ∂α + ∂ ∂f To complete the proof of Theorem 5.11.1, we must show that every Dolbeault class contains ¯ , then exactly one d-closed form. But if α1 and α2 are d-closed 1-forms with α2 − α1 = ∂f ¯ 0 = d(α2 − α1 ) = ∂ ∂f. Thus f is harmonic, hence constant, so α1 = α2 . The proof of Theorem 5.11.1 is now complete.

165

166

Chapter 10 Uniformization 10.1

Automorphisms of the plane and the disk

10.1.1

Aut(C)

We begin with the following result. P ROPOSITION 10.1.1. Any injective holomorphic map f : C → C is affine linear. Proof. We look at the singularity at ∞. If this singularity is a pole, then f extends to a rational mapping of P1 to itself, and being holomorphic, f must be a polynomial. By the fundamental theorem of algebra, f is affine linear. We claim that f cannot have an essential singularity. Indeed, if it does, then the image of any of the sets AR := {|z| > R} is open (since by the maximum principle holomorphic functions are open maps) and dense by the remark following Theorem 1.4.5. Fix zo ∈ C and let a = f (zo ). Then the image under f of some neighborhood |z − zo | < ε is a neighborhood of a in C. Take AR with R > ε + |zo |. Since f (AR ) is open and dense, it meets f ({|z − zo | < ε}). It follows that f cannot be injective. The proof is complete. C OROLLARY 10.1.2. Every holomorphic diffeomorphism f : C → C is affine linear.

10.1.2

Aut(D)

T HEOREM 10.1.3. Every holomorphic diffeomorphism f : D → D is of the form √

f (z) = e

−1θ

b−z 1 − ¯bz

for some θ ∈ [0, 2π) and b ∈ D. Proof. Fix f ∈ Aut(D), and write a = f (0). Consider the map g(w) := the origin and satisfies (1 − |w|2 )(1 − |a|2 ) 1 − |g(w)|2 = . |1 − a ¯w|2 167

a−w , 1−¯ aw

which maps a to

It follows that g fixes both the unit disk and its boundary, and since g(g(w)) = w, g is invertible. Now, g◦f is an automorphism of the unit disk √ √ fixing the origin. By the Schwarz Lemma, g◦f (z) = −1θ − −1θ e z for some θ ∈ R. Thus with b = e a, f has the desired form.

10.2

A review of covering spaces

10.2.1

Definition

D EFINITION 10.2.1. A covering space of V is a space U , together with a continuous surjective map F : U → V such that for each v ∈ V there is a neighborhood W of v in V such that F −1 (W ) is a disjoint union of sets Uα , α ∈ A each of which is homeomorphic, via F , to W . R EMARK . Note that if F : U → V is a covering space, then F is a surjective local homeomorphism. However, the converse is not true. There are many examples of local homeomorphisms that are not uniform, like a covering map is. In fact, there are so many examples that triples (F, U, V ) with F : U → V a surjective local homeomorphism go by a special name: we call U a sheaf over V . We will not discuss sheaves in this book.

10.2.2

Curve lifting property

Covering spaces have a curve lifting property: given a path γ : [0, 1] → V and a point p ∈ F −1 (γ(0)), there is a unique path γ˜ : [0, 1] → U such that γ˜ (0) = p and F ◦ γ˜ = γ. The existence of the lift is established by successively lifting neighboring small pieces of the curve using the local isomorphism property of the covering map. Using the curve lifting property, one can show that, more generally, any map g : S → U from a simply connected space has a lift g˜ : S → V , i.e., g = F ◦ g˜. Even more generally, a map h : W → U has a lift to V if and only if h∗ π1 (W ) < F∗ π1 (U ). Two covers F1 : U1 → V and F2 : U2 → V are said to be isomorphic if there is an isomorphism (in whatever category) G : U1 → U2 such that F1 = F2 ◦ G.

10.2.3

Universal cover

There exists a universal covering, i.e., a simply connected space U0 and a covering map F0 : U0 → V . Moreover, U0 is unique up to isomorphism. The word universal refers to the following universal property: If F : U → V is any other cover, then F factors through F0 uniquely, in the sense that there is a unique cover G : U0 → U such that F0 = F ◦ G. The fundamental group π1 (V, q) acts on the universal cover F0 : U0 → V as follows. Fix a point p ∈ F −1 (q). Choose a point u ∈ U0 and a loop γ in V with base point q. We shall now define [γ]u. Fix a path α from p to u. Its projection under F0 is a path F0 ◦ α starting at q and ending at F0 (u). Now let γ˜ be the unique lift of γ starting at p, and let β be the unique lift of F0 ◦ α starting at γ˜ (1). We define [γ]u := β(1). 168

It is not hard to see that the point β(1) does not depend on the choices, but only on the point u and the homotopy class [γ] of γ. (The independence on α is due to simple connectivity, and the lifts are unique.) This gives an action of π1 (V, q) on U0 , and the action preserves the fibers of the covering map F0 . Moreover, the orbit space U0 /π1 (V, q) is naturally isomorphic to the original space V .

10.2.4

Intermediate covers

Given any subgroup H < π1 (V, q), the action defined above on U0 , when restricted to H, defines an action of H on the universal cover. The orbit space U0 /H is itself a covering space of V , and two such covers are isomorphic if and only if the corresponding subgroups are conjugate. Thus there is a 1-1 correspondence between isomorphism classes of connected coverings and conjugacy classes of subgroups of π1 (V, q). To see how the reverse of this correspondence works, let F : U → V be a covering, and fix a point p ∈ F −1 (U ). We define H := {[γ] ∈ π1 (V, q) ; [γ]p = p} = (π1 (V, q))p to be the stabilizer at p. Since the orbit of π1 (V, q) is the whole fiber, different choices of p will yield different but conjugate subgroups. D EFINITION 10.2.2. The degree of a covering F : U → V is the index [π1 (V, q) : H] of the subgroup H < π1 (V, q) corresponding to the cover F . E XAMPLE 10.2.3. 1. Consider the torus X = C/L obtained from C after taking a quotient by a lattice. The natural quotient map π : C → X is the universal cover. The fundamental group of X is the free Abelian group on two generators, and it is isomorphic to the lattice L. (The generating loops are the two (non-parallel) edges of a fundamental parallelogram.) Any proper subgroup of L is isomorphic either to a (sub)lattice L0 or to Z. The corresponding quotients are tori C/L0 or C∗ . The degree of the cover of the former is just the degree of the induced map F : C/L0 → C/L and the degree of the cover C∗ → X is infinite. 2. Suppose we begin with the upper half plane H = {z ∈ C ; Im z > 0}. The map z 7→ e2πiz maps H surjectively onto the punctured unit disk, and it is easy to check that this map is a covering map. This exponential map is the projection for the quotient by the group of translations by (real) integers. Any subgroup is cyclic, and generated by some translation z 7→ √ z + N . The resulting quotient is still a punctured disk, and the quotient map is z 7→ 2π −1z/N . Thus this punctured disk is just an N -fold cover of the original punctured disk. e √ Indeed, (e2π −1z/N )N = e2πiz .

10.3

The Uniformization Theorem

10.3.1

Statement

T HEOREM 10.3.1 (Uniformization). Let X be a connected Riemann surface. The universal cover of X is exactly one of the following Riemann surfaces: 1. The Riemann sphere P1 = C ∪ {∞} 169

2. The complex plane C 3. The unit disk D Before turning to the proof, we will discuss some of the implications of the Uniformization Theorem.

10.3.2

Potential-theoretic classification of Riemann surfaces

There are several ways to classify Riemann surfaces. For compact Riemann surfaces it is natural to use the genus of the surface as an invariant, as we shall see later on. For more general Riemann surfaces, one of the earliest classification of Riemann surfaces is in terms of their subharmonic functions. D EFINITION 10.3.2. A Riemann surface is said to be 1. potential-theoretically hyperbolic if it has a non-constant bounded subharmonic function; 2. potential-theoretically elliptic if it is compact; 3. potential-theoretically parabolic otherwise. R EMARK . We call this classification potential-theoretic because the condition of having a bounded subharmonic function is equivalent to the existence of a Green’s Function. The Riemann sphere is clearly elliptic. The unit disk is hyperbolic, as the function |z|2 shows. On the other hand, the complex plane is parabolic. This parabolicity can be shown by a Liouvilletype theorem for subharmonic functions, but we will show it in a slightly different way. P ROPOSITION 10.3.3. Let Y be a compact Riemann surface, and p1 , ..., pk ∈ Y distinct points. Then the surface X = Y − {p1 , ..., pk } is potential-theoretically parabolic. Proof. Suppose f is a bounded subharmonic function on X. Let f˜ : Y → R ∪ {−∞} be an extension of f defined by f˜(pk ) := lim sup f (z). z→pk

Observe that f˜ has the same bound as f . Since ∆f ≥ 0 in the sense of distributions on X and f˜ is bounded on Y , ∆f˜ ≥ 0 in the sense of distributions on Y . But then f˜ is subharmonic on Y , hence constant. Thus f is constant, and hence X is potential-theoretically parabolic.

10.3.3

Riemann surfaces covered by P1

Observe that since P1 is compact, any surface covered by P1 is also compact. It follows from the Riemann-Hurwitz Formula that the only Riemann surface covered by P1 is itself P1 . We could also argue as follows. Suppose P1 covers a compact Riemann surface X. Then the arithmetic genus of X must be zero, for otherwise we could pull back a holomorphic form from X to P1 , and the latter has no holomorphic forms. But then it follows that X is simply connected. Thus the covering map is an isomorphism. 170

10.3.4

Riemann surfaces covered by C

Observe that if a Riemann surface X is hyperbolic, it cannot be covered by C. Indeed, we could then pull back a bounded subharmonic function by the covering map and obtain a contradiction. Thus any Riemann surface covered by C is either parabolic or elliptic. The converse, however, is not true. In fact, we have the following result. P ROPOSITION 10.3.4. The only Riemann surfaces covered by C are 1. C, 2. C − {0}, and 3. the complex tori C/Λ Proof. Let X be covered by C. Then the group Γ ⊂ Aut(C) of deck transformations is a discrete subgroup of C. It follows that if the group is non-trivial then it is generated by either one or two independent translations. In the latter case the deck group is a lattice, and we obtain a torus. On the other hand, if the group is generated by one translation, then we can conjugate the group to the √group of translation {z 7→ z + n ; n ∈ Z}. Then the map π : C → C − {0} given by π(z) = e2π −1z is the covering map. The proof is complete.

10.3.5

Poincaré classification

Thus we come to another type of classification of Riemann surfaces. D EFINITION 10.3.5. A Riemann surface is said to be 1. Poincaré hyperbolic if it is covered by the unit disk, 2. Poincaré parabolic if it is covered by the complex plane, and 3. Poincaré elliptic if it is P1 . In fact, this classification is geometric. We begin with the following proposition. P ROPOSITION 10.3.6. If a Riemann surface is Poincaré hyperbolic (resp. parabolic, elliptic) then its holomorphic tangent bundle has a metric of constant negative (resp. zero, constant positive) curvature. Let us start with the unit disk. On it, one has the metric gD , the so-called Poincaré metric, defined by dz · d¯ z gD := . (1 − |z|2 )2 171

The curvature of gD is √

−1∂ ∂¯ log(1 − |z|2 )2 √ −2zd¯ z = −1∂ 1 − |z|2 √ dz ∧ d¯ z |z|2 dz ∧ d¯ z = −2 −1 − 1 − |z|2 (1 − |z|2 )2 √ −1 ωD , = −4 · 2

ΘgD :=

where ωD is the area (1, 1)-form associated to gD . Thus D has a metric of constant negative curvature. P ROPOSITION 10.3.7. Every automorphism of D is an isometry of gD . Proof. Fix f ∈ Aut(D). Recall that f is the composition of a rotation and the map g(w) := for some a. As we already observed, 1 − |g(w)|2 = g 0 (w) = − and thus

(1−|w|2 )(1−|a|2 ) . |1−¯ aw|2

a−w 1−¯ aw

Moreover,

1 − |a|2 , (1 − a ¯w)2

(1 − |a|2 )2 1 |g 0 (w)|2 = = . 2 2 2 2 4 (1 − |g(w)| ) (1 − |g(w)| ) |1 − a ¯w| (1 − |w|2 )2

Thus g is an isometry of the Poincaré metric. Since a rotation clearly preserves the Poincaré metric, the proof is complete. Now, since the deck group of any Riemann surface X covered by the disk is a group of automorphisms of the unit disk, and hence by Proposition 10.3.7 a group of isometries of the Poincaré metric, the latter metric descends to a metric of gX , also called the Poincaré metric. The definition 1,0 of gX is as follows: Let π : D → X denote the covering map and fix v ∈ TX,x . Since π is locally −1 invertible, we can select a local inverse π defined on a small neighborhood of x. Then we define gX (v, v¯) =

|dπ −1 (v)|2 . (1 − |π −1 (x)|2 )2

By covering theory, any two branches of π −1 are conjugate by a deck transformation, which is an automorphism of D, and hence an isometry of gD . Thus gX is well-defined. It is now clear that √ − −1∂ ∂¯ log gX = −4ωX where ωX is the (1,1)-form associated to gX . Indeed, √ √ −π ∗ −1∂ ∂¯ log gX = − −1∂ ∂¯ log π ∗ gX √ = − −1∂ ∂¯ log gD = −4ωD = −4π ∗ ωX , 172

and since π is locally invertible, thus an immersion, we have our claim. Next we turn to the complex plane. The Euclidean metric |dz|2 has zero curvature. Moreover, since the deck group of any Riemann surface covered by the plane consists of translations, which are isometries of the Euclidean metric, we can push the Euclidean metric down to any such surface. Finally, we deal with the Riemann sphere. Consider the open cover U0 , U1 by affine charts. Then z0 = z11 in U0 ∩ U1 . We define the metric gF S = Observe that

|dzj |2 (1 + |zj |2 )2

on Uj ,

j = 0, 1.

|dz1 |2 |dz0 |2 1 |dz1 |2 = = , (1 + |z0 |2 )2 |z1 |4 (1 + |z1 |−2 )2 (1 + |z1 |2 )2

so that gF S is a well-defined metric for the holomorphic tangent bundle on P1 . This metric is called the Fubini-Study metric. The curvature of the Fubini-Study metric is √ √ √ z d¯ z zj −1 dzj ∧ d¯ j j −1∂ ∂¯ log(1 + |zj |2 ) = −1∂ =2· = 2ωF S , 2 1 + |zj | 2 (1 + |zj |2 )2 and thus we have proved Proposition 10.3.6. R EMARK . On P1 , the hyperplane line bundle H → P1 and the tangent bundle TP1 have metrics, both of which are called the Fubini-Study metric. I have found that this fact causes confusion in discussions; Riemannian geometers tend to think of metric as “metric for the tangent bundle”, while complex geometers also deal with metrics for line bundles, which often arise as weights in some L2 -theory (see, for example, Chapter 11). The relationship between these metrics is the following: the curvature of the Fubini-Study metric for H → P1 is a positive multiple of the metric form associated to the Fubini-Study metric for TP1 → P1 .

10.3.6

Existence of Positive line bundles: another approach

Modulo the proof of the Uniformization Theorem, the work of the previous section almost gives a second proof of the existence, on any Riemann surface, of line bundles with metrics having strictly positive curvature. T HEOREM 10.3.8. Every Riemann surface admits a positive holomorphic Hermitian line bundle. Proof. Indeed, if X is covered by the disk then its canonical bundle KX , being the dual of the holomorphic tangent bundle, is positively curved. For C and C − {0}, the restriction of the FubiniStudy metric provides the desired metric. The only Riemann surfaces we have not handled are tori. In fact, the canonical bundle of any complex torus X is trivial, since one can use the form dπ, where π : C → X is the universal cover. (Recall that π is multi-valued, but that dπ is a 173

well-defined, nowhere zero differential form on X.) It follows that a metric for KX is in (1, 1)correspondence with functions of the form e−ϕ , the correspondence being −ϕ

e

e−ϕ 7 → . |dπ|2

Thus a metric is strictly positively curved on X if and only if ϕ is strictly plurisubharmonic. Since X is compact, and thus has no non-constant subharmonic functions, neither the canonical nor the anti-canonical bundle admits a metric of positive curvature. We have to go elsewhere for such a metric. There are several ways to construct positive line bundles on complex tori, and we have already seen two such ways in Chapter 9. Here we take yet another approach. Recall that in Chapter 3 we showed, using θ-functions, that on every complex torus X there exists a non-constant meromorphic function, i.e., a holomorphic map f : X → P1 . The map f certainly has branching, since otherwise it would define a cover of P1 , and consequently an isomorphism of X and P1 , which is topologically impossible. Let D be the divisor of zeros of f , and D0 the divisor of poles of f . Then D and D0 are effective divisors defining the same line bundle L on X, and whose supports have empty intersection. Let s and s0 be the global sections of L whose zero divisors are D and D0 respectively. These sections are determined up to a constant, and in fact s/s0 is a constant multiple of f . We now define a Hermitian form for L as follows: if v ∈ Lx , then h(v, v¯) :=

|v|2 . |s(x)|2 + |s0 (x)|2

In other words, locally h = e−ϕ with ϕ = log(|s|2 + |s0 |2 ). We then find that √

¯ = −1∂ ∂ϕ =

√ √

−1∂ −1

sds + s0 ds0 |s|2 + |s0 |2

sds + s¯0 ds0 ) ∧ sds + s0 ds0 ds ∧ ds + ds0 ∧ ds0 (¯ + |s|2 + |s0 |2 (|s|2 + |s0 |2 )2

!

Letting F = (s, s0 ), we see that for a (1, 0)-tangent vector ξ, √

This shows that

2 2 2 ¯ ¯ = |F | |dF (ξ)| − |F · dF (ξ)| . −1∂ ∂ϕ(ξ, ξ) |F |4

√ ¯ is strictly positive away from the set −1∂ ∂ϕ B := {x ∈ X ; for some ξ ∈ TX,x , dF (ξ) is proportional to F }.

Since s/s0 = cf for some constant c, we see that B is exactly the branching locus of f . In particular, B is non-empty. 174

R EMARK . The above conclusion is not surprising: we leave it to the reader to check that L is the pullback by f of the hyperplane line bundle on P1 , and h is the metric on X obtained by pulling back the Fubini-Study metric for the hyperplane line bundle on P1 . To deal with the absence of strict positivity along B, we choose a second meromorphic function whose branching locus is different from B. We could appeal to θ-functions to construct such a function, but instead we exploit the group property of tori: we compose our function f with a translation [z] 7→ [z + ε] arbitrarily close to the identity, to obtain a new function fε . The branching locus of this function is clearly Bε = {b − ε ; b ∈ B}. If ε is a sufficiently small complex number then B ∩ Bε = ∅. We now get a new line bundle Lε associated with the meromorphic function fε as above, and with metric hε = e−ϕε whose curvature is strictly positive away from Bε . R EMARK . We will see in Chapter 14 that, for small but non-zero ε, the line bundle Lε is not isomorphic to L as a holomorphic line bundle, although it is isomorphic to L as a complex line bundle. Now we take the line bundle Λ := L ⊗ Lε with the metric e−(ϕ+ϕε ) . Evidently this metric has curvature √ √ ¯ + −1∂ ∂ϕ ¯ ε −1∂ ∂ϕ which is everywhere positive. The proof of Theorem 10.3.8 is complete.

10.4

Proof of The Uniformization Theorem

L EMMA 10.4.1. Let X be a Riemann surface and Y ⊂⊂ X a connected, simply connected domain with non-empty smooth boundary. Then Y is biholomorphic to the unit disk. Proof. Let p ∈ Y . Consider the line bundle L associated to the divisor 1 · p. Then there is a holomorphic section s of L over X with Ord(s) = 1 · p. By Theorem 8.2.1 there is a holomorphic section t of L over a small neighborhood of Y (which is an open Riemann surface by assumption) with no zeros. The function f := s/t is then a holomorphic function on a neighborhood of Y with no zeros other than a simple zero at p. Moreover, f is smooth up to ∂Y , where it also does not vanish. Since ∂Y is smooth and compact, we can solve the Dirichlet problem on Y . Let u be a continuous function on Y that is harmonic on Y , such that u = − log |f | on ∂Y. Since Y is simply connected, there is a holomorphic function g : Y → C such that Re g = u. Let F := f eg . 175

Note that f (z) = 0 if and only if z = p. Moreover |F | = |f |e− log |f | = 1 on ∂Y, and thus for each z ∈ Y |F (z)| < sup |F | = 1. ∂Y −1

Thus F : Y → D and F (∂D) = ∂Y . It follows that F is proper. Indeed, if K ⊂⊂ D is a compact set, then f −1 (K) is closed and contained in Y ⊂⊂ X, thus f −1 (K) is compact. Since F is proper, the number n(z) of preimages of a point z in D is finite. Counting multiplicity, this number n(z) is independent of z, as we explained in the Remark following Theorem 3.3.14. Since F −1 (0) = {p}, n(0) = 1. It follows that F is 1-1. Conclusion of the proof of Theorem 10.3.1. Let X be simply connected Riemann surface. If X is compact, we already know that X must be P1 . Thus suppose X is open. Let Y1 ⊂⊂ Y2 ⊂⊂ ... ⊂ X be a normal exhaustion. By Lemmas 7.2.8 and 10.4.1, each Yj is simply connected and biholomorphic to the unit disk, say Fn : Yn → D. 1,0 Now fix p ∈ Y1 and a tangent vector ξ ∈ TX,p . After composing from the left with a disk automorphism, we may assume that Fn (p) = 0 for all n. By scaling Fn to Gn := Rn · Fn for some constant Rn , we may also assume that dGn (ξ) = 1. Consider the maps fn : D(Rn ) → D(Rn−1 ) defined by fn := Gn−1 G−1 n . Then by the Schwarz Lemma, 1 > |fn0 (0)| =: Rn−1 /Rn . It follows that Rn is increasing, and thus converges to some R ∈ (0, ∞]. Let D0 (∞) := C. We will show that, perhaps after passing to a subsequence, Gn converges to a biholomorphic map F : X → D(R). To this end, consider the maps gn := Gn ◦ F1−1 : D → C. Then gn (0) = 0 and gn0 (0) = ∂ . By Köbe’s Compactness Theorem, a subsequence G0n (p)ξ = 1, provided we take ξ = dF1−1 (0) ∂ζ of the gn converges, and thus so does a subsequence of the Gn on Y1 . Replace {Gn } by this subsequence, and now consider Y2 . Repeating the same argument, we obtain a subsequence that converges on Y2 . Continuing in this way, we may assume, after disposing of the unnecessary Gi and Yi , that {Gn |Ym }n≥m converges uniformly. it follows that we have a holomorphic limit map F : X → D0 (R) with |F 0 (0)| = R. An application of the argument principle shows that F is injective. We need only show that F is surjective, and for surjectivity it suffices to show that F is proper. To this end, let {pk } ⊂ X be a sequence such that for each compact set K, all but a finite number of the pk lie outside K. We wish to prove that given any disk D0 (r) with r < R, then |F (pk )| > r for all but finitely many k. But let Ym ⊃ K. If m is sufficiently large, then Gn → F on Ym uniformly, and thus |F (pk ) − Gn (pk )| < (Rm − r)/2 for n > m >> 0. Since each Gn is proper, for each k sufficiently large there exists n such that |Gn (pk )| > r + (Rm − r)/2. But then |F (pk )| > |Gn (pk )| − (Rm − r)/2 > r, as claimed. The proof of Theorem 10.3.1 is complete.

176

Chapter 11 Hörmander’s Theorem In this chapter we will solve the ∂¯ equation with L2 estimates.

11.1

Hilbert Spaces of Sections

11.1.1

The Hilbert space of sections

Let H → X be a complex line bundle with Hermitian metric e−ϕ and let ω be an area form on X. Given two sections f and g of H, we define Z (f, g) := f g¯e−ϕ ω X

to be the L2 inner product of f and g. The space Γ(X, H) of smooth sections, with the norm || || induced by the inner product ( , ), is not complete. It’s completion with respect to || || is denoted L2 (ϕ, ω).

11.1.2

The Hilbert space of line bundle-valued (0, 1)-forms

We will also need a Hilbert space of H-valued (0, 1)-forms. Of course, H-valued (0, 1)-forms are just sections of the line bundle TX∗0,1 ⊗ H, so in some sense we have dealt with this situation; we can simply replace H byTX∗0,1 ⊗ H. However, we are then required to have a metric for the latter line bundle. Instead of following this more general situation, we exploit the fact that TX∗0,1 is a special bundle. Indeed, if α is a (0, 1)-form, then locally α = f d¯ z , which which it follows that the (1, 1)form √ α∧α ¯ 2 −1 √ = |f | dz ∧ d¯ z 2 2 −1 is locally a measure, and can be integrated without reference to a metric for TX∗0,1 and an area form for X. More generally, if e−ϕ is a metric for a holomorphic line bundle H and α, β ∈ 177

Γ(X, T ∗0,1 X ⊗ H), then

α ∧ β¯ −ϕ √ e 2 −1

is a global (1, 1)-form. We define Z (α, β) := X

α ∧ β¯ −ϕ √ e . 2 −1

(11.1)

The Hilbert space closure of the set Γ(X, TX∗0,1 ⊗ H) of smooth, H-valued (0, 1)-forms with respect to the norm || || obtained from the inner product (11.1) is denoted L20,1 (ϕ). R EMARK . There is an alternative way to explain what we have done. Instead of proceeding as above, we could also introduce a metric for the bundle TX∗0,1 , given in the local frame d¯ z by, say, e−η . In this setting, our L2 -norm would be Z 1 2 α∧α ¯ e−(ϕ+η) ω. ||α|| := √ 2 −1 X √ However, a metric for TX∗0,1 is the inverse of an area element. Thus Ω = eη −1dz ∧d¯ z is a globally √ −ψ −1dz ∧ d¯ z in the same local frame, defined, nowhere-zero (1, 1)-form. It follows that if ω = e then eη+ψ is a globally defined function, and Z 1 2 e−(η+ψ) α ∧ α ¯ e−ϕ . ||α|| = √ 2 −1 X In our setting, we have decided to simplify things by taking η = −ψ.

11.1.3

The ∂¯ operator on L2

The operator ∂¯ : Γ(X, H) → Γ(X, TX∗0,1 ⊗ H) can be defined on all of L2 (ϕ, ω), in the sense of distributions. However, to be most useful to us, we need the image of ∂¯ to be contained in L20,1 (ϕ), which is never the case. 2

E XAMPLE 11.1.1. Let X = C and H the trivial line bundle with metric e−|z| . Consider the function f defined in polar coordinates by 0, r ∈ [0, 1) f (r) := 1, r ≥ 1 ¯ , defined in the sense of distributions, is not represented by any Then f is clearly in L2 , but ∂f 2 ¯ L -form. Indeed, suppose ∂f = α ∈ L2 . A simple calculation shows that in polar coordinates, ! √ −1θ √ √ ∂g 1 ∂g e ∂g = e −1θ + −1 . ∂ z¯ 2 ∂r r ∂θ 178

Let h be a real-valued function with compact support on (0, ∞), and let β be the (0, 1)-form 2

β(z) = e|z| e−

√

−1θ

h(|z|)d¯ z.

Since h has compact support, β is clearly in L2 . Thus Z Z √ √−1 ∂ 1 2 ¯ ) ∧ βe−|z| = − f (r) √ e− −1θ h(r) dz ∧ d¯ z (∂f ∂ z¯ 2 2 −1 C C Z ∞ Z 2π ∂h h(r) = − f (r) + dθrdr ∂r r 0 0 Z ∞ ∂ = −2π (rh(r)) dr ∂r 1 1

If we take h(r) = r−1 e− a (r−1) χ(r) for some smooth function χ : [0, ∞) → [0, 1] with compact 1 support such that χ ≡ 2π on [1/2, 2] then on the one hand, Z 1 ¯ ) ∧ βe−|z|2 = 1. √ (∂f 2 −1 C On the other hand,

1 √ 2 −1

Z

¯ ) ∧ βe−|z|2 ≤ ||∂f ¯ || · ||β||, (∂f

C

and

Z 2 Z β ∧ β¯ −|z|2 β ∧ β¯ √ e √ . e−2(r−1)/a dr → 0 as a → 0. . 2 −1 2 −1 C 1/2 C ¯ cannot be represented by an element of L2 This contradiction shows that the current ∂f Z

(0,1) (e

−|z|2

).

Since we cannot define ∂¯ on L2 , we settle for defining it on the largest possible subset of L2 . ¯ denoted Domain(∂), ¯ is defined to be the set of all f ∈ L2 D EFINITION 11.1.2. The domain of ∂, ¯ ∈ L2 . such that ∂f 0,1 ¯ is a proper subset of L2 . On the other hand, since it contains all We have seen that Domain(∂) ¯ is dense in L2 by the definition of L2 . of the smooth functions, Domain(∂)

11.1.4

The formal adjoint of ∂¯

If α = hd¯ z is a smooth H-valued (0, 1)-form with compact support then for any smooth section f of H we have √ Z Z

∂f −ϕ −1 ψ ϕ ∂ −ϕ ¯ e dz ∧ d¯ z=− e e e h f¯e−ϕ ω, α, ∂f = h ∂ z ¯ 2 ∂z X X where locally the area form ω is given by ω=e

−ψ

√ −1 dz ∧ d¯ z. 2 179

D EFINITION 11.1.3. The operator ϑ defined on smooth compactly supported forms α = hd¯ z by −ϕ ψ ϕ ∂ e h ϑα := −e e ∂z ¯ is called the formal adjoint of ∂.

11.2

The Basic Identity in the L2 method

¯ In this section we develop an identity that plays the central role in solving ∂.

11.2.1

An integration-by-parts identity

Let X be a relatively compact open subset in a Riemann surface M . Let β be a smooth, H-valued (0, 1)-form with compact support, written locally as β = f d¯ z . Then integration by parts yields no boundary terms, and we have

= = = = = = =

||ϑβ||2 ¯ (∂ϑβ, β) Z ∂ ψ − e (fz − ϕz f ) f¯e−ϕ ∂ z¯ X Z Z Z ψ −ϕ ψ −ϕ ¯ ¯ − e ψz¯(fz − ϕz f )f e − e (fzz¯ − ϕz fz¯)f e + eψ ϕzz¯|f |2 e−ϕ X X Z Z ZX eψ ϕzz¯|f |2 e−ϕ eψ (e−ϕ fz¯)z f¯ + eψ ψz¯(e−ϕ f )z f¯ − − Z X Z X Z X e−ϕ fz¯(eψ f¯)z + eψ ϕzz¯|f |2 e−ϕ e−ϕ f (eψ ψz¯f¯)z + X X Z ZX eψ ψz ψz¯|f |2 + ψz¯f fz¯ + fz¯ψz¯f + fz¯fz¯ e−ψ eψ (ϕ + ψ)zz¯|f |2 e−ϕ + ZX ZX eψ (ϕ + ψ)zz¯|f |2 e−ϕ + eψ |fz¯ + ψz¯f |2 e−ϕ X

X

If we define |β|2ω := eψ |f |2

and ∇β := (fz¯ + ψz¯f )d¯ z ⊗2 ,

Then we have proved the following theorem. T HEOREM 11.2.1 (The Basic Identity). The identity √ Z Z 1 2 ψ−ϕ 2 −ϕ −1 ¯ ||ϑβ|| = √ ∇β ∧ ∇βe + |β|ω e ∂ ∂(ϕ + ψ). 2 2 −1 X X holds for all smooth compactly supported H-valued (0, 1)-forms β. 180

(11.2)

11.2.2

Geometric interpretation of the basic identity

Two new objects, namely ∇ and | |ω , appear in the formula (11.2). In this short paragraph we explain the meaning of both. 1. The (1, 1)-form ω identifies the (0, 1)-form β with a (1, 0)-vector field ξβ , i.e., a section of TX1,0 , as follows: for any (0, 1)-vector v, ω(ξβ , v) = β(v). Locally, ∂ . ∂z We can apply ∂¯ to this (1, 0)-vector field, and obtain a (0, 1)-form with values in the line bundle TX1,0 . Extending to line-bundle valued objects the isomorphism of sections of TX1,0 and those of TX∗0,1 induced by ω, we recover a TX∗0,1 -valued (0, 1)-form which is precisely ∇β. ξβ = eψ f

Alternatively, we can observe that the operator ∇ defined by the expression ∇(f d¯ z ) := (fz¯ + ψz¯f )d¯ z ⊗2 ∗(0,1)

defines the (0, 1)-part of a connection on H ⊗ TX , provided the line bundle H is holo∗(0,1) morphic. (The 0th -order term ψz¯f arises because TX is not holomorphic.) The L2 -norm ∗(0,1) for sections Υ of the bundle H ⊗ TX is given by Z √ −1Υ ∧ Υ −ϕ e , ω X which is a (1, 1)-form. (Previously we defined the (0, 1)-part of a connection as ∇0,1 , but for the Basic Identity, the notation ∇ is very common. Unfortunately this notation is inconsistent with the notation ∇ for H, but since all objects are explicitly defined here, we hope the confusion will be manageable.) 2. Since β is an H-valued (0, 1)-form, denoted

√1 β 2 −1

¯ −ϕ is a multiple of ω. This multiple is ∧ βe

|β|2ω e−ϕ . Locally, if β = f d¯ z then √ 1 −1 −ϕ −ϕ ¯ √ β ∧ βe = f f¯e dz ∧ d¯ z = eψ−ϕ |f |2 ω, 2 2 −1 so that |β|2ω = eψ |f |2 . 181

In terms of these quantities, we can recast our integration by parts identity in more geometric terms, giving a proof that looks more formal. To this end, recalling the definition of ∇, we have Z ∂ ψ 2 ¯ e (fz − ϕz f ) f¯e−ϕ ω ||ϑβ|| = (∂ϑβ, β) = − ∂ z ¯ X ¯ β) + ([∇, ∇]β, β) = (∇β, ∇β) + ([∇, ∇]β, β). = (∇∇β, β) = (∇∇β, The last equality uses the facts that (i) the connection is metric-compatible and (ii) the form β vanishs on the boundary. The commutator term is explicitly calculated as [∇, ∇]β = −{(fz − ϕz f )z¯ + ψz¯(fz − ϕz f )} −{(−(fz¯ + ψz¯f ))z − ϕz (−(fz¯ + ψz¯f ))} = −{fzz¯ − ϕzz¯f − ϕz fz¯ + ψz¯fz − ψz¯ϕz f } +{fzz¯ + ψzz¯f + ψz¯fz − ϕz fz¯ − ϕz ψz¯f } = (ϕzz¯ + ψzz¯)f, and thus grants us our curvature term. This presentation carries us closer to the geometric meaning of curvature as the failure of directional derivatives to commute.

11.3

Hörmander’s Theorem √

z T HEOREM 11.3.1 (Hörmander’s Theorem). Fix a smooth Hermitian metric ω = e−ψ 2−1 dz ∧ d¯ for X. Let H → X be a holomorphic line bundle with Hermitian metric e−ϕ whose curvature satisfies √ ¯ + ψ) ≥ cω −1∂ ∂(ϕ (11.3) for some positive constant c. Then for every α ∈ L2(0,1) (ϕ) there exists u ∈ L2 (ϕ, ω) such that ¯ = α in the sense of currents and ∂u Z Z 1 2 −ϕ √1 α ∧ α |u| e ω ≤ ¯ e−ϕ . 2 −1 c X X Proof. Let Γ0 (X, H) ⊂ L20,1 (ϕ, ω) denote the set of all smooth forms with compact support in X. By the basic identity, the hypothesis (11.3) implies the estimate ||ϑβ||2 ≥ c||β||2 . It follows that for all β ∈ Γ0 (X, H), 1 |(α, β)|2 ≤ ||α||2 ||β||2 ≤ ||α||2 ||ϑβ||2 . c Consider the anti-linear functional λ : ϑ(Γ0 (X, H)) → C defined by λ(ϑβ) := (α, β). 182

(11.4)

The estimate (11.4) implies that λ is well-defined and continuous on the subspace ϑ(Γ0 (X, H)) and has norm at most c−1/2 ||α||. By the Hahn-Banach Theorem, λ extends to a continuous linear functional on L2 (ω, ϕ) with the same norm. By the Riesz Representation Theorem λ is represented by inner product against some u ∈ L2 (e−ϕ , ω), i.e., λ(f ) = (u, f ), such that ||u|| = ||λ|| ≤ c−1/2 ||α||. It follows that (u, ϑβ) = (α, β)

for all β ∈ Γ0 (X, H).

¯ = f in the sense of currents. The proof is finished. But the latter says precisely that ∂u

11.4

Proof of the Korn-Lichtenstein Theorem

In Chapter 2 we used the following result to prove that every orientable Riemann surface supports a complex structure. T HEOREM 2.1.11 Let g be a C 2 -smooth Riemannian metric in √ a neighborhood of 0 in R2 . There exists a positively oriented change of coordinates z = u(x, y) + −1v(x, y) fixing the origin, such that g = a(z) dzd¯ z.

We end this chapter with a proof of Theorem 2.1.11.

11.4.1

Derivation of the PDE

Since the result is local, we allow ourselves to shrink neighborhoods when necessary. Let us choose an orthonormal basis of 1-forms in R2 , i.e., α and β are 1-forms such that g = α2 + β 2 . (Such a choice can be made by choosing any two independent √ 1-forms and then applying the GramSchmidt Algorithm from linear algebra.) Let ω = α + −1β. Observe that, with · denoting the complexification of the symmetric product, ω·ω ¯ = α2 + β 2 = g. Thus if we can find function F : R2 → C − {0} and z : R2 → C such that ω = F dz, then we have g = |F |2 |dz|2 , then the function z gives us the change of coordinates we seek. In turn, ω = F dz if and only if we can find a non-vanishing function G such that d log G ∧ ω = −dω.

(11.5)

Indeed, the equation 0 = dG ∧ ω + Gdω = d(Gω) means that Gω = dz by the Poincaré Lemma, and then we take F = 1/G. 183

¯ Equation (11.5) is a partial differential equation analogous to the ∂-equation. To see how this is so, choose tangent vectors ξ and η such that αξ = βη = 1 and αη = βξ = 0. Letting ξω =

ξ−

√

−1η

and ξω¯ =

2

ξ+

√ 2

−1η

,

we find that ωξω = 1,

ω ¯ ξω¯ = 1,

ωξω¯ = 0 and ω ¯ ξω = 0.

It follows that df = df (ξω )ω + df (ξω¯ )¯ ω, and thus, with φ defined by dω = φω ∧ ω ¯, d log G ∧ ω = −dω ⇐⇒ d log G(ξω¯ ) = φ. In other words, if we want to find our function G, we must solve the PDE ξω¯ γ = φ

(11.6)

and then take G = eγ . R EMARK . Note that if our original metric g were the Euclidean metric and we choose the standard basis as our orthogonal vectors, then (11.6) is exactly the Cauchy-Riemann equations.

11.4.2

The operator ∂¯ω

To simplify the situation with the weights, it is convenient to work with differential forms of the form f ω ¯ , rather than with functions. Accordingly, we define the operator ∂¯ω by on a function f by ∂¯ω f = (ξω¯ f )¯ ω.

11.4.3

Hilbert space structures

R EMARK . We note that

√

−1 ω 2

∧ω ¯ =α∧β

is the area element associated associated to the metric g. Let begin with the Hilbert space of functions. An inner product of functions f and h on U is defined by Z √ ¯ −ψ −1 ω ∧ ω (f, h)0 := f he ¯. 2

U

We then take H0 (ω, e−ψ ) to be the Hilbert space completion of the set of smooth functions with finite norm. Next we define the weighted Hilbert space structure on the set of smooth multiples of ω ¯ by Z 1 (f ω ¯ , h¯ ω ) := √ fω ¯ ∧ h¯ ω e−ψ , 2 −1 U and let H1 (ω, e−ψ ) be the Hilbert Space completion of the space of all such smooth multiples of ω ¯. 184

11.4.4

The formal adjoint of ∂¯ω

Suppose f ω ¯ and g ω ¯ are compactly supported on U . Then Z √ −1 e−ψ f ξω¯ hω ∧ ω ¯ (f ω ¯ , ∂¯ω h)1 = 2 U Z √ −eψ ξω (e−ψ f ω ∧ ω ¯) −1 hω ∧ ω ¯ = 2 ω∧ω ¯ U and thus we see that the formal adjoint of ∂¯ω is the operator ϑω (f ω ¯) =

−eψ ξω (e−ψ ω ∧ (f ω ¯ )) . ω∧ω ¯

If we write ω = e−δ dx ∧ dy, then we find that ϑω (f ω ¯ ) = −ξω f + [ξω (ψ + δ)]f.

11.4.5

The basic estimate

¯ Guided by the derivation of the basic identity for the ∂-operator, We now establish an a priori ¯ estimate suitable to solving ∂ω . First, we calculate that ∂¯ω ϑω (f ω ¯ ) = −(ξω¯ ξω f )¯ ω + ξω¯ ξω (ψ + δ)f ω ¯ + ξω (ψ + δ)(ξω¯ f )¯ ω. Assuming f has compact support, we compute that Z Z ¯ ∧ω ω ¯ ∧ω −ψ ω ¯ ¯ √ (∂ω ϑω f ω ¯, f ω ¯ )0 = −(ξω¯ ξω f )f e + ξω¯ ξω (ψ + δ)|f |2 e−ψ √ 2 −1 2 −1 U U Z ω ¯ ∧ω + ξω (ψ + δ)(ξω¯ f )f¯e−ψ √ 2 −1 U Z Z ω ¯ ∧ω ω ¯ ∧ω −([ξω¯ , ξω ]f )f¯e−ψ √ −(ξω ξω¯ f )f¯e−ψ √ + = 2 −1 2 −1 U U Z ω ¯ ∧ ω + ξω¯ ξω (ψ + δ)|f |2 e−ψ √ 2 −1 U Z ω ¯ ∧ω + ξω (ψ + δ)(ξω¯ f )f¯e−ψ √ 2 −1 U Z Z ω ¯ ∧ ω ω ¯ ∧ω = |ξω¯ f |2 e−ψ √ + −([ξω¯ , ξω ]f )f¯e−ψ √ 2 −1 2 −1 U U Z ω ¯ ∧ω + ξω¯ ξω (ψ + δ)|f |2 e−ψ √ . 2 −1 U Now, one can easily obtain the estimates Z Z Z ω ¯ ∧ ω ¯ ∧ω C ω ¯ ∧ω −ψ 2 −ψ ω ([ξω¯ , ξω ]f )f¯e √ ≤ ε |[ξω¯ , ξω ]f | e √ + |f |2 e−ψ √ , ε U 2 −1 2 −1 2 −1 U U 185

and

Z ω ¯ ∧ω ω ¯ ∧ω √ |[ξω¯ , ξω ]f | e ≤C |ϑω (f ω ¯ )|2 √ , 2 −1 2 −1 U U where the latter estimate follows because [ξω , ξω¯ ] is a first order differential operator, and any first order derivative can be estimated in L2 by the differential operator ϑω . Combining with the identity above, we have the basic estimate: Z ω ¯ ∧ω 2 ||ϑ(f ω ¯ )||0 ≥ c ξω¯ ξω (ψ + δ)|f |2 e−ψ √ , (11.7) 2 −1 U Z

2 −ψ

where c > 0 is a (possibly quite small) constant that is independent of f .

11.4.6

Positively curved weights

Finally, we need to know that there are functions ψ that have arbitrarily large ξω ξω¯ -derivative. To this end we use the following trick. We note that at least at one point, we can see our metric g as Euclidean in some chosen√coordinate system. Assume the point is the origin and the coordinates are (u, v). Write ζ = u + −1v. It follows that the operators ξω and ξω¯ are given by ∂ ∂ + a(ζ) ¯ ∂ζ ∂ζ

and

∂ ∂ + b(ζ) ∂ζ ∂ ζ¯

respectively, with a(ζ), b(ζ) = O(|ζ|). It follows that

∂2 ∂2 ∂2 ξω ξω¯ = (1 + a(ζ)b(ζ)) + a(ζ) ¯2 + b(ζ) 2 + F ∂ζ ∂ζ∂ ζ¯ ∂ζ for some first order differential operator F . If we now let ψ(ζ) = C|ζ|2 for some large constant C > 0, then |ξω ξω¯ ψ − C| = O(|ζ|). It follows that for C >> 0, ξω¯ ξω (ψ + δ) ≥ 1 in a small neighborhood of 0. We thus obtain from (11.7) the estimate ||ϑ(f ω ¯ )||20 ≥ ||f ω ¯ ||21 ,

11.4.7

fω ¯ ∈ H1 (ω, e−ψ ).

(11.8)

Conclusion of the proof of Theorem 2.1.11

We can apply the method of proof of Hörmander’s Theorem 11.3.1 with the estimate (11.8) to conclude that the equation ∂¯ω (γ) = φ¯ ω has a solution (with estimates, which we do not use here). The proof of Theorem 2.1.11 is complete. 186

Chapter 12 Embedding Riemann Surfaces In this chapter, we aim to show the following fundamental facts about Riemann surfaces: 1. Every holomorphic line bundle admits a non-zero meromorphic section. 2. Every Riemann surface admits a non-constant meromorphic function. 3.

(a) Every compact Riemann surface embeds in P3 . (b) Every open Riemann surface embeds in C3

All of these facts rely on the existence of many holomorphic sections of holomorphic line bundles admitting smooth metrics with sufficiently positive curvature. In Section 12.1 we will use Hörmander’s Theorem to obtain sections of sufficiently positive line bundle on a Riemann surface X, with control over the values of the sections and their derivatives at a finite number of points. In Section 12.4 we will use those sections to construct an embedding of a compact Riemann surface X in PN for some large N , after which a generic projection argument will show that X can be biholomorphically projected to a 3-dimensional projective subspace. In Section 12.5 we will carry out a similar argument to embed open Riemann surfaces in C3 , only this time, because of the absence of compactness, we will carry out the projection argument in conjunction with the proof of the embedding theorem, using the technique of exhausting X by compact, smoothly bounded subsets.

12.1

Controlling the derivatives of sections

Let X be a Riemann surface, not necessarily compact. By Proposition 9.4.1 or Theorem 9.4.4 ¯ is a there is a line bundle H → X and a smooth Hermitian metric e−ϕ whose curvature ∂ ∂ϕ strictly positive form. We fix such a line bundle H → X and metric e−ϕ , as well as a second holomorphic line bundle E → X with smooth Hermitian metric e−λ whose curvature we know nothing about. 187

Let us fix an area form ω on X. (For example, we could take ω = matter what area form we take.) Locally we can write √ −ψ −1 dz ∧ d¯ z. ω=e 2

√ ¯ but it does not −1∂ ∂ϕ,

Note that ω is the area form of a metric for the tangent bundle, the curvature form for this metric ¯ is ∂ ∂ψ. Fix a point xo ∈ X. Let z be a local coordinate in a relatively compact neighborhood U of xo , such that z(x) = 0. Fix open sets V ⊂⊂ W ⊂⊂ U with xo ∈ V , and let χ : X → [0, 1] be a smooth function such that Support(χ) ⊂ U and χ|W ≡ 1. For each x ∈ V consider the function η = ηN,ε,x := χ · log(|z − z(x)|2 + ε2 )N . Then e−ηN,ε,x is a singular Hermitian metric for the trivial bundle on X. This metric is smooth with curvature √ N ε2 ¯ N,ε,x = + θ ≥ θ, −1∂ ∂η (|z − z(x)|2 + ε2 )2 where θ is a smooth, possibly negative (1, 1)-from on all of X. If X is compact, then for some integer ko >> 0, the curvature of the metric e−(kϕ+λ+ηN,ε,x )

(12.1)

√ ¯ for all k ≥ ko . Moreover, compactness of X implies for H ⊗k ⊗ E is greater than ω − −1∂ ∂ψ that ko can be chosen independent of x ∈ V . If on the other hand X is open, then we can replace ϕ by h ◦ ϕ for some convex, rapidly increasing function h. Then ¯ ◦ ϕ) = h00 ◦ ϕ∂ϕ ∧ ∂ϕ ¯ + h0 ◦ ϕ∂ ∂ϕ ¯ ≥ h0 ◦ ϕ∂ ∂ϕ. ¯ ∂ ∂(h Thus for sufficiently rapidly increasing h we can guarantee that the curvature of the metric (12.1), √ ¯ this time independently of x ∈ X. with k = 1, is greater than ω − −1∂ ∂ψ, Now assume that the line bundles H and E (hence H ⊗k ⊗ E) are trivial on U . Let σ = χ · p(z − z(x)) where p is a polynomial of degree at most N . We can think of σ as a section of H ⊗k ⊗ E, and the Taylor polynomial of σ at x to order N in the local coordinate z is precisely p. Unfortunately, σ is not holomorphic on X, although it is holomorphic on the neighborhood W of x. ¯ Then β is a ∂-closed ¯ Let β = ∂σ. (0, 1)-form with values in H ⊗k ⊗ E, supported on U but identically zero in W . It follows that Z 1 ¯ −(kϕ+λ+ηN,ε,x ) ≤ C √ β ∧ βe 2 −1 X √ ¯ for some constant C > 0 independent of ε. But since −1∂ ∂(kϕ + λ + ηN,ε,x + ψ) ≥ ω > 0, ⊗k Hörmander’s Theorem 11.3.1 tells us that there is a section uε of H ⊗ E such that Z ¯ ∂uε = β and |uε |2 e−(kϕ+λ+ηN,ε,x ) ω ≤ C. X

188

It follows from the Lebesgue Dominated Convergence Theorem that uε → u in L2 and Z |u|2 e−(kϕ+λ+ηN,0,x ) ω ≤ C < +∞. X

Now, the finiteness of the integral implies in particular that on V the function |u|2 e−(kϕ+λ+ηN,x +ψ) |V =

|u|2 e−(kϕ+λ+ψ) |z − z(x)|2N

is locally integrable. Since ϕ, λ and ψ are smooth functions, it must be the case that the Taylor polynomial of u to order N − 1 at x must be zero. Indeed, in local polar coordinates with center x, the area form is a multiple of rdrdθ so that if u vanishes to order ` at x, it must be the case that r2`+1−2N dr is integrable near the origin on the non-negative real line, and this is only so if ` ≥ N . ¯ = β − ∂u ¯ = 0, so that s is holomorphic. Now consider the section s = σ − u. Then ∂s Moreover, the Taylor polynomial of s at x is p. Finally, if X is compact, we can cover X by a finite number of neighborhoods Vj such that Vj ⊂⊂ Wj ⊂⊂ Uj are neighborhoods as above. Thus we can choose the number ko above independent of the point x ∈ X. We have proved the following theorem. T HEOREM 12.1.1. Let X be a Riemann surface, and let p be a polynomial of a complex variable. Let H → X be a positive holomorphic line bundle (taken to be trivial if X is open) and E → X any holomorphic line bundle. Then there is an integer ko with the following property. For any x ∈ X and any k ≥ ko there is a holomorphic section s of H ⊗k ⊗ E → X whose Taylor polynomial in some local coordinate near x is p. In particular, there is an integer ko such that for any x ∈ X and any k ≥ ko there is a holomorphic section s of H ⊗k ⊗ E whose derivative in any local coordinate does not vanish at x. By a similar, but slightly more delicate argument, we get the following result. T HEOREM 12.1.2. Let H → X be a holomorphic line bundle on a Riemann surface. Suppose there is a positively curved Hermitian metric for H. Then there is an integer ko > 0 with the following property. For any integer k ≥ ko , any distinct points x, y ∈ X and any a ∈ Hx⊗k and b ∈ Hy⊗k there is a holomorphic section s of H ⊗k → X such that s(x) = a and s(y) = b. Proof. The reason the argument of Theorem 12.1.1 does not immediately give the present result is that we have to worry about what happens when the points x and y get close together. Thus we will distinguish those cases by slightly modifying the argument. We begin by choosing a locally finite cover by relatively compact coordinate neighborhoods Uj on which H is trivial, and relatively compact open subsets Vj ⊂⊂ Wj ⊂⊂ Uj such that {Vj } is again a locally finite subcover. We also fix coordinates zj on Uj and smooth functions χj with support on Uj such that χj ≡ 1 on Wj . Now define for x ∈ Vj and y ∈ W` the singular function ηx,y,ε := χj log(|zj − zj (x)|2 + ε2 ) + χ` log(|z` − z` (y)|2 + ε2 ). 189

Of course there is some choice involved, since a given point may be in more than one neighborhood, but since any point can lie in at most a finite number of the neighborhoods Uj , we will see that the argument can be made independent of the choice of j and `. Let ϕ be a positively curved metric for H. Consider the metric e−(kϕ+ηx,y,ε ) . We calculate that for some positive constant C that is independent of x and y and of the possible choices of j and `, √

¯ x,y,ε ≥ −1∂ ∂η

ε2 ε2 + − Cω. (|z − z(x)|2 + ε2 )2 (|z − z(y)|2 + ε2 )2

It follows that there is a constant ko such that for all k ≥ ko , √

¯ −1∂ ∂(kϕ + ηx,y,ε ) +

√ ¯ ≥ cω > 0, −1∂ ∂ψ

again independent of the points x and y and the possible choices of j and `. (If X is not compact, we replace ϕ by h ◦ ϕ for some convex, rapidly increasing function h, and take ko = k = 1.) The rest of the argument resembles more closely the argument in the proof of Theorem 12.1.1. Since H is trivial on Uj and U` , the constant sections a and b are well-defined on Uj and U` respectively. We then define the global section σ = χj · a + χ` · b. This section is smooth on X, supported on Uj ∪ U` , and holomorphic on Wj ∪ W` . It follows that ¯ is supported on (Uj − Wj ) ∪ (U` − W` ). By Hörmander’s Theorem 11.3.1 there is a section uε ∂σ of H ⊗k such that Z ¯ ε = ∂σ ¯ ∂u and |uε |2 e−(kϕ+ηx,y,ε ) ω < +∞. X

As in the proof of Theorem 12.1.1, we can let ε → 0 and obtain from the Lebesgue Dominated Convergence Theorem an integrable section u, this time with respect to the metric e−(kϕ+ηx,y,0 ) . The latter integrability implies that u(x) = 0 and u(y) = 0. The section s = σ − u therefore has the desired properties.

12.2

Meromorphic sections of line bundles

In Chapter 4 we discussed the correspondence between divisors and line bundles. At that time, we were able only to complete the correspondence between divisors on the one hand and holomorphic line bundles with meromorphic sections on the other. At that time, we stated (as Theorem 4.5.7) the fact that every holomorphic line bundle has a non-trivial meromorphic section. We can now complete the classification by proving Theorem 4.5.7. 190

Proof of Theorem 4.5.7. Let L → X be a holomorphic line bundle. Consider the line bundle H associated to the divisor of a point p ∈ X, and the associated global holomorphic section sp of H such that Ord(sp ) = p. (Recall the discussion in Section 4.5.) By Theorem 6.1.19, c(H) = 1. Therefore, by Proposition 9.4.2, the line bundle H has a metric e−ϕo of strictly positive curvature. By Theorem 12.1.1, for all m >> 0 the line bundle Em := H ⊗m ⊗ L has a non-trivial global holomorphic section, say tm . It follows that σ :=

tm ∈ ΓM (X, L) − {0}. s⊗m p

The proof of Theorem 4.5.7 is complete.

12.3

Plenitude of meromorphic functions

In this section we demonstrate the existence of many meromorphic functions on compact Riemann surfaces. D EFINITION 12.3.1. A family F of differentiable functions on a manifold X is said to separate points if for each pair of distinct point x, y ∈ X there exists f ∈ F such that f (x) 6= f (y). We say F separates tangents if for each x ∈ X there exists f ∈ F such that dF (x) 6= 0. Since every holomorphic line bundle on an open Riemann surface is trivial, Theorems 12.1.2 and 12.1.1 already imply the following result. C OROLLARY 12.3.2. The family O(X) of holomorphic functions on an open Riemann surface separates points and tangents. For compact Riemann surfaces we have the following easy corollary of Theorems 12.1.2 and 12.1.1. P ROPOSITION 12.3.3. The family M (X) of all meromorphic functions on a compact Riemann surface X separates points and tangents. Proof. In view of Theorem 12.1.2, can find sections s and σ of some line bundle L1 such that s(x), σ(x), and σ(y) are non-zero while s(y) = 0. Consider the meromorphic function f = s/σ. Then f (x) 6= 0 while f (y) = 0. Next, we can find sections s and σ of some line bundle L2 such that s(x) and σ(x) are non-zero, 0 s (x) is non-zero and σ 0 (x) = 0 (in some local trivialization and local coordinates). Consider the meromorphic function f = s/σ. Then

s0 (x) s0 (x) s(x)σ 0 (x) − = σ(x) σ(x)2 σ(x) is non-zero at x. The proof is complete. f 0 (x) =

191

12.4

Kodaira’s Embedding Theorem

12.4.1

Embedding Riemann surfaces in PN

We are now ready to prove the following result. T HEOREM 12.4.1. Every compact Riemann surface X can be holomorphically embedded in some PN . Proof. Fix a positive line bundle H → X, which exists by Theorem 9.4.1. Let k be the larger of the integers ko given by Theorem 12.1.1 for the case of Taylor polynomial p(z) = z, and by Theorem 12.1.2. Now consider ΓO (X, H ⊗k ). Then, as in Section 4.7, we have the map φ|H ⊗k | : X → P(ΓO (X, H ⊗k )∗ ). In terms of a basis s0 , ..., s` of ΓO (X, H ⊗k ), φ|H ⊗k = [s0 , ..., s` ]. If all of the sections si did not separate a given pair x and y, then neither would any linear combination. Thus by Theorem 12.1.2, φ|H ⊗k | separates any pair of distinct points x and y. Next, perhaps after renumbering, we can assume that s0 (x) 6= 0. If all of the functions fi = si /s0 had vanishing derivatives at a given point x, then so would every linear combination of these, and this would contradict Theorem 12.1.1. It follows that the map φ|H ⊗k | has full rank at each point of X. Thus we have proved that φ|H ⊗k | is an embedding, as desired.

12.4.2

Embedding Riemann surfaces in P3

T HEOREM 12.4.2. Every compact Riemann surface X can be holomorphically embedded in P3 . Proof. The result will be proved by descending induction. To this end, suppose n ≥ 4 and we have an embedding φ : X → Pn . For dimension reasons, φ(X) 6= Pn . Let Q ∈ Pn − φ(X). Suppose we pick a projective hyperplane Π ⊂ Pn − {Q}. Then we can define a map ProjQ,Π : X → Π as follows. If x ∈ φ(X), then there is a unique projective line L in Pn containing x and Q. Since L 6⊂ Π, L ∩ Π consists of a single point, which we call ProjQ,Π (x). Because the dimension n of the ambient space is at least 4, after small perturbations of Q and Π we can ensure that the map ProjQ,Π : X → Π is actually an embedding. To see this, we can introduce the so-called chordal set. Let I := {(x, y, p) ∈ Pn × Pn × Pn ; x, y ∈ φ(X) and p lies on the line through x and y}. Note that x and y can lie in φ(X) freely (apart from coalescing), which gives us two complex dimensions, and once x and y have been chosen, we have one more line (so one more complex 192

dimension) for p. It follows that I is 3-dimensional, and thus, with π3 : I → Pn the projection sending (x, y, p) to p, π3 (I) is 3-dimensional. Hence for n ≥ 4, Pn − π3 (I) has interior, so it is not empty. It follows from the definition that if we choose Q ∈ Pn − π3 (I), then ProjQ,Π : X → Π is 1-1. Moreover, if we choose Q ∈ Pn − π3 (I), then no vector tangent to φ(X) will be annihilated by ProjQ,Π , and thus ProjQ,Π : X → Π ∼ = Pn−1 is an embedding. The proof is thus complete by descending induction.

12.5

Embedding open Riemann surfaces

12.5.1

Statement and outline of the proof

The goal of this section is to prove the following theorem. T HEOREM 12.5.1. Let X be a non-compact connected Riemann surface. Then there is a proper injective holomorphic map φ : X → C3 . In the case of open Riemann surfaces, the absence of compactness complicates matters, and we cannot simply imitate the proof of Theorem 12.4.1. In fact, the proof consists of three parts. In the first part, we being by exhausting X by certain compact sets. On each of these sets we produce N functions separating points and tangents, where N may depend on the compact set we are on. We then employ the analogue of the projection method used in the proof of 12.4.2 to produce 3 functions that separate points and tangents on the compact set. Passing to the limit will give us a 1-1 map F : X → C3 that is an immersion. Unfortunately, we cannot conclude that this map is proper. The second and hardest step is to produce a map G : X → C3 that is proper. This step will occupy the majority of the section. Finally a simple argument will combine these two maps to produce a proper injective immersion φ : X → C3 .

12.5.2

The injective immersion of X in C3

Let ρ : X → (−∞, a) be a subharmonic exhaustion function, where a ∈ R ∪ {∞}. Fix a sequence of numbers aj % a and set Kj := ρ−1 ((−∞, aj ]). (1) (1) First, we claim that we can find global holomorphic functions f1 , ..., fN that separate points and tangents on K1 . To see this, we argue as follows. First, let x ∈ K1 . Then there exists a function f1 defined on all of X, whose derivative does not vanish at x. It follows that f˜x separates any pair of distinct points near x. Now, since K1 is compact, we can cover it by a finite number of neighborhoods, and for each such neighborhood Uj we get a function f˜j that separates the points of Uj . By taking fj = f˜j + Cj for appropriately chosen constants Cj , we can guarantee that in fact (1) (1) the functions f1 , ..., fN thus obtained separate the points of K1 . We let φ(1) : X → CN be the map (1) (1) φ(1) = (f1 , ..., fN ). The map φ(1) separates points and tangents on K1 . 193

Now consider the set I := {(x, y, x + ty) ∈ CN × CN ; x, y ∈ φ(1) (X), t ∈ C} and the map π3 : I → CN sending (x, y, p) to p. Since I is 3-dimensional, the closure π3 (I) of its image π3 (I) is at most three dimensional. If N ≥ 4, then the projection from any point not on π3 (I) onto any hyperplane Π restricts to φ(1) (X) injectively. This amounts to saying that if N ≥ 4 (1) (1) (1) (1) then N − 1 generic linear combinations F1 , ..., FN −1 of the functions f1 , ..., fN produce a map (1) ΦN −1 : X → CN −1 that still separates points and tangents on K1 . (j) (j) (j) Now suppose we have found functions F1 , F2 , F3 such that the map Φ(j) : X → C3 separates points and tangents of Kj . Let Lj := Kj+1 − Kj . Then Lj is compact, and the same (j) (j) argument as above shows that there are functions g1 , g2 ,(j) , g3 : X → C3 that separate points (j) (j) (j) and tangents on Lj . We can put these together to make a map (Φ(j) , g1 , g2 , g3 ) : X → C6 that separates points and tangents on Kj+1 . By taking projections close to the identity, it follows that the map (j) (j) (j) (j) (j) (j) Φ(j+1) := Φ(j) + ·(ε2 g1 , ε2 g2 , ε3 g3 ) : X → C3 separates points and tangents on Kj+1 . If we take ε(j) ∈ C3 sufficiently small, evidently the process converges to give us the desired injective immersion. Thus we have proved the following result. T HEOREM 12.5.2. Let X be an open Riemann surface. Then there is an injective immersion X → C3 . R EMARK . In fact, the above argument can easily be modified to show that the set S of holomorphic maps X → C3 that are not injective immersions is of the first category in the sense of Baire, i.e., S is a countable union of closed sets without interior. Indeed, since X can be exhausted by a discrete family of compact sets, it suffices to show that the generic map separates points and tangents on some fixed compact set K, i.e., the set SK of maps that do not separate tangents and points of K is closed and has no interior. To see that SK is closed, consider such a sequence Φj : X → C3 converging to Φ. Then there are points xj ∈ K such that dΦj (xj ) = 0. After passing to a convergent subsequence, we can assume xj → x. Evidently dΦ(x) = 0. By a similar argument there are distinct x, y ∈ X such that Φ(x) = Φ(y). To see that SX has no interior, let Ψ ∈ SK and a the map Φ : X → C3 separating the points and tangents of K. Then (Ψ, Φ) : X → C6 is an injective immersion, and thus for generic enough ε ∈ C3 , Ψ + (ε1 Φ1 , ε2 Φ2 , ε3 Φ3 ) : X → C3 is again an injective immersion, which shows that Ψ is not in the interior of SK .

12.5.3

Analytic polyhedra

D EFINITION 12.5.3. Let X be an open Riemann surface. An open set P ⊂⊂ X is called an analytic polyhedron of order N if there exist analytic functions f1 , ..., fN ∈ O(X) such that P is a union of components of the set {z ∈ X ; |fi (z)| < 1, i = 1, ..., N }. 194

In this paragraph we prove two crucial lemmas on the construction of analytic polyhedra. L EMMA 12.5.4. Let X be an open Riemann surface, K ⊂⊂ X a holomorphically convex compact set and U ⊃ K an open subset of X. Then there is an analytic polyhedron P such that K ⊂ P ⊂⊂ U. Proof. We may assume that U is relatively compact. Since K is holomorphically convex, for each p ∈ ∂U we can find a function fp ∈ O(X) such that |fp | < 1 on K and |fp (p)| > 1. The latter also holds in a neighborhood of p by continuity. By the compactness of ∂U we can find finitely many functions f1 , ..., fN such that the sets Vj := {|fj | > 1} cover ∂U and miss K. It follows that the intersection P of U with {|fj | < 1} does not meet the boundary of U and contains K. The proof is complete. L EMMA 12.5.5. Let K be a compact set in an open Riemann surface X and P an analytic polyhedron of order N ≥ 3. Then there exists an analytic polyhedron P 0 of order N − 1 such that K ⊂ P0 ⊂ P. Proof. Suppose P is a union of components of the set {|fj | < 1 ; 1 ≤ j ≤ N }. We fix positive numbers εo < ε1 < ε2 < ε3 < 1 such that |fj (z)| < εo for all z ∈ K and 1 ≤ j ≤ N . By the methods used in the proof of Theorem 12.5.2 we can choose functions fj0 , 1 ≤ j ≤ N − 1 such that (i) the map (f10 /fN , ..., fN0 −1 /fN ) separates tangents at the points of P ∩ {|fN | ≥ ε2 }, and (ii) fj0 is so close to fj that |fj0 (z)| < εo for z ∈ K and U := P ∩ {|fj0 (z)| < ε3 , 1 ≤ j ≤ N } ⊂⊂ P. (Since we do not have to separate points, we can handle the case N ≥ 3 rather than being forced to take N ≥ 4.) Now consider the open set ∆ν := {x ∈ X ; |fj0 (x)ν − fN (x)ν | < εν1 , 1 ≤ j ≤ N − 1} for some positive integer ν to be chosen. We will prove that for ν >> 0, the components of ∆ν that meet K form the desired analytic polyhedron. Let us denote by Pν0 the collection of all these components. First, if (ε1 /ε0 )ν > 2 then for any z ∈ K, |fj0 (z) − fN (z)| < εν1 . It follows that for ν >> 0, K ⊂ ∆ν , and thus K ⊂ Pν0 . We now show that Pν0 ∩ ∂U = ∅. If not, let y ∈ Pν0 ∩ ∂U . If |fN (y)| < ε2 , then |fj0 (y)|ν = |fj0 (y)ν − fN (y)ν + fN (y)ν | < εν1 + εν2 < εν3 195

for ν >> 0, which contradicts that y ∈ ∂U . It follows that y is contained in the compact set L := ∂U ∩ {|fN | ≥ ε3 }. We assert that no component of ∆ν that meets L can also meet K. We do this in local coordinates: let L1 be the intersection of L with some coordinate patch containing y and having local coordinate z. Then with Fj = fj0 /fN , 1 ≤ j ≤ N − 1, we have |Fj (y)ν − 1| < (ε1 /ε2 )ν . We now show that for any ζ with |z − y| = ν −2 , max |fj0 (z)ν − fN (z)ν | > εν1 ,

1≤j≤N −1

(12.2)

from which our assertion clearly follows. To prove (12.2), note first that |fj0 (z)ν − fN (z)ν | = |Fj (z)ν − 1| · |fN (z)ν |. Since |fN (z)| ≥ c2 (1 + O(ν −2 )), |fN (z)ν | ≥ εν2 (1 + O(ν −1 )) > εν2 /2 for ν >> 0. Now, by Taylor’s Theorem, Fj (z) = Fj (y)(1 + `j (z − y) + O(ν −4 )),

1 ≤ j ≤ N − 1.

Since the map (F1 , ..., FN −1 ) separates tangents, the numbers `1 , ..., `N −1 do not all vanish. Thus maxj |`j | > c > 0. We have (Fj (z)/Fj (y))ν = 1 + ν`j (z − y) + O(ν −2 )), Now write ν

ν

Fj (z) − 1 = Fj (y)

Fj (z) Fj (y)

ν

1 ≤ j ≤ N − 1.

− 1 + Fj (y)ν − 1.

It follows that max |fj0 (z)ν − fN (z)ν | > max |Fj (z)ν − 1|εν2 /2 j j ν ν ν Fj (z) ν = max ε2 |Fj (y)| − 1 − |Fj (y) − 1| j Fj (y) ν ν ν Fj (z) > max ε2 |Fj (y)| − 1 − εν1 j Fj (y) ν ε > 2 (c/ν + O(ν −2 )) > εν1 , 4 again for ν >> 0. By the compactness of L, the constant c can be taken uniform, and thus the final ν we choose is independent of the point y ∈ L. The proof of the lemma is complete. 196

12.5.4

Conclusion of the proof of Theorem 12.5.1

By Theorem 12.5.2 there is a holomorphic map g : X → C3 that separates points and tangents. We are going to construct a map f : X → C3 with the following property: for every k ∈ N, {x ∈ X ; |f (x)| ≤ k + |g(x)|} ⊂⊂ X.

(12.3)

If we have such a map f , then the map (f, g) : X → C6 is an injective immersion. The projection argument used above then shows that there are numbers aij , 1 ≤ i, j ≤ 3 such that 3 X

|aij | < 1,

i = 1, 2, 3,

j=1

and the map f˜ =

f1 +

3 X

a1j gj , f2 +

j=1

3 X

a2j gj , f3 +

j=1

3 X

! a3j gj

j=1

separates points and tangents on X. But then {x ∈ X ; |f˜(x)| ≤ k} ⊂ {x ∈ X ; |f (x)| ≤ k + |g(x)|} ⊂⊂ X, and Theorem 12.5.1 follows. Let Kj ⊂⊂ interior(Kj+1 ) ⊂ Kj+1 , j ≥ 1 be a sequence of compact sets such that b j = Kj K

[

and

Kj = X.

j≥1

By Lemmas 12.5.4 and 12.5.5, there are analytic polyhedra Pj , j ≥ 1, such that Kj ⊂ Pj ⊂ Kj+1 . Let Mj := sup |g|. Pj

Condition (12.3) is implied by the following condition: for all k ≥ N , |f | ≥ k + Mk+1

in Pk+1 − Pk .

(12.4)

Indeed, (12.4) implies that |f | + k ≥ |g| in Pk+1 − Pk , and thus by holomorphic convexity |f | ≥ k + |g| in X − Pk . We begin by constructing f1 , f2 , f3 ∈ O(X) such that for every k ∈ N, max |fj (x)| > k + Mk+1

1≤j≤3

197

on ∂Pk .

(12.5)

By the definition of analytic polyhedron of order 3, we can find h1,k , h2,k , h3,k ∈ O(X) so that maxj |hj,k | < 1 on Pk−1 but maxj |hj,k | = 1 on ∂Pk . Fix numbers ak slightly larger than 1 and large integers mk , and let fj,k = (ak hj,k )mk , j = 1, 2, 3 and fj =

∞ X

fj,k ,

j = 1, 2, 3.

k=1

With appropriate choices, we can ensure that max |fj,k | ≤ 2−k in Pk−1 j

and

k−1 X fj,` in ∂Pk . max |fj,k | > mk+1 + k + 1 + max j j `=1

These conditions imply that the series defining the fj converge to holomorphic functions on X satisfying (12.5). Now set Gk := {x ∈ Pk+1 − Pk ; max |fj (x)| ≤ k + Mk+1 } j

and Hk := {x ∈ Pk ; max |fj (x)| ≤ k + Mk+1 }. j

By (12.5), these disjoint sets are compact. The O(X)-hull G\ k ∪ Hk of Gk ∪ Hk is contained in Kk+2 , and inspection shows that G\ k ∪ Hk = Gk ∪ Hk ∪ Lk for some Lk ∈ X − Pk . (In fact it is easy to deduce that Lk = ∅, but we will not use this fact.) By approximating the functions x ∈ Hk ∪ Lk ˜ k (x) := 0 h Ak x ∈ Gk for appropriate large constants Ak , using the Runge Approximation Theorem 7.2.9, we obtain functions hk ∈ O(X) such that X |hk | < 2−k in Hk and |hk | ≥ 1 + k + Mk+1 + hj in Gk , k ≥ 1. ` 0. If ω ∈ ΓO (X, KX ) is not identically zero, then g X Im Ak (ω)Bk (ω) < 0. j=1

In particular, if all of the aj -periods of ω are zero, then ω ≡ 0. ¯ we have Proof. Applying Lemma 14.2.1 with α = ω = β, 1 √ 2 −1

Z fω ω ¯ = Im

g X

∂U

Ak (ω)Bk (ω).

j=1

On the other hand, by Stokes’ Theorem we have Z √ Z 1 −1 √ fω ω ¯=− ω∧ω ¯. 2 2 −1 ∂U X The right hand side is non-positive, and negative unless ω = 0. The proof is complete. C OROLLARY 14.2.3. If ω1 , ..., ωg ∈ ΓO (X, KX ) is a basis, then with Z aij := ωj , ai

the matrix A = (aij ) is invertible. In particular, we can choose a basis ω1 , ..., ωg of ΓO (X, KX ) such that Z 1 i=j ωj = δij := 0 i 6= j ai (We call such a basis a normalized basis). 214

P Proof. Let c ∈ Cg with Ac = 0. Then with ω = cj ωj , we have Z ω = 0, i = 1, ..., g. ai

Thus by Proposition 14.2.2, ω = 0. Since ω1 , ..., ωg is a basis, c = 0. T HEOREM 14.2.4 (Riemann Bilinear Relations). Let X be a compact Riemann surface of positive genus g and ω1 , ..., ωg ∈ ΓO (X, KX ) a normalized basis. Then with Z Bjk := ωj , bk

the matrix B = (Bjk ) is symmetric and Im B is positive definite. Proof. First we show B is symmetric. To this end, Stokes’ Theorem and Lemma 14.2.1 give Z ωj ∧ ωk 0 = ZX = fωj ωk ∂U

=

g X

A` (ωj )B` (ωk ) − B` (ωj )A` (ωk )

`=1

= Bj (ωk ) − Bk (ωj ). P Now let c = (c1 , ..., cg ) ∈ Rg − {0} and set ω = j cj ωj . Then ω 6≡ 0, so by Proposition 14.2.2 X X X 0 > Im Aj (ω)Bj (¯ ω ) = Im cj Bjk ck = −Im Bjk cj ck . j

j,k

jk


14.3

Reciprocity

14.3.1

Abelian differentials of the second and third kind

We are going to need the following fact. P ROPOSITION 14.3.1. Let X be a compact Riemann surface, x and y distinct points of X, n ≥ 1 an integer, and z a local coordinate vanishing at x. 1. There exists a unique meromorphic 1-form ηx,y all of whose ak -periods are 0, that is holomorphic on X − {x, y}, such that Resx (ηx,y ) = 1 and Resy (ηx,y ) = −1. 2. There exists a unique meromorphic 1-form ηxn all of whose ak -periods are 0, that is holomorphic on X − {x}, such that ηxn − z −(n+1) dz is holomorphic near x. 215

Proof. Fix an open cover {Ui } of X by coordinate charts. For part 1, choose meromorphic 1-forms ηi in Ui that are holomorphic on Ui − {x, y}, have simple poles at x and y, and have residue 1 at x if x ∈ Ui and residue −1 at y if y ∈ Ui . (For example, one could arrange that each Ui contains at most one element of the set {x, y}, and if Ui 3 x (resp. Ui 3 y) take ηi = (zi − zi (x))−1 dzi (resp. ηi = −(zi − zi (x))−1 dzi ).) For part 2, choose meromorphic 1-forms ηi in Ui that are holomorphic on Ui − {x}, such that if x ∈ Ui then ηi − (zi − zi (x))−(n+1) dzi is holomorphic. In both cases, deg(Res({ηi })) = 0. (In the second case Res({ηi }) = 0.) An application of Theorem 8.5.6 shows the existence of a meromorphic 1-form with the desired polar structure. Any two such meromorphic sections clearly differ by a holomorphic 1-form. In view of Corollary 14.2.3, we can annihilate the periods of our meromorphic 1-forms by adding an appropriate holomorphic 1-form. Proposition 14.2.2 then implies that these 1-forms are uniquely determined. R EMARK . Meromorphic 1-forms of the types appearing in 1 and 2 of Proposition 14.3.1 are classically respectively called (normalized) Abelian differentials of the second and third kind. Abelian differentials of the first kind are just holomorphic 1-forms. It is an exercise to show that any meromorphic 1-form can be expressed as a linear combination of abelian differentials of the first, second and third kind.

14.3.2

The Reciprocity Theorem

We retain the notation of the previous section. T HEOREM 14.3.2. Fix a compact Riemann surface X of genus g. Let ω1 , ..., ωg ∈ ΓO (X, KX ) be a normalized basis, and let ηx,y and ηxn be normalized Abelian differentials of the second and third kind respectively. 1. If γx,y is a curve joining x to y in X − {a1 , ..., ag , b1 , ..., bg } then Z Z √ ωk , k = 1, .., g. ηx,y = 2π −1 γx,y

bk

2. If z is the local coordinate used to define ηxn and ωk = fk (z)dz, then √ Z 2π −1 (n−1) ηxn = fk (z(x)), k = 1, .., g. n! bk Proof. We identify X − {a1 , ..., ag , b1 , ..., bg } with the interior U of the polygon used to construct the topological surface X. Let Z z

gk (z) :=

ωk . xo

Then by Lemma 14.2.1 (note the remark after the proof of Lemma 14.2.1) together with the normalization condition, Z Z gk ηx,y = Bk (ηx,y ) = ∂U

ηx,y . bk

216

On the other hand, by the Residue Theorem we have Z Z √ √ gk ηx,y = 2π −1(gk (x) − gk (y)) = 2π −1 ∂U

ωk .

γx,y

This proves 1. Establishing 2 is similar: Z Z √ √ ∂n (n−1) n gk ηxn = 2π −1Resx (gk ηxn ) = n gk = 2π −1fk (z(x)). ηx = ∂z ∂U bk The proof is complete.

14.4

Proof of Abel’s Theorem

Fix a normalized basis ω1 , ..., ωg ∈ ΓO (X, KX ). We begin with the following reduction. P L EMMA 14.4.1. A divisor D = j (xj − yj ) is the divisor of a meromorphic function f if and only if there exist constants ck ∈ C, k = 1, ..., N , such that the meromorphic 1-form α :=

N X

ηxj ,yj +

j=1

g X

ck ω k

(14.2)

k=1

√ has 2π −1-commensurate periods, i.e., √ Ak (α), Bk (α) ∈ 2π −1Z,

k = 1, ..., g.

Proof. Suppose there is a meromorphic function f with Ord(f ) = D. Then g

N

X X df = ηxj ,yj + ck ω k f j=1 k=1 for some constants ck , and we compute that Z √ df ∈ 2π −1Z γ f for all closed curves γ in X − Support(D). Conversely, if there exist c1 , ..., cg such that, with α defined by (14.2), Z √ α ∈ 2π −1Z γ

for all closed curves γ not passing through the support of D, then the (well-defined) function f given by Rx f (x) := e xo α has the property that Ord(f ) = D. 217

Conclusion of the proof of Abel’s Theorem. Let α be the form given by (14.2). By definition of ηx,y (and especially that the ak -periods are all 0), Ak (α) = ck , and by the Reciprocity Theorem, Bk (α) =

N X

√

Z

yj

2π −1

ωk + xj

j=1

where

g X

c` B`k ,

`=1

Z ωk .

B`k = b`

Thus N Z X j=1

! g X 1 √ Bk (α) − Bk` A` (α) = 2π −1 `=1 Z Z g X 1 1 √ Bk (α) ωk − √ A` (α) ωk . = 2π −1 2π −1 a b ` ` `=1

yj

ωk

xj

Now, Lemma 14.4.1 tells us that D is the divisor of a meromorphic function if and only if 1 √

2π −1

Ak (α),

1 √

2π −1

Bk (α) ∈ Z.

It follows that D is the divisor of a meromorphic function if and only if there are integers nk` and mk` such that Z Z g N Z yj X X ωk = nk` ωk + mk` ωk , k = 1, ..., g. j=1

xj

a`

`=1

b`

But the latter holds if and only if for every holomorphic form θ, X Z yj θ ∈ Π(X, θ). j

xj


14.5

A discussion of Jacobi’s Inversion Theorem

In Paragraph 14.1.2 we defined a map A : X → J(X) from a compact Riemann surface X to its Jacobian, and an associated map µ : Div0 (X) → J(X) from the set of divisors of degree 0 on 218

X to the Jacobian J(X). To recall the definition of µ, note that any degree-zero divisor D can be written as a finite sum X D= xj − y j j

where xi 6= yj for all i, j. (However the xj are not necessarily distinct, and similarly for the yj .) We define the map µ by X µ(D) := A (xj ) − A (yj ). j

The map µ is clearly a group homomorphism. Abel’s Theorem tells us that the kernel of this homomorphism is the set of linearly trivial divisors, i.e., divisors of a meromorphic function. Recall that the quotient of the set of divisors Div(X) by the set of divisors of meromorphic functions is Pic(X), the group of all holomorphic line bundles on X. The subgroup of equivalence classes of degree-0 line bundles is called Pic0 (X). Modding out by the kernel of µ, we obtain a map µ ˜ : Pic0 (X) → J(X). Abel’s Theorem is precisely the statement that µ ˜ is injective. The Jacobi Inversion Theorem can be stated in this context as follows. T HEOREM 14.5.1 (Jacobi Inversion). The map µ ˜ : Pic0 (X) → J(X) is surjective. We will not give a complete proof of Jacobi’s Inversion Theorem. Instead we content ourselves with an argument that we hope is convincing. Let X be a compact Riemann surface of genus g. Consider the quotient X (g) := X g /Sg of the g-fold Cartesian product of X by the group of permutations of g elements. That is to say, X (g) is the set of unordered g-tuples of points of X. It is not hard to see that at a g-tuple of distinct points, X (g) looks like a manifold. Amazingly, the points of X (g) with repetitions are also smooth. To establish smoothness of X (g) at “multiple points”, the reader can work locally with a disk. The idea is then to think of the points of D(g) as roots of a polynomial of degree g, and using the symmetric functions of the roots as local coordinates. 0 (g) By choosing a base point Pg z ∈ X, we can identify X with a subset of Div (X) by sending {x1 , ..., xg } to the divisor i=1 (xi − z). Now define µ ˜(g) : X (g) → J(X) by µ ˜

(g)

g X ({x1 , ..., xg }) = (A (xi ) − A (z)). i=1

The goal is then to show that µ ˜(g) (X (g) ) = J(X). This goal is achieved in two steps. 219

1. First, one shows that the image of µ is open and dense, as follows. We choose local coordinates zi centered at xi . We can also choose torus coordinates on J(X) as follows. By fixing a basis a1 , b1 , ..., ag , bg for H1 (X, Z) and a corresponding normalized basis ω1 , ..., ωg of ΓO (X, KX ), we identify (ΓO (X, KX ))∗ with Cg , and identify Π(H1 (X, Z)) with the lattice Λ in Cg ∼ = R2g generated by the 2g real vectors e2j−1 ,

g Z X k=1

ωj e2k ,

j = 1, ..., g.

bk

The quotient of Cg by this lattice is naturally isomorphic to the complex torus J(X). In the coordinates chosen above, (g)

Z

µ ˜ {z1 , ..., zg } =

z1

Z

zg

ωg .

ω1 , ..., z

z

(The square brackets indicate that the vector is taken modulo periods.) Thus d˜ µ(g) = (ω1 , ..., ωg ). Since the dimension of (ΓO (X, KX ))∗ is g, the rank of d˜ µ(g) must be g at some point. But the points of maximal rank occupy an open dense set. Thus since X (g) and J(X) have the same dimension, the implicit function theorem shows that the image of µ ˜(g) is open and dense. 2. The map µ ˜(g) , being a holomorphic (and thus continuous) map of equidimensional compact complex manifolds whose Jacobian is not identically zero, must be a proper map, and in particular, its image is closed. It follows that µ ˜(g) is surjective, as claimed. In conjunction with Abel’s Theorem we deduce the following fact. T HEOREM 14.5.2 (Structure of the moduli space of degree zero line bundles). The collection Pic0 (X) of holomorphic line bundles of degree 0 on a compact Riemann surface X of positive genus is naturally identified with the Jacobian J(X). In particular, Pic0 (X) has the structure (group structure included) of a compact complex torus of dimension equal to the genus of X.

220

Bibliography [Ahlfors-1978] Ahlfors, L., Complex Analysis. An introduction to the theory of analytic functions of one complex variable. McGraw-Hill, 1978. [Ahlfors & Sario-1960] Ahlfors, L.; Sario, L., Riemann Surfaces. Princeton Math. Series, No. 26. Princeton University Press, Princeton, NJ 1960. [D’Angelo-1993] D’Angelo, J. P., Several Complex Variables and the Geometry of Real Hypersurfaces. CRC Press, 1993. [Farkas & Kra-1990] Farkas, H.; Kra, I, Riemann Surfaces. 2nd Edition. Springer GTM-71, Springer Verlag, 1990. [Forster-1991] Forster, O. Lectures on Riemann Surfaces. Springer GTM 81, Springer Verlag, 1991. [Griffiths & Harris-1978] Griffiths, P., Harris, J., Principles of Algebraic Geometry Wiley Interscience, 1978. [Hörmander-1990] Hörmander, L., An introduction to complex analysis in several variables. Third edition, North-Holland, 1990. [Narasimhan-1992] Narasimhan, R., Compact Riemann Surfaces. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1992. [Rudin-1991] Rudin, W., Functional Analysis. Second edition. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, 1991.

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