An Introduction to Riemann Surfaces

Cornerstones Series Editors Charles L. Epstein, University of Pennsylvania, Philadelphia, PA, USA Steven G. Krantz, Washington University, St. Louis, MO, USA Advisory Board Anthony W. Knapp, Emeritus, State University of New York at Stony Brook, Stony Brook, NY, USA

For further volumes: www.springer.com/series/7161

Terrence Napier  Mohan Ramachandran

An Introduction to Riemann Surfaces

Terrence Napier Department of Mathematics Lehigh University Bethlehem, PA 18015 USA [email protected]

Mohan Ramachandran Department of Mathematics SUNY at Buffalo Buffalo, NY 14260 USA [email protected]

ISBN 978-0-8176-4692-9 e-ISBN 978-0-8176-4693-6 DOI 10.1007/978-0-8176-4693-6 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011936871 Mathematics Subject Classification (2010): 14H55, 30Fxx, 32-01 © Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.birkhauser-science.com)

To Raghavan Narasimhan

Preface

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

viii

Preface

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

Preface

ix

Much of the background material required for study of this book is provided in detail in Part III (Chaps. 7–11). It is strongly recommended that, rather than first studying all of Part III, the reader instead focus on Parts I and II, and consult the appropriate sections in Part III only as needed (tables providing the interdependence of the sections appear after table of contents). In fact, the background material has been placed in the last part of the book instead of the first in order to make such consultation more convenient. The main prerequisite for the book is some knowledge of point-set topology (as in, for example, [Mu]) and elementary measure theory (as in, for example, Chap. 1 of [Rud1]), although parts of these subjects are reviewed in Sects. 7.1 and 9.1. On the other hand, smooth manifolds (as in Chap. 9 and in, for example, [Mat], [Ns3], and [Wa]) and Hilbert spaces (as in Chap. 7 and in, for example, [Fol] and [Rud1]) are essential objects in this book, so it would be to the reader’s advantage to have had some previous experience with these objects. It may surprise the reader to learn that previous experience with complex analysis in the plane (as in, for example, [Ns5]), while helpful, is not necessary. In fact, as indicated earlier, Chaps. 1 and 2 and Sect. 5.2 (along with the required background material and the corresponding exercises) together actually provide the material for a fairly complete course on complex analysis on domains in the plane along with the analogous study of complex analysis on Riemann surfaces (other suggested course outlines appear after the tables listing the interdependence of the sections). Exercises are included for most of the sections, and some of the theory is developed in the exercises. Acknowledgements The authors would first like to especially thank Raghavan Narasimhan for his comments on the first draft of this book, and for all he has taught them. Next, for their helpful comments and corrections, the authors would like to thank Charles Epstein and Dror Varolin, as well as the following students: Breeanne Baker, Qiang Chen, Tom Concannon, Patty Garmirian, Spyro Roubos, Kathleen Ryan, Brittany Shelton, Jin Yi, and Yingying Zhang. On the other hand, any mistakes or incoherent statements in the book are entirely the fault of the authors. Finally, the authors would like to thank their families and friends for their patience during the preparation of this book. Bethlehem, PA, USA Buffalo, NY, USA

T. Napier M. Ramachandran

Contents

Interdependence of Sections . . . . . . . . . . . . . . . . . . . . . . . . . .

xv

Suggested Course Outlines . . . . . . . . . . . . . . . . . . . . . . . . . . xvii Part I

Analysis on Riemann Surfaces

1

Complex Analysis in C . . . . . . . . . . . . . . . . . 1.1 Holomorphic Functions . . . . . . . . . . . . . . 1.2 Local Solutions of the Cauchy–Riemann Equation 1.3 Power Series Representation . . . . . . . . . . . . 1.4 Complex Differentiability . . . . . . . . . . . . . 1.5 The Holomorphic Inverse Function Theorem in C 1.6 Examples of Holomorphic Functions . . . . . . .

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3 3 6 12 16 18 21

2

¯ Riemann Surfaces and the L2 ∂-Method for Scalar-Valued Forms . 2.1 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . 2.2 Holomorphic Functions and Mappings . . . . . . . . . . . . . . 2.3 Holomorphic Attachment . . . . . . . . . . . . . . . . . . . . . 2.4 Holomorphic Tangent Bundle . . . . . . . . . . . . . . . . . . . 2.5 Differential Forms on a Riemann Surface . . . . . . . . . . . . . 2.6 L2 Scalar-Valued Differential Forms on a Riemann Surface . . . 2.7 The Distributional ∂¯ Operator on Scalar-Valued Forms . . . . . . 2.8 Curvature and the Fundamental Estimate for Scalar-Valued Forms ¯ 2.9 The L2 ∂-Method for Scalar-Valued Forms of Type (1, 0) . . . . 2.10 Existence of Meromorphic 1-Forms and Meromorphic Functions 2.11 Radó’s Theorem on Second Countability . . . . . . . . . . . . . ¯ for Scalar-Valued Forms of Type (0, 0) . . . . 2.12 The L2 ∂-Method 2.13 Topological Hulls and Chains to Infinity . . . . . . . . . . . . . 2.14 Construction of a Subharmonic Exhaustion Function . . . . . . . 2.15 The Mittag-Leffler Theorem . . . . . . . . . . . . . . . . . . . . 2.16 The Runge Approximation Theorem . . . . . . . . . . . . . . .

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25 26 30 35 38 45 53 60 63 65 67 71 73 77 82 88 91

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xi

xii

3

Contents

¯ The L2 ∂-Method in a Holomorphic Line Bundle . . . . . . . . . . 3.1 Holomorphic Line Bundles . . . . . . . . . . . . . . . . . . . . 3.2 Sheaves Associated to a Holomorphic Line Bundle . . . . . . . . 3.3 Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 The ∂¯ Operator and Dolbeault Cohomology . . . . . . . . . . . 3.5 Hermitian Holomorphic Line Bundles . . . . . . . . . . . . . . . 3.6 L2 Forms with Values in a Hermitian Holomorphic Line Bundle . 3.7 The Connection and Curvature in a Line Bundle . . . . . . . . . 3.8 The Distributional ∂¯ Operator in a Holomorphic Line Bundle . . ¯ 3.9 The L2 ∂-Method for Line-Bundle-Valued Forms of Type (1, 0) . ¯ 3.10 The L2 ∂-Method for Line-Bundle-Valued Forms of Type (0, 0) . 3.11 Positive Curvature on an Open Riemann Surface . . . . . . . . . 3.12 The Weierstrass Theorem . . . . . . . . . . . . . . . . . . . . .

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101 101 115 119 124 128 131 134 139 140 142 146 151

4

Compact Riemann Surfaces . . . . . . . . . . . . . . . . . . . . . . . 4.1 Existence of Holomorphic Sections on a Compact Riemann Surface 4.2 Positive Curvature on a Compact Riemann Surface . . . . . . . . . 4.3 Equivalence of Positive Curvature and Positive Degree . . . . . . . 4.4 A Finiteness Theorem . . . . . . . . . . . . . . . . . . . . . . . . 4.5 The Riemann–Roch Formula . . . . . . . . . . . . . . . . . . . . 4.6 Statement of the Serre Duality Theorem . . . . . . . . . . . . . . ¯ 4.7 Statement of the ∂-Hodge Decomposition Theorem . . . . . . . . ¯ 4.8 Proof of Serre Duality and ∂-Hodge Decomposition . . . . . . . . 4.9 Hodge Decomposition for Scalar-Valued Forms . . . . . . . . . .

157 157 160 162 164 165 167 177 181 185

5

Uniformization and Embedding of Riemann Surfaces . . . . . . 5.1 Holomorphic Covering Spaces . . . . . . . . . . . . . . . . 5.2 The Riemann Mapping Theorem in the Plane . . . . . . . . . 5.3 Holomorphic Attachment of Caps . . . . . . . . . . . . . . . 5.4 Exhaustion by Domains with Circular Boundary Components 5.5 Koebe Uniformization . . . . . . . . . . . . . . . . . . . . . 5.6 Automorphisms and Quotients of C . . . . . . . . . . . . . . 5.7 Aut (P1 ) and Uniqueness of the Quotient . . . . . . . . . . . 5.8 Automorphisms of the Disk . . . . . . . . . . . . . . . . . . 5.9 Classification of Riemann Surfaces as Quotient Spaces . . . . 5.10 Smooth Jordan Curves and Homology . . . . . . . . . . . . 5.11 Separating Smooth Jordan Curves . . . . . . . . . . . . . . . 5.12 Holomorphic Attachment and Removal of Tubes . . . . . . . 5.13 Tubes in a Compact Riemann Surface . . . . . . . . . . . . . 5.14 Tubes in an Arbitrary Riemann Surface . . . . . . . . . . . . 5.15 Nonseparating Smooth Jordan Curves . . . . . . . . . . . . . 5.16 Canonical Homology Bases in a Compact Riemann Surface . 5.17 Complements of Connected Closed Subsets of P1 . . . . . . 5.18 Embedding of an Open Riemann Surface into C3 . . . . . . .

191 192 204 207 209 211 216 218 220 224 228 236 241 249 254 264 267 275 279

Part II

Further Topics

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Contents

5.19 5.20 5.21 5.22

xiii

Embedding of a Compact Riemann Surface into Pn Finite Holomorphic Branched Coverings . . . . . . Abel’s Theorem . . . . . . . . . . . . . . . . . . . The Abel–Jacobi Embedding . . . . . . . . . . . .

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. . . .

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290 294 302 306

Holomorphic Structures on Topological Surfaces . . . . . . . . 6.1 Almost Complex Structures on Smooth Surfaces . . . . . . . 6.2 Construction of a Special Local Coordinate . . . . . . . . . . 6.3 Regularity of Solutions on an Almost Complex Surface . . . ¯ Connection, and Curvature . . . . . . . 6.4 The Distributional ∂, 2 6.5 L Solutions on an Almost Complex Surface . . . . . . . . . 6.6 Proof of Integrability . . . . . . . . . . . . . . . . . . . . . . 6.7 Statement of Schönflies’ Theorem . . . . . . . . . . . . . . . 6.8 Harmonic Functions and the Dirichlet Problem . . . . . . . . 6.9 Proof of Schönflies’ Theorem . . . . . . . . . . . . . . . . . 6.10 Orientable Topological Surfaces . . . . . . . . . . . . . . . . 6.11 Smooth Structures on Second Countable Topological Surfaces

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311 311 320 322 326 328 329 332 338 350 356 364

7

Background Material on Analysis in Rn and Hilbert Space Theory 7.1 Measures and Integration . . . . . . . . . . . . . . . . . . . . . 7.2 Differentiation and Integration in Rn . . . . . . . . . . . . . . . 7.3 C ∞ Approximation . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Differential Operators and Formal Adjoints . . . . . . . . . . . . 7.5 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Weak Sequential Compactness . . . . . . . . . . . . . . . . . .

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375 375 382 390 392 397 403

8

Background Material on Linear Algebra . . . . . . . . . . 8.1 Linear Maps, Linear Functionals, and Complexifications 8.2 Exterior Products . . . . . . . . . . . . . . . . . . . . 8.3 Tensor Products . . . . . . . . . . . . . . . . . . . . .

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407 407 409 412

9

Background Material on Manifolds . . . . . . . . . . . . . . . . 9.1 Topological Spaces . . . . . . . . . . . . . . . . . . . . . . 9.2 The Definition of a Manifold . . . . . . . . . . . . . . . . . 9.3 The Topology of Manifolds . . . . . . . . . . . . . . . . . . 9.4 The Tangent and Cotangent Bundles . . . . . . . . . . . . . 9.5 Differential Forms on Smooth Curves and Surfaces . . . . . . 9.6 Measurability in a Smooth Manifold . . . . . . . . . . . . . 9.7 Lebesgue Integration on Curves and Surfaces . . . . . . . . . 9.8 Linear Differential Operators on Manifolds . . . . . . . . . . 9.9 C ∞ Embeddings . . . . . . . . . . . . . . . . . . . . . . . . 9.10 Classification of Second Countable 1-Dimensional Manifolds 9.11 Riemannian Metrics . . . . . . . . . . . . . . . . . . . . . .

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415 415 419 423 428 437 447 449 462 465 469 475

6

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Part III Background Material

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xiv

Contents

10 Background Material on Fundamental Groups, Covering Spaces, and (Co)homology . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 The Fundamental Group . . . . . . . . . . . . . . . . . . . . . 10.2 Elementary Properties of Covering Spaces . . . . . . . . . . . 10.3 The Universal Covering . . . . . . . . . . . . . . . . . . . . . 10.4 Deck Transformations . . . . . . . . . . . . . . . . . . . . . . 10.5 Line Integrals on C ∞ Surfaces . . . . . . . . . . . . . . . . . . 10.6 Homology and Cohomology of Second Countable C ∞ Surfaces 10.7 Homology and Cohomology of Second Countable C 0 Surfaces

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477 477 483 489 492 498 503 511

11 Background Material on Sobolev Spaces and Regularity 11.1 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . 11.2 Uniform Convergence of Derivatives . . . . . . . . . 11.3 The Strong Friedrichs Lemma . . . . . . . . . . . . . 11.4 The Sobolev Lemma . . . . . . . . . . . . . . . . . . 11.5 Proof of the Regularity Theorem . . . . . . . . . . .

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531 532 534 535 541 542

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 Notation Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553

Interdependence of Sections

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

Part I

Require(s)

Part II

Require(s)

1.1–1.6 2.1–2.2 2.3 2.4 2.5 2.6 2.7 2.8, 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 3.1–3.10 3.11, 3.12

7.1–7.4 Chap. 1, 9.1 2.1, 2.2 2.1, 2.2, 8.1, 9.2, 9.4 2.4, 8.2, 9.5–9.7 2.5, 7.5 2.5, 9.8 2.6, 2.7 2.9 or 3.9 2.10 2.10 9.1–9.3 2.8, 2.11, 2.13 2.12, 2.14 2.12, 2.14 8.3, 2.5, 2.7, 2.8 2.14, 3.10

4.1 4.2

3.10 2.14 (2.11 is not required), 4.1 4.2 4.2 10.1–10.5, 2.3, 2.16 5.5 9.9, 10.1–10.6 5.5, 5.10, 5.11 5.10, 5.11, 10.7 5.13, 5.15 (Chap. 4 for Corollary 5.16.4) 5.2–5.4, 5.15 5.17 Chap. 4 5.1 Chap. 4, 5.20, 10.6, 10.7 2.9, 9.11, Chap. 11, 7.6 5.15, 5.17, 9.10 5.10, 5.11, 10.1–10.7 6.7–6.10

4.3 4.4–4.9 5.1–5.5 5.6–5.9 5.10, 5.11 5.12–5.14 5.15 5.16 5.17 5.18 5.19 5.20 5.21–5.22 6.1–6.6 6.7–6.9 6.10 6.11

xv

Suggested Course Outlines

A typical one-semester course would begin with careful study of Sects. 1.1 and 1.2, a quick reading of Sects. 1.3–1.6, careful study of Sects. 2.1 and 2.2, some (possibly brief) consideration of Sect. 2.3, and careful study of Sects. 2.4 and 2.5. The following are some of the natural directions in which the course could then proceed: • Holomorphic and meromorphic functions on an open Riemann surface: Sections 2.6–2.16 and, possibly, 5.2. • Holomorphic line bundles on an open Riemann surface: Sections 2.7, 2.8, 3.1–3.10, 2.11, 2.13, 2.14, 3.11, and 3.12. • Uniformization: Sections 2.6–2.14, 2.16, and 5.1–5.9. • Compact Riemann surfaces: Sections 2.7, 2.8, 3.1–3.10, 2.13, 2.14 (here, one needs the results of Sect. 2.14 only for an open subset of a compact Riemann surface, which is automatically second countable, so Radó’s theorem from Sect. 2.11 is not required for this case), 4.1–4.9, and 5.19. • Integrability of almost complex structures on a surface: Sections 2.6–2.9 and 6.1–6.6. Suggested Topics for Further Study Beyond a one-semester course, the reader may wish to consider, for example: • The characterization of a Riemann surface in terms of holomorphic attachment of tubes: Sections 5.10–5.14. • Further topics concerning compact Riemann surfaces: Sections 5.10–5.13, 5.15, 5.16, 5.19–5.22. • The embedding theorem for an open Riemann surface: Sections 5.17 and 5.18. • The existence of smooth structures on second countable topological surfaces: Sections 5.17 and 6.7–6.11.

xvii

Part I

Analysis on Riemann Surfaces

Chapter 1

Complex Analysis in C

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

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

and

  ∂ 1 ∂ ∂ ≡ +i . ∂ z¯ 2 ∂x ∂y

The operators ∂/∂z and ∂/∂ z¯ obey the standard sum, product, and quotient rules. Moreover, if γ : (a, b) → C is a (real) differentiable function on an interval (a, b) and u is a C 1 function on a neighborhood of the image, then ∂u dγ ∂u d dγ u(γ (t)) = (γ (t)) + (γ (t)) . dt ∂z dt ∂ z¯ dt T. Napier, M. Ramachandran, An Introduction to Riemann Surfaces, Cornerstones, DOI 10.1007/978-0-8176-4693-6_1, © Springer Science+Business Media, LLC 2011

3

4

1

Complex Analysis in C

Observe that if f is a C 1 function on an open set in C, and u = Re f and v = Im f , then ∂f =0 ∂ z¯

⇐⇒

∂u ∂v = ∂x ∂y

and

∂u ∂v =− . ∂y ∂x

We refer to the above as the homogeneous Cauchy–Riemann equation(s). We use a similar notation if elements of C are represented by other names (for example, ζ = s + it). Definition 1.1.1 Let  be an open subset of C. A function f ∈ C 1 () is called holomorphic (or analytic or complex analytic) on  if ∂f/∂ z¯ ≡ 0 on . We denote the set of all holomorphic functions on  by O(), and for each function f ∈ O(), we define d ∂f ∂f ∂f f = [f (z)] ≡ = = −i . dz ∂z ∂x ∂y We also say that f is a primitive for the function f . A holomorphic function on C is also called an entire function. The proof of the following is left to the reader (see Exercise 1.1.1): Theorem 1.1.2 Any sum, product, quotient, or composition of holomorphic functions is holomorphic on its domain. Moreover, if f and g are holomorphic functions, then (f + g) = f + g , (f/g) =

f g − fg , g2

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

Consequently, every rational function of z is holomorphic on its domain, and for every n ∈ Z, d(zn )/dz = nzn−1 . An important tool in this book (and in the general study of analysis on manifolds) is the machinery of differential forms. The required definitions and facts appear in Chap. 9. For now, we take an informal approach to differential forms on open subsets of C, which will suffice for this chapter. We have the real differentials dx and dy on the complex plane C (identified with R2 ) and the complex differentials dz = dx + i dy

and d z¯ = dx − i dy.

Any complex 1-form α may be written α = P dx + Q dy = a dz + b d z¯ ,

1.1 Holomorphic Functions

5

for some (unique) pair of complex-valued functions P and Q and for a = 12 (P − iQ) and b = 12 (P + iQ). If α is continuous (i.e., if the coefficients P and Q are continuous) on a neighborhood of the image of a C 1 path γ = (u + iv) : [r, s] → C, with u = Re γ and v = Im γ , then   s  du dv P (γ (t)) α= + Q(γ (t)) dt dt dt γ r   s dγ dγ + b(γ (t)) dt. = a(γ (t)) dt dt r The exterior product of dx and dy is denoted by dx ∧ dy = −dy ∧ dx (and dx ∧ dx = dy ∧ dy = 0). Any complex 2-form β may be written β = f dx ∧ dy = f · (i/2) dz ∧ d z¯ , for some (unique) function f . If f is integrable on a Lebesgue measurable set E ⊂ C, then    i f dλ = f dx ∧ dy = f · dz ∧ d z¯ , 2 E E E where λ denotes Lebesgue measure (see Sect. 7.1). If f is a C 1 function on an open set in C, then the differential (or exterior derivative) of f satisfies df =

∂f ∂f ∂f ∂f ¯ dx + dy = dz + d z¯ = ∂f + ∂f, ∂x ∂y ∂z ∂ z¯

where ∂f dz ∂z

¯ ≡ ∂f d z¯ . ∂f ∂ z¯ ¯ = 0 (equivalently, df = h dz for In particular, f is holomorphic if and only if ∂f some function h). If α = P dx + Q dy is a 1-form of class C 1 (i.e., the coefficients P and Q are of class C 1 ), then the exterior derivative of α is given by   ∂Q ∂P dα = dP ∧ dx + dQ ∧ dy = − dx ∧ dy. ∂x ∂y ∂f ≡

and

Equivalently, for α = a dz + b d z¯ (with a = 12 (P − iQ) and b = 12 (P + iQ)),   ∂b ∂a ¯ dα = da ∧ dz + db ∧ d z¯ = − dz ∧ d z¯ = ∂α + ∂α, ∂z ∂ z¯ where ∂α = (∂b) ∧ d z¯ =

∂b dz ∧ d z¯ ∂z

and

¯ = (∂a) ¯ ∧ dz = − ∂a dz ∧ d z¯ . ∂α ∂ z¯

6

1

Complex Analysis in C

According to Stokes’ theorem, if   C is a smooth domain and α is a C 1 1-form on a neighborhood of , then   α= dα, ∂



where the left-hand side is given by the sum of the integrals of α over the finitely many boundary curves of , each parametrized so that  lies to the left (i.e., each boundary curve is positively oriented relative to ). Exercises for Sect. 1.1 1.1.1. Prove Theorem 1.1.2.

1.2 Local Solutions of the Cauchy–Riemann Equation For 0 < r < R < ∞ and z0 ∈ C, we set (z0 ; R) ≡ {z ∈ C | |z − z0 | < R}, (z0 ; r, R) ≡ {z ∈ C | r < |z − z0 | < R}, ∗ (z0 ; R) ≡ {z ∈ C | 0 < |z − z0 | < R} = (z0 ; 0, R), C∗ ≡ C \ {0}. Lemma 1.2.1 Let  be a smooth relatively compact domain in C, and let f be a C 1 function on a neighborhood of . (a) Cauchy integral formula. For each point z ∈ , we have 1 f (z) = 2πi

 ∂

f (ζ ) dζ + ζ −z

 

 ∂f/∂ ζ¯ ¯ dζ ∧ d ζ . ζ −z

(b) Cauchy’s theorem. We have   ∂f f (ζ ) dζ + dζ ∧ d ζ¯ = 0. ∂  ∂ ζ¯ Remarks 1. In particular, for f holomorphic on a neighborhood of , we get (the more standard versions)   1 f (ζ ) f (z) = f (ζ ) dζ = 0. dζ ∀ z ∈  and 2πi ∂ ζ − z ∂ 2. The formula in part (a) is also called the Cauchy–Pompeiu integral formula or ¯ the ∂-Cauchy integral formula.

1.2 Local Solutions of the Cauchy–Riemann Equation

7

Proof of Lemma 1.2.1 Observe that the function ζ → (ζ − z)−1 is locally integrable in C, so the integrals in (a) are defined. For z ∈  and 0 < r < dist(z, ∂), Stokes’ theorem, together with the product rule and the holomorphicity of ζ → (ζ − z)−1 , gives     ∂f/∂ ζ¯ f (ζ ) dζ ∧ d ζ¯ = dζ d − ζ −z \ (z;r) ζ − z \ (z;r)   f (ζ ) f (ζ ) =− dζ + dζ ∂ ζ − z ∂ (z;r) ζ − z  2π  f (ζ ) dζ + =− f (z + reiθ )i dθ. 0 ∂ ζ − z Letting r → 0+ , we get (a). Part (b) follows from Stokes’ theorem (alternatively, one may fix a point z ∈  and apply (a) to the function ζ → f (ζ ) · (ζ − z)).  Lemma 1.2.2 (Local solution of the inhomogeneous Cauchy–Riemann equation) Let D be a disk in C, let α ∈ C 0 (∂D), let k ∈ Z>0 ∪ {∞}, let β be a C k function on a neighborhood of D, and, for each point z ∈ D, let    α(ζ ) β(ζ ) 1 ¯ f (z) = dζ + dζ ∧ d ζ . 2πi ∂D ζ − z D ζ −z Then f ∈ C k (D) and ∂f/∂ z¯ = β on D. In particular, f is holomorphic on D \ supp β. Proof Setting D = (z0 ; R), we may fix a number r with 0 < r < R, we may fix a C k function β1 with compact support in D such that β1 = β on a neighborhood of (z0 ; r), and we may set β2 ≡ β − β1 . Then f = g + h, where for each point z ∈ D,  2π  1 α(z0 + Reiθ ) β2 (ζ ) 1 iθ · Re dλ(ζ ) g(z) ≡ dθ − iθ 2π 0 z0 + Re − z π D ζ −z and h(z) ≡ −

1 π

 D

1 β1 (ζ ) dλ(ζ ) = − ζ −z π

 C

β1 (ζ ) dλ(ζ ). ζ −z

The functions (z, θ ) → α(z0 − z) and (z, ζ ) → β2 (ζ )/(ζ − z), 0 and their derivatives of arbitrary order in x and y (z = x + iy), are continuous and bounded on (z0 ; r) × [0, 2π] and (z0 ; r) × D, respectively. Thus differentiation past the integral (Proposition 7.2.5) implies that g is of class C ∞ on (z0 ; r) and      2π  ∂g ∂ α(z0 + Reiθ ) ∂ β2 (ζ ) 1 1 iθ (z) = · Re dλ(ζ ) = 0. dθ − ∂ z¯ 2π 0 ∂ z¯ z0 + Reiθ − z π D ∂ z¯ ζ − z + Reiθ )/(z

+ Reiθ

8

1

Complex Analysis in C

Thus g is holomorphic and of class C ∞ on (z0 ; r). Similarly, setting R ≡ R + r + |z0 |, we get, for each point z ∈ (z0 ; r), h(z) = −

1 π

=−

1 π

 C

β1 (ζ + z) · ζ −1 dλ(ζ ) = −



(z0 ;R )

1 π

 (z0 −z;R)

β1 (ζ + z) · ζ −1 dλ(ζ )

β1 (ζ + z) · ζ −1 dλ(ζ );

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

∂h 1 (z) = − ∂ z¯ π



Since supp β1 ⊂ D, the Cauchy integral formula (Lemma 1.2.1) now gives ∂h/∂ z¯ = β1 = β on (z0 ; r), and since the choice of r ∈ (0, R) was arbitrary, the claim follows.  Theorem 1.2.3 If  ⊂ C is an open set and f ∈ O(), then f ∈ C ∞ () and f ∈ O(). Proof For any disk D  , the Cauchy integral formula (Lemma 1.2.1) and the local solution of the Cauchy–Riemann equation (Lemma 1.2.2) together imply that f is of class C ∞ on D. It follows that f ∈ O(), because     ∂f ∂ ∂f ∂ ∂f = = = 0. ∂ z¯ ∂ z¯ ∂z ∂z ∂ z¯



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

∀p ∈ [1, ∞], f ∈ O().

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

1.2 Local Solutions of the Cauchy–Riemann Equation

9

and we may choose a function η ∈ D( (z0 ; R2 )) with η ≡ 1 on (z0 ; R1 ). For each f ∈ O() and each point z ∈ (z0 ; R1 ), the Cauchy integral formula gives  ∂η 1 f (ζ ) (ζ )(ζ − z)−1 dλ(ζ ). f (z) = η(z)f (z) = − π (z0 ;R2 )\ (z0 ;R1 ) ∂ ζ¯ The function (z, ζ ) → f (ζ ) · (∂η/∂ ζ¯ )(ζ ) · (ζ − z)−1 , and its derivatives of arbitrary order in x and y (for z = x +iy), are bounded on (z0 ; r)×[ (z0 ; R2 )\ (z0 ; R1 )]. Differentiation past the integral now gives, for z ∈ (z0 ; r),  1 ∂η [Af ](z) = − f (ζ ) (ζ )A[(ζ − ·)−1 ](z) dλ(ζ ). π (z0 ;R2 )\ (z0 ;R1 ) ∂ ζ¯ Hence sup |Af | ≤ Cz0 f L1 ( (z0 ;R2 )\ (z0 ;R1 )) ≤ Cz0 f L1 ()

(z0 ;r)

for some constant Cz0 independent of the choice of f . Covering K by finitely many such disks (z0 ; r), we get the desired constant C.  Corollary 1.2.5 For every open set  ⊂ C and every nonempty compact set K ⊂ , there is a constant C = C(, K) such that |f (w) − f (z)| ≤ Cf Lp () |w − z| for all w, z ∈ K, p ∈ [1, ∞], and f ∈ O() (cf. Definition 7.2.3). Proof Given a disk D  , Theorem 1.2.4 provides a (real) constant B such that supD |∂f/∂z| ≤ Bf Lp () for every f ∈ O() and p ∈ [1, ∞]. Hence, for every pair of points w, z ∈ D, for every function f ∈ O(), and for every number p ∈ [1, ∞], we have  1    d |f (w) − f (z)| =  [f (z + t (w − z))] dt  0 dt   1   ∂f (z + t (w − z)) · (w − z) dt  =  0 ∂z ≤ Bf Lp () |w − z|. Covering K by finitely many such disks and fixing a Lebesgue number δ > 0 for the covering, we get a constant B0 such that |f (w) − f (z)| ≤ B0 f Lp () |w − z| for all w, z ∈ K with |w − z| < δ, all f ∈ O(), and all p ∈ [1, ∞]. There is also a constant B1 such that maxK |f | ≤ B1 f Lp () for all f ∈ O() and all p ∈ [1, ∞].  Thus the constant C ≡ max(B0 , 2B1 /δ) has the required property. Corollary 1.2.6 If {fn } is a sequence of holomorphic functions converging uniformly on compact subsets of an open set  ⊂ C to a function f , then f ∈ O(), and for each k ∈ Z≥0 , the sequence of kth derivative functions {fn(k) } converges uniformly to f (k) on compact subsets of .

10

1

Complex Analysis in C

Proof For each disk D   and each point z ∈ D, the Cauchy integral formula gives   1 1 fn (ζ ) f (ζ ) dζ = dζ. f (z) = lim fn (z) = lim n→∞ n→∞ 2πi ∂D ζ − z 2πi ∂D ζ − z Thus Lemma 1.2.2 implies that f is holomorphic on D. That fn(k) → f (k) uniformly on each compact set K ⊂  follows easily from Theorem 1.2.4.  Corollary 1.2.7 (Montel’s theorem) If {fn } is a sequence of holomorphic functions that is uniformly on each compact subset of an open set  ⊂ C, then  bounded some subsequence fnk converges uniformly on compact subsets of  to a function f ∈ O(). Proof For any compact set K ⊂ , Corollary 1.2.5 implies that there is a constant C > 0 such that |fn (w) − fn (z)| ≤ C|w − z| for all w, z ∈ K and for every n ∈ Z>0 . Ascoli’s theorem (together with Cantor’s diagonal process) now provides a subsequence converging uniformly on compact subsets of , and Corollary 1.2.6 implies that the limit is holomorphic.  According to Definition 7.4.2, for L1loc functions u and v on an open set  ⊂ C, we have     ∂ ∂u ≡ u=v ∂ z¯ distr ∂ z¯ distr if and only if 

   ∂ϕ u − v ϕ¯ dλ ∀ϕ ∈ D(). dλ = ∂z  

Theorem 1.2.8 (Regularity theorem) If  is an open subset of C, u ∈ L1loc (), k ∈ Z>0 ∪ {∞}, β ∈ C k (), and (∂u/∂ z¯ )distr = β, then u ∈ C k () (i.e., there is a unique C k function that is equal to u almost everywhere in ). In particular, if (∂u/∂ z¯ )distr = 0, then u ∈ O() (i.e., there is a unique holomorphic function that is equal to u almost everywhere in ). Proof It suffices to show that u is of class C k on every disk D  . Thus, by subtracting the C k solution v of ∂v/∂ z¯ = β in D provided by Lemma 1.2.2, we may assume without loss of generality that β ≡ 0. To show that u is holomorphic on D, we fix a disk D with D  D   and a nonnegative C ∞ function κ with compact support in the unit disk (0; 1) in C such

that C κ dλ = 1. By Lemma 7.3.1 and Lemma 7.4.4, for each sufficiently large n ∈ Z>0 , the function un given by  u(ζ )n2 κ(n(z − ζ )) dλ(ζ ) ∀z ∈ D un (z) ≡ 

1.2 Local Solutions of the Cauchy–Riemann Equation

11

is of class C ∞ and ∂un /∂ z¯ = 0; that is, un ∈ O(D ). Moreover, un − uL1 (D ) → 0 as n → ∞. Hence Theorem 1.2.4 implies that the sequence {un |D } is Cauchy in (C 0 (D),  · L∞ (D) ) and therefore uniformly convergent. Corollary 1.2.6 now implies that u is holomorphic on D and therefore that u ∈ C k (D).  Theorem 1.2.9 (Mean value property) If f ∈ O( (z0 ; R)), then for every r ∈ (0, R), f (z0 ) =

1 2π



 1 f z0 + reiθ dθ = 2 πr

2π 0

 (z0 ;r)

f (ζ ) dλ(ζ ).

Proof Lemma 1.2.1 gives the first equality. For the second, we integrate to get r2 f (z0 ) = 2 =



r

0

1 2π

f (z0 )s ds =  (z0 ;r)

1 2π



r

0



2π

 f z0 + seiθ s dθ ds

0

f (ζ ) dλ(ζ ).



Theorem 1.2.10 (Riemann’s extension theorem) Let  ⊂ C be an open set, let z0 ∈ , and let f be a function that is holomorphic on  \ {z0 }. If lim (z − z0 )f (z) = 0

z→z0

or f ∈ L2 (),

then there exists a (unique) function f˜ ∈ O() such that f˜ = f on  \ {z0 }. Proof Let a(z) = z − z0 for each z ∈ C and fix R > 0 with D ≡ (z0 ; R)  . Assuming first that a(z)f (z) → 0 as z → z0 , we get f ∈ L1loc (). If ϕ ∈ D(D), then for every r ∈ (0, R), Stokes’ theorem gives    ∂ϕ f d[f ϕ dz] = f (z)ϕ(z) dz. dz ∧ d z¯ = − ∂ z¯ D\ (z0 ;r) D\ (z0 ;r) ∂ (z0 ;r) Letting r → 0+ , we get

 f D

∂ϕ dλ = 0. ∂ z¯

Thus (∂f/∂ z¯ )distr = 0 in , and the regularity theorem (Theorem 1.2.8) implies that f ∈ O(). For f ∈ L2 () and for z ∈ ∗ (z0 ; R/2), the mean value property (Theorem 1.2.9) and the Schwarz (or Hölder) inequality give, for r = |z − z0 |/2,     1  |a(z)f (z)| =  2 a(ζ )f (ζ ) dλ(ζ ) πr (z;r) ≤

1 aL2 ( (z;r)) · f L2 ( (z;r)) πr 2

12

1

Complex Analysis in C

1 aL2 ( (z0 ;3r)) · f L2 ( (z0 ;3r)) πr 2 9 = √ f L2 ( (z0 ;3r)) → 0 as r → 0+ . 2π ≤

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



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

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

n Theorem 1.3.1 (Abel) For any power series ∞ n=0 cn (z − z0 ) , either there is an R ∈ (0, ∞] such that the series converges uniformly and absolutely on compact subsets of D ≡ {z ∈ C | |z − z0 | < R} to a holomorphic function f and the series diverges for |z − z0 | > R, or the series converges only when z = z0 , in which case we set R = 0. Furthermore, the power series ∞  n=1

n · cn (z − z0 )n−1

and

∞  cn (z − z0 )n+1 n+1 n=0

have this same radius of convergence R; and if R > 0, then on the above set D, the sum of the first is equal to f , and the sum of the second has derivative f . Proof Let R = sup{|z − z0 | | the series converges at z}. Then, clearly, the series diverges for |z − z0 | > R. If 0 < r < R, then the series converges at some point z1 ∈ C

1.3 Power Series Representation

13

with r < t = |z1 − z0 | < R, and hence in particular, the sequence {|cn (z1 − z0 )n |} is bounded above. For each z ∈ (z0 ; r) and each n ∈ Z≥0 , we have |cn (z − z0 )n | ≤ |cn (z1 − z0 )n |(r/t)n . Therefore, by the Weierstrass M-test, the series converges uniCorollary 1.2.6 implies that the sum f (z) is formly and absolutely on (z0 ; r). n−1 . In particular, this seholomorphic on D and that f (z) = ∞ n=1 n · cn (z − z0 ) ries representing f (z) converges for |z − z0 | < R. The proofs of the remaining claims are left to the reader (see Exercise 1.3.1).  Conversely, we have the following: Theorem 1.3.2 If R ∈ (0, ∞], z0 ∈ C, and f is a holomorphic function on the set D = {z ∈ C | |z − z0 | < R}, then f (z) =

∞  f (n) (z0 ) n=0

n!

(z − z0 )n

∀z ∈ D.

f (ζ ) 1 Proof If 0 < r < s < R, then f (z) = 2πi ∂ (z0 ;s) ζ −z dζ for every point z in (z0 ; r). On the other hand, for all z ∈ (z0 ; r) and ζ ∈ ∂ (z0 ; s), we have |(z − z0 )/(ζ − z0 )| < r/s < 1 and hence ∞

 f (ζ )/(ζ − z0 ) f (ζ ) f (ζ )(z − z0 )n (ζ − z0 )−n−1 , = = ζ − z 1 − [(z − z0 )/(ζ − z0 )] n=0

and for each n ∈ Z≥0 , |f (ζ )(z − z0 )n (ζ − z0 )−n−1 | ≤ max |f | · r −1 · (r/s)n+1 . ∂ (z0 ;s)

Thus the Weierstrass M-test convergence in ζ and, integrating term

gives uniform n by term, we get f (z) = ∞ n=0 cn (z − z0 ) for each z ∈ (z0 ; r), where  f (ζ ) 1 cn = dζ ∀n ∈ Z≥0 . 2πi ∂ (z0 ;s) (ζ − z0 )n+1 For n ∈ Z≥0 , according to Theorem 1.3.1, differentiation of the above power series term by term n times gives f (n) (z). Evaluating at z0 , we get f (n) (z0 ) = cn · n!, as claimed.  Corollary 1.3.3 Let  be a domain (i.e., a connected open set) in C, and let f be a nonconstant holomorphic function on . Then: (a) For every point z0 ∈ , f (m) (z0 ) = 0 for some m ∈ Z≥0 . (b) For every point z0 ∈ , there are a unique m ∈ Z≥0 and a unique function g ∈ O() such that g(z0 ) = 0 and f (z) = (z − z0 )m g(z) for every point z ∈ . (c) Identity theorem. The set f −1 (0) has no limit points in .

14

1

Complex Analysis in C

(d) For u = Re f and v = Im f , the sets u−1 (0) and v −1 (0) are nowhere dense in . Remark Suppose f is a holomorphic function on an open set  ⊂ C and z0 ∈ C. It follows from part (b) that if f is not identically zero on some connected neighborhood of z0 in , then there exist a unique nonnegative integer m and a function g ∈ O() such that g(z0 ) = 0 and f (z) = (z − z0 )m g(z) for every point z ∈ . The nonnegative integer m is called the order of f at z0 and is denoted by ordz0 f . For m = ordz0 f > 0, z0 is called a zero of order m. For m = 1, we also say that f has a simple zero at p. If f vanishes on some neighborhood of z0 , then we say that f has order ordz0 f = ∞ at z0 .  (n) ]−1 (0) is closed in , and by TheProof of Corollary 1.3.3 The set A ≡ ∞ n=0 [f orem 1.3.2, A is also open. Part (a) now follows. For (b), observe that by (a), the set {n ∈ Z≥0 | f (n) (z0 ) = 0} is nonempty. Taking m to be the minimum of this set and applying Theorem 1.3.2, we get the claim. For the proof of (c), suppose z0 ∈ . By part (b), there are a nonvanishing holomorphic function g on a neighborhood U of z0 in  and an integer m ∈ Z≥0 such that f (z) = (z − z0 )m g(z) for each z ∈ U . Hence f −1 (0) ∩ U ⊂ {z0 } and the claim follows. Finally, for the proof of (d), observe that if u ≡ 0 on a disk D ⊂ , then on D, we have ∂v/∂z = ∂v/∂ z¯ = i∂u/∂ z¯ ≡ 0, and hence dv ≡ 0 and v is constant on D. But by the identity theorem (part (c) above), this is impossible. Thus u−1 (0) is nowhere dense. Since v is the real part of the holomorphic function −if , it follows that v −1 (0) is also nowhere dense.  Theorem 1.3.4 (Maximum principle) If the modulus |f | of a holomorphic function f on a domain  ⊂ C attains a local maximum at some point z0 ∈ , then f is constant on . Proof Let u = Re f and let D = (z0 ; R)   be a disk on which |f | ≤ |f (z0 )|. By Corollary 1.3.3, it suffices to show that u is constant on D. For this, we may assume without loss of generality that f (z0 ) = 0 (otherwise, we have f ≡ 0 on a neighborhood) and hence that f (z0 ) = 1 (divide by f (z0 )). The mean value property (Theorem 1.2.9) gives      1 1 1 f (ζ ) dλ(ζ ) = u(ζ ) dλ(ζ ) ≤ dλ(ζ ) = 1. 1 = Re πR 2 D πR 2 D πR 2 D Hence the nonnegative function 1 − u integrates to 0, and therefore u ≡ 1 on D.  Theorem 1.3.5 (Open mapping theorem) If f is a nonconstant holomorphic function on a domain  ⊂ C, then the image U ≡ f () is open. Proof Suppose w0 = f (z0 ) ∈ ∂U for some z0 ∈ . By the identity theorem (Corollary 1.3.3), there is a disk D   containing z0 such that w0 is not in the compact

1.3 Power Series Representation

15

set K ≡ f (∂D). Choosing a point w1 ∈ C \ U with |w1 − w0 | < dist(w1 , K), we get a function g ≡ (f − w1 )−1 ∈ O() whose maximum modulus on D is not attained at any point in ∂D. This contradicts the maximum principle (Theorem 1.3.4), so U must be open.  For a holomorphic function on a punctured disk, a punctured plane, or an annulus, we have the Laurent series representation provided by the following theorem, the proof of which is left to the reader (see Exercise 1.3.2): Theorem 1.3.6 (Laurent series representation) Let z0 ∈ C. constants, and let r and R be the (a) Let {cn }n∈Z be a collection of complex

∞ n and n radii of convergence for the power series ∞ c ζ −n n=1 n=0 cn ζ , respec ∞ n tively. If 1/r < R, series n=−∞ cn (z − z0 ) is absolutely

then the Laurent n | < ∞) and uniformly convergent (i.e., the summable (i.e., |c (z − z ) n 0 n∈Z

−n and ∞ c (z − z )n are uniformly convergent) series ∞ 0 n=1 c−n (z − z0 ) n=0 n on compact subsets of the set A ≡ {z ∈ C | 1/r < |z − z0 | < R}, and the sum z → f (z) ≡ =

∞ 

cn (z − z0 )n ≡



cn (z − z0 )n

n=−∞

n∈Z

∞ 

∞ 

c−n (z − z0 )−n +

n=1

cn (z − z0 )n

n=0

is a holomorphic function on A. Moreover, (i) For any constant t ∈ (1/r, R),  f (ζ ) 1 cn = dζ ∀n ∈ Z; 2πi ∂ (z0 ;t) (ζ − z0 )n+1

∞

n+1 and n−1 have radii of (ii) The power series ∞ n=1 (−n)c−n ζ n=1 ncn ζ convergence r and R, respectively, and f (z) =

∞ 

ncn (z − z0 )n−1

∀z ∈ A;

n=−∞

and

∞

n−1 /(−n + 1) and n+1 /(n + 1) (iii) The power series ∞ n=2 c−n ζ n=0 cn ζ have radii of convergence r and R, respectively, and if c−1 = 0, then the holomorphic function z → F (z) ≡

∞ 

cn (z − z0 )n+1 n + 1 n=−∞

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

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Complex Analysis in C

(b) Let s and S be constants with 0 ≤ s < S ≤ ∞; let f be a holomorphic function on the set B ≡ {z ∈ C | s < |z − z0 | < S}; let t ∈ (s, S); let  1 f (ζ ) dζ ∀n ∈ Z; cn ≡ 2πi ∂ (z0 ;t) (ζ − z0 )n+1 and let r and R be the radii of convergence of the power series n and ∞ n=0 cn ζ , respectively. Then 1/r ≤ s < S ≤ R and f (z) =



cn (z − z0 )n

∞

n=1 c−n ζ

n

∀z ∈ B.

n∈Z

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

1.4 Complex Differentiability Complex differentiability provides a third characterization of holomorphic functions. Definition 1.4.1 A function f defined on a neighborhood of a point z0 ∈ C is complex differentiable at z0 if lim

z→z0

f (z) − f (z0 ) z − z0

exists. We say that f is complex differentiable on an open set  ⊂ C if f is complex differentiable at each point in . If f is a function that is holomorphic on a neighborhood of a point z0 ∈ C, then applying part (b) of Corollary 1.3.3 to the holomorphic function f − f (z0 ), we get a holomorphic function h on a neighborhood of z0 such that f (z) − f (z0 ) = (z − z0 )h(z) for all z. It follows that f is complex differentiable at z0 . Conversely, if a function f is complex differentiable at a point z0 ∈ C, then f is continuous at z0 , and computing the limit in Definition 1.4.1 separately along the x and y directions, we get ∂f ∂f ∂f f (z) − f (z0 ) (z0 ) = −i (z0 ) = (z0 ) = lim z→z ∂x ∂y ∂z z − z0 0

and

∂f (z0 ) = 0. ∂ z¯

Consequently, if a function f is of class C 1 and complex differentiable on an open set, then f is holomorphic with derivative given by the limit appearing in Definition 1.4.1. Thus we have the following:

1.4 Complex Differentiability

17

Lemma 1.4.2 Let f be a C 1 function on an open set  ⊂ C. Then f ∈ O() if and only if f is complex differentiable on . Moreover, if this is the case, then f (z0 ) = lim

z→z0

f (z) − f (z0 ) z − z0

∀z ∈ .

It is not evident that a complex differentiable function on an open set is of class C 1 , so we cannot conclude from the above that such a function is holomorphic. Nevertheless, this is the case (see Exercise 1.4.2). In fact, by a theorem of Looman and Menchoff, any continuous function that satisfies the Cauchy–Riemann equations on an open set is holomorphic (see [Ns5]). On the other hand, in this book, we will need only the C 1 case addressed in Lemma 1.4.2. Exercises for Sect. 1.4 1.4.1 Prove Goursat’s theorem: If S ≡ (a, b) × (c, d) ⊂ C is a bounded open rectangular region and f is a function that is complex differentiable on a neighborhood of S, then ∂S f (z) dz = 0 (cf. Lemma 1.2.1).

Outline of a proof. One must show that M0 ≡ | ∂S f (z) dz| = 0. Let S0 = S. Breaking up S into four congruent rectangular regions, one sees that for one of

the rectangles S1 , M1 ≡ | ∂S1 f (z) dz| ≥ M40 . Proceeding inductively, one gets a decreasing sequence of open rectangular regions {Sν } such that for each ν ∈ Z>0 , Sν has side lengths 2−ν (b − a) and 2−ν (d − c) and      f (z) dz ≥ 4−ν M0 . Mν ≡  ∂Sν

There exists a point z0 in the intersection of the closures, and the function  f (z)−f (z0 ) − ∂f z−z0 ∂z (z0 ) if z ∈ S \ {z0 }, g(z) ≡ 0 if z = z0 , is continuous. For a constant r > 0, one gets M0 ≤ supSν |g| · r. Letting ν → ∞, one gets the claim. 1.4.2 Let f be a complex differentiable function on an open rectangular region S ⊂ C, let z0 = x0 + iy0 ∈ S with x0 , y0 ∈ R, and let F : S → C be the function given by  x  y z = x + iy → f (t + iy0 ) dt + i f (x + it) dt; x0

y0

that is, F (z) is obtained by integration of f (z)dz along the horizontal line segment from z0 to x + iy0 followed by integration along the vertical line segment from x + iy0 to z. (a) Show that ∂F /∂y = if on S. Applying Goursat’s theorem (see Exercise 1.4.1), express F (z) in terms of similar integrals that yield ∂F /∂x = f on S. Conclude from this that F is holomorphic and F = f .

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Complex Analysis in C

(b) Using part (a), prove that a function on an open set is holomorphic if and only if it is complex differentiable. 1.4.3 Show that complex differentiability provides a more direct proof that the sum of a power series is holomorphic than that proof of Theo appearing in the n is a power series c (z − z ) rem 1.3.1. More precisely, suppose that ∞ 0 n=0 n with radius of convergence R ∈ (0, ∞], and f (z) is the sum of this series for each point z ∈ D ≡ {z ∈ C | |z − z0 | < R}. The first part of the proof of Abel’s theorem on power series (Theorem 1.3.1) implies that the series converges absolutely on D and uniformly on any disk of radius < R centered at z0 . (a) Prove directly, without invoking Abel’s theorem, that the power series ∞ 

ncn (z − z0 )n−1

n=1

also converges for z ∈ D (and therefore that this power series has radius of convergence at least R). (b) Let g(z) be the sum of the series in part (a) for each point z ∈ D. Prove directly from Definition 1.4.1 (without invoking Corollary 1.2.6) that f is complex differentiable on D with f = g. Using the fact that the limit of a uniformly convergent sequence of continuous functions is continuous, show that f is of class C 1 , and hence that f is holomorphic.

1.5 The Holomorphic Inverse Function Theorem in C A holomorphic function f mapping an open set  bijectively onto an open set U is called a biholomorphism (or a biholomorphic mapping) if the inverse function f −1 is holomorphic. We also say that f maps  biholomorphically onto U and that  and U are biholomorphic. Observe that the chain rule implies that f is nonvanishing and [f −1 ] = 1/(f ◦ f −1 ). A function that maps a neighborhood of each point biholomorphically onto an open set is called a local biholomorphism (or a locally biholomorphic mapping). The following fact is fundamental in the study of Riemann surfaces: Theorem 1.5.1 (Holomorphic inverse function theorem in C) Let f be a holomorphic function on an open set  ⊂ C. (a) If z0 ∈  and f (z0 ) = 0, then f maps some neighborhood of z0 biholomorphically onto a neighborhood of w0 ≡ f (z0 ). (b) If f is one-to-one, then f maps  biholomorphically onto an open set. The main step of the proof is the following inequality:

1.5 The Holomorphic Inverse Function Theorem in C

19

Lemma 1.5.2 Let D be a disk in C, let f be a holomorphic function on D, and let C1 and C2 be positive constants with |f | ≥ C1 and |f | ≤ C2 on D. Then |f (z2 ) − f (z1 )| ≥ C1 |z2 − z1 | −

C2 |z2 − z1 |2 2

∀z1 , z2 ∈ D.

In particular, if diam D < 2 C1 /C2 , then f is injective. Proof For every pair of points z1 , z2 ∈ D, we have f (z2 ) − f (z1 ) − f (z1 )(z2 − z1 )  1  d  = f (z1 + t (z2 − z1 )) dt − f (z1 )(z2 − z1 ) 0 dt  1   f (z1 + t (z2 − z1 )) − f (z1 ) dt = (z2 − z1 ) 

0

1 1

= (z2 − z1 ) 0

 = (z2 − z1 )2 0

0

 d  f (z1 + st (z2 − z1 )) ds dt ds

1 1

t · f (z1 + st (z2 − z1 )) ds dt.

0

Therefore |f (z2 ) − f (z1 )| ≥ |f (z1 )| · |z2 − z1 |  1 − |z2 − z1 |2 0

1

t · |f (z1 + st (z2 − z1 ))| ds dt,

0



and the claim follows.

Proof of Theorem 1.5.1 For the proof of part (a), observe that we may choose a constant R > 0 so small that D ≡ (z0 ; R) ⊂ , C1 ≡ infD |f | > 0, and C2 ≡ supD |f | < C1 /R. Lemma 1.5.2 then implies that f D is injective and hence, by the open mapping theorem (Theorem 1.3.5), f maps D homeomorphically onto a neighborhood U of w0 in C with inverse function h ≡ [f D ]−1 : U → D. Furthermore, for w1 ∈ U and w ∈ U \ {w1 } sufficiently close to w1 , we have h(w) − h(w1 ) h(w) − h(w1 ) = → 1/f (h(w1 )) as w → w1 ; w − w1 f (h(w)) − f (h(w1 )) so h is a complex differentiable function. Moreover, h has continuous partial derivatives, since ∂h ∂h 1 ∂h = −i = = ; ∂x ∂y ∂z f ◦ h and hence by Lemma 1.4.2, h ∈ O(U ).

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Complex Analysis in C

For the proof of (b), observe that by the open mapping theorem (Theorem 1.3.5), f maps  homeomorphically onto an open set . Moreover, by the identity theorem (Corollary 1.3.3), the zero set Z ≡ (f )−1 (0) of the holomorphic function f has no limit points in , and hence its image Y ≡ f (Z) has no limit points in . Therefore, by part (a), f maps  \ Z biholomorphically onto  \ f (Z), and Riemann’s  extension theorem (Theorem 1.2.10) implies that f −1 is holomorphic on . Exercises for Sect. 1.5 1.5.1 Give a proof of the holomorphic inverse function theorem (Theorem 1.5.1) that uses the C ∞ inverse function theorem (Theorem 9.9.1). 1.5.2 A subset S of Rn is convex if (1 − t)x + ty = x + t (y − x) ∈ S for all points x, y ∈ S and every number t ∈ [0, 1] (in other words, S contains every line segment with endpoints in S). A real-valued function ϕ on a convex set S ⊂ Rn is convex if ϕ((1−t)x +ty) ≤ (1−t)·ϕ(x)+t ·ϕ(y) for all points x, y ∈ S and every number t ∈ [0, 1] (in other words, the graph of the restriction of the function to a line segment in S lies below the line segment between the endpoints on the graph). (a) Prove that if ϕ is a convex function on a convex set S ⊂ Rn , then for every a ∈ R, {x ∈ S | ϕ(x) < a} is a convex set. (b) For a real-valued C 2 function ϕ on an open set  ⊂ Rn , the Hessian at a point p ∈  is the bilinear function (Hess(ϕ))p (·, ·) on Rn given by (Hess(ϕ))p (u, v) =

n  i,j =1

∂ 2ϕ (p)ui vj ∂xi ∂xj

for all u = (u1 , . . . , un ), v = (v1 , . . . , vn ) ∈ Rn . Prove that if  is convex and (Hess(ϕ))p (u, u) ≥ 0

∀u ∈ Rn , p ∈ ,

then ϕ is convex. Hint. First consider the case n = 1. (c) Let f be a holomorphic function on a neighborhood of 0 with f (0) = 0 and f (0) = 0. Prove that for every sufficiently small  > 0, f maps the disk (0; ) biholomorphically onto a convex neighborhood of 0. Hint. After applying Theorem 1.5.1 to obtain a local holomorphic inverse g in a neighborhood of 0, show that the function |g|2 is convex near 0. 1.5.3 Let  be a convex domain in C (see Exercise 1.5.2), f ∈ O(), and let C1 and C2 be positive constants with |f | ≥ C1 and |f | ≤ C2 on . (a) Prove that |f (z2 ) − f (z1 )| ≥ C1 |z2 − z1 | −

C2 |z2 − z1 |2 2

∀z1 , z2 ∈ .

(b) Prove that if diam  < 2C1 /C2 , then f maps  biholomorphically onto a domain in C.

1.6 Examples of Holomorphic Functions

21

1.6 Examples of Holomorphic Functions In this section, we consider some examples of holomorphic functions. Example 1.6.1 (Rational functions) As noted in Theorem 1.1.2, every rational function an zn + · · · + a1 z + a0 z → bm zm + · · · + b1 z + b0 is holomorphic (on its domain). Example 1.6.2 (Exponential, logarithmic, and power functions) The exponential function on C is defined by z = x + iy → exp(z) ≡ ex cos y + iex sin y = ex · eiy (see Example 7.2.8). One may verify directly that the exponential function is nonvanishing and holomorphic with derivative function z → exp z (see Exercise 1.6.1) and that exp(z) = exp(¯z) for each z ∈ C. Fixing w ∈ C, one may also verify directly that exp(w + x) = exp w · exp x for x ∈ R, and the identity theorem (Theorem 1.3.3) applied to the holomorphic function z → exp(w + z) then implies that exp(w + z) = exp w · exp z for all w, z ∈ C. The power series representation centered at 0 is given by exp(z) =

∞ n  z n=0

n!

∀z ∈ C.

We also denote the exponential function by z → ez . The holomorphic map exp : C → C∗ is surjective, and by the holomorphic inverse function theorem (Theorem 1.5.1), the map is locally biholomorphic. Thus the local inverses determine a multiple-valued holomorphic function, which is called the logarithmic function on C∗ . This multiple-valued function is denoted by z → log z, provided there is no danger that it will be confused with the real-valued (and singlevalued) logarithmic function x → log x on (0, ∞). The derivative of the logarithmic function is the (single-valued) holomorphic function z → 1/ exp(log z) = 1/z. A local holomorphic inverse of the exponential function is a single-valued holomorphic branch of the logarithmic function. The difference of any two single-valued holomorphic branches on a domain is a holomorphic function with values in the discrete set 2πiZ, and hence this difference must be constant. We may describe the logarithmic function more explicitly as follows. Recall that polar coordinates in R2 are given by C ∞ local inverses of the surjective local diffeomorphism (0, ∞) × R → R2 given by (r, θ ) → reiθ (see Examples 7.2.8, 9.2.8, and 9.7.20). Thus the local inverses provide a multiple-valued C ∞ function z = reiθ → θ , which is denoted by z → arg z and is called the argument function. For any C ∞ local inverse  = (ρ, α) :  → () ⊂ (0, ∞) × R of the map (r, θ ) → reiθ on a domain  ⊂ C∗ (in particular, ρ : z → |z|), the composition

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Complex Analysis in C

of the exponential function with the C ∞ function z → L(z) = log |z| + iα(z) is the identity. Thus L is a holomorphic branch of the logarithmic function. Conversely, given a holomorphic branch L = τ + iα of the logarithmic function on a domain  ⊂ C∗ with τ = Re L and α = Im L, we have z = eτ (z) · eiα(z) for each z ∈ . It follows that τ : z → log |z| and that (eτ , α) is a C ∞ local inverse of the map (r, θ ) → reiθ . Thus the logarithmic function is given by z → log |z| + i arg z. In particular, the real-valued logarithmic function on (0, ∞) is the restriction of any branch L = τ + iα of the logarithmic function with α ≡ 0 on (0, ∞). For any λ > 0 with λ = 1, the exponential function with base λ is the holomorphic function given by z → λz ≡ ez log λ for z ∈ C, and the logarithmic function with log z base λ is the multiple-valued holomorphic function given by z → logλ z ≡ log λ for ∗ z ∈ C (i.e., given by the local holomorphic inverses of the exponential function with base λ). For any ζ ∈ C∗ , the corresponding power function is the multiple-valued holomorphic function given by z → zζ ≡ exp(ζ log z) for z ∈ C∗ . Example 1.6.3 (Trigonometric functions) The trigonometric functions are defined, at each suitable z, by i sin z ≡ − (eiz − e−iz ), 2 sin z , tan z ≡ cos z 1 , sec z ≡ cos z

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

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

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

1.6 Examples of Holomorphic Functions

23

z1 , . . . , zj , . . . , zn ∈ C, the function ζ → f (z1 , . . . , zj −1 , ζ, zj +1 , . . . , zn ) is a holomorphic function on the open set {ζ ∈ C | (z1 , . . . , zj −1 , ζ, zj +1 , . . . , zn ) ∈ } (with the obvious adjustments for j = 1 or n). A C 1 map F = (f1 , . . . , fm ) :  → Cm is called a holomorphic map if fi is a holomorphic function for each i = 1, . . . , m. For each j = 1, . . . , n, we set     ∂ 1 ∂ ∂ 1 ∂ ∂ ∂ and . = −i = +i ∂zj 2 ∂x j ∂y j ∂ z¯ j 2 ∂x j ∂y j (a) Prove that a C 1 function f on an open set  ⊂ Cn is holomorphic if and only if ∂f ≡0 ∂ z¯ j

∀j = 1, . . . , n.

(b) Prove that any composition of holomorphic mappings (of open subsets of complex Euclidean spaces) is holomorphic.

Chapter 2

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

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

T. Napier, M. Ramachandran, An Introduction to Riemann Surfaces, Cornerstones, DOI 10.1007/978-0-8176-4693-6_2, © Springer Science+Business Media, LLC 2011

25

26

2

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

2.1 Definitions and Examples In this section, we consider the definition of a Riemann surface and some examples. Definition 2.1.1 Let X be a Hausdorff space. (a) A homeomorphism  : U → U  of an open set U ⊂ X onto an open set U  ⊂ C is called a local complex chart of dimension 1 (or a 1-dimensional local complex chart) in X. We also denote this local complex chart by (U, , U  ). (b) Two 1-dimensional local complex charts (U1 , 1 , U1 ) and (U2 , 2 , U2 ) in X are holomorphically compatible if the coordinate transformations 1 ◦ −1 2 : 2 (U1 ∩ U2 ) → 1 (U1 ∩ U2 ) and

−1  1 ◦ −1 = 2 ◦ −1 2 1 : 1 (U1 ∩ U2 ) → 2 (U1 ∩ U2 )

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

Fig. 2.1 Holomorphic coordinate transformations

2.1 Definitions and Examples

27

Remarks 1. Higher-dimensional complex manifolds are also considered in this book, but only in some of the exercises (see, for example, Exercise 2.2.6). For this reason, all local complex charts, all local holomorphic charts, and all holomorphic atlases should be assumed to be of dimension 1 unless otherwise indicated. 2. A holomorphic structure on X determines a unique underlying real 2-dimensional C ∞ (in fact, real analytic) structure, with C ∞ atlas consisting of local C ∞ coordinates given by (x, y) = (Re z, Im z) for any local holomorphic coordinate z (see Chap. 9). A map from (to) an open subset of X from (respectively, to) an open subset of a manifold or complex 1-manifold is said to be of class C k if the map is of class C k with respect to the underlying C k structures. 3. In many treatments, manifolds are assumed to be second countable (often without comment). In fact, according to a theorem of Radó that will be proved in Sect. 2.11, every Riemann surface is automatically second countable. 4. A local holomorphic chart (U, , V ) and a local holomorphic coordinate neighborhood (U, z = ) are really the same object, but one generally uses the former terminology when emphasizing the mapping properties, and the latter when emphasizing the coordinate properties. 5. We will call a local holomorphic coordinate neighborhood (U, z = ) in which (U ) is a disk a holomorphic coordinate disk (or simply a coordinate disk). We will use the analogous terminology in similar contexts (for example, a holomorphic coordinate annulus will refer to a local holomorphic coordinate neighborhood with image an annulus). Similarly, given a set S ⊂ U and a property of (U ), we will often say that S has this property in (or with respect to) the local coordinate neighborhood. For example, if (S) is a closed rectangle in R2 , then we will say that S is a closed rectangle in (U, z). We will also use terminology analogous to the above in the context of topological and C ∞ manifolds. 6. It turns out that the theory of compact Riemann surfaces and that of noncompact Riemann surfaces differ. A noncompact Riemann surface is also called an open Riemann surface. 7. For convenience, most of the main theorems, as well as some of the elementary facts, in this book are stated as applying to Riemann surfaces. However, analogues, with the appropriate modifications, also hold for complex 1-manifolds (see Example 2.1.4 below). We now consider some examples. The verifications of some details are left to the reader. Example 2.1.2 (The complex plane) The complex plane C together with the holomorphic structure determined by the holomorphic atlas {(C, z → z, C)} is a Riemann surface. Example 2.1.3 An open set  in a complex 1-manifold X has a natural induced holomorphic structure in which each triple of the form (U ∩ , U ∩ , (U ∩ )), where (U, , U  ) is a local holomorphic chart in X, is a local holomorphic chart in  (see Exercise 2.1.1). Unless otherwise indicated, we take this to be the holomorphic structure on such a given open subset .

28

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

Fig. 2.2 The Riemann sphere

 Example 2.1.4 Let X ≡ α∈A Xα be a disjoint union of complex 1-manifolds {Xα }α∈A , and let ια : Xα → X be the inclusion map for each index α ∈ A. Then X has a unique (natural) holomorphic structure in which each triple of the form   (ια (U ),  ◦ ι−1 α , U ), where (U, , U ) is a local holomorphic chart in Xα for some α ∈ A, is a local holomorphic chart in X (see Exercise 2.1.2). Unless otherwise indicated, we take this to be the holomorphic structure on such a given disjoint union. In particular, every complex 1-manifold is equal to a disjoint union of Riemann surfaces, that is, of its connected components. Consequently, most statements about Riemann surfaces may be easily modified to give an analogous statement about complex 1-manifolds. Example 2.1.5 (The Riemann sphere) The Riemann sphere (or extended complex plane) is the one-point compactification P1 = C ∪ {∞} of C (see Definition 9.1.11) together with the holomorphic structure determined by the holomorphic atlas {(C, z → z, C), (C∗ ∪ {∞}, z → 1/z, C)} (where 1/∞ = 0). The Riemann sphere is diffeomorphic to the unit sphere S2 ⊂ R3 (see Examples 9.2.3) under the stereographic projection S2 → P1 (see Fig. 2.2 and x1 x2 + i 1−x ((0, 0, 1) → ∞). Exercise 2.1.3), i.e., the mapping (x1 , x2 , x3 ) → 1−x 3 3 Example 2.1.6 (Complex tori) A lattice in C is a subgroup of the form = Zξ1 + Zξ2 , where ξ1 , ξ2 ∈ C are complex numbers that are linearly independent over R. We may associate to an equivalence relation ∼ given by z∼w

⇐⇒

z−w∈

(in other words, the equivalence class of each element z ∈ C is z + ). Let us denote the corresponding quotient space and quotient mapping by ϒ : C → X. That is, X is the set of equivalence classes for ∼, ϒ is the mapping z → z + , and X is given the quotient topology (i.e., U ⊂ X is open if and only if its inverse image ϒ −1 (U ) is open). Observe that ϒ is an open mapping. For if U ⊂ C is open, then ϒ −1 (ϒ(U )) is the union of the open sets {U + ξ }ξ ∈ . For each point z = u1 ξ1 + u2 ξ2 ∈ C with

2.1 Definitions and Examples

29

Fig. 2.3 A complex torus

u1 , u2 ∈ R, the set

  1 1 Uz ≡ z + t1 ξ1 + t2 ξ2 | − < t1 , t2 < 2 2

is the image of the open square (u1 − 1/2, u1 + 1/2) × (u2 − 1/2, u2 + 1/2) ⊂ R2 under the real linear isomorphism R2 → C given by (t1 , t2 ) → t1 ξ1 +t2 ξ2 , and hence Uz is a relatively compact neighborhood of z in C (see Fig. 2.3). It is also easy to check that Uz ∩ (Uz + ( \ {0})) = ∅

and ϒ(U z ) = X.

In particular, the openness of ϒ, together with the first equality, implies that ϒ maps Uz homeomorphically onto ϒ(Uz ); and the second equality implies that X is compact. Furthermore, X is Hausdorff. For it is clear that each of the sets ϒ(Uz ) is Hausdorff, while if w ∈ ∂Uz , then the neighborhoods   1 1 V ≡ z + t1 ξ1 + t2 ξ2 | − < t1 , t2 < 4 4 and



1 1 W ≡ w + t1 ξ1 + t2 ξ2 | − < t1 , t2 < 4 4



of z and w, respectively, have disjoint images in X. If U  ⊂ C is an open set with U  ∩ (U  + ξ ) = ∅ for each ξ ∈ \ {0}, then ϒ maps U  homeomorphically onto an open set U ⊂ X. Thus we get a local complex chart (U, (ϒU  )−1 , U  ) in X. The collection of such local complex charts determines a holomorphic structure on X. For if (V , (ϒV  )−1 , V  ) is another such

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

α  C

ϒ   β  X T2 where α is the real linear isomorphism (and therefore diffeomorphism) given by (t1 , t2 ) → t1 ξ1 + t2 ξ2 , ϒ0 is the C ∞ mapping onto the real torus T2 = S1 × S1 given by (t1 , t2 ) → (e2πit1 , e2πit2 ), and the induced mapping β is a well-defined diffeomorphism. For it is easy to verify that β is a well-defined bijection. Moreover, given a point t = (t1 , t2 ) ∈ R2 , ϒ0 maps the neighborhood V ≡ (t1 −1/2, t1 +1/2)× (t2 − 1/2, t2 + 1/2) diffeomorphically onto a neighborhood of ϒ0 (t) (with inverse given by the product of 1/2π and a C ∞ argument function in each coordinate). Since locally, β is a composition of the C ∞ map ϒ , the C ∞ map α, and a local C ∞ inverse of ϒ0 , β must be a C ∞ map. Similarly, β −1 is also a C ∞ map, and therefore β is a diffeomorphism. ϒ0

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

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

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(c) A bijective holomorphic mapping : X → Y with holomorphic inverse is called a biholomorphism (or a biholomorphic mapping). If such a biholomorphism exists, then we say that X and Y are biholomorphically equivalent (or simply biholomorphic). For Y = X, we also call an automorphism of X. (d) A holomorphic mapping : X → Y is a local biholomorphism (or a locally biholomorphic mapping) if maps a neighborhood of each point in X biholomorphically onto an open subset of Y . Remarks 1. A holomorphic function on an open subset  of a complex 1-manifold X is precisely a holomorphic mapping of  into C. 2. A function f on an open subset  of a complex 1-manifold X is holomorphic if and only if for each point p ∈ , there exists a local holomorphic chart (U, , U  ) in X such that p ∈ U and f ◦ −1 ∈ O(( ∩ U )). Similarly, a continuous mapping of complex 1-manifolds : X → Y is holomorphic if and only if for each point p ∈ X, there exists a local holomorphic chart (U, , U  ) in Y such that (p) ∈ U and ◦ ∈ O( −1 (U )) (see Exercise 2.2.1). Furthermore, a holomorphic mapping of complex 1-manifolds is of class C ∞ with respect to the underlying C ∞ structures (in fact, the mapping is real analytic with respect to the underlying real analytic structures). 3. Biholomorphic equivalence is an equivalence relation (see Exercise 2.2.2), and we usually identify any two biholomorphic complex 1-manifolds. 4. Any sum or product of holomorphic functions is holomorphic, and any composition of holomorphic mappings is holomorphic (see Exercise 2.2.3). In particular, the set of holomorphic functions on an open set is an algebra. 5. In Example 2.1.6, the quotient mapping ϒ : C → X of C to the complex torus X is locally biholomorphic, because the local holomorphic charts are given by local inverses of ϒ. Although all complex tori are diffeomorphic, they are not all biholomorphic (see Exercise 5.9.1). Many of the elementary theorems for holomorphic functions on domains in the plane immediately give analogues for holomorphic mappings on Riemann surfaces. For example, we have the following: Theorem 2.2.2 Let X and Y be Riemann surfaces. (a) Identity theorem. If  : X → Y is a nonconstant holomorphic mapping, then the fiber −1 (p) over each point p ∈ Y is discrete in X (i.e., −1 (p) has no limit points in X). Moreover, if : X → Y is a holomorphic map that is not identically equal to , then {x ∈ X | (x) = (x)} is discrete in X. (b) Riemann’s extension theorem. A continuous mapping  : X → Y that is holomorphic on the complement of a discrete subset of X is holomorphic. (c) Maximum principle. If f ∈ O(X) and |f | attains a local maximum at some point p ∈ X, then f is constant. In particular, if X is compact, then every holomorphic function on X is constant. (d) Open mapping theorem. If  : X → Y is a nonconstant holomorphic mapping, then the image (X) is open.

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

f ◦ ≡ p, and therefore U ⊂ F . Hence the interior F of F is nonempty, and by the ◦

above argument, ∂(F ) = ∅. It follows that F = X, and hence that  is constant.  Lemma 2.2.3 (Local representation of holomorphic mappings) Let p be a point in a Riemann surface X. (a) If f is a nonconstant holomorphic function on X, then there exist a positive integer m and a local holomorphic coordinate neighborhood (U, z) of p such that z(p) = 0 and f = f (p) + zm on U . Moreover, m = m(f, p) is unique, and for any local holomorphic coordinate neighborhood (W, w) of p, the function (w − w(p))−m (f − f (p)) ∈ O(W \ {p}) extends to a holomorphic function on W that does not vanish at p. (b) If : X → Y is a nonconstant holomorphic mapping of X to a Riemann surface Y , then for some m ∈ Z>0 and for every local holomorphic coordinate neighborhood (V , ζ ) of (p) in Y , there is a local holomorphic coordinate neighborhood (U, z) of p in X such that z(p) = 0 and ζ ( ) = ζ ( (p)) + zm on U ∩ −1 (V ). Moreover, the integer m = m( , p) is unique, and for any local holomorphic coordinate neighborhood (W, w) of p, the holomorphic function (w − w(p))−m (ζ ( ) − ζ ( (p))) on W ∩ −1 (V ) \ {p} extends to a holomorphic function on W ∩ −1 (V ) that does not vanish at p. Proof For the proof of (a), we may fix a local holomorphic chart (U0 ,  = w, V0 ) with p ∈ U0 . Corollary 1.3.3 then provides a unique integer m ∈ Z>0 and a unique function g ∈ O(U0 ) such that g(p) = 0 and f − f (p) = (w − w(p))m · g on U0 . Choosing U0 to be sufficiently small, we also get a holomorphic branch L of the logarithmic function on a neighborhood of g(U0 ) (that is, eL(ζ ) = ζ ), and hence we have the holomorphic mth root function eL/m . The holomorphic function z ≡ (w − w(p)) · eL(g)/m then satisfies z(p) = 0 and f = f (p) + zm on U0 . Moreover,   z ◦ −1 (w(p)) = eL(g(p))/m = 0, so the holomorphic inverse function theorem (Theorem 1.5.1) implies that the function z maps some neighborhood U of p in U0 biholomorphically onto a neighborhood of 0 in C. Finally, we have uniqueness of m. For if f − f (p) = ζ n on Q for some n ∈ Z>0 and some local holomorphic chart (Q, = ζ, Q ) with p ∈ Q, then by Corollary 1.3.3, since (z ◦ −1 ) (0) = 0, there is a holomorphic function h on

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33

a neighborhood of 0 in C with h(0) = 0 and z( −1 (ξ )) = ξ h(ξ ) for each ξ ∈ Q near 0. Hence, for each ξ near 0,      m ξ n = f −1 (ξ ) − f (p) = z −1 (ξ ) = ξ m (h(ξ ))m . The uniqueness part of Corollary 1.3.3 implies that n = m (and hm ≡ 1). Thus (a) is proved. The proof of (b) is left to the reader (see Exercise 2.2.5).  Definition 2.2.4 Let  be an open subset of a complex 1-manifold X, and let p ∈ . (a) If f is a holomorphic function on  and m is a positive integer such that f = zm g on a neighborhood p for some local holomorphic coordinate z with z(p) = 0 and some nonvanishing holomorphic function g (i.e., f (p) = 0 and m is the integer provided by part (a) of Lemma 2.2.3), then we say that f has a zero of order m at p. For m = 1, we also say that f has a simple zero at p. (b) If :  → Y is a holomorphic mapping into a complex 1-manifold Y and m is a positive integer such that ζ ( ) − ζ ( (p)) has a zero of order m at p for some local holomorphic coordinate ζ on a neighborhood of (p) (i.e., m is the integer provided by part (b) of Lemma 2.2.3), then we say that has multiplicity m at p, and we write multp = m. Definition 2.2.5 A meromorphic function on an open subset  of a complex 1-manifold X is a function f :  \ P → C on the complement  \ P of a discrete subset P of  such that for each point p ∈ P , |f (x)| → ∞ as x → p. Each point p ∈ P is called a pole of f . We denote the set of meromorphic functions on  by M(). Proposition 2.2.6 For any holomorphic function f on the complement X \ P of a discrete subset P of a complex 1-manifold X, the following are equivalent: (i) The function f is a meromorphic function on X with pole set P . (ii) There exists a holomorphic mapping h : X → P1 such that h = f on X \ P and P = h−1 (∞). (iii) For each point p ∈ P and for every local holomorphic coordinate neighborhood (U, z) of p in X, there exist a nonvanishing holomorphic function g on a neighborhood V of p in U and a positive integer νp such that f = (z − z(p))−νp g on V \ {p}. (iv) For each point p ∈ P , there exist a local holomorphic coordinate neighborhood (U, z) of p in X, a nonvanishing holomorphic function g on a neighborhood V of p in U , and a positive integer νp such that f = (z − z(p))−νp g on V \ {p}. (v) For each point p ∈ P , there exist a local holomorphic coordinate neighborhood (U, z) of p in X and a positive integer νp such that z(p) = 0 and f = z−νp on U \ {p}.

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Moreover, in the above, for each point p ∈ P , the integer νp in (iii)–(v) is equal to the multiplicity at p of the holomorphic mapping h in (ii). Proof Clearly, any one of the conditions (ii)–(v) implies (i). Conversely, if f is meromorphic with pole set P , then f extends to a unique continuous mapping h :  → P1 with h−1 (∞) = P , and Riemann’s extension theorem (Theorem 2.2.2) implies that h is a holomorphic mapping; that is, (ii) holds. If p ∈ P and νp = multp h, then by the local representation of holomorphic mappings (Lemma 2.2.3), for every local holomorphic coordinate neighborhood (U, z) of p, there is a nonvanishing holomorphic function u on a neighborhood V of p in U such that h−1 (∞) ∩ V = {p} and u = (z − z(p))−νp · (1/ h) on V \ {p}. Setting g ≡ 1/u, we get (iii), and (iv) follows. Moreover, we may choose (U, z) so that z(p) = 0 and  1/ h = zνp on U , so we get (v) as well. Definition 2.2.7 We identify any meromorphic function f on an open subset  of a complex 1-manifold X with the associated holomorphic mapping  → P1 . If p ∈ f −1 (∞) is a pole of f at which this mapping has multiplicity m, then we say that f has a pole of order m at p. We say that f has a zero of order m at q ∈  if q ∈ f −1 (0) and the holomorphic function f \f −1 (∞) has a zero of order m at q (note that this is consistent with the above identification). For any point p ∈ , the order of f at p is given by ⎧ 0 if p ∈ / f −1 ({0, ∞}), ⎪ ⎪ ⎪ ⎨m if f has a zero of order m at p, ordp f ≡ ⎪ −m if f has a pole of order m at p, ⎪ ⎪ ⎩ ∞ if f ≡ 0 on a neighborhood of p. We say that f has a simple zero (simple pole) at p if ordp f = 1 (respectively, ordp f = −1). Remarks 1. If f and g are meromorphic functions on a connected neighborhood of a point p in a complex 1-manifold X and neither f nor g is identically zero, then in terms of some local holomorphic coordinate z on a neighborhood of p, we have f = zm f0 and g = zn g0 for some pair of integers m and n and some pair of nonvanishing holomorphic functions f0 and g0 on a neighborhood of p. It follows that each of the functions f + g, fg, and f/g is holomorphic on a neighborhood of p or has a removable singularity or a pole at p. Consequently, any sum, product, or quotient (provided the denominator does not vanish identically on any open set) of meromorphic functions is a meromorphic function, provided we holomorphically extend the resulting function over any removable singularities. In other words, for any connected open subset  of a complex 1-manifold, M() is a field. 2. Suppose X is a complex 1-manifold, (U, z) is a local holomorphic coordinate  n neighborhood of a point p ∈ X, f ∈ O(X \ {p}), and f = ∞ n=−∞ cn (z − z(p)) is the corresponding Laurent series representation of f about p provided by Theorem 1.3.6. Then f has a pole of order m ∈ Z>0 at p if and only if cn = 0 for all n < −m and c−m = 0.

2.3 Holomorphic Attachment

35

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

2.3 Holomorphic Attachment One may produce infinitely many examples of Riemann surfaces by holomorphic attachment. In fact, one goal of this book is a proof (appearing in Chap. 5) that every Riemann surface may be obtained by holomorphic attachment of tubes to a domain

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in the Riemann sphere P1 . Holomorphic attachment is not required until Chap. 5, so a reader who prefers to skip this section and move on to other topics in this chapter may do so without fear of missing a required concept. It will be convenient to have holomorphic attachment formulated as follows: Proposition 2.3.1 (Holomorphic attachment) Let X and X  be complex 1-manifolds, and let : G → G be a biholomorphism of an open set G ⊂ X onto an open set G ⊂ X such that −1 (K ∩ G) is closed in X for every compact set K ⊂ X (i.e., (x) → ∞ in the one-point compactification of X as x → p ∈ ∂G ) and (K  ∩ G ) is closed in X for every compact set K  ⊂ X (i.e., −1 (x) → ∞ in the one-point compactification of X as x → p ∈ ∂G). Let ι : X → X  X and ι : X  → X  X  be the inclusion mappings of X and X  into the disjoint union X  X , let ∼ be the equivalence relation in the disjoint union X  X determined by ι (x) ∼ ι( (x)) for every point x ∈ G , and let  : X  X → Y = X ∪ X  ≡ (X  X  )/∼ be the associated quotient space. Then there is a unique 1-dimensional holomorphic structure on Y with respect to which  ◦ ι and  ◦ ι are biholomorphisms of X and X  , respectively, onto open subsets of Y (equivalently, there is a unique holomorphic structure on Y with respect to which  is a local biholomorphism). The proof is left to the reader (see Exercise 2.3.1). Definition 2.3.2 For X ⊃ G, X ⊃ G , and : G → G as in Proposition 2.3.1, the complex 1-manifold X ∪ X  is called the holomorphic attachment of X and X along . Remarks 1. It is customary and convenient to identify X and X  with their (disjoint) images in X  X  . However, although it is customary to leave out any explicit mention of the inclusion maps ι and ι , we will often mention the inclusion maps when considering specific mappings of the above spaces. This will allow us to avoid any danger of confusion (for example, there is danger of confusion whenever X = X  ). 2. The natural identification of X  X with X   X gives a natural identification of X ∪ X  with X  ∪ −1 X. In applications, usually one first removes a set and then holomorphically attaches a new set of a desired type. For example, several key arguments in Chap. 5 will require that we replace sets with caps (i.e., disks) or tubes (i.e., annuli). For now, we consider the following: Example 2.3.3 (Holomorphic attachment of a tube) Let Y be a complex 1-manifold; let R0 , R1 > 1; let {(Dν , ν , (0; Rν ))}ν∈{0,1} be disjoint local holomorphic charts in Y ; let Aν ≡ −1 ν ((0; 1, Rν )) ⊂ Dν for ν = 0, 1; let T ≡ (0; 1/R0 , R1 ); and

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37

Fig. 2.4 Holomorphic attachment of a tube

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

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Observe that if Y is connected, then X is connected; and if Y is compact, then X is compact. If X is compact, then Y is compact. However, X may be connected even if Y is not connected (see Exercise 2.3.2). Fixing Rν∗ with 1 < Rν∗ ≤ Rν for ν = 0, 1, we may form the holomorphic attachment X ∗ = Z  T ∗ /∼ of the tube T ∗ ≡ (0; 1/R0∗ , R1∗ ) ⊂ T to Y at the coordinate ∗ ∗ ∗ disks {(Dν∗ ≡ −1 ν ((0; Rν )), ν ≡ ν Dν∗ , (0; Rν ))}ν∈0,1 . The natural inclusion ∼ =

Z  T ∗ ⊂ Z  T then descends to a natural biholomorphism X∗ −→ X (see Exercise 2.3.4), so we may identify X ∗ with X. In particular, we may always choose R0 , R1 ∈ (1, ∞) to be equal and arbitrarily close to 1. By holomorphically attaching tubes, one obtains examples of complex 1-manifolds with complicated topology. In fact, one of the main goals of Chap. 5 will be to show that every Riemann surface may be obtained by holomorphic attachment of tubes at elements of a locally finite family of disjoint coordinate disks in a domain in P1 (see Theorem 5.13.1 and Theorem 5.14.1). Exercises for Sect. 2.3 2.3.1 Prove Proposition 2.3.1. 2.3.2 Let X be a complex 1-manifold obtained by holomorphically attaching a tube to a complex 1-manifold Y as in Example 2.3.3. Prove that X is connected if Y is connected. Give an example that shows that X may be connected even if Y is not. 2.3.3 Write out a specific example of holomorphic attachment of a tube to the Riemann sphere (observe that the resulting Riemann surface appears to have the topological type of a torus). 2.3.4 In the notation of Example 2.3.3, verify that the natural inclusion of Z  T ∗ into Z  T descends to a biholomorphism of X∗ onto X. 2.3.5 Let X and X  be complex 1-manifolds, let : G → G be a biholomorphism of an open set G ⊂ X  onto an open set G ⊂ X, let ι : X → X  X  and ι : X  → X  X  be the inclusion mappings of X and X  into the disjoint union X  X  , let ∼ be the equivalence relation in the disjoint union X  X determined by ι (x) ∼ ι( (x)) for every point x ∈ G , and let Y ≡ (X  X )/∼ be the associated quotient space. Prove that if there exists a compact set K ⊂ X for which −1 (K ∩G) is not closed in X  , then Y is not Hausdorff.

2.4 Holomorphic Tangent Bundle A complex 1-manifold X has an underlying real 2-dimensional C ∞ structure, and therefore a tangent bundle and a complexified tangent bundle T X : T X → X

and

(T X)C : (T X)C → X,

respectively (see Sect. 9.4). We recall that for p ∈ X, a tangent vector v ∈ (Tp X)C is a linear derivation on the germs of C ∞ functions at p; that is, for every pair

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of C ∞ functions f, g on a neighborhood of p, we have v(fg) = v(f ) · g(p) + f (p) · v(g) ∈ C. Given a local holomorphic coordinate neighborhood (U, z = x + iy), the vector fields ∂/∂x and ∂/∂y form a basis for the tangent space at each point in U . Furthermore, the corresponding complex vector fields ∂/∂z and ∂/∂ z¯ as in Chap. 1 form a different complex basis for the complexified tangent space at each point, and their spans yield a natural decomposition into a sum of 1dimensional vector spaces. Moreover, a C 1 function g on U is holomorphic if and only if ∂g/∂ z¯ = ∂ g/∂z ¯ = 0. Similarly, the differentials dz and d z¯ form a dual basis that yields a decomposition of the complexified cotangent space at each point. The precise definitions and verifications appear below. Definition 2.4.1 Let X be a complex 1-manifold. (a) For each point p ∈ X, a tangent vector v ∈ (Tp X)C is of type (1, 0) (of type (0, 1)) if d f¯(v) = v(f¯) = 0 (respectively, df (v) = v(f ) = 0) for every holomorphic function f on a neighborhood of p. The subspace (Tp X)1,0 of (Tp X)C formed by the tangent vectors at p of type (1, 0) is called the holomorphic tangent space (or (1, 0)-tangent space) at p. The subspace (Tp X)0,1 ⊂ (Tp X)C formed by the tangent vectors at p of type (0, 1) is called the (0, 1)tangent space at p. The spaces  (T X)1,0 = (T X)C (T X)1,0 : (T X)1,0 ≡ (Tp X)1,0 → X p∈X

and (T X)0,1 = (T X)C (T X)0,1 : (T X)0,1 ≡



(Tp X)0,1 → X

p∈X

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

and (T ∗ X)0,1 = (T ∗ X)C (T ∗ X)0,1 : (T ∗ X)0,1 ≡



(Tp∗ X)0,1 → X

p∈X

are called the holomorphic cotangent bundle (or (1, 0)-cotangent bundle) and (0, 1)-cotangent bundle, respectively. The holomorphic cotangent bundle

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(T ∗ X)1,0 is also called the canonical line bundle (or canonical bundle) and is also denoted by KX : KX → X. (c) Given a local holomorphic coordinate neighborhood (U,  = z = x + iy) in X (with x = Re z and y = Im z), we call the vector fields     1 ∂ ∂ ∂ 1 ∂ ∂ ∂ ≡ −i and ≡ +i = ∂/∂z ∂z 2 ∂x ∂y ∂ z¯ 2 ∂x ∂y the partial derivative operators with respect to z and z¯ , respectively. In other words, denoting the standard complex coordinate on C by w, we have, for any suitable function f , ∂(f ◦ −1 ) ∂f = ◦ ∂z ∂w

and

∂f ∂(f ◦ −1 ) = ◦ . ∂ z¯ ∂ w¯

Proposition 2.4.2 Let X be a complex 1-manifold. (a) For each point p ∈ X, we have direct sum decompositions (Tp X)C = (Tp X)1,0 ⊕ (Tp X)0,1

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

and we have isomorphisms ∼ =

(Tp∗ X)1,0 → ((Tp X)1,0 )∗

and

∼ =

(Tp∗ X)0,1 → ((Tp X)0,1 )∗

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

(Tp X)0,1 = C · (∂/∂ z¯ )p ,

(Tp∗ X)1,0 = C · (dz)p ,

(Tp∗ X)0,1 = C · (d z¯ )p ,

and dz((∂/∂z)p ) = 1,

d z¯ ((∂/∂ z¯ )p ) = 1,

dz((∂/∂ z¯ )p ) = 0,

d z¯ ((∂/∂z)p ) = 0.

Moreover, for every complex number ζ = a + ib with a, b ∈ R, we have         ∂ ∂ ∂ ∂ ¯ a +b =ζ +ζ ∂x p ∂y p ∂z p ∂ z¯ p and a(dx)p + b(dy)p =

ζ¯ ζ (dz)p + (d z¯ )p . 2 2

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41

(d) A C 1 function f on an open set  ⊂ X is holomorphic if and only if for each point p ∈ , ∂f/∂ z¯ ≡ 0 on U ∩  for some (equivalently, for every) local holomorphic coordinate neighborhood (U, z) of p. If : X → Y is a C 1 mapping of X into a complex 1-manifold Y and (r, s) ∈ {(1, 0), (0, 1)}, then the following are equivalent: (i) is holomorphic. (ii) ∗ (T X)r,s ⊂ (T Y )r,s . (iii) ∗ (T ∗ Y )r,s ⊂ (T ∗ X)r,s . Proof Let (U,  = z, U  ) be a local holomorphic chart in X and let p ∈ U . A function f ∈ C 1 (U ) is holomorphic if and only if f ◦ −1 ∈ O(U  ). Let w denote the standard complex coordinate on C. Since ∂f ∂(f ◦ −1 ) ◦ −1 = , ∂ z¯ ∂ w¯ we see that f is holomorphic if and only if ∂ f¯/∂ z¯ = ∂f/∂ z¯ ≡ 0, and hence that (∂/∂z)p ∈ (Tp X)1,0 \ {0} and (∂/∂ z¯ )p ∈ (Tp X)0,1 \ {0}. Moreover,             ∂ ∂ ∂ ∂ ∂ ∂ = + and =i −i . ∂x p ∂z p ∂ z¯ p ∂y p ∂z p ∂ z¯ p Therefore, since dimC (Tp X)C = 2, we have the direct sum decomposition     ∂ ∂ (Tp X)C = C · ⊕C· = (Tp X)1,0 ⊕ (Tp X)0,1 . ∂z p ∂ z¯ p We also have     1 ∂ i ∂ 1 i 1 i ∂ = (dx + i dy) − = − · 0 + i · · 0 − i · · 1 = 1. dz ∂z 2 ∂x 2 ∂y 2 2 2 2 A similar computation gives dz(∂/∂ z¯ ) = 0, and it follows that d z¯ (∂/∂ z¯ ) = dz(∂/∂z) = 1

and d z¯ (∂/∂z) = dz(∂/∂ z¯ ) = 0.

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



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

∂f ∂f (p)(dz)p + (p)(d z¯ )p . ∂z ∂ z¯

In particular, f is holomorphic if and only if df = (∂f/∂z) · dz. If : X → Y is a holomorphic mapping into a Riemann surface Y , (V , w) is a local holomorphic coordinate neighborhood of (p) in Y , and g = w ◦ , then for all a, b ∈ C,           ∂ ∂ ∂ ∂g ∂ ∂g +b + b (p) . ( ∗ )p a = a (p) ∂z p ∂ z¯ p ∂z ∂w (p) ∂z ∂ w¯ (p) Consequently, if any one of the following linear mappings is nontrivial (i.e., not the zero mapping), then all of the mappings are isomorphisms: ( ∗ )p : (Tp X)C → (T(p) Y )C , ( ∗ )p : (Tp X)1,0 → (T(p) Y )1,0 , (dg)p : (Tp X)1,0 → C.

( ∗ )p : Tp X → T(p) Y, ( ∗ )p : (Tp X)0,1 → (T(p) Y )0,1 ,

2. We express the decomposition of the complexified tangent and cotangent bundles by writing (T X)C = (T X)1,0 ⊕ (T X)0,1 and (T ∗ X)C = (T ∗ X)1,0 ⊕ (T ∗ X)0,1 , and for each pair (r, s) ∈ {(1, 0), (0, 1)}, we let P r,s : (T X)C → (T X)r,s

and

P r,s : (T ∗ X)C → (T ∗ X)r,s

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

2.4 Holomorphic Tangent Bundle

43

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

−1 ∗ ∗ r,s bijections ι∗ : (T )r,s → −1 (T X)r,s () and ι : (T ∗ X)r,s () → (T ) , respectively.

Guided by Definition 9.4.5, we make the following definition: Definition 2.4.3 Let X be a complex 1-manifold. (a) The coefficient functions (or simply the coefficients) of a vector field v on a set S ⊂ X with respect to (or in) a local holomorphic coordinate neighborhood (U, z) are the functions dz(v) = v(z) and d z¯ (v) = v(¯z) on S ∩ U . (b) We call a vector field v of type (1, 0) on an open set  ⊂ X (that is, vp is of type (1, 0) for each point p) a holomorphic vector field if the coefficient function f = dz(v) = v(z) :  ∩ U → C in every local holomorphic coordinate neighborhood (U, z) is holomorphic (we have d z¯ (v) = v(¯z) = 0, since v is of type (1, 0)); that is, v = f · (∂/∂z) on  ∩ U for some function f ∈ O( ∩ U ). Remark A vector field on a set S ⊂ X is of class C k if and only if its coefficient functions with respect to every local holomorphic coordinate neighborhood (or equivalently, for each point p ∈ S, with respect to some local holomorphic coordinate neighborhood of p) are of class C k (see Exercise 2.4.1). We close this section with the observation that the holomorphic inverse function theorem for domains in the plane (Theorem 1.5.1) gives the following analogue for Riemann surfaces: Theorem 2.4.4 (Holomorphic inverse function theorem for Riemann surfaces) Let  : X → Y be a holomorphic mapping of Riemann surfaces. (a) If p ∈ X and (∗ )p = 0, then  maps some neighborhood of p biholomorphically onto a neighborhood of q ≡ (p). In particular, if f is a holomorphic function on a neighborhood of a point p ∈ X and (df )p = 0, then (U, f U ) is a local holomorphic coordinate neighborhood for some neighborhood U of p. (b) If  is one-to-one, then  maps X biholomorphically onto an open subset (X) of Y . The proof is left to the reader (see Exercise 2.4.2). Corollary 2.4.5 Let f be a holomorphic function on a Riemann surface X, let r be a positive regular value (see Definition 9.4.6) of the function ρ ≡ |f | (|f | is of class C ∞ on the complement of its zero set), and let  ≡ {p ∈ X | |f (p)| < r}. Then every point p ∈ ∂ admits a local holomorphic coordinate neighborhood (U, z = x + iy) in which  ∩ U = {q ∈ U | x(q) < 0}.

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Remark In particular,  is a C ∞ open set (see Definition 9.7.14). Proof We have 2ρ dρ = dρ 2 = f¯ df + f d f¯ on X \ f −1 (0), and hence, given a point p ∈ ∂, we have (df )p = 0. Therefore, by Theorem 2.4.4, there exists a local holomorphic chart of the form (U, f U , U  ) with p ∈ U . Moreover, we may choose U and U  so small that there exists a holomorphic branch g of the logarithmic function on U  (see Example 1.6.2). Since g is a biholomorphism (with holomorphic inverse function ζ → eζ ), we get a local holomorphic coordinate neighborhood (U, z = x + iy ≡ − log r + g ◦ f ) in which x = Re z = log |f/r|U . In particular,  ∩ U = {q ∈ U | x(q) < 0}.



Exercises for Sect. 2.4 2.4.1 Let X be a Riemann surface. (a) For a vector field v on X and for k ∈ Z≥0 ∪ {∞}, prove that the following are equivalent (cf. Exercise 9.4.2): (i) The vector field v is of class C k . (ii) The coefficients of v in every local holomorphic coordinate neighborhood are of class C k . (iii) For every point in X, there exists a local holomorphic coordinate neighborhood with respect to which the coefficients of v are of class Ck . (iv) The (1, 0) and (0, 1) parts of v are of class C k . (b) For a vector field v of type (1, 0) on X, prove that the following are equivalent: (i) The vector field v is holomorphic. (ii) The coefficient dz(v) of v in some local holomorphic coordinate neighborhood (U, z) of each point in X is holomorphic. (iii) For every holomorphic function f on an open set U ⊂ X, the function df (v) : p → df (vp ) = vp (f ) is holomorphic. (c) Prove that any sum of holomorphic vector fields, and any product of a holomorphic function and a holomorphic vector field, are holomorphic (in particular, the set of holomorphic vector fields on X is a complex vector space). 2.4.2 Prove Theorem 2.4.4. 2.4.3 Let X be a Riemann surface. (a) Prove that there is a unique 2-dimensional holomorphic structure on (T X)1,0 such that for each local holomorphic chart (U,  = z, U  ) in X, the triple   −1 (T X)1,0 (U ), ( ◦ (T X)1,0 , dz), U  × C is a local holomorphic chart in (T X)1,0 (see Exercise 2.2.6 for the definition of a complex manifold, and cf. Exercise 9.4.3).

2.5 Differential Forms on a Riemann Surface

45

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

for each point p ∈ U and each element α ∈ (Tp∗ X)1,0 , is a local holomorphic chart in (T ∗ X)1,0 . (d) Find (natural) C ∞ structures in (T X)0,1 and (T ∗ X)0,1 .

2.5 Differential Forms on a Riemann Surface We recall that a differential form α of degree r on a subset S of a C ∞ manifold M is a mapping of S into r (T ∗ M)C with αp ∈ r (Tp∗ M)C for each point p ∈ S (see Sect. 9.5). On a Riemann surface, the decomposition of the complexified tangent space into (1, 0) and (0, 1) parts induces a decomposition of differential forms. Definition 2.5.1 Let X be a complex 1-manifold, and let (r, s) ∈ Z≥0 × Z≥0 . (a) We set ⎧ 0 ∗  (T X)C = X × C ⎪ ⎪ ⎪ ⎨(T ∗ X)r,s r,s T ∗ X ≡ ⎪ 2 (T ∗ X)C ⎪ ⎪ ⎩ X × {0}

if (r, s) = (0, 0), if (r, s) = (1, 0) or (0, 1), if (r, s) = (1, 1), if r ≥ 2 or s ≥ 2,

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

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by ⎧ α ⎪ ⎪ ⎪ ⎪ ⎨a ≡ α(∂/∂z),

if q = 0, b ≡ α(∂/∂ z¯ ) (i.e., α = a dz + b d z¯ ) if q = 1, α a ≡ α(∂/∂z, ∂/∂ z¯ ) = (i.e., α = a dz ∧ d z¯ ) if q = 2, ⎪ ⎪ ⎪ dz ∧ d z¯ ⎪ ⎩ 0 if q > 2. Remarks 1. For each point p ∈ X, we have α ∧ β = α¯ ∧ β¯ = 0 for all α, β ∈ 1,0 Tp∗ X. Observe also that ξ ∧ ζ is of type (r + t, s + u) if ξ ∈ r,s Tp∗ X and ζ ∈ t,u Tp∗ X. 2. On any local holomorphic coordinate neighborhood (U, z = x + iy), we have (i/2) dz ∧ d z¯ = dx ∧ dy. 3. For all r, s ∈ Z≥0 , conjugation gives r,s T ∗ X = s,r T ∗ X; that is, α¯ ∈ s,r  T ∗ X for each element α ∈ r,s T ∗ X (see Exercise 2.5.1). 4. Clearly, a differential form of type (r, s) is of degree r + s. 5. Exercises 9.5.4 and 2.4.3 provide a natural C ∞ structure on r,s T ∗ X. Guided by Definition 9.5.2 and Definition 2.4.3, we make the following definition: Definition 2.5.2 Let  be an open subset of a complex 1-manifold X. (a) A holomorphic 1-form on  is a differential form θ of type (1, 0) on  such that for every local holomorphic coordinate neighborhood (U, z) in X, the coefficient function θ (∂/∂z) = θ/dz is holomorphic on  ∩ U ; that is, for some function f ∈ O( ∩ U ), we have θ = f dz on  ∩ U . The vector space of holomorphic 1-forms on  is denoted by X () or (). (b) A meromorphic 1-form on  is a differential form θ of type (1, 0) defined on the complement in  of a discrete subset P such that for every local holomorphic coordinate neighborhood (U, z) in X, the coefficient function θ/dz is meromorphic on  ∩ U with pole set P ∩ U ; that is, for some function f ∈ M( ∩ U ), we have θ = f dz on  ∩ U \ f −1 (∞) =  ∩ U \ P (we normally say simply θ = f dz on  ∩ U ). (c) A holomorphic (meromorphic) function on  is also called a holomorphic (respectively, meromorphic) 0-form. (d) A meromorphic 1-form θ on  has a zero (a pole) of order ν at a point p ∈  if for every local holomorphic coordinate neighborhood (U, z) of p, the meromorphic function θ/dz has a zero (respectively, a pole) of order ν at p. If ν = 1, then we also say that θ has a simple zero (respectively, a simple pole) at p. Remarks 1. A holomorphic 1-form is of class C ∞ (see Proposition 2.5.3 below). 2. For a nontrivial (i.e., not everywhere zero) meromorphic 1-form on a Riemann surface, the set of zeros (i.e., the set of points at which the 1-form is equal to 0) is discrete (see Exercise 2.5.2).

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47

The proof of the following characterization of continuous, C k , holomorphic, and meromorphic differential forms is left to the reader (see Exercise 2.5.3): Proposition 2.5.3 (Cf. Proposition 9.5.3 and Definition 9.7.12) Let α be a differential form of degree r defined at points of a subset S of a Riemann surface X, let k ∈ Z≥0 ∪ {∞}, and let d ∈ [1, ∞]. Then: (a) The following are equivalent: (i) The differential form α is continuous (of class C k , measurable, in Ldloc ). (ii) The coefficients of α in every local holomorphic coordinate neighborhood are continuous (respectively, of class C k , measurable, in Ldloc ). (iii) For every point in S, there exists a local holomorphic coordinate neighborhood with respect to which the coefficients of α are continuous (respectively, of class C k , measurable, in Ldloc ). Moreover, for r = 1, α is continuous (of class C k , measurable, in Ldloc ) if and only if the (1, 0) and (0, 1) parts of α are continuous (respectively, of class C k , measurable, in Ldloc ). (b) For S open and α of type (1, 0), α is holomorphic if and only if for every point in S, there exists a local holomorphic coordinate neighborhood (U, z) with respect to which the coefficient α(∂/∂z) = α/dz is holomorphic. (c) Suppose α is of type (1, 0) and S =  \ P for some discrete set P in an open set  ⊂ X. Then α is a meromorphic 1-form on  with pole set P if and only if for each point p ∈ , there exists a local holomorphic coordinate neighborhood (U, z) of p such that the coefficient α(∂/∂z) = α/dz is a meromorphic function on  ∩ U with pole set  ∩ U ∩ P . Moreover, if α is a meromorphic 1-form on , then α has a zero (a pole) of order ν at a point p ∈  if and only if there exists a local holomorphic coordinate neighborhood (U, z) of p such that the coefficient α/dz has a zero (respectively, a pole) of order ν at p. The decomposition of 1 (T ∗ X)C yields a decomposition of the exterior derivative operator d (see Definition 9.5.5): Definition 2.5.4 Let X be a complex 1-manifold, let α be a differential form of degree r on an open subset  of X, and let P p,q : 1 (T ∗ X)C → p,q T ∗ X be the associated projection for each pair (p, q) ∈ {(1, 0), (0, 1)}. (a) If α is of class C 1 , then we define ⎧ 1,0 ⎪ ⎨P (dα) if r = 0, ∂α = d(P 0,1 α) if r = 1, ⎪ ⎩ 0 if r ≥ 2,

⎧ 0,1 ⎪ ⎨P (dα) if r = 0, ¯ = d(P 1,0 α) if r = 1, and ∂α ⎪ ⎩ 0 if r ≥ 2.

¯ = 0 (∂α = 0), then we say that α is ∂-closed ¯ (b) If α is of class C 1 and we have ∂α (respectively, ∂-closed).

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¯ (α = ∂β) for some C 1 differential form β of degree r − 1 on  (in (c) If α = ∂β ¯ particular, r > 0), then we say that α is ∂-exact (respectively, ∂-exact). If β may k be chosen to be of class C with k ∈ Z>0 ∪ {∞}, then we also say that α is ¯ C k ∂-exact (respectively, C k ∂-exact). If for each point in , there exists a C 1 ¯ 0 = αU differential form β0 of degree r − 1 on a neighborhood U0 with ∂β 0 ¯ (∂β0 = αU0 ), then we say that α is locally ∂-exact (respectively, locally ∂exact). If for some k ∈ Z>0 ∪ {∞}, each of the local forms β0 may be chosen to ¯ be of class C k , then we also say that α is locally C k ∂-exact (respectively, locally C k ∂-exact). It is also convenient to consider the trivial 0-form α ≡ 0 to be C ∞ ¯ ¯ and to write 0 = ∂0 = ∂0. ∂-exact and C ∞ ∂-exact, ¯ Remark If α is a ∂-exact (∂-exact) r-form on a Riemann surface X, then we may ¯ = α (respectively, ∂β = α). For r = 1, choose a C 1 differential form β with ∂β β is of type (0, 0) and α is of type (0, 1) (respectively, α is of type (1, 0)). For ¯ = α (respectively, ∂P 0,1 β = ¯ 1,0 β = ∂β r = 2, α is of type (1, 1) and we have ∂P ∂β = α). Thus, in either case (r = 1 or 2), α is of type (p, q + 1) (respectively, of type (p + 1, q)) with p + q + 1 = r, and we may choose β to be of type (p, q). Moreover, according to part (c) of Proposition 2.5.5 below, if α is of class C k for some k ∈ Z>0 ∪ {∞}, then any such β of type (p, q) is also of class C k . Proposition 2.5.5 For any C 1 differential form α of degree r on an open subset  of a Riemann surface X, we have the following: ¯ and if α is of class C 2 , then (a) The exterior derivative satisfies dα = ∂α + ∂α, 2 2 2 ¯ ¯ ¯ d α = ∂ α = ∂ α = ∂ ∂α + ∂∂α = 0; in other words, d = ∂ + ∂¯ on C 1 forms and ¯ = 0 on C 2 forms. d 2 = ∂ 2 = ∂¯ 2 = ∂ ∂¯ + ∂∂ ¯ = ∂ α. ¯ ¯ Consequently, α is ∂-closed (∂-closed) if We also have ∂α = ∂¯ α¯ and ∂α ¯ ¯ (α = ∂β) for and only if α¯ is ∂-closed (respectively, ∂-closed); and if α = ∂β ¯ some differential form β of class C 2 , then α is ∂-closed (respectively, ∂-closed). ¯ is of type (p, q + 1). (b) If α is of type (p, q), then ∂α is of type (p + 1, q) and ∂α ¯ = 0 if q ≥ 1. In particular, ∂α = 0 if p ≥ 1 and ∂α ¯ (c) If α is ∂-exact (∂-exact) and of class C k for some k ∈ Z>0 ∪ {∞}, then α is ¯ ∂-closed (respectively, ∂-closed) and of type (p, q + 1) (respectively, of type (p + 1, q)) for some pair of integers (p, q). Moreover, there exists a C 1 differ¯ (respectively, α = ∂β), and any ential form β of type (p, q) such that α = ∂β such form β is actually of class C k . (d) Let (U, z) be a local holomorphic coordinate neighborhood in . If r = 0, then on U , dα =

∂α ∂α dz + d z¯ , ∂z ∂ z¯

∂α =

∂α dz, ∂z

¯ = ∂α d z¯ . and ∂α ∂ z¯

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49

If r = 1 and α = a dz + b d z¯ on U , then on U , we have   ∂b ∂a dα = − dz ∧ d z¯ , ∂z ∂ z¯ and ∂α =

∂b dz ∧ d z¯ ∂z

and

¯ = ∂a d z¯ ∧ dz = − ∂a dz ∧ d z¯ . ∂α ∂ z¯ ∂ z¯

(e) For any C 1 differential form β on , we have ∂(α ∧ β) = (∂α) ∧ β + (−1)r α ∧ ∂β and ¯ ¯ ∧ β) = (∂α) ¯ ∧ β + (−1)r α ∧ ∂β. ∂(α (f) If α is of type (0, 0) or (1, 0), then α is a holomorphic r-form if and only if ¯ = 0. If α is a holomorphic 0-form (i.e., a holomorphic function), then ∂α is ∂α a holomorphic 1-form. (g) If  : Y → X is a holomorphic mapping of a Riemann surface Y into X, then ¯ ∗ α = ∗ ∂α ¯ on −1 (). In particular, if α is a holo∂∗ α = ∗ ∂α and ∂ ∗ morphic r-form, then  α is also a holomorphic r-form. If α is a meromorphic r-form and (Y ) is not contained in the pole set of α, then ∗ α is a meromorphic r-form. Proof Let (U, z) be a local holomorphic coordinate neighborhood in . If r = 0, then, by (the remarks following) Proposition 2.4.2, dα =

∂α ∂α dz + d z¯ ; ∂z ∂ z¯

and since the first summand on the right-hand side is of type (1, 0) and the second ¯ If r = 1 and α = is of type (0, 1), we get the expressions in (d) for ∂α and ∂α. 2 2 a dz + b d z¯ on U , then since d = 0 (on class C forms), dz ∧ dz = d z¯ ∧ d z¯ = 0, and dz ∧ d z¯ = −d z¯ ∧ dz, we have ¯ dα = d(P 0,1 α + P 1,0 α) = d(P 0,1 α) + d(P 1,0 α) = ∂α + ∂α, ∂α = d(P 0,1 α) = d(bd z¯ ) = db ∧ d z¯   ∂b ∂b ∂b = dz + d z¯ ∧ d z¯ = dz ∧ d z¯ , ∂z ∂ z¯ ∂z and ¯ = d(P 1,0 α) = d(adz) = da ∧ dz ∂α   ∂a ∂a ∂a dz + d z¯ ∧ dz = − dz ∧ d z¯ . = ∂z ∂ z¯ ∂ z¯

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In particular, we get (d). The verifications of the remaining claims are left to the reader (see Exercise 2.5.5).  We have the following analogue of the Poincaré lemma (Lemma 9.5.7) for the ¯ operators ∂ and ∂: Lemma 2.5.6 (Dolbeault lemma) Let p be a point in a Riemann surface X. Then there exists a neighborhood D of p in X such that for every k ∈ Z>0 ∪ {∞}, all r, s ∈ Z≥0 , and every C k differential form β of type (r, s + 1) (respectively, (r + 1, s)) on a neighborhood of D, there is a C k differential form α of type (r, s) on D with ¯ = β (respectively, ∂α = β) on D. Consequently, every C k differential form of ∂α ¯ (respectively, locally C k ∂type (r, s + 1) (of type (r + 1, s)) is locally C k ∂-exact exact). Proof We may fix a local holomorphic coordinate neighborhood (U, z) of p in X and a neighborhood D  U such that z(D) is a disk in C. Suppose k ∈ Z>0 ∪ {∞}, r, s ∈ Z≥0 , and β is a C k differential form of type (r, s + 1) on a neighborhood ¯ if r > 1 or s > 0, so we of D, which we may assume to be U . Clearly, β ≡ 0 = ∂0 may also assume that r = 0 or 1 and that s = 0. Thus we have either β = b d z¯ or β = b dz ∧ d z¯ on U for some function b ∈ C k (U ). By Lemma 1.2.2, there exists a function a ∈ C k (D) with ∂a/∂ z¯ = b on D. Thus ¯ = b d z¯ ∂a

and

¯ ∂(−a dz) = b dz ∧ d z¯ ,

¯ on D, where α = a if r = 0 and α = −a dz if r = 1. and it follows that β = ∂α If instead, we take β to be of type (r + 1, s), then β¯ is of type (s, r + 1). Hence, ¯ = β¯ on D. by the above, there is a C k differential form α of type (s, r) such that ∂α k ¯ = β¯ = β on D.  Thus the C form α¯ is of type (r, s) and ∂ α¯ = ∂α We close this section with some remarks concerning integration on a Riemann surface. We first observe that line integrals (Definition 9.7.18) may be expressed in terms of local holomorphic coordinates. For example, if (U, z) is a local holomorphic coordinate neighborhood in a complex 1-manifold X, α = f dz + g d z¯ is a continuous differential form of degree 1 on U , and γ : [a, b] → U is a piecewise C 1 path, then   b  d d f (γ (t)) α= z(γ (t)) + g(γ (t)) [z(γ (t))] dt. dt dt γ a For integration of 2-forms, observe that there is a natural orientation (see Sect. 9.7) in a complex 1-manifold X determined by the local holomorphic charts. For if (U,  = z = x + iy ↔ (x, y), U  ) and (V , = w = u + iv ↔ (u, v), V  ) are two local holomorphic charts, then on U ∩ V , J ◦−1 ◦  =

  du ∧ dv dw ∧ d w¯  ∂w 2 = = > 0. dx ∧ dy dz ∧ d z¯ ∂z 

2.5 Differential Forms on a Riemann Surface

51

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

m  

α,

j =1 Sj

where the supremum is taken over all choices of disjoint measurable subsets S1 , . . . , Sm of S each of which is contained in a local holomorphic coordinate neighborhood. Remark By definition, a positive C ∞ (1, 1)-form on a complex manifold of arbitrary dimension is called a Kähler form if it is d-closed. This condition holds automatically in complex dimension 1, since every 2-form on a C ∞ manifold of real dimension 2 is closed. We make the following observation: Proposition 2.5.7 (Cf. Exercise 2.5.8 and Exercise 6.7.1) For a nontrivial meromorphic function f (i.e., f is not everywhere zero) on a compact Riemann surface X, counting multiplicities, the number of zeros is equal to the number of poles. More precisely, if Z is the set of zeros, P is the set of poles, μp is the order of the zero at each point p ∈ Z, and νp is the order of the pole at each point p ∈ P , then 

ordp f =

p∈Z∪P



μp −

p∈Z



νp = 0.

p∈P

In particular, if f has exactly one (simple) pole, then f maps X biholomorphically onto P1 . Proof For r > 0 sufficiently small, the local representation of meromorphic functions provided by Proposition 2.2.6 allows us to fix disjoint local holomorphic charts {(Up , p = zp , (0; 2r))}p∈Z∪P in X such that for each point p ∈ Z ∪ P , we have zp (p) = 0 and on Up ,  μ zp p if p ∈ Z, f = −νp if p ∈ P . zp Thus the meromorphic 1-form θ ≡ df/f satisfies, on Up ,  θ=

μp · zp−1 · dzp −νp · zp−1 · dzp

if p ∈ Z, if p ∈ P .

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2

Stokes’ theorem now gives 

2πiμp −

p∈Z

 p∈P

2πiνp =

  p∈Z∪P

∂−1 p ((0;r))

θ



=−



X\

p∈Z∪P

−1 p ((0;r))

dθ = 0.

In particular, if f is holomorphic except for a simple pole at p ∈ X (i.e., P = {p} and νp = 1), then f −1 (∞) = {p}, and for each point q ∈ X \ {p}, the meromorphic function f − f (q) is nonvanishing except for a simple zero at q. Thus f : X → P1 is injective, and by the holomorphic inverse function theorem (Theorem 2.4.4), f maps X biholomorphically onto an open subset of P1 . On the other hand, f (X) is compact, hence closed, so we must have f (X) = P1 .  Exercises for Sect. 2.5 2.5.1 Verify that for all r, s ∈ Z≥0 , conjugation gives r,s T ∗ X = s,r T ∗ X on a Riemann surface X. 2.5.2 Let θ be a nontrivial meromorphic 1-form on a Riemann surface X. Verify that {x ∈ X | θx = 0} is discrete. 2.5.3 Prove Proposition 2.5.3. 2.5.4 Prove that P1 does not admit any nontrivial holomorphic 1-forms. 2.5.5 Prove parts (a)–(c) and (e)–(g) of Proposition 2.5.5. 2.5.6 Let α be a differential form of type (1, 0) on a Riemann surface X. Prove that α is a holomorphic 1-form if and only if α is holomorphic as a mapping of X into the 2-dimensional complex manifold (T ∗ X)1,0 (see Exercise 2.4.3 for a description of the natural holomorphic structure on (T ∗ X)1,0 ). 2.5.7 Let X be a complex torus (Example 2.1.6). Prove that the vector space of holomorphic 1-forms on X is 1-dimensional. 2.5.8 Let  be a (nonempty) smooth relatively compact domain in a Riemann surface X, and for any given nontrivial meromorphic function h on X that does not have any zeros or poles in ∂, let μh denote the number of zeros of h in , and νh the number of poles of h in , counting multiplicities. (a) Prove the following version of the argument principle (cf. Proposition 2.5.7 and Exercises 5.1.6, 5.1.7, 5.9.5, 6.7.1, and 6.7.6): If f is a nontrivial meromorphic function on X that does not have any zeros or poles in ∂, then  1 df μf − νf = . 2πi ∂ f (b) Prove the following version of Rouché’s theorem: If f and g are nontrivial meromorphic functions on X that do not have any zeros or poles in ∂, and |g| < |f | on ∂, then μf − νf = μf +g − νf +g . Hint. Using part (a), show that the integer-valued function t → μf +tg − νf +tg is continuous on the interval [0, 1].

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53

2.5.9 If X is a complex 1-manifold, p ∈ X, θ is a holomorphic 1-form on V \{p} for some neighborhood V of p in X, (U,  = z, (z0 ; R)) is a local holomorphic coordinate neighborhood of p with  (p) = z0 andn U ⊂ V , f = θ/dz (i.e., θ = f dz on U \ {p}), and f = ∞ n=−∞ cn (z − z0 ) is the corresponding Laurent series representation of f , then the coefficient c−1 is called the residue of θ at p and is denoted by resp θ . Equivalently, if r ∈ (0, R) and D ≡ −1 ((0; r)), then  1 θ. resp θ = 2πi ∂D (a) Prove that the residue of a holomorphic 1-form at an isolated singularity is well defined (that is, prove that the above definition is independent of the choice of the local holomorphic coordinate). (b) Prove the following version of the residue theorem (cf. Exercises 5.1.6, 5.1.7, 5.9.5, 6.7.1, and 6.7.6): If  is a (nonempty) smooth relatively compact domain in a Riemann surface X, S is a finite subset of , and θ is a holomorphic 1-form on X \ S, then   1 θ= resp θ. 2πi ∂ p∈S

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

2.6 L2 Scalar-Valued Differential Forms on a Riemann Surface Throughout this section, X denotes a complex 1-manifold. The goal of this section is the development of a suitable L2 space of differential forms. Since the objects that we integrate on oriented surfaces are the 2-forms (see Sect. 9.7) and we wish to pair forms of the same degree and integrate in order to get an inner product, it is natural to consider a pointwise pairing that gives a 2-form. Let us first √ consider 2) dz and integration on C. We have the natural unit-length (1, 0)-form θ = (1/ √ √ the natural positive (1, 1)-form ω = −1θ ∧ θ¯ = ( −1/2) dz ∧ d z¯ . In the notation of Sect. 9.7, the Lebesgue measure λ in C is equal to the measure λω associated to ω (see Definition 9.7.10). If α = aθ and β = bθ are differential forms of type (1, 0) with L2 coefficients, then     ¯ a, bL2 (C) = a b¯ dλ = a b¯ ω = a b¯ iθ ∧ θ¯ = iα ∧ β. C

C

C

C

If α and β are L2 differential forms of type (0, 0) (i.e., L2 functions), then   ¯ α, βL2 (C) = α β dλ = α β¯ ω. C

C

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Finally, if α = aω and β = bω are differential forms of type (1, 1) with L2 coefficients, then    α β a, bL2 (C) = a b¯ dλ = a b¯ ω = · · ω. C C C ω ω The right-hand sides in the above point to a reasonable definition for the L2 inner product of differential forms on the complex 1-manifold X. Observe that the expression for the inner product of a pair of forms of type (0, 0) and (1, 1) involves the Kähler form ω, although that for the inner product of a pair of (1, 0)-forms does not. Thus, on X, for forms of type (0, 0) and (1, 1), we must define the inner product with respect to some choice of a positive (1, 1)-form. For functions, it turns out to be useful to weaken the requirement on the (1, 1)-form by allowing it to be only nonnegative. After all, it is not even a priori clear that X admits a global Kähler form; although, as will be shown in Sect. 2.11, it turns out that every Riemann surface is second countable and therefore that X does admit a Kähler form (see Corollary 2.11.3). As will be seen in Sect. 2.9, it is also useful to include a weight function e−ϕ (in an abuse of language, we also call ϕ a weight function). For example, a Riemann surface need not admit any L2 holomorphic 1-forms, but given a holomorphic 1-form, one may construct a C ∞ function ϕ that grows so large at infinity that the holomorphic 1-form becomes L2 with respect to the weight function e−ϕ . Finally, observe that iα ∧ α¯ > 0 for every nonzero α ∈ 1,0 T ∗ X. Based on these considerations, we make the following definition: Definition 2.6.1 Let S be a measurable subset of X, let ϕ be a measurable realvalued function that is defined on S, and let α and β be measurable differential forms of type (p, q) that are defined on S. (a) For (p, q) = (1, 0), we define  αL2

1,0 (S,ϕ)

If

√

≡

√ −1α ∧ α¯ · e−ϕ

1/2 ∈ [0, ∞].

S

−1α ∧ β¯ · e−ϕ is integrable on S, then we define  √ −1α ∧ β¯ · e−ϕ ∈ C. α, βL2 (S,ϕ) ≡ 1,0

S

(b) For (p, q) = (0, 1), we define αL2

0,1

1/2   √ −ϕ ≡ − −1α ∧ α ¯ · e ∈ [0, ∞]. (S,ϕ) S

√

If − −1α ∧ β¯ · e−ϕ is integrable on S, then we define  √ α, βL2 (S,ϕ) ≡ − −1α ∧ β¯ · e−ϕ ∈ C. 0,1

S

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L2 Scalar-Valued Differential Forms on a Riemann Surface

55

(c) If (p, q) = (0, 0) and ω is a nonnegative measurable form of type (1, 1) defined on S, then we define  αL2

0,0

(S,ω,ϕ) ≡

|α|2 e−ϕ ω

1/2 ∈ [0, ∞].

S

If α β¯ · e−ϕ ω is integrable on S, then we define  α, βL2 (S,ω,ϕ) ≡ α β¯ · e−ϕ ω ∈ C. 0,0

S

(d) If (p, q) = (1, 1) and ω is a positive measurable form of type (1, 1) defined on S, then we define   2   α  −ϕ 1/2   e ω αL2 (S,ω,ϕ) ≡ = α/ωL2 (S,ω,ϕ) ∈ [0, ∞].   1,1 0,0 S ω If the form α β −ϕ · ·e ω ω ω is integrable on S, then we define  α, βL2

1,1 (S,ω,ϕ)

≡ S

  α β α β −ϕ · ·e ω= , ω ω ω ω L2

∈ C.

0,0 (S,ω,ϕ)

(e) When there is no danger of confusion (for example, when the choice of (p, q), S, ω, or ϕ is understood from the context), we will suppress parts of the notation. For example, we will often write αL2p,q (S,ω,ϕ) simply as αS,ω,ϕ , αS,ϕ , αS,ω , αω,ϕ , αϕ , αω , αL2 (S,ω,ϕ) , αL2 (S,ϕ) , αL2 (ω,ϕ) , αL2 (ϕ) , αL2 (ω) , or α. We will also use the analogous simplified notation for α, βL2p,q (S,ω,ϕ) , for αL2p,q (S,ϕ) , and for α, βL2p,q (S,ϕ) . Moreover, when no mention of a weight function appears in the context, then the weight function will be assumed to be ϕ ≡ 0 and we will write αL2p,q (S,ω,0) simply as αL2p,q (S,ω) and so on. (f) Let γ and η be measurable differential 1-forms defined on S with (r, s) parts γ r,s and ηr,s , respectively, for each (r, s) ∈ {(1, 0), (0, 1)}. Then we set γ 2L2 (S,ϕ) ≡ γ 1,0 2ϕ + γ 0,1 2ϕ . 1

We also set γ , ηL2 (S,ϕ) ≡ γ 1,0 , η1,0 ϕ + γ 0,1 , η0,1 ϕ , provided each of the 1 summands on the right-hand side is defined. We also use the simplified notation analogous to that appearing in (e). Definition 2.6.2 Let S be a measurable subset of X, and let ϕ be a measurable real-valued function that is defined on S.

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(a) For (p, q) = (1, 0) or (0, 1), L2p,q (S, ϕ) consists of all equivalence classes of measurable differential forms α of type (p, q) on S with αL2 (S,ϕ) < ∞, where we identify any two elements that are equal almost everywhere. The set L21 (S, ϕ) consists of all equivalence classes of measurable 1-forms α with αL2 (S,ϕ) < ∞, where again, we identify any two elements that are equal almost everywhere. (b) For ω a nonnegative measurable differential form of type (1, 1) that is defined on S, L20,0 (S, ω, ϕ) consists of all equivalence classes of measurable functions α on S with αL2 (S,ω,ϕ) < ∞, where we identify any two elements that are equal almost everywhere. (c) For ω a positive measurable differential form of type (1, 1) that is defined on S, L21,1 (S, ω, ϕ) consists of all equivalence classes of measurable differential forms α of type (1, 1) on S with αL2 (S,ω,ϕ) < ∞, where we identify any two elements that are equal almost everywhere. (d) When no mention of a weight function appears in the context and there is no danger of confusion, then the weight function will be assumed to be ϕ ≡ 0 and we will write L2p,q (S, 0) simply as L2p,q (S), and L2p,q (S, ω, 0) simply as L2p,q (S, ω). Remark For our purposes, we will need only weight functions ϕ and nonnegative (or positive) (1, 1)-forms ω that are defined and of class C ∞ on an open subset of X. Proposition 2.6.3 Let S be a measurable subset of X, and let ϕ be a continuous real-valued function that is defined on S. (a) The pair (L21 (S, ϕ), ·, ·L2 (S,ϕ) ), and the pair (L2p,q (S, ϕ), ·, ·L2 (S,ϕ) ) for (p, q) = (1, 0) or (0, 1), are Hilbert spaces (where the inner product of any two equivalence classes is given by the pairing of any representatives). Moreover, we have the Hilbert space orthogonal decomposition L21 (S, ϕ) = L21,0 (S, ϕ) ⊕ L20,1 (S, ϕ). (b) For ω a continuous positive differential form of type (1, 1) that is defined on S and for (p, q) = (0, 0) or (1, 1), (L2p,q (S, ω, ϕ), ·, ·L2 (S,ω,ϕ) ) is a Hilbert space (where the inner product of any two equivalence classes is given by the pairing of any representatives). Moreover, in each of the above spaces, any sequence converging to an element α admits a subsequence that converges to α pointwise almost everywhere in S. Proof Let p, q ∈ {0, 1}. We set  L2p,q ≡

L2p,q (S, ϕ) as in (a) L2p,q (S, ω, ϕ) as in (b)

if (p, q) = (1, 0) or (0, 1), if (p, q) = (0, 0) or (1, 1);

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L2 Scalar-Valued Differential Forms on a Riemann Surface

57

and for each point r ∈ S and each pair of elements η, θ ∈ p,q Tr∗ X, we set ⎧ ⎪ if (p, q) = (0, 0), ηθ¯ · e −ϕ(r) · ωr ⎪ ⎪ ⎪ ⎪ if (p, q) = (1, 0), ⎨iη ∧ θ¯ · e −ϕ(r) H (η, θ) ≡ −iη ∧ θ¯ · e−ϕ(r) if (p, q) = (0, 1), ⎪ ⎪ ⎪ ⎪ η θ ⎪ ⎩ · · e−ϕ(r) ωr if (p, q) = (1, 1). ωr ωr We first show that (L2p,q , ·, ·) is an inner product space. For this, observe that if α, β ∈ L2p,q , then the (1, 1)-form H (α, β) : r → H (αr , βr ) is measurable. Suppose (U,  = z = x + iy, U  ) is a local holomorphic chart and ω ≡ (i/2) dz ∧ d z¯ = dx ∧ dy. Then the map h : (η, θ ) → h(η, θ ) ≡

H (η, θ) ∈C (ω )r

∀η, θ ∈ p,q Tr∗ X

is a Hermitian inner product in p,q Tr∗ X for each point r ∈ S ∩ U . We also have the corresponding pointwise norm η → |η|h = [h(η, η)]1/2 . Hence, for each measurable set R ⊂ S ∩ U and each complex number ζ with |ζ | = 1, we have (by the Schwarz inequality)    + + [Re(ζ H (α, β))] = [Re(ζ h(α, β))] dλω ≤ |α|h |β|h dλω R

R



R

1/2 

≤ R

 =

|α|2h dλω

R

1/2

|β|2h dλω

1/2  1/2 H (α, α) · H (β, β)

R

R

(where λω is the positive measure associated to ω as in Definition 9.7.10). Thus, if S1 , . . . , Sm are disjoint measurable subsets of S each of which lies in some local holomorphic coordinate neighborhood, then, by the Schwarz inequality for sums, 1/2  1/2 m  m    + [Re(ζ H (α, β))] ≤ H (α, α) · H (β, β) j =1 Sj

j =1

 ≤

Sj

m  

Sj

1/2  ·

H (α, α)

j =1 Sj



m  

H (β, β)

j =1 Sj

1/2  1/2 H (α, α) · H (β, β) = α · β.

≤ S

S

Passing to the supremum, we get  [Re(ζ H (α, β))]+ ≤ α · β < ∞. S

1/2

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

 ζ H (α, β) =

|α, β| = 

S

≤

[Re(ζ H (α, β))]+ −



S

[Re(ζ H (α, β))]− S

[Re(ζ H (α, β))]+ ≤ α · β.

S

For each constant s ∈ C, we have H (sα + β, sα + β) = |s|2 H (α, α) + 2 Re(sH (α, β)) + H (β, β), so H (sα + β, sα + β) is integrable. If α = α  a.e. (almost everywhere) in S, then clearly, H (α  , α  ) = H (α, α) and H (α  , β) = H (α, β) a.e. Thus L2p,q is a welldefined vector space, and as is now easy to verify, ·, · is a well-defined Hermitian inner product. It remains to show that (L2p,q , ·, ·) is complete with respect to the norm α = α, α1/2 . We again apply (appropriately modified) standard arguments (cf., for example, [Rud1]). We must show that a given Cauchy sequence {αν } in L2p,q converges. For this, it suffices to show that some subsequence converges. Hence, after replacing the sequence with a suitable subsequence, we may assume without loss  of generality that αν+1 −αν  < 1. For a local holomorphic chart (U,  = z, U  ), for ω = dx ∧ dy = (i/2) dz ∧ d z¯ and for h(·, ·) = H (·, ·)/ω , as before, let us set ϕN ≡

N 

|αν+1 − αν |h

∀N ∈ Z>0

and ϕ ≡

∞ 

ν=1

|αν+1 − αν |h .

ν=1

For each N ∈ Z>0 , we have ϕN L2 (S∩U,λω



)≤

N 

|αν+1 − αν |h L2 (S∩U,λω

ν=1



)≤

N 

αν+1 − αν  < 1.

ν=1

Fatou’s lemma now implies that ϕL2 (S∩U,λω ) ≤ 1. In particular, ϕ < ∞ almost  everywhere in S ∩ U . It follows that the series α1 +

∞ 

(αν+1 − αν )

ν=1

converges pointwise almost everywhere in S to a measurable differential form α of type (p, q), and hence αν → α pointwise a.e.

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L2 Scalar-Valued Differential Forms on a Riemann Surface

59

It remains to show that α ∈ L2p,q and that α − αν  → 0. Given  > 0, we may choose N ∈ Z>0 so that αμ − αν  <  for all μ, ν > N . For any ν > N , Fatou’s lemma (Theorem 9.7.11) gives   α − αν 2 = H (α − αν , α − αν ) ≤ lim inf H (αμ − αν , αμ − αν ) ≤  2 . μ→∞

S

S

Thus α = (α − αν ) + αν ∈ L2p,q and α − αν  ≤ . Therefore αν → α in L2p,q (and pointwise almost everywhere in S) and L2p,q is a Hilbert space. The proposition (including the orthogonal decomposition in (a)) now follows.  We have the following useful version of Theorem 1.2.4: Theorem 2.6.4 Let ϕ be a continuous real-valued function and let ω be a continuous positive differential form of type (1, 1) on X. Then, for every compact set K ⊂ X, there is a constant C = C(X, K, ω, ϕ) > 0 such that max |f | ≤ Cf L2 (X,ω,ϕ) K

∀ f ∈ O(X).

Consequently, O(X) ∩ L20,0 (X, ω, ϕ) is a closed subspace of L20,0 (X, ω, ϕ). Proof Given a point p ∈ K, we may fix a relatively compact local holomorphic coordinate neighborhood (U, z) on which the function (i/2)(dz ∧d z¯ )/ω is bounded. Therefore, for some constant R > 0, we have, for every holomorphic function f on U ,   (i/2) dz ∧ d z¯ −ϕ 2i · e · ω ≤ Rf 2L2 (U,ω,ϕ) . |f | dz ∧ d z¯ = |f |2 eϕ 2 ω U U Fixing a relatively compact neighborhood Vp of p in U and applying Theorem 1.2.4, we get a constant Cp > 0 such that sup |f | ≤ Cp f L2 (U,ω,ϕ) ≤ Cp f L2 (X,ω,ϕ)

∀f ∈ O(X).

Vp

Covering K by finitely many such neighborhoods Vp and taking C to be the maximum of the associated constants Cp , we get the claim. The proof that O(X) ∩  L20,0 (X, ω, ϕ) is a closed subspace is left to the reader (see Exercise 2.6.3). Exercises for Sect. 2.6 2.6.1 Show that there are no nontrivial L2 holomorphic 1-forms on C (cf. Exercise 2.5.4). 2.6.2 For the function ϕ : z → |z|2 on C, show that there exists a nontrivial holomorphic 1-form in L21,0 (C, ϕ).

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2.6.3 Let ϕ be a continuous real-valued function on a Riemann surface X. Prove that the vector space of holomorphic 1-forms in L21,0 (X, ϕ) is a closed subspace of L21,0 (X, ϕ). Also prove that if ω is a continuous positive (1, 1)-form on X, then the vector space of holomorphic functions in L20,0 (X, ω, ϕ) is a closed subspace (i.e., prove the second part of Theorem 2.6.4).

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

Definition 2.7.1 Let α and β be locally integrable differential forms on an open set  ⊂ X. We write ∂¯ distr α = β if for every local holomorphic coordinate neighborhood (U, z), one of the following holds: (i) On U ∩ , α is a 0-form, β = b d z¯ , and (∂α/∂ z¯ )distr = b; (ii) On U ∩ , α = a1 dz + a2 d z¯ , β = b dz ∧ d z¯ , and (∂a1 /∂ z¯ )distr = −b; (iii) The form α is of degree > 1 and β ≡ 0. Remark Given a locally integrable differential form α, ∂¯distr α need not exist as a form (see the remarks following Definition 7.4.2). For example, let u be the char¯ ≡ 0 on the complement acteristic function of the unit disk  ≡ (0; 1). Then ∂u ¯ C \ ∂ of the measure-zero set ∂. Thus, if ∂distr u were to exist on C, then it would be the zero form, and hence by the regularity theorem (Theorem 1.2.8), u would be holomorphic, which it is not. It also follows that for the measurable differential form α = u dz, ∂¯distr α does not exist as a form. It is often more convenient to work with an intrinsic form of an operator, so we look for an intrinsic description of ∂¯ distr . Distributional differential operators in Rn are defined via integration against test functions (as in Sect. 7.4.2). On a surface, the objects that we integrate on open sets (or, more generally, measurable sets) are 2-forms, and for a differential form, its wedge product with a form of complementary degree is a 2-form. Thus, in the present context, it is natural to

2.7 The Distributional ∂¯ Operator on Scalar-Valued Forms

61

integrate against test forms of complementary degree. For example, suppose α and ¯ = β. Since β are C ∞ differential forms of type (1, 0) and (1, 1), respectively, and ∂α ∞ we integrate C forms of type (1, 1) (i.e., 2-forms) on open sets in X, it is natural to integrate the wedge product of α and the conjugate of a (1, 0)-form, and the scalar product (i.e., the wedge product) of β and a function (i.e., a (0, 0)-form). Given a function f ∈ D(X), we have ¯ · f¯ = (dα) · f¯ = d(α · f¯) + α ∧ d f¯ β · f¯ = (∂α) = d(α · f¯) + α ∧ ∂ f¯ + α ∧ ∂¯ f¯ = d(α · f¯) + α ∧ ∂f . Integrating and applying Stokes’ theorem, we get   ¯ β ·f = α ∧ ∂f . X

X

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

 X

β · f¯ · e−ϕ =



α ∧ eϕ ∂(e−ϕ f ) · e−ϕ .

X

The above suggests a natural intrinsic form of the definition of ∂¯ distr on forms of type (1, 0). A similar computation applies to forms of type (0, 0). Based on the above, we make the following definition: Definition 2.7.2 Given a real-valued function ϕ ∈ C ∞ (X), the associated canonical connection (or the Chern connection) is the operator D = Dϕ = D  + D  , where for each C 1 differential form α on an open subset of X, we define D  α = Dϕ α ≡ eϕ ∂(e−ϕ α) = ∂α + eϕ (∂e−ϕ ) ∧ α = ∂α − ∂ϕ ∧ α and ¯ D  α = Dϕ α ≡ ∂α. We call D  and D  , respectively, the (1, 0) part and (0, 1) part of the connection.

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

2

Remarks 1. Observe that D  = ∂¯ does not depend on the choice of ϕ. 2. We have D = d + eϕ (∂e−ϕ ) ∧ (·) = d − ∂ϕ ∧ (·). 3. If α is of type (p, q), then D  α is of type (p + 1, q) (in particular, D  α = 0 if p ≥ 1) and D  α is of type (p, q + 1) (D  α = 0 if q ≥ 1). 4. If α and β are C 1 differential forms and α is of degree p, then (see Exercise 2.7.1) D(α ∧ β) = (dα) ∧ β + (−1)p α ∧ Dβ = (Dα) ∧ β + (−1)p α ∧ dβ. 5. Dϕ , Dϕ , and Dϕ are defined in the same way as above for ϕ ∈ C 1 (X). We now get the following equivalent form for the definition of ∂¯distr (which we  ): also denote by Ddistr Proposition 2.7.3 Let ϕ be a real-valued C ∞ function on X, and let α and β be locally integrable differential forms on an open set  ⊂ X. (a) If α is of type (1, 0) and β is of type (1, 1), then ∂¯distr α = β if and only if   −ϕ  α ∧D fe = β · f¯ · e−ϕ ∀f ∈ D(). 



(b) If α is of type (0, 0) and β is of type (0, 1), then ∂¯distr α = β if and only if   α · (−D  γ ) · e−ϕ = β ∧ γ · e−ϕ ∀γ ∈ D0,1 (). 



(c) If α is of type (p, q) with p ≥ 2 or q ≥ 1, then ∂¯distr α = 0. Remark It will follow from the proof that if the conditions in Definition 2.7.1 hold for the forms α and β in some local holomorphic coordinate neighborhood of every point, then they hold in every local holomorphic coordinate neighborhood. Proof of Proposition 2.7.3 Suppose α = a dz and β = b dz ∧ d z¯ in some local holomorphic coordinate neighborhood (U, z) with U ⊂ . Then, for every function f ∈ D(U ) (which we may view as a C ∞ function with compact support in ), we have   β f¯e−ϕ = b(e−ϕ f ) dz ∧ d z¯ 

U

and  

α ∧ D  f · e−ϕ =

 U

α ∧ ∂(e−ϕ f ) =

 a· U

∂ −ϕ (e f ) dz ∧ d z¯ . ∂z

= eϕ u, we see that if the above left-hand sides are always

Given u ∈ D(U ), setting f equal, then we have ∂¯distr α = β. Conversely, if ∂¯distr α = β and  f ∈ D(), then, choosing finitely many C ∞ functions {ην }nν=1 on  such that ην ≡ 1 on supp f

2.8 Curvature and the Fundamental Estimate for Scalar-Valued Forms

63

and such that for each ν, the support of ην is contained in some local holomorphic coordinate neighborhood Uν , the above gives    −ϕ −ϕ   α∧D f ·e = α ∧ D (ην · f ) · e = α ∧ D  (ην · f ) · e−ϕ 

ν

=



 ν

βην f e−ϕ =

Uν

 ν

Uν

ν

βην f e−ϕ =





β f¯ · e−ϕ .



Thus part (a) is proved. Part (c) is obvious and the proof of part (b) is left to the reader (see Exercise 2.7.2).  The regularity theorem (Theorem 1.2.8) gives the following in this context: Theorem 2.7.4 If p ∈ {0, 1}, α is a locally integrable form of type (p, 0) on X, and ∂¯distr α = β for a form β of class C k for some k ∈ Z>0 ∪ {∞}, then α is also of class C k . In particular, if ∂¯distr α = 0, then α is a holomorphic p-form. Exercises for Sect. 2.7 2.7.1 Let ϕ be a C ∞ real-valued function on a Riemann surface X. Show that if α and β are C 1 differential forms on X and α is of degree p, then Dϕ (α ∧ β) = (dα) ∧ β + (−1)p α ∧ Dϕ β = (Dϕ α) ∧ β + (−1)p α ∧ dβ. 2.7.2 Prove part (b) of Proposition 2.7.3. 2.7.3 Let X be a Riemann surface, let ϕ be a real-valued C ∞ function on X, let {θν } be a sequence of holomorphic 1-forms in L21,0 (X, ϕ), and let θ ∈ L21,0 (X, ϕ). Assume that for each element α ∈ L21,0 (X, ϕ), we have lim θν , αL2

ν→∞

1,0 (X,ϕ)

= θ, αL2

1,0 (X,ϕ)

(that is, {θν } converges weakly to θ in L21,0 (X, ϕ)). Prove that θ is a holomorphic 1-form.

2.8 Curvature and the Fundamental Estimate for Scalar-Valued Forms Throughout this section, X again denotes a complex 1-manifold. Suppose ϕ is a real-valued C ∞ function on X. We recall that for any C ∞ differential form α on an open subset of X, we have Dϕ α = Dα = D  α + D  α, where D  α = eϕ ∂[e−ϕ α] = ∂α − (∂ϕ) ∧ α

¯ and D  α = ∂α.

64

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

2

¯ = 0, we Consequently, (D  )2 = (D  )2 = 0 and D 2 = D  D  + D  D  . Since ∂ ∂¯ + ∂∂ have ¯ − (∂ϕ) ∧ ∂α ¯ + ∂∂α ¯ − ∂[(∂ϕ) ¯ ∧ α] D 2 α = ∂ ∂α ¯ − (∂∂ϕ) ¯ ¯ = ϕ ∧ α, = −(∂ϕ) ∧ ∂α ∧ α + (∂ϕ) ∧ ∂α ¯ is a differential form of type (1, 1) (of course, θϕ ∧ α = 0 if where ϕ ≡ ∂ ∂ϕ deg α > 0). Definition 2.8.1 The curvature (or curvature form or Levi form) associated to a ¯ In other real-valued C ∞ function ϕ on X is the differential form  = ϕ ≡ ∂ ∂ϕ. words,  is defined by D 2 = D  D  + D  D  =  ∧ (·). The function ϕ is called subharmonic (strictly subharmonic, harmonic, superharmonic, strictly superharmonic) if the real (1, 1)-form iϕ satisfies iϕ ≥ 0 (respectively, iϕ > 0, iϕ = 0, iϕ ≤ 0, iϕ < 0). In a slight abuse of language, we also say that ϕ has nonnegative (respectively, positive, zero, nonpositive, negative) curvature. Remarks 1. The above terminology, with the same definitions, is also applied to C 2 functions. There is also a natural and useful notion of a continuous subharmonic function (see, for example, [Ns5]), but continuous subharmonic functions are not considered in this book. 2. For any local holomorphic coordinate neighborhood (U, z = x + iy) in X and for any real-valued C ∞ function ϕ on U , we have   ∂ 2ϕ ∂ 2ϕ 1 ∂ 2ϕ ∂ 2ϕ + 2 dx ∧ dy. iϕ = i dz ∧ d z¯ = 2 dx ∧ dy = ∂z∂ z¯ ∂z∂ z¯ 2 ∂x 2 ∂y Thus ϕ is subharmonic (strictly subharmonic, harmonic) on U if and only if ∂ 2ϕ ≥0 ∂z∂ z¯

(respectively, > 0,

= 0).

3. It is easy to see that a strictly subharmonic function cannot attain a local maximum (see Exercise 2.8.2). In fact, one can show that subharmonic functions satisfy a strong maximum principle (see, for example, [Ns5]). In particular, every subharmonic function on a compact Riemann surface is constant. Proposition 2.8.2 (Fundamental estimate for scalar-valued forms) Let ϕ be a real¯ Then, for all valued C ∞ function on X, and let D  = Dϕ ,  = ϕ , and D  = ∂. functions u, v ∈ D(X), we have      iuve ¯ −ϕ . D u, D vL2 (X,ϕ) = D u, D vL2 (X,ϕ) + In particular,

D  u2L2 (X,ϕ)

= D  u2L2 (X,ϕ)

+

X

 X

i|u|2 e−ϕ

≥



X i|u|

2 e−ϕ .

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

65

Remark If i ≥ 0, then we get D  u, D  vL2 (X,ϕ) = D  u, D  vL2 (X,ϕ) + u, vL2 (X,i,ϕ) and D  u2L2 (X,ϕ) = D  u2L2 (X,ϕ) + u2L2 (X,i,ϕ) ≥ u2L2 (X,i,ϕ) . Proof of Proposition 2.8.2 For each pair of functions u, v ∈ D(X), Proposition 2.7.3 gives   D  u, D  vL2 (X,ϕ) = iD  u ∧ D  ve−ϕ = i(D  D  u) · ve ¯ −ϕ X

X



i(D  D  u) · ve ¯ −ϕ +

=− X





iuve ¯ −ϕ X





iuve ¯ −ϕ

vD  D  u · e−ϕ +

= −i X





(D  v) ∧ D  u · e−ϕ

=i

X



(D  u) ∧ D  v · e−ϕ +

X

= D  u, D  vL2 (X,ϕ) +



iuve ¯ −ϕ

+

X

= −i





X

iuve ¯ −ϕ X

iuve ¯ −ϕ .

X



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

¯ for Scalar-Valued Forms of Type (1, 0) 2.9 The L2 ∂-Method The following theorem (in various guises) is the main tool in this book: Theorem 2.9.1 Let X be a Riemann surface, let ϕ be a real-valued C ∞ func¯ ≥ 0, and let Z = {x ∈ X | x = 0}. Then, tion on X with i = iϕ = i∂ ∂ϕ for every measurable (1, 1)-form β on X with βX\Z ∈ L21,1 (X \ Z, i, ϕ) and β = 0 a.e. in Z (in particular, β is in L2loc on X), there exists a form α ∈ L21,0 (X, ϕ) such that  α = ∂¯distr α = β Ddistr

and

αL2 (X,ϕ) ≤ βL2 (X\Z,i,ϕ) .

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

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

Remark The form β in the above theorem is in L2loc because β vanishes on Z, while for any compact set K in any local holomorphic coordinate neighborhood (U, z), we have  · L2

1,1 (K\Z,(i/2) dz∧d z¯ )

where

 A ≡ max K

≤ A · L2

1,1 (K\Z,i,ϕ)

i eϕ (i/2) dz ∧ d z¯

,

1/2 < ∞.

Proof of Theorem 2.9.1 Let N ≡ βL2 (X\Z,i,ϕ) = β/(i)L2 (X\Z,i,ϕ) . For each function f ∈ D(X), the Schwarz inequality and the fundamental estimate (Proposition 2.8.2) give     β¯ −ϕ f· · e · i = f, β/(i)L2 (X\Z,i,ϕ)  i X\Z

       f βe ¯ −ϕ  =     X

≤ f L2 (X\Z,i,ϕ) · β/(i)L2 (X\Z,i,ϕ) ≤ N · f L2 (X,i,ϕ) ≤ N · D  f L2 (X,ϕ) .  ¯ −ϕ is a bounded complex linear It follows that the mapping ϒ : [D  f ] → −i X f βe functional on the subspace D  [D(X)] of L21,0 (X, ϕ). For by the above inequality, ϒ is well defined, and for each f ∈ D(X), we have |ϒ[D  f ]| ≤ N · D  f L2 (X,ϕ) . In particular, ϒ ≤ N . By the Hahn–Banach theorem (Theorem 7.5.11), there  on L2 (X, ϕ) such that ϒ  D  [D(X)] = ϒ exists a bounded linear functional ϒ 1,0   = ϒ. Therefore, by Theorem 7.5.10, there exists a (unique) element and ϒ   ≤ N and ϒ  (·) = ·, αL2 (X,ϕ) . Moreα ∈ L21,0 (X, ϕ) such that αL2 (X,ϕ) = ϒ over, for each f ∈ D(X), we have 

iα ∧ D  f e−ϕ = ϒ(D  f ) = X



¯ −ϕ = (−i)f βe X



iβ f¯e−ϕ .

X

 α = β, as required. Finally, the regularity Therefore, by Proposition 2.7.3, Ddistr statement at the end follows from Theorem 2.7.4. 

It is often more convenient to apply Theorem 2.9.1 in one of the following forms, the proofs of which are left to the reader (see Exercises 2.9.1 and 2.9.2): Corollary 2.9.2 Suppose that X is a Riemann surface, ω is a Kähler form on X, ϕ is a real-valued C ∞ function on X, ρ is a nonnegative measurable function on X with iϕ ≥ ρω, and Z = {x ∈ X | ρ(x) = 0}. Then, for every measurable (1, 1)-form β on X with β = 0 a.e. in Z and βX\Z ∈ L21,1 (X \ Z, ρω, ϕ) = L21,1 (X \ Z, ω, ϕ + log ρ)

2.10

Existence of Meromorphic 1-Forms and Meromorphic Functions

67

(in particular, β is in L2loc on X), there exists a form α ∈ L21,0 (X, ϕ) such that  Ddistr α = ∂¯distr α = β

and

αL2 (X,ϕ) ≤ βL2 (X\Z,ρω,ϕ) .

In particular, if β is of class C k for some k ∈ Z>0 ∪ {∞}, then α is also of class C k ¯ = β. and ∂α Remark The main point is that βL2 (X\Z,iϕ ,ϕ) ≤ βL2 (X\Z,ρω,ϕ) if iϕ ≥ ρω. Corollary 2.9.3 Suppose that X is a Riemann surface, ω is Kähler form on X, ϕ is a real-valued C ∞ function on X, and C is a positive constant with iϕ ≥ C 2 ω. Then, for every form β ∈ L21,1 (X, ω, ϕ), there exists a form α ∈ L21,0 (X, ϕ) such that  α = ∂¯distr α = β Ddistr

and

αL2 (X,ϕ) ≤ C −1 βL2 (X,ω,ϕ) .

In particular, if β is of class C k for some k ∈ Z>0 ∪ {∞}, then α is also of class C k ¯ = β. and ∂α Remark In Sect. 2.11, we will show that every Riemann surface admits a Kähler form. Exercises for Sect. 2.9 2.9.1 Prove Corollary 2.9.2. 2.9.2 Prove Corollary 2.9.3. 2.9.3 Let X be a Riemann surface, let ϕ be a real-valued C ∞ function on X with i = iϕ ≥ 0, and let Z = {x ∈ X | x = 0}. Prove that for every C ∞ (1, 1)form β on X with β = 0 a.e. in Z and βX\Z ∈ L21,1 (X \ Z, i, ϕ), there exists a C ∞ form α ∈ L20,1 (X, ϕ) such that ∂α = β

and αL2

0,1 (X,ϕ)

≤ βL2

1,1 (X\Z,i,ϕ)

.

2.9.4 Let X be a Riemann surface, let ϕ be a real-valued C ∞ function on X, let ω be a nonnegative C ∞ (1, 1)-form on X, let Z = {x ∈ X | ωx = 0}, and let β be a C ∞ (1, 1)-form on X such that β = 0 a.e. in Z and βX\Z ∈ L21,1 (X \ Z, ω, ϕ). Prove that β ≡ 0 on Z (hence in the C ∞ case of Theorem 2.9.1 and in Exercise 2.9.3, there is no loss of generality if we assume that β ≡ 0 on Z).

2.10 Existence of Meromorphic 1-Forms and Meromorphic Functions The power of Theorem 2.9.1 is demonstrated by the following important application, which will play a crucial role in the proof of such central facts as Radó’s theorem on second countability (see Sect. 2.11) and the Riemann mapping theorem (see Chap. 5):

68

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

Theorem 2.10.1 For every point in a Riemann surface, there exists a meromorphic 1-form that is holomorphic except for a pole of arbitrary prescribed order ≥ 2 at the point. In fact, for each integer m ≥ 2, there exists a universal constant Cm > 0 such that if X is any Riemann surface, p is any point in X, and (D,  = z, (0; 1)) is any local holomorphic chart with p ∈ D and (p) = z(p) = 0, then there exists a meromorphic 1-form θ on X with the following properties: (i) The meromorphic 1-form θ is holomorphic on X \ {p} and has a pole of order m at p; (ii) We have θ L2 (X\D) ≤ Cm ; (iii) The meromorphic 1-form θ − z−m dz on D has at worst a simple pole at p; and (iv) We have zθ − z−m+1 dzL2 (D) ≤ Cm . Remark The value for the constant Cm that we will obtain is far from optimal. Lemma 2.10.2 For any choice of constants a, b, c ∈ R with a < b < c, there exists a C ∞ function χ : R → R such that (i) For each t ∈ R, χ  (t) ≥ 0 and χ  (t) ≥ 0; (ii) For each t ≤ a, χ(t) = 0; and (iii) For each t ≥ c, χ(t) = t − b. Proof For s > 0, the function ⎧ s ⎪ ⎪ ⎨ s+exp( 1 + 1 ) = t−a t−c ρs (t) ≡ 0 ⎪ ⎪ ⎩ 1

1 s exp( a−t ) 1 1 s exp( a−t )+exp( t−c )

if a < t < c, if t ≤ a, if c ≤ t,

c is of class C ∞ , ρs ≥ 0, and ρs ≥ 0. Moreover, the function μ : s → a ρs (t) dt is continuous on (0, ∞), μ(s) → 0 as s → 0+ , and μ(s) → c − a > c − b as s → ∞ (for example, by the dominated convergence theorem). Therefore, by the intermediate value theorem, there is a number s0 > 0 such that μ(s0 ) = c − b. The function t t → χ(t) ≡ a ρs0 (u) du then has the required properties.  Lemma 2.10.3 Let r > 0 be a constant and let ψ be a C ∞ strictly subharmonic function on (0; r). Then there is a constant b0 = b0 (r, ψ) > 0 such that for every constant b > b0 , there exist a constant R = R(b, r, ψ) ∈ (0, r) and a nonnegative C ∞ subharmonic function ϕ on C∗ with ϕ ≡ 0 on a neighborhood of C \ (0; r) and ϕ(z) = ψ(z) − log |z|2 − b

∀z ∈ ∗ (0; R).

Proof Setting ρ(z) ≡ ψ(z) − log |z|2 for all z ∈ ∗ (0; r), we may fix positive constants R0 and R1 with 0 < R0 < R1 < r and a positive constant b0 > sup(0;R0 ,R1 ) ρ. Given a constant b > b0 , we may fix constants a and c with

2.10

Existence of Meromorphic 1-Forms and Meromorphic Functions

69

c > b > a > b0 > 0, and applying Lemma 2.10.2, we get a C ∞ function χ : R → R such that χ  ≥ 0 and χ  ≥ 0 on R, χ(t) = 0 for t ≤ a, and χ(t) = t − b for t ≥ c. For R ∈ (0, r) sufficiently small, we have ρ > c on ∗ (0; R). This constant R and the function ϕ on C∗ given by  χ(ρ) on ∗ (0; R1 ), ϕ≡ 0 on C \ (0; R1 ), then have the required properties. For the choice of χ guarantees that ϕ is nonnegative and of class C ∞ , ϕ vanishes on C \ (0; R0 ), and ϕ = ρ − b on ∗ (0; R). Furthermore, on ∗ (0; R1 ), we have iρ = iψ > 0 and hence iϕ = χ  (ρ) · iψ + χ  (ρ) · i∂ρ ∧ ∂ρ ≥ 0.



Proof of Theorem 2.10.1 The idea of the proof (an idea that is applied in various guises throughout this book) is to first produce a C ∞ solution, that is, a C ∞ 1-form τ that has all of the required properties except that it is not meromorphic (away ¯ = ∂τ ¯ , one gets a from p). Forming a suitable L2 solution α of the equation ∂α meromorphic 1-form θ = τ − α with the required properties. Letting ζ denote the (standard) coordinate function on C, setting ψ0 ≡ |ζ |2 , and applying Lemma 2.10.3, we get positive constants b, R0 , and R1 and a nonnegative C ∞ subharmonic function ϕ0 on C∗ such that R0 < R1 < 1, ϕ0 ≡ 0 on C \ (0; R1 ), and ϕ0 = |ζ |2 − log |ζ |2 − b

on ∗ (0; R0 ).

In particular, iϕ0 = iψ0 = i dζ ∧ d ζ¯ > 0 on ∗ (0; R0 ). We may also fix a constant R2 ∈ (0, R0 ) and a function η0 ∈ D((0; R0 )) such that η0 ≡ 1 on (0; R2 ). Given an integer m ≥ 2, we have the meromorphic 1-form γm ≡ ζ −m dζ on C and the C ∞ form τm ≡ η0 γm of type (1, 0) on C∗ . In particular, τm ≡ 0 on a neigh¯ m= borhood of C \ (0; R0 ) and τm is holomorphic on ∗ (0; R2 ). Hence βm ≡ ∂τ ¯ 0 ∧ γm is a C ∞ 2-form that vanishes on C∗ \ (0; R2 , R0 ). Observe that ∂η Am ≡ βm L2 (∗ (0;R0 ),iϕ

0

,ϕ0 )

= βm L2 ((0;R2 ,R0 ),idζ ∧d ζ¯ ,ϕ0 ) < ∞

and Bm ≡ ζ τm − ζ γm L2 (∗ (0;1)) = (η0 − 1)dζ /ζ m−1 L2 ((0;R2 ,1)) < ∞. By construction, the function ϕ0 +log |ζ |2 is bounded on ∗ (0; 1), so we may define a positive constant Cm by Cm ≡ Am + Bm + sup e(ϕ0 +log |ζ |

2 )/2

∗ (0;1)

· Am < ∞.

Suppose now that p is a point in a Riemann surface X (D,  = z, (0; 1)) is a local holomorphic chart with p ∈ D and (p) = z(p) = 0, and Y ≡ X \ {p}. The

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form τ on Y that is equal to ∗ τm on D \ {p} and 0 elsewhere is a C ∞ differential form of type (1, 0) that vanishes on X \ −1 ((0; R0 )) and that is holomorphic on −1 (∗ (0; R2 )). The function ϕ on Y given by ϕ = ϕ0 (z) on D \ {p} and 0 elsewhere is nonnegative, subharmonic, and of class C ∞ . Moreover, ϕ vanishes on X \ −1 ((0; R1 )) and iϕ = i dz ∧ d z¯ > 0 on −1 (∗ (0; R0 )). The C ∞ (1, 1)¯ on Y is equal to ∗ βm on D \ {p}, vanishes on form β ≡ ∂τ   Y \ −1 ((0; R2 , R0 )) ⊃ Z ≡ q ∈ Y | i(ϕ )q = 0 , and satisfies βL2 (Y \Z,iϕ ,ϕ) = βm L2 (∗ (0;R0 ),iϕ

= Am < ∞.

0 ,ϕ0 )

Therefore, by Theorem 2.9.1, there exists a C ∞ (1, 0)-form α on Y such that ¯ =β ∂α

αL2 (Y,ϕ) ≤ Am .

and

¯ = 0 and is therefore holomorThus C ∞ (1, 0)-form θ ≡ τ − α on Y satisfies ∂θ phic on Y . Since τ and therefore α are holomorphic on −1 (∗ (0; R2 )), we have α−1 (∗ (0;R2 )) = f (z)dz for some function f ∈ O(∗ (0; R2 )). In particular, 

i |ζf (ζ )|2 dζ ∧ d ζ¯ = 2 ∗ (0;R2 )

 −1 (∗ (0;R2 ))

i 2 α ∧ α¯ · e−ϕ · e|z| −b 2

1 2 ≤ eR2 −b · α2L2 (Y,ϕ) < ∞. 2 Hence, by Riemann’s extension theorem (Theorem 1.2.10), the function ζ → ζf (ζ ) extends to a (unique) holomorphic function on (0; R2 ); that is, f extends to a meromorphic function with at worst a simple pole at 0. Therefore, since η0 ≡ 1 on (0; R2 ) and γm is a meromorphic 1-form with a pole of order m > 1 at 0, we see that θ = τ − α determines a meromorphic 1-form on X that is holomorphic on Y and that has a pole of order m at p. For the bounds, we observe that θ L2 (X\D) =  − αL2 (X\D) = αL2 (X\D,ϕ) ≤ Am ≤ Cm and that zθ − z−m+1 dzL2 (D) ≤ zτ − z−m+1 dzL2 (D) + zαL2 (D) ≤ Bm + sup e(ϕ0 +log |ζ |

· αL2 (D,ϕ)

≤ Bm + sup e(ϕ0 +log |ζ |

· αL2 (Y,ϕ)

≤ Bm + sup e(ϕ0 +log |ζ |

· Am ≤ C m .

2 )/2

∗ (0;1)

2 )/2

∗ (0;1)

2 )/2

∗ (0;1)



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Corollary 2.10.4 Every Riemann surface X admits a nonconstant meromorphic function. Proof We may choose two distinct points p and q in X. Applying Theorem 2.10.1, we get two meromorphic 1-forms η and θ that are holomorphic except for poles of order 2 at p and q, respectively. The quotient f = η/θ then determines a nonconstant meromorphic function on X.  Exercises for Sect. 2.10 2.10.1 Show that there exists a constant A > 0 such that for each integer m ≥ 2, the constant Cm ≡ Am has the properties described in Theorem 2.10.1. 2.10.2 Prove the following generalization of Theorem 2.10.1. Let X0 be a Riemann surface, let θ0 be a meromorphic 1-form on X0 that is holomorphic except for a pole of order m ≥ 2 at some point p0 ∈ X0 , let 0 be a relatively compact neighborhood of p0 in X0 , and let f0 be a holomorphic function on X0 that vanishes at p0 . Then there exists a constant C = C(X0 , θ0 , 0 , f0 ) > 0 such that if X is any Riemann surface, p is a point in X, and  :  → 0 is a biholomorphic mapping of a neighborhood  of p onto 0 with (p) = p0 , then there exists a meromorphic 1-form θ on X with the following properties: (i) The meromorphic 1-form θ is holomorphic X \ {p} and has a pole of order m at p; (ii) We have θL2 (X\) ≤ C; (iii) The meromorphic 1-form θ − ∗ θ0 on  has at worst a simple pole at p; and (iv) We have f0 () · θ − f0 () · ∗ θ0 L2 () ≤ C.

2.11 Radó’s Theorem on Second Countability The goal of this section is the following important theorem: Theorem 2.11.1 (Radó) Every Riemann surface X is second countable; that is, the topology in X admits a countable basis. The proof considered here is similar to that in [Sp]. Lemma 2.11.2 Every nonempty open subset  of a Riemann surface X has only countably many connected components. Proof Fixing a point p ∈  and a connected neighborhood U of p in , and applying Theorem 2.10.1, we get a nontrivial (i.e., not everywhere zero) holomorphic 1-form θ on X \ {p} with θL2 (X\U ) < ∞. If {i }i∈I is the family of (distinct)

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connected components of  that do not meet (i.e., which do not contain) U and I = ∅, then for every finite set J ⊂ I , we have  θ 2L2 ( ) . θ 2L2 (X\U ) ≥ θ2L2 (  ) = i∈J

i

i

i∈J

 Therefore ∞ > i∈I θ2L2 ( ) , and hence, since the right-hand side is an uni ordered sum of (strictly) positive terms (see Example 7.1.3), the index set I must be countable.  Proof of Radó’s theorem Corollary 2.10.4 provides a nonconstant meromorphic function, that is, a nonconstant holomorphic mapping f : X → P1 . Let P be the countable collection of all sets D ⊂ P1 , where D is either a disk in C with rational radius and center in Q + iQ, or D = P1 \ (0; r) for some r ∈ Q>0 . Let B be the collection of all relatively compact open subsets B of X such that B is a connected component of f −1 (D) for some set D ∈ P. It follows that B is a basis for the topology in X (see Exercise 2.11.1). Moreover, by Lemma 2.11.2, the set f −1 (D) has only countably many connected components for each D ∈ P, and therefore the basis B must be countable.  Radó’s theorem and Proposition 9.7.6 together give the following: Corollary 2.11.3 Every Riemann surface X admits a Kähler form (i.e., a positive C ∞ differential form of type (1, 1)). In fact, given a continuous real (1, 1)-form τ on X, there exists a Kähler form ω with ω ≥ τ on X. Corollary 2.11.4 (Montel’s theorem for a Riemann surface) If {fn } is a sequence of holomorphic functions on a Riemann surface X and {fn } is uniformly bounded on each compact subset of X, then some subsequence of {fn } converges uniformly on compact subsets of X to a holomorphic function on X. Proof We may choose a countable open covering {Uν } of X such that for each ν, Uν is relatively compact in some local holomorphic coordinate neighborhood. Montel’s theorem in the plane (Corollary 1.2.7) and Cantor’s diagonal process together yield a subsequence {fnk } such that the sequence {fnk Uν } converges uniformly to a holomorphic function on Uν for each ν. It follows that {fnk } converges uniformly on compact subsets of X to some holomorphic function on X.  Remark A C ∞ surface need not be second countable. Moreover, Radó’s theorem is false in higher dimensions; that is, there exist connected 2-dimensional complex manifolds that are not second countable, provided, of course, one does not include second countability as part of the definition (see, for example, [Hu]). Exercises for Sect. 2.11 In Exercises 2.11.1 and 2.11.2 below, as in the proof of Radó’s theorem (Theorem 2.11.1), X denotes a Riemann surface; f : X → P1 denotes a nonconstant holomorphic mapping; P denotes the collection of all sets

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

¯ 2.12 The L2 ∂-Method for Scalar-Valued Forms of Type (0, 0) It is also useful to consider solutions of the inhomogeneous Cauchy–Riemann equa¯ = β in which α is a function and β is a differential form of type (0, 1). In tion ∂α fact, this problem is essentially equivalent to the problem considered in Sect. 2.9. In order to obtain the L2 solution, we will have to consider the curvature form associated to a Kähler form ω (i.e., a C ∞ positive differential form of type (1, 1)) on a complex 1-manifold X. Suppose θ1 and θ2 are two nonvanishing holomorphic 1-forms on an open set U ⊂ X and Gj ≡ ω/(iθj ∧ θ¯j ) for j = 1, 2. Then the difference (− log G1 ) − (− log G2 ) = log |θ1 /θ2 |2 ¯ log G1 ] = ∂ ∂[− ¯ log G2 ]. Thus we may make is a harmonic function; that is, ∂ ∂[− the following definition: Definition 2.12.1 For any Kähler form ω on a complex 1-manifold X, the associated curvature is the unique differential form (X,ω) = ω that satisfies ¯ log(ω/(iθ ∧ θ¯ ))] ω U = − log(ω/(iθ ∧θ)) ¯ = ∂ ∂[− for every nonvanishing local holomorphic 1-form θ on an open set U ⊂ X. Remarks 1. Equivalently, if ω = G · (i/2) dz ∧ d z¯ (i.e., G = −2iω(∂/∂z, ∂/∂ z¯ ) = ω(∂/∂x, ∂/∂y)) in a local holomorphic coordinate neighborhood (U, z = x + iy), ¯ log G] on U . then ω = − log G = ∂ ∂[− 2. If ϕ is a real-valued C ∞ function on X, then e−ϕ ω = ω + ϕ . Example 2.12.2 For the Euclidean Kähler form ω ≡ (i/2) dz ∧ d z¯ on C, ω ≡ 0. Example 2.12.3 The chordal Kähler form on the Riemann sphere P1 is the unique Kähler form ω for which ωC = 2i(1 + |z|2 )−2 dz ∧ d z¯ . The corresponding curvature form satisfies iω = ω (see Exercise 2.12.1).

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The main goal of this section is the following: Theorem 2.12.4 Let X be a Riemann surface, let ω be a Kähler form on X, let ϕ be a real-valued C ∞ function on X with i ≡ ie−ϕ ω = iω + iϕ ≥ 0, let ρ ≡ i/ω, and let Z = {x ∈ X | x = 0} = {x ∈ X | ρ(x) = 0}. Then, for every measurable (0, 1)-form β on X with β = 0 a.e. in Z and βX\Z ∈ L20,1 (X \ Z, ϕ + log ρ) (in particular, β is in L2loc on X), there exists a function α ∈ L20,0 (X, ω, ϕ) such that  α = ∂¯distr α = β Ddistr

and

αL2 (X,ω,ϕ) ≤ βL2 (X\Z,ϕ+log ρ) .

In particular, if β is of class C k for some k ∈ Z>0 ∪ {∞}, then α is also of class C k ¯ = β. and ∂α Remarks 1. In order to apply Theorem 2.12.4, one needs a Kähler form ω. Fortunately, by Corollary 2.11.3, a Kähler form always exists. In fact, this version is actually equivalent to Theorem 2.9.1. 2. Although it is possible to obtain the theorem directly, we will instead prove the theorem by first forming the exterior product of the given form and a meromorphic 1-form (provided by Theorem 2.10.1), and then applying the solution for forms of type (1, 0) (Theorem 2.9.1). 3. The proof that β is in L2loc in the above (as well as in Corollary 2.12.5 below) is similar to that in the remark following the statement of Theorem 2.9.1. It is often more convenient to apply Theorem 2.12.4 in one of the following forms, the proofs of which are left to the reader (see Exercises 2.12.2 and 2.12.3): Corollary 2.12.5 Suppose that X is a Riemann surface, ω is a Kähler form on X, ϕ is a real-valued C ∞ function on X, ρ is a nonnegative measurable function on X with iω + iϕ ≥ ρω, and Z = {x ∈ X | ρ(x) = 0}. Then, for every measurable (0, 1)-form β on X with β = 0 a.e. in Z and βX\Z ∈ L20,1 (X \ Z, ϕ + log ρ) (in particular, β is in L2loc on X), there exists a function α ∈ L20,0 (X, ω, ϕ) such that  α = ∂¯distr α = β Ddistr

and

αL2 (X,ω,ϕ) ≤ βL2 (X\Z,ϕ+log ρ) .

In particular, if β is of class C k for some k ∈ Z>0 ∪ {∞}, then α is also of class C k ¯ = β. and ∂α Corollary 2.12.6 Suppose that X is a Riemann surface, ω is a Kähler form on X, ϕ is a real-valued C ∞ function on X, and C is a positive constant for which iω + iϕ ≥ C 2 ω on X. Then, for every form β ∈ L20,1 (X, ϕ), there exists a func-

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75

tion α ∈ L20,0 (X, ω, ϕ) such that  α = ∂¯distr α = β Ddistr

and

αL2 (X,ω,ϕ) ≤ C −1 βL2 (X,ϕ) .

In particular, if β is of class C k for some k ∈ Z>0 ∪ {∞}, then α is also of class C k ¯ = β. and ∂α Proof of Theorem 2.12.4 According to Theorem 2.10.1, we may fix a nontrivial meromorphic 1-form θ on X, and we may let Q be the (discrete) set of zeros and poles of θ . On the Riemann surface Y ≡ X \ Q, the function τ ≡ ω/(iθ ∧ θ¯ ) is then positive and of class C ∞ , and we have − log τ = ω ; that is, iϕ−log τ = i = ρω. The form β0 ≡ θ ∧ β is a measurable differential form of type (1, 1) that vanishes on Z ∩ Y and that satisfies (since the set Q is discrete, and therefore of measure 0) β0 2L2 (Y \Z,i

ϕ−log τ ,ϕ−log τ )

   θ ∧ β 2 −(ϕ−log τ )   e = ρω   Y \Z ρω 

= β0 2L2 (Y \Z,ρω,ϕ−log τ )  =

Y \Z

 =

Y \Z

 =

X\Z

ω θ ∧ β θ¯ ∧ β¯ · · · e−(ϕ+log ρ) ω ω ω iθ ∧ θ¯ iθ ∧ β ·

iθ ∧ θ¯ β¯ −(ϕ+log ρ) · ·e iθ ∧ θ¯ θ

(−i)β ∧ β¯ · e−(ϕ+log ρ) = β2L2 (X\Z,ϕ+log ρ) < ∞.

Therefore, β0 is in L2loc on Y , and by Theorem 2.9.1, there exists a form α0 ∈  α = ∂¯ L21,0 (Y, ϕ − log τ ) such that Ddistr 0 distr α0 = β0 in Y and α0 L2 (Y,ϕ−log τ ) ≤ β0 L2 (Y \Z,iϕ−log τ ,ϕ−log τ ) = βL2 (X\Z,ϕ+log ρ) . The measurable function α ≡ −α0 /θ on X (again, Q is a set of measure 0) then satisfies    iα0 ∧ α¯ 0 −ϕ α2L2 (X,ω,ϕ) = |α|2 e−ϕ ω = |α0 /θ |2 e−ϕ ω = e ω Y Y Y iθ ∧ θ¯  = iα0 ∧ α¯ 0 · e−(ϕ−log τ ) Y

= α0 2L2 (Y,ϕ−log τ ) ≤ β2L2 (X\Z,ϕ+log ρ) . In particular, α is in L2loc on X, and it remains to show that ∂¯distr α = β on X. For each point p ∈ X, we may choose a local holomorphic coordinate neighborhood (U,  = z, (0; 2)) such that U ∩ Q ⊂ {p} and z(p) = 0. The meromorphic function f ≡ θ/dz on U is then nonvanishing and holomorphic on U \ {p} ⊂ Y , and

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the measurable functions a ≡ α0 /dz and b ≡ β/d z¯ on U are in L2loc ⊂ L1loc . In particular, α = −a/f and β0 = f b dz ∧ d z¯ on U . We may also choose a nonnegative C ∞ function χ such that χ ≡ 1 on C \ (0; 2) and χ ≡ 0 on (0; 1). For every C ∞ function η with compact support in D ≡ −1 ((0; 1)), the dominated convergence theorem gives (here, we denote the coordinate on C by w)  ∂η i α· · dz ∧ d z¯ ∂ z¯ 2 D  ∂η i χ(z/)α · = lim · dz ∧ d z¯ ∂ z¯ 2 →0+ D  −a ∂η i · · dz ∧ d z¯ χ(z/) = lim f ∂ z¯ 2 →0+ D     ∂ η i = lim − a · χ(z/) · dz ∧ d z¯ ∂ z¯ f 2 →0+ D   η i 1 ∂χ (z/) · dz ∧ d z¯ a + ¯ f 2 D  ∂w   i = lim − b · χ(z/)η · dz ∧ d z¯ + 2 →0 D   1 ∂χ i α − (z/)η · dz ∧ d z¯ ¯ 2 D  ∂w    i 1 ∂χ i α = − bη · dz ∧ d z¯ − lim (z/)η · dz ∧ d z¯ . 2 ¯ 2 →0+ D  ∂ w D On the other hand, since α ∈ L2loc (X) and ∂χ/∂ w¯ ≡ 0 on C \ (0; 1, 2), we have, for some constant C > 0,     +  α 1 ∂χ (z/)η · i dz ∧ d z¯  ≤ Cα ◦ −1  2 L ((0;,2)) → 0 as  → 0 .    ∂ w ¯ 2 D It follows that ∂¯distr α = β.



Exercises for Sect. 2.12 2.12.1 Verify that (see Example 2.12.3) there is a unique Kähler form ω on P1 such that ωC = 2i(1 + |z|2 )−2 dz ∧ d z¯ . Verify also that iω = ω. 2.12.2 Prove Corollary 2.12.5. 2.12.3 Prove Corollary 2.12.6. 2.12.4 Let X be a Riemann surface, let ω be a Kähler form on X, let ϕ be a real-valued C ∞ function on X with i ≡ ie−ϕ ω = iω + iϕ ≥ 0, let

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ρ ≡ i/ω, and let Z = {x ∈ X | x = 0} = {x ∈ X | ρ(x) = 0}. Prove that for every C ∞ (1, 0)-form β on X with β = 0 a.e. in Z and βX\Z ∈ L21,0 (X \ Z, ϕ + log ρ), there exists a C ∞ function α ∈ L20,0 (X, ω, ϕ) such that ∂α = β

and αL2 (X,ω,ϕ) ≤ βL2 (X\Z,ϕ+log ρ)

(cf. Exercises 2.9.3 and 2.9.4).

2.13 Topological Hulls and Chains to Infinity As suggested by Theorem 2.9.1 and its applications (see also Theorem 3.9.1 and its applications), it is useful to have available a large supply of C ∞ strictly subharmonic functions on an (open) Riemann surface, i.e., C ∞ functions ϕ for which iϕ > 0 (see Definition 2.8.1). In Sect. 2.14, it will be shown that every open Riemann surface admits a C ∞ strictly subharmonic exhaustion function. Recall that a function ρ on a Hausdorff space X is an exhaustion function if {x ∈ X | ρ(x) < a}  X for each a ∈ R (see Definition 9.3.10). The first proof of the existence of a C ∞ strictly subharmonic exhaustion function on a connected noncompact Riemannian manifold was obtained by Greene and Wu [GreW]. Demailly [De2] provided an elementary proof using a local construction. Demailly’s proof may be modified (as well as simplified) to give an exhausting C ∞ strict subsolution for an arbitrary second-order linear elliptic differential operator with continuous coefficients on a second countable noncompact C ∞ manifold (for the details see [NR]). The proof of the existence of a C ∞ strictly subharmonic exhaustion function on an open Riemann surface appearing in this book is adapted from [NR]. In this section, we first consider some of the topological facts required for the proof. Definition 2.13.1 For a subset A of a Hausdorff space X, the topological hull hX (A) of A in X is the union of A with all of the connected components of X \ A that are relatively compact in X. An open subset  of X is called topologically Runge in X if hX () = . Remark Intuitively, one obtains hX (A) by filling in the holes of A (see Fig. 2.5). The proofs of the following properties are left to the reader (see Exercise 2.13.1): Lemma 2.13.2 Let X be a Hausdorff space. (a) For every set A ⊂ X, we have hX (hX (A)) = hX (A) ⊃ A. (b) If A ⊂ B ⊂ X, then hX (A) ⊂ hX (B). (c) If X is locally connected and A is a closed subset of X, then hX (A) is closed.

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Fig. 2.5 A subset and its topological hull

For a compact subset of a suitable topological space (such as a manifold), the topological hull has a nice form (cf. [Mal] and [Ns5]): Lemma 2.13.3 If K is a compact subset of a connected, noncompact, locally connected, locally compact Hausdorff space X, then hX (K) is compact; X \ hX (K) has only finitely many components; and for each point p ∈ X \ hX (K), there is a connected noncompact closed subset C of X with p ∈ C ⊂ X \ hX (K). Remark One gets a version in which X has more than one, but only finitely many, connected components by applying the lemma to each connected component (see Exercise 2.13.2). Proof of Lemma 2.13.3 The lemma is trivial for K empty, so we may assume that K = ∅. Since X is locally connected, hX (K) is a closed set whose complement has no relatively compact components (by Lemma 2.13.2). Since X is locally compact Hausdorff, we may choose a relatively compact neighborhood  of K in X. The components of X \ K are open and disjoint, so only finitely many meet the compact set ∂ ⊂ X \ K. By replacing  with the union of  and all of the relatively compact components of X \ K meeting ∂, we may assume that no relatively compact component of X \ K meets ∂. On the other hand, every component E of X \ K must satisfy E ∩ K = ∂E = ∅. For E is open and closed in X \ K, so ∂E ⊂ K, while E = X, so ∂E = E \ E = ∅ (E cannot be both open and closed in the connected space X). It follows that if E meets X \ , then E meets ∂, and hence E is not relatively compact in X. Thus X \  ⊂ E1 ∪ · · · ∪ Em for finitely many components E1 , . . . , Em of X \ K, none of which are relatively compact in X. It follows that hX (K) ⊂  and X \ hX (K) = E1 ∪ · · · ∪ Em . Applying the above argument with K  =  in place of K, we get a relatively compact neighborhood  of K  in X such that X \  ⊂ X \ hX (K  ) = E1 ∪ · · · ∪ Ek , where E1 , . . . , Ek are distinct components of X \ K  , none of which are relatively compact in X. For each i = 1, . . . , m, we get Ej ⊂ Ei for some

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79

j ∈ {1, . . . , k}. Thus, if Ai is the set of points in Ei that lie some connected noncompact closed subset of X that is contained in Ei , then Ai = ∅. Given a point p ∈ Ei ∩ Ai , we may choose a relatively compact connected neighborhood V of p in Ei . We may then choose a point q ∈ Ai ∩ V and a connected noncompact closed subset B of X with q ∈ B ⊂ Ei . The set C ≡ V ∪ B is then a connected noncompact closed subset of X that lies in Ei and that contains V . Thus V ⊂ Ai , and hence the nonempty set Ai is both open and closed in the connected set Ei . Therefore, Ai = Ei , and the lemma follows.  Lemma 2.13.4 Let X be a second countable, noncompact, connected, locally connected, locally compact Hausdorff space. Then there is a sequence of compact sets ◦ ∞ {Kν }∞ ν=1 Kν , and for each ν, Kν ⊂ K ν+1 and hX (Kν ) = Kν . ν=1 such that X = Proof By Lemma 9.3.6, we may fix a sequence of compact sets {Hν } with X =  ∞  ν=1 Hν . Set K1 ≡ hX (H1 ). Given Kν , we may choose a compact set Kν+1 with ◦

  Hν ∪ Kν ⊂ Kν+1 , and we may set Kν+1 ≡ hX (Kν+1 ). This yields the desired sequence. 

Lemma 2.13.5 Let X be a connected, locally connected, locally compact Hausdorff space; let B be a countable collection of connected open subsets that is a basis for the topology in X; let K be a compact subset of X; and let U be a connected component of X \ hX (K). Then, for each point p ∈ U , there exists a sequence of basis elements {Bj } that tends to infinity (i.e., {Bj } is a locally finite family in X) such that p ∈ B1 , and for each j , Bj  U and Bj ∩ Bj +1 = ∅. Proof Lemma 2.13.3 provides a connected noncompact closed subset C of X with p ∈ C ⊂ U , and Lemma 9.3.6 provides a countable locally finite (in X) covering A of C by basis elements that are relatively compact in U . For each point q ∈ C, there is a finite sequence of elements B1 , . . . , Bk of A that forms a chain from p to q; that is, p ∈ B1 , q ∈ Bk , and Bj ∩ Bj +1 = ∅ for j = 1, . . . , k − 1 (we will call k the length of the chain). For the set E of points q in C for which there is a chain from p to q is clearly nonempty and open in C. On the other hand, E is also closed, because if q ∈ E, then q ∈ B for some set B ∈ A and there must be some point r ∈ B ∩ E. A chain B1 , . . . , Bk from p to r yields the chain B1 , . . . , Bk , B from p to q. Thus E = C. Observe that if q ∈ C and B1 , . . . , Bk is a chain of minimal length from p to q, then the sets B1 , . . . , Bk are distinct. Now since C is noncompact and closed, we may choose a locally finite sequence of points {qν } in C; i.e., qν → ∞ in X (for example, we may fix an increasing sequence {ν } of relatively compact open subsets of X with union X and choose qν ∈ C \ ν for each ν). For each ν, we may choose a chain B1(ν) , . . . , Bk(ν) of ν minimal length from p to qν . Since the elements of A are relatively compact in U and A is locally finite in X, there are only finitely many possible choices for Bj(ν) for each j (only finitely many elements of A are in some chain of length j from p). Moreover, for each fixed j ∈ Z>0 , we have kν > j for ν  0, because the set of

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points in C joined to p by a chain of length ≤ j is relatively compact in C while qν → ∞. Therefore, after applying a diagonal process and passing to the associated subsequence of {qν }, we may assume that for each j , there is an element Bj ∈ A with Bj(ν) = Bj for all ν  0. Thus we get an infinite chain of distinct elements {Bj } from p to infinity as required in (ii) (local finiteness in X is guaranteed since A is  locally finite and the elements {Bj } are distinct). Remark The above lemma actually holds even if the basis elements are not necessarily connected, but it is easier to picture chains of connected sets. Exercises for Sect. 2.13 2.13.1 Prove Lemma 2.13.2. 2.13.2 Let K be a compact subset of a noncompact, locally connected, locally compact Hausdorff space X. Prove that if X has only finitely many connected components, then hX (K) is compact, X \ hX (K) has only finitely many components, and for each point p ∈ X \ hX (K), there is a connected noncompact closed subset C of X with p ∈ C ⊂ X \ hX (K). 2.13.3 Let K be a compact subset of a connected, noncompact, locally connected, locally compact Hausdorff space X. (a) Prove that hX (K) is equal to the intersection of all topologically Runge neighborhoods of K. (b) Prove that for every neighborhood U of hX (K) in X, there exists a topologically Runge open set  with hX (K) ⊂  ⊂ U . Hint. Show that one may assume without loss of generality that U  X and that there exists a finite covering of ∂U by connected noncompact closed subsets of X that lie in X \ hX (K). Then set  equal to the complement of these sets in U . 2.13.4 Prove that if X is a locally connected Hausdorff space and A is a connected subset of X, then hX (A) is connected. 2.13.5 Let  be a relatively compact open subset of a connected, noncompact, locally connected, locally compact Hausdorff space X. Prove that hX () is a relatively compact open subset of X and that X \ hX () has only finitely many connected components (cf. Exercise 2.13.6 below). 2.13.6 This exercise will be applied in Exercise 2.16.5 (cf. [Ns5]). (a) Prove that if Y is a locally compact Hausdorff space, C is a compact connected component of Y , and U is a neighborhood of C in Y , then there exists a set A such that C ⊂ A ⊂ U and A is open and closed in Y . Hint. First work in a relatively compact neighborhood V of C in Y . Let D be the intersection of all subsets of V that are open and closed in V and that contain C. Show that each neighborhood of D in V contains a set A ⊃ D that is open and closed in V . Using this observation, prove that D is connected and conclude that D = C. Finally, construct the desired open and closed subset of Y for a given neighborhood U .

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2.13.7

2.13.8 2.13.9 2.13.10 2.13.11 2.13.12

81

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

Hint. First show that there is a countable locally finite covering A of D by connected open relatively compact subsets of X \ K each of which meets D, that X \ K contains the closure C ≡ V of the union V

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

2.14 Construction of a Subharmonic Exhaustion Function This section contains the construction, alluded to in the beginning of Sect. 2.13, of a C ∞ strictly subharmonic exhaustion function on an open Riemann surface. The main goal is the following: Theorem 2.14.1 Suppose X is an open Riemann surface, K is a compact subset of X satisfying hX (K) = K, τ is a continuous real-valued function on X, θ is a continuous real (1, 1)-form on X, and  is a neighborhood of K in X. Then there exists a C ∞ exhaustion function ϕ on X such that (i) On X, ϕ ≥ 0 and iϕ ≥ 0; (ii) On X \ , ϕ > τ and iϕ ≥ θ ; and (iii) We have supp ϕ ⊂ X \ K. Remark It suffices to consider the case τ ≥ 0 and θ ≥ 0, since in general, one may construct ϕ with the nonnegative function |τ | ≥ τ and the nonnegative form θ + + θ − ≥ θ in place of τ and θ , respectively. Setting K =  = ∅ and choosing θ > 0 (as we may by Corollary 2.11.3), we get the following: Corollary 2.14.2 Every open Riemann surface admits a C ∞ strictly subharmonic exhaustion function. After fixing a Kähler form, one may work with a convenient 0-form in place of the curvature form: Definition 2.14.3 The Laplace operator associated to a Kähler form ω on a complex 1-manifold X is the second-order linear differential operator ω with C ∞ co-

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efficients given by ω ϕ ≡

¯ i∂ ∂ϕ ω

for every C 2 function ϕ. In particular, for ϕ real-valued, we have ω ϕ = iϕ /ω. Remarks 1. Writing ω = G · (i/2) dz ∧ d z¯ = G dx ∧ dy with respect to a local holomorphic coordinate z = x + iy, we get   2 2 ∂2 ∂2 1 ∂ . + ω = = G ∂z∂ z¯ 2G ∂x 2 ∂y 2 For example, if ω = (i/2) dz ∧ d z¯ = dx ∧ dy is the standard Euclidean volume form on C, then   1 ∂2 ∂2 ∂2 = ω = 2 , + ∂z∂ z¯ 2 ∂x 2 ∂y 2 which is 1/2 the standard Laplace operator. 2. Clearly, a real-valued C ∞ (or C 2 ) function ϕ on an open set  ⊂ X is subharmonic (strictly subharmonic) if and only if ω ϕ ≥ 0 (respectively, ω ϕ > 0) on . For the rest of this section, X will denote an open Riemann surface. According to Radó’s theorem, X is second countable, and applying Corollary 2.11.3, we get a Kähler form ω on X. Instead of proving Theorem 2.14.1 directly, we will prove the following equivalent version (in Exercise 2.14.1, the reader is asked to verify that the two versions are equivalent): Theorem 2.14.4 (Cf. [GreW], [De2], and [NR]) Suppose K is a compact subset of X satisfying hX (K) = K, ρ is a continuous real-valued function on X, and  is a neighborhood of K in X. Then there exists a C ∞ exhaustion function ϕ on X such that (i) On X, ϕ ≥ 0 and ω ϕ ≥ 0; (ii) On X \ , ϕ > ρ and ω ϕ > ρ; and (iii) We have supp ϕ ⊂ X \ K. The main step in the proof of Theorem 2.14.4 is the following: Proposition 2.14.5 Let K be a compact subset of X satisfying hX (K) = K. Then, for each point p ∈ X \ K, there is a nonnegative C ∞ function α on X such that ω α ≥ 0 on X, supp α ⊂ X \ K, α(p) > 0, and ω α(p) > 0. Remark According to the maximum principle for subharmonic functions (see, for example, [Ns5]), a nonconstant subharmonic function on a domain cannot attain a maximum value (in particular, any subharmonic function on a compact Riemann

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surface is constant). Therefore, the converse of the proposition also holds; that is, if such a function α exists for some point p ∈ X \ K, then the component of X \ K containing p is not relatively compact in X. Assuming Proposition 2.14.5 for now, we may prove Theorem 2.14.4. Proof of Theorem 2.14.4 By Proposition 9.3.11, we may assume that ρ is a positive exhaustion function. Let K0 = K. By Lemma 2.13.4,  we may choose a sequence of nonempty compact sets {Kν } such that X = ∞ ν=1 Kν and such that for each ◦

ν = 1, 2, 3, . . . , Kν−1 ⊂ K ν and hX (Kν ) = Kν . Given a point p ∈ X \ , there is a unique ν = ν(p) > 0 with p ∈ Kν \ Kν−1 , and Proposition 2.14.5 provides a nonnegative C ∞ function αp and a relatively compact neighborhood Vp of p in X \ Kν−1 such that ω αp ≥ 0 on X, supp αp ⊂ X \ Kν−1 , and αp > ρ and ω αp > ρ on Vp (one obtains the last two conditions by multiplying by a sufficiently large positive constant). We may choose a sequence of points {pk } in X, and corresponding functions {αpk } and neighborhoods {Vpk }, so that {Vpk } forms a locally finite covering of X \  (for example, we may take {pk } to be an enumeration of the countable set ∞ ν=0 Zν , where for each ν, Zν is a finite set ◦

◦

of points in X \ [K ν−1 ∪ ] such that {Vp }p∈Zν covers Kν \ [K ν−1 ∪ ]). The collection {supp αpk } is then locally  finite in X, since supp αpk ⊂ X \ Kν−1 whenever pk ∈ Kν−1 . Hence the sum ∞ k=1 αpk is locally finite and therefore convergent to a C ∞ function ϕ on X satisfying ϕ ≥ αpk > ρ and ω ϕ ≥ ω αpk > ρ on Vpk for each k. Therefore, since {Vpk } covers X \ , we get ϕ > ρ and ω ϕ > ρ on X \ . In particular, ϕ is an exhaustion function. Similarly, we also have supp ϕ ⊂ X \ K,  and ϕ ≥ 0 and ω ϕ ≥ 0 on X. It remains to prove Proposition 2.14.5. The idea is to form a sequence of functions, each of which is subharmonic outside a small set and strictly subharmonic on the bad set of the previous function. Multiplying each function by a sufficiently large positive constant (obtained inductively), one pushes these small bad sets off to infinity. Lemma 2.14.6 There exists a nonnegative C ∞ function χ on R such that χ ≡ 0 on (−∞, 0] and χ, χ  , χ  > 0 on (0, ∞). Proof For example, the C ∞ function  0 χ(t) = exp(t − (1/t))

if t ≤ 0, if t > 0,

has the required properties. The verification is left to the reader (see Exercise 2.14.3).  Lemma 2.14.7 For every r ∈ (0, 1), there exists a nonnegative C ∞ function α on P1 such that

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(i) We have α > 0 on (0; 1) and α ≡ 0 on P1 \ (0; 1); and (ii) The function α is strictly subharmonic on the annulus (0; r, 1) (and therefore subharmonic on P1 \ (0; r)). Proof Letting χ : R → [0, ∞) be the function provided by Lemma 2.14.6, it is easy to see that the function given by α(∞) = 0 and  2 2 2 ∀z ∈ C α(z) ≡ χ e−|z| /r − e−1/r has the required properties. Again, the verification is left to the reader (see Exercise 2.14.4).  Proposition 2.14.8 Let c ∈  ≡ (0; 1), and let  : P1 → P1 be the map given by z → (z − c)/(1 − cz) ¯ (with (1/c) ¯ = ∞ and (∞) = −1/c). ¯ Then we have the following: (i)  is an automorphism of P1 (i.e., a biholomorphic mapping of P1 onto itself ); (ii) (c) = 0,  (0) = 1 − |c|2 , and  (c) = 1/(1 − |c|2 ); and (iii) () =  and (∂) = ∂. The proof is left to the reader (see Exercise 2.14.5). It follows from the above that  is an automorphism of . In Chapter 5, we will see that in fact, every automorphism of  is of the form b for some b ∈ ∂ and some  as above (see Theorem 5.8.2). Lemma 2.14.9 For every nonempty relatively compact open subset U of the unit disk  ≡ (0; 1), there exists a nonnegative C ∞ function β on P1 such that (i) We have β > 0 on  and β ≡ 0 on P1 \ ; and (ii) The function β is strictly subharmonic on the set  \ U (and therefore subharmonic on P1 \ U ). Proof Fixing a point c ∈ U , we get the automorphism  : z → (z − c)/(1 − cz) ¯ of P1 mapping c to 0 as in Proposition 2.14.8, and for r ∈ (0, 1) sufficiently small, we have −1 ((0; r)) ⊂ U . By Lemma 2.14.7, there exists a nonnegative C ∞ function α on P1 such that α > 0 on , α ≡ 0 on P1 \ , and α is strictly subharmonic on (0; r, 1). The function β ≡ α() then has the required properties.  Proof of Proposition 2.14.5 By Lemma 2.13.5, given a point p ∈ X \ K, there is a locally finite (in X) sequence of relatively compact open subsets {Vm } of X \ K such that p ∈ V1 and such that for each m, we have Vm ∩ Vm+1 = ∅ and there is a biholomorphism of a neighborhood of the closure V m of Vm onto a neighborhood of the closure  of the unit disk  ≡ (0; 1) that maps Vm onto . Hence we may choose a sequence of nonempty open sets {Wm }∞ m=0 with disjoint closures such that p ∈ W0  V1 and such that for each m ≥ 1, Wm  Vm ∩ Vm+1 (see Fig. 2.6). By Lemma 2.14.9, there is a sequence of nonnegative C ∞ functions {βm }∞ m=1 such that for each m, βm ≡ 0 on X \ Vm , ω βm ≥ 0 on X \ Wm , and

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

βm > 0 and ω βm > 0 on W m−1 . We will choose positive constants {Rm }∞ m=1 inductively so that for each m = 1, 2, 3, . . . , ! m  ≥ 0 on X \ Wm , Rj βj ω > 0 on W 0 . j =1 Fix R1 > 0. Given R1 , . . . , Rm−1 > 0 with the above property, using the fact that ω βm > 0 on W m−1 , we get, for Rm  0, ! ! m m−1   ω Rj βj = Rm ω βm + ω Rj βj > 0 on W m−1 . j =1

j =1

On X \ (Wm−1 ∪ Wm ), we have ω βm ≥ 0 and hence ! ! m m−1   Rj βj ≥ ω Rj βj ≥ 0. ω j =1

j =1

On W 0 ⊂ X \ (Wm−1 ∪ Wm ), the above  middle expression, and hence the expression on the left, is positive. Moreover,  m j =1 Rj βj ≥ R1 β1 > 0 on W 0 . Proceeding, we get the sequence {Rm }. The sum Rm βm is locally finite in X and the sequence of sets {Wm } is locally finite in X, so the sum converges to a function α with the required properties.  Remark When second countability of a particular open Riemann surface X is evident (for example, when X is a proper nonempty open subset of a compact Riemann surface), one gets Theorem 2.14.4 without any reliance on Radó’s theorem, and hence with almost no reliance on Sects. 2.6–2.12. The existence of strictly subharmonic exhaustion functions allows one to use the ¯ For example, we have the following: full power of the L2 ∂-method. Theorem 2.14.10 Let p ∈ {0, 1} and let β be a locally square-integrable differential form of type (p, 1) on the open Riemann surface X. Then there exists a locally

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square-integrable differential form α of type (p, 0) with ∂¯distr α = β. In particular, if ¯ = β. β is of class C k for some k ∈ Z>0 ∪ {∞}, then α is also of class C k and ∂α Proof We may fix a Kähler form ω on X. Applying Theorem 2.14.4, with the compact set given by K = ∅ and the function ρ chosen (with the help of Lemma 9.7.13) so large that β ∈ L20,1 (X, ρ) or L21,1 (X, ω, ρ) and ρ > 1+|iω /ω|, we get a positive C ∞ strictly subharmonic (exhaustion) function ϕ on X such that iω + iϕ ≥ ω iϕ ≥ ω

and and

βL2 (X,ϕ) < ∞ if p = 0, βL2 (X,ω,ϕ) < ∞ if p = 1.

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



In Sects. 2.15 and 2.16, we will consider other applications to open Riemann surfaces. Exercises for Sect. 2.14 2.14.1 Prove that Theorem 2.14.1 and Theorem 2.14.4 are equivalent. 2.14.2 Let ϕ be a real-valued C ∞ function on a Riemann surface X. (a) Prove that if χ is a real-valued C ∞ function on R, then ¯ χ(ϕ) = χ  (ϕ)χ(ϕ) + χ  (ϕ)∂ϕ ∧ ∂ϕ.

2.14.3 2.14.4 2.14.5 2.14.6

2.14.7

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

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(a) Prove that for each connected component U of X \ K and each point p ∈ X \K, there is a nonnegative C ∞ function α on X such that ω α ≥ 0 on X, supp α ⊂ U , α(p) > 0, and ω α(p) > 0 (cf. Proposition 2.14.5). (b) Suppose that U is a connected component of X \ K, C is a connected noncompact closed subset of X contained in U , and ρ is a continuous real-valued function on X. Prove that there exists a C ∞ function ϕ on X such that (i) We have ϕ ≥ 0 and ω ϕ ≥ 0 on X; (ii) We have ϕ > ρ and ω ϕ > ρ on C; and (iii) We have supp ϕ ⊂ U . Hint. First show that there exists a connected locally connected closed set D in X with C ⊂ D ⊂ U (for example, by forming the union of the closures of elements of a suitable locally finite covering of C by coordinate disks each of which meets C). Then  show that there is a sequence of compact sets {Kν } such that X = ν Kν and such that for each ν, ◦

Kν ⊂ K ν+1 and hD (Kν ∩ D) = Kν ∩ D. Proceed now as in the proof of Theorem 2.14.4. (c) Suppose that D ⊂ X \ K is a closed subset of X with no compact connected components and ρ is a real-valued continuous function on X. Prove that there exists a C ∞ function ϕ on X such that (i) We have ϕ ≥ 0 and ω ϕ ≥ 0 on X; (ii) We have ϕ > ρ and ω ϕ > ρ on D; and (iii) We have supp ϕ ⊂ X \ K.

2.15 The Mittag-Leffler Theorem ¯ is the construction of a holoOne of the main applications of the L2 ∂-method morphic or meromorphic function (or section of a holomorphic line bundle) with prescribed values on a given discrete set, or as in the generalization of the classical Mittag-Leffler theorem below, with some prescribed Laurent series terms (see Theorem 1.3.6). This generalization is due to Behnke–Stein [BehS] (see also Florack [Fl]): Theorem 2.15.1 (Mittag-Leffler theorem) Let X be an open Riemann surface, let P be a discrete subset of X (i.e., a closed set with no limit points in X), and for each point p ∈ P , let Up be a neighborhood of p with Up ∩ P = {p}, let fp be a holomorphic function on Up \ {p}, and let mp be a positive integer. Then there exists a holomorphic function f on X \ P such that for each point p ∈ P , f − fp extends to a holomorphic function on Up that either vanishes on a neighborhood of p or has a zero of order at least mp at p (in other words,  if z is a local nholomorphic coordinate on a neighborhood of p, and f = n∈Z an (z − z(p)) and  fp = n∈Z bn (z − z(p))n are the corresponding Laurent series expansions centered at p, then amp −n = bmp −n for n = 1, 2, 3, . . . ).

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89

Remark It follows that one may actually choose the above function f ∈ O(X \ P ) so that for each point p ∈ P , f − fp extends to a holomorphic function on Up with a zero of order equal to mp at p (see Exercise 2.15.1). The proofs of Corollaries 2.15.2, 2.15.3, and 2.15.4 below are left to the reader (see Exercises 2.15.2, 2.15.3, and 2.15.4). Corollary 2.15.2 Let P be a discrete subset of an open Riemann surface X. (a) If fp is a meromorphic function on a neighborhood of p and mp ∈ Z>0 for each point p ∈ P , then there exists a function f ∈ M(X) such that f is holomorphic on X \ P and ordp (f − fp ) ≥ mp for every p ∈ P . (b) If fp is a holomorphic function on a neighborhood of p and mp ∈ Z>0 for each point p ∈ P , then there exists a function f ∈ O(X) with ordp (f − fp ) ≥ mp for every p ∈ P . (c) If ζp ∈ C for each point p ∈ P , then there exists a function f ∈ O(X) with f (p) = ζp for every p ∈ P . Corollary 2.15.3 (Behnke–Stein theorem [BehS]) Every open Riemann surface X is Stein; that is, X has the following properties: (i) (Holomorphic convexity) If P is any infinite discrete subset of X, then there exists a holomorphic function on X that is unbounded on P ; (ii) (Separation of points) If p, q ∈ X and p = q, then there exists a holomorphic function f on X such that f (p) = f (q); and (iii) (Global functions give local coordinates) For each point p ∈ X, there exists a holomorphic function f on X such that (df )p = 0. Corollary 2.15.4 Let X be an open Riemann surface, let P be a discrete subset of X, and for each point p ∈ P , let Up be a neighborhood of p with Up ∩ P = {p}, let θp be a holomorphic 1-form on Up \ {p}, and let mp be a positive integer. Then there exists a holomorphic 1-form θ on X \ P such that for each point p ∈ P , θ − θp extends to a holomorphic 1-form on Up that either vanishes on a neighborhood of p or has a zero of order at least mp at p. Proof of Theorem 2.15.1 Let Y = X \ P . We may choose a Kähler form ω on X, and we may choose a locally finite family of disjoint local holomorphic coordinate neighborhoods {(Vp , zp )}p∈P and a family of open sets {Wp }p∈P such that for each  point  p ∈ P , we have p ∈ Wp  Vp  Up and zp (p) = 0. Let V ≡ p∈P Vp and W ≡ p∈P Wp . By cutting off, we may construct a real-valued C ∞ function ρ0 on Y such that supp ρ0 ⊂ V and ρ0 = mp log |zp |2 on Wp \ {p} for each p ∈ P . We may also fix a C ∞ function γ on Y such that supp γ ⊂ V and γ = fp on Wp \ {p} for each ¯ is then a C ∞ differential form of type (0, 1) on Y with p ∈ P . The form β = ∂γ supp β ⊂ V \ W . Since log |zp |2 is a harmonic function on Vp \ {p} for each p ∈ P , by applying Theorem 2.14.4 (or Theorem 2.14.1), with the compact set equal to the

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empty set and the function ρ chosen (with the help of Lemma 9.7.13) so large that β ∈ L20,1 (Y, ρ − |ρ0 |) and ρ > 1 + |iω /ω| + |ω ρ0 | on Y , we get a positive C ∞ strictly subharmonic (exhaustion) function ρ1 on X such that iω + iρ0 + iρ1 ≥ ω on Y and such that βL2 (Y,ρ0 +ρ1 ) < ∞. Corollary 2.12.6 now provides a C ∞ function α on Y such that ¯ =β ∂α

and

αL2 (Y,ω,ρ0 +ρ1 ) < ∞.

In particular, the C ∞ function f ≡ γ − α is holomorphic on Y . Moreover, for each p ∈ P , we have f − fp = −α on Wp \ {p}. On the other hand, for some positive constant C (depending on p), we have −mp

zp

αL2 (Wp \{p},(i/2)dzp ∧d z¯ p )

= αL2 (Wp \{p},(i/2) dzp ∧d z¯ p ,ρ0 ) ≤ CαL2 (Wp \{p},ω,ρ0 +ρ1 ) ≤ CαL2 (Y,ω,ρ0 +ρ1 ) < ∞. −m

Hence the holomorphic function zp p α on Wp \ {p} is square-integrable, and therefore, by Riemann’s extension theorem (Theorem 1.2.10), this function extends to a holomorphic function on Wp . Thus α extends holomorphically past p with order at least mp at p, and therefore f − fp extends to a holomorphic function on Up with order at least mp at p.  Exercises for Sect. 2.15 2.15.1 Prove that in the Mittag-Leffler theorem (Theorem 2.15.1), one may actually choose the function f ∈ O(X \ P ) so that for each point p ∈ P , f − fp extends to a holomorphic function on Up with a zero of order equal to mp at p. 2.15.2 Prove Corollary 2.15.2. 2.15.3 Prove the Behnke–Stein theorem (Corollary 2.15.3). 2.15.4 Prove Corollary 2.15.4. 2.15.5 Let f be a meromorphic function on an open Riemann surface X. Prove that there exist holomorphic functions g and h on X such that h is not the zero function and f = g/ h. 2.15.6 The goal of this exercise is a generalization of the Mittag-Leffler theorem that, in higher dimensions, is known as the solution of the additive Cousin problem (or the Cousin problem I). Let X be an open Riemann surface, let P be a discrete subset of X, let {mp }p∈P be a collection of positive integers, let {Ui }i∈I be an open covering of X, and for each pair of indices i, j ∈ I , let fij be a holomorphic function on Ui ∩ Uj with ordp fij ≥ mp for each point p ∈ P ∩ Ui ∩ Uj . Assume that the family {fij } satisfies the (additive) cocycle relation fik = fij + fj k on Ui ∩ Uj ∩ Uk Prove that there exist functions {gi }i∈I such that

∀i, j, k ∈ I.

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

2.16 The Runge Approximation Theorem According to the Mittag-Leffler theorem (Theorem 2.15.1), on an open Riemann surface X, one may prescribe values for a holomorphic function (to arbitrary order) at the points in a given discrete set. The identity theorem implies that it is not possible to prescribe values on a set that is not discrete. However, for a compact set K with hX (K) = K, one can uniformly approximate a holomorphic function on a neighborhood of K by a global holomorphic function. In other words, we have the following Riemann surface analogue, due to Behnke and Stein [BehS], of the classical Runge approximation theorem [Run] for domains in the plane: Theorem 2.16.1 (Runge approximation theorem) Suppose K is a compact subset of an open Riemann surface X with hX (K) = K, f0 is a holomorphic function on a neighborhood of K in X, and  > 0. Then there exists a holomorphic function f on X such that |f − f0 | <  on K. The converse is also true (see Exercise 2.16.4). Until now, we have not applied, in an essential way, the L2 estimate in Theorem 2.9.1; but this estimate will play an important role in the proof of this approximation theorem. We will also consider the more general version Theorem 2.16.3, so the reader may wish to skip the proof of Theorem 2.16.1 below and instead, consider only the proof of Theorem 2.16.3. Proof of Theorem 2.16.1 Multiplying f0 by a C ∞ function that has support contained in a small neighborhood of K but that is equal to 1 on some smaller neighborhood, and then extending by 0 to all of X and choosing suitable neighborhoods, we get a C ∞ function τ on X and open sets 0 and 1 such that K ⊂ 0  1  X, τ = f0 on 0 , and supp τ ⊂ 1 . We may also fix a Kähler form ω on X (by Corollary 2.11.3). Applying Theorem 2.14.1 (with the compact set and its neighborhood given by the empty set, and the nonnegative (1, 1)-form chosen so that its sum with iω is greater than or equal to ω), we get a positive C ∞ strictly subharmonic (exhaustion) function ρ0 on

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X such that iω + iρ0 ≥ ω on X. Applying the theorem again (this time with the compact set given by K), we get a nonnegative C ∞ subharmonic (exhaustion) function ρ1 on X such that supp ρ1 ⊂ X \ K and ρ1 > 0 on X \ 0 . In particular, r ≡ inf1 \0 ρ1 > 0. ¯ is a compactly For each positive integer ν, let ϕν ≡ ρ0 + νρ1 . The form β ≡ ∂τ supported C ∞ differential form of type (0, 1) on X. Since we have iω + iϕν = iω + iρ0 + νiρ1 ≥ ω, ¯ = β and Corollary 2.12.6 provides a C ∞ function α on X such that ∂α ¯ 2 2 α2L2 (X,ω,ϕ ) ≤ β2L2 (X,ϕ ) = ∂τ L ( ν

1 \0 ,ϕν )

ν

¯ 2 2 ≤ e−νr ∂τ L (

1 \0 ,ρ0 )

.

By construction, we have ρ1 ≡ 0 on some relatively compact neighborhood 2 of K in 0 , and hence α2L2 (

2 ,ω,ρ0 )

= α2L2 (

2 ,ω,ϕν )

≤ α2L2 (X,ω,ϕ ) . ν

Therefore, by choosing ν sufficiently large, we can make αL2 (2 ,ω,ρ0 ) arbitrarily small. Furthermore, the C ∞ function f ≡ τ − α on X is actually holomorphic ¯ = 0), and we have f − f0 L2 ( ,ω,ρ ) = αL2 ( ,ω,ρ ) . Applying The(since ∂f 2 0 2 0 orem 2.6.4 and choosing ν  0, we get |f − f0 | <  on K.  We have the following consequence (the converse, which also holds, is considered in Exercise 2.16.5): Corollary 2.16.2 Let  be a topologically Runge open subset of an open Riemann surface X. Then, for every holomorphic function f0 on , for every compact set K ⊂ , and for every  > 0, there exists a holomorphic function f on X such that |f − f0 | <  on K. Proof For every compact set K ⊂ , we have hX (K) ⊂ hX () = . Theorem 2.16.1 now gives the claim.  We also have the following more general version of Theorem 2.16.1: Theorem 2.16.3 (Runge approximation with poles at prescribed points) Suppose K is a compact subset of a Riemann surface X, P is a finite subset of X \K, Y = X \P , hY (K) = K, f0 is a holomorphic function on a neighborhood of K in X, and  > 0. Then there exists a meromorphic function f on X such that f is holomorphic on X \ P = Y , f has a pole at each point in P , and |f − f0 | <  on K. For the proof, we will need the following combined version of Lemma 2.10.3 and Theorem 2.14.1:

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Lemma 2.16.4 Suppose X is a Riemann surface, K is a compact subset of X, P is a finite subset of X \ K, Y = X \ P , hY (K) = K, {(Up , zp )}p∈P is a collection of disjoint local holomorphic coordinate neighborhoods in X with p ∈ Up and zp (p) = 0 for each p ∈ P , ω is a Kähler form on X, and  is a neighborhood of K in X. Then, for every sufficiently large positive constant b, there exist open sets {Vp }p∈P with p ∈ Vp  Up for each point p ∈ P such that for every sufficiently large positive constant R (depending on the above choices), there exists a nonnegative C ∞ subharmonic exhaustion function ϕ on Y with the following properties: (i) On Y \ , ϕ > 0 and iϕ ≥ ω; (ii) We have supp ϕ ⊂ Y \ K; and (iii) For each p ∈ P , we have ϕ = R · (|zp |2 − log |zp |2 − b) on Vp \ {p}. Proof If P = ∅ and X is compact, then we have K = X and the claim is trivial. Thus we may assume without loss of generality that Y is noncompact. By shrinking  and the sets {(Up , zp )}p∈P if necessary, we may also assume without loss of generality that Up  X \  for each point p ∈ P . We have hY (K) = K, so Theorem 2.14.1 provides a nonnegative C ∞ subharmonic exhaustion function α on Y such that supp α ⊂ Y \ K and such that α > 0 and iα ≥ ω on Y \ . For b  0, Lemma 2.10.3 provides, for each point p ∈ P , a nonnegative C ∞ subharmonic function βp on X \ {p} such that βp ≡ 0 on X \ Up and βp = |zp |2 − log |zp |2 − b > 0 on Wp \ {p} for some relatively compact neighborhood Wp of p in Up (the finiteness of P allows us to choose a single sufficiently large constant b that works for each of the points p ∈ P ). Choosinga relatively compact  neighborhood Vp of p in Wp for each p ∈ P , setting V ≡ p∈P Vp  W ≡ p∈P Wp , and choosing a nonnegative C ∞ function η on X such that η ≡ 1 on a neighborhood of X \ W and supp η ⊂ X \ V , we see that if R  0, then the function ϕ = η · α + p∈P R · βp has the required properties.  Proof of Theorem 2.16.3 We have Y = X \ P and hY (K) = K, and as in the proof of Lemma 2.16.4, we may assume without loss of generality that Y is noncompact. Multiplying f0 by a C ∞ function that has support contained in a small neighborhood of K but that is equal to 1 on some smaller neighborhood, and then extending by 0 to all of X and choosing suitable neighborhoods, we get a C ∞ function τ on X and open sets 0 and 1 such that K ⊂ 0  1  X \ P , τ = f0 on 0 , and supp τ ⊂ 1 . We may also choose disjoint local holomorphic coordinate neighborhoods {(Up , zp )}p∈P in X such that for each p ∈ P , we have p ∈ Up  X \ 1 and zp (p) = 0. We may also fix a Kähler form ω on X (by Corollary 2.11.3). By applying Lemma 2.16.4 (with the compact set and its neighborhood given by the empty set, and the Kähler form chosen so that its sum with iω is greater than or equal to ω), we get a positive constant b0 , open sets {Vp }p∈P with p ∈ Vp  Up for each point p ∈ P , a positive integer k, and a positive C ∞ strictly subharmonic (exhaustion) function ρ0 on Y such that iω + iρ0 ≥ ω on Y and such that for each point p ∈ P , ρ0 = k(|zp |2 − log |zp |2 − b0 ) on Vp \ {p}. Applying Lemma 2.16.4 again (this time with the compact set given by K) and shrinking each of the neigh-

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borhoods Vp for p ∈ P , we get a positive constant b1 and a nonnegative C ∞ subharmonic (exhaustion) function ρ1 on Y such that supp ρ1 ⊂ Y \ K, ρ1 > 0 on X \ 0 , and ρ1 = |zp |2 − log |zp |2 − b1 on Vp \ {p} for each p ∈ P (here, we apply the lemma to get constants b and R, and we then divide the resulting function by R and ∞ set b1 ≡ b/R). In particular, r ≡ inf1 \0 ρ1 > 0. We may also fix a C function η with compact support in the neighborhood p∈P Vp of P such that η ≡ 1 on a neighborhood of P . Now for each positive integer ν, let ϕν ≡ ρ0 + νρ1 . Given a positive integer μ (μ+ν+k) and a constant δ > 0, we get a C ∞ function γ on Y by setting γ = δη/zp on ¯ is then a compactly Up \ {p} for each p ∈ P and γ = τ elsewhere. The form β ≡ ∂γ supported C ∞ differential form of type (0, 1) on Y . Since we have iω + iϕν = iω + iρ0 + νiρ1 ≥ ω, ¯ = β and Corollary 2.12.6 provides a C ∞ function α on Y such that ∂α  ¯ 22 ¯ 2 2 α2L2 (Y,ω,ϕ ) ≤ β2L2 (Y,ϕ ) = ∂τ + δ 2 zp−(μ+ν+k) ∂η L ( \ ,ϕ ) L (V ν

1

ν

¯ 2 2 ≤ e−νr ∂τ L (

1 \0 ,ρ0 )

+

0



p ,ϕν )

ν

p∈P

¯ 22 δ 2 zp−(μ+ν+k) ∂η L (V

p ,ϕν )

.

p∈P

By construction, we have ρ1 ≡ 0 on some relatively compact neighborhood 2 of K in 0 , and hence α2L2 (

2 ,ω,ρ0 )

= α2L2 (

2 ,ω,ϕν )

≤ α2L2 (Y,ω,ϕ ) . ν

Therefore, by choosing ν sufficiently large and then choosing δ > 0 sufficiently small (depending on ν), we can make αL2 (2 ,ω,ρ0 ) arbitrarily small. ¯ = 0. Near The C ∞ function f ≡ γ − α on Y is actually holomorphic, since ∂f −μ ν+k ν+k each point p ∈ P , the function zp α = δzp − zp f is holomorphic except for an isolated singularity at p, and |zpν+k α|2 = |α|2 e−(−(ν+k) log |zp | ) is locally integrable near p by the choice of ϕν . Therefore, by Riemann’s extension theorem (Theorem 1.2.10), zpν+k α extends to a holomorphic function in a neighborhood of p, 2

−(μ+ν+k)

near p, and hence the function −α = f − γ , which is equal to f − δzp is meromorphic on a neighborhood of p with at worst a pole of order ν + k at p. It follows that f extends to a meromorphic function on X that is holomorphic on X \ P and that has a pole of order μ + ν + k at each point p ∈ P . Finally, since f − f0 L2 (2 ,ω,ρ0 ) = αL2 (2 ,ω,ρ0 ) , Theorem 2.6.4 implies that for ν  0 and δ sufficiently small (depending on ν), we have |f − f0 | <  on K.  Remarks 1. Note that we may choose the function f in the above proof to have a pole of any order > ν + k at each point in P (see Exercise 2.16.3). 2. A version in which the discrete set P may be infinite is considered in Exercise 2.16.6.

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We close this section with a consequence of Theorem 2.16.3 that will play an important role in the proof of the Riemann mapping theorem in Chap. 5 (see Sect. 5.4). Lemma 2.16.5 Given a compact subset K of an open Riemann surface X, there exist a holomorphic function f on a neighborhood Y of K in X, a positive regular value r of the function |f |, and a C ∞ domain  such that  is a connected component of {x ∈ Y | |f (x)| < r} and K ⊂   Y . Proof Clearly, we may assume that K is nonempty and connected. Thus we may choose a compact set K1  X with K ⊂ K1 and hX (K1 ) = K1 (in fact, we could replace K with hX (K) and set K = K1 , but this would require the fact, considered in Exercise 2.13.4, that the topological hull of a connected set in a manifold is connected, and is not really necessary here). Similarly, we may choose a relatively compact neighborhood 0 of K1 in X and a compact set K2 such that ∂0 ⊂ K2 ⊂ X \ K1 and hX\K1 (K2 ) = K2 (see Exercise 2.13.2). In particular, the set X \ (K1 ∪ K2 ) = (X \ K1 ) \ K2 has only finitely many components, and hence we may choose a finite set P ⊂ X \ (K1 ∪ K2 ) that meets each of these components. The domain Y ≡ X \ P then contains K1 ∪ K2 , and furthermore, hY (K1 ∪ K2 ) = K1 ∪ K2 . For each component of Y \ (K1 ∪ K2 ) is of the form U \ P , where U is a component of X \ (K1 ∪ K2 ), and by construction, P must meet U . Thus U \ P must contain points arbitrarily close to P = X \ Y , and hence U \ P cannot be relatively compact in Y . Now, by applying Theorem 2.16.3 to a locally constant function on a neighborhood of K1 ∪ K2 that is equal to 0 on K1 and 3 on K2 , we get a meromorphic function g on X such that g is holomorphic on X \ P = Y , g has a pole at each point in P , |g| < 1 on K1 ⊂ 0 , and |g| > 2 on K2 ⊃ ∂0 . Since g is nonconstant, the set of positive critical values of the function ρ ≡ |g|Y is countable (dg = 0 at any critical point of ρ in Y \ g −1 (0) = X \ g −1 ({0, ∞}) since 2ρ dρ = dρ 2 = g¯ dg + g d g¯ on this set). Thus we may fix a regular value r ∈ (1, 2). The connected component  of {x ∈ Y | ρ(x) < r} containing the connected compact set K must then be a nonempty C ∞ (by Corollary 2.4.5) domain that is relatively compact in 0 \ P = 0 ∩ Y , since ρ → ∞ at P and ρ > 2 on ∂0 . Setting f ≡ gY , we get the desired objects.  Exercises for Sect. 2.16 2.16.1 Let X be an open Riemann surface. Using the Runge approximation theorem (not the results of Sect. 2.15), prove the following (cf. Corollary 2.15.3): (i) Separation of points. If p, q ∈ X and p = q, then there exists a holomorphic function f on X such that f (p) = f (q); and (ii) Global holomorphic functions give local coordinates. For each point p ∈ X, there exists a holomorphic function f on X such that (df )p = 0. 2.16.2 Let P be a finite subset of a compact Riemann surface X, and let U be a neighborhood of P . Prove that there is a meromorphic function f on X

96

2.16.3 2.16.4

2.16.5

2.16.6

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

2.16

The Runge Approximation Theorem

2.16.7

2.16.8

2.16.9

2.16.10

97

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

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(c) Prove that if f ∈ C ∞ (C) and  is a smooth relatively compact domain in C, then       ∂f  "  f (ζ )  f (z) − 1 dζ  ≤ sup  · 8λ()/π.  2πi ∂ ζ − z  ∂ ζ¯ (d) Hartogs–Rosenthal theorem (see [HarR]). Suppose K is a compact set of measure 0 in C. Prove that if f0 is a continuous function on K and  > 0, then there exists a holomorphic function f on a neighborhood of K in C such that |f − f0 | <  on K. Prove also that if in addition, C \ K is connected, then there actually exists an entire function f such that |f − f0 | <  on K. (e) Prove that given an open set  ⊂ C and a compact set K ⊂ , there exists a constant C = C(K, ) > 0 such that     ∂f  max |f | ≤ C f L2 () + sup  ∀f ∈ C ∞ (). K  ∂ ζ¯ Hint. First consider the case in which K = (z0 ; R) and  = (z0 ; 4R) for some point z0 ∈ C and some R > 0. For this, apply the Cauchy integral formula (Lemma 1.2.1) to the disk (z0 ; 3R) and to the function ηf , where η ∈ D((z0 ; 3R)) and η ≡ 1 on (z0 ; 2R). For the general case, cover K by finitely many disks of the form (z0 ; R) with (z0 ; 4R) ⊂ . (f) Let X be a Riemann surface. Prove that given an open set  ⊂ X, a compact set K ⊂ , a Kähler form ω on X, and a real-valued C ∞ function ϕ on X, there exists a constant C = C(K, , ω, ϕ) > 0 such that      ∂f ∧ ∂f 1/2   max |f | ≤ C f L2 (,ω,ϕ) + sup ∀f ∈ C ∞ ().  K ω  (g) Bishop–Kodama localization theorem (see [Bis] and [Kod]). Let X be an open Riemann surface, let K be a compact subset of X, and let f0 be a continuous function on K. Assume that each point p ∈ K admits a neighborhood U in X such that for every  > 0, there exists a holomorphic function f on a neighborhood of the compact set K  ≡ K ∩ U in X with |f − f0 | <  on K  . Prove that for every  > 0, there exists a holomorphic function f on a neighborhood of K in X such that |f − f0 | <  on K. Hint. By replacing X with a large domain, one may assume that there are a Kähler metric ω and a C ∞ strictly subharmonic function ϕ on X as in Corollary 2.12.6. Fix a finite covering of K by relatively compact open subsets of X with the above approximation property, and fix a suitable partition of unity. Given δ > 0, one may form local approximations to within δ on each of these sets. Patching these local

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99

approximations using the partition of unity, one gets a C ∞ function τ ; ¯ is controlled by δ at points in K. Multiplying ∂τ ¯ and the (0, 1)-form ∂τ ∞ by a C (cutoff) function that is equal to 1 on K and that vanishes outside a small neighborhood of K (this function depends on the choice of δ), one gets a (1, 0)-form β that is controlled by δ everywhere in X. ¯ = β along Corollary 2.12.6 then provides a solution of the equation ∂α 2 with an L estimate. Guided by part (f), one sees that if δ > 0 is sufficiently small, then the restriction of τ − α to a small neighborhood of K (which depends on the choice of δ) has the required properties. (h) Let X be an open Riemann surface, and let K ⊂ X be a compact set of measure 0. Prove that for every continuous function f0 on K and for every  > 0, there exists a holomorphic function f on a neighborhood of K in X such that |f − f0 | <  on K. Prove also that if in addition, hX (K) = K, then there actually exists a function f ∈ O(X) such that |f − f0 | <  on K. Remarks According to Mergelyan’s theorem (see [Me] and [Rud1]), given a compact set K ⊂ C with connected complement, a continuous function f0 on K that is holomorphic on the interior of K, and a constant  > 0, there exists an entire function f with |f − f0 | <  on K. The Mergelyan–Bishop theorem includes the natural analogue for an open Riemann surface X: Given a compact set K ⊂ X with hX (K) = K, a continuous function f0 on K that is holomorphic on the interior of K, and a constant  > 0, there exists a holomorphic function f on X with |f − f0 | <  on K. The proof of the Hartogs– Rosenthal theorem outlined in parts (b)–(d) is due to Hartogs and Rosenthal. The proof of the Bishop–Kodama localization theorem outlined in parts (e)–(g) is due to Jarnicki and Pflug (see [JP] and [Ga]).

Chapter 3

¯ in a Holomorphic Line Bundle The L2 ∂-Method

In this chapter, we consider a useful generalization of the notion of a holomorphic function, namely, that of a holomorphic section of a holomorphic line bundle. We first consider the basic properties of holomorphic line bundles as well as those of sheaves and divisors. We then proceed with a discussion of the solution of the inhomogeneous Cauchy–Riemann equation with L2 estimates in this more general setting. In this setting, there is a natural generalization of Theorem 2.9.1 for Hermitian holomorphic line bundles (E, h) with positive curvature; that is, ih > 0, where h ¯ considered in Sect. 2.8. is a natural generalization of the curvature form ϕ = ∂ ∂ϕ In fact, Sects. 3.6–3.9 may be read in place of most of the material in Sects. 2.6–2.9. We then consider applications, mostly to the study of holomorphic line bundles on open Riemann surfaces (holomorphic line bundles on compact Riemann surfaces are considered in greater depth in Chap. 4). For example, in Sect. 3.11, we prove that every holomorphic line bundle on an open Riemann surface admits a positivecurvature Hermitian metric (this follows easily from the results of Sect. 2.14); and we then obtain a slightly more streamlined proof of (a generalization of) the MittagLeffler theorem (Theorem 2.15.1). In Sect. 3.12, we prove the Weierstrass theorem (Theorem 3.12.1), according to which every holomorphic line bundle on an open Riemann surface is actually holomorphically trivial.

3.1 Holomorphic Line Bundles Throughout this section, X denotes a complex 1-manifold. In this section, we consider the basic properties of holomorphic line bundles and differential forms with values in a holomorphic line bundle. In order to introduce some of the terminology, we first consider a set-theoretic version. Definition 3.1.1 Let Y be a topological space. (a) A (set-theoretic) complex line bundle over (or on) Y consists of a set E, a surjective mapping  : E → Y , and a choice of a 1-dimensional complex vector T. Napier, M. Ramachandran, An Introduction to Riemann Surfaces, Cornerstones, DOI 10.1007/978-0-8176-4693-6_3, © Springer Science+Business Media, LLC 2011

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

  U 

  U ×C     prU

in which  is a bijection that is linear on each fiber. A local trivialization of the form (X, ) is called a global trivialization. (b) Given two local trivializations (U1 , 1 = (, 1 )) and (U2 , 2 = (, 2 )) of a complex line bundle E over Y , the map g : U1 ∩ U2 → C∗ determined by 1 ◦ 2−1 (x, ζ ) = (x, g(x) · ζ ) for all (x, ζ ) ∈ (U1 ∩ U2 ) × C (that is, for each point x ∈ U1 ∩ U2 , g(x) = (1 )x /(2 )x is the nonzero scalar representing the linear isomorphism ζ → 1 (2−1 (x, ζ ))) is called a transition function (from 2 to 1 or from 2 to 1 ). (c) A collection of local  trivializations A = {(Ui , i )}i∈I of a complex line bundle E on Y with Y = i Ui is called a line bundle atlas for E. Remark A more standard approach is to consider only line bundles for which the total space E is a topological space, the projection  is continuous, and the local trivializations are homeomorphisms. The above weaker definition is more convenient for our purposes. A holomorphic structure allows one to consider a holomorphic version: Definition 3.1.2 Let  : E → X be a complex line bundle over X. (a) Two local trivializations (U1 , 1 = (, 1 )) and (U2 , 2 = (, 2 )) of E are holomorphically compatible if the transition functions g12 , g21 : U1 ∩ U2 → C∗ , given by g12 (p) =

(1 )p 1 = g21 (p) (2 )p

∀p ∈ U1 ∩ U2 ,

are holomorphic. (b) A line bundle atlas for E for which the transition functions are holomorphically compatible is called a holomorphic line bundle atlas. (c) Two holomorphic line bundle atlases A1 and A2 for E are holomorphically equivalent if A1 ∪ A2 is a holomorphic line bundle atlas (this is an equivalence relation).

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(d) An equivalence class S of holomorphic line bundle atlases for E is called a holomorphic line bundle structure in E over X. The pair (E, S) (usually denoted simply by E) is called a holomorphic line bundle over (or on) X. (e) If E is equipped with a holomorphic line bundle structure, then any local trivialization (U, ) in any holomorphic line bundle atlas in the holomorphic line bundle structure is called a local holomorphic trivialization for E. If we have U = X, then (X, ) is a (global) holomorphic trivialization for E and E is holomorphically trivial. The trivial line bundle over X is the line bundle 1X ≡ X × C → X. Remarks 1. A topological line bundle over a topological space and a C ∞ line bundle over a C ∞ manifold are defined analogously. The natural higher-rank analogues, in which the local trivializations are fiberwise linear bijections of the form −1 (U ) → U × Cr , are called vector bundles. 2. A holomorphic line bundle on X has a natural underlying C ∞ structure, an underlying C ∞ line bundle structure, and a 2-dimensional holomorphic structure (see Exercise 2.2.6 for the definition of a complex manifold and Exercise 3.1.9 for the verification). Definition 3.1.3 Let  : E → X be a holomorphic line bundle over X. (a) A section of E on a set B ⊂ X is a mapping s : B → E such that  ◦ s = IdB . For each point p ∈ B, we usually denote the value s(p) ∈ Ep by sp . If B is an open set, then we also call s a local section of E. If B = X, then we also call s a global section. (b) Let s be a section of E on a set B. Given a local trivialization (U, (, )), the function (s) : B ∩ U → C (i.e., the function p → (sp )) is called the representation of s in the local trivialization. (c) We say that a section s of E on a set B is continuous (of class C k with k ∈ Z≥0 ∪ {∞}, holomorphic, measurable, Ldloc with d ∈ [1, ∞]) if the representation (s) of s in every local holomorphic trivialization (U, (, )) (equivalently, in some local holomorphic trivialization in a neighborhood of each point in B) is a continuous (respectively, C k , holomorphic, measurable, Ldloc ) function. (d) For B ⊂ X an open set, we call a section s of E on the complement in B of a discrete subset P of B a meromorphic section of E on B if the representation (s) of s in every local holomorphic trivialization (U, (, )) (equivalently, in some local holomorphic trivialization in a neighborhood of each point in B) is a meromorphic function on B ∩ U with set of poles P ∩ U . We say that s has a zero (a pole) of order m at a point p ∈ B if the representation (s) of s in every (equivalently, in some) local holomorphic trivialization (U, (, )) over a neighborhood U of p has a zero (respectively, a pole) of order m at p.

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For any point p ∈ B, the order of s at p is given by ⎧ 0 if p is neither a zero nor a pole of s, ⎪ ⎪ ⎪ ⎨m if s has a zero of order m at p, ordp s = ⎪ −m if s has a pole of order m at p, ⎪ ⎪ ⎩ ∞ if s ≡ 0 on a neighborhood of p. We say that s has a simple zero (simple pole) at p if ordp s = 1 (respectively, ordp s = −1). (e) For any open set U ⊂ X, the set of holomorphic sections of E on U is denoted 0 (U, E)); the set of meromorphic sections by O(E)(U ) or (U, O(E)) (or HDol of E on U is denoted by M(E)(U ) or (U, M(E)); and the set of C ∞ sections of E on U is denoted by E(E)(U ) or (U, E(E)). The set of C ∞ sections with compact support in U is denoted by D(E)(U ) or D(U, E). Remarks 1. It is not hard to verify that, as noted in the above definition, if a section of a holomorphic line bundle on X has a continuous (C k , holomorphic, meromorphic, measurable, Ldloc ) representation in some local holomorphic trivialization in a neighborhood of each point, then the representation in every local holomorphic trivialization is continuous (respectively, C k , holomorphic, meromorphic, measurable, Ldloc ). The analogous statement holds for zeros and poles of a meromorphic section. The verifications are left to the reader (see Exercise 3.1.1). 2. For a nontrivial (i.e., not everywhere zero) meromorphic section of a holomorphic line bundle over a Riemann surface, the set of zeros (i.e., the set of points at which the section is equal to the zero element of the fiber) is discrete (see Exercise 3.1.2). 3. One goal of this chapter is a proof that every holomorphic line bundle on an open Riemann surface admits a great many (nontrivial) holomorphic sections. On the other hand, a holomorphic line bundle on a compact Riemann surface need not admit a nontrivial holomorphic section (see Exercise 3.1.6). Much of Chap. 4, and part of this chapter, are concerned with determining when (and how many) nontrivial holomorphic sections exist. 4. We may identify a section x → (x, f (x)) of the trivial line bundle 1X = X × C → X with the complex-valued function f . Example 3.1.4 The holomorphic tangent bundle (T X)1,0 : (T X)1,0 → X and holomorphic cotangent bundle (T ∗ X)1,0 : (T ∗ X)1,0 → X have natural holomorphic line bundle structures. For given two local holomorphic coordinate neighborhoods (U1 , z1 ) and (U2 , z2 ) in X, we have the local trivialization (Uj , ((T X)1,0 , dzj )) of (T X)1,0 over Uj for j = 1, 2, and we have holomorphic transition functions g12 =

dz1 ∂z1 = dz2 ∂z2

and

g21 =

dz2 ∂z2 = = 1/g12 . dz1 ∂z1

For each j = 1, 2, we also have the local trivialization (Uj , ((T ∗ X)1,0 , ρj )) of (T ∗ X)1,0 over Uj , where ρj (α) = α((∂/∂zj )p ) for each point p ∈ Uj and each

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element α ∈ (Tp∗ X)1,0 . The transition functions are the holomorphic functions g12 =

ρ1 ∂/∂z1 ∂z2 = = ρ2 ∂/∂z2 ∂z1

and

g21 =

∂/∂z2 ∂z1 = = 1/g12 . ∂/∂z1 ∂z2

The local holomorphic sections of (T X)1,0 are precisely the local holomorphic vector fields, and the local holomorphic sections of (T ∗ X)1,0 are precisely the local holomorphic 1-forms (and the analogous statements hold for continuous, C k , measurable, meromorphic, and Ldloc vector fields and forms). The holomorphic cotangent bundle (T ∗ X)1,0 is also called the canonical line bundle (or canonical bundle) and is denoted by KX : KX → X. Example 3.1.5 For the open subsets U0 ≡ C and U∞ ≡ P1 \ {0} of P1 , let E ≡ [(U0 × C)  (U∞ × C)]/∼, where for (z, ζ ) ∈ (U0 \ {0}) × C and (w, ξ ) ∈ (U∞ \ {∞}) × C, (z, ζ ) ∼ (w, ξ )

⇐⇒

z=w

and ζ = z · ξ ;

let ρ : (U0 × C)  (U∞ × C) → E be the quotient map; and let E : E → P1 be the (well-defined) surjective mapping given by E : ρ(z, ζ ) → z. Then E : E → P1 is a line bundle, which is called the hyperplane bundle. Moreover, for the local trivializations (E , 0 ) ≡ [ρU0 ×C ]−1 : −1 E (U0 ) = ρ(U0 × C) → U0 × C and (E , ∞ ) ≡ [ρU∞ ×C ]−1 : −1 E (U∞ ) = ρ(U∞ × C) → U∞ × C, we have the holomorphic transition functions g0∞ from ∞ to 0 and g∞0 from 0 to ∞ determined by g0∞ =

0 1 = : ∞ g∞0

z → z.

Thus E, together with these local trivializations, is a holomorphic line bundle. The holomorphic section s of E on U∞ determined by z → ρ(z, 1) = (E , ∞ )−1 (z, 1) for (z, 1) ∈ U∞ × C (i.e., the section that is represented by the constant function ∞ (s) : z → 1 in the local holomorphic trivialization (U∞ , (E , ∞ ))) is represented in the local holomorphic trivialization (U0 , (E , 0 )) by the holomorphic function 0 (s) : z → z. It follows that s extends to a unique holomorphic section of E on P1 that is nonvanishing except for a simple zero at the point 0. Section 3.3 contains a simple method for producing many (in fact, it turns out, all) examples of holomorphic line bundles on a Riemann surface.

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Given a local holomorphic trivialization (U,  = (, )) of a holomorphic line bundle  : E → X, we get a nonvanishing holomorphic section t on U given by −1 tp = −1 p (1) =  (p, 1) for all p ∈ U ; that is, t is the section with local representation p → 1. Conversely, according to Proposition 3.1.6 below, a nonvanishing local holomorphic section provides a local holomorphic trivialization. In other words, local holomorphic trivializations are equivalent to nonvanishing local holomorphic sections; and the latter point of view is often more convenient than the former. Further characterizations of continuity, continuous differentiability, etc., of sections in terms of local holomorphic sections are also contained in the proposition. Proposition 3.1.6 Suppose that  : E → X is a holomorphic line bundle, k ∈ Z≥0 ∪ {∞}, and d ∈ [1, ∞]. Then we have the following: (a) If t is a nonvanishing holomorphic section of E on an open set U ⊂ X, then the mapping  : −1 (U ) → C given by ξ → ξ/t (p) for all p ∈ U and ξ ∈ Ep determines a local holomorphic trivialization (U,  = (, )) of E (with (t) ≡ 1). (b) The product f s of a continuous (C k , holomorphic, meromorphic, measurable) function f and a continuous (respectively, C k , holomorphic, meromorphic, measurable) section s of E and the sum s1 + s2 of two continuous (respectively, C k , holomorphic, meromorphic, measurable) sections s1 and s2 of E are continuous (respectively, C k , holomorphic, meromorphic, measurable) sections. The sum of two Ldloc sections is in Ldloc , and for d  ∈ [1, ∞] with (1/d) + (1/d  ) = 1, the  product of an Ldloc function and an Ldloc section is in L1loc . (c) A section s : S → E on a set S ⊂ X is continuous (C k , holomorphic, meromorphic, measurable, Ldloc ) if and only if the quotient s/t : p → s(p)/t (p) is continuous (respectively, C k , holomorphic, meromorphic, measurable, Ldloc ) for every nonvanishing local holomorphic section t of E (equivalently, for some nonvanishing local holomorphic section t on a neighborhood of each point in S). Furthermore, if s is meromorphic, then s has a zero (a pole) of order m at a point p ∈ S if and only if the meromorphic function s/t has a zero (respectively, a pole) of order m at p for every (equivalently, for some) nonvanishing local holomorphic section t of E on a neighborhood of p. Proof Given a nonvanishing holomorphic section t of E on an open set U ⊂ X, the mapping  :  −1 (U ) → C given by ξ → ξ/t (p) for all p ∈ U and ξ ∈ Ep determines a bijection  = (, ) : −1 (U ) → U × C. Moreover, if (V , = (, ϒ)) is a local holomorphic trivialization of E, then (by Definition 3.1.3) the representing function g = ϒ(t) : U ∩ V → C∗ = C \ {0} is holomorphic and we have ϒ(ξ ) = g(p) · (ξ )

∀p ∈ U ∩ V , ξ ∈ Ep .

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

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Definition 3.1.7 Let E : E → X and E  : E  → X be holomorphic line bundles over complex 1-manifolds X and X  , let ϒ : X → X be a holomorphic map, and let : E → E  be a mapping such that E  ◦ = ϒ ◦ E and such that for each  is linear. Then: point p ∈ X, the mapping Ep : Ep → Eϒ(p) (a) The map is called a line bundle homomorphism (or line bundle map) along ϒ . Unless otherwise indicated, we will assume that a given line bundle homomorphism is taken along the identity map. (b) We call a continuous (C k , holomorphic) line bundle homomorphism if the function  ◦ (s) is a continuous (respectively, C k , holomorphic) function for every local holomorphic section s of E and every local holomorphic trivialization (U, (E  , )) of E  . (c) A bijective holomorphic line bundle homomorphism along the identity with holomorphic inverse line bundle homomorphism is called a holomorphic line bundle isomorphism (and the two bundles are said to be isomorphic). Example 3.1.8 If  : X → Y is a holomorphic mapping (a C k mapping with k ∈ Z>0 ∪ {∞}) of X into a complex 1-manifold Y , then the associated tangent mapping ∗ : (T X)1,0 → (T Y )1,0 and pullback mapping ∗ : (T ∗ Y )1,0 → (T ∗ X)1,0 are holomorphic (respectively, C k−1 ) line bundle homomorphisms (see Exercise 3.1.3). Remarks 1. A holomorphic line bundle E → X is holomorphically trivial if and only if E ∼ = 1X . 2. As we will see, every holomorphic line bundle on an open Riemann surface is holomorphically trivial (Theorem 3.12.1). However, even in that context, the abstract point of view is still useful, since, for example, there is usually no natural choice of a global holomorphic trivialization. 3. A real linear (or conjugate linear) homomorphism of holomorphic line bundles is defined analogously, with the mapping real linear (respectively, conjugate linear) on each fiber. Continuous and C k real linear and conjugate linear line bundle homomorphisms and isomorphisms are defined analogously. 4. We often identify two isomorphic line bundles without comment, although, occasionally one must proceed with some caution in doing so. Definition 3.1.9 For holomorphic line bundles E : E → X and E  : E  → X:  (a) The dual bundle of E is given by E ∗ ≡ p∈X Ep∗ , and the corresponding projection E ∗ : E ∗ → X is determined by α → p for each point p ∈ X and each linear functional α ∈ Ep∗ .  (b) The tensor product bundle of E and E  is given by E ⊗ E  ≡ p∈X Ep ⊗ Ep (see Sect. 8.3), and the corresponding projection E⊗E  : E ⊗ E  → X is determined by ξ → p for each point p ∈ X and each element ξ ∈ Ep ⊗ Ep . We will assume that any given dual or tensor product bundle as above has the holomorphic line bundle structure provided by the following:

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Proposition 3.1.10 Let E : E → X, E  : E  → X, and E  : E  → X be holomorphic line bundles. Then we have the following: (a) There is a (unique) holomorphic line bundle structure in E ∗ : E ∗ → X such that for each nonvanishing holomorphic section t of E on an open set U ⊂ X, the map  : −1 E ∗ (U ) → U × C given by α → (p, α(t (p))) for all p ∈ U and α ∈ Ep∗ is a local holomorphic trivialization. (b) There is a unique holomorphic line bundle structure in E ⊗ E  such that for each pair of holomorphic sections t of E and t  of E  on an open set U ⊂ X, the tensor product section t ⊗ t  (where (s ⊗ s  )p = sp ⊗ sp for each point p and each pair of sections s, s  ) is a local holomorphic section of E ⊗ E  . (c) For the above holomorphic line bundle structures, we have the (natural) holomorphic line bundle isomorphisms (i) E → (E ∗ )∗ given by ξ → λξ , where λξ (α) = α(ξ ) for all ξ ∈ Ep and α ∈ Ep∗ with p ∈ X; (ii) E ∗ ⊗ E → 1X = X × C given by α ⊗ ξ → α(ξ ) for all α ⊗ ξ ∈ Ep∗ ⊗ Ep with p ∈ X; (iii) 1X ⊗ E → E given by (p, ζ ) ⊗ ξ → ζ ξ for all p ∈ X, ζ ∈ C, and ξ ∈ Ep ; (iv) E ⊗ E  → E  ⊗ E given by η ⊗ ξ → ξ ⊗ η for all η ⊗ ξ ∈ Ep ⊗ Ep with p ∈ X; and (v) (E ⊗ E  ) ⊗ E  → E ⊗ (E  ⊗ E  ) given by (η ⊗ ξ ) ⊗ τ → η ⊗ (ξ ⊗ τ ) for all η ⊗ ξ ∈ Ep , ξ ∈ Ep , and τ ∈ Ep with p ∈ X. Proof For the proof of (a), we consider two nonvanishing holomorphic sections t and u of E on open sets U and V , respectively. Proposition 3.1.6 then implies that the function g ≡ t/u : U ∩ V → C∗ is holomorphic. Thus, for each element α ∈ Ep∗ with p ∈ U ∩ V , we have α(t (p)) = g(p)α(u(p)). It follows that the expression in (a) yields a bijection and that the transition functions for any two such bijections are holomorphic. Thus these mappings form a holomorphic line bundle atlas that determines a holomorphic line bundle structure on E ∗ , and (a) follows. The proofs of (b) and (c) are left to the reader (see Exercise 3.1.4).  Example 3.1.11 The canonical line bundle KX = (T ∗ X)1,0 is the dual of the holomorphic tangent bundle (T X)1,0 . Example 3.1.12 The dual bundle E ∗ → P1 of the hyperplane bundle E → P1 (Example 3.1.5) is called the tautological bundle. Remarks Let E, E  , E  be holomorphic line bundles on X. 1. Guided by part (c) of Proposition 3.1.10, we identify (E ∗ )∗ with E, E ∗ ⊗ E with 1X , 1X ⊗ E with E, E ⊗ E  with E  ⊗ E, and (E ⊗ E  ) ⊗ E  with E ⊗ (E  ⊗ E  ). 2. We denote (E ⊗ E  ) ⊗ E  = E ⊗ (E  ⊗ E  ) simply by E ⊗ E  ⊗ E  ; we set 0 E = 1X ; and for any positive integer r, we set r factors

 

E ≡ E ⊗ · · · ⊗ E. r

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3. One may treat E ∗ as a multiplicative inverse of E, and the trivial line bundle 1X as a multiplicative identity. The proof of the following is left to the reader (see Exercise 3.1.5): Proposition 3.1.13 Let E : E → X and E  : E  → X be holomorphic line bundles, let S ⊂ X, let k ∈ Z≥0 ∪ {∞}, and let d, d  ∈ [1, ∞] with (1/d) + (1/d  ) = 1. Then we have the following: (a) For any section α of E ∗ on S, the following are equivalent: (i) The section α is continuous (of class C k , holomorphic, measurable). (ii) The function α(s) is continuous (respectively, of class C k , holomorphic, measurable) for every local continuous (respectively, class C k , holomorphic, measurable) section s of E. (iii) For each point p ∈ S, the function α(s) is continuous (respectively, of class C k , holomorphic, measurable) for some nonvanishing continuous (respectively, C k , holomorphic, measurable) section s of E on a neighborhood of p. (b) For any section α of E ∗ on S, the following are equivalent: (i) The section α is in Ldloc . (ii) The function α(s) is in Ldloc for every local continuous section s of E. (iii) For each point p ∈ S, the function α(s) is in Ldloc for some nonvanishing continuous section s of E on a neighborhood of p.  Moreover, if α is in Ldloc and s is a section of E in Ldloc on S, then the function α(s) is in L1loc . (c) The tensor product s ⊗ s  of continuous (C k , holomorphic, meromorphic, measurable) sections s of E and s  of E  on S is continuous (respectively, C k , holomorphic, meromorphic, measurable). Furthermore, if s is nonvanishing, then the section s −1 of E ∗ is continuous (respectively, C k , holomorphic, meromorphic, measurable). If t is a meromorphic section of E on X that is not identically zero on any connected component of X, then t −1 is a meromorphic section  of E ∗ . Finally, if u is a section of E in Ldloc on S and u is a section of E  in Ldloc on S, then u ⊗ u is in L1loc . (d) Any section s of E ∗ ⊗ E  on X determines a line bundle homomorphism  : E → E  given by ξ → (sp /t)(ξ ) · t for each point p ∈ X, each element t ∈ Ep \ {0}, and each element ξ ∈ Ep ; and  is continuous (of class C k , holomorphic) if and only if the section s is continuous (respectively, of class C k , holomorphic). Conversely, any line bundle homomorphism  : E → E  determines a section s of E ∗ ⊗ E  on X given by sp ≡ t −1 ⊗ (t) for each point p ∈ X and each element t ∈ Ep \ {0}; and s is continuous (of class C k , holomorphic) if and only if  is continuous (respectively, of class C k , holomorphic). Moreover, the above associations are inverse mappings. We now consider differential forms with values in a holomorphic line bundle.

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Definition 3.1.14 Let E : E → X be a holomorphic line bundle. For each pair p, q ∈ Z≥0 and for r = p + q, we define  

p,q T ∗ X ⊗ E ≡

p,q Tx∗ X ⊗ Ex ⊂ r (T ∗ X)C ⊗ E ≡

r (Tx∗ X)C ⊗ Ex x∈X

x∈X

( p,q T ∗ X ⊗ E = X × {0} for p > 1 or q > 1, r (T ∗ X)C ⊗ E = X × {0} for r > 2). The corresponding projections are the (surjective) mappings  p,q T ∗ X⊗E : p,q T ∗ X ⊗ E → X

and  r (T ∗ X)C ⊗E : r (T ∗ X)C ⊗ E → X

given by α → x for each x ∈ X and each α ∈ p,q Tx∗ X ⊗ E or r (Tx∗ X)C ⊗ E. The elements of p,q T ∗ X ⊗ E are said to be of type (p, q). For each point x ∈ X, we also set [ p,q T ∗ X ⊗ E]x ≡ p,q Tx∗ X ⊗ Ex ⊂ [ r (T ∗ X)C ⊗ E]x ≡ r (Tx∗ X)C ⊗ Ex . Given a nonnegative integer s, another holomorphic line bundle F : F → X, and a point x ∈ X, the mapping [ r (T ∗ X)C ⊗ E]x × [ s (T ∗ X)C ⊗ F ]x → [ r+s (T ∗ X)C ⊗ E ⊗ F ]x given by α β ∧ ⊗ t ⊗ u, t u for any choice of t ∈ Ex \ {0} and u ∈ Fx \ {0}, is called the exterior product (or wedge product). (α, β) → α ∧ β ≡

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

0,0 T ∗ X ⊗ E = 1X ⊗ E ∼ =E

and

KX ⊗ E = 1,0 T ∗ X ⊗ E

are actually tensor products of holomorphic line bundles, so they have natural holomorphic line bundle structures (provided by Proposition 3.1.10). For p > 1 or q > 1,

p,q T ∗ X ⊗ E = X × {0} ∼ =X has the natural holomorphic structure inherited from X. 2. p,q T ∗ X ⊗ E and r T ∗ X ⊗ E have natural C ∞ structures (see Exercise 3.1.10). Moreover, p,q T ∗ X ⊗ E and r T ∗ X ⊗ E are C ∞ vector bundles (see, for example, [Wel]). 3. The above definition of the exterior product is independent of the choice of the nonzero elements of the fibers of the line bundles (see Exercise 3.1.7). Guided by Definition 9.5.2, Definition 9.7.12, Definition 2.5.1, Definition 2.5.2, and Proposition 3.1.6, we make the following definition:

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Definition 3.1.15 Let E : E → X be a holomorphic line bundle, let k ∈ Z≥0 ∪ {∞}, and let d ∈ [1, ∞]. (a) For each r ∈ Z≥0 , an E-valued differential form of degree r (or an r-form with values in E or an E-valued r-form) in X on a set S ⊂ X is a mapping α of S into

r (T ∗ X)C ⊗ E such that  r (T ∗ X)C ⊗E ◦ α = IdS (in particular, α/s is a scalarvalued differential form of degree r for every nonvanishing local holomorphic section s of E). We usually denote the value of α at x by αx for each point x ∈ S. If αx ∈ p,q Tx∗ X ⊗ Ex for each point x ∈ S, then we say that α is of type (p, q) or that α is an E-valued (p, q)-form. (b) An E-valued differential form α is continuous (of class C k , holomorphic, meromorphic, measurable, in Ldloc ) if the scalar-valued differential form α/s is continuous (respectively, of class C k , holomorphic, meromorphic, measurable, in Ldloc ) for every nonvanishing local holomorphic section s of E. A sequence of Ldloc differential forms {αν } with values in E converges in Ldloc to an E-valued differential form α if {αν /s} converges to α/s in Ldloc for every nonvanishing local holomorphic section s of E. (c) For each open set U ⊂ X and each nonnegative integer r, the set of C ∞ E-valued r-forms on U is denoted by E r (U, E) or E r (E)(U ). The set of C ∞ E-valued r-forms with compact support in  is denoted by D r (U, E) or D r (E)(U ). Similarly, for p, q ∈ Z≥0 , the set of C ∞ E-valued (p, q)forms on U is denoted by E p,q (U, E) or E p,q (E)(U ), and the set of C ∞ Evalued (p, q)-forms with compact support in U is denoted by Dp,q (U, E) or D p,q (E)(U ). The set of E-valued holomorphic 1-forms on U is denoted by (U, E) or (E)(U ). Remarks 1. Let α be an r-form with values in a holomorphic line bundle E on a subset S of X. Given a nonvanishing holomorphic section s of E on an open set U , we get the scalar-valued form β ≡ α/s with αS∩U = β ⊗ s. If α is of type (1, 0), then α is actually a section of the holomorphic line bundle 1,0 T ∗ X ⊗ E. If r = 1, then for each point x ∈ S, we may also identify αx with the linear mapping of the tangent space at x into Ex given by v →

αx (v) · t, t

for any choice of t ∈ Ex \ {0} (the above is independent of the choice of t). For r = 2 and x ∈ S, we may identify αx with the skew-symmetric bilinear pairing of the tangent space at x into Ex given by (u, v) →

αx (u, v) · t t

for any choice of t ∈ Ex \ {0}. The existence of the above identifications is the reason one calls α a differential form with values in E. For s a nonvanishing holomorphic section on an open set U and β = α/s on S ∩ U , the above also leads us to denote αS∩U = β ⊗ s simply by β · s or by s ⊗ β or s · β (see Sect. 8.3).

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2. For α and β differential forms with values in holomorphic line bundles E and F , respectively, we may form the exterior product α ∧ β with values in E ⊗ F . Identifying E ⊗ F with F ⊗ E, we see that if α and β are of degree r and m, respectively, then α ∧ β = (−1)rm β ∧ α. 3. Let α be an E-valued r-form on a set S, let (U, z) be a local holomorphic coordinate neighborhood, and let s be a nonvanishing holomorphic section of E on an open set V . Then, on S ∩ U ∩ V , we have α = as, where a = α/s, if r = 0; α = a dz ⊗ s + b d z¯ ⊗ s, where a and b are the coefficients of α/s, if r = 1; and α = a dz ∧ d z¯ ⊗ s, where a is the coefficient of α/s, if r = 2. 4. A holomorphic 1-form θ with values in a holomorphic line bundle E over X may be viewed as a holomorphic section of the holomorphic line bundle KX ⊗ E. We also have an identification of 1,1 T ∗ X ⊗ E with 0,1 T ∗ X ⊗ KX ⊗ E under the mapping α = θ ∧ β · s = −β ∧ θ · s → −β · (θ ⊗ s); so a (1, 1)-form with values in E may be identified with a (0, 1)-form with values in KX ⊗ E (see Sect. 3.10). On the other hand, certain operators and pairings to be considered in this chapter and Chap. 4 are not completely preserved under this mapping, so one must proceed with some caution when making this identification. The proof of the following is left to the reader (see Exercise 3.1.8): Proposition 3.1.16 Let E and F be holomorphic line bundles on X, let k ∈ Z>0 ∪ {∞}, and let d, d  ∈ [1, ∞]. Then we have the following: (a) A differential form α on a set S ⊂ X with values in E is continuous (of class C k , holomorphic, meromorphic, measurable, in Ldloc ) if and only if for each point in S, the scalar-valued differential form α/s is continuous (respectively, of class C k , holomorphic, meromorphic, measurable, in Ldloc ) for some nonvanishing local holomorphic section s of E on a neighborhood of the point. A sequence of Ldloc (S) differential forms {αν } with values in E converges in Ldloc (S) to an E-valued differential form α if and only if for each point in S, {αν /s} converges to α/s in Ldloc for some nonvanishing local holomorphic section s of E on a neighborhood of the point. (b) Let α and β be differential forms with values in E and F , respectively. If α and β are continuous (of class C k , holomorphic, meromorphic, measurable), then α + β and α ∧ β are continuous (respectively, of class C k , holomorphic, meromorphic, measurable). If α and β are in Ldloc , then α + β is in Ldloc . If α  and β are in Ldloc and Ldloc , respectively, and (1/d) + (1/d  ) = 1, then α ∧ β is locally integrable (i.e., α ∧ β is in L1loc ). Remark It follows from the above proposition that the set of continuous (C k , holomorphic, meromorphic, measurable, Ldloc ) differential forms with values in a given holomorphic line bundle over a subset of a complex 1-manifold is a vector space.

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

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each local holomorphic trivialization (U, (E , ϒ)), we get a local C ∞ chart

−1  1 (T ∗ X) ⊗E (U ), , U  × C2 = U  × R4 C

in

1 (T ∗ X)C

⊗ E, where        ∂ ∂ , ϒ(ξ ) · α  : α ⊗ ξ → z(c), ϒ(ξ ) · α ∂z c ∂ z¯ c

for each point c ∈ U , each element ξ ∈ Ec , and each element α ∈

1 (Tc∗ X)C . In other words, setting ξ = (E , ϒ)−1 (c, 1), we have ((a(dz)c + b(d z¯ )c ) ⊗ ξ ) = (z(c), a, b)

∀(a, b) ∈ C2 .

(b) Prove that there is a (unique) structure of a C ∞ manifold on 0,1 T ∗ X ⊗ E such that for each local holomorphic chart (U,  = z, U  ) in X and each local holomorphic trivialization (U, (E , ϒ)), we get a local C ∞ chart −1

 0,1 T ∗ X⊗E (U ), , U  × C = U  × R2 in 0,1 (T ∗ X)C ⊗ E, where   : α ⊗ ξ → z(c), ϒ(ξ ) · α



∂ ∂ z¯

  c

for each point c ∈ U , each element ξ ∈ Ec , and each element α ∈

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

(U ), , U  × C = U  × R2 −1

1,1 T ∗ X⊗E

in 1,1 T ∗ X ⊗ E, where       ∂ ∂  : α ⊗ ξ → z(c), ϒ(ξ ) · α , ∂z c ∂ z¯ c for each point c ∈ U , each element ξ ∈ Ec , and each element α ∈

1,1 Tc∗ X. (d) Prove that the projection mappings to X in the above are C ∞ mappings. (e) Prove that an E-valued differential form is continuous (of class C k ) if and only if it is continuous (respectively, of class C k ) as a mapping.

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3.2 Sheaves Associated to a Holomorphic Line Bundle Sheaf theory is a convenient and powerful tool for describing local objects and for passing to global objects. In this book, rather than consider the general theory of sheaves, we instead mostly consider and apply a few specific types of sheaves associated to a holomorphic line bundle. We first consider some examples before considering a formal definition. Throughout this section,  : E → X denotes a holomorphic line bundle over a complex 1-manifold X. Example 3.2.1 The sheaf O(E) of holomorphic sections of E consists of the assignment U → O(E)(U ) = (U, O(E)) of the collection of holomorphic sections O(E)(U ) to each open set U ⊂ X together with the associated restriction maps ρVU : (U, O(E)) → (V , O(E)) given by s → sV for open sets U ⊃ V (O(E) is also called a locally free analytic sheaf of rank 1). The sheaf of holomorphic functions O is given by U → O(U ), and we identify O with O(1X ). Let p ∈ X and let ∼p be the equivalence relation on the set of local holomorphic sections of E that are defined on a neighborhood of p determined by s ∼p t

⇐⇒

s = t on some neighborhood of p.

Each of the associated equivalence classes is called a germ of a holomorphic section of E at p. The set O(E)p of germs at p is called the stalk of O(E) at p. We denote by germp s the germ represented by a local holomorphic section s of E on a neighborhood of p. For each open set U ⊂ X, (U, O(E)) is a module over (U, O) (we view (∅, O(E)) as the trivial module {0}). The operations descend to the level of germs, making O(E)p a module over the ring Op for each point p ∈ X. More precisely, for neighborhoods U , V , and W of p, sections s ∈ (U, O(E)) and t ∈ (V , O(E)), and a function f ∈ O(W ), we define germp f · germp s ≡ germp (f · s), germp s + germp t ≡ germp (s + t), germp s − germp t ≡ germp (s − t). Example 3.2.2 The definitions of the sheaf M = M(1X ) of meromorphic functions, the sheaf M(E) of meromorphic sections of E (a sheaf of modules over M), the germ of a meromorphic section of E, and the stalk M(E)p of M(E) at p ∈ X are analogous to the definitions in Example 3.2.1. ∼ O(KX ) of holomorphic Example 3.2.3 The definitions of the sheaf  = (1X ) = 1-forms (a sheaf of O-modules), the sheaf (E) ∼ = O(KX ⊗ E) of E-valued holomorphic 1-forms (a sheaf of O-modules), the sheaf of meromorphic 1-forms (which

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is isomorphic to M(KX ) as a sheaf of M-modules), the sheaf of E-valued meromorphic 1-forms (which is isomorphic to M(KX ⊗ E) as a sheaf of M-modules), and the associated germs and stalks, are analogous to the definitions in Example 3.2.1. Example 3.2.4 The definitions of the sheaves E = E 0 , E r , E p,q , E(E), E r (E), and E p,q (E) (which are sheaves of modules over E), and the associated germs and stalks, are analogous to the definitions in Example 3.2.1. Example 3.2.5 Given a holomorphic line bundle homomorphism  : E → F (along the identity), we get corresponding sheaf morphisms ∗ : O(E) → O(F ) and ∗ : M(E) → M(F ) given by s → (s). Similarly, a C ∞ line bundle homomorphism  : E → F induces sheaf morphisms E r (E) → E r (F ) and E p,q (E) → E p,q (F ) (locally, the mappings are given by α → (α/t) ⊗ (t) for any nonvanishing local holomorphic section t of E). Example 3.2.6 Let p ∈ X. The set mp of germs at p of holomorphic functions that vanish at p is the maximal ideal in the ring Op , and for each positive integer r, mrp is precisely the set of germs at p of holomorphic functions that vanish at p to order at least r. For each open set U ⊂ X, let F (U ) be the O(U )-submodule of O(E)(U ) consisting of the holomorphic sections that vanish at p to order at least r (in particular, F (U ) = O(E)(U ) if p ∈ / U ). Together with the given restriction mappings, these submodules form a subsheaf F of O(E). For each point x ∈ X, the stalk Fx at x is the collection of germs that are represented by functions in F (U ) for some neighborhood U of x. Thus  O(E)x if x = p, Fx = mrp · O(E)p if x = p. The Op -submodule mrp · O(E)p of O(E)p consists of the germs at p of holomorphic sections s of E with a zero of order at least r at p. The quotient O(E)p /mrp · O(E)p = {ξ + mrp · O(E)p | ξ ∈ O(E)p } is a complex vector space of dimension r. For in terms of a local holomorphic coordinate z vanishing at r and a nonvanishing holomorphic section t of E in a neighborhood of p, the elements ξ0 , . . . , ξr−1 of V represented by t, zt, z2 t, . . . , zr−1 t, respectively, form a basis. For each open set U ⊂ X, let  O(E)p /mrp · O(E)p if p ∈ U, S(U ) ≡ 0 if p ∈ / U, and for any open set V ⊂ U , let ρVU : S(U ) → S(V ) be the obvious restriction map; i.e., ρVU is the identity if p ∈ V (so that S(U ) = S(V )) and ρVU is identically 0 if

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p∈ / V . Then S is a sheaf of modules over O. We may form germs and stalks at a point x by identifying elements that have the same  image under some restriction map in some neighborhood of x. Thus Sx = O(E)x Fx at each point x ∈ X. Because there is this single nonzero r-dimensional vector space stalk at p, S is called a skyscraper sheaf. The induced maps O(E)(U ) → S(U ) given by  germp t + mrp · O(E)p ∈ O(E)p /mrp · O(E)p if p ∈ U, t → 0 if p ∈ / U, determine a sheaf morphism, which we denote by O(E) → S. This morphism  may also be identified with the (quotient) module maps O(E)x → Sx = O(E)x Fx at the level of stalks. Example 3.2.7 Let Y be a topological space, and let R be a ring. Then the associated constant sheaf R is given by R(U ) ≡ R for each nonempty open set U ⊂ Y , and R(∅) ≡ 0. We also denote R by R. In particular, we have the zero sheaf U → {0}, which is denoted by 0. Constant sheaves of modules, groups, etc., are defined analogously. We now consider the formal definitions. Definition 3.2.8 Let Y be a topological space. (a) A sheaf F on Y is an assignment of a set F(U ) to each open set U ⊂ Y and a restriction mapping ρVU : F (U ) → F(V ) to each pair of open sets U ⊃ V such that (i) For each open set U ⊂ Y , we have ρUU = IdU on U ; U V (ii) For open sets U ⊃ V ⊃ W , we have ρW = ρW ◦ ρVU ; (iii) If {Ui }i∈I is a family of open subsets of Y , U = i∈I Ui , and s, t ∈ F (U ) with ρUUi (s) = ρUUi (t) for each i ∈ I , then s = t; and  (iv) If {Ui }i∈I is a family of open subsets of Y , U = i∈I Ui , si ∈ F (Ui ) for U

U

each i ∈ I , and ρUii∩Uj (si ) = ρUij∩Uj (sj ) for every pair of indices i, j ∈ I , then there exists an s ∈ F(U ) such that ρUUi (s) = si for each i ∈ I . The elements of F(U ) are called sections of F over U . The set F (U ) is also denoted by (U, F) or H 0 (U, F ). (b) A sheaf R on Y together with the assignment of a ring structure to R(U ) for each open set U such that the restriction mappings are homomorphisms with respect to these ring structures is called a sheaf of rings. Given a sheaf of rings R on Y , a sheaf of R-modules on Y is a sheaf F on Y together with the assignment of a module structure to F (U ) over the ring R(U ) for each open set U ⊂ Y such that the restriction mappings commute with the associated sum and product operations. Sheaves of groups, ideals, vector spaces, and other algebraic structures are defined analogously.

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

 F(V )

ρVU  V  G(V )

If each of the above maps U is an inclusion, then we call F a subsheaf of G and we call  an inclusion. If each of the above maps is an isomorphism, then we call  a sheaf isomorphism. (d) Let F be a sheaf on Y , and for each point p ∈ Y , let ∼p be the equivalence relation on the set of sections of F over neighborhoods of p defined as follows: Given s ∈ F (U ) and t ∈ F(V ) with p ∈ U ∩ V , we have s ∼p t if and only U V if there is an open set W ⊂ U ∩ V with ρW (s) = ρW (t). For each section s of F over a neighborhood U of p, we call the equivalence class represented by s the germ of s at p and we denote this germ by germp s. The collection of germs at p is called the stalk at p and is denoted by Fp . If F is a sheaf of R-modules on Y for some sheaf of rings R, then for each point p ∈ Y , the stalk Rp is a ring and the stalk Fp is a module over Rp with the natural induced operations. A sheaf morphism  : F → G induces a well-defined morphism p : Fp → Gp given by p (germp t) ≡ germp U (t) for each point p ∈ Y , each neighborhood U , and each section t ∈ F(U ). These induced morphisms are inclusions (isomorphisms) if and only if U is injective (respectively, bijective) for each open set U (see Exercise 3.2.2). If U is surjective for each open set U , then p : Fp → Gp is surjective for each point p ∈ M. However, the converse is false (see Exercise 3.2.3). Remarks 1. Given a sheaf morphism  : F → G, the kernel K is the subsheaf with sections over a nonempty open set U given by ker U and the restriction mappings inherited from F . In general, the assignment U → im U need not be a sheaf. It is an example of what is called a presheaf, and it therefore determines a sheaf (called the image), but we will not consider presheaves and their associated sheaves in this book. 2. The skyscraper sheaf is an example of a quotient sheaf. Again, to define quotient sheaves in general, one must consider presheaves and their associated  sheaves. 3. For a sheaf F , since we may write ∅ as a union of the form i∈∅ Ui , and since for s, t ∈ F(∅), the equality ρU∅ i (s) = ρU∅ i (t) then holds vacuously for i ∈ ∅, the axiom (iii) implies that F(∅) contains at most one element. Similarly, the condition (iv) implies that F(∅) is nonempty and therefore that F(∅) is a singleton. In particular, for F a sheaf of modules, we have F(∅) = 0. In many contexts, it is often convenient to consider exact sequences of sheaves. However, this is done at the level of stalks, not sections:

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Definition 3.2.9 A sequence of sheaf morphisms 1

2

3

m

F0 −→ F1 −→ F2 −→ · · · −→ Fm on a topological space Y is called exact if for each point p ∈ Y and each index j = 2, . . . , m, we have ker(j )p = im(j −1 )p . An exact sequence of sheaves does not always yield an exact sequence at the level of sections on nonempty open sets (see Exercise 3.2.3). Exercises for Sect. 3.2 3.2.1 Verify that the examples given in this section are indeed sheaves. 3.2.2 Verify that the morphisms on stalks induced by a sheaf mapping  are well defined and that they preserve the given algebraic structures (group, ring, or module). Also verify that these morphisms are inclusions (isomorphisms) if and only if U is injective (respectively, bijective) for each open set U . 3.2.3 Prove that if 0 → F → G → H is an exact sequence of sheaves on a topological space Y , then for every nonempty open set U ⊂ Y , the sequence 0 → F(U ) → G(U ) → H(U ) is exact. Give an example to show that the analogous statement for an exact sequence F → G → 0 need not hold.

3.3 Divisors Divisors are convenient objects for encoding zeros and poles of meromorphic functions and sections. In particular, they allow one to produce many (in fact, it turns out, all) examples of holomorphic line bundles on a Riemann surface. Throughout this section, X denotes a complex 1-manifold. Definition 3.3.1 For a given complex 1-manifold X: (a) A divisor on X is a mapping D : X → Z with discrete support; that is, the set D −1 (Z \ {0}) ∩ K is finite for every compact set K ⊂ X. The Abelian group consisting of the set of divisors on X (together with the natural addition) is denoted by Div(X). (b) Given a meromorphic section s of a holomorphic line bundle on X, the function div(s) : X → Z ∪ {+∞} is given by p → ordp s. In particular, if s does not vanish identically on any nonempty open subset of X, then D = div(s) is a divisor that is called the divisor of s and s is called a defining section for D. The divisor D = div(f ) of a meromorphic function f that does not vanish identically on any nonempty open subset of X is called a principal divisor and f is called a solution of D (or a defining function for D).

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

and div(s/t) = div(s) − div(t)

(where s/t = s ⊗ t −1 is the associated meromorphic section of E ⊗ F ∗ ). Consequently, linear equivalence of divisors is an equivalence relation (see Exercise 3.3.1). 3. If  : E → F is a holomorphic isomorphism of holomorphic line bundles and s ∈ (X, O(E)), then div((s)) = div(s). Proposition 3.3.2 Let D be a divisor on X; let F be the set of pairs λ = (U, f ) consisting of an open set U ⊂ X and a meromorphic function f on U with div(f ) = DU (i.e., f is a local defining  function for D); let λ ≡ U × C for each element λ = (U, f ) ∈ F ; let [D] ≡ λ∈F λ /∼, where ∼ is the equivalence relation determined by (p0 , ζ0 ) ∼ (p1 , ζ1 )

⇐⇒

p0 = p1

and

ζ0 =

f0 (p0 ) · ζ1 f1

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

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

∀p ∈ X, ξ ∈ Ep .

Moreover, this is the unique isomorphism mapping t to s. (c) For any holomorphic line bundle E on X and any meromorphic section t of E with divisor D = div(t), we have E ∼ = [D] if and only if D  ∼ D. Proof It is easy to see that  is a well-defined map, and for each λ = (U, f ) ∈ F , (, λ ) : −1 (U ) = ρ(λ ) → U × C is a well-defined bijection. Since supp D is discrete, a local defining function for D exists in a neighborhood of each point, and hence  is surjective. Moreover, for each point p ∈ X, [D]p = −1 (p) has a natural well-defined 1-dimensional complex vector space structure determined by ρ(p, ζ ) + η · ρ(p, ξ ) = ρ(p, ζ + η · ξ ) for each λ = (U, f ) ∈ F with p ∈ U , each pair of elements (p, ζ ), (p, ξ ) ∈ λ , and each scalar η ∈ C; and for each λ = (U, f ) ∈ F with p ∈ U , λ [D]p : [D]p → C is a complex linear isomorphism. Thus [D] is a complex line bundle with local trivializations {(U, (, λ ))}λ=(U,f )∈F . If λj = (Uj , fj ) ∈ F for j = 0, 1, then the corresponding transition functions are the nonvanishing holomorphic functions λ 0 f0 = : U0 ∩ U1 → C λ1 f1

and

λ1 f1 = : U0 ∩ U1 → C, λ0 f0

so these local trivializations determine a holomorphic line bundle structure in [D]. Moreover, for each point p ∈ U0 ∩ U1 with D(p) ≥ 0, we have λ0 ((, λ1 )−1 (p, f1 (p))) =

f0 (p) · f1 (p) = f0 (p). f1

Thus the section s is well defined, and since λ (s) = f ∈ M(U ) for each λ = (U, f ) ∈ F , s is a meromorphic section with div(s) = D (div(s)U = div(f ) = DU for each such λ). Thus (a) is proved, and part (b) follows. For the proof of part (c), suppose E is a holomorphic line bundle on X, t is a meromorphic section of E on X that does not vanish identically on any open subset of X, and D  = div(t). If E ∼ = [D], then the image of t is a meromorphic section t  of [D] with divisor D  , and t  /s is a meromorphic function with divisor div(t  /s) = D  − D, and hence D  ∼ D. Conversely, if D  ∼ D and f is a defining function for D  −D, then t/f is a meromorphic section of E with divisor div(t/f ) =  D  − (D  − D) = D, and part (b) implies that E ∼ = [D]. Definition 3.3.3 Given a divisor D on X, the holomorphic line bundle [D] and the defining section provided by Proposition 3.3.2 are called the holomorphic line bun-

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dle associated to D and the associated defining section, respectively. Given a holomorphic line bundle E over X, OD (E) is the subsheaf of M(E), considered as a sheaf of O-modules, for which for every open set U ⊂ X, OD (E)(U ) ⊂ M(E)(U ) is the O(U )-submodule of sections s ∈ M(E)(U ) with div(s) + DU ≥ 0 (in particular, for D ≥ 0, O−D (E) is a subsheaf of O(E)). For E the trivial line bundle, we let OD ≡ OD (E). Remarks 1. Given two divisors D and D  on X with associated holomorphic line bundles [D] and [D  ], respectively, and associated defining sections s and s  , respectively, s ⊗s  and s −1 are meromorphic sections of [D]⊗[D  ] and [D]∗ , respectively, with divisors div(s ⊗ s  ) = D + D  and div(s −1 ) = −D. Thus Proposition 3.3.2 gives natural isomorphisms [D] ⊗ [D  ] ∼ = [−D]. = [D + D  ] and [D]∗ ∼ 2. According to Proposition 3.3.2, if E is a holomorphic line bundle on X and t is a meromorphic section with divisor D = div(t) (note that the product of t and any nonvanishing holomorphic function on X also has divisor D), then we may identify the pair (E, t) with ([D], s), where s is the associated defining section for D. Thus a divisor on X may be viewed as a holomorphic line bundle together with a choice of a meromorphic section. In particular, a holomorphic line bundle on a Riemann surface is isomorphic to the line bundle associated to some divisor if and only if E admits a nontrivial meromorphic section. It turns out that every holomorphic line bundle on a Riemann surface has this property (see Corollary 3.11.7 and Theorem 4.2.3). 3. Let D be a divisor on a Riemann surface X, and let s be the associated defining section of [D]. Then OD and O([D]) are isomorphic as sheaves of O-modules under the sheaf morphism f → f · s. If E is a holomorphic line bundle on X, then OD (E) and O(E ⊗ [D]) are isomorphic as sheaves of O-modules under the sheaf morphism t → t ⊗ s (see Exercise 3.3.4). Example 3.3.4 The hyperplane bundle E → P1 (see Example 3.1.5) is equal to the line bundle associated to the divisor D with D(0) = 1 and D(q) = 0 for q ∈ P1 \ {0}. In fact, for any point p ∈ P1 \ {0}, the meromorphic function f given by z → (z − p)/z if p ∈ C∗ and z → 1/z if p = ∞ has a simple pole at 0 and a simple zero at p. Thus, for Dp = 1 · p, we have Dp = D + div(f ), and hence [Dp ] ∼ = E. The tautological bundle is E ∗ = [−Dp ]. Definition 3.3.5 Let D be an effective divisor and let E be a holomorphic line bundle on X. The skyline sheaf of E along D is the sheaf QD (E) of O-modules with sections over any nonempty open set U given by QD (E)(U ) = 0 if U ∩ supp D = ∅ and otherwise by  QD (E)(U ) = O(E)p /O−D (E)p p∈U ∩supp D

=



 D(p) O(E)p mp · O(E)p

p∈U ∩supp D

∼ =



p∈U ∩supp D



CD(p) ∼ =C

p∈U ∩supp D (D(p))

,

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and with each restriction mapping ρVU : QD (E)(U ) → QD (E)(V ) for U ⊃ V given by the module projection s = {sp }p∈U ∩supp D → {sp }p∈V ∩supp D for supp V ∩ D = ∅, and s → 0 otherwise. We also denote QD (E) by O(E)/ O−D (E). Remarks 1. It is easy to verify that QD (E) is a sheaf of O-modules (see Exercise 3.3.3). At the level of stalks, for each point p ∈ X, we have the natural isomorphism QD (E)p ∼ = O(E)p /O−D (E)p . 2. A skyscraper sheaf is a skyline sheaf for a divisor D = p for some point p. 3. The sheaf QD (E) is an example of a quotient sheaf. Observation 3.3.6 Let E be a holomorphic line bundle on X. Given an effective divisor D on X with associated holomorphic defining section s, we get the holomorphic line bundle map  : E → E ⊗ [D] given by ξ → (ξ ) = ξ ⊗ sp for p ∈ X and ξ ∈ Ep . We also get the corresponding sheaf morphism ∗ : O(E) → O(E ⊗ [D]) given by ξ → ξ ⊗ s for any local holomorphic section ξ . We also denote D = div(s) by div(). We also get the natural induced sheaf morphism O(E ⊗ [D]) → QD (E ⊗ [D]). This yields the corresponding exact sequence of sheaves (see Exercise 3.3.5) 0 → O(E) → O(E ⊗ [D]) → QD (E ⊗ [D]) → 0. If G is any holomorphic line bundle on X and t is any holomorphic section of G with div(t) = D, then we may identify (G, t) with ([D], s), and hence we may identify the above exact sequence with the exact sequence 0 → O(E) → O(E ⊗ G) → QD (E ⊗ G) → 0 associated to the holomorphic line bundle map given by ξ → (ξ ) = ξ ⊗ tp for p ∈ X and ξ ∈ Ep . In fact, any holomorphic homomorphism  : E → F ∼ = E ⊗ E ∗ ⊗ F of holomorphic line bundles that is not indentically zero over any nonempty open subset of X may be viewed as above. For Proposition 3.1.13 provides a corresponding holomorphic section t of E ∗ ⊗ F , and  may be identified with the holomorphic line bundle mapping  : ξ → ξ ⊗ t (p) for p ∈ X and ξ ∈ Ep . Setting D = div() = div(t), we get the exact sequence 0 → O(E) → O(F ) → QD (F ) → 0, which we may identify with the exact sequence 0 → O(E) → O(E ⊗ E ∗ ⊗ F ) → QD (E ⊗ E ∗ ⊗ F ) → 0, given by multiplication by t , as well as with the exact sequence 0 → O(E) → O(E ⊗ [D]) → QD (E ⊗ [D]) → 0, given by multiplication by the associated defining section for D.

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

Remarks 1. The mapping D → deg D gives a homomorphism deg : Div(X) → Z for X a compact Riemann surface. 2. By Proposition 2.5.7, for X a compact Riemann surface, any principal divisor has degree 0. Consequently, if D and D  are two divisors with isomorphic associated line bundles, then deg D = deg D  . Thus, for any holomorphic line bundle E on X that admits a nontrivial meromorphic section s, we may define the degree of E to be the integer deg E ≡ deg(div(s)), which is independent of the choice of s (and depends only on the isomorphism equivalence class of E). 3. If X is a compact Riemann surface and D is a divisor of negative degree on X, then [D] has no nontrivial global holomorphic sections, since the divisor of such a section must have nonnegative degree. In particular, if D is a nontrivial effective divisor, then D has a nontrivial holomorphic section, while [−D] = [D]∗ has no nontrivial holomorphic sections (of course, for the trivial case D ≡ 0, (X, O([D])) = (X, O([−D])) = (X, O) ∼ = C). Exercises for Sect. 3.3 3.3.1 Verify that linear equivalence of divisors is an equivalence relation. 3.3.2 Let G be the set of isomorphism equivalence classes of holomorphic line bundles on a Riemann surface X. (a) Show that G together with the product ⊗ and the identity element 1X is an Abelian group. (b) Show that the mapping in D → [D] determines an injective homomorphism into G of the quotient group of Div(X) by the subgroup of principal divisors (it will later follow that this mapping is actually an isomorphism). 3.3.3 Verify that for any divisor D and any holomorphic line bundle E, OD (E) (see Definition 3.3.3) is a subsheaf of O-modules of M(E). Verify also that for D ≥ 0, QD (E) (see Definition 3.3.5) is a sheaf of O-modules. 3.3.4 Prove that if D is a divisor on a Riemann surface X, s is a meromorphic section of [D] with div(s) = D, and E is a holomorphic line bundle on X, then OD (E) and O(E ⊗ [D]) are isomorphic as sheaves of O-modules under the sheaf morphism t → t ⊗ s (in particular, OD ∼ = O([D]) under f → f s). 3.3.5 Verify that the (first) associated sequence of sheaf mappings in Observation 3.3.6 is exact.

3.4 The ∂¯ Operator and Dolbeault Cohomology Throughout this section, E and F denote holomorphic line bundles over a complex 1-manifold X. There is no canonical way to generalize the exterior derivative oper-

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125

ator d and the operator ∂ on scalar-valued forms to an operator on E-valued forms, although there is a very useful generalization of the canonical connection that depends on the choice of a Hermitian metric in E (see Sect. 3.7). On the other hand, ¯ the holomorphic structure in E provides a natural generalization of the operator ∂. Definition 3.4.1 Let α be an E-valued r-form on an open set  ⊂ X. ¯ is the unique E-valued (r + 1)-form that satisfies (a) If α is of class C 1 , then ∂α ¯ = ∂(α/s) ¯ ∂α ·s

on U

for every nonvanishing holomorphic section s of E on an open subset U of . ¯ = 0, then we say that α is ∂-closed. ¯ (b) If α is of class C 1 and ∂α ¯ for some C 1 E-valued (r − 1)-form β on , then we say that α is (c) If α = ∂β ¯ ∂-exact. If β may be chosen to be of class C k for some k ∈ Z>0 ∪ {∞}, then we ¯ also say that α is C k ∂-exact. If for each point in , there exists a C 1 E-valued ¯ 0 = αU , then we say that (r − 1)-form β0 on a neighborhood U0 such that ∂β 0 ¯ α is locally ∂-exact. If for some k ∈ Z>0 ∪ {∞}, each of the local forms β0 may ¯ be chosen to be of class C k , then we also say that α is locally C k ∂-exact. It is ∞ ¯ also convenient to consider the trivial 0-form α ≡ 0 to be C ∂-exact and to ¯ write 0 = ∂0. The operator ∂¯ is well defined because if α is a C 1 differential form and s and t are nonvanishing holomorphic sections of E on an open set U , then s/t is a holomorphic function and hence     ¯∂(α/t) · t = ∂¯ s · α · t = s · ∂¯ α · t = ∂(α/s) ¯ · s. t s t s The Dolbeault lemma (applied to a local representation of the form) implies that for ¯ (p, q + 1)-form of class k ∈ Z>0 ∪ {∞} and p, q ∈ Z≥0 , every E-valued ∂-closed k k ¯ C is locally C ∂-exact. Proposition 2.5.5 has the following analogue, the proof of which is left to the reader (see Exercise 3.4.1): Proposition 3.4.2 For any C 1 E-valued r-form α on an open set  ⊂ X, we have the following: (a) If α is of class C 2 , then ∂¯ 2 α = 0. ¯ is of type (p, q + 1). In particular, ∂α ¯ = 0 if q ≥ 1. (b) If α is of type (p, q), then ∂α k ¯ ¯ (c) If α is ∂-exact and of class C for some k ∈ Z>0 ∪ {∞}, then α is ∂-closed and of type (p, q + 1) for some pair of integers (p, q). Moreover, there exists a ¯ and any such C 1 E-valued differential form β of type (p, q) such that α = ∂β, form β is actually of class C k . (d) If (U, z) is a local holomorphic coordinate neighborhood, s is a nonvanishing holomorphic section of E on U , and a, b ∈ C 1 (U ), then ∂a ¯ d z¯ · s ∂(as) = ∂ z¯

¯ dz + b d z¯ ) · s = − ∂a dz ∧ d z¯ · s. and ∂(a ∂ z¯

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(e) If α is of type (0, 0) or (1, 0), then α is an E-valued holomorphic r-form if and ¯ = 0. only if ∂α Definition 3.4.3 For any q ∈ Z≥0 , the qth Dolbeault cohomology of E over X is the quotient vector space



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

The Dolbeault Exact Sequence Let  : E → F be a holomorphic line bundle mapping that is not identically zero over any nonempty open subset of X. Equivalently, for the corresponding holomorphic section s of E ∗ ⊗ F and the effective divisor D = div() = div(s) (s is the associated defining section for D under the identification of E ∗ ⊗ F with [D]), we have F = E ⊗ [D] and (ξ ) = ξ ⊗ s(p) for each point p ∈ X and each vector ξ ∈ Ep (see Observation 3.3.6). We get the associated exact sequence of sheaves ∗

0 → O(E) → O(F ) → QD (F ) → 0. We will now see that  also induces an exact sequence of linear maps of vector spaces 0 → (X, O(E)) → (X, O(F )) → (X, QD (F )) 1 1 → HDol (X, E) → HDol (X, F ) → 0.

The second map (X, O(E)) → (X, O(F )) is given by t → (t) = t ⊗ s, which is clearly injective. The third map  D(p) O(F )p /mp O(F )p (X, O(F )) → (X, QD (F )) = p∈supp D

is given by D(p)

t → {germp t + mp

O(F )p }p∈supp D .

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

ξ ≡ {germp t + mp

O(F )p }p∈supp D ∈ (X, QD (F )),

we get ξ → η. Conversely, if t is any C ∞ section of F that is holomorphic near supp D and that represents an element ξ of (X, QD (F )), θ is the unique extension ¯ ¯ on X, and of ∂t/s to a C ∞ form on X, and η = [θ]Dol , then we have θ ⊗ s = ∂t hence η → [θ ⊗ s]Dol = 0. 1 (X, E) → H 1 (X, F ) is surjective. SupIt remains to show that the map HDol Dol ∞ pose τ is a C form of type (0, 1) with values in F on X and λ ≡ [τ ]Dol . By applying the Dolbeault lemma and cutting off as above, we may assume that τ vanishes on some neighborhood of supp D in X. Hence τ/s extends to a unique C ∞ form θ of type (0, 1) with values in E on X, and we have η ≡ [θ ]Dol → λ. We may summarize the above remarks as follows: Theorem 3.4.4 Any holomorphic line bundle mapping  : E → F over X with divisor D = div() (in particular,  is not identically zero over any nonempty open

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subset of X) induces an exact sequence of linear maps 0 → (X, O(E)) → (X, O(F )) → (X, QD (F )) 1 1 → HDol (X, E) → HDol (X, F ) → 0.

Definition 3.4.5 We call the sequence provided by Theorem 3.4.4 the Dolbeault exact sequence corresponding to the line bundle map. Exercises for Sect. 3.4 3.4.1 Prove Proposition 3.4.2 (cf. Proposition 2.5.5 and Exercise 2.5.5).

3.5 Hermitian Holomorphic Line Bundles Throughout this section, X denotes a complex 1-manifold. In order to consider L2 forms with values in a holomorphic line bundle, we need some notion of a pointwise norm (or length) of the values. This is provided by a Hermitian metric. Definition 3.5.1 A Hermitian metric h in a holomorphic line bundle  : E → X on X is a choice of a Hermitian inner product hp (·, ·) in the fiber Ep of E over each point p ∈ X such that for each pair of local C ∞ sections s and t of E, the function h(s, t) : p → hp (sp , tp ) is of class C ∞ . For each point p ∈ X and each pair of elements u, v ∈ Ep , we also write h(u, v) ≡ hp (u, v) and |u|2h ≡ h(u, u). If E is equipped with a Hermitian metric h, then we call (E, h) or E a Hermitian holomorphic line bundle. Remarks 1. According to Proposition 3.11.1 in Sect. 3.11 below, every holomorphic line bundle on a Riemann surface admits a Hermitian metric. The main point is that according to Radó’s theorem (Theorem 2.11.1), every Riemann surface is second countable, after which the construction of a Hermitian metric (using a partition of unity) is standard. For now, we will avoid using Proposition 3.11.1 (until Sect. 3.11), and thereby avoid any reliance on Radó’s theorem. This will also allow us to avoid using (for now) most of the material from Sects. 2.6–2.9 (in fact, Sects. 3.6–3.9 below may be read in place of most of the material in Sects. 2.6–2.9). 2. If (E, h) is a Hermitian holomorphic line bundle on X and s is a nonvanishing local holomorphic section of E, then, setting ϕ ≡ − log |s|2h , we get     u v h(u, v) = · · e−ϕ . s s Thus such a nonvanishing local holomorphic section s allows one to locally represent the Hermitian metric h by the weight function ϕ (i.e., by e−ϕ ). In particular, it follows that if u and v are sections of E that are continuous (C k with k ∈ Z≥0 ∪ {∞}, measurable), then the function h(u, v) is continuous (respectively, C k , measurable).

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For the trivial line bundle 1X = X × C, the Hermitian metrics are precisely the Hermitian metrics of the form e−ϕ h0 , where ϕ is a real-valued C ∞ function on X and h0 is the standard Hermitian metric given by h0 ((x, ζ ), (x, ξ )) = ζ · ξ¯ for all x ∈ X and ζ, ξ ∈ C; in other words, a Hermitian metric in the trivial line bundle is equivalent to a C ∞ weight function ϕ. Example 3.5.2 Let  : E → P1 be the hyperplane bundle, and let s be the holomorphic section of E considered in Example 3.1.5, which is nonvanishing except for a simple zero at 0. Then there is a unique Hermitian metric h in E with |sz |2h = |z|2 /(1 + |z|2 ) for z ∈ P1 (|s∞ |2h = 1). For the section s/z extends to a holomorphic section t on P1 that is nonvanishing except for a simple zero at ∞ (in the notation of Example 3.1.5, tz = (E , 0 )−1 (z, 1) for z ∈ C). For z ∈ P1 and u, v ∈ Ez , we may then set h(u, v) = hz (u, v) ≡

⎧ (z)) ⎨ (u/t (z))·(v/t 2

if z ∈ C,

⎩ (u/s(z))·(v/s(z))

if z ∈ P1 \ {0}

1+|z|

1+|1/z|2

(with 1/z = 0 for z = ∞). One may now verify that h is well defined on the overlap C∗ and that h is a Hermitian metric with the required properties (see Exercise 3.5.1). Lemma 3.5.3 A Kähler form on X is equivalent to a Hermitian metric in the holomorphic tangent bundle. More precisely: (a) If ω is a Kähler form on X, then there is a (unique) Hermitian metric g in (T X)1,0 such that g(u, v) = −iω(u, v) ¯ for each point p ∈ X and each pair of tangent vectors u, v ∈ (Tp X)1,0 . (b) If g is a Hermitian metric in (T X)1,0 , then there is a (unique) Kähler form ω on X such that ω(u, v) ¯ = ig(u, v) for each point p ∈ X and each pair of tangent vectors u, v ∈ (Tp X)1,0 . Proof Given a Kähler form ω, we may define g (as in (a)) by g(u, v) = gp (u, v) ≡ −iω(u, v) ¯ for each p ∈ X and all u, v ∈ (Tp X)1,0 . Given a point p ∈ X and a local holomorphic coordinate neighborhood (U, z) of p, we have ω = G(i/2) dz ∧ d z¯ on U for some positive C ∞ function G, and hence for each pair of tangent vectors u0 , v0 ∈ (Tp X)1,0 , we have gp (u0 , v0 ) = (G(p)/2) · dz(u0 )dz(v0 ). It follows easily that gp is a Hermitian inner product and that the function g(u, v) is of class C ∞ for each choice of local C ∞ vector fields u and v of type (1, 0). Thus g is a Hermitian metric, and it is clear that g is uniquely determined. Conversely, if g is a Hermitian metric in (T X)1,0 , then we may define a C ∞ differential 2-form ω by ω(u, v) = ωp (u, v) ≡ ig(u1,0 , v 0,1 ) − ig(v 1,0 , u0,1 ) for each point p ∈ X and each pair of complex tangent vectors u and v at p with (1, 0) parts u1,0 and v 1,0 , respectively, and (0, 1) parts u0,1 and v 0,1 , respectively.

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Clearly, for p ∈ X and u, v ∈ (Tp X)1,0 , we have ω(u, v) ¯ = ig(u, v). Moreover, in any local holomorphic coordinate neighborhood (U, z = x + iy), we have  2  2 ∂  ∂  ω = ω(∂/∂z, ∂/∂ z¯ ) dz ∧ d z¯ =   idz ∧ d z¯ = 2  dx ∧ dy, ∂z g ∂z g so ω is a C ∞ positive real (1, 1)-form, that is, a Kähler form. Again, uniqueness follows.  Definition 3.5.4 A Hermitian metric g in the holomorphic tangent bundle (T X)1,0 = KX∗ is called a Kähler metric on X. According to Lemma 3.5.3, a Kähler metric has a corresponding Kähler form and vice versa. According to the following lemma, the proof of which is left to the reader (see Exercise 3.5.2), Hermitian metrics in holomorphic line bundles induce Hermitian metrics in dual bundles and tensor product bundles: Lemma 3.5.5 Let (E, h), (E  , h ), and (E  , h ) be Hermitian holomorphic line bundles on X. Then: (a) There exists a unique Hermitian metric h∗ in the dual bundle E ∗ such that h∗ (α, β) = h∗p (α, β) ≡

α(s)β(s) , |s|2h

for each point p ∈ X, each pair α, β ∈ Ep∗ , and each element s ∈ Ep \ {0}. Equivalently, for each nonzero element s ∈ E, |1/s|2h∗ = 1/|s|2h . (b) There exists a unique Hermitian metric h ⊗ h in the tensor product bundle E ⊗ E  such that (h ⊗ h )(u ⊗ u , v ⊗ v  ) = h(u, v) · h (u , v  ) for each point p ∈ X, each pair u, v ∈ Ep , and each pair u , v  ∈ Ep . (c) Under the identification of (E ∗ )∗ with E, we have (h∗ )∗ = h (where (h∗ )∗ is the Hermitian metric obtained by applying part (a) to (E ∗ , h∗ )); under the identification of E  ⊗ E with E ⊗ E  , we have h ⊗ h = h ⊗ h (where h ⊗ h and h ⊗ h are the Hermitian metrics obtained by applying part (b)); and under the identification of (E ⊗ E  ) ⊗ E  with E ⊗ (E  ⊗ E  ), we have (h ⊗ h ) ⊗ h = h ⊗ (h ⊗ h ). Definition 3.5.6 Let (E, h), (E  , h ), and (E  , h ) be Hermitian holomorphic line bundles on X. The Hermitian metric h∗ in E ∗ given by Lemma 3.5.5 is called the dual Hermitian metric. The Hermitian metric h ⊗ h in E ⊗ E  given by Lemma 3.5.5 is called the tensor product Hermitian metric. Under the natural identification of (E ⊗ E  ⊗ E  , (h ⊗ h ) ⊗ h ) with (E ⊗ E  ⊗ E  , h ⊗ (h ⊗ h )), we

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131

set h ⊗ h ⊗ h ≡ (h ⊗ h ) ⊗ h = h ⊗ (h ⊗ h ). For any r ∈ Z>0 , we also denote the Hermitian metric h ⊗ · · · ⊗ h in E r by hr . If (E, h) and (E  , h ) are Hermitian holomorphic line bundles over a complex 1manifold X, and s and s  are nonvanishing local holomorphic sections of E and E  , respectively, then, setting ϕ ≡ − log |s|2h and ϕ  ≡ − log |s  |2h , we get − log |s −1 |2h∗ = log |s|2h = −ϕ

and

− log |s ⊗ s  |2h⊗h = ϕ + ϕ  .

Thus h∗ and h ⊗ h are locally represented by the weight functions −ϕ and ϕ + ϕ  , respectively. Exercises for Sect. 3.5 3.5.1 Verify that the function h constructed in Example 3.5.2 is a Hermitian metric with the required properties. 3.5.2 Prove Lemma 3.5.5.

3.6 L2 Forms with Values in a Hermitian Holomorphic Line Bundle Throughout this section, (E, h) denotes a Hermitian holomorphic line bundle on a complex 1-manifold X. The goal of this section is the development of a suitable L2 space of differential forms. As discussed in Sect. 2.6, since the objects that we integrate on oriented real 2-dimensional manifolds are the 2-forms and we wish to pair forms of the same degree and integrate in order to get an inner product, it is natural to consider a pointwise pairing that gives a 2-form. For this, the following notation is convenient (see [De3]): Definition 3.6.1 Given a point x ∈ X, integers m, n ∈ Z≥0 , and elements α ∈ [ m (T ∗ X)C ⊗ E]x and β ∈ [ n (T ∗ X)C ⊗ E]x , we set  ( α ) ∧ ( βs ) · |s|2h if m + n ≤ 2, {α, β}(E,h) = {α, β}h = {α, β} ≡ s 0 if m + n > 2, in m+n (Tx∗ X)C for any nonzero element s ∈ Ex . In other words, {α, β} is equal to the product of the exterior product of the form parts (with the second factor conjugated) and the inner product of the line bundle parts. Observe that the above definition does not depend on the choice of s. For m = n = 0, we have {α, β}h = h(α, β). Moreover, if ϕ = − log |s|2 for a nonvanishing local holomorphic section s of E, and α and β are differential forms with values in E and αs = α/s and βs = β/s, then {α, β}h = αs ∧ β¯s · e−ϕ ,

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which is the pointwise pairing that appears in the definition of the L2 inner product of scalar-valued forms (see Sect. 2.6). In particular, if α and β are continuous (C k with k ∈ Z≥0 ∪ {∞}, measurable), then the scalar-valued differential form {α, β}h is continuous (respectively, C k , measurable). Moreover, if d, d  ∈ [1, ∞], (1/d) +  (1/d  ) = 1, and α and β are in Ldloc and Ldloc , respectively, then {α, β}h is in L1loc . Based on the above, we make the following definition (cf. Definition 2.6.1): Definition 3.6.2 Let S be a measurable subset of X, and let α and β be E-valued measurable differential forms of type (p, q) that are defined on S. (a) For (p, q) = (1, 0), we define  αL2

1,0 (S,E,h)

1/2

≡

∈ [0, ∞].

i{α, α}h S

If {α, β}h is integrable on S, then we define  α, βL2 (S,E,h) ≡ i{α, β}h ∈ C. 1,0

S

(b) For (p, q) = (0, 1), we define αL2

0,1

1/2   i{α, α}h ∈ [0, ∞]. (S,E,h) ≡ − S

If {α, β}h is integrable on S, then we define  α, βL2 (S,E,h) ≡ − i{α, β}h ∈ C. 0,1

S

(c) If (p, q) = (0, 0) and ω is a nonnegative measurable form of type (1, 1) defined on S, then we define  αL2

0,0

(S,E,ω,h) ≡

S

1/2 |α|2h · ω



1/2

=

{α, α}h · ω

∈ [0, ∞].

S

If h(α, β) · ω = {α, β}h · ω is integrable on S, then we define   α, βL2 (S,E,ω,h) ≡ h(α, β) · ω = {α, β}h · ω ∈ C. 0,0

S

S

(d) If (p, q) = (1, 1) and ω is a positive measurable form of type (1, 1) defined on S, then we define  1/2   2 1/2   α α α   , αL2 (S,E,ω,h) ≡ = ω   ω 1,1 S ω h S ω ω h = α/ωL2

0,0 (S,E,ω,h)

∈ [0, ∞].

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133

If {α/ω, β/ω}h ω is integrable on S, then we define   α, βL2

1,1

(S,E,ω,h) ≡

S

α β , ω ω



 ω= h

α β , ω ω

 L20,0 (S,E,ω,h)

∈ C.

(e) When there is no danger of confusion (for example, when the choice of (p, q), S, E, ω, or h is understood from the context), we will suppress parts of the notation. For example, we will often write αL2p,q (S,E,ω,h) simply as αS,E,ω,h , αS,ω,h , αω,h , αh , αE , αω , αS,h , αL2 (S,h) , αS,E , αL2 (S,E,ω,h) , αL2 (S,ω,h) , αL2 (S,ω,h) , αL2 (ω,h) , αL2 (S,h) , αL2 (S,E) , αL2 (h) , αL2 (E) , or α. The analogous simplified notation will be used for α, βL2p,q (S,E,ω,h) , for αL2p,q (S,E,h) , and for α, βL2p,q (S,E,h) . (f) Let γ and η be measurable E-valued 1-forms defined on S with (r, s) parts γ r,s and ηr,s , respectively, for each (r, s) ∈ {(1, 0), (0, 1)}. Then we set ≡ γ 1,0 2h + γ 0,1 2h . We also set γ 2L2 1 (S,E,h)

γ , ηL2 (S,E,h) ≡ γ 1,0 , η1,0 h + γ 0,1 , η0,1 h , 1

provided each of the summands on the right-hand side is defined. We also use the simplified notation analogous to that appearing in (e). Definition 3.6.3 Let S be a measurable subset of X. (a) For (p, q) = (1, 0) or (0, 1), the set L2p,q (S, E, h) consists of all equivalence classes of measurable E-valued differential forms α of type (p, q) on S with αL2 (S,h) < ∞, where we identify any two elements that are equal almost everywhere. The set L21 (S, E, h) consists of all equivalence classes of measurable E-valued 1-forms α on S with αL2 (S,h) < ∞, where again, we identify any two elements that are equal almost everywhere. (b) For ω a nonnegative measurable differential form of type (1, 1) that is defined on S, L20,0 (S, E, ω, h) consists of all equivalence classes of measurable sections α of E on S with αL2 (S,ω,h) < ∞, where we identify any two elements that are equal almost everywhere. (c) For ω a positive measurable differential form of type (1, 1) that is defined on S, L21,1 (S, E, ω, h) consists of all equivalence classes of E-valued measurable differential forms α of type (1, 1) on S with αL2 (S,ω,h) < ∞, where we identify any two elements that are equal almost everywhere. (d) Suppose E = 1X = X × C → X is the trivial holomorphic line bundle, h0 is the standard Hermitian metric in E, and ϕ is the real-valued C ∞ function on X for which h = e−ϕ h0 . Then, for a suitable pair (p, q), we set L2p,q (S, ϕ) ≡ L2p,q (S, E, h), and for a suitable ω, L2p,q (S, ω, ϕ) ≡ L2p,q (S, E, ω, h) (cf. Definition 2.6.2). The analogous notation is used for L21 , and for ·, ·L2 and  · L2 (cf. Definition 2.6.1).

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Remarks 1. In Definitions 3.6.2 and 3.6.3, we could allow the Hermitian metric h to be only measurable (in an appropriate sense) and defined only on the given measurable set S, but this is not necessary for our purposes. 2. If ω is a positive measurable (1, 1)-form defined on a measurable set S ⊂ X, α ∈ L2p,q (S, E, ω, h) with p + q = 0 or 2 or α ∈ L2p,q (S, E, h) with p + q = 1, β ∈ L2r,s (S, E, ω, h) with r + s = 0 or 2 or β ∈ L2r,s (S, E, h) with r + s = 1, and (p, q) = (r, s), then we set α, βL2 = 0. The proof of the following proposition, which is left to the reader (see Exercise 3.6.1) is similar to that of Proposition 2.6.3: Proposition 3.6.4 Let S be a measurable subset of X. (a) The pair (L21 (S, E, h), ·, ·h ), and the pair (L2p,q (S, E, h), ·, ·h ) for (p, q) = (1, 0) or (0, 1), are Hilbert spaces (where the inner product on any two equivalence classes is given by the pairing of any representatives). Moreover, we have the Hilbert space orthogonal decomposition L21 (S, E, h) = L21,0 (S, E, h) ⊕ L20,1 (S, E, h). (b) For ω a continuous positive differential form of type (1, 1) that is defined on S and for (p, q) = (0, 0) or (1, 1), (L2p,q (S, E, ω, h), ·, ·ω,h ) is a Hilbert space (where the inner product on any two equivalence classes is given by the pairing of any representatives). Moreover, in each of the above spaces, any sequence converging to an element α admits a subsequence that converges to α pointwise almost everywhere in S. Exercises for Sect. 3.6 3.6.1 Prove Proposition 3.6.4. 3.6.2 Prove a version of Theorem 2.6.4 for sections of a Hermitian holomorphic line bundle. In other words, prove that if X is a Riemann surface, ω is a Kähler form on X, and (E, h) is a Hermitian holomorphic line bundle on X, then for every compact set K ⊂ X, there is a constant C = C(X, K, ω, h) > 0 such that max |s|h ≤ CsL2 (X,ω,h) K

∀s ∈ (X, O(E)).

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

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135

suitable correction term to the exterior derivative given by a local trivialization. For a Hermitian holomorphic line bundle, there is a canonical choice. Theorem 3.7.1 There exists a unique operator D = DE = Dh = D(E,h) with the following properties: (i) If α is a C 1 r-form with values in E on an open set  ⊂ X, then Dα is a C 0 (r + 1)-form with values in E on ; (ii) If α is a C 1 E-valued differential form on an open set  ⊂ X and s is any nonvanishing holomorphic section of E on an open set U ⊂ , then on U , 2 2 Dα = d(α/s) · s + |s|−2 h ∂(|s|h ) ∧ α = d(α/s) · s + (∂ log |s|h ) ∧ α

(that is, for ϕ ≡ − log |s|2h , Dh α = [Dϕ (α/s)] · s); (iii) If ρ is a C 1 p -form and α is a C 1 E-valued r-form on an open set  ⊂ X, then D(ρ ∧ α) = dρ ∧ α + (−1)p ρ ∧ Dα; (iv) If α and β are C 1 E-valued differential forms on an open set  ⊂ X with α of degree r, then d{α, β}h = {Dα, β}h + (−1)r {α, Dβ}h . Proof Let α and β be C 1 E-valued differential forms on an open set  ⊂ X, let r ≡ deg α, let s be a nonvanishing holomorphic section of E on an open set U ⊂ , and let αs ≡ α/s and βs ≡ β/s on U . We first show that D is well defined by (ii). If t is another nonvanishing holo¯ = 0, morphic section on U , then setting αt ≡ α/t and f ≡ t/s, we get, since ∂f 2 (dαt ) · t + |t|−2 h ∂(|t|h ) ∧ α 2 2 = d(f −1 αs ) · f s + |f |−2 |s|−2 h ∂(|f | |s|h ) ∧ α 2 = −f −2 df ∧ (αs ) · f s + d(αs ) · s + |f |−2 f¯∂f ∧ α + |s|−2 h ∂(|s|h ) ∧ α 2 = (dαs ) · s + |s|−2 h ∂(|s|h ) ∧ α.

Thus Dα is well defined and (i) follows easily. If ρ is a C 1 p-form on , then 2 D(ρ ∧ α) = d(ρ ∧ αs ) · s + |s|−2 h ∂(|s|h ) ∧ ρ ∧ α 2 = dρ ∧ α + (−1)p ρ ∧ (dαs ) · s + (−1)p ρ ∧ |s|−2 h ∂(|s|h ) ∧ α

= dρ ∧ α + (−1)p ρ ∧ Dα on U . Thus (iii) holds. For the proof of (iv), observe that d{α, β}h = d[αs ∧ βs · |s|2h ] = (dαs ) ∧ βs · |s|2h + (−1)r αs ∧ dβs · |s|2h + (d|s|2h ) ∧ αs ∧ βs

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

136

= (dαs ) ∧ βs · |s|2h + (−1)r αs ∧ dβs · |s|2h + (∂|s|2h ) ∧ αs ∧ βs + (−1)r αs ∧ (∂|s|2h ) ∧ βs 2 2 = [(dαs ) + |s|−2 h (∂|s|h ) ∧ αs ] ∧ βs · |s|h 2 2 + (−1)r αs ∧ [dβs + |s|−2 h (∂|s|h ) ∧ βs ] · |s|h

= [(Dα)/s] ∧ βs · |s|2h + (−1)r αs ∧ [(Dβ)/s] · |s|2h = {Dα, β}h + (−1)r {α, Dβ}h 

on U . Thus (iv) is proved.

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

  D  = DE = Dh = D(E,h) ,

 and ∂¯ = D  = DE .

For E = 1X and ϕ the weight function representing h (i.e., h = e−ϕ h0 , where h0 is the standard Hermitian metric), we set Dϕ ≡ Dh and Dϕ ≡ Dh (cf. Definition 2.7.2). Let α be a C ∞ differential form with values in E, let s be a nonvanishing local holomorphic section of E, let αs = α/s, and let ϕ = − log |s|2h . Then we have D  α = ∂αs · s + eϕ ∂(e−ϕ ) ∧ α = eϕ ∂[e−ϕ αs ] · s = [Dϕ αs ] · s and ¯ = [∂α ¯ s ] · s. D  α = ∂α Consequently, (D  )2 = (D  )2 = 0 and D 2 = D  D  + D  D  . Locally, since ∂ ∂¯ + ¯ = 0, we have ∂∂ ¯ s ) · s + eϕ ∂(e−ϕ ) ∧ ∂α ¯ s ·s D 2 α = (∂ ∂α ¯ ϕ ∂(e−ϕ ) ∧ αs ) · s ¯ s ) · s + ∂(e + (∂∂α ¯ ϕ ∂(e−ϕ )) ∧ α = ∂ ∂ϕ ¯ ∧ α = h ∧ α, = ∂(e ¯ is a well-defined scalar-valued form of type (1, 1). where h = ∂ ∂ϕ Definition 3.7.3 (Cf. Definition 2.8.1) The curvature (or curvature form) of (E, h) is the scalar-valued (1, 1)-form , which we also denote by E , h , or (E,h) , ¯ log |s|2 ) for every nonvanishing local holomorphic secgiven locally by  = ∂ ∂(− h tion s of E. In other words, D 2 = D  D  + D  D  =  ∧ (·).

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137

¯ For E = 1X and ϕ the weight function representing h, we set ϕ ≡ h = i∂ ∂ϕ (cf. Definition 2.8.1). Remarks 1. If g is a Kähler metric with Kähler form ω, and (U, z) is a local holomorphic coordinate neighborhood, then ω = G(i/2) dz ∧ d z¯ and |∂/∂z|2g = G/2 ¯ log G) = ω (see Defion U for some positive C ∞ function G. Hence g = ∂ ∂(− nition 2.12.1). 2. In a slight abuse of language, we say that (E, h) has nonnegative (positive, zero, nonpositive, negative) curvature if ih ≥ 0 (respectively, ih > 0, ih = 0, ih ≤ 0, ih < 0). A holomorphic line bundle E that admits a Hermitian metric of positive curvature is called positive. 3. For any nonvanishing local holomorphic section s of E and any Kähler form ω, we have ih = i− log |s|2 = [ω (− log |s|2h )] · ω h

(see Definitions 2.8.1 and 2.14.3). Combined with the results of Sect. 2.14, this observation will later allow us to construct Hermitian metrics of positive curvature (see Sects. 3.11 and 4.2). 4. The dual Hermitian metric h∗ in E ∗ satisfies ih∗ = −ih , and if (E  , h ) is a second Hermitian holomorphic line bundle on X, then ih⊗h = ih + ih . Example 3.7.4 Recall that the hyperplane bundle E → P1 is the holomorphic line bundle E = [D] for the divisor D with D(0) = 1 and D(q) = 0 for q ∈ P1 \ {0} (see Examples 3.1.5, 3.3.4, and 3.5.2). If s is the holomorphic section of E with div(s) = D considered in Example 3.1.5, and h is the Hermitian metric in E with |s|2h = |z|2 /(1 + |z|2 ) (|s∞ |2h = 1) considered in Example 3.5.2, then dz ∧ d z¯ z¯ z · dz ∧ d z¯ dz ∧ d z¯ h = −∂ ∂¯ log |s|2h = − = . 1 + |z|2 (1 + |z|2 )2 (1 + |z|2 )2 In particular, ih > 0. In fact, 2ih is the chordal Kähler form considered in Example 2.12.3. The following will play a role analogous to that of Proposition 2.8.2 in the L2 ¯∂-method: ¯ Proposition 3.7.5 (Fundamental estimate) Let D  = Dh ,  = h , and D  = ∂. Then, for every t ∈ D(E)(X), we have    2  2 2 i|t|h ≥ i|t|2h . D tL2 (X,h) = D tL2 (X,h) + X

X

Remarks 1. In particular, if in the above, i ≥ 0, then D  t2L2 (X,h) = D  t2L2 (X,h) + t2L2 (X,i,h) ≥ t2L2 (X,i,h) .

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2. The corresponding equality for inner products of C ∞ compactly supported sections also holds (see Exercise 3.7.2). Proof of Proposition 3.7.5 Given t ∈ D(E)(X), we may choose a finite covering {Uν }m ν=1 of supp t by relatively compact open subsets of X such that for each ν, We may also choose there exists a nonvanishing holomorphic section sν of E on Uν .  such that supp η ⊂ U for each ν and ην ≡ 1 on supp t . C ∞ functions {ην }m ν ν ν=1 Thus, setting ϕ ≡ − log |sν |2h and fν ≡ t/sν ∈ C ∞ (Uν ) for each ν, and applying Proposition 2.8.2, we get    2   i{Dh t, Dh t}h = i{Dh t, Dh (ην t)}h Dh tL2 (X,h) = X

= =



Uν

Uν

=− =−

i{[Dϕ ν fν ] · sν , [Dϕ ν (ην fν )] · sν }h i[Dϕ ν fν ] ∧ Dϕ ν (ην fν ) · e−ϕν

 

¯ ν ) ∧ ∂(η ¯ ν fν ) · e−ϕν + i(∂f Uν

Uν



i{D  t, D  (ην t)}h +



=−

X

ν



i{D t, D t}h + X

= D  t2L2 (X,h) +







 X

X



Uν

Uν

iϕν fν ην fν e−ϕν

η¯ ν · ih · |t|2h

ih |t|2h

ih |t|2h .



Exercises for Sect. 3.7 3.7.1 Let (E, h) be a Hermitian holomorphic line bundle on a Riemann surface X. Prove that for every point p ∈ X, there exists a nonvanishing holomorphic section t of E on a neighborhood of p such that |tp |2h = 1 and (d|t|2h )p = 0. Conclude from this that for every C 1 E-valued differential form α on a neighborhood of p, we have (D(E,h) α)p = (d(α/t) · t)p . 3.7.2 Prove that in Proposition 3.7.5, the corresponding equality for inner products of C ∞ compactly supported sections also holds (cf. Proposition 2.8.2). That is, prove that if (E, h) is a Hermitian holomorphic line bundle on a Riemann ¯ and s, t ∈ D(E)(X), then surface X, D  = Dh ,  = h , D  = ∂,      h(s, t)i. D s, D tL2 (X,h) = D s, D tL2 (X,h) + X

3.7.3 Prove that if (E, h) is a Hermitian holomorphic line bundle on a compact Riemann surface X with negative curvature (i.e., −ih > 0), then (X, O(E)) = 0.

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3.8 The Distributional ∂¯ Operator in a Holomorphic Line Bundle Throughout this section, X denotes a complex 1-manifold, (E, h) denotes a Hermitian holomorphic line bundle on X, and D = D  + D  = D  + ∂¯ denotes the associated canonical connection. The following is a natural definition for the distributional ∂¯ operator in E: Definition 3.8.1 Let α and β be locally integrable differential forms on X with val α = ∂¯ ues in E. We write Ddistr distr α = β if for every nonvanishing local holomorphic section s of E, we have ∂¯distr (α/s) = (β/s) (see Definition 2.7.1). The following proposition gives an equivalent, and more intrinsic, version (cf. Proposition 2.7.3): Proposition 3.8.2 Let α and β be locally integrable differential forms with values in E on an open set  ⊂ X. (a) If α is of type (1, 0) and β is of type (1, 1), then ∂¯distr α = β if and only if    {α, D t}h = {β, t}h ∀t ∈ D(E)(). 



(b) If α is of type (0, 0) and β is of type (0, 1), then ∂¯distr α = β if and only if   − {α, D  γ }h = {β, γ }h ∀γ ∈ D0,1 (E)(). 



(c) If α is of type (p, q) with p ≥ 2 or q ≥ 1, then ∂¯distr α = 0. Remark The proof will show that if the condition in Definition 3.8.1 holds for the forms α/s and β/s for some nonvanishing holomorphic section s on a neighborhood of each point, then it holds for every choice of s. Proof of Proposition 3.8.2 Suppose α is of type (1, 0) and β is of type (1, 1). If s is a nonvanishing holomorphic section of E on an open set U ⊂ X, αs = α/s, βs = β/s, and ϕ ≡ − log |s|2h , then given a C ∞ section t of E with compact support in U , we get, for f ≡ t/s ∈ D(U ) (which we may view as a function in D()),   {β, t}h = βs · f¯ · e−ϕ 

and

 

{α, Dh t}h

 = U

U

{α, (Dϕ f ) · s}h

 = U

αs ∧ Dϕ f · e−ϕ .

This observation and Proposition 2.7.3 together imply that if the above left-hand sides are always equal, then ∂¯distr α = β. Conversely, if ∂¯distr α = β and t ∈ D(E)(),

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 then, choosing finitely many C ∞ functions {ην }m ην ≡ 1 on ν=1 on  such that supp t and such that for each ν, the support of ην is contained in some neighborhood Uν on which there exists a nonvanishing holomorphic section of E, we get, by the above,       {α, Dh t}h = {α, Dh (ην t)}h = {β, ην t}h = {β, t}h . 

ν

Uν

ν

Uν



Thus part (a) is proved. Part (c) is obvious, and the proof of part (b) is left to the reader (see Exercise 3.8.1).  Theorem 2.7.4 immediately gives the following in this context: Theorem 3.8.3 If p ∈ {0, 1}, α is a locally integrable form of type (p, 0) with values in E, and ∂¯distr α = β for an E-valued form β of class C k for some k ∈ Z>0 ∪ {∞}, then α is also of class C k . In particular, if ∂¯distr α = 0, then α is a holomorphic p-form with values in E. It is sometimes convenient to consider L2 versions of Dolbeault cohomology (cf. Definition 3.4.3). Definition 3.8.4 Let ω be a Kähler form on X; let Z 1 = L20,1 (X, E, h); and let B 1 ⊂ Z 1 be the subspace consisting of elements of the form ∂¯distr α, where α ∈ L20,0 (X, E, ω, h) is an L2 section for which ∂¯distr α exists and is in L20,1 (X, E, h). The first L2 Dolbeault cohomology of (X, E, ω, h) is then the quotient vector space 1 1 1 HDol,L 2 (X, E, ω, h) ≡ Z /B .

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

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

¯ for Line-Bundle-Valued Forms 3.9 The L2 ∂-Method of Type (1, 0) The following is a direct generalization of Theorem 2.9.1: Theorem 3.9.1 Let X be a Riemann surface, let (E, h) be a Hermitian holomorphic line bundle on X with i = ih ≥ 0, and let Z = {x ∈ X | x = 0}.

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Then, for every measurable E-valued (1, 1)-form β on X with β = 0 a.e. in Z and βX\Z ∈ L21,1 (X \ Z, E, i, h) (in particular, β is in L2loc on X), there exists a form α ∈ L21,0 (X, E, h) such that  α = ∂¯distr α = β Ddistr

and

αL2 (X,E,h) ≤ βL2 (X\Z,E,i,h) .

In particular, if β is of class C k for some k ∈ Z>0 ∪ {∞}, then α is also of class C k ¯ = β. and ∂α Remark The form β is in L2loc because β vanishes on Z, and hence for any compact set K in any local holomorphic coordinate neighborhood (U, z), and for any nonvanishing holomorphic section s of E on U , we have β/sL2

1,1 (K,(i/2) dz∧d z¯ )

where

 A = sup K

≤ AβL2

1,1 (K\Z,i,h)

i |s|−2 (i/2) dz ∧ d z¯ h

< ∞,

1/2 < ∞.

It is often more convenient to apply Theorem 3.9.1 in one of the following forms (cf. Corollary 2.9.2 and Corollary 2.9.3), the proofs of which are left to the reader (see Exercises 3.9.1 and 3.9.2): Corollary 3.9.2 Suppose that X is a Riemann surface, (E, h) is a Hermitian holomorphic line bundle on X, ω is a Kähler form on X, ρ is a nonnegative measurable function on X with ih ≥ ρω, and Z = {x ∈ X | ρ(x) = 0}. Then, for every measurable E-valued (1, 1)-form β on X with β = 0 a.e. in Z and βX\Z ∈ L21,1 (X \ Z, E, ρω, h) = L21,1 (X \ Z, E, ω, e− log ρ h) (in particular, β is in L2loc on X), there exists a form α ∈ L21,0 (X, E, h) such that  α = ∂¯distr α = β Ddistr

and

αL2 (X,h) ≤ βL2 (X\Z,ρω,h) .

In particular, if β is of class C k for some k ∈ Z>0 ∪ {∞}, then α is also of class C k ¯ = β. and ∂α Corollary 3.9.3 Suppose that X is a Riemann surface, (E, h) is a Hermitian holomorphic line bundle on X, ω is a Kähler form on X, and C is a positive constant with ih ≥ C 2 ω. Then, for every form β ∈ L21,1 (X, E, ω, h), there exists a form α ∈ L21,0 (X, E, h) such that  Ddistr α = ∂¯distr α = β

and

αL2 (X,h) ≤ C −1 βL2 (X,ω,h) .

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

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Proof of Theorem 3.9.1 Let N ≡ βL2 (X\Z,i,h) = β/(i)L2 (X\Z,i,h) , and let V ≡ D(E)(X). For each section t ∈ V, the Schwarz inequality and the fundamental estimate (Proposition 3.7.5) give             β  = t, β/(i) 2  {t, β}h  =   · i h t, L (X\Z,i,h)     i X X\Z ≤ tL2 (X\Z,i,h) · β/(i)L2 (X\Z,i,h) ≤ N · tL2 (X,i,h) ≤ N · D  tL2 (X,h) .  It follows that the mapping ϒ : [D  t] → −i X {t, β}h is a well-defined bounded complex linear functional on the subspace D  V of L21,0 (X, E, h). For by the above inequality, ϒ is well defined, and for each t ∈ V, we have |ϒ[D  t]| ≤ N · D  tL2 (X,h) . In particular, ϒ ≤ N . By the Hahn–Banach theorem (The on L2 (X, E, h) such orem 7.5.11), there exists a bounded linear functional ϒ 1,0   = ϒ. Therefore, by Theorem 7.5.10, there exists a  D  V = ϒ and ϒ that ϒ   ≤ N and ϒ  (·) = (unique) element α ∈ L21,0 (X, E, h) such that αL2 (X,h) = ϒ ·, αL2 (X,h) . Moreover, for each t ∈ V, we have 



i{α, D t}h

= ϒ(D  t) =

X



 X

(−i){t, β}h =

X

i{β, t}h .

 α = β, as required. Finally, the regularity Therefore, by Proposition 3.8.2, Ddistr statement at the end follows from Theorem 3.8.3. 

Exercises for Sect. 3.9 3.9.1 Prove Corollary 3.9.2. 3.9.2 Prove Corollary 3.9.3.

¯ 3.10 The L2 ∂-Method for Line-Bundle-Valued Forms of Type (0, 0) Throughout this section, X denotes a Riemann surface (although all of the results to be stated also hold for X a complex 1-manifold), g denotes a Kähler metric on X with Kähler form ω, and (E, h) denotes a Hermitian holomorphic line bundle on X. The goal of this section is a proof that with appropriate curvature assumptions, one ¯ = β for β a differmay solve the inhomogeneous Cauchy–Riemann equation ∂α ential form of type (0, 1) with values in E (cf. Sect. 2.12). We first consider the following: Proposition 3.10.1 For q = 0, 1, let q : (1,q) T ∗ X ⊗ E→ (0,q) T ∗ X ⊗ KX ⊗ E be the map given by q : α = θ ∧ γ · s = (−1)q γ ∧ θ · s → (−1)q γ · (θ ⊗ s)

3.10

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for each x ∈ X, γ ∈ (0,q) Tx∗ X, θ ∈ (KX )x , and s ∈ Ex . Then we have the following: (a) For each q = 0, 1, q is a well-defined mapping for which the restriction

(1,q) Tx∗ X ⊗ Ex → (0,q) Tx∗ X ⊗ (KX ⊗ E)x is a linear isomorphism for each point x ∈ X (in fact, 0 is equal to the identity). (b) If α is an E-valued differential form of type (1, q), then α is continuous (C k with p k ∈ Z≥0 ∪ {∞}, measurable, in Lloc with p ∈ [1, ∞]) if and only if the KX ⊗ Evalued differential form q (α) of type (0, q) is continuous (respectively, C k , p measurable, in Lloc ). For q = 0, α is a local holomorphic 1-form with values in E if and only if 0 (α) is a local holomorphic section of KX ⊗ E. (c) For each point x ∈ X and all α, β ∈ (1,q) Tx∗ X ⊗ Ex , we have {0 (α), 0 (β)}g∗ ⊗h · ω = i{α, β}h for q = 0,   α β · ω for q = 1. −i{1 (α), 1 (β)}g ∗ ⊗h = , ω ω h (d) For each q = 0, 1, q preserves L2 inner products and L2 norms. More precisely, suppose α and β are measurable E-valued differential forms of type (1, q) on a measurable set S. Then we have 0 (α)L2 (S,ω,g∗ ⊗h) = αL2 (S,h) 1 (α)L2 (S,g ∗ ⊗h) = αL2 (S,ω,h)

for q = 0, for q = 1.

Moreover, for q = 0, 0 (α), 0 (β)L2 (S,ω,g∗ ⊗h) = α, βL2 (S,h) , where both of the above terms are defined if either one is defined. For q = 1, 1 (α), 1 (β)L2 (S,g∗ ⊗h) = α, βL2 (S,ω,h) , where again, both of the above terms are defined if either one is defined. (e) For any locally integrable E-valued differential form α of type (1, 0) on X, ∂¯ distr α exists if and only if ∂¯distr [0 (α)] exists. Moreover, if these objects do exist, then ∂¯distr [0 (α)] = 1 [∂¯distr α]. Proof In terms of a local holomorphic coordinate z and a nonvanishing local holomorphic section s of E, the maps are given by 0 : a dz · s → a dz ⊗ s, 1 : a dz ∧ d z¯ · s = −a d z¯ ∧ dz · s → −a d z¯ · (dz ⊗ s); and parts (a) and (b) follow. Fixing a point r ∈ X and choosing the local holomorphic coordinate z so that |dz|2g ∗ = 2 at r, we get ωr = (i/2)(dz ∧ d z¯ )r . For

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α = a dz · s and β = b dz · s at r, we have {0 (α), 0 (β)}g∗ ⊗h · ω = a b¯ · |dz ⊗ s|2g∗ ⊗h · ω = a b¯ · |dz|2g ∗ · |s|2h · ω = a b¯ · |s|2h · i dz ∧ d z¯ = i{α, β}h . For α = a dz ∧ d z¯ · s and β = b dz ∧ d z¯ · s at r, we have −i{1 (α), 1 (β)}g∗ ⊗h = −i{−a d z¯ · (dz ⊗ s), −b d z¯ · (dz ⊗ s)}g∗ ⊗h ¯ z¯ ∧ dz) · |dz ⊗ s|2g ∗ ⊗h = −ia b(d ¯ = ia b(dz ∧ d z¯ ) · 2|s|2h     β α = (i/2) · (i/2) · 4ω · |s|2h ω·s ω·s   α β = · ω. , ω ω h Parts (c) and (d) now follow. Finally, for the proof of (e), we may assume without loss of generality that X is an open subset of C. If α = a dz · s is a locally integrable (1, 0)-form with values in E, then, according to Definition 2.7.1 and Definition 3.8.1, the existence of ∂¯distr 0 (α) and the existence of ∂¯distr α are each equivalent to the existence of   ∂a ∈ L1loc (X). b≡ ∂ z¯ distr Moreover, if this function does exist, then 1 (∂¯distr α) = 1 (−b dz ∧ d z¯ · s) = b d z¯ · (dz ⊗ s) = (∂¯distr a) · (dz ⊗ s) = ∂¯ distr [0 (α)].



Remarks 1. 1 is an example of a C ∞ line bundle isomorphism. 2. By the above proposition, the operator D = ∂¯ and its distributional version are preserved (in the appropriate sense) under the identifications given by 0 and 1 . However, the operators D  , D, and  (associated to the Hermitian metrics in the holomorphic line bundles E and KX ⊗ E) are not preserved under these identifications; in fact, even the resulting types do not correspond. For example, in the notation of Proposition 3.10.1, for a C ∞ (1, 0)-form α ≡ a dz ∈ E 1,0 (1C )(C) with values in the trivial line bundle over C, we have ∂α = D  α = 0 (as a (2, 0)-form), 1,0 (K )(C). but D  0 (α) = ∂a C ∂z dz · dz ∈ E 3. The above proposition also gives the mapping ( (1,q) T ∗ X) ⊗ KX∗ ⊗ E→( (0,q) T ∗ X) ⊗ KX ⊗ KX∗ ⊗ E = ( (0,q) T ∗ X) ⊗ E for q = 0, 1.

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We have the following facts, which are equivalent to, respectively, Theorem 3.9.1, Corollary 3.9.2, and Corollary 3.9.3: Corollary 3.10.2 Assume that i ≡ ig + ih ≥ 0 on X, and let ρ ≡ i/ω and Z ≡ {x ∈ X | ρ(x) = 0} = {x ∈ X | x = 0}. Then, for every measurable E-valued (0, 1)-form β on X with β = 0 a.e. in Z and βX\Z ∈ L20,1 (X \ Z, E, e− log ρ h), there exists a section α ∈ L20,0 (X, E, ω, h) such that  α = ∂¯distr α = β Ddistr

and

αL2 (X,E,ω,h) ≤ βL2 (X\Z,E,e− log ρ h) .

In particular, if β is of class C k for some k ∈ Z>0 ∪ {∞}, then α is also of class C k ¯ = β. and ∂α Corollary 3.10.3 Suppose that ρ is a nonnegative measurable function on X with ig + ih ≥ ρω, and Z = {x ∈ X | ρ(x) = 0}. Then, for every measurable E-valued (0, 1)-form β on X with β = 0 a.e. in Z and βX\Z ∈ L20,1 (X \ Z, E, e− log ρ h), there exists a section α ∈ L20,0 (X, E, ω, h) such that  α = ∂¯distr α = β Ddistr

and

αL2 (X,E,ω,h) ≤ βL2 (X\Z,E,e− log ρ h) .

In particular, if β is of class C k for some k ∈ Z>0 ∪ {∞}, then α is also of class C k ¯ = β. and ∂α Corollary 3.10.4 Suppose that C is a positive constant with ig + ih ≥ C 2 ω on X. Then, for every β ∈ L20,1 (X, E, h), there exists a section α ∈ L20,0 (X, E, ω, h) such that  Ddistr α = ∂¯distr α = β

and

αL2 (X,ω,h) ≤ C −1 βL2 (X,h) .

In particular, if β is of class C k for some k ∈ Z>0 ∪ {∞}, then α is also of class C k ¯ = β. and ∂α Corollary 3.10.4 immediately gives the following example of a vanishing theorem (see Definitions 3.4.3 and 3.8.4): Corollary 3.10.5 Let X be a Riemann surface and let E be a holomorphic line bundle on X. (a) If g is a Kähler metric on X with Kähler form ω and h is a Hermitian metric in E satisfying ih ≥ C 2 ω on X for some positive constant C, then 1 ∗ 1 ∗ HDol,L 2 (X, KX ⊗ E, ω, g ⊗ h) = HDol,L2 ∩E (X, KX ⊗ E, ω, g ⊗ h) = 0.

Equivalently, if ig + ih ≥ C 2 ω, then 1 1 HDol,L 2 (X, E, ω, h) = HDol,L2 ∩E (X, E, ω, h) = 0.

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(b) Assume that X is compact. If E is positive (i.e., there exists a Hermitian metric h 1 (X, K ⊗ E) = 0. Equivalently, if there in E with ih > 0 on X), then HDol X exist a Kähler metric g on X and a Hermitian metric h in E with ig +ih > 0, 1 (X, E) = 0. Consequently, if E is positive, then, given a holomorphic then HDol 1 (X, F ⊗ E r ) = 0 for all r  0. line bundle F on X, we have HDol The proof of Corollary 3.10.2 appears below, while the proofs of Corollary 3.10.3, Corollary 3.10.4, and Corollary 3.10.5 are left to the reader (see Exercises 3.10.1–3.10.3). Proof of Corollary 3.10.2 Let βˆ be the form of type (1, 1) with values in KX∗ ⊗ E corresponding to β (see the third remark following Proposition 3.10.1); that is, if, in terms of a local holomorphic coordinate z and a nonvanishing local holomorphic section s of E, we have β = b d z¯ · s, then       ∂ ∂ ˆ β = b d z¯ ∧ dz · ⊗ s = −b dz ∧ d z¯ · ⊗s . ∂z ∂z We have ˆ L2 (X\Z,ω,e− log ρ g⊗h) = β ˆ L2 (X\Z,ω,g⊗(e− log ρ h)) ˆ L2 (X\Z,ρω,g⊗h) = β β = βL2 (X\Z,e− log ρ h) < ∞ and i(KX∗ ⊗E,g⊗h) = i(KX∗ ,g) + ih = i = ρω. Therefore, by Theorem 3.9.1, there exists a form αˆ of type (1, 0) with values in ˆ is of class C k for some k ∈ KX∗ ⊗ E (with αˆ of class C k if β, and therefore β,  ¯ ˆ ˆ L2 (X\Z,ρω,g⊗h) . ˆ L2 (X,g⊗h) ≤ β Z>0 ∪ {∞}) such that Ddistr αˆ = ∂distr αˆ = β and α The identification given by (the third remark following) Proposition 3.10.1 gives the desired solution α.  Exercises for Sect. 3.10 3.10.1 Prove Corollary 3.10.3. 3.10.2 Prove Corollary 3.10.4. 3.10.3 Prove Corollary 3.10.5 (for the proof of the last statement in part (b), the reader may wish to apply Proposition 3.11.1 from Sect. 3.11).

3.11 Positive Curvature on an Open Riemann Surface According to Radó’s theorem (Theorem 2.11.1), every Riemann surface is second countable. From second countability, one gets the following:

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Proposition 3.11.1 Every holomorphic line bundle  : E → X on a Riemann surface X admits a Hermitian metric. Proof Since X is second countable, there exist a locally finite collection of local holomorphic trivializations {(Uν , (, ν ))}ν∈N and a C ∞ partition of unity {λν } with supp λν ⊂ Uν (and λν ≥ 0) for each ν ∈ N (see Sect. 9.3). The pairing  λν (p)ν (u)ν (v) ∀p ∈ X, u, v ∈ Ep h(u, v) = hp (u, v) ≡ ν∈N

is then a Hermitian metric in E. The verification is left to the reader (see Exercise 3.11.1).  We now (begin to) address the natural question as to when a holomorphic line bundle on a Riemann surface admits a positive-curvature Hermitian metric. For a holomorphic line bundle on an open Riemann surface, the facts proved in Sect. 2.14 provide a Hermitian metric with arbitrarily large positive curvature. In other words, we have the following: Theorem 3.11.2 Let X be an open Riemann surface, let E be a holomorphic line bundle on X, and let θ be a continuous real (1, 1)-form on X. Then E admits a Hermitian metric h with ih ≥ θ . In particular, if g is a Kähler metric on X with associated Kähler form ω, then E admits a Hermitian metric h with ih ≥ ω

and

ig + ih ≥ ω.

Proof Let us fix a Hermitian metric k in E and a Kähler metric g with Kähler form ω on X (provided by Proposition 3.11.1). Given a continuous real (1, 1)-form θ on X, Theorem 2.14.4 provides a C ∞ (exhaustion) function ϕ such that        θ   ik   ig    .    + ω ϕ ≥ 1 +   +  ω ω   ω  Thus the Hermitian metric h ≡ e−ϕ k satisfies ih = [ω ϕ] · ω + ik ≥ θ , ih ≥ ω,  and ig + ih ≥ ω. Remarks 1. When second countability of a particular Riemann surface X is evident (for example, when X is an open subset of a compact Riemann surface), one may construct Hermitian metrics and, for X an open Riemann surface, C ∞ strictly subharmonic exhaustion functions (as in Sect. 2.14) and positive-curvature Hermitian metrics (as above) without any reliance on Radó’s theorem, and hence with almost no reliance on Sects. 2.6, 2.7, 2.9–2.12, 3.6, and 3.8–3.10. 2. A holomorphic line bundle E on a compact Riemann surface need not admit a positive-curvature Hermitian metric. In Chap. 4, it will be shown that in fact, a holomorphic line bundle E on a compact Riemann surface admits a Hermitian metric of positive curvature if and only if E is the line bundle associated to a divisor of positive degree (Theorem 4.3.1).

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The above considerations, together with the facts proved in Sects. 3.9 and 3.10, give the following (cf. Theorem 2.14.10): Theorem 3.11.3 Let X be an open Riemann surface, let E be a holomorphic line bundle on X, let p ∈ {0, 1}, and let β be a locally square-integrable differential form of type (p, 1) with values in E on X. Then there exists a locally square-integrable differential form α of type (p, 0) with values in E such that ∂¯distr α = β. If β is of ¯ = β. class C k for some k ∈ Z>0 ∪ {∞}, then α is also of class C k and ∂α Proof We may fix a Kähler metric g with Kähler form ω, and a Hermitian metric k in E. Applying Theorem 2.14.4, with the compact set given by K = ∅ and the function ρ chosen (with the help of Lemma 9.7.13) so large that β ∈ L20,1 (X, E, e−ρ k) or L21,1 (X, E, ω, e−ρ k) and ρ > 1 + |ik /ω| + |ig /ω|, we get a positive C ∞ strictly subharmonic (exhaustion) function ϕ on X such that the Hermitian metric h = e−ϕ k satisfies ig + ih ≥ ω ih ≥ ω

and βL2 (X,E,h) < ∞ if p = 0, and βL2 (X,E,ω,h) < ∞ if p = 1.

Corollary 3.10.4 and Corollary 3.9.3 now provide the desired E-valued differential form α.  In particular, we have the following vanishing theorem: Corollary 3.11.4 If E is a holomorphic line bundle on an open Riemann surface X, 1 (X, E) = 0. then HDol The construction of a holomorphic section of a holomorphic line bundle with prescribed values on a given discrete set (or prescribed Laurent series about each of the ¯ For an points in the discrete set) is one of the main applications of the L2 ∂-method. open Riemann surface, the above considerations give the following generalization of the Mittag-Leffler theorem (Theorem 2.15.1): Theorem 3.11.5 (Mittag-Leffler theorem for a holomorphic line bundle) Let X be an open Riemann surface, let E be a holomorphic line bundle on X, let P be a discrete subset of X, and for each point p ∈ P , let Up be a neighborhood of p with Up ∩ P = {p}, let tp be a holomorphic section of E on Up \ {p}, and let mp be a positive integer. Then there exists a holomorphic section s of E on X \ P such that for each point p ∈ P , s − tp extends to a holomorphic section on Up that either vanishes on a neighborhood of p or has a zero of order at least mp at p (in other words, if z is a local holomorphic coordinate and  u is a nonvanishing holomorphic section of E on a neighborhood of p, and s/u = n∈Z an (z − z(p))n and tp /u = n∈Z bn (z − z(p))n are the corresponding Laurent series expansions centered at p, then amp −n = bmp −n for n = 1, 2, 3, . . . ).

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Remarks 1. Recall that we also denote the value of a section v of a line bundle E at a point p by vp . The section tp in the above theorem should not be confused with such a value. 2. It follows that one may actually choose the above section s ∈ (X \ P , O(E)) so that for each point p ∈ P , s − tp extends to a holomorphic section on Up with a zero of order equal to mp at p (see Exercise 3.11.3). 3. In Exercise 3.11.2, the reader is asked to give a proof that is a direct generalization of the proof of Theorem 2.15.1. The proof provided below, which uses the language of divisors, is more efficient (in particular, the proof is, in part, a demonstration of the quiet power of divisors). It should also be noted, however, that according to the Weierstrass theorem (Theorem 3.12.1), every holomorphic line bundle on an open Riemann surface is holomorphically trivial; so it turns out that the MittagLeffler theorem (Theorem 2.15.1) and its generalization for sections of a line bundle (Theorem 3.11.5) are actually equivalent. The proof of the following is left to the reader (see Exercise 3.11.4): Corollary 3.11.6 (Cf. Corollary 2.15.2) Let X be an open Riemann surface, let E be a holomorphic line bundle on X, and let P be a discrete subset of X. (a) If tp is a meromorphic section of E on a neighborhood of p and mp ∈ Z>0 for each point p ∈ P , then there exists a section s ∈ (X, M(E)) such that s is holomorphic on X \ P and ordp (s − tp ) ≥ mp for every p ∈ P . (b) If tp is a holomorphic section of E on a neighborhood of p and mp ∈ Z>0 for each point p ∈ P , then there exists a holomorphic section s ∈ (X, O(E)) with ordp (s − tp ) ≥ mp for every p ∈ P . (c) If ζp ∈ Ep for each point p ∈ P , then there exists a section s ∈ (X, O(E)) with s(p) = ζp for every p ∈ P . Of course, for a given holomorphic line bundle on a compact Riemann surface X, a holomorphic section with prescribed values on a given discrete set need not exist (for example, (X, O(1X )) = O(X) = C). Much of Chap. 4 is devoted to determining when (and how many) such sections exist. Proof of Theorem 3.11.5 We may assume without loss of generality that P = ∅;  we may let D be the (nontrivial) effective divisor given by D ≡ p∈P mp · p (i.e., D(p) = mp for each point p ∈ P and D vanishes on X \ P ); we may fix a holomorphic section u ∈ (X, O([D])) with div(u) = D; and finally, we may fix a C ∞ section γ of the holomorphic line bundle E ⊗ [−D] on the open set Y ≡ X \ P such that γ = tp /u near each point p ∈ P . ¯ extends to a C ∞ E ⊗ [−D]-valued (0, 1)The E ⊗ [−D]-valued (0, 1)-form ∂γ form β on X that vanishes on a neighborhood of P . Theorem 3.11.3 provides a C ∞ ¯ = β. In particular, α is holomorphic near P , section α of E ⊗ [−D] on X with ∂α and the section s ≡ (γ − α) ⊗ u = γ ⊗ u − α ⊗ u of E ⊗ [−D] ⊗ [D] = E is holomorphic on Y . Finally, near each point p ∈ P , we have s − tp = −α ⊗ u, and

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therefore, since α ⊗ u is holomorphic near p with ordp (α ⊗ u) ≥ ordp u = mp , s has the required properties.  Corollary 3.11.7 (Cf. Theorem 4.2.3) Any holomorphic line bundle E on an open Riemann surface X is equal to the line bundle associated to some nontrivial effective divisor D on X; in other words, E admits a nontrivial (i.e., not everywhere zero) holomorphic section with at least one zero. Further generalizations of the Mittag-Leffler theorem, and also of the Runge approximation theorem, are considered in the exercises. Exercises for Sect. 3.11 3.11.1 Verify that the function h constructed the proof of Proposition 3.11.1 is a Hermitian metric. 3.11.2 Give a proof of Theorem 3.11.5 that is analogous to the proof of Theorem 2.15.1 given in Sect. 2.15 (and that does not use divisors). 3.11.3 Prove that in Theorem 3.11.5, one may actually choose the holomorphic section s on X \ P so that for each point p ∈ P , s − tp extends to a holomorphic section on Up with a zero of order equal to mp at p. 3.11.4 Prove Corollary 3.11.6 3.11.5 The goal of this exercise is a generalization of the Behnke–Stein theorem (Corollary 2.15.3). Let (E, h) be a Hermitian holomorphic line bundle on an open Riemann surface X. Prove that: (i) Holomorphic convexity. If S is any infinite discrete subset of X, then there exists a holomorphic section s of E on X such that |s|h is unbounded on S; (ii) Separation of points. If p, q ∈ X and p = q, then there exists a holomorphic section s of E on X such that s(p) = 0 and s(q) = 0; and (iii) Global sections give local coordinates. For each point p ∈ X, there exists a holomorphic section s of E on X such that sp = 0 and (d(s/t))p = 0 for each nonvanishing holomorphic section t of E on a neighborhood of p. 3.11.6 The goal of this exercise is a version of Theorem 3.11.5 that is analogous to the version of the Mittag-Leffler theorem considered in Exercise 2.15.6. Let X be an open Riemann surface, let E be a holomorphic line bundle on X, let P be a discrete subset of X, let {mp }p∈P be a collection of positive integers, let {Ui }i∈I be an open covering of X, and for each pair of indices i, j ∈ I , let tij ∈ (Ui ∩ Uj , O(E)) with ordp tij ≥ mp for each point p ∈ P ∩ Ui ∩ Uj . Assume that the family {tij } satisfies the (additive) cocycle relation tik = tij + tj k

on Ui ∩ Uj ∩ Uk

∀i, j, k ∈ I.

Prove that there exist sections {si }i∈I such that (i) For each index i ∈ I , si ∈ (Ui , O(E)) and ordp si ≥ mp for each point p ∈ P ∩ Ui ; and

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(ii) For each pair of indices i, j ∈ I , we have tij = sj − si on Ui ∩ Uj . Prove also that the above implies Theorem 3.11.5. Hint. Using a C ∞ partition of unity {λν } such that each point in P lies in supp λν for exactly one index ν and such that for each ν, supp λν ⊂ Ukν for some index kν ∈ I , one may form a C ∞ solution of the problem of the form ¯ i } agree on the overvi = ν λν · tkν i . In particular, the E-valued forms {∂v laps and therefore determine a well-defined E-valued (0, 1)-form β on X. 3.11.7 The goal of this exercise is a version of the Runge approximation theorem (Theorem 2.16.1) for sections of a holomorphic line bundle. Let X be an open Riemann surface, let (E, h) be a Hermitian holomorphic line bundle on X, and let K ⊂ X be a compact set satisfying hX (K) = K. Prove that for every holomorphic section t of E on a neighborhood of K in X and for every  > 0, there exists a section s ∈ (X, O(E)) such that |s − t|h <  on K. 3.11.8 The goal of this exercise is a version of Theorem 2.16.3 for sections of a holomorphic line bundle. Suppose K is a compact subset of a Riemann surface X, (E, h) is a Hermitian holomorphic line bundle on X, P is a finite subset of X \ K, Y = X \ P , hY (K) = K, t is a holomorphic section of E on a neighborhood of K in X, and  > 0. Prove that there exists a meromorphic section s of E on X such that s is holomorphic on Y , s has a pole at each point in P , and |s − t|h <  on K. 3.11.9 The goal of this exercise is a generalization of Exercise 2.16.7. Let X be an open Riemann surface, let (E, h) be a Hermitian holomorphic line bundle on X, let K ⊂ X be a compact subset with hX (K) = K, let P ⊂ X be a discrete subset with P ⊂ X \ K, let t0 be a holomorphic section of E on a neighborhood of K in X, and for each point p ∈ P , let tp be a holomorphic section of E on Up \ {p} for some neighborhood Up of p in X with Up ∩ P = {p}, and let mp be a positive integer. Prove that for every  > 0, there exists a holomorphic section s of E on X \ P such that |s − t0 |h <  on K and such that for each point p ∈ P , s − tp extends to a holomorphic section on Up that either vanishes on a neighborhood of p or has a zero of order at least mp at p.

3.12 The Weierstrass Theorem The main goal of this section is the following analogue, due to Florack [Fl], of the classical theorem of Weierstrass: Theorem 3.12.1 (Weierstrass theorem) Every holomorphic line bundle E on an open Riemann surface X is (holomorphically) trivial; that is, equivalently, every divisor on X has a solution. Recall that a solution of a divisor D on a complex 1-manifold X is a meromorphic function f such that f does not vanish identically on any open set and

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div(f ) = D (see Definition 3.3.1). To prove the theorem, we must show that E admits a nonvanishing holomorphic section. For this, the main step will be to show ¯ that E is C ∞ trivial; i.e., E admits a nonvanishing C ∞ section. The ∂-method will then provide a correction that yields a suitable holomorphic section. For a divisor D with [D] = E and a meromorphic section s of E with div(s) = D, E is C ∞ trivial if and only if there exists a C ∞ function ρ on X \ D −1 ((−∞, 0)) such that s is nonvanishing on X \ supp D and s/ρ extends to a nonvanishing C ∞ section of E on X (equivalently, ρ is nonvanishing on X \ supp D and for each point p ∈ supp D, there are a local holomorphic coordinate neighborhood (U, z) with z(p) = 0 and a nonvanishing C ∞ function ψ on U such that ρ = zD(p) · ψ on U \ {p}). Such a function ρ is called a weak solution of the divisor D. The construction of a weak solution very closely parallels that of a strictly subharmonic exhaustion function provided in Sect. 2.14. The first step is the following local construction: Lemma 3.12.2 Let (V , ,  ≡ (0; 1)) be a local holomorphic chart in a Riemann surface X. Then, for each pair of distinct points p, q ∈ V , there exists a weak solution ρ of the divisor D = p − q with ρ ≡ 1 on X \ V . Proof Given a pair of distinct points p, q ∈ V , setting ζ = (p) and ξ = (q), we may choose a holomorphic branch of the function   z−ζ z → log z−ξ on the complement  \ [ζ, ξ ] of the line segment [ζ, ξ ] from ζ to ξ (see Exercise 3.12.2). In other words, there exists a holomorphic function g on  \ [ζ, ξ ] such that z−ζ eg(z) = ∀z ∈  \ [ζ, ξ ]. z−ξ We may also choose R with |ζ |, |ξ | < R < 1 and a C ∞ function λ with compact support in  such that λ ≡ 1 on (0; R). The function ρ given by ρ ≡ 1 on X \ V and  exp(λ(z) · g(z)) if z ∈ (0; 1) \ (0; R), ρ(−1 (z)) ≡ z−ζ if z ∈ (0; R) \ {ξ }, z−ξ is then a weak solution of the divisor D = p − q on X.



Lemma 3.12.3 Let X be an open Riemann surface, let K ⊂ X be a compact set with hX (K) = K, and let p ∈ X \ K. Then there exists a weak solution of the divisor D = 1 · p on X with ρ ≡ 1 on K. Proof By Lemma 2.13.5, there exists a locally finite (in X) sequence of relatively compact open subsets {Vm }∞ m=1 of X \ K such that p ∈ V1 and such that for each m, Vm is biholomorphic to a disk and Vm ∩ Vm+1 = ∅. Hence we may choose a sequence of points {pm }∞ m=0 such that p0 = p and pm ∈ Vm ∩ Vm+1 for each m ≥ 1.

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153

By Lemma 3.12.3, for each m = 1, 2, 3, . . . , there exists a weak solution ρm of the divisor Dm = pm−1 − pm with ρm ≡ 1 on X \ Vm . The locally finite product ρm then yields a weak solution ρ of the divisor D = 1 · p with ρ ≡ 1 on K.  Lemma 3.12.4 Every divisor D on an open Riemann surface X has a weak solution. ∞ Proof By Lemma 2.13.4, ∞there exists a sequence of compact sets {Kν }ν=0 such that K0 = ∅ and X = ν=0 Kν , and such that for each ν = 0, 1, 2, . . . , we have ◦

Kν ⊂ K ν+1 and hX (Kν ) = Kν . For each point p ∈ supp D, there is a unique index ν(p) such that p ∈ Kν(p)+1 \ Kν(p) . Moreover, by Lemma 3.12.3, there exists a weak solution ρp of the divisor Dp = 1 · p on X such that ρp ≡ 1 on Kν(p) . Since the collection of points in supp D and the collection of sets {X \ Kν } are locally finite in X, the product ! D(p) ρp p∈supp D

is locally finite and yields a weak solution of the divisor D.



Proof of Theorem 3.12.1 By Lemma 3.12.4, E admits a nonvanishing C ∞ section v. ¯ The quotient β = (∂v)/v is then a C ∞ scalar-valued form of type (0, 1) (which, locally, may be thought of as ∂¯ log v). By Theorem 3.11.3 (or Theorem 2.14.10 ¯ = β. The or Corollary 3.11.4), there exists a C ∞ function α on X such that ∂α nonvanishing C ∞ section s ≡ e−α · v of E is then holomorphic because ¯ ¯ = −e−α · ∂v · v + e−α · ∂v ¯ = 0. ∂s v



Remarks 1. Since, according to the Weierstrass theorem, a holomorphic line bundle on an open Riemann surface is holomorphically trivial, one need only really consider scalar-valued differential forms (along with weights) as in Chap. 2 in this case. 2. Suitable higher-dimensional analogues of the Mittag-Leffler and Weierstrass theorems, due to Oka and Cartan, hold on Stein manifolds (see, for example, [Hö] or [KaK]). The problem of obtaining an analogue of the Mittag-Leffler theorem (the additive Cousin problem or the Cousin problem I) and the problem of obtaining an analogue of the Weierstrass theorem (the multiplicative Cousin problem or the Cousin problem II) played central roles in the development of several complex variables; and these analogues are fundamental tools in the subject. Exercises for Sect. 3.12 3.12.1 Let D be a divisor on a Riemann surface X, let E = [D], and let s be a meromorphic section of E with div(s) = D. Verify that E is C ∞ trivial if and only if there exists a C ∞ function ρ on X \ D −1 ((−∞, 0)) such that s

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is nonvanishing on X \ supp D and s/ρ extends to a nonvanishing C ∞ section of E on X. Verify also that a C ∞ function ρ on X \ D−1 ((−∞, 0)) has the above property if and only if ρ is nonvanishing on X \ supp D and for each point p ∈ supp D, there are a local holomorphic coordinate neighborhood (U, z) with z(p) = 0 and a nonvanishing C ∞ function ψ on U such that ρ = zD(p) · ψ on U \ {p}. 3.12.2 Verify that given a pair of distinct points ζ, ξ ∈ C, there exists a holomorphic branch of the function   z−ζ z → log z−ξ on the complement C \ [ζ, ξ ] of the line segment [ζ, ξ ] from ζ to ξ . In other words, there exists a holomorphic function g on C \ [ζ, ξ ] such that eg(z) =

z−ζ z−ξ

∀z ∈ C \ [ζ, ξ ]

(this fact was used in the proof of Lemma 3.12.2). 3.12.3 Let D be a divisor on a compact Riemann surface X. Prove that the associated holomorphic line bundle E = [D] is C ∞ trivial (i.e., E admits a nonvanishing C ∞ section) if and only if deg D = 0. 3.12.4 As is the case for the Mittag-Leffler theorem (see Exercises 2.15.6 and 3.11.6), the Weierstrass theorem may be stated in terms of cocycles (in higher dimensions, this is known as the solution of the multiplicative Cousin problem or the Cousin problem II). Let X be a Riemann surface. (a) Suppose  : E → X is a holomorphic line bundle and {fij = i / j }i,j ∈I are the transition functions associated to some holomorphic line bundle atlas {(Ui , i = (, i ))}i∈I for E. Prove that E is (holomorphically) trivial if and only if there exists a collection {fi }i∈I such that fi is a nonvanishing holomorphic function on Ui for each i ∈ I and fij = fj /fi for all i, j ∈ I . (b) Suppose {Ui }i∈I is an open covering of X and fij is a nonvanishing holomorphic function on Ui ∩ Uj for each pair of indices i, j ∈ I . Assume that the family {fij }i,j ∈I satisfies the (multiplicative) cocycle relation fik = fij fj k

on Ui ∩ Uj ∩ Uk

∀i, j, k ∈ I.

Prove that there exists a unique holomorphic line bundle  : E → X for which there exists a holomorphic line bundle atlas {(Ui , i = (, i ))}i∈I with transition functions given by i /j = fij for each pair i, j ∈ I . Conclude that in particular, if X is an open Riemann surface, then there exists a collection {fi }i∈I such that fi is a nonvanishing holomorphic function on Ui for each i ∈ I and fij = fj /fi for all i, j ∈ I .

Part II

Further Topics

Chapter 4

Compact Riemann Surfaces

In this chapter, we consider some facts concerning holomorphic line bundles (and their holomorphic sections) on compact Riemann surfaces. We first consider conditions that guarantee the existence of holomorphic sections with prescribed values. Unlike the open Riemann surface case (in which one has Theorem 3.11.5), a holomorphic line bundle need not have the positivity required for such a section to exist. For example, the space of holomorphic functions on a compact Riemann surface X has dimension 1, and a negative holomorphic line bundle on X has no nontrivial global holomorphic sections. After the above considerations, we consider the fact that a holomorphic line bundle is positive if and only if its degree is positive, and we then consider finiteness of Dolbeault cohomology. The Riemann–Roch formula is then proved (using finiteness). Finally, we consider the Serre duality theorem and the Hodge decomposition theorem, and some of their consequences. Different approaches to the above facts appear in (for example) [For] and [Ns4].

4.1 Existence of Holomorphic Sections on a Compact Riemann Surface In this section, we consider consequences of the following: Theorem 4.1.1 Let E be a holomorphic line bundle on a compact Riemann surface X. If there exist a Kähler metric g on X and a Hermitian metric h in E with ig + ih > 0 on X, then for every effective divisor D on X, the linear map (X, O(E ⊗ [D])) −→(X, QD (E ⊗ [D])) = (X, O(E ⊗ [D])/O−D (E ⊗ [D])) T. Napier, M. Ramachandran, An Introduction to Riemann Surfaces, Cornerstones, DOI 10.1007/978-0-8176-4693-6_4, © Springer Science+Business Media, LLC 2011

157

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=



Compact Riemann Surfaces

 D(p) O(E ⊗ [D])p (mp · O(E ⊗ [D])p )

p∈suppD

∼ =



CD(p) ∼ = Cdeg D

p∈suppD

is surjective. In other words, given a holomorphic section tp of E ⊗ [D] on a neighborhood of p in X for each point p ∈ suppD, there exists a holomorphic section s of E ⊗ [D] on X such that ordp (s − tp ) ≥ D(p) for each point p ∈ suppD. In particular, dim (X, O(E ⊗ [D])) ≥ deg D. 1 (X, E) = 0, and therefore the Dolbeault Proof By Corollary 3.10.5, we have HDol exact sequence (see Theorem 3.4.4 and Definition 3.4.5) is

0 → (X, O(E)) → (X, O(E ⊗ [D])) → (X, QD (E ⊗ [D])) → 0. Furthermore, any choice of a holomorphic section tp of E ⊗ [D] on a neighborhood of p in X for each point p ∈ suppD determines a section ξ ∈ (X, QD (E ⊗ [D])). Every section s ∈ (X, O(E ⊗ [D])) with image ξ in (X, QD (E ⊗ [D])) then satisfies ordp (s − tp ) ≥ D(p) for each point p ∈ suppD.  Remark One may also prove a stronger version (see Exercise 4.1.4) that is analogous to the Mittag-Leffler theorem (Theorem 2.15.1 and Theorem 3.11.5). The proof of the following equivalent version of Theorem 4.1.1 is left to the reader (see Exercise 4.1.1): Corollary 4.1.2 Let E be a positive holomorphic line bundle on a compact Riemann surface X. Then, for every effective divisor D on X, the associated linear map (X, O(KX ⊗ E ⊗ [D])) −→ (X, QD (KX ⊗ E ⊗ [D])) is surjective. In particular, dim (X, O(KX ⊗ E ⊗ [D])) ≥ deg D. Corollary 4.1.3 Let X be a compact Riemann surface, let F be a holomorphic line bundle on X, and let E be a positive holomorphic line bundle on X. Then for every integer r  0 and for every effective divisor D on X (r does not depend on the choice of D), the linear map (X, O(F ⊗ E r ⊗ [D])) −→ (X, QD (F ⊗ E r ⊗ [D])) is surjective. In particular, dim (X, O(F ⊗ E r ⊗ [D])) ≥ deg D. The proof is left to the reader (see Exercise 4.1.2).

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Corollary 4.1.4 Given an effective divisor D on a compact Riemann surface X, there exists a Kähler form ω0 on X such that for every Hermitian holomorphic line bundle (E, h) on X with ih ≥ ω0 , the associated linear map (X, O(E)) −→ (X, QD (E)) ∼ = Cdeg D is surjective (in particular, dim (X, O(E)) ≥ deg D). In fact, we may choose ω0 so that if (E, h) is a Hermitian holomorphic line bundle on X with ih ≥ ω0 and tp is a holomorphic section of E on a neighborhood of p in X for each point p ∈ suppD, then there exists a holomorphic section s of E on X such that ordp (s − tp ) = D(p) for each point p ∈ suppD. Proof We may fix a Kähler metric g on X with associated Kähler form ω and a Hermitian metric k in [D] on X. We may now choose a Kähler form ω0 such that ω0 + ig − ik > 0 (for example, we may take ω0 = Cω for C  0). Hence, if (E, h) is a Hermitian holomorphic line bundle on X with ih ≥ ω0 , then the Hermitian holomorphic line bundle (E ⊗ [−D], h ⊗ k ∗ ) satisfies ig + ih⊗k ∗ = ig + ih − ik ≥ ig + ω0 − ik > 0. Applying Theorem 4.1.1 to the line bundle E ⊗ [−D] ⊗ [D] ∼ = E, we get surjectivity of the mapping (X, O(E)) −→ (X, QD (E)). Now, for each point p ∈ suppD, we may fix a holomorphic section up of E on a neighborhood of p such that ordp up = D(p). Applying the above to the divisor 2D, we get a Kähler metric ω0 such that if (E, h) is a Hermitian holomorphic line bundle on X with ih ≥ ω0 and tp is a holomorphic section of E on a neighborhood of p for each point p ∈ suppD, then there exists a section s ∈ (X, O(E)) with ordp (s − tp − up ) ≥ 2D(p) > D(p), and hence ordp (s − tp ) = D(p), for each point p ∈ suppD.  Corollary 4.1.5 Let D be an effective divisor on a compact Riemann surface X, let F be a holomorphic line bundle on X, and let E be a positive holomorphic line bundle on X. Then, for r  0 (depending on the choice of D, F , and E), the associated linear map (X, O(F ⊗ E r )) −→ (X, QD (F ⊗ E r )) is surjective (in particular, dim (X, O(F ⊗ E r )) ≥ deg D). In fact, if r  0 and tp is a holomorphic section of F ⊗ E r on a neighborhood of p in X for each point p ∈ suppD, then there exists a holomorphic section s of F ⊗ E r on X such that ordp (s − tp ) = D(p) for each point p ∈ suppD. The proof is left to the reader (see Exercise 4.1.3).

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Exercises for Sect. 4.1 4.1.1 4.1.2 4.1.3 4.1.4

Prove Corollary 4.1.2. Prove Corollary 4.1.3. Prove Corollary 4.1.5. The goal of this problem is a generalization of Theorem 4.1.1 that is analogous to the Mittag-Leffler theorem (Theorem 2.15.1 and Theorem 3.11.5). Let E be a holomorphic line bundle on a compact Riemann surface X. Assume that there exist a Kähler metric g on X and a Hermitian metric h in E with ig + ih > 0 on X. Let D be an effective divisor on X, and for each point p ∈ suppD, let tp be a holomorphic section of E ⊗ [D] on Up \ {p} for some neighborhood Up of p with Up ∩ suppD = {p}. Prove that there exists a holomorphic section s of E ⊗ [D] on X \ suppD such that for each point p ∈ suppD, s − tp extends to a holomorphic section up of E ⊗ [D] on Up with ordp up ≥ D(p).

4.2 Positive Curvature on a Compact Riemann Surface According to Theorem 3.11.2, every holomorphic line bundle E on an open Riemann surface is positive (i.e., E admits a positive-curvature Hermitian metric). This is not true in general for a holomorphic line bundle E on a compact Riemann surface. It will be shown in Sect. 4.3 (see Theorem 4.3.1) that in fact, a holomorphic line bundle E on a compact Riemann surface is positive if and only if E is the line bundle associated to a divisor of positive degree. For now, we consider the following weaker fact: Theorem 4.2.1 Let X be a compact Riemann surface. If E = [D] is the holomorphic line bundle associated to a nontrivial effective divisor D on X (i.e., E is a holomorphic line bundle that admits a nontrivial global holomorphic section with at least one zero), then E is positive. For the proof, it will be convenient to have the following construction of a nonnegative-curvature Hermitian metric in the holomorphic line bundle associated to a divisor given by a single point: Lemma 4.2.2 Let X be a Riemann surface, let p ∈ X, let D be the divisor given by D(p) = 1 and D(q) = 0 for each point q ∈ X \ {p} (i.e., D = 1 · p), let s be a holomorphic section of the associated holomorphic line bundle E = [D] on X with div(s) = D, and let U be a neighborhood of p. Then there exists a Hermitian metric h in E on X such that (i) We have |s|2h = 1 and ih = 0 at each point in X \ U , and (ii) We have ih ≥ 0 on X and ih > 0 at p.

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161

Proof Clearly, we may assume that there is a holomorphic coordinate z on U with z(p) = 0. Applying Lemma 2.10.3, we get a constant b > 0, a nonnegative C ∞ subharmonic ϕ on the set Y ≡ X \ {p}, and a relatively compact neighborhood V of p in U such that ϕ ≡ 0 on a neighborhood of X \ U and ϕ = |z|2 − log |z|2 − b on V . We now get a Hermitian metric h in X by requiring that |s|2h = e−ϕ . More precisely, for any ξ ∈ EY , we set |ξ |2h = |ξ/s|2 e−ϕ . We may extend s/z to a nonvanishing holomorphic section t of E on U , and for any ξ ∈ EV , we set 2 |ξ |2h = |ξ/t|2 e−|z| +b . It is now easy to see that h is well defined and that h has the properties (i) and (ii).   Proof of Theorem 4.2.1 Suppose E = [D], where D = m j =1 νj pj is a nontrivial effective divisor with νj > 0 for each j . For each j = 1, . . . , m, Lemma 4.2.2 provides a Hermitian metric hj in [pj ] with ihj ≥ 0 on X and ihj > 0 on some neighborhood Vj of pj in X. The Hermitian metric k ≡ hν11 ⊗ · · · ⊗ hνmm in E then satisfies ik ≥ 0 on X and ik > 0 on the neighborhood V ≡ V1 ∪ · · · ∪ Vm of suppD. Applying Corollary 2.14.2 in X \ suppD and cutting off, we get a C ∞ function α on X such that iα > 0 on a neighborhood of X \ V . Thus, for > 0 sufficiently small, the Hermitian metric h ≡ e− α k satisfies ih = iα + ik > 0.  Remark Since the Riemann surface X in the above theorem is compact, one gets second countability without appealing to Radó’s theorem. Therefore, although a fact from Sect. 2.14 (Corollary 2.14.2) was applied in the above proof, there was no real dependence on Radó’s theorem and therefore almost no dependence on Sects. 2.6, 2.7, 2.9–2.12, 3.6, and 3.8–3.10. It follows from Theorem 4.2.1 that every compact Riemann surface admits a positive holomorphic line bundle. Moreover, we have the following consequence, which, when combined with Corollary 3.11.7, implies that holomorphic line bundles and divisors on Riemann surfaces are actually equivalent: Theorem 4.2.3 Any holomorphic line bundle E on a compact Riemann surface X is equal to the line bundle associated to some nontrivial divisor D. In fact, given any finite set S ⊂ X and any choice of integers {mp }p∈S , there exists a meromorphic section s of E such that ordp s = mp for each point p ∈ S. Proof Suppose S ⊂ X is a nonempty finite set and mp ∈ Z for each point p ∈ S. Fixing a positive holomorphic line bundle (provided by Theorem 4.2.1) and applying Corollary 4.1.5, we get a holomorphic line bundle F on X such that E ⊗ F and F admit global holomorphic sections u and v, respectively, with ordp u = max(mp , 0) and ordp v = max(−mp , 0) for each point p ∈ S. The quotient s ≡ u/v is then a nontrivial meromorphic section of E, ordp s = mp for each point p ∈ S, and E = [D] for the nontrivial divisor D ≡ div(s). 

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Exercises for Sect. 4.2 4.2.1 Let X be a compact Riemann surface, let E be a holomorphic line bundle on X, let p ∈ X, and let m ∈ Z≥2 . Prove that if E admits a Hermitian metric with ih ≥ 0, then there exists a meromorphic 1-form θ with values in E on X such that θ is holomorphic on X \ {p} and θ has a pole of order m at p (cf. Theorem 2.10.1). Hint. Apply Corollary 4.1.2 to the effective divisor D = m · p, the positive holomorphic line bundle E ⊗ [p], and a local holomorphic section tp of KX ⊗ E ⊗ [(m + 1)p] with a zero of order 1 at p. Divide by a suitable power of a section of [p] with divisor p.

4.3 Equivalence of Positive Curvature and Positive Degree In this section we consider the more complete characterization of positivity of holomorphic line bundles on compact Riemann surfaces alluded to in Sects. 3.11 and 4.2. Theorem 4.3.1 Let D be a divisor on a compact Riemann surface X, and let E = [D] be the associated holomorphic line bundle. Then E is positive if and only if deg D > 0. Proof If E is positive, then by Corollary 4.1.5, for r  0, E r admits a nontrivial holomorphic section s that has at least one zero. Setting D  = div(s), we get deg D =

1 1 deg(rD) = deg D  > 0. r r

Conversely, suppose deg D > 0. To show that E is positive, it suffices to show that E r = [rD] admits a nontrivial holomorphic section for some positive integer r. For the divisor of the section will be nontrivial (since deg E = r deg D > 0), and hence it will follow from Theorem 4.2.1 that E r admits a positive-curvature Hermi2/r tian metric hr . The Hermitian metric h in E determined by |ξ |2h = |ξ r |hr for each ξ ∈ E (see Exercise 4.3.1) will then satisfy ih = r −1 · ihr > 0. If D ≥ 0, then clearly, E admits a nontrivial holomorphic section. Assuming now that D is not effective, we have D = D+ − D− , where D± are (unique) nontrivial (i.e., not everywhere zero) effective divisors with disjoint supports, and E = E+ ⊗ E− with E± ≡ [D± ]. By Theorem 4.2.1 and Corollary 4.1.3, for m a sufficiently large positive integer and n an arbitrary positive integer, we have m+n m dim (X, O(E+ )) = dim (X, O(E+ ⊗ [nD+ ])) ≥ deg(nD+ ) = n deg D+ .

For any effective divisor G, multiplication by a holomorphic section of [G] with divisor G yields the exact sequence 0 → (X, O([(m + n)D+ − G])) → (X, O([(m + n)D+ ])) → (X, O([(m + n)D+ ])/O−G ([(m + n)D+ ])) ∼ = Cdeg G .

4.3 Equivalence of Positive Curvature and Positive Degree

163

Setting G = (m + n)D− , we get m+n dim (X, O(E m+n )) ≥ dim (X, O(E+ )) − (m + n) deg D−

≥ n deg D+ − (m + n) deg D− = n deg D − m deg D− . Fixing m and choosing n  0, we get the claim.



Remark One may also prove the theorem by applying the Riemann–Roch formula, which will be considered in Sect. 4.5 (see Exercise 4.5.1). Corollary 4.3.2 If D is a divisor of positive degree on a compact Riemann surface X and E = [D], then E r admits a nontrivial holomorphic section for r  0 and no nontrivial holomorphic sections for r < 0. Proof By Theorem 4.3.1, E > 0, and hence E r admits a nontrivial holomorphic section s for r  0 (for example, by Corollary 4.1.5). For r < 0, we have deg E r = r deg D < 0, so (X, O(E r )) = 0.  Observe also that facts considered earlier concerning positive holomorphic line bundles on compact Riemann surfaces have equivalent forms in terms of divisors of positive degree. For example, Theorem 4.3.1 and Corollary 3.10.5 together give the following vanishing theorem: Corollary 4.3.3 If D is a divisor of positive degree on a compact Riemann sur1 (X, K ⊗ [D]) = 0. Moreover, for any holomorphic line bundle face X, then HDol X 1 (X, F ⊗ [rD]) = 0 for r  0. F on X, we have HDol Some other examples, which follow from the results of Sect. 4.1, are summarized in the following corollary, the proof of which is left to the reader (see Exercise 4.3.2): Corollary 4.3.4 Let D0 be a divisor of positive degree on a compact Riemann surface X. (a) For every effective divisor D on X, the associated linear map (X, O(KX ⊗ [D0 + D])) −→ (X, QD (KX ⊗ [D0 + D])) is surjective (in particular, dim (X, O(KX ⊗ [D0 + D])) ≥ deg D). (b) If F is any holomorphic line bundle on X, then for every sufficiently large positive integer r and every effective divisor D on X (r is independent of the choice of D), the linear map (X, O(F ⊗ [rD0 + D])) → (X, QD (F ⊗ [rD0 + D])) is surjective (in particular, dim (X, O(F ⊗ [rD0 + D])) ≥ deg D).

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(c) If F is any holomorphic line bundle on X and D is any effective divisor on X, then for every sufficiently large positive integer r (depending on the choice of D0 , F , and D), the linear map (X, O(F ⊗ [rD0 ])) −→ (X, QD (F ⊗ [rD0 ])) is surjective (in particular, dim (X, O(F ⊗ [rD0 ])) ≥ deg D). In fact, if r  0 and tp is a holomorphic section of F ⊗ [rD0 ] on a neighborhood of p in X for each point p ∈ suppD, then there exists a holomorphic section s of F ⊗ [rD0 ] on X such that ordp (s − tp ) = D(p) for each point p ∈ suppD. Exercises for Sect. 4.3 4.3.1 As in the proof of Theorem 4.3.1, let E be a holomorphic line bundle, let r be a positive integer, and let hr be a Hermitian metric in the r-fold tensor power E r . For each element ξ ∈ E, let ξ r = ξ ⊗ · · · ⊗ ξ ∈ E r , and let |ξ |2h = 2/r |ξ r |hr . Verify that this determines a Hermitian metric h in E with h = (1/r)hr . 4.3.2 Prove Corollary 4.3.4. 4.3.3 Prove that if E is a holomorphic line bundle on a compact Riemann surface X 1 (X, E) = 0. and deg E − deg KX > 0, then HDol

4.4 A Finiteness Theorem The vector space of holomorphic sections of a holomorphic line bundle on a compact Riemann surface is always finite-dimensional. In fact, we have the following: Theorem 4.4.1 (Finiteness theorem) If E is a holomorphic line bundle on a compact Riemann surface X, then 0 (X, E) ≤ max(deg E + 1, 0) < ∞ dim (X, O(E)) = dim HDol

and 1 (X, E) < ∞. dim HDol

Proof Fix a point p ∈ X and let G be the divisor with supp G = {p} and G(p) = 1 (i.e., G = 1 · p). For each positive integer r, we have the linear map (X, O(E)) → (X, Q(r−1)G (E)) ∼ = Cr−1

(= {0}

if r = 1).

If dim (X, O(E)) ≥ r, then the kernel contains a nontrivial section. On the other hand, such a section determines an effective divisor D such that E = [D] and deg E = deg D ≥ r − 1.

4.5 The Riemann–Roch Formula

165

Thus we get the desired bound on dim (X, O(E)). Multiplication by a holomorphic section of [rG] with divisor rG gives the exact sequence of sheaves 0 → O(E) → O(E ⊗ [rG]) → QrG (E ⊗ [rG]) → 0 and the corresponding Dolbeault exact sequence 0 → (X, O(E)) → (X, O(E ⊗ [rG])) → (X, QrG (E ⊗ [rG])) 1 1 (X, E) → HDol (X, E ⊗ [rG]) → 0. → HDol

By Theorem 4.2.1 and Corollary 3.10.5 (or by Corollary 4.3.3), for r  0, we have 1 (X, E ⊗ [rG]) = 0, and hence the map HDol 1 (X, E) Cr ∼ = (X, QrG (E ⊗ [rG])) → HDol 1 (X, E) < ∞. is surjective. Thus dim HDol



Definition 4.4.2 For a compact Riemann surface X, the genus of X is the nonnegative integer 1 1 genus(X) ≡ dim HDol (X, 1X ) = dim HDol (X).

Exercises for Sect. 4.4 4.4.1 Prove that if E is a holomorphic line bundle on a compact Riemann surface X, 1 (X, E) ≤ max(deg K − deg E + 1, 0) (cf. Exercise 4.3.3). then dim HDol X

4.5 The Riemann–Roch Formula In this section, we consider the following formula for the dimension of the space of holomorphic sections of a holomorphic line bundle on a compact Riemann surface (this formula is much more precise than the estimates for the dimension considered in previous sections): Theorem 4.5.1 (Riemann–Roch formula) If E is a holomorphic line bundle on a compact Riemann surface X, then 1 (X, E) = 1 − genus(X) + deg E. dim (X, O(E)) − dim HDol

Remark For a divisor D on a compact Riemann surface X, we set q

hq (D) ≡ dim HDol (X, [D])

for q = 0, 1.

Setting g ≡ genus(X) and d ≡ deg D, we get the following equivalent form for the Riemann–Roch formula, which is easier to remember: h0 (D) − h1 (D) = 1 − g + d.

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The proof given here, which is standard, is similar to that in [For]. The main point is the following induction step: Lemma 4.5.2 Suppose A and B are divisors on a compact Riemann surface X such that B = A + p for some point p ∈ X and such that the Riemann–Roch formula holds for A or for B. Then the Riemann–Roch formula holds both for A and for B. Proof Multiplication by a holomorphic section of the line bundle [p] with divisor p = B − A yields the Dolbeault exact sequence 0 → (X, O([A])) → (X, O([B])) → (X, QB−A ([B])) 1 1 → HDol (X, [A]) → HDol (X, [B]) → 0.

Setting V ≡ im[(X, O([B])) → (X, QB−A ([B]))] 1 = ker[(X, QB−A ([B])) → HDol (X, [A])] ⊂ (X, QB−A ([B])) ∼ =C

and 1 (X, [A])] W ≡ im[(X, QB−A ([B])) → HDol 1 1 (X, [A]) → HDol (X, [B])], = ker[HDol

we get dim V + dim W = dim (X, QB−A (B)) = 1, and therefore h0 (B) − h1 (B) − deg B = (h0 (A) + dim V) − (h1 (A) − dim W) − deg A − 1 = h0 (A) − h1 (A) − deg A. 

The lemma now follows. Proof of Theorem 4.5.1 For D = 0, we have h0 (D) − h1 (D) = 1 − g = 1 − g + d. For an arbitrary divisor D on X, we have D = B − A for effective divisors A = p1 + · · · + pm

and

B = q1 + · · · + qn .

Setting B0 = 0 and Bν = q1 + · · · + qν for ν = 1, . . . , n, and applying Lemma 4.5.2 inductively on ν, we get the Riemann–Roch formula for the divisor B = Bn . Setting D0 = B and Dμ = B − p1 − · · · − pμ for μ = 1, . . . , m, and applying Lemma 4.5.2 inductively on μ, we get the Riemann–Roch formula for the divisor D = Dm . 

4.6 Statement of the Serre Duality Theorem

167

Exercises for Sect. 4.5 4.5.1 Using the Riemann–Roch formula in place of Theorem 4.3.1, give a different proof of Corollary 4.3.2 (i.e., a different proof that for any divisor D of positive degree on a compact Riemann surface X, [rD] admits a nontrivial holomorphic section for r  0). Use this to then give a different proof of Theorem 4.3.1.

4.6 Statement of the Serre Duality Theorem Throughout this section, E denotes a holomorphic line bundle on a Riemann surface X. Recall that we have the natural identifications E ⊗ E ∗ ∼ = 1X = E∗ ⊗ E ∼ ∗ determined by s ⊗ ξ ↔ ξ ⊗ s → ξ · s = ξ(s) for ξ ∈ E and s ∈ E. Combining this with the wedge product, one gets a natural pointwise bilinear pairing sE (·, ·) of forms with values in E and forms with values in E ∗ that is related to the pointwise pairing {·, ·}h of forms with values in E associated to a Hermitian metric h as described in Definition 3.6.1 ({·, ·}h is not, however, bilinear; it is linear in the first entry and conjugate linear in the second). One then also gets an associated integration pairing SE (·, ·) that is related to the L2 pairing. The precise definitions are as follows: Definition 4.6.1 Let m, n ∈ Z≥0 . (a) For each point x ∈ X, the bilinear pairing sE (·, ·) : [m (T ∗ X)C ⊗ E]x × [n (T ∗ X)C ⊗ E ∗ ]x → [m+n (T ∗ X)C ]x is given by sE (α, β) ≡ α0 ∧ β0 · s · ξ = α0 ∧ β0 · ξ(s) for all α = α0 ⊗ s ∈ [m (T ∗ X)C ⊗ E]x and β = β0 ⊗ ξ ∈ [n (T ∗ X)C ⊗ E ∗ ]x with s ∈ Ex and ξ ∈ Ex∗ . (b) If m ∈ {0, 1, 2} and α and β are measurable differential forms of degree m and 2 − m, respectively, with values in E and E ∗ , respectively, on X, then we define  SE (α, β) ≡ sE (α, β), X

provided the above integral is defined. Remarks 1. It is easy to see that sE (·, ·) is well defined. In fact, under the identification of E ⊗ E ∗ ∼ = 1X , we have sE (α, β) = α ∧ β as in Definition 3.1.14 (see Exercise 4.6.1). 2. The pairings sE (·, ·) and SE (·, ·) are bilinear, unlike the pairings {·, ·}h and ·, ·L2 , which are linear in the first entry, but conjugate linear in the second.

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3. It is easy to see that sE (α, β) = (−1)mn sE ∗ (β, α) and SE (α, β) = (−1)mn · SE ∗ (β, α) for all (suitable) α and β of degree m and n, respectively. Furthermore, if α is of type (p, q) and β is of type (r, s), then sE (α, β) is of type (p + r, q + s) (the value being 0 if p + r > 1 or q + s > 1). 4. Suppose α and β are differential forms with values in E and E ∗ , respectively. It is easy to verify that if α and β are continuous (C k , measurable), then sE (α, β) is continuous (respectively, C k , measurable). Lemma 4.6.2 For C ∞ differential forms α and β with values in E and E ∗ , respectively, on X: ¯ E (α, β)) = sE (∂α, ¯ β) + (−1)deg α sE (α, ∂β). ¯ (a) We have ∂(s (b) If α is of type (p, 0) and β is of type (1 − p, 0) with p ∈ {0, 1}, and α or β has ¯ β) + (−1)p SE (α, ∂β) ¯ = 0. In particular, if α is compact support, then SE (∂α, ∞ a C section of E with compact support and β is a holomorphic 1-form with values in E ∗ (i.e., β ∈ (X, O(KX ⊗ E ∗ )) under the usual identification given ¯ β) = 0. by Proposition 3.10.1), then SE (∂α, Proof The local representation of sE (·, ·) gives Part (a) (see Exercise 4.6.2). For the proof of (b), observe that sE (α, β) is a form of type (1, 0) and therefore ¯ E (α, β)). Thus, integrating the expressions in (a) and applying d(sE (α, β)) = ∂(s Stokes’ theorem, we get (b).  The above lemma allows us to make the following definition: Definition 4.6.3 For X a compact Riemann surface, the Serre pairing 1 (X, E) × (X, O(KX ⊗ E ∗ )) → C SE (·, ·) : HDol

is given by SE ([θ ]Dol , s) ≡ SE (θ, s) for each form θ ∈ E 0,1 (E)(X) and each section s ∈ (X, O(KX ⊗ E ∗ )). The Serre mapping is the linear mapping ∗ 1 ∗ ιE S : (X, O(KX ⊗ E )) → [HDol (X, E)]

given by s → SE (·, s). Remarks 1. Lemma 4.6.2 implies that the Serre pairing is well defined. 2. In a slight abuse of notation, we denote both the general integration pairing in Definition 4.6.1 and the Serre pairing by SE (·, ·). 3. Here, we are identifying each global holomorphic section s ∈ (X, O(KX ⊗ E ∗ )) of KX ⊗ E ∗ with the corresponding holomorphic 1-form with values in E ∗ . 4. If α = a d z¯ · t and β = b dz · ξ for a local holomorphic coordinate z and for local holomorphic sections t and ξ of E and E ∗ , respectively, then sE (α, β) = ab · d z¯ ∧ dz · ξ(t) = −ab · dz ∧ d z¯ · ξ(t).

4.6 Statement of the Serre Duality Theorem

169

One goal of the remainder of this chapter is the following: Theorem 4.6.4 (Serre duality theorem) If E is a holomorphic line bundle on a compact Riemann surface X, then the Serre mapping ιE S : s → SE (·, s) gives an isomorphism ∼ =

∗ 1 ∗ ιE S : (X, O(KX ⊗ E )) −→ [HDol (X, E)] .

Remark Equivalently, we have an isomorphism ∼ =

∗

1 X ⊗E : (X, O(E)) = (X, O(KX ⊗ (KX ⊗ E ∗ )∗ )) −→ [HDol (X, KX ⊗ E ∗ )]∗ . ιK S

Letting K and D be divisors on X with KX = [K] and E = [D] (i.e., K is the divisor of some meromorphic 1-form η and we identify (KX , η) with ([K], s), where s is the associated defining section for K, and the analogous identification holds for E and D), we also get the equivalent forms ∼ =

1 ∗ ι[D] S : (X, O([K − D])) −→ [HDol (X, [D])] , ∼ =

1 ι[K−D] : (X, O([D])) −→ [HDol (X, [K − D])]∗ . S 1 (X)]∗ , which Corollary 4.6.5 For X compact, the Serre map ι1SX : (X) → [HDol is given by  1X θ ∧α ιS (α)([θ ]Dol ) = X

1 ∀α ∈ (X) = (X, O(KX )), [θ]Dol ∈ HDol (X),

is an isomorphism. In particular, genus(X) = dim (X). The Serre duality theorem and the Riemann–Roch formula have many applications. We consider a few of the immediate applications here. We first observe that Serre duality gives the following equivalent version of the Riemann–Roch formula: Theorem 4.6.6 For any holomorphic line bundle E on a compact Riemann surface X, we have dim (X, O(E)) − dim (X, O(KX ⊗ E ∗ )) = 1 − genus(X) + deg E. Equivalently, for any divisor D of degree d on a compact Riemann surface X of genus g and for any divisor K with [K] = KX , we have h0 (D) − h0 (K − D) = 1 − g + d. Corollary 4.6.7 For any compact Riemann surface X of genus g, we have deg KX = 2g − 2.

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Proof The above Riemann–Roch formula (applied to E = KX ) and Corollary 4.6.5 together imply that g = dim (X, O(KX )) = dim (X, O) + 1 − g + deg KX = 1 + 1 − g + deg KX . 

The claim now follows.

The following application will play a role in the proof of the Abel–Jacobi embedding theorem (Theorem 5.22.2): Theorem 4.6.8 Up to biholomorphism, the only compact Riemann surface of genus 0 is P1 . In fact, if X is a compact Riemann surface that is not biholomorphic to P1 , then for every point p ∈ X, there is a holomorphic 1-form θ on X with θp = 0. Proof The verification that genus(P1 ) = 0 (i.e., that P1 has no nontrivial holomorphic 1-forms) is left to the reader (cf. Exercise 2.5.4). Suppose now that X is a compact Riemann surface of genus g, and p ∈ X is a point at which every holomorphic 1-form on X vanishes. Fixing a divisor K with [K] = KX , letting D = 1 · p, and fixing a section s ∈ (X, O([D])) with div(s) = D, we get an isomorphism ∼ =

(X, O([K])) −→ (X, O([K − D])) given by θ → θ/s. The Riemann–Roch formula (in the form of Theorem 4.6.6) then gives h0 (D) − g = h0 (D) − h0 (K) = h0 (D) − h0 (K − D) = 1 − g + deg D = 1 − g + 1, and hence h0 (D) = 2. Thus there exists a section t ∈ (X, O([D])) \ C · s, and f ≡ t/s : X → P1 is a nonconstant meromorphic function that is holomorphic except for a simple pole at p (t (p) = 0, since otherwise, f would be a nonconstant holomorphic function on X). Therefore, by Proposition 2.5.7, f is a biholomorphism of X onto P1 , and the claim follows.  Remarks 1. The canonical line bundle of a compact Riemann surface X of genus 1 is trivial (see Exercise 4.6.3). 2. It follows from the above theorem that if θ1 , . . . , θg is a basis for (X) for a compact Riemann surface X of genus g > 0, then we get a Kähler form ω≡

g 

iθj ∧ θ¯j .

j =1

We recall that if a holomorphic line bundle E is the line bundle associated to a nontrivial effective divisor, then E is positive. However, as the following example shows, the converse is false (see [Ns4]): Observation 4.6.9 Any compact Riemann surface X of genus g > 1 admits a positive holomorphic line bundle with no nontrivial holomorphic sections. To see this,

4.6 Statement of the Serre Duality Theorem

171

we first observe that there exists an effective divisor D such that deg D = g and h0 (D) = 1. For there exists a nontrivial holomorphic 1-form on X and therefore a point p1 ∈ X at which the form does not vanish. Thus the vector space V1 ≡ {θ ∈ (X, O(KX )) | θp1 = 0} is of dimension g − 1 > 0. Given points p1 , . . . , pm−1 ∈ X and subspaces (X, O(KX )) = V0 ⊃ V1 ⊃ V2 ⊃ · · · ⊃ Vm−1 , where 1 < m ≤ g and Vj = {θ ∈ (X, O(KX )) | θp1 = · · · = θpj = 0} and dim Vj = g − j for j = 1, . . . , m − 1, we may choose a nontrivial element of Vm−1 and a point pm ∈ X \ {p1 , . . . , pm−1 } at which this element does not vanish. Setting Vm ≡ {θ ∈ (X, O(KX )) | θp1 = · · · = θpm = 0} = {θ ∈ Vm−1 | θpm = 0} ⊂ Vm−1 , we get dim Vm = g − (m − 1) − 1 = g − m. Thus, by induction, we get distinct points p1 , . . . , pg ∈ X at which no nontrivial holomorphic 1-form can simultaneously vanish. We now let K be a divisor with [K] = KX , we set D = p1 + · · · + pg , and we fix a section s ∈ (X, O([D])) with div(s) = D. The Riemann–Roch formula and Serre duality give h0 (D) = h0 (K − D) + 1. On the other hand, if θ ∈ (X, O([K − D])), then θ · s is a holomorphic 1-form that vanishes at the points p1 , . . . , pg and hence θ ≡ 0. Thus h0 (D) = 1, as desired. Fixing a point p ∈ X \ {p1 , . . . , pg } and setting F = D − p, we get deg F = g − 1 > 0, so E ≡ [F ] is positive by Theorem 4.3.1. However, letting P be the divisor with P (p) = 1 and P (x) = 0 for x = p (i.e., P = 1 · p), and choosing a section t ∈ (X, O([P ])) with div(t) = P , we get the injective linear mapping (X, O(E)) → (X, O([D])) given by multiplication by t. Since s(p) = 0, s cannot be in the image of this mapping. Therefore dim (X, O(E)) < h0 (D) = 1, and hence E has no nontrivial holomorphic sections. Remark The above observation is false in genus 1 (see Exercise 4.6.3). We record here for later use the observation that Lemma 4.6.2 leads one to the following equivalent characterization of the operator ∂¯distr , which is completely intrinsic, unlike Definition 3.8.1, which involves the choice of a local holomorphic coordinate and a nonvanishing local holomorphic section, and unlike the characterization in Proposition 3.8.2, which involves the choice of a Hermitian metric: Lemma 4.6.10 Let p ∈ {0, 1} and let α and β be locally integrable differential forms of type (p, 0) and type (p, 1), respectively, with values in E on X. Then ∂¯ distr α = β if and only if ¯ ) + (−1)p SE (β, γ ) = 0 SE (α, ∂γ

¯ , α) − (−1)p SE ∗ (γ , β) = 0) (i.e., SE ∗ (∂γ

for every form γ ∈ D 1−p,0 (E ∗ )(X). In particular, α is holomorphic if and only if ¯ ) = 0 (i.e., SE ∗ (∂γ ¯ , α) = 0) SE (α, ∂γ

∀γ ∈ D1−p,0 (E ∗ )(X).

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Proof For p = 1, suppose α = a dz ⊗ s and β = b dz ∧ d z¯ ⊗ s on U for some local holomorphic coordinate neighborhood (U, z) and some nonvanishing holomorphic section s of E on U . For every function f ∈ D(U ), we may extend f/s by 0 to a C ∞ differential form γ of type (0, 0) with values in E ∗ on X and compact support in U . Thus  bf dz ∧ d z¯ SE (β, γ ) = U

and ¯ )= SE (α, ∂γ

 U

¯ = a · dz ∧ ∂f

 a· U

∂f · dz ∧ d z¯ . ∂ z¯

¯ )−SE (β, τ ) = 0 Definition 3.8.1 and the above together now imply that if SE (α, ∂τ 0,0 ∗ ¯ for every form τ ∈ D (E )(X), then ∂distr α = β. Conversely, suppose ∂¯ distr α = β and γ is a C ∞ compactly supported differential ∞ form of type (0, 0) with values in E ∗ on X. We may form  a finite collection of C m compactly supported functions {ην }ν=1 on X such that ην ≡ 1 on supp γ and such that for each ν, the support of ην is contained in some local holomorphic coordinate neighborhood (Uν , zν ) on which there is a nonvanishing holomorphic section sν of E. Setting aν ≡ α/(dzν ⊗ sν ) ∈ C ∞ (Uν ), bν ≡ β/(dzν ∧ d z¯ ν ⊗ sν ) ∈ C ∞ (Uν ), and fν ≡ ην γ · sν ∈ D(Uν ) for each ν, we get   ∂fν ¯ )= ¯ ν γ )) = SE (α, ∂(η aν · · dzν ∧ d z¯ ν SE (α, ∂γ ∂ z¯ ν Uν ν ν   = bν fν · dzν ∧ d z¯ ν = SE (β, ην γ ) = SE (β, γ ). ν

Uν

ν

The proof of the case p = 0 is left to the reader (see Exercise 4.6.4).



Remark Recall that we have the natural identification of an E-valued (1, 1)-form with a KX ⊗ E-valued (0, 1)-form provided by Proposition 3.10.1. In the present context, this identification can lead to some confusion, since the above pairings are preserved only up to sign. To see this, let (U, z) be local a holomorphic coordinate neighborhood, let s be a nonvanishing holomorphic section of E on U , and let α = dz ⊗ s and β = d z¯ ⊗ s −1 . Viewing α as a (1, 0)-form with values in E and β as a (0, 1)-form with values in E ∗ , we get sE (α, β) = dz ∧ d z¯ . Viewing α as a (0, 0)-form with values in KX ⊗ E and β = −dz ∧ d z¯ ⊗ ((∂/∂z) ⊗ s −1 ) as a (1, 1)-form with values in KX∗ ⊗ E ∗ , we get sKX ⊗E (α, β) = −dz ∧ d z¯ = −sE (α, β). Fortunately, confusion can be avoided by specifying the reference line bundle (E or KX ⊗ E) in the subscript for s(·, ·) and S(·, ·). Moreover, for our purposes, the sign is not crucial. The behavior of the pairings with respect to these identifications is summarized in Proposition 4.6.11 below (cf. Proposition 3.10.1), the proof of which is left to the reader (see Exercise 4.6.5). The full proposition is not used in this book, so the reader may wish to omit it.

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Proposition 4.6.11 For q ∈ {0, 1} and for any holomorphic line bundle F on X, let Fq : (1,q) T ∗ X ⊗ F →(0,q) T ∗ X ⊗ KX ⊗ F denote the mapping given by Fq : α = θ ∧ γ · s = (−1)q γ ∧ θ · s → (−1)q γ · (θ ⊗ s) for each x ∈ X, θ ∈ (KX )x , γ ∈ (0,q) Tx∗ X, and s ∈ Ex . In particular, we have K ∗ ⊗F

q X : (1,q) T ∗ X ⊗ (KX∗ ⊗ F )→(0,q) T ∗ X ⊗ KX ⊗ KX∗ ⊗ F = (0,q) T ∗ X ⊗ F . (a) For each q = 0, 1 and x ∈ X, we have  KX∗ ⊗E ∗ −1   (β) sE (α, β) = (−1)1−q sKX ⊗E E q (α), 1−q if α ∈ ((1,q) T ∗ X ⊗ E)x and β ∈ ((0,1−q) T ∗ X ⊗ E ∗ )x ; and we have   K ∗ ⊗E −1 ∗ (α), E sE (α, β) = (−1)1−q sKX∗ ⊗E q X 1−q (β) if α ∈ ((0,q) T ∗ X ⊗ E)x and β ∈ ((1,1−q) T ∗ X ⊗ E ∗ )x . (b) If q ∈ {0, 1} and α is a measurable differential form of type (1, q) with values in E and β is a measurable differential form of type (0, 1 − q) with values in E ∗ on X, then  KX∗ ⊗E ∗ −1   SE (α, β) = (−1)1−q SKX ⊗E E (β) ; q (α), 1−q here the left-hand side is defined if and only if the right-hand side is defined. If q ∈ {0, 1} and α is a measurable differential form of type (0, q) with values in E and β is a measurable differential form of type (1, 1 − q) with values in E ∗ on X, then   K ∗ ⊗E −1 ∗ (α), E SE (α, β) = (−1)1−q SKX∗ ⊗E q X 1−q (β) ; here again, the left-hand side is defined if and only if the right-hand side is defined. Exercises for Sect. 4.6 Exercises 4.6.3 and 4.6.6–4.6.10 require the Serre duality theorem (and its consequences), so the reader may wish to postpone consideration of these exercises until after consideration of the proof in Sect. 4.8. 4.6.1 Verify that in the notation of Definition 4.6.1, under the identification of E ⊗ E∗ ∼ = 1X , we have sE (α, β) = α ∧ β as in Definition 3.1.14. 4.6.2 Prove part (a) of Lemma 4.6.2. 4.6.3 Let X be a compact Riemann surface of genus g. (a) Prove that if g = 1, then KX is trivial and dim (X, O(E)) = deg E for every holomorphic line bundle E of positive degree on X. (b) Prove that if g > 1, then dim (X, O(KXm )) = (2m − 1)(g − 1) for every m ∈ Z>1 .

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4.6.4 Prove Lemma 4.6.10 for p = 0. 4.6.5 Prove Proposition 4.6.11. Some facts concerning hyperelliptic Riemann surfaces and Weierstrass gaps are developed in Exercises 4.6.6–4.6.10 below (see also Exercises 5.20.3–5.20.6 and 5.20.8 and, for example, [FarK] for further discussion). 4.6.6 A compact Riemann surface is called hyperelliptic if it admits a divisor D such that deg D = 2 and h0 (D) ≥ 2. (a) Prove that a compact Riemann surface X is hyperelliptic if and only if there exists a meromorphic function on X with exactly two poles (counting multiplicity). (b) Prove that every compact Riemann surface of genus g ≤ 2 is hyperelliptic. 4.6.7 Given a point p in a compact Riemann surface X, a positive integer ν is called a gap (or a Weierstrass gap) at p if there does not exist a function f ∈ M(X) that is holomorphic on X \ {p} and that has a pole of order ν at p. A positive integer that is not a gap is called a nongap (or a Weierstrass nongap) at p. The goal of this exercise is the following: Theorem (Weierstrass gap theorem) At any point p in a compact Riemann surface X of genus g, (i) The sum of any two nongaps is a nongap; and (ii) There are exactly g gaps, and for g > 0, the gaps are given by positive integers ν1 , . . . , νg (the gap sequence) satisfying 1 = ν1 < ν2 < · · · < νg < 2g. Prove the Weierstrass gap theorem following the outline below. (a) Prove that the sum of any two nongaps at p is a nongap. (b) Prove that for g = 0, there are no gaps at p. (c) Let g > 0 and let ν ∈ Z>0 . Fixing a holomorphic section s of [p] with div(s) = p, we get the Dolbeault exact sequence (Sect. 3.4) α

0 → (X, O([(ν − 1)p])) → (X, O([νp])) → (X, Qp ([νp])) 1 1 → HDol (X, [(ν − 1)p]) → HDol (X, [νp]) → 0,

where α is the map given by u → u ⊗ s. Prove that ν is a gap if and only if α is surjective, and conclude from this that h0 ((ν − 1)p) if ν is a gap, 0 h (νp) = 0 h ((ν − 1)p) + 1 if ν is nongap. (d) Let g > 0 and let K be a divisor with [K] = KX . Use the Riemann–Roch formula in the form of Theorem 4.6.6 to prove that h0 (K − νp) + 1 if ν is a gap, h0 (K − νp + p) = 0 h (K − νp) if ν is nongap.

4.6 Statement of the Serre Duality Theorem

175

(e) Let g > 0. Prove that 1 is a gap and that each gap is at most 2g − 1. (f) Let g > 0. Prove that if ν is a gap, then there exists a holomorphic section t of [K − (ν − 1)p] with t (p) = 0. Forming such a section and multiplying by s ν−1 for each gap ν (for s as in (c)), show that one gets linearly independent holomorphic 1-forms θ1 , . . . , θr , where r is the number of gaps. Conclude that r ≤ g. (g) Let g > 0. Given a nontrivial holomorphic 1-form θ with a zero of order ν − 1 at p (θp = 0 if ν = 1), show that ν must be a gap. Show also that by subtracting a suitable multiple of one of the holomorphic 1-forms θ1 , . . . , θr produced in part (f), one gets a holomorphic 1-form that is either trivial or that has a zero of order ≥ ν at p. Using this observation, complete the proof of the theorem. 4.6.8 Let α1 < α2 < · · · < αg = 2g be the nongaps in (1, 2g] at a point p in a compact Riemann surface of genus g > 1 (as in Exercise 4.6.7). (a) Prove that αj + αg−j ≥ 2g for j = 1, . . . , g − 1. Hint. We have j + 1 nongaps αg−j < α1 + αg−j < α2 + αg−j < · · · < for each j . αj + αg−j  g (b) Prove that j =1 αj ≥ g(g + 1), and prove that equality holds if and only if αj + αg−j = 2g for j = 1, . . . , g − 1. g (c) Prove that j =1 αj = g(g + 1) if and only if (α1 , α2 , . . . , αg ) = (2, 4, . . . , 2g). Hint. If αj + αg−j = 2g for j = 1, . . . , g − 1, then in particular, we have α1 + αg−1 = 2g = αg . Hence, if g > 2, then the nongap α1 + αg−2 lies in (αg−2 , αg ). Apply this observation and induction to get α1 + αj −1 = αj for j = 2, . . . , g. 4.6.9 Let X be a compact Riemann surface of genus g > 0. (a) For each point p ∈ X, the weight (or Weierstrass weight) at p is the ing teger w(p) ≡ j =1 (νj − j ), where (ν1 , . . . , νg ) (with 1 = ν1 < · · · < νg < 2g) is the gap sequence (as in Exercise 4.6.7). Prove that for each point p ∈ X, we have 0 ≤ w(p) ≤ g(g − 1)/2. Prove also that w(p) = 0 if and only if the gap sequence is (1, 2, 3, . . . , g) (this gap sequence is called the generic gap sequence) and w(p) = g(g − 1)/2 if and only if the gap sequence is (1, 3, 5, . . . , 2g − 1) (this gap sequence is called the hyperelliptic gap sequence, and we then say that p is a hyperelliptic point). Hint. Use Exercise 4.6.8. (b) Given a basis θ = (θ1 , . . . , θg ) for (X, O(KX )), the associated Wrong(g+1)/2 )) given by skian is the section W (θ ) ∈ (X, O(KX





... fg f1



(∂/∂z)(f1 )

. . . (∂/∂z)(f ) g



W (θ ) =

· (dz)g(g+1)/2 .. .. ..



. . .



(∂/∂z)g−1 (f1 ) . . . (∂/∂z)g−1 (fg )

on each local holomorphic coordinate neighborhood (U, z), where fj ≡ θj /dz for j = 1, . . . , g. Prove that the Wronskian W (θ ) is well

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defined by the above (i.e., W (θ ) does not depend on the choice of the local holomorphic coordinate z). Prove also that if θ  = (θ1 , . . . , θg ) is another basis for (X, O(KX )), then there is a constant ζ ∈ C∗ such that W (θ  ) = ζ · W (θ ) on X. (c) A point p ∈ X is called a Weierstrass point if w(p) > 0. Prove that if θ = (θ1 , . . . , θg ) is a basis for (X, O(KX )), then the set of Weierstrass points in X is equal to the set of zeros of the Wronskian W (θ ) and the order of the zero of W (θ ) at each Weierstrass point p is equal to w(p). Hint. For g = 1, one may prove this by using the triviality of the canonical line bundle (Exercise 4.6.3). Assume that g > 1, let p ∈ X, and let (ν1 , . . . , νg ) be the gap sequence at p. Choose the basis θ = (θ1 , . . . , θg ) as in part (f) of Exercise 4.6.7. Fix a local holomorphic coordinate neighborhood (U, z) with z(p) = 0, and replace each θj with its product with a suitable constant, so that fj ≡ θj /dz = zνj −1 + higher-order terms for each j = 1, . . . , g. Given a k-tuple of holomorphic functions h = (h1 , . . . , hk ) on U with k > 1, set





... hk h1



(∂/∂z)(h1 )

. . . (∂/∂z)(h ) k



W (h) ≡

∈ O(U ). .. .. ..



. . .



(∂/∂z)k−1 (h1 ) . . . (∂/∂z)k−1 (hk )

Show that for arbitrary positive integers μ1 < · · · < μk , for hj = zμj −1  for j = 1, . . . , k, and for m = kj =1 (μj − j ), we have W (h) =



(μj − μi )zm .

1≤i<j ≤k

Conclude from this that for f = (f1 , . . . , fg ) and h = (zν1 −1 , . . . , zνg −1 ), we have ordp (W (f ) − W (h)) > w(p), and then that W (f ) has a zero of order w(p) at p. (d) Prove that p∈X w(p) = (g − 1)g(g + 1) (in particular, every compact Riemann surface of genus g > 1 has a Weierstrass point). (e) Assume that g > 1 and let m be the number of Weierstrass points. Prove the following: (i) We have 2g + 2 ≤ m ≤ (g − 1)g(g + 1). (ii) We have m = 2g + 2 if and only if X has at least 2g + 2 hyperelliptic points. In particular, m = 2g + 2 if and only if every Weierstrass point in X is a hyperelliptic point. (iii) If m = 2g + 2, then X is a hyperelliptic Riemann surface (see Exercise 4.6.6 for the definition of a hyperelliptic Riemann surface, and see Exercise 5.20.4 for the converse). (f) Prove that if g = 2, then every Weierstrass point in X is a hyperelliptic point (in particular, this gives a proof of the fact that a compact Riemann

¯ 4.7 Statement of the ∂-Hodge Decomposition Theorem

177

surface of genus 2 is hyperelliptic that differs slightly from the proof suggested by the hint in Exercise 4.6.6). 4.6.10 Prove that every nontrivial automorphism of a compact Riemann surface X of genus g > 1 has at most 2g + 2 fixed points. Hint. Given  ∈ Aut(X) \ {IdX }, fix a point p ∈ X that is not a fixed point of . Show that there exist a nongap α at p and a function f ∈ M(X) such that 1 < α ≤ g + 1, f is holomorphic on X \ {p}, and f has a pole of order α at p. Show that the meromorphic function f − f ◦  has exactly 2α zeros (counting multiplicities).

¯ 4.7 Statement of the ∂-Hodge Decomposition Theorem Throughout this section, E denotes a holomorphic line bundle on a Riemann surface X. One may relate the pairing SE (·, ·) and the L2 inner product associated to a Hermitian metric h in E by considering the unique operator ∗¯  satisfying SE (α, ∗¯  β) = α, βL2 for suitable L2 E-valued 1-forms α and β. To see this, we first observe that a Hermitian metric induces a natural conjugate linear isomorphism of E and E ∗ . Definition 4.7.1 For a Hermitian metric h in E with dual metric h∗ in E ∗ , the associated flat operator is the C ∞ conjugate linear isomorphism  = (E,h) = h = E : E → E ∗ given by

s → (s) = s ≡ h(·, s) = 

|s|2h s −1 0

if s = 0, if s = 0.

The associated sharp operator is the (C ∞ conjugate linear) inverse mapping ∗ # = #(E,h) = #h = #E ≡ −1 (E,h) : E → E,

for which we write ξ → ξ # . Remarks 1. It is easy to verify that  and # are C ∞ conjugate linear isomorphisms (see Exercise 4.7.1); that is, their restrictions to each fiber are conjugate linear isomorphisms, and s  and ξ # are C ∞ sections for all local holomorphic (or C ∞ ) sections s and ξ of E and E ∗ , respectively. It follows that if s is a section of E which is continuous (C k with k ∈ Z≥0 ∪ {∞}, measurable, Lsloc with s ∈ [1, ∞]), then s  is continuous (respectively, C k , measurable, Lsloc ). 2. One can form analogous operators for a general finite-dimensional inner product space (of arbitrary dimension). 3. This notation originated in the classical tensor calculus, in which the operators  and # denoted, respectively, the lowering and raising of indices (in analogy with the corresponding musical notation).

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Proposition 4.7.2 For any Hermitian metric h in E with corresponding dual Hermitian metric h∗ in E ∗ : (a) Under the identification ((E ∗ )∗ , (h∗ )∗ ) = (E, h), we have E ∗ = #E . (b) We have h(s, ξ # ) = ξ · s = h∗ (ξ, s  ) for all s ∈ Ex and ξ ∈ Ex∗ with x ∈ X. (c) We have h∗ (s  , t  ) = h(t, s) = h(s, t) for all s, t ∈ Ex with x ∈ X. In particular, |s  |h∗ = |s|h for all s ∈ E. (d) We have h(ξ # , ζ # ) = h∗ (ζ, ξ ) = h∗ (ξ, ζ ) for all ξ, ζ ∈ Ex∗ with x ∈ X. In particular, |ξ # |h = |ξ |h∗ for all ξ ∈ E ∗ . Proof Suppose x ∈ X, s ∈ Ex , and ξ ∈ Ex∗ with s = 0 and ξ = 0. Then h(s, ξ # ) = (#ξ ) · s = ξ · s and h∗ (ξ, s  ) = (ξ · s)h∗ (s −1 , |s|2h s −1 ) = (ξ · s)|s|2h · |s −1 |2h∗ = ξ · s. Thus (b) is proved. Part (a) follows, since (b) gives h(s, ξ  ) = s · ξ = ξ · s = h(s, ξ # ). For s and t as in (c), we have h∗ (s  , t  ) = s  · t = h(t, #s) = h(t, s). Part (d) now follows from parts (a) and (c). Definition 4.7.3 given by



The Hodge star operator ∗ : (T ∗ X)C → (T ∗ X)C is the mapping ¯ → −iα + i β¯ ∗ : (α + β)

for all α, β ∈ (1,0) Tx∗ X with x ∈ X. The conjugate Hodge star operator is the mapping ∗¯ : (T ∗ X)C → (T ∗ X)C given by ∗¯ : γ → ∗γ = ∗γ for all γ ∈ (T ∗ X)C . In other words, for a local holomorphic coordinate z, we have ∗ dz = −i dz and ∗ d z¯ = ∗ dz = i d z¯ . The proof of the following is left to the reader (see Exercise 4.7.2). Proposition 4.7.4 The Hodge star operator ∗ and conjugate Hodge star operator ∗¯ on X have the following properties: (a) For each point r ∈ X, the restrictions of ∗ and ∗¯ to (Tr∗ X)C are linear and conjugate linear, respectively, isomorphisms. If α is a continuous (C k with k ∈ Z≥0 ∪ {∞}, measurable, Lsloc with s ∈ [1, ∞]) 1-form, then ∗α and ∗¯ α are continuous (respectively, C k , measurable, Lsloc ).

¯ 4.7 Statement of the ∂-Hodge Decomposition Theorem

179

(b) We have ∗∗ = ∗¯ ∗¯ = −Id. (c) For (p, q) = (1, 0) or (0, 1), ∗ maps (p,q) T ∗ X onto (p,q) T ∗ X and ∗¯ maps (p,q) T ∗ X onto (q,p) T ∗ X. (d) Each of the operators ∗ and ∗¯ maps the real cotangent bundle T ∗ X onto itself and ∗ = ∗¯ on T ∗ X. (e) If (p, q) = (1, 0) or (0, 1), x ∈ X, and α, β ∈ (p,q) Tx∗ X, then ¯ α ∧ ∗¯ β = α ∧ ∗β = (−1)q iα ∧ β. In particular, α ∧ ∗¯ α ≥ 0. (f) Let α and β be measurable forms of degree 1 on a measurable set S ⊂ X, and let ϕ be a measurable real-valued function on S. Then (see Definition 2.6.1)  2 αL2 (S,ϕ) = α ∧ ∗¯ α · e−ϕ =  ∗ α2L2 (S,ϕ) = ¯∗α2L2 (S,ϕ) . S

Moreover, if α, β ∈ L21 (S, ϕ) (which is equivalent to ∗α, ∗β ∈ L21 (S, ϕ) as well as to ∗¯ α, ∗¯ β ∈ L21 (S, ϕ)), then  α, βL2 (S,ϕ) = ∗α, ∗βL2 (S,ϕ) = ¯∗β, ∗¯ αL2 (S,ϕ) = α ∧ ∗¯ β · e−ϕ . S

For forms with values in a holomorphic line bundle, one may combine the conjugate Hodge star operator with the flat and sharp operators as follows: Definition 4.7.5 Let h be a Hermitian metric in E, let  = (E,h) be the associated flat operator, let # = #(E,h) be the associated sharp operator, and let ∗¯ : (T ∗ X)C → (T ∗ X)C be the conjugate Hodge star operator. Then the associated conjugate Hodge star-flat operator ∗¯  = ∗¯ (E,h) = ∗¯ h = ∗¯ E : 1 (T ∗ X)C ⊗ E → 1 (T ∗ X)C ⊗ E ∗ 





and the associated conjugate Hodge star-sharp operator ∗¯ # = ∗¯ #(E,h) = ∗¯ #h = ∗¯ #E : 1 (T ∗ X)C ⊗ E ∗ → 1 (T ∗ X)C ⊗ E are the mappings given by ∗¯  (α ⊗ s) = (¯∗α) ⊗ s 

and ∗¯ # (α ⊗ ξ ) = (¯∗α) ⊗ ξ #

for α ∈ 1 (Tx∗ X)C = (Tx∗ X)C , s ∈ Ex , and ξ ∈ Ex∗ with x ∈ X. Remark The conjugate linearity of ∗¯ , , and # make the above operators well defined (and conjugate linear on the fibers), since for ζ ∈ C\{0} and for α, s, and ξ as above, we have (¯∗(ζ α)) ⊗ (s/ζ ) = ζ¯ · (1/ζ¯ ) · (¯∗α) ⊗ s  = (¯∗α) ⊗ s 

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and (¯∗(ζ α)) ⊗ (ξ/ζ )# = ζ¯ · (1/ζ¯ ) · (¯∗α) ⊗ ξ # = (¯∗α) ⊗ ξ # . 

The basic properties of ∗¯ E and ∗¯ #E and the relationship between the pairings SE (·, ·) and ·, ·L2 (X,E,h) are contained in the following proposition, the proof of which is left to the reader (see Exercise 4.7.3). Proposition 4.7.6 Given a Hermitian metric h in E with the associated Hermitian metric h∗ in E ∗ , and given a pair of indices (p, q) = (1, 0) or (0, 1), the corresponding conjugate Hodge star-flat and conjugate Hodge star-sharp operators have the following properties: (a) We have ∗¯ E ∗ = ∗¯ #E , ∗¯ #E ◦ ∗¯ E = −Id, ∗¯ E ◦ ∗¯ #E = −Id, ∗¯ E ((p,q) T ∗ X ⊗ E) = (q,p) T ∗ X ⊗ E ∗ , and ∗¯ #E ((p,q) T ∗ X ⊗ E ∗ ) = (q,p) T ∗ X ⊗ E. If α is an E-valued 1-form that is continuous (C k with Z≥0 ∪ {∞}, measurable, Lsloc with  s ∈ [1, ∞]), then ∗¯ E α is continuous (respectively, C k , measurable, Lsloc ). (b) If x ∈ X and α, β ∈ ((p,q) T ∗ X ⊗ E)x , then 













sE (α, ∗¯ E β) = (−1)q i{α, β}h = (−1)p i{¯∗E β, ∗¯ E α}h∗ . In particular, sE (α, ∗¯ E α) ≥ 0. If x ∈ X, α ∈ ((p,q) T ∗ X ⊗ E)x , and β ∈ ((q,p) T ∗ X ⊗ E ∗ )x , then 

sE (α, β) = −(−1)q i{α, ∗¯ #E β}h . (c) If α is a measurable differential form of degree 1 with values in E on X, then (see Definition 3.6.2) 



α2L2 (X,E,h) = SE (α, ∗¯ E α) = ¯∗E α2L2 (X,E ∗ ,h∗ ) . (d) The conjugate Hodge star-flat and conjugate Hodge star-sharp operators determine conjugate linear isomorphisms ∗¯ E : L2p,q (X, E, h) → L2q,p (X, E ∗ , h∗ ), 

∗¯ E : L21 (X, E, h) → L21 (X, E ∗ , h∗ ), 

∗¯ #E = −(¯∗E )−1 : L2q,p (X, E ∗ , h∗ ) → L2p,q (X, E, h), 

∗¯ #E = −(¯∗E )−1 : L21 (X, E ∗ , h∗ ) → L21 (X, E, h), 

satisfying   (i) ¯∗E α, ∗¯ E βL2 = β, αL2 , for all α, β ∈ L21 (X, E, h); (ii) ¯∗#E α, ∗¯ #E βL2 = β, αL2 , for all α, β ∈ L21 (X, E ∗ , h∗ );  (iii) α, βL2 = SE (α, ∗¯ E β), for all α, β ∈ L21 (X, E, h) (in particular, the  right-hand side SE (α, ∗¯ E β) is defined);

¯ 4.8 Proof of Serre Duality and ∂-Hodge Decomposition

181

(iv) α, ∗¯ #E βL2 = −SE (α, β) for all α ∈ L21 (X, E, h) and β ∈ L21 (X, E ∗ , h∗ ) (in particular, SE (α, β) is defined). Another goal of the remainder of this chapter is the following: ¯ Theorem 4.7.7 (∂-Hodge decomposition theorem) For every Hermitian holomorphic line bundle (E, h) on a compact Riemann surface X, we have the orthogonal decomposition ¯ 0,0 (E)(X)) E 0,1 (E)(X) = ∗¯ #E ((X, O(KX ⊗ E ∗ ))) ⊕ ∂(E with respect to the restriction of the L2 inner product ·, ·L2 (X,E,h) to E 0,1 (E)(X) 0,1 (here, we treat elements of (X, O(KX ⊗ E ∗ )) as (1, 0)-forms with values in E ∗ ). 

Remark There is also a natural definition for ∗¯ E and ∗¯ #E on forms of degree 0 and 2, but it depends on the choice of a Kähler metric. The operators map a 0-form to a 2-form and a 2-form to a 0-form. One may also define such an operator on a complex manifold of higher dimension n. The operator then maps a (p, q)-form to an (n − p, n − q)-form (see, for example, [De3] or [MKo]). In our case, we have n = 1 and (p, q) = (1, 0) or (0, 1), so (n − p, n − q) = (q, p). One should not view the conjugate Hodge star operator as simply switching the indices (p, q) in general, only in the special (1-dimensional) case considered in this book. Exercises for Sect. 4.7 4.7.1 Verify that for any Hermitian holomorphic line bundle, the associated operators  and # are C ∞ conjugate linear isomorphisms. 4.7.2 Prove Proposition 4.7.4. 4.7.3 Prove Proposition 4.7.6.

¯ 4.8 Proof of Serre Duality and ∂-Hodge Decomposition ¯ The Serre duality theorem and the ∂-Hodge decomposition theorem will be proved simultaneously using orthogonal decomposition in a Hilbert space. Throughout this section, E denotes a holomorphic line bundle on a compact Riemann surface X. By Theorem 4.2.3, we may choose divisors D and K in X with [D] = E and [K] = KX . Finally, we may fix a Hermitian metric h in E. According to Proposition 4.7.6, we then have ∗¯ #E

(X, O(KX ⊗ E ∗ )) → L21,0 (X, E ∗ , h∗ ) −→ L20,1 (X, E, h), where ∗¯ #E is a (surjective) conjugate linear isomorphism that satisfies ¯∗#E α, ∗¯ #E βL2

0,1 (X,E,h)

= β, αL2

1,0 (X,E

∗ ,h∗ )

∀α, β ∈ L21,0 (X, E ∗ , h∗ ).

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Lemma 4.8.1 In L20,1 (X, E, h), we have ¯ 0,0 (E)(X)))⊥ . ∗¯ #E ((X, O(KX ⊗ E ∗ ))) = (∂(E 1 (X, E) given by Moreover, the conjugate linear map (X, O(KX ⊗ E ∗ )) → HDol

α → [¯∗#E α]Dol ∗ 1 ∗ and the Serre mapping ιE S : (X, O(KX ⊗ E )) → (HDol (X, E)) are injective.

Proof If α ∈ (X, O(KX ⊗ E ∗ )) and t ∈ E 0,0 (E)(X), then by Proposition 4.7.6 and Lemma 4.6.2 (or Lemma 4.6.10), ¯ ∗¯ #E αL2 = −SE (∂t, ¯ α) = SE (t, ∂α) ¯ = 0. ∂t, ¯ 0,0 (E)(X)))⊥ . We have β = ∗¯ # α, where α ≡ −¯∗ β ∈ Conversely, suppose β ∈ (∂(E E E L21,0 (X, E ∗ , h∗ ), and for every section t ∈ E 0,0 (E)(X), we have ¯ ∗¯ #E αL2 = −SE (∂t, ¯ α). 0 = ∂t, Lemma 4.6.10 now implies that α is a holomorphic 1-form with values in E ∗ . That the map α → [¯∗#E α]Dol is injective follows easily. Finally, if α ∈ ker ιE S , then by Proposition 4.7.6, ∗#E α]Dol = SE (¯∗#E α, α) = −¯∗#E α2L2 . 0 = (ιE S (α)) · [¯ Thus ∗¯ #E α = 0 and hence α = 0. Therefore ιE S is injective.



Lemma 4.8.2 The following are equivalent: ¯ 0,0 (E)(X)) is closed in the subspace E 0,1 (E)(X) of the Hilbert (i) The range ∂(E 2 space L0,1 (X, E, h). ¯ 0,0 (E)(X)) (i.e., the ∂-Hodge ¯ (ii) E 0,1 (E)(X) = ∗¯ #E ((X, O(KX ⊗ E ∗ ))) ⊕ ∂(E decomposition theorem holds for E). 1 (X, E) given by (iii) The conjugate linear map (X, O(KX ⊗ E ∗ )) → HDol α → [¯∗#E α]Dol is bijective. (iv) The Serre mapping ∗ 1 ∗ ιE S : (X, O(KX ⊗ E )) → (HDol (X, E))

is bijective (i.e., the Serre duality theorem holds for E).

¯ 4.8 Proof of Serre Duality and ∂-Hodge Decomposition

183

Proof The equivalence of (i)–(iii) follows from Lemma 4.8.1 and general Hilbert space theory (see Corollary 7.5.7). For the proof that (i)–(iii) and (iv) are equivalent, we observe that given a lin1 (X, E), we get the linear functional ρ : t → −τ ([¯ ∗#E t]Dol ) ear functional τ on HDol 2 ∗ ∗ ∗ on the finite-dimensional subspace (X, O(KX ⊗ E )) ⊂ L1,0 (X, E , h ). Thus there exists an element s ∈ (X, O(KX ⊗ E ∗ )) such that ρ = ·, sL2 (X,E∗ ,h∗ ) . If 1,0

the orthogonal decomposition (ii) holds, then for each α ∈ E 0,1 (E)(X), we have ¯ for some holomorphic section t ∈ (X, O(KX ⊗ E ∗ )) and some C ∞ α = ∗¯ #E t + ∂β section β ∈ E 0,0 (E)(X). Therefore, by Proposition 4.7.6, τ ([α]Dol ) = τ ([¯∗#E t]Dol ) = −ρ(t) = −t, sL2

1,0 (X,E

= −¯∗#E t, ∗¯ #E sL2

0,1 (X,E,h)

∗ ,h∗ )

= −α, ∗¯ #E sL2

= −s, tL2

0,1 (X,E,h)

1,0 (X,E

∗ ,h∗ )

= SE (α, s)

= ιE S (s) · [α]Dol . E Thus the Serre map ιE S is surjective, and therefore, by Lemma 4.8.1, ιS is bijective. Conversely, if there exists an element

¯ 0,0 (E)(X))) \ ∂(E ¯ 0,0 (E)(X)) α ∈ E 0,1 (E)(X) ∩ cl(∂(E ⊂ (¯∗#E ((X, O(KX ⊗ E ∗ ))))⊥ , 1 (X, E) is not in the image V of ∗ ¯ #E ((X, O(KX ⊗ E ∗ ))). Hence then [α]Dol ∈ HDol 1 there exists a linear functional τ on HDol (X, E) such that τ ([α]Dol ) = 1 and τ = 0 on V. On the other hand, for each s ∈ (X, O(KX ⊗ E ∗ )), we have

¯ #E sL2 (X,E,h) = 0, ιE S (s) · [α]Dol = SE (α, s) = −α, ∗ E so ιE S (s) = τ . Thus ιS is not surjective if this is the case.



Next, we observe that in order to obtain Serre duality for the divisor D, it suffices to obtain Serre duality for some divisor ≤ D. Lemma 4.8.3 If there exists an effective divisor F on X for which the Serre map [D−F ] 1 : (X, O([K − D + F ])) → (HDol (X, [D − F ]))∗ ιS

is bijective, then the Serre map 1 ∗ ι[D] S : (X, O([K − D])) → (HDol (X, [D]))

is also bijective. Proof Fix a nontrivial section u ∈ (X, O([F ])). Given a nontrivial linear func1 (X, [D]))∗ , we may convert τ into a linear functional τ ∈ tional τ ∈ (HDol u

184

4

Compact Riemann Surfaces

1 (X, [D − F ]))∗ by setting (HDol

τu ([θ ]Dol ) ≡ τ ([θ · u]Dol )

1 ∀[θ]Dol ∈ HDol (X, [D − F ]).

In other words, τu is the composition of τ and the surjective linear mapping 1 1 1 HDol (X, [D − F ]) = HDol (X, [D] ⊗ [F ]∗ ) → HDol (X, [D])

induced by multiplication by u (as in the Dolbeault exact sequence). By hypothesis, there exists a section t ∈ (X, O([K − D + F ])) ⊂ E 1,0 ([−D + F ])(X) such that [D−F ] τu = ιS (t) = S[D−F ] (·, t).

We will now show that the meromorphic section t/u of [K − D] is actually holo1 morphic and that ι[D] S (v) = τ . Given an element [θ ]Dol ∈ HDol (X, [D]), we may ¯ = θ near supp div(u) and cut off to get a C ∞ section α form local solutions of ∂γ ¯ = θ near supp div(u). Thus, by replacing θ with θ − ∂α, ¯ of [D] on X such that ∂α we see that we may choose the representing class θ to vanish on a neighborhood of supp div(u). Therefore, θ/u extends by 0 to a well-defined C ∞ form θu of type (0, 1) with values in [D − F ] (recall that this is how we proved surjectivity of such a map in the Dolbeault exact sequence in Sect. 3.4) and we have τ ([θ ]Dol ) = τ ([θu · u]Dol ) = τu ([θu ]Dol ) = S[D−F ] (θu , t)   s[D−F ] (θ/u, t) = s[D] (θ, t/u). = X

X

In fact, for E ≡ [D], the meromorphic 1-form v ≡ t/u with values in E ∗ = [−D] is holomorphic. For given a pole p of v, we may choose a local holomorphic coordinate neighborhood (U,  = z) of p in X such that z(p) = 0, u and t are nonvanishing on U \ {p}, and there exist a nonvanishing holomorphic section s of E and a meromorphic function f with v = f dz/s on U . We may also fix a C ∞ function λ with compact support in U such that λ ≡ 1 on P ≡ −1 ((0; R))  U for some R > 0. Since 1/f has a zero at p, the section λs/(zf ) of E on U \ {p} extends ¯ = (∂λ) ¯ · s/(zf ) to a unique C ∞ section β of E on X. Moreover, the form γ ≡ ∂β vanishes outside a compact subset of U and on P . Thus   ¯ ∂λ sE (γ , v) = ∧ dz 0 = τ (0) = τ ([γ ]Dol ) = X U \P z

  dz dz =− = −2πi = 0. = d λ z U \P ∂P z We have therefore arrived at a contradiction, and hence v = t/u has no poles. Therefore, for θ as above, we get  τ ([θ]Dol ) = sE (θ, v) = SE ([θ ]Dol , v). X

Thus τ = SE (·, v) = ιE S (v) as required.



4.9 Hodge Decomposition for Scalar-Valued Forms

185

Proof of Theorem 4.6.4 and Theorem 4.7.7 According to the above lemmas, it suffices to show that for the divisor D on X, Serre duality holds for the line bundle [D − F ] for some effective divisor F . Let E = [D]. Fixing a point p ∈ X and applying Theorem 4.2.1 (or Theorem 4.3.1), we see that [−D + mp] will admit a positive-curvature Hermitian metric provided m  0. Fixing a Kähler form ω and replacing D with D − mp and E with [D − mp], we may assume without loss of generality that E admits a Hermitian metric k with −ik ≥ ω (after choosing m  0 or replacing ω with the product of ω and a small positive constant). ¯ 0,0 (E)(X)) According to Lemma 4.8.2, it remains to show that the range ∂(E 2 0,1 is closed in the subspace E (E)(X) of L0,1 (X, E, k). For this, suppose β ∈ ¯ ν − βL2 (X,E,k) → 0. E 0,1 (E)(X) and {αν } is a sequence in E 0,0 (E)(X) with ∂α Observe that the fundamental estimate (Proposition 3.7.5) gives  ¯ 2k = Dk α2k − ∂α |α|2k · ik ≥ α2L2 (X,E,ω,k) ∀α ∈ E 0,0 (E)(X). 0,0

X

It follows that the sequence {αν } is Cauchy, and hence that the sequence converges to some α ∈ L20,0 (X, E, ω, k). If γ ∈ E 0,1 (E)(X), then by Proposition 3.8.2,  −

    D γ D γ {α, Dk γ }k = − α, k = − lim αν , k ν→∞ ω L2 (X,E,ω,k) ω L2 X   ¯ ν , γ }k = − lim {αν , Dk γ }k = lim {∂α ν→∞ X

ν→∞ X

¯ ν , γ L2 = iβ, γ L2 = = i lim ∂α ν→∞

 X

{β, γ }k .

Applying Proposition 3.8.2 (again), we get ∂¯distr α = β. The regularity theorem (The¯ = β. Thus the range is closed orem 3.8.3) now implies that α is of class C ∞ and ∂α in E 0,1 (E)(X), and the theorems follow. 

4.9 Hodge Decomposition for Scalar-Valued Forms Applying Theorem 4.7.7 to the trivial line bundle, we get the following: Theorem 4.9.1 (Hodge decomposition for scalar-valued forms) If X is a compact Riemann surface, then we have the orthogonal decompositions ¯ 0,0 (X)), E 0,1 (X) = (X) ⊕ ∂(E E 1,0 (X) = (X) ⊕ ∂(E 0,0 (X)),

186

4

Compact Riemann Surfaces

E 1 (X) = (X) ⊕ (X) ⊕ d(E 0 (X)) ⊕ ∗¯ d(E 0 (X)),   d ker E 1 (X) → E 2 (X) = (X) ⊕ (X) ⊕ d(E 0 (X)), where (X) = (X, O(KX )) and (X) = {θ¯ | θ ∈ (X)} (not the closure of (X)). Remark Recall that we think of a (1, 0)-form and a (0, 1)-form as orthogonal. Proof of Theorem 4.9.1 The first equality follows from Theorem 4.7.7 and the second follows by conjugation of the spaces in the first. For the third and fourth equalities, we first observe that ∗¯ d(E 0 (X)) ⊥ ker d in 1 E (X), and hence ∗¯ d(E 0 (X)) is orthogonal to (X) ⊕ (X) ⊕ d(E 0 (X)) ⊂ ker d. For if α ∈ ker d and v ∈ E 0 (X), then Proposition 4.7.4 and Stokes’ theorem give    α, ∗¯ dvL2 = α ∧ ∗¯ ∗¯ dv = − α ∧ dv = d(αv) = 0. X

X

X

Next, we observe that the first two equalities yield the orthogonal decomposition ¯ 0,0 (X)). E 1 (X) = (X) ⊕ (X) ⊕ ∂(E 0,0 (X)) ⊕ ∂(E ¯ and ∗¯ dρ = ∗¯ ∂ρ + ∗¯ ∂ρ ¯ = ∂(i ¯ ρ) Given ρ ∈ E 0,0 (X), we have dρ = ∂ρ + ∂ρ ¯ + ∂(−i ρ). ¯ Thus ¯ 0,0 (X)). d(E 0 (X)) ⊕ ∗¯ d(E 0 (X)) ⊂ ∂(E 0,0 (X)) ⊕ ∂(E Conversely, given α, β ∈ E 0,0 (X), by setting i 1 ¯ ρ ≡ (α + β) and τ ≡ − (α¯ − β), 2 2 ¯ = dρ + ∗¯ dτ . Thus the third equality holds. The fourth equality we get ∂α + ∂β follows since the right-hand side is contained in ker d and ∗¯ d(E 0 (X)) ⊥ ker d.  For a compact Riemann surface X, the space of harmonic 1-forms is given by Harm1 (X) ≡ (X) ⊕ (X). Clearly, the space Harm1 (X) ∩ E 1 (X, R) of real harmonic 1-forms consists of all elements of the form α + α¯ with α ∈ (X). Recall that for each r ∈ Z≥0 , the rth complex de Rham cohomology (see Definition 10.6.1) of a C ∞ surface M is given by     d d r r (M) = HdeR (M, C) = ker E r (M) → E r+1 (M) /im E r−1 (M) → E r (M) , HdeR and the rth real de Rham cohomology is given by     d d r HdeR (M, R) = ker E r (M, R) → E r+1 (M, R) /im E r−1 (M, R) → E r (M, R)

4.9 Hodge Decomposition for Scalar-Valued Forms

187

(where we set d = 0 on E −1 (M) = 0). Thus we get the following consequence of Hodge decomposition (Theorem 4.9.1), the proof of which is left to the reader (see Exercise 4.9.1): Corollary 4.9.2 For any compact Riemann surface X, we have the following: 1 (X) has a unique (a) Every complex de Rham cohomology class [θ ]deR ∈ HdeR 1 1 (X) given by harmonic representative; that is, the map Harm (X) → HdeR θ → [θ ]deR is a (surjective) complex linear isomorphism. 1 (X, R) has a unique har(b) Every real de Rham cohomology class [θ]deR ∈ HdeR 1 (X, R) 1 monic representative; that is, the map Harm (X) ∩ E 1 (X, R) → HdeR given by θ → [θ ]deR is a (surjective) real linear isomorphism. 1 (X, R) = dim H 1 (X) = 2 · genus(X). Consequently, dimR HdeR C deR

Remark In particular, for any compact Riemann surface X, we have 2 · genus(X) = dimC H 1 (X, C) = dimR H 1 (X, R) = rank H 1 (X, Z) = dimC H1 (X, C) = dimR H1 (X, R) = rank H1 (X, Z) (see Sects. 10.6 and 10.7). Hence the (holomorphic) genus of a compact Riemann surface is actually a topological invariant; i.e., it does not depend on the holomorphic structure or even on the C ∞ structure. In other words, any two compact Riemann surfaces that are homeomorphic have the same genus. The converse, though true, is not proved in this book (see, for example, [T] for a proof). Theorem 4.9.3 (Poincaré duality theorem) Let X be a compact Riemann surface, and let r ∈ {0, 1, 2}. Then the map  ([α]deR , [β]deR ) → P([α]deR , [β]deR ) ≡ α∧β X

r (X) × H 2−r (X) → C. Furgives a well-defined bilinear pairing P(·, ·) : HdeR deR thermore, the map [β]deR → ιP ([β]deR ) ≡ P(·, [β]deR ) gives a (surjective) lin2−r r (X))∗ . In particular, dim H 2 (X) = (X) → (HdeR ear isomorphism ιP : HdeR C deR 0 dimC HdeR (X) = 1.

Proof The pairing is well defined. For if α ∈ E r (X) ∩ ker d, β ∈ E 2−r (X) ∩ ker d, ρ ∈ E r−1 (X) (we set ρ = 0 if r = 0), and τ ∈ E 2−r−1 (X) (we set τ = 0 if r = 2), then by Stokes’ theorem,  (α + dρ) ∧ (β + dτ ) X





=



α∧β + X



α ∧ dτ + X

dρ ∧ β + X

dρ ∧ dτ X

188

4



 =

α ∧ β + (−1) X

d(α ∧ τ ) +

r X

 =

Compact Riemann Surfaces



 d(ρ ∧ β) +

X

d(ρ ∧ dτ ) X

α ∧ β. X

1 (X), we may apply Corollary 4.9.2 to get α 1,0 ∈ (X) Now, given [α]deR ∈ HdeR 0,1 and α ∈ (X) with [α]deR = [α 1,0 ]deR + [α 0,1 ]deR , and we may choose the representative α so that α = α 1,0 + α 0,1 . For (p, q) = (1, 0) or (0, 1), we have

(−1)p i α p,q ∧ α = (−1)p i α p,q ∧ α p,q ≥ 0, p,q

with equality at a point x if and only if αx

= 0. Thus, if α p,q is nontrivial, then

P([(−1)p i α p,q ]deR , [α]deR ) > 0. It follows that ιP ([α]deR ) = 0 if and only if [α]deR = 0, and hence that ιP maps 1 (X) isomorphically onto (H 1 (X))∗ . HdeR deR 0 2 (X), let us fix a Kähler (X) = C. To determine HdeR It is easy to verify that HdeR metric g with Kähler form ω on X. To a differential form θ ∈ E 1,1 (X), we may ¯ associate a differential form θˆ of type (0, 1) with values in KX . Applying the ∂∞ Hodge decomposition theorem, we get a C section ρˆ of KX and a holomorphic section ζ of KX ⊗ KX∗ = 1X (i.e., a complex constant) with θˆ = ∗¯ #(KX ,g∗ ) ζ + ∂¯ ρ. ˆ ¯ ¯ Letting ρ be the scalar-valued (1, 0)-form associated to ρ, ˆ we get θ = −  ζ ω + ∂ρ = −ζ¯ ω + dρ (see Exercise 4.9.2). We also have [ω]deR = 0, since v ≡ X ω > 0, so 2 (X) = C[ω] ∼ HdeR deR = C. For any ξ, η ∈ C, we have (here we identify ξ and η with the corresponding constant functions x → ξ and x → η) P([ξ ]deR , [ηω]deR ) = P([ηω]deR , [ξ ]deR ) = ξ η · v, which vanishes if and only if ξ = 0 or η = 0. It now follows that the linear mappings 0 2 (X))∗ and ι : H 2 (X) → (H 0 (X))∗ are (surjective) isoιP : HdeR (X) → (HdeR P deR deR morphisms.  Exercises for Sect. 4.9 4.9.1 Prove Corollary 4.9.2. 4.9.2 Let X be a Riemann surface. According to Proposition 3.10.1, we may identify a (1, 1)-form with a corresponding (0, 1)-form with values in KX , and we may identify a (1, 0)-form with values in KX∗ with a corresponding section of KX ⊗ KX∗ = 1X , i.e., with a function. Let g be a Kähler metric with associated Kähler form ω on X, and let ζ ∈ C. Viewing the constant function ζ as a holomorphic 1-form with values in KX∗ , show that the scalar-valued (1, 1)form associated to the KX -valued (0, 1)-form ∗¯ #(KX ,g∗ ) ζ is equal to −ζ¯ ω (this fact was used in the proof of the Poincaré duality theorem).

4.9 Hodge Decomposition for Scalar-Valued Forms

189

4.9.3 Let (E, h) be a Hermitian holomorphic line bundle on a compact Riemann surface X. Prove the Dh -Hodge decomposition theorem: E 1,0 (E)(X) = (X, O(KX ⊗ E)) ⊕ Dh (E 0,0 (E)(X)). ¯ = D  (#E (iu)) for each section u ∈ Hint. First prove that ∗¯ #E (∂u) h ¯ E 0,0 (E ∗ )(X). Then apply the operator ∗¯ #E to the summands in the ∂-Hodge 0,1 ∗ decomposition for E (E )(X).

Chapter 5

Uniformization and Embedding of Riemann Surfaces

In this chapter, we consider certain complex analytic characterizations of Riemann surfaces (some topological and C ∞ characterizations appear in Chap. 6). The first goal is the following Riemann surface analogue of the classical Riemann mapping theorem in the plane: Theorem 5.0.1 (Riemann mapping theorem) A simply connected Riemann surface is biholomorphic to the Riemann sphere P1 , to the complex plane C, or to the unit disk  = {z ∈ C | |z| < 1}. Exactly one of the above outcomes will hold, since no two of the Riemann surfaces P1 , C, and  are biholomorphic (see Exercise 2.2.7). Theorem 5.0.1 is also called the uniformization theorem. Parts of the theorem were first obtained by Riemann. The first complete proofs were given by Koebe [Koe1], [Koe2] and Poincaré [P]. The proof given in this chapter is essentially due to Simha [Sim] and Demailly [De3], although the idea of forming compactifications of exhausting subdomains is due to Koebe. The theorem allows one to classify Riemann surfaces as quotients of the Riemann sphere, the plane, or the disk (see Sect. 5.9). The second goal of this chapter is the fact that every Riemann surface X may be obtained by holomorphic attachment of tubes at elements of a locally finite sequence of coordinate disks in a domain in P1 . In particular, for X compact, this allows one to form a canonical homology basis. Finally, we consider finite holomorphic branched coverings (see Sect. 5.20) as well as the following embedding theorems: • Every open Riemann surface admits a holomorphic embedding into C3 (see Sect. 5.18). This fact is due to Narasimhan [Ns1] and is the 1-dimensional version of the Bishop–Narasimhan–Remmert embedding theorem for Stein manifolds. • Every compact Riemann surface admits a holomorphic embedding into some (higher-dimensional) complex projective space (see Sect. 5.19). This fact is the 1-dimensional version of the Kodaira embedding theorem. T. Napier, M. Ramachandran, An Introduction to Riemann Surfaces, Cornerstones, DOI 10.1007/978-0-8176-4693-6_5, © Springer Science+Business Media, LLC 2011

191

192

5

Uniformization and Embedding of Riemann Surfaces

• Every compact Riemann surface of positive genus admits a holomorphic embedding into some (higher-dimensional) complex torus (see Sect. 5.22). This fact is the Abel–Jacobi embedding theorem.

5.1 Holomorphic Covering Spaces The general theory of covering spaces is reviewed in Chap. 10 (see Sects. 10.1– 10.5). In this section, we consider covering spaces of Riemann surfaces. Definition 5.1.1 A holomorphic covering space (or a 1-dimensional complex cov connected, a covering Riemann surface) of a complex ering manifold or, for X  → X in which X  is a complex 1-manifold 1-manifold X is a covering space ϒ : X and the mapping ϒ is locally biholomorphic (see Definition 10.2.1). In other words, each point in X has a connected neighborhood U such that ϒ maps each connected component of ϒ −1 (U ) biholomorphically onto U . The holomorphic cov → X is holomorphically equivalent to a holomorphic coverering space ϒ : X ˇ  that is a ˇ ing space ϒ : X → X if there exists a biholomorphism  : Xˇ → X ˇ lifting of ϒ ; i.e.,  is a fiber-preserving biholomorphism. A simply connected  → X of a Riemann surface is also called holomorphic covering space ϒ : X the holomorphic universal covering space (or universal covering Riemann surface). Remark Holomorphic equivalence of coverings is an equivalence relation (see Exercise 5.1.1).  → X be a (topoProposition 5.1.2 Let X be a complex 1-manifold and let ϒ : X logical) covering space.  is a complex 1-manifold and ϒ is holomorphic, then ϒ : X  → X is a (a) If X holomorphic covering space.  with respect to which (b) In general, there is a unique holomorphic structure on X  ϒ : X → X is a holomorphic covering space. Proof Part (a) follows from the holomorphic inverse function theorem (Theo is Hausdorff, since X is rem 2.4.4). For the proof of (b), first observe that X Hausdorff (see Proposition 10.2.10). Next observe that we may form a holomorphic atlas A = {(Ui , i , Ui )}i∈I on X such that Ui is connected and evenly covi ≡ ϒ −1 (Ui ) is the union of a colered by ϒ for each i ∈ I . For each i ∈ I , U (λ)  each of which is mapped lection of disjoint connected open sets {Ui }λ∈i in X (λ) homeomorphically onto Ui . Setting i = i ◦ ϒU (λ) , we get a complex ati

 ≡ {(U (λ) , (λ) , U  )}λ∈i ,i∈I in X.  Suppose i, j ∈ I , λ ∈ i , μ ∈ j , and las A i i i (μ) (λ) Ui ∩ Uj = ∅. Then the coordinate transformation

5.1 Holomorphic Covering Spaces

193

i ◦ [j ]−1 = i ◦ −1 j (μ) (U (λ) ∩U (μ) ) : j (Ui (λ)

(μ)

(μ)

j

i

j

(λ)

(μ)

∩ Uj )

(μ)

(λ) −→ (λ) i (Ui ∩ Uj ) ⊂ i (Ui ∩ Uj )

 Moreis holomorphic, and hence the atlas determines a holomorphic structure on X. over, ϒ is holomorphic with respect to this holomorphic structure, since −1 i ◦ ϒ ◦ [(λ) = IdUi ∈ O(Ui ) i ]

∀i ∈ I, λ ∈ i .

 with respect to which Finally, given an arbitrary holomorphic structure on X (λ) ϒ is a local biholomorphism, and given i ∈ I and λ ∈ i , the mapping i = i ◦ ϒU (λ) is a composition of a biholomorphism on Ui ⊂ X and a biholomori

(λ)

(λ)

phism on Ui , and therefore i is biholomorphic. Thus the elements of the atlas  are holomorphically compatible with the local holomorphic charts on X,  and A uniqueness follows.   → X of a complex 1-manifold X, we will Remark Given a covering space ϒ : X  assume that X has the induced holomorphic structure unless otherwise indicated.  → X be a covering RieTheorem 5.1.3 (Holomorphic lifting theorem) Let ϒ : X mann surface of a Riemann surface X, let  : Y → X be a holomorphic mapping of a Riemann surface Y to X, let y0 ∈ Y , let x0 = (y0 ), and let xˆ0 ∈ ϒ −1 (x0 ).  is a lifting of , then  : Y → X  is holomorphic. If  is a local biholo(a) If   is a local biholomorphism morphism (a holomorphic covering map), then  (respectively, a holomorphic covering map).  xˆ0 ), then there is a unique lifting of  to a holomor(b) If ∗ π1 (Y, y0 ) ⊂ ϒ∗ π1 (X,   (y0 ) = xˆ0 . Conversely, if such a lifting   exists, phic map  : Y → X with   then ∗ π1 (Y, y0 ) ⊂ ϒ∗ π1 (X, xˆ0 ). Proof The proof of (a) is left to the reader (see Lemma 10.2.4 and Exercise 5.1.2). Part (b) follows from the (topological) lifting theorem (Theorem 10.2.5) together with part (a).   → X and ϒˇ : Xˇ → X be covering Riemann surfaces Corollary 5.1.4 Let ϒ : X of a Riemann surface X, let x0 ∈ X, let xˆ0 ∈ ϒ −1 (x0 ), and let xˇ0 ∈ ϒˇ −1 (x0 ). If ˇ xˇ0 ), then there exists a unique commutative diagram of  xˆ0 ) ⊂ ϒˇ ∗ π1 (X, ϒ∗ π1 (X, holomorphic covering maps Xˇ     ϒ  ϒˇ    ϒ   X X

194

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ˇ xˇ0 ), then ϒ  xˆ0 ) = ϒˇ ∗ π1 (X,  (xˆ0 ) = xˇ0 . Moreover, if ϒ∗ π1 (X,  is a biholomorwith ϒ phism; that is, the two coverings are holomorphically equivalent. In particular, up to holomorphic equivalence of holomorphic coverings, every Riemann surface has a unique holomorphic universal covering space. Proof The holomorphic lifting theorem gives the existence and uniqueness of the commutative diagram. Corollary 10.2.6 then implies that if the image groups are  is a biholomorphism. equal, then ϒ  Recall the following (see Definition 2.2.1): Definition 5.1.5 A biholomorphism of a Riemann surface X onto itself is called an automorphism of X. The group of automorphisms of X (with product given by composition) is denoted by Aut (X). Recall that a deck transformation (or covering transformation) of a covering  → X is a homeomorphism  : X →X  such that ϒ ◦  = ϒ , and space ϒ : X the group of deck transformations is denoted by Deck(ϒ) (see Sect. 10.4). If the above is a Riemann surface covering, then the holomorphic lifting theorem (Theo Moreover, the holomorrem 5.1.3) implies that Deck(ϒ) is a subgroup of Aut (X). phic lifting theorem and Theorem 10.4.5 together give the following (the details of the proof are left to reader in Exercise 5.1.3):  → X be the (holomorphic) universal covering space of Theorem 5.1.6 Let ϒ : X  that acts propa Riemann surface X. Then  ≡ Deck(ϒ) is a subgroup of Aut(X)  (see Definition 10.4.3), and there is a unique erly discontinuously and freely on X  giving rise to a commutative diagram holomorphic structure on \X  X

 ϒ  ϒ    ∼   \X X  =  → X is a biholoin which ϒ is a holomorphic covering map and the map \X morphism.  → \X  is a holomorphic covering that we may identify In other words, ϒ : X  → X. It follows that if ϒ →X  : X with the original holomorphic covering ϒ : X  and Deck(ϒ  ) = Deck(ϒ), is the universal covering space of a Riemann surface X ∼ ∼ →X  : X then X = \X = X and we may identify the holomorphic coverings ϒ  → X. and ϒ : X Conversely, a group of automorphisms acting properly discontinuously and freely gives rise to a holomorphic universal covering space (cf. Theorem 10.4.6):  be a simply connected Riemann surface, let  be a subTheorem 5.1.7 Let X  that acts properly discontinuously and freely on X,  and let group of Aut (X)

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 → X ≡ \X  be the corresponding quotient space mapping. Then we ϒ = ϒ : X have the following: (a) There is a unique holomorphic structure on X with respect to which the quotient  → X is a holomorphic universal covering space and  = Deck(ϒ). map ϒ : X (b) If   is a subgroup of , then we have a commutative diagram  X

 ϒ        ≡  X  ϒ X  \X

ϒ

 x0 = ϒ(x˜0 ), and xˆ0 = of holomorphic covering maps. Moreover, if x˜0 ∈ X, ϒ ( x ˜ ), and  0 χ :  = Deck(ϒ) → π1 (X, x0 )

 and χ :   = Deck(ϒ  ) → π1 (X, xˆ0 )

are the corresponding group isomorphisms, then  xˆ0 ) → π1 (X, x0 ). ∗ : π1 (X, χ ◦χ  −1 = ϒ  xˆ0 ) on X  is the same as that of In other words, the action of [γˆ ] ∈ π1 (X,   ϒ∗ [γˆ ] = [ϒ (γˆ )] ∈ π1 (X, x0 ). Proof For the proof of part (a), we first observe that Theorem 10.4.6 implies that  → X is a covering map and Deck(ϒ) = . Let A be the collection of local ϒ: X complex charts in X of the form {(U,  ◦ (ϒU  )−1 , U  )}, where U is an evenly covered connected open subset of X and (U  , , U  ) is a local holomorphic chart  for which U  is a connected component of ϒ −1 (U ). We will show that A in X is a holomorphic atlas in X. Clearly, the neighborhoods in A cover X. If (U,  ◦ (ϒU  )−1 , U  ) and (V ,  ◦ (ϒV  )−1 , V  ) are two local complex charts in A and p ∈ U ∩ V , then we may choose a connected evenly covered neighborhood W of p in U ∩ V . Setting W  ≡ (ϒU  )−1 (W ), we get an element γ ∈  such that the connected open set γ (W  ) ⊂ ϒ −1 (V ) meets, and hence is contained in, V  . Thus the coordinate transformation on (W  ) =  ◦ (ϒU  )−1 (W ) is given by  ◦ (ϒV  )−1 ◦ [ ◦ (ϒU  )−1 ]−1 =  ◦ γ ◦ (ϒU  )−1 ◦ ϒ ◦ −1 =  ◦ γ ◦ −1 , and is therefore holomorphic as a composition of holomorphic maps. Thus A is a  → X is holomorphic with respect to this holoholomorphic atlas. The map ϒ : X morphic structure, since given a local holomorphic chart (U,  ◦ (ϒU  )−1 , U  ) ∈ A, we have  ◦ (ϒU  )−1 ◦ ϒ =  on U  . Finally, if A is a holomorphic atlas on X with respect to which the covering map ϒ is holomorphic, (Q, , Q ) ∈ A , and (U,  ◦ (ϒU  )−1 , U  ) ∈ A, then the coordinate transformation on  ◦ (ϒU  )−1 (U ∩ Q) is given by  ◦ [ ◦ (ϒU  )−1 ]−1 =  ◦ ϒ ◦ −1 ,

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and is therefore holomorphic as a composition of holomorphic maps. The inverse function theorem then implies that its inverse map is also holomorphic. Thus A and A determine the same holomorphic structure on X, and we have uniqueness. For the proof of (b), suppose   is a subgroup of . Then Theorem 10.4.6 gives the required commutative diagram of topological covering maps, and part (a) gives the  with respect to which ϒ holomorphic structure on X  is a holomorphic covering  → X is also holomorphic, because : X map. It follows that the covering map ϒ locally, the map may be written as the composition of the holomorphic map ϒ with a local holomorphic inverse of ϒ  . Corollary 5.1.8 Let X be a Riemann surface, let x0 ∈ X, and let  be a subgroup of π1 (X, x0 ). Then, up to holomorphic equivalence of holomorphic covering spaces,  → X and a point xˆ0 ∈ ϒ  −1 (x0 ) : X there are a unique covering Riemann surface ϒ    such that the homomorphism ϒ∗ : π1 (X, xˆ0 ) → π1 (X, x0 ) maps π1 (X, xˆ0 ) isomor → X is the universal covering space, then we have phically onto . In fact, if ϒ : X the commutative diagram ϒ    X = \X ϒ   ϒ

X  X

(here, we identify π1 (X, x0 ) ⊃  with Deck(ϒ)). Proof This follows immediately from Theorem 10.3.3 and Theorem 5.1.2 (which give the existence of the holomorphic universal cover), Theorem 5.1.7 (the construction of a holomorphic covering from a properly discontinuous free group action by automorphisms), and Corollary 5.1.4 (holomorphic equivalence of holomorphic coverings with the same image fundamental group).  We now consider some fundamental examples: Example 5.1.9 The holomorphic universal covering map of the punctured plane C∗ is the map ϒ : C → C∗ given by z → exp(2πiz). For ϒ is a surjective locally biholomorphic mapping with local inverses given by holomorphic branches of the multiple-valued holomorphic function ζ → (log ζ )/(2πi) (see Example 1.6.2). Given a point z0 = x0 + iy0 ∈ C with x0 , y0 ∈ R, we may form the connected neighborhood U ≡ {ρe2πit | 0 < ρ < ∞, x0 − (1/2) < t < x0 + (1/2)} = ϒ({x + iy | y ∈ R, x0 − (1/2) < x < x0 + (1/2)}) of ζ0 = ϒ(z0 ). The inverse image ϒ −1 (U ) is then the union of the collection of disjoint domains {Un }n∈Z , where for each n, Un is the infinite strip Un = {z = x + iy | y ∈ R, x0 − (1/2) + n < x < x0 + (1/2) + n}, which is mapped biholomorphically onto U by ϒ (see Fig. 5.1).

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Fig. 5.1 The universal covering of the punctured plane

Fix a point z0 ∈ C, let ζ0 ≡ ϒ(z0 ) ∈ C∗ , and for each t ∈ [0, 1], let γ˜ (t) = z0 + t and γ (t) = ϒ(γ˜ (t)) = ζ0 e2πit . Clearly, the map 0 : z → z + 1 is a deck transformation, so we have an injective homomorphism Z → Deck(ϒ) given by n → n0 (where n0 is the translation z → z + n). Moreover, the fiber ϒ −1 (ζ0 ) is equal to z0 + Z, so we also have surjectivity. Thus π1 (C∗ ) ∼ = Deck(ϒ) ∼ = Z. In fact, since the lifting γ˜ of γ satisfies γ˜ (0) = z0 and γ˜ (1) = z0 + 1 = 0 (z0 ), π1 (C∗ , ζ0 ) is the infinite cyclic group generated by [γ ]ζ0 . That is, [γ ]ζ0 → 1 under the composition of the isomorphism Deck(ϒ) → Z (given by  → (z0 ) − z0 ) and the isomorphism π1 (X, ζ0 ) → Deck(ϒ) (given by [α]ζ0 → , where (z0 ) = α(1) ˜ for α˜ the lifting of α with α(0) ˜ = z0 ). Any proper subgroup  of Z ∼ = Deck(ϒ) ∼ = π1 (C∗ , ζ0 ) must be of the form  = mZ for some positive integer m; that is,  is the infinite cyclic subgroup gener∗ ated by [γ ]m ζ0 . We have a holomorphic universal covering map ϒ : C → C given 2πiz/m . In fact, ϒ is the composition of the holomorphic universal covby z → e ering map C → C∗ given by z → e2πiz and the biholomorphism C → C given by z → z/m. The universal covering map ϒ has associated deck transformation group Deck(ϒ ) =  ⊂ Deck(ϒ), because  ⊂ Deck(ϒ ), and for any point z ∈ C,  · z = z + mZ = ϒ−1 (ϒ (z)). Therefore, C∗ ∼ = \C and the holomorphic covering space of C∗ associated to the subgroup  as in Corollary 5.1.8 is the finite holomorphic covering C∗ → C∗ given by exp(2πiz/m) → exp(2πiz); that is, the mapping is given by z → zm . Example 5.1.10 Let us consider complex tori from the point of view of holomorphic covering spaces (see Example 2.1.6). Recall that a lattice in C is a subgroup of the form  = Zξ1 + Zξ2 , where ξ1 , ξ2 ∈ C are linearly independent over R. We have an injective homomorphism  → Aut (C) under which each element ξ ∈  maps to the automorphism given by z → z + ξ . Thus we may identify  with a subgroup of Aut(C). Clearly,  acts freely. Moreover,  is the image of Z2 under the real linear isomorphism α : R2 → C given by (t1 , t2 ) → t1 ξ1 + t2 ξ2 , and hence  is a discrete subset of C. Hence, if K is a compact subset of C, then the set K = {ξ ∈  | |ξ | ≤ diam K} is finite. Since K ∩ (( \ K ) · K) = ∅, it follows that  must also act properly discontinuously. Thus we have the holomorphic covering map ϒ : C → X = \C, where X is a complex torus.

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Topologically, X is the torus T2 = S1 × S1 . In fact, we have a commutative diagram of C ∞ maps α  R2 C ϒ0

 T2

ϒ  β  X

where α is the real linear isomorphism (and therefore a diffeomorphism) given by (t1 , t2 ) → t1 ξ1 + t2 ξ2 (we may consider Z2 , which is mapped onto , as a group of diffeomorphisms of R2 that is given by translations and that acts properly discontinuously and freely), ϒ0 is the C ∞ covering map onto the real torus T2 = S1 × S1 given by (t1 , t2 ) → (e2πit1 , e2πit2 ), and β is the induced diffeomorphism. Example 5.1.11 The holomorphic universal covering of the punctured unit disk ∗ = ∗ (0; 1) = {ζ ∈ C | 0 < |ζ | < 1} is the holomorphic mapping ϒ : H → ∗ given by ϒ(z) = exp(2πiz) ∈ ∗ (0; 1) for each point z in the upper half-plane H ≡ {z ∈ C | Im z > 0}. For the inverse image of ∗ under the universal covering map C → C∗ given by z → exp(2πiz) (Example 5.1.9) is H, so the restriction of the covering map to H gives the universal covering of ∗ . Given a point z0 = x0 + iy0 ∈ H with x0 , y0 ∈ R, the inverse image ϒ −1 (U ) of the connected neighborhood   1 1 U ≡ ρ exp(2πit) | 0 < ρ < 1, x0 − < t < x0 + 2 2   1 1 = ϒ z = x + iy | y > 0, x0 − < x < x0 + 2 2 (an open sector) of ζ0 ≡ ϒ(z0 ) in ∗ (0; 1) is equal to the union of the collection of disjoint domains {Un }n∈Z , where for each n ∈ Z, Un is the infinite half-strip Un ≡ {z = x + iy | y > 0, x0 − (1/2) + n < x < x0 + (1/2) + n}, which is mapped biholomorphically onto U by ϒ (see Fig. 5.2). Fix a point z0 ∈ H, let ζ0 ≡ ϒ(z0 ) ∈ ∗ (0; 1), and for each t ∈ [0, 1], let γ˜ (t) = z0 + t and γ (t) = ϒ(γ˜ (t)) = ζ0 e2πit . As in Example 5.1.9, the fundamental group π1 (∗ , ζ0 ) ∼ = Deck(ϒ) ∼ = Z is the infinite cyclic (free) group generated by [γ ]ζ0 , and Deck(ϒ) is precisely the group of automorphisms given by z → z + n for n ∈ Z (i.e., the restrictions to H of deck transformations of the universal covering C → C∗ ). Any proper subgroup  of Z ∼ = Deck(ϒ) ∼ = π1 (C∗ , ζ0 ) must be of the form  = mZ for some positive integer m (that is,  is the infinite cyclic subgroup generated by [γ ]m ζ0 ), and the corresponding intermediate covering spaces are the mappings H → ∗ = \H given by z → e2πiz/m

and ∗ → ∗ given by z → zm .

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Fig. 5.2 The universal covering of the punctured unit disk

The mapping z → 1/z gives a biholomorphism ∗ → C \ (0; 1) = (0; 1, ∞), so the universal covering of (0; 1, ∞) is the mapping H → (0; 1, ∞) given by z → 1/e2πiz = e−2πiz . Example 5.1.12 For r ∈ (0, 1), the holomorphic universal covering of the annulus A ≡ (0; r, 1) = {ζ ∈ C | r < |ζ | < 1} is the holomorphic mapping ϒ : H → A given by z → exp(2πiL(z)), where   log r π =− L : H → B ≡ ζ ∈ C | 0 < Im ζ < log λ 2π is the holomorphic branch of the logarithmic function z → logλ z with base λ ≡ exp(−2π 2 / log r) > 1 (see Example 1.6.2) given by z →

1 · (log |z| + i arccot(Re z/ Im z)) log λ

(and the arccotangent function is given by arccot ≡ (cot (0,π) )−1 : R → (0, π)). For B is the inverse image of A under the universal covering map C → C∗ given by ζ → e 2πiζ (as in Examples 5.1.9 and 5.1.11), L is a biholomorphism of H onto B (with inverse map given by the exponential function ζ → eζ log λ = λζ with base λ), and ϒ is the composition of the two mappings. Fix z0 = R0 eiα0 ∈ H with R0 > 0 and 0 < α0 < π , and let ζ0 ≡ ϒ(z0 ) = ρ0 eiθ0 with ρ0 = |ζ0 | = e−2πα0 / log λ = e(α0 /π) log r

and θ0 =

(log R0 )(log r) 2π log R0 =− . log λ π

For each t ∈ [0, 1], let γ˜ (t) ≡ z0 · λt = L−1 (L(z0 ) + t)

and γ (t) ≡ ϒ(γ˜ (t)) = ζ0 · e2πit .

As in Examples 5.1.9 and 5.1.11, the deck transformations of the universal covering map B → A are precisely the translations ζ → ζ + n for n ∈ Z. Pulling back to H, we see that the deck transformations of ϒ : H → A are given by z → L−1 (L(z) + n) = exp((L(z) + n) log λ) = λn z

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Fig. 5.3 The universal covering of an annulus

for n ∈ Z. Thus π1 (A) ∼ = Deck(ϒ) ∼ = Z with generator 0 : z → λz, and in fact, π1 (A, ζ0 ) is the infinite cyclic (free) group generated by [γ ]ζ0 . Observe also that the inverse image ϒ −1 (U ) of the connected neighborhood U ≡ {ρeiθ | r < ρ < 1, θ0 − π < θ < θ0 + π} ⊂ A of ζ0 is equal to the union of the disjoint domains {Un }n∈Z , where for each n ∈ Z, Un ≡ {z ∈ C | Im z > 0, λn−(1/2) R0 < |z| < λn+(1/2) R0 } and ϒ maps Un biholomorphically onto U (see Fig. 5.3). Any proper subgroup  of Z ∼ = Deck(ϒ) ∼ = π1 (A, ζ0 ) must be of the form  = mZ for some positive integer m; that is,  is the infinite √ cyclic subgroup m m z. Setting A ˇ = (0; m r, 1), we have generated by [γ ]m , that is, by  : z →  λ 0 √ζ0 exp(−2π 2 / log m r) = λm , and hence we have a holomorphic universal covering map ϒ : H → Aˇ given by   2πi z → exp L(z) . m The universal covering map ϒ has associated deck transformation group Deck(ϒ ) =  ⊂ Deck(ϒ). Therefore, Aˇ ∼ = \H and the holomorphic covering space of A associated to the subgroup  as in Corollary 5.1.8 is the finite holomorphic covering Aˇ → A given by exp(2πiL(z)/m) → exp(2πiL(z)); that is, the mapping is given by z → zm . For s, t ∈ R with 0 < s < t, we have a biholomorphism (0; s/t, 1) → (0; s, t) given by ζ → tζ . Hence, for λ ≡ exp(−2π 2 / log(s/t)), the map H → (0; s, t) given by z → t exp(2πiL(z)/ log λ) is the holomorphic universal covering of (0; s, t). As in the case t = 1, every finite covering of (0; s, t) is √ √ holomorphic equivalent to a covering of the form (0; m s, m t) → (0; s, t) given by z → zm for some m ∈ Z≥0 . This characterization of finite coverings of annuli will be applied in the proof of the Riemann mapping theorem in Sect. 5.5. Moreover, it turns out that two annuli (0; r, 1) and (0; s, 1) with r, s ∈ (0, 1) are biholomorphic if and only if r = s. Equivalently, two annuli (0; r, s) and (0; R, S) with 0 < r < s < +∞ and 0 < R < S < +∞ are biholomorphic if and only if r/s = R/S (see Exercise 5.8.2).

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Exercises for Sect. 5.1 5.1.1 Prove that holomorphic equivalence of holomorphic covering spaces is an equivalence relation. 5.1.2 Prove part (a) of Theorem 5.1.3. 5.1.3 Prove Theorem 5.1.6. 5.1.4 Prove that for r ∈ (0, 1), the universal covering map ϒ : H → (0; r, 1) appearing in Example 5.1.12 extends to a universal covering map H \ {0} → (0; r, 1) (where H and (0; r, 1) are the closures of H and (0; r, 1), respectively). 5.1.5 For each loop γ in C and each point z0 ∈ C \ γ ([0, 1]), the winding number of γ around z0 is given by n(γ ; z0 ) ≡ γ˜ (1) − γ˜ (0), where γ˜ is a lifting of γ under the universal covering mapping C → C \ {z0 } given by ζ → z0 + e2πiζ (see Example 5.1.9). (a) Show that the winding number is an integer that is independent of the choice of the lifting of the loop to the universal cover. (b) Show that for distinct points z0 , z1 ∈ C, the mapping [γ ]z1 → n(γ ; z0 ) ∼ =

determines a well-defined group isomorphism π1 (C \ {z0 }, z1 ) −→ Z. (c) Prove that for each loop γ in C and each point z0 ∈ C \ γ ([0, 1]),  1 dz (see Definition 10.5.3). n(γ ; z0 ) = 2πi γ z − z0 5.1.6 Consideration of winding numbers (see Exercise 5.1.5) yields versions of the Cauchy integral formula and Cauchy’s theorem (cf. Lemma 1.2.1 and Exercise 6.7.1), the residue theorem (cf. Exercises 2.5.9, 5.9.5, 6.7.1, and 6.7.6), and the argument principle and Rouché’s theorem (cf. Exercises 2.5.8, 2.9.5, 6.7.1, and 6.7.6) for general loops. Let  be a domain in C, let γ be loop in  that is path homotopic in  to a constant loop, and let C ≡ γ ([0, 1]). (a) Cauchy’s theorem. Prove that for each function f ∈ O(),  f (z) dz = 0. γ

Hint. Apply Theorem 10.5.6. (b) Cauchy integral formula. Prove that for each function f ∈ O() and each point z0 ∈  \ C,  f (z) 1 dz = f (z0 )n(γ ; z0 ). 2πi γ z − z0 (c) Prove that there is a domain 0 such that C ⊂ 0   and γ is path homotopic in 0 to a constant loop. Conclude that in particular, n(γ ; z0 ) = 0

∀z0 ∈ C \ 0 .

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(d) Residue theorem. Prove that if S is a discrete subset of  that is contained in  \ C, and θ is a holomorphic 1-form on  \ S, then 1 2πi

 θ= γ



n(γ ; p)resp θ

p∈S

(note that the right-hand side is actually a finite sum by part (c)). Hint. Using part (c), show that one may assume without loss of generality that S is a finite set. Writing θ = f (z) dz, and letting gp be the principal part of f about p (i.e., the sum of the negative-exponent terms of the Laurent

series about p) for each p ∈ S, one gets a holomorphic 1-form θ − [ p∈S gp (z)] dz on . (e) Argument principle. Prove that if f is a nontrivial meromorphic function on  with zero set Z and pole set P both contained in  \ C, then 1 2πi

 γ

df n(γ ; z0 ) ordz0 f. = f z0 ∈Z∪P

(f) Rouché’s theorem. Suppose f and g are nontrivial meromorphic functions on  that do not have any zeros or poles in C, and |g| < |f | on C. Let Zf denote the zero set of f , Pf the pole set of f , Zf +g the zero set of f + g, and Pf +g the pole set of f + g. Prove that



n(γ ; z0 ) ordz0 f =

z0 ∈Zf ∪Pf

n(γ ; z0 ) ordz0 (f + g).

z0 ∈Zf +g ∪Pf +g

5.1.7 The goal of this problem to obtain more general (homological) versions of the facts considered in Exercise 5.1.6. Let  be a domain in C, let γ1 , . . . , γm  be loops in , let C ≡ m γ j =1 j ([0, 1]), and let a1 , . . . , am ∈ C. Assume that  m aj β =0 γj

j =1

for every C ∞ closed

1-form β on  (in the language of Sects. 10.6 and 10.7, the 1-cycle ξ = aj γj is homologous to 0 in ). Note that according to Theorem 10.5.6, this is the case if, for example, γj is path homotopic in  to a constant loop for each j . (a) Cauchy’s theorem. Prove that for each function f ∈ O(), m j =1

 f (z) dz = 0.

aj γj

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(b) Cauchy integral formula. Prove that for each function f ∈ O() and each point z0 ∈  \ C,  m m aj f (z) dz = f (z0 ) aj n(γj ; z0 ). 2πi γj z − z0 j =1

j =1

(c) Prove that there is an open set 0 such that C ⊂ 0   and m

aj n(γj ; z0 ) = 0

∀z0 ∈ C \ 0 .

j =1

Hint. Observe that the left-hand side of the above equation is a continuous function of z0 . (d) Residue theorem. Prove that if S is a discrete subset of  that is contained in  \ C, and θ is a holomorphic 1-form on  \ S, then  m m aj θ= aj n(γj ; p)resp θ 2πi γj j =1

p∈S j =1

(note that the right-hand side is actually a finite sum by part (c)). Hint. In the proof of part (d) of Exercise 5.1.6, one reduced to the case in which S is finite; but it is not clear that one can do so here (although it is possible to do so using, for example, the homological fact considered in Exercise 5.17.5). Assuming that S is infinite, let {pn } be an enumeration of S, and writing θ = f (z) dz, let gn be the principal part of f about pn for each n (see part (d) of Exercise 5.1.6). Fix a sequence of compact sets {Kν } such that ◦  Kν ⊂ K ν+1 and h (Kν ) = Kν for each ν and  = ∞ ν=1 Kν . For each n, apply the Runge approximation theorem to get a holomor−n on K whenever phic function hn on  such ν

∞ that |gn − hn | < 2 pn ∈ Kν . Show that θ − [ n=1 (gn − hn )] dz is a holomorphic 1-form on . (e) Argument principle. Prove that if f is a nontrivial meromorphic function on  with zero set Z and pole set P both contained in  \ C, then  m m aj df aj n(γj ; z0 ) ordz0 f. = 2πi γj f j =1

z0 ∈Z∪P j =1

(f) Rouché’s theorem. Suppose f and g are nontrivial meromorphic functions on  that do not have any zeros or poles in C, and |g| < |f | on C. Let Zf denote the zero set of f , Pf the pole set of f , Zf +g the zero set of f + g, and Pf +g the pole set of f + g. Prove that

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aj n(γj ; z0 ) ordz0 f

z0 ∈Zf ∪Pf j =1

=



m

aj n(γj ; z0 ) ordz0 (f + g).

z0 ∈Zf +g ∪Pf +g j =1

5.2 The Riemann Mapping Theorem in the Plane The goal of this section is the following: Theorem 5.2.1 (Riemann mapping theorem in the plane) Any simply connected domain  in C with  = C is biholomorphic to the unit disk  = (0; 1). The proof given here is due to Fejér and F. Riesz. We first consider the following: Proposition 5.2.2 Let f be a nonvanishing holomorphic function on a simply connected Riemann surface X. Then there exists a holomorphic function h such that e h = f on X (i.e., h is a single-valued holomorphic branch of the function log f ). In particular, for every positive integer q, there is a holomorphic function k = e h/q such that k q = f (i.e., k is a single-valued holomorphic qth root of f ). Proof By Corollary 10.5.7, the holomorphic 1-form θ = df/f is exact, and hence ¯ is equal to the (0, 1) there is a holomorphic function h on X such that dh = θ (∂h part of θ , which is 0). Fixing a point p ∈ X and a single-valued holomorphic branch g of the function log f on a connected neighborhood U of p, we may choose h so that h(p) = g(p). Since dh = θ = dg on U , it follows that the holomorphic functions eh and f agree on U and therefore on X.  The biholomorphism will be obtained as a limit of a sequence of holomorphic mappings, so the following observation will be required: Lemma 5.2.3 Let X be a Riemann surface, and let {fν } be a sequence of injective holomorphic functions on X (i.e., a sequence of biholomorphic mappings of X onto open sets in C) that converges uniformly on compact subsets of X to a nonconstant holomorphic function f . Then f is injective. Proof Given a point a ∈ X, the identity theorem (Theorem 2.2.2) implies that we may choose a relatively compact connected neighborhood D of a with f (∂D) ⊂ C \ {f (a)}. Hence there exists a δ > 0 such that |f − f (a)| > δ on ∂D, and for ν  0, we also get |fν − fν (a)| > δ on ∂D. For ν  0, we have δ > |fν (a) − f (a)|, and it follows that f (a) ∈ fν (D), because any point ζ in the complement of the domain fν (D) satisfies |fν (a) − ζ | ≥ dist(fν (a), ∂(fν (D))) = dist(fν (a), fν (∂D)) ≥ δ.

5.2 The Riemann Mapping Theorem in the Plane

205

Now if a0 and a1 are distinct points in X, then we may choose disjoint neighborhoods D0 and D1 of a0 and a1 , respectively, as above with f (aj ) ∈ fν (Dj ) for j = 1, 2 and ν  0. For such an index ν, we also have fν (D0 ) ∩ fν (D1 ) = ∅ (since fν is injective), and hence f (a0 ) = f (a1 ).  The first step in the proof of Theorem 5.2.1 is the following: Lemma 5.2.4 Let  ⊂ C be a domain such that  = C and such that any nonvanishing holomorphic function on  has a single-valued holomorphic square root, and let a ∈ . Then there exists an injective holomorphic map f :  →  ≡ (0; 1) with f (a) = 0. Proof Fixing a point z0 ∈ C \ , we may form a function h ∈ O() such that [h(z)]2 = z − z0 for each point z ∈ . In particular, h is injective. Moreover, no two points w, z ∈  can satisfy h(w) = −h(z). For if such a pair exists, then squaring, we get w = z. But then h(w) = h(z) = 0, and squaring again, we get / . By the open mapping theorem, h() z − z0 = 0, which is impossible, since z0 ∈ is a domain, and hence for some δ > 0, we have (h(a); δ) ⊂ h(). It follows that h() ∩ (−h(a); δ) = ∅. For if ζ ∈ (−h(a); δ), then −ζ ∈ (h(a); δ) ⊂ h(). Since h() ∩ (−h()) = ∅, we must have ζ ∈ / h(). It is now easy to check that for any sufficiently large constant R > 0, the function f on  given by f (z) =

1 h(z) − h(a) · R h(z) + h(a)

∀z ∈ 

has the required properties (see Exercise 5.2.1).



Theorem 5.2.1 is an immediate consequence of Proposition 5.2.2 and the following version: Theorem 5.2.5 Let  ⊂ C be a domain such that  = C and such that any nonvanishing holomorphic function on  has a single-valued holomorphic square root. Then  is biholomorphic to the unit disk  ≡ (0; 1). Proof Fixing a point a ∈ , Lemma 5.2.4 implies that F ≡ {f ∈ O() | f is injective, f () ⊂ , and f (a) = 0} = ∅. In particular, M ≡ supf ∈F |f  (a)| ∈ (0, ∞]. We may choose a sequence {fν } in F such that |fν (a)| → M as ν → ∞, and after applying Montel’s theorem (Corollary 1.2.7) and passing to a subsequence, we may assume without loss of generality that {fν } converges uniformly on compact subsets of  to a holomorphic function f . Corollary 1.2.6 then implies that |f  (a)| = M. In particular, M ∈ (0, ∞), f is nonconstant, and therefore by Lemma 5.2.3, f is injective. We also have |f | ≤ 1, and hence the maximum principle implies that |f | < 1 on .

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It remains to show that f () = . For each point c ∈ , let c be the automorphism of the disk given by c (z) =

z−c 1 − cz ¯

∀z ∈ 

(see Proposition 2.14.8). If there exists a point c ∈  \ f () ⊂ C∗ , then the injective holomorphic function c ◦ f :  →  is nonvanishing, and hence there exists an injective holomorphic function h :  →  such that h2 = c ◦ f . Differentiation at a then gives 2|c|1/2 |h (a)| = (1 − |c|2 )M. The injective holomorphic function g ≡ h(a) ◦ h :  →  satisfies g(a) = 0, and hence g ∈ F . However, we have |g  (a)| = (1 − |c|)−1 · (1 − |c|2 )M/(2 |c|) = (1 + |c|)M/(2 |c|) > M. Thus we have arrived at a contradiction, and therefore f is a biholomorphism of  onto the disk .  Corollary 5.2.6 If  is a simply connected domain in P1 , and P1 \  contains more than one point, then  is biholomorphic to (0; 1). Proof Fixing a point ξ ∈ C \  and replacing  with its image under the automorphism P1 → P1 given by z → 1/(z − ξ ) if necessary, we may assume that  is a proper domain in C. The claim now follows from Proposition 5.2.2 and Theorem 5.2.5.  The proof of the Riemann mapping theorem gives the following: Corollary 5.2.7 If  is a domain in P1 and every closed C ∞ 1-form on  is exact, then  is biholomorphic to P1 , to C, or to (0; 1). In particular,  is simply connected. The proof is left to the reader (see Exercise 5.2.2). Remark According to the Osgood–Taylor–Carathéodory theorem (see [OT] and [C]), if  ⊂ P1 is a simply connected region and ∂ is a Jordan curve (see Sect. 5.10), then any biholomorphism of  onto the disk  may be extended to a homeomorphism of  onto . This fact is useful (and appealing), but it is not proved (or applied) in this book. However, a related fact called Schönflies’ theorem (Theorem 6.7.1) is considered in Chap. 6. Exercises for Sect. 5.2 5.2.1 Verify that the function f constructed in the proof of Lemma 5.2.4 has the required properties. 5.2.2 Prove Corollary 5.2.7.

5.3 Holomorphic Attachment of Caps

207

Fig. 5.4 Three disks (or caps) and a domain with three boundary coordinate annuli

5.3 Holomorphic Attachment of Caps As considered in Sect. 2.3, one may construct examples of Riemann surfaces by holomorphic attachment. In this section, we consider holomorphic attachment of caps (i.e., disks), which in particular, may be used to compactify a suitable domain. Such compactifications will play an important role in the proof of the Riemann mapping theorem for a Riemann surface. Let X be a complex 1-manifold; let  ⊂ X be an open set; let {R0j }j ∈J and {R1j }j ∈J be collections of numbers in (1, ∞); let {(Uj , j , (0; 1/R0j , R 1j ))}j ∈J be a locally finite family of local holomorphic charts in X with ∂ ⊂ j ∈J Uj ; R1j )), Dj ≡ (0; R1j ), and Aj ≡ and for each j ∈ J , let Aj ≡ −1 j ((0; 1, (0; 1, R1j ) ⊂ Dj . Assume that W ∩  ⊂ j ∈J Aj for some neighborhood W of ∂ in X, and that for each index j ∈ J , Aj ⊂ , Aj ∩ Ak = ∅ for each k ∈ J \ {j }, and Uj ∩ ∂ = −1 j (∂(0; 1)) (note that the sets {Uj } need not be disjoint; for example, we may let  ≡ C \ ∂(0; 1) ⊂ X ≡ C, U0 = U1 ≡ (0; 1/2, 2), 0 : z → 1/z, 1 : z → z, A0 ≡ (0; 1/2, 1), and A1 ≡ (0; 1, 2)). Then we have the biholomorphism : A≡

j ∈J

∼ =

Aj −→ A ≡

 j ∈J

Aj ⊂ D ≡



Dj ,

j ∈J

where for every index j ∈ J and every point p ∈ Aj , (p) is the image of j (p) under the holomorphic inclusion Aj → A (see Fig. 5.4, in which J = {1, 2, 3}). The set  −1 (K  ∩ A ) is closed in  for every compact set K  ⊂ D, and the set (K ∩A) is closed in D for every compact set K ⊂ . Thus we get the holomorphic

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Fig. 5.5 Holomorphic attachment of the caps

attachment Y ≡ D ∪  = D  /∼, where for j ∈ J , p ∈ Aj , and z ∈ Aj , the images p0 of p and z0 of z under the inclusions Aj , Aj → D   satisfy p0 ∼ z0 if and only if j (p) = z. In particular, for the holomorphic inclusions  :  → Y and Dj : Dj → Y for j ∈ J , we have

Dj ((0; 1)).  () = Y \ j ∈J

In other words, Y is a complex 1-manifold obtained by gluing caps (i.e., disks) at the boundary of  (see Fig. 5.5). We say that Y is obtained by holomorphic attachment of caps (or disks) to  at the boundary. Observe that Y is compact if and only if   X. If Y is connected, then  is connected, but the converse is false. Observe also that if in some local holomorphic coordinate neighborhood of some boundary component,  is the complement of a concentric circle in an annulus, then we are separating the two pieces and gluing a cap over each piece (as pictured in Fig. 5.5, one cuts a tube into two new tubes, and then places a cap over each of the two new ends). This process of holomorphic tube removal will be considered in more detail in Sect. 5.12. It will be applied in the characterization of a Riemann surface by holomorphic tube attachment (see Sects. 5.13 and 5.14).

5.4 Exhaustion by Domains with Circular Boundary Components

209

Although it is natural to picture D as being glued onto  (instead of  being glued onto D),  is chosen above to map a subset of  into the union of disks D (instead of the reverse direction). The mapping is chosen in the above direction in order to make the notation fit with the notation of holomorphic tube attachment and removal (see Sect. 2.3 and Sect. 5.12). The points of view are, of course, equivalent, since we have D ∪  =  ∪ −1 D. ∗ with 1 < R ∗ ≤ R for each ν ∈ {0, 1} and j ∈ J , setting Fixing a number Rνj νj νj  ∗ ∗ Dj ≡ (0; R1j ) ⊂ Dj for each j ∈ J , and setting D ∗ ≡ j ∈J Dj∗ ⊂ D, we may form the holomorphic attachment Y ∗ = D ∗  / ∼. The natural inclusion D ∗   ⊂ ∼ =

D   then descends to a natural biholomorphism Y ∗ −→ Y (see Exercise 5.3.1), so we may identify Y ∗ with Y . In particular, we may always choose R0j , R1j ∈ (1, ∞) to be equal and arbitrarily close to 1 for each j ∈ J . Exercises for Sect. 5.3 5.3.1 In the notation of this section, verify that the natural inclusion of D ∗   into D   descends to a biholomorphism of Y ∗ onto Y (cf. Exercise 2.3.4).

5.4 Exhaustion by Domains with Circular Boundary Components In this section, we see that every Riemann surface admits an exhaustion by relatively compact domains for which each boundary component is a concentric circle in a coordinate annulus, a fact that will play an important role in the proof of the Riemann mapping theorem. More precisely, we obtain the following: Proposition 5.4.1 Given a compact subset K of a Riemann surface X, there exist a compact Riemann surface X  , a finite (possibly empty) collection of disjoint local holomorphic charts {(Dj , j , (0; Rj ))}nj=1 in X  with Rj > 1 for each j , and a biholomorphism of a connected relatively compact neighborhood  of K in X onto  X  \ nj=1 j−1 ((0; 1)).  Remark By replacing  with the inverse image of X \ nj=1 j−1 ((0; Rj∗ )), where Rj∗ ∈ (1, Rj ) is sufficiently close to 1 for each j , we see that we may choose  so that there exist constants {Rj }nj=1 in (1, ∞) and disjoint local holomorphic charts {(Uj , j , (0; 1/Rj , Rj ))}nj=1 in X such that ∂ ⊂ U1 ∪ · · · ∪ Un

and such that for each j = 1, . . . , n, we have Uj ∩ ∂ = −1 j (∂(0; 1)) and

Aj ≡ −1 j ((0; 1, Rj )) = Uj ∩  (in particular,  is a smooth domain). In fact, in the proof of Proposition 5.4.1, X is constructed by holomorphic attachment of caps to such a domain. We first consider the following:

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Lemma 5.4.2 Suppose f is a holomorphic function on a Riemann surface X, s0 is a positive regular value of the function ρ ≡ |f |,  is a connected component of {x ∈ X | ρ(x) < s0 }, and S is a nonempty compact connected component of ∂. Then √ there exist m ∈ Z>0 and r, t ∈ R with 0 < r < s ≡ m s0 < t, and a local holomorphic chart (A,  = z, (0; r, t)), such that f A = zm and A ∩  = −1 ((0; r, s))

and S = A ∩ ∂ = −1 (∂(0; s)).

Proof At each point x ∈ S, we have f (x)(df )x + f (x)(df )x = (dρ 2 )x = 2ρ(x)(dρ)x = 0. Hence (df )x = 0, and by the holomorphic inverse function theorem (Theorem 2.4.4), we may form a local holomorphic chart of the form (Ux , x = f Ux , Ux ) such that x ∈ Ux and the sets Ux ∩ (0; s0 ) and Ux ∩ ∂(0; s0 ) are connected. In particular, the connected set x−1 (Ux ∩ (0; s0 )) = Ux ∩ f −1 ((0; s0 )) = Ux ∩ ρ −1 ((−∞, s0 )) must meet, and therefore lie in, the connected component  of ρ −1 ((−∞, s0 )). Thus Ux ∩  = x−1 (Ux ∩ (0; s0 )), and similarly, x−1 (Ux ∩∂(0; s0 )) = Ux ∩f −1 (∂(0; s0 )) = Ux ∩ρ −1 (s0 ) = Ux ∩∂ = Ux ∩S.  Fixing a relatively compact connected neighborhood M of S in x∈S Ux , we see that f M is a local biholomorphism and M ∩  = M ∩ f −1 ((0; s0 ))

and

S = M ∩ ∂ = M ∩ f −1 (∂(0; s0 )).

In particular, the restriction f S : S → ∂(0; s0 ) is a local diffeomorphism, and hence by part (a) of Lemma 10.2.11, a finite C ∞ covering map. Therefore f (S) = ∂(0; s0 ), and by part (b) of Lemma 10.2.11, there exist constants r0 and t0 and a relatively compact connected neighborhood A of S in M such that 0 < r0 < s0 < t0 and such that the restriction f A : A → (0; r0 , t0√ ) is a finite holomorphic covering √ map. As in Example 5.1.12, for r = m r0 and t = m t0 for some choice of m ∈ Z>0 , we have a commutative diagram   (0; r, t) A f A

 (0; r0 , t0 ) in which  is a biholomorphism and the map (0; r, t) → (0; r0 , t0 ) is the finite holomorphic covering map given by z → zm . Thus  has the required properties.  Proof of Proposition 5.4.1 Clearly, we may assume that X is noncompact and that K is nonempty. According to Lemma 2.16.5, there exist a holomorphic function f on a domain Y ⊃ K, a positive regular value s0 of the function |f |, and a

5.5 Koebe Uniformization

211

C ∞ domain  that is a connected component of {x ∈ Y | |f (x)| < s0 } satisfying K ⊂   Y . In particular, ∂ is a compact C ∞ submanifold of Y . For each connected component S of ∂, Lemma 5.4.2 provides a positive in√ teger m, constants r and t with 0 < r < s ≡ m s0 < t, and a local holomorphic −1 ((0; r, s)), , (0; r, t)) such that S =  −1 (∂(0; s)), V ∩  =  chart (V ,  m  . Fixing R with s/t < 1/R < 1 < R < s/r and setting  ≡ and f V =  )U , where U ≡  −1 ((0; s/R, sR)), we then get a local holomorphic chart (s/ (U, , (0; 1/R, R)) such that S = −1 (∂(0; 1)) and U ∩  = −1 ((0; 1, R)). Applying this to each boundary component, we get constants {Rj }nj=1 in (1, ∞) and disjoint local holomorphic charts {(Uj , j , (0; 1/Rj , Rj ))}nj=1 in X such that ∂ ⊂ U1 ∪ · · · ∪ Un and such that for each j = 1, . . . , n, we have Uj ∩ ∂ = −1 −1 j (∂(0; 1)) and Aj ≡ j ((0; 1, Rj )) = Uj ∩ . As in Sect. 5.3, we may now let Dj ≡ (0; Rj ) for each j = 1, . . . , n, and we may let X  be the holomorphic attachment given by 

X ≡

n 

Dj  /∼,

j =1

where for each j = 1, . . . , n, for p ∈ Aj , and for z ∈ Aj ≡ (0; 1, Rj ) ⊂ Dj ,  the images p0 of p and z0 of z under the inclusions Aj , Aj → nj=1 Dj   satisfy p0 ∼ z0 if and only if j (p) = z. The natural inclusion of  is a bi holomorphism of  onto X  \ nj=1 j−1 ((0; 1)), where for each j = 1, . . . , n,

 j ≡ −1 Dj : Dj (Dj ) → Dj is the local holomorphic chart in X given by the in verse of the holomorphic inclusion Dj : Dj → X .

5.5 Koebe Uniformization We are now ready to prove the Riemann mapping theorem for a Riemann surface (Theorem 5.0.1). In fact, we will prove something stronger. Definition 5.5.1 An orientable C ∞ surface M is planar if every closed C ∞ 1-form with compact support in M is exact. Remarks 1. Of course, every exact 1-form θ of class C ∞ is C ∞ -exact; in fact, every potential for θ is of class C ∞ . 2. Planarity of second countable orientable C ∞ surfaces is actually a topological condition (see Exercise 5.5.2). Lemma 5.5.2 Let M be an orientable C ∞ surface. (a) If M is simply connected, then M is planar. (b) If M is planar, then every domain in M is planar. (c) Every domain in the Riemann sphere P1 is planar.

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Proof Part (a) follows immediately from Corollary 10.5.7. For part (b), we simply observe that a closed C ∞ 1-form with compact support in a domain  in an orientable planar C ∞ surface M extends by 0 to a C ∞ compactly supported closed differential 1-form on M, and therefore the form is exact. Finally, part (c) follows from (a) and (b).  In particular, the Riemann mapping theorem (Theorem 5.0.1) is an immediate consequence of the Riemann mapping theorem in the plane (Theorem 5.2.1) and the following: Theorem 5.5.3 (Koebe uniformization theorem [Koe1], [Koe2], [Koe3], [Koe4]) Any planar Riemann surface is biholomorphic to a domain in P1 . The proof given here is essentially due to Simha [Sim] and Demailly [De3], although the idea of forming compactifications of exhausting subdomains is due to Koebe. We first consider the compact case. Lemma 5.5.4 Any compact planar Riemann surface is biholomorphic to P1 . In fact, there exists a universal constant C > 0 such that if X is any compact planar Riemann surface, p is any point in X, and (D,  = z, (0; 1)) is any local holomorphic chart with p ∈ D and (p) = z(p) = 0, then there exists a biholomorphism F : X → P1 for which (i) The meromorphic function F − (1/z) has a removable singularity at p (in particular, F (p) = ∞); and (ii) We have dF L2 (X\D) ≤ C and zdF + z−1 dzL2 (D) ≤ C. Proof By Theorem 2.10.1, there exists a universal constant C > 0 such that if X is any compact planar Riemann surface, p is a point in X, and (D,  = z, (0; 1)) is a local holomorphic chart with p ∈ D and (p) = z(p) = 0, then there exists a meromorphic 1-form θ on X with the following properties: (i) The meromorphic 1-form θ is holomorphic on X \ {p} and has a pole of order 2 at p; (ii) We have θL2 (X\D) ≤ C; (iii) The meromorphic 1-form θ + z−2 dz on D has at worst a simple pole at p; and (iv) We have zθ + z−1 dzL2 (D) ≤ C (note that the meromorphic 1-form provided by Theorem 2.10.1 has been multiplied by −1). For some constant a ∈ C and for some holomorphic function g on (0; 1), we have   a 1 θ = − 2 + + g(z) · dz on D. z z In particular, by Stokes’ theorem,  2πia =

∂−1 ((0;1/2))

 θ =−

X\−1 ((0;1/2))

dθ = 0.

5.5 Koebe Uniformization

213

The holomorphic function g has a primitive, i.e., a function G ∈ O((0; 1)) with G = g (one may see this by observing that the closed 1-form g(z) dz must be exact, since (0; 1) is simply connected; or one may apply Theorems 1.3.1 and 1.3.2). The meromorphic function h0 on D given by h0 = z−1 + G(z) then satisfies dh0 = ∂h0 = θ on D \ {p}. Fixing a C ∞ function η with compact support in D that is equal to 1 on −1 ((0; 1/2)), we see that θ −d(ηh0 ) extends from X \{p} to a closed C ∞ 1-form θ1 on X (with θ1 ≡ 0 near p). Since X is planar, there exists a C ∞ function h1 on X with dh1 = θ1 . Thus F ≡ ηh0 + h1 is a C ∞ function on X \ {p} that satisfies ¯ = 0 (the (0, 1) part of θ is 0), so F is holomorphic on dF = θ . In particular, ∂F X \ {p}. Furthermore, the holomorphic function F − (1/z) on D \ {p} extends to a C ∞ , and therefore holomorphic, function on D (which is equal to G(z) + h1 on −1 ((0; 1/2))). Hence F is a meromorphic function on X that is holomorphic on X \ {p} and that has a simple pole at p, and Proposition 2.5.7 now implies that F is a biholomorphism of X onto P1 . The L2 bounds for dF = θ follow from the construction of θ .  Lemma 5.5.5 Let M be an orientable C ∞ surface, let r, R ∈ R with 0 < r < R, let (D, , (0; R)) be a local C ∞ chart in M, and let N = M \  −1 ((0; r)). Then N is planar if and only if M is planar. Proof It is easy to see that N is connected, and Lemma 5.5.2 implies that N is planar if M is planar. For the converse, suppose N is planar and θ is a closed C ∞ 1-form with compact support in M. Corollary 10.5.7 provides a C ∞ function u on D with du = θ D . Fixing a C ∞ function η with compact support in D such that η ≡ 1 on  −1 ((0; s)) for some s ∈ (r, R), we see that (after extending ηu by 0 to all of M), θ0 ≡ θ − d(ηu) is a closed C ∞ 1-form with compact support in N . Therefore, since N is planar, there exists a C ∞ function v on N with dv = θ0 N . On the other hand, since dv = θ0 ≡ 0 on the coordinate annulus  −1 ((0; r, s)), the function v must be equal to a constant on this set, and we may assume without loss of generality that this constant is 0. Thus v extends by 0 to a C ∞ function v0 on M with dv0 = θ0 . Therefore θ = θ0 + d(ηu) = d(v0 + ηu) is exact and M is planar.  Proof of Theorem 5.5.3 By Lemma 5.5.4, it remains to prove that any given noncompact planar Riemann surface X is biholomorphic to a domain in P1 . We may applyProposition 5.4.1 to get a sequence of nonempty domains {ν }∞ ν=1 such that X= ∞ ν=1 ν and such that for each ν, we have ν  ν+1 and there exist a compact Riemann surface Xν , a finite collection of disjoint local holomorphic charts (ν) (ν) (ν) (ν) ν ν in Xν with {Rj }nj =1 in (1, ∞), and a biholomor{(Dj , j , (0; Rj ))}nj =1 nν (ν) −1 phism ν of ν onto Xν \ j =1 (j ) ((0; 1)). Let us fix a point p ∈ 1 and a local holomorphic chart (D,  = z, (0; 1)) with p ∈ D  1 and (p) = z(p) = 0. For each ν, Lemma 5.5.2 implies that ν is planar and therefore, by Lemma 5.5.5, Xν is planar (by downward induction  (ν) on k, one sees that the Riemann surface Xν \ kj =1 (j )−1 ((0; 1)) is planar for k = 1, . . . , nν ). Therefore, by Lemma 5.5.4, for some constant C > 0 (independent of ν) and for each ν, there exists a biholomorphism Fν : Xν → P1 such that for

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fν ≡ Fν ◦ ν : ν → P1 , the meromorphic function fν − (1/z) on D has a removable singularity at p and the meromorphic 1-form θν ≡ dfν satisfies θν L2 (ν \D) ≤ C

and zθν + z−1 dzL2 (D) ≤ C.

Fixing a point q ∈ 1 \ {p}, we may also assume that fν (q) = 0 for each ν ∈ Z>0 (simply by replacing Fν with Fν − Fν (ν (q))). We will show that by passing to a subsequence, we may assume that the sequence of meromorphic functions {fν } converges to the desired biholomorphism of X onto a domain in P1 . For this, we observe that for each compact set K ⊂ X \ {p} and each ν ∈ Z>0 with ν ⊃ K, we have θν L2 (K) = θν L2 (K\D) + θν L2 (K∩D)   1  −1  ≤ θν L2 (ν \D) +  + z dz) + z−2 dzL2 (K∩D) (zθ z ν  2 L (K∩D) ≤C+

C + z−2 dzL2 (K∩D) = AK , minK∩D |z|

where clearly, the positive constant AK is independent of the choice of ν. Given a point s ∈ X \ {p}, we may fix a local holomorphic chart (U,  = ζ, (0; 2)) with U  X \ {p} and s = −1 (0). For some ν0 ∈ Z>0 , we have U ⊂ ν and θν = (∂fν /∂ζ )dζ on U for all ν ≥ ν0 . The above L2 -estimate and the L∞ /Lp estimate (Theorem 1.2.4) together imply that the moduli of the holomorphic functions {∂fν /∂ζ }ν≥ν0 are uniformly bounded by some positive constant B (depending on (U, ζ )) on the set V ≡ −1 ((0; 1)). Setting s = q, we see that q lies in the set Q of points x ∈ X \ {p} for which there exist a neighborhood W and a positive integer μ such that {fν W }ν≥μ is uniformly bounded. Clearly, Q is open, and given a point x ∈ Q, we may choose the above point s and local holomorphic chart (U,  = ζ, (0; 2)) so that s ∈ Q and x ∈ V . It follows that Q = X \ {p}, and hence {fν } is uniformly bounded on each compact subset of X \ {p}. Thus, after applying Montel’s theorem (Corollary 2.11.4) and passing to a suitable subsequence, we may assume without loss of generality that the sequence {fν } converges uniformly on compact subsets of X \ {p} to a holomorphic function f on X \ {p} with f (q) = 0. Furthermore, for each ν, the holomorphic function gν ≡ fν − (1/z) on D (recall that fν − (1/z) has a removable singularity at p) satisfies      ∂fν 1   ∂gν    z z · + =  ∂z  ∂z  2 z L2 (D,(i/2) dz∧d z¯ ) L (D,(i/2) dz∧d z¯ )   1  1   = √ zθν + dz ≤ C. z L2 (D) 2 Thus, after again passing to a subsequence, we may also assume that the sequence of holomorphic functions {z · ∂gν /∂z} converges uniformly on compact subsets of

5.5 Koebe Uniformization

215

D to a holomorphic function h with h(p) = 0. In particular, on D \ {p}, h ∂fν ∂gν 1 1 ∂f ← = − 2→ − 2 ∂z ∂z ∂z z z z

as ν → ∞.

The function h/z (with the singularity at p removed) is holomorphic on D, and therefore this function has a primitive. It follows that f is a meromorphic function on X, f is holomorphic on X \ {p}, f has a simple pole at p, and f − (1/z) has a removable singularity at p. In particular, f is nonconstant and Lemma 5.2.3 implies that f X\{p} : X \ {p} → C is injective. Since the holomorphic mapping f : X → P1 satisfies f (p) = ∞, f is injective on all of X, and therefore f is a biholomorphism of X onto a domain in P1 .  The Koebe uniformization theorem and Corollary 5.2.7 together give the following: Corollary 5.5.6 If X is Riemann surface and every closed C ∞ 1-form on X is exact, then X is biholomorphic to P1 , to C, or to (0; 1). In particular, X is simply connected. We also have the following equivalent version (see Sect. 10.6): 1 (X, C) = 0, then X is biholoCorollary 5.5.7 If X is a Riemann surface and HdeR 1 morphic to P , to C, or to (0; 1). In particular, X is simply connected.

Remark As shown in Sects. 10.6 and 10.7, the vanishing of the complex (de Rham) 1 (X, C) ∼ H 1 (X, C) is equivalent to the vanishing of cohomology vector space HdeR = 1 (X, R) ∼ H 1 (X, R), as well as to the real (de Rham) cohomology vector space HdeR = the vanishing of H1 (X, R) (in fact, for any choice of a subring A of C containing Z, it is equivalent to the vanishing of H1 (X, A)). Exercises for Sect. 5.5 5.5.1 The goal of this problem is a stronger version of the Koebe uniformization theorem. Prove that there exists a universal constant C > 0 such that if X is any planar Riemann surface, p is a point in X, and (D, z = , (0; 1)) is a local holomorphic chart with p ∈ D and (p) = z(p) = 0, then there exists a biholomorphism f of X onto a domain in P1 such that (i) The meromorphic function f − (1/z) has a removable singularity at p (in particular, f (p) = ∞); and (ii) We have df L2 (X\D) ≤ C and zdf + z−1 dzL2 (D) ≤ C. 5.5.2 This problem requires the topological characterization of cohomology in ˇ terms of Cech 1-forms appearing in Sect. 10.7. Let M and N be homeomorphic second countable orientable C ∞ surfaces. Prove that M is planar if and only if N is planar.

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5.6 Automorphisms and Quotients of C The uniformization theorem allows one to obtain a characterization of Riemann surfaces as quotient spaces of P1 , C, and  = (0; 1). This characterization is considered in Sects. 5.6–5.9. We first determine, in this section, all of the Riemann surfaces with universal covering space C. For this, we first observe the following: Theorem 5.6.1 Let (a,b) : C → C be the mapping z → az + b for each pair (a, b) ∈ C∗ × C. Then we have the following: (a) For each pair (a, b) ∈ C∗ × C, the map (a,b) : C → C is an automorphism of C. This automorphism has a fixed point (given by b/(1 − a)) if and only if a = 1. (b) Aut (C) = {(a,b) | (a, b) ∈ C∗ × C}. Proof Part (a) is easily verified. For the proof of (b), suppose  ∈ Aut (C). Then (z) → ∞ as z → ∞, so  extends to an automorphism of P1 that fixes ∞. In particular, the function z → ((z) − (0))/z is holomorphic on P1 (since the meromorphic function  − (0) on P1 has a simple pole at ∞ and a simple zero at 0) and therefore this function is constant.  Remark The pair (G ≡ C∗ × C, ·), where (a, b) · (c, d) = (ac, ad + b) ∀(a, b), (c, d) ∈ G, is a group, and the map G → Aut (C) given by (a, b) → (a,b) (for (a,b) as above) is a (surjective) group isomorphism (see Exercise 5.6.1). Examples 5.1.9 and 5.1.10, along with the following lemma, imply that any Riemann surface with holomorphic universal cover C must be (up to biholomorphism) C, C∗ , or a complex torus (this will be examined further in Sect. 5.9). Lemma 5.6.2 For any subgroup  of Aut (C) that acts properly discontinuously and freely, and for ϒ = ϒ : C → X = \C the associated Riemann surface quotient, we have the following: (a)  is a group of translations, and either  = {0}, or  = Zξ for some ξ ∈ C∗ , or  is a lattice (here, we identify any ζ ∈ C with the translation z → z + ζ ). (b) If  = {0}, then X = C and ϒ = IdC . (c) If  = Zξ for some ξ ∈ C∗ , then ϒ is holomorphically equivalent to the covering ϒ0 : C → C∗ given by z → e2πiz (as in Example 5.1.9). More precisely, there is a commutative diagram C ϒ0

 C∗

α  C ϒ β   X

5.6 Automorphisms and Quotients of C

217

in which α is the linear isomorphism given by z → ξ z and β is a biholomorphism. (d) If  is a lattice, then there exists a complex number τ such that Im τ > 0 and such that the holomorphic covering space ϒ0 : C → X0 = 0 \C corresponding to the lattice 0 ≡ Z + Zτ is holomorphically equivalent to ϒ . More precisely, for some (nonunique) τ ∈ C with Im τ > 0, there is a commutative diagram C ϒ0

 X0

α  C ϒ β   X

in which α is a linear isomorphism (with α(0 ) = ) and β is a biholomorphism. Moreover, if X  =   \C is any complex torus that is biholomorphic to X, then ϒ0 : C → X0 is also equivalent in the above sense to the covering ϒ  : C → X . Proof By Theorem 5.6.1, every automorphism  of C must be of the form z → az + b for constants a ∈ C∗ and b ∈ C. Moreover,  has a fixed point if and only if a = 1. Therefore, every element of  must be a translation, and hence we may identify  with a subgroup of C. Now  must be a discrete set in C. For if there is a sequence {ξν } in  that converges to a point ξ that is not an element of the sequence, then choosing a compact set K containing a neighborhood of {0, ξ }, we see that 0 + ξν ∈ K ∩ (K + ξν ) for all large ν, which is impossible since  acts properly discontinuously. In particular, if  = {0}, then we may choose an element ξ1 ∈  \ {0} of minimal modulus. We then have  ∩ (Rξ1 ) = Zξ1 . For if r ∈ R and rξ1 ∈ , then rξ1 − rξ1 (where r denotes the floor of r) is an element of  with modulus (r − r)|ξ1 | < |ξ1 |, and hence r ∈ Z. The complement of Zξ1 in  must be a closed set that does not meet Rξ1 . Hence, if this complement is nonempty, then we may choose an element ξ2 ∈  with ξ2 ∈ Zξ1 of minimal distance from [0, 1] · ξ1 , and an element uξ1 ∈ [0, 1] · ξ1 at which this minimal distance is attained. For any ζ ∈  \ Zξ1 and t ∈ R, we have |ζ − tξ1 | = |(ζ − tξ1 ) − (t − t)ξ1 | ≥ |ξ2 − uξ1 | > 0, so ξ2 is actually of minimal distance from Rξ1 , and this minimal distance is attained at uξ1 . In particular, ξ1 and ξ2 are linearly independent over R. It also follows that  = Zξ1 + Zξ2 . For given an element ξ ∈ , there must exist real numbers t1 , t2 ∈ R with ξ = t1 ξ1 + t2 ξ2 . In order to show that ξ ∈ Zξ1 + Zξ2 , we may assume without loss of generality that t2 ∈ [0, 1), since we may replace ξ with t1 ξ1 + (t2 − t2 )ξ2 . Thus ξ is an element of  with distance from Rξ1 at most |ξ − (t1 + t2 u)ξ1 | = t2 |ξ2 − uξ1 | < |ξ2 − uξ1 |. Therefore, by the choice of ξ2 , we must have ξ ∈ Zξ1 . Thus the claim (a) is proved.

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The claim (b) is trivial, and the claim (c) is easy to verify (see Exercise 5.6.2). For the proof of (d), let us assume that  is a lattice generated by ξ1 , ξ2 ∈ C. Since we may replace ξ2 with −ξ2 if necessary, we may choose the generators so that the number τ ≡ ξ2 /ξ1 satisfies Im τ > 0. The linear isomorphism α : C → C given by z → ξ1 z then maps the lattice 0 ≡ Z + Zτ onto . It is now easy to verify that we get the desired commutative diagram. Finally, suppose ϒ  : C → X  is the universal covering of a complex torus X  =   \C, and γ : X → X is a biholomorphism. Then γ lifts to an automorphism γ˜ of the universal covering C, and hence γ˜ must be of the form z → az + b with a ∈ C∗ and b ∈ C. For each element ξ ∈ , we have aξ = γ˜ (ξ ) − γ˜ (0) ∈   and hence a ⊂   . By symmetry, we then have a −1   ⊂  and hence a =   . Thus ξ1 ≡ aξ1 , ξ2 ≡ aξ2 generate   , and ξ2 /ξ1 = ξ2 /ξ1 = τ . Thus we get the anal ogous commutative diagram for C → X . Exercises for Sect. 5.6 5.6.1 In the notation of Theorem 5.6.1, show that the pair (G ≡ C∗ × C, ·), where (a, b) · (c, d) = (ac, ad + b) for all (a, b), (c, d) ∈ G, is a group, and that the map G → Aut (C) given by (a, b) → (a,b) is a (surjective) group isomorphism. 5.6.2 Prove part (c) of Lemma 5.6.2.

5.7 Aut (P1 ) and Uniqueness of the Quotient Suppose  : P1 → P1 is an automorphism of P1 . The mapping  : P1 → P1 given by (0) = 0, (∞) = ∞, and, for z ∈ C∗ , ⎧ (z)−(0) if (0), (∞) ∈ C, ⎪ ⎨ (z)−(∞) (z) = (z) − (0) if (∞) = ∞, ⎪ ⎩ 1 if (0) = ∞ (z)−(∞) (for (0), (∞) ∈ C, we set (−1 (∞)) = 1) is then an automorphism of P1 . For  is a bijective continuous mapping that is holomorphic on C \ {−1 (∞)}, so Riemann’s extension theorem (Theorem 2.2.2) implies that  is a bijective holomorphic mapping. In particular, the restriction C : C → C is an automorphism of C with (0) = 0, and hence by Theorem 5.6.1, there exists a constant α ∈ C∗ such that (z) = αz for each z ∈ P1 . Solving this equation for (z), we see that there exist constants a, b, c, d ∈ C such that for each point z ∈ C \ {−1 (∞)}, we have cz + d ∈ C∗ and (z) =

az + b . cz + d

5.7 Aut (P1 ) and Uniqueness of the Quotient

219

Given two points z1 , z2 ∈ C \ {−1 (∞)}, we have z1 = z2

⇐⇒

az1 + b az2 + b = cz1 + d cz2 + d

⇐⇒

(ad − bc)(z1 − z2 ) = 0.

Choosing the two points to be distinct, we see that ad − bc = 0. Furthermore, for z ∈ C \ {−1 (∞)}, we have az + b → a/c cz + d

(= ∞

if c = 0)

as z → ∞

and (since a(−d/c) + b = 0 for c = 0 and a(−d/c) + b = ∞ for c = 0) az + b →∞ cz + d

as z → −d/c

(= ∞ if c = 0).

Therefore, (∞) = a/c and (−d/c) = ∞. We express this by saying that the equation (z) =

az + b cz + d

holds for all z ∈ P1 . Definition 5.7.1 A Möbius transformation is a mapping  : P1 → P1 = C ∪ {∞} of the form az + b  : z → , cz + d where a, b, c, d ∈ C with ad − bc = 0 (here, we set (∞) = a/c and (−d/c) = ∞). Observe that the constants a, b, c, d giving a Möbius transformation  as above are not unique, since for any constant λ ∈ C∗ , the constants λa, λb, λc, λd give the same Möbius transformation . As shown above, every automorphism of P1 is a Möbius transformation. The converse, as well as other appealing facts, also hold. For the statements, it will be convenient to have the following terminology: Definition 5.7.2 Let F = R or C, and let n ∈ Z>0 . (a) The general linear group GL(n, F) over F is the group of nonsingular n × n matrices with entries in F (with product given by matrix multiplication). (b) The special linear group SL(n, F) over F is the subgroup of GL(n, F) given by the nonsingular n × n matrices with entries in F and determinant 1. (c) The projectivized special linear group PSL(n, F) over F is the quotient of the group SL(n, F) by the normal subgroup {I, −I }; that is, PSL(n, F) = SL(n, F)/A ∼ −A.

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Remark It is easy to see that GL(n, R), SL(n, R), and PSL(n, R) are subgroups of GL(n, C), SL(n, C), and PSL(n, C), respectively. Theorem 5.7.3 The Möbius transformations have the following properties: (a) The Möbius transformations together with the product given by composition form a group that is equal to the automorphism group of P1 . (b) We have a surjective homomorphism GL(2, C) → Aut (P1 ) given by   az + b a b ∀z ∈ P1 . A= → A , where A (z) = c d cz + d (c) The map [A] →  A (for A as in (b)) is a well-defined (surjective) isomorphism of PSL(2, C) onto Aut (P1 ). (d) Every Möbius transformation that is not the identity has exactly one or two fixed points in P1 . In particular, up to biholomorphism, the only Riemann surface with universal covering P1 is P1 itself. (e) Given ζ1 , ζ2 , ζ3 , ξ1 , ξ2 , ξ3 ∈ P1 with ζi = ζj and ξi = ξj for all i, j ∈ {1, 2, 3} with i = j , there exists a unique Möbius transformation  with (ζj ) = ξj for j = 1, 2, 3. (f) If C is a circle in P1 (i.e., C is a circle in C or C = L ∪ {∞} for some line L in C), then the image of C under any Möbius transformation is also a circle in P1 . The proof is left to the reader (see Exercise 5.7.1). Exercises for Sect. 5.7 5.7.1 Prove Theorem 5.7.3 (part of it was already proved in the discussion preceding the statement of the theorem).

5.8 Automorphisms of the Disk The following fact will allow us to characterize the automorphism group of the disk: Theorem 5.8.1 (Schwarz’s lemma) Let f :  →  be a holomorphic mapping of the unit disk  ≡ (0; 1) to itself with f (0) = 0. Then we have |f (z)| ≤ |z| for all z ∈  and |f  (0)| ≤ 1. Moreover, if |f (z)| = |z| for some point z ∈  \ {0} or |f  (0)| = 1, then there is a constant c ∈ C such that |c| = 1 and f (z) = cz for all z ∈ . Proof Riemann’s extension theorem (Theorem 1.2.10) implies that the (continuous) function h on  given by  f (z)/z if z ∈  \ {0}, h(z) = f  (0) if z = 0,

5.8 Automorphisms of the Disk

221

is holomorphic. Moreover, we have lim supz→c |h(z)| ≤ 1 for each point c ∈ ∂ and therefore, by the maximum principle, |h| ≤ 1 on . Furthermore, if |h(z)| = 1 for some point z ∈ , then h is equal to a constant c of modulus 1.  Theorem 5.8.2 (Cf. Proposition 2.14.8) For each point c ∈  ≡ (0; 1), let c : P1 → P1 be the map given by c (z) =

z−c 1 − cz ¯

∀z ∈ P1 .

Then we have the following: (a) For each point c ∈ , c is a Möbius transformation, c (c) = 0, c (0) = 1 − |c|2 , and c (c) = 1/(1 − |c|2 ). (b) For each choice of c ∈  and b ∈ ∂, the Möbius transformation  ≡ bc ¯ = b ¯ −bc (z),  maps  onto , and the has inverse z → −1 (z) = −c (bz) restriction  :  →  is an automorphism of . Furthermore, every automorphism of  is of this form. Proof For c ∈ , we have 1 · 1 − c¯ · c = 1 − |c|2 > 0, so c is a Möbius transformation. The remaining properties in (a) are left to the reader (see Exercise 2.14.5). We have  −1   1 1 −c 1 c = · , −c¯ 1 c¯ 1 1 − |c|2 and hence for each point z ∈ P1 , −1 c (z) =

z+c = −c (z). 1 + cz ¯

Therefore, for b ∈ ∂ and  = bc , we have ¯ = −1 (z) = −c (bz)

¯ +c bz ¯ 1 + c¯bz

= b¯

z + bc 1 + bcz

¯ −bc (z) = b

∀z ∈ P1 .

It is easy to check that (∂) ⊂ ∂, and hence by the above, we have (∂) = ∂. In particular, () must be either  or P1 \ , and it follows that () = , since (c) = 0 ∈ . Therefore,  :  →  is an automorphism. Finally, suppose  is an arbitrary automorphism of . Then 0 ≡ (0) ◦  is an automorphism of  that maps 0 to 0. But then −1    (−1 0 ) (0) · 0 (0) = (0 ◦ 0 ) (0) = 1.

Since by the Schwarz lemma, each of the factors in the first of the above expressions has modulus at most 1, each must have modulus equal to 1. Therefore, again by the Schwarz lemma, there is a constant b ∈ ∂ such that 0 (z) = bz for all z ∈ . Therefore, for all z ∈ , (z) = −(0) (bz) = b−b(0) (z).  ¯

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It is often more convenient to work with the upper half-plane H = {z ∈ C | Im z > 0}, which is biholomorphic to the unit disk . Theorem 5.8.3 For H = {z ∈ C | Im z > 0} and  = (0; 1), we have the following: −iz+i (a) The mapping  : P1 → P1 given by z → i 1−z 1+z = z+1 is a Möbius transformation that maps  onto H and ∂ onto R ∪ {∞}. (b) The set GL+ (2, R) ≡ {A ∈ GL(2, R) | det A > 0} is a subgroup of GL(2, R), and we have a surjective homomorphism GL+ (2, R) → Aut (H) given by   a b A= → A H , c d

where A is the Möbius transformation given by A (z) =

az + b cz + d

∀z ∈ P1 .

(c) The map [A] →  A H (for A as in (b) and for [A] the equivalence class represented by A) is a well-defined surjective isomorphism of PSL(2, R) onto Aut (H). Proof We have (−i)(1) − (1)(i) = −2i = 0, so  is a Möbius transformation. Moreover, (1) = 0, (i) = 1, and (−1) = ∞. Therefore, by Theorem 5.7.3,  maps ∂, which is the unique circle containing 1, i, −1, onto the unique circle containing 0, 1, ∞, i.e., onto R ∪ {∞}. In particular,  must map  onto exactly one of the two connected components of P1 \ (R ∪ {∞}), and therefore, since (0) = i ∈ H, we have () = H. Thus (a) is proved. If A = ac db ∈ GL+ (2, R), then clearly, the corresponding Möbius transformation az + b A : z → cz + d maps the circle R ∪ {∞} onto the circle R ∪ {∞}, and therefore A maps H onto one of the connected components of P1 \ (R ∪ {∞}). On the other hand, since ad − bc = det A > 0, we have A (i) =

ai + b ac + bd ad − bc = 2 +i 2 ∈ H. 2 ci + d c +d c + d2

Thus A H ∈ Aut (H). For the converse, observe that any automorphism of H is of the form  ◦  ◦  −1 H for some automorphism  of , and hence by Theorem 5.8.2 and Theorem 5.7.3, the automorphism must be of the form H for some Möbius transformation  with (H) = H and (R ∪ {∞}) = R ∪ {∞}. Guided by the arguments

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223

in Sect. 5.7, let us set  A=

a c

⎧ μ −μ(0) ⎪ ⎪ 1 −(∞) ⎪  ⎪ ⎨ b μ −μ(0) = 0 1 d ⎪ ⎪ ⎪  ⎪ ⎩ 0 μ  1 −(∞)

if (0), (∞) ∈ R, if (∞) = ∞, if (0) = ∞,

where μ = ±1 is chosen so that det A > 0, i.e., so that A ∈ GL+ (2, R). Then A ◦  is a Möbius transformation that maps 0 to 0, ∞ to ∞, R ∪ {∞} to R ∪ {∞}, and H to H, and hence A ◦  must be of the form z → rz for some r ∈ (0, ∞) ⊂ R.  Therefore, for B = A−1 0r 10 ∈ GL+ (2, R), we have  = B , and applying Theorem 5.7.3, we get (b) and (c).  Remark Clearly, Theorem 5.7.3 provided the recipe for the construction of the biholomorphism  of  onto H; we only needed to choose a Möbius transformation mapping three points arranged in counterclockwise order on ∂ to three corresponding points arranged in the same order on the circle R ∪ {∞} oriented with H on the left. On the other hand, one can also verify the claims in the theorem directly without applying Theorem 5.7.3. Next, we see that Example 5.1.11 and Example 5.1.12 are the only examples of Riemann surfaces with universal covering  ∼ = H and infinite cyclic fundamental group. Lemma 5.8.4 If X is a Riemann surface with universal covering  ≡ (0; 1) ∼ = H and fundamental group π1 (X) ∼ = Z, then X is biholomorphic to the punctured disk ∗ ≡ ∗ (0; 1) or to the annulus (0; r, 1) for some r ∈ (0, 1). Proof We have the universal covering map ϒ : H → X = \H for some infinite cyclic subgroup  of Aut (H) that acts properly discontinuously and freely. By Theorem 5.8.3,  is generated by H for some Möbius transformation  of the form  : z → (az + b)/(cz + d) with a, b, c, d ∈ R and ad − bc = 1. In particular,  has one or two fixed points in P1 , and since  has no fixed points in H and (¯z) = (z) for each point z ∈ P1 , the fixed point or points must lie in R ∪ {∞}. Suppose  has exactly one fixed point λ ∈ R ∪ {∞}. The Möbius transformation  −1 if λ ∈ R, 1 : z → z−λ z if (∞) = ∞, restricts to an automorphism of H and maps λ to ∞. Hence the Möbius transformation 1 ◦  ◦ 1−1 also restricts to an automorphism of H and has the unique fixed point ∞. It follows that 1 ◦  ◦ 1−1 is a translation z → z + β for some constant β ∈ R∗ = R \ {0}. By replacing  with −1 (which has the same unique fixed point λ) if necessary, we may assume that β > 0. Thus the Möbius transformation

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 : z → β −1 1 (z) restricts to an automorphism of H and the Möbius transformation 0 =  ◦  ◦  −1 is given by z → z + 1. Letting 0 ∼ = Z be the subgroup of Aut(H) generated by 0 H , we get a commutative diagram of holomorphic mappings H ϒ

 X

  H ϒ0 0   ∗

where ϒ0 is the universal covering map z → e2πiz as in Example 5.1.11, and 0 is a biholomorphism. Let us now assume that  has exactly two fixed points μ, ν ∈ R ∪ {∞}, which we may order so that μ > ν if μ, ν ∈ R. The Möbius transformation ⎧ z−μ ⎪ if μ, ν ∈ R, ⎨ z−ν  : z → z − μ if ν = ∞, ⎪ ⎩ −1 if μ = ∞, z−ν restricts to an automorphism of H and maps μ to 0 and ν to ∞. Hence the Möbius transformation 0 =  ◦  ◦  −1 also restricts to an automorphism of H and has fixed points 0 and ∞. It follows that 0 is a dilation z → λz for some constant λ > 0, and by replacing  with −1 (which has the same fixed points) if necessary, we may assume that λ > 1. Letting 0 ∼ = Z be the subgroup of Aut (H) generated by 0 H and r ≡ exp(−2π 2 / log λ) ∈ (0, 1), we get a commutative diagram of holomorphic mappings H ϒ

 X

  H ϒ0 0   (0; r, 1)

where ϒ0 is the universal covering map z → exp(2πiL(z)) (where L is the branch of logλ z given by z → log1 λ (log |z| + i arccot(x/y)) for each z = x + iy ∈ H) as in  Example 5.1.12, and 0 is a biholomorphism. Exercises for Sect. 5.8 5.8.1 Prove part (a) of Theorem 5.8.2. 5.8.2 Prove that two annuli (0; r, 1) and (0; s, 1) with r, s ∈ (0, 1) are biholomorphic if and only if r = s.

5.9 Classification of Riemann Surfaces as Quotient Spaces The results of the previous sections may be summarized as follows:

5.9 Classification of Riemann Surfaces as Quotient Spaces

225

Theorem 5.9.1 Up to biholomorphism, every Riemann surface X is biholomorphic to exactly one of the following: (i) (ii) (iii) (iv) (v)

P1 ; C; C∗ ; \C, where  = Z + τ Z and τ ∈ C with Im τ > 0; \, where  is a subgroup of   z−c Aut () = bc | b ∈ S1 , c ∈ , and c : z → 1 − cz ¯ that acts properly discontinuously and freely (equivalently, \H, where  is a subgroup of   az + b | a, b, c, d ∈ R with ad − bc = 1 ∼ Aut (H) =  : z → = PSL(2, R) cz + d that acts properly discontinuously and freely).

The Riemann mapping theorem also allows one to characterize Riemann surfaces according to their fundamental groups. For example, the Riemann mapping theorem, Lemma 5.6.2, and Lemma 5.8.4 give the following characterization of Riemann surfaces with infinite cyclic fundamental group: Theorem 5.9.2 If X is a Riemann surface with fundamental group π1 (X) ∼ = Z, then X is biholomorphic to exactly one of the following: (i) C∗ = C \ {0}; (ii) ∗ = ∗ (0; 1) = {z ∈ C | 0 < |z| < 1}; (iii) the annulus (0; r, 1) = {z ∈ C | r < |z| < 1} for some r ∈ (0, 1). Remark According to Exercise 5.8.2, the constant r in (iii) is unique (for a given X).  Then X is Definition 5.9.3 Let X be a Riemann surface with universal covering X. called ∼ (i) elliptic if X = P1 ; ∼ (ii) parabolic if X = C; ∼ (iii) hyperbolic if X =∼ = H. Remarks 1. By the Riemann mapping theorem, every Riemann surface is of exactly one of the above types. 2. A complex torus, that is, a compact Riemann surface with universal covering C, is also called an elliptic curve (which should not be confused with the elliptic Riemann surface P1 ).

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Exercises for Sect. 5.9 5.9.1 Let  = Z + Zτ and   = Z + Zτ  be lattices in C with Im τ, Im τ  > 0 (as in Lemma 5.6.2). Prove that the complex tori X ≡ \C and X ≡   \C are +b biholomorphic if and only if τ  = aτ cτ +d for some matrix 

a c

b d

 ∈ SL(2, Z) = {2 × 2 integral matrices in SL(2, R)}.

Find an a example of a pair of complex tori that are not biholomorphic. 5.9.2 Prove that if a Riemann surface X has noncyclic Abelian fundamental group, then X is biholomorphic to a complex torus. 5.9.3 A Kähler form ω is said to be of constant curvature λ ∈ R if iω = λω (see Definition 2.12.1). (a) Show that the Kähler form ω ≡ y −2 dx ∧ dy on H is of constant curvature −1. Show also that if  is a group of automorphisms acting properly discontinuously and freely on H, then ∗ ω = ω for each element  ∈ . (b) Let X be a Riemann surface. Using part (a) along with Examples 2.12.2 and 2.12.3, conclude that: (i) If X is elliptic, then X admits a Kähler form of constant curvature 1. (ii) If X is parabolic, then X admits a Kähler form of constant curvature 0. (iii) If X is hyperbolic, then X admits a Kähler form of constant curvature −1. (c) Let X be a compact Riemann surface. Prove that the converse of each of the statements (i)–(iii) in part (b) holds for X; that is, prove that X is elliptic (parabolic, hyperbolic) if and only if X admits a Kähler form ω of constant curvature 1 (respectively, 0, −1). Show that the converses do not hold in general for open Riemann surfaces. (d) Let X be a compact Riemann surface of genus g. Using part (c) above and the results of Chap. 4 (other approaches will be considered in Exercise 5.16.1 and Sect. 5.22), prove that (i) X is elliptic if and only if g = 0. (ii) X is parabolic if and only if g = 1. (iii) X is hyperbolic if and only if g > 1. Hint. According to Exercise 4.6.3, the canonical line bundle of a compact Riemann surface of genus 1 is trivial. 5.9.4 An element g of a group is called a torsion element if g is of finite order (i.e., g m = e for some positive integer m). A group in which only the identity is a torsion element is said to be torsion-free. (a) Prove that every torsion element of Aut (H) must have fixed point. (b) Prove that the fundamental group of every Riemann surface is torsionfree. 5.9.5 The uniformization theorem leads to a natural generalization of the notion of the winding number of a loop around a point (see Exercises 5.1.5, 5.1.6, and  → X be the universal 5.1.7). Let X be an open Riemann surface, let ϒ : X

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227

covering of X, let γ be a loop in X that is path homotopic to a constant loop,  In particular, we let C = γ ([0, 1]), and let γ˜ be a lifting of γ to a loop in X.  have X = H or C. Given a point p ∈ X \ C, we set

nX (γ ; p) ≡ z0

n(γ˜ ; z0 ) ∈ Z

∈ϒ −1 (p)

(the right-hand side is a finite sum by Exercise 5.1.6). (a) Prove that if p ∈ X \ C and θ is a holomorphic 1-form on X \ {p} with resp θ = 1, then  1 nX (γ ; p) = θ. 2πi γ Show that such a 1-form θ always exists (see Corollary 2.15.4 and Exercise 2.15.4), and conclude that nX (γ ; p) is independent of the choice of the holomorphic universal covering map ϒ and the lifting γ˜ . (b) Prove that there is a domain  such that C ⊂   X and γ is path homotopic in  to a constant loop. Conclude that in particular, nX (γ ; p) = 0 for each point p ∈ X \ . (c) Residue theorem. Prove that if S is a discrete subset of X that is contained in X \ C, and θ is a holomorphic 1-form on X \ S, then  1 θ= nX (γ ; p) resp θ 2πi γ p∈S

(note that the right-hand side is actually a finite sum by part (b)). (d) Argument principle. Prove that if f is a nontrivial meromorphic function on X with zero set Z and pole set P both contained in X \ C, then  1 df nX (γ ; p) ordp f. = 2πi γ f p∈Z∪P

(e) Rouché’s theorem. Suppose f and g are nontrivial meromorphic functions on X that do not have any zeros or poles in C, and |g| < |f | on C. Let Zf denote the zero set of f , Pf the pole set of f , Zf +g the zero set of f + g, and Pf +g the pole set of f + g. Prove that p∈Zf ∪Pf

nX (γ ; p) ordp f =



nX (γ ; p) ordp (f + g).

p∈Zf +g ∪Pf +g

(f) Suppose X is simply connected, S is a discrete subset of X, and θ is a holomorphic 1-form on X \ S. Prove that there exists a function f ∈ O(X \ S) with df = θ (i.e., θ is exact) if and only if resp θ = 0 for each point p ∈ S.

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5.10 Smooth Jordan Curves and Homology Sections 5.10–5.17 rely on the Koebe uniformization theorem (Theorem 5.5.3) as well as elementary facts concerning the first homology and cohomology groups of a second countable C ∞ surface (as reviewed in Sects. 10.6 and 10.7), but we will avoid using the classification of Riemann surfaces as quotient spaces considered in Sects. 5.6–5.9. The main goal is the fact that every Riemann surface may be obtained by holomorphic attachment of tubes to a planar domain (as in Sects. 2.3 and 5.12). The idea of the proof is to find holomorphic annuli (i.e., tubes) in the Riemann surface, and then to inductively remove them. Reversing the process then gives the desired construction of the given Riemann surface. Each annulus to be removed is found in a small neighborhood of some diffeomorphic image of the circle S 1 (that is, a smooth Jordan curve) that represents a nonzero homology class (in fact, one may take the holomorphic annulus to be the small neighborhood, but it will be slightly easier to allow for other possibilities). In this section, we consider the first step, which is to show that the first homology group of a second countable C ∞ surface is generated by smooth Jordan curves. Definition 5.10.1 Let M be a topological surface. By a Jordan curve (or a Jordan loop or a simple loop or a simple closed curve) in M, we will mean either a loop γ : [a, b] → M for which γ [a,b) is injective or the image C = γ ([a, b]) of such a loop. We will often identify γ with the homeomorphism S1 → C given by e2πit → γ (a + t (b − a)) for each t ∈ [0, 1] (recall that a bijective continuous mapping of compact Hausdorff spaces is a homeomorphism). Definition 5.10.2 Let M be a C ∞ surface and let γ : [a, b] → M be a path. (a) For c ∈ (a, b), we say that γ is smooth at c if on some neighborhood of c, γ is a C ∞ map with nonvanishing tangent vector γ˙ . We say that γ is smooth from the right at a if γ extends to a path in M that is defined on some interval [a − , b] for some  > 0 and that is smooth at a. Similarly, we say that γ is smooth from the left at b if γ extends to a path that is defined on some interval [a, b + ] for some  > 0 and that is smooth at b. We call γ a smooth path (or a smooth curve) if γ is smooth at each point in (a, b), smooth from the right at a, and smooth from the left at b. We also call the image of a smooth path a smooth path (or a smooth curve). (b) We say that γ is loop-smooth at a (or, equivalently, at b) if γ (a) = γ (b) (i.e., γ is a loop) and on some neighborhood of 1 = e 2πi0 = e2πi(1) ∈ C in S1 , the associated map γˆ : S1 → M given by e2πit → γ (a + t (b − a)) for t ∈ [0, 1] is C ∞ with nonvanishing tangent map γˆ∗ . Equivalently, on some neighborhood of t−a a + Z(b − a), the map R → M given by t → γ (t −  b−a (b − a)) (where x is ∞ the floor of x ∈ R) is C with nonvanishing tangent vector (see Exercise 5.10.1). We call γ a smooth loop (or a smooth closed curve) if γ is a smooth path that is loop-smooth at a and b (in particular, γ is a loop); that is, the associated map S1 → M is a C ∞ immersion (see Definition 9.9.3). We also call the image of a smooth loop a smooth loop (or a smooth closed curve).

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(c) We call γ a piecewise smooth path (or piecewise smooth curve) in M if for some partition a = t0 < t1 < · · · < tm = b, γ [tj −1 ,tj ] is a smooth path for each j = 1, . . . , m. If, in addition, γ (a) = γ (b), then we also call γ a piecewise smooth loop (or a piecewise smooth closed curve). We will also call the image of a piecewise smooth path a piecewise smooth path (or a piecewise smooth curve) and the image of a piecewise smooth loop a piecewise smooth loop (or a piecewise smooth closed curve). (d) We call γ a smooth Jordan curve (or a smooth Jordan loop or a smooth simple loop or a smooth simple closed curve) in M if γ is a smooth loop that is also a Jordan curve (that is, the associated map S1 → γ ([a, b]) ⊂ M is a C ∞ embedding). In this case, we will also call the image γ ([0, 1]) a smooth Jordan curve (or a smooth Jordan loop or a smooth simple loop or a smooth simple closed curve). We call γ a piecewise smooth Jordan curve (or a piecewise smooth Jordan loop or a piecewise smooth simple loop or a piecewise smooth simple closed curve) if γ is a piecewise smooth loop that is also a Jordan curve. In this case, we also call the image γ ([0, 1]) a piecewise smooth Jordan curve (or a piecewise smooth Jordan loop or a piecewise smooth simple loop or a piecewise smooth simple closed curve). Remarks 1. As usual, we assume that all given loops and paths have domain [0, 1] or S1 unless otherwise noted. 2. The Jordan curves in a surface M are precisely those subsets that are homeomorphic to S1 . 3. The smooth Jordan curves in a smooth surface M are precisely the 1-dimensional compact connected C ∞ submanifolds (see Theorem 9.10.1). 4. A smooth path γ : [a, b] → M may be a loop and still not be a smooth loop. There is the added requirement that γ meet itself smoothly at the base point γ (a) = γ (b). 5. If a path γ : [a, b] → M is smooth from the right at c, then we may define the right-hand tangent vector γ˙+ (c) to be the tangent vector at c of some smooth extension of γ [c,c+) for some  > 0 to a smooth path on a neighborhood of c. Similarly, if γ is smooth from the left at c, then we get a left-hand tangent vector γ˙− (c). Actually, one only needs γ to be C 1 from the right at c in the former case, and from the left at c in the latter case. If γ is a loop that is loop-smooth at a (and b), then we may define the tangent vector at the base point by γ˙ (a) = γ˙ (b) ≡ γ˙+ (a) = γ˙− (b). 6. The same terminology is applied to paths in 2-dimensional topological and C ∞ manifolds. We will now see that one may smooth a piecewise smooth injective or Jordan curve. For the proof, it will be convenient to first make the pieces of the curve transverse at the nonsmooth points, and to then choose special coordinates in a neighborhood of a corner at which two pieces of the curve meet transversely. Two curves in a C ∞ surface are transverse at a point of intersection that is a smooth point for each of the curves if their tangent vectors at the point are linearly independent. The following two lemmas accomplish these steps.

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Lemma 5.10.3 Let α : [a, b] → M and β : [c, d] → M be smooth paths in a C ∞ ˙ surface M such that 0 ∈ (a, b) ∩ (c, d), α(0) = β(0), and α(0) ˙ and β(0) are linearly independent (i.e., α and β are transverse at this point). Then there exists a local C ∞ chart (W, , W  ) in M such that for some r > 0, we have [−r, r] ⊂ α −1 (W ) ∩ β −1 (W ) and (α(t)) = (t, 0)

and

(β(t)) = (0, t)

∀t ∈ [−r, r].

Proof The claim is local, so we may assume without loss of generality that M is an open subset of R2 with α(0) = β(0) = (0, 0) ∈ M. Writing α = (α1 , α2 ) and β = (β1 , β2 ), we see that since α  (0) = 0 (α  (0) and β  (0) are linearly independent), we may apply a linear change of coordinates to get α1 (0) = 0. Hence, by the 1dimensional C ∞ inverse function theorem (the easy case of Theorem 9.9.1), we may assume that α1 extends to a diffeomorphism of an open interval containing [a, b] onto an open interval containing 0. The map (x, y) → (α1−1 (x), y − α2 (α1−1 (x))) is then a diffeomorphism of neighborhoods of (0, 0) in R2 (with C ∞ inverse mapping (u, v) → (α1 (u), v + α2 (u))) and hence, applying this diffeomorphism to get a change of coordinates, we may assume that α(t) = (t, 0) ∈ R2 for each t ∈ [a, b]. Similarly, since α  (0) and β  (0) are linearly independent, we have β2 (0) = 0, and hence the mapping (x, y) → (x − β1 (β2−1 (y)), β2−1 (y)) is a diffeomorphism of neighborhoods of (0, 0) (with C ∞ inverse mapping (u, v) → (u + β1 (v), β2 (v))). This diffeomorphism maps the point (t, 0) to (t − β1 (β2−1 (0)), β2−1 (0)) = (t, 0). Therefore, after applying this diffeomorphism, we get α(t) = (t, 0) and β(t) = (0, t) for all t near 0.  Lemma 5.10.4 Let M be a C ∞ surface, let α : I = [a, b] → M be an injective piecewise smooth path for which 0 ∈ (a, b) and α is smooth at each point in (a, b) \ {0}, let V be a neighborhood of p = α(0) in M, and let U be a neighborhood of 0 in (a, b) ∩ α −1 (V ). Then there exists an injective piecewise smooth path β : [a, b] → M such that for some partition a = t0 < t1 = 0 < t2 < · · · < tm = b of I , (i) The path β is smooth at each point in (a, b) \ {t1 , . . . , tm−1 }, while β˙− (tj ) and β˙+ (tj ) are linearly independent for each j = 1, . . . , m − 1; and (ii) We have {t1 , . . . , tm−1 } ⊂ U , β = α on {0} ∪ (I \ U ), β(U ) ⊂ V , and β is path homotopic to α. Remark In fact, as will be clear from the proof, we may choose β so that m ≤ 4 and β = α on [a, 0] (see Fig. 5.6). Proof of Lemma 5.10.4 Since α is injective, we may fix a constant R > 0 and a local C ∞ chart (W,  = (1 , 2 ), (−R, R) × (−R, R)) in M with p ∈ W ⊂ V , (p) = (0, 0), and α −1 (W ) ⊂ U . For some r > 0 with [−r, r] ⊂ α −1 (W ), we have smooth paths α1 , α2 : [−r, r] → W with α1 [−r,0] = α[−r,0] and α2 [0,r] = α[0,r] . Setting τ ≡ (α1 ) = (τ1 , τ2 ) and ρ ≡ (α2 ) = (ρ1 , ρ2 ) and applying Lemma 5.10.3 (to α1 and an arbitrary smooth path transverse to α1 at p), we see that we may choose  so that τ (t) = (τ1 (t), τ2 (t)) = (t, 0) ∈ R2 for each t ∈ [−r, r]

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Fig. 5.6 Modifying an injective piecewise smooth path to get transversality

Fig. 5.7 The two possible local representations of the path and their local modifications

(in particular, r < R). We may also fix numbers s and S such that 0 < s < S < r and α([−s, s]) ⊂ α1 ([−s, s]) ∪ α2 ([−s, s]) ⊂ −1 ((−S, S) × (−S, S)) ⊂ W \ α([a, −r] ∪ [r, b]). The lemma is trivial if α˙ − (0) and α˙ + (0) are linearly independent, so we may assume that ρ  (0) = (ρ1 (0), 0). Since α2 is smooth, we then have ρ1 (0) = 0, and hence we may assume that ρ1 is nonvanishing on [−r, r]. We now consider the two possible cases (see Fig. 5.7). If ρ2 = 0 at all points in [0, r] that are near 0, then, after shrinking r, we may assume that ρ2 ≡ 0 on [0, r], and since α is injective, we must then have ρ1 > 0 on [−r, r]. In this case, we let γ : [0, s] → [0, ρ1 (s)] × [0, s] ⊂ [−S, S] × [−S, S] be the injective piecewise smooth path from (0, 0) to ρ(s) = (ρ1 (s), 0) given by the vertical line segment path in R2 from (0, 0) to (0, s), followed by the line segment path from (0, s) to ρ(s), where each segment is parametrized so as to be smooth. If there exist nonnegative points arbitrarily close to 0 at which ρ2 is nonzero, then we may choose s so that ρ2 (s) = 0 and ρ2 (s) = 0. We may then let γ : [0, s] → [−|ρ1 (s)|, |ρ1 (s)|] × [−|ρ2 (s)|, |ρ2 (s)|] ⊂ [−S, S] × [−S, S] be the injective piecewise smooth path from (0, 0) to ρ(s) given by the vertical line segment path from (0, 0) to (0, ρ2 (s)), followed by the horizontal line segment path from (0, ρ2 (s)) to ρ(s), where again, each segment is parametrized so as to be smooth. In particular, in this case, since ρ1 is either strictly increasing on [−r, r] or strictly decreasing on [−r, r], γ will meet ρ([s, r]) only at the point γ (s) = ρ(s). In either case, it is easy to verify that the piecewise smooth path β : [a, b] → M

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Fig. 5.8 Smoothing a piecewise smooth Jordan curve

given by β ≡ α on [a, 0] ∪ [s, b] and β ≡ −1 (γ ) on [0, s] has the required properties.  Lemma 5.10.5 Let M be a C ∞ surface, let α : I = [a, b] → M be an injective piecewise smooth path or a piecewise smooth Jordan curve that is loop-smooth at a and b, let a = t0 < t1 < · · · < tm = b be a partition of I such that α is smooth at each point in (a, b) \ {t1 , . . . , tm−1 }, and for each j = 1, . . . , m − 1, let Vj be a neighborhood of α(tj ) in M and let Uj be a neighborhood of tj in (a, b) ∩ α −1 (Vj ). Then there exists a path β : I → M such that (i) The path β is an injective smooth path if α is an injective path, and β is a smooth Jordan curve if α is a Jordan curve; and (ii) We have β = α on {t1 , . . . , tm−1 } ∪ (I \ (U1 ∪ · · · ∪ Um−1 )), β(Uj ) ⊂ Vj for each j = 1, . . . , m − 1, and β is path homotopic to α. In other words, an injective piecewise smooth path (or a piecewise smooth Jordan curve that is loop-smooth at the base point) is path homotopic to an injective smooth path (respectively, a smooth Jordan curve) that agrees with the original path outside an arbitrarily small neighborhood of the nonsmooth points, as well as at the nonsmooth points (see Fig. 5.8). Proof of Lemma 5.10.5 Clearly, we may assume that the neighborhoods {Vj } are disjoint. Next we observe that we may assume without loss of generality that α˙ − (tj ) and α˙ + (tj ) are linearly independent (i.e., that we have transversality) for each j = 1, . . . , m − 1. For we may choose a constant u > 0 so that 2u < min1≤j ≤m (tj − tj −1 ) and [tj − u, tj + u] ⊂ Uj for each j = 1, . . . , m − 1. For j = 1, . . . , m − 1, Lemma 5.10.4, when applied to the curve t → α(tj + t) for t ∈ [−u, u] along with a neighborhood Vj of α(tj ) in Vj \ α(I \ (tj − u, tj + u)) and a neighborhood Uj of 0 in (Uj + (−tj )) ∩ (−u, u) with α(Uj + tj ) ⊂ Vj , yields an injective curve βj : [−u, u] → M with the properties described in the lemma. The new curve  α,  [t − u, t + u] and given by t →  β (t − t ) on which is equal to α on I \ m−1 j j j j j =1 [tj − u, tj + u] for each j , agrees with α off of Uj for each j , as well as near the endpoints and at the points t1 , . . . , tm−1 . Moreover, αˆ is an injective piecewise smooth path if α is such a path, and a piecewise smooth Jordan curve that is loopsmooth at a if α is such a Jordan curve, αˆ is homotopic to α, αˆ maps Uj into Vj for each j , and αˆ has the desired transversality property at those interior points {s1 , . . . , sk } ⊃ {t1 , . . . , tm−1 } at which it is not smooth. Thus, by replacing α with α, ˆ i }k and {Vj } replacing {t1 , . . . , tm−1 } with {s1 , . . . , sk }, and replacing {Uj } with {U i=1

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i }k , where if si − tj is one of the points in U  at which βj is not smooth, with {V j i=1 i ⊂ V  are suitable sufficiently small neighborhoods of si i ⊂ U  + tj and V then U j j and α(s ˆ i ), respectively, we may assume that α has the desired transversality property. By Lemma 5.10.3, for some r > 0, we may choose disjoint local C ∞ charts {(Wj , j , (tj − r, tj + r) × (tj − r, tj + r))}m−1 j =1 such that for each j = 1, . . . , m − 1, we have Wj  Vj , (tj − r, tj + r) ⊂ (tj −1 , tj +1 ) ∩ α −1 (Wj ) ∩ Uj , and

 j (α(t)) =

(t, tj ) (tj , t)

if t ∈ (tj − r, tj ], if t ∈ [tj , tj + r)

(one obtains j by applying the lemma to the curve t → α(t + tj ) and then letting j be the sum of the resulting local chart with the point (tj , tj )). Since either α[a,b] is injective or α[a,b) is injective with α(a) = α(b), we may choose a constant s ∈ (0, r) such that for each j = 1, . . . , m − 1, −1 j ([tj − s, tj + s] × [tj − s, tj + s]) ∩ α([a, tj − r] ∪ [tj + r, b]) = ∅. Fixing a nondecreasing C ∞ function λ : R → [0, ∞) such that λ ≡ 0 on a neighborhood of (−∞, 0] and λ ≡ 1 on a neighborhood of [s, ∞) (for example,  s for a nonnegative C ∞ function χ with compact support in (0, s) such that q = 0 χ(v) dv > 0, t the function λ : t → q −1 0 χ(v) dv has the required properties), we see that the path β : [a, b] → M given by ⎧  ⎪ if t ∈ [a, b] \ m−1 ⎨ α(t) j =1 [tj , tj + s], t → −1 ((1 − λ(t − t )) · (t, t ) j j j ⎪ ⎩ + λ(t − t ) · (t , t)) if t ∈ [tj , tj + s] for some j, j j has the required properties. The details of the verification are left to the reader (see Exercise 5.10.2).  We record here for later use (see Sect. 5.15) the following consequence of Lemma 5.10.5: Lemma 5.10.6 If p and q are distinct points in a C ∞ surface M, then there exists an injective smooth path in M from p to q. Proof Fixing a point p ∈ M, we must show that the set N of points x ∈ M \ {p} for which there is an injective smooth path in M from p to x is equal to the set M \ {p}. Clearly, N = ∅. Given a point x0 ∈ M \ {p}, we may choose a local C ∞ chart (U, , (0; 2)) in M with (x0 ) = 0 and p ∈ / U , and we may set D ≡ −1 ((0; 1)). If D meets N , then there exists an injective smooth path α

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in M from p to a point in D, and we may set t0 ≡ min α −1 (D) ∈ (0, 1). Hence, given a point x ∈ D, the product of suitable reparametrizations of the paths α[0,t0 ] and t → −1 ((1 − t)(α(t0 )) + t(x)) is an injective piecewise smooth path in M from p to x. Applying Lemma 5.10.5, we get an injective smooth path in M from p to x. Thus D ⊂ N . From this, it follows that N is open with empty boundary in the connected set M \ {p}, and hence that N = M \ {p}.  We now come to the main goal of this section: Proposition 5.10.7 Let M be a C ∞ surface. Then, for every (continuous) loop α in M, there exist smooth Jordan curves α1 , . . . , αn in M such that  θ= α

n 

θ

j =1 αj

for every closed C ∞ 1-form θ on M. Consequently, if M is second countable and A is a subring of C containing Z, then every 1-cycle in M with coefficients in A is homologous (as a 1-cycle over C) to an A-linear combination of smooth Jordan curves. Remark The second part of the above proposition follows immediately from the first part and Lemma 10.6.6. The only reason for including the second countability condition in this second part is that in this book, homology is considered only for second countable surfaces. Second countability does not play an essential role in the proof. It also follows that for M second countable, H1 (M, A) is generated by those homology classes that are represented by smooth Jordan curves, provided M is orientable (or M admits an orientable C ∞ surface structure) if A = R and A = C (see Sects. 10.6 and 10.7). Proof of Proposition 5.10.7 Let α : [0, 1] → M be a loop in M. We may assume without loss of generality that M is second countable, since (for example) we may replace M with a connected relatively compact neighborhood of α([0, 1]) (second countability will allow us to use the convenient language of homology). We may choose a partition 0 = t0 < t1 < · · · < tm = 1, local C ∞ charts m {(Uj , j , Uj )}m j =1 , and open sets {Dj }j =1 such that for each j = 1, . . . , m, Uj is simply connected, α([tj −1 , tj ]) ⊂ Dj ⊂ Uj , j (Dj ) is a disk in R2 , and Dj ⊂ Uk for any k ∈ {1, . . . , m} with Dj ∩ Dk = ∅ (to get the last property, one may apply Lemma 9.3.6). We may also assume that α is piecewise smooth and that α(tj −1 ,tj ] is injective for j = 1, . . . , m (for example, we may replace the segment α[tj −1 ,tj ] : [tj −1 , tj ] → Dj with a suitable smooth path in Dj from α(tj −1 ) to α(tj ) that is injective on (tj −1 , tj ]). We now proceed by induction on m to show that α is homologous to a sum of smooth Jordan curves. The cases m = 1 and m = 2, in which α lies in the simply connected open set U1 , are trivial. Assume now that m > 2 and that the claim holds for partitions of [0, 1] into fewer than m intervals.

5.10

Smooth Jordan Curves and Homology

235

If α(r) = α(s) for some r, s, and j with t0 ≤ r ≤ tj −1 < tj ≤ s ≤ tm−1

or

t1 ≤ r ≤ tj −1 < tj ≤ s ≤ tm ,

then α is homologous to the sum of the two piecewise smooth loops α1 ∗ α3 and α2 , where α1 , α2 , and α3 are suitable reparametrizations of α[t0 ,r] , α[r,s] , and α[s,tm ] , respectively. For each of these loops, we get a partition as above containing fewer than m subintervals, and therefore, by the induction hypothesis, each is homologous to a sum of smooth Jordan curves. Thus we may assume that no such points r and s exist. Given an index j ∈ {1, . . . , m − 1}, we may set r ≡ min([tj −1 , tj ] ∩ α −1 (α([tj , tj +1 ]))) ∈ (tj −1 , tj ] and s ≡ max([tj , tj +1 ] ∩ α −1 (α(r))) ∈ [tj , tj +1 ). If r < s, then α is homologous to the sum of the (path homotopically trivial) loop β1 in Uj given by a suitable reparametrization of α[r,s] and the piecewise smooth loop β2 : [0, 1] → M determined by β2 ◦ ϕ = α[t0 ,r]∪[s,tm ] , where ϕ : [t0 , r] ∪ [s, tm ] → [0, 1] is the map given by  1 t−t 0 if t ∈ [t0 , r], 2 · r−t ϕ(t) = 1 1 0 t−s if t ∈ [s, tm ]. 2 + 2 · tm −s By replacing α with β2 , we get a path in which the corresponding segment satisfies r = s (the partition corresponding to β2 has the m intervals given by ϕ([tk−1 , tk ]) ⊂ β2−1 (Dk ) for k ∈ {1, . . . , m}\{j, j +1}, ϕ([tj −1 , r]) ⊂ β2−1 (Dj ), and ϕ([s, tj +1 ]) ⊂ β2−1 (Dj +1 )). Performing this procedure inductively on j , we see that we may assume that α([tj −1 , tj )) and α([tj , tj +1 ]) are disjoint and that α([tj −1 , tj ]) and α((tj , tj +1 ]) are disjoint for each j = 1, . . . , m − 1. A similar argument shows that we may assume without loss of generality that α([tm−1 , tm )) and α([t0 , t1 ]) are disjoint and that α([tm−1 , tm ]) and α((t0 , t1 ]) are disjoint. Thus we may assume that α is a piecewise smooth Jordan curve. Finally, fixing a point a ∈ (0, t1 ) at which α is smooth, we see that α is homologous to the piecewise smooth Jordan curve γ given by t → α(t + a) on [0, 1 − a] and t → α(t − 1 + a) on [1 − a, 1]. Since γ is also loop-smooth at 0, Lemma 5.10.5 implies that γ is homologous to a smooth Jordan curve, and the claim follows.  Exercises for Sect. 5.10 5.10.1 Let γ : [a, b] → M be a loop in a C ∞ surface M. Verify that γ is loopsmooth at a (i.e., on some neighborhood of 1 = e2πi·0 , the associated map γˆ : S1 → M given by e2πit → γ (a + t (b − a)) for t ∈ [0, 1] is C ∞ with nonvanishing tangent map) if and only if on some neighborhood of t−a a + Z(b − a) ⊂ R, the map R → M given by t → γ (t −  b−a (b − a)) ∞ is C with nonvanishing tangent vector.

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5.10.2 Verify that the path β constructed in the proof of Lemma 5.10.5 has the required properties. 5.10.3 Prove that every Jordan curve α : [0, 1] → M (injective path) in a C ∞ surface M is path homotopic to a smooth Jordan curve (respectively, a smooth injective path) β. Prove also that if 0 = t0 < · · · < tn = 1 is a partition and Uj is a neighborhood of γ ([tj −1 , tj ]) for j = 1, . . . , n, then β may be chosen so that β([tj −1 , tj ]) ⊂ Uj for j = 1, . . . , n. Hint. One may form a partition of the curve into segments, each of which lies in a coordinate disk. One may then replace a small segment of the curve about each of the partition points with a smooth segment that agrees with the original curve at the partition points. The complement of a suitable small neighborhood of the partition points is equal to a finite collection of disjoint segments, and Lemma 5.10.6 allows one to replace each of these segments with a smooth segment. One may then apply Lemma 5.10.5.

5.11 Separating Smooth Jordan Curves In this section, we consider some properties of smooth Jordan curves that allow us to find and remove annuli. The main goals are to show that a smooth Jordan curve in an oriented smooth surface admits a neighborhood that is diffeomorphic to an annulus, and to show that the de Rham pairing of a separating smooth Jordan curve and the cohomology class of a compactly supported C ∞ closed 1-form is zero. Lemma 5.11.1 Let γ : [0, 1] → M be an injective smooth path or a smooth Jordan curve in an oriented C ∞ surface M, and let C = γ ([0, 1]). (a) If γ is a smooth Jordan curve, then for some R > 1, there exists an orientation-preserving diffeomorphism  of some neighborhood A of C in M onto the annulus (0; 1/R, R) with (γ (t)) = e2πit for each t ∈ [0, 1] (see Fig. 5.9). Consequently, M \ C has either one or two connected components, and each of these connected components has boundary equal to C. (b) If γ is an injective smooth path, then for some  > 0, there exists an orientationpreserving diffeomorphism  of some neighborhood R of C in M onto the open rectangle (−, 1 + ) × (−, ) with (γ (t)) = (t, 0) for each t ∈ [0, 1]. Consequently, M \ C is connected. Remarks 1. For a smooth Jordan curve in a nonorientable surface, one may find a neighborhood that is diffeomorphic to either an annulus or a Möbius band. This fact will be proved and applied in the discussion of C ∞ structures on surfaces in Chap. 6 (see Sect. 6.10). 2. As indicated in part (a), the existence of annular neighborhoods implies that the complement M \ C of a smooth Jordan curve C in an oriented smooth surface M has exactly one or two connected components, and each of these connected components has boundary equal to C. On the other hand, one may also obtain this fact directly (see Exercise 5.11.3).

5.11

Separating Smooth Jordan Curves

237

Fig. 5.9 A smooth annular neighborhood of a smooth Jordan curve

3. For our purposes, the easier fact that any smooth Jordan curve is separating in some neighborhood of the curve would suffice. However, it will be convenient to have the annular neighborhood provided by Lemma 5.11.1. 4. Given a, b, c, d ∈ R with a < 1 < b and c < 1 < d, there exists an orientation≈ preserving diffeomorphism  : (0; a, b) → (0; c, d) such that S1 = Id S1 (see Exercise 5.11.4). Consequently, part (a) of the lemma actually holds for any R > 1. Similarly, part (b) holds for any  > 0. Proof of Lemma 5.11.1 We prove part (a) and leave the proof of part (b), which is similar, to the reader (see Exercise 5.11.1). Assuming that γ is a smooth Jordan curve, we may form the associated mapping γ0 : S1 → C given by e2πit → γ (t) and the mapping α : R → C given by t → γ0 (e2πit ) = γ (t − t) (which is actually the extension of γ to a periodic mapping with period 1). By Theorem 9.9.2, C is a compact C ∞ submanifold of M, γ0 is a diffeomorphism, and α is a local diffeomorphism (in fact, α is a C ∞ covering map). Observe that C ∞ local extensions of local inverses of these maps have nonvanishing differentials at points in C. Hence we may form a finite covering of C by positively oriented local C ∞ charts {(Uj , j = (xj , yj ), Rj = Ij × (−δj , δj ))}m j =1 in M such that for each j = 1, . . . , m, Ij is an open interval, δj > 0, C ∩ Uj = {p ∈ Uj | yj (p) = 0} = α(Ij ), and xj (α(t)) = t for each point t ∈ Ij . In particular, since α∗ maps (T R)t bijectively onto (T C)γ (t) for each point t ∈ R, we have dxj T (C∩Uj ∩Uk ) = dxk T (C∩Uj ∩Uk ) for each pair of indices j, k. On the other hand, we have dyj T (C∩Uj ) = 0 for each j , so we have (dxj ∧ dyj )p = (dxk ∧ dyj )p

∀p ∈ C ∩ Uj ∩ Uk .

For we have (dxk )p = a(dxj )p + b(dyj )p for some scalars a, b ∈ R, and restricting to T C, we see that a = 1. We may also form nonnegative C ∞ functions {λj }m j =1 such that supp λj ⊂ Uj for

each j and λj ≡ 1 on a neighborhood of C. Thus we may define a C ∞ mapping  : M → C = R2 by =

m j =1

λj · e2πi(xj +iyj ) =

m j =1

λj · e−2πyj · (cos 2πxj , sin 2πxj )

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5

 =

m

λj · e

−2πyj

Uniformization and Embedding of Riemann Surfaces

· cos 2πxj ,

j =1

m

 λj · e

−2πyj

· sin 2πxj

= (1 , 2 )

j =1

(as usual, we have extended λj · e2πi(xj +iyj ) by 0 to all of M for each j ). For p ∈ C and ζ = γ0−1 (p) = u + iv ↔ (u, v), we have ζ = e2πixj (p) (i.e., (u, v) = (cos(2πxj (p)), sin(2πxj (p)))) for any index j with p ∈ Uj . Hence (p) =

m

λj (p) · e2πixj (p) =

j =1

and (since



m

λj (p) · γ0−1 (p) = γ0−1 (p)

j =1

dλj = d( λj ) = 0 near C) (d)p = ((d1 )p , (d2 )p )  m (−2π)λj (p)(v(dxj )p + u(dyj )p ), = j =1 m

 2πλj (p)(u(dxj )p − v(dyj )p ) .

j =1

Therefore, since u2 + v 2 = |ζ |2 = 1, we have (d1 ∧ d2 )p = 4π 2

m

λj (p)λk (p)v 2 (dxj ∧ dyk )p

j,k=1

+ 4π 2

m

λj (p)λk (p)u2 (dxk ∧ dyj )p

j,k=1

= 4π 2

m

λj (p)λk (p)v 2 (dxj ∧ dyk )p

j,k=1

+ 4π 2

m

λj (p)λk (p)u2 (dxj ∧ dyk )p

j,k=1

= 4π 2

m

λj (p)λk (p)(dxj ∧ dyk )p .

j,k=1

On the other hand, if P is the set of indices j for which p ∈ Uj , then we have (dxj ∧ dyk )p = (dxk ∧ dyk )p for all j, k ∈ P . Therefore λj (p)λk (p)(dxk ∧ dyk )p (d1 ∧ d2 )p = 4π 2 j,k∈P

= 4π

2



k∈P

λk (p)(dxk ∧ dyk )p > 0.

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Separating Smooth Jordan Curves

239

Therefore, by continuity, d1 ∧ d2 > 0 on some neighborhood  of C, and hence by the C ∞ inverse function theorem (Theorem 9.9.1 and Theorem 9.9.2),  is an orientation-preserving local diffeomorphism. Furthermore, the intersection of  −1 (S1 ) with a sufficiently small neighborhood of C must be equal to C. For if this were not the case, then we could choose a sequence {pν } in M \ C that converged to a point p ∈ C and that satisfied (pν ) ∈ S1 for each ν. Setting qν ≡ γ0 ((pν )) ∈ C for each ν, we would get qν = γ0 ((pν )) → γ0 ((p)) = p. But then, for large ν, both pν and qν would lie in a neighborhood of p on which  was injective. Since (qν ) = γ0−1 (qν ) = (pν ), we have arrived at a contradiction. Lemma 10.2.11 now implies that  maps a neighborhood H of C diffeomorphically onto a neighborhood of S1 . Choosing R > 1 sufficiently close to 1, we see that the restriction  of  to (H )−1 ((0; 1/R, R)) has the required properties. Finally, setting Q1 ≡ (0; 1/R, 1) and Q2 ≡ (0; 1, R), we see that −1 (Qj ) must lie in some connected component Pj of M \ C for each j = 1, 2. Any connected component of M \ C containing neither −1 (Q1 ) nor −1 (Q2 ) would have to be a connected component of M, since it would have no boundary points in M. Therefore, since M is connected, the set of connected components of M \ C is  {P1 , P2 }. Finally, observe that ∂Pj = −1 (∂Qj ) = C for j = 1, 2. Definition 5.11.2 A closed subset of a connected topological space X is called separating in X if X \ K is not connected. If X \ K is connected, then K is called nonseparating in X. For example, a circle in C is separating; in fact, as will be proved in Chap. 6, every Jordan curve in C is separating (see Corollary 6.7.2). The smooth Jordan curve C = γ ([0, 1]) appearing in Fig. 5.9 is nonseparating. Lemma 5.11.3 If  C is a separating smooth Jordan curve in an oriented smooth surface M, then C θ = 0 for every closed C ∞ 1-form θ with compact support. Remark The converse, which is also true, will be proved in Sect. 5.15 (see Proposition 5.15.2); and the proof of the converse will be applied in the construction of a canonical homology basis for a compact Riemann surface in Sect. 5.16. Proof of Lemma 5.11.3 By Lemma 5.11.1, M \ C must have exactly two connected components M1 and M2 , and we must have ∂M1 = ∂M2 = C. Orienting C positively with respect to M1 and applying Stokes’ theorem (Theorem 9.7.17), we get    θ= dθ = 0 = 0.  C M1 M1 We close this section with the following elementary observation, which will allow us to choose disjoint tubes for removal from a Riemann surface, thereby reducing its homology.

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 Lemma 5.11.4 Let M be a topological surface; let K = α∈A Kα , where {Kα } is a countable locally finite collection of disjoint compact sets each of which is mapped onto a closed disk in the plane in some local chart; and let N = M \ K. Then every path γ : [0, 1] → M with γ (0), γ (1) ∈ N is path homotopic (in M) to a path in N . In particular, N is connected. Proof Suppose first that for some constants r and R with 0 < r < R < ∞, (D, , (0; R)) is a local chart in M and K =  −1 ((0; r)). Fixing r0 ∈ (r, R) with γ (0), γ (1) ∈ M \  −1 ((0; r0 )), we get real numbers {aj }kj =1 and {bj }kj =1 such that 0 < a1 < b1 < a2 < b2 < · · · < ak < bk < 1, Ij ≡ [aj , bj ] ⊂  γ −1 ( −1 ((0; r0 ))) for j = 1, . . . , k, and γ ([0, 1]\ kj =1 Ij ) ⊂ N (the compact set γ −1 ( −1 ((0; r))) admits a finite covering by disjoint open intervals that are relatively compact in γ −1 ( −1 ((0; r0 ))), and we may take {(aj , bj )}kj =1 to be these intervals). In particular, for some r1 ∈ (r, r0 ) and for each j = 1, . . . , k, γ (aj ) and γ (bj ) lie in the topological annulus  −1 ((0; r1 , r0 )), and hence we have a path homotopy Hj : γ [aj ,bj ] ∼ λj to a path in λj : [aj , bj ] →  −1 ((0; r1 , r0 )) ⊂ N . The map H : [0, 1] × [0, 1] → M given by  Hj (t, s) if (t, s) ∈ [aj , bj ] × [0, 1] for some j, H (t, s) =  γ (t) if (t, s) ∈ ([0, 1] \ kj =1 [aj , bj ]) × [0, 1], is then a path homotopy from γ to a path in N . Now for the general case, let A0 = {α1 , . . . , αm } ⊂ A be the finitely many indices α for which γ ([0, 1]) meets K α . By the above, we may homotope γ within the connected component of M \ α∈A\A  0 Kα containing the connected set  γ ([0, 1]) ∪ α∈A0 Kα to a path γ1 in M \ α∈{α1 }∪(A\A0 ) Kα . Proceeding inductively, we get the claim.  Exercises for Sect. 5.11 5.11.1 Prove part (b) of Lemma 5.11.1. 5.11.2 Prove that in part (a) of Lemma 5.11.1, if 0 : V → W is an orientationpreserving diffeomorphism of a neighborhood V of γ ([s0 , s1 ]) onto a neighborhood W of the arc E ≡ {e2πit | s0 ≤ t ≤ s1 } for some pair s0 , s1 with 0 < s0 < s1 < 1, and 0 (γ (t)) = e2πit for each t ∈ γ −1 (V ), then we may choose R and  so that S ≡ −1 ({rζ | 1/R < r < R, ζ ∈ E}) ⊂ V and  = 0 on S. Prove also that in part (b), if 0 : V → W is an orientationpreserving diffeomorphism of a neighborhood V of γ ([s0 , s1 ]) onto a neighborhood W of the line segment L = [s0 , s1 ] × {0} for some pair s0 , s1 with 0 ≤ s0 < s1 ≤ 1 and 0 (γ (t)) = (t, 0) for each t ∈ γ −1 (V ), then we may choose  and  so that S ≡ −1 ([s0 , s1 ] × (−, )) ⊂ V and  = 0 on S. 5.11.3 Prove directly from local descriptions (without applying the existence of annular neighborhoods as in Lemma 5.11.1) that the complement M \ C of a smooth Jordan curve C in a smooth (not necessarily orientable) surface M has exactly one or two connected components, and each of these connected components has boundary equal to C.

5.12

Holomorphic Attachment and Removal of Tubes

241

5.11.4 Prove that if a < 1 < b and c < 1 < d, then there exists an orientation≈ preserving diffeomorphism  : (0; a, b) → (0; c, d) such that S1 = Id S1 (hence part (a) of Lemma 5.11.1 holds for any R > 1). 5.11.5 Let γ : R → M be a C ∞ embedding of R into an oriented C ∞ surface M (i.e., a proper C ∞ mapping for which the tangent vector γ˙ is nonvanishing), and let C = γ (R). Prove that there exists an orientation-preserving diffeomorphism  of some neighborhood of C in M onto the open strip R × (−1, 1) with (γ (t)) = (t, 0) for each t ∈ R. Conclude that in particular, M \ C has either one or two connected components. 5.11.6 Prove that there exists a universal constant C > 0 such that if X is a Riemann surface, p is a point in X, and (D, z = , (0; 1)) is a local holomorphic chart with p ∈ D and (p) = z(p) = 0, then there exists a meromorphic 1-form θ on X such that (cf. Exercise 5.5.1) (i) The meromorphic 1-form θ is holomorphic on X \ {p}; (ii) We have θL2 (X\D) ≤ C; (iii) The meromorphic 1-form θ + z−2 dz on D has a removable singularity at p; −1 (iv) We have zθ  + z dzL2 (D) ≤ C; and (v) We have γ θ = 0 for every separating smooth Jordan curve γ in X \ {p}.

5.12 Holomorphic Attachment and Removal of Tubes In this section we develop notation for the holomorphic attachment and removal of a locally finite family of disjoint tubes (cf. Example 2.3.3).

5.12.1 Holomorphic Attachment of Tubes Let Y be a complex 1-manifold; let {R0j }j ∈J and {R1j }j ∈J be numbers in the interval (1, ∞); let {(D0j , 0j , (0; R0j ))}j ∈J

and {(D1j , 1j , (0; R1j ))}j ∈J

be locally finite families of disjoint local holomorphic charts in Y such that for all j, k ∈ J , D0j ∩ D1k = ∅, and for each j ∈ J , let Aνj ≡ −1 νj ((0; 1, Rνj )) ⊂ Dνj for ν = 0, 1, let Tj ≡ (0; 1/R0j , R1j ), let A0j ≡ (0; 1/R0j , 1) ⊂ Tj , and let A1j ≡ (0; 1, R1j ) ⊂ Tj . Setting Aν ≡

j ∈J

Aνj ⊂ Dν ≡

j ∈J

Dνj ⊂ Y

and Aν ≡

 j ∈J

Aνj ⊂ T ≡

 j ∈J

Tj

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Fig. 5.10 An annulus (or tube) and two disjoint disks

for ν = 0, 1, we get a biholomorphism  : A0 ∪ A1 → A0 ∪ A1 , where for each index ν ∈ {0, 1}, each index j ∈ J , and each point z ∈ Aνj , the image p of z under the inclusion Aνj → Aν satisfies   if ν = 0, −1 0j (1/z) ∈ A0j (p) ≡ −1  1j (z) ∈ A1j if ν = 1. Setting

 Z≡Y \



−1 νj ((0; 1))

⊃ A0 ∪ A1 ,

ν∈{0,1}, j ∈J

we see that  −1 (K  ∩ (A0 ∪ A1 )) is closed in T for every compact set K  ⊂ Z, and (K ∩ (A0 ∪ A1 )) is closed in Z for every compact set K ⊂ T (see Fig. 5.10). We may therefore form the holomorphic attachment X ≡ Z ∪ T = Z  T /∼, where for ν ∈ {0, 1}, j ∈ J , p ∈ Aνj , and z ∈ Aνj , the images p0 of p and z0 of z in Z  T under the inclusions Aνj , Aνj → Z  T satisfy p0 ∼ z0 if and only if z · 0j (p) = 1 for ν = 0 and z = 1j (p) for ν = 1 (see Fig. 5.11, in which J = {1, 2, 3}). In other words, X is a complex 1-manifold obtained by removing the unit disks in each of the coordinate disks D0j and D1j , and gluing in a tube (i.e., an annulus) Tj (equivalently, the boundaries of the unit disks are glued together). We call X the complex 1-manifold obtained by holomorphic attachment of tubes at elements of the locally finite family of disjoint coordinate disks {(Dνj , νj , (0; Rνj ))}ν∈{0,1},j ∈J (or simply the complex 1-manifold obtained by holomorphic attachment of tubes at {Dνj }ν∈{0,1},j ∈J ). Observe that X is compact if and only if Y is compact. If Y is connected, then X is connected. However, X may be connected even if Y is not (see Exercise 2.3.2).

5.12

Holomorphic Attachment and Removal of Tubes

243

Fig. 5.11 Holomorphic attachment of tubes ∗ with 1 < R ∗ ≤ R for each ν ∈ {0, 1} and j ∈ J , setting Fixing a number Rνj νj νj  ∗ ∗ ≡ (0; 1/R0j , R1j ) ⊂ Tj for each j ∈ J , and setting T ∗ ≡ j ∈J Tj∗ ⊂ T , we may form the holomorphic attachment X∗ ≡ Z  T ∗ /∼ of the tubes {Tj∗ }j ∈J at the coordinate disks  ∗  ∗ ∗ ∗ , (0; R ) Dνj ≡ −1 . νj νj ((0; Rνj )), νj Dνj ν∈{0,1},j ∈J

Tj∗

The natural inclusion Z  T ∗ ⊂ Z  T then descends to a natural biholomorphism ∼ =

X ∗ −→ X (see Exercise 5.12.1), so we may identify X∗ with X. In particular, for each j ∈ J , we may always choose R0j and R1j to be equal and arbitrarily close to 1.

5.12.2 Holomorphic Removal of Tubes The inverse operation of tube attachment is tube removal. Let X be a complex 1-manifold, let {R0j }j ∈J and {R1j }j ∈J be collections of numbers in (1, ∞), let {(Tj , j , (0; 1/R0j , R1j ))}j ∈J be a locally finite collection of disjoint local holomorphic charts in X, and for each j ∈ J , let A0j ≡ −1 j ((0; 1/R0j , 1)) ⊂ Tj

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and A1j ≡ −1 j ((0; 1, R1j )) ⊂ Tj . For each ν = 0, 1 and each j ∈ J , let Dνj ≡ (0; Rνj ) and Aνj ≡ (0; 1, Rνj ) ⊂ Dνj . Setting Aν ≡

Aνj ⊂ T ≡

j ∈J

Tj

j ∈J

and

Aν ≡



Aνj ⊂ Dν ≡

j ∈J



Dνj

j ∈J

for ν = 0, 1, we get a biholomorphism  : A0 ∪A1 → A0 A1 , where for each index ν ∈ {0, 1}, each index j ∈ J , and each point p ∈ Aνj , (p) is the image in A0  A1 of the point  1/j (p) ∈ A0j for ν = 0, j (p) ∈ A1j for ν = 1, under the inclusion Aνj → A0  A1 ⊂ D0  D1 . Setting Z ≡ X \ [T ∩ ∂(A0 ∪ A1 )] = X \ (T ∩ ∂A0 ) = X \ (T ∩ ∂A1 )

−1 =X\ j (∂(0; 1)) ⊃ A0 ∪ A1 , j ∈J

we see that  −1 (K  ∩ (A0 A1 )) is closed in Z for every compact set K  ⊂ D0 D1 , and (K ∩ (A0 ∪ A1 )) is closed in D0  D1 for every compact set K ⊂ Z. Thus we get the holomorphic attachment Y ≡ (D0  D1 ) ∪ Z = D0  D1  Z/∼, where for ν ∈ {0, 1}, j ∈ J , p ∈ Aνj , and z ∈ Aνj , the images p0 of p and z0 of z in D0  D1  Z under the inclusions Aνj , Aνj → D0  D1  Z satisfy p0 ∼ z0 if and only if z · j (p) = 1 for ν = 0 and z = j (p) for ν = 1. In other words, Y is the complex 1-manifold obtained by removing the unit circles in each of the tubes (i.e., coordinate annuli) Tj and gluing in caps (i.e., disks) D0j and D1j on each end of the remaining two pieces. That is, Y is obtained by attaching caps to the open subset Z of X. We call Y the complex 1-manifold obtained by holomorphic removal of the locally finite family of disjoint tubes {(Tj , j , (0; 1/R0j , R1j ))}j ∈J from X (or simply by holomorphic removal of {Tj }j ∈J ). Observe that X is compact if and only if Y is compact. If Y is connected, then X is connected, but connectivity of X does not imply connectivity of Y (see Exercise 5.12.2). As is the case for holomorphic attachment of tubes, shrinking the annuli about the unit circle does not affect the outcome. More precisely, fixing a num∗ with 1 < R ∗ ≤ R ∗ ber Rνj νj for each ν ∈ {0, 1} and j ∈ J , setting Tj ≡ νj

∗ ∗ ∗ ≡ (0; R ∗ ) for each −1 j ∈ J , setting Dνj νj j ((0; 1/R0j , R1j )) ⊂ Tj for each ∗ ⊂ D for ν = 0, 1, we may form ν ∈ {0, 1} and j ∈ J , and setting Dν∗ ≡ j ∈J Dνj ν the complex 1-manifold Y ∗ ≡ D0∗  D1∗  Z/ ∼ obtained by holomorphic removal of the tubes {Tj∗ }j ∈J . The natural inclusion D0∗  D1∗  Z ⊂ D0  D1  Z then descends ∼ =

to a natural biholomorphism Y ∗ −→ Y (see Exercise 5.12.3), so we may identify Y ∗ with Y . In particular, for each j ∈ J , we may always choose R0j and R1j to be equal and arbitrarily close to 1.

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5.12.3 Holomorphic Reattachment of Tubes Tube attachment and tube removal are inverse operations (up to biholomorphism). In other words, if one removes tubes in a complex 1-manifold X as in Sect. 5.12.2, and then (re)attaches tubes with the disks and constants as in Sect. 5.12.1, then the resulting complex 1-manifold is biholomorphic to the original complex 1-manifold X. In fact, if one reattaches only those tubes in some subfamily, then the resulting complex 1-manifold is biholomorphic to the complex 1-manifold obtained by removal from X of those tubes in the complementary subfamily. Similarly, if one attaches tubes to a complex 1-manifold Y as in Sect. 5.12.1, and then removes tubes with the annuli and constants as in Sect. 5.12.2, then the resulting complex 1-manifold is biholomorphic to the original complex 1-manifold Y . The details of holomorphic reattachment of tubes are provided in this section. The details of the inverse operation (holomorphic attachment followed by holomorphic removal of tubes) are left to the reader (see Exercise 5.12.5). In the notation of Sect. 5.12.2, let Y = (D0  D1 ) ∪ Z, let Z : Z → Y be the holomorphic inclusion of Z, and for each ν ∈ {0, 1} and j ∈ J , let Dνj : Dνj → Y νj = Dνj (Dνj ), be the holomorphic inclusion of the disk Dνj = (0; Rνj ), let D ∼

= νj → νj ≡ −1 : D and let  Dνj . Given a set of indices J ⊂ J , holomorphic atDνj tachment of tubes at elements of the locally finite family of disjoint coordinate disks  More preνj ,  νj , (0; Rνj ))}ν∈{0,1},j ∈J in Y yields a complex 1-manifold X. {(D  cisely, for each j ∈ J , we may set

0j ≡ (0; 1/R0j , 1), A 1j ≡ (0; 1, R1j ) ⊂ Tj ≡ (0; 1/R0j , R1j ), A and νj ⊂ Y νj  ≡ Dνj (Aνj ) = Z (Aνj ) =  −1 ((0; 1, Rνj )) ⊂ D A νj

for ν = 0, 1.

As in Sect. 5.12.1, setting ν ≡ A

νj ⊂ D ν ≡ A

j ∈J

νj ⊂ Y D

ν ≡ and A

j ∈J



νj ⊂ T ≡ A

j ∈J



Tj

j ∈J

0 ∪ A 1 → A  ∪ A  , where for each : A for ν = 0, 1, we get a biholomorphism  0 1 νj , the image p of z under index ν ∈ {0, 1}, each index j ∈ J, and each point z ∈ A ν satisfies νj → A the inclusion A  (p) ≡ 

 −1   (1/z) = D0j (1/z) ∈ A  0j   −1 1j (z) = D1j (z) ∈ A1j

0j

if ν = 0, if ν = 1.

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Setting ≡ Y \ Z



    −1  νj (0; 1) = Y \

ν∈{0,1}, j ∈J

= Z (Z) ∪



  Dνj (0; 1)

ν∈{0,1}, j ∈J

0 ∪ D 1 ) = A 0 ∪ A 1 , νj ⊃ Z (Z) ∩ (D D

ν∈{0,1},j ∈J \J

, with inclusion ≡Z  ∪ we may form the holomorphic attachment (of tubes) X  T  → X  and T : Tj → X  for each j ∈ J. We call X  the complex mappings Z : Z j 1-manifold obtained by holomorphic reattachment of the tubes {Tj }j ∈J to Y . Setting J ≡ J \ J, holomorphic removal of the locally finite family of disjoint  tubes {(Tj , j , (0; 1/R0j , R1j ))}j ∈J from X yields a complex 1-manifold X. More precisely, setting

  ν ≡ ν ≡ ν ≡ Aνj ⊂ T ≡ Tj ⊂ T and A Aνj ⊂ D Dνj A j ∈J

j ∈J

j ∈J

j ∈J

0 ∪ A 1 → A   A  , where for each index : A for ν = 0, 1, we get a biholomorphism  0 1   A   (p) is the image in A ν ∈ {0, 1}, each index j ∈ J, and each point p ∈ Aνj ,  0 1 of the point  1/j (p) ∈ A0j for ν = 0, j (p) ∈ A1j for ν = 1,   A  ⊂ D 0  D 1 . Setting under the inclusion Aνj → A 0 1  ≡ X \ [T ∩ ∂(A 0 ∪ A 1 )] = X \ (T ∩ ∂ A 0 ) = X \ (T ∩ ∂ A 1 ) Z

0 ∪ A 1 , −1 Tj ⊃ Z ∩ T = A =X\ j (∂(0; 1)) = Z ∪ j ∈J

j ∈J

1 ) ∪  ≡ (D 0  D  with inclusion we may form the holomorphic attachment Y  Z,  → Y  and   for each ν ∈ {0, 1} and j ∈ J. One  Dνj : Dνj → Y mappings Z : Z → X  given by may now verify that the natural mapping Y Z(p) → Z(Z (p)) Z(p) → Tj (j (p))  Dνj (z) → Z(Dνj (z))

∀p ∈ Z, ∀j ∈ J, p ∈ Tj , ∀ν ∈ {0, 1}, j ∈ J, z ∈ Dνj ,

is a well-defined biholomorphism that maps Z(Z) onto Z(Z (Z)), Z(Tj ) νj ) for each in Dνj (Dνj ) onto Z(D onto Tj (Tj ) for each index j ∈ J, and   dex ν ∈ {0, 1} and each index j ∈ J (see Exercise 5.12.4). Thus we may identify Y  with X. In other words, holomorphic reattachment of the tubes {Tj }j ∈J to Y yields the same complex 1-manifold as holomorphic removal of the tubes {Tj }j ∈J from X.

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5.12.4 Finite Sequential Holomorphic Removal/Attachment of Tubes If a locally finite family of disjoint tubes in a complex 1-manifold is divided into finitely many disjoint subfamilies, and the subfamilies are then inductively removed (i.e., if the subfamilies are removed one at a time), then the resulting complex 1-manifold will be naturally biholomorphic to the complex 1-manifold obtained by removing the tubes simultaneously. More precisely, in the notation of Sect. 5.12.2, suppose {Jk }m k=1 are nonempty disjoint subsets of the index set J with union J ; that is a partition of J . For each k = 1, . . . , m, let is, {Jk }m k=1 A(k) ν ≡

Aνj ⊂ T (k) ≡

j ∈Jk

Tj



A(k) ν ≡

and

j ∈Jk (k)

Aνj ⊂ Dν(k) ≡

j ∈Jk (k) 

(k)

for ν = 0, 1, and let k : A0 ∪ A1 → A0 phism as in Sect. 5.12.2. Now let (1)



(k) 

 A1

(1)



Dνj

j ∈Jk

be the associated biholomor(1)

(1)

Z1 ≡ X \ [T (1) ∩ ∂(A0 ∪ A1 )] = X \ (T (1) ∩ ∂A0 ) = X \ (T (1) ∩ ∂A1 ) =X\

−1 j (∂(0; 1)) ⊃ A0 ∪ A1 ∪ (1)

(1)

j ∈J1 (1)

m

T (k) ,

k=2 (1)

let Y1 ≡ (D0  D1 ) ∪1 Z1 be the complex 1-manifold obtained by holomorphic removal of the tubes {Tj }j ∈J1 , and let Z1 : Z1 → Y1 , and D (1) : Dν(1) → Y1 for ν ν = 0, 1, be the corresponding inclusion mappings. We then have the locally finite family of disjoint tubes {(Z1 (Tj ), j ◦ −1 Z1 , (0; 1/R0j , R1j ))}j ∈J2 in Y1 , and holomorphic removal of these tubes yields the complex 1-manifold (2)

(2)

Y2 ≡ (D0  D1 ) ∪2 ◦−1 Z2 Z1

(2)

and the corresponding inclusion mappings Z2 : Z2 → Y2 and D (2) : Dν → Y2 ν for ν = 0, 1, where   m

(2) (2) −1 (k) Z1 (j (∂(0; 1))) ⊃ Z1 A0 ∪ A1 ∪ T Z2 ≡ Y1 \ . j ∈J2

k=3

Proceeding inductively, we get complex 1-manifolds X = Y0 , Y1 , . . . , Ym and, for each k = 1, . . . , m, holomorphic inclusion mappings Zk : Zk → Yk of an open (k) set Zk ⊂ Yk−1 and D (k) : Dν → Yk for ν = 0, 1. Moreover, setting Z0 ≡ X, ν

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Z0 ≡ IdX , Zm+1 ≡ Ym , and Zm+1 ≡ IdYm , we see that for each k = 1, . . . , m, the composition Zk−1 ◦ · · · ◦ Z0 is defined on the open set

(k) −1 , X\ j (∂(0; 1)) ⊃ T j∈

we have Zk = Yk−1 \

k−1 i=1

Ji

Zk−1 ◦ · · · ◦ Z0 (−1 j (∂(0; 1))),

j ∈Jk

Zm+1 ◦ · · · ◦ Zk+1 is defined on D (k) (Dν(k) ) for ν = 0, 1, and ν

(k) Yk = (D0

(k)  D1 ) ∪k ◦(Z ◦···◦Z )−1 k−1 0

(k)

(k)

Zk = D0  D1  Zk /∼

is the complex 1-manifold obtained by holomorphic removal of the tubes {Zk−1 ◦ · · · ◦ Z0 (Tj )}j ∈Jk from Yk−1 (in other words, under the appropriate identifications, Yk is the complex 1-manifold obtained by holomorphic removal of the tubes {Tj }j ∈Jk from Yk−1 ). On the other hand, as in Sect. 5.12.2, holomorphic removal of the tubes {Tj }j ∈J yields the complex 1-manifold Y ≡ (D0  D1 ) ∪ Z = D0  D1  Z/∼, where

−1 Z≡X\ j (∂(0; 1)), j ∈J ∼ =

and  : A0 ∪ A1 → A0  A1 is the biholomorphism for which the restriction 



(k)   A(k) ∪A(k) is equal to the composition of the inclusion A(k) 0  A1 → A0  A1 0

1

(k)

= (k) ∼

(k) 

(k) 

and the biholomorphism k : A0 ∪ A1 → A0  A1 for each k = 1, . . . , m. Let Z : Z → Y and Dν : Dν → Y for ν = 0, 1 be the corresponding inclusion mappings. One may now verify that the natural mapping Y → Ym , which sends the image Z (p) in Y of any point p ∈ Z to the point Zm ◦ · · · ◦ Z0 (p) ∈ Ym , and which for ν = 0, 1 and k = 1, . . . , m, sends the image in Y of any point (k) (k) q ∈ Dν under the composition of the inclusions Dν and Dν → Dν to the point Zm+1 ◦ · · · ◦ Zk+1 ◦ D(k) (q) in Ym , is a well-defined biholomorphism (see Exerν cise 5.12.6). Thus we may identify Ym with Y . Similarly, one may consider finite sequential holomorphic attachment of tubes, and verify that the resulting complex 1-manifold is naturally biholomorphic to the complex 1-manifold obtained by simultaneous holomorphic attachment of the tubes (see Exercise 5.12.7). Exercises for Sect. 5.12 5.12.1 In the notation of Sect. 5.12.1, verify that the natural inclusion of Z  T ∗ into Z  T descends to a biholomorphism of X∗ onto X.

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249

5.12.2 Let X be a complex 1-manifold obtained by holomorphic attachment of tubes to a complex 1-manifold Y as in Sect. 5.12.1. Prove that X is connected if Y is connected. Give an example that shows that X may be connected even if Y is not (cf. Exercise 2.3.2). 5.12.3 In the notation of Sect. 5.12.2, verify that the natural inclusion of D0∗  D1∗  Z into D0  D1  Z descends to a biholomorphism of Y ∗ onto Y . → X  in Sect. 5.12.3 is a well-defined biholomor5.12.4 Verify that the mapping Y phism. 5.12.5 Verify that holomorphic tube removal undoes tube attachment. That is, in analogy with Sect. 5.12.3, verify that if one holomorphically attaches tubes to a complex 1-manifold Y as in Sect. 5.12.1, and then holomorphically removes some of the tubes as in Sect. 5.12.2, then the resulting complex 1-manifold is biholomorphic to the complex 1-manifold obtained by holomorphic attachment of the remaining tubes to Y . 5.12.6 Verify that the mapping Y → Ym in Sect. 5.12.4 is a well-defined biholomorphism. 5.12.7 In analogy with the description of finite sequential holomorphic removal of tubes in Sect. 5.12.4, provide details for the process of finite sequential holomorphic attachment of tubes to a complex 1-manifold. Include a verification that the resulting complex 1-manifold is (naturally) biholomorphic to the complex 1-manifold obtained by simultaneous holomorphic attachment of the tubes. 5.12.8 Describe, with proofs, C ∞ and C 0 versions of attachment, cap attachment, tube attachment, and tube removal in which the biholomorphisms are replaced with diffeomorphisms in second countable 2-dimensional C ∞ manifolds and homeomorphisms in second countable 2-dimensional topological manifolds, respectively. 5.12.9 Let T ∼ = (0; 1/R0 , R1 ) be a holomorphic coordinate annulus (i.e., a tube) in a Riemann surface X as in Sect. 5.12.2 (with R0 , R1 > 1). Assume that the complex 1-manifold obtained by holomorphic removal of T is connected, but not simply connected. Prove that there is an infinite covering Riemann  → X (cf. Exercise 5.9.4) such that surface ϒ : X (i) T is evenly covered by ϒ; and (ii) For any connected component T0 of ϒ −1 (T ), we have Deck(ϒ) · T0 = ϒ −1 (T ).

5.13 Tubes in a Compact Riemann Surface Recall that we are working toward a proof that any Riemann surface may be obtained by holomorphic attachment of tubes to a planar domain. We consider the compact case in this section. Theorem 5.13.1 Up to biholomorphism, every compact Riemann surface may be obtained by holomorphic attachment of tubes at finitely many disjoint coordinate

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Fig. 5.12 Representation of a compact Riemann surface via holomorphic attachment of finitely many tubes to the Riemann sphere

disks in the Riemann sphere P1 . More precisely, suppose X is a compact Riemann surface and K ⊂ X is a union of finitely many disjoint compact sets, each of which is a closed disk in some local holomorphic chart (i.e., some local holomorphic chart maps the set onto a closed disk in C). Then, in the notation of Sect. 5.12.1, X is biholomorphic to the compact Riemannsurface Z ∪ T obtained by holomorphic attachment of tubes {Tj }j ∈J (with T = j ∈J Tj ) to the Riemann surface Y = P1 at elements of a finite family of disjoint coordinate disks {Dνj }ν∈{0,1},j ∈J (see Fig. 5.12). Moreover, the biholomorphism may be chosen so that the image of K is disjoint from the image of T in Z ∪ T . According to Sect. 5.12.3, the following theorem, which is in a form that is more convenient for the proof, is equivalent to Theorem 5.13.1: Theorem 5.13.2 For every compact Riemann surface, one may obtain a Riemann surface that is biholomorphic to P1 by holomorphic removal of finitely many disjoint tubes. More precisely, suppose X is a compact Riemann surface and K ⊂ X is a union of finitely many disjoint compact sets, each of which is a closed disk in some local holomorphic chart. Then, in the notation of Sect. 5.12.2, one may apply (finite) holomorphic tube removal to get a compact Riemann surface Y ≡ (D0  D1 ) ∪ Z ∼ = P1 , and one may choose the tubes to be disjoint from K. Lemma 5.13.3 Let X be a compact Riemann surface, and let K ⊂ X be a union of finitely many disjoint compact sets, each of which is a closed disk in some local holomorphic chart. If H1 (X, R) = 0, then there exists a local holomorphic chart (A, , (0; 1/R ∗ , R ∗ )) in X \ K with R ∗ > 1 and a closed C∞ 1-form θ on X such that the smooth Jordan curve γ : t → −1 (e2πit ) satisfies γ θ = 1. In particular, the image C ≡ γ ([0, 1]) is nonseparating in X. Proof By hypothesis, there exists a nonzero homology class in H1 (X, R), and since every class contains a linear combination of loops based at a point in X \ K (Proposition 10.6.7), we may assume that this class is represented by a loop in X with

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251

base point in X \ K. By Lemma 5.11.4, we may also assume that this representing loop lies in X \ K. According to Proposition 5.10.7, this loop is homologous in X \ K, and therefore, in X, to a sum of smooth Jordan loops. It follows that there exist a smooth Jordan curve β : [0, 1] → X \ K and a closed C ∞ real 1-form τ on X with β τ = 0. Lemma 5.11.1 provides a diffeomorphism of a relatively compact neighborhood U of the smooth Jordan curve B ≡ β([0, 1]) in X \ K onto an annulus that maps B onto a concentric circle. By Proposition 5.4.1, there exists a smooth relatively compact domain  in U such that B ⊂ , and such that each of the finitely many connected components L of ∂ has a neighborhood that admits a biholomorphism onto an annulus that maps L onto a concentric circle. Moreover, by Lemma 5.11.1,  \ B = 1 ∪ 2 , where 1 and 2 are disjoint smooth relatively compact domains in U , and for each j = 1, 2, B is a boundary component of j and (∂j ) \ B is a union of boundary components of ∂ (B is separating in U  and therefore in   U ). By Stokes’ theorem, (∂1 )∩(∂) τ = ± β τ = 0 (the sign depends on the choice of orientation). Hence the integral of τ along some boundary component of  (which is a boundary component of 1 ) is nonzero. Choosing a suitable local holomorphic chart (A, , (0; 1/R∗ , R ∗ )) in X \ K mapping this boundary component onto S1 , letting γ : t → −1 (e2πit ), letting θ be a suitable rescaling of τ , and applying Lemma 5.11.3, we get the claim.  Remark In the above proof, we used Proposition 5.4.1 in order to get a holomorphic coordinate annulus in X with nonseparating boundary components. In fact, Theorem 5.9.2 implies that a slight shrinking of the neighborhood U is itself biholomorphic to an annulus (U is itself biholomorphic to an annulus, to ∗ , or to C∗ ). A reader who has worked through the discussion in Sect. 5.9 of the classification of Riemann surfaces as quotients by groups of automorphisms may prefer to apply Theorem 5.9.2 in the above proof in place of Proposition 5.4.1 (see Exercise 5.13.1). Proof of Theorem 5.13.2 We proceed by induction on d ≡ dimR H1 (X, R), which, according to Proposition 10.6.7, is finite, since X is compact. The case d = 0 follows from the Koebe uniformization theorem (Theorem 5.5.3) or just Lemma 5.5.4. Assuming that d > 0, Lemma 5.13.3 provides a number R∗ > 1, a local holomorphic chart (A, , (0; 1/R ∗ , R ∗ )) in X \ K, and a closedC ∞ 1-form θ on X such that the smooth Jordan curve γ : t → −1 (e2πit ) satisfies γ θ = 1, and (hence) the image C ≡ γ ([0, 1]) is nonseparating in X. Fixing a coordinate annulus T = −1 ((0; 1/R, R))  A for some R ∈ (1, R ∗ ), holomorphic removal of the tube T as in Sect. 5.12.2 yields a compact Riemann surface Y . More precisely, setting Aν ≡ (0; 1, R) ⊂ Dν ≡ (0; R) for ν = 0, 1, A0 ≡ −1 ((0; 1/R, 1)) ⊂ T , and A1 ≡ −1 ((0; 1, R)) ⊂ T , we get a biholomorphism  : A0 ∪ A1 → A0  A1 , where for each index ν ∈ {0, 1} and each point p ∈ Aν , (p) is the image in A0  A1 of the point 

1/(p) ∈ A0 (p) ∈ A1

for ν = 0, for ν = 1,

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Fig. 5.13 Reduction of the dimension of the first homology via holomorphic removal of a tube

under the inclusion Aν → A0  A1 ⊂ D0  D1 . The set Z ≡ X \ C is connected, since C is nonseparating. Thus we get the compact Riemann surface Y0 ≡ (D0  D1 ) ∪ Z = (D0  D1 )  Z/∼, where for ν ∈ {0, 1}, z ∈ Aν , and p ∈ Aν , the images z0 of z and p0 of p in D0  D1  Z under the inclusions Aν , Aν → D0  D1  Z satisfy z0 ∼ p0 if and only if z · (p) = 1 for ν = 0 and z = (p) for ν = 1. In other words, Y0 is a compact Riemann surface obtained by removing the unit circle C in the tube (i.e., annulus) T and gluing in caps (i.e., disks) D0 and D1 on each end of the remaining two pieces (see Fig. 5.13). That is, Y0 is obtained by attaching caps to the open subset Z of X. Letting Z : Z → Y0 and Dν : Dν → Y0 for ν = 0, 1 be the corresponding inclusion mappings, we get Z (T \ C) = Z (A0 ∪ A1 ) = D0 (A0 ) ∪ D1 (A1 ) and

  Z (K) ⊂ Z (X \ T ) = Y0 \ D0 (D0 ) ∪ D1 (D1 ) .

Setting Dν∗ ≡ (0; R ∗ ) ⊃ D ν for ν = 0, 1, we also see that the inclusion D0 D1 → Y0 extends to a biholomorphism of D0∗  D1∗ onto a neighborhood of D0 (D0 ) ∪ D1 (D1 ) (see the remarks at the end of Sect. 5.12.2). It now suffices to show that dim H1 (Y0 , R) < d. For if this is the case, then, by the induction hypothesis, one may obtain a Riemann surface Y ∼ = P1 from Y0 by holomorphic removal of finitely many disjoint tubes that do not meet the image of K as well as the images of the caps D0 and D1 . Thus, by simultaneously removing these tubes along with the tube removed from X above, and applying Sect. 5.12.4, we get a Riemann surface that is biholomorphic to Y ∼ = P1 . In order to show that dim H1 (Y0 , R) < d = dim H 1 (X, R), it suffices to produce an injective linear map H1 (Y0 , R) → H1 (X, R) V, where V ≡ R · [γ ]H1 is the (1-dimensional) subspace spanned by [γ ]H1 . Let ρ : H1 (X, R) → H1 (X, R)/V

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253

be the quotient map. Given an element [ξ ]H1 ∈ H1 (Y0 , R), Lemma 5.11.4 implies that we may choose the representative ξ to be a real linear combination of loops in   Y0 \ D0 (D0 ) ∪ D1 (D1 ) = Z (X \ T ), and hence we may form the mapping [ξ ]H1 (Y0 ,R) → ρ([(−1 Z )∗ ξ ]H1 (X,R) ). To see that this mapping is well defined, linear, and injective, let us consider a 1-cycle ξ that is a linear combination of loops in Z (X \ T ), and let us set  −1  −1 ξ ≡ (Z )∗ ξ = (Z )∗ ξ and t ≡ θ ∈ R. ξ

  Given a closed C ∞ 1-form η on X, for s ≡ γ η, we have γ (η − sθ ) = 0, and hence, since the path homotopy class of γ generates the fundamental group of A ⊃ T , (η − sθ )A is exact. Cutting off a potential, we get a real-valued C ∞ function λ with compact support in A such that η − sθ = dλ on a neighborhood of T . Thus we get ∗ a well-defined closed C ∞ 1-form τ on Y0 by setting τ = (−1 Z ) (η − sθ − dλ) at each point in Z (X \ T ) and τ = 0 at each point in D0 (D0 ) ∪ D1 (D1 ). We then have        τ= η−s θ= η − st = η−t η= η. ξ

ξ

ξ

ξ

ξ

γ

ξ  −tγ

In particular, if [ξ ]H1 (Y0 ,R) = 0, then [ξ  ]H1 (X,R) = t[γ ]H1 (X,R) ∈ V, and it follows that the mapping is well defined. Linearity is easy to verify. For injectivity, observe that if, for ξ as above, we have [ξ  ]H1 (X,R) = u[γ ]H1 (X,R) ∈ V for some u ∈ R, then we must have u = t . Given a closed C ∞ 1-form τ on Y0 , τ is exact on a neighborhood of D0 (D0 ) ∪ D1 (D1 ), and hence for some real-valued C ∞ function λ0 with compact support in some small neighborhood of D0 (D0 ) ∪ D1 (D1 ), we have τ + dλ0 ≡ 0 on some (smaller) neighborhood of this set. Thus we may define a closed C ∞ 1-form η0 on X by setting η0 = ∗Z (τ + dλ0 ) at each point in X \ T , and η0 = 0 at each point in T . We then have     τ = (τ + dλ0 ) = η0 = t η0 = 0 ξ

ξ

ξ

(since γ lies in T ), and hence [ξ ]H1 (Y0 ,R) = 0.

γ



Remarks 1. The above proof does not really use the full Koebe uniformization theorem (or the Riemann mapping theorem), just the more elementary Proposition 5.4.1 and Lemma 5.5.4. The proof of the general case in Sect. 5.14 will use the full Koebe uniformization theorem (but not the Riemann mapping theorem). 2. It turns out that the injective linear map H1 (Y0 , R) → H1 (X, R)/R · [γ ]H1 in the above proof is never surjective, because dim H1 (X, R) is even for any compact Riemann surface X. This is a consequence of the existence of a canonical homology basis (see Sect. 5.16), as well as of the Hodge decomposition theorem (see Sect. 4.9).

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Exercises for Sect. 5.13 5.13.1 Give a slightly different (and shorter) proof of Lemma 5.13.3 by applying Theorem 5.9.2 in order to show that U may be chosen to be biholomorphic to an annulus (see the remark following the proof of Lemma 5.13.3). 5.13.2 Prove that every compact oriented C ∞ surface M may be obtained by C ∞ oriented attachment (i.e., the local charts are positively oriented and the attaching maps are orientation-preserving diffeomorphisms) of a finite family of disjoint tubes (see Exercise 5.12.8) to a compact oriented C ∞ surface N with H1 (N, R) = 0 (and hence H1 (N, A) = 0 for any subring A of C containing Z). Using this fact together with Lemma 5.5.4 and Theorem 5.9.2, give a slightly different proof of Theorem 5.13.1. Note. It will follow from the results of Chap. 6 that in fact, N is diffeomorphic to a sphere.

5.14 Tubes in an Arbitrary Riemann Surface The goal of this section is the following general tube-attachment theorem (see [Ri] for a more complete classification of noncompact second countable topological surfaces): Theorem 5.14.1 Up to biholomorphism, every Riemann surface may be obtained by holomorphic attachment of tubes at elements of a locally finite family of disjoint coordinate disks in a domain in P1 . More precisely, suppose X is a Riemann surface, and K ⊂ X is a union of a locally finite family of disjoint compact sets, each of which is a closed disk in some local holomorphic chart. Then, in the notation of Sect. 5.12.1, X is biholomorphic to the Riemann surface Z ∪ T obtained by holomorphic attachment of tubes {Tj }j ∈J (with T = j ∈J Tj ) to a domain Y in P1 at elements of a locally finite family of disjoint coordinate disks {Dνj }ν∈{0,1},j ∈J . Moreover, the biholomorphism may be chosen so that the image of K is disjoint from the image of T in Z ∪ T . As in the compact case, it is more convenient to prove the following version, which, according to Sect. 5.12.3, is equivalent to Theorem 5.14.1: Theorem 5.14.2 For every Riemann surface X, one may obtain a planar Riemann surface by holomorphic removal of a locally finite family of disjoint tubes. More precisely, suppose X is a Riemann surface, and K ⊂ X is a union of a locally finite family of disjoint compact sets, each of which is a closed disk in some local holomorphic chart. Then, in the notation of Sect. 5.12.2, one may apply holomorphic tube removal to get a Riemann surface Y ≡ (D0  D1 ) ∪ Z that is biholomorphic to a domain in P1 . Moreover, the tubes may be chosen to be disjoint from K. The idea of the proof is as follows. We first form an exhaustion of X by relatively compact smooth domains {ν }∞ ν=1 such that each boundary component is a

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255

concentric circle in a coordinate annulus (i.e., in a tube). For each connected component  of ν+1 \ ν , by holomorphically removing the tubes about all but one of the connected components of ∂ ∩ ∂ν , we see that we may assume that ∂ ∩ ∂ν is connected (see Lemma 5.14.4 and Fig. 5.14). Holomorphic removal of the tubes about the boundary components of the domains {ν } then yields a disjoint union of compact Riemann surfaces (see Fig. 5.16). By Theorem 5.13.2, we may then get a disjoint union of copies of the Riemann sphere by holomorphic removal of tubes in each of these components (see Fig. 5.17). Finally, by holomorphically reattaching the boundary tubes about the boundary components of the domains {ν }, we get a planar Riemann surface (see Fig. 5.18). Lemma 5.14.3 Suppose X is an open Riemann surface, and K ⊂ X is a union of a locally finite family of disjoint compact sets, each of which is a closed disk in some local holomorphic chart. Then there exists a sequence of smooth domains {ν }∞ ν=1 in X such that  (i) ν  ν+1 for each ν ∈ Z>0 and X = ∞ ν=1 ν ; and (ii) For each ν ≥ 1 and for each connected component C of ∂ν , there is a local holomorphic chart (A, , (0; 1/R, R)) for some R = R(C) > 0 with C ⊂ A ⊂ X \ K, (A ∩ ν ) = (0; 1, R), and C = −1 (∂(0; 1)).  Proof We may write K = ∞ j =1 Kj for disjoint connected compact sets {Kj }, each of which is either the empty set or a closed disk  in some local holomorphic chart. We may also choose compact sets {Lν }∞ with ν Lν = X and Lν ⊂ Lν+1 for each ν, ν=1 such that for each ν, jν < jν+1 and Lν ∩ Kj = ∅ for all and positive integers {jν }∞ ν=1 j ≥ jν . By Proposition5.4.1, we may choose a relatively compact smooth domain 1 in the domain X \ ∞ j =j1 Kj (this set is connected by Lemma 5.11.4) such that K1 ∪ · · · ∪ Kj1 −1 ⊂ 1 and 1 has the property (ii). Given a domain ν−1  L1 ∪ X\ ∞ j =jν−1 Kj , we may choose a relatively compact smooth domain ν in the  domain X \ ∞ j =jν Kj such that ν−1 ∪ Lν ∪ K1 ∪ · · · ∪ Kjν −1 ⊂ ν and ν has the property (ii). Thus, by induction, we get the desired sequence of domains {ν }.  Lemma 5.14.4 Suppose X is an open Riemann surface, and K ⊂ X is a union of a locally finite family of disjoint compact sets, each of which is a closed disk in some local holomorphic chart. Then, in the notation of Sect. 5.12.2, one may apply holomorphic tube removal to get a Riemann surface Y ≡ (D0  D1 ) ∪ Z with holomorphic inclusion Z : Z → Y and a sequence of smooth domains {ν }∞ ν=1 in Y such that setting 0 = ∅, we have the following (see Fig. 5.14): (i) The closures of the removed tubes are disjoint from K (in particular, K ⊂ Z); (ii) ν  ν+1 for each ν ≥ 1 and Y = ∞ ν=1 ν ; (iii) For each ν ≥ 1 and for each connected component C of ∂ν , there is a local holomorphic chart (A, , (0; 1/R, R)) for some R = R(C) > 0 with C ⊂ A  Z (Z \ K), (A ∩ ν ) = (0; 1, R), and C = −1 (∂(0; 1)); (iv) For each ν ≥ 0, the (finitely many) connected components of ν+1 \ ν lie in distinct connected components of Y \ ν , and for ν ≥ 1, if  is a connected

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Fig. 5.14 Holomorphic removal of certain boundary tubes for an exhaustion yielding an exhaustion with separating boundary components

component of ν+1 \ ν and V is the connected component of Y \ ν containing , then one of the boundary components of  is a boundary component C of ν satisfying ∂V = ∂ ∩ ∂ν = C; and (v) For each ν ≥ 1, each boundary component C of ∂ν is separating in Y . Proof By Lemma 5.14.3, there exist a family of smooth domains {ν }∞ ν=1 , disjoint local holomorphic charts  (ν) (ν)  Aj , j , (0; 1/Rν∗ , Rν∗ ) ν∈Z ,j ∈{1,...,m } >0

ν

with mν ∈ Z>0 and Rν∗ > 1 for each ν, disjoint smooth Jordan curves  (ν)  Cj ν∈Z ,j ∈{1,...,m } , ν

>0

and disjoint domains {j(ν) }ν∈Z≥0 , j ∈{1,...,kν } with kν ∈ {1, . . . , mν } (setting m0 = 1) for each ν ≥ 0 such that setting 0 ≡ ∅, we have  ν ; (i) ν  ν+1 for each ν ≥ 0 and X = ∞ mν (ν) ν=1 (ii) For each ν ≥ 1, ∂ν ⊂ j =1 Aj  X \ K, and for each j = 1, . . . , mν , j (Aj ∩ ν ) = (0; 1, Rν∗ ) and Cj = (j )−1 (∂(0; 1)) (in particular,  ν (ν) ∂ν = m j =1 Cj ); (ν)

(ν)

(ν)

(ν)

(ν)

(ν)

(iii) For each ν ≥ 0, 1 , . . . , kν are the connected components of ν+1 \ ν (in (0) 1

= 1 ); and particular, k0 = 1 and (ν) (ν) (iv) For each ν ≥ 1 and each j = 1, . . . , kν , Cj is a connected component of ∂j (thus we have chosen exactly one of the common connected components of

5.14

Tubes in an Arbitrary Riemann Surface (ν)

∂ν and j

257

for each j = 1, . . . , kν , and holomorphic attachment of caps at

ν the remaining boundary curves {Cj(ν) }m j =kν +1 will yield a Riemann surface in

(ν)

which the images of the boundary curves {Cj }kjν=1 will be separating). (ν)

In particular, we have Aj  ν+1 \ ν−1 for each ν ≥ 1 and each j = 1, . . . , mν , so the family {A(ν) j } is locally finite in X. Finally, choosing constants {Rν } with 1 < Rν < Rν∗ for each ν ≥ 1, we may set (ν)

Tj

  (ν) (ν) ≡ (j )−1 (0; 1/Rν , Rν )  Aj

and

(ν)

(ν)

D0j = D1j ≡ (0; Rν )

for each ν ≥ 1 and j = 1, . . . , mν , C≡

mν ∞

(ν) Cj

⊂T ≡

ν=1 j =kν +1

mν ∞

(ν)

Tj ,

ν=1 j =kν +1

Z ≡ X \ C ⊃ X \ T ⊃ K (i.e., the condition (i) in the statement of the lemma holds), and  (ν) Dμj for μ = 0, 1. Dμ ≡ ν≥1,kν <j ≤mν (ν)

Holomorphic removal of the tubes {Tj }ν≥1, kν <j ≤mν as in Sect. 5.12.2 yields a complex 1-manifold Y = (D0  D1 ) ∪ Z (for the appropriate biholomorphism ). Denoting the associated inclusions by Z : Z → Y and Dμ : Dμ → Y for (ν) ∗

μ = 0, 1, we get Z (X \ T ) = Y \ (D0 (D0 ) ∪ D1 (D1 )). Setting Dμj (ν) (0; Rν∗ ) ⊃ Dμj

≡

for each choice of ν, j , and μ, and setting 

Dμ∗ ≡

(ν) ∗

Dμj

for μ = 0, 1,

ν≥1,kν <j ≤mν

we also see that the inclusion D0  D1 → Y extends to a biholomorphism of D0∗  D1∗ onto a neighborhood of D0 (D0 ) ∪ D1 (D1 ) (see the remarks at the end of Sect. 5.12.2). (0) Now Z ∩ 1 = 1 = 1 is connected. Assuming that Z ∩ ν is connected for some ν ≥ 1, we see that Z ∩ ν+1 is equal to the union of the connected set Z ∩ ν with the connected open sets (ν)

(ν)

(ν)

(ν)

1 ∪ T1 , . . . , kν ∪ Tkν , each of which meets Z ∩ ν . Thus  it follows by induction that Z ∩ ν is connected for each ν, and hence that Z = ∞ ν=1 (Z ∩ ν ) is connected. Since Y \ Z (Z) is covered by a union of a locally finite family of (disjoint) domains each of which meets Z (Z) (and is biholomorphic to a disk), it follows that Y is connected.

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Next we observe that if 0 ≤ ν < μ, then Z ∩ μ \ ν has exactly kν distinct con(ν) (ν) nected components, each of which contains exactly one of the sets 1 , . . . , kν . For the proof of this fact, we proceed by induction on μ, the case μ = ν + 1 being trivial. Assuming that the claim holds for Z ∩ μ \ ν , each of the disjoint con(μ) (μ) (μ) (μ) nected sets 1 ∪ T1 , . . . , kμ ∪ Tkμ must meet exactly one of the kν connected (μ)

components of Z ∩ μ \ ν , because for each j = 1, . . . , kμ , Tj ∩ μ meets, and therefore lies in, exactly one such connected component. Thus the claim holds for μ + 1, and therefore for all μ > ν. It also follows that for each ν ≥ 0, the set Z \ ν has exactly kν distinct connected components; and we may denote these connected (ν) (ν) (ν) components by {Vj }kjν=1 , where j ⊂ Vj for each j = 1, . . . , kν . Furthermore, (ν)

for each ν ≥ 1 and each j = 1, . . . , kν , Cj Z

\ Cj(ν)

is a separating curve in Z. In fact,

has the two connected components (ν)

Vj

and (Z ∩ ν ) ∪

(ν)

(Vi

(ν)

∪ Ti ).

1≤i≤kν ,i =j

We now let 0 = ∅, and for each ν ≥ 1, we let ν be the union of Z (Z ∩ ν ) with those (finitely many) connected components of D0 (D0 ) ∪ D1 (D1 ) that meet this set. For each ν ≥ 1, ν is then a relatively compact domain in Y with distinct boundary components Z (C1(ν) ), . . . , Z (Ck(ν) ), and since Z maps Z biholomorν phically onto Z (Z), ν satisfies the conditions (ii) and (iii) in the statement of the lemma. Furthermore, by construction, for each ν ≥ 0, ν+1 \ ν is the union of the (ν) (ν) disjoint connected open sets {j }kjν=1 , where for each ν and j , j is the union of Z (j(ν) ) with those connected components of D0 (D0 ) ∪ D1 (D1 ) that meet Z (j(ν) ), and Y \ ν is the union of disjoint connected open sets {Wj(ν) }kjν=1 , where (ν)

for each ν and j , Wj

(ν)

(ν)

⊃ j is the union of Z (Vj ) with those connected com-

ponents of D0 (D0 ) ∪ D1 (D1 ) that meet Z (Vj(ν) ). Finally, for each ν ≥ 1 and (ν)

(ν)

(ν)

each j ∈ {1, . . . , kν }, the boundary component Z (Cj ) = ∂j ∩ ∂ν = ∂Wj (ν)

(ν)

of ν (and of j ) is separating in Y , since Y \ Z (Cj ) has (only) the two connected components

 (ν) (ν) (ν)  and ν ∪ Wi ∪ Z (Ti ) . Wj  1≤i≤kν ,i =j Proof of Theorem 5.14.2 The compact case is Theorem 5.13.2, so we may assume that X is noncompact. Step 1. Choice of a suitable exhaustion by domains. We may holomorphically remove relatively compact tubes with closures disjoint from K to get a Riemann surface X with the properties considered in Lemma 5.14.4. Moreover, we may assume that the local holomorphic chart on each tube actually extends to a neighborhood of the closure. Thus, if we let K  be the union of the image of K with the closures of the holomorphically attached caps in X  , and holomorphic removal of

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Tubes in an Arbitrary Riemann Surface

259

Fig. 5.15 The initial (modified) Riemann surface X

tubes in X  with closures disjoint from K  yields a planar Riemann surface Y , then, as in Sect. 5.12.4, we may identify Y with a Riemann surface obtained by holomorphic removal of tubes in X that are disjoint from K. Therefore, by replacing X with X  and K with K  , we may assume without loss of generality that there exist a family of (nonempty) smooth domains {ν }∞ ν=1 , disjoint local holomorphic charts  (ν) (ν)  Aj , j , (0; 1/Rν∗ , Rν∗ ) ν∈Z

>0 , j ∈{1,...,mν }

with mν ∈ Z>0 and Rν∗ > 1 for each ν, disjoint smooth Jordan curves {Cj(ν) }ν∈Z>0 , j ∈{1,...,mν } , (ν)

disjoint domains {j }ν∈Z≥0 , j ∈{1,...,mν } with m0 = 1, and for each ν ≥ 0, disjoint domains {Vj(ν) }j ∈{1,...,mν } such that setting 0 ≡ ∅, we have (see Fig. 5.15)  (i) ν  ν+1 for each ν ≥ 0 and X = ∞ ν=1 ν ; mν (ν) (ii) For each ν ≥ 1, ∂ν ⊂ j =1 Aj  X \ K, and for each j = 1, . . . , mν , j (Aj ∩ ν ) = (0; 1, Rν∗ ) (ν)

(ν)

(iv)

Cj = (j )−1 (∂(0; 1)) (ν)

(ν)

mν

(ν) j =1 Cj ); (ν) For each ν ≥ 0, 1(ν) , . . . , m ν are the distinct connected components of (0) ν+1 \ ν (in particular, 1 = 1 ), V1(ν) , . . . , Vm(ν) ν are the distinct connected (ν) (ν) components of X \ ν , and j ⊂ Vj for each j = 1, . . . , mν ; and For each ν ≥ 1 and each j = 1, . . . , mν , the smooth Jordan curve Cj(ν) is sepa(ν) (ν) (ν) rating in X and Cj = ∂j ∩ ∂ν = ∂Vj is a boundary component of ν (ν) (ν) and of j (and of Vj ).

(in particular, ∂ν =

(iii)

and

(ν) In particular, we have Aj  ν+1 \ ν−1 for each ν ≥ 1 and each j = 1, . . . , mν ,

so the family {A(ν) j } is locally finite in X. Finally, choosing constants {Rν } with 1 < Rν < Rν∗ for each ν ≥ 1, we may set

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Fig. 5.16 The complex 1-manifold Y0 obtained by holomorphic removal of the boundary tubes for the exhaustion of X (ν)

Tj

≡ (j )−1 ((0; 1/Rν , Rν ))  Aj (ν)

(ν)

(ν)

(ν)

and D0j = D1j ≡ (0; Rν )

for each ν ≥ 1 and each j = 1, . . . , mν , and

(ν)

(ν) C0 ≡ Cj ⊂ T0 ≡ Tj ⊂ X \ K, ν,j

Z 0 ≡ X \ C0 = Dμ ≡



ν,j (ν) j

⊃ K,

ν,j (ν)

Dμj

for μ = 0, 1.

ν,j

Step 2. Removal of the boundary tubes, removal of the interior tubes, and reattachment of the boundary tubes. Holomorphic removal of the tubes {Tj(ν) } as in Sect. 5.12.2 yields a second countable complex 1-manifold Y0 = (D0  D1 ) ∪0 Z0 (for the appropriate biholomorphism 0 ). Denoting the associated inclusions by Z0 : Z0 → Y0 and Dμ : Dμ → Y0 for μ = 0, 1, we get Z0 (X \ ∗

(ν) (ν) T 0 ) = Y0 \ (D0 (D0 ) ∪ D1 (D1 )). Setting Dμj ≡ (0; Rν∗ ) ⊃ Dμj for each choice of ν, j , and μ, and setting  (ν) ∗ Dμ∗ ≡ Dμj for μ = 0, 1, ν,j

we also see that the inclusion D0  D1 → Y0 extends to a biholomorphism of D0∗  D1∗ onto a neighborhood of D0 (D0 ) ∪ D1 (D1 ) (see the remarks at the end of Sect. 5.12.2). (ν) The complex 1-manifold Y0 has distinct connected components {Yj }ν≥0,1≤j ≤mν (ν)

(ν)

with Z0 (j ) ⊂ Yj

(ν)

for each ν and j ; in fact, Yj

may be viewed as the compact

Riemann surface obtained by holomorphic attachment of caps to j(ν) at the boundary as in Sect. 5.3 (see Fig. 5.16). Therefore, by Theorem 5.13.2, we may apply holomorphic tube removal in each of these connected components to get a Riemann surface that is biholomorphic to P1 . Thus simultaneous holomorphic removal of these tubes from Y0 yields a complex 1-manifold Y1 in which each connected component is biholomorphic to P1 (see Fig. 5.17). Moreover, we may choose the tubes

5.14

Tubes in an Arbitrary Riemann Surface

261

Fig. 5.17 The complex 1-manifold Y1 obtained by holomorphic removal of tubes in each connected component of Y0

Fig. 5.18 The desired Riemann surface Y obtained by holomorphic reattachment of the boundary tubes to Y1 (equivalently, via holomorphic removal of tubes from X)

so that their closures do not meet the set Z0 (K) ∪ D0 (D0 ) ∪ D1 (D1 ). As in Sect. 5.12.4, we may identify Y1 with the Riemann surface obtained by holomorphic removal of all of the above tubes (those removed in the construction of Y0 together with those removed in the construction of Y1 ) from X. Hence holomorphic reattachment of the tubes {Tj(ν) } to Y1 as in Sect. 5.12.3 yields a Riemann surface Y that is biholomorphic to the Riemann surface obtained by holomorphic removal of a locally finite collection of disjoint tubes in X whose closures do not meet T 0 ∪ K (see Fig. 5.18). Step 3. Planarity of Y . By the Koebe uniformization theorem (Theorem 5.5.3), it now suffices to show that Y is planar. For this, we denote by T the union of the tubes in X \ (T 0 ∪ K) that were removed in order to get Y , we denote by C ⊂ T the union of the corresponding (unit) circles in each of the tubes in T (which were cut out in the tube-removal process), we set Z = X \ C, and we let Z : Z → Y denote the corresponding holomorphic inclusion. The complement Y \ Z (Z) is a union of a locally finite family of disjoint compact sets, each of which is a closed disk in some local holomorphic chart. By construction, Y is connected, and hence Z ∼ = Z (Z) is connected (for example, by Lemma 5.11.4). Moreover, for each ν = 0, 1, 2, . . . , the (ν) (ν) (ν) (ν) set Z ∩ ν = ν \ C, and the sets Z ∩ j = j \ C and Z ∩ Vj = Vj \ C for j = 1, . . . , mν , are connected (see Exercise 5.14.1).

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Let 0 = ∅, and for each ν = 1, 2, 3, . . . , let ν ⊂ Y be the connected component of Y \ Z (∂ν ) containing Z (Z ∩ ν ) = Z (ν \ C) (which we may identify with the Riemann surface obtained by holomorphic removal from ν of the tubes given by those connected components of T that are contained in ν ). Similarly, for each ν ≥ 0 and each j = 1, . . . , mν , let (ν) j be the connected component of Y \ Z (C0 ) containing Z (Z ∩ j(ν) ) = Z (j(ν) \ C) (which we may identify with the Riemann surface obtained by holomorphic removal from j(ν) of the tubes given by those connected components of T that are contained in j(ν) ), and let Wj(ν) be the connected component of Y \ Z (∂ν ) containing Z (Z ∩ Vj(ν) ) = Z (Vj(ν) \ C) (which we may identify with the Riemann surface obtained by holomorphic re(ν) moval from Vj of the tubes given by those connected components of T which are  contained in Vj(ν) ). Then {ν } is a sequence of smooth domains with Y = ν ν (ν)

(ν)

and ν  ν+1 for each ν ≥ 0. Moreover, for each ν ≥ 0, {j } and {Wj } are the distinct connected components of ν+1 \ ν and Y \ ν , respectively, and, for (ν) each j = 1, . . . , mν , we have (ν) j ⊂ Wj . Furthermore, for each ν ≥ 1, the distinct (ν)

(ν)

(ν)

boundary components of ν are given by Z (Cj ) = ∂j ∩ ∂ν = ∂Wj

for

(ν) 1, . . . , mν , Z (Cj )

j = 1, . . . , mν . Moreover, for each ν ≥ 1 and j = is separating in Y with complement equal to the union of the disjoint connected open sets

(ν) (ν) (ν) Wj and ν ∪ Wi ∪ Z (Ti ). 1≤i≤mν ,i =j

To verify that Y is planar, we must show that every closed C ∞ 1-form θ with compact support on Y is exact. For each ν ≥ 1 and each j = 1, . . . , mν , Z (Cj(ν) ) is separating in Y , so  θ = 0 (for either orientation) (ν)

Z (Cj )

by Lemma 5.11.3. Hence, since there is a biholomorphism of a neighborhood of Z (Tj(ν) ) onto an annulus that maps Z (Cj(ν) ) onto a concentric circle, θ is exact on this neighborhood. Therefore, by replacing θ with the difference of θ and the differential of a C ∞ compactly supported function on Y with differential equal to θ on a neighborhood of Z (T0 ) (note that we may choose the function to be compactly supported because θ is identically zero on a neighborhood of all but finitely many connected components of Z (T0 )), we may assume that

(ν) j . supp θ ⊂ Y \ Z (T0 ) ⊂ ν,j

Identifying Y1 with the complex 1-manifold obtained by holomorphic removal (ν) of the tubes {Z (Tj )} from Y (see Sect. 5.12.4), we get the holomorphic inclusion Y \Z (C0 ) : Y \ Z (C0 ) → Y1 . Thus the restriction θ Y \Z (C0 ) determines a

5.14

Tubes in an Arbitrary Riemann Surface

263

1-form on the image Y \Z (C0 ) (Y \ Z (C0 )) in Y1 that extends by 0 to a closed compactly supported C ∞ 1-form τ on Y1 . On the other hand, each connected component P of Y1 is a copy of P1 in which, for some unique choice of ν and j , P ∩ Y \Z (C0 ) (Y \ Z (C0 )) = Y \Z (C0 ) ((ν) j ), and the complement P \ Y \Z (C0 ) ((ν) j ) is a union of finitely many disjoint connected compact sets, each of which is mapped onto a closed disk by some local holomorphic chart. Hence τ P = dλ for some C ∞ function λ on P , and in partic(ν) (ν) (ν) ular, λ is constant on the domain Y \Z (C0 ) (Tj ∩ j ). Pulling back to j and extending by the associated constant, we see that for each ν ≥ 1 and each (ν) (ν) (ν) j = 1, . . . , mν , there exists a C ∞ function ρj on the domain j ∪ Z (Tj ) such that dρj(ν) = θ(ν) ∪ j

(ν) Z (Tj )

. Similarly, we also have a C ∞ function ρ1

(0)

on

(0) 1

= 1 with dρ1(0) = θ(0) . 1 ∞ Now let η1 = ρ1(0) on 1 = (0) 1 . Given a function ην ∈ C (ν ) with dην = θ ν for some ν ≥ 1 (in particular, ην is locally constant on T0 ∩ ν ), by replacing (ν) (ν) each function ρj with the sum of ρj and some constant, we may assume that ρj(ν) = ην on the domain (ν)

(ν)

(ν)

[j ∪ Z (Tj )] ∩ ν = Z (Tj ) ∩ ν . The function ην+1 given by ην+1 ν = ην and ην+1 (ν) ∪ j

(ν) Z (Tj )

(ν)

= ρj

for j =

1, . . . , mν is then a C ∞ function on ν+1 with dην = θ ν+1 . Thus, by induction, we get a sequence of C ∞ functions {ην }, and we get a well-defined C ∞ function η  on Y with dη = θ by setting ην = ην for each ν = 1, 2, 3, . . . . Remark One mayinstead complete the proof of the above theorem by showing more directly that γ θ = 0 for every loop γ in Y . For any such loop is homologous (ν)

to a sum of loops, each of which lies in one of the sets j (here, one uses the fact that the smooth Jordan curves {Cj(ν) } are separating in Y ). One may also choose the loops to avoid C0 . Integrating the form τ on Y1 over the images of these loops in Y1 , one gets the claim. Exercises for Sect. 5.14 5.14.1 Verify that in the proof of Theorem 5.14.2, for each ν = 0, 1, 2, . . . , the set Z ∩ ν = ν \ C, and the sets Z ∩ j(ν) = j(ν) \ C and Z ∩ Vj(ν) = Vj(ν) \ C for j = 1, . . . , mν , are connected. 5.14.2 According to Sard’s theorem (see, for example, [Mi]), the set of critical values of a C ∞ function on a second countable C ∞ manifold is a set of measure 0. Using Sard’s theorem, Theorem 9.10.1, Lemma 5.11.1, and arguments similar to those appearing in this section, prove that every second

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countable oriented C ∞ surface may be obtained by C ∞ oriented attachment of a locally finite family of disjoint tubes to an oriented planar C ∞ surface N (cf. Exercises 5.12.8 and 5.13.2). Using this fact together with Theorem 5.5.3 and Theorem 5.9.2, construct a slightly different proof of Theorem 5.14.1.

5.15 Nonseparating Smooth Jordan Curves The goal of this section is a proof of equivalence of the topological (as in Definition 5.11.2) and analytic (as in Lemma 5.11.3) characterizations of separating smooth Jordan curves in an oriented smooth surface M; that is, the goal is a proof that for every nonseparating smooth Jordan curve C in M, there exists a compactly supported closed C ∞ 1-form with nonzero integral along C. This fact will not be applied directly until Sect. 5.17, but the construction of the associated 1-form will be a key element in the construction in Sect. 5.16 of a canonical homology basis for a compact Riemann surface. The main point of this section is the following: Lemma 5.15.1 For any smooth Jordan curve β : [0, 1] → M with nonseparating image B = β([0, 1]) in an oriented C ∞ surface M, we have the following: (a) There exists a smooth Jordan curve α : [0, 1] → M with image A ≡ α([0, 1]) such that A ∩ B = {α(0)} = {β(0)} and such that the pair of tangent vectors ˙ α(0), ˙ β(0) is a positively oriented basis for the tangent space Tα(0) M. (b) For any choice of a smooth Jordan curve α with the properties in (a), the sets A ≡ α([0, 1]) and A ∪ B are nonseparating. Moreover, for any choice of neighborhoods U and V of A and B, respectively, there exist closed compactly supported C ∞ real 1-forms θ and   τ on M  such that supp θ ⊂ V \ B, supp τ ⊂ U \ A, α θ = 1, β θ = 0, α τ = 0, β τ = 1, and for any loop γ   in M, we have γ θ ∈ Z and γ τ ∈ Z (hence, for M second countable, we have [θ ]H 1 , [τ ]H 1 ∈ H 1 (M, Z)). Proof According to Lemma 5.11.1, there exists a positively oriented local C ∞ chart (T , , (0; 1/R, R)) with R > 1, B ⊂ T , and (β(t)) = e2πit for each t ∈ [0, 1]. Fixing R0 ∈ (1, R), we may let α1 : [−1, 1] → T be the path from  −1 (1/R0 ) to  −1 (R0 ) given by t →  −1 (R0t ) =  −1 (et log R0 ) =  −1 (et log R0 + 0i). By Lemma 5.10.6, there is an injective smooth path α2 in the domain M \ B from  −1 (R0 ) to  −1 (1/R0 ), and we may set s0 ≡ max α2−1 (α1 ([0, 1])) ∈ [0, 1), s1 ≡ min([s0 , 1] ∩ α2−1 (α1 ([−1, 0]))) ∈ (s0 , 1],

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265

Fig. 5.19 Transverse nonseparating smooth Jordan curves and a smooth annular neighborhood

t0 ≡ α1−1 (α2 (s0 )) ∈ (0, 1], t1 ≡ α1−1 (α2 (s1 )) ∈ [−1, 0). We then get a piecewise smooth Jordan curve α3 in M by setting ⎧ ⎪ if t ∈ [0, t0 /3], ⎨α1 (3t) s1 −s0 (3t − t )) if t ∈ [t0 /3, 1 + (t1 /3)], α3 (t) = α2 (s0 + 3+t 0 1 −t0 ⎪ ⎩ if t ∈ [1 + (t1 /3), 1]. α1 (3(t − 1)) Moreover, α3 is loop-smooth at α3 (0) = β(0) =  −1 (1), α3 is smooth at each point in (0, 1) \ {t0 /3, 1 + (t1 /3)}, and α3 ((0, 1)) ⊂ M \ B. Applying Lemma 5.10.5, we get a smooth Jordan curve α that agrees with α3 outside a small neighborhood of the set {t0 /3, 1 + (t1 /3)}, and that maps that small neighborhood into a small neighborhood of {α3 (t0 /3), α3 (1 + (t1 /3))}. In particular, we may choose α so that ˙ ˙ β(0) forms a posiα([0, 1]) ∩ B = {α(0)} = {−1 (1)} = {β(0)}, and the pair α(0), tively oriented basis. Thus part (a) is proved. For the proof of part (b), suppose that α is an arbitrary smooth Jordan curve for which the image A ≡ α([0, 1]) meets B only in the point α(0) = β(0) and ˙ the pair α(0), ˙ β(0) forms a positively oriented basis for Tα(0) M. Let αˆ : R → M be the 1-periodic C ∞ immersion given by t → α(t − t). Given a neighborhood V of B in M, we may choose the annular neighborhood T to be relatively compact in V . Finally, let us denote the two connected components of T \ B by T0 ≡  −1 ((0; 1, R)) and T1 ≡  −1 ((0; 1/R, 1)) (see Fig. 5.19). Since polar coordinates (r, θ ) (see Example 9.7.20) near (β(0)) = 1 form a positively oriented local C ∞ chart, we have   ˙ 0 < (dr ∧ dθ ) ∗ α(0), ˙ ∗ β(0)     ∂ ˙ 2π = (dr ∧ dθ ) ∗ α(0), ∂θ 1 d ˆ = 2πdr(∗ α(0)) ˙ = 2π r((α(t))) . t=0 dt It follows that the function t → |(α(t))| ˆ is strictly increasing on a neighborhood of each integer. Hence we may fix a number s ∈ (0, 1/2) such that α([0, s]) ⊂ T0 and α([1 − s, 1]) ⊂ T1 .

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We may choose a real-valued C ∞ function ϕ on M \ B such that supp ϕ  V , ∞ ϕ ≡ 0 on T0 , and ϕ ≡ 1 on T1 . Hence dϕ extends to a unique closed  C 1-form θ on M with supp θ  V \ T ⊂ V \ B, and in particular, we have β θ = 0. Since θ = 0 on T ⊃ α([0, 1] \ [s, 1 − s]) and θ = dϕ on M \ B ⊃ α([s, 1 − s]), we have   θ= θ = ϕ(α(1 − s)) − ϕ(α(s)) = 1. α

α[s,1−s]

It also follows that θ has an integer-valued integral along each loop γ in M. To see this, we may assume that γ (0) = γ (1) ∈ T \ B, and we may choose a partition 0 = t0 < t1 < · · · < tm = 1 of [0, 1] such that for each j = 1, . . . , m, we have γ ([tj −1 , tj ]) ⊂ M \ B or T and γ (tj ) ∈ T \ B. In particular, we have ϕ(γ (tj )) ∈ {0, 1} for each j , and, letting J ≡ {j ∈ {1, . . . , m} | γ ([tj −1 , tj ]) ⊂ M \ B}, we get   θ= θ= [ϕ(γ (tj )) − ϕ(γ (tj −1 ))] ∈ Z. γ

j ∈J

γ [tj −1 ,tj ]

j ∈J

 Now since α θ = 0, Lemma 5.11.3 implies that A is nonseparating in M. Moreover, the reverse curve α − is a smooth Jordan curve with nonseparating image A, ˙ ˙ and the pair β(0), α˙ − (0) (i.e., the pair β(0), −α(0)) ˙ is positively oriented. Hence, given a neighborhood U of A, we may construct (as above) a closed C ∞ 1-form τ   such that supp τ ⊂ U \ A, α τ = 0, β τ = 1, and γ τ ∈ Z for each loop γ . It remains to show that the set A ∪ B is nonseparating in M. For this, let  be a connected component of M \ (A ∪ B), and let p = α(0) = β(0). For each point q ∈ (A ∪ B) \ {p}, there is a local C ∞ chart (Uq , q , Uq = (−1, 1) × (−1, 1)) such that q ∈ Uq , Uq meets exactly one of the sets A, B, and Uq ∩ (A ∪ B) = −1 q ((−1, 1) × {0}). It follows that the set (∂) \ {p} ⊂ (A ∪ B) \ {p} is both open and closed in (A ∪ B) \ {p}. Since ∂ is not a singleton (otherwise, a connected neighborhood punctured at the boundary point would lie in , and hence the connected set A ∪ B would be contained in the singleton ∂, which is impossible), ∂ must meet (A ∪ B) \ {p} and therefore contain one of the connected sets A \ {p}, B \ {p}. On the other hand, according to Lemma 5.10.3, there exists a local C ∞ coordinate neighborhood in which p is represented by the origin, and A and B are represented by the x-axis and y-axis, respectively, near the origin. It follows that  must contain the part of some open coordinate quadrant near the origin, and therefore that A ∪ B ⊂ ∂. Thus ∂ = A ∪ B. Similarly, the set of points q ∈ (A ∪ B) \ {p} for which Uq \ (A ∪ B) ⊂  is open and closed in (A ∪ B) \ {p}. If the intersection of this set with A \ {p} is empty, then  is a smooth open subset of M \ B. Since θ has compact support in M \ B, Stokes’ theorem gives    θ= dθ = 0, 1= θ =± α

(M\B)∩∂



and we have arrived at a contradiction. Similarly, we must have Uq \ (A ∪ B) ⊂  for each point q ∈ B \ {p}. The local coordinate representation about p considered above implies that the complement of A ∪ B in some neighborhood of p also lies in . Thus M \ (A ∪ B) = , and A ∪ B is nonseparating in M. 

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267

Proposition 5.15.2 For any orientable C ∞ surface M:

 (a) A smooth Jordan curve C in M is separating if and only if C θ = 0 for every closed C ∞ 1-form θ with compact support in M. (b) M is planar (i.e., every C ∞ compactly supported closed 1-form on M is exact) if and only if every smooth Jordan curve C in M is separating. Proof Part (a) follows from Lemma 5.11.3 and Lemma 5.15.1. Part (b) follows from Proposition 5.10.7, Proposition 10.5.5, and part (a).  Exercises for Sect. 5.15 5.15.1 Let M be an oriented second countable C ∞ surface, let M ∗ = M ∪ {∞} be the one-point compactification of M (see Definition 9.1.11), and let β : [0, 1] → M be a smooth Jordan curve with image B ≡ β([0, 1]). (a) Assuming that B is separating in M but nonseparating in M ∗ , prove that: (i) There exists a C ∞ embedding α : R → M with image A ≡ α(R) such that A ∩ B = {α(0)} = {β(0)} and such that the pair of tangent ˙ vectors α(0), ˙ β(0) is a positively oriented basis for the tangent space Tα(0) M; and (ii) If α : R → M is any C ∞ embedding with the properties in (a), then the set A ≡ α(R) is nonseparating in M, the set A ∪ B is nonseparating in M ∗ , and for any choice of a neighborhood U of A, there ∞ exists  a closed C real 1-form  τ on M such that supp τ ⊂ U \ A and β τ = 1, and such that γ τ ∈ Z for every loop γ in M (cf. Exercise 5.11.5).  (b) Assuming that B is separating in M ∗ , prove that β θ = 0 for every closed C ∞ 1-form θ on M (i.e., [β]H1 = 0).

5.16 Canonical Homology Bases in a Compact Riemann Surface In this section, we construct a canonical homology basis associated to holomorphically attached tubes in a compact Riemann surface. The required facts concerning homology and cohomology appear in Sects. 10.6 and 10.7. Theorem 5.16.1 Let g ∈ Z≥0 , and let X be a compact Riemann surface obtained by holomorphic attachment of g disjoint tubes to P1 (as in Sect. 5.12.1 and Theorem 5.13.1). Then we have the following (the case g = 3 is illustrated in Fig. 5.20): (a) For F = R or C, we have 2g = rank H1 (X, Z) = rank H 1 (X, Z) = dimF H1 (X, F) = dimF H 1 (X, F). (b) There exist smooth Jordan curves α1 , β1 , . . . , αg , βg in X with images Aj = αj ([0, 1]) and Bj = βj ([0, 1]) for j = 1, . . . , g such that

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Fig. 5.20 A canonical homology basis

(i) For all i, j = 1, . . . , g with i = j , Ai ∩ Aj = Ai ∩ Bj = Bi ∩ Bj = ∅; (ii) For each j = 1, . . . , g, Aj ∩ Bj = {αj (0)} = {βj (0)}; and (iii) For each j = 1, . . . , g, the pair of tangent vectors α˙ j (0), β˙j (0) is a posi˙ tively oriented basis for the tangent space Tαj (0) X (i.e., ω(α(0), ˙ β(0)) >0 for every positive 2-form ω). (c) If α1 , β1 , . . . , αg , βg are smooth Jordan curves in X satisfying the conditions (i)–(iii) in (b), then the homology classes [α1 ]H1 , [β1 ]H1 , . . . , [αg ]H1 , [βg ]H1 form a basis for the A-module H1 (X, A) for any subring A of C containing Z. Given neighborhoods Uj of Aj and Vj of Bj in X for j = 1, . . . , g, there exg g ist C ∞ closed 1-forms {θj }j =1 and {τj }j =1 such that supp θj ⊂ Vj \ Bj and supp τj ⊂ Uj \ Aj for j = 1, . . . , g, and for any subring A of C containing Z, g g the cohomology classes {[θj ]H 1 }j =1 ∪ {[τj ]H 1 }j =1 form a basis for H 1 (X, A) g g that is dual to the basis {[αj ]H1 }j =1 ∪ {[βj ]H1 }j =1 ; that is, for i, j = 1, . . . , g, ([θi ]H 1 , [βj ]H1 )deR = ([τi ]H 1 , [αj ]H1 )deR = 0 and

 ([θi ]H 1 , [αj ]H1 )deR = ([τi ]H 1 , [βj ]H1 )deR =

1 if i = j, 0 if i = j .

(d) Suppose T1 , . . . , Tg are open subsets of X with disjoint closures such that X \ (T 1 ∪ · · · ∪ T g ) is connected and for each j = 1, . . . , g, Tj is a relatively compact subset of a local holomorphic chart that maps Tj onto an annulus. Then holomorphic removal of the tubes T1 , . . . , Tg yields a compact Riemann surface that is biholomorphic to P1 . Moreover, we may choose smooth Jordan

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269

curves α1 , β1 , . . . , αg , βg as in (b) so that for each j = 1, . . . , g, Bj is a concentric circle in Tj with respect to the biholomorphism onto an annulus and Ai ∩ T j = ∅ for i = 1, . . . , jˆ, . . . , g. (e) Suppose T1 , . . . , Tg are open subsets of X with disjoint closures such that for each j = 1, . . . , g, Tj is a relatively compact subset of a local holomorphic chart that maps Tj onto an annulus. Assume that there exist smooth Jordan curves α1 , β1 , . . . , αg , βg as in (b) such that for each j = 1, . . . , g, Bj ⊂ Tj . Then [βj ]βj (0) generates the fundamental group π1 (Tj , βj (0)) for each j = 1, . . . , g, and holomorphic removal of the tubes T1 , . . . , Tg yields a compact Riemann surface that is biholomorphic to P1 . Remarks 1. Part (a) of the theorem (together with the discussion of homology in Sects. 10.6 and 10.7) implies that g, the number of tubes that one attaches to P1 in order to construct a given compact Riemann surface X, depends only on the topology of X, not on the holomorphic structure or even the C ∞ structure. 2. It will follow from the results of Chap. 6 that any compact orientable topological surface M (see Sect. 6.10) may be obtained by attaching g tubes to the sphere, where 2g = rank H1 (M, Z). One may also obtain this fact directly (see, for example, [T]). 3. Actually, any two compact orientable topological surfaces are homeomorphic if and only if they have isomorphic homology groups (again, see [T]). Of course, two homeomorphic compact Riemann surfaces need not be biholomorphic. For example, two complex tori need not be biholomorphic, although all complex tori are diffeomorphic (see Exercise 5.9.1). 4. Part (e) is a partial converse of part (d), but in (e), one may not be able to choose each coordinate annulus Tj so that the given curve Bj becomes a concentric circle. However, any choice of such a concentric circle in Tj will represent [βj ]H1 or −[βj ]H1 , and one may homotope αj slightly to get transversality. 5. For α1 , β1 , . . . , αg , βg as in part (b), choosing a path γj from a fixed base point x0 to αj (0) = βj (0) and letting uj = [γj ∗ αj ∗ γj− ], vj = [γj ∗ βj ∗ γj− ] ∈ π1 (X, x0 ) for each j = 1, . . . , g, one can show that u1 , v1 , . . . , ug , vg generate π1 (X, x0 ) (see Exercise 5.17.3), and that in fact, π1 (X, x0 ) is equal to the free group on these −1 −1 −1 generators modulo the relation u1 v1 u−1 1 v1 · · · ug vg ug vg = e (see, for example, [Hat]). However, we will not use this fact. Before addressing the proof of Theorem 5.16.1, we consider a definition and some consequences. Definition 5.16.2 Let X be a compact Riemann surface. A homology basis [α1 ]H1 , [β1 ]H1 , . . . , [αg ]H1 , [βg ]H1 for H1 (X, Z) (which will also be a basis for H1 (X, A) for any subring A of C containing Z) with homology classes represented by smooth Jordan curves α1 , β1 , . . . , αg , βg satisfying the conditions in part (b) of Theorem 5.16.1 is called a canonical homology basis. Applying Theorem 5.13.1, Lemma 5.11.1, Theorem 5.9.2, and Theorem 5.16.1, we get the following:

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Corollary 5.16.3 Let X be a compact Riemann surface, let 2g = rank H1 (X, Z), and let α1 , β1 , . . . , αg , βg be smooth Jordan curves satisfying the conditions in part (b) of Theorem 5.16.1 (in particular, [α1 ]H1 , [β1 ]H1 , . . . , [αg ]H1 , [βg ]H1 is a canonical homology basis). Then there exist open subsets T1 , . . . , Tg of X with disjoint closures such that X \ (T 1 ∪ · · · ∪ T g ) is connected and such that for each j = 1, . . . , g, Tj is a relatively compact subset of a local holomorphic chart that maps Tj onto an annulus, Bj ⊂ Tj , [βj ]βj (0) generates the fundamental group π1 (Tj , βj (0)), Ai ∩ T j = ∅ for all i = 1, . . . , jˆ, . . . , g, and holomorphic removal of the tubes T1 , . . . , Tg yields a Riemann surface that is biholomorphic to P1 . Theorem 5.13.1, Theorem 5.16.1, Corollary 4.9.2, and Corollary 4.6.5 together give the following: Corollary 5.16.4 Up to biholomorphism, every compact Riemann surface X may be obtained by holomorphic attachment of g disjoint tubes to P1 , where 1 g = dimC HDol (X) = dim (X, O(KX )) = genus(X).

Proof of Theorem 5.16.1 Throughout this proof, A will denote a subring of C containing Z. Step 1. We construct smooth Jordan curves satisfying (b). By hypothesis, we g g may choose open subsets {Tj }j =1 of X with disjoint closures, constants {R0j }j =1 g and {R1j }j =1 in (1, ∞), and a biholomorphism j of a neighborhood of T j onto a domain in C with j (Tj ) = (0; 1/R0j , R1j ) for each j = 1, . . . , g such that if 2πit ) for each t ∈ [0, 1], B = β ([0, 1]) ⊂ T for each j = 1, . . . , g, βj (t) = −1 j j j j (e and Z = X \ (B1 ∪ · · · ∪ Bg ), then there is a biholomorphism Z of Z onto a domain in P1 with complement P1 \ Z (Z) equal to the union of a finite collection of disjoint compact sets, each of which is mapped onto a closed disk by some local holomorphic chart in P1 . Observe that X1 ≡ X \ (B2 ∪ · · · ∪ Bg ) = Z ∪ T1 is connected, since Z and T1 are connected and Z ∩ T1 = ∅. Moreover, since Z = X1 \ B1 is connected, B1 is nonseparating in X1 . Therefore, by Lemma 5.15.1, there exists a smooth Jordan curve α1 with image A1 = α1 ([0, 1]) in X1 such that A1 ∩ B1 = {α1 (0)} = {β1 (0)} and the pair of tangent vectors α˙ 1 (0), β˙1 (0) is a positively oriented basis for the tangent space Tα1 (0) X. Moreover, the sets A1 , B1 , and A1 ∪ B1 are nonseparating in X1 (for any choice of α1 ). Suppose k ∈ {2, . . . , g} and we have constructed smooth Jordan curves α1 , . . . , αk−1 such that for each j = 1, . . . , k − 1, we have j ∪ · · · ∪ Bg ), Aj ≡ αj ([0, 1]) ⊂ Xj ≡ X \ (A1 ∪ · · · ∪ Aj −1 ∪ B1 ∪ · · · ∪ B Xj is connected, Aj , Bj , and Aj ∪Bj are nonseparating in Xj , Aj ∩Bj = {αj (0)} = {βj (0)}, and the pair of tangent vectors α˙ j (0), β˙j (0) is a positively oriented basis for the tangent space Tαj (0) X. Applying Lemma 5.15.1 to the smooth Jordan curve βk in the Riemann surface k ∪ · · · ∪ Bg ) Xk ≡ X \ (A1 ∪ · · · ∪ Ak−1 ∪ B1 ∪ · · · ∪ B

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271

(Bk is nonseparating in Xk because Ak−1 ∪ Bk−1 is nonseparating in Xk−1 and Xk \ Bk = Xk−1 \ (Ak−1 ∪ Bk−1 )), we get a smooth Jordan curve αk such that Ak ≡ αk ([0, 1]) ⊂ Xk , Ak ∩ Bk = {αk (0)} = {βk (0)}, the pair of tangent vectors α˙ k (0), β˙k (0) is a positively oriented basis for the tangent space Tαk (0) X, and (for any choice of such an αk ) the sets Ak , Bk , and Ak ∪ Bk are nonseparating in Xk . Thus, by induction, we get smooth Jordan curves αj and βj for j = 1, . . . , g satisfying the conditions in (b). Step 2. We show that the homology classes [α1 ]H1 , [β1 ]H1 , . . . , [αg ]H1 , [βg ]H1 represented by the Jordan curves α1 , β1 , . . . , αg , βg constructed in Step 1 are linearly independent in H1 (X, A). We may fix domains U1 , V1 , . . . , Ug , Vg in X such that for each j = 1, . . . , g, we have Aj ⊂ Uj , Bj ⊂ Vj , and for each i = 1, . . . , jˆ, . . . , g, (Ui ∪ Vi ) ∩ (Uj ∪ Vj ) = ∅. By Lemma 5.15.1, for each j = 1, . . . , g, ∞ we may choose closed  C 1-forms θj and τj on  X such that  supp  θj ⊂ Vj \ Bj , supp τj ⊂ Uj \ Aj , αj θj = βj τj = 1, βj θj = αj τj = 0, γ θ, γ τ ∈ Z for every loop γ in X, and (clearly)     θj = θj = τj = τj = 0 for i = 1, . . . , jˆ, . . . , g. αi

βi

αi

βi

, . . . , [αg ]H1 , [βg ]H1 are linearly It follows that the homology classes 1 ]H1 , [β1 ]H1

[α g g independent over A. For if ξ = j =1 rj · αj + j =1 sj · βj ∈ Z1 (X, A), with rj , sj ∈ A for j = 1, . . . , g, then for each j = 1, . . . , g, we have   rj = θj and sj = τj . ξ

ξ

So if ξ is homologous to 0, then rj = sj = 0 for each j = 1, . . . , g. Similarly, the cohomology classes [θ1 ]H 1 , [τ1 ]H 1 , . . . , [θg ]H 1 , [τg ]H 1 ∈ H 1 (X, A) are linearly independent. Step 3. We show that the cohomology classes [θ1 ]H 1 , [τ1 ]H 1 , . . . , [θg ]H 1 , [τg ]H 1 represented by the 1-forms θ1 , τ1 , . . . , θg , τg constructed in Step 2 span H 1 (X, A). For this, we must show that each closed C ∞ 1-form ρ on X with integrals along loops taking values in A is cohomologous to a linear combination over A of the differential forms θ1 , τ1 , . . . , θg , τg . By replacing ρ with the 1-form ρ−

  g  ρ · θj − ρ · τj ,

g  j =1

αj

j =1

βj

we may assume without loss of generality that   ρ= ρ = 0 for j = 1, . . . , g. αj

βj

In this case, we will show that in fact, ρ is exact.

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Uniformization and Embedding of Riemann Surfaces

 βj

ρ = 0, and hence, since the path homotopy

class [βj ] generates the fundamental group of a neighborhood of Tj , ρ is exact on this neighborhood. Choosing a function with differential ρ on this neighborhood and subtracting from ρ the differential of a suitable cut-off of the function, we may assume that ρ ≡ 0 on Tj for each j = 1, . . . , g. It follows thatthere is a unique g closed C ∞ 1-form ρˆ on P1 with ρˆ = 0 on the open set W ≡ Z ( j =1 (Tj \ Bj )) ∪ (P1 \ Z (Z)) and ∗Z ρˆ = ρ on Z. Since P1 is simply connected, there is a C ∞ function λˆ on P1 with d λˆ = ρ. ˆ In particular, λˆ is locally constant on W . On the other hand, for each j = 1, . . . , g, choosing s ∈ (0, 1/2) with αj ([0, s] ∪ [1 − s, 1]) ⊂ Tj , we get    ˆ Z (αj (1 − s))) − λ( ˆ Z (αj (s))). ρ= ρ= ρˆ = λ( 0= αj

αj [s,1−s]

Z (αj [s,1−s] )

Thus λˆ has the same constant value on the two connected components Z (−1 j ((0; 1/R0j , 1)))

Z (−1 j ((0; 1, R1j )))

and

of the set Z (Tj \ Bj ) ⊂ W . It follows that the function λˆ ◦ Z extends to a unique C ∞ function λ on X (which is constant on Tj for each j = 1, . . . , g), and Tj . Thus we have dλ = ρ, since dλ = ∗Z ρˆ = ρ on Z and dλ = 0 = ρ on [θ1 ]H 1 , [τ1 ]H 1 , . . . , [θg ]H 1 , [τg ]H 1 is a basis for H 1 (X, A). Step 4. We show that the classes [α1 ]H1 , [β1 ]H1 , . . . , [αg ]H1 , [βg ]H1 form a basis for H1 (X, A). Given a 1-cycle ξ ∈ Z1 (X, A), the 1-cycle ζ≡

g  j =1

satisfies

 θj

ξ

g  j =1

 τj

· βj ∈ Z1 (X, A)

ξ



 ξ −ζ

· αj +

θj =

ξ −ζ

τj = 0 for j = 1, . . . , g,

and it follows that ξ is homologous to ζ . Thus [α1 ]H1 , [β1 ]H1 , . . . , [αg ]H1 , [βg ]H1 is a basis for H1 (X, A). In particular, part (a) is proved. Step 5. We prove part (c). Suppose that α1 , β1 , . . . , αg , βg are arbitrary smooth Jordan curves in X satisfying the conditions (i)–(iii) in (b), and for each j = 1, . . . , g, Uj is a neighborhood of Aj and Vj is a neighborhood of Bj . Then, for each j = 1, . . . , g, αj and βj are nonseparating in X (since by Lemma 5.11.1 and Lemma 5.10.3, the connected set βj ((0, 1)) meets each connected component of X \Aj and the connected set αj ((0, 1)) meets each connected component of X \Bj ). By Lemma 5.15.1, there exist C ∞ closed 1-forms θ1 , τ1 , . . . , θg , τg on X such that for i, j = 1, . . . , g, supp θj ⊂ Vj \ Bj , supp τj ⊂ Uj \ Aj ,    1 if i = j, θj = τj = 0 if i = j , αi βi

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Canonical Homology Bases in a Compact Riemann Surface

273

 = βi θj = 0, and [θj ]H 1 , [τj ]H 1 ∈ H 1 (X, Z). The argument in Step 2 shows that the homology classes [α1 ]H1 , [β1 ]H1 , . . . , [αg ]H1 , [βg ]H1 are linearly independent in H1 (X, A), and the cohomology classes [θ1 ]H 1 , [τ1 ]H 1 , . . . , [θg ]H 1 , [τg ]H 1 are linearly independent in H 1 (X, A). In particular, [α1 ]H1 , [β1 ]H1 , . . . , [αg ]H1 , [βg ]H1 is a basis for the 2g-dimensional vector space H1 (X, R) (as well as for H1 (X, C)) with dual basis [θ1 ]H 1 , [τ1 ]H 1 , . . . , [θg ]H 1 , [τg ]H 1 for H 1 (X, R) (as well as for H 1 (X, C)). The argument in Step 4 shows that [α1 ]H1 , [β1 ]H1 , . . . , [αg ]H1 , [βg ]H1 must also span H1 (X, A), and hence the dual cohomology classes [θ1 ]H 1 , [τ1 ]H 1 , . . . , [θg ]H 1 , [τg ]H 1 span H 1 (X, A). Thus part (c) is proved. Step 6. We prove part (d). The proof of (d) is similar to the proof of Theorem 5.13.1. Let Bj be a concentric circle in the holomorphic coordinate annulus Tj g g for j = 1, . . . , g. By hypothesis, X \ j =1 T j and therefore X \ j =1 Bj are connected. Holomorphic removal of the tube Tg yields a compact Riemann surface with real first homology space of dimension < 2g. However, since the first homology of a compact Riemann surface is of even dimension, the dimension of the homology must have actually dropped by at least 2. On the other hand, removal of Tg still leaves (the images of) the remaining tubes, each of which contains a nonseparating smooth Jordan curve (given by the image of Bj for some j ∈ {1, . . . , g − 1}). Proceeding inductively, we will not get a planar Riemann surface before we have removed all of the g tubes T1 , . . . , Tg , at which point we get a compact Riemann surface with zero homology, that is, a Riemann surface that is biholomorphic to P1 . Observe also that Step 1 provides the desired Jordan curves satisfying (b). Step 7. We prove part (e). Observe that for each j , the path homotopy class [βj ] in π1 (Tj ) cannot be the identity, because by (c), [βj ]H1 (X,Z) = 0. Thus we have [βj ] = [γj ]kj for some kj ∈ Z \ {0}, where γj is a loop corresponding to a concentric circle in Tj . By Lemma 5.15.1, there exist C ∞ closed 1-forms η1 , . . . , ηg on X such that for all i, j = 1, . . . , g,   1 if i = j, ηj = 0 if i = j, βi 

αi τj

 and such that γ ηj ∈ Z for each loop γ in X. In particular, for each j , we have   1 = βj ηj = kj · γj ηj ∈ kj Z, and hence we must have kj = ±1. Thus [βj ] generates π1 (Tj ), and Cj ≡ γj ([0, 1]) is nonseparating. Consequently, by subtracting the differential of a suitable C ∞ function, we may also assume that supp ηj ⊂ X \ (T 1 ∪ · · · ∪ T!j ∪ · · · ∪ T g ). It follows by induction that for j = 2, . . . , g, X \ (T 1 ∪ · · · ∪ T j −1 ) is a domain in which Cj and T j are nonseparating, and hence X \ (T 1 ∪ · · · ∪ T g ) is connected. Part (d) now gives the claim (e).  Remark The above proof did not use Theorems 10.7.16 and 10.7.18 in an essential way. In fact, a proof of these theorems for the special case of a compact Riemann surface is actually contained within the above.

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Exercises for Sect. 5.16 5.16.1 Using Exercise 5.9.2 and the results of this section, prove that every compact Riemann surface of genus 1 is biholomorphic to a complex torus (a different approach was considered in Exercise 5.9.3, and yet another approach will be considered in Sect. 5.22). 5.16.2 Let X be a compact Riemann surface of genus 2. Prove that there is a cover → X such that ing Riemann surface ϒ : X  is obtained by holomorphic attachment of a locally finite infinite (i) X sequence of disjoint tubes {Tν }∞ ν=1 to C; 2 ∼ (ii) We have Deck(ϒ) = Z ; (iii) We have Tν ∩ (Tν ) = ∅ for each  ∈ Deck(ϒ) \ {IdX  } and each ν ∈ Z>0 ; and  (iv) We have Deck(ϒ) · T1 = ∞ ν=1 Tν . Hint. Apply Exercise 5.16.1 (cf. Exercise 5.12.9). 5.16.3 The goal of this exercise is a proof of a theorem of Abel (see part (d) below). Let [α1 ]H1 , [β1 ]H1 , . . . , [αg ]H1 , [βg ]H1 be a canonical homology basis for a compact Riemann surface X of genus g > 0. (a) Prove that if η1 and η2 are two closed C ∞ 1-forms on X, then  η 1 ∧ η2 = X

g  j =1

       η1 · η2 − η1 · η2 .

αj

βj

βj

αj

Hint. Choose the representatives α1 , β1 , . . . , αg , βg to be smooth Jordan curves as in part (b) of Theorem 5.16.1. For each j = 1, . . . , g, construct C ∞ annular neighborhoods Jj ⊃ Aj and Tj ⊃ Bj , as in Lemma 5.11.1. Also choose the neighborhoods so that (J i ∪ T i ) ∩ (J j ∪ T j ) = ∅ for i = j . Form dual closed 1-forms θj and τj with supp θj ⊂ Tj \ Bj and supp  τj ⊂ Jj \ Aj for j = 1, . . . , g.  Observe that for all i, j , X θi ∧ θj = X τi ∧ τj = 0, and for i = j , X θi ∧ τj = 0. Show that θj has a potential λj on Tj (in particular, λj is locally constant near Bj ∪ ∂Tj ), andhence θj ∧ τj = d(λj τj ) on Tj . Applying Stokes’ theorem, prove that X θj ∧ τj = 1, and then prove the claim. (b) Prove that for every holomorphic 1-form η on X, η2L2

=2

g j =1

   η · η .

 Im βj

αj

(c) Prove that there is a unique basis η1 , . . . , ηg for (X) such that for all i, j = 1, . . . , g,   1 if i = j , ηi = 0 if i = j . αj

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275

(d) For the basis η1 , . . . , ηg in part (c), let Z = (zij ) be the complex g × g matrix with entries  ηi ∀i, j = 1, . . . , g. zij ≡ βj

zj i for all i and j ) and Im Z = Prove that Z is symmetric (i.e., zij =

g (Im zij ) is positive definite (i.e., i,j =1 Im zij ui uj > 0 for all (u1 , . . . , ug ) ∈ Rg \ {0}). Hint. For symmetry, first observe that ηi ∧ ηj ≡ 0 for all i and j . Then apply part (a). In order to prove that Im Z is positive definite, first prove that "ηi , ηj #L2 = 2 Im zij for all i and j . 5.16.4 Prove a C ∞ version of Theorem 5.16.1 for an oriented compact C ∞ surface M obtained by C ∞ oriented attachment of g tubes to a compact planar oriented C ∞ surface N (cf. Exercises 5.12.8 and 5.13.2).

5.17 Complements of Connected Closed Subsets of P1 The following fact will be applied in the construction of embeddings of open Riemann surfaces (in Sect. 5.18) and in the proof of Schönflies’ theorem (in Sects. 6.7–6.9). Lemma 5.17.1 The complement and boundary of a simply connected domain in P1 are connected, and each connected component of the complement of any connected closed subset of P1 is simply connected. Remark Each connected component of the complement of a nonempty proper open subset of a topological surface must meet, and therefore contain, a boundary component of the set. Hence the complement is connected if the boundary is connected (it is easy to see that the converse is false). Proof of Lemma 5.17.1 Let  be a nonempty domain in P1 . Suppose first that  is simply connected, and that A and B are disjoint (and possibly empty) subsets of ∂ that are open in ∂ and that satisfy ∂ = A ∪ B. Then A = (∂) \ B and B = (∂) \ A are also compact, so we may choose disjoint open subsets U and V of P1 with A ⊂ U and B ⊂ V . In particular, ∂U ∩ ∂ = ∂V ∩ ∂ = ∅. The Riemann mapping theorem implies that  is equal to the union of an increasing sequence of relatively compact domains with connected boundary, so the compact set (∂U ∪ ∂V ) ∩  = (∂U ∪ ∂V ) ∩  is contained in such a domain 0 . Consequently, each of the sets ( \ 0 ) ∩ U = ( \ 0 ) ∩ U and ( \ 0 ) ∩ V = ( \ 0 ) ∩ V is both open and closed in the connected open set  \ 0 . If A = ∅, then the first of the above sets is nonempty, and hence  \ 0 ⊂ U ⊂ P1 \ V ; so B ⊂ (∂) ∩ V = ∅. Thus ∂ is connected, and it follows that P1 \  is also connected.

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Assuming now that the domain  is a connected component of the complement P1 \ K of a nonempty connected compact set K  P1 , we will show that  must be simply connected. For this, suppose that θ is an arbitrary closed C ∞ 1-form on . Given a loop γ in , Proposition 5.4.1 provides a domain 0 such that γ ([0, 1]) ⊂ 0  , ∂0 is the union of disjoint smooth Jordan curves C1 , . . . , Cm , and for some r > 1 and for each j = 1, . . . , m, there is a local holomorphic chart (Uj , j , (0; 1/r, r)) in  with Cj = −1 j (∂(0; 1)) and

1 0 ∩ Uj = −1 j ((0; 1, r)). Let Ej be the connected component of P \ 0 containing Cj for each j . Since Cj is a separating smooth Jordan curve in P1 (for example, by Proposition 5.15.2), it follows that Ej ∩ 0 = Cj (if this were not the case, then there would exist a path in the set (Ej \ Cj ) ∪ 0 ⊂ P1 \ Cj from −1 −1 j ((0; 1/r, 1)) to j ((0; 1, r))). In particular, the connected components E1 , . . . , Em are distinct and therefore disjoint. Reordering so that K ⊂ E1 , we see that the set 1 ≡ 0 ∪ E2 ∪ · · · ∪ Em is a smooth domain in P1 that lies in the connected component  of P1 \ K and that satisfies ∂1 = C1 ⊂  and 1 ∩ U1 = −1 1 ((0; 1, r)).   Now Stokes’ theorem implies that ∂1 θ = 1 dθ = 0. Hence, since ∂1 =

C1 = −1 1 (∂(0; 1)), the restriction of θ to U1 must be exact. Forming a potential and cutting off, we get a C ∞ function ρ on  such that dρ = θ on a neighborhood of ∂1 . Hence the 1-form τ that is equal to θ − dρ at each point in 1 and 0 at each point in P1 \ 1 is closed and of class C ∞ , and therefore 





θ= γ

(θ − dρ) = γ

τ = 0. γ

It follows that θ is exact (on ), and therefore, by Corollary 5.2.7,  must be simply connected.  Exercises for Sect. 5.17 5.17.1 Two additional approaches to the proof of the second part of Lemma 5.17.1 (each connected component of the complement of any connected closed subset of P1 is simply connected) are outlined in parts (a) and (b) below. Let K be a connected closed subset of P1 , and let  be nonempty connected component of P1 \ K. (a) According to Sard’s theorem (see, for example, [Mi]), the set of critical values of a C ∞ function on a second countable C ∞ manifold is a set of measure 0. Prove that  is simply connected by modifying the proof of Lemma 5.17.1 so that Sard’s theorem, Theorem 9.10.1, and Lemma 5.11.1 are applied in place of Proposition 5.4.1. (b) Suppose θ is a closed C ∞ 1-form on . Applying Lemma 5.15.2, show that θ integrates to 0 along every smooth Jordan curve C in , and applying Proposition 5.10.7, conclude that θ is exact. From this, conclude that  is simply connected.

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Complements of Connected Closed Subsets of P1

277

5.17.2 A Riemann surface is said to be of finite topological type if its fundamental group is finitely generated. Prove that a Riemann surface X is of finite topological type if and only if X is biholomorphic to the complement X ∼ = Y \K in some compact Riemann surface Y of some (possibly empty) compact set K ⊂ Y that is equal to a union of finitely many disjoint compact sets, each of which is either a singleton or a closed disk in some local holomorphic chart (X is said to be of finite type if K may be chosen to be a finite set). Hint. Assuming that π1 (X) is finitely generated, show that after holomorphic removal of finitely many tubes, one may assume that X  P1 (see Sect. 5.13). Suppose   X is a topologically Runge smooth domain, and U1 , . . . , Um are the connected components of P1 \ . Applying Proposition 5.15.2, prove that ∂ has exactly m connected components. Assuming that ∞ ∈ U1 and fixing a point zj ∈ Uj \ X for each j = 2, . . . , m, show that the closed C ∞ 1-forms θj = dz/(z − zj ) for j = 2, . . . , m on X represent linearly independent cohomology classes, and from this, obtain a bound !j ∪ · · · ∪ Um is simply connected, and on m. Also show that  ∪ U1 ∪ · · · ∪ U use this observation to obtain Y and K. 5.17.3 Let X be a compact Riemann surface of genus g; let α1 , β1 , . . . , αg , βg be smooth Jordan curves in X with the properties in part (b) of Theorem 5.16.1; ([0, 1]) for j = 1, . . . , g; let K be a conlet Aj ≡ αj ([0, 1]) and Bj ≡ βj g nected compact subset of X \ [ j =1 (Aj ∪ Bj )] with connected complem ment X \ K in X; let {Dν }ν=1 be relatively compact smooth domains in g X \ [K ∪ j =1 (Aj ∪ Bj )] with disjoint closures and connected boundaries; and for each ν = 1, . . . , m, let Kν be a connected compact subset of Dν and 1]) = ∂Dν (cf. Exercise 5.17.2). γν a smooth Jordan curve with Cν ≡ γν ([0, (a) Prove that the open set  ≡ X \ [K ∪ m ν=1 D ν ] is connected.  (b) Prove that if Y is the connected component of X \ [K ∪ m ν=1 Kν ] conˆj = λj ∗ βj ∗ λ− for some path taining , p ∈ , αˆ j = λj ∗ αj ∗ λ− and β j j λj in  from p to αj (0) = βj (0) for j = 1, . . . , g, and γˆν = ην ∗ γν ∗ ην− for some path ην in  \ K from p to γν (0) for ν = 1, . . . , m, then the path homotopy classes of the loops αˆ 1 , . . . , αˆ g , βˆ1 , . . . , βˆg , γˆ1 , . . . , γˆm in Y generate π1 (Y, p). In particular, the path homotopy classes of the loops αˆ 1 , . . . , αˆ g , βˆ1 , . . . , βˆg in X (in X \ K) generate π1 (X, p) (respectively, generate π1 (X \ K, p)). Hint. Set K0 ≡ K. For the proof of (b), first consider the case g = 0. That is, for X = P1 , prove that [γˆ1 ], . . . , [γˆm ] generate π1 (Y, p). For this, prove first that for each ν = 0, . . . , m, Kν has a small simply connected neighborhood in which the complement of Kν is biholomorphic to a punctured disk or an annulus (the associated neighborhood of K0 will be denoted by D0 ). Deduce from this that fixing a point zν ∈ Kν for each ν, there is a continuous mapping of X \ {z0 , . . . , zm } into Y that fixes points in an arbitrarily large compact subset of Y . Prove also that on the other hand, every loop in Y with base point p is homotopic within X \ {z0 , . . . , zm } to a loop in X \ ν Dν . Using these observations, reduce to the case in which Kν = {zν } for each ν.

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5.17.4 Let X be a Riemann surface. The commutator of elements a and b of a group G is the element [a, b] ≡ aba −1 b−1 . The commutator subgroup of G is the subgroup generated by the set of commutators and is denoted by [G, G]. (a) Prove that for any group G, [G, G] is a normal subgroup of G and the quotient group G/[G, G] is an Abelian group. Prove also that [G, G] is the smallest normal subgroup of G with this property, that is, if H is any normal subgroup of G for which G/H is Abelian, then H ⊃ [G, G]. (b) Prove that for any point p ∈ X, the map  π1 (X, p) [π1 (X, p), π1 (X, p)] → H1 (X, Z) given by [γ ]π1 (X,p) · [π1 (X, p), π1 (X, p)] → [γ ]H1 (X,Z) is a well-defined group isomorphism. Hint. In exhausting smooth topologically Runge subdomains, construct generating loops as in Exercise 5.17.3 along with dual closed 1-forms. In order to obtain the dual closed 1-forms associated to the boundary Jordan curves, apply Lemma 5.15.1 and Exercise 5.15.1. (c) Prove that for any subring A of C containing Z, we have H1 (X, A) ∼ = H1 (X, A) (see the remarks at the end of Sect. 10.7 for the definition of the singular homology group H1 (X, A), and the exercises for Sect. 10.7 for some of its properties). (d) Prove that for any subring A of C containing Z, the mapping [ξ ]H1 (X,A) → ([ξ ]H1 (X,A) , ·)deR gives an injective homomorphism H1 (X, A) → Hom(H 1 (X, A), A) (according to Theorem 10.7.18, this homomorphism is also surjective if π1 (X) is finitely generated). 5.17.5 Let K be a compact subset of a Riemann surface X. Prove that there exist domains  and  such that K ⊂     X and such that for every C ∞ closed 1-form ρ on  , there exist a C ∞ closed 1-form τ on X and a function λ ∈ D( ) such that ρ − τ = dλ on . Conclude from this that 1 1 1 1 (X) → HdeR ()] = im[HdeR ( ) → HdeR ()]. im[HdeR

Hint. For a large smooth topologically Runge subdomain  , construct generating boundary loops along with dual closed 1-forms as in the above exercises. Also form a large subdomain . For ρ as above, one can then form a C ∞ closed 1-form θ on X such that ρ − θ is exact near ∂ . Forming a potential near ∂ and cutting off, one gets a function λ ∈ D( ) such that ρ − θ − dλ vanishes near ∂. The restriction of ρ − θ − dλ to  then extends to a C ∞ closed 1-form τ0 on X. The 1-form τ ≡ τ0 + θ and the function λ then have the required properties.

5.18

Embedding of an Open Riemann Surface into C3

279

5.18 Embedding of an Open Riemann Surface into C3 So far in this chapter, we have considered two characterizations of Riemann surfaces: as quotients of the planar domains P1 , C, and (0; 1), and as planar domains to which tubes have been holomorphically attached. In the remaining sections of this chapter, we will consider other characterizations. Specifically, we will see in this section that every open Riemann surface admits a holomorphic embedding into C3 , in Sect. 5.19 that every compact Riemann surface admits a holomorphic embedding into a higher-dimensional complex projective space, in Sect. 5.20 that every compact Riemann surface admits a finite holomorphic branched covering mapping onto P1 , and in Sect. 5.22 that every compact Riemann surface of positive genus admits a holomorphic embedding into a higher-dimensional complex torus. Definition 5.18.1 Let F = (f1 , . . . , fn ) : X → Cn be a mapping of a complex 1-manifold X into some complex Euclidean space Cn . (a) F is holomorphic if fj is a holomorphic function for each j = 1, . . . , n. (b) F is a holomorphic immersion if for each point p ∈ X, we have (dfj )p = 0 for some j ∈ {1, . . . , n}, that is, the linear map (dF )p ≡ ((df1 )p , . . . , (dfn )p ) : (Tp X)1,0 → Cn is nonsingular. (c) F is called a holomorphic embedding if F is a proper injective holomorphic immersion. The goal of this section is a proof of the following theorem of Narasimhan [Ns1]: Theorem 5.18.2 Every open Riemann surface X admits a (proper) holomorphic embedding into C3 . Remarks 1. Given a holomorphic embedding F : X → Cn of a Riemann surface X into Cn , we may identify X with its image Y ≡ F (X). In fact, it is a consequence of the theory of several complex variables that every holomorphic function on X is the restriction of some holomorphic function on Cn (see, for example, [Hö] or [KaK]). 2. The higher-dimensional analogue for Stein manifolds (and Stein spaces) was obtained independently by Bishop, Narasimhan, and Remmert. The proof given here for a Riemann surface is essentially Narasimhan’s original proof (with some modifications), which may be generalized to higher-dimensional Stein manifolds [Ns2]. One may adapt Bishop’s proof (see [Hö]) and Remmert’s proof of the higherdimensional case to the Riemann surface case. 3. Compact Riemann surfaces do not admit holomorphic embeddings into Euclidean spaces, since every holomorphic function on a compact Riemann surface is constant. However, every compact Riemann surface admits a holomorphic embedding into a complex projective space (see Sect. 5.19). Moreover, every compact Riemann surface of positive genus admits a holomorphic embedding into a higherdimensional complex torus (see Sect. 5.22).

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The main tool in the proof of the embedding theorem is the following uniform version of the Runge approximation theorem for a locally finite family of disjoint coordinate disks (see [Ns1]): Theorem 5.18.3 Let  be an open subset of an open Riemann surface X such that  is equal to the union of a locally finite family of disjoint relatively compact open subsets of X, each of which is biholomorphic to a disk. Then, for every holomorphic function g on , every closed subset K of X with K ⊂ , and every positive continuous function ρ on , there is a holomorphic function f on X such that |f − g| < ρ on K. The idea of Narasimhan’s proof of the embedding theorem is as follows. First, one forms a covering of the Riemann surface as in the following: Proposition 5.18.4 In any open Riemann surface X, there exist three nonempty open subsets 0 , 1 , and 2 such that X = 0 ∪ 1 ∪ 2 and such that for j = 0, 1, 2, j is the union of a locally finite family of disjoint relatively compact open subsets of X, each of which is biholomorphic to a disk. By applying Theorem 5.18.3 in order to uniformly approximate a suitable locally constant function on j for each j = 0, 1, 2, one gets a proper holomorphic mapping of X into C3 . A Baire category argument then provides an approximation of this mapping that is a holomorphic embedding. Our first goal is the proof of Theorem 5.18.3. Recall that the topological hull hY (A) of a subset A of a Hausdorff space Y is the union of A with all of the connected components of Y \ A that are relatively compact in Y (see Definition 2.13.1). Lemma 5.18.5 Let X be a connected, locally connected, locally compact Hausdorff space; let K0 be a compact subset of X; and let {Cν }∞ ν=1 be a locally finite family of disjoint connected compact subsets of X. Then there exists a compact subset K ◦

of X such that hX (K) = K, K0 ⊂ K, and for each ν, Cν ⊂ K or Cν ∩ K = ∅. Proof By local finiteness, there exists a positive integer μ such that we may choose a relatively compact neighCν ∩ K0 = ∅ for each ν > μ. Thus   μ C borhood U of the compact set K0 ∪ ν=1 Cν in the open set X \ ∞ ν=μ+1 ν . In particular, for each ν, Cν is contained in U , or in a connected component of X \ U that is relatively compact in X, or in a connected component of X \ U that is not relatively compact in X. Lemma 2.13.2 and Lemma 2.13.3 together now imply that  the set K ≡ hX (U ) has the required properties. Lemma 5.18.6 Let {Kν }m ν=0 be a family of disjoint compact subsets of a topological surface M such that hM (K0 ) = K0 and such that for each index ν = 1, . . . , m, there is a local chart on a neighborhood of Kν that maps Kν onto a closed disk. Then the compact set K ≡ m ν=0 Kν satisfies hM (K) = K.

5.18

Embedding of an Open Riemann Surface into C3

281

Proof If U is a connected component of M \ K0 , then U is not relatively compact in M, and hence by Lemma 5.11.4, U \ (K1 ∪ · · · ∪ Km ) is a connected open set that is not relatively compact in X. Thus each of the connected components of M \ K is of this form, and therefore hM (K) = K.  ¯ For the proof of Theorem 5.18.3, we will apply the L2 ∂-method with a weight function provided by the following: Lemma 5.18.7 Let X be an open Riemann surface. Suppose that  ⊂ X is the union of a locally finite family of disjoint relatively compact open subsets of X, each of which is biholomorphic to a disk, ω is a positive continuous (1, 1)-form on X, ρ is a positive continuous function on X, and K is a closed subset of X with K ⊂ . Then there exists a C ∞ strictly subharmonic function ϕ on X such that (i) On X, iϕ ≥ ω; (ii) On K, ϕ < −ρ; and (iii) On X \ , ϕ > ρ. Proof We may assume without loss of generality that  has infinitely many connected components {ν }∞ ν=1 . Replacing each of these connected components with a slightly smaller coordinate disk and replacing K with a suitable (larger) subset of , we may also assume that the components of  have disjoint closures and that for each ν, the biholomorphism of ν onto a disk extends to a biholomorphism on a neighborhood of ν , and the image of K ∩ ν is a closed disk. We may also fix a neighborhood  of K with  ⊂ . By applying Lemma 5.18.5 inductively (together with Lemma 9.3.6 and Radó’s theorem), we get a sequence of nonempty compact sets {Km }∞ m=1 such that ∞ m=1 Km = X and 1 ⊂ K1 , and such that for each m, we have hX (Km ) = Km , ◦

Km ⊂ K m+1 , each connected component ν of  lies either in Km or in X \ Km , and  ∩ (Km+1 \ Km ) = ∅. Thus by reordering, we may assume that for some strictly increasing sequence of positive integers {νm }, we have, for each m, 1 ∪ · · · ∪ ν m ⊂ K m

and

X \ Km ⊃

∞

ν .

ν=νm +1

m ν ⊂ Km \ Km−1 for each m = Set K0 = 0 ≡ ∅, ν0 ≡ 0, and m ≡ νν=ν m−1 +1 ∞ 1, 2, 3, . . . . Theorem 2.14.1 provides a C strictly subharmonic function ϕ0 on X such that iϕ0 ≥ ω and ϕ0 > ρ. Given C ∞ subharmonic functions ϕ0 , . . . , ϕm−1 on X, we may choose a real-valued C ∞ function αm on X such that supp αm ⊂ m and on  ∩ m , αm + ϕ0 + · · · + ϕm−1 < −ρ and αm is subharmonic. Lemma 5.18.6 and Theorem 2.14.1 provide a nonnegative C ∞ subharmonic function ψm on X such that ψm ≡ 0 on Km−1 ∪ (K ∩ m ), and ψm + αm ≥ 0 and iψm +αm ≥ 0 on m \ . αm is subharmonic on  X, ϕm ≥ 0 on X \ , Thus the C ∞ function ϕm ≡ ψm +

νm ϕm ≡ 0 on Km−1 , and by induction, m j =0 ϕj < −ρ on K ∩ ν=1 ν . Thus we get

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a sequence of functions {ϕm }, and it follows that the locally finite sum ∞ m=0 ϕm determines a C ∞ strictly subharmonic function ϕ with the required properties.  Proof of Theorem 5.18.3 We may assume without loss of generality that  has infinitely many connected components {ν }∞ ν=1 . Hence we have local holomorphic charts {(ν , ν , (0; 1))}∞ , and we may fix nonempty neighborhoods  and D ν=1 of K in  such that  ⊂ D and, for each ν, ν (D ∩ ν ) is a relatively compact disk in (0; 1). We may also fix a C ∞ function τ on X such that supp τ ⊂  and ¯ lies in  \ D. τ ≡ g on D. In particular, the support of the C ∞ (0, 1)-form γ ≡ ∂τ Fixing a Kähler form ω on X and applying Theorem 2.14.1, we get a C ∞ strictly subharmonic function ϕ0 on X such that iω + iϕ0 ≥ ω. By Theorem 2.6.4, for each ν, there exists a constant δν ∈ (0, 1) such that |h| < ρ on K ∩ ν for every function h ∈ O( ∩ ν ) with hL2 (∩ν ,ω,ϕ0 ) < δν . By applying Lemma 5.18.7, we may form a C ∞ subharmonic function ϕ1 on X such that for each ν, ϕ1 < 2 log δν on  ∩ ν , and for ϕ ≡ ϕ0 + ϕ1 , γ 2L2 ( In particular, γ 2L2 (X,ϕ) =

ν ,ϕ)

= γ 2L2 (

ν \D,ϕ)

< 2−ν .

∞

2 ν=1 γ L2 (ν ,ϕ)

< 1, and by Corollary 2.12.6, there ¯ = γ and αL2 (X,ω,ϕ) < 1. The function exists a C ∞ function α on X such that ∂α f ≡ τ − α is then a holomorphic function on X. Furthermore, since τ = g on D, α must be holomorphic on D ⊃ , and for each ν, αL2 (∩ν ,ω,ϕ0 ) ≤ δν · αL2 (∩ν ,ω,ϕ) ≤ δν · αL2 (X,ω,ϕ) < δν .

It follows that |f − g| = |α| < ρ on K.



We now turn to the construction of the three subsets in Proposition 5.18.4. For this, we form a graph that divides X into small simply connected sets (actually, as the reader is asked to prove in Exercise 6.11.4, X admits a triangulation; but we will not use this more general fact). The three sets then consist of a union of small neighborhoods of the vertices in the graph, a union of small neighborhoods of the open edges, and finally, the complement of the graph. For the construction of the graph, we will need some elementary facts, the first of which is the following analogue of the identity theorem (Corollary 1.3.3) for real analytic functions: Lemma 5.18.8 If the zero set Z = f −1 (0) of a real-valued real analytic function f on an open interval I ⊂ R has a limit point in I , then f ≡ 0. Proof Given a point a ∈ Z,

for some  >k 0, the function f may be written as a power series x → f (x) = ∞ k=0 ck (x − a) on the interval (a − , a + ) ⊂ I . This series converges absolutely, so we may define a holomorphic function g on the disk

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283

(a; ) in C by g(z) =

∞

ck (z − a)k

∀z ∈ (a; ).

k=0

If a is a limit point of Z, then the identity theorem implies that g is constant. ◦

Thus the interior Z is nonempty. The above also implies that the boundary of the ◦

◦

nonempty open set Z does not meet I , and hence that Z = Z = I .



The desired graph will be constructed as the union of coordinate circles and line segments, and the following lemma will give local finiteness of intersections: Lemma 5.18.9 Suppose f : 0 → 1 is a biholomorphism of domains 0 and 1 in C, and for each j = 0, 1, zj ∈ C, rj > 0, and (zj ; rj )  j . Then the intersection f (∂(z0 ; r0 )) ∩ ∂(z1 ; r1 ) is infinite if and only if f (∂(z0 ; r0 )) = ∂(z1 ; r1 ). Proof Let ρ : R → C be the periodic function t → |f (z0 + r0 e2πit ) − z1 |. If the above intersection is infinite, then the set ρ −1 (r1 ) has a limit point t0 ∈ R. On the other hand, on some neighborhood of t0 , the function log ρ is the restriction to R of the real part of the composition of a holomorphic branch of the logarithmic function with the holomorphic function ζ → f (z0 + r0 e2πiζ ) − z1 , and hence log ρ is real analytic (in fact, the composition of real analytic functions is real analytic, so one can show that ρ is itself real analytic near t0 ). Lemma 5.18.8 then implies that ρ is constant near t0 , and hence the interior of the set ρ −1 (r1 ) in R is nonempty; and, moreover, this interior has no boundary points. Thus ρ ≡ r1 on R, and hence f (∂(z0 ; r0 )) ⊂ ∂(z1 ; r1 ). The reverse containment follows by symmetry.  In order to get simply connected neighborhoods of the edges in a graph, we will apply the following elementary fact: Lemma 5.18.10 If I is an open interval in R and W is a neighborhood of the set I × {0} in R2 , then there exists a simply connected neighborhood V of I × {0} in W ∩ (I × R) such that V ∩ (R × {0}) = I × {0} and V \ R = V \ (I × {0}) has exactly two connected components: V− ≡ {(x, y) ∈ V | y < 0}

and V+ ≡ {(x, y) ∈ V | y > 0},

both of which are nonempty. In particular, I × {0} = (I × R) ∩ ∂V− ∩ ∂V+ . Proof The sets V− ≡ {(x, y) ∈ R2 | x ∈ I, y < 0, and {x} × [y, 0] ⊂ W } and V+ ≡ {(x, y) ∈ R2 | x ∈ I, y > 0, and {x} × [0, y] ⊂ W }

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are open connected sets. For if (x1 , y1 ), (x2 , y2 ) ∈ V− with x1 < x2 , then for  > 0 sufficiently small, we have − > yj for j = 1, 2, and the connected set ({x1 } × [y1 , 0)) ∪ ([x1 , x2 ] × {−}) ∪ ({x2 } × [y2 , 0)) lies in V− . Similar arguments show that V+ is also connected, and that V− and V+ are open. The set V ≡ V− ∪ V+ ∪ (I × {0}) ⊂ W ∩ (I × R) is a connected neighborhood of I × {0} with V ∩ R = I × {0}, because each point in I × {0} has an open rectangular neighborhood of the form J × (−δ, δ) ⊂ W for some open interval J ⊂ I and some δ > 0, and hence this neighborhood is contained in V and meets both V− and V+ . Finally, the set V is simply connected because if α = (u, v) is a loop in V with base point (t0 , 0) ∈ I × {0}, then the map (t, s) → (u(t), (1 − s)v(t)) is a path homotopy in V from α to a loop β in I × {0}, and the map (t, s) → (u(t) + s(t0 − u(t)), 0) is a path homotopy in I × {0} from β to the trivial loop.



Proof of Proposition 5.18.4 We first construct a suitable graph. Since X is second countable (by Radó’s theorem), Lemma 9.3.6 provides a locally finite sequence of local holomorphic charts {(Uν , ν , Uν )} such that (i) For each ν ∈ Z>0 , we have  ≡ (0; 1)  Uν ; (ii) The family {Dν } ≡ {−1 ν ()} is a (locally finite) covering of X; and (iii) If μ, ν ∈ Z>0 with D μ ∩ D ν = ∅, then Dμ  Uν .  In particular, the subset M0 ≡ ∞ ν=1 ∂Dν of X is closed, and by Lemma 5.18.9, the set ∂Dμ ∩ ∂Dν is finite for all μ, ν ∈ Z>0 for which ∂Dμ = ∂Dν . We will now construct additional sets so that the union becomes a connected set that has connected intersection with each of the sets {D ν }. For all μ, ν ∈ Z>0 , the set Dν ∩ ∂Dμ has only finitely many connected components, because each nonempty connected component is the image under −1 μ of an open arc of the circle S1 = ∂, and any endpoint of the arc must map into the finite set ∂Dμ ∩ ∂Dν . It follows that for each ν, the set M0 ∩ D ν has only finitely many connected components (and each of these connected components is compact). Setting L0 = ∅, we now construct sequences of closed sets {Mk }∞ k=0 and compact sets {Lk }∞ inductively as follows. Suppose we have constructed closed k=0 sets M0 , . . . , Mk−1 and compact sets L1 , . . . , Lk−1 so that for each j = 1, . . . , k − 1, Mj = Mj −1 ∪ Lj and Mj ∩ D ν has only finitely many connected components for each ν. If Mk−1 ∩ D ν is connected for each ν, then we set Lk = ∅ and Mk = Mk−1 . Otherwise, letting ν be the minimal index in Z>0 for which the set Mk−1 ∩ D ν is not connected, we may choose a set Lk ⊂ D ν such that ν (Lk ) is the line segment from some point in a connected component A of ν (Mk−1 ∩ D ν ) to a point

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Embedding of an Open Riemann Surface into C3

285

Fig. 5.21 Local construction of the graph

in ν (Mk−1 ∩ D ν ) \ A with length (ν (Lk )) = dist(A, ν (Mk−1 ∩ D ν ) \ A) (see Fig. 5.21). Thus the set Mk ≡ Mk−1 ∪ Lk is closed, and the number of connected components of Mk ∩ D ν is strictly less than that of Mk−1 ∩ D ν . Observe also that for any μ ∈ Z>0 with Lk ∩ D μ = ∅, either Lk ⊂ Dμ (and hence the endpoints of Lk lie in Mk−1 ∩ Dμ ) or every connected component of Lk ∩ D μ meets ∂Dμ (and hence meets M0 ∩ D μ ). Thus, in either case, the number of connected components of Mk ∩ D μ is not more than that of Mk−1 ∩ D μ . Thus we get sequences of sets {Mk } and {Lk }, and it follows from the construction that for each ν, the set Mk ∩ D ν is connected for each k  0. Moreover, by local finiteness of the family {Dν }, the family of compact sets {Lk } is locally finite, and hence for every compact S, Mk+1 ∩ S = Mk ∩ S for k  0. Thus the set X1 ≡

∞

k=1

Mk = M0 ∪

∞

Lk

k=1

is a closed subset of X, and X1 ∩ D ν is connected for each ν. We may also choose a discrete subset X0 of X1 such that X0 meets ∂Dν for each ν, X0 contains ∂Dμ ∩ ∂Dν whenever μ, ν ∈ Z>0 with ∂Dμ = ∂Dν , and X0 contains the endpoints of Lk for each k. In particular, the collection of connected components of X1 \ X0 is locally finite in X, and each connected component is relatively compact in X. Observe also that for each k, in some local holomorphic chart, Lk is a line segment whose interior points do not meet Mk−1 ⊃ M0 . Thus the sets Lk ∩ ∂Dν for each ν, and Lk ∩ Lj for each j < k, must consist of endpoints of Lk (or be empty), and hence these sets must be contained in X0 . Furthermore, since every disk is mapped onto the upper half-plane by some automorphism of P1 (Theorem 5.8.3), the closure of each connected component L of X1 \ X0 is contained in some local holomorphic coordinate neighborhood for which L is mapped onto an open interval in R. Now Lemma 9.3.6 provides a neighborhood 0 of the discrete set X0 in X such that 0 is a union of a locally finite family of disjoint relatively compact open subsets of X, each of which is biholomorphic to a disk and contains exactly one point from X0 (see Fig. 5.22). Lemma 9.3.6, Lemma 5.18.10, and the Riemann mapping theorem in the plane (Theorem 5.2.1) provide a neighborhood 1 of X1 \ X0 in X \ X0 such that 1 is a union of a locally finite family of disjoint relatively

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Fig. 5.22 The three desired open sets

compact connected open subsets of X, and such that if V is one of these subsets (i.e., one of the connected components of 1 ), then V is biholomorphic to a disk, V contains exactly one connected component L of X1 \ X0 , and V \ L has exactly two connected components (see Fig. 5.22). We will now show that the third set may be taken to be the open set 2 ≡ X \ X1 (see Fig. 5.22). Given a connected component  of 2 , we have  ∩ Dν = ∅ for some ν. Since ∂Dν ⊂ X1 , we must have  ⊂ Dν \ X1  X. Thus ν () ⊂  is a connected component of the complement of the connected compact set ν (X1 ∩ D ν ) in P1 . Therefore, Lemma 5.17.1 and the Riemann mapping theorem in the plane imply that  is biholomorphic to a disk. Furthermore, ∂ is a connected subset of X1 ∩ D ν that cannot be a single point (for if p ∈ ∂, then P1 \ {ν (p)} is a connected open set that meets, but is not contained in, ν () ⊂ , so P1 \ {ν (p)} must meet ∂ν () = ν (∂)). Hence ∂ meets some connected component L of X1 \ X0 with L ⊂ D ν . On the other hand, if V is the connected component of 1 containing L, then V \ L = V− ∪ V+ , where V− and V+ are nonempty disjoint connected open subsets of X \ X1 , and hence  must contain V− or V+ . It follows that the collection of connected components of 2 must be locally finite in X.  Remarks 1. As is clear from the above proof, the closure of each connected component of X1 \ X0 is the image of a smooth injective path with endpoints in X0 . Thus the set X1 may be viewed as a graph embedded in X with vertices equal to the points of X0 . 2. The above decomposition of X into manifolds of increasing dimension X0 , X1 \ X0 , X \ X1 (i.e., X2 \ X1 for X2 = X) is an example of a stratification of X. 3. One may also push the construction further to get a triangulation of X (see Exercise 6.11.4). 4. The set X1 \ X0 is a real analytic submanifold of X \ X0 . On the other hand, for our purposes, it is not really necessary that the local charts, in which the connected components of X1 \ X0 become intervals, be holomorphic. Continuous local charts would suffice, and in fact, the existence of suitable local C ∞ charts would follow from smoothness of the components, Theorem 9.10, and a natural analogue of Lemma 5.11.1. However, that one may choose the local charts to be holomorphic follows directly from the construction. The above facts will allow us to produce a proper holomorphic mapping into C3 , and the following lemma will allow us to approximate the mapping by an embedding:

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Embedding of an Open Riemann Surface into C3

287

Lemma 5.18.11 Let X be an open Riemann surface, and let ω be a Kähler form on X. Then there exists a Kähler form ω on X such that every C ∞ function ϕ on X with iϕ ≥ ω has the following properties: (i) For every pair of distinct points p, q ∈ X and every constant ζ ∈ C, there exists a function f ∈ O(X) ∩ L2 (X, ω, ϕ) with f (p) − f (q) = ζ . (ii) For every point p ∈ X and every cotangent vector η ∈ 1,0 Tp∗ X, there exists a function f ∈ O(X) ∩ L2 (X, ω, ϕ) with (df )p = η. Proof We may fix a locally finite collection of local holomorphic charts {(Dν , zν = ν , (0; 3))}∞ ν=1

 with X = ν −1 ν ((0; 1)), and a nonnegative function λ ∈ D((0; 3)) with λ ≡ 1 on (0; 2). For each point p ∈ −1 ν ((0; 1)), we have 2 ¯ i∂ ∂[λ(z ν ) log |zν − zν (p)| ]   2  (∂λ/∂z)(zν ) ∂ λ 2 = (zν ) log |zν − zν (p)| + 2 Re i dzν ∧ d z¯ ν ≥ −Cν ω ∂z∂ z¯ z¯ ν − z¯ ν (p)

on Dν \ {p}, where the constant   i dzν ∧ d z¯ ν ∂ 2λ ∂λ + 2 max · max Cν ≡ (log 16) max −1 ∂z∂ z¯ ∂z ω ν (supp λ) is independent of the choice of the point p. Moreover, a standard partition of unity argument yields a C ∞ function ρ on X such that ρ > 1 + 2Cν + |ω /ω| on Dν for each ν, and we may set ω ≡ ρ · ω. Suppose now that ϕ is a C ∞ function on X with iϕ ≥ ω . Given two (not necessarily distinct) points p, q ∈ X and a constant ζ ∈ C, we may fix indices μ and −1 ν with p ∈ −1 μ ((0; 1)) and q ∈ ν ((0; 1)) (and with μ = ν if p = q) and a function τ ∈ D(X) such that τ ≡ ζ on a neighborhood of p and τ ≡ 0 on a neighborhood of q if p = q, and τ ≡ ζ · (zμ − zμ (p)) on a neighborhood of p ¯ then has compact support in the Riemann if p = q. The C ∞ (0, 1)-form γ ≡ ∂τ surface Y ≡ X \ {p, q}, and the C ∞ function ψ ≡ ϕ + λ(zμ ) log |zμ − zμ (p)|2 + λ(zν ) log |zν − zν (q)|2 on Y satisfies iω + iψ ≥ ω. ¯ =γ Therefore, by Corollary 2.12.6, there exists a C ∞ function α on Y such that ∂α and αL2 (Y,ω,ψ) < ∞. In particular, α is holomorphic near p and q. Moreover, if p = q, then for some constant C > 0, we have  |α/(zμ − zμ (p))|2 i dzμ ∧ d z¯ μ ≤ C · α2 2 −1 < ∞, −1 μ ((0;1))

L (μ ((0;1)),ω,ψ)

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and the analogous inequality holds near q. Therefore, by Riemann’s extension theorem (Theorem 1.2.10), α extends to a C ∞ function on X (which we will also denote by α) that vanishes at p and q. Thus f ≡ τ − α is then a holomorphic function on X, −1 f (p) − f (q) = ζ , and since ϕ = ψ on X \ (−1 μ (supp λ) ∪ ν (supp λ)) ⊂ Y , we 2 have f ∈ L (X, ω, ϕ). Thus the property (i) holds. If p = q and η = ζ · (dzμ )p , then  |α/(zμ − zμ (p))2 |2 i dzμ ∧ d z¯ μ ≤ C · α2 2 −1 <∞ −1 μ ((0;1))

L (μ ((0;1)),ω,ψ)

for some constant C > 0, and hence α extends to a C ∞ function on X (which we will also denote by α) that is holomorphic near p with a zero of order at least 2 at p. Hence f ≡ τ − α ∈ L2 (X, ω, ϕ) is then a holomorphic function on X satisfying (df )p = η, and the property (ii) also holds.  Lemma 5.18.12 Let X be an open Riemann surface, let  be an open subset that is equal to the union of a locally finite family of disjoint relatively compact open subsets of X, each of which is biholomorphic to a disk, let K be a closed subset of X with K ⊂ , let C be a countable subset of X, let ηp ∈ 1,0 Tp∗ X for each point p ∈ C, let D be a countable set of pairs of distinct points (p, q) in X, let ζp,q ∈ C for each pair (p, q) ∈ D, and let ρ be a positive continuous function on X. Then there exists a holomorphic function f on X such that |f | < ρ on K, (df )p = ηp for each point p ∈ C, and f (p) − f (q) = ζp,q for each pair (p, q) ∈ D. Proof We may assume without loss of generality that  has infinitely many connected components {ν }∞ ν=1 , we may fix a neighborhood  of K with  ⊂ , and we may fix Kähler forms ω and ω on X with the properties described in Lemma 5.18.11. Theorem 2.14.1 then provides a C ∞ strictly subharmonic function α on X with iα ≥ ω . By Theorem 2.6.4, for each ν, there exists a constant δν ∈ (0, 1) such that |h| < ρ on K ∩ ν for every function h ∈ O( ∩ ν ) with hL2 (∩ν ,ω,α) < δν . Moreover, Lemma 5.18.7 provides a C ∞ subharmonic function β on X such that β < 2 log δν on  ∩ ν for each ν = 1, 2, 3, . . . . Setting ϕ ≡ α + β, we see that if h ∈ O(X) and hL2 (X,ω,ϕ) < 1, then for each ν, hL2 (∩ν ,ω,α) ≤ δν · hL2 (∩ν ,ω,ϕ) < δν , and hence |h| < ρ on K. According to Theorem 2.6.4, O(X) ∩ L2 (X, ω, ϕ) is a closed subspace of 2 L (X, ω, ϕ), and is therefore itself a Hilbert space. By construction, for each point p ∈ X, each cotangent vector η ∈ 1,0 Tp∗ X, each point q ∈ X \ {p}, and each constant ζ ∈ C, the sets Ap,η ≡ {f ∈ O(X) ∩ L2 (X, ω, ϕ) | (df )p = η}, Bp,q,ζ ≡ {f ∈ O(X) ∩ L2 (X, ω, ϕ) | f (p) − f (q) = ζ }, are nonempty, and it follows from Theorem 2.6.4 and Theorem 1.2.4 that the above sets are also open in O(X) ∩ L2 (X, ω, ϕ). Moreover, these sets are dense. For if

5.18

Embedding of an Open Riemann Surface into C3

289

f ∈ O(X) ∩ L2 (X, ω, ϕ) with (df )p = η (respectively, f (p) − f (q) = ζ ), then fixing h ∈ Ap,0p (respectively, h ∈ Bp,q,0 ), we get f + h ∈ Ap,η (respectively, f + h ∈ Bp,q,ζ ) for every  ∈ C∗ , and (f + h) − f  → 0 as  → 0. According to the Baire category theorem (see for example, [Mu] or [Rud1]), the intersection of any countable collection of dense open subsets of a Hilbert space (or any complete metric space) is dense. Hence there must exist an element f of " " Ap,ηp ∩ Bp,q,ζp,q p∈C

(p,q)∈D

with f L2 (X,ω,ϕ) < 1, and the claim follows.



Proof of Theorem 5.18.2 By Proposition 5.18.4, there exists a covering of X by three open sets 0 , 1 , and 2 such that for each j = 0, 1, 2, the connected com∞ ponents {(ν) j }ν=1 of j form a locally finite collection of relatively compact open subsets of X and each connected component is biholomorphic to a disk. By shrinking these sets slightly and applying Theorem 5.18.3, we may also assume that there exists a holomorphic function gj on X such that |gj − ν| < 1 on (ν) j for ν = 1, 2, 3, . . . . In particular, the holomorphic mapping g = (g0 , g1 , g2 ) : X → C3 is proper. We will approximate g by a (proper) holomorphic embedding. We may fix closed subsets K1 and K2 of X such that K1 ⊂ 1 , K2 ⊂ 2 , and X = 0 ∪ K1 ∪ K2 , and we may set f0 ≡ g0 . By the identity theorem, the set C  ≡ {p ∈ X | (df0 )p = 0} is discrete, and fixing a countable dense subset C of X containing C  , we see that the set D of pairs of distinct points (p, q) in X with p ∈ C and f0 (p) = f0 (q) is countable. Applying Lemma 5.18.12, we get a holomorphic function h1 on X such that (dh1 )p = −(dg1 )p for each point p ∈ C, h1 (p) − h1 (q) = g1 (q) − g1 (p) for each pair (p, q) ∈ D, and |h1 | < 1 on K1 . Thus the holomorphic function f1 ≡ g1 + h1 satisfies |f1 − ν| < 2 on K1 ∩ (ν) 1 for each ν, (df1 )p = 0 for each point p ∈ C ⊃ C  (in particular, (f0 , f1 ) is a holomorphic immersion into C2 ), and f1 (p) = f1 (q) for each pair (p, q) ∈ D. It now suffices to show that the set E ⊂ X × X of pairs of distinct points (p, q) with (f0 (p), f1 (p)) = (f0 (q), f1 (q)) is countable. For the above argument will then provide a holomorphic function f2 such that |f2 − ν| < 2 on K2 ∩ (ν) 2 for each ν and f2 (p) = f2 (q) for each pair (p, q) ∈ E. The mapping f = (f0 , f1 , f2 ) : X → C3 will then be a holomorphic embedding. Since the set of points in X at which df0 = 0 or df1 = 0 is countable, and E ∩ ({p} × X) is countable for each point p ∈ X, it suffices to show that the set of points (p, q) ∈ E at which (df0 )p = 0 and (df1 )p = 0 is countable. Therefore, since X × X is second countable, is suffices to show that each such point (p, q) is an isolated point in E. By the open mapping theorem, the holomorphic inverse function theorem, and the identity theorem, there exist disjoint connected neighborhoods U and V of p and q, respectively, such that for j = 0, 1, fj maps U biholomorphically onto a domain Uj ⊂ C, fj (V ) ⊂ Uj , and fj−1 (fj (p)) ∩ V = {q}. For each point (r, s) ∈ E ∩ (U × V ), we have (f0 U )−1 (f0 (s)) = r = (f1 U )−1 (f1 (s)).

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Since C is dense in X, we may fix a point r ∈ C ∩ U ∩ f0−1 (f0 (V )) and a point s ∈ V ∩ f0−1 (f0 (r)). We then get (r, s) ∈ D, and hence f1 (r) = f1 (s); that is, (f0 U )−1 ◦ f0 − (f1 U )−1 ◦ f1 is not identically zero on the domain V . It follows that we may choose V so small that q is the only zero of the above holomorphic function, that is, so that E ∩ (U × V ) ⊂ U × {q}. Hence E ∩ (U × V ) =  {((f0 U )−1 (f0 (q)), q)} = {(p, q)}, and the claim follows. Exercises for Sect. 5.18 5.18.1 Let X be an open Riemann surface. Using only Theorem 5.18.3 and Proposition 5.18.4, prove directly (without appealing to the Baire category theorem) that there exists a proper holomorphic immersion of X into C3 . 5.18.2 Prove Theorem 5.18.3 directly using the Runge approximation theorem ¯ (Theorem 2.16.1) in place of the L2 ∂-method. 5.18.3 The proof of Theorem 5.18.2 shows that every open Riemann surface admits a holomorphic immersion into C2 . Prove this fact directly by applying the Mittag-Leffler theorem (Theorem 2.15.1).

5.19 Embedding of a Compact Riemann Surface into Pn In this section, we consider an embedding theorem for a compact Riemann surface X. As mentioned in Sect. 5.18, X cannot admit a holomorphic embedding into a Euclidean space, because O(X) = C. However, X does admit a holomorphic embedding into a complex projective space. Definition 5.19.1 Let n ∈ Z>0 . The n-dimensional complex projective space Pn is the quotient space Pn ≡ Cn+1 \ {0}/∼, where for all points w, z ∈ Cn+1 \ {0}, we have w ∼ z if and only if w = λz for some scalar λ ∈ C∗ . We call any representative (ζ0 , . . . , ζn ) ∈ Cn+1 \ {0} for a point ζ ∈ Pn homogeneous coordinates for ζ , and we write ζ = [ζ0 , . . . , ζn ]. Remarks 1. In other words, Pn is the space of complex lines in Cn+1 that pass through the origin. 2. For each j = 0, . . . , n, the set Uj ≡ {[ζ0 , . . . , ζn ] ∈ Pn | ζj = 0} is well defined and open (in the quotient topology), and the mapping Uj → Cn given by [ζ0 , . . . , ζn ] → (ζ0 /ζj , . . . , ζ j /ζj , . . . , ζn /ζj )  is a well-defined homeomorphism (see Exercise 5.19.1). Moreover, Pn = nj=0 Uj . 3. For n = 1, P1 (as defined above) is a connected compact Riemann surface with local holomorphic charts (U0 , [ζ0 , ζ1 ] →  ζ1 /ζ0 , C) and (U1 , [ζ0 , ζ1 ] →  ζ0 /ζ1 , C)

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291

given by the mappings in Remark 2 above, and the quotient mapping C2 \ {0} → P1 is holomorphic. Moreover, although we also denote the Riemann sphere C ∪ {∞} by P1 , there is no ambiguity because we have a biholomorphism C2 \ {0}/∼ → C ∪ {∞} given by [ζ0 , ζ1 ] → ζ0 /ζ1 ([ζ0 , 0] → ∞). 4. For an arbitrary n ≥ 1, Pn is a connected compact complex manifold of dimension n. The local holomorphic charts are provided by the mappings in Remark 2 above, and the quotient map Cn+1 \ {0} → Pn is holomorphic (see Exercise 5.19.1). We will not use this fact directly, but it justifies the use of the terminology appearing in Definition 5.19.2 below (cf. Definition 5.18.1). Definition 5.19.2 Let X be a complex 1-manifold, and let F : X → Pn be a continuous mapping into a complex projective space Pn . For each j = 0, . . . , n, let j : Uj → Cn be the homeomorphism of the open set Uj ≡ {[ζ0 , . . . , ζn ] ∈ Pn | ζj = 0} onto Cn given by [ζ0 , . . . , ζn ] → (ζ0 /ζj , . . . , ζ j /ζj , . . . , ζn /ζj ). (a) F is called holomorphic if the mapping j ◦F : F −1 (Uj ) → Cn is holomorphic for each j = 0, . . . , n. (b) F is called a holomorphic immersion if the mapping j ◦ F : F −1 (Uj ) → Cn is a holomorphic immersion for each j = 0, . . . , n. (c) F is called a holomorphic embedding if X is compact and F is an injective holomorphic immersion (i.e., F is a proper injective holomorphic immersion). One may obtain holomorphic mappings into complex projective spaces from holomorphic sections of a holomorphic line bundle as follows: Lemma 5.19.3 Let E be a holomorphic line bundle on a Riemann surface X, and let s = (s0 , . . . , sn ) be an (n + 1)-tuple of holomorphic sections of E with no common zeros. Then the mapping κs : X → Pn given by   s0 (x) sn (x) ,..., ∀x ∈ X and ξ ∈ Ex \ {0x } κs (x) ≡ ξ ξ is well defined and holomorphic. Proof For x ∈ X and ξ, η ∈ Ex \ {0x }, we have     s0 (x) sn (x) η s0 (x) sn (x) ,..., = · ,..., , ξ ξ ξ η η so the mapping is well defined. Moreover, given a point x0 ∈ X and an index j with sj (x0 ) = 0, then sj is nonvanishing on some neighborhood U of x0 in X and the composition of the mapping [ζ0 , . . . , ζn ] →  (ζ0 /ζj , . . . , ζ j /ζj , . . . , ζn /ζj ) with the mapping κs U is the mapping    x → s0 (x)/sj (x), . . . , sj (x)/s j (x), . . . , sn (x)/sj (x) , which is holomorphic.



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Definition 5.19.4 Let E be a holomorphic line bundle on a Riemann surface X, and let s = (s0 , . . . , sn ) be an (n + 1)-tuple of holomorphic sections of E with no common zeros. Then the associated holomorphic mapping X → Pn given by   s0 (x) sn (x) x → ,..., ∀x ∈ X and ξ ∈ Ex \ {0x } ξ ξ is denoted by κs or by [s0 , . . . , sn ]. Example 5.19.5 Let E → P1 be the hyperplane bundle (see Examples 3.1.5 and 3.3.4); that is, E = [D], where D is the divisor with D(0) = 1 and D(x) = 0 for each point x ∈ P1 \ {0}. Let s0 be a holomorphic section with div(s0 ) = D. We also have the holomorphic section s1 ≡ s0 /z (s1 (∞) = 0), and the corresponding mapping κ(s0 ,s1 ) = [s0 , s1 ] is equal to inverse of the biholomorphism (C2 \ {0})/∼ → C ∪ {∞} described in the remarks following Definition 5.19.1 (see Exercise 5.19.2). Recall that the vector space of holomorphic sections of a holomorphic line bundle on a compact Riemann surface is finite-dimensional (Theorem 4.4.1). Lemma 5.19.6 Let E be holomorphic line bundle on a compact Riemann surface X, and let s = (s0 , . . . , sn ) for a basis s0 , . . . , sn for (X, O(E)). (a) A point p ∈ X satisfies s(p) = (0, . . . , 0) if and only if every holomorphic section of E on X vanishes at p. (b) Suppose that the sections s0 , . . . , sn have no common zeros. Then the associated map κs = [s0 , . . . , sn ] is injective if and only if for each pair of distinct points p, q ∈ X, there exists a holomorphic section t of E with t (p) = 0 and t (q) = 0. (c) Suppose (again) that the sections s0 , . . . , sn have no common zeros. Then the associated map κs = [s0 , . . . , sn ] is a holomorphic immersion if and only if for each point p ∈ X, there is a holomorphic section t with a simple zero at p. Remark In the standard terminology (see, for example, [GriH]), the point p in part (a) is a base point for the holomorphic sections (or, more precisely, for the linear system of divisors associated to the nontrivial holomorphic sections of E). In (b), the holomorphic sections are said to separate points in X. In (c), the holomorphic sections are said to give local coordinates in X. Proof of Lemma 5.19.6 If p ∈ X with s(p) = (0, . . . , 0), then every section t ∈ (X, O(E)) must vanish at p (since t is a linear combination of the sections s0 , . . . , sn ). Part (a) now follows. Let us assume for the rest of this proof that the sections s0 , . . . , sn have no common zeros. Given distinct points p, q ∈ X, there exists an index j ∈ {0, . . . , n} with sj (q) = 0. If sj (p) = 0, then the section sj separates the points p and q as in (b), and we have κs (p) = κs (q). Assuming that sj (p) = 0 and that κs is injective, the corresponding points    s0 (p)/sj (p), . . . , sj (p)/s j (p), . . . , sn (p)/sj (p)

5.19

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293

  s0 (q)/sj (q), . . . , sj (q)/s j (q), . . . , sn (q)/sj (q)



in Cn must be distinct, and therefore si (p) si (q) = sj (p) sj (q)

for some i ∈ {0, . . . , jˆ, . . . , n}.

Thus the holomorphic section t ≡ si −

si (p) · sj sj (p)

satisfies t (p) = 0 and t (q) = 0. For the converse, suppose that sj (p) = 0 and sj (q) = 0, and that there exists a holomorphic section

t with t (p) = 0 and t (q) = 0. We then have constants ζ0 , . . . , ζn ∈ C with t = ni=0 ζi si , and hence n i=0

ζi

si (p) = 0 and sj (p)

n

ζi

i=0

si (q) = 0. sj (q)

It follows that the points    s0 (p)/sj (p), . . . , sj (p)/s j (p), . . . , sn (p)/sj (p) and

   s0 (q)/sj (q), . . . , sj (q)/s j (q), . . . , sn (q)/sj (q)

in Cn must be distinct and therefore that κs (p) = κs (q). The claim (b) now follows. For the proof of (c), let p ∈ X and let j ∈ {0, . . . , n} with sj (p) = 0. If κs is a holomorphic immersion, then for some i ∈ {0, . . . , jˆ, . . . , n}, we have (d(si /sj ))p = 0, and hence the holomorphic section t ≡ si −

si (p) · sj sj (p)

has a simple zero at p. Conversely, given a section t

∈ (X, O(E)) with a simple zero at p, we have constants ζ0 , . . . , ζn ∈ C with t = ni=0 ζi si , and hence 0 = (d(t/sj ))p =

n

ζi · (d(si /sj ))p .

i=0

Thus (d(si /sj ))p = 0 for some i ∈ {0, . . . , jˆ, . . . , n}, and (c) follows.



Definition 5.19.7 A holomorphic line bundle E on a compact Riemann surface X is called ample if E is positive; that is, E admits a positive-curvature Hermitian metric or, equivalently (by Theorem 4.3.1), deg E > 0. The line bundle E is called

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very ample if the map κs : X → Pn associated to any basis s for (X, O(E)) exists and is a holomorphic embedding. Remark Since the line bundle associated to any nontrivial effective divisor has positive degree, if E is very ample, then E is ample. The results of Chap. 4 and the above observations now give us the following: Theorem 5.19.8 Let E be a holomorphic line bundle on a compact Riemann surface X. Then we have the following: (a) If E is ample, then E d is very ample for every d  0. (b) If deg E > 2, then KX ⊗ E is very ample. (c) If deg E > 2 · genus(X), then E is very ample. Proof Clearly, part (a) follows from part (c). Moreover, part (c) follows from part (b) applied to the holomorphic line bundle KX∗ ⊗ E, since this line bundle has degree −2 genus(X) + 2 + deg E by Corollary 4.6.7. Thus it suffices to prove part (b). Assuming that deg E > 2, suppose p, q ∈ X are two (possibly equal) points and D is the divisor given by D = p + q. Then, applying Corollary 4.1.2 to the positive holomorphic line bundle E ⊗ [−D] (we have deg E ⊗ [−D] > 2 − 2 = 0) and the effective divisor D, we get a holomorphic section t of KX ⊗ E ⊗ [−D] ⊗ [D] = KX ⊗ E such that t (p) = 0 and t (q) = 0 if p = q, and t has a simple zero at p if p = q. Thus, by Lemma 5.19.6, the map κs is defined for any basis s of  (X, O(KX ⊗ E)), and κs is a holomorphic embedding. Remark According to the higher-dimensional analogue of Theorem 5.19.8, which is called the Kodaira embedding theorem, if E is an ample holomorphic line bundle on a connected compact complex manifold X, then E d is very ample for d  0 (see, for example, [GriH], [MKo], or [Wel]). Exercises for Sect. 5.19 5.19.1 Prove that Pn is a complex manifold of dimension n with local holomorphic charts given by the mappings in Definition 5.19.2, and the quotient map Cn+1 \ {0} → Pn is holomorphic (see Exercise 2.2.6 for the required definitions). 5.19.2 Verify that the mapping κ(s0 ,s1 ) = [s0 , s1 ] in Example 5.19.5 is equal to the inverse of the biholomorphism (C2 \ {0})/∼ → C ∪ {∞} described in the remarks following Definition 5.19.1.

5.20 Finite Holomorphic Branched Coverings As we will see in this section, any nonconstant proper holomorphic mapping of Riemann surfaces is actually a finite holomorphic covering map away from a discrete

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Finite Holomorphic Branched Coverings

295

set of branch points (see Proposition 5.20.1 and Definition 5.20.2 below). We first recall some terminology and facts concerning the topology of manifolds (Sect. 9.3). A continuous mapping of Hausdorff spaces is called proper if the inverse image of every compact subset of the target is compact (see Definition 9.3.9). A sequence in a manifold converges to a point p if for each neighborhood U of p, all but finitely many terms of the sequence lie in U . According to Theorem 9.3.2, a compact subset K of a manifold has the Bolzano–Weierstrass property; that is, every sequence in K admits a convergent subsequence. Proposition 5.20.1 Let  : X → Y be a nonconstant proper holomorphic mapping of Riemann surfaces X and Y , and let mq = multq  ∈ Z>0 denote the multiplicity of  at q for each point q ∈ X. Then we have the following: (a)  is surjective, and counting multiplicities, the cardinalities of the fibers of  are all equal to some finite constant n ∈ Z>0 ; that is, mq = n ∀p ∈ Y. q∈−1 (p)

V of p (b) If p ∈ Y and Z ≡ −1 (p), then there exist a connected neighborhood  and disjoint connected open sets {Uq }q∈Z such that −1 (V ) = q∈Z Uq and such that, for each q ∈ Z, we have q ∈ Uq , Uq : Uq → V is a surjective proper holomorphic mapping, and, counting multiplicities, the number of points in each fiber of Uq is equal to mq . (c) The set of critical values C ≡ ({q ∈ X | (∗ )q = 0}) = ({q ∈ X | multq  > 1}) is discrete, and the mapping X\−1 (C) : X \ −1 (C) → Y \ C is a finite (proper) holomorphic covering map. (d) If −1 (p) is a singleton for some point p ∈ Y \ C, then  is a biholomorphism of X onto Y . Proof By the open mapping theorem (Theorem 2.2.2),  ≡ (X) is an open subset of Y . If p ∈ , then there is a sequence {qν } in X such that (qν ) → p, and by replacing this sequence with a suitable subsequence, we may assume that {qν } converges to some point q ∈ X. But we then have (qν ) → (q), and hence p = (q) ∈ . Thus  is both open and closed in the Riemann surface Y , and hence  is surjective. Once again, let us fix a point p ∈ Y and let Z ≡ −1 (p). Since Z is both compact (because  is proper) and discrete (by the identity theorem), Z must be finite. The local representation of holomorphic mappings (Lemma 2.2.3) provides a local holomorphic coordinate neighborhood (V ,  = ζ, V  ) with p ∈ V and ζ (p) = 0, disjoint local holomorphic coordinate neighborhoods {(Uq , q = zq , Uq )}q∈Z in X, and positive integers {mq }q∈Z such that for each q ∈ Z, we have q ∈ Uq , zq (q) = 0, m (Uq ) ⊂ V , and ζ () = zq q on Uq . For  > 0 sufficiently small, we have

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 (0; ) ⊂ V  and −1 ( −1 ((0; ))) ⊂ U ≡ q∈Z Uq . For if this were not the case, then there would exist a sequence {qν } in X \ U such that (qν ) → p and qν → q ∈ X \ U with (q) = p. But we would then have q ∈ Z \ U , which is impossible. We may also choose  so small that (0;  1/mq ) ⊂ Uq for each q ∈ Z. Therefore, by replacing V  and V with (0; ) and  −1 ((0; )), respectively, 1/mq )), respectively, for and replacing Uq and Uq with (0;  1/mq ) and −1 q ((0;  each q ∈ Z, we may assume that the local holomorphic charts are given by (V ,  = ζ, V  = (0; ))

and

{(Uq , zq , Uq = (0;  1/mq ))}q∈Z .

In particular, for each point q ∈ Z,  is represented on Uq by the holomorphic map (0;  1/mq ) → (0; ) given by z → zmq , and hence, counting multiplicities, the cardinality of the fiber of Uq over y is equal to mq for each point y ∈ V , and part (b) follows. Furthermore, counting multiplicities,

the number of points in each fiber over each point in V is equal to the constant n = q∈Z mq . It follows that the function on Y given by the cardinality of each fiber, again counting multiplicities, is locally constant and therefore constant. The claim (c) now follows from Lemma 10.2.11. The claim (d) follows from the above (or from part (a) and Theorem 2.4.4).  Definition 5.20.2 A proper holomorphic mapping  : X → Y of a complex 1manifold X onto a Riemann surface Y for which the restriction to each connected component is surjective and for which almost every fiber consists of n points (i.e., every fiber has n points counting multiplicities) is called a finite holomorphic branched covering map (or a finite holomorphic ramified covering map) with n sheets. Each critical point p of  is called a branch point (or ramification point) of order m − 1, where m = multp  > 1. The sum of all of the orders of the branch points of  is called the branching order of . According to Corollary 2.10.4, every Riemann surface admits a nonconstant meromorphic function. Thus we have the following: Theorem 5.20.3 Every compact Riemann surface admits a finite holomorphic branched covering map onto P1 . We now record the following fact for use in the proof of Abel’s theorem (see Sect. 5.21): Proposition 5.20.4 Let X be a complex 1-manifold, let Y be a Riemann surface, let  : X → Y be a finite holomorphic branched covering map with n sheets, and let C ⊂ Y be the set of critical values. (a) If γ : [0, 1] → Y is an injective path with γ ((0, 1)) ⊂ Y \ C, then there exist exactly n distinct liftings γ1 , . . . , γn of γ to paths in X. Moreover, the sets {γj ((0, 1))}nj=1 are disjoint, and if p ∈ −1 (γ (0)) and m = multp , then exactly m of the liftings have initial point p.

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(b) If f is a holomorphic function on X, then the function Y \ C → C given by f (x) y → x∈−1 (y)

has a unique extension to a holomorphic function on Y . (c) Given a holomorphic 1-form α on X, there is a unique holomorphic 1-form β on Y such that if γ is any injective path in Y with γ ((0, 1)) ⊂ Y \ C, and γ1 , . . . , γn are the distinct liftings of γ to paths in X (as given by part (a)), then  β= γ

n 

α.

j =1 γj

Proof If γ is an injective path in Y with γ ((0, 1)) ⊂ Y \ C, then, since the restriction X \ −1 (C) → Y \ C is an (unbranched) covering map with exactly n points in each fiber, there exist exactly n distinct liftings δ1 , . . . , δn of γ (0,1) to paths in X \−1 (C). Moreover, the images of these liftings are disjoint. For if δi (t0 ) = δj (t1 ) for some pair of indices i, j ∈ {1, . . . , n} and some pair of numbers t0 , t1 ∈ (0, 1), then γ (t0 ) = (δi (t0 )) = (δj (t1 )) = γ (t1 ), and the injectivity of γ implies that t0 = t1 . Uniqueness of liftings in a covering space then implies that δi = δj , and hence i = j . Now let p1 , . . . , pk be the distinct points in −1 (γ (0)), and let mi ≡ multpi  for each i = 1, . . . , k (in particular,

n = ki=1 mi ). Then, by Proposition 5.20.1, there exist a connected neighborhood V of γ (0) in Y and disjoint connected open sets U1 , . . . , Uk such that −1 (V ) =  k −1 i=1 Ui and such that for each i = 1, . . . , k, we have pi ∈ Ui , Ui ∩  (C) ⊂ {pi }, and Ui : Ui → V is a finite branched holomorphic covering map with mi sheets. Let J be the connected component of γ −1 (V ) containing 0. Then, for each j = 1, . . . , n, δj (J ∩ (0, 1)) lies in Ui for some unique index i, and properness implies that δj (t) → pi as t → 0+ . A similar argument applies at t = 1. Thus δj extends to a unique lifting γj : [0, 1] → X of γ , and this lifting satisfies γj (0) = pi . Furthermore, fixing a point t0 ∈ J ∩ (0, 1), we see that for each i = 1, . . . , k, each of the mi points in −1 (γ (t0 )) ∩ Ui will be equal to the value at t = t0 for exactly one of the paths δ1 , . . . , δn . Thus exactly mi of the paths γ1 , . . . , γn will have initial value pi , and part (a) is proved. For the

proof of (b), suppose f ∈ O(X) and h : Y \ C → C is the function given by y → x∈−1 (y) f (x). If U ⊂ Y \ C is a domain that is evenly covered by the holomorphic covering map X \ −1 (C) → Y \ C, and U1 , . . . , Un are the

distinct connected components of the inverse image −1 (U ), then we have hU = nj=1 fj , where for each j = 1, . . . , n, fj ≡ f ◦ (Uj )−1 : U → C. It follows that h is a holomorphic function on Y \ C. Moreover, by properness, h must also be bounded on K \ C for each compact set K ⊂ Y , and hence by Riemann’s extension theorem (Theorem 1.2.10), h extends to a unique holomorphic function on Y .

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Finally, for the proof of (c), suppose α is a holomorphic 1-form on X. In analogy with part (b), the corresponding holomorphic 1-form on Y is obtained by averaging over the fibers. More precisely, if U ⊂ Y \ C is a domain that is evenly covered by the holomorphic covering map X \−1 (C) → Y \C, and U1 , . . . , Un are the distinct connected components of −1 (U ), then we let ηU be the holomorphic 1-form on U

n given by ηU ≡ j =1 αj , where for each j = 1, . . . , n, αj is the unique holomorphic 1-form on U satisfying (Uj )∗ αj = αUj . These holomorphic 1-forms agree on the overlaps, and therefore they determine a well-defined holomorphic 1-form η on Y \ C. Given a point q ∈ C, we may form neighborhoods U and V of −1 (q) and q, respectively, such that U : U → V is a finite holomorphic branched covering map and such that for some function f ∈ O(U ), we have αU = df (for we may choose V so that the connected components of −1 (V ) lie in a union of disjoint simply connected open sets). By part (b), the function y → x∈−1 (y) f (y) on V \ C extends to a unique holomorphic function g on V , and as is clear from the construction of η, we have dg = η on V \ C. It follows that η extends to a holomorphic 1-form β on Y . Suppose γ is an injective path in Y with γ ((0, 1)) ⊂ Y \ C, and γ1 , . . . , γn are the distinct liftings of γ . For each  ∈ (0, 1/2), there is a partition  = t0 < · · · < tk = 1 −  such that for each ν = 1, . . . , k, γ ([tν−1 , tν ]) is contained in an evenly covered domain in Y \ C. The paths {γj [tν−1 ,tν ] }nj=1 are then the distinct liftings of γ [tν−1 ,tν ] , and hence  γ [,1−]

β=

k  ν=1 γ [tν−1 ,tν ]

β=

n  k ν=1 j =1 γj [tν−1 ,tν ]

α=

n  j =1 γj [,1−]

α.

Letting  → 0+ , we get  β= γ

n 

α.

j =1 γj

Finally, suppose β  is another holomorphic 1-form with the above property. Each point p ∈ Y \C has a connected neighborhood U in Y \C on which β = df for some function f ∈ O(U ). Since f may be obtained by integration of β along injective paths in U with initial point p, and each of these integrals must agree with the corresponding integral of β  , we have β  = df = β on U . Thus β  = β on Y \ C and therefore on Y . 

Exercises for Sect. 5.20 5.20.1 Suppose X is a complex 1-manifold, Y is a Riemann surface,  : X → Y is a finite holomorphic branched covering map with n sheets, C ⊂ Y is the set of critical values, f ∈ O(X), and P (z1 , . . . , zn ) is a complex polynomial in n variables that is symmetric (that is, P (zσ (1) , . . . , zσ (n) ) = P (z1 , . . . , zn ) for every point (z1 , . . . , zn ) ∈ Cn and every permutation σ of {1, . . . , n}).

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For each point y ∈ Y \ C, let g(y) = P (x1 , . . . , xn ), where x1 , . . . , xn are the distinct points in −1 (y). Prove that there is a unique holomorphic extension of g to a holomorphic function on Y . Exercises 5.20.2–5.20.9 below require facts stated in Sect. 4.6 (cf. [FarK]). 5.20.2 Riemann–Hurwitz formula. Let  : X → Y be a finite holomorphic branched covering mapping of compact Riemann surfaces X and Y , let b be the branching order of , and let n be the number of sheets. Prove that genus(X) =

b + n · (genus(Y ) − 1) + 1. 2

Hint. We may fix a nontrivial meromorphic 1-form β on Y (Theorem 2.10.1), and we may let α be the nontrivial meromorphic 1-form on X given by α ≡ ∗ β. Set D = div(α) and E = div(β). Given a point q ∈ Y and a point p ∈ −1 (q), apply the local representation of holomorphic maps to see that D(p) = m − 1 + mE(q), where m ≡ multp . Now sum over all of the critical values of  and nonzero points of E, and apply Corollary 4.6.7. 5.20.3 Prove that a compact Riemann surface X is hyperelliptic (see Exercise 4.6.6) if and only if X admits a 2-sheeted holomorphic branched covering mapping onto P1 . 5.20.4 Let X be a hyperelliptic Riemann surface of genus g, and let  : X → P1 be a 2-sheeted holomorphic branched covering map (see Exercises 4.6.6 and 5.20.3). (a) Prove that  has exactly 2g + 2 branch points, and that each branch point is of order 1. Hint. Apply the Riemann–Hurwitz formula (Exercise 5.20.2). (b) Prove that if g > 1, then every Weierstrass point in X is a hyperelliptic point (see Exercise 4.6.9) and the set of Weierstrass points is precisely the set of branch points of  (in particular, X has exactly 2g + 2 Weierstrass points, and combining this with Exercise 4.6.9, we see that a compact Riemann surface of genus g > 1 is hyperelliptic if and only if it has exactly 2g + 2 Weierstrass points). Hint. Show that the branch points are hyperelliptic (Weierstrass) points and apply Exercise 4.6.9. (c) Prove that if g > 1 and  : X → P1 is any 2-sheeted holomorphic branched covering map, then  =  ◦  for some automorphism  ∈ Aut (P1 ). Hint. Show that after composing with automorphisms of P1 (i.e., with Möbius transformations), we may assume that there are Weierstrass points p, q, r ∈ X with (p) = (p) = 0, (q) = (q) = 1, and (r) = (r) = ∞. Then consider the meromorphic function /. 5.20.5 Let X be a hyperelliptic Riemann surface of genus g > 1, and let  : X → P1 be a 2-sheeted holomorphic branched covering map (see Exercise 5.20.3).

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(a) Prove that there exists a unique nontrivial automorphism (i.e., an automorphism that is not the identity) J of X such that  ◦ J = . Prove also that J 2 = Id, that the fixed points of J are precisely the 2g + 2 branch points of  (i.e., the 2g + 2 Weierstrass points in X), and that J is independent of the choice of  (see Exercise 5.20.4). The associated automorphism J is called the hyperelliptic involution or the sheet interchange. (b) Let J be the hyperelliptic involution on X. Prove that if  ∈ Aut (X) \ {Id, J } (i.e.,  is not in the subgroup generated by J ), then  has at most four fixed points (in particular, J is the unique nontrivial automorphism of X with at least 2g + 2 fixed points). Hint. Applying Exercise 5.20.4, one sees that ◦ = ◦ for some Möbius transformation . Show that if m is the number of fixed points of , then the number of fixed points of  is at least m/2. (c) Prove that the hyperelliptic involution J commutes with every automorphism of X (i.e., J is in the center of Aut (X)). 5.20.6 Let X be a compact Riemann surface of genus g > 1. In this exercise, the theorem of Schwarz on finiteness of the automorphism group Aut (X) (with a bound on the order of the group) is obtained as an application of Exercises 4.6.6–4.6.10 and Exercises 5.20.2–5.20.5 (a more precise bound on the order of Aut (X) is obtained in Exercise 5.20.9 below). (a) Let S be the set of Weierstrass points in X. Prove that (S) = S for each  ∈ Aut (X). Conclude from this that the map  → S is a group homomorphism of Aut (X) into the permutation group of S. (b) Prove that if X is not hyperelliptic, then the homomorphism considered in (a) is injective, and the order of Aut (X) is at most [(g − 1)g(g + 1)]!. Hint. Apply Exercises 4.6.9 and 4.6.10. (c) Prove that if X is hyperelliptic with hyperelliptic involution J (see Exercise 5.20.5), then the kernel of the homomorphism considered in (a) is equal to {1, J }, and the order of Aut (X) is at most 2 · [(2g + 2)!]. 5.20.7 One way to obtain examples of finite holomorphic branched coverings is to form quotients of Riemann surfaces by finite groups of automorphisms. Throughout this exercise, X will denote a Riemann surface, and  will denote a (finite) subgroup of order n in Aut (X). The quotient space by  (see Definition 10.4.3), Y ≡ \X = X/[p∼q ⇐⇒ (p) = q for some  ∈ ], with quotient map ϒ : X → Y , will be assumed to be equipped with the quotient topology. For each point p ∈ X, p will denote the subgroup of  consisting of those elements that fix p (i.e., p is the isotropy subgroup at p). (a) Let p ∈ X and let m be the order of p . Prove that there exist a local holomorphic chart (D, , (0; 1)) and a primitive mth root of unity ω (i.e., ωm = 1 but ωk = 1 if 1 ≤ k < m) such that (D) = D for each  ∈ p , p = −1 (0), and the group { ◦  ◦ −1 |  ∈ p } is precisely

5.20

Finite Holomorphic Branched Coverings

301

the (rotation) group of automorphisms of (0; 1) given by z → ωk z for k = 0, 1, 2, . . . , m − 1. Prove also that p is a cyclic group. Hint. Fix a local holomorphic chart (U, , U  ) with p = −1 (0). Applying Exercise 1.5.2, show that for every sufficiently small  > 0,  ◦  ◦ −1 maps the disk (0; )  U  biholomorphically onto a relatively compact convex neighborhood of 0 in U  for each  ∈ p , and # −1 ∈p  ◦  ◦  ((0; )) is a convex neighborhood of 0 on which

the restriction of  ◦  ◦ −1 is an automorphism for each  ∈ p . Show that any convex domain is simply connected, apply the Riemann mapping theorem (Theorem 5.2.1) in order to obtain the desired local holomorphic chart, and finally, apply Theorem 5.8.2 in order to obtain the desired primitive mth root. (b) Prove that ϒ : X → Y is an open mapping and that Y is a Hausdorff space. (c) Prove that there is a unique holomorphic structure on Y (i.e., a unique structure of a Riemann surface) for which ϒ : X → Y is a finite holomorphic branched covering map with n sheets. Prove also that a point p ∈ X is a branch point of order m − 1 if and only if p is of order m > 1. Hint. Given p ∈ X, fix a local holomorphic chart (D,  = z, (0; 1)) as in (a) in which p = −1 (0) and the restrictions of the distinct elements of p are represented by the rotations z → ωk z for k = 0, . . . , m − 1. Show that D may be chosen so that D ∩ [( \ p ) · D] = ∅. Then  = ϒ(D),   = ζ, (0; 1)) show that this yields a local complex chart (D (ϒ(q)) = ((q))m for q ∈ D (in other words, the in Y determined by  restriction of ϒ to D is represented by the finite holomorphic branched covering map (0; 1) → (0; 1) given by z → ζ = zm ). Show that the corresponding atlas is a holomorphic atlas in Y and that ϒ is an nsheeted finite holomorphic branched covering map with the stated properties. 5.20.8 Let X be a compact Riemann surface of genus g > 1. Prove that X is hyperelliptic (see Exercises 4.6.6 and 5.20.3–5.20.5) if and only if X admits a holomorphic involution  (i.e.,  ∈ Aut(X) and  2 = Id) with exactly 2g + 2 fixed points. Hint. Given a nontrivial holomorphic involution  of X, Exercise 5.20.7 provides a 2-sheeted branched holomorphic covering map X → Y ≡ \X, where  = {1, } is the group generated by . Now apply Exercises 5.20.2 and 5.20.3 and Theorem 4.6.8. 5.20.9 According to Exercise 5.20.6, every compact Riemann surface of genus g > 1 has finite automorphism group. In this exercise, the following bound on the order of the automorphism group is obtained as an application of Exercises 5.20.2–5.20.8: Theorem (Hurwitz) Let X be a compact Riemann surface of genus g > 1, and let n be the order of the automorphism group  ≡ Aut (X). Then n ≤ 84(g − 1).

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For the proof, let ϒ : X → Y ≡ \X be the n-sheeted finite holomorphic branched covering provided by Exercise 5.20.7, let h ≡ genus(Y ), and let q1 , . . . , qk be the distinct critical values of ϒ . (a) Prove that for each j = 1, . . . , k, the groups p ≡ { ∈  | (p) = p} for p ∈ ϒ −1 (qj ) have the same order mj > 1 and that mj divides n. (b) Prove that 2g − 2 = n · (2h − 2) + n ·

 k  1 1− mj j =1

(c) (d) (e) (f) (g) (h)

(which is interpreted as 2g − 2 = n · (2h − 2) if ϒ has no critical values). Prove that if h ≥ 2, then n ≤ g − 1. Prove that if h = 1, then n ≤ 4(g − 1). Prove that if h = 0 and k ≥ 5, then n ≤ 4(g − 1). Prove that if h = 0 and k = 4, then n ≤ 12(g − 1). Prove that if h = 0 and k = 3, then n ≤ 84(g − 1). Prove that if h = 0, then k > 2.

5.21 Abel’s Theorem According to the Weierstrass theorem (Theorem 3.12.1), every holomorphic line bundle on an open Riemann surface is holomorphically trivial. Equivalently, every divisor on an open Riemann surface has a solution; that is, there exists a meromorphic function with arbitrary prescribed discrete zeros and poles. This is not the case for a compact Riemann surface. For example, any divisor that is linearly equivalent to the zero divisor must have degree 0. In fact, most divisors of degree zero on a compact Riemann surface of positive genus do not have a solution. The goal of this section is Abel’s theorem (Theorem 5.21.2), which provides a necessary and sufficient condition for a divisor of degree 0 on a compact Riemann surface to have a solution. Abel’s theorem will also allow us to form a holomorphic embedding into a higher-dimensional complex torus (see Sect. 5.22). The approach taken here is similar to the approach in, for example, [For]. Observe that for a Riemann surface X, we may identify the group of divisors with finite support in X with the group of integral 0-chains C0 (X, Z) (see Definition 10.6.3) under the isomorphism D → D(p) · p. p∈supp D

Thus we have the homomorphism ∂ : C1 (X, Z) → Div(X) (the boundary operator) and the group of integral 1-cycles Z1 (X, Z) = ker ∂. Moreover, we have the following fact, the proof of which is left to the reader (see Exercise 5.21.1). Lemma 5.21.1 For any divisor D on a compact Riemann surface X, we have deg D = 0 if and only if D ∈ im(∂ : C1 (X, Z) → Div(X)).

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303

Recall that every holomorphic line bundle on a Riemann surface is equal to the line bundle associated to some nontrivial divisor (Corollary 3.11.7 and Theorem 4.2.3). Moreover, a divisor D has a solution (that is, D is the divisor of some meromorphic function) if and only if the associated holomorphic line bundle is holomorphically trivial (Proposition 3.3.2). The goal of this section is the following: Theorem 5.21.2 (Abel’s theorem) Suppose D is a divisor of degree 0 on a compact Riemann surface X. Then D has a solution (i.e., the associated holomorphic line bundle E = [D] is holomorphically trivial) if and  only if there exists an integral 1-chain ξ ∈ C1 (X, Z) in X such that ∂ξ = D and ξ θ = 0 for every θ ∈ (X). We first recall the notion of a weak solution of a divisor D on a Riemann surface X (see Sect. 3.12). Let E = [D] and fix a meromorphic section s of E with div(s) = D (in particular, s is nontrivial). Then E is C ∞ trivial if and only if there is a C ∞ function ρ on X \ D −1 ((−∞, 0)) such that ρ is nonvanishing on X \ supp D and s/ρ extends to a nonvanishing C ∞ section v of E on X. Such a function ρ is called a weak solution of the divisor D. Equivalently, a C ∞ function ρ on X \ D −1 ((−∞, 0)) is a weak solution if ρ is nonvanishing on X \ supp D and for each point p ∈ supp D, there are a local holomorphic coordinate neighborhood (U, z) with z(p) = 0 and a nonvanishing C ∞ function ψ on U such that ρ = zD(p) · ψ on U \ {p}. Observe that the logarithmic differential dρ/ρ is locally integrable on X, since writing ρ = zD(p) · ψ on a coordinate neighborhood (U, z) with z(p) = 0 as above, we get, near p, dρ dz dψ = D(p) · + . ρ z ψ Observe also that for v = s/ρ as above, we have ¯ ¯ ∂ρ ∂v =− v ρ

on X \ supp D,

and for (U, z) as above, we have ¯ ¯ ∂ψ ∂v =− v ψ

on U.

Lemma 5.21.3 If U is a disk in C, z0 , z1 ∈ U , τ is a weak solution of the divisor F = z1 − z0 with τ ≡ 1 on C \ U , and f is a holomorphic function on a neighborhood of U , then  1 dτ ∧ df = f (z1 ) − f (z0 ). 2πi U τ Proof The function z → τ (z) · (z − z0 )/(z − z1 ) extends to a nonvanishing C ∞ function ψ on C. For  > 0 sufficiently small, Stokes’ theorem (Theorem 9.7.17)

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and the Cauchy integral formula and Cauchy’s theorem (Lemma 1.2.1) together give  dτ 1 ∧ df 2πi U \((z0 ;)∪(z1 ;)) τ   dτ dτ 1 1 f· f· = + 2πi ∂(z1 ;) τ 2πi ∂(z0 ;) τ   f (z) f (z) 1 1 dz − dz = 2πi ∂(z1 ;) z − z1 2πi ∂(z0 ;) z − z0  dψ 1 f· + 2πi ∂(z0 ;)∪∂(z1 ;) ψ  1 dψ . = f (z1 ) − f (z0 ) + df ∧ 2πi (z0 ;)∪(z1 ;) ψ Letting  → 0+ , we get the claim.



Lemma 5.21.4 Let W be a relatively compact neighborhood of the image C = γ ([0, 1]) of a path γ in a Riemann surface X. Then there exists a weak solution ρ of the divisor D = ∂γ = γ (1) − γ (0) such that ρ ≡ 1 on X \ W and   dρ 1 ∧ θ ∀θ ∈ (X). θ= 2πi X ρ γ Proof By Lemma 3.12.2, there exist a partition 0 = t0 < t1 < t2 < · · · < tm = 1, m local holomorphic charts {(Uj , j , (0; 1))}m j =1 in W , connected open sets {Vj }j =1 with γ ([tj −1 , tj ]) ⊂ Vj  Uj for each j = 1, . . . , m, and for each j = 1, . . . , m, a weak solution ρj of the divisor γ (tj ) − γ (tj −1 ) with ρj ≡ 1 on X \ Vj . The product ρ1 · · · ρm then extends to a weak solution ρ of D that satisfies ρ ≡ 1 on X \  j Vj ⊃ X \ W . Furthermore, if θ is a holomorphic 1-form on X, then for each j = 1, . . . , m, we have θUj = dfj for some function fj ∈ O(Uj ) (by Corollary 10.5.7). Therefore, by Lemma 5.21.3, 1 2πi



1 dρ ∧θ = ρ 2πi m

X

j =1

=

m j =1

 Vj

dρj ∧ dfj ρj 

(fj (γ (tj )) − fj (γ (tj −1 ))) =

θ. γ



Proof of Theorem 5.21.2 For the proof, we let D be a divisor of degree 0, which we may assume to be nontrivial. Suppose D has a solution, that is, there exists a nonconstant meromorphic function f with div(f ) = D. Thus f : X → P1 is an n-sheeted finite holomorphic branched covering map with finite set of critical values C, and we may choose an injective path γ : [0, 1] → P1 with γ (0) = ∞, γ (1) = 0, and γ ((0, 1)) ⊂ P1 \ (C ∪ {0, ∞}). According to Proposition 5.20.4, γ has exactly n

5.21

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305

distinct liftings γ1 , . . . , γn to paths in X. Moreover, for each point p ∈ f −1 (∞), exactly −D(p) of these liftings have initial point p, and for each point q ∈ f −1 (0), exactly D(q)

of these liftings have terminal point q. It follows that the integral 1chain ξ ≡ nj=1 γj satisfies ∂ξ = D. Furthermore, if θ is a holomorphic 1-form on X, then Proposition 5.20.4 provides a holomorphic 1-form β on P1 such that 

 θ= ξ

β. γ

However, P1 has no nontrivial holomorphic 1-forms (see, for example, Exercise 2.5.4 and Theorem 4.6.8), so ξ θ = 0. Conversely, suppose we have an integral 1-chain ξ in X such that ∂ξ = D and such that every holomorphic 1-form on X integrates to 0 along ξ . Applying Lemma 5.21.4, we get a weak solution ρ of D such that  X

 ¯ ∂ρ dρ ∧θ = ∧θ =0 ρ X ρ

∀θ ∈ (X)

(see Exercise 5.21.2). Fixing a meromorphic section s of E = [D] with div(s) = D, the section s/ρ extends to a nonvanishing C ∞ section v of E. Hence ¯ ¯ the C ∞ differential form η ≡ ∂v/v of type (0, 1), which agrees with −∂ρ/ρ on X \ supp D, satisfies  "η, τ #L2

0,1

(X) =

(−i)η ∧ τ¯ = 0

∀τ ∈ (X).

X

Applying the Hodge decomposition theorem for scalar-valued forms (Theo¯ on X. The nonvanishing C ∞ rem 4.9.1), we get a C ∞ function ϕ with η = ∂ϕ −ϕ section t ≡ e v of E then satisfies ¯ = −e−ϕ η · v + e−ϕ ∂v ¯ = 0, ∂t and therefore t is holomorphic. Thus E = [D] is holomorphically trivial and D has a solution.  Exercises for Sect. 5.21 5.21.1 Prove Lemma 5.21.1. 5.21.2 Verify the claim in the proof of Abel’s theorem that for any integral 1-chain ξ in a compact Riemann surface X such that ∂ξ = D and such that every holomorphic 1-form on X integrates to 0 along ξ , there exists a weak solution ρ of D such that  X

 ¯ dρ ∂ρ ∧θ = ∧θ =0 ρ X ρ

∀θ ∈ (X).

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5.22 The Abel–Jacobi Embedding The goal of this section is the fact that every compact Riemann surface of positive genus admits a holomorphic embedding into a higher-dimensional complex torus. We first consider the basic facts and definitions concerning such higher-dimensional complex tori, which are analogous to those corresponding to 1-dimensional complex tori considered in Example 2.1.6 (see also Example 5.1.10). The verifications of the basic facts are left to the reader. For n ∈ Z>0 , a lattice in Cn is a subgroup of the form  = Zλ1 + · · · + Zλ2n , where λ1 , . . . , λ2n ∈ Cn are elements that are linearly independent over R. We have an isomorphism of Cn onto the group of translations in Cn (which is, of course, a subgroup of the group of self-diffeomorphisms of Cn ) under which each element ζ ∈ Cn is identified with the translation given by z → z + ζ . Thus we may identify  with a subgroup of the group of translations in Cn . Clearly,  acts freely. Moreover,  is the image of Z2n under the real linear isomorphism α : R2n → Cn given by (t1 , . . . , t2n ) → t1 ξ1 + · · · + t2n ξ2n , and hence  is a discrete subset of Cn . We have the quotient map ϒ : Cn → T ≡ \Cn , and the quotient space (which is a quotient group) T is called a complex torus of (complex) dimension n. As in Example 5.1.10, Theorem 10.4.6 implies that T admits a unique structure of a 2n-dimensional compact smooth manifold for which ϒ is the C ∞ universal covering map. In fact, one may show that the local charts given by local inverses of ϒ form a holomorphic atlas, and hence that T is a (fundamental) example of a complex manifold of complex dimension n and that ϒ is a holomorphic covering map (see Exercise 5.22.1). We will not use this fact directly, but it justifies the use of the terminology appearing in Definition 5.22.1 below. As in the 1-dimensional case, we also have a commutative diagram of C ∞ maps R2n

α  n C

ϒ0

ϒ  β  T



T2n

where α is the real linear isomorphism given by (t1 , . . . , t2n ) → t1 λ1 + · · · + t2n λ2n , ϒ0 is the C ∞ covering map onto the real torus T2n = S1 × · · · × S1 given by (t1 , . . . , t2n ) → (e2πit1 , . . . , e2πit2n ), and β is the induced diffeomorphism. Definition 5.22.1 Let n ∈ Z>0 , let  be a lattice in Cn , let ϒ : Cn → T be the associated quotient map onto the complex torus T ≡ \Cn , and let  : X → T be a continuous mapping of a complex 1-manifold X into T. (a)  is called holomorphic if the mapping  ◦  is holomorphic for each local inverse  of ϒ.

5.22

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307

(b)  is called a holomorphic immersion if the mapping  ◦  is a holomorphic immersion for each local inverse  of ϒ. (c)  is called a holomorphic embedding if X is compact and  is an injective holomorphic immersion (i.e.,  is a proper injective holomorphic immersion). The main goal of this section is the following: Theorem 5.22.2 (Abel–Jacobi embedding theorem) Let X be a compact Riemann surface of genus g > 0, and let θ = (θ1 , . . . , θg ) be a (complex) basis for (X). Then we have the following: (a) The mapping



 [ξ ]H1 (X,Z) →

ξ





θ≡

θ1 , . . . , ξ

θg ξ

determines a well-defined injective group homomorphism H1 (X, Z) → Cg , and the image θ of this homomorphism is a lattice in Cg . In fact, the image of any integral basis for H1 (X, Z) is a real basis for Cg . (b) Let ϒθ : Cg → Tθ be the covering map associated to the complex torus Tθ ≡ θ \Cg . Given a point p0 ∈ X, the mapping      θ = ϒθ θ1 , . . . , θg , p → ϒθ γ

γ

γ

where γ is an arbitrary path in X from p0 to p, determines a well-defined holomorphic embedding Jθ,p0 : X → Tθ . Definition 5.22.3 Given a compact Riemann surface X of genus g > 0 and a basis θ = (θ1 , . . . , θg ) for (X), the lattice θ in Cg provided by the Abel–Jacobi embedding theorem is called the period lattice associated to θ . The corresponding complex torus θ \Cg is called the Jacobi variety of X and is denoted by Jac(X) or Jac(X, θ ). Given a point p0 ∈ X, the holomorphic embedding Jθ,p0 : X → Jac(X) provided by the Abel–Jacobi embedding theorem is called the Abel–Jacobi embedding associated to the basis θ and the point p0 . Remarks 1. As defined above, the period lattice, Jacobi variety, and Abel–Jacobi embedding depend on the choice of a basis for (X) (and the Abel–Jacobi embedding depends on the choice of an initial point p0 ∈ X). A more intrinsic, but equivalent, point of view also exists (see, for example, [GriH]). 2. For a higher-dimensional smooth complex projective variety (or even for a connected compact complex manifold) with nontrivial holomorphic 1-forms, one may form the analogous holomorphic mapping, which is called the Albanese map. The target complex torus is called the Albanese variety. This mapping is important in algebraic geometry (see, for example, [GriH]). In higher dimensions, the Albanese map need not be an embedding (for example, for Y a compact Riemann surface of positive genus, X ≡ Y × P1 is a smooth projective variety that admits

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nontrivial holomorphic 1-forms, but the corresponding Albanese map is constant on {y} × P1 for each point y ∈ Y ). For genus 1, the Abel–Jacobi embedding theorem immediately gives the following (cf. Exercises 5.9.3 and 5.16.1): Corollary 5.22.4 Every compact Riemann surface of genus 1 is biholomorphic to a complex torus (of dimension 1). Proof of Theorem 5.22.2 It follows from the definition of integral homology (Definition 10.7.9) that the mapping H1 (X, Z) → Cg in (a) is a well-defined group homomorphism. Moreover, since (θ, θ¯ ) ≡ (θ1 , . . . , θg , θ¯1 , . . . , θ¯g ) 1 (X, C) (Corollary 4.9.2), the (well-defined) is a basis for (X) ⊕ (X) ∼ = HdeR 2g linear map H1 (X, C) → C given by            [ξ ]H1 → θ, θ¯ = θ1 , . . . , θg , θ¯1 , . . . , θ¯g = θ, θ ξ

ξ

ξ

ξ

ξ

ξ

ξ

ξ

is injective, and by Theorem 10.7.18, the images under this map of the elements of any integral basis ξ1 , . . . , ξ2g for H1 (X, Z) ⊂ H1 (X, C) form a complex basis for C2g . Setting λj ≡ ξj θ ∈ Cg for each j = 1, . . . , 2g, it follows that λ1 , . . . , λ2g

2g is a real basis for Cg . For if aj ∈ R for j = 1, . . . , 2g and j =1 aj λj = 0, then we have      2g 2g 2g aj θ, θ = aj λj , aj λj = (0, 0), j =1

ξj

ξj

j =1

j =1

and therefore a1 = · · · = a2g = 0. Part (a) now follows. For the proof of (b), let us fix a point p0 ∈ X. We will let p and q denote points in X, and we will let α and β denote two paths in X with α(0) = β(0) = p0 , α(1) = p, and β(1) = q.      If p = q, then α θ − β θ = α∗β − θ ∈ θ , and hence ϒθ ( α θ ) = ϒθ ( β θ ). Thus the associated map Jθ,p0 : X → Tθ is well defined. Conversely, if p and q satisfy Jθ,p0 (p) = Jθ,p0 (q), then    θ − θ = λ = θ ∈ θ α

β

ξ

for some integral 1-cycle ξ ∈ Z1 (X, Z). Hence the 1-chain ζ ≡ α − β − ξ ∈ C1 (X, Z) has boundary divisor D = ∂ζ = p − q and satisfies  η = 0 ∀η ∈ (X). ζ

5.22

The Abel–Jacobi Embedding

309

Abel’s theorem (Theorem 5.21.2) now implies that p = q. For if D were nontrivial (i.e., if p and q were distinct points), then Abel’s theorem would provide a nontrivial meromorphic function f : X → P1 with div(f ) = D, and by Proposition 2.5.7 (or Proposition 5.20.1), f would be a biholomorphism. Thus Jθ,p0 is injective. We may fix a holomorphic mapping F = (f1 , . . . , fg ) : U → Cg on a connected  neighborhood U of p in X such that dF = (df1 , . . . , dfg ) = θ on U , F (p) = α θ , and F (U ) is contained in a connected neighborhood V of F (p) that ϒθ maps homeomorphically onto its image ϒθ (V ). Therefore, if W is a connected neighborhood of Jθ,p0 (p) = ϒθ (F (p)) in ϒθ (V ),  : W → (W ) ⊂ Cg is a local inverse of ϒθ , and Q ≡ ϒθ−1 (W ) ∩ V , then the homeomorphism  ◦ ϒθ Q : Q → (W ) is given by z → z + λ for some (constant) element λ ∈ θ . Thus, on the neighborhood F −1 (Q) of p, we have G ≡  ◦ Jθ,p0 =  ◦ ϒθ ◦ F = F + λ. Thus G is holomorphic and (dG)p = ((θ1 )p , . . . , (θg )p ). In particular, since the nontrivial holomorphic 1-forms θ1 , . . . , θg have no common zeros (by Theorem 4.6.8), we have (dG)p = 0. Thus the mapping Jθ,p0 : X → Tθ is a holomorphic embedding.  Exercises for Sect. 5.22 5.22.1 Verify that an n-dimensional complex torus X = \Cn admits the structure of an n-dimensional complex manifold (see Exercise 2.2.6) for which the quotient mapping Cn → X is holomorphic with local holomorphic inverses.

Chapter 6

Holomorphic Structures on Topological Surfaces

In this chapter, we address the natural problem of determining conditions for a topological surface to admit a holomorphic structure. One necessary condition is, of course, that the surface be orientable. According to Radó’s theorem (Theorem 2.11.1), another necessary condition is that the surface be second countable. It turns out that these two conditions are also sufficient. Sections 6.1–6.6 consist of a proof of the theorem of Korn and Lichtenstein, according to which every almost complex structure on a smooth surface is integrable; that is, it is induced by a holomorphic structure. It then follows that every orientable second countable smooth surface admits a holomorphic structure. The proof of integrability appearing in this chapter is based on the proofs of the higher-dimensional analogue appearing in [Hö] and [De3]. Sections 6.7–6.11 consist of a proof that every second countable topological surface admits a smooth structure. The first part of the chapter then implies that any second countable orientable topological surface admits a holomorphic structure. One of the main tools in the proof of the existence of smooth structures is Schönflies’ theorem, and a proof (due to Kneser and Radó) of this fact appears in Sects. 6.7–6.9. The proof of integrability and the proof of the existence of smooth structures may be read independently.

6.1 Almost Complex Structures on Smooth Surfaces The main goal of Sects. 6.1–6.6 is the existence of holomorphic structures on second countable oriented smooth surfaces. One obtains a holomorphic structure by first producing an almost complex structure. Definition 6.1.1 Let X be a 2-dimensional smooth manifold. (a) An almost complex structure on X is a choice of subsets   (Tp X)1,0 and (T X)0,1 = (Tp X)0,1 (T X)1,0 = p∈X

p∈X

T. Napier, M. Ramachandran, An Introduction to Riemann Surfaces, Cornerstones, DOI 10.1007/978-0-8176-4693-6_6, © Springer Science+Business Media, LLC 2011

311

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6 Holomorphic Structures on Topological Surfaces

of (T X)C such that (i) For each point p ∈ X, (Tp X)1,0 and (Tp X)0,1 are complex subspaces of (Tp X)C with (Tp X)C = (Tp X)1,0 ⊕ (Tp X)0,1 and (Tp X)1,0 = (Tp X)0,1 ; and (ii) For each point in X, there is a nonvanishing C ∞ vector field Z on a neighborhood U such that (Tp X)1,0 = C · Zp (hence (Tp X)0,1 = C · Zp ) for each point p ∈ U . We write (T X)C = (T X)1,0 ⊕(T X)0,1 . The manifold X together with an almost complex structure is called an almost complex manifold of dimension 2 (or an almost complex surface if X is connected). (b) For any almost complex structure decomposition (T X)C = (T X)1,0 ⊕ (T X)0,1 , we have the associated decomposition (T ∗ X)C = (T ∗ X)1,0 ⊕ (T ∗ X)0,1 of the cotangent bundle with summands   (Tp∗ X)1,0 and (T ∗ X)0,1 = (Tp∗ X)0,1 , (T ∗ X)1,0 = p∈X

p∈X

where for each point p ∈ X, (Tp∗ X)1,0 ≡ {α ∈ (Tp∗ X)C | α(v) = 0 ∀v ∈ (Tp X)0,1 } ∼ = [(Tp X)1,0 ]∗ and (Tp∗ X)0,1 ≡ (Tp∗ X)1,0 = {α ∈ (Tp∗ X)C | α(v) = 0 ∀v ∈ (Tp X)1,0 } ∼ = [(Tp X)0,1 ]∗ (in particular, (Tp∗ X)C = (Tp∗ X)1,0 ⊕ (Tp∗ X)0,1 ). (c) Let Z be a nonvanishing C ∞ vector field with values in (T X)1,0 on an open set U ⊂ X, and let θ be the 1-form on U determined by θ (Z) ≡ 1 and θ (Z) ≡ 0. For any vector field v, we call the functions a ≡ θ (v) and b ≡ θ¯ (v) (i.e., v = aZ + bZ) the coefficients of v with respect to Z and Z. For any 1-form α, we call the functions A ≡ α(Z) and B ≡ α(Z) (i.e., α = Aθ + B θ¯ ) the coefficients of α with respect to θ and θ¯ . (d) An almost complex structure (T X)C = (T X)1,0 ⊕ (T X)0,1 on X is integrable if this decomposition is equal to the decomposition of (T X)C into (1, 0) and (0, 1) parts associated to some holomorphic structure on X; that is, the almost complex structure is induced by the holomorphic structure. Remark For an almost complex structure (T X)C = (T X)1,0 ⊕ (T X)0,1 , we use the notation and terminology analogous to that used in the integrable case. For example, an element of (T X)1,0 is of type (1, 0). Recall that by definition, a vector field or differential form on a smooth manifold is continuous (C k with k ∈ Z≥0 ∪ {∞}, measurable, in Ldloc with d ∈ [1, ∞]) if its coefficients in local C ∞ charts are continuous (respectively, C k , measurable, in Ldloc ). We have the following analogous characterization on an almost complex surface:

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313

Proposition 6.1.2 Let (T X)C = (T X)1,0 ⊕(T X)0,1 be an almost complex structure on a smooth surface X, let k ∈ Z≥0 ∪ {∞}, and let d ∈ [1, ∞]. (a) If v is a vector field on a set S ⊂ X, then the following are equivalent: (i) The vector field v is continuous (C k , measurable, in Ldloc ). (ii) The coefficients of v with respect to Z and Z are continuous (respectively, C k , measurable, in Ldloc ) for every nonvanishing C ∞ local vector field Z of type (1, 0). (iii) For every point p ∈ S, the coefficients of v with respect to Z and Z are continuous (respectively, C k , measurable, in Ldloc ) for some nonvanishing C ∞ vector field Z of type (1, 0) on a neighborhood of p. (iv) The (1, 0) and (0, 1) parts of v are continuous (respectively, C k , measurable, in Ldloc ). (b) If α is a 1-form on a set S ⊂ X, then the following are equivalent: (i) The 1-form α is continuous (C k , measurable, in Ldloc ). (ii) For every nonvanishing C ∞ local vector field Z of type (1, 0), the coefficients of α with respect to the corresponding dual 1-forms θ and θ¯ (determined by θ (Z) ≡ 1 and θ (Z) ≡ 0) are continuous (respectively, C k , measurable, in Ldloc ). (iii) For every point p ∈ S, there exists a nonvanishing C ∞ local vector field Z of type (1, 0) on a neighborhood of p such that the coefficients of α with respect to the corresponding dual 1-forms θ and θ¯ are continuous (respectively, C k , measurable, in Ldloc ). (iv) The (1, 0) and (0, 1) parts of α are continuous (respectively, C k , measurable, in Ldloc ). (c) A nonvanishing local vector field Z of type (1, 0) is continuous (C k , measurable) if and only if its dual 1-form θ (of type (1, 0)) is continuous (respectively, C k , measurable). In particular, for each point in p ∈ X, there exists a nonvanishing C ∞ differential form of type (1, 0) on a neighborhood of p. Proof Let Z be a nonvanishing C ∞ vector field of type (1, 0) on an open set U ⊂ X, and let θ and θ¯ be the corresponding dual 1-forms. Suppose also that we have a local C ∞ chart (U,  = (x, y), (U )). Then the corresponding coefficients a = dx(Z) and b = dy(Z) of Z are of class C ∞ , and we have Z =a Hence the matrix C ≡

∂ ∂ +b ∂x ∂y

 a b a¯ b¯

and Z = a¯

∂ ∂ + b¯ . ∂x ∂y

is nonsingular at each point and

C −1 =

1 a b¯ − ab ¯



b¯ −a¯

−b a



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has C ∞ entries. Setting A ≡ θ (∂/∂x) and B ≡ θ (∂/∂y), we get θ = A dx + B dy and       θ (Z) A 1 =C . = B 0 θ (Z) Hence the column vector

    1 A −1 =C · 0 B

has C ∞ entries, and therefore θ is a C ∞ form of type (1, 0). Armed with this fact, it is now easy to verify (a)–(c) (see Exercise 6.1.1).  Proposition 6.1.3 Let X be a smooth surface. (a) If X admits an almost complex structure (T X)C = (T X)1,0 ⊕ (T X)0,1 , then X is orientable. In fact, there is a unique orientation for which iα ∧ α¯ > 0 for each α ∈ (T ∗ X)1,0 \ {0}. (b) If X is oriented and second countable, then X admits an almost complex structure that induces (as in (a)) the given orientation. Proof For the proof of (a), suppose (T X)C = (T X)1,0 ⊕ (T X)0,1 is an almost complex structure. If (U, (x, y)) is a connected local C ∞ coordinate neighborhood, then either iα ∧ α¯ >0 (dx ∧ dy)p

∀p ∈ U, α ∈ (Tp∗ X)1,0 \ {0},

iα ∧ α¯ <0 (dx ∧ dy)p

∀p ∈ U, α ∈ (Tp∗ X)1,0 \ {0}.

or

For if θ is a nonvanishing C ∞ differential form of type (1, 0) on an open set V ⊂ U , then i(θ ∧ θ¯ )p iα ∧ α¯ = |α/θp |2 (dx ∧ dy)p (dx ∧ dy)p

∀p ∈ V , α ∈ (Tp∗ X)1,0 \ {0},

and the real-valued C ∞ function i(θ ∧ θ¯ )/(dx ∧ dy) is nonvanishing. So if, for some choice of θ and V , i(θ ∧ θ¯ )/(dx ∧ dy) > 0 at some point in V , then the set of points p ∈ U at which the inequality i(α ∧ α)/(dx ¯ ∧ dy)p > 0 holds for every α ∈ (Tp∗ X)1,0 \ {0} is nonempty with empty boundary in U , and is therefore equal to U . Similarly, if (x, y) and (x , y ) are two choices of local C ∞ coordinates for which the above quotients are positive, then we have (dx ∧ dy)/(dx ∧ dy ) > 0, and hence they have compatible orientations. The claim (a) now follows. As motivation for the proof of (b), we first examine how the orientation in C = R2 relates to the standard (integrable) almost complex structure. For the standard coor-

6.1 Almost Complex Structures on Smooth Surfaces

315

dinate z = x + iy ↔ (x, y), we have, at each point p,        ∂ 1 ∂ ∂ . =C· −i (Tp R2 )1,0 = (Tp C)1,0 = C · ∂z p 2 ∂x p ∂y p Under the identification of Tp R2 with {p} × R2 , we see that (∂/∂y)p is obtained by multiplication of (∂/∂x)p by i, that is, by rotation of the vector counterclockwise by 90°. On a general surface, one may define a counterclockwise 90° rotation if one has a Riemannian metric (to determine the rotational angle measure) and an orientation (to determine the counterclockwise sense). Suppose now that X is a second countable oriented smooth surface. By Proposition 9.11.2, X admits a Riemannian metric g. For each point p ∈ X, we may let (Tp X)1,0 be the collection of all vectors of the form w = 12 (u − iv), where u, v ∈ Tp X have the following properties: (i) We have g(u, v) = 0 (i.e., u ⊥ v) and |u|g = |v|g ; and (ii) We have α(u, v) ≥ 0 for every positive α ∈ 2 Tp∗ X. Setting (Tp X)0,1 ≡ (Tp X)1,0 for each point p ∈ X, we now show that the spaces   (T X)1,0 ≡ (Tp X)1,0 and (T X)0,1 ≡ (Tp X)0,1 p∈X

p∈X

determine an almost complex structure (T X)C = (T X)1,0 ⊕ (T X)0,1 on X. For this, let us fix a positively oriented local C ∞ coordinate neighborhood (U, (x, y)) in X. Then we have C ∞ real vector fields u≡

∂/∂x |∂/∂x|g

and

v≡

(∂/∂y) − g(∂/∂y, u)u |(∂/∂y) − g(∂/∂y, u)u|g

that satisfy g(u, v) ≡ 0 and |u|g = |v|g ≡ 1 on U . Moreover, for every point p ∈ U and every positive α ∈ 2 Tp∗ X, we have α(up , vp ) = =

α · (dx ∧ dy)(up , vp ) (dx ∧ dy)p 1 1 α · · > 0. (dx ∧ dy)p |(∂/∂x)p |g |(∂/∂y)p − g((∂/∂y)p , up )up |g

Thus the values of the nonvanishing C ∞ vector field Z ≡ (1/2)(u − iv) on U lie in (T X)1,0 . For each point p ∈ U and each complex number ζ = a + ib with a, b ∈ R, we have 1 ζ · Zp = ((aup + bvp ) − i(−bup + avp )), 2 g(aup + bvp , −bup + avp ) = −ab + ba = 0, |aup + bvp |2g = |ζ |2 = |−bup + avp |2g , and for each positive α ∈ 2 (Tp∗ X), α(aup + bvp , −bup + avp ) = |ζ |2 α(up , vp ) ≥ 0.

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6 Holomorphic Structures on Topological Surfaces

Thus C · Zp ⊂ (Tp X)1,0 . Conversely, given an element w = (1/2)(r − is) ∈ (Tp X)1,0 , we have r = aup + bvp with a, b ∈ R, s ⊥ r, and |s|2g = |r|2g = a 2 + b2 . Thus s = ±(−bup + avp ). On the other hand, we also have (dx ∧ dy)(r, s) ≥ 0 and (dx ∧ dy)(r, −bup + avp ) = (a 2 + b2 )(dx ∧ dy)(up , vp ) ≥ 0 (with equality if and only if r = s = 0). Thus s = −bup + avp , and hence for ζ = a + ib, we have w = ζ · Zp . Therefore, (Tp1,0 X)1,0 = C · Zp . It also follows that Zp and Zp are linearly independent over C, since (dx ∧ dy)(up , −vp ) < 0. Thus we have produced an almost complex structure (T X)C = (T X)1,0 ⊕ (T X)0,1 . The verification that this almost complex structure induces the given orientation is left to the reader (see Exercise 6.1.2).  A holomorphic structure induces an almost complex structure. The main goal of Sects. 6.1–6.6 is the following: Theorem 6.1.4 (Integrability theorem) Every almost complex structure on a smooth surface is integrable. In fact, every almost complex structure is induced by a unique (1-dimensional) holomorphic structure. Radó’s theorem (Theorem 2.11.1) and the above integrability theorem together give the following: Corollary 6.1.5 Any smooth surface that admits an almost complex structure must be second countable. Proposition 6.1.3 and the integrability theorem together give the following: Corollary 6.1.6 Every oriented second countable smooth surface admits a holomorphic structure that induces the given orientation. Definition 6.1.7 Let (T X)C = (T X)1,0 ⊕ (T X)0,1 be an almost complex structure on a 2-dimensional smooth manifold X, let α be a C 1 differential form of degree r on an open set  ⊂ X, and let P p,q : 1 (T ∗ X)C → p,q T ∗ X be the associated projection for each pair (p, q) ∈ {(1, 0), (0, 1)}. Then we define ⎧ ⎧ 1,0 0,1 ⎪ ⎪ ⎨P (dα) if r = 0, ⎨P (dα) if r = 0, 0,1 ¯ ≡ d(P 1,0 α) if r = 1, ∂α ≡ d(P α) if r = 1, and ∂α ⎪ ⎪ ⎩ ⎩ 0 if r ≥ 2, 0 if r ≥ 2. Remarks Proposition 6.1.2 allows us to make the above definition because according to this proposition, the (r, s) part of a C k differential form is of class C k . We also recall that we use the notation and terminology analogous to that used in the inte¯ ¯ = 0. According to grable case. For example, a differential form α is ∂-closed if ∂α Proposition 6.1.8 below, the operators ∂ and ∂¯ retain at least some of the properties that these operators have in the integrable case (cf. Proposition 2.5.5).

6.1 Almost Complex Structures on Smooth Surfaces

317

Proposition 6.1.8 Let (T X)C = (T X)1,0 ⊕(T X)0,1 be an almost complex structure on a smooth surface X. Then (a) The operators d, ∂, and ∂¯ satisfy d = ∂ + ∂¯ on C 1 forms and d 2 = ∂ 2 = ∂¯ 2 = ¯ = 0 on C 2 forms. ∂ ∂¯ + ∂∂ ¯ (b) If α is a C 1 differential form of type (p, q), then ∂α is of type (p + 1, q) and ∂α ¯ = 0 if q ≥ 1. is of type (p, q + 1). In particular, ∂α = 0 if p ≥ 1 and ∂α (c) If α and β are C 1 differential forms, then ∂(α ∧ β) = (∂α) ∧ β + (−1)deg α α ∧ ∂β and ¯ ∧ β) = (∂α) ¯ ∧ β + (−1)deg α α ∧ ∂β. ¯ ∂(α The proof is left to the reader (see Exercise 6.1.3). Note that we have not included an analogue of the property (d) of Proposition 2.5.5 (as well as (c), (f), and (g)). The formation of such an analogue is at the heart of the proof of the integrability theorem. As a first step toward the proof of the integrability theorem, we show that the integrability condition is completely local. For this, we will need the following: Lemma 6.1.9 Let X be an almost complex surface, let f = u + iv : X → C be a C 1 function with u = Re f and v = Im f (which we may also view as a C 1 map¯ )p = 0. Then the comping f = (u, v) : X → R2 ), and let p ∈ X. Assume that (∂f plex linear extension (f∗ )p : (Tp X)C → (Tf (p) C)C of the real linear tangent map (f∗ )p : Tp X → Tf (p) C = Tf (p) R2 preserves the almost complex structures; that is, (f∗ )p (Tp X)r,s ⊂ (Tf (p) C)r,s

∀(r, s) ∈ {(1, 0), (0, 1)}.

Moreover, if, in addition, (df )p = 0 (i.e., (∂f )p = 0), then the real linear maps ((du)p , (dv)p ) : Tp X → R2

and

(f∗ )p : Tp X → Tp R2 = Tp C,

and the complex linear maps (f∗ )p : (Tp X)C → (Tp C)C , (df )p (Tp X)1,0 = dz ◦ (f∗ )p (Tp X)1,0 : (Tp X)1,0 → C, (d f¯)p (Tp X)0,1 = d z¯ ◦ (f∗ )p (Tp X)0,1 : (Tp X)0,1 → C, are bijective. Proof To avoid confusion, when thinking of the complex-valued function f as a mapping of X into the Riemann surface C, let us denote this mapping by F . Letting z = x + iy denote the complex coordinate in C = R2 with Re z = x and Im z = y, we then have f = z ◦ F and f¯ = z¯ ◦ F . Note also that for each point a ∈ C, (dz)a

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6 Holomorphic Structures on Topological Surfaces

maps (Ta C)1,0 isomorphically onto C and vanishes on (Ta C)0,1 , while (d z¯ )a maps (Ta C)0,1 isomorphically onto C and vanishes on (Ta C)1,0 . Given a tangent vector v ∈ (Tp X)1,0 , we have ¯ (v) 0 = ∂f ¯ = df (v) ¯ = dz(F∗ v) ¯ and 0 = df (v) ¯ = d f¯(v) = d z¯ (F∗ v). It follows that (F∗ )p (Tp X)r,s ⊂ (TF (p) C)r,s for (r, s) ∈ {(1, 0), (0, 1)}. Furthermore, if (df )p = 0, then we may choose a tangent vector w ∈ (Tp X)C with 0 = df (w) = ∂f (w) = df (P 1,0 w). Thus (df )p : (Tp X)1,0 → C is a nontrivial linear mapping of 1-dimensional complex vector spaces, and is therefore an isomorphism. It follows that the complex linear maps

−1 ◦ (df )p (Tp X)1,0 : (Tp X)1,0 → (TF (p) C)1,0 (F∗ )p (Tp X)1,0 = dz(TF (p) C)1,0 and

−1 ◦ (d f¯)p (Tp X)0,1 : (Tp X)0,1 → (TF (p) C)0,1 (F∗ )p (Tp X)0,1 = d z¯ (TF (p) C)0,1 are bijective. In particular, the complexified tangent map (F∗ )p : (Tp X)C → (TF (p) C)C is bijective, and hence, by Proposition 8.1.3, the corresponding real tangent map (F∗ )p : Tp X → TF (p) C must also be bijective, as must be the map ((du)p , (dv)p ) = ((dx)p , (dy)p ) ◦ F∗ : Tp X → R2 .



We also have the following uniqueness property: Lemma 6.1.10 Let X be a Riemann surface with induced almost complex structure (T X)C = (T X)1,0 ⊕ (T X)0,1 and corresponding (0, 1) part of the d operator ¯ denoted by ∂. X)1,0 ⊕ (T X)0,1 be a second almost complex structure on X (a) Let (T X)C = (T ¯ If p ∈ X ∂. with corresponding (0, 1) part of the d operator denoted by 

and there exists a local holomorphic chart (U, , U ) in X with p ∈ U and ¯ p = 0, then the two almost complex structures agree at p; that is, ∂) ( 1,0 = (T X)1,0 (T p X) p

and

0,1 = (T X)0,1 . (T p X) p

(b) Any holomorphic structure on the smooth surface underlying X that induces the same given almost complex structure is equal to the given holomorphic structure on X. Proof In the situation of part (a), by Lemma 6.1.9, (∗ )p maps (Tp X)C isomorphi1,0 and (T X)1,0 isomorphically onto (T(p) C)C , and both of the subspaces (T p X) p 1,0 cally onto T(p) C ⊂ (T(p) C)C . Part (a) now follows.

6.1 Almost Complex Structures on Smooth Surfaces

319

For proof of (b), we observe that on a smooth surface, any two 1-dimensional holomorphic structures with the same underlying smooth structures and induced (almost) complex structures have the same ∂¯ operator. Hence, if (U, z) is a local holomorphic coordinate neighborhood with respect to one of the holomorphic struc¯ = 0, and hence z is a local holomorphic coordinate in the other holotures, then ∂z morphic structure. Thus, holomorphic atlases for the two holomorphic structures are holomorphically equivalent; i.e., the holomorphic structures are the same.  Lemma 6.1.11 Let X be an almost complex surface. If for each point p ∈ X, there ¯ differential form of type (1, 0) on a neighborhood exists a nonvanishing C ∞ ∂-closed of p, then the almost complex structure is integrable. Proof By hypothesis, there exists a covering {Uj } of X by open sets such that for each j , there is a nonvanishing C ∞ form αj of type (1, 0) on Uj with ¯ j = 0. By the Poincaré lemma (Lemma 9.5.7), we may also assume that dαj = ∂α for each j , there exists a C ∞ function j on Uj with dj = αj , and therefore ¯ j = 0. According to Lemma 6.1.9, for uj ≡ Re j and vj ≡ Im j , duj ∧ dvj ∂ is nonvanishing, and hence by the C ∞ inverse function theorem (Theorem 9.9.1 and Theorem 9.9.2), we may also assume that j is a C ∞ diffeomorphism of Uj onto an open subset Uj of the plane. For each pair of indices j, k, applying Lemma 6.1.9, we see that for the C ∞ coordinate transformation fj k ≡ j ◦ −1 k on k (Uj ∩ Uk ), we have ¯ j k = (dj ) ◦ (−1 )∗ ◦ P 0,1 = (∂j ) ◦ ((k )∗ )−1 ◦ P 0,1 ∂f k = (∂j ) ◦ P 0,1 ◦ ((k )∗ )−1 ≡ 0, where P r,s denotes the projection of tangent vectors onto their (r, s) parts for (r, s) ∈ {(1, 0), (0, 1)}; that is, fj k is holomorphic. Thus the atlas {(Uj , j , Uj )} determines a holomorphic structure on X, and by Lemma 6.1.10, the induced complex structure is equal to the given almost complex structure.  Remark An almost complex structure on a higher-dimensional C ∞ manifold is induced by a holomorphic structure if and only if for every C ∞ differential form β of type (0, 1), the 2-form dβ has no (2, 0) part (see, for example, [Hö]). Of course, on a Riemann surface, this condition holds automatically. Exercises for Sect. 6.1 6.1.1 Complete the proof of Proposition 6.1.2. 6.1.2 Verify that the almost complex structure constructed in the proof of part (b) of Proposition 6.1.3 induces the given orientation. 6.1.3 Prove Proposition 6.1.8. 6.1.4 Let g1 and g2 be two Riemannian metrics on an oriented smooth surface X. Prove that g1 and g2 induce the same almost complex structure (as in the proof of Proposition 6.1.3) if and only if there exists a C ∞ real-valued function ϕ on X such that g2 = eϕ g1 .

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6 Holomorphic Structures on Topological Surfaces

6.2 Construction of a Special Local Coordinate According to Lemma 6.1.11, to obtain integrability in an almost complex surface, it ¯ suffices to obtain a local ∂-closed C ∞ differential form of type (1, 0) that is nonzero ¯ provides a technique at a given point p. On a Riemann surface, the L2 ∂-method for producing nonconstant holomorphic (or meromorphic) functions and forms (and ¯ on a Riemann sursections of holomorphic line bundles). Most of the L2 ∂-method face as considered in Chap. 2 works without change on an almost complex surface, but there are three difficulties. The first is that on a Riemann surface, the technique relied on the (obvious) existence of nonconstant local holomorphic functions and forms. But on an almost complex surface, it is the existence of such local functions and forms that is the problem to be solved. The second difficulty is that the construction of a positive-curvature weight function (or Hermitian metric) in Sect. 2.14 used a local holomorphic coordinate. Again, in the present context, the existence of a local holomorphic coordinate is the problem to be solved. The third difficulty is that the proof of regularity for ∂¯ on a Riemann surface (Theorem 1.2.8 and Theorem 2.7.4) relied on normal families of holomorphic functions, which again, are not a priori available on an almost complex surface. The proof of regularity provided in this chapter, which is similar to the proofs in [De3] and [Hö], requires elements of Sobolev space theory (see Chap. 11). For the first two difficulties, following Demailly [De3], the idea is to work with a local C ∞ complex coordinate z such that ¯ does not vanish identically, ∂z ¯ does vanish to high order at the point p. although ∂z The existence of such a coordinate is given by the following lemma, which is the main goal of this section: Lemma 6.2.1 Let X be an almost complex surface, let p ∈ X, and let m ∈ Z≥0 . Then there exists a C ∞ function f on a neighborhood of p in X such that ¯ has vanishing derivatives of order ≤ m at p (see Defi(∂f )p = (df )p = 0 and ∂f nition 9.5.8). Proof Since the claim is local, we may assume that X is an open subset of C to which an almost complex structure (T X)C = (T X)1,0 ⊕ (T X)0,1 has been assigned. This almost complex structure may be different from that induced by C. In particular, ∂ and ∂¯ will be the operators associated to the almost complex structure on X, not the standard holomorphic structure on C. We may also assume that p = 0 ∈ C. By replacing z with z¯ and X with a small neighborhood of p if necessary, we may assume without loss of generality that the coordinate function z in C satisfies ∂z = 0 on X. Thus we may form the C ∞ function μ≡

¯ (∂z) (∂z)

=

¯ (∂z) ¯ (∂ z¯ )

on X.

Note that the above denotes a quotient of (0, 1)-forms, not a partial derivative. In ¯ = μ · ∂z = μ · ∂¯ z¯ . other words, μ is the C ∞ function defined by ∂z

6.2 Construction of a Special Local Coordinate

321

We proceed by induction on m. For the case m = 0, we simply set f (z) = z − μ(0)¯z. We then get ¯ 0 − μ(0)(∂¯ z¯ )0 = 0, ¯ )0 = (∂z) (∂f and hence (∂f )0 = (df )0 = (dz)0 − μ(0)(d z¯ )0 = 0 (since dz and d z¯ are pointwise linearly independent). Thus the claim is proved for m = 0. Assume now that m > 0 and that the desired function exists for lower orders. In other words, there exists a C ∞ function h on a neighborhood of 0 such that ¯ has vanishing derivatives of order ≤ m − 1 at 0 and (dh)0 = 0. In particular, ∂h Lemma 6.1.9 and the C ∞ inverse function theorem imply that (Re h, Im h) gives local C ∞ coordinates in a neighborhood of 0. Therefore, after replacing the coordinate z with the local coordinate h − h(0), we may assume that the C ∞ function μ has vanishing derivatives of order ≤ m − 1 at 0 (see Proposition 9.5.9). Equivalently, the degree-m Taylor polynomial P for μ centered at 0 is either homogeneous of degree m or trivial. Thus, expressing P as a polynomial in z and z¯ (which we may do, since x = Re z = 2−1 (z + z¯ ) and y = Im z = (2i)−1 (z − z¯ )), we get P (z) = A0 zm + A1 zm−1 z¯ + · · · + Am z¯ m , for some complex constants A0 , . . . , Am . Setting 1 1 Q(z) ≡ A0 zm z¯ + A1 zm−1 z¯ 2 + · · · + Am z¯ m+1 , 2 m+1 we get ∂Q ∂Q ∂Q = P and (0) = (0) = 0. ∂ z¯ ∂z ∂ z¯ We will show that the function f (z) ≡ z − Q(z) has the required properties. We have   ¯ = df ◦ P 0,1 = dz − ∂Q dz − ∂Q d z¯ ◦ P 0,1 ∂f ∂z ∂ z¯ ∂Q ¯ − ¯ − P · ∂¯ z¯ = ∂z · ∂z ∂z   ∂Q μ · ∂z. = μ−P − ∂z Since μ − P has vanishing derivatives of order ≤ m at 0, ∂Q/∂z = 0 at 0, and μ has ¯ has vanishing derivatives vanishing derivatives of order ≤ m − 1 at 0, we see that ∂f of order ≤ m at 0. Similarly, we have (df )0 = (dz)0 −

∂Q ∂Q (0) · (dz)0 − (0) · (d z¯ )0 = (dz)0 = 0. ∂z ∂ z¯

Thus f has the required properties and the lemma follows.



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6 Holomorphic Structures on Topological Surfaces

6.3 Regularity of Solutions on an Almost Complex Surface For regularity, we will use Theorem 11.0.1, which is a general regularity theorem for first-order differential operators satisfying a certain estimate. In order to obtain the required (1, 0)-form in Lemma 6.1.11 on a neighborhood of a given point in an almost complex surface, it suffices to work locally. Thus it suffices to consider a nonempty relatively compact domain X in R2 (obtained by replacing the given surface with a connected local C ∞ coordinate neighborhood) on which we have a given almost complex structure T XC = (T X)1,0 ⊕(T X)0,1 and in which p = (0, 0). Letting (x, y) denote the real coordinates in R2 , we get a second almost complex structure under the identification R2 = C given by (x, y) ↔ z = x + iy. To avoid ¯ r,s T X, and r,s T ∗ X only for the confusion, we will use the notation P r,s , ∂, ∂, operators and spaces associated to the given almost complex structure on X, not the standard holomorphic structure on C. According to Lemma 6.2.1, Lemma 6.1.9, and the C ∞ inverse function theorem (Theorem 9.9.1 and Theorem 9.9.2), we may ¯ has vanishing derivatives of order ≤ 2 at p = 0 also assume that the (1, 0)-form ∂z (actually, for the results considered in this section, order 0 would suffice, but we will need vanishing derivatives of order ≤ 2 in later sections). In particular, we have (∂z)0 = (dz)0 = 0, so we may also assume that ∂z = 0 at every point in X (replacing X with a small neighborhood of 0 if necessary). ¯ Given a local C 1 function f , we have We now derive local formulas for ∂ and ∂.   ¯ = df ◦ P 0,1 = ∂f dz + ∂f d z¯ ◦ P 0,1 ∂f ∂z ∂ z¯ ∂f ¯ ∂f ¯ ∂f ¯ ∂f ∂z ∂z + ∂ z¯ = ∂z + ∂z ∂ z¯ ∂z ∂ z¯   ∂f ∂f = +μ· · ∂z = Zf · ∂z, ∂ z¯ ∂z =

where μ is the C ∞ function on X given by μ≡

¯ (∂z) (∂z)

=

¯ (∂z) ¯ (∂ z¯ )

¯ = μ · ∂¯ z¯ ), (i.e., ∂z

and Z is the C ∞ vector field (and therefore linear differential operator of order 1 with C ∞ coefficients) on X given by Z≡

∂ ∂ +μ· . ∂ z¯ ∂z

Furthermore, by the choice of the local representation of X, the function μ has vanishing derivatives of order ≤ 2 at p = 0. In particular, we may assume that |μ|2 < 3/16 on X. We also have ∂f = (∂¯ f¯) = Z f¯ · ∂z = Zf · ∂z,

where Z ≡ Z =

∂ ∂ + μ¯ · . ∂z ∂ z¯

6.3 Regularity of Solutions on an Almost Complex Surface

323

Given a local C 1 differential form α of type (1, 0), we have α = f ∂z for the C 1 function f ≡ α/(∂z). Thus   ¯ ¯∂α = (∂f ¯ ) ∧ ∂z + f ∂∂z ¯ = ∂f + μ · ∂f · ∂z ∧ ∂z + f ∂∂z ∂ z¯ ∂z √  √  √ √ ∂f −1 −1 ¯ ∂f +μ· · ∂z ∧ ∂z + 2 −1f ∂ ∂z = 2 −1 ∂ z¯ ∂z 2 2   √ ∂f ∂f ¯ · ω, = 2 −1 +μ· + τ · f ω = Af ∂ z¯ ∂z where ω is the nonvanishing real C ∞ differential form of type (1, 1) given by √ −1 ∂z ∧ ∂z, ω≡ 2 τ is the C ∞ function on X given by √ ¯ ¯ ∂ ∂z −1 ∂ ∂z · = τ≡ 2 ω ∂z ∧ ∂z (see Exercise 6.3.1 for a more explicit expression for τ in terms of μ), and A and A¯ are the linear differential operators of order 1 with C ∞ coefficients on X given by   √ √ ∂ ∂ A ≡ −2 −1 + μ¯ · + τ¯ = −2 −1(Z + τ¯ ) ∂z ∂ z¯ and

  √ √ ∂ ∂ ¯ A ≡ 2 −1 +μ· + τ = 2 −1(Z + τ ). ∂ z¯ ∂z

¯ has vanishing derivatives of order ≤ 2 at 0, τ has vanishing In particular, since ∂z derivatives of order ≤ 1 at 0. For β = f ∂z a C 1 differential form of type (0, 1), we have ¯ f¯∂z) = (A¯ f¯ · ω) = Af · ω. ∂β = ∂¯ β¯ = ∂( We may summarize the above as follows: Lemma 6.3.1 For any given point p in an almost complex surface X, in order to ¯ show that there exists a nonvanishing ∂-closed C ∞ differential form of type (1, 0) on a neighborhood of p, it suffices to consider the case in which: (i) X is a relatively compact domain in R2 together with a given almost complex structure T XC = (T X)1,0 ⊕ (T X)0,1 , as well as the holomorphic structure associated to the identification R2 = C given by (x, y) ↔ z = x + iy, and p = 0; (ii) The (1, 0)-form ∂z is nonvanishing on X;

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6 Holomorphic Structures on Topological Surfaces

(iii) We have C ∞ functions μ and τ and a nonvanishing C ∞ real differential form ω of type (1, 1) on X such that μ and τ have vanishing derivatives of order ≤ 2 and of order ≤ 1, respectively, at 0, |μ|2 < 3/16 on X, and √ √ −1 −1 ¯ ¯∂z = μ · ∂z, ω= ∂z ∧ ∂z, and ∂ ∂z = τ · ω; 2 2 and (iv) For every local C 1 function f , we have ¯ = Zf · ∂z ∂f

and

∂f = Zf · ∂z,

∂ ∂ +μ· ∂ z¯ ∂z

and

Z≡

where Z=

∂ ∂ + μ¯ · , ∂z ∂ z¯

and we have ¯ ∂z) = Af ¯ ·ω ∂(f

and ∂(f ∂z) = Af · ω,

where A¯ and A are the linear differential operators of order 1 with C ∞ coefficients on X given by   √ √ ∂ ∂ ¯ A ≡ 2 −1 +μ· + τ = 2 −1(Z + τ ) ∂ z¯ ∂z and

  √ √ ∂ ∂ + μ¯ · + τ¯ = −2 −1(Z + τ¯ ). A ≡ −2 −1 ∂z ∂ z¯

Fixing the choices and notation in the above lemma for the rest of this section, ¯ the main goal is the following regularity property for the operator A: Proposition 6.3.2 (Regularity) If  is a relatively compact open subset of X, and f ∈ L2loc () with A¯ distr f ∈ C ∞ (), then f ∈ C ∞ (). Observe that for a local C 1 function f on X, we have  2  2  2  2  ∂f          +  ∂f  = 2 ∂f  + 2 ∂f  .  ∂x   ∂y   ∂z   ∂ z¯  Furthermore, if f ∈ D(X), then (denoting Lebesgue measure by λ) we get  2     2   ∂f   ∂f  ∂f ∂f     dλ = = dλ  ∂z  2   X ∂z X ∂z ∂z L (X)   ∂ 2 f¯ ∂f ∂ f¯ · dλ = − f · dλ = ∂z∂ z¯ X ∂z ∂ z¯ X

6.3 Regularity of Solutions on an Almost Complex Surface

 = X

325

 2     ∂f  ∂f ∂ f¯ ∂f ∂f  · dλ = dλ =   ∂ z¯  2 . ∂ z¯ ∂z X ∂ z¯ ∂ z¯ L (X)

Thus Proposition 6.3.2 will follow immediately from the general first-order regularity theorem (Theorem 11.0.1) together with the following: Lemma 6.3.3 For every open set   X, there is a constant C > 0 such that  2  ∂f    2   ¯ 2  ∂ z¯  2 ≤ C · f L2 () + Af L2 () ∀f ∈ D(). L () Proof Given an open set   X and a function f ∈ D(), we have  2  2  ∂f    ∂f    ¯ 4  = Af − 2i · μ · − 2i · τ · f   2 ∂ z¯ L2 () ∂z L ()  2   ¯ 2 2 + 16 · sup |μ|2 ·  ∂f  ≤ 2Af  ∂z  2 L ()  L () + 16 · sup |τ |2 · f 2L2 () . 

Thus, since |μ|2 < 3/16 on X (by construction) and τ is bounded on , there is a constant C > 0 independent of the choice of f such that  2  2  ∂f   ∂f 

 2 2    ¯ ≤ C · f L2 () + Af L2 () + 3 ·  4   ∂z  2 ∂ z¯ L2 () L ()  2  ∂f 

  ¯ 2 2 = C f 2L2 () + Af + 3  ∂ z¯  2 . L () L () 

The desired inequality now follows.

Finally, we note that the real volume forms ω and dx ∧ dy = (i/2) dz ∧ d z¯ induce the same orientation in X, since X is connected and the quotient η≡

ω dx ∧ dy

is a nonvanishing C ∞ function that is equal to 1 at 0 (in fact, as the reader is asked to verify in Exercise 6.3.1, η = 1/(1 − |μ|2 ). We will use this orientation when integrating 2-forms. Exercises for Sect. 6.3 6.3.1 Show that η = 1/(1 − |μ|2 ) and   ∂|μ|2 ∂μ ∂μ ∂|μ|2 1 2 + − · |μ| . + μ · τ= ∂z ∂z ∂z 1 − |μ|2 ∂ z¯

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6 Holomorphic Structures on Topological Surfaces

¯ Connection, and Curvature 6.4 The Distributional ∂, We continue in this section with the notation of Sect. 6.3. In particular, we have the positive (1, 1)-form ω = (i/2)∂z ∧ ∂z on X. We also fix a real-valued C ∞ function ϕ on X. For every measurable set S ⊂ X and every continuous real (1, 1)form θ defined at points of S, we may define ·, ·L2 (S,ϕ) , ·, ·L2 (S,θ,ϕ) ,  · L2 (S,ϕ) , and  · L2 (S,θ,ϕ) on scalar-valued differential forms and the associated L2 (Hilbert) spaces L2p,q (S, ϕ) and L2p,q (S, θ, ϕ) (with θ ≥ 0 or θ > 0 wherever appropriate) with respect to the almost complex structure (T X)C = (T X)1,0 ⊕ (T X)0,1 exactly as in Sect. 2.6. We may also define the corresponding canonical connection (or Chern connection) D = Dϕ = D + D

= Dϕ + Dϕ

≡ eϕ ∂[e−ϕ (·)] + ∂¯ on scalar-valued differential forms exactly as in Sect. 2.7, and the corresponding curvature ¯  = ϕ ≡ ∂ ∂ϕ as in Sect. 2.8. In particular, we have D 2 = D D

+ D

D =  ∧ (·). Definition 6.4.1 Let α and β be locally integrable differential forms on an open subset U of X. We write ∂¯distr α = β if one of the following holds: (i) On U , α is a 0-form, β = b∂z, and Z distr α = b; (ii) On U , α = a1 ∂z + a2 ∂z, β = bω, and A¯ distr (a1 ) = b; (iii) The form α is of degree > 1 and β ≡ 0. In analogy with Proposition 2.7.3, we have the following equivalent form for the definition of ∂¯distr : Proposition 6.4.2 Let α and β be locally integrable differential forms on an open set U ⊂ X. (a) If α is of type (1, 0) and β is of type (1, 1), then ∂¯distr α = β if and only if   α ∧ D f e−ϕ = β · f¯ · e−ϕ ∀f ∈ D(U ). U

U

(b) If α is of type (0, 0) and β is of type (0, 1), then ∂¯distr α = β if and only if   α · (−D γ ) · e−ϕ = β ∧ γ · e−ϕ ∀γ ∈ D0,1 (U ). U

U

(c) If α is of type (p, q) with p ≥ 2 or q ≥ 1, then ∂¯distr α = 0. Proof Part (c) is trivial and the proof of (b) is left to the reader (see Exercise 6.4.1). For the proof of (a), suppose α = a∂z and β = bω are locally integrable forms of

¯ Connection, and Curvature 6.4 The Distributional ∂,

327

type (1, 0) and (1, 1), respectively, on an open set U ⊂ X. By the (weak) Friedrichs lemma (Lemma 7.3.1), we may choose a sequence of C ∞ functions {aν } converging to a in L1loc . Setting αν = aν ∂z for each ν, we get αν → α in L1loc . Recall also that we have ω = η dx ∧ dy. If ∂¯distr α = β, then A¯ distr a = b. Hence, for every function f ∈ D(U ), we have    α ∧ D f · e−ϕ = lim αν ∧ D f · e−ϕ = lim αν ∧ ∂(e−ϕ f ) ν→∞ U

U



= lim

ν→∞ U



= lim

ν→∞ U



= lim

ν→∞ U



= 

ν→∞ U

ν→∞ U

¯ ν ) · (e−ϕ f ) (∂α



¯ ν ) · e−ϕ f · ω = lim A(a

ν→∞ U

¯ ν ) · e−ϕ f η · dx ∧ dy A(a

aν · A¯ ∗ (e−ϕ f η) · dx ∧ dy

a · A¯ ∗ (e−ϕ f η) · dx ∧ dy

U

b · e−ϕ f η · dx ∧ dy =

=



¯ −ϕ f ) = lim αν ∧ ∂(e



U

β f¯e−ϕ .

U

Conversely, suppose   −ϕ

α∧D f ·e = β f¯e−ϕ U

∀f ∈ D(U ).

U

Given u ∈ D(U ), setting f = eϕ u/η, we get   a · A¯ ∗ u · dx ∧ dy = a · A¯ ∗ (e−ϕ f η) · dx ∧ dy U

U



= lim

ν→∞ U

aν · A¯ ∗ (e−ϕ f η) · dx ∧ dy



= lim

ν→∞ U



= U

αν

∧ D f

·e

−ϕ

 =

b · e−ϕ f η · dx ∧ dy =

α



∧ D f

·e

−ϕ

U

 =

β f¯e−ϕ

U

b · u¯ · dx ∧ dy. U

Hence A¯ distr a = b and therefore ∂¯distr α = β. Thus (a) is proved.



Remark The main point of the above proof is that the characterization holds for C ∞ forms, and the C ∞ forms are dense in L1loc . One may also prove Proposition 6.4.2 more directly without density of the C ∞ forms (see Exercise 6.4.2). The proof of Proposition 2.8.2 (with Proposition 6.4.2 in place of Proposition 2.7.3) gives the following identical fundamental estimate:

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6 Holomorphic Structures on Topological Surfaces

Proposition 6.4.3 (Fundamental estimate) For all functions u, v ∈ D(X), we have 



D u, D vL2 (X,ϕ) = D u, D vL2 (X,ϕ) + iϕ uve ¯ −ϕ . X

In particular, D u2L2 (X,ϕ) = D

u2L2 (X,ϕ) +



iϕ |u|2 e−ϕ ≥ X



iϕ |u|2 e−ϕ .

X

Remark If iϕ ≥ 0 in the above, then we get D u, D vL2 (X,ϕ) = D

u, D

vL2 (X,ϕ) + u, vL2 (X,iϕ ,ϕ) and D u2L2 (X,ϕ) = D

u2L2 (X,ϕ) + u2L2 (X,i

ϕ ,ϕ)

≥ u2L2 (X,i

ϕ ,ϕ)

.

Exercises for Sect. 6.4 6.4.1 Prove part (b) of Proposition 6.4.2. 6.4.2 Prove part (a) of Proposition 6.4.2 by direct computation without using the density of the space of C ∞ forms. Hint. Apply the formulas for η and τ appearing in Exercise 6.3.1.

6.5 L2 Solutions on an Almost Complex Surface Carrying over the notation of Sects. 6.3 and 6.4, we get the following analogue of Theorem 2.9.1, which gives local existence of solutions of the Cauchy–Riemann equation on an almost complex surface along with L2 estimates. The proof, which is left to the reader (see Exercise 6.5.1), is identical to the proof of Theorem 2.9.1, except that one must use Proposition 6.4.2 in place of Proposition 2.7.3, Proposition 6.4.3 in place of Proposition 2.8.2, and Proposition 6.3.2 in place of Theorem 2.7.4. Theorem 6.5.1 Assume that i = iϕ ≥ 0, and let Z = {x ∈ X | x = 0}. Then, for every measurable (1, 1)-form β on X with β = 0 a.e. in Z and βX\Z ∈ L21,1 (X \ Z, i, ϕ), there exists a form α ∈ L21,0 (X, ϕ) such that

Ddistr α = ∂¯distr α = β

and

αL2 (X,ϕ) ≤ βL2 (X\Z,i,ϕ) .

¯ = β. In particular, if β is of class C ∞ , then α is also of class C ∞ and ∂α The proof of the following corollary is also left to the reader (see Exercise 6.5.2):

6.6 Proof of Integrability

329

Corollary 6.5.2 (Cf. Corollary 2.9.3) Suppose that C is a positive constant with iϕ ≥ C 2 ω on X. Then, for every β ∈ L21,1 (X, ω, ϕ), there exists a form α ∈ L21,0 (X, ϕ) such that

Ddistr α = ∂¯distr α = β

and

αL2 (X,ϕ) ≤ C −1 βL2 (X,ω,ϕ) .

¯ = β. In particular, if β is of class C ∞ , then α is also of class C ∞ and ∂α Exercises for Sect. 6.5 6.5.1 Prove Theorem 6.5.1. 6.5.2 Prove Corollary 6.5.2.

6.6 Proof of Integrability The integrability theorem (Theorem 6.1.4) follows from Lemma 6.1.11 together with the following: Lemma 6.6.1 In the notation of Sects. 6.3–6.5, there exists a nonvanishing C ∞ ¯ ∂-closed differential form of type (1, 0) on a neighborhood of p = 0 in X ⊂ C. Proof Let ϕ(z) = |z|2 + log |z|2 for each point z ∈ C∗ , and for each  > 0, let ϕ (z) = |z|2 + log(|z|2 + )

∀z ∈ C.

Then ϕ ≥ ϕ on C∗ for each  > 0. On X (with the choices and notation given in Lemma 6.3.1), we have ¯ = (z + z¯ μ) · ∂z, ¯ 2 = z∂z + z¯ ∂z ∂|z| and hence i ¯ 2 = i |z + z¯ μ|2 ∂z ∧ ∂z = |z + z¯ μ|2 · ω ∂|z|2 ∧ ∂|z| 2 2 and i i i ¯ 2 i ¯ + i z¯ ∂ ∂z ¯ ∂ ∂|z| = ∂z ∧ ∂z + z∂∂z + ∂ z¯ ∧ ∂z 2 2 2 2 2 = (1 + zτ¯ + |μ|2 + z¯ τ ) · ω. Thus, for each  > 0, we have 2 ¯ 2 ¯ 2 ¯  = i∂ ∂|z| ¯ 2 + i ∂ ∂|z| − i ∂|z| ∧ ∂|z| iϕ = i∂ ∂ϕ (|z|2 + ) (|z|2 + )2   |z + z¯ μ|2 1 + 2 Re(zτ¯ ) + |μ|2 − = 2 · (1 + 2 Re(zτ¯ ) + |μ|2 ) + ·ω |z|2 +  (|z|2 + )2

= 2 · (1 + ρ ) · ω,

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6 Holomorphic Structures on Topological Surfaces

where (using |z + z¯ μ|2 = |z|2 (1 + |μ|2 ) + 2 Re(z2 μ)) ¯ ρ ≡ 2 Re(zτ¯ ) + |μ|2 + = 2 Re(zτ¯ ) + |μ|2 +

(|z|2 + )(1 + 2 Re(zτ¯ ) + |μ|2 ) − |z + z¯ μ|2 (|z|2 + )2 ¯ (1 + |μ|2 ) + (|z|2 + )2 Re(zτ¯ ) − 2 Re(z2 μ) . 2 2 (|z| + )

Since μ and τ have vanishing derivatives of orders ≤ 2 and ≤ 1, respectively, at 0, after replacing X with a relatively compact neighborhood of 0, we may assume that there exists a constant C > 0 such that |μ| ≤ C|z|3 and |τ | ≤ C|z|2 on X (see Exercise 9.5.10). Therefore ρ ≥ −2C|z|3 − 2C

|z|5 |z|3 − 2C ≥ −2C(|z|3 + 2|z|). |z|2 +  (|z|2 + )2

Replacing X again with a sufficiently small neighborhood of 0, we see that we may assume without loss of generality that ρ ≥ −1/2, and hence iϕ ≥ ω on X, for every  > 0. Finally, recall that we have the positive C ∞ function η = ω/(dx ∧ dy), which we may assume to be bounded above and below by positive constants. Now let β be the C ∞ form of type (1, 1) given by ¯ = 2iτ · ω = 2iτ · η · dx ∧ dy. β ≡ ∂∂z For every  > 0, we have    2 β2L2 (X,ω,ϕ ) = 4|τ |2 e−ϕ ω ≤ 4|τ |2 e−ϕ η dλ ≤ 4C 2 |z|2 e−|z| η dλ. 

X

X

X

Replacing X again with a sufficiently small neighborhood of 0, we may assume without loss of generality that βL2 (X,ω,ϕ ) ≤ 1 for every  > 0. According to Corollary 6.5.2, for every  > 0, there exists a C ∞ form α of type (1, 0) on X such that ¯  =β ∂α

and α L2 (X,ϕδ ) ≤ α L2 (X,ϕ ) ≤ 1

∀δ > .

Fix a sequence of positive numbers {δν } with δν  0. Applying weak sequential compactness in a Hilbert space (see Theorem 7.6.1) and Cantor’s diagonal process, we get a sequence of positive numbers {ν } such that ν  0 and such that for each μ ∈ Z>0 , there is a form θμ ∈ L21,0 (X, ϕδμ ) with θμ L2 (X,ϕδμ ) ≤ 1 and γ , αν L2 (X,ϕδμ ) → γ , θμ L2 (X,ϕδμ )

as ν → ∞

∀γ ∈ L21,0 (X, ϕδμ )

(for each μ, we have, for ν  0, ν < δμ and hence αν L2 (X,ϕδμ ) ≤ 1). Since X  R2 , the set of (equivalence classes of) forms in L21,0 (X, ϕδ ) is actually equal to L21,0 (X) for each δ > 0 (although the inner products differ). In particular, given a form γ0 ∈ L21,0 (X), the above (with γ = eϕδμ · γ0 ) shows that for each μ ∈ Z>0 ,

6.6 Proof of Integrability

331

we have γ0 , αν L2 (X) → γ0 , θμ L2 (X) . Thus in fact, we have a single form α such

θ = β (for example, that α = θμ for every μ. In particular, we have ∂¯distr α = Ddistr μ by Proposition 6.4.2), and by Fatou’s lemma, we have   2 1 iα ∧ α¯ e−|z| = iα ∧ α¯ e−ϕ ≤ 1. 2 |z| X X The regularity property (Proposition 6.3.2) implies that α is of class C ∞ , and finiteness of the above integral implies that α = 0 at the point p = 0. Thus γ ≡ ∂z − α ¯ is a ∂-closed C ∞ form of type (1, 0) on X, and at 0, we have γ = dz − 0 = 0. Thus the lemma is proved (and integrability follows).  Exercises for Sect. 6.6 6.6.1 The integrability theorem (together with Sard’s theorem) may be used to construct a proof of the Koebe uniformization theorem (Theorem 5.5.3) that differs slightly from the proof appearing in Chap. 5. (a) Let  = ∅ be a C ∞ relatively compact domain in an oriented smooth surface M. By applying Theorem 9.10.1 and Lemma 5.11.1, prove (directly, without using Proposition 5.4.1) that there are an oriented compact C ∞ surface M , a finite (possibly empty) collection of disjoint positively oriented local C ∞ charts {(Dj , j , (0; Rj ))}nj=1 in M with Rj > 1 for each j , and an orientation-preserving diffeomorphism  :  → ( )

of  in M onto a of a connected relatively compact neighborhood n  −1

domain ( ) in M such that () = M \ j =1 j ((0; 1)). Prove also that M is planar if and only if  is planar. (b) For the objects in part (a), prove that for any almost complex structure (T M)C = (T M)1,0 ⊕ (T M)0,1 inducing the given orientation in M, there exists an almost complex structure (T M )C = (T M )1,0 ⊕ (T M )0,1 in M such that ∗ (Tp M)1,0 = (T(p) M )1,0

and

∗ (Tp M)0,1 = (T(p) M )0,1

for every point p in some neighborhood of  in  . (c) According to Sard’s theorem (see, for example, [Mi]), the set of critical values of a C ∞ function on a second countable smooth manifold is a set of measure zero. Using this fact together with Proposition 9.3.11 and Theorem 9.9.2, prove that each compact subset of a second countable C ∞ surface M lies in some C ∞ relatively compact domain in M. (d) Let K be a compact subset of a Riemann surface X. By applying parts (a)–(c) above together with Theorem 6.1.4, prove that there exist a C ∞ domain  with K ⊂   X and a biholomorphism  :  → ( ) of a connected relatively compact neighborhood  of  in X onto a domain ( ) in a compact Riemann surface X . Prove also that if X is planar, then one may choose X to be planar.

332

6 Holomorphic Structures on Topological Surfaces

(e) Give a proof of the Koebe uniformization theorem that is analogous to the proof appearing in Sect. 5.5 but that uses parts (a)–(d) above in place of Proposition 5.4.1. Exercises 6.6.2–6.6.4 below require the discussion of homology and cohomology appearing in Sects. 10.6 and 10.7, as well as some of the facts and exercises from Chap. 5. 6.6.2 Prove that if M is a compact oriented C ∞ surface with H1 (M, R) = 0, then M is diffeomorphic to a sphere. 6.6.3 Prove that if M is a second countable noncompact oriented C ∞ surface with H1 (M, R) = 0, then M is diffeomorphic to R2 . 6.6.4 Let M be a second countable C ∞ surface. (a) Prove that the universal cover of M is diffeomorphic to the plane R2 or the sphere S2 . Hint. Apply integrability, uniformization, and Exercise 6.6.8. (b) Prove that if M is orientable, then π1 (M) is torsion-free and  H1 (M, Z) ∼ = π1 (M) [π1 (M), π1 (M)], while if M is nonorientable, then every torsion element of π1 (M) has order 1 or 2 and π1 (M)/  ∼ = Z/2Z for some torsion-free normal subgroup  of π1 (M). Hint. Apply Exercise 5.9.4 and Exercise 5.17.4. (c) Let A be a subfield of C containing Z, and let F = R or C. Prove that if M is orientable, then H1 (M, A) ∼ = H1 (M, A), while if M is nonorientable,  ∼ then H1 (M, F) = H1 (M, F). (d) Prove that if M is orientable, then for any subring A of C containing Z, the mapping   [ξ ]H1 (M,A) → [ξ ]H1 (M,A) , · deR gives an injective homomorphism H1 (M, A) → Hom(H 1 (M, A), A) (according to Theorem 10.7.18, this homomorphism is also surjective if π1 (X) is finitely generated). (e) Assume that M is orientable, and let K ⊂ M be a compact set. Prove that there exist domains  and  such that K ⊂     M and such that for every C ∞ closed 1-form ρ on  , there exist a C ∞ closed 1-form τ on M and a function λ ∈ D( ) such that ρ − τ = dλ on . Conclude from this that 1 1 1 1 im[HdeR (M) → HdeR ()] = im[HdeR ( ) → HdeR ()].

Hint. See Exercise 5.17.5.

6.7 Statement of Schönflies’ Theorem The main goal of the rest of this chapter is the fact that every second countable topological surface admits a C ∞ structure. For the proof, one forms a C 0 atlas and

6.7 Statement of Schönflies’ Theorem

333

Fig. 6.1 The homeomorphism provided by Schönflies’ theorem

then inductively modifies the local charts on the overlaps to get C ∞ compatibility. For the modification, one applies the following (see Fig. 6.1): Theorem 6.7.1 (Schönflies’ theorem) Let γ : S1 → P1 be a Jordan curve with image C ≡ γ (S1 ) in the Riemann sphere P1 . Then there exists a homeomorphism  : P1 → P1 such that S1 = γ and such that P1 \S1 : P1 \ S1 → P1 \ C is a diffeomorphism. In particular, we have the following: Corollary 6.7.2 (Jordan curve theorem) If γ : S1 → P1 is a Jordan curve with image C ≡ γ (S1 ), then P1 \ C has exactly two connected components. Moreover, the boundary of each of these connected components is equal to C. Remark One may choose the homeomorphism  in Schönflies’ theorem to map the unit disk (0; 1) onto either of the connected components of P1 \ C by replacing  with the homeomorphism z → (z/|z|2 ) = (1/¯z) (0 → (∞), ∞ → (0)) if necessary. A proof of Schönflies’ theorem, which is due to Kneser and Radó [KnR], appears in Sect. 6.9 (a different approach, as well as a more complete treatment of triangulations and the classification of surfaces, can be found in [T]). In fact, their proof allows one to choose the diffeomorphism  so that the restriction of −1 to one of the connected components of P1 \ C, and the restriction of 1/−1 to the other connected component, are harmonic. For parts of the proof, we will apply the Riemann mapping theorem in the plane (Theorem 5.2.1), although it is not really necessary. For now, we consider a preliminary observation, namely, that a Jordan curve in P1 that is piecewise smooth away from some point on the curve is separating. Lemma 6.7.3 Let α : [a, b] → P1 be a Jordan curve in P1 such that for some point c ∈ (a, b), α[a,r] and α[s,b] are piecewise smooth paths for all r ∈ (a, c) and s ∈ (c, b). Then the image C ≡ α([a, b]) is separating in P1 . Proof We have C = P1 (for example, C is not simply connected), so by applying a suitable Möbius transformation, we may assume without loss of generality that ∞∈ / C. By choosing a smooth point u ∈ (a, b) and replacing α with the path given by t → α(t − a + u) for t ∈ [a, b + a − u] and t → α(t − b + u) for t ∈ [b + a − u, b]

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6 Holomorphic Structures on Topological Surfaces

(and modifying c accordingly), we may assume that α is loop-smooth at a (see Definition 5.10.2). We may also fix r0 , r1 ∈ R such that a < r0 < r1 < c and such that α[a,r1 ] is a smooth path. Theorem 9.9.2 then implies that there is a connected neighborhood U of α(r0 ) such that U ∩ C ⊂ α((a, r1 )) and U \ C has exactly two connected components U0 and U1 . Moreover, if P1 \ C is connected, then there exists a loop β : [0, 1] → P1 such that β(0) = β(1) = α(r0 ), β((0, 1)) ⊂ P1 \ C, and for some  > 0, β((0, )) ⊂ U0 and β((1 − , 1)) ⊂ U1 . Consequently, we may choose R > 0 so small that for D ≡ (α(c); R), we have D  P1 \ (U ∪ β([0, 1]) ∪ α([a, r1 ])), and setting r ≡ min{t ∈ [a, c] | α(t) ∈ D}

and s ≡ max{t ∈ [c, b] | α(t) ∈ D},

we get r1 < r < c < s < b and α([a, b] \ (r, s)) ⊂ P1 \ D. Thus the loop γ : [a, b] → P1 given by  α(t) if t ∈ [a, b] \ (r, s), γ (t) = t−r α(r) + s−r · (α(s) − α(r)) if t ∈ [r, s], is a piecewise smooth Jordan curve that is loop-smooth at a. Lemma 5.10.5 provides a smooth Jordan curve η : [a, b] → P1 such that the image A ≡ η([a, b]) satisfies A ∩ U = C ∩ U and β((0, 1)) ⊂ P1 \ A. But β meets both connected components U0 and U1 of U \ A = U \ C, and therefore, since A is separating in P1 (by Proposition 5.15.2) and each connected component of P1 \ A has boundary equal to A (by Lemma 5.11.1), we have arrived at a contradiction. Thus C is separating.  Exercises for Sect. 6.7 The exercises for this section require Schönflies’ theorem (and its consequences), so the reader may wish to postpone consideration of these exercises until after consideration of the proof in Sect. 6.9. 6.7.1 Let γ : [0, 1] → C be a (continuous) Jordan curve in C with interior  (i.e.,  is the unique bounded connected component of C \ C provided by Corollary 6.7.2), and let C = γ ([0, 1]). Assume that γ is oriented so that  is on the left; that is, for the corresponding homeomorphism γ0 : S1 → C given by e2πit → γ (t), after choosing the homeomorphism  : P1 → P1 with S1 = γ0 provided by Schönflies’ theorem (Theorem 6.7.1) to map the unit disk  ≡ (0; 1) diffeomorphically onto  (see the remark following the statement of Corollary 6.7.2), γ and γ0 are directed so that this diffeomorphism is orientation-preserving (to ensure this, one simply replaces γ with the reverse curve and  with the mapping z → (¯z) if necessary). (a) Prove that γ is homotopic in  to a trivial loop. (b) Prove the following version of Cauchy’s theorem (cf. Lemma 1.2.1 and Exercises 5.1.6 and  5.1.7): If f is a holomorphic function on a neighborhood of , then γ f (z) dz = 0. (c) Prove the following version of the Cauchy integral formula (cf. Lemma 1.2.1 and Exercises 5.1.6 and 5.1.7): If f is a holomorphic function

6.7 Statement of Schönflies’ Theorem

335

on a neighborhood of  and z0 ∈ , then  f (z) 1 f (z0 ) = dz. 2πi γ z − z0 (d) Prove the following version of the residue theorem (cf. Exercises 2.5.9, 5.1.6, 5.1.7, 5.9.5, and 6.7.6): If S is a finite subset of  and θ is a holomorphic 1-form on  \ S for some neighborhood  of , then   1 θ= resp θ. 2πi γ p∈S

(e) Prove the following version of the argument principle (cf. Exercises 2.5.8, 5.1.6, 5.1.7, 5.9.5, and 6.7.6): If f is a nontrivial meromorphic function on a neighborhood of  with no zeros or poles in ∂, and counting multiplicities, μ is the number of zeros of f in  and ν is the number of poles in , then  1 df . μ−ν = 2πi γ f (f) Prove the following version of Rouché’s theorem (cf. Exercises 2.5.8, 5.1.6, 5.1.7, 5.9.5, and 6.7.6): If f and g are nontrivial meromorphic functions on a neighborhood of  that do not have any zeros or poles in ∂, and |g| < |f | on ∂, then μf − νf = μf +g − νf +g , where counting multiplicities, μf is the number of zeros of f in , νf is the number of poles of f in , μf +g is the number of zeros of f + g in , and νf +g is the number of poles of f + g in . 6.7.2 Let M be a topological surface. (a) Let α : [0, 1] → M be an injective path with image A ≡ α([0, 1]). Prove that for every t0 ∈ (0, 1), there exist numbers a, b and a local C 0 chart (U, , U = (−1, 1)×(a, b)) in M such that 0 ≤ a < t0 < b ≤ 1, A∩U = γ ((a, b)), and (α(t)) = (0, t) for every t ∈ (a, b). Prove also that if M is a C ∞ surface, then the local chart may be chosen to map U \ A diffeomorphically onto U \ ({0} × R). (b) Prove that if B is a Jordan curve in M, then M \ B has exactly one or two connected components. Prove also that for each connected component U of M \ B, we have ∂U = B. (c) Suppose M is an oriented smooth surface, γ : [0, 1] → M is a Jordan in M if and only if curve, and C = γ ([0, 1]). Prove that C is separating ∞ 1-form θ on M. Prove θ = 0 for every closed compactly supported C γ also that M is planar if and only if every Jordan curve in M is separating. Hint. Apply Exercise 5.10.3 and Proposition 5.15.2. 6.7.3 Let α : [0, 1] → M be an injective path in a topological surface M. Prove that there exists a Jordan curve β : [0, 2] → M with β[0,1] = α. Hint. Apply Exercise 6.7.2.

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6.7.4 Prove that for any nonempty relatively compact domain   M in a topological surface M, the following are equivalent: (i) Each point in ∂ admits a local coordinate neighborhood (U,  = (x, y)) in M such that  ∩ U = {q ∈ U | x(q) < 0}. (ii) There exist disjoint Jordan curves C1 , . . . , Cn such that ∂ = C1 ∪ · · · ∪ Cn and such that for each j = 1, . . . , n, Cj is separating in the conj ∪ · · · Cn ) containing Cj . nected component of M \ (C1 ∪ · · · ∪ C (iii) There exist Jordan curves γj : [0, 1] → M, with corresponding continuous 1-periodic extensions γ˜j : R → M, for j = 1, . . . , n, such that the images Cj ≡ γj ([0, 1]) for j = 1, . . . , n are disjoint; ∂ = C1 ∪· · ·∪Cn ; and for each index j = 1, . . . , n and for each point t0 ∈ R, there is a local coordinate neighborhood (U,  = (x, y), U = (−1, 1) × (a, b)) in M with  ∩ U = {q ∈ U | x(q) < 0}, t0 ∈ (a, b) ⊂ γ˜j−1 (U ), and (γ˜j (t)) = (0, t) for every t ∈ (a, b). Prove also that if M is a smooth surface, then each of the local charts (U, ) in (i) and (iii) may be chosen so that the restriction U \∂ is a diffeomorphism of U \ ∂ onto its image. Hint. Apply Exercise 6.7.2 and Theorem 9.10.1. 6.7.5 In this exercise, we consider some facts concerning orientation preservation. (a) Let M be a C ∞ surface, let  ⊂ M be a C ∞ open set, let p ∈ ∂, and let (U,  = (x, y), U ) and (V ,  = (u, v), V ) be local C ∞ charts in M with p ∈ U ∩ V ∩ ∂ and  ∩ U = {p ∈ U | x(p) < 0}

and  ∩ V = {p ∈ V | u(p) < 0}.

Assume that  and  induce the same orientation in ∂ at p; that is, for the inclusion mapping ι : U ∩ V ∩ ∂ → U ∩ V , we have (ι∗ dv)p /(ι∗ dy)p > 0. Prove that  and  induce the same orientation in some neighborhood of p in U ∩ V (and hence in the connected component of U ∩ V containing p). (b) Let  = (u, v) : U → V be a homeomorphism of neighborhoods U and V of (0, 0) in R2 such that (i) (0, 0) = (0, 0); (ii) U ∩ ((−∞, 0) × R) = {(x, y) ∈ U | u(x, y) < 0}; (iii) U \({0}×R) maps U \ ({0} × R) diffeomorphically onto V \ ({0} × R); and (iv) The function t → v(0, t) is increasing. Prove that there exists a neighborhood W of (0, 0) in U for which the restriction W \({0}×R) is orientation-preserving. Hint. Produce a smooth Jordan curve C bounding a smooth domain   U ∩ ((−∞, 0) × R) such that part of C is a directed line segment with  lying to the left, and another part is the inverse image under  of a directed line segment with () lying to the left. Then apply part (a). (c) Let M be an oriented smooth surface, and let (U,  = (x, y), U = (−1, 1) × (a, b)) be a local C 0 chart in M for which the restriction 0 ≡ −1 (U \({0}×(a,b))) : −1 (U \({0}×(a, b))) → U \({0}×(a, b))

6.7 Statement of Schönflies’ Theorem

337

is a diffeomorphism. Prove that 0 must be either orientation-preserving or orientation-reversing. Hint. Show that it suffices to consider the case M  C ⊂ P1 . Fix a suitable disk D  U that meets the y-axis. Applying Schönflies’ theorem, one may form a homeomorphism  : P1 → P1 that agrees with −1 on ∂D and that maps D diffeomorphically onto its image (D) = −1 (D)  U . In particular, D is orientation-preserving or -reversing. Now apply part (b) to the homeomorphism 0 ◦  near points in (∂D) \ ({0} × R). (d) Prove that for any Jordan curve γ : S1 → P1 with image C, the restriction P1 \S1 : P1 \ S1 → P1 \ C of the homeomorphism provided by Schönflies’ theorem must be either an orientation-preserving or an orientationreversing diffeomorphism (in particular, by replacing  with the homeomorphism z → (z/|z|2 ) = (1/¯z), one may always choose the homeomorphism so that this restriction is orientation-preserving). (e) Let M be an oriented smooth surface, and let α : [0, 1] → M be an injective path with image A ≡ α([0, 1]). Prove that for every t0 ∈ (0, 1), there exist numbers a, b and a local C 0 chart (U, , U = (−1, 1) × (a, b)) in M such that 0 ≤ a < t0 < b ≤ 1, A ∩ U = γ ((a, b)), (α(t)) = (0, t) for every t ∈ (a, b), and the restriction U \A : U \ A → U \ ({0} × R) is an orientation-preserving diffeomorphism. (f) Let M be an oriented smooth surface, and let   M be a domain with the properties (i)–(iii) listed in Exercise 6.7.4. Prove that each of the boundary curves γj and the local charts (U, ) in (iii) may be chosen so that the restriction U \∂ is an orientation-preserving diffeomorphism of U \ ∂ onto its image. Prove also that (with the above choice of orientation), if θ   is a C 1 closed 1-form on M, then (see Definition 10.5.3) nj=1 γj θ = 0. 6.7.6 One may apply Exercise 6.7.5 to obtain versions of the residue theorem (cf. Exercises 2.5.9, 5.1.6, 5.1.7, 5.9.5, and 6.7.1), and the argument principle and Rouché’s theorem (cf. Exercises 2.5.8, 5.1.6, 5.1.7, 5.9.5, and 6.7.1), for a suitable nonsmooth domain in a Riemann surface. Let X be a Riemann surface, let  be a relatively compact domain in X, and let γj : [0, 1] → X be a Jordan curve with 1-periodic extension γ˜j : R → X for j = 1, . . . , n. Assume that: (i) The images Cj ≡ γj ([0, 1]) for j = 1, . . . , n are disjoint; (ii) We have ∂ = C1 ∪ · · · ∪ Cn ; and (iii) For each index j = 1, . . . , n and for each point t0 ∈ R, there is a local C 0 coordinate neighborhood (U,  = (x, y), U = (−1, 1) × (a, b)) in X such that  ∩ U = {q ∈ U | x(q) < 0}, t0 ∈ (a, b) ⊂ γ˜j−1 (U ), (γ˜j (t)) = (0, t) for every t ∈ (a, b), and U \Cj : U \ Cj → U \ ({0} × (a, b)) is an orientation-preserving diffeomorphism.

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(a) Residue theorem. Prove that if S is a finite subset of  and θ is a holomorphic 1-form on X, then  n   1 θ= resp θ. 2πi γj j =1

p∈S

(b) Argument principle. Prove that if f is a nontrivial meromorphic function on X with no zeros or poles in ∂, and counting multiplicities, μ is the number of zeros of f in  and ν is the number of poles in , then μ−ν =

 n  1 df . 2πi γj f j =1

(c) Rouché’s theorem. Prove that if f and g are nontrivial meromorphic functions on X that do not have any zeros or poles in ∂, and |g| < |f | on ∂, then μf − νf = μf +g − νf +g , where counting multiplicities, μf is the number of zeros of f in , νf is the number of poles of f in , μf +g is the number of zeros of f + g in , and νf +g is the number of poles of f + g in . 6.7.7 Prove that if M is an orientable C ∞ surface and γ : [0, 1] → M is a Jordan curve with image C = γ ([0, 1]), then for some R > 1, there exists a neighborhood  of C in M and a homeomorphism  :  → (0; 1/R, R) with (γ (t)) = e2πit for each t ∈ [0, 1]. Hint. Apply Exercises 6.7.2 and 6.7.5. 6.7.8 State and prove an analogue of Lemma 5.15.1 for a continuous Jordan curve with nonseparating image in an orientable smooth surface M (see Exercise 6.7.7). Using this fact (and Exercise 6.7.5),prove that the image C of a Jordan curve γ in M is separating if and only if γ θ = 0 for every C ∞ closed 1-form θ with compact support in M (a different proof of this analogue of Proposition 5.15.2 was suggested in Exercise 6.7.2).

6.8 Harmonic Functions and the Dirichlet Problem Kneser and Radó’s proof of Schönflies’ theorem relies on the solution of the Dirichlet problem, but only for harmonic functions on simply connected domains in P1 with complement containing at least two points. In this section, we consider an approach to the Dirichlet problem that is an ad hoc version of the important Perron method. In particular, by restricting our attention to simply connected domains, we are able to avoid some work by applying the Riemann mapping theorem in the plane. However, it should be noted that the general Perron method can be developed entirely from elementary results, and it leads to a more complete solution. Moreover, most of the results concerning open Riemann surfaces appearing in this book can be ¯ (see, for example, obtained by using the Perron method in place of the L2 ∂-method [AhS] or [For]).

6.8 Harmonic Functions and the Dirichlet Problem

339

A complex-valued C 2 function u on a complex 1-manifold X is harmonic if ¯ ∂ ∂u = 0 (cf. Definition 2.8.1). A real-valued C 2 function ρ on X is subharmonic ¯ ≥ 0 (respectively, i∂ ∂ρ ¯ > 0). (strictly subharmonic) if i∂ ∂ρ Lemma 6.8.1 Let X be a Riemann surface. (a) If u is a real-valued harmonic function on X and X is simply connected, then u = Re f for some holomorphic function f on X. In particular, the real-valued harmonic functions on a Riemann surface are precisely the functions that are locally equal to the real part of a local holomorphic function. (b) Strong maximum principle for harmonic functions. If u is a real-valued harmonic function on X that attains a local maximum or local minimum value, then u is constant. (c) If {un } is a sequence of real-valued harmonic functions on X that is uniformly bounded on each compact subset of X, then some subsequence {unk } converges uniformly on compact subsets of X to a harmonic function on X. (d) Weak maximum principle for subharmonic functions. If   X is a relatively compact domain in X and ϕ is a continuous real-valued function on  that is C 2 and subharmonic on , then max ϕ = max∂ ϕ. Proof If u is a real-valued harmonic function on X, then θ ≡ ∂u is a (closed) holo¯ = −∂ ∂u ¯ = 0. Therefore, if X is simply connected, then by morphic 1-form, since ∂θ Corollary 10.5.7, 2θ = df for some C ∞ function f on X. Since θ is of type (1, 0), ¯ = 0, so f is holomorphic. On the other hand, we have ∂f ∂(u − Re f ) = θ −

1 ¯ − Re f ) = ∂(u − Re f ) = 0, df = 0 and ∂(u 2

so u − Re f is constant. Choosing f to agree with u at some point, we get u = Re f on X in this case, and (a) follows. For a real-valued harmonic function u on X, part (a) implies that eu is locally the modulus of a holomorphic function (for u = Re f , we have eu = |ef |). Therefore, if u attains a local maximum value of c at some point, then the maximum principle for holomorphic functions (Theorem 1.3.4) implies that u ≡ c in a neighborhood of the point. Hence the interior  of u−1 (c) is nonempty. Corollary 1.3.3 implies that u ≡ c near any point in . Hence  is both open and closed in X, and therefore  = X. Suppose {un } is a sequence of real-valued harmonic functions on X that is uniformly bounded on each compact subset of X. Each point has a neighborhood D  X such that D is biholomorphic to a disk and for some constant R = R(D) > 0, |un | ≤ R on D for each n ∈ Z>0 . In particular, by part (a), for each n, we have un D = Re fn for some function fn ∈ O(D). The holomorphic function Fn ≡ efn then satisfies 0 < e−R ≤ |Fn | = eun ≤ eR . Montel’s theorem (Corollary 2.11.4) implies that some subsequence of {Fn } converges uniformly on compact subsets of D to a holomorphic function F . Uniform continuity of the logarithmic function on the interval [e−R , eR ] ⊂ (0, ∞) then implies that some subsequence of {un D } = {log |Fn |} converges uniformly on compact subsets of D to the har-

340

6 Holomorphic Structures on Topological Surfaces

Fig. 6.2 Local representation of the zero set of a harmonic function

monic function log |F |. Covering X by a locally finite collection of such coordinate disks D and applying Cantor’s diagonal process, we get the claim (c). Finally, for the proof of (d), assuming that max ϕ > max∂ ϕ, we reason to a contradiction. By replacing  with a sufficiently large relatively compact open subset of , we may assume without loss of generality that  = X. Fixing a point p ∈ X \  and applying Corollary 2.14.2, we get a positive C ∞ strictly subharmonic function ρ on X \ {p}. For  > 0 sufficiently small and for ψ ≡ ϕ + ρ, we have max ψ > max∂ ψ ; and hence ψ attains its maximum value at some interior point q ∈ . On the other hand, for any local holomorphic coordinate neighborhood (U, z = x + iy) of q in , we have ∂ 2ψ ∂ 2ψ (q) + (q) > 0; ∂x 2 ∂y 2 so we have arrived at a contradiction.



The following local description of the zero set of a harmonic function is an easy consequence of Lemma 6.8.1 and the local description of holomorphic mappings given by Lemma 2.2.3. The proof is left to the reader (see Exercise 6.8.2). Lemma 6.8.2 Let Z = {p ∈ X | u(p) = 0} be the zero set of a nonconstant realvalued harmonic function u on a Riemann surface X. Then, for each point p ∈ Z, there exist a constant Rp > 0, a positive integer mp , and a local holomorphic chart mp (Up , p = zp , (0; Rp )) such that p = −1 p (0) and u = Im(zp ) on Up . In particular, (Z ∩ Up ) = (L0 ∪ · · · ∪ Lmp −1 ) ∩ (0; Rp ), where Lj ≡ {reij π/mp | r ∈ R} for each j = 0, . . . , mp − 1. Moreover, mp = 1 if and only if (du)p = 0, and the set of points p ∈ Z at which mp > 1 is discrete in X. Remark The case mp = 6 is pictured in Fig. 6.2. Clearly, the zero set Z in the above lemma is closed. The above characterization implies that in particular, Z is a locally finite graph. For a nonempty open subset  of a Riemann surface X and a continuous function ρ : ∂ → C, the associated (classical) Dirichlet problem is that of finding a

6.8 Harmonic Functions and the Dirichlet Problem

341

continuous function u :  → C that is harmonic on  and that satisfies u∂ = ρ. This problem is not always solvable. For example, let  ≡ (0; 1), let  ≡  \ {0}, and suppose u is a continuous function on  =  that is harmonic on  and that satisfies u(0) = 1 and u ≡ 0 on ∂. By replacing u with Re u, we may assume that u is real-valued. The maximum principle (Lemma 6.8.1) implies that 0 < u < 1 on . But then, for  > 0 sufficiently small, the harmonic function v : z → u(z) +  log |z|2 on , which approaches −∞ at 0 and which vanishes at ∂, will have some positive values and therefore will attain a local maximum at some point in . This contradicts the maximum principle, so no such solution u of the given Dirichlet problem can exist. For a nonempty relatively compact domain  in a Riemann surface and a continuous function on ∂, the maximum principle implies that the associated Dirichlet problem has at most one solution. For if u and v are solutions, then w ≡ u − v is a continuous function on  that vanishes on ∂ and that is harmonic on . Hence, by the maximum principle, the maximum and minimum values of Re w and Im w must be 0, and therefore w ≡ 0. By applying the Perron method, one can prove that the Dirichlet problem is solvable on any relatively compact domain in a Riemann surface for which each boundary component contains more than one point (see, for example, [AhS]). For our purposes, the following will suffice: Proposition 6.8.3 The Dirichlet problem is solvable on each connected component  of the complement P1 \ K of any connected closed set K  P1 . We now work toward the proof. We first consider the (Poisson) formula for the solution on a disk. As motivation, we consider the following consequence of the Cauchy integral formula: Lemma 6.8.4 (Mean value property) If R > 0 and u : (0; R) → R is a continuous function that is harmonic on (0; R), then    2π  R 2 − |z|2 1 ζ + z dζ 1 iθ u(Re ) · dθ = Re u(ζ ) · u(z) = 2π 0 |Reiθ − z|2 2πi ∂(0;R) ζ −z ζ for each point z ∈ (0; R). In particular, we have the mean value property  2π  u(ζ ) 1 1 iθ u(Re ) dθ = u(0) = dζ. 2π 0 2πi ∂(0;R) ζ Proof By Lemma 6.8.1, u(0;R) = Re f for some function f ∈ O((0; R)). Let v ≡ Im f . For each r ∈ (0, R), the Cauchy integral formula (Lemma 1.2.1) gives    1 1 1 f (ζ ) u(ζ ) v(ζ ) dζ = dζ + dζ f (0) = 2πi ∂(0;r) ζ 2πi ∂(0;r) ζ 2π ∂(0;r) ζ  2π  2π 1 i iθ = u(re ) dθ + v(reiθ ) dθ. 2π 0 2π 0

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6 Holomorphic Structures on Topological Surfaces

Comparing real parts, we see that u(0) =



1 2πi

∂(0;r)

1 u(ζ ) dζ = ζ 2π



2π

u(reiθ ) dθ,

0

and letting r → R − (and observing that u(reiθ ) → u(Reiθ ) uniformly), we get u(0) =

1 2πi

 ∂(0;R)

1 u(ζ ) dζ = ζ 2π



2π

u(Reiθ ) dθ.

0

Given a point z ∈ (0; R), Proposition 2.14.8 provides the automorphism (i.e., the Möbius transformation) 0 : ζ →

ζ − (z/R) 1 − (¯z/R)ζ

of P1 , which maps (0; 1) onto itself. Setting (ζ ) ≡ R0 (ζ /R) =

R2ζ − R2 z R 2 − z¯ ζ

∀ζ ∈ P1 ,

we get an automorphism  ∈ Aut (P1 ) that maps (0; R) onto itself. Applying the above mean value property at 0 to the harmonic function u(−1 ), we get u(z) = u ◦ −1 (0) = =

1 2πi

 

∂(0;R)

1 2πi

 ∂(0;R)

u(−1 (ξ )) dξ ξ

u(ζ ) ·  (ζ ) dζ (ζ )

1 R 2 − |z|2 dζ u(ζ ) · 2πi ∂(0;R) (ζ − z)(R 2 − z¯ ζ )  2π 1 R 2 − |z|2 Reiθ dθ = u(Reiθ ) · 2π 0 (Reiθ − z)(R 2 − z¯ Reiθ )  2π R 2 − |z|2 1 u(Reiθ ) · = dθ iθ 2π 0 (Re − z)(Re−iθ − z¯ )  2π R 2 − |z|2 1 u(Reiθ ) · dθ. = 2π 0 |Reiθ − z|2 =

Combining the above with the equality   |ζ |2 − |z|2 ζ +z , = Re ζ −z |ζ − z|2 we get the desired formula.



6.8 Harmonic Functions and the Dirichlet Problem

343

Lemma 6.8.5 (Poisson formula) Given a continuous function ρ : ∂(0; R) → C for some R > 0, the function u : (0; R) → C given by ⎧  2π R 2 − |z|2 ⎨ 1 ρ(Reiθ ) · dθ u(z) ≡ 2π 0 |Reiθ − z|2 ⎩ ρ(z)

if z ∈ (0; R), if z ∈ ∂(0; R),

is the unique solution of the associated Dirichlet problem (that is, u is continuous on (0; R), u is harmonic on (0; R), and u∂(0;R) = ρ). Remark The function K: (ζ, z) → (R 2 − |z|2 )/(2πR|ζ − z|2 ) is called the Poisson kernel. We have u(z) = ∂(0;R) ρ(ζ )K(ζ, z) ds(ζ ) for z ∈ (0; R). Proof of Lemma 6.8.5 Clearly, we may assume without loss of generality that ρ is 2 −|z|2 ζ +z real-valued. As in the proof of Lemma 6.8.4, the equality |ζ|ζ| −z| 2 = Re[ ζ −z ] then implies that u(0;R) is equal to the real part of the function z →

1 2πi

 ρ(ζ ) · ∂(0;R)

1 = 2πi

 ∂(0;R)

ζ + z dζ ζ −z ζ

z ρ(ζ ) dζ + ζ −z 2πi

 ∂(0;R)

ρ(ζ )/ζ dζ. ζ −z

Moreover, by Lemma 1.2.2 (or by differentiation past the integral), the above function is holomorphic, and therefore u is harmonic on (0; R). It remains to show that u is continuous at each boundary point z0 ∈ ∂(0; R). Let λ denote the Lebesgue measure on R, and for each δ > 0, let Nδ ≡ {θ ∈ [0, 2π] | |Reiθ − z0 | < 2δ}. For each δ > 0 and each z ∈ (0; R) ∩ (z0 ; δ), by applying Lemma 6.8.4 to the constant (and therefore harmonic) function ζ → 1, we get    1  |u(z) − u(z0 )| =  2π

R 2 − |z|2 dθ |Reiθ − z|2 0   2π  1 R 2 − |z|2 − ρ(z0 ) · dθ  iθ 2 2π 0 |Re − z|

≤

1 2π +

2π

ρ(Reiθ ) ·

 |ρ(Reiθ ) − ρ(z0 )| · Nδ

1 2π

R 2 − |z|2 dλ(θ ) |Reiθ − z|2

 [0,2π]\Nδ

|ρ(Reiθ ) − ρ(z0 )| ·

R 2 − |z|2 dλ(θ ) |Reiθ − z|2

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6 Holomorphic Structures on Topological Surfaces

 2π R 2 − |z|2 1 dθ 2π 0 |Reiθ − z|2 τ ∈Nδ  2π 1 −2 (R 2 − |z|2 ) dθ + δ · 2 sup |ρ| · 2π 0 ∂(0;R)

≤ sup |ρ(Reiτ ) − ρ(z0 )| ·

=

sup

(z0 ;2δ)∩∂(0;R)

|ρ − ρ(z0 )| + δ −2 · 2 sup |ρ| · (R 2 − |z|2 ). ∂(0;R)

Now, given  > 0, the continuity of ρ implies that we may choose δ > 0 so small that the first term on the right-hand side is less than /2. For z ∈ (0; R) ∩ (z0 ; δ) sufficiently close to z0 , the second term will then also be less than /2. Thus u is continuous at z0 .  The solution of the Dirichlet problem on more general domains requires the following fundamental tool from the Perron method: Definition 6.8.6 Let  be an open subset of a complex 1-manifold X, and let p ∈ ∂. A real-valued function β on  is called a C ∞ barrier at p on  if β is a C ∞ subharmonic function and lim β(z) = 0

z→p

but

lim sup β(z) < 0 ∀q ∈ (∂) \ {p}. z→q

Remark One may also define a notion of a continuous subharmonic function and consider continuous barriers. For the construction of a barrier in our case, we will need some elementary geometric facts. We first recall that any holomorphic function f on a neighborhood of a point z0 ∈ C with f (z0 ) = 0 is conformal at z0 . That is, if α and β are two smooth paths in C with α(s0 ) = β(t0 ) = z0 , then f preserves the angle measure   Re(α (s0 )β (t0 )) arccos |α (s0 )| · |β (t0 )| between the (tangent vectors of) the paths, because (f ◦ α) (s0 )(f ◦ β) (t0 ) = α (s0 )β (t0 )|f (z0 )|2

and |f (z0 )|2 > 0.

We also have the following fact, the proof of which is left to the reader (see Exercise 6.8.3): Lemma 6.8.7 Let C be a circle of radius R in the plane, let M be a line in the plane that meets C in two points, let l be the length of the line segment given by the intersection of M with the closed disk bounded by C, and let d be the distance from the center of the circle to the line M. Then, at each of the two points of intersection, the two angles between M and C are given by     l/2 l/2 ∈ (0, π/2] and θ1 = π − θ0 = 2 · arctan . θ0 = 2 · arctan R+d R−d

6.8 Harmonic Functions and the Dirichlet Problem

345

Fig. 6.3 The image  of the domain  under a branch of the logarithmic function

Lemma 6.8.8 If K is a connected closed subset of P1 containing more than one point,  is a connected component of P1 \ K, and p ∈ ∂, then there exists a C ∞ barrier β at p on . Proof This proof is based on the arguments in [AhS] (see also [BerG]). By applying an automorphism of P1 , we may assume that p = ∞ and that 0 ∈ K. By Lemma 5.17.1,  is simply connected, and hence by Proposition 5.2.2, there exists a (single-valued) holomorphic function L on  with eL(z) = z for each point z ∈  ⊂ C∗ (i.e., L is a single-valued branch of the logarithmic function). Setting u ≡ Re L and v ≡ Im L, we have u(z) = log |z| and v(z) is an argument of z for each point z ∈ . The function L maps  biholomorphically onto a domain  ⊂ C. Since u → −∞ at 0 and +∞ at ∞, the intermediate value theorem implies that for each r ∈ R, we have  ∩ (r + iR) = r + iAr , where Ar is a nonempty open subset of R in which no two distinct points differ by a multiple of 2π . In particular, Ar is equal to the union of a (nonempty) countable collection of disjoint open intervals {Irj }j ∈Jr = {(arj , brj )}j ∈Jr (the connected components) of total length (i.e., Lebesgue measure)  (brj − arj ) ≤ 2π, λ(Ar ) = j ∈Jr

with Jr = Z>0 or Jr = {1, . . . , mr } for some mr ∈ Z>0 (see Fig. 6.3). We will construct a subharmonic function γ on  such that γ (ζ ) is bounded above by a negative constant for ζ in the intersection with any half-plane {Re ξ < r}, but γ (ζ ) → 0 as Re ζ → ∞. The function β = γ (L) will then be a barrier at p = ∞ on . For the construction of γ , we will first construct, for each r ∈ R, a negative subharmonic function γr such that γr = −1 on {ζ ∈  | Re ζ ≤ r} and γr (ζ ) → 0 as Re ζ → +∞. For this, we will construct a harmonic function on {ζ ∈ C | Re ζ > r} for which the value at any point ζ will be the product of −2/π and a sum of angle measures. Each of these angle measures will be for an angle formed by the vertical line Re ξ = r and, for some j ∈ Jr , the circle passing through ζ and the endpoints of the line segment r + iI rj . As ζ approaches any point in r + iIrj , the associated circle will approach the line r + iR and the angle measure will approach π . Thus the limit of the function at points in r + iAr will be at most −2. We will get the function γr on  as an extension of the composition of a suitable nondecreasing convex function

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6 Holomorphic Structures on Topological Surfaces

and the above function. As Re ζ → ∞, the above circles will grow large and the above angle measures will shrink to 0; and it will follow that γr (ζ ) → 0. To get a function γ that is bounded away  from 0 on the intersection of  with any left half-plane, we will then let γ = 2−ν γrν for some sequence rν → ∞. We now consider the details of the construction. For each number r ∈ R and each index j ∈ Jr , the Möbius transformation rj : P1 → P1 given by ζ →

r + iarj − ζ r + ibrj − ζ

(r + iarj → 0, r + ibrj → ∞, ∞ →  1) is an automorphism of P1 = C ∪ {∞} that maps the segment r + iI rj onto {∞} ∪ (−∞, 0], the segment {∞} ∪ (r + i(R \ Irj )) onto [0, ∞) ∪ {∞} (here, of course, for a ∈ R, (−∞, a] = {x ∈ R | x ≤ a} and [a, ∞) = {x ∈ R | x ≥ a}, while {∞} denotes the singleton consisting of the point at infinity in P1 ), and the half-plane Hr = {ζ ∈ C | Re ζ > r} onto the upper halfplane H = {ζ ∈ C | Im ζ > 0}. We may also form a single-valued argument function that is the unique harmonic function α : C \ i(−∞, 0] → (−π/2, 3π/2) satisfying eiα(ξ ) = ξ/|ξ | for each ξ ∈ C \ i(−∞, 0] (i.e., α is the imaginary part of a branch of the logarithmic function on the simply connected domain C \ i(−∞, 0]). Thus 2 −1 αrj ≡ − α ◦ rj : rj (C \ i(−∞, 0]) → (−3, 1) π is a harmonic function, −2 < αrj < 0 on Hr , αrj ≡ −2 on r + iIrj , and αrj ≡ 0 on r + i(R \ I rj ). Moreover, the fact that a Möbius transformation maps circles in P1 to circles (Theorem 5.7.3), the fact that (local) biholomorphisms are conformal, Lemma 6.8.7, and the inequality 0 < arctan x < x for x > 0 together give the following estimate:   brj − arj 2 brj − arj 4 − · < − arctan ≤ αrj (ζ ) < 0 ∀ζ ∈ Hr π Re ζ − r π 2(Re ζ − r) (as pictured in Fig. 6.4, for ζ ∈ Hr , the inverse image under rj of the ray M ≡ [0, ∞) · rj (ζ ) is an arc of a circle that meets the ray r + i(−∞, arj ] at the vertex −1 (0), thus determining an angle of the same measure as the angle r + iarj = rj formed by M and the nonnegative x-axis). The Weierstrass  M-test and Lemma 6.8.1 together now imply that the (possibly finite) series j ∈Jr αrj converges uniformly on compact subsets of Hr to a harmonic function αr satisfying 0 > αr (ζ ) > −

 2 brj − arj 2 λ(Ar ) 4 · =− · ≥− π Re ζ − r π Re ζ − r Re ζ − r

∀ζ ∈ Hr .

j ∈Jr

Now, according to Lemma 2.10.2, we may fix a C ∞ function χ : R → [−1, ∞) such that χ , χ

≥ 0 on R, χ ≡ −1 on (−∞, −3/2], and χ(t) = t for each t ≥ −1/2 (simply form the function as in the lemma with a = −3/2, b = −1, and c = −1/2,

6.8 Harmonic Functions and the Dirichlet Problem

347

Fig. 6.4 Harmonic functions associated to two components of  ∪ (r + iR)

and then subtract 1 to get χ ). On the other hand, for each r ∈ R, each index j ∈ Jr , and each point ζ ∈ r + iIrj ⊂ r + iAr =  ∩ ∂Hr , we have αrj (ζ ) = −2 and hence αr ≤ αrj < −3/2 at all points in Hr near ζ . It follows that the function γr :  → [−1, 0) given by  −1 if ζ ∈  \ Hr , γr (ζ ) = χ(αr (ζ )) if ζ ∈  ∩ Hr , sequence of real numis a C ∞ subharmonic function. Choosing a strictly increasing  −ν γ :  → [−1, 0) bers {rν } with rν → ∞, we see that the function γ ≡ ∞ 2 rν ν=1 is equal to the finite sum −2

−μ+1

+

μ−1 

2−ν γrν ≤ −2−μ+1

ν=1

on the open set  \ H rμ for each μ. Since these open sets form an increasing sequence with union , it follows that γ is a C ∞ subharmonic function on . Furthermore, γ is bounded above by a negative constant on the intersection of  with any left half-plane {ζ ∈ C | Re ζ < r} for r ∈ R. On the other hand, 0 as γ (ζ ) →−ν Re ζ → +∞. For given  > 0, we may choose μ ∈ Z>0 so large that ∞ = ν=μ+1 2 −μ 2 < /2. Hence, if ζ ∈  with Re ζ > rμ , then   μ μ   −ν   −ν 4 0 > γ (ζ ) > − + . 2 γrν (ζ ) ≥ − + 2 χ − 2 2 Re ζ − rν ν=1

ν=1

Since χ(0) = 0, we see that for each ζ ∈  with Re ζ  rμ , we have 0 > γ (ζ ) > −. It follows that the function β ≡ γ (L) :  → [−1, 0) is a C ∞ barrier at p = ∞ on  (note that the function z → log |z| = Re L(z) is bounded above on each compact subset of C).  The last fact we will need for the proof of Proposition 6.8.3 is the following (for a much more general version, see, for example, [Mu]):

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6 Holomorphic Structures on Topological Surfaces

Theorem 6.8.9 (Tietze extension theorem in C) If ρ : K → R is a continuous function on a nonempty compact subset K of C, then there exists a continuous function ρ0 : C → R such that ρ0 K = ρ, infC ρ0 = minK ρ, and supC ρ0 = maxK ρ. Proof We may fix a number R > 0 so that K ⊂ (0; R); a sequence {Dν }, where Dν = (ζν ; rν )  (0; 2R)\ K for each ν, the collection {Dν } is locally finite in C \ K, and (0; R) \ K ⊂ Dν ; a sequence of nonnegative  continuous functions {λν } with supp λν ⊂ Dν for each ν, λν ≤ 1 on C, and λν ≡ 1 on (0; R) \ K; and for each ν, a point zν ∈ K with dist(zν , ζν ) = dist(K, ζν ) > rν . Clearly, the function α : C → R given by  ρ on K, α≡  ρ(zν ) · λν on C \ K, is continuous on C \ K and satisfies αK = ρ. We now show that α is continuous at each point z0 ∈ K. Given  > 0, we may choose δ1 > 0 so that |α(z) − α(z0 )| = |ρ(z) − ρ(z0 )| < 

∀z ∈ (z0 ; 3δ1 ) ∩ K.

By local finiteness, there is an N ∈ Z>0 such that Dν lies in the δ1 -neighborhood of K each ν > N . In particular, for each ν > N , we have rν < dist(K, ζν ) = |zν − ζν | < δ1 . We may also choose a number δ such that 0 < δ < δ1 , (z0 ; δ) ⊂ (0; R), and (z0 ; δ) ∩ Dν = ∅ for each ν = 1, . . . , N . Therefore, if z ∈ C \ K with |z − z0 | < δ, then we have z ∈ (0; R), z ∈ / Dν for each ν = 1, . . . , N , and for each ν > N with z ∈ Dν , we have |zν − z0 | ≤ |zν − ζν | + |ζν − z| + |z − z0 | < 3δ1 . Therefore

     |α(z) − α(z0 )| =  λν (z) · (ρ(zν ) − ρ(z0 )) ≤ λν (z) · |ρ(zν ) − ρ(z0 )| < .

Thus the extension α is continuous at z0 . Hence the function ρ0 : C → R given by ⎧ ⎪ if minK ρ ≤ α(z) ≤ maxK ρ, ⎨α(z) z → minK ρ if α(z) < minK ρ, ⎪ ⎩ maxK ρ if α(z) > maxK ρ, has the required properties.



Proof of Proposition 6.8.3 Given a connected closed set K  P1 , a connected component  of P1 \ K, and a continuous function ρ : ∂ → C, we show that the associated Dirichlet problem is solvable. By considering Re ρ and Im ρ and applying

6.8 Harmonic Functions and the Dirichlet Problem

349

a suitable automorphism of P1 , we may assume without loss of generality that ρ is real-valued and ∞ ∈ . Any Dirichlet problem on the complement of a singleton has the constant solution, so we may also assume that K contains more than one point. By the Tietze extension theorem (Theorem 6.8.9), there exists a continuous function ρ0 : C → R with |ρ0 | ≤ M ≡ max |ρ| on C ⊃ ∂ and ρ0 = ρ on ∂. We will produce solutions on elements of an exhausting sequence of coordinate disks for  (with boundary values given by ρ0 ). We will then pass to a convergent subsequence; and finally, we will use barriers to show that the limit is a solution of the Dirichlet problem. By Lemma 5.17.1 and the Riemann mapping theorem in the plane (see Corollary 5.2.6), there exists a biholomorphism  :  →  of  onto the unit disk  ≡ (0; 1). We may choose a sequence of numbers {Rν } in (0, 1) converging to 1 such that ∞ ∈ ν ≡ −1 ((0; Rν ))  

∀ν = 1, 2, 3, . . . .

For each ν, Lemma 6.8.5 provides a (unique) real-valued continuous function uν on ν such that uν ν is harmonic and uν ∂ν = ρ0 ∂ν . In particular, |uν | ≤ M on ν by the maximum principle (part (b) of Lemma 6.8.1). Applying part (c) of Lemma 6.8.1 and passing to a subsequence, we may assume that the sequence {uν } converges uniformly on compact subsets of  to a real-valued harmonic function on  (more precisely, setting uˆ ν = uν on ν and uˆ ν = 0 on  \ ν , we see that for each μ, a subsequence of {uˆ ν } will converge uniformly on μ , and hence Cantor’s diagonal process provides a subsequence converging uniformly on compact subsets of ). It now suffices to show that the function u :  → [−M, M] ⊂ R given by  u≡

limν→∞ uν ρ

on , on ∂,

is continuous at each point z0 ∈ ∂. Given  > 0, there is a δ1 > 0 such that |ρ0 (z) − ρ0 (z0 )| <  for each z ∈ C with |z − z0 | < δ1 . By Lemma 6.8.8, there exists a barrier β at z0 on . Since β < 0 on  and lim supz→ζ β(z) < 0 at each point ζ ∈ (∂) \ {z0 }, we have sup\(z0 ;δ1 ) β < 0; and hence, by replacing β with the product of β and a sufficiently large positive constant, we may assume that β < −2M on  \ (z0 ; δ1 ). For each ν, we have uν + β ≤ −M ≤ −|ρ(z0 )| < ρ(z0 ) + 

on (∂ν ) \ (z0 ; δ1 )

and uν + β ≤ ρ0 < ρ(z0 ) + 

on (∂ν ) ∩ (z0 ; δ1 ).

Therefore, by the weak maximum principle for subharmonic functions (part (d) of Lemma 6.8.1), we have uν + β < ρ(z0 ) +  on ν for each ν, and passing to the

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6 Holomorphic Structures on Topological Surfaces

limit, we get u + β ≤ ρ(z0 ) +  on . Therefore, since β → 0 at z0 and ρ is continuous, we must have lim sup u(z) ≤ ρ(z0 ) +  z→z0

∀ > 0;

and hence lim supz→z0 u(z) ≤ ρ(z0 ) = u(z0 ). The same argument applied to −ρ0 , −ρ, and −uν → −u gives lim sup(−u(z)) ≤ −u(z0 ), z→z0

and hence lim infz→z0 u(z) ≥ u(z0 ). Thus limz→z0 u(z) = u(z0 ), and therefore u is continuous.  Exercises for Sect. 6.8 6.8.1 State and prove a (Poisson) formula for the solution of the Dirichlet problem on a disk (z0 ; R) with center z0 . 6.8.2 Prove Lemma 6.8.2. 6.8.3 Prove Lemma 6.8.7.

6.9 Proof of Schönflies’ Theorem We are now ready to prove Schönflies’ theorem (following [KnR]). Throughout this section,  denotes the unit disk (0; 1). We begin with some preliminary observations. Lemma 6.9.1 Let γ : S1 → P1 be a Jordan curve with image C ≡ γ (S1 ), and let  be a connected component of P1 \ C. Then ∂ is an infinite connected subset of C (that is, γ −1 (∂) is either S1 or a closed arc in S1 ), the solution  :  → C of the Dirichlet problem in  with boundary values given by ∂ = (γ −1 )∂ : ∂ → S1 exists, and  is a proper C ∞ mapping of  into . Proof Observe that C = P1 , since (for example) C is not simply connected. According to Lemma 5.17.1, ∂ is a nonempty connected compact subset of C. Moreover, if ∂ were a singleton {q}, then we would have  = P1 \ {q} (the connected set P1 \ {q} would meet , but not ∂), which is clearly impossible because C ⊂ P1 \ . Thus γ −1 (∂) is either S1 or a closed arc in S1 . According to Proposition 6.8.3, the solution  :  → C of the Dirichlet problem in  with boundary values given by ∂ = (γ −1 )∂ : ∂ → S1 exists. Let u ≡ Re  and v ≡ Im . We may choose a point p ∈  with |(p)| = R ≡ max ||, a line L in C with L ∩ (0; R) = {(p)}, and constants a, b, c ∈ R such that the function λ : (x + iy) → ax + by + c for x, y ∈ R satisfies L = λ−1 (0) and λ < 0 on (0; R). Thus the continuous function λ() ≡ au + bv + c :  → R, which is harmonic

6.9 Proof of Schönflies’ Theorem

351

on , attains its maximum at p. On the other hand, λ() is nonconstant because if λ() were constant, then we would have (∂) ⊂ () = {(p)}, which is impossible since ∂ = (γ −1 )∂ : ∂ → S1 is injective and ∂ is not a singleton. Therefore, by the maximum principle (Lemma 6.8.1), p ∈ ∂, and it follows that R = 1 (one may also see that R = 1 by applying Lemma 6.8.1 to the C ∞ subharmonic function ||2 on ) and () ⊂ . Finally, since (∂) ⊂ ∂, the restriction  :  →  must be a proper C ∞ mapping; that is, the inverse image of every compact subset of  is compact.  Lemma 6.9.2 Suppose that γ : S1 → P1 is a Jordan curve, C ≡ γ (S1 ), and for each connected component  of P1 \ C, there exists a continuous mapping  :  →  such that ( )∂ = (γ −1 )∂ : ∂ → S1 ,  () ⊂ , and ( ) :  →  is a proper local diffeomorphism. Then P1 \C has exactly two connected components 0 and 1 (in particular, the Jordan curve theorem holds for γ ), ∂0 = ∂1 = C, and for each ν = 0, 1, ν : ν →  is a homeomorphism for which the restriction (ν )ν : ν →  is a diffeomorphism. Consequently, the mapping  : P1 → P1 given by  −1 (z) if z ∈ ,  0 (z) ≡ −1  (1/¯ z ) if z ∈ P1 \ , 1 is a homeomorphism for which S1 = γ and P1 \S1 : P1 \ S1 → P1 \ S1 is a diffeomorphism (in particular, Schönflies’ theorem holds for γ ). Proof Given a connected component 0 of P1 \ C, Lemma 10.2.11 implies that the proper local diffeomorphism (0 )0 : 0 →  is a (finite) C ∞ covering map and therefore a diffeomorphism (since  is simply connected). Thus  = 0 (0 ) ⊂ 0 (0 ) ⊂ , and hence 0 : 0 →  is a continuous bijection, and therefore by compactness a homeomorphism. In particular, γ −1 (∂0 ) = 0 (∂0 ) = ∂ = S1 , and hence ∂0 = C. Furthermore, P1 \ C is not connected. For if 0 = P1 \ C, then  would be a homeomorphism of P1 onto . But these two spaces are not homeomorphic. For example, if we remove two distinct points from P1 , then we get a domain that is biholomorphic to C∗ and therefore is not simply connected. However, if we remove any two distinct points in ∂ from , the resulting space is still simply connected. Now, letting 1 be a connected component of P1 \ C with 1 = 0 , we get a homeomorphism  : 0 ∪ 1 = 0 ∪ C ∪ 1 → P1 by setting  = 0 on 0 and  = 1/ 1 on 1 (observe that the homeomorphism  → P1 \  given by z → 1/¯z is equal to the identity on ∂); and  maps 0 ∪ 1 diffeomorphically onto P1 \ C. Suppose there exists a third connected component  of P1 \ C. By applying a Möbius transformation, we may assume that 0 ∈ 0 and ∞ ∈ 1 , and we may fix r > 0 with (0; r)  0 . Letting γ0 (t) = γ (e2πit ) for each t ∈ [0, 1] and fixing a path α in 0 \ {0} ≈  \ {0 (0)} from γ0 (0) = γ0 (1) to r, we see that the loop β = α − ∗ γ0 ∗ α generates the fundamental group π1 (0 \ {0}, r) ∼ =

352

6 Holomorphic Structures on Topological Surfaces

π1 (0 \ {0}, r) (the verification is left to the reader). Thus the loop σ : t → re2πit represents the element [β]m r ∈ π1 (0 \ {0}, r) for some m ∈ Z, and hence    1 1 1 dz = m dz = m dz. 2πi = σ z β z γ0 z However, γ0 is path homotopic to the constant loop in  ≈  and therefore in C∗ ⊃ , so γ0 (1/z) dz = 0. Thus we have arrived at a contradiction, and hence 0 ∪ C ∪ 1 = P1 .  Proof of Theorem 6.7.1 Let γ : S1 → P1 be a Jordan curve, let C ≡ γ (S1 ), and let  be a connected component of P1 \ C. According to Lemma 6.9.1, ∂ is an infinite connected subset of C, the solution  :  → C of the Dirichlet problem in  with boundary values given by ∂ = (γ −1 )∂ : ∂ → S1 exists, and the restriction  :  →  is a proper C ∞ mapping. Let u ≡ Re  and v ≡ Im . According to Lemma 6.9.2, it suffices to show that  is a local diffeomorphism. By the C ∞ inverse function theorem (Theorem 9.9.1 and Theorem 9.9.2), this is the case if and only if du ∧ dv is nowhere 0 in . So, for the rest of this proof, we will assume that there exists a point p ∈  at which (du ∧ dv)p = 0, and we will reason to a contradiction. This condition implies that there exist constants a, b ∈ R that are not both 0, but that satisfy a(du)p + b(dv)p = 0. The continuous function ρ ≡ au + bv − au(p) − bv(p) :  → R is then harmonic on  and satisfies ρ(p) = 0 and (dρ)p = 0. The argument will proceed (in several steps) as follows (see Fig. 6.5). We will first show that ρ is nonconstant. The local description of zero sets of nonconstant harmonic functions in Lemma 6.8.2 will then imply that for Z ≡ ρ −1 (0), Z ∩  is a graph with at least four edges emanating from p. These edges cannot be rejoined by a sequence of edges in Z ∩  \ {p}, since if this were possible, then we would get a Jordan curve in Z enclosing a region in . The maximum principle would then imply that ρ vanishes on this region. Similar arguments will then imply that the (at least four) connected components of Z ∩  \ {p} approach at least four distinct points in ∂. But these four boundary points must then map into the intersection of the line L ≡ {ax + by − au(p) − bv(p) = 0} with ∂, which consists of only two points. Since ∂ is injective, this will yield the desired contradiction. The steps below contain the details of the above argument. Step 1. Proof that ρ is nonconstant. The line L in C given by L ≡ {z ∈ C | a · Re z + b · Im z − au(p) − bv(p) = 0} contains the point (p) ∈ , and hence L ∩ ∂ contains exactly two points, which we will denote by ξ1 and ξ2 . If ρ is constant, then () ⊂ L, and hence (∂) ⊂ {ξ1 , ξ2 }, which, according to Lemma 6.9.1, is impossible. Thus ρ is nonconstant, and in particular, the set S ≡ {q ∈  | ρ(q) = 0 and (dρ)q = 0} is a discrete subset of .

6.9 Proof of Schönflies’ Theorem

353

Fig. 6.5 Three possibilities for the zero set Z, the first two of which are ruled out by the maximum principle, the third by the injectivity of  on the boundary

Step 2. Proof of the nonexistence of a separating Jordan curve that lies in the zero set of ρ and that does not meet ∂ in more than one point. Let Z ≡  −1 (L ∩ ) = {z ∈  | ρ(z) = 0}. Suppose there exists a separating Jordan curve A in P1 such that A ⊂ Z and A ∩ ∂ contains at most one point. Then P1 \ A has a connected component  that does not meet the connected subset C \ (A ∩ ∂) of P1 \ A. In particular,  ∩ ∂ = ∅ (otherwise, since ∂ is connected and not a singleton,  would contain points in (∂) \ (A ∩ ∂) ⊂ C \ (A ∩ ∂)). On the other hand,  ⊂ P1 \  (otherwise, the set ∂, which, according to Lemma 6.9.1, is infinite, would be contained in the set A \  = A ∩ ∂, which contains at most one point), so we must have  ⊂ . But then, since ∂ ⊂ A ⊂ Z, the maximum principle implies that ρ is constant on the open set  and therefore on . This contradicts Step 1, so no such Jordan curve A exists. Step 3. Local description of certain injective paths near a point of Z ∩ . Given a point q ∈ Z ∩ , Lemma 6.8.2 implies that for some R > 0 and some m ∈ Z>0 , there is a local holomorphic chart (D,  = ζ, (0; R)) such that D ⊂ , q = −1 (0), and (Z ∩ D) = (L0 ∪ · · · ∪ Lm−1 ) ∩ (0; R) for m distinct lines L0 , . . . , Lm−1 through 0. Suppose α : [a, b] → Z \ {q} is an injective path, α([a, b]) ∩ ∂ contains at most one point, and for some point c ∈ (a, b), α[a,r] and α[s,b] are piecewise smooth paths (in P1 ) for all r ∈ (a, c) and s ∈ (c, b). Then α([a, b]) meets at most one of the 2m connected components of Z ∩ D \ {q}. For if A and B are two distinct connected components of Z ∩D \{q} meeting α([a, b]), then since α is injective and the functions |ζ |A and |ζ |B are homeomorphisms onto the interval (0, R), there are unique numbers r, s ∈ [a, b] such that r ∈ α −1 (A), |α(r)| = minα−1 (A) |ζ (α)|, s ∈ α −1 (B), and |α(s)| = minα −1 (B) |ζ (α)|. By exchanging A with B and r with s if necessary, we may assume that r < s. Therefore, the path β given by α[r,s] , followed by a suitable parametrization of the inverse image under  of the line segment from ζ (α(s)) to 0, and then by a suitable parametrization of the inverse image under  of the line segment from 0 to ζ (α(r)), is a Jordan curve in  that satisfies the conditions in Lemma 6.7.3. Thus β’s image is a separating Jordan curve in P1 that satisfies the conditions of the Jordan curve in Step 2, and hence we have arrived at a contradiction. Thus the image of α cannot meet more than one connected component of Z ∩ D \ {q}.

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Step 4. Proof that the closure of each connected component of Z ∩  \ S is the image of an injective path that is smooth at each point in . The (closed) zero set Z of ρ satisfies Z ∩ ∂ ⊂  −1 ({ξ1 , ξ2 }) (a set with at most two points), Z ∩  is the zero set of the nonconstant harmonic function ρ , and p is a point in the discrete set S = {q ∈ Z ∩  | (dρ)q = 0}. By Lemma 6.8.2 (or Theorem 9.9.2), Z ∩  \ S is a (properly embedded) 1-dimensional smooth submanifold of the open set  \ S. Thus, by Step 2, Lemma 6.7.3 (or Proposition 5.15.2), and Theorem 9.10.1, each connected component A of Z ∩  \ S is a noncompact connected 1-dimensional smooth submanifold of  \ S that is the image of a diffeomorphism α : (0, 1) → A. We may choose a sequence {sν } in (0, 1) such that sν → 0 and such that the sequence {α(sν )} converges to a point q0 ∈ A ⊂ . Since α : (0, 1) →  \ S is a proper mapping, we have q0 ∈ / A and therefore q0 ∈ S ∪ (Z ∩ ∂). Similarly, we may choose a sequence {tν } in (0, 1) such that tν → 1 and such that the sequence {α(tν )} converges to a point q1 ∈ S ∪ (Z ∩ ∂). If q0 ∈ S, then by Step 3 and Lemma 6.8.2, there is a local holomorphic chart (D0 , 0 = ζ0 , (0; R0 )) such that D0  , D 0 ∩ S = {q0 } = −1 0 (0), 0 (Z ∩ D0 ) = (L0 ∪ · · · ∪ Lm−1 ) ∩ (0; R0 ) for m > 1 distinct lines L0 , . . . , Lm−1 through 0, and A meets (and hence contains) exactly one of the 2m connected components of Z ∩ D0 \ {q0 } = Z ∩ D0 \ S. After applying a rotation, we may assume that A ∩ D0 = −1 0 ((0, R0 )). It fol−1 (−R /2), the set A ≡ A ∪  ((−R lows that for w0 ≡ −1 0 0 0 /2, R0 )) is a con0 0 nected noncompact 1-dimensional smooth submanifold of the open subset 0 ≡  \ [(S \ {q0 }) ∪ {w0 }] of P1 , and that the connected set A is open in A0 . Moreover, since 0 (α(sν )) → 0, it follows that the connected open set α −1 (D0 ) = α −1 (−1 0 ((0, R0 ))) must be equal to an interval of the form (0, r0 ) for some point r0 ∈ (0, 1). In particular, α(t) → q0 as t → 0+ (0 (α) is a strictly increasing realvalued C ∞ function on (0, r0 )) and q1 ∈ α([r0 , 1)) \ D 0 . As above, if q1 ∈ S, then there exist a connected neighborhood D1   with D 1 ∩ S = {q1 }, a point w1 ∈ D1 \ {q1 }, and a connected noncompact 1-dimensional smooth submanifold A1 of the open subset 1 ≡  \ [(S \ {q1 }) ∪ {w1 }] of P1 such that q1 ∈ A1 ⊂ A ∪ D1 and A is a connected open set in A1 . We then have α −1 (D1 ) = (r1 , 1) for some point r1 ∈ (0, 1) and α(t) → q1 as t → 1− . Furthermore, we may choose the neighborhood D1 so that D 1 ∩ D 0 = ∅ if q0 ∈ S (and hence 0 < r0 < r1 < 1 in this case). If q0 ∈ ∂, then α(t) → q0 as t → 0+ . For if this is not the case, then after fixing a sufficiently small neighborhood D of q0 with D ∩ Z ∩ ∂ = {q0 } (as we may, since Z ∩∂ contains at most two points), we may form a sequence {aν } in (0, 1) such that aν → 0 but α(aν ) ∈  \ D for each ν. In fact, since the connected image under α of the interval with endpoints aν and sν must meet ∂D for ν  0, we may assume that α(aν ) ∈ ∂D for each ν. Moreover, by passing to a subsequence, we may assume that the sequence {α(aν )} converges to a point q ∈ [(Z ∩ ∂) ∪ S] ∩ ∂D. We then cannot have q ∈ S (otherwise, by the above arguments for the case q0 ∈ S, we would have α(t) → q as t → 0+ ), so q ∈ Z ∩ ∂ ∩ ∂D = ∅, which is clearly impossible. Thus α(t) → q0 as t → 0+ . A similar argument shows that if q1 ∈ ∂, then α(t) → q1 as t → 1− . Observe also that we cannot have q0 = q1 ∈ ∂, because if this were the

6.9 Proof of Schönflies’ Theorem

355

case, then α would extend to a Jordan curve [0, 1] →  with separating image (by Lemma 6.7.3) and Step 2 would be violated. Now, by the above, if both q0 and q1 lie in S, then A is a connected open relatively compact set in the connected noncompact 1-dimensional smooth submanifold A0 ∪ A1 of the open subset (0 ∪ 1 ) \ {w0 , w1 } =  \ [(S \ {q0 , q1 }) ∪ {w0 , w1 }] of P1 , α(t) → q0 as t → 0+ , and α(t) → q1 as t → 1− . Since A0 ∪ A1 is diffeomorphic to R (by Theorem 9.10.1), it follows that there is a smooth injective path β : [0, 1] →  with A = β((0, 1)), β(0) = q0 , and β(1) = q1 . If q0 ∈ S but q1 ∈ ∂, then A is a connected open set in the manifold A0 , and the restriction of a suitable diffeomorphism of the interval (−∞, 1) onto A0 then determines an injective path β : [0, 1] →  such that β((0, 1)) = A, β is smooth from the right at 0, β is smooth at each point in (0, 1), β(0) = q0 , and β(1) = q1 . A similar injective path β exists if q0 ∈ ∂ and q1 ∈ S. If q0 , q1 ∈ ∂, then α extends to an injective path β : [0, 1] →  with β(0) = q0 and β(1) = q1 . To summarize, if A is any connected component of Z ∩  \ S, then there exists an injective path β : [0, 1] → Z such that β(0,1) is a diffeomorphism of (0, 1) onto A, β(0), β(1) ∈ S ∪ (Z ∩ ∂), β is smooth from the right at 0 if β(0) ∈ S, and β is smooth from the left at 1 if β(1) ∈ S. Step 5. Formation of a piecewise smooth path to the boundary. For our point p ∈ S, we have a local holomorphic chart (D,  = ζ, (0; R)) such that D ⊂ , p = −1 (0), and (Z ∩ D) = (L0 ∪ · · · ∪ Lm−1 ) ∩ (0; R) for m ≥ 2 distinct lines L0 , . . . , Lm−1 through 0. Given a connected component A of Z ∩ D \ {p}, we may form an injective path α : [0, 1] → Z from p to ∂ as follows. If B1 is the connected component of Z ∩  \ S containing A, then by Step 6, there exists an injective path βj : [0, 1] → Z such that β1 (0) = p, β1 ((0, 1)) = B1 , β1 (1) ∈ S ∪ (Z ∩ ∂), β1 is smooth from the right at 0 and smooth at each point in (0, 1), and β1 is smooth from the left at 1 if β1 (1) ∈ S. If β1 (1) ∈ ∂, then we set α = β1 . If / ∂, then we proceed by induction as follows. Suppose we have constructed β1 (1) ∈ smooth injective paths β1 , . . . , βk−1 in  such that β1 (0) = p, Bj ≡ βj ((0, 1)) is a connected component of Z ∩  \ S, for each j = 1, . . . , k − 1, the connected components B1 , . . . , Bk−1 are distinct, βj −1 (1) = βj (0) ∈ S for j = 2, . . . , k − 1, and βk−1 (1) ∈ S. By Step 3, there is a connected component Bk of Z ∩  \ S with βk−1 (1) ∈ B k and Bk = Bk−1 . In fact, Step 3 and Step 4 imply that B k ∩ B k−1 = {βk−1 (1)} and B k ∩B j = ∅ for j = 1, . . . , k −2. As above, we may form an injective path βk : [0, 1] → Z such that βk (0) = βk−1 (1), βk ((0, 1)) = Bk , βk (1) ∈ S ∪ (Z ∩ ∂), βk is smooth from the right at 0 and smooth at each point in (0, 1), and βk is smooth from the left at 1 if βk (1) ∈ S. If βk (1) ∈ ∂, then we terminate the process and we set α = β1 ∗ · · · ∗ βk . If βk (1) ∈ S, we continue to form the next path. Thus, if the process eventually terminates, then we get an injective path α from p = α(0) to a point α(1) ∈ ∂ such that α is piecewise smooth (with values in Z ∩ ) on each compact subinterval of [0, 1). If the process never terminates, then we get a sequence of smooth paths {βk }, and we may define a continuous injective map β : [0, 1) → Z ∩  that is piecewise smooth on each compact subinterval by setting    1 k(k + 1) β(t) = βk t − 1 + k

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6 Holomorphic Structures on Topological Surfaces

for each point t ∈ [1 − k −1 , 1 − (k + 1)−1 )] and each index k = 1, 2, 3, . . . . We may fix a sequence {tν } in (0, 1) for which tν → 1 and {β(tν )} converges to a point q ∈ . Since the collection of connected components of Z ∩  \ S is locally finite in  (by the local characterization of Z ∩ ), q must lie in Z ∩ ∂, a set of at most two points. Given a neighborhood D of q with D ∩ Z ∩ ∂ = {q}, we see that for δ ∈ (0, 1) sufficiently small, we have β((1 − δ, 1)) ∩ ∂D = ∅ (otherwise, there would exist a sequence in (0, 1) converging to 1 with image converging to a point in the set Z ∩ ∂D ∩ ∂, which is empty). Thus β((1 − δ, 1)) ⊂ D, and it follows that β(t) → q as t → 1− . Thus we may extend β to a path α : [0, 1] → Z with α(1) = q ∈ Z ∩ ∂. Step 6. Completion of the proof. Continuing with the notation of Step 5, and letting A1 , . . . , A2m be the distinct connected components of Z ∩ D \ {p}, we see that for each j = 1, . . . , 2m, we may form an injective path αj : [0, 1] → Z such that Aj ⊂ αj ((0, 1)) ⊂  \ {p}, αj (0) = p, αj (1) ∈ Z ∩ ∂, and the restriction of αj to each compact subinterval of [0, 1) is piecewise smooth. Step 3 implies that the sets α1 ((0, 1]), . . . , α2m ((0, 1]) are disjoint. In particular, the 2m ≥ 4 points α1 (1), . . . , α2m (1) ∈ Z ∩ ∂ must be distinct. Thus we have arrived at a contradiction, and the theorem follows.  The above proof also gives the following: Theorem 6.9.3 (Kneser–Radó [KnR]) If γ : S1 → P1 is a Jordan curve, C ≡ γ (S1 ), and  is a connected component of P1 \ C, then the solution  of the Dirichlet problem in  with boundary values given by ∂ = γ −1 : C → S1 is a homeomorphism of  onto  that maps  diffeomorphically onto . Exercises for Sect. 6.9 6.9.1 Show that in the proof of Schönflies’ theorem, we could have formed a smooth path to the boundary. In other words, prove that if ρ is a nonconstant real-valued harmonic function on a simply connected domain   C and p ∈ Z ≡ ρ −1 (0), then there exists a proper C ∞ embedding α : R →  such that α(0) = p and α(R) ⊂ Z.

6.10 Orientable Topological Surfaces In this section we consider a natural notion of orientability for a second countable topological surface. The simplest example of a nonorientable C ∞ surface is the Möbius band (Example 9.7.1). So one natural way in which to define orientability of a (second countable) topological surface is to require that the surface not contain a Möbius band. In order to verify that the definition is consistent with the definition of orientability of a C ∞ surface (Definition 9.7.2), we will need some preliminary facts. The first is a version of Lemma 5.11.1 for a nonorientable C ∞ surface:

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Orientable Topological Surfaces

357

Lemma 6.10.1 Let γ : [0, 1] → M be a smooth Jordan curve in a C ∞ surface M, and let C = γ ([0, 1]). If C does not admit an orientable neighborhood in M, then there exists a diffeomorphism  :  → B of some neighborhood  of C in M onto the Möbius band B ≡ [0, 1] × (0, 1)/(0, t) ∼ (1, 1 − t) ∀t ∈ (0, 1), with the quotient map : [0, 1] × (0, 1) → B, such that (γ (t)) = (t, 1/2) for each point t ∈ [0, 1] (here, (t, 1/2) denotes an ordered pair, not an interval). Proof We proceed as in the proof of Lemma 5.11.1, but here, we must put a twist in the constructed band. We have the associated diffeomorphism γ0 : S1 → C given by e2πit → γ (t), and the C ∞ covering map α : R → C given by t → γ0 (e2πit ) = γ (t − t) (where t denotes the floor of t). We may form local C ∞ charts {(Uj , j = (xj , yj ), Rj = Ij × (−δj , δj ))}m j =1 in M and a partition 0 = t0 < t1 < t2 < · · · < tm = 1 such that (Um , m = (xm , ym ), Rm ) = (U1 , (1 + x1 , −y1 ), (1 + I1 ) × (−δ1 , δ1 )) and such that for each j = 1, . . . , m, Ij is an open interval containing [tj −1 , tj ], 0 < δj < 1/2, C ∩ Uj = {p ∈ Uj | yj (p) = 0} = α(Ij ), and xj (α(t)) = t for each point t ∈ Ij . We may also choose the local charts so that !(I1 ) = !(Im ) < 1/2 and Ij ⊂ (0, 1) for j = 2, . . . , m − 1. We may fix disjoint m connected open sets {Wj }m j =1 and connected open sets {Vj }j =1 such that (i) We have γ ([tj −1 , tj ]) ⊂ Vj ⊂ Uj for each j = 1, . . . , m; (ii) We have Vm ∩ V1 ⊂ Wm ⊂ Um = U1 , and for each j = 1, . . . , m − 1, we have Vj ∩ Vj +1 ⊂ Wj ⊂ Uj ∩ Uj +1 ; (iii) We have Vi ∩ Vj = ∅ whenever γ ([ti−1 , ti ]) ∩ γ ([tj −1 , tj ]) = ∅ (that is, whenever i and j are indices with 1 ≤ i < j − 1 < m − 1 or 1 < i < j − 1 = m − 1); and −1 (iv) Vj ∩ −1 1 ({0} × (−δ1 , δ1 )) = Vj ∩ m ({1} × (−δm , δm )) = Vj ∩ Wm = ∅ for each j = 2, . . . , m − 1 (for this, we must first shrink δ1 , . . . , δm slightly so that −1 1 ({0} × (−δ1 , δ1 )) ∩ C = {γ (0)} = {γ (1)}). Let V ≡ V1 ∪ · · · ∪ Vm . We now alter the induced orientations inductively as follows. Suppose that 1 ≤ k ≤ m − 2 and that for j = 2, . . . , k, the local coordinates (xj −1 , yj −1 ) and (xj , yj ) induce the same orientation on the connected open subset Wj −1 of Uj −1 ∩ Uj ; that is, dxj ∧ dyj > 0 on Wj −1 dxj −1 ∧ dyj −1

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(we assume that this holds vacuously for k = 1). Then, by replacing yk+1 with −yk+1 if necessary, we may assume that the above holds for j = k + 1 as well. Proceeding inductively, we see that we may assume that dxj ∧ dyj >0 dxj −1 ∧ dyj −1

on Wj −1 ⊃ Vj −1 ∩ Vj

∀j = 2, . . . , m − 1.

In fact, the above condition holds for j = m as well. For if this were not the case, then the local coordinates (xm , −ym ) = (1 + x1 , y1 ) (on Um = U1 ) would induce the same orientation as the local coordinates (xm−1 , ym−1 ) on the set Wm−1 ⊃ Vm−1 ∩ Vm and the same orientation as the local coordinates (x1 , y1 ) on Wm ⊃ Vm ∩ V1 . But then V would be a connected orientable neighborhood of C, so we would arrive at a contradiction. Thus the condition holds for j = m. m−1 Let us now fix C ∞ functions λ and {λj }m−1 j =2 λj ≡ 1 on a j =2 such that λ + neighborhood 0 of C, 0 ≤ λ ≤ 1, supp λ ⊂ V1 ∪ Vm , and for each j = 2, . . . , m − 1, 0 ≤ λj ≤ 1 and supp λj ⊂ Vj . We may define the characteristic functions  χ0 ≡ and

1 on V ∩ −1 1 ([0, 1/2) × (−δ1 , δ1 )), 0 on V \ −1 1 ([0, 1/2) × (−δ1 , δ1 )),

 1 on V ∩ −1 m ((1/2, 1) × (−δm , δm )), χ1 ≡ −1 0 on V \ m ((1/2, 1) × (−δm , δm ))

−1 (observe that −1 1 ((−1/2, 0) × (−δ1 , δ1 )) = m ((1/2, 1) × (−δm , δm )), so χ0 · χ1 ≡ 0 but χ0 + χ1 ≡ 1 on V ∩ U1 = V ∩ Um ), and the mapping  : 0 → B given by 

≡

χ0 · λ · (x1 , (1/2) + y1 ) + χ1 · λ · (xm , (1/2) + ym )

+

m−1 

 λj · (xj , (1/2) + yj ) .

j =2

The Möbius band B has the C ∞ atlas {(Qj , j , R = (0, 1) × (0, 1))}2j =1 , where Q1 ≡ (R), 1 ≡ ( R )−1 , Q2 ≡ (([0, 1] \ {1/2}) × (0, 1)), and  (s + (1/2), t) if (s, t) ∈ [0, 1/2) × (0, 1), 2 ( (s, t)) ≡ (s − (1/2), 1 − t) if (s, t) ∈ (1/2, 1] × (0, 1). If  is a sufficiently small connected neighborhood of C in 0 , then −1 Wm ∩  ⊃  ∩ −1 1 ({0} × (−δ1 , δ1 )) =  ∩ m ({1} × (−δm , δm )).

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359

Moreover, we have λ ≡ 1 and λj ≡ 0 on Wm ∩  for j = 2, . . . , m − 1, (Wm ∩ ) ⊂ Q2 , and on Wm ∩ , we have 2 ◦  = ( 12 + x1 , 12 + y1 ). It follows that  is a C ∞ mapping on , and in fact, Wm ∩ is a diffeomorphism. We also have ( \ Wm ) ⊂ Q1 , and on a neighborhood of the set  \ Wm , we have (ρ1 , ρ2 ) ≡ 1 ◦   = λ · (χ0 x1 + χ1 xm ) +

m−1 

λ j · xj ,

j =2 −1

λ · (χ0 · (2

−1

+ y1 ) + χ1 · (2

+ ym )) +

m−1 

 λj · (2

−1

+ yj ) ,

j =2

which is of class C ∞ . Given a point p ∈ C \ Wm , we have (dρ1 ∧ dρ2 )p = λ2 (p) · ((χ0 (p) dx1 + χ1 (p) dxm ) ∧ (χ0 (p) dy1 + χ1 (p) dym ))p +

m−1 

λ(p)λj (p)((χ0 (p) dx1 + χ1 (p) dxm ) ∧ dyj )p

j =2

+

m−1 

λ(p)λj (p)(dxj ∧ (χ0 (p) dy1 + χ1 (p) dym ))p

j =2

+

m−1 

λi (p)λj (p)(dxi ∧ dyj )p

i,j =2

(here we  have used the fact that for p = γ (t), we have (xj (p), yj (p)) = (t, 0) and d(λ + j dλj )p = 0). Therefore, since (dxj ∧ dyj )p = (dxk ∧ dyj )p for all j and k with p ∈ Uj ∩ Uk , we have (dρ1 ∧ dρ2 )p = λ2 (p)χ0 (p)(dx1 ∧ dy1 )p + λ2 (p)χ1 (p)(dxm ∧ dym )p +

m−1 

λ(p)λj (p)(dxj ∧ dyj )p

j =2

+

m−1 

λ(p)λj (p)χ0 (p)(dx1 ∧ dy1 )p

j =2

+

m−1 

λ(p)λj (p)χ1 (p)(dxm ∧ dym )p

j =2

+

m−1  i,j =2

λi (p)λj (p)(dxj ∧ dyj )p .

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6 Holomorphic Structures on Topological Surfaces

Since the local charts induce the same orientations on the sets Vi ∩ Vj for all i = 1, . . . , m and j = 2, . . . , m − 1, the above 1-form is nonzero. Thus, by continuity, we may choose  so that dρ1 ∧ dρ2 is nonvanishing on a neighborhood of  \ Wm , and therefore, by the C ∞ inverse function theorem (Theorem 9.9.1 and Theorem 9.9.2),  is a local diffeomorphism of  onto a neighborhood of ([0, 1] × {1/2}) in B with (γ (t)) = ((t, 1/2)) for each point t ∈ [0, 1]. As in the proof of Lemma 5.11.1, we may choose  so small that  is a diffeomorphism onto (). Moreover, we may assume that for some  ∈ (0, 1/2), we have () = ([0, 1] × ((1/2) − , (1/2) + )). On the other hand, the mapping  : B → () given by (s, (1/2) + t) →

(s, (1/2) + 2t)

∀t ∈ (−1/2, 1/2)

is a well-defined diffeomorphism that is equal to the identity on ([0, 1] × {1/2}). Thus the mapping  ≡ −1 ◦  :  → B is a diffeomorphism with the required properties.  Lemma 6.10.2 Let M be a C ∞ surface. If every smooth Jordan curve in M admits an orientable neighborhood, then M is orientable. Proof We first show that if α : [a, b] → M is a path for which the inverse image α−1 (x) of each point x ∈ M contains at most two points, then the image α([0, 1]) admits an orientable neighborhood in M. For this, we let d ≡ sup{t ∈ (a, b] | α([a, t]) admits an orientable neighborhood} ∈ (a, b], c ≡ inf{t ∈ [a, d) | α([t, d]) admits an orientable neighborhood} ∈ [a, d). If α(c) = α(d), then we have α([c, c + ]) ∩ α([d − , d]) = ∅ for some (sufficiently small)  ∈ (0, (d − c)/2); and we may choose connected open sets 0 , 1 , and  such that 0 ∩ 1 = ∅, α([c, c + ]) ⊂ 0 , α([d − , d]) ⊂ 1 , α([c + , d − ]) ⊂ , and the connected open sets 0 ∪  and  ∪ 1 are orientable. We may choose orientations on each of these sets that agree on their (connected) intersection , and hence 0 ∪  ∪ 1 is an orientable neighborhood of α([c, d]), and in particular, [c, d] = [a, b]. Thus we may assume without loss of generality that α(c) = α(d), and we will denote this point by p. Next, we observe that the connected set α((c, d)) admits an orientable neighborhood in M. For α −1  (p) = {c, d}, so there exists a sequence of open intervals {Iν } such that (c, d) = ν Iν and such that for each ν = 1, 2, 3, . . . , we have Iν  Iν+1  (c, d)

and

α −1 (α(I ν )) ∩ [c, d] ⊂ Iν+1 .

By the choice of c and d, for each ν there exists a nonvanishing C ∞ real 2-form ων on a relatively compact connected neighborhood Uν of α(Iν+1 ) in M. Moreover, by inductively replacing ων with −ων whenever necessary, we may assume that (ων+1 /ων )q > 0

∀q ∈ α(Iν+1 ) (a connected set)

∀ν = 1, 2, 3, . . . .

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361

For each ν = 1, 2, 3, . . . , we may also choose a nonnegative C ∞ function λν on M such that supp λν ⊂ Uν , λν ≡ 1 on α(Iν ), and λν ≡ 0 on α([c, d] \ Iν+1 ). If {ν } is a sequence of positive numbers converging sufficiently fast to 0, then the series ∞ 

ν λν ων

ν=1 ∞ converges uniformly on compact subsets of M  (that is, for every nonvanishing C real 2-form ω on an open set U , the series ν λν ων /ω converges uniformly on compact subsets of U ) to a continuous real 2-form ω. Moreover, for each index μ = 1, 2, 3, . . . and each point q ∈ α(Iμ ), we have

(ω/ωμ )q =

∞ 

ν λν (q)(ων /ωμ )q ≥ μ > 0.

ν=1

Thus ω is nonzero at each point in α((c, d)) and therefore at each point in some connected neighborhood  of α((c, d)) in M \ {p}. Hence this neighborhood is orientable by Proposition 9.7.4. We may now fix a local C ∞ chart (U, , (0; 1)) with p = −1 (0) and α([c, d]) ⊂ U , a constant δ ∈ (0, (d − c)/2) with α([c, c + δ] ∪ [d − δ, d]) ⊂ U , and a constant r ∈ (0, 1) so small that for D ≡ −1 ((0; r)), D ∩ α([c + δ, d − δ]) = ∅. Let 0 and 1 be the connected components of  ∩ U containing α((c, c + δ]) and α([d − δ, d)), respectively. By replacing the neighborhood  with the connected component of the neighborhood ( \ D) ∪ 0 ∪ 1 containing α((c, d)), we may assume that  ∩ D = (0 ∪ 1 ) ∩ D and  ∩ D = (0 ∪ 1 ) ∩ D. We may also choose the orientation in  so that it agrees with the orientation induced by  in the connected open set 0 ⊂ U ∩ . For the proof of the existence of an orientable neighborhood of α([a, b]), it suffices to show that the two orientations also agree in 1 . For if they do agree, then  ∪ D is an orientable neighborhood of α([c, d]), and the claim follows. In particular, we may assume without loss of generality that 0 = 1 (i.e., that 0 ∩ 1 = ∅). Setting P ≡ −1 ((0; r/2)) and applying Lemma 5.10.6, we get an injective smooth path β : [0, 1] →  with β(0) ∈ α((c, d)) ∩ 0 ∩ P and β(1) ∈ α((c, d)) ∩ 1 ∩ P . Setting s ≡ max{t ∈ [0, 1] | β(t) ∈ 0 ∩ P } ∈ (0, 1), u ≡ min{t ∈ [s, 1] | β(t) ∈ 1 ∩ P } ∈ (s, 1), we get the piecewise smooth Jordan curve γ obtained by joining the path β[s,u] (with β((s, u)) ⊂  \ P ) and the path in P ⊂ D with image under  equal to the line segment from (β(u)) to (β(s)). Applying Lemma 5.10.5, we get a smooth Jordan curve C in  ∪ D that is the union of two arcs A and B (i.e., A and B are each the homeomorphic image of a compact interval) such that A ⊂ , B ⊂ D, and A ∩ B = {p0 , p1 } with pi ∈ i ∩ D for i = 0, 1 (see Fig. 6.6). In particular, since

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Fig. 6.6 Construction of an orientable neighborhood of the curve

by hypothesis, every smooth Jordan curve admits an orientable neighborhood, there exist connected open sets V and W such that A ⊂ V ⊂ , B ⊂ W ⊂ D, V ∩ W is equal to the union of two disjoint connected open sets Q0 and Q1 with pi ∈ Qi ⊂ i ∩ D for i = 0, 1, and V ∪ W is orientable. Fixing the orientation in V ∪ W so that the orientation induced in V agrees with that induced by , we see that it induces the same orientation in Q0 ⊂ V ∩ 0 ∩ D as . But then V ∪ W and  must induce the same orientation in the connected open set W ⊂ D and therefore in Q1 ⊂ V ∩ 1 ∩ D. Similarly, since V ∪ W and  induce the same orientation in V ⊃ Q1 ,  and  must induce the same orientation in Q1 ⊂ 1 ∩ D and therefore in 1 . Therefore, since  ∩ D = (0 ∪ 1 ) ∩ D, the neighborhood  ∪ D of α([c, d]) is orientable, and the claim concerning α follows. We now get an orientation in M as follows. Let us fix a point p0 ∈ M and a local C ∞ chart (U0 , 0 = (x0 , y0 ), (0; 1)) with p0 = −1 0 (0). Given a point p1 ∈ M \ {p0 }, we may choose an injective smooth path α1 in M from p0 to p1 and, by the above, a nonvanishing C ∞ real 2-form ω1 on a neighborhood of α1 ([0, 1]) such that (ω1 )p0 /(dx0 ∧ dy0 )p0 > 0. By Theorem 9.9.2, we may now fix a local C ∞ chart (U1 , 1 = (x1 , y1 ), (0; 1)) such that p1 = −1 1 (0), (dx1 ∧ dy1 )p1 / (ω1 )p1 > 0, and α1 ([0, 1]) ∩ U1 is connected. Consequently, (dx1 ∧ dy1 )/ω1 > 0 at each point in α1 ([0, 1]) ∩ U1 . We may cover M by such local C ∞ charts (note that even p0 ∈ U1 for a suitable choice of p1 ∈ M \ {p0 }). In order to check compatibility, let us suppose that p2 ∈ M \ {p0 } with p2 = p1 (the proof for p2 = p1 is similar) and that we have chosen an associated path α2 , form ω2 , and local chart (U2 , 2 = (x2 , y2 ), (0; 1)). If p2 ∈ U1 \ {p0 }, then we may form an injective smooth path β : [0, 1] → U1 \ {p0 } from p1 to p2 , and we may set r ≡ max β −1 (α1 ([0, 1])) ∈ [0, 1]

and

s ≡ α1−1 (β(r)).

The product of suitable reparametrizations of α1 [0,s] , β[r,1] , and α2− (omitting β if r = 1) is then a piecewise smooth loop γ . Moreover, γ takes any given value at most

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363

twice, so by the above, there is a nonvanishing real C ∞ 2-form ω on a neighborhood of γ ([0, 1]) with ωp0 /(dx0 ∧ dy0 )p0 > 0. Consequently, for q = α1 (s), we have (dx1 ∧ dy1 )q (ω1 )q (dx1 ∧ dy1 )q = · > 0. ωq (ω1 )q ωq Similarly, we have (dx2 ∧ dy2 )p2 /ωp2 > 0, and hence (dx1 ∧ dy1 )p2 ωp2 (dx1 ∧ dy1 )p2 = · > 0. (dx2 ∧ dy2 )p2 ωp2 (dx2 ∧ dy2 )p2 So the orientations agree in a neighborhood of p2 ∈ U1 ∩ U2 . If p2 ∈ M \ {p0 } is arbitrary and p ∈ U1 ∩ U2 , then we may form a local C ∞ coordinate neighborhood (U,  = (x, y), (0; 1)) of p as above but with U ⊂ U1 ∩ U2 . By the above, each of the local charts 1 and 2 induces the same orientation in U as  and therefore as the other. Thus the orientations are compatible on the overlap, and hence M is orientable.  Proposition 6.10.3 Let M be a C ∞ surface. Then M is nonorientable if and only if some open subset of M is homeomorphic to the Möbius band. Proof If M is nonorientable, then by Lemma 6.10.2, there exists a smooth Jordan curve C in M that does not admit an orientable neighborhood. Lemma 6.10.1 then provides a homeomorphism (in fact, a diffeomorphism) of some neighborhood onto the Möbius band. Conversely, suppose M is orientable and  :  → B is a homeomorphism of some open subset  of M onto the Möbius band B ≡ [0, 1] × (0, 1)/(0, t) ∼ (1, 1 − t)

∀t ∈ (0, 1)

with quotient map : [0, 1] × (0, 1) → B (in particular,  is second countable). Let γ : [0, 1] → B be the Jordan curve given by s → (s, 1/2), and let C ≡ γ ([0, 1]). For the continuous mapping R : B × [0, 1] → B given by R( (s, t), u) =

(s, (1/2) + (1 − u)(t − 1/2))

∀(s, t, u) ∈ [0, 1] × (0, 1) × [0, 1],

we have R(p, u) = p for each point (p, u) ∈ (C × [0, 1]) ∪ (B × {0}), and R(p, 1) ∈ C for each point p ∈ B; that is, R is a strong deformation retraction of B onto C. Hence, since [γ ] generates π1 (C), [γ ] must also generate π1 (B) (given a loop τ in B with base point τ (0) = γ (0), the mapping (t, u) → R(τ (t), u) is a path homotopy from τ to a loop in C). Moreover, the differential of the C ∞ function given by (s, t) → s for all (s, t) ∈ (0, 1) × (0, 1) extends to a unique closed C ∞ 1-form θ on B with γ θ = 1. It follows that H1 (B, R) = R · [γ ]H1 (B,R) ∼ = R, and hence the ∼ R (see Sect. 10.7). Jordan curve β ≡  −1 (γ ) satisfies H1 (, R) = R · [β]H1 (,R) = Now β is homologous in  to a sum of smooth Jordan curves (by Proposition 5.10.7), so there exists a smooth Jordan curve α : [0, 1] →  with nonzero

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homology class [α]H1 (,R) . On the other hand, the image A ≡ α([0, 1]) is nonseparating in . For Stokes’ theorem implies that any separating smooth Jordan curve for which the complement has a relatively compact connected component in  must be homologous to 0, while the complement of any compact subset of B has exactly one connected component that is not relatively compact in B (any connected component that is not relatively compact must contain the connected open set ([0, 1] × ((0, ) ∪ (1 − , 1))) for some sufficiently small  ∈ (0, 1/2)). Therefore, by Lemma 5.15.1, there exists a second smooth Jordan curve σ in  for which the homology classes [α]H1 (,R) and [σ ]H1 (,R) are linearly independent, which is impossible because [α]H1 spans H1 (, R). Thus we have arrived at a contradiction, and the proposition follows.  The above proposition implies that the following definition is not in conflict with the definitions of orientability and nonorientability for a smooth surface given in Sect. 9.7: Definition 6.10.4 A second countable topological surface M is called nonorientable if some open subset of M is homeomorphic to the Möbius band. The surface M is called orientable if no open subset of M is homeomorphic to the Möbius band. Exercises for Sect. 6.10 6.10.1 Let γ : [0, 1] → M be a (continuous) Jordan curve in a C ∞ surface M, and let C ≡ γ ([0, 1]). Assume that C does not admit an orientable neighborhood in M. Prove that there exists a homeomorphism  :  → B of some neighborhood  of C in M onto the Möbius band B ≡ [0, 1] × (0, 1)/(0, t) ∼ (1, 1 − t) ∀t ∈ (0, 1) with quotient map : [0, 1] × (0, 1) → B such that (γ (t)) = (t, 1/2) for each point t ∈ [0, 1] (cf. Exercise 6.7.7). 6.10.2 Let B be the Möbius band with associated C ∞ quotient map : [0, 1] × (0, 1) → B (see Exercise 6.10.1), let γ (t) ≡ (t, 1/2) for each point t ∈ [0,  1], and let C ≡ γ ([0, 1]). Prove that C is nonseparating in B, but γ θ = 0 for every closed C ∞ 1-form θ with compact support in B (cf. Proposition 5.15.2).

6.11 Smooth Structures on Second Countable Topological Surfaces The main goal of this section is the following: Theorem 6.11.1 Every second countable topological surface admits a smooth (surface) structure.

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Smooth Structures on Second Countable Topological Surfaces

365

Note that it is redundant to refer to a smooth surface structure, since no topological surface is homeomorphic to a manifold of dimension n = 2 (see Exercise 6.11.1). It follows from the above theorem and the results of Sect. 6.10 that the definition of the homology group H1 (M, A) provided in Sect. 10.7 (Definition 10.7.9) applies to any orientable second countable topological surface M and any subring A of C containing Z. That this group is isomorphic to the singular homology group is considered in Exercise 6.11.9. Theorem 6.11.1, Proposition 6.10.3, and Corollary 6.1.6 together give the following: Corollary 6.11.2 Every second countable orientable topological surface admits a (1-dimensional) holomorphic structure. We now address the proof of the theorem. The idea of the proof is to inductively form smooth structures on successively larger open sets. In the inductive step, one attaches a topological disk D ≈ (0; r) to an open set  with a smooth structure S, after first using Schönflies’ theorem to modify the homeomorphism on the overlap so that it becomes smooth with respect to S. We first consider the following case, in which one attaches an inner disk to a topological annulus . Lemma 6.11.3 Let K be a compact subset of  ≡ (0; 1), and let S0 be a (2-dimensional) smooth structure on  \ K. Then there exists a smooth structure S on  such that S = S0 near ∂; that is, S and S0 induce the same restricted smooth structure on the complement  \ K of some compact subset K ⊃ K of . Proof We may assume without loss of generality that K = (0; r) for some r ∈ (0, 1). A generator for the fundamental group of the annulus  \ K = (0; r, 1) is homologous to a sum of Jordan loops that are smooth with respect to the smooth structure S0 (by Proposition 5.10.7). Hence there is a Jordan curve γ : [0, 1] →  \ K that is not homologous to 0 and that is smooth with respect to S0 . The image C ≡ γ ([0, 1]) is separating in P1 (by the Jordan curve theorem) and therefore in  \ K. Moreover,  \ K is orientable (as shown in Sect. 6.10, orientability of surfaces is a topological property), so Stokes’ theorem implies that no connected component of ( \ K) \ C is relatively compact in  \ K (one may also see this directly). Therefore, choosing  ∈ (0, (1 − r)/2) so small that C ⊂ (0; r + , 1 − ), we see that the set ( \ K) \ C has exactly two connected components U0 and U1 , where U0 ⊃ (0; r, r + ) and U1 ⊃ (0; 1 − , 1). Hence the domain 0 ≡ (0; r + ) ∪ U0   satisfies ∂0 = C. By Lemma 5.11.1, for some R > 1, there are a relatively compact connected neighborhood A of C in  \ K and a homeomorphism 0 : A → (0; 1/R, R) such that 0 (C) = S1 , 0 is a diffeomorphism with respect to the smooth structure on the domain A induced by S0 and the standard smooth structure on the target (0; 1/R, R), and A∩0 = A∩U0 = 0−1 ((0; 1/R, 1)). By Schönflies’ theorem, there exists a homeomorphism 1 : P1 → P1 such that 1 C = 0 C : C → S1 and 1 0 maps 0 diffeomorphically (with respect to the standard smooth structure

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Fig. 6.7 Construction of the local chart (, , (0; R))

Fig. 6.8 Extension of (a modification of) the smooth structure S0 on  to a smooth structure S on a neighborhood of  ∪ K

on both the domain and the target) onto  (and 1 P1 \0 maps P1 \ 0 diffeomorphically onto P1 \ ). Setting  ≡ 0 ∪ A  0 and  1 (z) for z ∈ 0 , (z) ≡ 0 (z) for z ∈ A \ 0 =  \ 0 = 0−1 ({ζ ∈ C | 1 ≤ |ζ | < R}), we get a homeomorphism  :  → (0; R) such that \0 = 0 \0 :  \ 0 → (0; 1, R) is a diffeomorphism with respect to the smooth structure induced by S0 on the domain and the standard smooth structure on the target (see Fig. 6.7). Now let A be the collection of local charts that are equal either to (, , (0; R)) or to (U \ 0 , U \0 , (U \ 0 )) for some local C ∞ chart (U, , (U )) in the smooth surface ( \ K, S0 ). Then A is a C ∞ atlas on , and the corresponding C ∞ structure S induces the same C ∞ structure as S0 on the open subset   \ 0 ⊂  \ K. The following lemma is the main step in the proof of Theorem 6.11.1: Lemma 6.11.4 Let M be a second countable topological surface, let  and  be open subsets with  ⊂ , let S0 be a (2-dimensional) smooth structure on , let (D, ,  ≡ (0; 1)) be a local C 0 chart in M, and let K be a compact subset of D. Then there exists a smooth structure S on a neighborhood of  ∪ K such that S and S0 induce the same smooth structure on  \ D (see Fig. 6.8).

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367

Proof We may fix an open set 0 with  ⊂ 0 ⊂ 0 ⊂  (by Lemma 9.3.6) and a constant r ∈ (0, 1) with K ⊂ D0 ≡ −1 ((0; r))  D. Given distinct points ζ1 , . . . , ζm ∈ ∂(0; r) arranged in counterclockwise order, we let ζ0 ≡ ζm , and for each j = 1, . . . , m, we let αj : [0, 1] → (0; r) be an injective parametrization of the line segment from ζj −1 to ζj , we let βj : [0, 1] → ∂(0; r) be an injective parametrization of the counterclockwise arc of the circle ∂(0; r) from ζj −1 to ζj , we let γj ≡ αj− ∗ βj , we let Cj ≡ γj ([0, 1]), and we let ηj : S1 → Cj be the associated homeomorphism given by e2πit → γj (t). Exactly one of the two connected components of P1 \ Cj , which we will denote by Vj , is contained in (0; r) and satisfies ∂Vj = Cj (this follows from Schönflies’ theorem, but it is also easy to verify directly in this case). We also set Uj ≡ −1 (Vj ) ⊂ D0 for j = 1, . . . , m and pj ≡ −1 (ζj ) for j = 0, . . . , m. We may choose such boundary points ζ0 , ζ1 , . . . , ζm = ζ0 so close together that if j ∈ {1, . . . , m} with U j ∩ 0 = ∅, then U j is contained in some local C ∞ chart (Wj , j , (0; 1)) in the C ∞ surface (, S0 ). Schönflies’ theorem (Theorem 6.7.1) then allows us to modify  into a diffeomorphism with respect to S0 on each such set Uj , while leaving the values elsewhere unchanged. More precisely, if j ∈ {1, . . . , m} with U j ∩ 0 = ∅, then Schönflies’ theorem provides a homeomorphism  → V j that maps  diffeomorphically onto Vj and that is equal to the homeomorphism ηj : S1 → Cj on S1 = ∂, as well as a homeomorphism j (U j ) = j (Uj ) →  that maps j (Uj ) diffeomorphically onto  and that 1 is equal to the homeomorphism ηj−1 ◦  ◦ −1 j : j (∂Uj ) → S on j (∂Uj ) = ∂j (Uj ). The associated composition U j → j (U j ) →  → V j then yields a homeomorphism j : U j → V j such that j ∂Uj = ∂Uj : ∂Uj → ∂Vj and j Uj : Uj → Vj is a diffeomorphism with respect to the smooth structure on Uj induced by S0 and the standard smooth structure on Vj ⊂ C (see Fig. 6.9). Thus the mapping  : D 0 → (0; r) given by ⎧ ⎪ ⎨(x) if x ∈ D 0 \ (U1 ∪ · · · ∪ Um ), (x) ≡ (x) if x ∈ U j with j ∈ {1, . . . , m} and U j ∩ 0 = ∅, ⎪ ⎩ j (x) if x ∈ U j with j ∈ {1, . . . , m} and U j ∩ 0 = ∅, is a well-defined homeomorphism that maps each of the sets Uj ⊂ 0 with closure meeting 0 diffeomorphically, with respect to S0 , onto its image Vj ⊂ C (which has the standard C ∞ structure). Setting     m m m    −1 −1 (0; r) (0; r) Q≡ Vj = V j = D0 Uj, j =1

j =1

we may now let A1 be the collection of local C 0 charts in the open set 1 ≡ (0 \ {p1 , . . . , pm }) ∪ D0 = (0 \ Q) ∪ D0 ,

j =1

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Fig. 6.9 Construction of a smooth structure outside the vertices of the polygonal region Q

which are equal either to (D0 , D0 , (0; r)) or to   W ∩ 0 \ Q, (W ∩0 \Q) , (W ∩ 0 \ Q) for some local C ∞ chart (W, , (W )) in (, S0 ) with W ∩ 0 = ∅. This collection A1 is then a C ∞ atlas, because for any local C ∞ chart (W, , (W )) in (, S0 ), we have  m   Vj (W ∩ 0 \ Q) ∩ D0 = W ∩ 0 ∩ −1  = W ∩ 0 ∩

j =1



 Uj ,

1≤j ≤m,U j ∩0 =∅

and with respect to the C ∞ structure S0 ,  maps this set diffeomorphically onto its image. Clearly, the C ∞ structure S1 in 1 determined by A1 agrees with S0 in the open subset 0 \ Q ⊃ 1 \ D ⊃  \ D. So far, we have obtained a smooth structure on the set 1 ⊃ ( \ {p1 , . . . , pm }) ∪ K. Letting J ⊂ {1, . . . , m} be the set of indices j for which pj ∈ 0 , we will expand the set of definition for our smooth structure so as to include these points. We may choose a collection of disjoint local C 0 charts {(Ej , j , )}j ∈J with pj = −1 j (0) and Ej ⊂ 0 ∩ D for each index j ∈ J . In particular, for each j ∈ J , we have Ej \{pj } ⊂ 1 . Hence the local charts in ∗ = \{0} given by the compositions of local C ∞ charts in (1 , S1 ) meeting Ej with −1 j ∗ determine a smooth

∗ structure Rj on ∗ for which the mapping −1 j ∗ : ( , Rj ) → (Ej \ {pj }, S1 ) is a diffeomorphism. Therefore, by Lemma 6.11.3, there is a smooth structure Qj on  that induces the same smooth structure as Rj on the annulus (0; uj , 1) for some uj ∈ (0, 1).  Setting N ≡ j ∈J −1 j ((0; uj )), we may now let A2 be the collection of local 0 C charts in the open set

2 ≡ 0 ∪ D0 = 1 ∪ {pj | j ∈ J } ⊃  ∪ K

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369

∞ that are equal either to (−1 j (W ),  ◦ j , P ) for some local C chart (W, , P ) in (, Qj ), or to (W \ N, W \N , (W \ N)) for some local C ∞ chart (W, , (W )) in (1 , S1 ) with W ⊂ N . The collection A2 is then a C ∞ atlas on 2 that induces the same C ∞ structure as S1 on the open set

2 \ N = 1 \ N ⊃ 1 \ D ⊃  \ D. Since S1 induces the same C ∞ structure as S0 on  \ D, the claim follows.



We are now ready for the proof of Theorem 6.11.1. In fact, we will prove the following slightly stronger version: Theorem 6.11.5 Let M be a second countable topological surface, let S0 be a (2-dimensional) smooth structure on some (nonempty) open subset  of M, and let K be a closed subset of M with K ⊂ . Then there exists a smooth structure S on M that induces the same smooth structure as S0 on some neighborhood of K. Proof We will assume that M \  is noncompact, the proof for M compact being similar. We may fix an open set 0 with K ⊂ 0 ⊂ 0 ⊂ , a locally finite collec∞ tion of local C 0 charts {(Uν , ν ,  ≡ (0; 1))}∞ ν=1 , and a sequence {rν }ν=1 in (0, 1) −1 such that Dν ≡ ν ((0; rν ))  Uν  M \ K for each ν and M \ 0 ⊂

∞ 

Dν .

ν=1

We now set 0 ≡ , 0 ≡ 0 , and ν ≡ 0 ∪ D1 ∪ · · · ∪ Dν for each ν = 1, 2, 3, . . . , and we proceed inductively. Suppose that for each μ = 0, . . . , ν − 1, we have constructed a (2-dimensional) C ∞ structure Sμ on an open set μ ⊃ μ . Assume also that for each μ = 1, . . . , ν −1, Sμ and Sμ−1 induce the same C ∞ structure on the set μ−1 \ U μ . Then, by Lemma 6.11.4, there exists a smooth structure Sν on an open set ν ⊃ ν−1 ∪ D ν = ν that induces the same smooth structure as Sν−1 on the set ν−1 \ U ν . Thus we get a sequence of sets with smooth structures {(ν , Sν )}. These smooth structures eventually stabilize on any given compact set to give a well-defined global smooth structure. More precisely, by local finiteness, given a point p ∈ M, there are an index ν0 and a neighborhood W of p such that W ∩ Uν = ∅ for all ν > ν0 . In particular, we have W ⊂ ν0 , and we may choose W so that there is a local C ∞ chart (W, , V ) for (ν0 , Sν0 ). By construction, this local chart is a local C ∞ chart with respect to Sν for every ν ≥ ν0 . It is now easy to see that the collection A of all such local charts is a smooth surface atlas on M, and that the associated  smooth structure S induces the same smooth structure on the  neighborhood M \ ν U ν of K as S0 . Exercises for Sect. 6.11 6.11.1 Let  be a nonempty open subset of R2 . Prove that  is not homeomorphic to an open subset of Rn for n = 2.

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Hint. The complement of a point in any connected neighborhood in R2 is not simply connected. 6.11.2 Prove that if M is a compact orientable topological surface with H1 (M, R) = 0, then M is homeomorphic to a sphere. 6.11.3 Prove that if M is a second countable noncompact orientable topological surface with H1 (M, R) = 0, then M is homeomorphic to R2 . 6.11.4 Let M be a second countable topological surface. The goal of this exercise is a proof of a theorem of Radó, according to which M admits a triangulation. A triangle in M is a homeomorphism  : T → T of a geometric closed triangle T ⊂ R2 onto a compact set T ⊂ M (in a slight abuse of notation, we also denote  by T ). The image of each vertex of T is called a vertex of  (or a vertex of T ), and the image of each edge of T is called an edge of  (or an edge of T ). A triangulation of M is a locally finite covering {Ti }i∈I of M by triangles such that for each pair of distinct indices i, j ∈ I , Ti ∩ Tj is either empty, a common vertex of Ti and Tj , or a common edge of Ti and Tj . ◦

◦

(a) Prove that if  : T → T is a triangle in M, then (T ) = T , ◦

(∂T ) = ∂T , ∂T is a separating Jordan curve in M, and T is a connected component of M \ ∂T . (b) Suppose {Ti }i∈I is a triangulation of M, i ∈ I , and E is an edge of Ti . Prove that there is a unique index j ∈ I \ {i} such that E is an edge of Tj . Prove also that Ti and Tj have exactly two vertices in common, namely, the endpoints of E. (c) Prove, in the following way, that if M is orientable, then M admits a triangulation (this is a special case of the above theorem of Radó). By the results of this chapter, M admits the structure of a Riemann surface. The proof of Proposition 5.18.4 gives a locally finite covering of M by triangles with disjoint interiors. Add more edges so that no vertex will lie in the interior of an edge, and no two distinct edges will meet in more than one point. (d) Prove, in the following way, that M admits a triangulation even if M is nonorientable (that is, prove the above theorem of Radó). By Theorem 6.11.1, M admits the structure of a C ∞ surface. Apply the 1-dimensional C ∞ version of Sard’s theorem (see Exercise 9.6.5) in order to construct a suitable graph in M as in the proof of Proposition 5.18.4, and then proceed as in the proof of part (c). 6.11.5 Let γ : [0, 1] → M be a Jordan curve with image C in a topological surface M. (a) Prove that if C admits a (topologically) orientable neighborhood, then for some R > 1, there exist a neighborhood  of C in M and a homeomorphism  :  → (0; 1/R, R) with (γ (t)) = e2πit for each t ∈ [0, 1] (cf. Exercise 6.7.7). (b) Prove that if C does not admit a (topologically) orientable neighborhood, then there exists a homeomorphism  :  → B of some neigh-

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371

borhood  of C in M onto the Möbius band B ≡ [0, 1] × (0, 1)/(0, t) ∼ (1, 1 − t)

6.11.6

6.11.7

6.11.8

6.11.9

∀t ∈ (0, 1),

with quotient map : [0, 1] × (0, 1) → B, such that (γ (t)) = (t, 1/2) for each point t ∈ [0, 1] (cf. Exercise 6.10.1). Let M be a second countable orientable topological surface, let γ : [0, 1] → M be a Jordan curve, and let C ≡ γ ([0, 1]). Prove that C is separating in M ˇ 1-form θ with compact support in M. if and only if γ θ = 0 for every Cech Prove also that every Jordan curve in M is separating if and only if every ˇ Cech 1-form θ with compact support in M is exact. Hint. Apply Exercise 6.7.2. Prove that every compact orientable topological surface M may be obtained by C 0 attachment of a locally finite family of disjoint tubes (see Exercise 5.12.8) to the unit sphere S2 . Let M be a nonorientable second countable topological surface. Prove that  → M for which there exists a connected orientable covering space ϒ : M →M each fiber contains exactly two elements (as in Exercise 9.7.8, ϒ : M is called the orientable double cover of M). This exercise requires facts considered in Exercises 6.6.2–6.6.4. Let M be a second countable topological surface. (a) Prove that the universal cover of M is homeomorphic to the plane R2 or the sphere S2 . (b) Prove that if M is orientable, then π1 (M) is torsion-free and H1 (M, Z) ∼ = π1 (M)/[π1 (M), π1 (M)], while if M is nonorientable, then every torsion element of π1 (M) has order 1 or 2 and π1 (M)/  ∼ = Z/2Z for some torsion-free normal subgroup  of π1 (M). (c) Let A be a subfield of C containing Z, and let F = R or C. Prove that if M is orientable, then H1 (M, A) ∼ = H1 (M, A), while if M is nonorientable, then H1 (M, F) ∼ = H1 (M, F). (d) Prove that if M is orientable, then for any subring A of C containing Z, the mapping [ξ ]H1 (M,A) → ([ξ ]H1 (M,A) , ·)deR gives an injective homomorphism H1 (M, A) → Hom(H 1 (M, A), A) (according to Theorem 10.7.18, this homomorphism is also surjective if π1 (X) is finitely generated). (e) Assume that M is orientable, and let K ⊂ M be a compact set. Prove that there exist domains  and  such that K ⊂     M and such that im[H 1 (M, C) → H 1 (, C)] = im[H 1 ( , C) → H 1 (, C)].

Part III

Background Material

Chapter 7

Background Material on Analysis in Rn and Hilbert Space Theory

In this chapter, we recall some basic definitions and facts concerning integration and Hilbert spaces.

7.1 Measures and Integration In this section, we recall some basic facts from measure theory. The proofs, which are omitted, can be found in, for example, [Fol], [Rud1], or [WZ]. Definition 7.1.1

Let X be a set.

(a) A σ -algebra in X is a collection M of subsets of X such that (i) We have ∅ ∈ M; (ii) X  \ S ∈ M for each set S ∈ M; and (iii) j ∈J Sj ∈ M for each countable family of sets {Sj }j ∈J in M. The pair (X, M), which is often denoted simply by X, is then called a measurable space, and any element of M is called a measurable set. A mapping f : S → Y of a measurable set S ⊂ X into a topological space Y (see Sect. 9.1) is called a measurable mapping if f −1 (U ) is measurable for each open set U ⊂ Y . For Y = [−∞, ∞] or C, we also call f a measurable function. (b) A positive measure on X (or on (X, M)) is a function μ: M → [0, ∞] on ∞  S ) = a σ -algebra M in X such that μ(∅) = 0 and μ( ∞ j j =1 j =1 μ(Sj ) for every sequence of disjoint measurable sets {Sj }. For each measurable set S, μ(S) is called the measure of S. The pair (X, μ) (or triple (X, M, μ)) is called a measure space. The measure μ is complete if every subset of a set of measure 0 is measurable (and hence of measure 0). We say that a property of points holds almost everywhere (abbreviated a.e.) in a set S ⊂ X if the property holds for all points in the complement S \ A of some set A ∈ M of measure 0. Remarks 1. If f is a sum or product of measurable functions, a composition of a continuous function with a measurable mapping, or a supremum, infimum, or limit of a sequence of measurable functions, then f is measurable. T. Napier, M. Ramachandran, An Introduction to Riemann Surfaces, Cornerstones, DOI 10.1007/978-0-8176-4693-6_7, © Springer Science+Business Media, LLC 2011

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376

2. We will occasionally consider measure spaces (X, M, μ) that are not com denote the collection of sets of the form plete (see Sect. 9.7). However, letting M A = E ∪ B, where E ∈ M and B is a subset of some set of measure 0, and letting μ(A) ˆ = μ(E) for each such set A, one gets a well-defined complete measure space  μ) (X, M, ˆ called the completion of (X, M, μ) (see, for example, [Rud1]). Definition 7.1.2 Let (X, μ) be a measure space, and let f be a measurable function on a measurable set S ⊂ X. (a) If f ≥ 0 a.e. in S, then the integral of f over S is given by 

 f (x) dμ(x) ≡

S

f dμ ≡ sup S

m 

rj μ(Sj ) ∈ [0, ∞],

j =1

where the supremum is taken over all choices of disjoint measurable subsets S1 , . . . , Sm of S and constants r1 , . . . , rm ∈ [0, ∞] with f ≥ rj a.e. in Sj for each j = 1, . . . , m. (b) We say that f is integrable on S if the nonnegative extended real numbers   R+ ≡ [Re f ]+ dμ, R− ≡ [Re f ]− dμ, S

 I+ ≡

S

[Im f ]+ dμ,

S

 I− ≡

[Im f ]− dμ,

S

are finite. If this is the case, then the integral of f over S is given by  f dμ ≡ R+ − R− + iI+ − iI− . S

Remarks 1. Every measurable subset Y of a measure space (X, μ) inherits a positive measure given by S → μ(S) for every measurable set S with S ⊂ Y . Integrals of functions over measurable subsets of S with respect to this induced measure then agree with the corresponding integrals with respect to μ. In an abuse of notation, we also denote the induced measure by μ. 2. A simple function ϕ on a measurable space X is a measurable complex-valued function  with finite range. Letting r1 , . . . , rn be the distinct values of ϕ, we have ϕ = nj=1 rj χSj , where Sj = ϕ −1 (rj ) for j = 1, . . . , n, and for any set E ⊂ X, χE denotes the corresponding characteristic function (i.e., χE ≡ 1 on E and χE ≡ 0 on X \ E). positive measure on X, S ⊂ X is measurable, and ϕ ≥ 0, then  If μ is a we have S ϕ dμ = nj=1 rj μ(Ej ∩ S). Consequently, if f is a measurable function on S and f ≥ 0 a.e., then   f dμ = sup ϕ dμ | ϕ is a nonnegative simple function and ϕ ≤ f a.e. in S . S

S

Example 7.1.3 (Counting measure) Let X be a set, let M be the collection of all subsets of X (which is clearly a σ -algebra), and let μ(S) ∈ Z≥0 ∪ {∞} ⊂ [0, ∞] be

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377

the number of elements of S for each set S ⊂ X. Then μ is a positive measure that is called the counting measure   on X. For any function f : X → [0, ∞] and any set S ⊂ X, we have f dμ = x∈S f (x), the (unordered) sum of f over S (recall that S   f (x) ≡ sup f F ∈F x∈S x∈F (x), where F is the set of all finite subsets of S). In particular, this sum is infinite if f −1 ((0, ∞]) is uncountable. A function f : X → C is integrable if and only if x∈S f (x) is absolutely summable, and if this is the case, then the integral of f over S is equal to the sum. Example 7.1.4 (Lebesgue measure) For n ∈ Z>0 , an n-cell is a Cartesian product Q = I1 × · · · × In ⊂ Rn of n

intervals in R. For any n-cell Q as above, the volume of Q is given by vol(Q) ≡ ni=1 (Ii ) ∈ (0, ∞] . For any set S ⊂ X, the Lebesgue outer measure of S is given by ∞  λ∗ (S) ≡ inf vol(Qj ) | {Qj } is a sequence of open n-cells that covers S . j =1

 ∞ ∗ This function is countably subadditive; that is, λ∗ ( ∞ j =1 Sj ) ≤ j =1 λ (Sj ) for any sequence of subsets {Sj } of Rn . A set S ⊂ Rn is Lebesgue measurable if it satisfies the Carathéodory criterion: λ∗ (A) = λ∗ (A ∩ S) + λ∗ (A \ S)

∀A ⊂ Rn .

The collection M of Lebesgue measurable sets is a σ -algebra, and the function λ ≡ λ∗ M : M → [0, ∞], which is called Lebesgue measure, is a positive measure with the following properties: (i) Completeness. Every subset of a set of Lebesgue measure 0 is measurable (in fact, any set of Lebesgue outer measure 0 is Lebesgue measurable). (ii) Regularity. Every open set is Lebesgue measurable (hence M contains the Borel σ -algebra, i.e., the smallest σ -algebra containing the open sets). Moreover, if a set S ⊂ Rn is Lebesgue measurable, then for every  > 0, there exist a closed set A and an open set B such that A ⊂ S ⊂ B and λ(B \ A) < . Consequently, there exist sets F and G such that F is a countable union of closed sets (i.e., F is an Fσ ), G is a countable intersection of open sets (i.e., G is a Gδ ), F ⊂ S ⊂ G, and λ(G \ F ) = 0 (the completeness property (i) and the measurability of Borel sets imply that the converse is also true). (iii) Normalization. For each n-cell Q, λ(Q) = vol(Q). (iv) Translation invariance. For each Lebesgue measurable set S ⊂ Rn and each point x ∈ Rn , the set S + x is Lebesgue measurable and λ(S + x) = λ(S). For n = 1 and for I an interval with endpoints a < b, we often denote the integral b b of a (nonnegative measurable or integrable) function over I by a f or a f (x) dx.

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Theorem 7.1.5 (Convergence theorems) Let (X, μ) be a measure space, and let {fν } be a sequence of measurable functions on a measurable set S ⊂ X. (a) Monotone convergence theorem. If 0 ≤ f1 ≤ f2 ≤ · · · , then   lim fν = lim fν . S ν→∞

ν→∞ S

(b) Fatou’s lemma. If fν ≥ 0 for each ν, then   lim inf fν ≤ lim inf fν . S ν→∞

ν→∞

S

(c) Lebesgue’s dominated convergence theorem. If fν → f and there is a nonnegative integrable function g such that |fν | ≤ g on S for each ν ∈ Z>0 , then f and fν for each ν are integrable and   f = lim fν . ν→∞ S

S

Definition 7.1.6 Let V be a vector space over the field F = R or C. (a) A function  ·  : V → [0, ∞) is called a norm if (i) For each vector v ∈ V \ {0}, we have v > 0; (ii) For each vector v ∈ V and each scalar a ∈ F, we have av = |a|v; and (iii) Triangle inequality. We have u + v ≤ u + v for all u, v ∈ V. The pair (V,  · ) (or simply V) is called a normed vector space. (b) A sequence {vj } in V converges to v in (V,  · ) if vj − v → 0 as j → ∞. We also call v the limit of {vj }, and we write v = lim vj j →∞

or

vj → v

as j → ∞.

A sequence that is not convergent is said to be divergent. (c) A sequence {vj } in (V,  · ) is a Cauchy sequence if for every  > 0, there exists an N ∈ Z>0 such that vi − vj  <  for all i, j > N . If every Cauchy sequence in V converges, then (V,  · ) is called a complete normed vector space or a Banach space. (d) A set U ⊂ V is open in (V,  · ) if for each vector v ∈ U , the ball of radius r centered at v, B(v; r) ≡ {u ∈ V | u − v < r}, is contained in U for some r > 0. A set D ⊂ V is closed if its complement V \ D is open. Equivalently, D is closed if the limit of every convergent sequence of vectors {vj } in V with ◦

vj ∈ D for each j lies in D. The interior S of a set S ⊂ V is the union of all open sets that are contained in S. The closure S (or cl(S)) of S is the intersection of all closed sets containing S; that is, S is the set of limits of sequences in S that converge in V. Remarks 1. If {uj } and {vj } are sequences in a normed vector space (V,  · ), and uj → u and vj → v, then uj + vj → u + v, and for each scalar c, cvj → cv.

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379

2. For vectors u and v in (V,  · ), we also have |u − v| ≤ u + v. In particular, if vj → v, then vj  → v. 3. The closure W of any (vector) subspace W of (V,  · ) is a subspace. 4. Any closed subspace of a Banach space is a Banach space with respect to the restriction of the given norm. 5. If V is a finite-dimensional vector space, then V is complete with respect to any norm. Moreover, any two norms in V are equivalent, so convergence of sequences, openness of sets, etc., are independent of the choice of norm. Moreover, a sequence {vj } in V converges to a vector v if and only if α(vj ) → α(v) for every linear functional α on V (equivalently, for some basis α1 , . . . , αn of the dual space V ∗ , αk (vj ) → αk (v) for k = 1, . . . , n). Example 7.1.7 For F = R or C, Fn together with the Euclidean norm

n 1/2  2 |ζj | ∀ζ = (ζ1 , . . . , ζn ) ∈ Fn ζ  = j =1

(in fact, with any norm) is a Banach space (we set F0 = {0}). Definition 7.1.8 Let (X, μ) be a measure space, and let p ∈ [1, ∞]. (a) Let f be a measurable function on X. If p < ∞, then for each measurable function f , the Lp norm of F is given by  1/p |f |p dμ . f Lp (X,μ) ≡ X

The L∞ norm of f is given by f L∞ (X,μ) ≡ inf{R ∈ [0, ∞] | |f | ≤ R a.e.}. We also denote the Lp norm by  · Lp (μ) or  · Lp (X) or  · Lp when the measure or space is clear from the context. (b) The Lp space is given by Lp (X, μ) ≡ {f | f is a measurable function with f Lp < ∞}/∼, where f ∼ g if and only if f = g a.e. We usually denote each equivalence class by a representative f . We also denote the Lp space by Lp (μ) or Lp (X) or Lp when the measure or the space are clear from the context. (c) For Lebesgue measure λ on Rn , we often denote Lp (Y, λ) simply by Lp (Y ) p for any Lebesgue measurable set Y ⊂ Rn . We let Lloc (Y ) denote the set of measurable functions f on Y such that each point in Y admits a neighborhood U p in Rn such that f U ∩Y ∈ Lp (U ∩ Y ) (in particular, for Y an open set, Lloc (Y ) is the set of functions for which the restriction to each compact subset of Y is in Lp ), where as in (b), we identify any two measurable functions that are equal almost everywhere. Elements of L1loc are called locally integrable.

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Theorem 7.1.9 If (X, μ) is a complete measure space and p ∈ [1, ∞], then (Lp (X, μ),  · Lp ) is a Banach space. Moreover, if {fν } is a sequence that converges in Lp (X, μ) to a function f ∈ Lp (X, μ), then some subsequence of {fν } converges pointwise almost everywhere to f . Remarks 1. In this book, the fact that  · Lp is a norm is used, in an essential  way, only for p ∈ {1, 2, ∞}. For p = 1 this follows easily from the inequality | f| ≤  |f |; for p = ∞, this follows easily from the triangle inequality for | · |; and for p = 2, this follows from the observation that  · L2 is the norm associated to an inner product (see Sect. 7.5). Completeness of Lp is used only for p = 2 or ∞. For the measure spaces of interest in this book, completeness of L2 follows from Proposition 2.6.3. The proof of completeness of Lp for p ∈ [1, ∞) is similar. The proof of completeness of L∞ is relatively easy (see, for example, [Rud1]). The claim concerning pointwise convergence of a subsequence may be verified directly as follows. We may choose a subsequence {fνk } such that ∞>

∞ 

p fνk − f Lp

=

k=1

  ∞

|fνk − f |p

k=1

(for example, by the monotone convergence theorem). Hence the integrand is finite almost everywhere, and the claim follows. 2. For any compact set K ⊂ Rn , the vector space C 0 (K) of continuous functions on K is a closed vector subspace of L∞ (K), since uniform convergence preserves continuity. 3. We also recall that for p, q ∈ [1, ∞] with p1 + q1 = 1 and for measurable functions f, g on (X, μ), we have the Hölder inequality f gL1 (X,μ) ≤ f Lp (X,μ) · gLq (X,μ) . In this book, this inequality is used, in an essential way, only for the (relatively easy) cases p = 1 and p = 2. 4. For a measurable set Y ⊂ Rn , we say that a sequence of measurable functions p {fν } on Y converges in Lloc (Y ) to a function f on Y if each point in Y admits a neighborhood U such that f U ∩Y and fν U ∩Y for each ν are in Lp (U ∩ Y ), and fν − f Lp (U ∩Y ) → 0. Theorem 7.1.10 (Fubini’s theorem) Let m, n ∈ Z>0 , and let f be a (Lebesgue) measurable function on Rm+n = Rm × Rn . Then m (a) For almost every point on Rn .  x ∈ R , the function f (x, ·) is measurable m (b) The function x → Rn |f (x, y)| dλ(y) is measurable on R . Moreover, f is integrable on Rm+n if and only if this function is integrable on Rm . (c) If f is nonnegative or integrable on Rm+n , then     f dλ = f (x, y) dλ(y) dλ(x) Rm+n

Rm

Rn

Rn

Rm

  =

 f (x, y) dλ(x) dλ(y).

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381

A proof is outlined in Exercise 7.1.3 (a proof for a general σ -finite product measure space can be found in, for example, [Fol] or [Rud1]). Exercises for Sect. 7.1 7.1.1 Let λ∗ and λ denote Lebesgue outer measure and Lebesgue measure, respectively, on Rn (see Example 7.1.4). (a) Verify that λ∗ is countably subadditive. (b) Verify that λ is a positive measure and that λ has the properties (i)–(iv) in Example 7.1.4. 7.1.2 Let f be a nonnegative measurable function on a measure space (X, μ). Prove that there is a sequence of simple functions {ϕν }∞ ν=1 on X such that 0 ≤ ϕν ≤ ϕν+1 for each ν ∈ Z>0 , ϕν → f as ν → ∞, and ϕν A → f A uniformly on any set A ⊂ X on which f is bounded. Hint. For example, consider ν

ν2  j −1 · χf −1 ([(j −1)/2ν ,j/2ν )) + ν · χf −1 ([ν,∞]) . ϕν ≡ 2ν j =1

7.1.3 The goal of this exercise is a proof of Fubini’s theorem (Theorem 7.1.10). Let m, n ∈ Z>0 , and for every set S ⊂ Rm × Rn = Rm+n and every point x ∈ Rm , let Sx ≡ {y ∈ Rn | (x, y) ∈ S}. Also fix an open (m + n)-cell R = A × B with A  Rm and B  Rn ; let S0 be the collection of all measurable sets S ⊂ R for which the set Sx ⊂ B ⊂ Rn is measurable for almost every point x ∈ A; and let S ⊂ S0 be the subcollection consisting of all sets S ∈ S0 for which the function on A given (almost everywhere) by x → λ(Sx ) is measurable and  λ(S) = λ(Sx ) dλ(x) A

(here, λ(S) and λ(Sx ) denote the Lebesgue measure in Rm+n and Rn , respectively). (a) Prove that S0 is a σ -algebra in R, and that S0 contains the collection of Borel subsets of R. (b) Prove that ∅ ∈ S and that R \ S ∈ S for each set S ∈ S. (c) Let {Sν } be a sequence of sets inS. Prove that if Sν ⊂ Sν+1 for each ν or the sets {Sν } are  disjoint, then ν Sν ∈ S. Prove also that if Sν ⊃ Sν+1 for each ν, then ν Sν ∈ S. (d) Prove that every Gδ and every Fσ in R are in S. Hint. First show that every open subset of Rd is a countable union of disjoint d-cells. (e) Prove that every set S ⊂ R of measure 0 is in S. Hint. S is contained in a Gδ of measure 0, and Lebesgue measure is complete. (f) Prove that S is precisely the collection of Lebesgue measurable subsets of R.

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7 Background Material on Analysis in Rn and Hilbert Space Theory

(g) Let E be a measurable subset of Rm+n . Prove that the set Ex ⊂ Rn is measurable for almost every point x ∈ Rm , the function x → λ(Ex ) is measurable, and  λ(E) = λ(Ex ) dλ(x). Rm

(h) Prove Fubini’s theorem (Theorem 7.1.10). Hint. First prove that the theorem holds for the characteristic function of a measurable set, and then that the theorem holds for a nonnegative simple function. Obtain the theorem for a nonnegative measurable function by applying Exercise 7.1.2. Finally, prove that the theorem holds for a measurable complex-valued function. 7.1.4 Let be an open subset of Rn and let p ∈ [1, ∞). The goal of this exercise is a proof that the vector space of continuous functions with compact support p in is dense in Lloc ( ). This fact will be applied in the proof of the Friedrichs lemma (Lemma 7.3.1). Let K be compact subset of . (a) Prove that if f is a nonnegative locally integrable function on , then for every  > 0, there exists a nonnegative simple function ϕ on Rn such that ϕ − f Lp (K) < . Hint. Apply Exercise 7.1.2. (b) Prove that if C is a compact subset of an open set U ⊂ Rn , then there exists a nonnegative continuous function η : Rn → [0, 1] such that η ≡ 1 on C and supp η ⊂ U . (c) Prove that if E is a measurable subset of Rn with characteristic function χE , then for every  > 0, there exists a nonnegative continuous function η : Rn → [0, 1] such that η − χE Lp (K) < . Hint. For δ > 0 sufficiently small, fix an open set U and a compact set C such that C ⊂ E ∩ K ⊂ U and λ(U \ C) < δ (see Example 7.1.4). Now apply part (b) above. p (d) Prove that if f ∈ Lloc ( ), then for every  > 0, there exists a continuous function g with compact support in such that g − f Lp (K) < . 7.1.5 Prove that if is an open subset of Rn and p ∈ [1, ∞), then the vector space of continuous functions with compact support in is dense in Lp ( ). Hint. Apply Exercise 7.1.4.

7.2 Differentiation and Integration in Rn In this section, we recall some facts concerning differentiation and integration on Rn that will allow us to consider differentiation and integration on manifolds (see Chap. 9). Definition 7.2.1 Let k ∈ Z≥0 ∪ {∞, ω}, let F = R or C, and let be an open subset of Rn . (a) A function f : → F is of class C k if one of the following holds:

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383

(i) We have k = 0 and f is continuous; (ii) We have k ∈ Z>0 and all of the partial derivatives of f of order ≤ k exist and are continuous on ; (iii) We have k = ∞ and f has continuous partial derivatives of all orders; (iv) We have k = ω and f is real analytic (that is, in a neighborhood of , . . . , an ) ∈ Rn , we may write f as a power seevery point a = (a1 m1 mn for ries f (x1 , . . . , xn ) = ∞ m1 ,...,mn =0 cm1 ,...,mn (x1 − a1 ) · · · (xn − an ) some choice of constants {cm1 ,...,mn } in F). It is convenient to consider any function on R0 = {0} to be of class C k . (b) The vector space of C k functions on with values in F is denoted by C k ( , F), or simply by C k ( ). We also denote C ∞ ( , F) by E( , F) or by E( ). The vector space of C ∞ F-valued functions with compact support in is denoted by D( , F) or by D( ). For an arbitrary set S ⊂ Rn , the vector space of continuous F-valued functions on S is denoted by C 0 (S, F) or by C 0 (S). (c) For f ∈ C 1 ( , F) and p ∈ , the differential of f at p is the F-linear map (df )p : Fn → F given by (df )p (v) =

n 

vj ·

j =1

∂f (p) ∀v = (v1 , . . . , vn ) ∈ Fn . ∂xj

We also denote by df the mappings p → (df )p and (p, v) → (df )p (v) (the latter mapping df : × Fn → F being of class C k−1 if f ∈ C k ( ) with k ∈ Z>0 ∪ {∞}). Definition 7.2.2 Let k ∈ Z≥0 ∪ {∞, ω}, let F = R or C, let be an open subset of Rn , and let F = (f1 , . . . , fm ) : → Fm be a mapping. (a) F is of class C k if fj ∈ C k ( ) for each j = 1, . . . , m. (b) If F is of class C 1 , then for each point p ∈ , the differential of F at p is the F-linear map (dF )p = ((df1 )p , . . . , (dfm )p ) : Fn → Fm given by (dF )p (v) =

n  j =1

vj ·

∂F (p) ∀v = (v1 , . . . , vn ) ∈ Rn . ∂xj

We also denote by dF the mapping p → (dF )p ∈ Hom(Fn , Fm ) and the mapping (p, v) → (dF )p (v) (the latter mapping dF : × Fn → Fm being of class C k−1 if F is of class C k with k ∈ Z>0 ∪ {∞}). (c) For F of class C 1 and m = n, the Jacobian determinant is the function   ∂fi JF = det(dF ) = det : → F. ∂xj (d) If F maps bijectively onto an open set  ⊂ Rm and both F and F −1 are of class C ∞ , then we call F a diffeomorphism of onto  (in particular, m = n).

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Remarks 1. If F : → Rm , then under the inclusion Rm ⊂ Cm , we may consider, for each point p ∈ , the differential (dF )p : Cn → Cm , which is the complex linear extension of the real linear map (dF )p : Rn → Rm . 2. Although we often identify C with R2 , one must proceed with caution when considering differentials. For if F is a C 1 mapping of an open set ⊂ Rn into R2m and p ∈ , then the complex linear extension (dF )p : Cn → C2m of the real linear differential (dF )p : Rn → R2m is clearly not the same as the differential mapping Cn → Cm that one obtains by considering F as a mapping of into Cm . In this book, the context will determine which differential mapping is being considered. 3. According to the chain rule, if F : → Rm and G :  → Fl are C k mappings with ⊂ Rn ,  ⊂ Rm , and k ∈ Z>0 ∪ {∞}, then G ◦ F is of class C k and d(G ◦ F )p = (dG)F (p) ◦ (dF )p : Cn → Cl for each point p ∈ F −1 (  ) (where for F = C, the above mapping (dF )p is the complex linear extension). In particular, if F : →  is a diffeomorphism, −1 ) then for each point p ∈ , (dF )−1 F (p) , and hence m = n in this p = d(F case. 4. By the intermediate value theorem, a continuous injective real-valued function f on an interval I ⊂ R is either strictly increasing or strictly decreasing, and f maps I homeomorphically onto an interval J ; that is, f −1 : J → I is continuous (in particular, the image of any endpoint of I is an endpoint of J ). It follows easily that if I is open and f is of class C ∞ with nonvanishing derivative, then f : I → J is a diffeomorphism. The higher-dimensional analogue is called the C ∞ inverse function theorem (Theorem 9.9.1). This (nontrivial) theorem and its consequences are applied in the study of the topological and holomorphic structure of Riemann surfaces in Sects. 5.10–5.17, and in the study of holomorphic structures on smooth surfaces in Chap. 6. Definition 7.2.3 A function f on a set S ⊂ Rn is said to be locally Lipschitz on S if for each point in S, there are a neighborhood U and a constant C = C(U ) > 0 such that |f (y) − f (x)| ≤ Cy − x for all x, y ∈ S ∩ U . We say that f is uniformly Lipschitz on S if there is a constant C > 0 such that |f (y) − f (x)| ≤ Cy − x for all x, y ∈ S. Lemma 7.2.4 If f is a C 1 function on an open set ⊂ Rn , then f is uniformly Lipschitz on every compact subset of . The proof is left to the reader (see Exercise 7.2.1). Proposition 7.2.5 (Differentiation past the integral) Let ⊂ Rn , let (X, μ) be a measure space, and let u be a complex-valued function on . Assume that the func-

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385

tion u(t, ·) is integrable on X for each fixed t ∈ , and let  u(t, x) dμ(x) ∀t ∈ . v(t) = X

(a) Suppose that t0 ∈ , u(·, x) is continuous at t0 for each fixed x ∈ X, and there exists a nonnegative integrable function g on X with |u(t, x)| ≤ g(x) for each point (t, x) ∈ × X. Then v is continuous at t0 . (b) Suppose that is open, j ∈ {1, . . . , n}, ∂u/∂tj is defined on × X, and there exists a nonnegative integrable function g on X with |(∂u/∂tj )(t, x)| ≤ g(x) for each point (t, x) ∈ × X. Then ∂v/∂tj is defined on and ∂v (t) = ∂tj

 X

∂u (t, x) dμ(x) ∂tj

∀t = (t1 , . . . , tn ) ∈ .

Proof Part (a) follows from the dominated convergence theorem. Clearly, for the proof of (b), we may assume that n = 1, that is an open interval, and that u is realvalued. As a limit of a sequence of measurable functions, ∂u/∂t is measurable in the variable x ∈ X. By hypothesis, |(∂u/∂t)(t, x)| ≤ g(x) for each point (t, x) ∈ × X, and hence ∂u/∂t is integrable in x ∈ X. For distinct points t, t0 ∈ and for x ∈ X, we have, by the mean value theorem,    u(t, x) − u(t0 , x) ∂u   (t0 , x) ≤ 2g(x). −  t − t0 ∂t Thus, once again, the dominated convergence theorem gives the claim.



Lemma 7.2.6 (Taylor’s formula to order one) If a = (a1 , . . . , an ) ∈ Rn and f is a C 2 function on a neighborhood of a with a + t (x − a) ∈ for each point x ∈ and each t ∈ [0, 1] (that is, is starlike about a), then for each point x = (x1 , . . . , xn ) ∈ , we have f (x) = f (a) +

n 

bj · (xj − aj ) +

j =1

n 

cij (x)(xi − ai )(xj − aj ),

i,j =1

where for all indices i, j = 1, . . . , n (and for coordinate functions u = (u1 , . . . , un )), bj ≡ and

 cij (x) ≡ 0

1

(1 − t)

 ∂  [f (u)] u=a ∂uj  ∂2  [f (u)] dt. u=a+t (x−a) ∂ui ∂uj

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7 Background Material on Analysis in Rn and Hilbert Space Theory

Proof For each point x = (x1 , . . . , xn ) ∈ , the fundamental theorem of calculus and integration by parts give 

1

f (x) − f (a) = 0

d [f (a + t (x − a))] dt dt

 d  [f (a + t (x − a))] t=1 dt  d  + (1 − t) [f (a + t (x − a))] t=0 dt  1 2 d + (1 − t) 2 [f (a + t (x − a))] dt. dt 0

= −(1 − t)

The chain rule now gives the desired expression for f (x).



Remark Differentiation past the integral (Proposition 7.2.5) implies that if f ∈ C k ( ) for some k ∈ Z≥2 ∪ {∞}, then each of the functions x → cij (x) in the lemma is of class C k−2 . We close this section with a consideration of the behavior of integrals under diffeomorphisms. We first recall that the image P = A(Q) of the compact n-cell Q = [0, s1 ] × · · · × [0, sn ] ⊂ Rn (with si > 0 for i = 1, . . . , n) under a linear map A : Rn → Rn is a compact parallelepiped of measure λ(P ) = |det A| · vol(Q) = |det A| · s1 · · · sn . This formula is trivial for n = 1, and relatively easy to verify for n = 2 (see Exercise 7.2.2), and ultimately, these are the only cases required in this book. For a general diffeomorphism, we have the following: Theorem 7.2.7 (Change of variables formula) Let F : →  be a diffeomorphism of an open set ⊂ Rn onto an open set  ⊂ Rn . Then the composition u ◦ F of any measurable function u on  with F is a measurable function on . Moreover, if u is a nonnegative measurable function on  , then   u dλ = (u ◦ F ) · |JF | dλ, 



where JF = det(dF ) is the Jacobian determinant of F . If u is a (complex-valued) integrable function on  , then the function (u ◦ F )|JF | is integrable on and the above equality holds. In particular, the image F (E) ⊂  of a set E ⊂ is measurable if E is measurable, and F (E) is of measure 0 if E is of measure 0. A proof is outlined in Exercise 7.2.3. The proof may be modified to give the change of variables formula for a C 1 diffeomorphism (i.e., a bijective C 1 mapping with C 1 inverse), but only the C ∞ case is applied in this book.

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Example 7.2.8 Polar coordinates. Polar coordinates play an important role in the study of complex analysis in the plane. We first recall one of the many equivalent constructions of the real trigonometric functions. The real arcsine function is the strictly increasing homeomorphism x → arcsin x of [−1, 1] onto [−π/2, π/2] x 1 given by x → 0 (1 − t 2 )−1/2 dt, where π ≡ −1 (1 − t 2 )−1/2 dt. The restriction to (−1, 1) is a diffeomorphism onto (−π/2, π/2); that is, the function and its inverse function (−π/2, π/2) → (−1, 1) are of class C ∞ . The real sine function is the unique extension θ → sin θ of the inverse function of the arcsine function to a 2π -periodic function that satisfies sin(θ + π) = − sin θ for each θ ∈ R. The real cosine function is the unique 2π -periodic function θ → cos θ determined by  cos θ = 1 − sin2 θ ∀θ ∈ [−π/2, π/2] and cos(θ + π) = − cos θ

∀θ ∈ R.

The real tangent function, the real cotangent function, the real secant function, and the real cosecant function are then given by   π sin θ tan θ ≡ ∀θ ∈ R +Z·π , cos θ 2 cos θ sin θ 1 sec θ ≡ cos θ cot θ ≡

csc θ ≡

1 sin θ

∀θ ∈ R \ (Z · π),   π ∀θ ∈ R +Z·π , 2 ∀θ ∈ R \ (Z · π).

The verification that these real trigonometric functions are of class C ∞ with the familiar derivative functions is straightforward (see Exercise 7.2.4). We also get the remaining real inverse trigonometric functions arccos ≡ (cos [0,π] )−1 , arctan ≡ (tan [−π/2,π/2] )−1 (arctan(±∞) ≡ ±π/2), etc., and their derivative functions. The familiar formulas for the sine and cosine of a sum will follow from consideration of the complex exponential function in Chap. 1 (see Example 1.6.2 and Exercise 1.6.2). For any fixed θ0 ∈ R, the mapping [0, ∞) × [θ0 , θ0 + 2π) → R2 given by (r, θ ) → (r cos θ, r sin θ ) is a continuous surjection. Moreover, the corresponding restrictions yield a bijection of the domain (0, ∞) × [θ0 , θ0 + 2π) onto the punctured plane R2 \ {(0, 0)}, and a diffeomorphism of the domain (0, ∞) × (θ0 , θ0 + 2π) onto the complement R2 \ P of the ray P ≡ {(r cos θ0 , r sin θ0 ) | r ≥ 0} in R2 (see Exercise 7.2.5). For any point (x, y) ∈ R2 and any pair (r, θ ) ∈ [0, ∞) × R with (r cos θ, r sin θ ) = (x, y), we call (r, θ ) polar coordinates for (x, y). Under the identification of R2 with C = R + iR,

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we also write eiθ ≡ cos θ + i sin θ = (cos θ, sin θ ) ∀θ ∈ R. The Jacobian determinant of the diffeomorphism (r, θ ) → (r cos θ, r sin θ ) described above is given by    cos θ −r sin θ     sin θ r cos θ  = r. Hence, if f is an integrable function or a nonnegative measurable function on R2 , then for any choice of θ0 ∈ R, Fubini’s theorem (Theorem 7.1.10) and the change of variables formula (Theorem 7.2.7) give   ∞ ∞ f (x, y) dx dy = f (x, y) dλ((x, y)) −∞ −∞

 = 

R2

(0,∞)×(θ0 ,θ0 +2π)

f (r cos θ, r sin θ )r dλ((r, θ ))

∞  θ0 +2π

=

f (r cos θ, r sin θ )r dθ dr. 0

θ0

In particular, the function (x, y) → r p = (x 2 + y 2 )p/2 is locally integrable on R2 for p > −2 (and integrable on the complement of any neighborhood of (0, 0) for p < −2). Remark Recall that, similarly, one may define the real logarithmic function to be x the diffeomorphism (0, ∞) → R given by x → log x ≡ 1 t −1 dt, and one may define the real exponential function x → exp(x) = ex to be the corresponding C ∞ inverse function. Clearly, these functions have derivative functions x → 1/x and x → 1/(1/ exp(x)) = exp(x), respectively. Moreover, the verifications of the basic algebraic properties of these functions are straightforward (see Exercise 7.2.6). Exercises for Sect. 7.2 7.2.1 Prove Lemma 7.2.4. 7.2.2 Prove that the image P = A(Q) of the compact 2-cell Q = [0, s1 ] × [0, s2 ] ⊂ R2 (with si > 0 for i = 1, 2) under a linear map A : R2 → R2 satisfies λ(P ) = |det A| · vol(Q) = |det A| · s1 · s2 . 7.2.3 The goal of this exercise is a proof of the change of variables formula (Theorem 7.2.7). Let F = (f1 , . . . , fn ) : →  be a diffeomorphism of open sets ,  ⊂ Rn , and for each point p ∈ , let Gp = (gp1 , . . . , gpn ) : → Gp ( ) ⊂ Rn be the diffeomorphism given by −1 )F (p) (F (x) − F (p)). x → (dF )−1 p (F (x) − F (p)) = (dF

Also, fix a relatively compact open subset U of .

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(a) Prove that there exists a positive constant C such that for each point a = (a1 , . . . , an ) ∈ U , each positive number s for which the compact cubic n-cell R = [a1 , a1 + s] × · · · × [an , an + s] is contained in U , and each point x = (x1 , . . . , xn ) ∈ R, we have −Cs 2 ≤ gai (x) ≤ s + Cs 2

∀i = 1, . . . , n.

Hint. Apply Lemma 7.2.6. (b) Prove that for C > 0 and R = [a1 , a1 + s] × · · · × [an , an + s] ⊂ U as in (a), we have F (R) ⊂F (a) − (dF )a (Cs 2 , . . . , Cs 2 ) + (dF )a ([0, s + 2Cs 2 ] × · · · × [0, s + 2Cs 2 ]), and conclude from this that the compact set F (R) satisfies λ(F (R)) ≤ |JF (a)| · vol(R) · (1 + 2Cs)n .  (c) Prove that the open set F (U ) ⊂  satisfies λ(F (U )) ≤ U |JF | dλ. Hint. First prove that for each ν0 ∈ Z>0 , U is equal to a countable union of disjoint n-cells of the form [i1 2−ν , (i1 + 1)2−ν ) × · · · × [in 2−ν , (in + 1)2−ν ) with i1 , . . . , in ∈ Z and ν ∈ Z>ν0 . (d) Prove that λ∗ (F (E)) ≤ E |JF | dλ for every measurable set E ⊂ . Conclude that in particular, the image of every set of measure 0 under F is a set of measure 0. Finally, prove that the image of every measurable set under F is measurable. (e) Prove the change of variables formula (Theorem 7.2.7). Hint. For u a nonnegative measurable function on  , show that u ◦ F is measurable. Given disjoint measurable sets E1 , . . . , Em ⊂  and constants r1 , . . . , rm ∈ [0, ∞] with u ≥ rj on Ej for each j = 1, . . . , m, prove that  m  rj λ(Ej ) ≤ (u ◦ F )|JF | dλ. j =1



  Pass to the supremum to get  u dλ ≤ (u ◦ F )|JF | dλ. Apply this to the diffeomorphism F −1 and the function (u ◦ F ) · |JF | on to get the reverse inequality. 7.2.4 Verify that the trigonometric functions (as defined in Example 7.2.8) are of class C ∞ with derivative functions

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d sin θ = cos θ, dθ

d cos θ = − sin θ, dθ

d tan θ = sec2 θ, dθ

d cot θ = − csc2 θ, dθ

d sec θ = sec θ tan θ, dθ

d csc θ = − csc θ cot θ. dθ

7.2.5 For a fixed number θ0 ∈ R, let  : [0, ∞) × [θ0 , θ0 + 2π) → R2 be the continuous mapping given by (r, θ ) → (r cos θ, r sin θ ). (a) Prove that  maps the set (0, ∞) × [θ0 , θ0 + 2π) bijectively onto the punctured plane R2 \ {(0, 0)} (as claimed in Example 7.2.8). (b) Prove that  maps the domain (0, ∞) × (θ0 , θ0 + 2π) diffeomorphically onto the complement R2 \ P of the ray P = {(r cos θ0 , r sin θ0 ) | r ≥ 0} in R2 . 7.2.6 Verify that the real logarithmic and exponential functions (as defined in the remark following Example 7.2.8) satisfy log(xy) = log x + log y

∀x, y ∈ R>0

and ex+y = ex · ey

∀x, y ∈ R.

7.3 C ∞ Approximation A convenient way in which to obtain a C ∞ approximation of a locally integrable function is to form a mollifier as follows: Lemma 7.3.1 (Friedrichs) Let κ be a nonnegative C ∞ function with compact sup n port in the unit ball B(0; 1) in R such that Rn κ dλ = 1, and let be an open subset of Rn . For each function u ∈ L1loc ( ) and for each δ > 0, let κ δ (x) ≡ δ −n κ(x/δ) for each point x ∈ Rn , let δ ≡ {x ∈ Rn | dist(x, Rn \ ) > δ}, and let  uδ (x) ≡

u(y)κ δ (x − y) dλ(y)

 =

u(x − y)κ δ (y) dλ(y) 

B(0;δ)

u(x − δy)κ(y) dλ(y)

= B(0;1)

for every x ∈ δ . Then, for every function u ∈ L1loc ( ), we have the following: (a) The function uδ belongs to C ∞ ( δ ) for every δ > 0. (b) For every compact set K ⊂ , uδ − uL1 (K) → 0 as δ → 0+ (i.e., uδ → u in L1loc ( )). Moreover, if u ∈ C 0 ( ), then uδ → u uniformly on K.

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C ∞ Approximation

391

Remarks 1. A stronger version of the lemma (which is applied in Chap. 6) appears in Sect. 11.3 (see Lemma 11.3.1). 2. An example of such a function κ is given by  C −1 exp(−1/(1 − 4x2 )) if x < 1/2, κ(x) = 0 if x ≥ 1/2, where



e−1/(1−4x ) dλ(x). 2

C= B(0;1/2)

3. The family of functions {κ δ } is called a (positive) mollifier (or an approximation of the identity), and {uδ } is called a mollification (or a regularization) of u. The convolution of two (suitable) functions f and g on Rn is the function  f ∗ g : x → f (x − y)g(y) dλ(y). Rn

Thus, in the above lemma, uδ is the convolution of κ δ and an extension of u to Rn . Proof of Lemma 7.3.1 Part (a) follows from differentiation past the integral (Proposition 7.2.5). For the proof of (b), we fix a compact set K ⊂ and we set K δ ≡ {x ∈ Rn | dist(x, K) ≤ δ} for each δ > 0. Fixing δ0 with 0 < δ0 < dist(K, Rn \ ), we get K δ0 ⊂ . Suppose first that u is continuous. If for 0 < δ ≤ δ0 , we set Mδ ≡ sup{|u(x) − u(y)| | x, y ∈ K δ0 , |x − y| < δ}, then by uniform continuity, we have Mδ → 0 as δ → 0+ . For each x ∈ K, we have       u(x − δy)κ(y) dλ(y) − u(x)κ(y) dλ(y) |uδ (x) − u(x)| =  

B(0;1)

≤

B(0;1)

|u(x − δy) − u(x)| · κ(y) dλ(y) ≤ Mδ . B(0;1)

Thus uδ → u uniformly on K as δ → 0+ , and hence in particular, uδ − uL1 (K) → 0 as δ → 0+ . For a general u ∈ L1loc ( ), given  > 0, we may choose a continuous function v on such that v − uL1 (K δ0 ) < /3 (see Exercise 7.1.4). For each δ ∈ (0, δ0 ), Fubini’s theorem (see Theorem 7.1.10 and Exercise 7.3.1) then implies that      (v(x − δy) − u(x − δy))κ(y) dλ(y) dλ(x) vδ − uδ L1 (K) =  K

B(0;1)

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  ≤

|(v(x − δy) − u(x − δy))κ(y)| dλ(y) dλ(x) K



B(0;1)



= 

B(0;1)

 |v(x − δy) − u(x − δy)|dλ(x) κ(y) dλ(y)

K



 |v(x) − u(x)| dλ(x) κ(y) dλ(y)

= 

B(0;1)

≤ B(0;1)



K+(−δy)

K δ0

 |v(x) − u(x)| dλ(x) κ(y) dλ(y)

= v − uL1 (K δ0 ) < /3. On the other hand, for δ > 0 sufficiently small, we have vδ − vL1 (K) < /3. Hence uδ − uL1 (K) ≤ uδ − vδ L1 (K) + vδ − vL1 (K) + v − uL1 (K) < . 

Thus (b) is proved.

One consequence of the Friedrichs lemma is the following uniqueness property: Lemma 7.3.2 If is an open subset of Rn , u, v ∈ L1loc ( ), and   uϕ dλ = vϕ dλ ∀ϕ ∈ D( ),



then u = v almost everywhere in . Proof We may assume without loss of generality that v ≡ 0. In the notation of Lemma 7.3.1, we have uδ ≡ 0 on δ for every δ > 0. On the other hand, for every compact set K ⊂ , we have uδ − uL1 (K) → 0, and therefore, for some sequence {δν }, 0 = uδν → u pointwise almost everywhere in K. Thus u = 0 almost everywhere in .  Exercises for Sect. 7.3 7.3.1 Verify that the function (x, y) → |v(x − δy) − u(x − δy)| for (x, y) ∈ K × B(0; 1) in the proof of Lemma 7.3.1 is measurable (hence Fubini’s theorem applies).

7.4 Differential Operators and Formal Adjoints Definition 7.4.1 For each multi-index α = (α1 , . . . , αn ) ∈ (Z≥0 )n , we set |α| = α1 + · · · + αn and  α     ∂ ∂ α1 ∂ αn ∂ |α| = ··· = α1 ∂x ∂x1 ∂xn ∂x1 · · · ∂xnαn

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393

(in particular, for α = (0, . . . , 0), (∂/∂x)α u = u for every function u). A linear differential operator of order k ∈ Z≥0 with coefficients {aα } on an open set ⊂ Rn is a linear map A from C k ( ) to the space of complex-valued functions on given by 

Au =

α∈(Z≥0

 aα

)n ,|α|≤k

∂ ∂x

α u

∀u ∈ C k ( ),

where aα is a complex-valued function on for each α ∈ (Z≥0 )n with |α| ≤ k. For any open set U ⊂ , we also denote the linear differential operator 



aα  U

α∈(Z≥0 )n ,|α|≤k

∂ ∂x

α : C k (U ) → C 0 (U )

by A, unless there is danger of confusion. We define the conjugate operator A¯ by 

¯ = Au¯ = Au

α∈(Z≥0 )n , |α|≤k

 a¯ α

∂ ∂x

α u

∀u ∈ C k ( ).

Remark In practice, we usually work with simpler expressions obtained by allowing the multi-indices to be in (Z≥0 )l for 1 ≤ l ≤ n. For example, if A is a second-order linear differential operator, then we may write A=

n  i,j =1

 ∂2 ∂ aij + bi + c, ∂xi ∂xj ∂xi n

i=1

and refer to {aij }, {bi }, and c as the coefficients of A. Observe also that by replacing {aij } with { 12 aij }, we may assume that aij = aj i for all i and j (i.e., that the matrix (aij ) is symmetric). One may consider an extension of a differential operator to a space of functions that are not necessarily differentiable in the following way: Definition 7.4.2 Let A be a linear differential operator of order k with (complex) coefficients {aα } on an open set ⊂ Rn such that aα ∈ C |α| ( ) for each multiindex α. (a) The formal transpose of A is the linear differential operator t A of order k given by  α  t |α| ∂ Au ≡ (−1) [aα · u] ∀u ∈ C k ( ). ∂x n α∈(Z≥0 ) , |α|≤k

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(b) The formal adjoint of A is the linear differential operator A∗ of order k given by  α  ∗ t ¯ |α| ∂ (−1) [a¯ α · u] ∀u ∈ C k ( ). A u ≡ Au = ∂x n α∈(Z≥0 ) , |α|≤k

(c) For u, v ∈ L1loc ( ), we say that Au is equal to v in the distributional (or weak) sense, and we write Adistr u = v, if   u · A∗ ϕ dλ = v · ϕ¯ dλ ∀ϕ ∈ D( ).

Equivalently,





 u · Aϕ dλ =

v · ϕ dλ ∀ϕ ∈ D( ).

t





Remarks 1. The function Adistr u is unique (if it exists) by Lemma 7.3.2. 2. For any function u ∈ L1loc ( ), Adistr  u always exists in a more general sense, namely, as the linear functional ϕ → u · t Aϕ dλ on D( ) (see, for example, [Rud2]). For this reason, to say Adistr u ∈ L1loc ( ) will mean that Adistr u exists as a function in L1loc ( ). Lemma 7.4.3 Let A and B be linear differential operators with C ∞ coefficients on an open set ⊂ Rn , and let k be the order of A. (a) If u ∈ C k ( ) (or k = 0 and u ∈ L1loc ( )), then Adistr u = Au. (b) For u, v ∈ L1loc ( ), we have Adistr u = v in if and only if each point p ∈ admits a neighborhood U ⊂ with Adistr [uU ] = vU . (c) We have t (t A) = A, (A∗ )∗ = A, t (ζ A + B) = ζ t A + t B, and (ζ A + B)∗ = ζ¯ A∗ + B ∗ for each ζ ∈ C, t (AB) = t B t A, and (AB)∗ = B ∗ A∗ . (d) Let u, v ∈ L1loc ( ) and let ζ ∈ C. Then (ζ A + B)distr u = ζ Adistr u + Bdistr u, provided Adistr u, Bdistr u ∈ L1loc ( ); and Adistr (ζ u + v) = ζ Adistr u + Adistr v provided Adistr u, Adistr v ∈ L1loc ( ). (e) Suppose u, Bdistr u, v ∈ L1loc ( ). Then we have (AB)distr u = v if and only if Adistr [Bdistr u] = v. (f) Suppose u, Adistr u ∈ L1loc ( ), k = 1, the 0th-order term of A vanishes, and ρ ∈ C ∞ ( ). Then Adistr [ρu] = ρAdistr u + uAρ. Proof Part (a) is trivial for k = 0, and it follows easily from integration by parts in each variable for k > 0. The details are left to the reader (see Exercise 7.4.1). For the proof of (b), suppose that u, v ∈ L1loc ( ) and each point admits a neighborhood in which Au is equal to v in the distributional sense. Then, given a function ϕ ∈ D( ), we may choose finitely many C ∞ functions {ην }m such that ην ≡ 1 ν=1 on supp ϕ and such that for each ν, supp ην lies in an open subset Uν of on

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395

which Au is equal to v in the distributional sense (see, for example, Sect. 9.3). Hence     u · A∗ ϕ dλ = u · A∗ [ην ϕ] dλ = v · ην ϕ dλ = v ϕ¯ dλ.

ν

Uν

ν

Uν



The converse is trivial. For (c), we observe that if ϕ, ψ ∈ D( ), then    ψ · [Aϕ] dλ = [t Aψ] · ϕ dλ = ψ · t (t A)ϕ dλ,





and 

 ψ · [t (AB)ϕ] dλ =



[(AB)ψ] · ϕ dλ

 =

[Bψ] · [t Aϕ] dλ 



ψ · [t B t Aϕ] dλ.

=

Lemma 7.3.2 now implies that t (t A)ϕ = Aϕ and t (AB)ϕ = t B t Aϕ. For any function u ∈ C ∞ ( ) and any point p ∈ , there exists a function ϕ ∈ D( ) such that ϕ ≡ u near p, so it follows that t (t A) = A and t (AB) = t B t A. The remaining claims in (c) are easy to verify directly. The proof of (d) is left to the reader (see Exercise 7.4.1). For the proof of part (e), let u ∈ L1loc ( ) with Bdistr u ∈ L1loc ( ), and let ϕ ∈ D( ). Then    u · [t (AB)ϕ] dλ = u · [t B t Aϕ] dλ = [Bdistr u] · t Aϕ dλ.





If (AB)distr u ∈ L1loc ( ), then the first expression is equal to  [(AB)distr u] · ϕ dλ,

and comparison with the last expression gives Adistr Bdistr u = (AB)distr u. Similarly, if Adistr Bdistr u ∈ L1loc ( ), then the last expression is equal to  [Adistr Bdistr u] · ϕ dλ,

and comparison with the first expression gives (AB)distr u = Adistr Bdistr u. For the proof of (f), we assume that k = 1, Adistr u = v ∈ L1loc ( ), the 0th-order term of A vanishes, and ρ ∈ C ∞ ( ). An easy computation shows that for ϕ ∈ D( ),

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396

¯ · ϕ). Hence we have t A(ρϕ) = ρ · (t Aϕ) − (Aρ) · ϕ (thus A∗ (ρϕ) = ρ · (A∗ ϕ) − (Aρ)     ρu · (t Aϕ) dλ = u · t A(ρϕ) dλ + u · (Aρ) · ϕ dλ = [ρv + u · Aρ] · ϕ dλ,









and (f) follows.

Remarks 1. The terminology in Definition 7.4.2 is motivated by the notion of the adjoint of a linear operator on a Hilbert space (see, for example, [Rud2]) and the equality ϕ, A∗ ψL2 ( ) = Aϕ, ψL2 ( )

∀ϕ, ψ ∈ D( )

(see Sect. 7.5) provided by Lemma 7.4.3. 2. Parts (a) and (b) of Lemma 9.8.3 hold even if A has C k coefficients. Lemma 7.4.4 Let κ be a nonnegative C ∞ function with compact support in the  n unit ball B(0; 1) in R such that Rn κ dλ = 1, and let be an open subset of Rn . For each u ∈ L1loc ( ) and δ > 0, let κ δ (x) ≡ δ −n κ(x/δ) for each point x ∈ Rn , let δ ≡ {x ∈ Rn | dist(x, Rn \ ) > δ}, and let   δ uδ (x) ≡ u(y)κ (x − y) dλ(y) = u(x − y)κ δ (y) dλ(y)

B(0;δ)

 =

u(x − δy)κ(y) dλ(y) B(0;1)

for every x ∈ δ . If A is a constant-coefficient linear differential operator on Rn , and u ∈ L1loc ( ) with Adistr u ∈ L1loc ( ), then A[uδ ] = [Adistr u]δ in δ for every δ > 0. In particular, for every compact set K ⊂ , we have A(uδ ) − Adistr uL1 (K) → 0

as δ → 0+ .

Proof Given δ > 0 and a function ϕ ∈ D( δ ), we have, by Fubini’s theorem (see Theorem 7.1.10 and Exercise 7.4.2),  uδ (x)[t Aϕ](x) dλ(x) δ

 

 u(x − y)κ δ (y) dλ(y) · [t Aϕ](x) dλ(x)

= δ



B(0;δ)



= B(0;δ)

 =

B(0;δ)

 =

B(0;δ)

 u(x − y) · [t Aϕ](x) dλ(x) κ δ (y) dλ(y)

δ



δ +(−y)



δ +(−y)

 u(x) · [t Aϕ](x + y) dλ(x) κ δ (y) dλ(y)  u(x) · [t A(ϕ(· + y))](x) dλ(x) κ δ (y) dλ(y)

7.5 Hilbert Spaces

397



 = 

B(0;δ)

δ +(−y)

 [Adistr u](x)ϕ(x + y) dλ(x) κ δ (y) dλ(y)

[Adistr u]δ (x)ϕ(x) dλ(x)

= δ

(that A has constant coefficients gives the fourth equality). Therefore, A(uδ ) = Adistr (uδ ) = [Adistr u]δ (by the uniqueness property provided by Lemma 7.3.2).  Exercises for Sect. 7.4 7.4.1 Prove parts (a) and (d) of Lemma 7.4.3. 7.4.2 Verify that the function (x, y) → u(x − y)κ δ (y)[t Aϕ](x) for (x, y) ∈ δ × B(0; δ) in the proof of Lemma 7.4.4 is integrable (hence Fubini’s theorem applies).

7.5 Hilbert Spaces Hilbert spaces (or complete Hermitian inner product spaces) are (possibly infinitedimensional) vector spaces in which one has many of the useful geometric tools (for example, orthogonal projection) that one has in Rn . They play a crucial role in this book. In this section, we recall some of the basic elements of Hilbert space theory. Proofs of most of the relevant facts required in this book are either provided or outlined in the exercises. Further facts concerning weak sequential compactness, which are applied only in Chap. 6, are considered in Sect. 7.6. We first recall the following: Definition 7.5.1 Let V be a complex vector space. (a) A Hermitian inner product (or simply an inner product) on V is a function ·, · : V × V → C such that for all u, v, w ∈ V and ζ ∈ C, we have (i) ζ u + v, w = ζ u, w + v, w, (ii) u, v = v, u, and (iii) v, v > 0 if v = 0. The pair (V, ·, ·) is called a Hermitian inner product space (or simply an inner product space). √ (b) The norm associated to ·, · is the function v → v ≡ v, v. (c) Two vectors u and v are orthogonal with respect to ·, · if u, v = 0 (equivalently, v, u = 0). We write u ⊥ v. If u ⊥ v and u = v = 1, then u and v are orthonormal. A nonempty set S ⊂ V is an orthonormal set if each pair of distinct vectors in S are orthonormal. (d) For every nonempty set S ⊂ V, we define S ⊥ ≡ {v ∈ V | v ⊥ u ∀u ∈ S}. Remark Similarly, a real inner product (or simply an inner product) on a real vector space V is a real symmetric bilinear form g on V such that g(v, v) > 0 for each v ∈ V \ {0}.

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Theorem 7.5.2 For any Hermitian inner product space (V, ·, ·), we have the following: (a) Law of cosines. We have u − v2 = u2 − 2 Reu, v + v2 for all u, v ∈ V. (b) Pythagorean theorem. We have u − v2 = u + v2 = u2 + v2 for every pair of orthogonal vectors u, v ∈ V. (c) Parallelogram law. We have u + v2 + u − v2 = 2u2 + 2v2 for all u, v ∈ V. (d) Schwarz inequality. We have |u, v| ≤ u · v for all u, v ∈ V. (e)  ·  is a norm on V. (f) The function ·, · is continuous; that is, if {uj } and {vj } are sequences converging to u and v, respectively, in (V,  · ), then uj , vj  → u, v. ⊥

(g) For any set S ⊂ V, S ⊥ is a closed subspace of (V,  · ), and S = S ⊥ , where S = cl(S) is the closure of S. Moreover, if S is a subspace of V, then S ∩ S ⊥ = {0}. Proof The proofs of (a)–(c) are left to the reader (see Exercise 7.5.1). It is easy to see that 0 = 0, that ζ v = |ζ | · v for all v ∈ V and ζ ∈ C, and that v > 0 if v ∈ V \ {0}. For the proof of (d), suppose u, v ∈ V \ {0}. Setting u = u/u and v  = v/v, and fixing ζ ∈ C with |ζ | = 1 and ζ u , v   = |u , v  |, we get, for each t ∈ R, 0 ≤ ζ u − tv  2 = 1 − 2t|u , v  | + t 2 . The right-hand side is a quadratic polynomial in t that attains its minimum value at t = |u , v  |. Substituting this value for t, we get 0 ≤ 1 − 2|u , v  |2 + |u , v  |2 , and hence |u, v| = |u , v  | ≤ 1. u · v Part (d) now follows. For the proof of (e), it remains to verify the triangle inequality. Given u, v ∈ V, we have u + v2 = u2 + 2 Reu, v + v2 ≤ u2 + 2|u, v| + v2 ≤ u2 + 2uv + v2 = (u + v)2 , and the triangle inequality follows. For the proof of (f), suppose uj → u and vj → v in V. Then |uj , vj  − u, v| ≤ |uj , vj  − uj , v| + |uj , v − u, v| = |uj , vj − v| + |uj − u, v| ≤ uj  · vj − v + uj − u · v → u · 0 + 0 · v = 0, and hence uj , vj  → u, v.

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399

For the proof of (g), suppose {uj } is a sequence in S ⊥ converging to u ∈ V. Then, for every vector v ∈ S, we have 0 = uj , v → u, v. Thus u, v = 0, and hence ⊥

u ∈ S ⊥ . Therefore, S ⊥ is closed. Since S ⊂ S, we also have S ⊥ ⊃ S . Conversely, if u ∈ S ⊥ and v ∈ S, then we may choose a sequence {vj } in S converging to v. ⊥

⊥

Hence 0 = u, vj  → u, v, and it follows that u ∈ S . Thus S ⊥ = S . It is easy to verify that S ⊥ is a subspace. Finally, if S is a (vector) subspace of V and v ∈ S ∩ S ⊥ , then v, v = 0, so v = 0. Thus (g) is proved.  Definition 7.5.3 A Hilbert space is a Hermitian inner product space (H, ·, ·) that is complete with respect to the corresponding norm  · . Example 7.5.4 Given a measure space (X, μ), we get the Hilbert space L2 (X, μ) with inner product and norm  u, vL2 (X,μ) ≡

uv¯ dμ and X

uL2 (X,μ) =

 u, uL2 (X,μ)

∀u, v ∈ L2 (X, μ).

For the inequality |uv| ≤ (1/2)(|u|2 + |v|2 ) implies that a product of two L2 functions is integrable (alternatively, one may use Hölder’s inequality). It is then easy to check that the above is an inner product. Completeness is the case p = 2 of Theorem 7.1.9. On the other hand, for the related Hilbert spaces considered in this book, completeness will be proved directly (see Proposition 2.6.3 and Proposition 3.6.4). Example 7.5.5 For any set X, 2 (X) denotes the L2 space associated to X together with the counting measure. The inner product and norm are then given by u, v2 (X) =



u(x)v(x)

and

u2 (X) =

x∈X



|u(x)|2

∀u, v ∈ 2 (X).

x∈X

In particular, taking X = Z>0 , we get u, v2 (Z>0 ) =

∞  n=1

u(n)v(n)

and

u2 (n)

 ∞  = |u(n)|2

∀u, v ∈ 2 (Z>0 ).

n=1

Theorem 7.5.6 If V is a closed subspace of a Hilbert space (H, ·, ·), then we have the direct sum decomposition H = V ⊕ V ⊥ . That is, for each vector u ∈ H, there are unique vectors v ∈ V and w ∈ V ⊥ such that u = v + w. Moreover, we have u − s > w for all s ∈ V \ {v}. The above decomposition H = V ⊕ V ⊥ is called the associated orthogonal decomposition. The surjective linear mapping u → v ∈ V (with w = u − v ∈ V ⊥ ) is called the orthogonal projection of H onto V.

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Proof of Theorem 7.5.6 Given a vector u ∈ H, we may choose a sequence {vj } in V such that u − vj  → r ≡ inf u − v ∈ [0, ∞). v∈V

According to the parallelogram law (see Theorem 7.5.2), vi − vj 2 = (u − vi ) − (u − vj )2 = 2u − vi 2 + 2u − vj 2 − (u − vi ) + (u − vj )2 = 2u − vi 2 + 2u − vj 2 − 4u − (1/2)(vi + vj )2 ≤ 2u − vi 2 + 2u − vj 2 − 4r 2 . Since we can make the last expression arbitrarily small by requiring that i and j be sufficiently large, {vj } is a Cauchy sequence, and therefore, {vj } converges to some vector v ∈ V. Setting w ≡ u − v, we see that for each vector s ∈ V, w = r ≤ u − s. Fixing ζ ∈ C with |ζ | = 1 and ζ w, s = |w, s|, we get, for each t ∈ R, u − (v + tζ −1 s)2 = ζ w − ts2 = w2 − 2t|w, s| + t 2 s2 . The quadratic polynomial on the right-hand side attains its minimum value at t = 0, so we must have w, s = 0. Thus w ∈ V ⊥ . The decomposition u = v + w is unique, since V ∩ V ⊥ = {0}. In particular, if s ∈ V with u − s = r, then the minimizing  sequence given by vj = s for all j must converge to v; that is, s = v. Corollary 7.5.7 For any subspace W of a Hilbert space (H, ·, ·), we have the following: (a) (W ⊥ )⊥ = W. (b) If V is a subspace of H with W ∪ W ⊥ ⊂ V, then W is closed in V (i.e., W ∩ V = W) if and only if V = W ⊕ W ⊥ . ⊥

⊥

Proof We have W ⊥ W and W = W ⊥ , so W ⊂ (W ⊥ )⊥ . Conversely, if u ∈ (W ⊥ )⊥ , then we have u = w + v with w ∈ W and v ∈ W ⊥ . Subtracting, we get v = u − w ∈ (W ⊥ )⊥ ∩ W ⊥ = {0}, so u ∈ W. Thus (a) is proved. Suppose W, W ⊥ ⊂ V as in (b). Given u ∈ V, we have u = w + v, where w ∈ W ⊥ and v ∈ W = W ⊥ ⊂ V. In particular, w = u − v ∈ W ∩ V. Thus, if W is closed in V, then w ∈ W. Conversely, if V = W ⊕ W ⊥ and u ∈ W ∩ V, then writing ⊥ ⊥ u = w + v with w ∈ W and v ∈ W ⊥ = W , we get v = u − w ∈ W ∩ W = {0}.  Thus u = w ∈ W and hence W ∩ V = W. Remark Theorem 7.5.6 and Corollary 7.5.7 do not hold in general for an inner product space (see Exercise 7.5.2).

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401

Definition 7.5.8 Let (V,  · ) be a normed vector space over F = R or C. A bounded linear functional α on V is a linear functional α : V → F (i.e., an F-valued linear map) for which the norm α ≡ inf{R ∈ [0, ∞] | |α(v)| ≤ Rv ∀v ∈ V} is finite. The set of all bounded linear functionals on V is called the (norm) dual space of V and is denoted by V ∗ . Remark On a finite-dimensional normed vector space V, every linear functional is bounded (see, for example, [Fol] or [Rud1]). Thus the algebraic dual space (see Sect. 8.1) is equal to the norm dual space of V if dim V < ∞. Proposition 7.5.9 For any normed vector space (V,  · ) with V = {0}, we have the following: (a) For each α ∈ V ∗ , α = supv∈V \{0} |α(v)|/v = supv∈V ; v=1 |α(v)|. (b) The dual space (V ∗ ,  · ) is a Banach space. The proof is left to the reader (see Exercise 7.5.3). The following important theorem allows one to identify a Hilbert space with its dual space up to conjugate linear isomorphism. Theorem 7.5.10 Let (H, ·, ·) be a Hilbert space. Then, for each vector v ∈ H, the mapping H → C given by u → u, v is a bounded linear functional with norm equal to v. Conversely, given a bounded linear functional α on H, there is a unique vector v ∈ H such that α(u) = u, v for all u ∈ H (in particular, α = v). Proof It follows immediately from the definition of an inner product that for each v ∈ H, the mapping α : H → C given by u → u, v is a linear functional. The Schwarz inequality implies that |α(u)| ≤ vu for each vector u ∈ H, so α is a bounded linear functional with α ≤ v. Setting u = v, we get α ≥ v, and hence we have equality. Observe also that if w ∈ H with ·, w = α = ·, v, then setting u = v − w, we get u2 = u, u = u, v − u, w = 0, and hence v = w. In particular, this observation gives the uniqueness claim in the theorem. Conversely, given an element α ∈ H∗ , the kernel V ≡ ker α = α −1 (0) is a closed subspace of H (see Exercise 7.5.4). If α ≡ 0, then α = ·, 0. If α = 0, then we may fix an element w ∈ V ⊥ \ {0} and we may set ζ ≡ α(w)/w2 ∈ C \ {0}. The vector v ≡ ζ w ∈ V ⊥ then satisfies α(v) = ζ α(w) =

|α(w)|2 = v, v. w2

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7 Background Material on Analysis in Rn and Hilbert Space Theory

Given a vector u ∈ H = V ⊕ V ⊥ , we have u = r + s with r ∈ V and s ∈ V ⊥ . In particular, s−

α(s) v ∈ V ∩ V ⊥ = {0}, v2

and hence α(u) = α(s) = α(s)

v, v = s, v = u, v. v2



Theorem 7.5.11 (Hahn–Banach theorem) If α is a bounded linear functional on a subspace V of a Hilbert space (H, ·, ·) (that is, α is bounded with respect to the norm obtained by restriction to V of the norm on H), then there exists a bounded linear functional β on H such that βV = α and β = α. Proof We first observe that there exists a unique extension of α to a bounded linear functional on the closure V. For if {vn } is a sequence in V that converges to a vector v, then for all m, n ∈ Z>0 , |α(vm ) − α(vn )| = |α(vm − vn )| ≤ αvm − vn . It follows that the sequence {α(vn )} in C is Cauchy and therefore convergent. Moreover, if {un } is another sequence in V converging to v, then |α(un ) − α(vn )| ≤ αun − vn  → 0. Thus we get a well-defined mapping γ : V → C by setting γ (v) = lim α(vn ), n→∞

for any v ∈ V and any sequence {vn } in V converging to v. In particular, γ V = α. The linearity properties of limits of sequences imply that γ is a linear functional, and since γ is an extension of α, we have γ  ≥ α. On the other hand, for v ∈ V and {vn } as above, we have |γ (v)| = lim |α(vn )| ≤ lim αvn  = αv, n→∞

n→∞

and hence γ  ≤ α. Thus γ is a bounded linear functional on V and γ  = α. Letting P be the orthogonal projection of H onto V, we get a linear functional β ≡ γ ◦ P : H → C with βV = γ . In particular, β ≥ γ  = α. Conversely, the Pythagorean theorem (see Theorem 7.5.2) gives |β(v)| = |γ ◦ P(v)| ≤ αPv ≤ αv ∀v ∈ H, and hence β ≤ α. Thus β ∈ H∗ and β = α.



Remark More general versions of the Hahn–Banach theorem can be found in, for example, [Rud1] and [Rud2].

7.6 Weak Sequential Compactness

403

Exercises for Sect. 7.5 7.5.1 Prove parts (a)–(c) of Theorem 7.5.2. 7.5.2 Give examples that show that Theorem 7.5.6 and Corollary 7.5.7 do not hold in general for an inner product space. 7.5.3 Prove Proposition 7.5.9. 7.5.4 Show that the kernel of any element of the dual space of an inner product space is a closed subspace. 7.5.5 Suppose {vn } is a sequence in an inner product space (V, ·, ·), v ∈ V, and u, vn  → u, v for every u ∈ V (i.e., {vn } converges weakly to v). Prove that {vn } converges (strongly) to v (i.e., vn − v → 0) if and only vn  → v.

7.6 Weak Sequential Compactness This section is required for the discussion of integrability of almost complex structures in Chap. 6. Throughout this section, (H, ·, ·) denotes a Hilbert space. The main goal is the following: Theorem 7.6.1 (Weak sequential compactness) The closed unit ball in a Hilbert space (H, ·, ·) is weakly sequentially compact; that is, for every bounded sequence {uν } in H, there is a subsequence {uνk } and a vector u ∈ H such that u ≤ supν uν  and uνk , v → u, v as k → ∞ ∀v ∈ H. Remark Conversely, by the uniform boundedness principle (see, for example, [Fol], [Rud1], or [Rud2]), a weakly convergent sequence in H (see Exercise 7.5.5) is bounded (we will not need this converse in this book). For the proof, we use complete orthonormal sets, a fundamental object in Hilbert space theory. Definition 7.6.2 An orthonormal set with dense span in H is called a complete orthonormal set (or a complete orthonormal basis). Theorem 7.6.3 If {uν } is a sequence in H \ {0}, then V ≡ Span{uν } has a countable orthonormal basis. Consequently, if H has a countable dense subset (i.e., H is separable), then H has a countable complete orthonormal set. Proof Let us assume that dim V = ∞, since the finite-dimensional case is similar, but easier. By passing to a subsequence (constructed inductively), we may assume that the vectors {uν } are linearly independent. The Gram–Schmidt orthonormalization process then yields an orthonormal basis {eν } for V determined inductively

7 Background Material on Analysis in Rn and Hilbert Space Theory

404

by u1 e1 ≡ u1 

and

eν ≡

uν − uν −

ν−1

μ=1 uν , eμ eμ

ν−1

μ=1 uν , eμ eμ 

for ν > 1.



Remark A Zorn’s lemma argument shows that in fact, every Hilbert space admits a complete orthonormal set. Lemma 7.6.4 (Bessel’s inequality) Let {ei }i∈I be an orthonormal set in H, and for each vector v ∈ H, let v(i) ˆ = v, ei  for each index i ∈ I . Then  |v(i)| ˆ 2. v2 ≥ i∈I

In particular, at most countably many of the complex numbers {v(i)} ˆ i∈I are nonzero. Remarks 1. {v(i)} ˆ i∈I is the collection of Fourier coefficients of v with respect to the orthonormal set. 2. One can show that in fact, equality in Bessel’s inequality holds if {ei }i∈I is a complete orthonormal set. In fact, v → vˆ is an isometric isomorphism onto 2 (I ) in this case (see, for example, [Rud1]). Proof of Lemma 7.6.4 Given a vector v ∈ H and a finite set of indices J ⊂ I , the Pythagorean theorem (Theorem 7.5.2) implies that  2  2       2    v, ej  · ej  |v(j ˆ )|2 . v =  v, ej  · ej  + v −  ≥ j ∈J

j ∈J

j ∈J



The claim now follows.

Proof of Theorem 7.6.1 Given a sequence {uν } in H with uν  ≤ 1 for each ν, we may assume without loss of generality that supν uν  = 1. We may also assume that V ≡ Span{uν } is dense in H. For if there exists a subsequence {uνk } converging weakly to a vector u with u ≤ 1 in the Hilbert space V, then letting P : H → V be the orthogonal projection, we get uνk , v = uνk , Pv → u, Pv = u, v ∀v ∈ H. Theorem 7.6.3 provides a countable orthonormal basis {ei }i∈I for V, which is then a complete orthonormal set in H, and we may let {v(i)} ˆ i∈I = {v, ei }i∈I be the Fourier coefficients of each v ∈ H. Applying Bessel’s inequality and Cantor’s diagonal process, we see that we may assume that the sequence {uˆ ν (i)}∞ ν=1 converges to some complex number ζi with |ζi | ≤ 1 for each i ∈ I . For each finite set J ⊂ I , Bessel’s inequality gives   |uˆ ν (j )|2 → |ζj |2 as ν → ∞. 1 ≥ uν 2 ≥ j ∈J

Thus

 i∈I

|ζi |2 ≤ 1.

j ∈J

7.6 Weak Sequential Compactness

405

We may define a unique linear functional τ on V by setting τ (ei ) = ζ¯i for each i ∈ I . For each vector v ∈ V, we then have   2       v(i) ˆ ζ¯i  ≤ |v(i)| ˆ 2 · |ζi |2 ≤ v2 |τ (v)|2 =  i∈I

i∈I

i∈I



(note that the sum i∈I v(i)ζ ˆ ˆ has at most i is actually a finite sum, since {v(i)} finitely many nonzero terms). Thus τ is a bounded linear functional of norm at most 1 on V. Applying the Hahn–Banach theorem (or the density of V) and Theorem 7.5.10, we get a (unique) vector u ∈ H such that u ≤ 1 and v, u = τ (v) for each v ∈ V. Now, given a vector v ∈ H and a number  > 0, we may choose a vector w ∈ V with v − w < /3. For some finite set J ⊂ I , we have wˆ ≡ 0 on I \ J , so   w(j ˆ ) · uˆ ν (j ) → w(j ˆ ) · ζ¯j = τ (w) = w, u as ν → ∞. w, uν  = j ∈J

j ∈J

Thus, for ν  0, we have |w, uν  − w, u| < /3, and hence |v, uν  − v, u| ≤ |v − w, uν | + |w, uν − u| + |w − v, u| < . The claim now follows.



Chapter 8

Background Material on Linear Algebra

In this chapter, we recall some basic definitions and facts concerning exterior products (which are essential in the discussion of differential forms in Sect. 9.5) and tensor products (which are essential in the discussion of holomorphic line bundles in Chap. 3). In this book, we mostly consider exterior and tensor products in vector spaces of dimension 1 or 2.

8.1 Linear Maps, Linear Functionals, and Complexifications The vector space of linear mappings V → W of vector spaces V and W over F = R or C is denoted by Hom(V, W), which is itself a vector space over F (in general, for modules M and N over a ring R, the set of module homomorphisms from M to N is an R-module that is denoted by Hom(M, N )). The (algebraic) dual space of V is the vector space V ∗ ≡ Hom(V, F). If dim V = n < ∞ and e1 , . . . , en is a basis, then there exist unique linear functionals λ1 , . . . , λn with λi (ej ) = δij for i, j = 1, . . . , n. These linear functionals form a basis for V ∗ that is called the associated dual basis. We also have the canonical isomorphism V ∼ = (V ∗ )∗ given by v(α) = α(v) for all ∗ v ∈ V and α ∈ V . If dim V = 1, then for every pair of vectors u, v ∈ V with v = 0, we denote the unique scalar c ∈ F with u = cv by u/v. We also denote the linear functional u → u/v by v −1 . Remarks 1. If V is infinite-dimensional and {eα }α∈A is a basis for V, then we get a collection of linearly independent linear functionals {λα }α∈A determined by λα (eβ ) = 1 if α = β, 0 if α = β. However, these linear functionals will not span the dual space, since the unique linear functional λ determined by λ(eα ) = 1 for all α ∈ A will not be in the span. Consequently, V  (V ∗ )∗ , since there exists a nontrivial linear functional τ on V ∗ with τ (λα ) = 0 for all α ∈ A (here, we have used the fact that every linearly independent subset of a vector space is contained in a basis). 2. For a normed vector space (V, · ), the associated norm dual space (see Definition 7.5.8) is equal to the algebraic dual space if and only if dim V < ∞. T. Napier, M. Ramachandran, An Introduction to Riemann Surfaces, Cornerstones, DOI 10.1007/978-0-8176-4693-6_8, © Springer Science+Business Media, LLC 2011

407

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For if {eν } is a sequence of linearly independent vectors, then there exists a linear functional λ with λ(eν ) = ν eν for each ν ∈ Z>0 . One obtains a real vector space from a complex vector space simply by restricting scalar multiplication to real scalars. More precisely, we have the following: Proposition 8.1.1 Let V be a complex vector space with vector addition + and scalar multiplication ·. Then the set V together with vector addition given by + and scalar multiplication given by ·R×V determines a real vector space VR of di= 2 dimC V. Moreover, for any complex basis {eν } for VC , the mension dimR VR √ collection {eν } ∪ { −1eν } is a real basis for VR . The proof is left to the reader (see Exercise 8.1.1). Definition 8.1.2 For a complex vector space V, the associated real vector space given by Proposition 8.1.1 is called the realification (or the underlying real vector space) of V and is denoted by VR or (in a slight abuse of notation) simply by V. One obtains a complex √ vector space VC from a real vector space V by forming all of the formal sums u + −1v for u, v ∈ V. More precisely, we have the following: Proposition 8.1.3 Let V and W be real vector spaces. (a) The set V ⊕ V, together with the standard direct sum vector addition and with scalar multiplication given by z · (u, v) = (xu − yv, yu + xv) for all (u, v) ∈ V ⊕ V and z = x + iy with x, y ∈ R, is a complex vector space VC of dimension dimC V = dimR V. Denoting the element (u, v) ∈ VC by u + iv for all u, v ∈ V, we have z · (u + iv) = (xu − yv) + i(yu + xv) for all z = x + iy with x, y ∈ R. We also have a real linear inclusion V → VC given by v → v + i0 ↔ (v, 0). (b) The map VC → VC given by u + iv → u + iv ≡ u − iv (i.e., (u, v) → (u, −v)) is a conjugate linear isomorphism. (c) Any real basis for V is a complex basis for VC . (d) If α, β ∈ Hom (V, W) are two real linear maps, then the complex linear extension of the map λ = α + iβ, i.e., the map λ : VC → WC (which, in an abuse of notation, we give the same name) given by λ(u + iv) = α(u) − β(v) + i(β(u) + α(v)) ∈ WC for all u, v ∈ V, is a complex linear mapping. Moreover, setting λ¯ ≡ ¯ w) ¯ for each w ∈ VC . The above correα − iβ = α + i(−β), we get λ(w) = λ( spondence gives a (canonical) isomorphism [Hom(V, W)]C ∼ = Hom(VC , WC ). In particular, we have (V ∗ )C ∼ = (VC )∗ . Furthermore, α ∈ Hom(V, W) is injective (surjective) if and only if the associated complex linear extension is injective (respectively, surjective). The proof is left to the reader (see Exercise 8.1.2).

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Definition 8.1.4 For a real vector space V, the associated complex vector space VC given by Proposition 8.1.3 is called the complexification of V. We denote each √ element (u, v) ∈ VC by w = u + iv = u + −1v, we call Re w = u its real part, we call Im w = v its imaginary part, and we call w¯ = u − iv its conjugate. We identify V with V + i0 ⊂ VC . Given a real vector space W, we identify each element λ ∈ [Hom (V, W)]C ⊃ Hom (V, W) with its complex linear extension λ ∈ Hom (VC , WC ) (for each λ ∈ Hom (VC , WC ), we have λ = Re λ + i Im λ, where (Re λ)(v) = Re(λ(v)) and (Im λ)(v) = Im(λ(v)) for each v ∈ V). We also use the identification and notation VC∗ ≡ (V ∗ )C = (VC )∗ . Example 8.1.5 (Rn )C = Cn under the identification x + iy ↔ (x1 + iy1 , . . . , xn + iyn )

∀x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) ∈ Rn .

Similarly, we have (Cn )R = R2n . Exercises for Sect. 8.1 8.1.1 Prove Proposition 8.1.1. 8.1.2 Prove Proposition 8.1.3. 8.1.3 Prove that the dual space of any infinite-dimensional vector space over F = R or C cannot have a countable basis.

8.2 Exterior Products Throughout this section, V denotes a vector space over F = R or C. In this book, we require exterior products only of degree ≤ 2, so we will restrict our attention to this case. Definition 8.2.1 For the vector space V: (a) A bilinear function on V (or V × V) is a map α : V × V → F that is linear in each entry; that is, for all (u1 , v1 ), (u2 , v2 ) ∈ V × V and for each ζ ∈ F, we have α(ζ u1 , v1 ) = α(u1 , ζ v1 ) = ζ α(u1 , v1 ), α(u1 + u2 , v1 ) = α(u1 , v1 ) + α(u2 , v1 ), α(u1 , v1 + v2 ) = α(u1 , v1 ) + α(u1 , v2 ). The vector space of bilinear functions on V × V (a vector subspace of the space of F-valued functions) is denoted by V ∗ ⊗ V ∗ and is called the tensor product of V ∗ with itself. (b) An element θ ∈ V ∗ ⊗ V ∗ is skew-symmetric (symmetric) if θ (u, v) = −θ (v, u) (respectively, θ (u, v) = θ (v, u)) for all u, v ∈ V. For α, β ∈ V ∗ , the exterior (or wedge) product is the skew-symmetric bilinear function α ∧ β given by (α ∧ β)(u, v) = α(u)β(v) − β(u)α(v) for all u, v ∈ V. The vector (sub)space of skew-symmetric bilinear functions is denoted by 2 V ∗ .

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Remarks 1. For vector spaces (or modules) U, V, W, a map α : U × V → W that is linear in each factor is called a bilinear pairing. 2. Another standard definition for ∧ includes a factor of 1/2. Proposition 8.2.2 For the vector space V, we have the following: (a) For all α, β, γ ∈ V ∗ and ζ ∈ F, (ζ α) ∧ β = α ∧ (ζβ) = ζ · (α ∧ β), (α + β) ∧ γ = α ∧ γ + β ∧ γ , α ∧ β = −β ∧ α. (b) If dim V = 1, then 2 V ∗ = {0}. If dim V = 2 and λ1 , λ2 is a basis for V ∗ , then λ1 ∧ λ2 is a basis for 2 V ∗ (in particular, dim 2 V ∗ = 1). In fact, if e1 , e2 is a basis for V with dual basis λ1 , λ2 , then α = α(e1 , e2 )λ1 ∧ λ2 for every α ∈ 2 V ∗ . (c) If F = R and α, β ∈ V ∗ ⊗ V ∗ are two (real) bilinear functions, then the complex bilinear extension of the function λ = α + iβ, i.e., the map λ : VC × VC → C (which we give the same name) given by λ(ζ, ξ ) =α(u, r) − α(v, s) − β(u, s) − β(v, r) + i(α(u, s) + α(v, r) + β(u, r) − β(v, s)) for ζ = u + iv and ξ = r + is with u, v, r, s ∈ V, is a (complex) bilinear func¯ ζ¯ , ξ¯ ) for all tion on VC . Setting λ¯ = α − iβ = α + i(−β), we get λ(ζ, ξ ) = λ( ζ, ξ ∈ VC . Moreover, if α and β are skew-symmetric, then λ is skew-symmetric. The above correspondence gives (canonical) isomorphisms [V ∗ ⊗ V ∗ ]C ∼ = VC∗ ⊗ VC∗

and

[2 V ∗ ]C ∼ = 2 (VC∗ ).

Proof The proofs of parts (a) and (c) are left to the reader (see Exercise 8.2.1). For the proof of (b), suppose dim V = 2, let λ1 , λ2 be a basis for V ∗ , and let e1 , e2 be the dual basis for V. For each τ ∈ 2 V ∗ , we have, for all u, v ∈ V, τ (u, v) = τ (λ1 (u)e1 + λ2 (u)e2 , λ1 (v)e1 + λ2 (v)e2 ) = (λ1 (u)λ2 (v) − λ2 (u)λ1 (v)) · τ (e1 , e2 ) = τ (e1 , e2 ) · (λ1 ∧ λ2 )(u, v). Thus 2 V ∗ = F · λ1 ∧ λ2 . On the other hand, (λ1 ∧ λ2 )(e1 , e2 ) = 1, so λ1 ∧ λ2 is a basis. It is clear that if dim V = 1, then 2 V ∗ = {0}.  Remarks 1. We set 0 V ∗ = F, 1 V ∗ = V ∗ , and ζ ∧ α = ζ · α ∈ p V ∗ if ζ ∈ 0 V ∗ and α ∈ p V ∗ with p ∈ {0, 1, 2}. If dim V ≤ 2, then we set p V ∗ = {0} for all p > 2, and we set α ∧ β = 0 for all α ∈ p V ∗ , β ∈ q V ∗ with p + q > 2. It

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follows that if dim V ≤ 2, p, q, r ∈ Z≥0 , α ∈ p V ∗ , β ∈ q V ∗ , and γ ∈ r V ∗ , then α ∧ β = (−1)pq β ∧ α and (α ∧ β) ∧ γ = α ∧ (β ∧ γ ) (we denote the latter simply by α ∧ β ∧ γ ). If dim V > 2, then we leave p V ∗ undefined for p > 2, since such spaces are not required in this book. 2. For F = R and p ∈ {0, 1, 2} or p > 2 ≥ dim V, Proposition 8.2.2 allows us to use the identification and notation p VC∗ ≡ [p V ∗ ]C = p (VC∗ ). Definition 8.2.3 Assume that F = R and that n ≡ dim V = 1 or 2. (a) We say that two nonzero elements λ1 , λ2 ∈ n V ∗ have equivalent orientations if λ1 /λ2 > 0 (this is an equivalence relation with exactly two equivalence classes). (b) An equivalence class as in (a) is called an orientation in V ∗ (or in V) and the other equivalence class is called the opposite orientation. If an element α ∈ 2 V ∗ is in the equivalence class determined by a given orientation, then α is said to be positive (or positively oriented) with respect to the orientation, we write α > 0, and we say α induces the orientation. If α induces the opposite orientation, then α is said to be negative (or negatively oriented) with respect to the given orientation, and we write α < 0. If α > 0 (< 0) or α = 0, then we say that α is nonnegative (respectively, nonpositive) and we write α ≥ 0 (respectively, α ≤ 0). An ordered basis (v1 , . . . , vn ) (= v1 or (v1 , v2 )) for V is positively oriented if each positive element α of n V ∗ is positive on the ordered basis (i.e., α(v1 ) > 0 if n = 1, α(v1 , v2 ) > 0 if n = 2). (c) Given an orientation in V ∗ and given two elements α, β ∈ n V ∗ , we write α > β (α ≥ β) if α − β > 0 (respectively, α − β ≥ 0). In this context, for a sequence {αν } in n V ∗ , we may define   αν inf αν ≡ inf · θ, ν ν θ   αν lim inf αν ≡ lim inf · θ, ν→∞ ν→∞ θ



 αν sup αν ≡ sup · θ, ν ν θ   αν lim sup αν ≡ lim sup · θ, ν→∞ ν→∞ θ

for an arbitrary positive element θ ∈ 2 V ∗ , provided the above defined coefficients exist in R. Remark It is easy see that for a sequence {αν } in n V ∗ as in (c) above, we have lim αν = α if and only if lim inf αν = lim sup αν = α. Example 8.2.4 For the standard dual basis λ1 , λ2 in (R2 )∗ , the orientation induced by λ1 ∧ λ2 is the right-handed orientation, while that represented by −λ1 ∧ λ2 = λ2 ∧ λ1 is the left-handed orientation. Exercises for Sect. 8.2 8.2.1 Prove parts (a) and (c) of Proposition 8.2.2.

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8.3 Tensor Products Tensor products are required for the study of holomorphic line bundles in Chaps. 3 and 4 (as well as in the last part of Chap. 5). Throughout this section, F denotes the field R or C. We consider tensor products from the point of view of bilinear functions (cf. Definition 8.2.1). Definition 8.3.1 Let U and V be vector spaces over F. (a) A bilinear function on U × V is a map α : U × V → F that is linear in each entry; that is, for all (u1 , v1 ), (u2 , v2 ) ∈ U × V and for each ζ ∈ F, we have α(ζ u1 , v1 ) = α(u1 , ζ v1 ) = ζ α(u1 , v1 ), α(u1 + u2 , v1 ) = α(u1 , v1 ) + α(u2 , v1 ), α(u1 , v1 + v2 ) = α(u1 , v1 ) + α(u1 , v2 ). We call the vector space of bilinear functions on U × V (a vector subspace of the space of F-valued functions) the tensor product of U ∗ and V ∗ and we denote it by U ∗ ⊗ V ∗ . For dim U, dim V < ∞, the tensor product of U and V is given by U ⊗ V ≡ (U ∗ )∗ ⊗ (V ∗ )∗ , under the identification of U with (U ∗ )∗ and V with (V ∗ )∗ . (b) The tensor product of α ∈ U ∗ and β ∈ V ∗ is the element α ⊗ β ∈ U ∗ ⊗ V ∗ defined by [α ⊗ β](u, v) ≡ α(u) · β(v) for each pair (u, v) ∈ U × V. Proposition 8.3.2 Let U , V, and W be finite-dimensional vector spaces over F. Then we have the following: (a) For all α, β ∈ U ∗ , γ , δ ∈ V ∗ , and ζ ∈ F, we have (ζ α) ⊗ γ = α ⊗ (ζ γ ) = ζ · (α ⊗ γ ), (α + β) ⊗ γ = α ⊗ γ + β ⊗ γ , α ⊗ (γ + δ) = α ⊗ γ + α ⊗ δ. n ∗ ∗ (b) If {θi }m i=1 and {λj }j =1 are bases for U and V , respectively, then the tensor products {θi ⊗ λj }1≤i≤m, 1≤j ≤n form a basis for U ∗ ⊗ V ∗ . In particular, dim U ∗ ⊗ V ∗ = mn. (c) There exist unique (surjective) isomorphisms ∼ =

(i) U ∗ ⊗ V ∗ → V ∗ ⊗ U ∗ satisfying α ⊗ β → β ⊗ α for all α ∈ U ∗ and β ∈ V ∗ ; ∼ =

(ii) (U ∗ ⊗V ∗ )⊗W ∗ → U ∗ ⊗(V ∗ ⊗W ∗ ) satisfying (α ⊗β)⊗γ → α ⊗(β ⊗γ ) for all α ∈ U ∗ , β ∈ V ∗ , and γ ∈ W ∗ ; ∼ =

(iii) V ∗ ⊗ V → F satisfying α ⊗ v → α(v) for all α ∈ V ∗ and v ∈ V; ∼ =

(iv) F ⊗ V ∗ → V ∗ satisfying ζ ⊗ α → ζ · v for all ζ ∈ F and α ∈ V ∗ ; ∼ =

(v)  : U ∗ ⊗ V → Hom(U , V) satisfying (α ⊗ v)(u) = α(u) · v for all α ∈ U ∗ , v ∈ V, and u ∈ U ; and

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413 ∼ =

(vi)  : U ∗ ⊗ V ∗ → (U ⊗ V)∗ satisfying (α ⊗ β)(u ⊗ v) = α(u) · β(v) for all α ∈ U ∗ , β ∈ V ∗ , u ∈ U , and v ∈ V. Proof The proof of (a) is left to the reader (see Exercise 8.3.1). For the proof of (b), let {ei } be the dual basis of U associated to {θi }, and let {fj } be the dual basis of ∗ ∗ V associated to {λj }. For each m n τ ∈ U ⊗ V and each pair (u, v) ∈ U × V, we have u = i=1 θi (u)ei and v = j =1 λj (v)fj , and hence τ (u, v) =

m  n 

θi (u)λj (v)τ (ei , fj ) =

i=1 j =1

m  n 

τ (ei , fj ) · (θi ⊗ λj )(u, v).

i=1 j =1

 n Thus τ = m i=1 j =1 τ (ei , fj ) · θi ⊗ λj , and it follows that the tensor products U ∗ ⊗ V ∗ . Moreover, if {ζij }1≤i≤m, 1≤j ≤n are elements {θi ⊗ λj }1≤i≤m,1≤j ≤n m span n of F and τ = i=1 j =1 ζij · θi ⊗ λj , then for each choice of indices i and j , evaluation of both of the above expressions for τ on (ei , fj ) gives ζij = τ (ei , fj ). Hence, if τ = 0, then we get ζij = 0 for all i and j . Thus we have linear independence. The proof of (c), which relies on (a) and (b), is left to the reader (see Exercise 8.3.1).  Remarks 1. The above proposition is stated mostly with respect to dual spaces, since this is a more convenient setup for the proofs. However, for finite-dimensional vector spaces, the analogous statements also hold with U , V, and W in place of U ∗ , V ∗ , and W ∗ , respectively, since U ∼ = (U ∗ )∗ , etc. 2. According to part (c-ii), the tensor product operation on finite-dimensional vector spaces is associative up to a canonical isomorphism. For this reason, we denote the k-fold tensor product of finite-dimensional vector spaces V1 , . . . , Vk by V1 ⊗ · · · ⊗ Vk . We also identify the pairs of spaces that are canonically isomorphic as in (c). 3. For any finite-dimensional vector   space V over F, we denote the k-fold tensor product of V by k V. We also set 0 V = F, and we set ζ ⊗ λ = ζ · λ for all ζ ∈ F and v ∈ V. 4. If U and V are finite-dimensional vector spaces over F, dim V = 1, and ∼ v ∈ V \ {0}, then for each k ξ ∈ U ⊗ V = V ⊗ U , there is a unique vector u ∈ U with ξ = u ⊗ v. If ξ = j =1 uj ⊗ vj with uj ∈ U and vj ∈ V for each j = 1, . . . , k, then u=

k  vj j =1

v

· uj .

∼U ⊗V ⊗ Equivalently, u = ξ ⊗ v −1 ∈ U ⊗ V ⊗ V ∗ under the identification U = V ∗ provided by Proposition 8.3.2. We also denote ξ by u · v or v · u, and u by ξ/v. The verification that ξ/v is well defined by the above is left to the reader (see Exercise 8.3.3). Given a vector w ∈ U , we may form the quotient w/v = w ⊗ v −1 ∈

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U ⊗ V ∗ . We may also view this as the quotient Finally, for any integer m, we set ⎧ m factors ⎪   ⎪ ⎪ ⎨V ⊗ · · · ⊗ V Vm ≡ F ⎪ ⎪ ⎪ ⎩ ∗ −m ∼ −m ∗ [V ] = [V ] and for any t ∈ V, we set ⎧ m factors ⎪ ⎪   ⎪ ⎨t ⊗ · · · ⊗ t ∈ V m tm ≡ 1 ∈ V0 = F ⎪ ⎪ ⎪ ⎩(t −1 )−m = (t −m )−1 ∈ V m

of w ∈ U ⊗ V ∗ ⊗ V ∼ = U and v.

if m > 0, if m = 0, if m < 0;

if m > 0, if m = 0 and t = 0, if m < 0and t = 0.

Exercises for Sect. 8.3 8.3.1 Prove parts (a) and (c) of Proposition 8.3.2. 8.3.2 Let U and V be two finite-dimensional real vector spaces. Prove that if α, β ∈ U ∗ ⊗ V ∗ are two (real) bilinear functions, then the complex bilinear extension of the function λ = α + iβ, i.e., the map λ : UC × VC → C (which we give the same name) given by λ(ζ, ξ ) =α(u, r) − α(v, s) − β(u, s) − β(v, r) + i(α(u, s) + α(v, r) + β(u, r) − β(v, s)) for ζ = u + iv and ξ = r + is with u, v ∈ U and r, s ∈ V, is a complex bilinear function on UC × VC . Setting λ¯ = α − iβ = α + i(−β), prove that λ(ζ, ξ ) = λ¯ (ζ¯ , ξ¯ ) for all ζ ∈ UC and ξ ∈ VC . Finally, prove that the above correspondence gives a (canonical) isomorphism [U ∗ ⊗ V ∗ ]C ∼ = UC∗ ⊗ VC∗ . 8.3.3 Verify that quotients ξ/v, as defined in Remark 4 following the proof of Proposition 8.3.2, are well-defined.

Chapter 9

Background Material on Manifolds

In this chapter, we recall some basic definitions and facts concerning analysis on manifolds (mainly of dimension 1 or 2).

9.1 Topological Spaces In this section, we recall some terminology and facts from point-set topology (see, for example, [Mu]). Definition 9.1.1 A topology on a set X is a collection of subsets T such that (i) X ∈ T and ∅ ∈ T ;  (ii) For every subcollection U of T , we have U ∈U U ∈ T ; and (iii) We have U ∩ V ∈ T for all U, V ∈ T . The pair (X, T ) (usually denoted simply by X) is called a topological space. Each element of T is called an open subset of X. For each point x ∈ X, any open set U ⊂ X containing x is called a neighborhood of x. Definition 9.1.2 Let (X, T ) be a topological space, and let A ⊂ X. (a) A is a closed subset of X if X \ A is open. ◦

(b) The interior A of A is the union of all open subsets of X contained in A, the closure A (also denoted by cl(A)) is the intersection of all closed sets containing A, ◦ and the boundary ∂A is the set A¯ \ A. (c) A limit point (or accumulation point) of A in X is a point p ∈ X such that p ∈ A \ {p}. A point in A that is not a limit point is called an isolated point of A. If A has no limit points in X, then A is called a discrete subset. (d) The collection TA ≡ {U ∩ A | U ∈ T } is called the subspace topology on A. (e) A mapping  : X → Y of X to a topological space Y is continuous if −1 (U ) ⊂ X is open for each open set U ⊂ Y . If  is bijective with continuous T. Napier, M. Ramachandran, An Introduction to Riemann Surfaces, Cornerstones, DOI 10.1007/978-0-8176-4693-6_9, © Springer Science+Business Media, LLC 2011

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inverse, then  is called a homeomorphism and we say that X and Y are homeomorphic. If  maps a neighborhood of each point in X homeomorphically onto an open subset of Y , then  is called a local homeomorphism. Depending on the context, C 0 (X) will denote either the vector space of real-valued continuous functions on X or the vector space of complex-valued continuous functions on X. (f) The set A is compact if every open covering of A admits a finite subcovering; that U = {Ui }i∈I of open subsets of X that satisfies  is, for every family  A ⊂ i∈I Ui , we have A ⊂ i∈J Ui for some finite set J ⊂ I . The set A is relatively compact in X if the closure A is compact, and if this is the case, then we write A  X. If every open covering of A admits a countable subcovering, then we say that A is σ -compact. (g) The set A is connected if for every pair of open sets U and V with A ⊂ U ∪ V , we have A ∩ U ∩ V = ∅ or A ∩ U = ∅ or A ∩ V = ∅. For the equivalence relation ∼ on A determined by x ∼ y if and only if x and y lie in a connected subset of A, each equivalence class is (a connected set) called a connected component (or simply a component) of A. The space X is locally connected if for every point p ∈ X and every neighborhood U of p, there is a connected neighborhood V of p that is contained in U (equivalently, every connected component of every open subset of X is open in X). (h) X is called a Hausdorff space if for each pair of distinct points x, y ∈ X, there exist open sets U and V with x ∈ U , y ∈ V , and U ∩ V = ∅. Remarks 1. An arbitrary intersection of closed sets is closed. 2. The interior of a set is the largest open set contained in the set, and the closure of a set is the smallest closed set containing the set. 3. The subspace topology on a subset is a topology on the subset. 4. Let A ⊂ X. Then A is compact (connected) if and only if A is compact (respectively, connected) with respect to the subspace topology on A. The restriction of any continuous mapping  : X → Y to A is continuous. Moreover, if A is compact (connected), then (A) is also compact (respectively, connected). 5. A compact subset of a Hausdorff space is closed. Consequently, a bijective continuous mapping of compact Hausdorff spaces is a homeomorphism. 6. For any set A ⊂ X, A = {x ∈ X | x is an isolated point or limit point of A}. In particular, A is discrete if and only if A is closed and each point in A is an isolated point of A (see Exercise 9.1.1). Definition 9.1.3 A collection B of subsets of a set X is a basis for a topology on X if B covers X and for each pair of sets B1 , B2 ∈ B and each point x ∈ B1 ∩ B2 , there is an element B ∈ B with x ∈ B ⊂ B1 ∩ B2 . The collection     B U ⊂ B T = B∈U

is called the topology generated by B in X, and B is called a basis for T . A topological space with a countable basis is called second countable.

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Remark The topology generated by a basis B as above is a topology on X. Moreover, this topology is equal to the intersection of all topologies on X containing the collection. Example 9.1.4 A normed vector space (V, · ) together with the associated open subsets is a topological space (see Definition 7.1.6). Example 9.1.5 Given topological spaces (Xi , Ti ) for i = 1, . . . , n, the collection B = {U1 × · · · × Un | Ui ∈ Ti for i = 1, . . . , n} is a basis for a topology in the Cartesian product X = X1 × · · · × Xn . The topology T in X generated by B is called the product topology. A mapping  = (1 , . . . , n ) : Y → X of a topological space Y into X is continuous if and only if i is continuous for each i = 1, . . . , n. The details are left to the reader (see Exercise 9.1.2). Example 9.1.6 The disjoint union of a family of sets {Aλ }λ∈ is the set  A≡ Aλ ≡ {(x, λ) | λ ∈ , x ∈ Aλ }. λ∈

 In other words, each element x ∈ λ∈ Aλ determines distinct copies {(x, λ)}λ∈,x∈Aλ of this element in λ∈ Aλ . For each λ ∈ , we identify Aλ with its image in A under the natural inclusion map ιλ : x → (x, λ), provided there is no danger of confusion. In fact,  given a set of indices  ⊂  and a subset Bλ ⊂ Aλ for each λ ∈ , we identify λ∈ Bλ with its image in A under the natural inclusion. )}λ∈ , the disjoint union X ≡  For a family of topological spaces {(Xλ , Tλ X inherits the (natural) topology T ≡ { λ λ∈ λ∈ Uλ | Uλ ∈ Tλ ∀λ ∈ }. In other words, T is the unique topology in X for which for each λ ∈ , the inclusion ιλ : Aλ → X maps Aλ homeomorphically onto an open set ιλ (Aλ ). Example 9.1.7 Let ∼ be an equivalence relation in a topological space X. Then the quotient topology in the corresponding quotient space Y ≡ X/∼ with quotient map  : X → Y ≡ X/∼ is the collection of all sets U ⊂ Y for which −1 (U ) is open in X (see Exercise 9.1.3). Observe that  is a surjective continuous mapping with respect to these topologies. Moreover, for an arbitrary surjective mapping

: X → Z of X onto a set Z, the fibers are equivalence classes for the equivalence relation given by x ∼ y ⇐⇒ (x) = (y). Hence we get an induced quotient topology in Z. If Z has a given topology TZ , then the quotient topology may differ from TZ . However, if is continuous with respect to TZ , then TZ is contained in the quotient topology. Definition 9.1.8 Let X be a topological space. (a) For p, q ∈ X, a path (or parametrized path or curve or parametrized curve) from p to q in X is a continuous mapping γ : [a, b] → X for some numbers a, b ∈ R with a < b with γ (a) = p and γ (b) = q. We call p the initial point and q the terminal point. We call γ a loop based at p (or a closed curve based at p) if p = q.

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(b) X is path connected if there is a path from p to q for each pair of points p, q ∈ X. The space X is locally path connected if for every point p ∈ X and every neighborhood U of p, there is a path connected neighborhood V of p that is contained in U . Remarks 1. We take the domain of a path to be the interval [0, 1] unless otherwise specified. 2. In some contexts, we also call the image of a parametrized path (with certain properties) a path (or curve). A 1-dimensional manifold (see Sect. 9.2) is also sometimes called a curve. 3. A path connected space is connected, and a connected locally path connected space is path connected (see Exercise 9.1.4). Example 9.1.9 The set ({0} × R) ∪ {(x, sin(1/x)) | x > 0} ⊂ R2 is connected, but not path connected. Definition 9.1.10 A Hausdorff space X is locally compact if for every point p ∈ X and every neighborhood U of p, there is a relatively compact neighborhood V of p in U . Remarks 1. A general topological space is called locally compact if each point has a neighborhood that lies in a compact set. For Hausdorff spaces, this condition is equivalent to the above (see, for example, [Mu]). 2. It is easy to see that if U is a neighborhood of a compact subset K in a locally compact Hausdorff space, then there exists a relatively compact neighborhood V of K in U (that is, V is compact and V ⊂ U , which we indicate by writing V  U ). Definition 9.1.11 The one-point compactification of a locally compact Hausdorff space (X, TX ) is the topological space with underlying set X ∪ {∞} obtained by adjoining a point ∞ (not in X) to X and with topology given by TX ∪ {(X \ K) ∪ {∞} | K is a compact subset of X}. The element ∞ is called the point at infinity. The one-point compactification X ∪ {∞} of a locally compact Hausdorff space X is a compact Hausdorff space, and if X is noncompact and connected, then X ∪ {∞} is connected (see Exercise 9.1.5). Exercises for Sect. 9.1 9.1.1 Prove that for any subset A of a topological space X, we have A = {x ∈ X | x is an isolated point or limit point of A}. Also show that A is discrete if and only if A is closed and each point in A is an isolated point of A.

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9.1.2 Verify that the product topology on the Cartesian product X = X1 × · · · × Xn of (finitely many) topological spaces X1 , . . . , Xn is a topological space (see Example 9.1.5). Also verify that a mapping  = (1 , . . . , n ) : Y → X of a topological space Y into X is continuous if and only if i is continuous for each i = 1, . . . , n. 9.1.3 Verify that the quotient topology in a quotient space of a topological space is a topology, and that the corresponding quotient map is continuous (see Example 9.1.7). 9.1.4 Prove that any path connected topological space is connected, and that any connected locally path connected topological space is path connected. 9.1.5 Verify that the one-point compactification X ∪ {∞} of a locally compact Hausdorff space X is a topological space and that X ∪ {∞} is compact Hausdorff. Prove also that if X is noncompact and connected, then X ∪ {∞} is connected.

9.2 The Definition of a Manifold Definition 9.2.1 Let M be a Hausdorff space, and let n ∈ Z≥0 . (a) A homeomorphism  : U → U  of an open set U ⊂ M onto an open set U  ⊂ Rn is called a local chart of dimension n (or an n-dimensional local chart) in M. We also denote this local chart by (U, , U  ). (b) For any two n-dimensional local charts (U1 , 1 , U1 ) and (U2 , 2 , U2 ) in M, the mapping 1 ◦ −1 2 : 2 (U1 ∩ U2 ) → 1 (U1 ∩ U2 ) is called a coordinate transformation (see Fig. 9.1). (c) A collection of n-dimensional local charts A = {(Ui , i , Ui )}i∈I that covers M  (i.e., for which M = i∈I Ui ) is called an atlas of dimension n on M. (d) If M admits an n-dimensional atlas, then M is called a manifold (or a topological manifold or a C 0 manifold) of dimension n. M is also called a (topological or C 0 ) n-manifold. (e) A connected 1-dimensional manifold is also called a curve (or a topological curve). A connected 2-dimensional manifold is also called a surface (or a topological surface or a C 0 surface).

Remarks 1. In this book, although we do not assume that a given manifold is second countable, we mostly consider second countable manifolds. The reader should also note that it is common to take second countability as a part of the definition of a manifold (without any specific comment). 2. Because we also call a path in a topological space a curve, we will mostly avoid referring to a connected 1-dimensional manifold as a curve unless there is no danger of confusion.

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Fig. 9.1 Coordinate transformations

Definition 9.2.2

Let M be Hausdorff space, let k ∈ Z≥0 ∪ {∞}, and let n ∈ Z≥0 .

(a) Two n-dimensional local charts (U1 , 1 , U1 ) and (U2 , 2 , U2 ) in M are C k compatible if the coordinate transformations 1 ◦ −1 2 : 2 (U1 ∩ U2 ) → 1 (U1 ∩ U2 ) and −1 = 2 ◦ −1 [1 ◦ −1 2 ] 1 : 1 (U1 ∩ U2 ) → 2 (U1 ∩ U2 )

(b) (c) (d)

(e)

(f) (g)

are of class C k . An atlas consisting of C k compatible local charts in M is called a C k atlas on M. Two n-dimensional C k atlases A1 and A2 on M are C k equivalent if A1 ∪ A2 is a C k atlas (this is an equivalence relation). An equivalence class S of n-dimensional C k atlases on M is called a C k structure of dimension n on M. The pair (M, S) (usually denoted simply by M) is called a C k manifold of dimension n (or a C k n-manifold). Any local chart (U, , U  ) in any C k atlas in the C k structure on a C k n-manifold M is called a local C k chart in M. Setting x = (x1 , . . . , xn ) = , we call x local C k coordinates, and for each point p ∈ U , we call (U, x) a local C k coordinate neighborhood of p. A C ∞ manifold is also called a smooth manifold. A connected 2-dimensional C k manifold (smooth manifold) is also called a C k surface (respectively, smooth surface). A connected 1-dimensional C k manifold (smooth manifold) is also called a C k curve (respectively, smooth curve).

Remark A real analytic manifold is defined analogously (i.e., the coordinate transformations are required to be real analytic). Examples 9.2.3 For n ∈ Z>0 , Rn is a smooth (in fact, real analytic) manifold of dimension n with C ∞ atlas {(Rn , x → x, Rn )}. The unit sphere Sn is a C ∞ (in fact, real analytic) manifold of dimension n with C ∞ atlas A = {(Ui+ , i , Vi )}n+1 i=1 ∪ − n+1 {(Ui , i , Vi )}i=1 , where for each i = 1, . . . , n + 1, i : Ui± ≡ {x = (x1 , . . . , xn+1 ) ∈ Sn | ±xi > 0} → Vi ≡ {t ∈ Rn | t < 1}

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is the mapping given by (x1 , . . . , xn+1 ) → (x1 , . . . , x i , . . . , xn+1 ). Any open subset of a C k manifold M is a C k manifold with C k atlas consisting of the restrictions to of local C k coordinates in M. The Cartesian product M1 × M2 of two C k manifolds M1 and M2 is a C k manifold, called the product manifold, with C k atlas consisting of local C k charts given by (U1 × U2 , (x, y) → (1 (x), 2 (y)), V1 × V2 ) for local C k charts (U1 , 1 , V1 ) and (U2 , 2 , V2 ) in M1 and M2 , respectively. For any family of C k manifolds {Mλ }λ∈of dimension n with corresponding atlases {Aλ }λ∈ , the disjoint union M ≡ λ∈ Mλ , with inclusion mappings ιλ : Mλ → M for λ ∈ , inherits the structure of a C k manifold of dimension n with atlas given by   A ≡ {(ιλ (U ),  ◦ ι−1 λ , U ) | (U, , U ) ∈ Aλ for some λ ∈ }.

The verifications of the above are left to the reader (see Exercise 9.2.1). Definition 9.2.4 For k ∈ Z≥0 ∪ {∞}, a continuous mapping : M → N of C k manifolds M and N is of class C k if for every choice of local C k charts (U1 , 1 , V1 ) and (U2 , 2 , V2 ) in M and N , respectively, the mapping −1 2 ◦ ◦ −1 1 : 1 (U1 ∩ (U2 )) → V2

is of class C k . The vector space of C k functions on M with values in F = R or C is denoted by C k (M, F). Depending on the context, C k (M) will denote either C k (M, R) or C k (M, C). We also denote C ∞ (M, F) by E(M, F) or by E(M). The vector space of C ∞ F-valued functions with compact support in M is denoted by D(M, F) or by D(M). A C ∞ bijective mapping with C ∞ inverse is called a diffeomorphism. A C ∞ map  for which the restriction U to some neighborhood U of each point maps U diffeomorphically onto an open set (U ) is called a local diffeomorphism. Remarks 1. The chain rule implies that if : M → N is a continuous mapping of C k manifolds with k ∈ Z≥0 ∪ {∞}, then is of class C k if and only if for each point p ∈ M, there exist local C k charts (U1 , 1 , V1 ) and (U2 , 2 , V2 ) on M and N , respectively, such that p ∈ U1 ∩ −1 (U2 ) and the mapping −1 2 ◦ ◦ −1 1 : 1 (U1 ∩ (U2 )) → V2

is of class C k (see Exercise 9.2.2). In other words, one need only check that the condition holds for elements of some C k atlas in the C k structure (not for every local C k chart). 2. For real analytic manifolds, one uses the analogous terminology. Moreover, the natural analogue of the above property, as well as analogues of many other properties of C k manifolds, also apply to real analytic manifolds. However, since

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a proof that compositions of real analytic mappings are real analytic requires some work (for example, one may form local extensions to complex manifolds and use the fact that compositions of holomorphic maps are holomorphic), and since general facts concerning real analytic manifolds are not required in this book, we will avoid consideration of such facts. 3. It is easy to verify that sums and products of C k functions are of class C k , so C k (M) is an algebra. Definition 9.2.5 For k ∈ Z≥0 ∪ {∞}, a path γ : [a, b] → M is a C k path (or a C k curve) if γ extends to a C k mapping of a neighborhood of [a, b] into M. The path is piecewise C k if there is a partition a = t0 < t1 < · · · < tm = b such that for each i = 1, . . . , m, γ [ti−1 ,ti ] is a C k path. We close this section with a consideration of subsets that inherit a C ∞ structure. Definition 9.2.6 Let M be a smooth manifold of dimension n, and let r ∈ {0, . . . , n}. A nonempty closed set N ⊂ M is a smooth (or C ∞ ) submanifold of dimension r in M if for each point a ∈ N , there exists a local C ∞ coordinate neighborhood (U, (x1 , . . . , xn )) of a in M such that N ∩ U = {p ∈ U | xr+1 (p) = xr+2 (p) = · · · = xn (p) = 0}. Lemma 9.2.7 A smooth submanifold N of dimension r in a smooth n-manifold M is itself a smooth manifold of dimension r with C ∞ atlas consisting of all local charts of the form (N ∩ U, (x1 , . . . , xr )), where (U, (x1 , . . . , xn )) is a local C ∞ chart in M with N ∩ U = {p ∈ U | xr+1 (p) = xr+2 (p) = · · · = xn (p) = 0}. Proof Clearly, N is Hausdorff. If (U,  = (x1 , . . . , xn )) and (V , = (y1 , . . . , yn )) are two local C ∞ charts in M with N ∩ U = {xr+1 = · · · = xn = 0}

and

N ∩ V = {yr+1 = · · · = yn = 0},

then each of the C ∞ maps 0 ≡ (x1 , . . . , xr ) and 0 ≡ (y1 , . . . , yr ) maps N ∩ U and N ∩ V , respectively, homeomorphically onto an open subset of Rr . Moreover, −1

0 ◦ −1 0 (t1 , . . . , tr ) = 0 ( (t1 , . . . , tr , 0, . . . , 0))

for every point (t1 , . . . , tr ) ∈ 0 (N ∩ U ∩ V ), and hence the coordinate transforma∞ tion 0 ◦ −1  0 : 0 (N ∩ U ∩ V ) → 0 (N ∩ U ∩ V ) is of class C . Remark We assume that a given smooth submanifold has the C ∞ structure given by the above lemma unless otherwise indicated. Example 9.2.8 For each R > 0, the circle ∂ (0; R) in R2 is a smooth submanifold of R2 . For given a point p ∈ ∂ (0; R), polar coordinates (r, θ ) (see Example 7.2.8) provide local C ∞ coordinates in some neighborhood U of p in R2 . Setting ρ = r − R, we get the local C ∞ coordinate neighborhood (U, (θ, ρ)) in which U ∩ ∂ (0; R) = {q ∈ U | ρ(q) = 0}.

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Exercises for Sect. 9.2 9.2.1 Verify the claims in Examples 9.2.3. 9.2.2 Verify that if : M → N is a continuous mapping of C k manifolds with k ∈ Z≥0 ∪ {∞}, then is of class C k if and only if for each point p ∈ M, there exist local C k charts (U1 , 1 , V1 ) and (U2 , 2 , V2 ) on M and N , respectively, such that p ∈ U1 ∩ −1 (U2 ) and the mapping 2 ◦ ◦ −1 1 : 1 (U1 ∩

−1 (U2 )) → V2 is of class C k .

9.3 The Topology of Manifolds Manifolds have many nice topological properties, especially second countable manifolds. When working in Rn , it is often convenient to phrase topological arguments in terms of sequences. Fortunately, in a manifold, it also often suffices to consider sequences (in a general topological space, the analogous arguments may be phrased in terms of more general objects called nets, as in, for example, [Ke]). Definition 9.3.1 Let {aν } be a sequence in a topological space X. We say that {aν } converges (in X) to a point a ∈ X if for every neighborhood U of a in X, there exists an integer μ > 0 such that aν ∈ U for all ν ≥ μ. We also say that {aν } is convergent, we call a the limit of {aν }, and we write lim aν = a

ν→∞

or aν → a as ν → ∞.

If {aν } does not have a limit, then we say that the sequence diverges (or the sequence is divergent). The following theorem, the proof of which is left to the reader (see Exercise 9.3.1), contains analogues of standard facts concerning topology in Rn : Theorem 9.3.2 Let D be a subset of a manifold M. Then (a) Sequentially closed sets. D is closed if and only if the limit of every convergent sequence {aν } in M with aν ∈ D for all ν lies in D. A point p ∈ M is a limit point of D if and only if p is the limit of a sequence of points in D \ {p}. The closure of D is equal to the limits of all sequences in D that converge in M. (b) Bolzano–Weierstrass property. If D is compact, then every sequence in D admits a subsequence that converges in M to a point in D. (c) Sequential continuity. A mapping  : D → N of D into a manifold N is continuous at a point a ∈ D if and only if (aν ) → (a) for every sequence {aν } in D with aν → a. We recall that a topological space is second countable if it has a countable basis. Lemma 9.3.3 If (X, T ) is a second countable topological space and B is any basis for T , then there is a countable basis B ⊂ B.

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Proof Fix a set B0 ∈ B. By hypothesis, there exists a countable basis A for T . For each pair a = (A1 , A2 ) ∈ A × A, we may fix a set Ba ∈ B with A1 ⊂ Ba ⊂ A2 if such a set exists; and we may set Ba = B0 if no such set exists. Then the collection B  = {Ba | a ∈ A × A} ⊂ B is countable. Given an open set U ⊂ X and a point p ∈ U , there exist an element A2 ∈ A with p ∈ A2 ⊂ U (since A is a basis for T ), an element B ∈ B with p ∈ B ⊂ A2 (since B is a basis for T ), and an element A1 ∈ A with p ∈ A1 ⊂ B ⊂ A2 . In particular, for a ≡ (A1 , A2 ) ∈ A × A, we have  p ∈ A1 ⊂ Ba ⊂ A2 ⊂ U . It follows that B  is a countable basis for T . Definition 9.3.4 Let X be a topological space. (a) A family of subsets A = {Ai }i∈I of X is locally finite if each point in X has a neighborhood that meets Ai for at most finitely many indices i ∈ I . (b) We call X paracompact if every open cover of X has a locally  finite refinement. That is, for every family of open sets {Ui }i∈I with X  = i∈I Ui , there is a locally finite family of open sets {Vj }j ∈J such that X = j ∈J Vj and such that for each j ∈ J , we have Vj ⊂ Ui for some i ∈ I . The proof of the following useful observation is left to the reader (see Exercise 9.3.2): Lemma 9.3.5 If {Ai }i∈I is a locally finite family of sets in a topological space X, then   Ai = Ai . i∈I

i∈I

The following lemma contains some useful facts concerning the topology of second countable locally compact Hausdorff spaces (for example, second countable manifolds): Lemma 9.3.6 For any second countable locally compact Hausdorff space X, we have the following: ∞ (a) There exists a sequence of open sets { ν }∞ ν=1 ν , ν=0 such that 0 = ∅, X = and ν−1  ν for each ν = 1, 2, 3, . . . . In particular, X is σ -compact. (b) If X is locally connected and connected, then we may choose the sequence of open sets { ν } in (a) so that ν is connected for each ν. (c) If {Uα }α∈A is a covering of a closed set K ⊂ X by open subsets of X and B is a basis for the topology in X, then there exists a countable locally finite (in X) covering {Bi }i∈I of K by elements of the basis B such that for each i ∈ I , we have Bi  Uα for some α ∈ A. In particular, X is paracompact. (d) If {Uα }α∈A is a covering of a closed set K ⊂ X by open subsets of X and for each α ∈ A, Bα and Bα are bases for the topology in Uα , then there exist countable locally finite (in X) coverings {Bi }i∈I and {Bi }i∈I of K such that (i) For each i ∈ I , we have Bi  Bi  Uα and Bi ∈ Bα , Bi ∈ Bα , for some α ∈ A; and

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(ii) If i, j ∈ I and B i ∩ B j = ∅, then Bi  Bj . (e) If {Vi }i∈I is a locally finite (in X) covering of a closed set K ⊂ X by relatively compact open subsets of X, then there exists a covering {Ki }i∈I of K by compact subsets of X with Ki ⊂ Vi for each i ∈ I . (f) If {Ki }i∈I is a locally finite family of closed subsets of X, then there exists a locally finite family of open sets {Vi }i∈I such that Ki ⊂ Vi for each index i ∈ I . Moreover, if the sets {Ki }i∈I are disjoint, then the sets {Vi }i∈I may also be chosen to be disjoint. Proof For the proof of (a), let us fix a countable basis B0 for the topology in X. We may assume that X = ∅, ∅ ∈ / B0 . Since X is locally compact, the collection of relatively compact elements of B0 is itself a basis (as one may easily check), so we may also assume that each element of B0 is relatively compact in X. Hence we , and for may choose a covering of X by relatively compact basis elements {Vj }∞ j =1  ∞ each j , we may let j ≡ V1 ∪ · · · ∪ Vj  X. In particular, X = ∞ V = j =1 j j =1 j . Let j0 = 0 and 0 = ∅. Given indices j0 < j1 < j2 < · · · < jν−1 , we may choose an index jν > jν−1 so that the compact set  jν−1 is contained in jν . Thus we get a sequence of indices {jν }, and setting ν ≡ jν for ν = 0, 1, 2, . . . , we get a sequence of open sets { ν } with the properties required in (a). If X is also locally connected and connected, then the collection of connected components of elements of B0 is countable, and hence by replacing B0 with this collection, we may assume that each element of the basis B0 is connected. For {Vj } as above, we may instead let j be the connected component of V1 ∪ · · · ∪ Vj con taining V1 for each j = 1, 2, 3, . . . . The union  ≡ j is then equal to X. For if p ∈ , then p ∈ Vj for some j and Vj must meet k for some k. Therefore, for l ≡ max(j, k), Vj ∪ k is a connected subset of V1 ∪ · · · ∪ Vl containing V1 , and hence p ∈ Vj ⊂ l ⊂ . Thus  is both open and closed in X and is therefore equal to X. A suitable subsequence of {j } (chosen inductively) will then have each term relatively compact in the next term, as required in (b). For the proof of (c), suppose B is an arbitrary basis for the topology in X. We may choose a sequence of open sets { ν }∞ ν=0 with the properties in (a). For each point p ∈ K, there is a basis element Cp ∈ B such that p ∈ Cp  Uα for some α ∈ A and such that for each index ν = 1, 2, 3, . . . , we have Cp  ν if and only if p ∈ ν , and we have C p ∩ ν = ∅ if and only if p ∈ ν . For each ν = 1, 2, 3, . . . , there exists a (possibly empty) finite collection of points {pi }i∈Iν in the compact set K ∩  \ such that K ∩ \ ⊂ ν ν−1 ν ν−1 i∈Iν Cpi . Letting I be the disjoint union ν∈Z>0 Iν and letting Bi = Cpi for each i ∈ I (here, we identify Iν with its image in I for each ν), we get a countable covering {Bi }i∈I of K by basis elements each of which is relatively compact in Uα for some α ∈ A. Moreover, for each ν, we have Bi ∩ ν = ∅ if i ∈ Iμ with μ > ν + 1, so the family {Bi } is locally finite. For the proof of (d), suppose we have two bases Bα and Bα for the topology in Uα for each α ∈ A. Again, we may assume that the empty set is not an element of any of these bases. Clearly, we may also assume that K = ∅. By applying part (c) to the covering {Uα } and the basis for the topology in X consisting of all open sets B such that either B ∩ K = ∅ or B ∈ Bα and B  Uα for some α ∈ A, we get a

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countable locally finite (in X) covering {Cm }m∈M of K such that for each m ∈ M,  for some α(m) ∈ A. Applying part (c) to the we have Cm  Uα(m) and Cm ∈ Bα(m) covering {Cm }, we get a countable locally finite covering {Vl }l∈L of K by nonempty open sets such that for each l ∈ L, we have Vl  Cm(l) for some m(l) ∈ M. Finally, we apply part (c) (one last time) to the covering {Vl } and the basis for the topology in X consisting of all open sets B such that either B ∩ K = ∅ or for some l ∈ L, we have B ⊂ Vl , B ∈ Bα(m(l)) , and B  Cm(k) whenever k ∈ L with B ∩ V k = ∅. Thus we get a countable locally finite (in X) open covering {Bi }i∈I of K such that for each i ∈ I , there is an index l(i) ∈ L with Bi  Vl(i) and Bi ∈ Bα(m(l(i))) , and Bi  Cm(k) whenever k ∈ L with B i ∩ V k = ∅. Setting Bi ≡ Cm(l(i)) for each i ∈ I , we get a second covering {Bi }i∈I of K such that for each i ∈ I , we have  , Bi = ∅, Bi  Bi  Uα(m(l(i))) , and Bi  Cm(l(j )) = Bj whenever Bi ∈ Bα(m(l(i))) j ∈ I with B i ∩ V l(j ) ⊃ B i ∩ B j = ∅. Finally, since {Bi } is a locally finite family of nonempty sets, each of the relatively compact subsets Cm of X will be equal to Cm(l(i)) (and therefore contain Bi ) for only finitely many indices i ∈ I . Therefore, since the family {Cm } is locally finite, it follows that the family {Bi } = {Cm(l(i)) } is also locally finite. The proofs of (e) and (f) are left to the reader (see Exercise 9.3.3).  Theorem 9.3.7 Let M be a second countable topological manifold. (a) If K is a closed subset of M and U = {Ui }i∈I is a covering of K by open subsets of M, then there exists a countable family of continuous functions {λj }j ∈J on M with

values in [0, 1] such that the family {supp λj }j ∈J is locally finite in M, j ∈J λj ≡ 1 on a neighborhood of K, and for each j ∈ J , we have supp λj ⊂ Ui for some i ∈ I . Moreover, if M is a C ∞ manifold, then we may choose the functions {λj } to be of class C ∞ . If U is locally finite and countable, then we may choose the family of functions so that J = I and supp λi ⊂ Ui for each index i. (b) If K0 and K1 are disjoint closed subsets of M, then there exists a continuous function λ : M → [0, 1] such that λ ≡ 0 on a neighborhood of K0 and λ ≡ 1 on a neighborhood of K1 . Moreover, if M is a C ∞ manifold, then we may choose λ to be of class C ∞ . Remark Recall that the support of a function is the closure of the complement of its zero set. The theorem holds for M a second countable locally compact Hausdorff space (see, for example, [Mu]), but the proof for M a manifold is easier. Proof of Theorem 9.3.7 Let n = dim M. For the proof of (a), we fix a closed set K ⊂ M and a covering U = {Ui }i∈I of K by open subsets of M. Applying Lemma 9.3.6, we see that we may assume without loss of generality that U is countable and locally finite. Applying Lemma 9.3.6 again, we get a countable locally finite family of local charts {(Bν , ν , ν (Bν ))}ν∈N in M and a covering {Bν }ν∈N of M such that for each ν ∈ N , Bν  Bν , ν (Bν ) = BRn (0; 1)  ν (Bν ), and either Bν ⊂ M \ K or Bν ⊂ Ui(ν) for some index i(ν) ∈ I . If M is a C ∞ manifold, then

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we may also assume that ν is a diffeomorphism for each ν. It is easy to verify that the function ρ : R → [0, ∞) given by  e1/(t−1) if t < 1, t → 0 if t ≥ 1, is of class C ∞ on R. Hence, for each ν ∈ N , ην ≡ ρ(|ν |2 ) is a nonnegative continuous function that has compact support contained in Bν and that is positive on Bν . Moreover, ην is of class C ∞ if M is a C ∞ manifold. The functions {λi }i∈I given by

ν∈N,i=i(ν) ην

∀i ∈ I λi ≡ ν∈N ην (note that in each of the above sums, locally, all but finitely many terms vanish) then have the properties required in (a). The verification of this and the proof of (b) are left to the reader (see Exercise 9.3.4).  Definition 9.3.8 If U = {Ui }i∈I is an open covering of a topological space M, then any family of continuous functions {λj }j ∈J on

M with values in [0, 1] such that the family {supp λj }j ∈J is locally finite in M, j ∈J λj ≡ 1 on M, and for each j ∈ J , supp λj ⊂ Ui for some i ∈ I , is called a partition of unity subordinate to the covering U . If M is a C ∞ manifold and λj is of class C ∞ for each j ∈ J , then {λj }j ∈J is called a C ∞ partition of unity. Definition 9.3.9 A continuous mapping  : X → Y of Hausdorff spaces is called proper if the inverse image of every compact subset of Y is compact. Definition 9.3.10 A real-valued function ρ on a Hausdorff space X is an exhaustion function if {x ∈ X | ρ(x) < a}  X for every a ∈ R. A sequence of sets {Dν } in X ◦  such that Dν  D ν+1 for each ν and ν Dν = X is called an exhaustion of X by the sets {Dν }. Remarks 1. If ρ is an exhaustion function, then any function τ ≥ ρ is also an exhaustion function. 2. A continuous function ρ : X → R is an exhaustion function if and only if ρ is bounded below and ρ is a proper mapping. Proposition 9.3.11 Every second countable C ∞ manifold M admits a positive C ∞ exhaustion function. In fact, for every continuous real-valued function τ on M, there exists a C ∞ exhaustion function ρ such that ρ > τ on M. Proof We may assume that M is noncompact, and we may choose a countable locally finite covering {Uν } by relatively compact open subsets and a C ∞ partition of unity {λν } with supp λν ⊂ Uν for each ν. Given a continuous function τ : M → R, ∞ exthe locally finite sum ∞ ν=1 (ν + maxsupp λν |τ |) · λν then gives a positive C haustion function that is greater than τ . 

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Exercises for Sect. 9.3 9.3.1 9.3.2 9.3.3 9.3.4

Prove Theorem 9.3.2. Prove Lemma 9.3.5. Prove parts (e) and (f) of Lemma 9.3.6. Verify that the functions {λi }i∈I constructed in the proof of part (a) of Theorem 9.3.7 have the required properties. Also prove part (b) of the theorem. 9.3.5 Prove that if D is a subset of a second countable manifold M, and every sequence in D admits a subsequence that converges to a point in D, then D is compact (this is a partial converse of part (b) of Theorem 9.3.2).

9.4 The Tangent and Cotangent Bundles Recall that if S = {g = 0} for a real-valued C ∞ function g with nonvanishing gradient on an open subset of R3 , then a tangent vector to S at a point p ∈ S is a vector v along which the total differential (dg)p vanishes. The tangent plane to S at p is the plane through p that is parallel to each such tangent vector v. For any such tangent vector v and any real-valued C ∞ function f on a neighborhood of p in R3 , the value of (df )p (v) depends only on the values of the restriction of f to a neighborhood of p in S. One may turn this around and view v as a linear functional f → (df )p (v) ∈ R that is actually defined on the germs of C ∞ functions at p. This point of view provides an efficient approach for defining tangent vectors (complex as well as real) for a smooth manifold. Definition 9.4.1 Let M be a smooth manifold and let F = R or C. (a) Let p ∈ M, and let ∼p be the equivalence relation on the set of F-valued C ∞ functions that are defined on a neighborhood of p determined by f ∼p g

⇐⇒

f = g on some neighborhood of p.

Each of the associated equivalence classes is called a germ of an (F-valued) C ∞ function at p. The set of germs of C ∞ functions at p is called the stalk of C ∞ at p and is denoted by Cp∞ or Ep . We denote by germp f the germ represented by a local C ∞ function f on a neighborhood of p; i.e., germp f is the germ of f at p. We also consider Ep as an algebra over F with the natural operations. More precisely, for neighborhoods U and V of p and functions f ∈ C ∞ (U ) and g ∈ C ∞ (V ), we define c · germp f ≡ germp (cf )

∀c ∈ F,

germp f · germp g ≡ germp (f U ∩V · gU ∩V ), germp f + germp g ≡ germp (f U ∩V + gU ∩V ), germp f − germp g ≡ germp (f U ∩V − gU ∩V ).

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429

(b) For each point p ∈ M, a tangent vector (over F) at p is a linear functional v : Ep → F that satisfies v(germp (f g)) = v(germp f · germp g) = v(germp f ) · g(p) + f (p) · v(germp g) for all germs germp f, germp g ∈ Ep (that is, v is a linear derivation on the algebra Ep ). For F = R (F = C), v is also called a real tangent vector (respectively, a complex tangent vector). Given a C ∞ F-valued function f on a neighborhood of p, we also write v(f ) ≡ v(germp f ). (c) For each point p ∈ M, the (real) vector space of real tangent vectors at p is called the tangent space (or real tangent space) to M at p and is denoted by Tp M. The complex vector space of complex tangent vectors at p is called the complexified tangent space to M at p and is denoted by (Tp M)C . (d) For each point p ∈ M and each F-valued C ∞ function f on a neighborhood of p, the differential of f at p is the linear functional (df )p on the tangent space at p given by (df )p (v) ≡ v(f ) for each tangent vector v at p. If F = (f1 , . . . , fm ) : U → Fm is a C ∞ mapping (equivalently, each of the functions f1 , . . . , fm is of class C ∞ ) of a neighborhood U into Fm , then we define (dF )p ≡ ((df1 )p , . . . , (dfm )p ), and for each tangent vector v to M at p, we set v(F ) ≡ (dF )p (v). (e) If (U,  = (x1 , . . . , xn ), U  ) is a local C ∞ chart in M, f is a function defined on a neighborhood of a point p ∈ U , and j ∈ {1, . . . , n}, then we define  ∂f ∂ (p) ≡ [f (−1 (t1 , . . . , tn ))](t ,...,t )=(p) , n 1 ∂xj ∂tj provided this partial derivative exists. The corresponding tangent vector

∂ ∂f (p) is denoted by . f → ∂xj ∂xj p (f) For each p ∈ M, the cotangent space and the complexified cotangent space to M at p are, respectively, the dual spaces Tp∗ M ≡ (Tp M)∗

and

(Tp∗ M)C ≡ [(Tp M)C ]∗ .

(g) If  : M → N is a C ∞ mapping of M into a C ∞ manifold N , then the induced tangent maps (or pushforward maps) at p ∈ M are the real linear map Tp M → T(p) N and the complex linear map (Tp M)C → (T(p) N )C , which are both denoted by (∗ )p and are given by (∗ )p (v)(f ) ≡ v(f ◦ ) for every C ∞ function f on a neighborhood of (p) and for every tangent vector v at p (with f R-valued and v ∈ Tp M if we are considering the tangent mapping of the real tangent spaces). The associated pullback mappings (or

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∗ N → T ∗ M and the complex cotangent mappings) are the real linear map T(p) p ∗ ∗ linear map (T(p) N)C → (Tp M)C , which are both denoted by (∗ )p and are given by the adjoints associated to the tangent mappings. That is,

(∗ )p (α)(v) ≡ α((∗ )p (v)) for every linear functional α on the tangent space to N at (p) and every tangent vector v to M at p (with α and v real when we are considering the real (co)tangent spaces, and complex when we are considering the complexified (co)tangent spaces). Remarks 1. If v is a tangent vector to a smooth manifold M at a point p ∈ M and f is a C ∞ function that is equal to a constant c on a neighborhood of p, then v(f ) = 0. For v(1) = v(1 · 1) = v(1) · 1 + 1 · v(1) = 2v(1), and hence v(c) = v(c · 1) = cv(1) = 0. 2. Product rule. It follows from the definition that if f and g are C ∞ functions on a neighborhood of a point p in a smooth manifold M, then (d(f g))p = g(p) · (df )p + f (p) · (dg)p . 3. Chain rule. Let  : M → N and : N → P be C ∞ mappings of C ∞ manifolds M, N , and P , and let p ∈ M. Then ([ ◦ ]∗ )p = ( ∗ )(p) ◦ (∗ )p

and

([ ◦ ]∗ )p = (∗ )p ◦ ( ∗ )(p) .

In particular, if  is a diffeomorphism (i.e.,  is bijective with C ∞ inverse), then (∗ )p and (∗ )p are linear isomorphisms with inverse mappings (∗ )−1 p = −1 )∗ ) = (( . If

: N → P = R or C, then (d(

◦ ((−1 )∗ )(p) and (∗ )−1 (p) p ))p = (d )(p) ◦ (∗ )p (on the real tangent space for P = R and on the complexified tangent space for P = C). For given a tangent vector v, we have (d( ◦ ))p (v) = v( ◦ ) = (∗ )p (v)( ) = (d )(p) ◦ (∗ )p (v). A similar argument shows that the above equality also holds if : N → P = Rm or Cm . Furthermore, taking N = P = Rm or Cm and = IdP , we see that the linear maps (∗ )p and (d)p have the same kernel and the same rank. Proposition 9.4.2 For any C ∞ manifold M of dimension n and any point p ∈ M, we have the following: (a) The map Tp M → (Tp M)C that associates to each real tangent vector v the complex tangent vector given by f → v(Re f ) + iv(Im f ) for every C ∞ function f on a neighborhood of p is an injective linear map. Identifying Tp M with a real vector subspace of (Tp M)C under the above injection, we get the real vector space direct sum decomposition (Tp M)C = Tp M ⊕ iTp M; and hence we may identify the complexified tangent space (Tp M)C with the complexification of the real tangent space Tp M (see Definition 8.1.4) with the real and

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431

imaginary projections given by Re : u + iv → u and Im : u + iv → v for each pair u, v ∈ Tp M. The map Tp∗ M → (Tp∗ M)C that associates to each real linear functional α ∈ Tp∗ M the complex linear functional in (Tp∗ M)C given by v → α(Re v) + iα(Im v) is an injective real linear map. Identifying Tp∗ M with a real vector subspace of (Tp∗ M)C under the above injection, we get the real vector space direct sum decomposition (Tp∗ M)C = Tp∗ M ⊕ iTp∗ M; and we may identify the complexified cotangent space (Tp∗ M)C with the complexification of the real cotangent space Tp∗ M with the real and imaginary projections given by Re : α + iβ → α and Im : α + iβ → β for each pair α, β ∈ Tp∗ M. (b) The tangent space Tp M is of real dimension n, and (Tp M)C is of complex dimension n. In fact, if (U,  = (x1 , . . . , xn )) is any local C ∞ coordinate neighborhood of p, then the partial derivative operators {(∂/∂xj )p }nj=1 form a real basis for Tp M and a complex basis for (Tp M)C . The differentials {(dxj )p }nj=1 form the real dual basis for Tp∗ M and the complex dual basis for (Tp∗ M)C . Proof That the mapping f → v(Re f ) + iv(Im f ) is a complex tangent vector for each v ∈ Tp M, and that the corresponding mapping Tp M → (Tp M)C is an injective real linear map, are left to the reader. Moreover, if v ∈ Tp M \{0}, then v(f ) ∈ R\{0} for some real-valued C ∞ function f on a neighborhood of p. Hence, if w ∈ Tp M, then iw(f ) ∈ iR, so iw = v. Thus Tp M ∩ iTp M = {0}. On the other hand, given a nonzero tangent vector w ∈ (Tp M)C , for each real C ∞ germ germp f , we may set u(f ) ≡ Re[w(f )] and v(f ) ≡ Im[w(f )]. The mappings u and v are real tangent vectors at p. For they are obviously real linear, and for each pair of real C ∞ germs germp f, germp g ∈ Ep , we have u(germp (f g)) = Re[w(f ) · g(p) + f (p)w(g)] = u(germp f ) · g(p) + f (p) · u(germp g), and the analogous equalities hold for v. Identifying u and v with their images in (Tp M)C , we also get w = u + iv. Thus we have the direct sum decomposition (Tp M)C = Tp M ⊕ iTp M of (Tp M)C into real subspaces. Letting V be the complexification of the real vector space Tp M, we have the real linear map V → (Tp M)C given by w → Re w + i Im w. It is now easy to verify that this map is a complex linear isomorphism. The verification that the analogous decomposition of the cotangent spaces holds is left to the reader. Suppose now that (U,  = (x1 , . . . , xn )) is a local C ∞ coordinate neighborhood of p. For each pair i, j = 1, . . . , n, we have  ∂xi 1 if i = j , (p) = δij = ∂xj 0 if i = j ; so the real tangent vectors {(∂/∂xj )p }nj=1 are linearly independent. Conversely,

given a real tangent vector v ∈ Tp M, we may set u ≡ nj=1 vj · (∂/∂xj )p ∈ Tp M, where vj ≡ v(xj ) ∈ R for each j = 1, . . . , n. Given a C ∞ function f on a neighbor-

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hood of p, Lemma 7.2.6 provides constants {bj }nj=1 and C ∞ functions {cij }ni,j =1 on a neighborhood of p such that f = f (p) +

n 

n 

bj (xj − xj (p)) +

j =1

cij (xi − xi (p))(xj − xj (p))

i,j =1

on a neighborhood of p. Thus, using the fact that u and v are derivations, one may now easily check that v(f ) =

n 

bj vj + 0 = u(f ).

j =1

Thus v = u, and hence {(∂/∂xj )p }nj=1 is a basis for Tp M. Clearly, (dxj )p ((∂/∂xi )p ) = δij for all i, j = 1, . . . , n, so {(dxj )p }nj=1 is the associated real dual basis for Tp M. The corresponding claims regarding the complexifications also follow.  Remarks 1. It follows from the above proposition that if (U, (x1 , . . . , xn )) is a local C ∞ coordinate neighborhood in M, v is a tangent vector to M at p ∈ U , α is an element of the cotangent space at p, and f is a C ∞ function a neighborhood of a point p, then v=

n  j =1



∂ (dxj )p (v) · ∂xj

= p

n  j =1



∂ v(xj ) · ∂xj

, p

n  ∂ · (dxj )p , α α= ∂xj p j =1

and (df )p =

n 

df

j =1

∂ ∂xj

(dxj )p = p

n  ∂f (p)(dxj )p . ∂xj j =1

2. Given a complex tangent vector w ∈ (Tp M)C , we have w = u + iv with u, v ∈ Tp M. Under the identification with the complexification of Tp M, we have Re w ≡ u and Im w = v. It should be noted that for a complex C ∞ function germ germp f , u(f ) = u(Re f ) + iu(Im f ) need not be equal to Re(w(f )) = u(Re f ) − v(Im f ). Definition 9.4.3 Let M be a smooth manifold. (a) The (real) tangent bundle and the complexified tangent bundle of M are given by   Tp M and (T M)C ≡ (Tp M)C , TM ≡ p∈M

p∈M

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433

respectively. The associated tangent bundle projections are the (surjective) mappings T M : T M → M

and

(T M)C : (T M)C → M

given by v → p for each point p ∈ M and each tangent vector v in Tp M or (Tp M)C . (b) The (real) cotangent bundle and the complexified cotangent bundle of M are given by   Tp∗ M and (T ∗ M)C ≡ (Tp∗ M)C , T ∗M ≡ p∈M

p∈M

respectively. The associated cotangent bundle projections are the (surjective) mappings T ∗ M : T ∗ M → M

and

(T ∗ M)C : (T ∗ M)C → M

given by α → p for each point p ∈ M and each linear functional α in Tp∗ M or (Tp∗ M)C . (d) Depending on the context, for each C ∞ function f on an open set U ⊂ M, the differential (or exterior derivative) of f is the function df : −1 T M (U ) → R or df : −1 (U ) → C with restriction to each tangent space equal to (df )p ; (T M)C that is, df (v) = v(f ) for each tangent vector v at each point p ∈ U . For F = (f1 , . . . , fm ) with f1 , . . . , fm ∈ C ∞ (U ), we define dF ≡ (df1 , . . . , dfm ). (e) If  : M → N is a C ∞ mapping into a C ∞ manifold N , then the associated tangent maps (or pushforward maps) are the maps ∗ : T M → T N and ∗ : (T M)C → (T N )C given by ∗ (v) = (∗ )p (v) for every tangent vector v to M at a point p ∈ M. The associated pullback mappings (or cotangent mappings) are the maps ∗ : T ∗ N → T ∗ M and ∗ : (T ∗ N )C → (T ∗ M)C given by ∗ N or (T ∗ N ) with p ∈ M. ∗ α = (∗ )p α for every element α of T(p) C (p) Remarks 1. The tangent and cotangent bundles of a smooth manifold M admit natural C ∞ (manifold) structures (see Exercise 9.4.3). They are also examples of C ∞ vector bundles (see, for example [Ns3] or [Wel]). 2. For any open subset of M with inclusion mapping ι : → M, we identify −1 ∗ T and T ∗ with the sets −1 T M ( ) ⊂ T M and T ∗ M ( ) ⊂ T M, respectively, −1 −1 ∗ under the bijections ι∗ : T → T M ( ) and ι : T ∗ M ( ) → T ∗ . We make the analogous identifications for the complexified tangent and complexified cotangent bundles. 3. For any open set ⊂ Rn , we identify the tangent bundles T and (T )C with the products × Rn and × Cn , respectively, under the bijections (T , d(Id )) : T → × Rn

and

((T )C , d(Id )) : (T )C → × Cn .

For any C ∞ map F : → Rm , we get the corresponding identification of the differential mapping dF on T with the differential mapping (p, v) → (dF )p (v) on

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× Rm (see Definition 7.2.2). We make the analogous identification for a C ∞ mapping F : → Cm . 4. For any C 1 function f on a neighborhood of a point p in a smooth manifold M, and any tangent vector v to M at p, we define df (v) = v(f ) ≡

n  j =1

vj

∂f (p), ∂xj

where (x1 , . . . , xn ) are local C ∞ coordinates in a neighborhood of p and vj = dxj (v) for j = 1, . . . , n. The verification that v(f ) is independent of the choice of local coordinates, and hence that we have an induced differential df : v → df (v), is left to the reader. Note also that this definition is consistent with the earlier definitions for f of class C ∞ . Similarly, if  : M → N is a C 1 mapping of smooth manifolds M and N , then we define ∗ : T M → T N and ∗ : (T M)C → (T N )C by (∗ v)(f ) ≡ v(f ◦ ) for each tangent vector v to M and each local C ∞ function f in N . It is easy to verify that ∗ v is a derivation and that we also have (∗ v)(f ) ≡ v(f ◦ ) for every C 1 function f . The pullback mappings are given by α → ∗ α ≡ α ◦ ∗ . The mappings ∗ and ∗ are well-defined mappings of the corresponding tangent and cotangent bundles, respectively. Moreover, if x = (x1 , . . . , xn ) and y = (y1 , . . . , ys ) are local C ∞ coordinates in a neighborhood of p

∈ M and a neighborhood of (p), respectively, then for every tangent vector v = j =1 vj · (∂/∂xj )p (i.e., vj = dxj (v) for each j = 1, . . . , n), we have ∗ v =

n s  

vj ·

k=1 j =1

∂(yk ◦ ) ∂ (p) · , ∂xj ∂yk p

and for every element α = sk=1 ak (dyk )(p) of the cotangent space to N at (p) (i.e., ak = α((∂/∂yk )p ) for each k = 1, . . . , s), we have ∗ α =

n  s 

ak ·

j =1 k=1

∂(yk ◦ ) (p) · (dxj )p . ∂xj

For any C 1 mapping : N → P of N into a C ∞ manifold P , we have ( ◦ )∗ = ∗ ◦ ∗

and

( ◦ )∗ = ∗ ◦ ∗ .

For any C 1 function g on N , we have d(g ◦ ) = (dg) ◦ ∗ = ∗ dg. 5. If  : M → N is a C 1 mapping of smooth manifolds M and N and ∗ ≡ 0, then  is locally constant (see Exercise 9.4.1). Definition 9.4.4 If γ : I → M is a C 1 mapping of an interval I into a smooth manifold M (i.e., γ is C 1 on the interior and γ admits local C 1 extensions at any endpoint contained in I ), then the tangent vector γ˙ (t0 ) to γ at each t0 ∈ I is given by d γ˙ (t0 ) ≡ γ∗ , dt t0

9.4 The Tangent and Cotangent Bundles

435

where t denotes the standard coordinate function on R (at an endpoint in I , the tangent map is that of a local C 1 extension). By the remarks preceding this definition, if  ◦ γ = (γ1 , . . . , γn ) in a local C ∞ coordinate neighborhood (U,  = (x1 , . . . , xn )) of γ (t0 ), then γ˙ (t0 ) =

n  j =1

∂ γj (t0 ) . ∂xj γ (t0 ) 

Definition 9.4.5 Let S be a subset of a smooth manifold M of dimension n. (a) A (real) vector field (complex vector field) on S in M is a map v : S → T M (respectively, v : S → (T M)C ) with T M ◦ v = IdS (respectively, (T M)C ◦ v = IdS ). We usually denote the value v(p) by vp ∈ Tp M (respectively, (Tp M)C ) for each point p ∈ S. For any C 1 function f on an open set U ⊂ M, df (v) = v(f ) denotes the function on S ∩ U given by p → df (vp ) = vp (f ) (that is, df (v) = df ◦ v). If  : U → N is a C 1 mapping of U into a smooth manifold N , then ∗ v denotes the mapping S ∩ U → (T N )C (or T N ) given by p → ∗ vp (that is, ∗ v = ∗ ◦ v). (b) The coefficient functions (or simply the coefficients) of a vector field v on S with respect to (or in) a local C ∞ coordinate neighborhood (U, (x1 , . . . , xn )) are the functions on S ∩ U given by vj ≡ dxj (v) = v(xj ) for j = 1, . . . , n (that

is, v = vj ∂x∂ j ). We say that v is continuous if v has continuous coefficients in every local C ∞ coordinate neighborhood. For S an open set and k ∈ Z≥0 ∪ {∞}, we say that v is of class C k if v has C k coefficients in every local C ∞ coordinate neighborhood. Remarks 1. For the purposes of this book, although it is occasionally convenient to consider vector fields, it is usually more convenient to consider differential forms instead (see Sect. 9.5). 2. A vector field v is of class C k if and only if each point admits a local C ∞ coordinate neighborhood in which the coefficients of v are of class C k (see Exercise 9.4.2). 3. For a function f and vector fields u and v, we define vector fields f v and u + v by p → f (p)vp and p → up + vp , respectively. We close this section with some terminology: Definition 9.4.6 Let  : M → N be a C ∞ mapping of smooth manifolds M and N . A point p ∈ M is a critical point of  if the linear mapping (∗ )p : Tp M → T(p) N is not surjective. The image of a critical point is called a critical value of . Every point in N that is not a critical value is called a regular value of  (in particular, each point in N \ (M) is a regular value).

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Exercises for Sect. 9.4 9.4.1 Prove that if  : M → N is a C 1 mapping of smooth manifolds M and N and ∗ ≡ 0, then  is locally constant. 9.4.2 Let M be a smooth manifold, let S ⊂ M, and let k ∈ Z≥0 ∪ {∞}. (a) For a vector field v on S, prove that the following are equivalent: (i) The vector field v is of class C k . (ii) For every point in S, there exists a local C ∞ coordinate neighborhood with respect to which the coefficients of v are of class C k . (iii) The function df (v) = v(f ) : p → vp (f ) is of class C k for each local C ∞ function f . (b) Prove that any sum of C k vector fields, and any product of a C k function and a C k vector field, is of class C k . 9.4.3 Let M be a smooth manifold of dimension n. (a) Prove that there are unique C ∞ (manifold) structures in T M and in (T M)C such that for each local C ∞ chart (U,  = (x1 , . . . , xn ), U  ) in M, the triples   −1 T M (U ), ( ◦ T M , d−1 (U ) ), U  × Rn TM

and  −1 (T M)C (U ), ( ◦ (T M)C , d−1

(T M)C

(U ) ), U



× Cn = U  × R2n



are local C ∞ charts in T M and (T M)C , respectively. That is, the associated local C ∞ chart in the tangent bundle is given by v → (x1 (p), . . . , xn (p), dx1 (v), . . . , dxn (v)) for each point p ∈ U and each tangent vector v to M at p. (b) Prove that there are unique C ∞ (manifold) structures in T ∗ M and in (T ∗ M)C such that for each local C ∞ chart (U,  = (x1 , . . . , xn ), U  ) in M, we get local C ∞ charts  −1  T ∗ M (U ), , U  × Rn and



 n  2n −1 (T ∗ M)C (U ), C , U × C = U × R



in T ∗ M and (T ∗ M)C , respectively, where the mappings and C are given by





∂ ∂ α → x1 (p), . . . , xn (p), α ,...,α ∂x1 p ∂xn p for each point p ∈ U and each element α of Tp∗ M and (Tp∗ M)C , respectively.

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437

(c) Prove that for each C ∞ function f on an open set ⊂ M, the differentials df : −1 T M ( ) → R

and df : −1 (T M)C ( ) → C

are of class C ∞ with respect to the smooth structures provided by (a). (d) Prove that if  : M → N is a C ∞ mapping of M into a C ∞ manifold N , then the tangent maps ∗ : T M → T N

and

∗ : (T M)C → (T N )C

and

∗ : (T ∗ N )C → (T ∗ M)C

and the pullback maps ∗ : T ∗ N → T ∗ M

are of class C ∞ with respect to the smooth structures provided by (a) and (b). (e) Prove that the inclusion mappings T M → (T M)C and T ∗ M → (T ∗ M)C and the real and imaginary projections T MC → T M and T ∗ MC → T ∗ M are C ∞ mappings. 9.4.4 Let M be a smooth manifold and let k ∈ Z>0 ∪ {∞}. Show that if f ∈ C k (M), then the mapping df : (T M)C → C is of class C k−1 , where (T M)C has the C ∞ structure provided by Exercise 9.4.3 above. Show also that if  : M → N is a C k mapping of M into a smooth manifold N , then the tangent mapping ∗ : (T M)C → (T N )C and the pullback mapping ∗ : (T ∗ N )C → (T ∗ M)C are of class C k−1 . 9.4.5 Let v be a vector field on a smooth manifold M, and let k ∈ Z≥0 ∪ {∞}. Prove that v is a C k vector field if and only if v is of class C k as a mapping of M into (T M)C , where (T M)C has the C ∞ structure provided by Exercise 9.4.3 above.

9.5 Differential Forms on Smooth Curves and Surfaces In this section, we consider differential forms (of degree ≤ 2), that is, objects that are locally exterior products in the cotangent bundle (see Sect. 8.2). For simplicity, we restrict our attention to smooth curves and surfaces (which are the only cases we will need). Definition 9.5.1 Let M be a smooth manifold of dimension n = 1 or 2. (a) For each r = 1, 2, we define r T ∗ M ≡

 p∈M

r Tp∗ M ⊂ r (T ∗ M)C ≡

 p∈M

r (Tp∗ M)C .

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We also set 0 T ∗ M ≡ M × R =



{p} × R =

p∈M 0



0 Tp∗ M

p∈M

∗

⊂  (T M)C ≡ M × C =



{p} × C =

p∈M

and for r ∈ Z>2 , we set r T ∗ M = r (T ∗ M)C ≡ M × {0} =

 p∈M



0 (Tp∗ M)C ,

p∈M

{p} × {0} =



(r Tp∗ M)C .

p∈M

For each r ∈ Z≥0 , the corresponding projections are the (surjective) mappings r T ∗ M : r T ∗ M → M

and

r (T ∗ M)C : r (T ∗ M)C → M

given by α → p for each point p ∈ M and each element α of r Tp∗ M or r (Tp∗ M)C . (b) Given a C 1 mapping  : M → N into a C ∞ manifold N of dimension ≤ 2 and a nonnegative integer r, the associated pullback mappings at each point p ∈ M ∗ N → r T ∗ M and the complex linear map are the real linear map r T(p) p ∗ r r ∗  (T(p) N )C →  (Tp M)C , which are both denoted by (∗ )p and are defined as follows. For r = 1, as in Definition 9.4.1, we set ∗p α(v) ≡ α(∗ v) for every linear functional α on the tangent space to N at (p) and every tangent vector v to M at p (with α and v real when we are considering the real (co)tangent spaces, and complex when we are considering the complexified (co)tangent spaces). For r = 2, we set ∗p α(u, v) ≡ α(∗ u, ∗ v) for every skew-symmetric bilinear function α on the tangent space to N at (p) and every pair of tangent vectors u, v to M at p (again, with α, u, and v real when we are considering the real (co)tangent spaces, and complex when we are considering the complexified (co)tangent spaces). For r = 0, we set ∗p ((p), ζ ) ≡ (p, ζ ) for each scalar ζ , and for r > 2, we set ∗p ≡ 0. The pullback mappings ∗ : r T ∗ N → r T ∗ M and ∗ : r (T ∗ N)C → r (T ∗ M)C are then the mappings with restriction to ∗ ∗ r T(p) N or r (T(p) N)C equal to ∗p for r = 0, 1, 2, . . . . Remarks 1. If  : M → N is a C 1 mapping of C ∞ manifolds of dimension ≤ 2, ∗ ∗ N)C , and β ∈ s (T(p) N )C , then ∗ (α ∧ β) = p ∈ M, r, s ∈ Z≥0 , α ∈ r (T(p) ∗ ∗  α ∧  β (see Exercise 9.5.1). 2. For a given smooth manifold M of dimension n = 1 or 2 and a given nonnegative integer r, r T ∗ M and r (T ∗ M)C have natural C ∞ (manifold) structures (see Exercise 9.5.4). They are also examples of C ∞ vector bundles (see, for example, [Ns3] or [Wel]). Definition 9.5.2 Let M be a smooth manifold of dimension n = 1 or 2, let S ⊂ M, and let r ∈ Z≥0 .

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(a) A real differential form (complex differential form) of degree r in M on S is a map α : S → r T ∗ M (respectively, α : S → r (T ∗ M)C ) with r T ∗ M ◦ α = IdS (respectively, r (T ∗ M)C ◦ α = IdS ). We usually denote the value of α at p by αp ∈ r Tp∗ M (respectively, r (Tp∗ M)C ) for each point p ∈ S. For r = 0, we identify α with the function p → ζ for αp = (p, ζ ). We also call a differential form of degree r an r-form. (b) Let α be a differential form of degree r on S. The coefficient functions (or simply the coefficients) of α with respect to (or in) a local C ∞ coordinate neighborhood (U, (x1 , . . . , xn )) are the functions on S ∩ U given by ⎧ α if r = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ aj = α(∂/∂xj ) for j = 1, . . . , n ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ (i.e., α = n aj dxj ) if r = 1, j =1 α ⎪ ⎪ a = α(∂/∂x1 , ∂/∂x2 ) = dx1 ∧dx ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ (i.e., α = a dx1 ∧ dx2 ) if n = r = 2, ⎪ ⎪ ⎪ ⎩ 0 if r > n.

(c) We say that a differential r-form α on S is continuous if its coefficients in every local C ∞ coordinate neighborhood are continuous. For k ∈ Z≥0 ∪ {∞}, we say that α is of class C k if the coefficients of α in every local C ∞ coordinate neighborhood are of class C k . (d) For S open, E r (S, F) denotes the set of C ∞ differential forms of degree r with values in F = R or C. Depending on the context, E r (S) will denote either E r (S, R) or E r (S, C). The set of C ∞ F-valued r-forms with compact support (where the support of a differential form is the closure of the set of points in S at which the form is nonzero) in S is denoted by D r (S, F) or by Dr (S). Remarks 1. Observe that if α and β are differential forms of degree r and s, respectively, on a subset S of a smooth manifold M of dimension 1 or 2, then the exterior product α ∧ β : p → (α ∧ β)p ≡ αp ∧ βp is an (r + s)-form, with α ∧ β ≡ 0 if r + s > dim M (see Sect. 8.2). For r = s, the sum α + β is the r-form given by p → αp + βp . 2. If α is a 1-form and v is a tangent vector at a point in a smooth manifold, then α(v) = α( ¯ v). ¯ If α is a 2-form and u and v are tangent vectors, then α(u, v) = α( ¯ u, ¯ v) ¯ (see Sects. 8.1 and 8.2). 3. If  : M → N is a C ∞ mapping of C ∞ manifolds of dimension ≤ 2 and α is an r-form on a set S ⊂ N , then ∗ α is an r-form on −1 (S) in M. For r = 0, viewing α as a function on S, we have ∗ α = α ◦ . We have the following description of C k differential forms and their behavior. Proposition 9.5.3 Suppose M is a smooth manifold of dimension n = 1 or 2 and k ∈ Z≥0 ∪ {∞}. Then we have the following:

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(a) Let α be a differential form of degree r on a set S ⊂ M. Then α is a continuous (class C k ) differential form if and only if for each point in S, there exists a local C ∞ coordinate neighborhood with respect to which the corresponding coefficient functions of α are continuous (respectively, of class C k ). (b) If  : N → M is a C ∞ mapping of a smooth manifold N into M, α is a differential form on a set in M, and α is continuous (of class C k ), then ∗ α is continuous (respectively, of class C k ). (c) The sum and exterior product of any two continuous (class C k ) differential forms is continuous (respectively, of class C k ). In particular, for any open set ⊂ M and for any pair of nonnegative integers r, s, the exterior product (α, β) → α ∧ β yields a mapping E r ( ) × E s ( ) → E r+s ( ). Proof Suppose α is a differential r-form defined at points in a set S ⊂ M and (U, x = (x1 , . . . , xn )) and (V , y = (y1 , . . . , yn )) are local C ∞ coordinate neighborhoods in M. For r = 1, we have, on S ∩ U ∩ V ,  n  n n  n    ∂yi dxj = ai dyi = ai bj dxj , α= ∂xj i=1

j =1 i=1

j =1

where {ai } and {bj } are the coefficients of α in (V , y) and (U, x), respectively. Since ∂yi /∂xj ∈ C ∞ (U ∩ V ) for each pair of indices i and j , it follows that if the coefficients {ai } are continuous (of class C k ), then each of the coefficients

n ∂yi bj = i=1 ai ∂xj is also continuous (respectively, of class C k ) for each j = 1, . . . , n. Similarly, for n = r = 2, we have α = a dy1 ∧ dy2 = aJ · dx1 ∧ dx2 , where J is the C ∞ function given by the (Jacobian) determinant    ∂y1 ∂y1   dy1 ∧ dy2  ∂x1 ∂x2  = . J≡ dx1 ∧ dx2  ∂y2 ∂y2  ∂x1

∂x2

Hence, if the coefficient a is continuous (of class C k ), then the coefficient aJ is continuous (respectively, of class C k ). Part (a) now follows. The proofs of (b) and (c) are left to the reader (see Exercise 9.5.2).  Remark In particular, the sets E r ( ) and D r ( ) associated to an open subset of a smooth manifold are vector spaces. It is easy to see that if f is a C k function on a smooth manifold M of dimension n = 1 or 2 with k ∈ Z>0 ∪ {∞}, then the differential, or exterior derivative, df is a C k−1 differential form of degree 1 on M. In particular, for every open set ⊂ M, we have d : E 0 ( ) → E 1 ( )

and

d : D0 ( ) → D 1 ( ).

The following lemma provides an exterior derivative operator d : E 1 ( ) → E 2 ( ).

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Lemma 9.5.4 Let M be a 2-dimensional smooth manifold. Then there is a unique operator d on local C 1 differential forms of degree 1 such that: (i) For every C 1 differential form α of degree 1 on an open set ⊂ M, dα is a C 0 differential form of degree 2 on ; (ii) For every pair of C 1 differential forms α and β of degree 1 on an open set ⊂ M and every constant ζ ∈ C, we have d(α + ζβ) = dα + ζ dβ; (iii) For every local C 2 function f , we have d 2 f = d(df ) = 0; (iv) For every C 1 function f and every C 1 differential form α of degree 1 on an open set ⊂ M, we have d(f α) = df ∧ α + f dα; and (v) For every C 1 differential form α of degree 1 on an open set ⊂ M and every open set U ⊂ , we have d(αU ) = (dα)U . Moreover, if α is a C 1 differential form of degree 1 on an open set ⊂ M, (U, (x, y)) is a local C ∞ coordinate neighborhood, and α = P dx + Qdy on U ∩ (i.e., P and Q are the coefficients of α in the coordinate neighborhood), then

∂Q ∂P − dx ∧ dy on U ∩ . dα = ∂x ∂y In particular, if α is of class C k with k ∈ Z>0 ∪ {∞}, then dα is of class C k−1 . Proof Suppose α is a C k 1-form on an open set ⊂ M with k ∈ Z>0 ∪ {∞}. Let us assume for the moment that an operator d satisfying the conditions (i)–(v) exists. If (U, (x, y)) is a local C ∞ coordinate neighborhood in M, then we have α = P dx + Q dy on U ∩ for unique functions P , Q ∈ C k (U ∩ ) (P ≡ α(∂/∂x), Q ≡ α(∂/∂y)). The conditions (i)–(v) then imply that on U ∩ , dα = d(P dx) + d(Q dy) = (dP ) ∧ dx + (dQ) ∧ dy ∂P ∂Q dy ∧ dx + dx ∧ dy ∂y ∂x

∂Q ∂P − dx ∧ dy. = ∂x ∂y

=

In particular, dα is of class C k−1 and uniqueness follows. Motivated by the above, given a C 1 1-form α on an open set ⊂ M, we now define the C 0 2-form dα on by setting

∂Q ∂P (dα)U ∩ = − dx ∧ dy ∂x ∂y whenever (U, (x, y)) is a local C ∞ coordinate neighborhood in M, and P and Q are the (unique) C 1 functions with α = P dx + Q dy on U ∩ . The 2-form dα is well defined because, if (V , (u, v)) is another local C ∞ coordinate neighborhood in M and α = R du + S dv on V ∩ , then on U ∩ V ∩ , we have     ∂ ∂ ∂ ∂u ∂v ∂Q = α = R+ S ∂x ∂x ∂y ∂x ∂y ∂y

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=

∂ 2v ∂u ∂R ∂v ∂S ∂ 2u R+ S+ + ∂x∂y ∂x∂y ∂y ∂x ∂y ∂x

=

∂ 2u ∂ 2v ∂u ∂u ∂R ∂u ∂v ∂R ∂v ∂u ∂S ∂v ∂v ∂S R+ S+ + + + . ∂x∂y ∂x∂y ∂y ∂x ∂u ∂y ∂x ∂v ∂y ∂x ∂u ∂y ∂x ∂v

Applying the analogous calculation to obtain ∂P /∂y and subtracting, we get





∂S ∂R ∂u ∂v ∂v ∂u ∂S ∂R ∂Q ∂P − = − · − = − · det J, ∂x ∂y ∂u ∂v ∂x ∂y ∂x ∂y ∂u ∂v where J is the (Jacobian) matrix  ∂u J≡ We also have

∂u ∂y ∂v ∂y

∂x ∂v ∂x

 .



∂x ∂y ∂y ∂x du + dv ∧ du + dv dx ∧ dy = ∂u ∂v ∂u ∂v

∂x ∂y ∂y ∂x = − · du ∧ dv ∂u ∂v ∂u ∂v

= det(J −1 ) · du ∧ dv. It follows that



∂Q ∂P − ∂x ∂y

· dx ∧ dy =

∂S ∂R − ∂u ∂v

· du ∧ dv,

and hence dα is well defined. As one may verify, the local representation of dα also gives the conditions (ii)–(v) (see Exercise 9.5.5).  Remark Actually, the conditions (i)–(iv) in the lemma suffice to uniquely determine the operator d (see Exercise 9.5.5). We now collect together the definition of the exterior derivative for a 0-form provided in Sect. 9.4 and the definition for a 1-form suggested by Lemma 9.5.4 above. Definition 9.5.5 Let α be a differential form of degree r ∈ Z≥0 on an open subset of a smooth manifold M of dimension n = 1 or 2. (a) If α is of class C 1 , then the exterior derivative of α is the continuous differential form dα of degree r + 1 on that is equal to (i) The differential of α if r = 0 (see Sect. 9.4); (ii) The 2-form provided by Lemma 9.5.4 if r = 1 and n = 2; (iii) The trivial (i.e., zero) differential form if r ≥ 2 or n = r = 1.

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(b) If α is of class C 1 and we have dα = 0, then we say that α is closed (or d-closed). (c) If α = dβ for some C 1 differential form β of degree r − 1 on (in particular, r > 0), then we say that α is exact (or d-exact). If β may be chosen to be of class C k for some k ∈ Z>0 ∪ {∞}, then we also say that α is C k -exact. For r = 1, we call the function β a potential for α. If for each point in , there exists a C 1 differential form β0 of degree r − 1 on a neighborhood U0 such that dβ0 = αU0 , then we say that α is locally exact (or locally d-exact) and we call β0 a local potential for α. If for some k ∈ Z>0 ∪ {∞}, each of the local forms β0 may be chosen to be of class C k , then we also say that α is locally C k -exact. It is also convenient to consider the trivial 0-form α ≡ 0 to be C ∞ d-exact and to write 0 = d0. Remarks Let M be a smooth manifold of dimension n = 1 or 2. 1. For n = 2, in terms of local C ∞ coordinates (x, y) on an open set in M, the exterior derivative of a C 1 differential form u of degree 0 (i.e., a C 1 function) is du =

∂u ∂u dx + dy, ∂x ∂y

and the exterior derivative of a C 1 differential 1-form α = P dx + Q dy is

∂Q ∂P − dx ∧ dy. dα = dP ∧ dx + dQ ∧ dy = ∂x ∂y 2. For k ∈ Z>0 ∪ {∞}, a C 1 function u on M is of class C k if and only if du is of class C k−1 . Consequently, any exact 1-form α of class C k−1 is automatically C k -exact, and in fact, any potential for α is of class C k . According to the Poincaré lemma (Lemma 9.5.7 below), any closed C k 2-form is locally C k -exact. For the purposes of this book, characterizations of global C k -exactness of a 2-form are not necessary (although a partial solution is considered in Exercise 9.5.9). 3. Any exact differential form α of class C 1 on M is closed. For dα = 0 automatically if deg α = 2. If deg α = 1, then any potential β for α must be of class C 2 , and hence dα = d 2 β = 0. 4. If α and β are C 1 differential forms on M, then (see Exercise 9.5.6) d(α ∧ β) = (dα) ∧ β + (−1)deg α α ∧ dβ. According to the following fact, the proof of which is left to the reader (see Exercise 9.5.7), exterior differentiation commutes with pullbacks: Proposition 9.5.6 If  : M → N is a C 2 mapping of smooth manifolds M and N of dimension 1 or 2 and β is a C 1 differential form on N , then d∗ β = ∗ dβ. If deg β = 0, then this also holds for  a C 1 mapping. Lemma 9.5.7 (Poincaré lemma) Let p be a point in a smooth surface M. Then there exists a neighborhood U of p in M such that for each k ∈ Z>0 ∪ {∞} and

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each closed C k differential form θ of degree r > 0 on U , there exists a C k differential form α on U with dα = θ (in particular, every closed C k form is locally C k -exact). Proof The statement is local, so we may assume without loss of generality that M = (−1, 1) × (−1, 1) ⊂ R2 . Suppose k ∈ Z>0 ∪ {∞} and θ is a C k differential form of degree r > 0 on M. If r = 1, then there are functions P , Q ∈ C k (M) such that

∂Q ∂P θ = P dx + Q dy and 0 = dθ = − dx ∧ dy. ∂x ∂y The function α on M given by  x  α(x, y) = P (t, 0) dt + 0

y

Q(x, t) dt

∀(x, y) ∈ M

0

then satisfies ∂α (x, y) = P (x, 0) + ∂x

 

y

∂ [Q(x, t)] dt ∂x

y

∂ [P (x, t)] dt = P (x, y) ∂t

0

= P (x, 0) + 0

(by Proposition 7.2.5) and ∂α (x, y) = Q(x, y) ∂y for (x, y) ∈ M. Thus dα = θ on M, and in particular, α ∈ C k (M) (in fact, α ∈ C k+1 (M)). If r = 2, then there is a function f ∈ C k (M) such that θ = f dx ∧ dy. The differential 1-form α = P dx + Q dy on M, where   1 y 1 x P (x, y) ≡ − f (x, t) dt and Q(x, y) ≡ f (t, y) dt ∀(x, y) ∈ M, 2 0 2 0 is of class C k (by Proposition 7.2.5) and satisfies ∂P /∂y = −f/2 and ∂Q/∂x = f/2. Thus dα = θ .  Remarks 1. In the last part of the above proof, the 1-forms 2P dx and 2Q dy also have exterior derivative θ . 2. The Poincaré lemma holds for a closed form of positive degree on a manifold of arbitrary dimension. The following terminology is used mainly in Chap. 6: Definition 9.5.8 Let M be a C ∞ manifold of dimension n, let p ∈ M, and let k ∈ Z≥0 . A C k function f on a neighborhood of p has vanishing derivatives of

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order less than or equal to k at p if α ∂ |α| f ∂ f (p) = α1 (p) = 0 ∂x ∂x1 · · · ∂xnαn for every local C ∞ coordinate neighborhood (U, x = (x1 , . . . , xn )) of p and every multi-index α = (α1 , . . . , αn ) ∈ (Z≥0 )n with |α| ≤ k. A C k vector field or differential form on a neighborhood of p has vanishing derivatives of order ≤ k at p if its coefficients in every local C ∞ coordinate neighborhood of p have vanishing derivatives of order ≤ k at p. A C ∞ vector field or differential form with vanishing derivatives of order ≤ m at p for every positive integer m is said to have vanishing derivatives of all orders at p. In fact, the above condition is independent of the choice of local coordinates. Proposition 9.5.9 Let M be an n-dimensional C ∞ manifold, let p ∈ M, let k ∈ Z≥0 , and let f be a C k function on a neighborhood of p. Then f has vanishing derivatives of order ≤ k at p if (and only if) there exists a local C ∞ coordinate neighborhood (V , y = (y1 , . . . , yn )) of p such that (∂/∂y)α f (p) = 0 for each multi-index α = (α1 , . . . , αn ) ∈ (Z≥0 )n with |α| ≤ k. Similarly, a C k vector field or differential form on a neighborhood of p has vanishing derivatives of order ≤ k at p if (and only if) its coefficients in some local C ∞ coordinate neighborhood of p have vanishing derivatives of order ≤ k at p. If θ = fβ, where θ is a C k vector field or differential form, f is a C k function, and β is a nonvanishing C k vector field or differential form, then θ has vanishing derivatives of order ≤ k at p if and only if f has vanishing derivatives of order ≤ k at p. Proof The claim concerning functions is clear for k = 0. We proceed by induction on k, assuming that k > 0 and that the claim holds for nonnegative integers less than k. Assume that we have a local C ∞ coordinate neighborhood (V , y = (y1 , . . . , ∂ α yn )) of p such that ( ∂y ) f (p) = 0 for each multi-index α with |α| ≤ k. In particular, by the induction hypothesis, for each j = 1, . . . , n, the function ∂f/∂yj has vanishing derivatives of order ≤ k − 1 at p. Given a local C ∞ coordinate neighborhood (U, x = (x1 , . . . , xn )) of p, a multi-index α ∈ (Z≥0 )n with |α| = k, and an index m with αm > 0, setting β ≡ (α1 , . . . , αm−1 , αm − 1, αm+1 , . . . , αn ) ∈ (Z≥0 )n , we get

∂ ∂x



α f (p) =

∂ ∂x

β



 n  ∂f ∂ β ∂yj ∂f (p). (p) = ∂xm ∂x ∂xm ∂yj j =1

The product rule yields a sum in which each of the terms contains, for some j , a derivative of ∂f/∂yj at p of order at most k − 1. Therefore, the above must vanish. The proofs of the claims concerning vector fields and differential forms are left to the reader (see Exercise 9.5.8). 

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Exercises for Sect. 9.5 9.5.1 Prove that if  : M → N is a C 1 mapping of C ∞ manifolds of dimension ≤ 2, ∗ N) , and β ∈ s (T ∗ N ) , then ∗ (α ∧ p ∈ M, r, s ∈ Z≥0 , α ∈ r (T(p) C C (p) β) = ∗ α ∧ ∗ β. 9.5.2 Prove parts (b) and (c) of Proposition 9.5.3. 9.5.3 Let M be a C ∞ manifold of dimension n = 1 or 2, let α be a differential form of degree r on an open set ⊂ M, and let k ∈ Z≥0 ∪ {∞}. Prove that for r = 1, α is of class C k if and only if the function α(v) : p → αp (vp ) is of class C k for each local C ∞ vector field v. Prove also that for r = 2, α is of class C k if and only if the function α(u, v) : p → αp (up , vp ) is of class C k for each pair of local C ∞ vector fields u and v. 9.5.4 Let M be a C ∞ manifold of dimension n = 1 or 2, and let r ∈ Z≥0 . For r = 0 or r > 2, r T ∗ M and r (T ∗ M)C have obvious C ∞ (manifold) structures. For r = 1, they have natural C ∞ structures provided by Exercise 9.4.3. We now consider the case r = 2. Assume that n = 2. (a) Prove that there are unique C ∞ (manifold) structures in 2 T ∗ M and in 2 (T ∗ M)C such that for each local C ∞ chart (U,  = (x1 , x2 ), U  ) in M, we get local C ∞ charts  −1    , U  × R and −12 ∗ C , U  × C , 2 T ∗ M (U ),  (U ),   (T M) C

and  C in 2 T ∗ M and 2 (T ∗ M)C , respectively, where the mappings  are given by    α → x1 (p), x2 (p), α (∂/∂x1 )p , (∂/∂x2 )p

α = x1 (p), x2 (p), (dx1 )p ∧ (dx2 )p for each point p ∈ U and each element α of 2 Tp∗ M and 2 (Tp∗ M)C , respectively. (b) Prove that if  : M → N is a C k mapping of M into a C ∞ manifold N of dimension ≤ 2 with k ∈ Z>0 ∪ {∞}, then the pullback mappings  ∗ : 2 T ∗ N →  2 T ∗ M

and ∗ : 2 (T ∗ N )C → 2 (T ∗ M)C

are of class C k−1 with respect to the smooth structures provided by (a). (c) Prove that the inclusion mapping 2 T ∗ M → 2 (T ∗ M)C and the real and imaginary projections 2 (T ∗ M)C → 2 T ∗ M are C ∞ mappings. 9.5.5 Let M be a 2-dimensional smooth manifold. (a) Verify that the conditions (ii)–(v) in Lemma 9.5.4 hold for the operator d constructed in the proof. (b) Prove that in fact, the conditions (i)–(iv) in Lemma 9.5.4 uniquely determine the operator d. 9.5.6 Prove that if α and β are C 1 differential forms on a smooth manifold M of dimension n = 1 or 2 and r = deg α, then d(α ∧β) = (dα)∧β +(−1)r α ∧dβ.

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9.5.7 Prove Proposition 9.5.6. 9.5.8 Verify the claims concerning vector fields and differential forms appearing in Proposition 9.5.9. 9.5.9 Let M be a second countable smooth surface, let k be a positive integer, and let θ be an exact differential form of degree 2 and class C k . Prove that there exists a differential form α of class C k such that dα = θ . Hint. By definition, there is a C 1 1-form β with dβ = θ . Applying the Poincaré lemma, one gets local C k 1-forms {βi } with dβi = α, and local C 2 ∞ functions {fi } with βi − β = df

i . Forming a suitable C partition of unity 2 {λi }, one gets a C 1-form γ ≡ (λi βi + fi dλi ) with dγ = θ . Now proceed by induction. 9.5.10 Let M be a smooth manifold, let p ∈ M, let k ∈ Z≥0 , and let f be a C k+1 function on M with vanishing derivatives of order ≤ k at p. Prove that for every local C ∞ coordinate neighborhood (U, x) of p, and every compact set K ⊂ U , there exists a constant C > 0 such that |f (q)| ≤ C x(q) − x(p) k+1 for all points q ∈ K.

9.6 Measurability in a Smooth Manifold In order to consider Lebesgue integration on manifolds, we must first extend the definition of Lebesgue measurability. Definition 9.6.1 Let M be a smooth manifold of dimension n, and let S ⊂ M. (a) S is called (Lebesgue) measurable if the set (S ∩ U ) ⊂ Rn is Lebesgue measurable for every local C ∞ chart (U, , U  ) in M. (b) S is of measure 0 if (S ∩ U ) is a set of Lebesgue measure 0 for every local C ∞ chart (U, , U  ) in M. (c) A statement (P) about points in S is said to hold almost everywhere if the set of points in S at which the statement does not hold is a set of measure 0. (d) A mapping : S → X of S into a topological space X is measurable if the inverse image −1 (U ) ⊂ S is measurable for every open set U ⊂ X (in particular, S is measurable). For X = [−∞, ∞] or C, is also called a measurable function. We have the following characterizations of measurable sets and mappings: Proposition 9.6.2 In any smooth manifold M, we have the following: (a) A set S ⊂ M is measurable (of measure 0) if and only if for each point p ∈ S, there exists a local C ∞ chart (U, , U  ) such that p ∈ U and (S ∩ U ) ⊂ Rn is measurable (respectively, of measure 0). The composition ρ ◦ of a continuous mapping ρ and a measurable mapping is measurable. The image of a measurable subset of M under a diffeomorphism is measurable.

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(b) The collection of measurable subsets of M is a σ -algebra that contains the Borel subsets of M. Moreover, any subset of a set of measure 0 is a set of measure 0. (c) Let S ⊂ M be a measurable set. Then any product or sum of measurable functions (with values in [−∞, ∞] or C) on S is measurable. A mapping = (ψ1 , . . . , ψm ) of S into a product X1 × · · · × Xm of topological spaces X1 , . . . , Xm is measurable if and only if ψj is measurable for each j = 1, . . . , m. Proof Part (a) follows from the definitions and the change of variables formula (Theorem 7.2.7), and parts (b) and (c) follow from standard arguments.  In analogy with Definition 9.5.2, we make the following: Definition 9.6.3 Let S be a measurable subset of a smooth manifold M of dimension n = 1 or 2. A differential form on S is measurable if its coefficients in every local C ∞ coordinate neighborhood are measurable. The proofs of the following facts concerning measurable differential forms are left to the reader (see Exercise 9.6.1): Proposition 9.6.4 (Cf. Proposition 9.5.3) Let M be a smooth manifold of dimension n = 1 or 2, and let α be a differential form on a measurable set S ⊂ M. Then we have the following: (a) The differential form α is measurable if and only if for every point in S, there exists a local C ∞ coordinate neighborhood with respect to which the coefficient functions of α are measurable. (b) If  : N → M is a diffeomorphism, then α is measurable if and only if the pullback differential form ∗ α on −1 (S) ⊂ N is measurable. (c) The sum and exterior product of any two measurable differential forms on S is measurable. Remark Similarly, a vector field on a measurable subset of a smooth manifold is measurable if its coefficients in any local C ∞ coordinate neighborhood (or in some local C ∞ coordinate neighborhood of each point) are measurable. Exercises for Sect. 9.6 9.6.1 Prove Proposition 9.6.4. 9.6.2 Let α be a differential form of degree r on a measurable subset S of a smooth manifold M of dimension n = 1 or 2. Prove that for r = 1, α is measurable if and only if the function α(v) : p → αp (vp ) is measurable for each local C ∞ vector field v. Also prove that for r = 2, α is measurable if and only if the function α(u, v) : p → αp (up , vp ) is measurable for every pair of local C ∞ vector fields u and v.

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9.6.3 Let v be a vector field on a measurable set S in a smooth manifold M. Prove that the following are equivalent: (i) The vector field v is measurable (i.e., its coefficients in any local C ∞ coordinate neighborhood are measurable). (ii) For any local C ∞ coordinate neighborhood (U, (x1 , . . . , xn )), the function

n dxj (v) is measurable on S ∩ U for each j = 1, . . . , n (i.e., v = j =1 vj · (∂/∂xj ) on S ∩ U with measurable coefficients v1 , . . . , vn on S ∩ U ). (iii) For every point p ∈ S, there exists a local C ∞ coordinate neighborhood (U, (x1 , . . . , xn )) in M such that the function dxj (v) is measurable on S ∩ U for each j = 1, . . . , n. (iv) The function df (v) = v(f ) is measurable for each local C ∞ function f . 9.6.4 Let v be a vector field on a measurable set S in a smooth manifold M. Prove that v is a measurable vector field (i.e., its coefficients in any local C ∞ coordinate neighborhood are measurable) if and only if v is measurable as a mapping of S into (T M)C , where (T M)C has the C ∞ (and hence, topological space) structure provided by Exercise 9.4.3. State and prove the analogous claim for a differential form. 9.6.5 Let f : M → N be a C ∞ mapping of second countable C ∞ manifolds M and N of dimension m and n, respectively. According to Sard’s theorem (see, for example, [Mi]), the set of critical values of f is a set of measure 0. Prove this theorem for the case m = n = 1. Hint. One may assume without loss of generality that M is a bounded open interval in R and N = R. Prove that for every interval I ⊂ M, f (I ) is measurable and λ(f (I )) ≤ supI |f  | · (I ). Show that one may cover any compact subset of the zero set of f  by a suitable finite collection of disjoint open intervals on which |f  | is small.

9.7 Lebesgue Integration on Curves and Surfaces A connected 1-dimensional smooth manifold M is orientable; that is, one may cover M by coordinate intervals with the same sense of left and right (more precisely, the coordinate transformations have positive derivative). The choice of such an atlas is then one of the two orientations in M. One gets the opposite orientation by reversing the direction of each of the intervals in the atlas. For M second countable, the existence of an orientation follows from the fact that every connected second countable 1-dimensional smooth manifold is diffeomorphic to either the circle or the real line (see Theorem 9.10.1). For the general case, one may apply a Zorn’s lemma argument. However, in two dimensions (or higher), no orientation need exist. Example 9.7.1 The Möbius band is the C ∞ surface M = [0, 1] × (0, 1)/(0, t) ∼ (1, 1 − t)

∀t ∈ (0, 1).

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The topology is the quotient topology for the quotient map  : [0, 1] × (0, 1) → M, and a C ∞ atlas is given by {(Uj , j , V )}2j =1 , where U1 = ((0, 1) × (0, 1)), V = (0, 1) × (0, 1), 1 = ((0,1)×(0,1) )−1 , U2 = (([0, 1] \ {1/2}) × (0, 1)), and  (s + (1/2), t) if (s, t) ∈ [0, 1/2) × (0, 1), 2 ((s, t)) = (s − (1/2), 1 − t) if (s, t) ∈ (1/2, 1] × (0, 1). We cannot choose an atlas so that on overlaps, the local charts agree on what a righthanded frame should be, because of the twist at the end. The details appear after the precise definitions below. Throughout this section, M denotes a smooth manifold of dimension n = 1 or 2 (for a discussion of orientation and integration in higher-dimensional manifolds, the reader may refer to, for example, [Mat], [Ns3], or [Wa]). We recall that the Jacobian determinant of a C 1 mapping F = (f1 , . . . , fm ) : U → Rm of an open subset U of Rm into Rm is the continuous function JF : U → R given by    ∂f1 · · · ∂f1   ∂x1 ∂xm   . ..  . .  .. JF ≡ det(dF ) =  .. .   ∂fm ∂fm   ∂x · · · ∂x 1

m

In particular, if F is a diffeomorphism, then JF = 0. For m = 1, we get JF = F  , and for m = 2, we get JF =

∂f1 ∂f2 ∂f2 ∂f1 − . ∂x1 ∂x2 ∂x1 ∂x2

Definition 9.7.2 In the manifold M, we have the following: (a) Two local C ∞ charts (U1 , 1 , U1 ) and (U2 , 2 , U2 ) in M have compatible orientations if each of the associated coordinate transformations has positive Jacobian determinant. That is, J

−1 1 ◦2

> 0 on 2 (U1 ∩ U2 )

(equivalently, J2 ◦−1 = (1/J1 ◦−1 ) ◦ (2 ◦ −1 1 ) > 0 on 1 (U1 ∩ U2 )). 1 2 (b) An atlas in M in which the local charts have compatible orientations is said to be oriented. (c) Two oriented atlases A1 and A2 in M have equivalent orientations if the atlas A1 ∪ A2 is oriented. The verification that this gives an equivalence relation is left to the reader (see Exercise 9.7.1). (d) An equivalence class of oriented atlases in M is called an orientation in M. If an orientation exists, then M is said to be orientable. Otherwise, M is called nonorientable. A manifold M together with an orientation is said to be oriented. A local chart in a representing atlas for the orientation is then said to be positively oriented.

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(e) A diffeomorphism  : M → N of M onto an oriented smooth manifold N of dimension n is orientation-preserving if for each positively oriented local C ∞ chart (V , , V  ) in N , the local C ∞ chart (−1 (V ), ◦ , V  ) is positively oriented in M (hence, −1 is also orientation-preserving). Lemma 9.7.3 The Möbius band is nonorientable. Proof In the notation of Example 9.7.1, the open subsets U1 and U2 are connected and orientable, since we have the diffeomorphism j : Uj → V ⊂ R2 for j = 1, 2. If M is orientable, then we may choose an oriented C ∞ atlas A. Since equivalence of orientations is an equivalence relation among oriented atlases, we may choose the orientation on M so that for some map ρ : R2 → R2 of the form (x, y) → (x, σy) with σ = ±1, the local C ∞ charts (U1 , ρ ◦ 1 , σ (V )) and (U2 , 2 , V ) induce orientations in U1 and U2 , respectively, that are compatible with the orientation on M. But U1 ∩ U2 = (((0, 1) \ {1/2}) × (0, 1)), and for each point (s, t) ∈ 2 (((0, 1/2) × (0, 1))) = (1/2, 1) × (0, 1), we have (ρ ◦ 1 ) ◦ −1 2 (s, t) = (s − (1/2), σ t), while for each point (s, t) ∈ 2 (((1/2, 1) × (0, 1))) = (0, 1/2) × (0, 1), we have (ρ ◦ 1 ) ◦ −1 2 (s, t) = (s + (1/2), σ (1 − t)). Thus the Jacobian determinant Jρ◦ ◦−1 is equal to σ on ((0, 1/2) × (0, 1)) 1 2 and −σ on ((1/2, 1) × (0, 1)), and hence the Jacobian determinant cannot be everywhere positive. Thus we have arrived at a contradiction, and therefore M is nonorientable.  Remarks 1. In the discussion of smooth structures on surfaces in Chap. 6, we will see that a (topological) Möbius band M is nonorientable with respect to any C ∞ structure on M (see Sect. 6.10). 2. If M is an oriented C ∞ manifold, then any open subset of M has a natural induced orientation. 3. If M is a connected orientable C ∞ manifold, then there are exactly two (opposite) orientations in M (see Exercise 9.7.2). Proposition 9.7.4 For each local C ∞ chart (U,  = (x1 , . . . , xn ), U  ) in M, let ω be the nonvanishing C ∞ differential form of degree n on U given by  if n = 1, dx1 ω ≡ dx1 ∧ dx2 if n = 2. (a) Given local C ∞ charts (U,  = (x1 , . . . , xn ), U  ) and (V , = (y1 , . . . , yn ), V  ) in M, we have ω

= (J ◦−1 ) ◦ . ω

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In particular, the two local C ∞ charts have compatible orientations if and only if ω /ω > 0 on U ∩ V . (b) If M is second countable and oriented, then there exists a nonvanishing C ∞ differential form ω of degree n on M such that for every positively oriented local C ∞ chart (U, , U  ) in M, we have ω/ω > 0 on U . (c) Given a nonvanishing continuous differential form ω of degree n on M, the collection A of all local C ∞ charts (U, , U  ) in M with ω/ω > 0 on U is an oriented atlas in M (and hence A determines a unique orientation in M). Proof The proof of part (a) and the proof of part (c) are left to the reader (see Exercise 9.7.3). For the proof of part (b), we observe that if M is second countable and oriented, then we may choose a countable locally finite covering of M by positively oriented local C ∞ charts {(Ui , i , Ui )}i∈I and a C ∞ partition of unity {λi } with supp λi ⊂ Ui for each i ∈ I . The C ∞ differential form ω≡



λi · ωi

i∈I

is then of degree n (following standard conventions, here we extend λi · ωi by 0 to a C ∞ n-form on M for each i ∈ I ), and given a positively oriented local C ∞ chart (U, , U  ) in M, part (a) implies that  ω ωi = λi · >0 ω ω i∈I

on U (in particular, ω is nonvanishing). Thus part (b) is proved.



Definition 9.7.5 (Cf. Definition 8.2.3) Assume that M is oriented, and let S ⊂ M. (a) A real differential form ω of degree n defined at points in S is said to be positive (nonnegative) if for every positively oriented local C ∞ coordinate neighborhood (U,  = (x1 , . . . , xn )) in M, we have ω > 0 (respectively, ≥ 0) ω

on S ∩ U,

where ω = dx1 if n = 1 and ω = dx1 ∧ dx2 if n = 2 (equivalently, each point in S has some coordinate neighborhood (U,  = (x1 , . . . , xn )) for which the above holds). We write ω > 0 or 0 < ω (respectively, ω ≥ 0 or 0 ≤ ω). Similarly, ω is said to be negative (nonpositive) and we write ω < 0 or 0 > ω (respectively, ω ≤ 0 or 0 ≥ ω) if −ω > 0 (respectively, −ω ≥ 0). For p ∈ M, an ordered basis (v1 , . . . , vn ) (= v1 or (v1 , v2 )) for Tp M is positively oriented if each positive α ∈ n Tp M is positive on the ordered basis (i.e., α(v1 ) > 0 if n = 1, α(v1 , v2 ) > 0 if n = 2). (b) For two real differential forms α and β of degree n defined at points in S, we write α ≥ β or β ≤ α if α − β ≥ 0.

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(c) Let α be a real differential form of degree n defined at points in a set S ⊂ M. Given a point p ∈ S, we define the positive part and negative part of α at p by   αp if αp ≥ 0, −αp if αp ≤ 0, + − and αp ≡ αp ≡ 0 if αp < 0, 0 if αp > 0, respectively. In other words, (α + )p =



αp ω

+

·ω

and

(α − )p =



αp ω

−

·ω

for an (arbitrary) positive element ω ∈ n Tp∗ M. (d) Given a sequence {αν } of real differential forms of degree n defined at points in S, and given a point p ∈ S, we define     (αν )p (αν )p inf(αν )p ≡ inf · ω, lim inf(αν )p ≡ lim inf · ω, ν ν ν→∞ ν→∞ ω ω     (αν )p (αν )p sup(αν )p ≡ sup · ω, lim sup(αν )p ≡ lim sup · ω, ω ω ν ν ν→∞ ν→∞ for an arbitrary positive element ω ∈ n Tp∗ M, provided the above defined coefficients exist in R. Remarks 1. If α is a measurable (continuous) real differential form of degree n = dim M as in (c) above, then α + and α − are measurable (respectively, continuous). The analogous statement concerning measurability holds for the infimum, supremum, limit inferior, limit superior, and limit of a sequence of real measurable differential forms of degree n as in (d) above. 2. It is easy see that for a sequence {αν } as in (d) above and for α as in (c), we have lim αν = α if and only if lim inf αν = lim sup αν = α. 3. A nonvanishing C ∞ real differential form of degree n on M is also called a volume form. Proposition 9.7.6 Assume that M is second countable and oriented. Then, for every continuous real differential n-form τ on M, there exists a positive C ∞ differential n-form ω such that ω ≥ τ on M. Proof Proposition 9.7.4 provides a positive C ∞ n-form ω on M, and Proposition 9.3.11 provides a C ∞ function ρ such that ρ > |τ/ω | on M. The positive C ∞ n-form ω ≡ ρω then satisfies ω ≥ τ on M.  Lemma 9.7.7 Suppose M is oriented, ω is a measurable nonnegative differential form of degree n defined at points in a measurable set S ⊂ M, (t1 , . . . , tn ) are the standard coordinates in Rn , ωR1 = dt1 , ωR2 = dt1 ∧ dt2 , and (U,  =

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(x1 , . . . , xn ), U  ) and (V , = (y1 , . . . , yn ), V  ) are two positively oriented local C ∞ charts in M. Then   [−1 ]∗ ω [ −1 ]∗ ω dλ = dλ. ωRn ωRn (S∩U ∩V )

(S∩U ∩V ) Proof This follows from the change of variables formula (Theorem 7.2.7). For if n = 2 and ω = f dx1 ∧ dx2 = g dy1 ∧ dy2 on U ∩ V , then f = g · J ◦−1 ◦  = g · |J ◦−1 ◦ | and [−1 ]∗ ω = (f ◦ −1 ) dt1 ∧ dt2 = (g ◦ −1 ) · |J ◦−1 | dt1 ∧ dt2 . Therefore  (S∩U ∩V )

[−1 ]∗ ω dλ = ωR2

 (S∩U ∩V )

(f ◦ −1 ) dλ



=

◦ −1 ( (S∩U ∩V ))

 =

(S∩U ∩V )

(g ◦ −1 ◦ ◦ −1 ) · |J ◦−1 | dλ

(g ◦ −1 ) dλ =



(S∩U ∩V )

[ −1 ]∗ ω dλ. ωR2

The proof for n = 1 is similar.



Definition 9.7.8 (Cf. [AhS], [Sp], [Wey]) Suppose M is oriented, ω is a measurable complex differential form of degree n defined at points in a measurable set S ⊂ M, (t1 , . . . , tn ) are the standard coordinates in Rn , ωR1 = dt1 , and ωR2 = dt1 ∧ dt2 . (a) If ω ≥ 0 on S and S ⊂ U for some positively oriented local C ∞ chart (U, , U  ) in M, then we define 

 ω≡ S

(S)

[−1 ]∗ ω dλ ∈ [0, ∞] ωRn

(which is independent of the choice of the local chart by Lemma 9.7.7). (b) If ω ≥ 0 on S, then we define the integral of ω over S by  ω ≡ sup S

m  

ω ∈ [0, ∞],

j =1 Sj

where the supremum is taken over all choices of disjoint measurable subsets S1 , . . . , Sm of S, each of which is contained in a (possibly different) local C ∞ chart.

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(c) We say that ω is integrable on S if the nonnegative extended real numbers   R+ ≡ [Re ω]+ , R− ≡ [Re ω]− ,  I+ ≡

S

S



[Im ω]+ ,

I− ≡

S

[Im ω]− ,

S

are finite. If this is the case, then we define the integral of ω over S by  ω ≡ R+ − R− + iI+ − iI− . S

(d) We say that ω is locally integrable on S if each point in S admits a neighborhood U in M such that ωU ∩S is integrable. Remarks 1. The definitions in (a) and (b) are consistent; that is, for S contained in a local C ∞ coordinate neighborhood, the integral of ω ≥ 0 defined in (a) is equal to its integral as defined in (b). 2. On a second countable oriented manifold, one may simplify the above by using a partition of unity (see, for example, [Mat], [Ns3], or [Wa]). Proposition 9.7.9 Assume that M is oriented, let S be a measurable subset of M, let ω be a nonnegative measurable differential form of degree n defined on S, and let α and β be measurable differential forms of degree n defined on S. (a) The function λω on the σ -algebra of measurable subsets of S defined by  λω : R → ω R

is a positive measure. (b) If f is a measurable function that satisfies 0 ≤ f < ∞ a.e. in S, then   f ω = f dλω . S

S

(c) If α = 0 a.e. in Z ≡ {x ∈ S | ωx = 0} and α ≥ 0 a.e. in S, then    α α · ω = · dλω . α= ω ω S S\Z S\Z (d) If α ≥ 0 and β ≥ 0 a.e. in S, then    (rα + β) = r α + β S

S

∀r ∈ [0, ∞).

S

(e) If α and β are integrable on S and r ∈ C, then rα + β is integrable on S and    (rα + β) = r α + β. S

S

S

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(f) If α ≥ β a.e. in S and each of the forms is either nonnegative or integrable on S, then   α ≥ β. S

S

(g) For any open covering B of M, we have  ω = sup S

m  

ω,

j =1 Sj

where the supremum is taken over all choices of disjoint measurable subsets S1 , . . . , Sm of S each of which is contained in an element of B. Proof Clearly, we have λω (∅) = 0. Suppose R is the union of a sequence {Sν } of disjoint measurable subsets of S. Given finitely many disjoint measurable subsets R1 , . . . , Rm of S, each of which is contained in a local C ∞ coordinate neighborhood in M, we have m   j =1 Rj

ω=

m  ∞   j =1 ν=1 Sν ∩Rj

ω=

∞  m   ν=1 j =1 Sν ∩Rj

ω≤

∞  

ω=

ν=1 Sν

∞ 

λω (Sν ).

ν=1



∞ Thus λω (R) ≤ ∞ ν=1 λω (Sν ). Conversely, given t ∈ [0, ∞) with ν=1 λω (Sν ) > t, we may choose N ∈ Z>0 and, for each ν = 1, . . . , N , disjoint measurable subsets ν ∞ coordinate neighborhood, such {Rj(ν) }m j =1 of Sν , each contained in some local C that mν  N   ω ≤ λω (R). t< (ν)

ν=1 j =1 Rj

Part (a) now follows. For (b), observe that by changing f on a set of measure 0, we may assume without loss of generality that 0 ≤ f < ∞ on S. The equality is clear if S is contained in a local C ∞ coordinate neighborhood (see the remarks following Definition 9.7.8). Applying (a), one gets the equality for a finite union of disjoint measurable subsets,  each of which has the above property. Passing to suprema, one gets  f ω ≤ S S f dλω . For the reverse inequality, observe that if S1 , . . . , Sm are disjoint measurable subsets of S and r1 , . . . , rm are nonnegative constants with f ≥ rj on Sj for each j , then m  j =1

rj λω (Sj ) =

m  j =1

 ω≤

rj Sj

m   j =1 Sj

 fω ≤

fω S

(one may verify the above inequalities directly). The proofs of (c)–(g) are left to the reader (see Exercise 9.7.4).



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Definition 9.7.10 Assume that M is oriented, and let ω be a nonnegative measurable n-form on a measurable set S ⊂ M. Then we denote by λω the positive measure on S provided by Proposition 9.7.9, and we call λω the measure associated to ω. Remark The measure λω is complete if ω > 0 almost everywhere (see Exercise 9.7.9). Applying the standard convergence theorems for measure spaces, we get the following versions for integrals of differential forms: Theorem 9.7.11 (Convergence theorems) Assume that M is oriented. Let ω and {ων }∞ ν=1 be measurable n-forms on a measurable set S ⊂ M. (a) Monotone convergence theorem. If 0 ≤ ω1 ≤ ω2 ≤ ω3 ≤ · · · and ων → ω, then   ω = lim ων . S

ν→∞ S

(b) Fatou’s lemma. If ων ≥ 0 for each ν and ω = lim infν→∞ ων , then   ω ≤ lim inf ων . S

ν→∞

S

(c) Dominated convergence theorem. Suppose ων → ω and there exists a nonnegative integrable differential form θ of degree n on S such that for each ν ∈ Z>0 and each point x ∈ S, we have either (ων )x = 0 or θx > 0 and |(ων )x /θx | ≤ 1. Then ω is integrable and   ω = lim ων . S

ν→∞ S

Remark It is not necessary to specify that the form ω be measurable, since the limit or limit inferior of a sequence of measurable differential forms of degree n is measurable. Proof of Theorem 9.7.11 For the proof of (a), we let P = {x ∈ S | ωx > 0}. By hypothesis, we have ων = 0 at each point in S \ P . Therefore, by the standard monotone convergence theorem (i.e., the monotone convergence theorem for integration of functions with respect to measures), we have       ων ων ω ·ω= · dλω → · dλω = ων = ω = ω as ν → ∞. S P ω P ω P ω P S The proofs of (b) and (c) are left to the reader (see Exercise 9.7.5). Definition 9.7.12 Let S be a measurable subset of M, and let p ∈ [1, ∞].



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(a) Let α be a measurable differential form of degree r in M on S. For r = 0, we p say that α is in Lloc (S) if for every local C ∞ chart (U, , U  ), α ◦ −1 is in p Lloc ((S ∩ U )) (that is, each point in (S ∩ U ) admits a neighborhood V such that the function |α ◦ −1 |p is integrable on V ∩ (S ∩ U )). For r > 0, we say p that α is in Lloc (S) if the coefficients of α in every local C ∞ chart (U, , U  ) p are in Lloc (S ∩ U ). For p = 1, we also say that α is locally integrable, and for p = 2, we also say that α is locally square integrable. p (b) Let {αν }∞ ν=1 and α be differential r-forms in Lloc (S). For r = 0, we say that {αν } p converges in Lloc (S) to α if for every local C ∞ chart (U, , U  ), the sequence p −1 {αν ◦  } converges to α ◦ −1 in Lloc ((S ∩ U )) (that is, for each point in  (S ∩ U ), there exists a neighborhood V in U  such that V ∩(S∩U ) |αν ◦ −1 − p α ◦ −1 |p dλ → 0). For r > 0, we say that {αν } converges in Lloc (S) to α if in ∞  every local C chart (U, , U ), the coefficients of the terms of {αν } converge p in Lloc (S ∩ U ) to the corresponding coefficients of α. Remarks 1. It is easy to check that the conditions in (a) and (b) hold precisely when they hold in some local C ∞ coordinate neighborhood of each point in S. 2. For M orientable, a measurable n-form α on S ⊂ M is locally integrable (in the above sense) if and only if each point in S admits a neighborhood U such that α is integrable on U ∩ S (as an n-form). The analogous statement for local L1 convergence also holds. p q 3. If α and β are differential forms in Lloc and Lloc , respectively, with p, q ∈ [1, ∞] and p −1 + q −1 = 1, then the exterior product α ∧ β is locally integrable; that is, α ∧ β is in L1loc . p 4. A measurable vector field is said to be in Lloc (S) if its coefficients in every p ∞  local C chart (U, , U ) are in Lloc (S ∩ U ). Lemma 9.7.13 Assume that M is second countable and oriented, and let ω be a nonnegative locally integrable measurable differential form of degree n on M. Then there exists a positive C ∞ function ρ on M so large that e−ρ ω is integrable. Proof We may assume that M is noncompact (otherwise take ρ ≡ 1) and we may choose a locally finite covering {Uν }∞ ν=1 by relatively compact open subsets and a C ∞ partition of unity {λν } with supp λν ⊂ Uν and 0 ≤ λν ≤ 1 for each ν. For each ν, we may choose a constant Rν such that  λν ω < 2−ν . 0 < Rν < 1 and Rν · M

∞

The locally finite sum ν=1 Rν · λν then gives a C ∞ function τ : M → (0, 1), and we may define ρ ≡ − log τ > 0. We then get  M

e−ρ ω =

∞  ν=1

 Rν

λν ω < M

∞  ν=1

2−ν = 1 < ∞.



9.7 Lebesgue Integration on Curves and Surfaces

459

Definition 9.7.14 An open subset of a 2-dimensional C ∞ manifold M is called C ∞ (or smooth) if for each point p ∈ M, there is a local C ∞ coordinate neighborhood (U, (x, y)) such that ∩ U = {q ∈ U | x(q) < 0}. Remark The boundary ∂ of a C ∞ open subset of M is either the empty set or a 1-dimensional C ∞ submanifold of M. Lemma 9.7.15 Let M be an oriented C ∞ manifold of dimension 2, and let be a C ∞ open subset of M with nonempty boundary ∂ . Then ∂ has a (natural) induced orientation with an oriented C ∞ atlas A consisting of all local C ∞ charts of the form (U ∩ ∂ , 2 U ∩∂ , 2 (U ∩ ∂ )), where (U,  = (1 , 2 ), U  ) is a positively oriented local C ∞ chart in M with ∩ U = {q ∈ U | 1 (q) < 0} (see Fig. 9.2). Furthermore, if  : M → N is an orientation-preserving diffeomorphism of M onto an oriented C ∞ manifold N and  ≡ ( ), then the diffeomorphism ∂ : ∂ → ∂ is orientation-preserving with respect to the induced orientations. Proof The collection A covers ∂ , since if (U, (1 , 2 ), U  ) is a local C ∞ chart that is not positively oriented, U is connected, and ∩ U = {1 < 0}, then the local C ∞ chart (U, (1 , −2 ), U  ), where U  ≡ {(x, −y) | (x, y) ∈ U  }, is positively oriented. Given two positively oriented local C ∞ charts (U,  = (1 , 2 ) = (x, y), U  ) and (V , = ( 1 , 2 ) = (u, v), V  ) with ∩ U = {x < 0} and ∩ V = {u < 0}, the coordinate transformation F ≡ 2 ◦ (2 U ∩∂ )−1 of the associated local C ∞ charts in ∂ satisfies

d ∂v −1 F  (t) = d 2 (2 U ∩∂ )−1 = ( (0, t)). ∗ dt ∂y On the other hand, we have u < 0 on U ∩ V ∩ = {p ∈ U ∩ V | x(p) < 0}, u ≡ 0 on U ∩ V ∩ ∂ = {p ∈ U ∩ V | x(p) = 0}, and u > 0 on U ∩ V \ = {p ∈ U ∩ V | x(p) > 0}. So, at each point in U ∩ V ∩ ∂ , we have ∂u ≥ 0 and ∂x and hence

 ∂u/∂x 0 < J ◦−1 ◦  =  ∂v/∂x

∂u = 0, ∂y  0  ∂u ∂v = · . ∂v/∂y  ∂x ∂y

It follows that F  (t) > 0 and therefore that the atlas is oriented. The proof of the last statement concerning the diffeomorphism  : M → N is left to the reader (see Exercise 9.7.6).  Definition 9.7.16 Let be a C ∞ open subset of an oriented 2-dimensional C ∞ manifold M with nonempty boundary ∂ . The orientation on ∂ given by Lemma 9.7.15 is called the induced orientation. Remark We assume that the boundary of a C ∞ open set in an oriented 2-dimensional C ∞ manifold has the induced orientation unless otherwise indicated.

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Fig. 9.2 The induced orientation on ∂

Theorem 9.7.17 (Stokes’ theorem) Let M be an oriented C ∞ surface, let be a nonempty smooth domain in M, and let α be a C 1 differential form of degree 1 on M such that ∩ supp α is compact. Then   α= dα. ∂

In particular, if supp α ∩ ∂ = ∅, then



dα

= 0.

  Remark Stokes’ theorem should actually be written ∂ ι∗ α = dα, where ι : ∂ → M is the inclusion map; but in an abuse of notation, we will usually leave out the pullback map ι∗ . Proof of Stokes’ theorem We may choose a finite collection {(Ui , i = (xi , yi ), Ui = (ai , bi ) × (0, 1))}m i=1 of positively oriented local C ∞ charts such that ∩supp α ⊂ U1 ∪· · ·∪Um  M and ∞ for each i = 1, . . . , m, ∩ U

i = {p ∈ Ui | xi (p) < 0} = ∅. We may also choose C m m functions {ηi }i=1 such that i=1 ηi ≡ 1 on ∩ supp α and supp ηi ⊂ Ui for each i = 1, . . . , m. By reordering if necessary, we may assume that there is an index k such that ai < 0 < bi for each i = 1, . . . , k and bi ≤ 0 for each i = k + 1, . . . , m. ∗  For each i = 1, . . . , m, we have βi ≡ (−1 i ) (ηi α) = Pi dx + Qi dy on Ui , where 1 2  Pi and Qi are C functions on R with compact support in Ui . Thus  dα =

  m

d(ηi α) =

i=1

=

m  

d(ηi α) =

i=1

m  

dβi =

i=1 i ( ∩Ui )

m  

d(ηi α)

i=1 ∩Ui

k  

dβi +

i=1 (ai ,0)×(0,1)

m  

dβi .

i=k+1 (ai ,bi )×(0,1)

For each i = 1, . . . , k, we have  0 1   1 0 ∂Qi ∂Pi (x, y) dx dy − (x, y) dy dx dβi = ∂x 0 ai ai 0 ∂y (ai ,0)×(0,1)  1  0 = [Qi (0, y) − Qi (ai , y)] dy − [Pi (x, 1) − Pi (x, 0)] dx 0

ai

9.7 Lebesgue Integration on Curves and Surfaces



1

=

461

 Qi (0, y) dy =

0



{0}×(0,1)

βi =

ηi α, ∂

and similarly, for i = k + 1, . . . , m, 



1

dβi = 0

(ai ,bi )×(0,1)

−

[Qi (bi , y) − Qi (ai , y)] dy



bi

[Pi (x, 1) − Pi (x, 0)] dx = 0.

ai



Summing, we get the claim. We close this section by briefly recalling the related notion of a line integral.

Definition 9.7.18 Let α be a continuous differential form of degree 1 on a C ∞ manifold M of dimension 1 or 2, and let γ : [a, b] → M be a piecewise C 1 path in M (see Definition 9.2.5). Then the (line) integral of α along γ is given by 

 α≡ γ

b

 α(γ˙ (t)) dt =

γ ∗ α.

(a,b)

a

Lemma 9.7.19 Let α be a continuous 1-form on a C ∞ manifold M of dimension n = 1 or 2, and let γ : [a, b] → M be a piecewise C 1 path in M. (a) If (U, (x1 , . . . , xn )) is a local C ∞ coordinate neighborhood in M, γ ([a, b]) ⊂ U , cj = xj ◦ γ for each j = 1, . . . , n, and α = a1 dx1 + · · · + an dxn on U , then  α= γ

n   j =1 a

b

aj (γ (t))cj (t) dt.

(b) If N ≡ γ ((a, b)) is itself a 1-dimensional C ∞ manifold for which the restriction and the inclusion map ι : N → M is γ (a,b) : (a, b) → N  is a diffeomorphism  of class C ∞ , then γ α = N ι∗ α, provided N is given the orientation induced by γ . The proof is left to the reader (see Exercise 9.7.7). Line integrals are considered in greater depth in Chap. 10. Example 9.7.20 For polar coordinates (r, θ ) in a suitable open subset of R2 , and for rectangular coordinates (x, y), we have dx ∧ dy = r dr ∧ dθ . Thus (r, θ ) gives positively oriented local C ∞ coordinates. Consequently, for each R > 0 and each θ0 ∈ R, the diffeomorphism θ → Reiθ of (θ0 , θ0 + 2π) onto (∂ (0; R)) \ {Reiθ0 } is orientation-preserving (where the boundary ∂ (0; R) of the smooth domain

(0; R) is given the induced orientation). Hence, for any continuous 1-form α =

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a dr + b dθ defined at points of ∂ (0; R), we have 

 α= ∂ (0;R)

θ0 +2π

b(Reiθ ) dθ.

θ0

Exercises for Sect. 9.7 9.7.1 Prove that equivalence of oriented C ∞ atlases on a C ∞ manifold M of dimension n = 1 or 2 is an equivalence relation. 9.7.2 Prove that any connected orientable C ∞ manifold M of dimension n = 1 or 2 admits exactly two orientations. 9.7.3 Prove parts (a) and (c) of Proposition 9.7.4. 9.7.4 Prove parts (c)–(g) of Proposition 9.7.9. 9.7.5 Prove parts (b) and (c) of Theorem 9.7.11. 9.7.6 Prove the last part of Lemma 9.7.15. That is, prove that if M and N are oriented 2-dimensional C ∞ manifolds, is a C ∞ open subset of M,  : M → N is an orientation-preserving diffeomorphism, and  ≡ ( ), then the diffeomorphism ∂ : ∂ → ∂ is orientation-preserving with respect to the induced orientations. 9.7.7 Prove Lemma 9.7.19. 9.7.8 Let M be a nonorientable C ∞ surface. Prove that there exists a surjective C ∞ → M of an orientable C ∞ surface M onto M such that each mapping ϒ : M point in M admits a connected neighborhood U for which ϒ −1 (U ) has exactly two connected components, each of which is mapped diffeomorphically → M is called the orientable double cover of M). onto U by ϒ (ϒ : M 9.7.9 Let ω be a nonnegative measurable n-form on an orientable C ∞ manifold M of dimension n = 1 or 2. Prove that if ω > 0 a.e., then the measure λω is complete.

9.8 Linear Differential Operators on Manifolds Throughout this section, M denotes a smooth manifold of dimension n. Linear differential operators on open sets in Rn were considered in Sect. 7.4. A linear differential operator on M is an operator that may be expressed in local C ∞ coordinates as a linear differential operator on an open subset of Rn . More precisely: Definition 9.8.1 Let be an open subset of M. A linear differential operator A of order k ∈ Z≥0 on in M is an operator that associates to each open set V ⊂ and each C k function u on V , a function Au ∈ C 0 (V ), and which has the following property. For each point in , there exist a local C ∞ coordinate neighborhood (U,  = (x1 , . . . , xn ), U  ) and a linear differential operator A of order k on ( ∩ U ) such that for each open set V ⊂ and each function u ∈ C k (V ),

9.8 Linear Differential Operators on Manifolds

463

we have [Au]V ∩U = A (u ◦ −1 ) ◦ V ∩U . In other words, there exist functions {aα }α∈(Z≥0 )n ; |α|≤k on ∩ U such that 

[Au]V ∩U =

α∈(Z≥0 )n ,|α|≤k

aα ·

∂ |α| u ∂x α

on for every open set V ⊂ and every function u ∈ C k (V ) (we have A =

V ∩ U −1 (aα ◦  ) · (∂/∂t)α , where (t1 , . . . , tn ) are the standard coordinates on Rn ). We call this a local representation of A with coefficients {aα }, and we write α  ∂ A= aα on ∩ U. ∂x n α∈(Z≥0 ) ,|α|≤k

We say that A has C d coefficients if it has local representations with C d coefficients in a neighborhood of each point. Remarks 1. It is easy to verify that if A is a linear differential operator of order k with C d coefficients on an open subset of M, then A has a local representation with C d coefficients in every local C ∞ coordinate neighborhood in M. 2. It follows from the definition that if A is a linear differential operator of order k on an open set ⊂ M, u ∈ C k ( ), and V is an open subset of , then [Au]V = A[uV ]. 3. Linear differential operators on differential forms are defined similarly (in terms of differential operators on the coefficients in local C ∞ coordinate neighborhoods). For distributional solutions of the differential equation Au = v, one considers distributional solutions of the corresponding equations in local C ∞ coordinates (cf. Definition 7.4.2). Definition 9.8.2 Let k ∈ Z≥0 , and let A be a linear differential operator of order k with C k coefficients on an open set ⊂ M. For u, v ∈ L1loc ( ), we say that Au is equal to v in the distributional (or weak) sense, and we write Adistr u = v (and Adistr u ∈ L1loc ( )), if for every local C ∞ coordinate neighborhood (U, , U  ) in M with corresponding linear differential operator A (i.e., Af = A (f ◦ −1 ) ◦  for f ∈ C k ), we have [A ]distr (u ◦ −1 ) = v ◦ −1 on ( ∩ U ). This property is local; that is, Adistr u = v on if and only if Adistr u = v on a neighborhood of each point in . Moreover, according to the following, this property is independent of the choice of local C ∞ coordinates: Proposition 9.8.3 Let A be a linear differential operator of order k with C k coefficients on an open set ⊂ M, and let u, v ∈ L1loc ( ). Then Adistr u = v if and only if for every point in , there exists a local C ∞ chart (V , , V  ) in M for which the

464

9

Manifolds

associated linear differential operator A satisfies [A ]distr (u ◦ −1 ) = v ◦ −1 on ( ∩ V ). In particular, Adistr u = v if and only if for each point in , there is a neighborhood U in such that Adistr (uU ) = vU . Proof Let (U, , U  ) be a local C ∞ coordinate neighborhood, and let A be the associated linear differential operator. Suppose (V , , V  ) is a local C ∞ coordinate neighborhood in M for which the associated linear differential operator A

satisfies [A ]distr (u ◦ −1 ) = v ◦ −1 on ( ∩ V ). We then have the coordinate transformation F ≡  ◦ −1 : (U ∩ V ) → (U ∩ V ), and for each function ϕ ∈ D(( ∩ U ∩ V )), the change of variables formula (Theorem 7.2.7) gives, for each f ∈ C ∞ (( ∩ U ∩ V )),  ( ∩U ∩V )

f

· A∗ (ϕ) dλ =

 

=

( ∩U ∩V )

( ∩U ∩V )

 =  =

( ∩U ∩V )

( ∩U ∩V )

 =  =

( ∩U ∩V )

( ∩U ∩V )

A (f ) · ϕ dλ (A(f ◦ ) ◦ −1 ) · ϕ dλ (A(f ◦ ) ◦ −1 ) · ϕ ◦ F |JF | dλ (A (f ◦ F )) · ϕ ◦ F |JF | dλ (f ◦ F ) · A∗ [|JF |(ϕ ◦ F )] dλ f · |JF −1 | · (A∗ [|JF |(ϕ ◦ F )] ◦ F −1 ) dλ.

Therefore, by Lemma 7.3.2, A∗ (ϕ) = |JF −1 | · (A∗ [|JF |(ϕ ◦ F )] ◦ F −1 ), and hence  ( ∩U ∩V )

(u ◦ −1 ) · A∗ (ϕ) dλ =

 

=  =

( ∩U ∩V )

( ∩U ∩V )

( ∩U ∩V )

(u ◦ −1 ) · A∗ [|JF |(ϕ ◦ F )] dλ (v ◦ −1 ) · ϕ ◦ F |JF | dλ (v ◦ −1 ) · ϕ dλ.

Thus [A ]distr (u ◦ −1 ) = v ◦ −1 on ( ∩ U ∩ V ). If each point in U admits such a local C ∞ coordinate neighborhood (V , , V  ), then given a function ϕ ∈ D(( ∩ U )), there exist functions η1 , . . . , ηm ∈

D(( ∩ U )) such that ηj ≡ 1 on supp ϕ and [A ]distr (u ◦ −1 ) = v ◦ −1

9.9

C ∞ Embeddings

465

on a neighborhood Vj of supp ηj in ( ∩ U ) for each j = 1, . . . , m. Thus 

−1

(u ◦  ( ∩U )

) · A∗ (ϕ) dλ = =

m   j =1 Vj m  

(u ◦ −1 ) · A∗ (ηj ϕ) dλ (v ◦ −1 ) · ηj ϕ dλ

j =1 Vj



(v ◦ −1 ) · ϕ dλ;

= ( ∩U )

and hence [A ]distr (u ◦ −1 ) = v ◦ −1 on ( ∩ U ).



9.9 C ∞ Embeddings In this section, we consider some criteria that guarantee that a C ∞ mapping is a (local) diffeomorphism, as well as criteria that guarantee that the zero set of a C ∞ function, or the image of a C ∞ mapping, is a smooth submanifold. We will apply these criteria in the study of the topological and holomorphic structure of Riemann surfaces in Sects. 5.10–5.17, and in the study of holomorphic structures on surfaces in Chap. 6. For our purposes, it will suffice to consider curves and surfaces. We first recall the following fact from analysis in Rn : Theorem 9.9.1 (C ∞ inverse function theorem) Suppose ⊂ Rn is an open set, F : → Rn is a C ∞ mapping with Jacobian determinant JF : → R, and a ∈ is a point at which JF (a) = 0. Then F maps some neighborhood U of a in diffeomorphically onto some open subset F (U ) of Rn . Note that conversely, the Jacobian determinant of a diffeomorphism F : U → V of open subsets U and V of Rn is nonvanishing, because for each point a ∈ U , JF (a) is the determinant of the linear operator (dF )a : Rn = Rn → Rn , an isomorphism with inverse linear map (dF −1 )F (a) . The proof of the theorem for n = 1 is straightforward. A proof for arbitrary n is outlined in Exercise 9.9.1. The proof can be modified to give a C k version, but only the C ∞ case is applied in this book. Theorem 9.9.2 Let M be a smooth manifold of dimension n = 1 or 2. (a) If u is a real-valued C ∞ function on an open set U ⊂ M and n = 1, then the restriction of u to some neighborhood of a point p ∈ U is a local C ∞ coordinate if and only if (du)p = 0. If u and v are two real-valued C ∞ functions on an open set U ⊂ M and n = 2, then the restriction of (u, v) to some neighborhood of a point p ∈ U gives local C ∞ coordinates if and only if (du ∧ dv)p = 0 (i.e., (du)p and (dv)p form a basis for Tp∗ M).

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(b) Suppose n = 2 and v ∈ C ∞ (M, R). If (dv)p = 0 at some point p ∈ M, then there exists a real-valued C ∞ function u on a neighborhood U of p such that (U, (u, v)) is a local C ∞ coordinate neighborhood in M. If dv = 0 at each point in the zero set Z ≡ {p ∈ M | v(p) = 0}, then Z is a smooth submanifold of dimension 1 in M. (c) Suppose r ∈ {1, 2} and  : N → M is a C ∞ mapping of an r-dimensional C ∞ manifold N into M. If q ∈ N and the tangent map (∗ )q : Tq N → T(p) M is injective, then  maps a neighborhood V of q in N diffeomorphically onto an r-dimensional submanifold (V ) of some neighborhood U of (q) in M. If  is injective and proper and (∗ )q : Tq N → T(q) M is injective for every point q ∈ N , then  maps N diffeomorphically onto an r-dimensional smooth submanifold (N ) of M. Conversely, if  maps N diffeomorphically onto a smooth submanifold of M, then ∗ : T N → T M is injective. If p ∈ M and (∗ )q : Tq N → Tp M is surjective for every point q ∈ L ≡ −1 (p) (i.e., p is a regular value), then L is a C ∞ submanifold of dimension r − n in N . Proof Let p ∈ M. For n = dim M = 1, we may fix a local C ∞ coordinate neighborhood (U,  = x). For any real-valued C ∞ function u on a neighborhood of p in U , the Jacobian determinant Ju ≡ ∂u/∂x (more precisely, Ju◦−1 ◦  = ∂u/∂x) satisfies du = Ju · dx. Similarly, for n = 2 and for (u, v) a pair of real-valued C ∞ functions on a neighborhood of p, the Jacobian determinant J(u,v) ≡

∂u ∂v ∂v ∂u − ∂x ∂y ∂x ∂y

in a local C ∞ coordinate neighborhood (U, (x, y)) of p satisfies du ∧ dv = J(u,v) dx ∧ dy. Part (a) now follows from the inverse function theorem. If n = 2, p ∈ M, and v is a real-valued C ∞ function on M with (dv)p = 0, then fixing a local C ∞ coordinate neighborhood (U, (x, y)) of p, one of the pairs (dx)p , (dv)p and (dy)p , (dv)p must be a basis for Tp∗ M. Setting u = x in the former case and u = y in the latter case, we see that by (a), (u, v) gives local C ∞ coordinates in a neighborhood of p. In particular, if dv = 0 at every point in the zero set Z of v, then Z is a C ∞ submanifold of dimension 1. Thus (b) is proved. Suppose r ∈ {1, 2} and  : N → M is a C ∞ mapping of an r-dimensional C ∞ manifold N into M. If r = n, then in local C ∞ charts, the Jacobian determinant of  is precisely the determinant of the tangent map ∗ (with respect to the bases provided by the partial derivative operators). So the inverse function theorem implies that if (∗ )q is an isomorphism for some point q ∈ N , then  maps a neighborhood of q diffeomorphically onto an open subset of M. Suppose r = 1 < 2 = n and q is a point at which (∗ )q is injective. Fixing a local C ∞ chart (U, = (x, y), V ⊂ R2 ) in a neighborhood of p ≡ (q) with (p) = (0, 0) ∈ V , we see that by exchanging x and y if necessary, we may assume that the function  ≡ x() satisfies (d)q = 0. Therefore, by part (a), we may assume that  determines a local C ∞ chart (W, , I ⊂ R) in N such that q ∈ W ⊂ −1 (U ) and I is an open interval containing 0 = (q). In fact, by fixing an open interval J with (0, 0) ∈ I × J ⊂ V , replacing I with an open subinterval I  containing 0 that

9.9

C ∞ Embeddings

467

is so small that W  ≡ −1 (I  ) ⊂ −1 (U  ) for U  = −1 (I  × J ), and replacing W with W  , U with U  , and V with I  × J , we may assume that V = I × J . The C ∞ function s → v(s) ≡ y(s) − y((−1 (x(s)))) on U satisfies   ∂  ∂  y((−1 (x))) · dx − y((−1 (x))) · dy ∂x ∂y  ∂  = dy − y((−1 (x))) · dx, ∂x

dv = dy −

and hence dx ∧ dv = dx ∧ dy = 0. Thus we get a local C ∞ chart (U0 , 0 = (x, v), I0 × J0 ) in M, where p ∈ U0 ⊂ U , and I0 and J0 are open intervals with 0 ∈ I0 ⊂ I and 0 ∈ J0 ⊂ J . By shrinking I0 if necessary, we may assume that the neighborhood W0 ≡ −1 (I0 ) of q in W satisfies (W0 ) ⊂ U0 . Now if t ∈ W0 , then (t) ∈ U0 and v((t)) = y((t)) − y((−1 ((t)))) = 0. Conversely, if s ∈ U0 with v(s) = 0, then setting t ≡ −1 (x(s)) ∈ W0 , we get x((t)) = (t) = x(s)

and v((t)) = 0 = v(s),

and hence (t) = s. Therefore, (W0 ) is the 1-dimensional submanifold N0 ≡

0−1 (I0 × {0}) of U0 . Furthermore, the map W0 = (xN0 )−1 ◦ W0 → N0 is a composition of diffeomorphisms, and is therefore a diffeomorphism. The verifications of the remaining claims in (c) are left to the reader.  Definition 9.9.3 A C ∞ mapping  : N → M of an r-dimensional C ∞ manifold N into an n-dimensional C ∞ manifold M with r, n ∈ {1, 2} is (i) An immersion if the tangent map (∗ )p : Tp N → Tp M is injective for each point p ∈ N ; (ii) An embedding (or imbedding) if  is a proper injective immersion; (iii) A submersion if the tangent map (∗ )p : Tp N → Tp M is surjective for each point p ∈ N . Remarks 1. In this book, we require embeddings to be proper, but it is also common practice not to require this as part of the definition. 2. If N is a smooth submanifold of a smooth manifold M, and ι : N → M is the inclusion map, then we identify T N with ι∗ T N ⊂ T M. If M and N are oriented and α is a differential form of degree   dim N on M, then, in a slight abuse of notation, we often write N ι∗ α simply as N α. Note that this does not conflict with the earlier definition of the integral over a measurable subset of M because there, we assumed α to be of degree dim M. 3. Theorem 9.9.2 implies that a C ∞ mapping  : N → M of C ∞ manifolds M and N of dimension n = 1 or 2 is a local diffeomorphism if and only if the tangent map is an isomorphism at each point. 4. It follows from Theorem 9.9.2 that an open subset of a C ∞ surface M is smooth if and only if for each point p ∈ ∂ , there exists a real-valued C ∞ function ϕ on a neighborhood U of p such that (dϕ)p = 0 and ∩ U = {x ∈ U | ϕ(x) < 0}.

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Exercises for Sect. 9.9 9.9.1 The goal of this exercise is a proof of the C ∞ inverse function theorem (Theorem 9.9.1). Let be an open subset of Rn , let F = (f1 , . . . , fn ) : → Rn be a C ∞ mapping with Jacobian matrix ⎛ ∂f

1

∂x1

⎜ . JF ≡ ⎜ ⎝ ..

∂fn ∂x1

··· .. . ···

∂f1 ∂xn

⎞

.. ⎟ ⎟ . ⎠:

→ Rn×n ∼ = Rn

2

∂fn ∂xn

and Jacobian determinant JF = det(JF ) : → R (clearly, both are of class C ∞ ), and let a = (a1 , . . . , an ) ∈ be a point at which JF (a) = 0. For each point b ∈ , let Rb = (rb1 , . . . , rbn ) : → Rn be the C ∞ map given by x → F (x) − F (b) − (dF )b (x − b). (a) Show that there is a constant C0 > 0 such that x ≤ C0 (dF )a (x) for every vector x ∈ Rn . (b) Prove that for every  > 0, there is a neighborhood U of a in such that Rb (y) − Rb (x) ≤  y − x for all points b, x, y ∈ U . (c) Prove that there are a constant C1 > 0 and a neighborhood U of a in such that F (y) − F (x) ≥ C1 y − x for all points x, y ∈ U . In particular, F U is injective. (d) Prove that if U is a neighborhood of a in on which JF is nonvanishing, then for every point b ∈ Rn , the C ∞ function F − b 2 cannot attain a positive local minimum value at a point in U . (e) Prove that if U is a neighborhood of a in such that JF U is nonvanishing and F U is injective, then F maps U homeomorphically onto an open set F (U ) in Rn . Hint. Assuming V  U is open, but W ≡ F (V ) is not open, there must exist a point p ∈ V with q = F (p) ∈ ∂W . Fix an open ball B centered at p with B  V . Then, by injectivity, q is not in the compact set K ≡ F (∂B). Now fix a point b ∈ Rn \ W with b − q < dist(q, K)/2 and apply part (d). (f) Prove that there exists a relatively compact neighborhood U of a in such that JF U is nonvanishing, F maps U homeomorphically onto an open set F (U ) in Rn , and, for G = (g1 , . . . , gn ) ≡ (F U )−1 : F (U ) → Rn , we have, for some constant C2 > 0, G(y) − G(x) ≤ C2 y − x for all x, y ∈ F (U ). (g) Prove that for U as in part (f), we have lim

x→F (b)

Rb (G(x)) =0 x − F (b)

∀b ∈ U.

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(h) Prove that for U as in part (f) and for each point b ∈ U , the first-order partial derivatives of G at F (b) exist and ⎞ ⎛ ∂g ∂g1 1 ∂x1 (F (b)) · · · ∂xn (F (b)) ⎟ ⎜ .. .. .. ⎟ = (JF (b))−1 . JG (F (b)) ≡ ⎜ . . . ⎠ ⎝ ∂gn ∂gn ∂x1 (F (b)) · · · ∂xn (F (b)) Hint. We have x = F (b) + (dF )b (G(x) − b) + Rb (G(x)) for each point x ∈ F (U ). Solve for G(x) (in terms of x and Rb (G(x))), and apply part (g). (i) Prove by induction on k that G is of class C k for each k = 0, 1, 2, . . . . 9.9.2 Let be an open subset of a second countable smooth manifold M. Prove that is a smooth open set if and only if there exists a C ∞ function ϕ : M → R such that = {x ∈ M | ϕ(x) < 0} and (dϕ)p = 0 for each point p ∈ ∂ .

9.10 Classification of Second Countable 1-Dimensional Manifolds The second countable curves are completely determined by the following: Theorem 9.10.1 Every second countable connected 1-dimensional manifold is homeomorphic to either the circle S1 or the line R. Any second countable connected 1-dimensional smooth manifold is diffeomorphic to either S1 or R. Remarks 1. It is easy to see that every open interval in R is diffeomorphic to R. For given a, b ∈ R with a < b, we have the diffeomorphisms of (a, ∞), (−∞, b), and (a, b) onto R given by t → log(t − a), t → − log(b − t), and t → log((t − a)/ (b − t)), respectively. 2. The theorem is completely believable, but the proof requires some work. One proof is provided in this section. Others may be found in, for example, [Mi] or [GuiP]. 3. Elements of the proof also appear in the proof of the existence of smooth structures on second countable topological surfaces in Sect. 6.11. Throughout this section, M denotes a second countable connected 1-dimensional topological or smooth manifold. The intermediate value theorem implies that a continuous injective real-valued function on an interval I is either strictly increasing or strictly decreasing, and that the function maps the interval homeomorphically onto an interval J (in particular, the image of any endpoint of I is an endpoint of J ). Furthermore, if the interval is open and the function is of class C ∞ with nonvanishing derivative, then the function is a diffeomorphism. Lemma 9.10.2 Let α : I → M be an injective continuous mapping of an open interval I ⊂ R into M with image A = α(I ). Then

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(a) The image A is a connected open subset of M onto which α maps I homeomorphically. (b) For M a smooth manifold and α of class C ∞ with nonvanishing tangent vector α, ˙ α maps I diffeomorphically onto A. Proof Given a point c ∈ I , there is a local chart (J, , R) in M with α(c) = −1 (0). The connected component H of α −1 (J ) containing c is then an open subinterval of I and  ◦ αH maps H homeomorphically onto an open interval. The claims (a) and (b) now follow.  Lemma 9.10.3 Let α : [a, b] → M and β : [c, d] → M be injective paths, and let A ≡ α([a, b]), A0 ≡ α((a, b)), B ≡ β([c, d]), and B0 ≡ β((c, d)). Assume that M is noncompact, A0 ∩ B0 = ∅, and A ⊂ B0 . Then β −1 (A) = [c, r] or [r, d] for some point r ∈ [c, d]. Proof Clearly, A = A¯ 0 and B = B 0 , and by Lemma 9.10.2, A0 and B0 are connected open subsets of M, α maps (a, b) homeomorphically onto A0 , and β maps (c, d) homeomorphically onto B0 . Of course, α maps [a, b] homeomorphically onto A and β maps [c, d] homeomorphically onto B, since these are compact Hausdorff spaces. Each connected component D of β −1 (A) is either a singleton or a closed interval, and hence α −1 (β(D)) is either a singleton or a closed interval contained in [a, b]. If D ⊂ (c, d), then β(D) is a compact subset of A ∩ B0 . Since A ⊂ B0 , it follows that α −1 (β(D)) is a connected compact subset of [a, b] that is not equal to the entire interval. Hence there is an open interval J such that J ⊂ (a, b) ∩ α −1 (B0 ) \ α −1 (β(D))

and ∂J ∩ α −1 (β(D)) = ∅.

Thus β −1 (α(J )) ∪ D  D is a connected subset of β −1 (A), and we have arrived at a contradiction. By hypothesis, β −1 (A) ⊂ [c, d] has a connected component with nonempty interior, and by the above, such a connected component must be a closed interval with at least one endpoint equal to c or d. Suppose that we have such a connected component of the form [r, d] with c < r < d but β −1 (A) = [r, d]. By the above, we then have either β −1 (A) = {u} ∪ [r, d] with u = c or β −1 (A) = [c, u] ∪ [r, d] for some u ∈ (c, r). By replacing α with the injective path t → α(−t) for t ∈ [−b, −a] (which has the same image A) if necessary, we may assume that the function α −1 ◦ β[r,d] is strictly increasing. Clearly, we must have β(u), β(r) ∈ A \ A0 (otherwise, β −1 (A) would contain a strictly larger subset of [c, d]), so we must have β(r) = α(a) and β(u) = α(b). In particular, if u ∈ (c, r), then A \ A0 = {α(a), α(b)} ⊂ B0 and B \ B0 = {β(c), β(d)} ⊂ A \ {β(u), β(r)} = A \ {α(a), α(b)} = A0 , and hence A ∪ B = A0 ∪ B0 is open. Suppose instead that u = c. Since α −1 ◦ β is a strictly increasing continuous mapping of [r, d] onto a closed subinterval of [a, b)

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471

and we have α −1 (β(c)) = b, we get a continuous injective mapping γ : (α −1 (β(d)), r − c + b) → A ∪ B by setting

 γ (t) =

α(t) if t ∈ (α −1 (β(d)), b], β(t − b + c) if t ∈ [b, r − c + b).

Hence, by Lemma 9.10.2, the image of γ is a neighborhood of α(b) = β(c) in M, and hence this point is an interior point of A ∪ B. We also have α(a) = β(r) ∈ B0 and β(d) = α(α −1 (β(d)) ∈ A0 . Thus A ∪ B is again open in this case. Therefore, in either case, since M is connected and noncompact, we have arrived at a contradiction. A similar argument shows that any connected component of β −1 (A) of the  form [c, r] with r > c must be equal to β −1 (A). Lemma 9.10.4 Let −∞ < a < b < ∞, and let f and g be real-valued continuous strictly increasing functions on [a, b] with f (a) < g(b). Then there exist numbers c and d with a < c < d < b and a continuous strictly increasing function h : [a, b] → R such that h = f on [a, c] and h = g on [d, b]. Moreover, if f and g are of class C ∞ with f  > 0 and g  > 0 on [a, b], then h may also be chosen to be of class C ∞ with h > 0. Proof We may fix points c, d ∈ (a, b) with c < d and f (c) < g(d). Hence the function h given by ⎧ ⎪ if t ∈ [a, c], ⎨f (t) t−c h(t) = f (c) + d−c · (g(d) − f (c)) if t ∈ (c, d), ⎪ ⎩ g(t) if t ∈ [d, b], is a continuous strictly increasing function with h = f on [a, c] and h = g on [d, b]. If f and g are of class C ∞ with f  > 0 and g  > 0, then fixing points u and v with c < u < v < d and f (u) < g(v), we may form a C ∞ partition of unity {λ1 , λ2 , λ3 } on R (with 0 ≤ λj ≤ 1 for j = 1, 2, 3) such that supp λ1 ⊂ (−∞, u), supp λ2 ⊂ (c, d), and supp λ3 ⊂ (v, ∞) (see Theorem 9.3.7). In particular, we have 

d

0 ≤ q ≡ f (a) +

(λ1 (s)f  (s) + λ3 (s)g  (s)) ds ≤ f (u) + g(d) − g(v) < g(d).

a

Setting r ≡

d c

λ2 (s) ds > 0, we see that the C ∞ function h given by 

h(t) = f (a) +

t

[λ1 (s)f  (s) + r −1 (g(d) − q)λ2 (s) + λ3 (s)g  (s)] ds

∀t ∈ [a, b]

a

satisfies h > 0 on (a, b), h = f on [a, c], and h = g on [d, b].



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Lemma 9.10.5 Suppose 0 and U are open subsets of M; I0 is an open interval; α0 : I0 → 0 and β : R → U are homeomorphisms; K0 is a compact subset of 0 ; and K is a compact subset of U that meets K0 . Assume that M is noncompact. Then there exists a homeomorphism α of an open interval I onto an open subset of M such that K0 ∪ K ⊂ ⊂ 0 ∪ U and α −1 K0 = α0−1 K0 . Moreover, if M is smooth and α0 and β are diffeomorphisms, then α may be chosen to be a diffeomorphism. Proof If K ⊂ 0 , then we may set α ≡ α0 : I ≡ I0 → ≡ 0 ; so we may assume without loss of generality that K ⊂ 0 . We may also assume that K is connected, and we may fix an interval I1 = [a, b]  I0 with K0 ⊂ α0 ((a, b)) and an interval J = [c, d]  R with K ⊂ β((c, d)). Let us first suppose that 0 ⊂ K. Then we may choose the intervals I1 and J so that α0 (I1 ) ⊂ β(J ); and by applying Lemma 9.10.3 (and reparametrizing β if necessary), we may assume that for some r ∈ (c, d), we have β −1 (α0 (I1 )) ∩ J = [c, r] with either a < α0−1 (β(c)) < α0−1 (β(r)) = b or a = α0−1 (β(r)) < α0−1 (β(c)) < b. In the former case, we may choose a number s ∈ (α0−1 (β(c)), b) so close to b that K0 ⊂ α0 ((a, s)). By reparametrizing β again, we may assume that s = α0−1 (β(s)) and r = α0−1 (β(r)) = b (in particular, a < α0−1 (β(c)) < s < b < d and c < s < b < d), and we may apply Lemma 9.10.4 to get constants u and v with s < u < v < b and a continuous strictly increasing function f : [s, b] → [s, b] such that f (t) = t for each point t ∈ [s, u], f (t) = α0−1 (β(t)) for each point t ∈ [v, b], and, for M smooth, f is of class C ∞ with f  > 0. The set ≡ α0 ((a, b]) ∪ β([b, d)) = α0 ((a, b)) ∪ β((c, d)) is then a domain in M, and K0 ∪ K  α0 ((a, b)) ∪ β((c, d)) = ⊂ 0 ∪ U. Moreover, we may define a homeomorphism α : I ≡ (a, d) → (which is a diffeomorphism if M is smooth) by setting ⎧ ⎪ ⎨α0 (t) α(t) = α0 (f (t)) ⎪ ⎩ β(t)

if t ∈ (a, s], if t ∈ [s, b], if t ∈ [b, d).

In particular, we have α −1 = α0−1 on the set α0 ((a, s)) ⊃ K0 . In the case in which a = α0−1 (β(r)) < α0−1 (β(c)) < b, we may apply the above arguments to β and the homeomorphism αˆ 0 : t → α0 (b − t + a) to get a domain and a homeomorphism αˆ : (a, d) → . The homeomorphism α : I ≡ (b − d + a, b) → given by t → α(b ˆ − t + a) then has the required properties. Finally, suppose 0 ⊂ K ⊂ U . Then we may choose the homeomorphism (or diffeomorphism) β : R → U and the constants c and d so that c < a < b < d,

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473

β([a, b]) = α0 ([a, b]), β(a) = α0 (a), and β(b) = α0 (b). After choosing numbers s0 and s1 with a < s0 < s1 < b and K0 ⊂ α0 ((s0 , s1 )), we have α0−1 (β(a)) = a < s0

and s1 < b = α0−1 (β(b)).

Therefore, Lemma 9.10.4 provides constants u0 , v0 , u1 , and v1 with a < u0 < v0 < s0 < s1 < u1 < v1 < b and continuous strictly increasing functions f0 : [a, s0 ] → [a, s0 ] and f1 : [s1 , b] → [s1 , b] such that f0 (t) = α0−1 (β(t)) for each point t ∈ [a, u0 ], f0 (t) = t for each point t ∈ [v0 , s0 ], f1 (t) = t for each point t ∈ [s1 , u1 ], and f1 (t) = α0−1 (β(t)) for each point t ∈ [v1 , b], and for M smooth, f0 and f1 are of class C ∞ with f0 > 0 and f1 > 0. We then have the domain ≡ β((c, d)) that satisfies K0 ∪ K ⊂ α0 ((s1 , s2 )) ∪ β((c, d)) = β((c, d)) = , and we have the homeomorphism (diffeomorphism for M smooth) α : I ≡ (c, d) → given by ⎧ β(t) if t ∈ (c, a] ∪ [b, d), ⎪ ⎪ ⎪ ⎨α (f (t)) if t ∈ [a, s ], 0 0 0 α(t) = ⎪ α (t) if t ∈ [s , s 0 0 1 ], ⎪ ⎪ ⎩ α0 (f1 (t)) if t ∈ [s1 , b].



Proof of Theorem 9.10.1 We first consider the case in which the manifold M is noncompact. By Lemma 9.3.6, there exist locally finite open coverings {Uν }∞ ν=1 and {Vν }∞ of X such that for each ν = 1, 2, 3, . . . , V  U  M, U is homeomorphic ν ν ν ν=1 to R, and for M smooth, Uν is also diffeomorphic to R. We may also choose the coverings so that for each ν, the set V1 ∪ · · · ∪ Vν meets Vν+1 . For we may let ν1 be the smallest index ν for which Vν = ∅, and given ν1 , . . . , νk , we may let νk+1 be the smallest index ν ∈ / {ν1 , . . . , νk } for which Vν meets Vν1 ∪ · · · ∪ Vνk (such an index exists because the union of the boundaries of the relatively compact open sets Vν1 , . . . , Vνk cannot be contained in the union of these sets). By induction, we obtain a sequence {Vνk } that covers M (otherwise, the union would have a boundary point that lies in Vν for some ν and we would have ν < νk for k  0). Thus we may replace the original coverings {Vν } and {Uν } with the coverings {Vνk } and {Uνk }, respectively. There exist domains { ν } in M and mappings {αν } such that for each ν = 1, 2, 3, . . . , ν

≡ V1 ∪ · · · ∪ Vν  ν ⊂ ν ≡ U1 ∪ · · · ∪ Uν ,

−1 and αν is a homeomorphism of an open interval Iν onto ν with αν−1  ν = αν+1  ν. Moreover, for M smooth, each mapping αν is a diffeomorphism. For the existence of 1 and α1 : I1 → 1 is clear, and one may then proceed to apply Lemma 9.10.5

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inductively. Since the sequence of functions {αμ−1 Vν }∞ μ=ν is constant for each ν, we may then define a homeomorphism (a diffeomorphism for M smooth) γ of  M = Vν onto an open interval by setting γ Vν = αν−1 Vν for each ν = 1, 2, 3, . . . . Fixing a diffeomorphism λ : R → γ (M), we get the desired homeomorphism (diffeomorphism for M smooth) γ −1 ◦ λ : R → M. Finally, suppose M is compact. We may fix a homeomorphism β : R → U onto a neighborhood U of a point p ∈ M, with β a diffeomorphism for M smooth. The set U \ {p} has exactly two connected components U0 and U1 , and each of the connected components of M \ {p} must meet, and therefore contain, at least one of these connected components. It follows that M \ {p} has at most two connected components. However, if N ≈ R is a connected component that contains only one of these connected components, say, U0 , then, fixing a point q ∈ U0 , we see that of the two connected components of N \ {q}, one is relatively compact in N , which is clearly impossible. Thus M \ {p} is connected, and we may choose a homeomorphism α : (0, 1) → M \ {p}, with α a diffeomorphism for M smooth. Since U0 and U1 are disjoint open connected subsets of M \ {p} that are not relatively compact in M \ {p}, we may choose U and α so that for some r ∈ (0, 1/4), we have α −1 (U0 ) = (0, 2r) and α −1 (U1 ) = (1 − 2r, 1). We may also choose the homeomorphism (or diffeomorphism) β so that β(−r) = α(1 − r) and β(r) = α(r). In particular, β([−r, r]) is a connected relatively compact subset of U that meets both U0 and U1 , and therefore p ∈ β((−r, r)), and we may set t0 ≡ β −1 (p). We have β((−∞, t0 )) = U1 , β((t0 , ∞)) = U0 , α −1 (β(t)) → 1 as t → t0− , and α −1 (β(t)) → 0 as t → t0+ ; so α −1 ◦ β(−∞,t0 ) and α −1 ◦ β(t0 ,∞) must be strictly increasing functions with α −1 ◦ β([−r, t0 )) = [1 − r, 1) and α −1 ◦ β((t0 , r]) = (0, r]. Fixing b0 ∈ (−r, t0 ) ∩ (−r, 0) and b1 ∈ (t0 , r) ∩ (0, r) and applying Lemma 9.10.4, we get continuous strictly increasing functions f0 : [−r, b0 ] → [1 − r, α −1 (β(b0 ))] and f1 : [b1 , r] → [1 + α −1 (β(b1 )), 1 + r] such that f0 (t) = 1 + t for t near −r, f0 (t) = α −1 (β(t)) for t near b0 , f1 (t) = 1 + α −1 (β(t)) for t near b1 , and f1 (t) = 1 + t for t near r. Moreover, f0 and f1 are of class C ∞ with f0 > 0 and f1 > 0 for M smooth. The mapping γ : (−r, 1 + b0 ) → M given by ⎧ ⎪ α(t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨α(f0 (t)) γ (t) = β(t) ⎪ ⎪ ⎪ α(−1 + f1 (t)) ⎪ ⎪ ⎪ ⎩α(f (t − 1)) 0

if t ∈ [r, 1 − r], if t ∈ (−r, b0 ], if t ∈ [b0 , b1 ], if t ∈ [b1 , r], if t ∈ [1 − r, 1 + b0 ),

is a surjective continuous mapping that is a local homeomorphism and that satisfies γ (t + 1) = γ (t) for each t ∈ (−r, b0 ). Moreover, γ is a local diffeomorphism if M is smooth. It follows that the mapping S1 → M given by e2πit → γ (t) for t ∈ (−r, 1 + b0 ) is a well-defined homeomorphism, and this mapping is a diffeomorphism if M is smooth. 

9.11

Riemannian Metrics

475

9.11 Riemannian Metrics In this section we consider Riemannian metrics (on smooth surfaces), which in this book are applied directly (only) in the construction of an almost complex structure on an orientable surface (Proposition 6.1.3) in Chap. 6. Definition 9.11.1 A Riemannian metric g on a 2-dimensional smooth manifold M is a choice of a real inner product gp (·, ·) on the real tangent space Tp M at each point p ∈ M such that for each pair of local C ∞ vector fields u and v, the function g(u, v) : p → gp (up , vp ) is of class C ∞ . For each point p ∈ M and each pair of real tangent vectors u, v ∈ Tp M, we also write g(u, v) = gp (u, v) and |u|2g = g(u, u). Remarks Riemannian metrics are fundamental objects in the study of the (differential) geometry of manifolds. A Riemannian metric provides a notion of distance, curvature, and volume in a manifold. In this book, we mostly consider related objects, such as Kähler metrics on Riemann surfaces and Hermitian metrics in holomorphic line bundles. Corresponding notions of volume and curvature do play important roles in the analytic techniques applied throughout this book, but the geometric meaning will not be the main consideration (for much more on Riemannian metrics, see, for example, [KobN1] and [KobN2]). Observe that if g is a Riemannian metric on a smooth surface M and (U, (x, y)) is a local C ∞ coordinate neighborhood in M, then the functions A ≡ g(∂/∂x, ∂/∂x), B ≡ g(∂/∂x, ∂/∂y) = g(∂/∂y, ∂/∂x), C ≡ g(∂/∂y, ∂/∂y), are of class C ∞ and g(u, v) = A dx(u) dx(v) + B dx(u) dy(v) + B dy(u) dx(v) + C dy(u) dy(v) for every pair of tangent vectors u, v at a point in U . In particular, it follows that if u and v are vector fields that are continuous (C k with k ∈ Z≥0 ∪ {∞}, measurable), then the function g(u, v) is continuous (respectively, C k , measurable). Proposition 9.11.2 Every second countable smooth surface M admits a Riemannian metric. Proof There exist a locally finite collection of local C ∞ charts {(Uν , ν = (xν , yν ), Uν )}ν∈N

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and a C ∞ partition of unity {λν } with supp λν ⊂ Uν (and λν ≥ 0) for each ν ∈ N . The pairing given by  λν (p)(dxν (u) dxν (v) + dyν (u) dyν (v)), g(u, v) = gp (u, v) ≡ ν∈N

for all p ∈ M and u, v ∈ Tp M, is then a Riemannian metric on M. The verification is left to the reader (see Exercise 9.11.1).  Exercises for Sect. 9.11 9.11.1 Verify that the pairing defined in the proof of Proposition 9.11.2 is a Riemannian metric.

Chapter 10

Background Material on Fundamental Groups, Covering Spaces, and (Co)homology

We recall that a path (or parametrized path or curve or parametrized curve) in a topological space X from a point x to a point y is a continuous mapping γ : [a, b] → X with γ (a) = x and γ (b) = y (see Sect. 9.1). We take the domain of a path to be [0, 1], unless otherwise indicated. A loop (or closed curve) with base point p ∈ X is a path in X from p to p. In this chapter, we consider the equivalence relation given by path homotopies. This leads to the fundamental group, which is the group given by the path homotopy equivalence classes of loops at a point, and to covering spaces, both of which are important objects in complex analysis and Riemann surface theory. We also consider homology groups, which are essentially Abelian versions of the fundamental group, and cohomology groups, which are groups that are dual to the homology groups.

10.1 The Fundamental Group Throughout this section, X denotes a nonempty topological space. Definition 10.1.1 A path homotopy in X is a continuous mapping H : [a, b] × [0, 1] → X, where −∞ < a < b < ∞ and the paths s → H (a, s) and s → H (b, s) are constant. We also call H a path homotopy (or simply a homotopy) from γ0 to γ1 , where for each s ∈ [0, 1], γs is the path given by γs (t) = H (t, s) for each t ∈ [a, b] (see Fig. 10.1). We also say that γ0 and γ1 are path homotopic (or simply homotopic), and we write H : γ0 ∼ γ1 or γ0 ∼ γ1 . Remarks 1. We take the domain of a path homotopy to be [0, 1] × [0, 1], unless otherwise indicated. T. Napier, M. Ramachandran, An Introduction to Riemann Surfaces, Cornerstones, DOI 10.1007/978-0-8176-4693-6_10, © Springer Science+Business Media, LLC 2011

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Fig. 10.1 A path homotopy from γ0 to γ1

2. Change of parameter. If γ : [a, b] → X is a path and ϕ : [a, b] → [a, b] is a continuous mapping with ϕ(a) = a and ϕ(b) = b, then H : γ ∼ γ ◦ ϕ,

where H (t, s) = γ (t + s(ϕ(t) − t)) ∀(t, s) ∈ [a, b] × [0, 1].

If δ : [a, b] → X is a path, H : δ ∼ γ , and ψ : [c, d] → [a, b] is a continuous mapping with ψ(c) = a and ψ(d) = b, then G : δ ◦ ψ ∼ γ ◦ ψ,

where G(t, s) = H (ψ(t), s) ∀(t, s) ∈ [c, d] × [0, 1].

For this reason, we lose nothing by restricting most of our attention to paths with domain [0, 1], since we may always reparametrize so that the domain becomes [0, 1]. 3. The path homotopy relation ∼ is an equivalence relation. To see this, suppose γj : [a, b] → X is a path in X with γj (a) = x and γj (b) = y for j = 0, 1, 2, and Hj : γj −1 ∼ γj for j = 1, 2. Then the mapping (t, s) → γ0 (t)

∀(t, s) ∈ [a, b] × [0, 1]

is a path homotopy from γ0 to itself; (t, s) → H1 (t, 1 − s) ∀(t, s) ∈ [a, b] × [0, 1] is a path homotopy from γ1 to γ0 ; and  ∀(t, s) ∈ [a, b] × [0, 1/2], H1 (t, 2s) (t, s) → H2 (t, 2s − 1) ∀(t, s) ∈ [a, b] × [1/2, 1], is a path homotopy from γ0 to γ2 . Definition 10.1.2 Let γ0 : [0, 1] → X and γ1 : [0, 1] → X be paths in X with γ0 (1) = γ1 (0). (a) The product path of γ0 and γ1 is the path γ0 ∗ γ1 : [0, 1] → X from γ0 (0) to γ1 (1) given by  γ0 (2t) if t ∈ [0, 1/2], γ0 ∗ γ1 (t) ≡ γ1 (2t − 1) if t ∈ [1/2, 1]. (b) The reverse path of γ0 is the path γ0− : [0, 1] → X from γ0 (1) to γ0 (0) given by γ0− (t) ≡ γ0 (1 − t)

∀t ∈ [0, 1].

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Remarks 1. The product path is obtained by tracing the first path at twice the speed, and then tracing the second path at twice the speed. The paths are traced at twice the speed so that the resulting domain is again [0, 1]. 2. The reverse path is obtained by retracing the path in the reverse direction. Clearly, for any path γ (with domain [0, 1]), we have (γ − )− = γ . 3. Occasionally, it is convenient to consider product and reverse paths with domains other than [0, 1]. If γj : [aj , bj ] → X is a path in X for j = 0, 1 and γ0 (b0 ) = γ1 (a1 ), then by γ0 ∗ γ1 : [a, b] → X we will mean a path obtained by tracing γ0 and then γ1 , after some (nonunique, noncanonical) linear reparametrizations. That is, after choosing an interval [a, b] and a point c ∈ (a, b), we have  t−a γ0 (a0 + c−a (b0 − a0 )) if t ∈ [a, c], γ0 ∗ γ1 (t) ≡ t−c γ1 (a1 + b−c (b1 − a1 ))) if t ∈ [c, b]. Similarly, by γ0− : [a, b] → X we will mean the path given by   t −a − ∀t ∈ [a, b]. γ0 (t) ≡ γ0 b0 + (a0 − b0 ) b−a 4. For paths γ0 and γ1 in X and a continuous mapping  : X → Y into a topological space Y , we have (γ0 ∗ γ1 ) = (γ0 ) ∗ (γ1 ) and (γ0− ) = ((γ0 ))− . 5. The product operation on paths is not associative. Given n paths γ1 , . . . , γn , we will set γ1 ∗ · · · ∗ γn ≡ (· · · ((γ1 ∗ γ2 ) ∗ γ3 ) · · · ) ∗ γn . On the other hand, the product is associative up to path homotopy. In fact, we have the following lemma, the proof of which is left to the reader (see Exercise 10.1.1). Lemma 10.1.3 Let γj : [0, 1] → X be a path with γj (0) = xj and γj (1) = yj for j = 0, 1, 2. For each point p ∈ X, let ep : [0, 1] → X denote the constant loop t → p. (a) We have γ0 ∗ γ0− ∼ ex0 , γ0− ∗ γ0 ∼ ey0 , ex0 ∗ γ0 ∼ γ0 , and γ0 ∗ ey0 ∼ γ0 . Furthermore, if γ0 ∼ γ1 , then γ0− ∼ γ1− . (b) If γ0 (1) = γ1 (0) and γ1 (1) = γ2 (0), then (γ0 ∗ γ1 ) ∗ γ2 ∼ γ0 ∗ (γ1 ∗ γ2 ). (c) If γ0 (1) = γ1 (1) = γ2 (0) and γ0 ∼ γ1 , then γ0 ∗ γ2 ∼ γ1 ∗ γ2 . (d) If γ0 (1) = γ1 (0) = γ2 (0) and γ1 ∼ γ2 , then γ0 ∗ γ1 ∼ γ0 ∗ γ2 . (e) If  : X → Y is a continuous mapping of topological spaces and H : γ0 ∼ γ1 , then (H ) : (γ0 ) ∼ (γ1 ). Lemma 10.1.3 allows us to associate a group to each point in X as follows: Definition 10.1.4 Let x0 ∈ X. (a) We denote the set of loops in X based at x0 with domain [0, 1] by L(X, x0 ). For each loop γ ∈ L(X, x0 ), we denote the path homotopy equivalence class of γ by [γ ] or [γ ]x0 or [γ ]π1 (X,x0 ) .

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(b) The fundamental group of X with base point x0 is the group π1 (X, x0 ) with elements given by the set of equivalence classes of loops in L(X, x0 ), with the product and inverse given by [γ0 ] · [γ1 ] ≡ [γ0 ∗ γ1 ] and [γ0 ]−1 ≡ [γ0− ], respectively, for all γ0 , γ1 ∈ L(X, x0 ), and the identity element given by [ex0 ], where ex0 ∈ L(X, x0 ) is the constant loop based at x0 . (c) If  : X → Y is a continuous mapping of topological spaces and y0 = (x0 ), then the map ∗ : π1 (X, x0 ) → π1 (Y, y0 ) given by [γ ]x0 → [(γ )]y0 is the induced pushforward homomorphism of fundamental groups. Remarks If  : X → Y and  : Y → Z are continuous mappings of topological spaces, x0 ∈ X, y0 = (x0 ), and z0 = (y0 ), then the induced homomorphisms ∗ : π1 (X, x0 ) → π1 (Y, y0 ) and ∗ : π1 (Y, y0 ) → π1 (Z, z0 ) satisfy ∗ ◦ ∗ = ( ◦ )∗ : π1 (X, x0 ) → π1 (Z, z0 ). In particular, if  is a homeomorphism, then ∗ : π1 (X, x0 ) → π1 (Y, y0 ) is an isomorphism with inverse mapping (−1 )∗ . Lemma 10.1.5 Let η : [0, 1] → X be a path from x0 to x1 in X. Then the map π1 (X, x0 ) → π1 (X, x1 ) given by [γ ]x0 → [η− ∗ γ ∗ η]x1 is a well-defined surjective isomorphism with inverse map given by [γ ]x1 → [η ∗ γ ∗ η− ]x0 . Moreover, if  : X → Y is a continuous mapping into a topological space Y , yj = (xj ) for j = 0, 1, and λ = (η), then the above isomorphism and the isomorphism [γ ]y0 → [λ− ∗ γ ∗ λ]y1 together give a commutative diagram π1 (X, x0 ) ∼ =

 π1 (X, x1 )

∗ 

∗ π1 (X, x0 ) ⊂ π1 (Y, y0 )

∼ ∼ = =   ∗  ∗ π1 (X, x1 ) ⊂ π1 (Y, y1 )

Proof This follows from Lemma 10.1.3.



Remark Because all fundamental groups of a path connected topological space X are isomorphic (although not canonically isomorphic), the associated groups are (or more precisely, the associated isomorphism class of groups is) often denoted by π1 (X). Definition 10.1.6 We say that X is simply connected if X is path connected and π1 (X) is trivial (that is, π1 (X, x0 ) = [ex0 ]x0 for some, hence for every, point x0 ∈ X). We say that X is locally simply connected if for every point p ∈ X and every neighborhood U of p, there exists a simply connected neighborhood V of p in U . For example, Rn is simply connected, and for n > 1, the unit sphere Sn is simply connected (see Exercise 10.1.3). We postpone consideration of examples of nontriv-

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ial fundamental groups until Sect. 10.4, since they are most easily considered within the context of covering spaces and deck transformations. The proof of the following consequence of Lemma 10.1.3 is left to the reader (see Exercise 10.1.2): Lemma 10.1.7 If X is simply connected, then any two paths in X with the same initial and terminal points are path homotopic. We will also need the following easy observation: Lemma 10.1.8 Let α be a path in X, let 0 = t0 < t1 < · · · < tn = 1 be a partition of [0, 1], and for each i = 1, . . . , n, let αi : [0, 1] → X be a reparametrization of the path α[ti−1 ,ti ] . Then the path β ≡ α1 ∗ · · · ∗ αn is a reparametrization of α. In particular, α ∼ β (see the remarks following Definition 10.1.1). Proof We prove by induction that βi ≡ α1 ∗ · · · ∗ αi is a reparametrization of α[0,ti ] for each i = 1, . . . , n, the case i = 1 being trivial. Assuming that βi−1 is a reparametrization of α[0,ti−1 ] , we have continuous mappings ϕ : [0, 1] → [0, ti−1 ] and ψ : [0, 1] → [ti−1 , ti ] such that ϕ(0) = 0, ϕ(1) = ti−1 , βi−1 = α ◦ ϕ, ψ(0) = ti−1 , ψ(1) = ti , and αi = α ◦ ψ . Thus βi = βi−1 ∗ αi = (α ◦ ϕ) ∗ (α ◦ ψ) = α ◦ ρ, where ρ : [0, 1] → [0, ti ] is the continuous mapping given by  ϕ(2t) if t ∈ [0, 1/2], ρ(t) ≡ ψ(2t − 1) if t ∈ [1/2, 1]. The claim now follows by induction, and the lemma is then the case i = n.



Lemma 10.1.9 Let X be a connected locally simply connected locally compact Hausdorff space. (a) For every compact set K ⊂ X, there is a path connected relatively compact neighborhood  of K in X such that im(π1 () → π1 (X)) is finitely generated. (b) If X is compact, then π1 (X) is finitely generated. (c) If X is second countable, then π1 (X) is countable. Proof For the proof of (a), suppose K is a compact subset of X and x0 is a point in K. By applying Lemma 9.3.6 and replacing K with the closure of a relatively compact connected neighborhood in X, we may assume that K is connected. The collection of relatively compact simply connected open subsets of X forms a basis for the topology. Therefore, by Lemma 9.3.6, there exist finite coverings of K by nonempty connected open sets {Bi }i∈I and {Bi }i∈I in X such that for each i ∈ I , Bi ∩ K = ∅, Bi is simply connected, and Bi  Bi , and for every pair of indices i, j ∈ I with B i ∩ B j = ∅, we have Bi  Bj . In particular, K is contained in the

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 domain  ≡ i∈I Bi  X. For each pair of indices i, j ∈ I with Bi ∩ Bj = ∅, we may choose a point pij = pj i ∈ Bi ∩ Bj and a path αij = αj i in  from x0 to pij . Let us also choose these paths so that αii lies in Bi whenever i ∈ I with x0 ∈ Bi . Finally, for each triple of indices i, j, k ∈ I with Bi ∩ Bj = ∅ and Bj ∩ Bk = ∅, we may choose a path βij k in Bj from pij to pj k . Given a loop λ in  with base point x0 , we may form a partition 0 = t0 < t1 < t2 < · · · < tm = 1 such that for each ν = 1, . . . , m, we have λ([tν−1 , tν ]) ⊂ Biν for some index iν ∈ I . In particular, for each ν = 1, . . . , m, we may choose a reparametrization λν : [0, 1] → Biν of λ[tν−1 ,tν ] . Set i0 = i1 and im+1 = im . For each ν = 1, . . . , m + 1 we may choose a path γν in Biν from piν−1 iν to λ(tν−1 ). For each ν = 1, . . . , m, since − is Biν ∪ Biν+1 is contained in the simply connected set Bi ν , the path γν ∗ λν ∗ γν+1 − path homotopic in X to the path βiν−1 iν iν+1 (which lies in ). Moreover, γ1 and αi0 i1 = αi1 i1 are paths in Bi1 with the same endpoints and are therefore path homotopic, while γm+1 and αi−m im+1 = αi−m im are paths in Bim with the same endpoints and are therefore also path homotopic. Hence we have the following path homotopy equivalences in X: λ ∼ γ1− ∗ γ1 ∗ λ1 ∗ γ2− ∗ γ2 ∗ λ2 ∗ γ3− ∗ · · · ∗ γm−1 − ∗ λm−1 ∗ γm− ∗ γm ∗ λm ∗ γm+1 ∗ γm+1

∼ γ1− ∗ βi0 i1 i2 ∗ βi1 i2 i3 ∗ · · · ∗ βim−1 im im+1 ∗ γm+1 ∼ αi0 i1 ∗ βi0 i1 i2 ∗ αi−1 i2 ∗ αi1 i2 ∗ βi1 i2 i3 ∗ αi−2 i3 ∗ · · · ∗ αim−1 im ∗ βim−1 im im+1 ∗ αi−m im+1 . It follows that the image im(π1 (, x0 ) → π1 (X, x0 )) is (finitely) generated by the finite set of elements {[αij ∗ βij k ∗ αj−k ]x0 | i, j, k ∈ I, Bi ∩ Bj = ∅, and Bj ∩ Bk = ∅}. Thus part (a) is proved. Part (b) follows immediately from (a). Finally, for the proof of part (c), we observe that part (a) and σ -compactness (Lemma 9.3.6) provide a sequence of do mains {ν } and a point x0 ∈ 1 such that X = ν ν and such that for each ν, ν  ν+1 and the image ν of π1 (ν , x0 ) in π1 (X, x0 ) is finitely generated  and therefore countable. Furthermore, π1 (X, x0 ) is equal to the countable set ∞ ν=1 ν . For given an element [γ ] ∈ π1 (X, x0 ), we have γ ([0, 1]) ⊂ ν for some ν  0, and hence [γ ] ∈ ν .  Exercises for Sect. 10.1 10.1.1 Prove Lemma 10.1.3. 10.1.2 Prove Lemma 10.1.7. 10.1.3 Prove that for n > 1, the unit sphere Sn is simply connected.

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Fig. 10.2 A covering space and an evenly covered set

10.2 Elementary Properties of Covering Spaces  → X be a surjective continuous mapping of topologiDefinition 10.2.1 Let ϒ : X cal spaces.   ≡ ϒ −1 (U ) = λ∈ Uλ , where (a) An open set U ⊂ X is evenly covered by ϒ if U  and ϒ maps Uλ homeo{Uλ }λ∈ is a collection of disjoint open subsets of X morphically onto U for each index λ ∈  (see Fig. 10.2).  is called a covering space of X if every point (b) ϒ is called a covering map and X in X has a neighborhood that is evenly covered by ϒ .  → X is a covering map that is a local  and X are C ∞ manifolds and ϒ : X (c) If X   ≡ ϒ −1 (U ) = diffeomorphism (i.e., for U λ∈ Uλ as in (a), ϒ maps each of  → X is called a C ∞ covering the sets Uλ diffeomorphically onto U ), then ϒ : X ∞ space (or a C covering manifold) of X. Remarks 1. The map R → R given by x → x 3 is a covering map (in fact, a homeomorphism) that is of class C ∞ , but this is not a C ∞ covering space because the map is not a local diffeomorphism.  is a locally connected topological space, ϒ : X  → X is a surjective con2. If X  onto a topological space X, and U is an evenly covered dotinuous mapping of X main in X, then ϒ maps each connected component of ϒ −1 (U ) homeomorphically onto U . Example 10.2.2 The map ϒ : R → S1 given by t → e2πit = (cos(2πt), sin(2πt)) is a C ∞ covering map. For as considered in Examples 7.2.8, 9.2.8, and 9.7.20, for each point t0 ∈ R, we get an open arc U ≡ S1 \ {e2πit0 } and we have  ϒ −1 (U ) = Un , n∈Z

where {Un } is the collection of disjoint open intervals given by Un ≡ (t0 + (n − 1)2π, t0 + n2π)

∀n ∈ Z,

and ϒ maps Un diffeomorphically onto U for each n. Definition 10.2.3 Let  : X → Z and  : Y → Z be continuous mappings of topo : X → Y such logical spaces. A lifting of  (relative to ) is a continuous map   = . In other words, the following diagram commutes: that  ◦ 

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Y            Z X  be connected topological spaces, let  : X → Y Lemma 10.2.4 Let X, Y , and Y  → Y be a covering space, and let   be a : X → Y be a continuous map, let ϒ : Y lifting of . (a) (b) (c) (d)

 is a lifting of  that agrees with   at some point in X, then   =  . If   If  is a local homeomorphism, then  is a local homeomorphism.  is a covering map. If  is a covering map and Y is locally connected, then  ∞ ∞  is a C ∞ covering Suppose X and Y are C manifolds,  is of class C , and Y ∞  space. Then the lifting  is of class C . Moreover, if  is a local diffeomor is a local diffeomorphism. If  : X → Y is a C ∞ covering space, phism, then   is a C ∞ covering space.  then  : X → Y

 be a lifting of  and assume that the set Proof For the proof of (a), let   (x)  (x) =  A≡ x∈X| is nonempty. Since X is connected, it suffices to show that A is both closed and open. Given a point x0 ∈ X, we may choose a neighborhood U of (x0 ) such that   ≡ ϒ −1 (U ) = Uλ , U λ∈

, and ϒ maps Uλ homeowhere {Uλ }λ∈ is a collection of disjoint open subsets of Y morphically onto U for each index λ ∈ . Hence there exist unique indices λ and (x0 ) ∈ Uλ and   (x0 ) ∈ Uλ , that is, λ with  (x0 ) = (ϒU )−1 ((x0 )) and  λ

 (x0 ) = (ϒU )−1 ((x0 )).  λ

−1 −1 (Uλ ) ∩   ) (Uλ ) is then a neighborhood of x0 . If x0 ∈ The set V ≡  / A, then (V ) ∩   (V ) = ∅, and hence V ⊂ X \ A. If x0 ∈ A, then λ = λ , and λ = λ , so   = (ϒU )−1 ◦  =   , and hence V ⊂ A. Thus A is both closed on V , we have  λ and open, and therefore A = X. The proofs of (b)–(d) are left to the reader (see Exercise 10.2.1).   → X be a connected covering space Theorem 10.2.5 (Lifting theorem) Let ϒ : X of a connected topological space X, let x0 ∈ X, and let  x0 ∈ ϒ −1 (x0 ). (a) Path lifting property. If γ : [a, b] → X is a path in X with γ (a) = x0 , then there  of γ to a path in X  with γˆ (a) = xˆ0 . is a unique lifting γˆ : [a, b] → X

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(b) Path homotopy lifting property. If H : [a, b] × [0, 1] → X is a path homotopy in X with H (a, ·) ≡ x0 , then there is a unique lifting  : [a, b] × [0, 1] → X  H (b, ·) is con with H (a, ·) ≡ xˆ0 (in particular, H of H to a path homotopy in X stant).  xˆ0 ) → π1 (X, x0 ) is injective. (c) The induced homomorphism ϒ∗ : π1 (X, (d) General lifting property. If  : Y → X is a continuous mapping of a connected locally path connected topological space Y to X, y0 ∈ −1 (x0 ), and  xˆ0 ), then there is a unique lifting   of  : Y → X ∗ π1 (Y, y0 ) ⊂ ϒ∗ π1 (X,  with (y0 ) = xˆ0 . In fact, for every point y ∈ Y and every path γ : [a, b] → Y  is the unique lifting ˆ (y) = λ(b), where λˆ : [a, b] → X from y0 to y, we have  ˆ  exists, then of the path λ ≡ (γ ) with λ(a) = xˆ0 . Conversely, if such a lifting   xˆ0 ). ∗ π1 (Y, y0 ) ⊂ ϒ∗ π1 (X, Proof We first prove uniqueness in part (a). Suppose γ : [a, b] → X is a path and ϒ(xˆ0 ) = x0 = γ (a) as in (a). For any evenly covered neighborhood U of x0 , it is clear that if γ lies in U , then there is a unique lifting of γ with initial point  x0 . In general, if α and β are two liftings with initial point xˆ0 , then letting c ≡ sup{t ∈ [a, b] | α = β on [a, t]} ∈ (a, b] and forming an evenly covered neighborhood U of γ (c), we may choose  > 0 so that a < c −  and γ ([c − , c + ] ∩ [a, b]) ⊂ U . But then, α(c − ) = β(c − ), and hence α = β on [c − , c + ] ∩ [a, b] and therefore on [a, c + ] ∩ [a, b]. It follows that c = b and α = β. Thus we have uniqueness. A similar argument shows that for d ≡ sup{t ∈ [a, b] | γ [a,t] admits a lifting with initial point xˆ0 }, we have d = b and d lies in the above set. Thus we have existence as well. For the proof of (b), observe that by part (a), for each s ∈ [0, 1], we have a unique s : [a, b] → X  of the path Hs : t → H (t, s) to a path in X  with H s (a) = xˆ0 . lifting H    Thus we may define a map H : [a, b] × [0, 1] → X by (t, s) → Hs (t); and we get (a, ·) ≡ xˆ0 and ϒ ◦ H  = H . If the set A ⊂ [a, b] × [0, 1] of points of discontinuity H  for H is nonempty, then we may set c ≡ inf{t ∈ [a, b] | ({t} × [0, 1]) ∩ A = ∅}, and we may choose r ∈ [0, 1] with (c, r) ∈ A. If  > 0 is sufficiently small and D ≡ ([c − , c + ] ∩ [a, b]) × ([r − , r + ] ∩ [0, 1]), then H (D) is contained in some evenly covered open set U . If c = a, then ϒ maps a neighborhood U0 of xˆ0 homeomorphically onto U , and we get a (continuous) lifting (ϒU0 )−1 ◦ H D : D → U0 that is equal to xˆ0 along {a} × ([r − , r + ] ∩ [0, 1]),  on D. However, the point (c, r) = (a, r) lies in the and therefore must be equal to H set ((c − , c + ) ∩ [a, b]) × ((r − , r + ) ∩ [0, 1]), which is open in [a, b] × [0, 1] and is contained in D. Since (c, r) ∈ A, this is not possible. Assuming that c > a,

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 is we may then choose the number  to be in the interval (0, c − a). But then H continuous on the connected subset [c − , c) × ([r − , r + ] ∩ [0, 1]) of D, and hence the image of this set must be contained in some open set U1 that ϒ maps homeomorphically onto U . Again, applying the local inverse, we get a continuous  along the set {c − } × ([r − , r + ] ∩ [0, 1]) lifting of H D that agrees with H  on all of D. Thus we have again and therefore by construction, must agree with H  must be continuous. Moreover, the arrived at a contradiction, and we see that H ({b} × [0, 1]) lies in the fiber ϒ −1 (H (b, 0)), and is therefore a connected set H  is a path homotopy. Uniqueness follows from (a). singleton. Thus H Part (c) now follows easily, and part (d) is left to the reader (see Exercise 10.2.2).   → X and ϒˇ : Xˇ → X be connected covering spaces Corollary 10.2.6 Let ϒ : X of a connected locally path connected topological space X, let x0 ∈ X, let xˆ0 ∈ ˇ xˇ0 ), then there exists a  xˆ0 ) ⊂ ϒˇ ∗ π1 (X, ϒ −1 (x0 ), and let xˇ0 ∈ ϒˇ −1 (x0 ). If ϒ∗ π1 (X, unique commutative diagram of covering maps Xˇ     ϒ  ϒˇ   ϒ    X X ˇ xˇ0 ), then ϒ  xˆ0 ) = ϒˇ ∗ π1 (X,  is a homeo (xˆ0 ) = xˇ0 . Moreover, if ϒ∗ π1 (X, with ϒ ∞ ∞ ˇ  ˇ morphism. If X, X, and X are C manifolds and ϒ and ϒ are C covering maps,  xˆ0 ) =  is a C ∞ covering map (and hence ϒ  is a diffeomorphism if ϒ∗ π1 (X, then ϒ ˇ xˇ0 )). ϒˇ ∗ π1 (X, Proof The lifting theorem (Theorem 10.2.5) and Lemma 10.2.4 give the existence and uniqueness of the commutative diagram. If the image groups are equal, then we  The composition with ϒ  is then also have a corresponding covering map Xˇ → X. →X  that fixes x0 , and therefore by uniqueness, a lifting of ϒ to a covering map X  is a homeomorphism in this case. The the composition must be the identity. Thus ϒ ∞ claims in the C case follow from the above and Lemma 10.2.4.   → X be a connected covering space of a connected Corollary 10.2.7 Let ϒ : X locally path connected topological space X, let U be a connected open subset of X, and let V be a connected component of ϒ −1 (U ). Then the restriction ϒV : V → U is a covering space. Proof Fix a point xˆ 0 ∈ V , and let x0 ≡ ϒ(xˆ0 ). Then, given a point x ∈ U , there is a  with γˆ (0) = xˆ0 path γ in U from x0 to x. Hence the unique lifting γˆ to a path in X must lie in V , and we have ϒ(γˆ (1)) = x. Thus ϒ(V ) = U , and it follows that  ϒV : V → U is a covering space.

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 → X and ϒˇ : Xˇ → X of a topologDefinition 10.2.8 Two covering spaces ϒ : X  → Xˇ that is a : X ical space X are equivalent if there exists a homeomorphism ϒ  lifting of ϒ ; i.e., ϒ is a fiber-preserving homeomorphism. Remarks 1. By Lemma 10.2.4, if the above are C ∞ covering manifolds, then the  is actually a diffeomorphism, and we say that the coverings are homeomorphism ϒ C ∞ equivalent. 2. Equivalence of coverings is an equivalence relation (see Exercise 10.2.3). 3. Corollary 10.2.6 immediately gives the following:  → X and ϒˇ : Xˇ → X be connected coverings of Proposition 10.2.9 Let ϒ : X a connected locally path connected topological space X, and let x0 ∈ X and xˆ0 ∈ ϒ −1 (x0 ). Then the two coverings are equivalent if and only if there exists a ˇ xˇ0 ).  x0 ) = ϒˇ ∗ π1 (X, point xˇ0 ∈ ϒˇ −1 (x0 ) such that ϒ∗ π1 (X, Any covering space of a C ∞ manifold has a natural induced C ∞ structure.  → M be a covering space. Proposition 10.2.10 Let ϒ : M  is a Hausdorff space. (a) If M is a Hausdorff space, then M  is a topological manifold of the same (b) If M is a topological manifold, then M dimension.  with respect (c) If M is a C ∞ manifold, then there is a unique C ∞ structure on M →M to which ϒ is a local diffeomorphism (i.e., with respect to which ϒ : M ∞ is a C covering manifold).  → M of a C ∞ manifold M, we assume that Remark Given a covering space ϒ : M ∞  M is a C manifold with the associated induced C ∞ structure unless otherwise indicated. Proof of Proposition 10.2.10 The proofs of parts (a) and (b) are left to the reader (see Exercise 10.2.4). For the proof of (c), observe that if M is a C ∞ mani is Hausdorff (by (a)), and we may form a C ∞ atfold of dimension n, then M las A = {(Ui , i , Ui )}i∈I on M such that Ui is connected and evenly covered i ≡ ϒ −1 (Ui ) is the union of a collection of for each i ∈ I . For each i ∈ I , U (λ)  each of which is mapped homeodisjoint connected open sets {Ui }λ∈i in M, (λ) morphically onto Ui . Setting i ≡ i ◦ ϒU (λ) , we get an n-dimensional atlas i

 ≡ {(U (λ) , (λ) , U )}λ∈i ,i∈I on M.  Suppose i ∈ I , j ∈ I , λ ∈ i , μ ∈ j , and A i i i (μ) (λ) Ui ∩ Uj = ∅. Then the coordinate transformation i ◦ [j ]−1 = i ◦ −1 j (μ) (U (λ) ∩U (μ) ) : j (Ui (λ)

(μ)

(μ)

j

(μ)

i

j

(λ) (λ) −→ i (Ui ∩ Uj ) ⊂ i (Ui ∩ Uj )

(λ)

(μ)

∩ Uj )

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 with respect to is of class C ∞ , and hence the atlas determines a C ∞ structure on M which ϒ is a local diffeomorphism.  with respect to which ϒ is a Moreover, given an arbitrary C ∞ structure on M local diffeomorphism, and given indices i ∈ I and λ ∈ i , the mapping (λ) i = i ◦ ϒU (λ) is a composition of a C ∞ diffeomorphism on Ui ⊂ M and a C ∞ difi

feomorphism on Ui , and is therefore a C ∞ diffeomorphism. Thus the elements  are C ∞ compatible with the local C ∞ charts on M,  and uniqueness of the atlas A follows.  (λ)

Recall that a continuous mapping of Hausdorff spaces is proper if the inverse image of every compact subset of the target is compact (see Definition 9.3.9). A sequence in a manifold converges to a point p if for every neighborhood U of p, all but finitely many terms of the sequence lie in U . Recall also that a compact subset of a manifold has the Bolzano–Weierstrass property (see Theorem 9.3.2). These properties allow one to prove the following useful fact concerning finite covering spaces (i.e., covering spaces with finite fibers) of manifolds: Lemma 10.2.11 For any local homeomorphism  : M → N of connected topological manifolds M and N , we have the following: (a) The mapping  is proper if and only if  is a finite covering map. (b) If K is a nonempty connected compact subset of (M) with connected compact  ≡ −1 (K),  is a sufficiently small connected neighborhood inverse image K  then  is the connected component of −1 () containing K, of K in N , and   the restriction ϒ ≡  :  →  is a finite covering map.  Proof For the proof of (a), let us first assume that  is a proper mapping. The map  is then surjective. For if y ∈ (M), then we may choose a sequence of points {xν } in M such that yν = (xν ) → y. Since  is proper, {xν } must lie in a compact subset of M, and hence some subsequence must converge to some point x ∈ M. Continuity then gives (x) = y. Thus (M) is a nonempty subset of N that is closed (by the above) and open (since  is a local homeomorphism); and therefore, since N is connected, we have (M) = N . Now given a point y0 ∈ N , we may choose a finite covering of the compact fiber F ≡ −1 (y0 ) by relatively compact connected open subsets V1 , . . . , Vn of M, each of which is mapped homeomorphically onto a connected neighborhood of y0 by . Each of these sets contains exactly one element of F , and since M is Hausdorff, we may choose the sets to be disjoint. We may also choose a connected neighborhood U of y0 in (V1 ) ∩ · · · ∩ (Vn ) such that −1 (U ) ⊂ V ≡ V1 ∪ · · · ∪ Vn . For if this were not the case, then as in the above proof of surjectivity, there would exist a sequence of points {xν } in M \ V that converges to a point x ∈ M \ V with (x) = y0 , which is impossible, since F ⊂ V . Thus the desired neighborhood U must exist. Hence −1 (U ) = U1 ∪ · · · ∪ Un , where for each i = 1, . . . , n, Ui ≡ −1 (U ) ∩ Vi = (Vi )−1 (U ) is a connected neighborhood that is mapped

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homeomorphically onto U by . Since the sets U1 , . . . , Un are disjoint, it follows that U is evenly covered, and hence that  : M → N is a finite covering map. For the proof of the converse, suppose  : M → N is a finite covering map. Then, for each point p ∈ N , the inverse image −1 (U ) of some (in fact, any) evenly covered neighborhood U of p has only finitely many connected components. Fixing a relatively compact neighborhood V of p in U , we see that −1 (V ) is compact. Since any compact set D ⊂ N admits a finite covering by such sets V , −1 (D) must be compact. For the proof of (b), it suffices to show that for any sufficiently small connected  the connected component of −1 () containneighborhood  of K and for   ≡ −1 (K), the restriction ϒ ≡  → ing the connected compact set K :  is proper; for we may then apply part (a). Clearly, for this, we may assume that  is connected and compact, we may choose a relatively compact K = N . Since K  in M. Since (M) is open, we may require that connected neighborhood V of K   (M). Moreover, if  is sufficiently small, then  will not meet the compact  will be  of −1 () containing K set (∂V ), and hence the connected component  ⊂V. a connected open subset of M that meets V , but not ∂V . Thus we will have  Moreover, if C is a compact subset of , then −1 (C) is a closed subset of M that  (since any point in −1 (C) ∩ cl( ) has a connected neighborhood does not meet ∂  −1  ). Thus −1 (C) ∩   is compact, in  () that meets, and hence is contained in,   the restriction ϒ ≡  :  →  is a proper mapping, and the claim follows.   Exercises for Sect. 10.2 10.2.1 10.2.2 10.2.3 10.2.4

Prove parts (b)–(d) of Lemma 10.2.4. Prove part (d) of Theorem 10.2.5. Prove that equivalence of coverings is an equivalence relation. Prove parts (a) and (b) of Proposition 10.2.10.

10.3 The Universal Covering  → X of a connected Definition 10.3.1 A simply connected covering space ϒ : X topological space X is called a universal covering space of X. Corollary 10.2.6 immediately gives the following: Lemma 10.3.2 (Uniqueness of universal coverings) A connected locally path connected topological space X has at most one universal covering space up to equivalence of coverings. In fact, if x0 ∈ X and for j = 1, 2, ϒj : Xj → X is a universal covering space of X and xj ∈ ϒj−1 (x0 ), then there is unique homeomorphism  : X1 → X2 such that ϒ2 ◦  = ϒ1 and (x1 ) = x2 . The main goal of this section is the following:

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Theorem 10.3.3 (Existence of universal coverings) Every connected locally simply  → X. connected topological space X has a universal covering space ϒ : X Proof The idea of the construction is to separate the terminal points of paths from a fixed initial point whenever the paths are not homotopic. Fix a point x0 ∈ X, and let P be the set of paths α : [0, 1] → X with initial point α(0) = x0 . Then path homotopy equivalence determines an equivalence relation ∼ on P, and we may set  ≡ P/∼. Denoting the path homotopy equivalence class of each path α by [α], we X  → X by [α] → α(1). may define a surjective map ϒ : X  with respect to which this is a universal We must define a topology on X  and U covering. Let A be the collection of all pairs ([α], U ), where [α] ∈ X is a simply connected neighborhood of ϒ([α]) = α(1) in X. For each element  with ξ (α(1)) = [α] ξ = ([α], U ) ∈ A, we get a well-defined map ξ : U → X by setting, for each point x ∈ U , ξ (x) ≡ [α ∗ β], where β : [0, 1] → U is an arbitrary path in U from α(1) to x. For if α ∼ α and β is a path in U from α(1) = α (1) to x ∈ U , then since U is simply connected, β and β , and therefore α ∗ β and α ∗ β , are path homotopic. Clearly, ϒ ◦ ξ = IdU , and hence ξ maps U bijectively onto ξ (U ) with inverse mapping ϒξ (U ) . Observe also that if ξ = ([α], U ), ζ = ([α], V ) ∈ A with U ⊂ V , then ξ = ζ U .  such that −1 (V ) One may verify that the collection T of subsets V ⊂ X ξ  (see Exeris an open set in X for every element ξ ∈ A is a topology on X cise 10.3.1). Moreover, ξ (U ) ∈ T for each ξ = ([α], U ) ∈ A. For given an element ζ = ([β], V ) ∈ A and a point x1 ∈ −1 ζ (ξ (U )) = ϒ(ζ (V ) ∩ ξ (U )) ⊂ U ∩ V , we may choose a simply connected neighborhood W of x contained in U ∩ V , a path γ : [0, 1] → U from α(1) to x1 , and a path δ : [0, 1] → V from β(1) to x1 . We then get [α ∗ γ ] = ξ (x1 ) = [ϒζ (V )∩ξ (U ) ]−1 (x1 ) = ζ (x1 ) = [β ∗ δ]. Hence if x ∈ W and η : [0, 1] → W is a path from x1 to x, then ζ (x) = [β ∗ δ ∗ η] = [α ∗ γ ∗ η] = ξ (x) ∈ ξ (U ). −1 Thus W ⊂ −1 ζ (ξ (U )), and hence ζ (ξ (U )) is open. Clearly, with respect to the topology T , ξ is a continuous bijection of U onto  is open, then the open set ξ (U ) for each ξ = ([α], U ) ∈ A. Moreover, if V ⊂ X   ϒ(V ∩ ξ (U )) = −1 ϒ(V ) = ξ (V ) ξ =([α],U )∈A

ξ ∈A

 → X is a surjective local homeois open; hence ϒ is an open mapping. Thus ϒ : X morphism (with local continuous inverses given by the mappings {ξ }ξ ∈A ). In fact,

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given a point x1 ∈ X and an element ([α], U ) ∈ A with x1 = α(1), ϒ −1 (U ) is equal to the union of disjoint connected open sets {([β∗α],U ) (U )}[β]x0 ∈π1 (X,x0 ) , each of which is mapped homeomorphically onto U by ϒ . For if [γ ] ∈ ϒ −1 (U ) and δ : [0, 1] → U is a path from x1 to γ (1), then β ≡ γ ∗ δ − ∗ α − is a loop based at x0 and [γ ] = [β ∗ α ∗ δ] = ([β∗α],U ) (γ (1)). If, on the other hand, β1 and β2 are loops based at x0 and there exist points y1 , y2 ∈ U with ([β1 ∗α],U ) (y1 ) = ([β2 ∗α],U ) (y2 ), then, applying ϒ , we see that y1 = y2 . Fixing a path δ : [0, 1] → U from x1 to y1 = y2 , we get [β1 ∗ α ∗ δ] = ([β1 ∗α],U ) (y1 ) = ([β2 ∗α],U ) (y2 ) = [β2 ∗ α ∗ δ], and hence [β1 ]x0 = [β1 ∗ α ∗ δ ∗ δ − ∗ α − ] = [β2 ∗ α ∗ δ ∗ δ − ∗ α − ] = [β2 ]x0 .  → X is a covering space. Thus the sets are disjoint, and ϒ : X  is simply connected. Let ex0 be the constant loop at x0 . It remains to show that X Given a path α ∈ P, we may choose a partition 0 = t0 < t1 < · · · < tn = 1 so that for each i = 1, . . . , n, α([ti−1 , ti ]) is contained in some simply connected open set Ui . For each i = 1, . . . , n, we may also choose a reparametrization αi : [0, 1] → Ui of α[ti−1 ,ti ] . In particular, by Lemma 10.1.8, α ∼ β ≡ α1 ∗· · ·∗αn . Let βi = α1 ∗· · ·∗αi for i = 1, . . . , n, and let xi = α(ti ) for i = 0, . . . , n. For i = 2, . . . , n, we have ([βi−1 ],Ui−1 ) (xi−1 ) = [βi−1 ] = [βi−1 ∗ αi ∗ αi− ] = ([βi ],Ui ) (xi−1 ). Thus we get a lifting of β to a path β˜ ≡ ([β1 ],U1 ) (α1 ) ∗ ([β2 ],U2 ) (α2 ) ∗ · · · ∗ ([βn ],Un ) (αn )  from in X ˜ β(0) = ([β1 ],U1 ) (x0 ) = [α1 ∗ α1− ] = [ex0 ] to ˜ β(1) = ([βn ],Un ) (xn ) = [βn ] = [β] = [α].  is path connected. Moreover, if γ is a loop in X  based at [ex0 ] and In particular, X we set α = ϒ(γ ) and form β as above, then by the path homotopy lifting property (Theorem 10.2.5), we have ˜ [α]x0 = [α] = [β] = β(1) = γ (1) = [ex0 ] = [ex0 ]x0 . Thus α is path homotopic to the constant loop ex0 , and therefore, again by the path  to the constant loop homotopy lifting property, the lifting γ is path homotopic in X  at [ex0 ].

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Exercises for Sect. 10.3 10.3.1 Verify that the collection of sets T constructed in the proof of Theorem 10.3.3 is a topology.

10.4 Deck Transformations Definition 10.4.1 A deck transformation (or covering transformation) of a covering  → X is a homeomorphism  : X →X  such that ϒ ◦  = ϒ (i.e.,  space ϒ : X preserves the fibers). The group determined by the set of deck transformations for the covering with multiplication given by composition is denoted by Deck(ϒ).  → X be a universal covering of a connected locally Theorem 10.4.2 Let ϒ : X path connected topological space X, let x0 ∈ X, let F ≡ ϒ −1 (x0 ), and let x˜0 ∈ F . We then have the following: (a) The map Deck(ϒ) → F given by  → (x˜0 ) is a bijection. (b) The map χ : Deck(ϒ) → π1 (X, x0 ) given by  → [ϒ(α)] ˜ x0 , where α˜ is an  from x˜0 to (x˜0 ), is a well-defined group isomorphism. arbitrary path in X  γ˜ (c) Let σ ≡ χ −1 : π1 (X, x0 ) → Deck(ϒ) be the inverse isomorphism. If x ∈ X, is a path from x˜0 to x, [α]x0 ∈ π1 (X, x0 ), α˜ is the unique lifting of α to a path in X˜ with α(0) ˜ = x˜0 , and γ˜1 is the unique lifting of γ ≡ ϒ(γ˜ ) to a path with ˜ then σ ([α]x0 )(x) = γ˜1 (1) (that is, σ ([α]x0 )(x) is equal to the γ˜1 (0) = α(1), terminal point of the lifting of α ∗ γ with initial point x˜0 ). (d) The composition π1 (X, x0 ) → Deck(ϒ) → F is the bijection given by [α]x0 → σ ([α]x0 )(x˜0 ). (e) Each element  ∈ Deck(ϒ) \ {Id} has no fixed points. In fact, if U is an evenly  ≡ ϒ −1 (U ), then the set covered domain in X and U0 is a component of U  U1 ≡ (U0 ) is a component of U that is distinct (hence disjoint) from U0 . Proof By uniqueness of universal coverings up to equivalence (Lemma 10.3.2), →X  such that for each point y ∈ F , there is a unique homeomorphism  : X ϒ ◦  = ϒ and (x˜0 ) = y; that is, the map Deck(ϒ) → F is bijective. Thus (a) is proved.  is Suppose that for j = 1, 2, j ∈ Deck(ϒ), yj = j (x˜0 ), and α˜ j : [0, 1] → X a path with initial point α˜ j (0) = x˜0 and terminal point α˜ j (1) = yj . In particular, αj ≡ ϒ(α˜ j ) is a loop based at x0 for j = 1, 2. If y1 = y2 (i.e., 1 = 2 ), then since  is simply connected, α˜ 1 and α˜ 2 , and therefore α1 and α2 , are path homotopic. X  from y1 to 1 ◦ 2 (x˜0 ), Hence χ is well defined. In general, 1 (α˜ 2 ) is a path in X and hence χ(1 ◦ 2 ) = [ϒ(α˜ 1 ∗ 1 (α˜ 2 ))]x0 = [α1 ∗ α2 ]x0 = [α1 ]x0 · [α2 ]x0 = χ(1 ) · χ(2 ).

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Thus χ is a homomorphism. If α1 is path homotopic to the trivial loop, then the lifting α˜ 1 is also a loop (by the path homotopy lifting property in Theorem 10.2.5) and hence 1 (x˜0 ) = y1 = x˜0 . Thus 1 = Id and it follows that χ is injective. Given  with α(0) ˜ = x˜0 and a loop α based at x0 , there are a unique lifting to a path α˜ in X ˜ Clearly, χ() = [α]x0 , so χ is a unique deck transformation  with (x˜0 ) = α(1). surjective and (b) is proved. The proofs of (c)–(e) are left to the reader (see Exercise 10.4.1).  Definition 10.4.3 Let  be a subgroup of the group of self-homeomorphisms of a locally compact Hausdorff space X. (a) We denote by ∼ the equivalence relation on X given by x ∼ y

⇐⇒

γ (x) = y for some γ ∈ ;

by \X the quotient space X/∼ , which is given the quotient topology; and by ϒ : X → \X the corresponding quotient map. (b) The group  acts properly discontinuously on X if for every compact set K ⊂ X, γ (K) ∩ K = ∅ for all but finitely many elements γ of . (c) The group  acts freely if for every element γ ∈  with γ = Id and every point x ∈ X, we have γ (x) = x. Lemma 10.4.4 Let  be a subgroup of the group of self-homeomorphisms of a locally compact Hausdorff space X. Then  acts properly discontinuously on X if and only if for each pair of points x, y ∈ X, there exist neighborhoods U of x and V of y such that γ (U ) ∩ V = ∅ for all but finitely many elements γ of . Proof If  acts properly discontinuously and x, y ∈ X, then fixing relatively compact neighborhoods U of x and V of y, we see that γ (U ) ∩ V ⊂ γ (U ∪ V ) ∩ (U ∪ V ) = ∅ for all but finitely many elements γ of . For the proof of the converse, suppose that K ⊂ X is compact and that for each pair of points x, y ∈ K, there exist neighborhoods Uxy of x and Vxy of y such that Gxy ≡ {γ ∈  | γ (Uxy ) ∩ Vxy = ∅} is finite.  For each point x ∈ K, we may choose a finite set Yx ⊂ K such that K ⊂ y∈Yx Vxy , and we may set Wx equal to the neighborhood y∈Yx Uxy of x.  Thus there exist points x1 , . . . , xn ∈ K such that K ⊂ nj=1 Wxj . If γ ∈  and γ (x) = y for some pair of points x, y ∈ K, then x ∈ Wxj ⊂ Uxj z , y ∈ Vxj z , and γ ∈ Gxj z for some index j ∈ {1, . . . , n} and some point z ∈ Yxj . It follows that {γ ∈  | γ (K) ∩ K = ∅} ⊂

n  

Gxj z

j =1 z∈Yxj

is a finite set.



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 → X be the universal covering space of a connected Theorem 10.4.5 Let ϒ : X locally simply connected locally compact Hausdorff space X. Then  ≡ Deck(ϒ)  → \X  is  the quotient map ϒ : X acts properly discontinuously and freely on X, a covering map, and we have a commutative diagram  X  ϒ  ϒ    ∼   \X X  =  → X is a homeomorphism. Moreover, if X is a C ∞ in which the mapping \X  (with respect to the induced manifold, then  is a group of self-diffeomorphisms of X  has a unique C ∞ structure with respect to which ϒ is a C ∞ structure) and \X  → X is a diffeomorphism. C ∞ covering map and the map \X  → \X  is a covering that we may identify with the origIn other words, ϒ : X  → X. It follows that if ϒ →X  is the universal covering : X inal covering ϒ : X space of a connected locally simply connected locally compact Hausdorff space (or  and Deck(ϒ ∼ ∼  ) = Deck(ϒ), then X a connected C ∞ manifold) X = \X = X and →X  and ϒ : X  → X. : X we may identify the coverings ϒ Proof of Theorem 10.4.5 Part (e) of Theorem 10.4.2 and Lemma 10.4.4 together imply that  acts properly discontinuously and freely. Part (a) of Theorem 10.4.2  → X. The inverse mapimplies that ϒ descends to a continuous bijection \X  ping X → \X is also continuous because ϒ is a surjective open mapping. Lemma 10.2.4 implies that ϒ is a covering map. Finally, if X is a C ∞ manifold  → X is the C ∞ universal covering, then Lemma 10.2.4 implies that every and ϒ : X deck transformation is a diffeomorphism. Moreover, we get a unique C ∞ structure  with respect to which the map \X  → X is a diffeomorphism (for example, on \X by Proposition 10.2.10) and Lemma 10.2.4 then implies that ϒ is a C ∞ covering map.  We also have the following partial converse:  be a simply connected locally simply connected locally comTheorem 10.4.6 Let X pact Hausdorff space, let  be a subgroup of the group of self-homeomorphisms  that acts properly discontinuously and freely on X,  and let ϒ = ϒ : X → of X  be the corresponding quotient space mapping. We then have the followX = \X ing: (a) The quotient space X is connected, locally simply connected, and locally com → X is the universal covering space; and  = Deck(ϒ). pact Hausdorff; ϒ : X (b) If   is a subgroup of , then we have a commutative diagram  X

 ϒ       ≡  X  \X X  ϒ

ϒ

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495

 x0 = ϒ(x˜0 ), and xˆ0 = ϒ of covering maps. Moreover, if x˜0 ∈ X,  (x˜ 0 ), and χ :  = Deck(ϒ) → π1 (X, x0 )

 and χ :   = Deck(ϒ  ) → π1 (X, xˆ0 )

are the corresponding group isomorphisms, then  xˆ0 ) → π1 (X, x0 ). ∗ : π1 (X, χ ◦χ −1 = ϒ  xˆ0 ) on X  is the same as that of In other words, the action of [γˆ ] ∈ π1 (X,  (γˆ )] ∈ π1 (X, x0 ). ∗ [γˆ ] = [ϒ ϒ  is a topological manifold, then X is a topological manifold of the same (c) If X dimension.  is a C ∞ manifold and each element of  is a diffeomorphism, then X has (d) If X  → X is a C ∞ (universal) a unique C ∞ structure with respect to which ϒ : X ∞   covering space. Moreover, for  ⊂  as in part (b), ϒ  and ϒ are C covering ∞  maps (with respect to a unique C structure on X).  ϒ −1 (ϒ(W )) = Proof ϒ is an open mapping because given an open set W ⊂ X,  is locally compact Haus · W is the union of the open sets {γ (W )}γ ∈ . Since X dorff and  acts properly discontinously, given a point x0 ∈ X, we may fix a point  and a finite set y0 ∈ ϒ −1 (x0 ), a relatively compact neighborhood V of y0 in X,  is / 1 . Since X 1 ⊂  such that γ (V ) ∩ V = ∅ for each element γ ∈  with γ ∈ Hausdorff, for each γ ∈ 1 with γ = Id, we may choose disjoint neighborhoods Vγ and Wγ of y0 and γ (y0 ), respectively. For γ = Id, we set Vγ = Wγ = V . Finally, we may choose a connected neighborhood U0 of y0 with

U0 ⊂ V ∩ Vγ ∩ γ −1 (Wγ ). γ ∈1

Setting U ≡ ϒ(U0 ), we get ϒ −1 (U ) =  · U0 =



γ (U0 );

γ ∈

and hence, in order to show that U is evenly covered by ϒ , it suffices to show that the sets {γ (U0 )}γ ∈ are disjoint and that ϒ maps each of these sets homeomorphically onto U . For this, we first observe that if α ∈  with U0 ∩ α(U0 ) = ∅, then α = Id. For we have V ∩ α(V ) = ∅, and hence α ∈ 1 and U0 ∩ α(U0 ) ⊂ Vα ∩ α −1 (Wα ) ∩ α(Vα ) ∩ Wα ⊂ Vα ∩ Wα . Thus α = Id, and it follows that the open sets {γ (U0 )}γ ∈ are disjoint and ϒ maps each of these sets bijectively and therefore homeomorphically (since ϒ is continuous and open) onto U . To see that X is Hausdorff, suppose x1 and x2 are distinct points in X. Then, for each j = 1, 2, we may choose an evenly covered connected neighborhood Uj of xj

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and a point yj ∈ ϒ −1 (xj ) such that the component Vj of ϒ −1 (Uj ) containing yj  Thus the set K ≡ V 1 ∪ V 2 is compact, and since  acts is relatively compact in X. properly discontinuously, we get a finite set 1 ⊂  such that γ (K) ∩ K = ∅ for  is Hausdorff and y2 ∈ /  · y1 , for each element each element γ ∈  \ 1 . Since X γ ∈ 1 we may choose neighborhoods Pγ and Qγ of γ (y1 ) and y2 , respectively, that are disjoint. Thus we get neighborhoods

γ −1 (Pγ ) and R2 ≡ V2 ∩ Qγ R1 ≡ V 1 ∩ γ ∈1

γ ∈1

of y1 and y2 , respectively, and neighborhoods W1 ≡ ϒ(R1 ) ⊂ U1 and W2 ≡ ϒ(R2 ) ⊂ U2 of x1 and x2 , respectively. The neighborhoods W1 and W2 are disjoint because ϒ −1 (W1 ) ∩ ϒ −1 (W2 ) ∩ V2 = ( · R1 ) ∩ R2 = (1 · R1 ) ∩ R2     ⊂ Pγ ∩ Qγ = ∅. γ ∈1

γ ∈1

Thus X is Hausdorff. Moreover, since ϒ is a surjective local homeomorphism, X is also connected, locally simply connected, and locally compact. For the proof that  = Deck(ϒ) (and for the proof of (b)), let us fix a point  and let x0 = ϒ(x˜0 ). Given an element  ∈ , we have ϒ ◦  = ϒ and x˜0 ∈ X therefore  ∈ Deck(ϒ). Conversely, if  ∈ Deck(ϒ), then ϒ((x˜0 )) = ϒ(x˜0 ), and hence there is an element  ∈  ⊂ Deck(ϒ) with (x˜0 ) = (x˜0 ). Therefore, by Theorem 10.4.2,  =  ∈ . Thus  = Deck(ϒ).     For the proof of (b), suppose   is a subgroup of , ϒ  : X → X =  \X is the corresponding quotient covering space, and xˆ0 = ϒ  (x˜ 0 ). The surjective map → X given by ϒ : X (x) →  ϒ(x) is well defined and therefore continuping ϒ  ous with respect to the quotient topology. If U is a nonempty domain in X that is  −1 (U ), then each conevenly covered by ϒ, and U0 is a connected component of ϒ −1 −1 (U0 ) ⊂ ϒ (U ) is equal to a connected component nected component V of ϒ  −1 of ϒ (U ). For if W is the component of ϒ −1 (U ) containing V , then ϒ  (W ) is a  (ϒ  −1 (U ) (ϒ connected subset of ϒ  (W )) = ϒ(W ) = U ) that meets, and is therefore contained in, U0 . So W = V . Since ϒ maps V homeomorphically onto U and  ϒ  maps V locally homeomorphically onto U0 (by Corollary 10.2.7), ϒ must map  is a covering map. U0 homeomorphically onto U . Thus ϒ  xˆ0 ), To complete the proof of (b), we observe that given an element [γˆ ] ∈ π1 (X,  (γˆ ) and we may form the common lifting γ˜ of γ and γˆ to a path we may set γ = ϒ  with γ˜ (0) = x˜0 . We then have in X −1 ([γˆ ])(x˜ 0 ) = [γˆ ] · x˜0 , [γ ] · x˜0 = χ −1 ([γ ])(x˜0 ) = γ˜ (1) = χ ∗ [γˆ ] = χ( χ −1 ([γˆ ])). so [γ ] = ϒ The proofs of (c) and (d) are left to the reader (see Exercise 10.4.2).



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Deck Transformations

497

Corollary 10.4.7 Let X be a connected locally simply connected locally compact Hausdorff space, let x0 ∈ X, and let  be a subgroup of π1 (X, x0 ). Then, up to equivalence of covering spaces, there is a unique connected covering space  xˆ0 ) → π1 (X, x0 )  → X such that for some point xˆ0 ∈ ϒ  −1 (x0 ), ϒ ∗ : π1 (X, : X ϒ  xˆ0 ) isomorphically onto . In fact, if ϒ : X  → X is the universal covmaps π1 (X, ering space, then we have the commutative diagram ϒ

 X  = \X  ϒ   ϒ

X  X

(here, we identify π1 (X, x0 ) ⊃  with Deck(ϒ)). Moreover, if X is a C ∞ manifold,  with  has the induced C ∞ structure, then there is a unique C ∞ structure on X and X ∞  are C covering maps. respect to which ϒ and ϒ Proof This follows immediately from Theorem 10.3.3 (existence of the universal cover), Theorem 10.4.6 (the construction of a covering from a properly discontinuous free group action), and Proposition 10.2.9 (equivalence of coverings with the same image fundamental group).  Example 10.4.8 For S1 = {(x, y) ∈ R2 | x 2 + y 2 = 1}, the C ∞ universal covering space is given by ϒ : R → S1 , where ϒ(t) = (cos(2πt), sin(2πt)) = e2πit for each t ∈ R (see Example 10.2.2). If  ∈ Deck(ϒ), then e2πi(t) = e2πit for all t ∈ R. Thus t → ((t) − t)/(2π) is a C ∞ function with values in Z, and is therefore a constant function. Conversely, any translation of the form t → t + n for n ∈ Z is clearly a deck transformation. Thus π1 (S1 ) ∼ = Deck(ϒ) ∼ = Z. In fact, for x0 = (1, 0), π1 (S1 , x0 ) is the infinite cyclic (free) group generated by [γ ]x0 , where γ (t) = e2πit for each t ∈ [0, 1] (since γ has the lifting γ˜ : t → t with γ˜ (0) = 0 and γ˜ (1) = 0 + 1 = 1). Any subgroup  of Z ∼ = π1 (S1 , x0 ) must be of the form  = mZ for some nonnegative integer m; that is,  is the infinite cyclic subgroup generated by [γ ]m x0 . We 1 2πit/m (the proof is simialso have a universal covering map R → S given by t → e lar to the above), and this covering map has deck transformation group mZ =  ⊂ Z. Therefore, S1 ∼ = \R and the covering space of S1 associated to the subgroup  as in Corollary 10.4.7 is the finite mapping S1 → S1 given by e2πit/m → e2πit , that is, by z → zm (see Example 1.6.2). Exercises for Sect. 10.4 10.4.1 Prove parts (c)–(e) of Theorem 10.4.2. 10.4.2 Prove parts (c) and (d) of Theorem 10.4.6.

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10.5 Line Integrals on C ∞ Surfaces Throughout this section, M denotes a 2-dimensional C ∞ manifold. We first recall that for a continuous differential form θ of degree 1 on M and a piecewise C 1 path b  γ : [a, b] → M, the (line) integral of θ along γ is given by γ θ = a θ (γ˙ (t)) dt (see  Definition 9.7.18). For any point a ∈ R and any map γ : {a} → M, we define γ θ ≡ 0. The chain rule and the fundamental theorem of calculus give the following (see Exercise 10.5.1): Lemma 10.5.1 For any continuous 1-form θ on M and any piecewise C 1 path γ : [a, b] → M:   (a) We have γ − θ = − γ θ .    (b) If a < c < b, then γ θ = γ  θ + γ  θ . [a,c]

[c,d]

(c) If θ = df for a C 1 function f on a neighborhood of γ ([a, b]), then  θ = f (γ (b)) − f (γ (a)). γ

Part (c) of the above lemma suggests that one may define integration of a locally exact 1-form (for example, a closed C 1 form) along a continuous path as a sum of changes of local potentials. The following lemma will imply that the integral is well defined. Lemma 10.5.2 Let θ be a locally exact 1-form on M (in particular, θ is continuous), let γ : [a, b] → M be a (continuous) path in M, let S : a = s0 < · · · < sm = b and T : a = t0 < · · · < tn = b be partitions of [a, b], and for each i = 1, . . . , m and each j = 1, . . . , n, let αi and βj be a C 1 function on a neighborhood Ui of γ ([si−1 , si ]) in M and on a neighborhood Vj of γ ([tj −1 , tj ]) in M, respectively, such that θ = dαi on Ui and θ = dβj on Vj . Then m  

n     αi (γ (si )) − αi (γ (si−1 )) = βj (γ (tj )) − βj (γ (tj −1 )) .

i=1

j =1

Furthermore, if γ is a piecewise C 1 path, then the above number is equal to the integral of θ along γ . Proof By considering a common refinement of the two partitions, we see that we may assume without loss of generality that T is a refinement of S; that is, {s0 , . . . , sm } ⊂ {t0 , . . . , tn }. Thus, for each j = 1, . . . , n, we have, for some unique index i, [tj −1 , tj ] ⊂ [si−1 , si ] and dβj = dαi = θ on some connected neighborhood W of γ ([tj −1 , tj ]) in Ui ∩ Vj . In particular, αi and βj differ by a constant on W , and hence αi (γ (tj )) − αi (γ (tj −1 )) = βj (γ (tj )) − βj (γ (tj −1 )). Therefore,

10.5

Line Integrals on C ∞ Surfaces

499

m  

 αi (γ (si )) − αi (γ (si−1 ))

i=1

=

=

m 



i=1

1≤j ≤n [tj −1 ,tj ]⊂[si−1 ,si ]

 αi (γ (tj )) − αi (γ (tj −1 ))



n  

 βj (γ (tj )) − βj (γ (tj −1 )) .

j =1

Finally, Lemma 10.5.1 implies that the above gives the integral when γ is a piece wise C 1 path. Definition 10.5.3 Let θ be a locally exact 1-form on M. For any (continuous) path γ : [a, b] → M, the integral of θ along γ is given by  n    θ≡ αi (γ (ti )) − αi (γ (ti−1 )) , γ

i=1

where a = t0 < t1 < t2 < · · · < tn = b is a partition of [a, b] and for each i = 1, . . . , n, αi is a C 1 function with dαi = θ on a neighborhood of γ ([ti−1 , ti ]) in M. Lemma 10.5.4 For any locally exact 1-form θ on M and any path γ : [a, b] → M:   (a) We have γ − θ = − γ θ .    (b) If a < c < b, then γ θ = γ  θ + γ  θ . [a,c] [c,d] (c) Suppose a ≤ c ≤ d ≤ b; I = {t ∈ R | c ≤ t ≤ d}; s0 , s1 , s2 , . . . , sm are arbitrary points in [a, b] with s0 = c and sm = d; and αi is C 1 function with dαi = θ on a neighborhood  of {γ (t) | si−1 ≤ t ≤ si or si−1 ≥ t ≥ si } in M for i = 1, . . . , m. Then γ  θ = m i=1 [αi (γ (si )) − αi (γ (si−1 ))]. I (d) Invariance under continuous reparametrization. If ϕ : [r, s] → [a, b] is a continuous map  with c ≡ ϕ(r) ≤ d ≡ ϕ(s) and I = {t ∈ R | c ≤ t ≤ d}, then  θ = γ ◦ϕ γ I θ . (e) If  : N → M is a C 1 mapping of a 2-dimensional C ∞ manifold N into M, then ∗ the   θ is a locally exact 1-form, and for every path τ in N , we have  pullback ∗ τ  θ = (τ ) θ . Proof The proofs of parts (a), (b), and (e) are left to the reader (see Exercise 10.5.2). We prove part (c) for any choice of points s0 , s1 , s2 , . . . , sm and functions α1 , . . . , αm by induction on m, the case m = 1 being trivial. Assume now that m > 1 and that the claim holds for positive integers less than m. If there is an index k ∈ {1, . . . , m − 1} with sk < s0 = c, then applying the induction hypothesis to the finite sequences sk , sk+1 , . . . , sm = d and sk , sk−1 , sk−2 , . . . , s1 , s0 = c, we get    θ= θ− θ γ I

γ [sk ,d]

γ [sk ,c]

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Fundamental Groups, Covering Spaces, and (Co)homology

m  

 αi (γ (si )) − αi (γ (si−1 ))

i=k+1 k 

−

 αk−i+1 (γ (sk−i )) − αk−i+1 (γ (sk−i+1 ))



i=1

=

m  

 αi (γ (si )) − αi (γ (si−1 )) .

i=1

If there is an index k ∈ {1, . . . , m − 1} with c = s0 ≤ sk ≤ sm = d, then for J ≡ {t ∈ R | c ≤ t ≤ sk } and K ≡ {t ∈ R | sk ≤ t ≤ d}, the induction hypothesis gives    θ= θ+ θ γ I

γ J

=

γ K

k  

m     αi (γ (si )) − αi (γ (si−1 )) + αi (γ (si )) − αi (γ (si−1 ))

i=1

=

i=k+1

m  

 αi (γ (si )) − αi (γ (si−1 )) .

i=1

Finally, if sm < sk for k = 1, . . . , m − 1, then the induction hypothesis gives    θ= θ− θ γ I

γ [c,sm−1 ]

=

m−1 

γ [d,sm−1 ]

   αi (γ (si )) − αi (γ (si−1 )) − αm (γ (sm−1 )) − αm (γ (sm ))



i=1

=

m  

 αi (γ (si )) − αi (γ (si−1 )) .

i=1

Thus (c) is proved. For the setup in (d), we may choose a partition r = t0 < t1 < · · · < tm = s such that for each i = 1, . . . , m, there is a C 1 function αi with dαi = θ on a neighborhood of γ (ϕ([ti−1 , ti ])) ⊃ γ ({s ∈ [a, b] | ϕ(ti−1 ) ≤ s ≤ ϕ(ti ) or ϕ(ti−1 ) ≥ s ≥ ϕ(ti )}). Setting si = ϕ(ti ) for i = 0, . . . , m and applying part (c), we get   m    θ= θ. αi (γ (ϕ(ti ))) − αi (γ (ϕ(ti−1 ))) = γ ◦ϕ γ I  i=1 Proposition 10.5.5 For any locally exact 1-form θ on M, the following are equivalent: (i) The  form θ is exact. (ii) γ θ = 0 for every loop γ in M.   (iii) The line integrals are independent of path; that is, γ0 θ = γ1 θ for every pair of paths γ0 and γ1 with the same endpoints.

Line Integrals on C ∞ Surfaces

10.5

501

Proof We may assume without loss of generality that M is connected. That (i) implies (ii) follows immediately from the definition of the line integral. That (ii) implies (iii) follows immediately from Lemma 10.5.4. Assuming now that the condition (iii) holds, and fixing a point p ∈ M, for each point x ∈ M we may define f (x) ≡ γ θ , where γ is an arbitrary path from p to x in M. Given q ∈ M, we may fix a C 1 function α with dα = θ on some connected neighborhood U of q, and a path η from p to q. Given a point x ∈ U , we may choose a path γ from q to x in U , and we get    f (x) = θ + θ = α(x) − α(q) + θ. γ

η

γ

Hence f and α differ by a constant on U , and therefore f ∈ C 1 (M) and df = θ .  One may greatly reduce the collection of loops along which one must check that the integral vanishes in order to get exactness. Theorem 10.5.6 Let θ be a locally  exact1-form on M, and let γ0 and γ1 be two path homotopic paths in M. Then γ0 θ = γ1 θ . Proof Let H : [a, b] × [0, 1] → M be a path homotopy from γ0 to γ1 , and for each s ∈ [0, 1], let γs ≡ H (·, s) : [a, b] → M. We may choose partitions a = t0 < t1 < t2 < · · · < tm = b

and 0 = s0 < s1 < s2 < · · · < sn = 1

such that for each pair (i, j ) ∈ {1, . . . , m} × {1, . . . , n}, there is a C 1 function αij with dαij = θ on a connected neighborhood Uij of H ([ti−1 , ti ] × [sj −1 , sj ]) in M (see Fig. 10.3). In particular, if i ≥ 2 and j ≥ 1, then αij and αi−1,j differ by a constant on some neighborhood of H ({ti−1 } × [sj −1 , sj ]). For j = 1, . . . , n,  θ= γsj −1

m  

 αij (γsj −1 (ti )) − αij (γsj −1 (ti−1 ))

i=1

=

m  



 αij (γsj −1 (ti )) − αij (γsj (ti )) + αij (γsj (ti )) − αij (γsj (ti−1 ))

i=1

 + αij (γsj (ti−1 )) − αij (γsj −1 (ti−1 ))

=

m−1 

 αij (γsj −1 (ti )) − αij (γsj (ti )) +



i=1

+ 

m   i=2

=

θ γsj

 θ γsj

 αi−1,j (γsj (ti−1 )) − αi−1,j (γsj −1 (ti−1 ))

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Fig. 10.3 Invariance of line integrals under path homotopy

(here we have used the fact that γs (a) and γs (b) are each constant in s). The claim now follows.  Remark One may view the equality

 αij (γsj −1 (ti )) − αij (γsj −1 (ti−1 )) = αij (γsj −1 (ti )) − αij (γsj (ti )) 

+ αij (γsj (ti )) − αij (γsj (ti−1 ))

 + αij (γsj (ti−1 )) − αij (γsj −1 (ti−1 )) as the replacement of the integral of θ along the segment γsj −1 [ti−1 ,ti ] with the integral along the path λi−1 ∗ γsj [ti−1 ,ti ] ∗ λ− i , where λi is the path s → H (ti , s) from γsj −1 (ti ) to γsj (ti ) for i = 0, . . . , m and j = 1, . . . , m (see Fig. 10.3). The integrals are equal because θ is exact on the neighborhood Uij . The integrals along the λi ’s cancel in the telescoping sum. Corollary 10.5.7 If M is simply connected, then any locally exact 1-form on M is exact. Corollary 10.5.8 If M is connected, x0 ∈ M, and θis a locally exact 1-form on M,  then the map π1 (X, x0 ) → (C, +) given by [γ ] → γ θ is a well-defined group homomorphism. Exercises for Sect. 10.5 10.5.1 Prove Lemma 10.5.1. 10.5.2 Prove parts (a), (b), and (e) of Lemma 10.5.4.

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Homology and Cohomology of Second Countable C ∞ Surfaces

503

10.5.3 Let θ = P dx + Q dy for continuous functions P , Q on an open set  ⊂ R2 . Show that if     ∂P ∂Q = ∈ L1loc (), ∂x distr ∂y distr then θ is locally exact.

10.6 Homology and Cohomology of Second Countable C ∞ Surfaces Differential forms and their appropriate generalizations allow one to associate certain Abelian groups and vector spaces to a C ∞ surface. In a sense, these groups measure the number of loops (and forms) in the surface when we identify loops (and forms) that are identical up to integration. We will need these groups only for second countable surfaces, so we restrict our attention to that case for simplicity. The approach taken here is similar to the approach in the book of Weyl [Wey]. Throughout this section, F denotes the field R or C. Definition 10.6.1 For a second countable C ∞ surface M and an integer r ∈ {0, 1, 2}, the rth de Rham cohomology group with coefficients in F is the quotient group (and quotient vector space)

 d r HdeR (M, F) ≡ ker E r (M, F) → E r+1 (M, F)

 d /im E r−1 (M, F) → E r (M, F) r (M, F) simply (here, we set d ≡ 0 on E −1 (M) ≡ 0). We also denote HdeR r r by HdeR (M). For each closed differential form α ∈ E (M, F), we call the correr (M, F) the de Rham cohomology class of α, sponding equivalence class in HdeR r and we denote this class by [α]HdeR (M,F) , by [α]deR , or simply by [α]. We also say that two differential forms are cohomologous if they represent the same de Rham cohomology class (i.e., if their difference is C ∞ -exact). We also let

 d 1 ZdeR (M, F) ≡ ker E 1 (M, F) → E 2 (M, F) ,

 d 1 BdeR (M, F) ≡ im E 0 (M, F) → E 1 (M, F) . r (M, F) is given by the closed C ∞ F-valued differential In other words, HdeR forms of degree r on M, where we identify two closed C ∞ differential forms α r (M, F) is an Abelian group and β if and only if α − β is C ∞ -exact. Clearly, HdeR with respect to the induced sum operation [α]deR + [β]deR = [α + β]deR . In fact, r (M, F) is a vector space over F with respect to the induced sum and the inHdeR duced scalar multiplication ζ · [α]deR = [ζ α]deR . For our purposes, we will mainly consider the case r = 1.

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Although we often use the same notation [α]deR to denote the de Rham cohor (M, R) or H r (M, C), there is no real mology class of a closed form α in HdeR deR danger of confusion, because according to the following proposition, we may view r (M, R) as a real vector subspace (and as a subgroup) of H r (M, C). HdeR deR Proposition 10.6.2 For a second countable C ∞ surface M and an integer r (M, R) → H r (M, C) given by [α] r r ∈ {0, 1, 2}, the map HdeR HdeR (M,R) → deR r (M,C) is a well-defined injective real linear map. In particular, identifying [α]HdeR r (M, R) with its image in H r (M, C), we get the real direct sum decomposition HdeR deR r (M, C) = H r (M, R) ⊕ iH r (M, R) given by HdeR deR deR [α]deR = [Re α]deR + i[Im α]deR

r ∀[α]deR ∈ HdeR (M, C)

r (M, C) with the complexification (H r (M, R)) ). (thus we may identify HdeR C deR

Proof Let α and β be closed C ∞ real differential forms of degree r. If α − β = dγ for some C ∞ real differential form γ of degree r − 1, then clearly, [α]deR = [β]deR r (M, C). Thus the map is well defined. It is also easy to check that the map in HdeR is linear. Finally, if α = dγ for some C ∞ complex differential form γ , then we have α = d Re γ (and d Im γ = 0). It follows that the map is injective. r (M, C), then For the direct sum decomposition, observe that if [α]deR ∈ HdeR d Re α = d Im α = 0 and [α]deR = [Re α]deR + i[Im α]deR . If ρ and τ are closed r (M, R) ∩ iH r (M, R), then we have C ∞ real r-forms with [ρ]deR = i[τ ]deR ∈ HdeR deR ∞ ρ − iτ = dβ for some C complex (r − 1)-form β (ρ − iτ = 0 for r = 0). But then  ρ = d(Re β) and τ = d(− Im β), and hence [ρ]deR = i[τ ]deR = 0. Remark In Sect. 10.7, we will see that for any second countable C ∞ surface M, 1 (M, F) is canonically isomorphic to a vector space H 1 (M, F) (the first cohoHdeR mology group) that depends only on the topology of M. In fact, we will see that 1 HdeR (M, F) ∼ = H 1 (M, F) ∼ = Hom(π1 (M), F)

(where Hom(π1 (M), F) is the group of homomorphisms of π1 (M) into F, which 1 (M, F) has a natural vector space structure). For this reason, we identify HdeR 1 with H (M, F). Although for the sake of convenience, we will define H 1 (M, F) ˇ as the space of classes of Cech 1-forms (which are natural topological analogues of closed 1-forms), the resulting space is actually canonically isomorphic to the singular cohomology group with coefficients in F (see the remarks at the end of Sect. 10.7 and Exercise 10.7.13). We will also consider, in Sect. 10.7, the cohomology group H 1 (M, A) with coefficients in an arbitrary subring A of C with 1 ∈ A, which is an A-submodule of H 1 (M, C). It turns out that under the above isomorphisms, H 1 (M, A) may be identified with the submodule Hom(π1 (M), A) of 1 (M, C) consisting of those classes Hom(π1 (M), C) and with the submodule of HdeR  1 [θ ]deR ∈ HdeR (M, C) for which γ θ ∈ A for every loop γ in M. In this section, we focus mainly on cohomology (and homology) over F, and set aside consideration of coefficients in other rings until Sect. 10.7.

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Homology and Cohomology of Second Countable C ∞ Surfaces

505

We now turn to homology. For this, we first consider the vector space consisting of all formal finite linear combinations of paths. It will later be convenient (in Sect. 10.7) to have considered formal finite linear combinations over more general rings. Definition 10.6.3 Let M be a second countable topological surface, and let A be a subring of C that contains 1 (i.e., that contains Z). (a) A 1-chain in M with coefficients in A is a formal finite linear combination ξ=

m 

ai · γi ,

i=1

where ai ∈ A and γi : [0, 1] → M is a (continuous) path for i = 1, . . . , m. Following standard  conventions, we identify the above 1-chain ξ with another 1-chain ζ = nj=1 bj βj in M if for each i = 1, . . . , m, 

al =

1≤l≤m γl =γi

⎧ ⎨ 1≤j ≤n bj ⎩ 0

if γi ∈ {β1 , . . . , βn },

βj =γi

if γi ∈ / {β1 , . . . , βn },

and for each j = 1, . . . , n, the analogous statement, with the roles of ξ and ζ switched, holds. A 0-chain in M with coefficients in A is a formal finite linear combination ξ=

m 

ai · pi ,

i=1

where ai ∈ A and pi ∈ X for i = 1, . . . , m (and we have the identifications analogous to those described above for 1-chains). For q = 0, 1, the q-chains in M, together with the obvious addition and scalar multiplication by elements of A, form an A-module (a vector space when A is a field), which we denote by Cq (M, A). (b) The boundary operator ∂ : C1 (M, A) → C0 (M, A) is the homomorphism (a linear map when A is a field) given by ξ=

m  i=1

ai · γi →

m 

ai · (γi (1) − γi (0))

i=1

(which, as one may easily check, is well defined). (c) A 1-cycle in M is a 1-chain ξ for which ∂ξ = 0. The set of 1-cycles in C1 (M, A) is a submodule (or subspace), which we denote by Z1 (M, A) ≡ ker(∂ : C1 (M, A) → C0 (M, A)).

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Remarks 1. Let P be the set of paths in M, and let G be the A-module consisting of the A-valued functions on P that vanish everywhere except at finitely many points. Then we may identify G with C1 (M, A) under the correspondence  f↔ f (γ ) · γ γ ∈P \f −1 (0)

(which we set equal to 0 if f ≡ 0). Similarly, we may identify the A-module D consisting of the A-valued functions on M that vanish everywhere except at finitely many points with C0 (M, A) under the correspondence  f↔ f (p) · p p∈M\f −1 (0)

(which we set equal to 0 if f ≡ 0). This point of view together with linearity makes it clear that the boundary operator ∂ is well defined by (b). 2. For any pair of subrings A1 and A2 of C with 1 ∈ A1 ⊂ A2 , Cr (M, A1 ) is a submodule of Cr (M, A1 ) for r = 0, 1 and Z1 (M, A1 ) is a submodule of Z1 (M, A2 ). Definition 10.6.4 Let M be a second countable C ∞ surface.  ∞ (a) For every 1-chain ξ = m i=1 ai · γi ∈ C1 (M, C) and every closed C 1-form (in fact, for every locally exact 1-form) θ on M, the integral (or line integral) of θ along ξ is given by   m  θ≡ ai θ ξ

i=1

γi

(which, as one may easily check, is well defined). (b) We denoteby B1 (M, F) the subspace of Z1 (M, F) consisting of all 1-cycles ξ for which ξ θ = 0 for every closed C ∞ 1-form θ on M (note that the definition yields the same space regardless of whether we require the forms θ to be real or complex). (c) The first homology group of M with coefficients in F is the quotient vector space H1 (M, F) ≡ Z1 (M, F)/B1 (M, F). We call the equivalence class in H1 (M, F) represented by ξ ∈ Z1 (M, F) the homology class of ξ , and we denote this class by [ξ ]H1 (M,F) , by [ξ ]H1 , or simply by [ξ ]. Two 1-cycles ξ, ζ ∈ Z1 (M, F) are said to  be homologous if they represent the same homology class, that is, if ξ θ = ζ θ for every closed C ∞ 1-form θ on M.  Remarks 1. If ξ ∈ C1 (M, F) and ξ θ = 0 for every closed C ∞ 1-form θ on M, then ξ is a 1-cycle and hence ξ ∈ B1 (M, F) (see Exercise 10.6.1). In other words, the assumption in part (b) that the 1-chain ξ is a 1-cycle is superfluous.

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2. It is more standard to define B1 (M, F) as the image of a certain boundary operator ∂ on a space of 2-chains (see the remarks at the end of Sect. 10.7), but it turns out that the above definition, which is in a more convenient form for our purposes, is equivalent to the standard definition (see the exercises for Chaps. 5 and 6). Consequently, H1 (M, F) is equal to the first singular homology group with coefficients in F. 3. We will consider homology and cohomology groups of topological surfaces in Sect. 10.7. In particular, we will see that the first homology group and the first de Rham cohomology group actually depend only on the topology of the surface. 4. Given a subring A of C with 1 ∈ A, the natural analogue of the above definition gives a submodule of Z1 (M, A) and a corresponding quotient; but without further hypotheses, the quotient need not coincide with the singular homology group. However, it turns out that the two do coincide for an orientable second countable C ∞ surface. With this in mind, for any orientable second countable C ∞ surface M and any subring A of C with 1 ∈ A, we define the A-modules     1 B1 (M, A) ≡ ξ ∈ Z1 (M, A)  θ = 0 ∀θ ∈ ZdeR (M, C) ξ

and H1 (M, A) ≡ Z1 (M, A)/B1 (M, A). On the other hand, in this section, we focus mostly on homology (and cohomology) over F, and set aside consideration of coefficients in other rings until Sect. 10.7. For a second countable C ∞ surface M, we have the real direct sum decomposition Z1 (M, C) = Z1 (M, R) ⊕ iZ1 (M, R) = (Z1 (M, R))C given by ξ = Re ξ +  i Im ξ for each ξ ∈ Z1 (M,  C), where for any q-chain η = m a λ , Re η ≡ (Re a )λ and Im η ≡ j j j j j j (Im aj )λj . According to the folj =1 lowing proposition, the proof of which is left to the reader (see Exercise 10.6.2), this also holds at the level of homology: Proposition 10.6.5 For a second countable C ∞ surface M, the real linear map H1 (M, R) → H1 (M, C) given by [ξ ]H1 (M,R) → [ξ ]H1 (M,C) is well defined and injective. In particular, we may identify H1 (M, R) with its image in H1 (M, C), and we have the real direct sum decomposition H1 (M, C) = H1 (M, R) ⊕ iH1 (M, R) given by [ξ ]H1 (M,C) = [Re ξ ]H1 (M,R) + i[Im ξ ]H1 (M,R)

∀[ξ ]H1 ∈ H1 (M, C)

(thus we may identify H1 (M, C) with the complexification (H1 (M, R))C ). Lemma 10.6.6 and Proposition 10.6.7 below often allow one to work with loops in place of general paths when considering homology:

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Lemma 10.6.6 Let M be a second countable C ∞ surface, let A be a subring of C with 1 ∈ A, and let x0 ∈ M. Then, for every 1-cycle ξ ∈ Z1 (M, A), there exists a 1-cycle η ∈ Z1 (M,  A) such that η is1 a linear combination over A of loops based (M, C) (that is, ξ and η are homologous as at x0 and ξ θ = η θ for every θ ∈ ZdeR 1-cycles over C). Moreover, for ξ an integral 1-cycle (that is, for A = Z), one may choose η itself to be a loop based at x0 .  Proof For ξ = m i=1 ai γi ∈ Z1 (M, A), we prove by induction on m that ξ is homologous to a linear combination of loops. For m = 1, we have 0 = ∂ξ = a1 · (γ1 (1) − γ1 (0)), and hence a1 = 0 or γ1 is a loop. Assume now that the claim holds for any linear combination of strictly  fewer than m paths. If aj = 0 or γj is a loop for some j , then ξ − aj γj = i=j ai γi is a 1-cycle, which, by the induction hypothesis, is homologous over C to a linear combination of loops. Thus we may assume that aj = 0 and γj is not a loop for each j = 1, . . . , m. Hence for we may fix distinct indices i1 , . . . , ik ∈ {1, . . . , m} and a path λν = γiν or γi− ν each ν = 1, . . . , k such that for all μ, ν = 1, . . . , k, we have λμ (0) = λν (0) and λμ (1) = λν (1) if μ = ν, and λμ (1) = λν (0) if and only if ν = μ + 1. Furthermore, we may choose k to be the largest integer for which it is possible to make such a choice. In particular, the point λk (1), which is equal to γik (0) or γik (1), is not an endpoint for any of the paths γi1 , . . . , γik−1 ; and hence, since ∂ξ = 0, λk (1) must be an endpoint for the path γik+1 for some index ik+1 ∈ {1, . . . , m} \ {i1 , . . . , ik }. Setting λk+1 equal to γik+1 if λk (1) = γik+1 (0), γi− if λk (1) = γik+1 (1), we see that by the k+1 maximality of k, λk+1 (1) = λμ (0) for some unique μ ∈ {1, . . . , k}. Letting λ be the loop λμ ∗ λμ+1 ∗ · · · ∗ λk+1 , and for each ν = μ, . . . , k + 1, letting σν be equal to 1 , we see that ξ is homologous over C to the 1-cycle if λν = γiν , −1 if λν = γi− ν η≡

 i∈{1,...,m}\{iμ ,...,ik+1 }

ai γi +

k+1 

(aiν − σμ aiμ σν )γiν + σμ aiμ λ.

ν=μ+1

Applying the induction hypothesis to the 1-cycle η − σμ aμ λ, we get the claim. Moreover, given a loop τ in M, we may fix a path ρ from x0 to τ (0), and τ will x0 . It follows be homologous over C to the loop ρ ∗ τ ∗ ρ − with base point at  that ξ is actually homologous over C to a linear combination η ≡ nj=1 bj ρj of loops {ρj }nj=1 based at x0 . Moreover, if A = Z, then may assume that bj = 1 for each j = 1, . . . , n (replacing bj with −bj and ρj with ρj− whenever bj < 0, we get   bj η = nj=1 k=1 ρj ). Hence ξ is then homologous over C to the loop ρ1 ∗ · · · ∗ ρn .  Proposition 10.6.7 Let M be a second countable C ∞ surface, and let x0 ∈ M. Then: (a) The mapping [α]x0 = [α]π1 (M,x0 ) → [α]H1 (M,F) gives a well-defined group homomorphism π1 (M, x0 ) → H1 (M, F). (b) The image of the fundamental group satisfies   SpanF im(π1 (M, x0 ) → H1 (M, F)) = H1 (M, F).

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(c) The image of any set of generators of the group π1 (M, x0 ) spans H1 (M, F) as a vector space, and consequently, H1 (M, F) has a countable basis. In particular, if π1 (M) is finitely generated (for example, if M is compact), then H1 (M, F) is finite-dimensional.  (d) ξ θ = 0 for every exact C ∞ 1-form θ on M and every 1-cycle ξ ∈ Z1 (M, C). Proof Theorem 10.5.6 implies that the mapping in (a) is well defined. Moreover, given two elements [α]x0 , [β]x0 ∈ π1 (M, x0 ) and a closed C ∞ 1-form θ on M, Lemma 10.5.4 implies that     θ= θ+ θ= θ. α∗β

α

1·α+1·β

β

Thus [α ∗ β]H1 (M,F) = [α]H1 (M,F) + [β]H1 (M,F) , and it follows that the mapping is a group homomorphism. Part (b) follows from Lemma 10.6.6, part (c) follows from part (b) and Lemma 10.1.9, and part (d) follows from part (b) and Proposition 10.5.5.  Definition 10.6.8 Let M be a second countable C ∞ surface. We call the bilinear 1 (M, F) × H (M, F) → F (or H (M, F) × H 1 (M, F) → F) given by pairing HdeR 1 1 deR  ([θ]deR , [ξ ]H1 )deR = ([ξ ]H1 , [θ]deR )deR ≡ θ ξ

1 (M, F) and [ξ ] for every [θ ]deR ∈ HdeR H1 ∈ H1 (M, F) the de Rham pairing on M (Definition 10.6.4 and Proposition 10.6.7 imply that this pairing is well defined). 1 (M, F) with the dual Remark In Sect. 10.7, we will see that we may identify HdeR ∗ space (H1 (M, F)) under the linear isomorphism [θ ]deR → ([θ ]deR , ·)deR (see Theorem 10.7.16).

For a C ∞ mapping  : M → N of second countable C ∞ surfaces M and N , recall that we have the pullback (linear) mapping ∗ : E r (N, F) → E r (M, F) for each r (see Definition 9.5.1), and because d ◦ ∗ = ∗ ◦ d (by Proposition 9.5.6), the pullback of a closed (exact, C ∞ -exact) differential form is also closed (respectively, exact, C ∞ -exact). For r = 0, 1, we denote by ∗ : Cr (M, F) → Cr (N, F) the induced pushforward linear map given by ξ=

m  i=1

ai γi →

m 

ai (γi ),

i=1

where a1 , . . . , am ∈ F and γ1 , . . . , γm are paths in M if q = 1, points in M if q = 0. As is easy to check, we have ∂ ◦∗ = ∗ ◦∂, and hence ∗ (Z1 (M, F)) ⊂ Z1 (N, F). ∞ It is also easy to check that if θ is a closed  C 1-form (or even a locally exact ∗ continuous 1-form) on N , then ξ  θ = ∗ (ξ ) θ for every 1-chain ξ ∈ C1 (M, F). In particular, ∗ B1 (M, F) ⊂ B1 (N, F). If  is a diffeomorphism, then ∗ and ∗ are isomorphisms with inverse mappings (∗ )−1 = (−1 )∗ and (∗ )−1 = (−1 )∗ .

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The above mappings induce linear mappings of the corresponding de Rham cohomology and homology spaces, which have the expected functoriality properties. Proposition 10.6.9 For any C ∞ mapping  : M → N of second countable C ∞ surfaces M and N , we have the following: (a) The mapping H1 (M, F) → H1 (N, F) given by [ξ ]H1 (M,F) → [∗ ξ ]H1 (N,F) is a well-defined linear map, and the diagram H1 (M, R)

 H (M, C) 1

 H1 (N, R)

  H (N, C) 1

commutes. If  is a diffeomorphism, then the map H1 (M, F) → H1 (N, F) is an isomorphism with inverse mapping [ζ ]H1 (N,F) → [−1 ∗ ζ ]H1 (M,F) . (b) If y0 = (x0 ) for some point x0 ∈ M, then the diagram of induced mappings π1 (M, x0 )

 H (M, F) 1

 π1 (N, y0 )

  H (N, F) 1

commutes. Moreover, the image in H1 (N, F) of (any generating set for) π1 (M, x0 ) spans the image (vector space) of H1 (M, F). 1 (N, F) → H 1 (M, F) given by [θ ] ∗ (c) The mapping HdeR deR → [ θ ]deR is well deR defined and linear, and the diagram 1 (N, R) HdeR

 H 1 (N, C) deR

 1 (M, R) HdeR

  H 1 (M, C) deR

1 (N, F) → H 1 (M, F) is an isomorphism if commutes. Moreover, the map HdeR deR  is a diffeomorphism. 1 (N, F) and each [ξ ] (d) For each [θ]H 1 (N,F) ∈ HdeR H1 (M,F) ∈ H1 (M, F), we have deR

∗ [ θ ]H 1

deR (M,F)

, [ξ ]H1 (M,F)

 deR

= [θ]H 1

deR (N,F)

, [∗ ξ ]H1 (N,F)

 deR

.

The proof is left to the reader (see Exercise 10.6.4). Note that the linear map in (c) may be defined in exactly the same way for de Rham cohomology in degrees 0 and 2. Definition 10.6.10 For  : M → N as in Proposition 10.6.9, we call the corresponding mappings of homology and de Rham cohomology the induced pushfor-

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ward and pullback mappings, respectively, and we denote these mappings by ∗ : H1 (M, F) → H1 (N, F)

and

1 1 ∗ : HdeR (N, F) → HdeR (M, F).

Remark If  : M → N and  : N → P are C ∞ mappings of second countable C ∞ surfaces, then ( ◦ )∗ = ∗ ◦ ∗ and ( ◦ )∗ = ∗ ◦  ∗ both at the level of q-chains and 1-forms, and at the level of homology and de Rham cohomology (see Exercise 10.6.5). Exercises for Sect. 10.6

 10.6.1 Prove that if ξ is a 1-chain on a C ∞ surface M and ξ θ = 0 for every closed C ∞ 1-form θ on M, then ξ is a 1-cycle. 10.6.2 Prove Proposition 10.6.5. 10.6.3 Let θ be an exact 1-form of class C 1 on a second countable C ∞ surface M. Prove that ξ θ = 0 for every 1-cycle ξ ∈ Z1 (M, C). 10.6.4 Prove Proposition 10.6.9. 10.6.5 Prove that if  : M → N and  : N → P are C ∞ mappings of second countable C ∞ surfaces, then ( ◦ )∗ = ∗ ◦ ∗ and ( ◦ )∗ = ∗ ◦  ∗ both at the level of q-chains and 1-forms and at the level of homology and de Rham cohomology. 10.6.6 Prove that the map ([α]deR , [β]deR ) → [α ∧ β]deR determines a well-defined 1 (M, C) × H 1 (M, C) → H 2 (M, C) for any second bilinear pairing HdeR deR deR ∞ countable C manifold M. 1 (R2 \ {0}, F) ∼ F. 10.6.7 Prove that for F = R or C, H1 (R2 \ {0}, F) ∼ = F and HdeR =

10.7 Homology and Cohomology of Second Countable C 0 Surfaces In Sect. 10.6, homology and cohomology groups were defined using a C ∞ structure on a surface. As we will see in this section, these groups are actually, up to isomorphism, topological; that is, one may define analogues on a second countable topological surface that are isomorphic to the corresponding C ∞ versions when M is given a C ∞ structure. This is particularly appealing because, as is shown in Chap. 6, every second countable topological surface admits a C ∞ structure. The fact that these groups do not depend on the choice of the C ∞ structure is not crucial for most of our purposes, but it does allow one to see that certain complex analytic objects associated to a Riemann surface are actually topological in nature. For this reason, a proof is outlined in this section. We also consider homology and cohomology with coefficients in an arbitrary subring A of C containing Z. When considering homology with coefficients in A, in order to guarantee that the natural generalization of Definition 10.6.4 (with A in place of F) yields a group that coincides with the first singular homology group (see the remarks at the end of this section and the exercises for Chaps. 5 and 6), we will

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restrict our attention to second countable topological surfaces that admit orientable smooth structures (in fact, as is shown in Chap. 6, this is actually a topological condition). There are many different standard ways of defining homology and cohomology on topological spaces, and with enough restrictions on the spaces, the resulting ˇ groups are isomorphic. Here, we consider a variant of Cech cohomology (see, for example, [Wa]) as considered in, for example, [Fa] (for an equivalent approach using potentials on covering spaces, see [For]). The idea is to produce a generalization of the notion of a locally exact 1-form, and to then proceed as before. The discussion in Sect. 10.5 of line integrals along continuous paths suggests that one may do this by considering the local potentials themselves, without mentioning the associated differential 1-form one gets by applying d. Throughout this section, F denotes the field R or C, and A denotes a subring of C that contains 1 (i.e., that contains Z). Definition 10.7.1 Let M be a second countable topological surface. (a) We denote by P 1 (M, A) (P stands for “potentials”) the collection of families of pairs {(fi , Ui )}i∈I for which {Ui }i∈I is an open covering of M; for each i ∈ I , fi : Ui → C is a continuous function on Ui , with fi real-valued if A ⊂ R; and the function fi − fj is locally constant on the overlap Ui ∩ Uj with values in A for each pair of indices i, j ∈ I . Given two elements F = {(fi , Ui )}i∈I , G = {(gj , Vj )}j ∈J ∈ P 1 (M, A) and a ring element ζ ∈ A, we denote by ζ F and F  G the elements of P 1 (M, A) given by ζ F ≡ {(ζfi , Ui )}i∈I and

 F  G ≡ (fi Ui ∩Vj + gj Ui ∩Vj , Ui ∩ Vj ) (i,j )∈I ×J .

(b) Let ∼ be the equivalence relation in P 1 (M, C) given by F = {(fi , Ui )}i∈I ∼ G = {(gj , Vj )}j ∈J if and only if fi − gj is locally constant on Ui ∩ Vj for all indices i ∈ I and j ∈ J ; that is, the union of the two families is an element of P 1 (M, C) (note that for F, G ∈ P 1 (M, A), we do not require that each of the complex-valued functions fi − gj take values in A, just that they be locally constant). (c) We denote the corresponding quotient mapping and quotient space by Z 1 (M,C) : P 1 (M, C) → Z 1 (M, C) ≡ P 1 (M, C)/∼, we let Z 1 (M, A) ≡ P 1 (M, A)/∼ = Z1 (M,C) (P 1 (M, A)) ⊂ Z 1 (M, C) and Z 1 (M,A) ≡ Z1 (M,C) P 1 (M,A) : P 1 (M, A) → Z 1 (M, A),

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ˇ and we call any element of Z 1 (M, A) a Cech 1-form (or a continuous closed ˇ ˇ 1-form) on M over A (or an integral Cech 1-form for A = Z or a real Cech ˇ 1-form for A = R or a complex Cech 1-form for A = C). We also equip Z1 (M, A) with the A-module structure with zero element Z 1 (M,A) ({(0, M)}) and multiplication (by elements of A) and addition given by ζ · Z1 (M,A) (F ) ≡ Z 1 (M,A) (ζ F ) and Z 1 (M,A) (F ) + Z 1 (M,A) (G) ≡ Z 1 (M,A) (F  G) for all F, G ∈ P 1 (M, A) and ζ ∈ A (the verification that the above give a well-defined module structure is left to the reader in Exercise 10.7.1). For F = {(fi , Ui )}i∈I ∈ P 1 (M, C) and θ = Z 1 (M,C) (F ), the support of θ is the closed subset supp θ of M with complement equal to the union of all open sets U ⊂ M for which fi U ∩Ui is locally constant for each i ∈ I (it is easy to see that this set is independent of the choice of the representative F ). Equivalently, M \ supp θ is the largest open set U0 for which there is a representative F = {(fi , Ui )}i∈I ∈ P 1 (M, C) (we may take F ∈ P 1 (M, R) if θ ∈ Z 1 (M, R)) with 0 ∈ I and f0 ≡ 0. ˇ (d) For any Cech 1-form θ on M and any path γ : [a, b] → M, the integral of θ along γ (or the line integral of θ along γ ) is given by  θ≡ γ

m  

 fiν (γ (tν )) − fiν (γ (tν−1 ))

ν=1

for any representative F = {(fi , Ui )}i∈I ∈ P 1 (M, C) of θ and any partition P : a = t0 < · · · < tm = b with γ ([tν−1 , tν ]) ⊂ Uiν for some iν ∈ I for each ν = 1, . . . , m (the verification that this number is independent of the choice of the representative F , the partition P, and the indices i1 , . . . , im is identical to the proof of Lemma 10.5.2, and the details  are left to the reader in Exercise 10.7.2). For any map γ : {a} → M, we set γ θ ≡ 0. ˇ (e) A Cech 1-form θ on M over A is exact if θ = Z1 (M,C) ({(f, M)}) for some continuous function f : M → C (which we may take to be real-valued if A ⊂ R simply by replacing f with Re f ); that is, θ is represented by the singleton family {(f, M)} (observe that {(f, M)} ∈ P 1 (M, A) and θ = Z1 (M,A) ({(f, M)}), ˇ provided we take f to be real-valued if A ⊂ R). The module of exact Cech 1 1-forms over A is denoted by B (M, A). Remarks 1. One may view the representing functions {(fi , Ui )}i∈I ∈ P 1 (M, A) for ˇ a Cech 1-form θ as generalized local potentials. The requirement that fi be realvalued if A ⊂ R is convenient for our purposes, but not absolutely necessary. Of course, it would be too restrictive to require that fi be A-valued (for example, any integer or rational-valued continuous function is locally constant).

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ˇ 2. For a Cech 1-form θ over A, one may not be able to choose a representative {(fi , Ui )}i∈I ∈ P 1 (M, A) with fi ≡ 0 and Ui = M \ supp θ for some i ∈ I . For example, choosing a real-valued continuous function f on R2 with f ≡ 0 ˇ on (0; 1) and f ≡ 1/2 on R2 \ (0; 2), we get the Cech 1-form θ represented 2 1 2 by {(f, R )} ∈ P (R , Z). If there is a representative {(fi , Ui )}i∈I ∈ P 1 (M, Z) as above with fi ≡ 0 and Ui = M \ supp θ for some i ∈ I , then according to Lemma 10.7.2 below, there is a constant ζ ∈ C such that f − (fi − ζ ) is integervalued. But then, evaluating on R2 \ (0; 2) and (0; 1), we see that both (1/2) − ζ and ζ must be integers, which is impossible. 3. By definition, two elements {(fi , Ui )}i∈I , {(gj , Vj )}j ∈J ∈ P 1 (M, A) represent ˇ the same Cech 1-form θ over A if fi − gj is locally constant as a C-valued function for all i ∈ I and j ∈ J . Since fi and gj roughly correspond to local potentials for θ (in a generalized sense), it is natural that the addition of constants not affect θ , even constants not in A. Thus, although it may seem more natural to require that fi − gj take values in A, such a requirement, which would introduce some complications (for example, the exact 1-forms represented by f ≡ 1/2 and g ≡ 0 would differ as ˇ ˇ integral Cech 1-forms, but not as real Cech 1-forms), is not necessary. Moreover, we have the following fact: Lemma 10.7.2 Given a second countable topological surface M and two repreˇ 1-form sentatives F = {(fi , Ui )}i∈I , G = {(gj , Vj )}j ∈J ∈ P 1 (M, A) for a Cech θ = Z 1 (M,A) (F ) = Z 1 (M,A) (G) over A, there exists a constant ζ ∈ C (with ζ ∈ R if A ⊂ R) such that the representative H ≡ {(gj − ζ, Vj )}j ∈J of θ is in P 1 (M, A) and the function fi − (gj − ζ ) is locally constant with values in A on Ui ∩ Vj for all i ∈ I and j ∈ J . Proof Fix a point x0 ∈ M and indices i0 ∈ I and j0 ∈ J with x0 ∈ Ui0 ∩ Vj0 , and set ζ ≡ gj0 (x0 ) − fi0 (x0 ). The set Q of points x ∈ M for which fi (x) − (gj (x) − ζ ) ∈ A for some (and therefore for every) choice of i ∈ I and j ∈ J with x ∈ Ui ∩ Vj is then nonempty. Moreover, given a point x ∈ M and indices i ∈ I and j ∈ J with x ∈ Ui ∩ Vj , the function fi − (gj − ζ ) will be constant on some neighborhood W of x in Ui ∩ Vj . It follows that Q is both open and closed in M, and hence Q = M.  Analogues of the arguments in Sect. 10.5 give the following: ˇ Theorem 10.7.3 For any Cech 1-form θ on a second countable topological surface M, we have the following: (a) For any path γ : [a, b] → M: (i) We have γ − θ = − γ θ .   (ii) If a < c < b, then γ θ = γ 

[a,c]

θ+

 γ [c,d]

θ.

(iii) If a ≤ c ≤ d ≤ b, J = {t ∈ R | c ≤ t ≤ d}, s0 , s1 , s2 , . . . , sm are points in [a, b] with s0 = c and sm = d, and F = {(fi , Ui )}i∈I ∈ P 1 (M, A) is a

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representative of θ such that for each ν = 1, . . . , m,

(b)

(c) (d) (e)

(f)

{γ (t) | sν−1 ≤ t ≤ sν or sν−1 ≥ t ≥ sν } ⊂ Uiν   for some iν ∈ I , then γ  θ = m ν=1 [fiν (γ (sν )) − fiν (γ (sν−1 ))]. J (iv) Invariance under continuous reparametrization. For any continuous mapping ϕ : [r, s] →[a, b] with  c ≡ ϕ(r) ≤ d ≡ ϕ(s) and for J ≡ {t ∈ R | c ≤ t ≤ d}, we have γ ◦ϕ θ = γ  θ . J The following are equivalent: ˇ (i) The  Cech 1-form θ is exact. (ii) γ θ = 0 for every loop γ in M.   (iii) The line integrals are independent of path; that is, γ0 θ = γ1 θ for every pair of paths γ0 and γ1 with the same endpoints.   If γ0 and γ1 are two path homotopic paths in M, then γ0 θ = γ1 θ . If M is simply connected, then θ is exact. Assuming that θ ∈ Z 1 (M, R) if A ⊂ R, given a point x0 ∈ M, we have θ ∈ Z 1 (M, A) if and only if γ θ ∈ A for every loop γ based at x0 . Moreover, if  θ ∈ Z 1 (M, A), then the mapping [γ ] →  γ θ determines a well-defined group homomorphism π1 (X, x0 ) → (A, +). If θ ∈ Z 1 (M, A) and B is a basis for the topology on M, then θ has a representative F = {(fi , Ui )}i∈I ∈ P 1 (M, A) such that {Ui }i∈I is a countable locally finite family of elements of B (which covers M).

Proof We prove part (e) and leave the proofs of the remaining parts to the reader (see Exercise 10.7.3). For this, let us fix a point x0 ∈ M and an element F = {(fi , Ui )}i∈I ∈ P 1 (M, C) representing θ , with F ∈ P 1 (M, R) if A ⊂ R. If we may choose F to be in P 1 (M, A), then given a loop γ based at x0 and a partition P : a = t0 < · · · < tm = b with γ ([tν−1 , tν ]) ⊂ Uiν for some iν ∈ I for each ν = 1, . . . , m, we have  θ= γ

m  

 fiν (γ (tν )) − fiν (γ (tν−1 ))

ν=1

= [fim (x0 ) − fi1 (x0 )] +

m−1 

 −fiν+1 (γ (tν )) + fiν (γ (tν )) ∈ A.



ν=1

For the proof of the converse, for each i ∈ I , let us assume that Ui is nonempty and connected (as we may, for example, by part (f)), let  us fix a point pi ∈ Ui and a path γi from x0 to pi , and let us set gi ≡ fi − fi (pi ) + γi θ . Then F ∼ G ≡ {(gi , Ui )}i∈I . Moreover, given indices i, j ∈ I and a point x ∈ Ui ∩ Uj , we may choose a path α in Ui from pi to p and β in Uj from pj to p. We then have        gi (x) − gj (x) = fi (x) − fi (pi ) + θ − fj (x) − fj (pj ) + θ = θ, γi

γj

η

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where η ≡ γi ∗ α ∗ β − ∗ γj− . Hence, if the integral of θ along each loop based 1 1 at x0 lies  in A, then G ∈ P (M, A) and therefore θ ∈ Z (M, A). That the mapping [γ ] → γ θ determines a well-defined group homomorphism π1 (X, x0 ) → (A, +) then follows from parts (a) and (c).  ˇ 1-form. On a second countable C ∞ surface, every locally exact 1-form is a Cech In fact, we have the following: Lemma 10.7.4 Let M be a second countable C ∞ surface. To any locally exact 1-form θ over F (i.e., θ is a real 1-form if F = R) on M, we may associate a unique ˇ Cech 1-form θˆ = Z1 (M,F) (F ) ∈ Z 1 (M, F), where F ≡ {(fi , Ui )}i∈I ∈ P 1 (M, F) for an arbitrary choice of an open covering {Ui }i∈I of M and functions {fi }i∈I with fi ∈ C 1 (Ui , F) and dfi = θUi for each i ∈ I (i.e., θ is independent of the choice of the collection F of local potentials for θ ). Moreover, we have the following: 1 (M, F) onto the vec(i) The map θ → θˆ determines a linear isomorphism of ZdeR 1 ˇ tor subspace of Z (M, F) consisting of all Cech 1-forms represented by a family of F-valued local C ∞ functions.   (ii) For every path γ in M, we have γ θ = γ θˆ for every choice of θ . ˇ (iii) The given 1-form θ is exact (as a 1-form) if and only if θˆ is exact (as a Cech 1 1-form), and if this is the case, then for some F-valued function f ∈ C (M), we have θ = df and θˆ = Z 1 (M,F) ({(f, M)}). (iv) The given 1-form θ has compact support if and only if θˆ has compact support.

The proof is left to the reader (see Exercise 10.7.4). On a second countable C ∞ ˇ surface, we will identify any locally exact 1-form θ with its corresponding Cech ˇ ˆ 1-form, and we will give the Cech 1-form the same name θ (instead of θ ). Definition 10.7.5 For a second countable topological surface M, the first cohomology group with coefficients in A is the quotient A-module H 1 (M, A) ≡ Z 1 (M, A)/B 1 (M, A) = Z 1 (M, A)/(B 1 (M, C) ∩ Z 1 (M, A)). ˇ For each Cech 1-form θ ∈ Z 1 (M, A), we call the corresponding equivalence class in H 1 (M, A) the cohomology class of θ , and we denote this class by [θ]H 1 (M,A) , ˇ 1-forms are cohomologous if by [θ ]H 1 , or simply by [θ]. We also say that two Cech they represent the same cohomology class (i.e., if their difference is exact). ˇ Given a Cech 1-form θ = Z 1 (M,C) (F ) with F = {(fi , Ui )}i∈I ∈ P 1 (M, C) on a ˇ second countable topological surface M, we get well-defined Cech 1-forms Re θ and Im θ in Z 1 (M, R) represented by Re F ≡ {(Re fi , Ui )}i∈I ∈ P 1 (M, R) and Im F ≡ {(Im fi , Ui )}i∈I ∈ P 1 (M, R), respectively, and we have θ = Re θ + i Im θ . It follows that we have a real direct sum decomposition Z 1 (M, C) = Z 1 (M, R) ⊕ iZ 1 (M, R), and that we may identify Z 1 (M, C) with the complexification (Z 1 (M, R))C (see Exercise 10.7.5). This also holds at the level of cohomology:

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Proposition 10.7.6 For a second countable topological surface M, the mapping H 1 (M, A) → H 1 (M, C) given by [θ]H 1 (M,A) → [θ ]H 1 (M,C) is a well-defined injective A-module homomorphism, and identifying H 1 (M, A) with its image in H 1 (M, C), we have     1 1 θ ∈ A for every loop γ in M . H (M, A) = [θ]H 1 (M,C) ∈ H (M, C)  γ

Moreover, we have the real direct sum decomposition H 1 (M, C) = H 1 (M, R) ⊕ iH 1 (M, R) given by [θ ]H 1 (M,C) = [Re θ]H 1 (M,R) + i[Im θ]H 1 (M,R) for each [θ ]H 1 (M,C) ∈ H 1 (M, C) (thus we may identify H 1 (M, C) with the complexification (H 1 (M, R))C ). Proof The claims regarding the map H 1 (M, A) → H 1 (M, C) follow from the 1 definitions and Theorem 10.7.3 (observe that  if A ⊂ R, [θ ] ∈ H (M, A), and  γ θ ∈ A for every loop γ in M, then γ Im θ = 0 for every loop γ , and hence [θ] has a real representative), so it remains to show that each cohomology class ξ in H 1 (M, R) ∩ iH 1 (M, R) is trivial. We have ξ = [θ ]H 1 (M,R) = i[η]H 1 (M,R) , where θ = Z1 (M,R) (F ) and η = Z1 (M,R) (G) for some choice of F = {(fj , Uj )}j ∈J , G = {(gn , Vn )}n∈N ∈ P 1 (M, R) (in particular, the functions {fj } and {gn } are real-valued). Consequently, there exists a continuous function u on M such that for each j ∈ J and n ∈ N , the function fj − ign − u is locally constant on Uj ∩ Vn . It follows that the functions fj − Re u and gn − Im u are locally constant on Uj and Vn , respectively, and hence that ξ = 0.  For a second countable surface, although the first cohomology with coefficients in F is clearly a topological object, for any C ∞ structure on the surface, it is canonically isomorphic to the first de Rham cohomology. Theorem 10.7.7 (de Rham) Let M be a second countable C ∞ surface. Then, for ˇ every Cech 1-form α over F on M, there exists a closed C ∞ 1-form β over F on M such that α − β is exact. Consequently, we have a well-defined (surjective) linear ∼ =

1 (M, F) → H 1 (M, F) given by [β] isomorphism HdeR deR → [β]H 1 , and under this isomorphism, assuming that A ⊂ F, we get an A-module isomorphism    ∼ =  1 [θ ]deR ∈ HdeR (M, F)  θ ∈ A for every loop γ in M −→ H 1 (M, A). γ

ˇ Moreover, if α is a Cech 1-form with compact support, then α − β is exact for some closed C ∞ 1-form β over F with compact support. ˇ Proof Suppose α is a Cech 1-form over F on M. Applying part (f) of Theorem 10.7.3, we get a representing family F = {(fi , Ui )}i∈I for α such that Ui  M

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for each i ∈ I and {Ui }i∈I is a countable locally finite open covering of M. We may also choose F so that fi ≡ 0 whenever i ∈ I and Ui ∩ (supp α) = ∅. Applying Theorem 9.3.7, we get a C ∞ partition of unity {λi }i∈I on M such that supp λi ⊂ Ui for each i ∈ I . Thus we get a continuous function f ≡ i∈I λi fi : M → F. For each i ∈ I , we have, on Ui ,  λj (fi − fj ) ∈ C ∞ (Ui ), fi − f = j ∈I

since fi − fj is locally constant on Ui ∩ Uj for each j ∈ J . Therefore, by ˇ Lemma 10.7.4, the Cech 1-form β ≡ α − Z 1 (M,F) ({(f, X)}), which is cohomologous to α, is actually a closed C ∞ 1-form on M (with compact support if α has compact support). Lemma 10.7.4 and Theorem 10.7.3 now give the remaining claims.  For a group G and an Abelian group A, the set of group homomorphisms from G to A, together with the natural zero element and addition, is an Abelian group denoted by Hom(G, A) (similarly, for modules M and N over a ring R, the set of module homomorphisms from M to N is an R-module, which is denoted by Hom(M, N )). In fact, Hom(G, A) is an A-module. The point of view provided by ˇ the Cech 1-forms leads to the following characterization of the first cohomology group: Theorem 10.7.8 (de Rham) Let M be a second countable topological surface, and let x0 ∈ M. Then the mapping H 1 (M, A) → Hom(π1 (M, x0 ), A) given by  [θ ]H 1 → ρ, where ρ : [γ ]x0 → γ θ , is a well-defined module isomorphism. Consequently, for M a second countable smooth surface, the analogous mapping 1 (M, F) → Hom(π (M, x ), F). [θ ]deR → ρ yields a vector space isomorphism HdeR 1 0 Proof Theorem 10.7.3 implies that the mapping is a well-defined injective module homomorphism, so it remains to verify surjectivity. For this, let us fix a locally finite covering {Ui }i∈I of M by simply connected open sets and a continuous partition of unity {ηi }i∈I with supp ηi ⊂ Ui for each i ∈ I , and for each index i ∈ I , let us fix a point pi ∈ Ui and a path γi from x0 to pi . Given a group homomorphism ρ : π1 (M, x0 ) → A, let us set, for each pair of indices i, j ∈ I and each point p ∈ Ui ∩ Uj , fij (p) ≡ ρ([γj ∗ β ∗ α − ∗ γi− ]x0 ) ∈ A, where α is an arbitrary path in Ui from pi to p, and β is an arbitrary path in Uj from pj to p. The functions {fij } are well defined because Ui is simply connected for each i ∈ I , and since ρ is a group homomorphism, they satisfy the cocycle relation fij + fj k = fik on Ui ∩ Uj ∩ Uk

∀i, j, k ∈ I.

Moreover, as is easy to verify, the functions are locally constant. For each j ∈ I ,  we get a continuous function fj ≡ i∈I ηi fij on Uj (here, we have extended the

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function ηi fij by 0 from a function on Ui ∩ Uj to a continuous function on Uj for each i ∈ I ), and we have   ηk (fkj − fki ) = ηk fij = fij : Ui ∩ Uj → A ∀i, j ∈ I. fj − fi = k∈I

k∈I

Moreover, fi is real-valued for each i ∈ I if A ⊂ R. Thus the family F ≡ ˇ {(fi , Ui )}i∈I is an element of P 1 (M, A), and hence F determines a Cech 1-form θ ≡ Z 1 (M,A) (F ). Given a loop λ based at x0 , we may fix a partition 0 = t0 < t1 < · · · < tm = 1 such that for each ν = 1, . . . , m, λ([tν−1 , tν ]) ⊂ Uiν for some iν ∈ I . In particular, for each ν = 1, . . . , m, we may fix paths βν and αν in Uiν from piν to λ(tν−1 ) and λ(tν ), respectively. Thus − ∗ αm λ ∼β1− ∗ α1 ∗ β2− ∗ α2 ∗ · · · ∗ βm

∼β1− ∗ γi− ∗ (γi1 ∗ α1 ∗ β2− ∗ γi− ) ∗ ··· 1 2 − ∗ γi− ) ∗ γim ∗ αm , ∗ (γim−1 ∗ αm−1 ∗ βm m

and hence ρ([λ]) = ρ([β1− ∗ γi− ]) + ρ([γi1 ∗ α1 ∗ β2− ∗ γi− ]) 1 2 − + · · · + ρ([γim−1 ∗ αm−1 ∗ βm ∗ γi− ]) + ρ([γim ∗ αm ]) m

]) + ρ([γi1 ∗ α1 ∗ β2− ∗ γi− ]) = ρ([γim ∗ αm ∗ β1− ∗ γi− 1 2 − + · · · + ρ([γim−1 ∗ αm−1 ∗ βm ∗ γi− ]) m

= (fim (λ(tm )) − fi1 (λ(t0 ))) + (fi1 (λ(t1 )) − fi2 (λ(t1 ))) + · · · + (fim−1 (λ(tm−1 )) − fim (λ(tm−1 )))  m  = (fiν (λ(tν )) − fiν (λ(tν−1 ))) = θ. ν=1

λ

Thus ρ is the image of [θ]H 1 , and we have surjectivity. The claim concerning a second countable C ∞ surface follows from the above together with Theorem 10.7.7.  The definitions of the homology groups with coefficients in F for a second countable topological surface are analogous to the definitions in the C ∞ case provided in Sect. 10.6. However, in this section, we will also consider homology over the ring A, and when doing so, we will add the condition that the surface admit an orientable smooth surface structure. Definition 10.7.9 (Cf. Definition 10.6.4) Let M be a second countable topological surface.

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 (a) For every 1-chain ξ = m i=1 ai ·γi ∈ C1 (M, C) (see Definition 10.6.3) and every ˇ Cech 1-form θ on M, the integral (or line integral) of θ along ξ is given by  θ≡ ξ

m  i=1

 ai

θ γi

(which, as one may easily check, is well defined). (b) We denote by  B1 (M, F) the subspace of Z1 (M, F) consisting of all 1-cycles ˇ 1-form θ on M (note that the definition ξ for which ξ θ = 0 for every Cech ˇ yields the same space regardless of whether we require the Cech 1-forms θ to be real or complex). The first homology group of M with coefficients in F is the quotient vector space H1 (M, F) ≡ Z1 (M, F)/B1 (M, F). We call the equivalence class in H1 (M, F) represented by ξ ∈ Z1 (M, F) the homology class of ξ , and we denote this class by [ξ ]H1 (M,F) , by [ξ ]H1 , or simply by [ξ ]. Two 1-cycles ξ, ζ ∈ Z1 (M, F) are said tobe homologous if they repreˇ sent the same homology class; that is, if ξ θ = ζ θ for every Cech 1-form θ on M. (c) Assuming that M admits an orientable 2-dimensional C ∞ structure, we denote by B1 (M,  A) the A-submodule of Z1 (M, A) consisting of all 1-cycles ξ for ˇ 1-form θ on M (again, the definition yields the which ξ θ = 0 for every Cech ˇ same module regardless of whether we require the Cech 1-forms θ to be real or complex). The first homology group of M with coefficients in A is the quotient A-module H1 (M, A) ≡ Z1 (M, A)/B1 (M, A). The equivalence class in H1 (M, A) represented by ξ ∈ Z1 (M, A) is the homology class of ξ , denoted by [ξ ]H1 (M,A) , by [ξ ]H1 , or simply by [ξ ]. Two 1-cycles ξ, ζ ∈ Z1 (M, A) are homologous if they represent the same homology class. Remark Although we defined H1 (M, A) only for M admitting an orientable C ∞ surface structure, this module is clearly a topological object (in particular, for most purposes, we may as well consider M to be an orientable C ∞ surface in this case). The extra condition is included in order to make H1 (M, A) coincide with the first singular homology group (see the remarks at the end of this section and the exercises for Chaps. 5 and 6). In Chap. 6, it is proved that every second countable topological surface admits a smooth structure, and that orientability is actually a topological condition. For a second countable topological surface M and for subrings A1 and A2 of C with Z ⊂ A1 ⊂ A2 , we have Z1 (M, A1 ) ⊂ Z1 (M, A2 ). We also have the real direct sum decomposition Z1 (M, C) = Z1 (M, R) ⊕ iZ1 (M, R) = (Z1 (M, R))C

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given by ξ = Re ξ + i Im ξ for each ξ ∈ Z1 (M,  C), where for any q-chain η =  m a λ , Re η ≡ (Re a )λ and Im η ≡ j j j j j j (Im aj )λj . According to the folj =1 lowing fact, the proof of which is left to the reader (see Exercise 10.7.6), these inclusions also hold at the level of homology: Proposition 10.7.10 (Cf. Proposition 10.6.5) For a second countable topological surface M, the real linear map H1 (M, R) → H1 (M, C) given by [ξ ]H1 (M,R) → [ξ ]H1 (M,C) is well defined and injective; and identifying H1 (M, R) with its image in H1 (M, C), we have the real direct sum decomposition H1 (M, C) = H1 (M, R) ⊕ iH1 (M, R) given by [ξ ]H1 (M,C) = [Re ξ ]H1 (M,R) + i[Im ξ ]H1 (M,R)

∀[ξ ]H1 ∈ H1 (M, C)

(thus we may identify H1 (M, C) with the complexification (H1 (M, R))C ). Furthermore, for M an orientable second countable smooth surface, the A-module homomorphism H1 (M, A) → H1 (M, C) given by [ξ ]H1 (M,A) → [ξ ]H1 (M,C) is well defined and injective. In particular, H1 (M, A) is torsion-free, and we may identify H1 (M, A) with its image in H1 (M, C). Recall that an element g of a group is called a torsion element if g is of finite order (i.e., g m = e for some positive integer m). A group in which only the identity is a torsion element is said to be torsion-free. Also as in the C ∞ case, one may often reduce questions regarding homology to questions regarding integration along loops by applying Lemma 10.7.11 and Proposition 10.7.12 below, the proofs of which are left to the reader (see Exercises 10.7.7 and 10.7.8): Lemma 10.7.11 (Cf. Lemma 10.6.6) Let M be a second countable topological surface, and let x0 ∈ M. Then, for every 1-cycle ξ ∈ Z1 (M, A), there exists a 1-cycle η ∈ Z1 (M,  A) such that η is1 a linear combination over A of loops based at x0 and θ = ξ η θ for every θ ∈ Z (M, C) (that is, ξ and η are homologous as 1-cycles over C). In particular, ξ θ ∈ A for every θ ∈ Z 1 (M, A). Moreover, for ξ an integral 1-cycle (that is, for A = Z), one may choose η itself to be a loop based at x0 . Proposition 10.7.12 (Cf. Proposition 10.6.7) Let M be a second countable topological surface, and let x0 ∈ M. Then: (a) The mapping [α]x0 = [α]π1 (M,x0 ) → [α]H1 (M,F) gives a well-defined group homomorphism π1 (M, x0 ) → H1 (M, F). For M an orientable second countable C ∞ surface, the mapping [α]x0 → [α]H1 (M,Z) gives a well-defined surjective group homomorphism π1 (M, x0 ) → H1 (M, Z), and in particular, the image of any set of generators of the group π1 (M, x0 ) generates H1 (M, Z) as a group (equivalently, as a Z-module) and H1 (M, Z) is countable. (b) The image of the fundamental group satisfies   SpanF im(π1 (M, x0 ) → H1 (M, F)) = H1 (M, F),

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and consequently, the image of any set of generators of the group π1 (M, x0 ) spans H1 (M, F) as a vector space and H1 (M, F) has a countable basis. In particular, if π1 (M) is finitely generated (for example, if M is compact), then H1 (M, F) is finite-dimensional. For M an orientable second countable C ∞ surface, we have   SpanA im(π1 (M, x0 ) → H1 (M, A)) = H1 (M, A), and consequently, the image of any set of generators of the group π1 (M, x0 ) generates H1 (M, A) as an A-module and H1 (M, A) is countably generated. In particular, if π1 (M) is finitely generated, then H1 (M, A) is finitely generated.  ˇ 1-form θ on M and every 1-cycle ξ ∈ Z1 (M, C). (c) ξ θ = 0 for every exact Cech Definition 10.7.13 Let M be a second countable topological surface. We call the bilinear pairing H 1 (M, F) × H1 (M, F) → F (or H1 (M, F) × H 1 (M, F) → F) given by  ([θ]H 1 , [ξ ]H1 )deR = ([ξ ]H1 , [θ]H 1 )deR ≡ θ ξ

for every [θ]H 1 ∈ and [ξ ]H1 ∈ H1 (M, F) the de Rham pairing over F on M (Definition 10.7.9 and Proposition 10.7.12 imply that this pairing is well defined). If M also admits an orientable 2-dimensional C ∞ structure, then we call the bilinear pairing of A-modules H1 (M, A) × H 1 (M, A) → A given by the restriction of the de Rham pairing over C the de Rham pairing over A. H 1 (M, F)

Remark Theorem 10.7.7 and Proposition 10.7.12 imply that although, in the C ∞ case, H1 (M, F) was defined (in Definition 10.6.4) using the C ∞ structure, the topological definition (Definition 10.7.9) actually yields the same vector space (the spaces are literally the same, not just canonically isomorphic). Moreover, under the 1 (M, F) ∼ H 1 (M, F), the versions of identification given by the isomorphism HdeR = the de Rham pairing (·, ·)deR in Definitions 10.6.8 and 10.7.13 coincide. A continuous mapping  : M → N of second countable topological surfaces M and N induces a pullback map ∗ : P 1 (N, A) → P 1 (M, A) given by {(fi , Ui )}i∈I → {(fi ◦ , −1 (Ui ))}i∈I . This map determines a well-defined pullback map ∗ : Z 1 (N, A) → Z 1 (M, A) (which we give the same name) given by Z 1 (N,A) (F ) → Z 1 (M,A) (∗ F ), and this map is an A-module homomorphism. Moreover, ∗ B 1 (N, A) ⊂ B 1 (M, A). For r = 0, 1, we denote by ∗ : Cr (M, A) → Cr (N, A) the induced pushforward module homomorphism given by ξ=

m  i=1

ai γi →

m  i=1

ai (γi ),

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where a1 , . . . , am ∈ A and γ1 , . . . , γm are paths in M if q = 1, points in M if q = 0. As is easy to check, we have ∂ ◦ ∗ = ∗ ◦ ∂, and hence ∗ (Z1 (M,  A)) ⊂ ˇ Z1 (N, A). It is also easy to check that if θ is a Cech 1-form on N , then ξ ∗ θ =  ∗ (ξ ) θ for every 1-chain ξ ∈ C1 (M, A). In particular, ∗ B1 (M, A) ⊂ B1 (N, A), assuming that M and N admit orientable C ∞ surface structures unless A = R or C. If  is a homeomorphism, then ∗ and ∗ are isomorphisms and (∗ )−1 = (−1 )∗ and (∗ )−1 = (−1 )∗ . The above mappings induce module homomorphisms of the corresponding cohomology and homology spaces that have the expected functoriality properties. Proposition 10.7.14 For any continuous mapping  : M → N of second countable topological surfaces M and N , we have the following: (a) The mapping H1 (M, A) → H1 (N, A) given by [ξ ]H1 (M,A) → [∗ ξ ]H1 (N,A) is a well-defined module homomorphism, and the diagram H1 (M, A)

 H (M, C) 1

 H1 (N, A)

  H (N, C) 1

commutes, provided M and N are orientable C ∞ surfaces if A is neither R nor C. If  is a homeomorphism, then the map H1 (M, A) → H1 (N, A) is an isomorphism with inverse mapping [ζ ]H1 (N,A) → [−1 ∗ ζ ]H1 (M,A) . (b) If y0 = (x0 ) for some point x0 ∈ M, then the diagram of induced mappings π1 (M, x0 )

 H (M, A) 1

 π1 (N, y0 )

  H (N, A) 1

commutes, provided M and N are orientable C ∞ surfaces if A is neither R nor C. Moreover, the image in H1 (N, A) of (any generating set for) π1 (M, x0 ) generates the image of H1 (M, A) as an A-module. (c) The mapping H 1 (N, A) → H 1 (M, A) given by [θ ]H 1 → [∗ θ ]H 1 is a welldefined module homomorphism, and the diagram H 1 (N, A) 

H 1 (M, A)

 H 1 (N, C)   H 1 (M, C)

commutes. Moreover, the map H 1 (N, A) → H 1 (M, A) is an isomorphism if  is a homeomorphism.

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(d) For each [θ ]H 1 (N,F) ∈ H 1 (N, F) and each [ξ ]H1 (M,F) ∈ H1 (M, F), we have

 

∗ [ θ ]H 1 (M,F) , [ξ ]H1 (M,F) deR = [θ]H 1 (N,F) , [∗ ξ ]H1 (N,F) deR . The proof is left to the reader (see Exercise 10.7.9). Definition 10.7.15 For  : M → N as in Proposition 10.7.14, we call the corresponding mappings of homology and cohomology the induced pushforward and pullback mappings, respectively, and we denote these module homomorphisms by ∗ : H1 (M, A) → H1 (N, A)

and ∗ : H 1 (N, A) → H 1 (M, A).

Remark If  : M → N and  : N → P are continuous mappings of second countable topological surfaces, then ( ◦ )∗ = ∗ ◦ ∗ and ( ◦ )∗ = ∗ ◦  ∗ both ˇ at the level of q-chains and Cech 1-forms and at the level of homology and cohomology (see Exercise 10.7.10). We may identify the cohomology space with the dual space of the homology space in a natural way: Theorem 10.7.16 (de Rham) For any second countable topological surface M: (a) The mapping [θ]H 1 (M,F) → ([θ]H 1 (M,F) , ·)deR gives a linear isomorphism ∼ =

H 1 (M, F) −→ (H1 (M, F))∗ = Hom(H1 (M, F), F). (b) The mapping [ξ ]H1 (M,F) → ([ξ ]H1 (M,F) , ·)deR gives an injective linear map H1 (M, F) → (H 1 (M, F))∗ = Hom(H 1 (M, F), F), and this map is surjective if and only if H1 (M, F) is finite-dimensional. If H1 (M, F) is infinite-dimensional, then H1 (M, F) has a countable basis, but H 1 (M, F) does not. Proof It is easy to see from the definitions that the mappings in (a) and (b) are linear maps. The injectivity of the mapping in (a) follows from Theorem 10.7.3, and the injectivity of the mapping in (b) follows from the definition of H1 (M, F). For the proof of surjectivity of the mapping in (a), observe that any linear functional τ ∈ (H1 (M, F))∗ determines a group homomorphism τˆ : π1 (M) → F given ˇ 1-form θ ∈ Z 1 (M, F) by [γ ] → τ ([γ ]H1 ). Thus Theorem 10.7.8 provides a Cech such that  τ ([γ ]H1 ) = τˆ ([γ ]) = θ ∀[γ ] ∈ π1 (M), γ

and Proposition 10.7.12 implies that τ ([ξ ]H1 ) = ([θ]H 1 , [ξ ]H1 )deR Thus part (a) is proved.

∀[ξ ]H1 ∈ H1 (M, F).

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For the proof of (b), observe that if H1 (M, F) is finite-dimensional, then by (a), H 1 (M, F) and H1 (M, F) must have the same dimension, and therefore the injective linear map in (b) must be surjective. Conversely, if H1 (M, F) is infinitedimensional, then by Proposition 10.7.12, H1 (M, F) has a countably infinite basis. But the dual space of any infinite-dimensional vector space cannot have a countable basis (see Exercise 8.1.3), so H 1 (M, F) does not have a countable basis. In particular, (H 1 (M, F))∗ also does not have a countable basis, so the linear map in (b) cannot be surjective.  Theorem 10.7.16 and Proposition 10.7.12 give the following: Corollary 10.7.17 If M is a compact topological surface, then dim H 1 (M, F) = dim H1 (M, F) < ∞. Remark According to Theorem 10.7.7, for M a second countable C ∞ surface, 1 (M, F); so Theorem 10.7.16 and Corolwe may identify H 1 (M, F) with HdeR 1 lary 10.7.17 hold in this case with HdeR (M, F) in place of H 1 (M, F). According to the fundamental theorem of finitely generated Abelian groups, every finitely generated torsion-free Abelian group (G, +) is isomorphic to Zn for some unique integer n called the rank of G; that is, as a Z-module, G has a basis with n elements. Consequently, a modification of the proof of Theorem 10.7.16 gives the following (weaker) version for coefficients in the subring A, the proof of which is left to the reader (see Exercise 10.7.12): Theorem 10.7.18 (de Rham) For any orientable second countable C ∞ surface M: (a) The mapping [θ]H 1 (M,A) → ([θ]H 1 (M,A) , ·)deR gives a module isomorphism ∼ =

H 1 (M, A) −→ Hom(H1 (M, A), A). (b) If π1 (M) is finitely generated (which is the case if, for example, M is compact), then we have the following: (i) The mapping [ξ ]H1 (M,A) → ([ξ ]H1 (M,A) , ·)deR gives a module isomorphism ∼ =

H1 (M, A) −→ Hom(H 1 (M, A), A). (ii) If {[ζi ]H1 }m i=1 are linearly independent (over Z) elements of H1 (M, Z), then {[ζi ]H1 }m i=1 are linearly independent (over A) in H1 (M, A). (iii) Every basis {[ξj ]H1 }nj=1 for H1 (M, Z) is also a basis for H1 (M, A), and the (dual) basis {[θj ]H1 }nj=1 for H 1 (M, Z) ∼ = Hom(H1 (M, Z), Z) determined by  1 if i = j , ([θi ]H 1 , [ξj ]H1 )deR = ∀i, j = 1, . . . , n 0 if i = j , is also a (dual) basis for H 1 (M, A).

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Remark In the exercises for Chaps. 5 and 6, the reader is asked to prove that the above homomorphism H1 (M, A) → Hom(H 1 (M, A), A) is injective for any orientable second countable C ∞ surface M (even if π1 (M) is not finitely generated). Theorem 10.7.18 and Proposition 10.7.12 give the following: Corollary 10.7.19 If M is a compact orientable C ∞ surface, then rank H1 (M, Z) = rank H 1 (M, Z) = dim H1 (M, F) = dim H 1 (M, F) < ∞. Remarks We close this section with some instructive, but nonessential, remarks concerning other (equivalent) versions of homology and cohomology on a second countable topological surface M. ˇ ˇ 1. Cech cohomology. A reader familiar with Cech (sheaf) cohomology (see, for 1 ˇ 1-forms on M as the example, [For]) will recognize the space Z (M, C) of Cech 0 0 0 space of global sections Hˇ (M, C /C), where C is the sheaf of germs of continuous functions and C is the sheaf of germs of locally constant functions. The space ˇ 1-forms is the image of Hˇ 0 (M, C 0 ). The exact sequence of B 1 (M, C) of exact Cech 0 ˇ sheaves 0 → C → C → C 0 /C → 0 yields the exact sequence of Cech cohomology spaces 0 → Hˇ 0 (M, C) → Hˇ 0 (M, C 0 ) → Hˇ 0 (M, C 0 /C) → Hˇ 1 (M, C) → Hˇ 1 (M, C 0 ) → Hˇ 1 (M, C 0 /C) → · · · . Since Hˇ 1 (M, C 0 ) = 0, we get the isomorphism Z 1 (M, C)/B 1 (M, C) = Hˇ 0 (M, C 0 /C)/im(Hˇ 0 (M, C 0 ) → Hˇ 0 (M, C 0 /C)) ∼ =

−→ Hˇ 1 (M, C). 2. Singular homology and cohomology. The standard 2-simplex is the closed triangular region 2 ≡ {(x, y) ∈ (R≥0 )2 | x + y ≤ 1} ⊂ R2 ; a singular 2-simplex in M is a continuous mapping σ : 2 → M; and a (singular) 2-chain in M with coefficients in A is any formal finite linear combination ξ=

m 

ai · σi ,

i=1

where ai ∈ A and σi : [0, 1] → M is a singular 2-simplex for i = 1, . . . , m. We  identify the above 2-chain ξ with another 2-chain ζ = nj=1 bj σˆ j in M if for each i = 1, . . . , m, ⎧ ⎨ 1≤j ≤n bj if σi ∈ {σˆ 1 , . . . , σˆ n },  al = σˆ j =σi ⎩ 1≤l≤m / {σˆ 1 , . . . , σˆ n }, 0 if σi ∈ σl =σi

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and for each j = 1, . . . , n, the analogous statement, with the roles of ξ and ζ switched, holds. The module formed by the 2-chains is denoted by C2 (M, A). For ν = 0, 1, 2, the νth edge of a singular 2-simplex σ in M is the path σ (ν) in M given by ⎧ ⎪ ⎨σ (1 − t, t) if ν = 0, t → σ (0, t) if ν = 1, ∀t ∈ [0, 1]. ⎪ ⎩ σ (t, 0) if ν = 2, The boundary operator ∂ : C2 (M, A) → C1 (M, A) is the homomorphism given by ξ=

m 

ai · σi →

i=1

m 

(0) (1) (2)  ai σi − σi + σi , i=1

the image is denoted by B1 (M, A) ≡ im(∂ : C2 (M, A) → C1 (M, A)), and each element of B1 (M, A) is called a (singular) 1-boundary. We have ∂ 2 = 0 (as is easy to verify), so B1 (M, A) is a submodule of Z1 (M, A), and we may define the first singular homology group of M with coefficients in A to be the quotient module  H1 (M, A) ≡ Z1 (M, A) B1 (M, A). For q = 0, 1, 2, we set q

C (M, A) ≡ Hom(Cq (M, A), A), and we call any element of the above module a q-cochain in M with coefficients q q+1 in A. For q = 0, 1, the homomorphism δ : C (M, A) → C (M, A) given by (δϕ)(ξ ) ≡ ϕ(∂ξ )

q

∀ϕ ∈ C (M, A), ξ ∈ Cq+1 (M, A)

is called the coboundary operator. We set 1 1 2 Z (M, A) ≡ ker(δ : C (M, A) → C (M, A)), 1 0 1 B (M, A) ≡ im(δ : C (M, A) → C (M, A)), 1 (M, A) a (singular) 1-cocycle, and we call each element we call each element of Z 1 (M, A) a (singular) 1-coboundary. Clearly, δ 2 = 0, so B 1 (M, A) is a subof B  1 (M, A), and we may define the first singular cohomology group of M module of Z with coefficients in A to be the quotient module  1 1 (M, A) B (M, A). H1 (M, A) ≡ Z

Further properties of H1 (M, A) and H1 (M, A), and how they relate to H1 (M, A) and H 1 (M, A), are considered in Exercises 10.7.13–10.7.16 and in the exercises for Chaps. 5 and 6.

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Exercises for Sect. 10.7 10.7.1 Verify that the operations assigned to Z 1 (M, A) (as in Definition 10.7.1) determine a well-defined module structure. ˇ 10.7.2 Verify that line integrals of closed Cech 1-forms, as defined in Definition 10.7.1, are well defined. 10.7.3 Prove parts (a)–(d) and part (f) of Theorem 10.7.3. 10.7.4 Prove Lemma 10.7.4. 10.7.5 Verify that for any second countable topological surface M, we have a real direct sum decomposition Z 1 (M, C) = Z 1 (M, R) ⊕ iZ 1 (M, R). 10.7.6 Prove Proposition 10.7.10. 10.7.7 Prove Lemma 10.7.11. 10.7.8 Prove Proposition 10.7.12. 10.7.9 Prove Proposition 10.7.14. 10.7.10 Prove that if  : M → N and  : N → P are continuous mappings of second countable topological surfaces, then ( ◦ )∗ = ∗ ◦ ∗ and ˇ 1-forms and at ( ◦ )∗ = ∗ ◦  ∗ both at the level of q-chains and Cech the level of homology and cohomology. 10.7.11 In the situation of Theorem 10.7.16, assuming that the homology vector space H1 (M, F) is infinite-dimensional, give a direct proof that the associated injection [ξ ]H1 → ([ξ ]H1 , ·)deR is not surjective using the fact that, for any infinite-dimensional vector space V over F, we have V  (V ∗ )∗ (see Sect. 8.1). 10.7.12 Prove Theorem 10.7.18. Some properties of H1 (M, A) and H1 (M, A), and how they relate to H1 (M, A) and H 1 (M, A), are considered in Exercises 10.7.13–10.7.16 below. 10.7.13 Let M be a second countable topological surface, and let A be a subring of C containing Z. (a) Suppose α and β are paths in M. (i) Prove that if α(1) = β(0), then α + β − α ∗ β = ∂σ for some singular 2-simplex σ in M (in particular, α + β − α ∗ β ∈ B1 (M, Z)). (ii) Prove that if α(1) = β(0), then −α + β − + α ∗ β = ∂σ for some singular 2-simplex σ in M (in particular, −α + β − + α ∗ β ∈ B1 (M, Z)). (iii) Prove that the constant loop ep0 : t → p0 at any point p0 ∈ M satisfies ep0 = ∂σ for the constant singular 2-simplex σ : (x, y) → p0 . (iv) Prove that if α and β are path homotopic and p1 ≡ α(1) = β(1), then there is a singular 2-simplex σ in M such that σ (0) = ep1 (the constant loop at p1 ), σ (1) = β, and σ (2) = α; and hence α − β = ∂(σ − σˆ ), where σˆ is the constant singular 2-simplex given by σˆ : (x, y) → p1 (in particular, α − β ∈ B1 (M, Z)). (v) Prove that α + α − ∈ B1 (M, Z). (vi) Prove that if σ is a singular 2-simplex with α = σ (2) and β = σ (0) , then σ (1) is path homotopic to α ∗ β.

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(b) Prove that for each point p0 ∈ M, the mapping [γ ]p0 → [γ ]H  (M,A) 1

gives a well-defined homomorphism π1 (M, p0 ) → H1 (M, A). Prove also that the image of this homomorphism generates H1 (M, A) as an A-module and that the map is surjective if A = Z. (c) Prove that H1 (M, A) is (naturally) isomorphic to H 1 (M, A). Hint. Fix a point p0 ∈ M, and applying part (a) above, show that the mapping H1 (M, A) → Hom(π1 (M, p0 ), A) determined by [ϕ]H 1 · [γ ] = ϕ(γ ) for all [ϕ]H 1 ∈ H1 (M, A) and [γ ] ∈ π1 (M, p0 )   is a well-defined injective homomorphism. For surjectivity, fix a path λx from p0 to x for each point x ∈ M, with λp0 = ep0 . Show that given a homomorphism F ∈ Hom(π1 (M, p0 ), A), there is a unique singular 1-cocycle ϕ determined by ϕ(γ ) ≡ F ([λγ (0) ∗ γ ∗ λ− γ (1) ]) for each path γ . (d) Prove that B1 (M, A) ⊂ B1 (M, A), provided M admits an orientable C ∞ surface structure if A = R and A = C. Conclude that there is a (natural) surjective homomorphism H1 (M, A) → H1 (M, A) (the reader is asked to prove bijectivity in the exercises for Chaps. 5 and 6). 10.7.14 Let M be a second countable topological surface, let σ be a singular 2-simplex in M, and let p0 ∈ M. Prove that: (a) If λ0 is a path in M from p0 to σ (0, 0), then there exists a singular (0) (1) (2) 2-simplex σ0 in M such that σ0 = σ (0) , σ0 = λ0 ∗ σ (1) , and σ0 = (2) λ0 ∗ σ . (b) If λ1 is a path in M from p0 to σ (1, 0), then there exists a singular 2-simplex σ1 in M such that σ1(0) = λ1 ∗ σ (0) , σ1(1) = σ (1) , and σ1(2) = σ (2) ∗ λ− 1. (c) If λ2 is a path in M from p0 to σ (0, 1), then there exists a singular (1) (1) ∗ λ− , and 2-simplex σ2 in M such that σ2(0) = σ (0) ∗ λ− 2 , σ2 = σ 2 σ2(2) = σ (2) . 10.7.15 Suppose α and β are path homotopic paths in a second countable topological surface M, σ is a singular 2-simplex in M, ν ∈ {0, 1, 2}, and α = σ (ν) . Prove that there exists a singular 2-simplex σˆ in M such that σˆ (ν) = β and σˆ (μ) = σ (μ) for μ ∈ {0, 1, 2} \ {ν}. Conclude that in particular, ∂σ = (−1)ν α − (−1)ν β + ∂ σˆ . 10.7.16 The real projective plane is the quotient space RP2 ≡ S2 /[x∼−x ∀x ∈ S2 ]. Let ϒ : S2 → RP2 be the corresponding quotient map. (a) Prove that RP2 admits a C ∞ surface structure for which ϒ is a C ∞ covering map. (b) Prove that RP2 is nonorientable (thus ϒ : S2 → RP2 is an orientable double cover as considered in Exercise 9.7.8). (c) Letting F = R or C, prove the following equalities (which demonstrate that the version of integral homology H1 (M, Z) considered in this section for orientable C ∞ surfaces would be inadequate for nonorientable

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surfaces): H1 (RP2 , Z) ∼ = π1 (RP2 ) ∼ = Z2 = Z/2Z, H1 (RP2 , F) = H1 (RP2 , F) = {0}, H 1 (RP2 , Z) = H1 (RP2 , Z) = H 1 (RP2 , F) = H1 (RP2 , F) = {0}. Hint. For the claim concerning the homology groups, let γ be a generating loop for π1 (RP2 ). If γ = ∂ξ for some integral 2-cochain ξ , then by applying Exercises 10.7.14 and 10.7.15, reduce to the case in which each of the boundary edges of the singular 2-simplices in ξ is either γ or the constant loop. Applying Exercise 10.7.13, obtain a contradiction (one may also prove this fact by observing that the first singular homology group is equal to the Abelianization of the fundamental group, as proved in, for example, [Hat]).

Chapter 11

Background Material on Sobolev Spaces and Regularity

This chapter is required for the proof of integrability of almost complex structures in Chap. 6. Throughout this chapter, we use the operator notation: Dj = and

 D = α

∂ ∂x

∂ ∂xj

for j = 1, . . . , n

α = (D1 )α1 · · · (Dn )αn =

∂x1α1

∂ |α| · · · ∂xnαn

for each multi-index α = (α1 , . . . , αn ) ∈ (Z≥0 )n (see Sect. 7.4). We also use the following notation. Let us denote the coordinates in Rn × Rn = R2n by (x, y) = ((x1 , . . . , xn ), (y1 , . . . , yn )) = (x1 , . . . , xn , y1 , . . . , yn ).  Given a linear differential operator L = bβ D β on an open set  ⊂ Rn , we denote by Lx and Ly the linear differential operators on  ×  given by Lx [h(x, y)] =



 β ∂ bβ (x) [h(x, y)] ∂x



 β ∂ bβ (y) [h(x, y)]. ∂y

and Ly [h(x, y)] = For example, for  = R2 , we have

Dx(1,2) [x12 x23 y14 y25 ] = (2x1 )(6x2 )y14 y25 and Dy(1,2) [x12 x23 y14 y25 ] = x12 x23 (4y13 )(20y23 ). T. Napier, M. Ramachandran, An Introduction to Riemann Surfaces, Cornerstones, DOI 10.1007/978-0-8176-4693-6_11, © Springer Science+Business Media, LLC 2011

531

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For L = x1 D (1,2) , we have L(x12 x23 ) = x1 (2x1 )(6x2 ), Lx [x12 x23 y14 y25 ] = x1 (2x1 )(6x2 )y14 y25 , Ly [x12 x23 y14 y25 ] = y1 x12 x23 (4y13 )(20y23 ). The main goal of this chapter is the following regularity theorem: Theorem 11.0.1 (General first-order regularity) Let A be a linear differential operator of order 1 with C ∞ coefficients on an open set  ⊂ Rn . Assume that there exists a constant C > 0 such that n    u2L2 () + Dj u2L2 () ≤ C · u2L2 () + Au2L2 () ∀u ∈ D(). (1) j =1

Then, for every function u ∈ L2loc () with Adistr u ∈ C ∞ (), we have u ∈ C ∞ (). Remarks 1. The analogous version holds in higher dimensions (see, for example, [Hö]) and for higher-order operators (see, for example, [GiT] or [Ns3]). 2. The term u2L2 () on the left-hand side is redundant. However, it will be convenient to have the theorem stated in this form, since the left-hand side represents the square of the first-order Sobolev norm of u (see Definition 11.1.1), and Sobolev spaces are applied in the proof. A natural approach to the regularity theorem (and to the study of differential equations in general) is to complete the space of C ∞ functions with respect to the norm obtained from Lp norms of derivatives. One obtains the so-called Sobolev spaces. One proves the regularity theorem by applying the Sobolev lemma, which gives an embedding of Sobolev spaces for a large number of derivatives into the Banach space C k . For our purposes, p = 2 suffices, but Sobolev spaces for arbitrary p ≥ 1 are, in general, very useful. The reader may refer to, for example, [Ad] and [GiT] for more about this important class of spaces.

11.1 Sobolev Spaces Definition 11.1.1 Let  be an open subset of Rn , and let k ∈ Z≥0 . (a) The associated Sobolev space of order k is given by α u ∈ L2 () ∀α ∈ Zn≥0 with |α| ≤ k}. W k () ≡ {u ∈ L2 () | Ddistr

(b) The Sobolev inner product and Sobolev norm are given by  α α

u, vW k () ≡

Ddistr u, Ddistr vL2 () |α|≤k 1/2

and uW k () ≡ u, uW k () , respectively, for all u, v ∈ W k ().

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533

(c) The local Sobolev space of order k is given by k α () ≡ {u ∈ L2loc () | Ddistr u ∈ L2loc () ∀α with |α| ≤ k}. Wloc k () and any measurable subset E of , we set For all u, v ∈ Wloc  α α

u, vW k (E) ≡

Ddistr u, Ddistr vL2 (E) |α|≤k 1/2

and uW k (E) ≡ u, uW k (E) , provided these numbers exist in C (or [0, ∞]). Remark Clearly, u, vW k (E) and uW k (E) are independent of the choice of the neighborhood  of E. Completeness of L2 () gives the following fact, the proof of which is left to the reader (see Exercise 11.1.1). Proposition 11.1.2 (W k (), ·, ·W k () ) is a Hilbert space for every open set  ⊂ Rn and every nonnegative integer k. The following lemma allows one to manipulate distributional values of differential operators in ways similar to the ways in which one treats standard values. In particular, it allows one to cut off in order to get compact support. Lemma 11.1.3 Let A and B be linear differential operators of order k ≥ 1 and l, respectively, with C ∞ coefficients on an open set  ⊂ Rn , and let m ≡ k + l − 1. l (a) If u ∈ Wloc (), then

Bdistr u =



β bβ · Ddistr u

∈ L2loc (),

|β|≤l



where B = |β|≤l bβ D β . In particular, for every compact set K ⊂ , there is l (), we have a constant C > 0 such that for every function u ∈ Wloc Bdistr uL2 (K) ≤ CuW l (K) . (b) The commutator [A, B] ≡ AB − BA is a linear differential operator of order m. k−1 m l (c) If u ∈ Wloc () and Adistr u ∈ Wloc (), then we have Bdistr u ∈ Wloc (), (AB)distr u ∈ L2loc (), (BA)distr u ∈ L2loc (), [A, B]distr u ∈ L2loc (), and (AB)distr (u) = Adistr Bdistr u = (BA)distr u + [A, B]distr u = Bdistr Adistr u + [A, B]distr u ∈ L2loc (). In particular, for any nonnegative integer s, the product of a function in C ∞ () s () is in W s (). and a function in Wloc loc

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(d) We have t B = (−1)l B + L for some linear differential operator L of order max(l − 1, 0) with C ∞ coefficients on  (t B = B if l = 0). Remark As noted above, the commutator of two linear differential operators A and B with C ∞ coefficients is given by [A, B] = AB − BA. If A and B are of order 0, then [A, B] = 0. Proof of Lemma 11.1.3 Part (a) follows immediately from Lemma 7.4.3 (note that L2loc ⊂ L1loc ). The product rule gives part (b) (see Exercise 11.1.2). For the proof m (). Parts (a) and (b) imply that [A, B]distr u ∈ L2loc (). of part (c), let u ∈ Wloc Furthermore, for each multi-index α with |α| ≤ k − 1, we have, by Lemma 7.4.3, α [B α 2 α Ddistr distr u] = (D B)distr u ∈ Lloc () (D B is of order k − 1 + l = m), and hence k−1 l Bdistr u ∈ Wloc (). If Adistr u ∈ Wloc (), then (a) and Lemma 7.4.3 together give (BA)distr u = Bdistr Adistr u ∈ L2loc (), and hence Adistr Bdistr u = (AB)distr u = (BA)distr u + [A, B]distr u ∈ L2loc (). The proof of (d) is left to the reader (see Exercise 11.1.2).



Exercises for Sect. 11.1 11.1.1 Prove Proposition 11.1.2. 11.1.2 Prove parts (b) and (d) of Lemma 11.1.3.

11.2 Uniform Convergence of Derivatives In the context of differential equations, it is also natural to consider uniform convergence of derivatives in place of Lp convergence. For this, it is useful to have the following well-known fact: Proposition 11.2.1 Let k be a nonnegative integer, let {uν } be a sequence of C k functions on an open set  ⊂ Rn , and let {vα }α∈(Z≥0)n , |α|≤k be a collection of functions such that D α uν → vα uniformly on compact subsets of  for each multi-index α with |α| ≤ k. Then, for v ≡ v(0,...,0) , we have v ∈ C k () and D α v = vα for each α with |α| ≤ k. The proof is left to the reader (see Exercise 11.2.1). Corollary 11.2.2 If k ∈ Z≥0 ,  is an open subset of Rn , and {uν } is a sequence of functions in C k () such that for every compact set K ⊂ , {D α uν K } is a Cauchy sequence in (C 0 (K),  ·  ≡ supK | · |) for each multi-index α with |α| ≤ k, then there exists a function u ∈ C k () such that D α uν → D α u uniformly on compact subsets of  for each α with |α| ≤ k.

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535

The proof is again left to the reader (see Exercise 11.2.2). Exercises for Sect. 11.2 11.2.1 Prove Proposition 11.2.1. 11.2.2 Prove Corollary 11.2.2.

11.3 The Strong Friedrichs Lemma The goal of this section is the following strong version of the Friedrichs regularization lemma (Lemma 7.3.1): Lemma 11.3.1 (Friedrichs) Let κ be a nonnegative C ∞ function with compact sup

port in the unit ball B(0; 1) in Rn such that Rn κ dλ = 1, let  be an open subset of Rn , let k be a nonnegative integer, and let A be a linear differential operator of order k with C ∞ coefficients on . For each function u ∈ L1loc () and for each δ > 0, let δ ≡ {x ∈  | dist(x, Rn \ ) > δ}, let κ δ (x) ≡ δ −n κ(x/δ) for each point x ∈ Rn , and let   uδ (x) ≡ u(y)κ δ (x − y) dλ(y) = u(x − y)κ δ (y) dλ(y) 

B(0;δ)

 =

u(x − δy)κ(y) dλ(y) B(0;1)

for every x ∈ δ . Finally, let K be a compact subset of , let δ0 be a constant with 0 < δ0 < dist(K, Rn \ ) (hence K ⊂ δ0 ), and let K δ ≡ {x ∈ Rn | dist(x, K) ≤ δ} for each δ > 0. Then we have the following: (a) (b) (c) (d) (e)

For every function u ∈ L1loc () and every δ > 0, we have uδ ∈ C ∞ (δ ). If u ∈ C k (), then A(uδ ) → Au uniformly on K as δ → 0+ . k () and δ ∈ (0, δ ). We have uδ W k (K) ≤ uW k (K δ0 ) for every u ∈ Wloc 0 k k If u ∈ Wloc (), then uδ − uW k (K) → 0 as δ → 0+ (i.e., uδ → u in Wloc ()). There is a constant C > 0 such that for every δ ∈ (0, δ0 ) and every function u ∈ W max(k−1,0) () with Adistr u ∈ L2loc (), we have A(uδ ) − (Adistr u)δ L2 (K) ≤ CuW max(k−1,0) () . max(k−1,0)

(f) If u ∈ Wloc as δ → 0+ .

() and Adistr u ∈ L2loc (), then A(uδ ) − Adistr uL2 (K) → 0

Proof The proof given here is similar to the proof of Lemma 5.2.2 in [Hö]. Part (a) follows from Lemma 7.3.1 (or differentiation past the integral). For the proof of (b), we observe that if u ∈ C k (), α is a multi-index with |α| ≤ k, and a is a C ∞ function on , then by Lemma 7.3.1 and Lemma 7.4.4, we have

536

11

Sobolev Spaces and Regularity

max |a · D α (uδ ) − a · D α u| = max |a · [D α u]δ − a · [D α u]| K

K

≤ max |a| · max |[D α u]δ − [D α u]| → 0 as δ → 0+ . K

K

The claim (b) now follows. For the proof of (c), suppose u ∈ L2loc () and δ ∈ (0, δ0 ). Then, for each point x ∈ K, the Schwarz inequality gives  |uδ (x)|2 =

B(0;1)

 ≤

2 u(x − δy)κ(y) dλ(y)  |u(x − δy)|2 κ(y) dλ(y) ·

B(0;1)

 =

κ(y) dλ(y)

B(0;1)

|u(x − δy)|2 κ(y) dλ(y). B(0;1)

Applying Fubini’s theorem (Theorem 7.1.10), we get |u(x − δy)| dλ(x) κ(y) dλ(y)



 uδ 2L2 (K)

2

≤ B(0;1)



K

|u(x)| dλ(x) κ(y) dλ(y)



2

= 

B(0;1)



≤

K+(−δy)

|u(x)| dλ(x) κ(y) dλ(y) = u2L2 (K δ0 ) . 2

B(0;1)

K δ0

k (), then applying Lemma 7.4.4, we get If u ∈ Wloc

uδ 2W k (K) = ≤

 |α|≤k



|α|≤k

D α (uδ )2L2 (K) =

 |α|≤k

[(D α )distr u]δ 2L2 (K)

(D α )distr u2L2 (K δ0 ) = u2W k (K δ0 ) .

Thus (c) is proved. For the proof of (d), we first observe that if v ∈ C 0 (), then vδ − vL2 (K) ≤ λ(K)1/2 max |vδ − v| → 0 as δ → 0+ K

by part (b) (or Lemma 7.3.1). Given u ∈ L2loc () and > 0, we may choose a continuous function v on  such that v − uL2 (K δ0 ) < /3 (see Exercise 7.1.4). Hence, by part (c), we have vδ − uδ L2 (K) = (v − u)δ L2 (K) < /3

11.3

The Strong Friedrichs Lemma

537

for each δ ∈ (0, δ0 ). On the other hand, for δ > 0 sufficiently small, we also have vδ − vL2 (K) < /3. Therefore uδ − uL2 (K) ≤ uδ − vδ L2 (K) + vδ − vL2 (K) + v − uL2 (K) < . k (), then applying Lemma 7.4.4 Thus uδ − uL2 (K) → 0 as δ → 0+ . If u ∈ Wloc (as above), we get  D α (uδ ) − (D α )distr u2L2 (K) uδ − u2W k (K) = |α|≤k



=

|α|≤k

[(D α )distr u]δ − [(D α )distr u]2L2 (K) → 0 as δ → 0+ .

Thus (d) is proved. We now prove (e) by induction on k. If k = 0, then the differential operator A is given by v → a · v for some C ∞ function a on . Therefore, if u ∈ L2 (), then for each point x ∈ K and each δ ∈ (0, δ0 ), the Schwarz inequality gives |A(uδ )(x) − (Adistr u)δ (x)|2  2 = (a(x) − a(x − δy))u(x − δy)κ(y) dλ(y) B(0;1)



 |u(x − δy)|2 κ(y) dλ(y) ·

≤ C2 · B(0;1)

 = C2 ·

|u(x − δy)|2 κ(y) dλ(y) ,

κ(y) dλ(y) B(0;1)

B(0;1)

where C ≡ 2 maxK δ0 |a|. Applying Fubini’s theorem (Theorem 7.1.10), we get A(uδ ) − (Adistr u)δ 2L2 (K)   2 2 ≤C · |u(x − δy)| dλ(x) κ(y) dλ(y) B(0;1)



K



|u(x)| dλ(x) κ(y) dλ(y)

=C · 2

2



B(0;1)



K+(−δy)

≤C · 2

|u(x)| dλ(x) κ(y) dλ(y) = C 2 u2L2 () . 2

B(0;1)



Assume now that k > 0 and that the inequality holds for operators of order < k. We have A = A + A , where A and A are linear differential operators, each term in A is of the form aD α with |α| = k, and A is of order k − 1. If u ∈ W k−1 () and Adistr u ∈ L2loc (), then in particular, Adistr u exists and is in L2loc () by Lemma 11.1.3. Hence, by the induction hypothesis, there is a constant

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Sobolev Spaces and Regularity

C  > 0 such that for each δ ∈ (0, δ0 ), we have A (uδ ) − (Adistr u)δ L2 (K) ≤ C  uW max(k−2,0) () ≤ C  uW max(k−1,0) () for every such function u. Furthermore, Adistr u = Adistr u − Adistr u ∈ L2loc () and A(uδ ) − (Adistr u)δ L2 (K) ≤ A (uδ ) − (Adistr u)δ L2 (K) + A (uδ ) − (Adistr u)δ L2 (K) .  Thus we may assume without loss of generality that A = A ; i.e., A = |α|=k aα D α with aα ∈ C ∞ () for each α. For each δ ∈ (0, δ0 ) and each point x ∈ K, differentiation past the integral (Proposition 7.2.5) gives, for u ∈ W k−1 () with Adistr u ∈ L2loc (), A(uδ )(x) − (Adistr u)δ (x)   = u(y)Ax [κ δ (x − y)] dλ(y) − B(x;δ)

 =

[Adistr u(y)]κ δ (x − y) dλ(y)

B(x;δ)

 u(y)Ax [κ (x − y)] dλ(y) −

u(y) · (t A)y [κ δ (x − y)] dλ(y).

δ

B(x;δ)

B(x;δ)

According to Lemma 11.1.3, we have t A = (−1)k A + L for some linear differential operator L of order k − 1 with C ∞ coefficients on . We also have  Ax [κ δ (x − y)] = aα (x)Dxα [κ δ (x − y)]  = (−1)k aα (x)Dyα [κ δ (x − y)] = (−1)k Ay [κ δ (x − y)] + M[κ δ (x − y)], where M is the linear differential operator on  ×  ⊂ Rn × Rn given by  (−1)k (aα (x) − aα (y))Dyα [h(x, y)]. M[h(x, y)] = α∈(Z≥0 )n , |α|=k

Thus A(uδ )(x) − (Adistr u)δ (x) = F (x, δ) − G(x, δ), where  u(y)M[κ δ (x − y)] dλ(y) F (x, δ) ≡ B(x;δ)

and

 G(x, δ) ≡

u(y)Ly [κ δ (x − y)] dλ(y). B(x;δ)

Since t L is of order k − 1 and u ∈ W k−1 (), Lemma 11.1.3 implies that t Ldistr u ∈ L2loc () and t Ldistr uL2 (K δ0 ) ≤ C0 uW k−1 () for some constant C0 > 0 that is

11.3

The Strong Friedrichs Lemma

539

independent of the choice of u. In particular, we have G(x, δ) = (t Ldistr u)δ (x) for each x ∈ K, and, applying part (c), we get G(·, δ)L2 (K) ≤ t Ldistr uL2 (K δ0 ) ≤ C0 uW k−1 () . It remains to bound the L2 norm of F (·, δ). For each multi-index α with |α| = k,  we have D α = D α Djα for some multi-index α  with |α  | = k − 1 and some index jα ∈ {1, . . . , n}. Thus, if h(x, y) is a C ∞ function on an open subset of  × , then    (−1)k (aα (x) − aα (y)) · Dyα (Djα )y (h(x, y)) M[h(x, y)] = |α|=k

=



  (−1)k Dyα (aα (x) − aα (y)) · (Djα )y (h(x, y))

|α|=k

+





  (−1)k (aα (x) − aα (y)), Dyα (Djα )y (h(x, y))

|α|=k

=



  (−1)k Dyα (aα (x) − aα (y)) · (Djα )y (h(x, y))

|α|=k

−





  (−1)k aα (y), Dyα (Djα )y (h(x, y))

|α|=k

=



  (−1)k Dyα (aα (x) − aα (y)) · (Djα )y (h(x, y)) + Ny (h(x, y)),

|α|=k

where [·, ·] denotes the commutator (see Lemma 11.1.3) and N is a linear differential operator of order k − 1 with C ∞ coefficients on  (N = 0 if k = 1). Note that the third equality holds because the function aα (x) is constant with respect to the variables y = (y1 , . . . , yn ) for each α. Therefore, since N = t (t N ) and u ∈ W k−1 (), Lemma 11.1.3 implies that for each point x ∈ K and each δ ∈ (0, δ0 ), we have  Hα (x, δ) + (t N distr u)δ (x), F (x, δ) = − |α|=k

where for each α,  Hα (x, δ) ≡ B(x;δ)



α (Ddistr u(y)) · (aα (x) − aα (y)) · (Djα )y [κ δ (x − y)] dλ(y).

A construction similar to that of the bound on the L2 norm of G(·, δ) yields a constant C1 > 0, which is independent of the choice of u and δ, such that (t N distr u)δ L2 (K) ≤ C1 uW k−1 () . To bound the L2 norm of each Hα (·, δ), we first observe that Lemma 7.2.4 implies that there is a constant C2 > 0 independent of the choice of u and δ such that for each

540

11

Sobolev Spaces and Regularity

multi-index α with |α| = k, for each point x ∈ K, and each point y ∈ B(x; δ) ⊂ K δ0 , we have |aα (x) − aα (y)| ≤ C2 δ and      x −y x −y 1 1 δ (Djα )y [κ (x − y)] = (Djα )y n κ = − n+1 (Djα κ) . δ δ δ δ Setting v ≡ vol(B(0; 1)) and C3 ≡ C2 maxx∈Rn , |α|=k |Djα κ(x)|, and applying the Schwarz inequality, we get, for each point x ∈ K and each multi-index α with |α| = k,  |Hα (x, δ)|

2

≤ C32

2

B(x;δ)

 = C32

B(0;1)

α |Ddistr u(y)|δ −n dλ(y) α |Ddistr u(x



≤ C32

2 − δy)| dλ(y)



·v· B(0;1)

α |Ddistr u(x − δy)|2 dλ(y).

Applying Fubini’s theorem (Theorem 7.1.10), we get   2 2 α 2 Hα (·, δ)L2 (K) ≤ C3 · v · |Ddistr u(x − δy)| dλ(x) dλ(y) B(0;1)

 = C32

K



·v· B(0;1)

K+(−δy)



α |Ddistr u(x)|2 dλ(x)

dλ(y)

≤ C32 · v 2 · u2W k−1 () . The claim (e) now follows. For the proof of (f), we fix an open set  with K ⊂    and a constant δ0 with 0 < δ0 < dist(K, Rn \  ). According to part (e), there is a constant C > 0 max(k−1,0) such that for every δ ∈ (0, δ0 ) and every function u ∈ Wloc () with Adistr u ∈ 2 Lloc (), we have A(uδ ) − (Adistr u)δ L2 (K) ≤ CuW max(k−1,0) ( ) . Given such a function u and a number > 0, part (d) implies that we may choose a function v ∈ C ∞ () such that u − vW max(k−1,0) ( ) < /(4C). For each δ ∈ (0, δ0 ), we get A(uδ ) − Adistr uL2 (K) ≤ A(uδ ) − (Adistr u)δ L2 (K) + (Adistr u)δ − Adistr uL2 (K) ≤ A((u − v)δ ) − (Adistr (u − v))δ L2 (K) + A(vδ ) − AvL2 (K) + (Av)δ − AvL2 (K) + (Adistr u)δ − Adistr uL2 (K)

11.4

The Sobolev Lemma

<

541

+ A(vδ ) − AvL2 (K) + (Av)δ − AvL2 (K) 4 + (Adistr u)δ − Adistr uL2 (K) .

By part (b), for δ > 0 sufficiently small, we have A(vδ ) − AvL2 (K) < /4, and by part (d), for δ > 0 sufficiently small, we also have (Av)δ − AvL2 (K) < /4

and (Adistr u)δ − Adistr uL2 (K) < /4.

Thus A(uδ ) − Adistr uL2 (K) < , and the claim (f) is proved.



11.4 The Sobolev Lemma The goal of this section is the following fact, which allows one to obtain differentiability of functions possessing a high number of L2 distributional derivatives: Lemma 11.4.1 (Sobolev lemma) For every open set  ⊂ Rn and every k ∈ Z≥0 , k+n k+n we have Wloc () ⊂ C k (); that is, each function u ∈ Wloc () is equal almost k everywhere to a function of class C . Proof Multiplying the given function by a C ∞ cutoff function and applying part (c) of Lemma 11.1.3, we see that we may assume without loss of generality that ◦

supp u ⊂ K ⊂ K ⊂  for some compact set K. Lemma 11.3.1 then provides a sequence of functions {uν } in D() such that supp uν ⊂ K for each ν and uν − uW k+n () = uν − uW k+n (K) → 0 as ν → ∞. Applying the inequality  ∂nϕ ∀ϕ ∈ D(Rn ) sup |ϕ| ≤ Rn ∂x1 · · · ∂xn (see Exercise 11.4.1), we see that for every multi-index α with |α| ≤ k and for all positive integers μ and ν, we have, for β = α + (1, . . . , 1), sup |D α uμ − D α uν | ≤ D β uμ − D β uν L1 (K) ≤ (λ(K))1/2 · D β uμ − D β uν L2 (K) 

≤ (λ(K))1/2 · uμ − uν W k+n () → 0 as μ, ν → ∞. Proposition 11.2.1 now implies that u ∈ C k ().



542

11

Sobolev Spaces and Regularity

Exercises for Sect. 11.4 11.4.1 Prove the inequality  ∂nϕ ∀ϕ ∈ D(Rn ). sup |ϕ| ≤ Rn ∂x1 · · · ∂xn

11.5 Proof of the Regularity Theorem We now come to the proof of Theorem 11.0.1. Throughout this section, A denotes a linear differential operator of order 1 with C ∞ coefficients on an open set  ⊂ Rn . The first step is the following observation: Lemma 11.5.1 If the inequality (1) in Theorem 11.0.1 holds for some constant C > 0 and for every function u ∈ D(), then for every positive integer k and every compact set K ⊂ , there is a constant C(K, k) > 0 such that   (2) u2W k (K) ≤ C(K, k) · u2L2 () + Au2W k−1 () ∀u ∈ D(). Proof We proceed by induction on k = 1, 2, 3, . . . . The case k = 1 follows from the (stronger) inequality (1). Thus we may assume that k > 1 and that the inequality (2) holds for every compact subset of  (for some choice of a constant) for strictly lower orders. We may fix a compact set K  with K ⊂ K  ⊂  and a function ρ ∈ D() with ρ ≡ 1 on K and supp ρ ⊂ K  . Let L denote the linear differential operator of order 0 given by multiplication by ρ. Given a multi-index α with |α| = k, we have  D α = D α Djα for some jα ∈ {1, . . . , n} and some multi-index α  with |α  | = k − 1. For each function u ∈ D(), we get A(Djα (ρu)) = Djα L(Au) + [A, Djα L]u, where Djα L and [A, Djα L] are of order 1. Applying part (a) of Lemma 11.1.3, we get a constant C1 (K, k) > 0 independent of the choice of u and α such that 

D α u2L2 (K) = D α Djα (ρu)2L2 (K)   ≤ C(K, k − 1) · Djα (ρu)2L2 () + ADjα (ρu)2W k−2 ()   = C(K, k − 1) · Djα (ρu)2L2 (K  ) + ADjα (ρu)2W k−2 (K  )   ≤ C1 (K, k) · u2W k−1 (K  ) + Au2W k−1 (K  )

  ≤ C1 (K, k) · C(K  , k − 1) u2L2 () + Au2W k−2 ()  + Au2W k−1 (K  )

 ≤ C1 (K, k) · (C(K  , k − 1) + 1) · u2L2 () + Au2W k−1 () . The claim now follows.



11.5

Proof of the Regularity Theorem

543

Theorem 11.0.1 now follows immediately from the Sobolev lemma (Lemma 11.4.1) together with the following fact: Lemma 11.5.2 If the inequality (1) in Theorem 11.0.1 holds for some constant C > 0 and for every function u ∈ D(), then for every nonnegative integer k, we have   max(k−1,0) k () ⊂ Wloc (). u ∈ L2loc () | Adistr u ∈ Wloc Proof Given a nonnegative integer k and a function u ∈ L2loc () with Adistr u ∈ max(k−1,0) k Wloc (), we must show that u ∈ Wloc (). We proceed by induction on k. The case k = 0 is trivial, so let us assume that k > 0 and the claim holds for the k−1 Sobolev spaces of order < k. In particular, u ∈ Wloc (). In order to show that k (), we may assume without loss of generality that u has compact support u ∈ Wloc in . For given a compact subset of , we may choose a (cutoff) function ρ ∈ D() such that ρ ≡ 1 on the subset. Lemma 11.1.3 then gives Adistr (ρu) = ρ · Adistr u + [A, ρ]u ∈ W k−1 (), k−1 since v → ρv and [A, ρ] are operators of order 0 and Adistr u, u ∈ Wloc (). Thus ◦

we may assume that supp u ⊂ K ⊂ K ⊂  for some compact set K. Applying the (strong) Friedrichs lemma (Lemma 11.3.1), we get a sequence of functions in {uν } in D() such that supp uν ⊂ K for each ν and, as ν → ∞, uν − uW k−1 () = uν − uW k−1 (K) → 0 and Auν − Adistr uW k−1 () → 0. Note that for the convergence of the first sequence, we have applied part (d) of the lemma, while for the second sequence, we have applied part (f) to the operator D α A for each multi-index α with |α| ≤ k − 1, and we have also used the fact that α A (D α A)distr u = Ddistr distr u (Lemma 7.4.3). By Lemma 11.5.1, for some constant  C > 0 and for all positive integers μ and ν,   uμ −uν 2W k () = uμ −uν 2W k (K) ≤ C  · uμ −uν 2L2 () +Auμ −Auν 2W k−1 () . Therefore, since the right-hand side converges to 0 as μ, ν → ∞ and W k () is a  Hilbert space, we have u ∈ W k (), and the lemma follows. Exercises for Sect. 11.5 11.5.1 Let A be a linear differential operator of order 1 with C ∞ coefficients on an open set  ⊂ Rn . Assume that the inequality (1) in Theorem 11.0.1 holds for some constant C > 0 and for every function u ∈ D(). Assume also that the coefficients of A and their partial derivatives of all orders are bounded on . Prove that for every positive integer k, there is a constant Ck > 0 such that   u2W k () ≤ Ck · u2L2 () + Au2W k−1 () ∀u ∈ D().

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546 [GuiP] [HarR]

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Notation Index

A Adistr , 394, 463 α ∗ β, 478 A∗ , 394 Aut(X), 194 B B1 (M, A), 506, 520 B 1 (M, A), 513 1 (M, F), 503 BdeR B1 (M, A), 527 1 (M, A), 527 B ∂S, 415 {·, ·}(E,h) , 131 [D], 121 r (M,F) , [ · ]deR , 503 [ · ]HdeR [ · ]H q (X,E) , [ · ]Dol , 126 Dol [·, ·], 278, 534 [ · ]x0 , [ · ]π1 (X,x0 ) , 479 B(x; r), 378

D r , 439 D r (E), 111

d, 429, 433, 442 ∂, 47 ¯ 47, 125 ∂, ∂¯distr , 60, 139 D, D  , D  , 61, 136  , 62, 139 Ddistr Deck(ϒ), 492 deg D, 124 deg E, 124 2 , 526 ω , 82 (z0 ; R), 6 (z0 ; r, R), 6 ∗ (z0 ; R), 6 div( ), 123 div(s), 119 Div(X), 119

C C 0 , 383, 416 C k , 382, 421

cl(S), 378, 415 cos z, 22 cot z, 22 Cq (M, A), 505, 527 q C (M, A), 527 csc z, 22 C∗ , 6

E E , 116, 421 E (E), 104, 116 E p,q , 45, 116 E p,q (E), 111, 116 E r , 116, 439 E r (E), 111, 116 exp(z), 21 ez , 21

D D , 421 D (E), 104 D p,q , 45 D p,q (E), 111

F

, 177 Fσ , 377

T. Napier, M. Ramachandran, An Introduction to Riemann Surfaces, Cornerstones, DOI 10.1007/978-0-8176-4693-6, © Springer Science+Business Media, LLC 2011

549

550

Notation Index ·, ·L2p,q (E,h) , ·, ·L2p,q (E,ω,h) , ·, ·L2 (E,h) , 132 1 · L2p,q (ϕ) , · L2p,q (ω,ϕ) , · L2 (ϕ) , 54 1 · L2p,q (E,h) , · L2p,q (E,ω,h) , · L2 (E,h) , 132

G γ˙ , 434 γ − , 478

(U, E (E)), 104

(U, F ), 117

(U, M(E)), 104

(U, O(E)), 104 Gδ , 377 genus(X), 165 GL(n, F), 219

1

L2p,q (ϕ), L2p,q (ω, ϕ), L21 (ϕ), 55 L2p,q (E, h), L2p,q (E, ω, h), L21 (E, h), 133 M M, 33, 115 M(E), 104, 115

multp , 33 H H, 198 H1 (M, A), 506, 520 H 1 (M, A), 516 H : α ∼ β, 477 H1 (M, A), 527 H1 (M, A), 527 r (M), 503 HdeR 1 1 , HDol, , 140 HDol, L2 L2 ∩E q q HDol (X), HDol (X, E),

N n(γ ; z0 ), 201 O O , 4, 30, 115 OD (E), 122 O (E), 104, 115

(E), 111, 115 X , 46, 115 ordp f , 14, 34 ordp s, 104

126

Hom(·, ·), 407, 518 hq , 165 h∗ , 130 h ⊗ h , 130 hX , 77 h∗X , 81

P P1 , 28 P 1 (M, A), 512 ⊥, 397 ∗ , 429, 433, 438, 509, 511, 522, 524 ∗ , 126, 429, 433, 480, 509, 510, 522, 524 π1 (X, x0 ), 479 T M , (T M)C , 433 T ∗ M , (T ∗ M)C , 433 PSL(n, F), 219

I ιE S , 168 J Jac(X), 307 JF , 383, 450 Jθ,p , 307

Q QD (E), 122

K KX , 40 κs , 292

R rank G, 525 RP2 , 529

L λω , 457 p,q T ∗ X, 45 p,q T ∗ X ⊗ E, 110 r T ∗ M, r (T ∗ M)C , 437 r V , 409 θ , 307 log z, 21 Lp , Lp (X, μ), 379 p Lloc , 379 ·, ·L2p,q (ϕ) , ·, ·L2p,q (ω,ϕ) , ·, ·L2 (ϕ) , 54 1

S S, 378, 415 sE (·, ·), SE (·, ·), 167, 168 sec z, 22 #, 177 σ (ν) , 527 ◦

S, 378, 415 sin z, 22 SL(n, F), 219 ∗, ∗¯ , 178

Notation Index ∗¯ , ∗¯ # , 179 , 416 T t A, 393 tan z, 22 h , 136 ω , 73 ϕ , 64 ⊗, 409, 412 T M, (T M)C , 429, 432 T ∗ M, (T ∗ M)C , 429, 433 Tθ , 307 (T X)p,q , 39 (T ∗ X)p,q , 39

551 V VC , 409 V ∗ , 401, 407

W ∧, 409 W k , 532 Z Z1 (M, A), 505 Z 1 (M, A), 513 1 (M, F), 503 ZdeR 1 (M, A), 527 Z zζ , 22

Subject Index

A Abel–Jacobi embedding theorem, 192, 307 Abel’s theorem, 303 Accumulation point, 415 Almost complex structure, 311 integrable, 311 surface, 311 Almost everywhere, 447 Ample, 293 very, 293 Approximation of the identity, 391 Argument function, 21 principle, 52, 202, 203, 227, 335, 337, 338 Atlas, 419 C k , 420 holomorphic, 26 holomorphically equivalent, 26 line bundle, 102 holomorphically equivalent, 102 Automorphism, 31, 194 group of , 221 of C, 216 of H, 222 of P1 , 218–220 B Banach space, 378 Barrier, 344 Basis for a topology, 416 Behnke–Stein theorem, 89 Bessel’s inequality, 404

Biholomorphic, 18, 31 Biholomorphism, 18, 31 local, 18, 31 Bilinear function, 409, 412 skew-symmetric, 409 symmetric, 409 Bishop–Kodama localization theorem, 98 Bishop–Narasimhan–Remmert embedding theorem, 191, 279 Bolzano–Weierstrass property, 423 Borel set, 377 Boundary of a set, 415 operator ∂, 505, 527 C Canonical homology basis, 269 Canonical line bundle, 40, 105 degree of, 169 Cauchy integral formula, 6, 201, 202, 334 Cauchy–Pompeiu integral formula, 6 Cauchy–Riemann equation homogeneous, 4 inhomogeneous, 7, 25 L2 solution for line-bundle-valued forms of type (0, 0), 145 for line-bundle-valued forms of type (1, 0), 140 for scalar-valued forms of type (0, 0), 74 for scalar-valued forms of type (1, 0), 65 local solution, 7 Cauchy’s theorem, 6, 201, 202, 334

T. Napier, M. Ramachandran, An Introduction to Riemann Surfaces, Cornerstones, DOI 10.1007/978-0-8176-4693-6, © Springer Science+Business Media, LLC 2011

553

554 ˇ Cech 1-form, 513 exact, 513 line integral of, 513 cohomology, 512, 526 Chain rule, 384, 430 Change of variables formula, 386 Characteristic function, 376 Chordal Kähler form, 73, 76, 137 C ∞ approximation, 390, 535 Ck

atlas, 420 differential form, 439 line-bundle-valued, 111 function, 382, 421 map, 383, 421 section, 103 vector field, 435 Closed differential form d, 443 ∂, 47 ¯ 47, 125 ∂, set, 378, 415 sequentially, 423 Closure, 378, 415 Coboundary operator δ, 527 Cocycle relation, 90, 150, 154, 518 Coefficients of a differential form, 45, 439 of a vector field, 43, 435 Cohomology, 516 ˇ Cech, 512, 526 de Rham, 503 Dolbeault, 126 Dolbeault L2 , 140 singular, 504, 527 Commutator, 534 subgroup, 278 Compact, 416 locally, 418 relatively, 416 Complete orthonormal basis, 403 set, 403 Complex differentiability, 16, 17 Complex manifold 1-dimensional, 26 n-dimensional, 35 Complex torus, 28, 197

Subject Index Complexification, 409 Connected, 416 component, 416 locally, 416 Connecting homomorphism, 127 Connection canonical for scalar-valued forms, 61 in a line bundle, 135, 136 Continuous, 415 differential form line-bundle-valued, 111 section, 103 sequentially, 423 Continuous closed 1-form, 513 Convolution, 391 Coordinate transformation, 419 C k , 420 holomorphic, 26 Coordinates homogeneous, 290 local C k , 420 local holomorphic, 26 Cotangent bundle, 433 (0, 1), 39 (1, 0), 39 complexified, 433 holomorphic, 39 projections, 433 real, 433 map, 430, 433 space, 429 (0, 1), 39 (1, 0), 39 holomorphic, 39 Covering branched, 294–298 map, 483 space, 483 C ∞ , 483 C ∞ equivalent, 487 equivalent, 487 holomorphic, 192 holomorphically equivalent, 192 universal, 489–491 Critical point, 435 value, 435 Curvature, 64, 136 form, 64, 136 negative, 64, 137 nonnegative, 64, 137 nonpositive, 64, 137

Subject Index Curvature (cont.) of a Kähler form, 73 positive, 64, 137, 147, 160, 162 zero, 64, 137 Cycle, 505 D ¯ ∂-Cauchy integral formula, 6 De Rham cohomology, 503 pairing, 509, 522 theorem, 524, 525 Deck transformation, 492 Defining function, 119 local, 120 section, 119 Degree of a divisor, 124 of a line bundle, 124 Diffeomorphism, 383, 421 Differential, 383, 429, 433 Differential form, 439 line-bundle-valued, 111 negative, 452 negative part, 453 on a Riemann surface, 45–52 positive, 452 positive part, 453 sequence infimum, 453 limit inferior, 453 limit superior, 453 supremum, 453 Differential operator, 392, 462 Dirichlet problem, 338–350 Discrete, 415 Disjoint union, 28, 417, 421 Distributional ¯ 60, 139 ∂, D  , 62, 139 differential operator, 394, 463 solution, 394, 463 Divisor, 119–124 defining section associated to, 121 effective, 120 line bundle associated to, 121 linearly equivalent, 120 of a line bundle map, 123 of a section, 119 principal, 119 solution of, 119, 152, 303 weak solution of, 152, 303

555 Dolbeault cohomology, 126 class, 126 exact sequence, 128 L2 cohomology, 140 lemma, 50 Dominated convergence theorem, 378, 457 Dual basis, 407 line bundle, 107 space algebraic, 407 norm, 401 E Edge of a singular 2-simplex, 527 Embedding Abel–Jacobi, 307 C ∞ , 467 holomorphic into Cn , 279–290 into Pn , 291 Entire function, 4 Evenly covered, 483 Exact differential form d, 443 C k , 48, 125, 443 ∂, 48 ¯ 48, 125 ∂, locally, 48, 125, 443 Exact sequence Dolbeault, 128 of sheaves, 119 Exhaustion by sets, 427 function, 77, 427 strictly subharmonic, 82 Exponential function, 21, 22 real, 388 Exterior derivative, 442 Exterior product, 409 F Fatou’s lemma, 378, 457 Finite type, 277 topological, 277 Finiteness theorem, 164 Flat operator, 177 Formal adjoint, 394 transpose, 393 Fourier coefficients, 404 Friedrichs lemma, 390 strong version, 535

556 Fubini’s theorem, 380 Fundamental estimate for scalar-valued forms, 64 in a line bundle, 137 on an almost complex surface, 328 Fundamental group, 479 Fundamental theorem of algebra, 35 G Gap, 174 sequence, 174 generic, 175 hyperelliptic, 175 General linear group, 219 Genus, 165, 169 Germ, 115, 117, 428 Goursat’s theorem, 17 Gram–Schmidt orthonormalization process, 403 Group action free, 493 properly discontinuous, 493 quotient by, 493 H Hahn–Banach theorem, 402 Harmonic, 64, 339 1-form, 186 Hartogs–Rosenthal theorem, 98 Hausdorff, 416 Hilbert space, 399 Hodge conjugate star, 178 conjugate star flat operator, 179 conjugate star sharp operator, 179 decomposition ¯ 181 for ∂, for scalar-valued forms, 185 star, 178 Holomorphic atlas, 26 attachment, 35–38 of caps, 207–209 of tubes, 36, 241–263 function in C, 4 on a Riemann surface, 30 map, 30 into Cn , 279 into Pn , 291 into a complex torus, 306 local representation, 32 1-form, 46 line-bundle-valued, 111

Subject Index removal of tubes, 243–248 section, 103 vector field, 43 Holomorphically compatible local charts, 26 local trivializations, 102 Homeomorphism, 416 Homology, 506, 519 canonical basis for, 267–273 singular, 278, 507, 511, 520, 527 Homotopy, 477 Hurwitz’s theorem, 301 Hyperelliptic gap sequence, 175 involution, 300 point, 175 Riemann surface, 174, 299 Hyperplane bundle, 105 curvature, 137 Hermitian metric in, 129 I Identity theorem, 13, 31 Immersion C ∞ , 467 holomorphic into Cn , 279 into Pn , 291 into a complex torus, 306 Inner product Hermitian, 397 real, 397 Integral, 376 differentiation past, 384 of a differential form, 454 Interior, 378, 415 Inverse function theorem C ∞ , 384, 465 holomorphic, 18, 43 J Jacobi variety, 307 Jacobian determinant, 383, 450 Jordan curve, 228 theorem, 333 K Kähler form, 51 metric, 130 Kodaira embedding theorem, 191, 294 Koebe uniformization, 212

Subject Index L Lattice, 28, 306 period, 307 Laurent series, 15 Law of cosines, 398 Lifting, 483 holomorphic, 193 theorem, 484 Limit point, 415 Line bundle associated to a divisor, 121 dual, 107 Hermitian, 128 holomorphic, 103 homomorphism, 107 isomorphism, 107 map, 107 set-theoretic, 101 tensor product, 107 Line integral, 461 along a 1-chain, 506, 520 along a continuous path, 499 Linear functional, 407 bounded, 401 Liouville’s theorem, 35 Lipschitz, 384 Local chart, 419 C k , 420 holomorphic, 26 Logarithmic function, 21, 22 real, 388 Loop, 417 Lp norm, 379 space, 379 p Lloc differential form, 458 line-bundle-valued, 111 function, 379 section, 103 L2 inner product of functions, 399 of line-bundle-valued forms, 132 of scalar-valued forms, 54 norm of a function, 399 of a line-bundle-valued form, 132 of a scalar-valued form, 54 space of functions, 399 of line-bundle-valued forms, 132, 133 of scalar-valued forms, 55

557 M Manifold, 419 C k , 420 complex of dimension 1, 26 product, 420 real analytic, 420 smooth, 420 Maximum principle, 14, 31 for harmonic functions, 339 for subharmonic functions, 339 Mean value property, 11, 341 Measurable differential form, 448 line-bundle-valued, 111 function, 375, 447 map, 375, 447 section, 103 set, 375, 447 vector field, 448 Measure complete, 375 completion of, 376 counting, 376 for a nonnegative form, 457 Lebesgue, 377 outer, 377 positive, 375 space, 375 Mergelyan–Bishop theorem, 97 Meromorphic function, 33 1-form, 46 existence of, 67 line-bundle-valued, 111 section, 103 Metric Hermitian, 128 dual, 130 tensor product, 130 Kähler, 130 Riemannian, 475 Mittag-Leffler theorem, 88, 97, 160 for a line bundle, 148, 160 Möbius band, 449 transformation, 219 Mollification, 391 Mollifier, 391 Monotone convergence theorem, 378, 457 Montel’s theorem, 10, 72 Multiplicity, 33

558 N Neighborhood, 415 Net, 423 Nongap, 174 Nonorientable, 450 Nonseparating, 239 Norm, 378 complete, 378 for an inner product, 397 O One-point compactification, 418 Open mapping theorem, 14, 31 Riemann surface, 27 set, 378, 415 Order, 14, 34, 46, 104 of a pole, 34, 46 of a zero, 14, 34, 46 Orientable, 450 double cover, 371, 462 topological surface, 364 Orientation, 450 compatible, 450 equivalent, 450 in a vector space, 411 induced on a boundary, 459 preserving, 451 Oriented, 450 positively, 450 Orthogonal, 397 decomposition, 399 projection, 399 Orthonormal, 397 Osgood–Taylor–Carathéodory theorem, 206 P Parallelogram law, 398 Partition of unity, 427 Path, 417 C k , 422 connected, 418 locally, 418 piecewise C k , 422 Path homotopy, 477 Perron method, 338 Planar, 211 Poincaré duality, 187 lemma, 443 Poisson formula, 343 kernel, 343 Polar coordinates, 387

Subject Index Pole, 33 of a meromorphic 1-form, 46 of a meromorphic section, 103 simple, 34, 46, 104 Potential, 443 local, 443 Power function, 22 Power series, 12, 13 Primitive, 4 Principal part, 202 Product path, 478 rule, 430 topology, 417 Projection map, 102 Projective space, 290 Projectivized special linear group, 219 Proper map, 427 Pullback, 429, 433, 438, 509, 511, 522, 524 Pushforward, 429, 433, 480, 509, 510, 522, 524 Pythagorean theorem, 398 Q q-boundary, 527 q-chain, 505, 526 q-coboundary, 527 q-cochain, 527 q-cocycle, 527 Quotient map, 417 space, 417 topology, 417 R Rank, 525 Real projective plane, 529 Realification, 408 Regular value, 435 Regularity first-order, 532 ¯ 63, 324 for ∂, for ∂/∂ z¯ , 10 Regularization, 391 Representation of a section, 103 Residue, 53 theorem, 53, 201, 203, 227, 335, 338 Reverse path, 478 Riemann mapping theorem, 191 in the plane, 204 Riemann sphere, 28 Riemann surface, 26 open, 27 Riemann–Hurwitz formula, 299 Riemann–Roch formula, 165, 169

Subject Index Riemann’s extension theorem, 11, 31 Rouché’s theorem, 52, 202, 203, 227, 335, 338 Runge approximation theorem, 91 for a line bundle, 151 approximation theorem with poles, 92, 97 for a line bundle, 151 open set holomorphically, 96 topologically, 77 S Sard’s theorem, 263, 276, 331, 449 Schönflies’ theorem, 333 proof of, 350–356 Schwarz inequality, 398 Second countable, 416 Section, 103 C k , 103 continuous, 103 defining associated to a divisor, 121 holomorphic, 103 p Lloc , 103 measurable, 103 meromorphic, 103 of a sheaf, 117 Separable, 403 Separating, 239 Sequence Cauchy, 378 convergent, 378, 423 divergent, 378, 423 limit of, 378, 423 Serre duality, 169 mapping, 168 pairing, 168 Sharp operator, 177 Sheaf, 115–119 constant, 117 definition, 117 isomorphism, 117 mapping, 117 of holomorphic 1-forms, 115 of holomorphic sections, 115 of meromorphic 1-forms, 115 of meromorphic sections, 115 skyline, 122 skyscraper, 116 Sheet interchange, 300 σ -algebra, 375 Simple function, 376 Simply connected, 480

559 Singular 2-simplex, 526 Sobolev lemma, 541 Sobolev space, 532 Special linear group, 219 Stalk, 115, 117, 428 Standard 2-simplex, 526 Stereographic projection, 28 Stokes’ theorem, 460 Stratification, 286 Strictly subharmonic, 64 Strictly superharmonic, 64 Strong deformation retraction, 363 Structure C k , 420 complex analytic, 26 holomorphic, 26 Subharmonic, 64, 339 Submanifold, 422 Submersion, 467 Superharmonic, 64 Support, 426 Surface, 419 C k , 420 Riemann, 26 smooth, 420 T Tangent bundle, 432 (0, 1), 39 (1, 0), 39 complexified, 432 holomorphic, 39 projections, 433 real, 432 map, 429, 433 space, 429 (0, 1), 39 (1, 0), 39 complexified, 429 holomorphic, 39 real, 429 vector, 429 (0, 1), 39 (1, 0), 39 complex, 429 holomorphic, 39 real, 429 to a path, 434 Tautological bundle, 108 Taylor’s formula, 385 Tensor product, 409, 412 line bundle, 107 Tietze extension theorem, 348

560 Topological hull, 77 extended, 81 Topological space, 415 Topology, 415 product, 417 subspace, 415 Torsion, 226, 332, 371, 521 free, 226, 332, 371, 521 Torus complex, 28, 306 real, 30 Total space, 102 Triangulation, 282, 286, 370 Trigonometric functions, 22 real, 387 Trivialization global, 102 local, 102 Type (p, q), 43, 110 U Upper half-plane, 198 V Vanishing derivatives, 444 Vanishing theorem, 145, 148, 163

Subject Index Vector field, 435 Volume form, 453 W Weak compactness, 403 convergence, 403 solution, 394, 463 Wedge product, 409 Weierstrass gap, 174 gap theorem, 174 nongap, 174 point, 176 theorem, 151 weight, 175 Weight function, 54 Winding number, 201, 226 Wronskian, 175 Z Zero of a holomorphic function, 14, 33 of a meromorphic 1-form, 46 of a meromorphic function, 34 of a meromorphic section, 103 simple, 14, 33, 34, 46, 104