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 . . . . . . .
. . . . . . .
. . . . . . .
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 . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
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 . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . .
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
. . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . .
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 . . . . . . . . . . . .
. . . .
. . . .
. . . .
. . . .
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
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
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 . . . . . . . . . . . . . . . . . .
. . . . . . .
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 . . . . . . . . . . . . . . . . . . . . .
. . . .
. . . .
. . . .
. . . .
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
. . . . . . . .
. . . . . . . .
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.
16
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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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(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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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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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
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¯ 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 .
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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
30
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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 .
2.2 Holomorphic Functions and Mappings
31
(c) A bijective holomorphic mapping : X → Y with holomorphic inverse is called a biholomorphism (or a biholomorphic mapping). If such a biholomorphism exists, then we say that X and Y are biholomorphically equivalent (or simply biholomorphic). For Y = X, 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
2.2 Holomorphic Functions and Mappings
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
2.3 Holomorphic Attachment
37
Fig. 2.4 Holomorphic attachment of a tube
let A0 ≡ (0; 1/R0 , 1) ⊂ T and A1 ≡ (0; 1, R1 ) ⊂ T . Thus we get a 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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39
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
2.4 Holomorphic Tangent Bundle
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).
2.5 Differential Forms on a Riemann Surface
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¯
2.5 Differential Forms on a Riemann Surface
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 .
52
¯ for Scalar-Valued Forms Riemann Surfaces and the L2 ∂-Method
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 α L2
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 α L2
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 α L2
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 α L2 (S,ω,ϕ) ≡ = α/ω L2 (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 α L2p,q (S,ω,ϕ) simply as α S,ω,ϕ , α S,ϕ , α S,ω , α ω,ϕ , α ϕ , α ω , α L2 (S,ω,ϕ) , α L2 (S,ϕ) , α L2 (ω,ϕ) , α L2 (ϕ) , α L2 (ω) , or α . We will also use the analogous simplified notation for α, βL2p,q (S,ω,ϕ) , for α L2p,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 α L2p,q (S,ω,0) simply as α L2p,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 α L2 (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 α L2 (S,ϕ) < ∞, where again, we identify any two elements that are equal almost everywhere. (b) For ω a nonnegative measurable differential form of type (1, 1) that is defined on S, L20,0 (S, ω, ϕ) consists of all equivalence classes of measurable functions α on S with α L2 (S,ω,ϕ) < ∞, where we identify any two elements that are equal almost everywhere. (c) For ω a positive measurable differential form of type (1, 1) that is defined on S, L21,1 (S, ω, ϕ) consists of all equivalence classes of measurable differential forms α of type (1, 1) on S with α L2 (S,ω,ϕ) < ∞, where we identify any two elements that are equal almost everywhere. (d) When no mention of a weight function appears in the context and there is no danger of confusion, then the weight function will be assumed to be ϕ ≡ 0 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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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 ϕ L2 (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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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 | ≤ C f 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 ≤ R f 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.
62
¯ 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 i ϕ satisfies i ϕ ≥ 0 (respectively, i ϕ > 0, i ϕ = 0, i ϕ ≤ 0, i ϕ < 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. i ϕ = 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 i uve ¯ −ϕ . D u, D vL2 (X,ϕ) = D u, D vL2 (X,ϕ) + In particular,
D u 2L2 (X,ϕ)
= D u 2L2 (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 u 2L2 (X,ϕ) = D u 2L2 (X,ϕ) + u 2L2 (X,i ,ϕ) ≥ u 2L2 (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
i uve ¯ −ϕ X
i uve ¯ −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) +
i uve ¯ −ϕ
+
X
= −i
X
i uve ¯ −ϕ X
i uve ¯ −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 = i ϕ = 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
α L2 (X,ϕ) ≤ β L2 (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
66
2
¯ 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 ≡ β L2 (X\Z,i ,ϕ) = β/(i ) L2 (X\Z,i ,ϕ) . For each function f ∈ D(X), the Schwarz inequality and the fundamental estimate (Proposition 2.8.2) give β¯ −ϕ 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 α L2 (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 i ϕ ≥ 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
α L2 (X,ϕ) ≤ β L2 (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 β L2 (X\Z,i ϕ ,ϕ) ≤ β L2 (X\Z,ρω,ϕ) if i ϕ ≥ 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 i ϕ ≥ 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
α L2 (X,ϕ) ≤ C −1 β L2 (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 = i ϕ ≥ 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 α L2
0,1 (X,ϕ)
≤ β L2
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
2
¯ 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 dz L2 (D) ≤ Cm . Remark The value for the constant Cm that we will obtain is far from optimal. Lemma 2.10.2 For any choice of constants a, b, c ∈ R with a < 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 i ρ = i ψ > 0 and hence i ϕ = χ (ρ) · i ψ + 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, i ϕ0 = i ψ0 = 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 ),i ϕ
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
70
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¯ for Scalar-Valued Forms Riemann Surfaces and the L2 ∂-Method
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 i ϕ = 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 β L2 (Y \Z,i ϕ ,ϕ) = βm L2 (∗ (0;R0 ),i ϕ
= Am < ∞.
0 ,ϕ0 )
Therefore, by Theorem 2.9.1, there exists a C ∞ (1, 0)-form α on Y such that ¯ =β ∂α
α L2 (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 · α 2L2 (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) = − α L2 (X\D) = α L2 (X\D,ϕ) ≤ Am ≤ Cm and that zθ − z−m+1 dz L2 (D) ≤ zτ − z−m+1 dz L2 (D) + zα L2 (D) ≤ Bm + sup e(ϕ0 +log |ζ |
· α L2 (D,ϕ)
≤ Bm + sup e(ϕ0 +log |ζ |
· α L2 (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 θ L2 (X\) ≤ C; (iii) The meromorphic 1-form θ − ∗ θ0 on has at worst a simple pole at p; and (iv) We have f0 () · θ − f0 () · ∗ θ0 L2 () ≤ C.
2.11 Radó’s Theorem on Second Countability The goal of this section is the following important theorem: Theorem 2.11.1 (Radó) Every Riemann surface X is second countable; that is, the topology in X admits a countable basis. The proof considered here is similar to that in [Sp]. Lemma 2.11.2 Every nonempty open subset of a Riemann surface X has only countably many connected components. Proof Fixing a point p ∈ and a connected neighborhood U of p in , and applying Theorem 2.10.1, we get a nontrivial (i.e., not everywhere zero) holomorphic 1-form θ on X \ {p} with θ L2 (X\U ) < ∞. If {i }i∈I is the family of (distinct)
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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 ) ≥ θ 2L2 ( ) = i∈J
i
i
i∈J
Therefore ∞ > i∈I θ 2L2 ( ) , 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 i ω = 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 ≡ i e−ϕ ω = i ω + i ϕ ≥ 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
α L2 (X,ω,ϕ) ≤ β L2 (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 i ω + i ϕ ≥ 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
α L2 (X,ω,ϕ) ≤ β L2 (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 i ω + i ϕ ≥ 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
α L2 (X,ω,ϕ) ≤ C −1 β L2 (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, i ϕ−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) = β 2L2 (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,i ϕ−log τ ,ϕ−log τ ) = β L2 (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 α 2L2 (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 ) ≤ β 2L2 (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¯ ≤ C α ◦ −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 i ω = 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 ≡ i e−ϕ ω = i ω + i ϕ ≥ 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 α L2 (X,ω,ϕ) ≤ β L2 (X\Z,ϕ+log ρ)
(cf. Exercises 2.9.3 and 2.9.4).
2.13 Topological Hulls and Chains to Infinity As suggested by Theorem 2.9.1 and its applications (see also Theorem 3.9.1 and its applications), it is useful to have available a large supply of C ∞ strictly subharmonic functions on an (open) Riemann surface, i.e., C ∞ functions ϕ for which i ϕ > 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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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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Topological Hulls and Chains to Infinity
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 i ϕ ≥ 0; (ii) On X \ , ϕ > τ and i ϕ ≥ 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 = i ϕ /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+|i ω /ω|, we get a positive C ∞ strictly subharmonic (exhaustion) function ϕ on X such that i ω + i ϕ ≥ ω i ϕ ≥ ω
and and
β L2 (X,ϕ) < ∞ if p = 0, β L2 (X,ω,ϕ) < ∞ if p = 1.
Corollary 2.12.6 and Corollary 2.9.3 now give the desired differential form α.
In Sects. 2.15 and 2.16, we will consider other applications to open Riemann surfaces. Exercises for Sect. 2.14 2.14.1 Prove that Theorem 2.14.1 and Theorem 2.14.4 are equivalent. 2.14.2 Let ϕ be a real-valued C ∞ function on a Riemann surface X. (a) Prove that if χ is a real-valued C ∞ function on R, then ¯ χ(ϕ) = χ (ϕ) χ(ϕ) + χ (ϕ)∂ϕ ∧ ∂ϕ.
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 i χ(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 i ϕ = 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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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 + |i ω /ω| + |ω ρ0 | on Y , we get a positive C ∞ strictly subharmonic (exhaustion) function ρ1 on X such that i ω + i ρ0 + i ρ1 ≥ ω on Y and such that β L2 (Y,ρ0 +ρ1 ) < ∞. Corollary 2.12.6 now provides a C ∞ function α on Y such that ¯ =β ∂α
and
α L2 (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
α L2 (Wp \{p},(i/2)dzp ∧d z¯ p )
= α L2 (Wp \{p},(i/2) dzp ∧d z¯ p ,ρ0 ) ≤ C α L2 (Wp \{p},ω,ρ0 +ρ1 ) ≤ C α L2 (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 i ω 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 i ω + i ρ0 ≥ 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 i ω + i ϕν = i ω + i ρ0 + νi ρ1 ≥ ω, ¯ = β and Corollary 2.12.6 provides a C ∞ function α on X such that ∂α ¯ 2 2 α 2L2 (X,ω,ϕ ) ≤ β 2L2 (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 α 2L2 (
2 ,ω,ρ0 )
= α 2L2 (
2 ,ω,ϕν )
≤ α 2L2 (X,ω,ϕ ) . ν
Therefore, by choosing ν sufficiently large, we can make α L2 (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 ) = α L2 ( ,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 i ϕ ≥ 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 i α ≥ 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 i ω 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 i ω + i ρ0 ≥ 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 i ω + i ϕν = i ω + i ρ0 + νi ρ1 ≥ ω, ¯ = β and Corollary 2.12.6 provides a C ∞ function α on Y such that ∂α ¯ 22 ¯ 2 2 α 2L2 (Y,ω,ϕ ) ≤ β 2L2 (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 α 2L2 (
2 ,ω,ρ0 )
= α 2L2 (
2 ,ω,ϕν )
≤ α 2L2 (Y,ω,ϕ ) . ν
Therefore, by choosing ν sufficiently large and then choosing δ > 0 sufficiently small (depending on ν), we can make α L2 (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 ) = α L2 (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
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2.16.7
2.16.8
2.16.9
2.16.10
97
(i) On Y \ , ϕ > 0 and i ϕ ≥ 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 i ω + i ϕ ≥ 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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123
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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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
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= (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 ,
