Lectures on riemann surfaces, otto forster

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Series QA333.F6713 515'.223 81-9054 AACR2 Title of the Original German Edition: Riemannsche Fliichen, Heidelberger Taschenbiicher 184, Springer-Verlag, Heidelberg, 1977 © 1981 by Springer-Verlag New York Inc Sottcover reprint of the hardcover 1st edition 1981 All rights reserved No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag, 175 Fifth Avenue, New York, New York 10010, U.s.A (Corrected fourth printing, 1999) ISBN-13: 978-1-4612-5963-3 001: 10.1007/978-1-4612-5961-9 e-ISBN-13: 978-1-4612-5961-9 Contents Preface Chapter Covering Spaces §1 §2 §3 §4 §5 §6 §7 §8 §9 §1O §11 The Definition of Riemann Surfaces Elementary Properties of Holomorphic Mappings Homotopy of Curves The Fundamental Group Branched and Unbranched Coverings The Universal Covering and Covering Transformations Sheaves Analytic Continuation Algebraic Functions Differential Forms The Integration of Differential Forms Linear Differential Equations Chapter Compact Riemann Surfaces §12 §13 §14 §15 §16 §17 §18 Cohomology Groups Dolbeault's Lemma A Finiteness Theorem The Exact Cohomology Sequence The Riemann-Roch Theorem The Serre Duality Theorem Functions and Differential Forms with Prescribed Principal Parts vii 1 10 13 20 31 40 44 48 59 68 81 96 96 104 109 118 126 132 146 v Contents VI §19 §20 §21 Harmonic Differential Forms Abel's Theorem The Jacobi Inversion Problem 153 159 166 Chapter Non-compact Riemann Surfaces 175 §22 §23 §24 §25 §26 §27 §28 §29 §30 §31 175 185 190 196 201 206 214 219 228 231 The Dirichlet Boundary Value Problem Countable Topology Weyl's Lemma The Runge Approximation Theorem The Theorems of Mittag- Leffler and Weierstrass The Riemann Mapping Theorem Functions with Prescribed Summands of Automorphy Line and Vector Bundles The Triviality of Vector Bundles The Riemann-Hilbert Problem Appendix 237 A B 237 238 Partitions of Unity Topological Vector Spaces References 243 Symbol Index 247 Author and Subject Index 249 Preface This book grew out of lectures on Riemann surfaces which the author gave at the universities of Munich, Regensburg and Munster Its aim is to give an introduction to this rich and beautiful subject, while presenting methods from the theory of complex manifolds which, in the special case of one complex variable, turn out to be particularly elementary and transparent The book is divided into three chapters In the first chapter we consider Riemann surfaces as covering spaces and develop a few basics from topology which are needed for this Then we construct the Riemann surfaces which arise via analytic continuation of function germs In particular this includes the Riemann surfaces of algebraic functions As well we look more closely at analytic functions which display a special multi-valued behavior Examples of this are the primitives of holomorphic i-forms and the solutions of linear differential equations The second chapter is devoted to compact Riemann surfaces The main classical results, like the Riemann-Roch Theorem, Abel's Theorem and the Jacobi inversion problem, are presented Sheaf cohomology is an important technical tool But only the first cohomology groups are used and these are comparatively easy to handle The main theorems are all derived, following Serre, from the finite dimensionality of the first cohomology group with coefficients in the sheaf of holomorphic functions And the proof of this is based on the fact that one can locally solve inhomogeneous CauchyRiemann equations and on Schwarz' Lemma In the third chapter we prove the Riemann Mapping Theorem for simply connected Riemann surfaces (or Uniformization Theorem) as well as the main theorems of Behnke-Stein for non-compact Riemann surfaces, i.e., the Runge Approximation Theorem and the Theorems of Mittag-Leffler and Weierstrass This is done using Perron's solution of the Dirichlet problem Vll Vlll Preface and Malgrange's method of proof, based on Weyl's Lemma, of the Runge Approximation Theorem In this chapter we also complete the discussion of Stein's Theorem, begun in Chapter 1, concerning the existence of holomorphic functions with prescribed summands of automorphy and present Rbhrl's solution of the Riemann-Hilbert problem on non-compact Riemann surfaces We have tried to keep the prerequisites to a bare minimum and to develop the necessary tools as we go along However the reader is assumed to be familiar with what would generally be covered in one semester courses on one complex variable, on general topology and on algebra Besides these basics, a few facts from differential topology and functional analysis have been used in Chapters and and these are gathered together in the appendix Lebesgue integration is not needed, as only holomorphic or differentiable functions (resp differential forms) are integrated We have also avoided using, without proof, any theorems on the topology of surfaces The material presented corresponds roughly to three semesters of lectures However, Chapters and presuppose only parts of the preceding chapters Thus, after §§1, and (the definitions of Riemann surfaces, sheaves and differential forms) the reader could go directly to Chapter And from here, only §§ 12-14 are needed in Chapter to be able to handle the main theorems on non-compact Riemann surfaces The English edition includes exercises which have been added at the end of every section and some additional paragraphs in §§8, 17 and 29 As well, the terminology concerning coverings has been changed Thanks are due to the many attentive readers of the German edition who helped to eliminate several errors; in particular to G Elencwajg, who also proposed some of the exercises Last but not least we would like to thank the translator, B Gilligan, for his dedicated efforts Munster May, 1981 O FORSTER Addendum to Fourth Corrected Printing For the second and fourth printing a number of misprints and errors have been corrected I wish to thank B Gilligan, B Elsner and O Hien for preparing lists of errata April 1999 O FORSTER CHAPTER Covering Spaces Riemann surfaces originated in complex analysis as a means of dealing with the problem of multi-valued functions Such multi-valued functions occur because the analytic continuation of a given holomorphic function element along different paths leads in general to different branches of that function It was the idea of Riemann to replace the domain of the function with a many sheeted covering of the complex plane If the covering is constructed so that it has as many points lying over any given point in the plane as there are function elements at that point, then on this" covering surface" the analytic function becomes single-valued Now, forgetting the fact that these surfaces are "spread out" over the complex plane (or the Riemann sphere), we get the notion of an abstract Riemann surface and these may be considered as the natural domain of definition of analytic functions in one complex variable We begin this chapter by discussing the general notion of a Riemann surface Next we consider covering spaces, both from the topological and analytic points of view Finally, the theory of covering spaces is applied to the problem of analytic continuation, to the construction of Riemann surfaces of algebraic functions, to the integration of differential forms and to finding the solutions of linear differential equations §1 The Definition of Riemann Surfaces In this section we define Riemann surfaces, holomorphic and meromorphic functions on them and also holomorphic maps between Riemann surfaces Riemann surfaces are two-dimensional manifolds together with an additional structure which we are about to define As is well known, an 240 Appendix then d: E x E > IR is a metric on E which induces the same topology as the semi-norms Pn, n EN A closed vector subspace FeE of a Frechet space is also a Frechet space If E;, i E J, is a countable family of Frechet spaces, then Il; I E; with the product topology is also a Frechet space E B.4 A typical example of a Frechet space is the vector space 0(X) of holomorphic functions on an open set X c C with the topology of uniform convergence on compact subsets This topology is induced by the seminorms PK, where PK(f):= sup IJ(x) I, XEK as K runs through the compact subsets of X This topology is also defined by countably many semi-norms PK , where Kn, n EN, is any sequence of compact subsets of X with UnEr, = x It: B.S Banach Spaces, Hilbert Spaces A complete normed vector space is called a Banach space Thus a Banach space is a Frechet space whose topology is defined by a single norm This is usually denoted I II A Hilbert space E is a Banach space whose norm is derived from a scalar product ( , ): E x E > C, J i.e., IIxil = (x, x) If A is a vector subspace of a Hilbert space E, then its orthogonal complement A~ :={y E E: (y, x) = for every x E A} is a closed vector subspace of E If A itself is closed, then E = A EB A~ B.6 Theorem of Banach Suppose E and Fare Frixhet spaces andJ: E a continuous linear surjective mapping Then J is open > F is B.7 Corollary Suppose E and F are Banach spaces andf: E > F is a continuous linear surjective mapping Then there exists a constant C > such that Jor every y E F there is an x E E with J(x) = y and IIxil ~ CilYII· Let U:= {x E E: IIxil < I} Since by the Theorem of BanachJis open, there exists an r > such that PROOF J(U) Let C := 2/B Now suppose y A.:= IIYII > O The element Y1 =:J V:={y E F is given If y := E F: IIyll < s} = 0, choose x = O Otherwise, (l/A.C)y lies in V and thus there exists Xl E U 241 B Topological Vector Spaces withf(xd = Yl Then for x := 1.Cx , one hasf(x) = yand Ilxll = 1.Qxlll :$ 1.C = Qyll· o B.8 Hahn-Banach Theorem Suppose E is a locally convex topological vector space, Eo c: E is a vector subspace and CfJo: Eo + C is a continuous linear functional Then there exists a continuous linear functional CfJ: E + C such that CfJ IEo = CfJo' B.9 Corollary Suppose E is a locally convex topological vector space and A c: B c: E are vector subspaces If every continuous linear functional CfJ: E + C such that CfJ IA = satisfies CfJ IB = 0, then A is dense in B ° PROOF Suppose A is not dense in B Then there exists b o E B such that b o ¢ A Let Eo:= A EB Cb o and define a linear functional CfJo: Eo + C by CfJo(a + 1.b o):= for a E A, E C It is easy to check that CfJo is continuous By the Hahn-Banach Theorem CfJo extends to a continuous linear functional CfJ: E + C Then CfJ IA = 0, but CfJ IB ;¢; 0, which is a contradiction B.10 Compact Mappings A linear mapping 1/1: E + F between two topological vector spaces E and F is called compact or completely continuous, if there exists a neighborhood U of zero in E such that I/I(U) is relatively compact in F In particular, a compact linear mapping is continuous Example Suppose X is an open subset of C and Y ~ X is a relatively compact open subset of X Then the restriction mapping fJ: @(X) + @(Y), is compact One sees this as follows Since U := {f E @(X): Y is compact in X, it follows that su~ I f(x) I < I} XEY is a neighborhood of zero in @(X) By Montel's Theorem the set M :={g E @(Y): sup Ig(y)1 :$ 1} YE Y is compact in @(Y) The claim now follows since fJ(U) c: M B.11 Theorem of L Schwartz Suppose E and Fare Frechet spaces and CfJ, 1/1: E + F are continuous linear mappings such that CfJ is surjective and 1/1 is compact Then the image of the mapping CfJ -1/1: E + F has finite codimension in F For the proof see [60] References (a) Complex Analysis in One Variable Ahlfors, L V.: Complex Analysis New York: McGraw-Hill, 1966 Behnke, H., and Sommer, F.: Theorie der analytischen Funktionen einer komplexen Veriinderlichen 3rd Ed Berlin-Heidelberg-New York: Springer-Verlag 1965 Cartan, H.: Elementary Theory of Analytic Functions of One or Several Complex Variables Reading, MA: Addison-Wesley, 1963 Hurwitz, A., and Courant, R.: Funktionentheorie (with an appendix by H Rohrl) 4th Ed Berlin-Heidelberg-New York: Springer-Verlag, 1964 Lang, S.: Complex Analysis Reading, MA: Addison-Wesley, 1977 The books of Behnke-Sommer and Hurwitz-Courant also contain large sections on Riemann surfaces (b) Riemann Surfaces 10 Ahlfors, L V., and Sario, L.: Riemann Surfaces Princeton, NJ: University Press, 1960 11 Chevalley, c.: Introduction to the Theory of Algebraic Functions of One Variable Amer Math Soc., 1951 12 Farkas, H M., and Kra, 1.: Riemann Surfaces New York-Heidelberg-Berlin: Springer-Verlag, 1980 13 Guenot, J., and Narasimhan, R: Introduction a la Theorie des Surfaces de Riemann Monographies de I'Enseignement Mathematique No 23, Geneve, 1976 14 Gunning, R c.: Lectures on Riemann surfaces, Princeton Math Notes (1966) 15 Gunning, R c.: Lectures on vector bundles over Riemann surfaces, Princeton Math Notes (1967) 16 Gunning, R c.: Lectures on Riemann surfaces: Jacobi varieties, Princeton Math Notes 12 (1972) 243 244 References 17 Nevanlinna, R.: Uni[ormisierung Berlin-Heidelberg-New York: SpringerVerlag, 1953 18 Pfluger, A., Theorie der Riemannschen F/(ichen, Berlin-Heidelberg-New York: Springer-Verlag, 1957 19 Serre, J-P.: Groupes Algebriques et Corps de Classes, Paris: Hermann, 1959 20 Springer, G.: Introduction to Riemann Surfaces Reading, MA: Addison-Wesley 1957 21 Weyl, H.: The Concept of a Riemann Surface Reading, MA: Addison-Wesley, 1955 The book of Hermann Weyl, which first appeared in 1913 (in German), was the first modern presentation of the theory of Riemann surfaces It is still well worth reading and contains many references to the older literature, particularly that of the nineteenth century The books of Chevalley and Serre treat the theory of Riemann surfaces from the algebraic standpoint (c) Complex Analysis in Several Variables, Complex Manifolds 30 Andreian Cazacu, C: Theorie der Funktionen mehrerer komplexer Veranderlichen Berlin: Dtsch VerI Wiss., 1975 31 Grauert, H and Remmert, R.: Theory of Stein spaces New York-HeidelbergBerlin: Springer-Verlag, 1979 32 Gunning, R C, and Rossi, H.: Analytic Functions of Several Complex Variables Englewood Cliffs, NJ: Prentice-Hall, 1965 33 Hirzebruch, F.: Topological Methods in Algebraic Geometry Berlin-HeidelbergNew York: Springer-Verlag, 1966 34 Hormander, L.: An Introduction to Complex Analysis in Several Variables 2nd Ed Amsterdam: North-Holland, 1973 35 Wells, R 0.: Differential Analysis on Complex Manifolds Englewood Cliffs, NJ: Prentice-Hall, 1973 2nd Ed New York-Heidelberg-Berlin: Springer-Verlag, 1980 (d) Topology, Differentiable Manifolds, Functional Analysis 40 Brocker, T., and Jiinich, K.: Einfuhrung in die Differentialtopologie Heidelberger Taschenbticher, Bd 143 Berlin-Heidelberg-New York: Springer-Verlag, 1973 41 Godement, R.: Topologie Aigebrique et Theorie des Faisceaux Paris: Hermann, 1958 42 Massey, W S.: Algebraic Topology: an Introduction New York-HeidelbergBerlin: Springer-Verlag, 1967 43 Narasimhan, R.: Analysis on Real and Complex Manifolds Amsterdam: NorthHolland, 1968 44 Schaeffer, H.: Topological Vector Spaces New York: MacMillan, 1966 45 Schubert, H.: Topologie Stuttgart: Teubner, 1964 46 Seifert, H., and Threlfall, W.: Lehrbuch der Topologie, Leipzig: Teubner 1934 Reprinted Chelsea, 1947 (English trans: Academic Press, 1980) 47 Treves, F.: Topological Vector Spaces, Distributions and Kernels New YorkLondon: Academic Press, 1967 48 Warner, F.: Foundations q[ Differentiable Manifolds and Lie Groups Glenview, IL-London: Scott-Foresman, 1971 References 245 (e) Special Topics 50 Ahlfors, L V.: The complex analytic structure of the space of closed Riemann surfaces, in Analytic Functions Princeton NJ, University Press, 1960, 45-66 51 Behnke, H., and Stein, K.: Entwicklungen analytischer Funktionen auf Riemannschen Fllichen Math Ann 120 (1948) 430-461 52 Bieberbach, L.: Theorie der gewohnlichen Differentialgleichungen auffunktionentheoretischer Grundlage dargestellt 2nd Ed Berlin-Heidelberg-New York: Springer-Verlag, 1965 53 Cartan, H.: Varietes analytiques complexes et cohomologie Colloque sur les fonctions de plusieurs variables, CBRM: Bruxelles, 1953,41-55 54 Florack, Herta: Reguliire und meromorphe Funktionen auf nicht geschlossenen Riemannschen Fliichen Schriftenreihe Math [nst Univ Munster, (1948) 55 Malgrange, B.: Existence et approximation des solutions des equations aux derivees partielles a convolution Annales de rInst Fourier (1955/56) 271-355 56 Meis, T.: Die minimale Bliitterzahl der Konkretisierungen einer kompakten Riemannschen Fliiche Schriftenreihe Math Inst Univ Munster 16 (1960) 57 Rohrl, H.: Das Riemann-Hilbertsche Problem der Theorie der linearen Differentialgleichungen Math Ann 133 (1957) 1-25 58 Ross, S L.: Differential Equations New York: Blaisdell, 1964 59 Serre, J-P.: Applications de la theorie generale a divers problemes globaux Seminaire H Cartan, E.N.S Paris 1951/52, Expose 20 Reprinted Benjamin, 1967 60 -: Deux theoremes sur les applications completement continues Seminaire H Cartan, E.N.S Paris 1953/54, Expose 16 Reprinted Benjamin, 1967 61 -: Quelques problemes globaux relatifs aux varietes de Stein Colloque sur les fonctions de plusieurs variables, CBRM: Bruxelles, 1953, 57-68 62 Stein, K.: Analytische Funktionen mehrerer komplexer Veriinderlichen zu vorgegebenen Periodizitiitsmodulin und das zweite Cousinsche Problem Math Ann 123, (1951) 201-222 63 Thimm, W.: Der Weierstrapsche Satz der algebraischen Abhiingigkeit von Abelschen Funktionen und seine Verallgemeinerungen in Festschrift zur Gedachtnisfeier fur Karl Weierstrass 1815-1965 (Hrsg H Behnke, K Kopfermann) Koln and Opladen: Westdeutscher Verlag, 1966 Symbol Index General Notation set of natural N numbers (including 0) ring of integers field of real numbers = {x E IR: x'" O} ={XEIR:X~O} = C* Re(z) Im(z) [pI [a, b] {x E IR: x :::; O} ={XEIR:X>O} field of complex numbers ={ZEC:Z",O} real part of the complex number z imaginary part of the complex number z = C v {oo}, the Riemann sphere N-dimensional projective space 142 = {x E IR: a:::; x :::; b}, where a, b E IR with a:::;b ]a, b] IR: a < x :::; b}, where a, b E IR with = {x E coefficients in the ring R Aii;B A is a relatively vA compact subset of B the boundary of A Sheaves and Function Spaces '?1 sheaf of continuous functions 41 g sheaf of differentiable functions 60 63, gl,O, gO, 63, 1,"(2) (0 t* n If jl* Harm l 1j>1 a