Graduate Texts in Mathematics 38 Editorial Board F W Gehring P R Halmos Managing Editor c C Moore H Grauert K Fritzsche Several Complex Variables Springer-Verlag New York Heidelberg Berlin H Grauert K Fritzsche Mathematischen Institut der Universitiit Bunsenstrasse - 34 Gottingen Federal Republic of Germany Mathematischen Institut der Universitat Bunsenstrasse 3-5 34 Gottingen Federal Republic of Germany Editorial Board P R Halmos F W Gehring C C Moore Managing Editor University of Michigan Department of Mathematics Ann Arbor, Michigan 48104 University of California at Berkeley Department of Mathematics Berkeley, California 94720 University of California Department of Mathematics Santa Barbara, California 93106 AMS Subject Classifications: 32-01, 32A05, 32A07, 32AIO, 32A20, 32BIO, 32CIO, 32C35, 32D05, 32DlO, 32ElO Library of Congress Cataloging in Publication Data Grauert, Hans, 1930Several complex variables (Graduate texts in mathematics; 38) Translation of Einftihrung in die Funktionentheorie mehrerer Veranderlicher Bibliography: p 201 Includes index Functions of several complex variables I Fritzsche, Klaus, joint author II Title III Series QA331.G69 515'.94 75-46503 All rights reserved No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag © 1976 by Springer-Verlag Inc Softcover reprint of the hardcover 1st edition 1976 ISBN-13: 978-1-4612-9876-2 e-ISBN-13: 978-1-4612-9874-8 DOl: 10.1007/978-1-4612-9874-8 Preface The present book grew out of introductory lectures on the theory offunctions of several variables Its intent is to make the reader familiar, by the discussion of examples and special cases, with the most important branches and methods of this theory, among them, e.g., the problems of holomorphic continuation, the algebraic treatment of power series, sheaf and cohomology theory, and the real methods which stem from elliptic partial differential equations In the first chapter we begin with the definition of holomorphic functions of several variables, their representation by the Cauchy integral, and their power series expansion on Reinhardt domains It turns out that, in l:ontrast to the theory of a single variable, for n ~ there exist domains G, G c en with G c G and G "# G such that each function holomorphic in G has a continuation on G Domains G for which such a G does not exist are called domains of holomorphy In Chapter we give several characterizations of these domains of holomorphy (theorem of Cartan-Thullen, Levi's problem) We finally construct the holomorphic hull H(G} for each domain G, that is the largest (not necessarily schlicht) domain over en into which each function holomorphic on G can be continued The third chapter presents the Weierstrass formula and the Weierstrass preparation theorem with applications to the ring of convergent power series It is shown that this ring is a factorization, a Noetherian, and a Hensel ring Furthermore we indicate how the obtained algebraic theorems can be applied to the local investigation of analytic sets One achieves deep results in this connection by using sheaf theory, the basic concepts of which are discussed in the fourth chapter In Chapter V we introduce complex manifolds and give several examples We also examine the different closures of en and the effects of modifications on complex manifolds Cohomology theory with values in analytic sheaves connects sheaf theory v Preface with the theory of functions on complex manifolds It is treated and applied in Chapter VI in order to express the main results for domains ofholomorphy and Stein manifolds (for example, the solvability of the Cousin problems) The seventh chapter is entirely devoted to the analysis of real differentiability in complex notation, partial differentiation with respect to z, z, and complex functional matrices, topics already mentioned in the first chapter We define tangential vectors, differential forms, and the operators d, d', d" The theorems of Dolbeault and de Rham yield the connection with cohomology theory The authors develop the theory in full detail and with the help of numerous figures They refer to the literature for theorems whose proofs exceed the scope of the book Presupposed are only a basic knowledge of differential and integral calculus and the theory of functions of one variable, as well as a few elements from vector analysis, algebra, and general topology TheI book is written as an introduction and should be of interest to the specialist and the nonspecialist alike Gottingen, Spring 1976 H Grauert K Fritzsche vi Contents Chapter I Holomorphic Functions Power Series Complex Differentiable Functions The Cauchy Integral Identity Theorems Expansion in Reinhardt Domains Real and Complex Differentiability Holomorphic Mappings 10 15 17 21 26 Chapter II Domains of Holomorphy The Continuity Theorem Pseudo convexity Holomorphic Convexity The Thullen Theorem Holomorphically Convex Domains Examples Riemann Domains over en Holomorphic Hulls 29 29 35 39 43 46 51 54 62 Chapter III The Weierstrass Preparation Theorem The Algebra of Power Series The Weierstrass Formula 68 68 71 vii Contents Convergent Power Series Prime Factorization Further Consequences (Hensel Rings, Noetherian Rings) Analytic Sets 74 78 81 84 Chapter IV Sheaf Theory 99 99 105 110 113 Sheaves of Sets Sheaves with Algebraic Structure Analytic Sheaf Morphisms Coherent Sheaves Chapter V Complex Manifolds Complex Ringed Spaces Function Theory on Complex Manifolds Examples of Complex Manifolds Closures of en 119 119 124 128 144 Chapter VI Cohomology Theory Flabby Cohomology The Cech Cohomology Double Complexes The Cohomology Sequence Main Theorem on Stein Manifolds 150 150 158 163 167 174 Chapter VII Real Methods Tangential Vectors Differential Forms on Complex Manifolds Cauchy Integrals Dolbeault's Lemma Fine Sheaves (Theorems of Dolbeault and de Rham) List of symbols Bibliography Index Vlll 179 179 185 188 191 193 199 201 203 CHAPTER I Holomorphic Functions Preliminaries Let e be the field of complex numbers If n is a natural number we call the set of ordered n-tuples of complex numbers the n-dimensional complex number space: Each component of a point E en can be decomposed uniquely into real and imaginary parts: Zv = Xv + iyv' This gives a unique 1-1 correspondence between the elements (Zl' ,zn) of en and the elements (Xl' , Xn, Yb' , Yn) of 1R 2n, the 2n-dimensional space of real numbers en is a vector space: addition of two elements as well as the multiplication of an element of en by a (real or complex) scalar is defined componentwise As a complex vector space en is n-dimensional; as a real vector space it is 2n-dimensional It is clear that the IR vector space isomorphism between en and [R2n leads to a topology on en: For = (Zl' , zn) = (Xl + iY1, , Xn + iYn) E en let 11311: 11311*: I2 I = Ct1 ZkZkY = Ct1 (xl + YDY , = Norms are defined on metrics given by max k= 1, ,n en (lxkl,IYkl)· by 31-+11311 and 31-+11311*, with corresponding dist(3b 32): dist*(3b 32): = = 1131 - 3211, 1131 - 3211*· I Holomorphic Functions In each case we obtain a topology on en which agrees with the usual topology for ~2n Another metric on en, defined by 131: = max IZkl and dist'(31) 32): = 131 - 321, induces the usual topology too k= • • n A region Been is an open set (with the usual topology) and a domain an open, connected set An open set G c en is called connected if one of the following two equivalent conditions is satisfied: a For every two points 31, 32 E G there is a continuous mapping cp: [0, 1J -+ en with cp(o) = 31> cp(l) = 32, and cp([O, IJ) c G b If B , B2 C G are open sets with Bl u B2 = G, Bl n B2 = and Bl =F 0, then B2 = 0· Definition Let Been be a region, 30 E B a point The set CB (30): = {3 E B:3 and 30 can be joined by a path in B} is called the component of 30 in B Remark Let Been be an open set Then: a b c d For each E B, CB (3) and B - CB (3) are open sets For each E B, CB (3) is connected From CB (3tl n CB (32) =F it follows that CB (3i) = CB (32)' B = CB (3) U 3eB e If G is a domain with E G c B, it follows that G c C B (3) f B has at most countably many components The proof is trivial Finally for 30 E en we define: U.(30): U:(30): U~(30): = {3 E en:dist(3, 30) < e}, = {3 E en:dist*(3, 30) < e}, = {3 E en:dist'(3, 30) < e} Power Series Let M be a subset of en A mapping f from M to function on M The polynomials ml, ,mn P(3) = L e is called a complex av, z'1' ' z~n, Vl, ,Vn=O are particularly simple examples, defined on all of en In order to simplify notation we introduce multi-indices: let Vi' ~ i ~ n, be non-negative integers and let = (ZI,' ,zn) be a point of en Then we define: Ivl: = n LVi' i= With this notation a polynomial has the form P(3) n Zii n 3v : = i= = m L v=O a v3v• Dolbeault's Lemma For w E H we also have Ch(B)(w) = _1_ h 2ni r f2(z) dz!\ dz = _1_ w 2ni JB z - r f2(z) dz JB-H z - !\ W dZ ' the integrand is continuous and bounded on B - H, as well as holomorphic with respect to w From the theory of parametric integrals it follows that Ch(1!IH is continuously differentiable and (Ch~lIH)w = ° Therefore glH is continuously differentiable and (gIH)z = flH D Remark If Bee, B* c [Rn are regions, Bee B open and f : B x B* ~ C arbitrarily often differentiable, then it follows from the theory of para .Integra Istat h C h(B)' f(z,x) dZ!\ d' metnc f wIth Chf(B) (w, x): = - Z IS arb'12m < L B Z - w trarily often differentiable on B x B*, and (Ch(B») (w x) f x,,, = _1_ 2ni r f~,(z, x) dz z _ JB W !\ dZ (Ch(B»)_ 'f w = f Dolbeault's Lemma (Dolbeault's lemma): Let Kv c C be compact sets for v = 1, ,n, Uv open neighborhoods of K v , K: = KI x K n , U: = U I x X Un Moreover, let