Graduate Texts in Mathematics 213 Editorial Board S Axler F.W Gehring K.A Ribet Springer New York Berlin Heidelberg Barcelona Hong Kong London Milan Paris Singapore Tokyo Graduate Texts in Mathematics IO 11 12 13 14 IS 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 TAKEtrrIlZARiNG Introduction to Axiomatic Set Theory 2nd ed OXTOBY Measure and Category 2nd ed ScHAEFER Topological Vector Spaces 2nded Hn.TONISTAMMBACH A Course in Homological Algebra 2nd ed MAc LANE Categories for the Working Mathematician 2nd ed HUGlIBSIPtPER Projective Planes SERRE A Course in Arithmetic TAKEtrrIlZARiNG Axiomatic Set Theory HUMPHREYS Introduction to Lie Algebras and Representation Theory COHEN A Course in Simple Homotopy Theory CONWAY Functions of One Complex Variable I 2nd ed BEALS Advanced Mathematical Analysis ANDERSONlFtJu.EJt Rings and Categories of Modules 2nd ed GoLUBITSKy/GUIU.EMIN Stable Mappings and Their Singularities BERBERIAN Lectures in Functional Analysis and Operator Theory WINTER The Structure of Fields ROSENBLAtT Random Processes 2nd ed HALMos Measure Theory HALMos A Hilbert Space Problem Book 2nd ed HUSEMOU.ER Fibre Bundles 3rd ed HUMPHREYS Linear Algebraic Groups BARNESIMAcK An Algebraic Introduction to Mathematical Logic GREUB Linear Algebra 4th ed HOLMES Geometric Functional Analysis and Its Applications HEwrrr/SmOMBERG Real and Abstract Analysis MANES Algebraic Theories KElLEy General Topology ZARIsKIlSAMUEL Commutative Algebra Vol.l ZARIsKIlSAMUEL Commutative Algebra V01.ll JACOBSON Lectures in Abstract Algebra I Basic Concepts JACOBSON Lectures in Abstract Algebra D Linear Algebra JACOBSON Lectures in Abstract Algebra m Theory of Fields and Galois Theory HIRsoi Differential Topology 34 SPITZER Principles of Random Walle 2nded 35 ALExANDERlWERMER Several Complex Variables and Banach Algebras 3rd ed 36 KEu.sy/NAMIOKA et al Linear Topological Spaces 37 MONK Mathematical Logic 38 GRAUERTIFRrrzsam Several Complex Variables 39 ARVESON An Invitation to c*-Algebras 40 KEMENY/SNEUiKNAPP Denumerable Markov Chains 2nd ed 41 APosroL Modular Functions and Dirichlet Series in Number Theory 2nded 42 SERRE Linear Representations of Finite Groups 43 GIU.MANlJERISON Rings of Continuous Functions 44 KENDIG Elementary Algebraic Geometry 45 LoEVE Probability Theory I 4th ed 46 LotiVE Probability Theory D 4th ed 47 MOISE Geometric Topology in Dimensions and 48 SAOfs/WU General Relativity for Mathematicians 49 GRUENBERGlWEIR Linear Geometry 2nd ed 50 EDWARDS Fermat's Last Theorem 51 KuNGENBERG A Course in Differential Geometry 52 HARTSHORNE Algebraic Geometry 53 MANJN A Course in Mathematical Logic 54 GRAVERIWATKINS Combinatorics with Emphasis on the Theory of Graphs 55 BRoWNIPEARCY Introduction to Operator Theory I: Elements of Functional Analysis 56 MASSEY Algebraic Topology: An Introduction 57 CRoWElllFox Introduction to Knot Theory 58 KOBUTZ p-adic Numbers p-adic Analysis and Zeta-Functions 2nd ed 59 LANG Cyclotomic Fields 60 ARNOLD Mathematical Methods in Classical Mechanics 2nd ed 61 WHITEHEAD Elements of Homotopy Theory 62 KARGAPOwvIMERuJAKOV Fundamentals of the Theory of Groups 63 BOu.DBAS Graph Theory (continued after index) Klaus Fritzsche Hans Grauert From Holomorphic Functions to Complex Manifolds With 27 Illustrations Springer Klaus Fritzsche Bergische Universitiit Wuppertal GauBstra6e 20 0-42119 Wuppertal Gennany Klaus.Fritzsche@math.uni-wuppertal.de Editorial Board S Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA axler@sfsu.edu Hans Grauert Mathematisches Institut Georg-August-Universitiit G6ttingen Bunsenstra6e 3-5 0-37073 G6ttingen Gennany F.W Gehring Mathematics Department East Hall University of Michigan Ann Arbor, MI 48109 USA fgehring@math.lsa umich.edu K.A Ribet Mathematics Department University of California, Berkeley Berkeley, CA 94720-3840 USA ribet@math.berkeley.edu Mathematics Subject Classification (2000): 32-01, 32Axx, 32005, 32Bxx, 32Qxx, 32E35 Library of Congress Cataloging-in-Publication Data Fritzsche Klaus From holomorphic functions to complex manifolds Klaus Fritzsche, Hans Grauert p cm - (Graduate texts in mathematics; 213) Includes bibliographical references and indexes ISBN 978-1-4419-2983-9 ISBN 978-1-4684-9273-6 (eBook) DOl 10.1007/978-1-4684-9273-6 I Complex manifolds III Series QA331.7 F75 2002 515'.98 -4c21 Holomorphic functions I Grauert Hans 1930- n Title 2001057673 © 2002 Springer-Verlag New York Inc All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York Inc • 175 Fifth Avenue New York NY 10010, 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 654 SPIN 10857970 Springer-Verlag New York Berlin Heidelberg A member of BertelsmannSpringer Science+Business Media GmbH This reprint has been authorized by Springer-Verlag (Berlin/Heidelberg/New York) for sale in the Mainland China only and not for export therefrom Preface The aim of this book is to give an understandable introduction to the theory of complex manifolds With very few exceptions we give complete proofs Many examples and figures along with quite a few exercises are included Our intent is to familiarize the reader with the most important branches and methods in complex analysis of several variables and to this as simply as possible Therefore, the abstract concepts involved with sheaves, coherence, and higher-dimensional cohomology are avoided Only elementary methods such as power series, holomorphic vector bundles, and one-dimensional cocycles are used Nevertheless, deep results can be proved, for example the Remmert-Stein theorem for analytic sets, finiteness theorems for spaces of cross sections in holomorphic vector bundles, and the solution of the Levi problem The first chapter deals with holomorphic functions defined in open subsets of the space en Many of the well-known properties of holomorphic functions of one variable, such as the Cauchy integral formula or the maximum principle, can be applied directly to obtain corresponding properties of holomorphic functions of several variables Furthermore, certain properties of differentiable functions of several variables, such as the implicit and inverse function theorems, extend easily to holomorphic functions In Chapter II the following phenomenon is considered: For n 2: 2, there are pairs of open subsets H c Peen such that every function holomorphic in H extends to a holomorphic function in P Special emphasis is put on domains G c en for which there is no such extension to a bigger domain They are called domains of holomorphy and have a number of interesting convexity properties These are described using plurisubharmonic functions If G is not a domain of holomorphy, one asks for a maximal set E to which all holomorphic functions in G extend Such an "envelope of holomorphy" exists in the category of Riemann domains, i.e., unbranched domains over en The common zero locus of a system of holomorphic functions is called an analytic set In Chapter III we use Weierstrass's division theorem for power series to investigate the local and global structure of analytic sets Two of the main results are the decomposition of analytic sets into irreducible components and the extension theorem of Remmert and Stein This is the only place in the book where singularities play an essential role Chapter IV establishes the theory of complex manifolds and holomorphic fiber bundles Numerous examples are given, in particular branched and unbranched coverings of en, quotient manifolds such as tori and Hopf manifolds, projective spaces and Grassmannians, algebraic manifolds, modifications, and toric varieties We not present the abstract theory of complex spaces, but provide an elementary introduction to complex algebraic geometry For example, we prove the theorem of Chow and we cover the theory of divi- vi Preface sors and hyperplane sections as well as the process of blowing up points and submanifolds The present book grew out of the old book of the authors with the title Seveml Complex Variables, Graduate Texts in Mathematics 38, Springer Heidelberg, 1976 Some of the results in Chapters I, II, III, and V of the old book can be found in the first four chapters of the new one However, these chapters have been substantially rewritten Sections on pseudoconvexity and on the structure of analytic sets; the entire theory of bundles, divisors, and meromorphic functions; and a number of examples of complex manifolds have been added Our exposition of Stein theory in Chapter V is completely new Using only power series, some geometry, and the solution of Cousin problems, we prove finiteness and vanishing theorems for certain one-dimensional cohomology groups Neither sheaf theory nor methods are required As an application Levi's problem is solved In particular, we show that every pseudoconvex domain in en is a domain of holomorphy Through Chapter V we develop everything in full detail In the last two chapters we deviate a bit from this principle Toward the end, a number of the results are only sketched We carefully define differential forms, higherdimensional Dolbeault and de Rham cohomology, and Kahler metrics Using results of the previous sections we show that every compact complex manifold with a positive line bundle has a natural projective algebraic structure A consequence is the algebraicity of Hodge manifolds, from which the classical period relations are derived We give a short introduction to elliptic operators, Serre duality, and Hodge and Kodaira decomposition of the Dolbeault cohomology In such a way we present much of the material from complex differential geometry This is thought as a preparation for studying the work of Kobayashi and the papers of Ohsawa on pseudoconvex manifolds In the last chapter real methods and recent developments in complex analysis that use the techniques of real analysis are considered Kahler theory is carried over to strongly pseudoconvex subdomains of complex manifolds We give an introduction to Sobolev space theory, report on results obtained by J.J Kohn, Diederich, Fornress, Catlin, and Fefferman (a-Neumann, subeUiptic estimates), and sketch an application of harmonic forms to pseudoconvex domains containing nontrivial compact analytic subsets The Kobayashi metric and the Bergman metric are introduced, and theorems on the boundary behavior of biholomorphic maps are added Prerequisites for reading this book are only a basic knowledge of calculus, analytic geometry, and the theory of functions of one complex variable, as well as a few elements from algebra and general topology Some knowledge about Riemann surfaces would be useful, but is not really necessary The book is written as an introduction and should be of interest to the specialist and the nonspecialist alike a Preface vii We are indebted to many colleagues for valuable suggestions, in particular to K Diederich, who gave us his view of the state of the art in a-Neumann theory Special thanks go to A Huckleberry, who read the manuscript with great care and corrected many inaccuracies He made numerous helpful suggestions concerning the mathematical content as well as our use of the English language Finally, we are very grateful to the staff of Springer-Verlag for their help during the preparation of our manuscript Wuppertal, Gottingen, Germany Summer 2001 Klaus Fritzsche Hans Grauert Contents Preface v I 1 Holomorphic FUnctions Complex Geometry Real and Complex Structures Hermitian Forms and Inner Products Balls and Polydisks Connectedness Reinhardt Domains Power Series Polynomials Convergence Power Series Complex Differentiable Functions The Complex Gradient Weakly Holomorphic Functions, Holomorphic Functions The Cauchy Integral The Integral Formula Holomorphy of the Derivatives The Identity Theorem The Hartogs Figure Expansion in Reinhardt Domains Hartogs Figures The Cauchy-Riemann Equations Real Differentiable Functions Wirtinger's Calculus The Cauchy-Riemann Equations Holomorphic Maps The Jacobian Chain Rules Tangent Vectors The Inverse Mapping Analytic Sets Analytic Subsets Bounded Holomorphic Functions Regular Points Injective Holomorphic Mappings 9 11 14 14 15 16 17 17 19 22 23 23 25 26 26 28 29 30 30 32 32 33 36 36 38 39 41 x II Contents Domains of Holomorphy l The Continuity Theorem General Hartogs Figures Removable Singularities The Continuity Principle Hartogs Convexity Domains of Holomorphy Plurisubharmonic Functions Subharmonic Functions The Maximum Principle Differentiable Subharmonic Functions Plurisubharmonic Functions The Levi Form Exhaustion Functions Pseudoconvexity Pseudoconvexity The Boundary Distance Properties of Pseudoconvex Domains Levi Convex Boundaries Boundary Functions The Levi Condition Affine Convexity A Theorem of Levi Holomorphic Convexity Affine Convexity Holomorphic Convexity The Cartan-Thullen Theorem Singular Functions Normal Exhaustions Unbounded Holomorphic Functions Sequences Examples and Applications Domains of Holomorphy Complete Reinhardt Domains Analytic Polyhedra Riemann Domains over Riemann Domains Union of Riemann Domains The Envelope of Holomorphy Holomorphy on Riemann Domains Envelopes of Holomorphy Pseudoconvexity Boundary Points Analytic Disks en 43 43 43 45 47 48 49 52 52 55 55 56 57 58 60 60 60 63 64 64 66 66 69 73 73 75 76 78 78 79 80 82 82 83 85 87 87 91 96 96 97 99 100 102 Index of Notation natural numbers, integers, etc = Nu {O} ={xE.R:x>O} = C - {O} imaginary unit A V* v' F(V) real dual space of a real (or complex) vector space complex dual space of a complex vector space C-valued real linear forms on a complex vector space Mp,q(k) Mn(k) GLn(C) , AB (or A· B) k-valued matrices with p rows and q columns k-valued square matrices of order n general linear group product of matrices z, zt vector z = (ZI' ,zn) and transposed vector standard Euclidean scalar product in am Euclidean norm and distance standard Hermitian scalar product sup-norm (or maximum-norm) (z IW)m Ilzll, dist(z, w) (zlw) Izl Tz open disk (in C) around a with radius r unit disk (0) open ball with radius r around z polydisk in en around z with polyradius r poiydisk with polyradius r = (r, , r) unit polydisk Pf{O) natural projection onto absolute value space polydisk pn{o, r(z» distinguished boundary (torus) distinguished boundary Tn{o, r(z» UccV U lies relatively compact in V Cj(z) Cauchy integral of f 'Vf(z) 'Vf(z) 'Vxf(z) 'Vyf(z) holomorphic gradient (/%1 (z), ,1%" (z» antiholomorphic gradient (hi (z), , h" (z» = (f"'1 (z), , I"'n (z» = (fYl (z), , IYn (z» OrCa) o Br(z) pn(z, r) P~(z) pn r: en -+ 1" Pz Tn(z,r) : T -+ C at z E pn 382 Index of Notation D V fez) Df(z) (&f)z(w) (8f)z(w) (df)z Jr(z) JR,r(z) g(G) O(G) higher partial derivative total real derivative of f at z = fZl (Z)W1 + + fZn (z)w n = h, (Z)W1 + + hn (z)w n = (&f)z + (8f)z complex (or holomorphic) Jacobian matrix real Jacobian matrix space of smooth functions on G space of holomorphic functions on G t=(z,w) Tz Q(O) tangent vector at z with direction w tangent space at z tangent vector (0:(0),0:'(0» N(It,.··, fq) A (or Reg(A» Sing(A) rkz(It,···, fq) common zero set of It, , fq set of regular points of A set of singular points of A rank of the Jacobian JUl> ,Jq) (pn,H) Euclidean Hartogs figure general Hartogs figure boundary of an analytic disk St = CPt (D) (P,H) bSt connected component of M containing z boundary distance (for a domain G) Lev(f)(z, w) Hz (&G) Hess(f)(z, w) ~ Levi form Lv,Jl fzvz" (z)WVwJl complex (or holomorphic) tangent space of &G Hessian Lv,Jl fxvx" (z)WVwJl KG holomorphically convex hull of K in G 91 -< 92 the Riemann domain 91 is contained in 92 $-hull of and envelope of holomorphy set of accessible boundary points of G H§(9), H(9) G ring of formal power series = {J m E c[zB : IIflit < oo} ring of convergent power series maximal ideal in H n set of nonzero element, set of units in an ideal I set of monic polynomials with coefficients in I germ of the holomorphic function f at z Index of Notation w(u, z) 6""D", pseudopolynomial algebraic derivative of w discriminant and discriminant set of w dimz(A), dim(A) local and global dimension of an analytic set D(w) Riemann sphere C U {oo} tangent space of a manifold tangential map XxzY Kx, reX) fiber product of manifolds canonical bundle and tangent bundle on X r(u, V) C(U,V) space of holomorphic sections in a bundle V space of smooth sections in V V$W, V®W V', pk j*V, VjW Nx(Y) Ox,O:X Whitney sum and tensor product of bundles complex dual bundle of V, tensor power of P pullback and quotient bundle normal bundle of Y in X trivial bundles X x C and X x C· Ci(OU, V) Zi(OU, V) Bi(OU, V) Hi(OU, V) Hi(X, V) Cech co chains with values in V Cech cocycles Cech coboundaries Cech cohomology group with values in V cohomology group with values in V singular homology and cohomology of X Ox ring of germs of holomorphic functions polar set of a meromorphic function set of meromorphic functions on X divisor of a meromorphic function line bundle associated to the divisor Z Picard group and set of all divisors = O(U, V) Hq(X), Hq(X) P(m) A(X) div(m) [Z] Pic(X), 9J(X) rn = cnjr pn St(k, n), Gk,n pI : Gk,n -+ pN 0(1),0(-1) n-dimensional complex torus complex projective space Stiefel manifold and Grassmannian Plucker embedding hyperplane bundle and tautological bundle Osgood space pI x X pI (m times) 383 384 Index of Notation e(h) = eeL) Chern class of a cocycle h (of a line bundle L) I\r F(X) bundle of r-dimensional differential forms bundle of holomorphic p-forms on X r-forms forms of type (p, q) on U wedge product n~ p;r(u) p;p,q(U) 1{)1\1/J dcp, 01{), 81{) d C total, holomorphic and antiholomorphic differential = i (8 - 0) (such that ddc = 2i08 ) fa f«()/{( - z) d( 1\ d( Chj(z) = (1/{27ri)) Hp,q(X) Dolbeault group of type (p, q) on X rth de Rham group of X WH fundamental form of the Hermitian metric H group of projective unitary transformations dV *1{) volume element Hodge star operator (p;r{x) + p;2n-r(x)) inner product f x I{) 1\ "* 'I/J, with "* 1/J = *'I/J Hr(x) PU(n + 1) (I{), 1/J) current associated to a form 1/J current associated to a submanifold M space of currents of degree 2n - r adjoint operators of d, 0, and = (1/2)1{) 1\ w, with Kahler form adjoint operator of L ~ o Jf'r{x) Jf'p,q{X) f3r(X), f3p ,q(X) W real Laplacian (eM + ~d) complex Laplacian {o{} + {}o) space of harmonic r-forms on X space of harmonic (p, q)-forms Betti numbers zero section of the bundle F generalized Siegel upper halfplane L~,q{n) dom(T) N Hilbert space closure of "'M(n) domain of the operator T Neumann operator N : L~,q(n) + dom(O) Index of Notation rapidly decreasing functions Sobolev space of index s tangential Sobolev norm N(n) Nebenhiille of n Fx· pseudo-differential metric on X Bergman kernel Bergman projection = Jd'n,O(O) n O(n) K(~,w) P: L2(n) ~£ AOO(n) 385 Index I-complete, 267 I-convex, 267 Abel's lemma, 12 abelian variety, 351 absolute space, accessible boundary point, 101 acyclic covering, 185 adjoint operator, 330 adjunction formula, 215 affine algebraic, 217 affine algebraic manifold, 224 affine algebraic set, 224 affine part, 210 algebraic derivative, 125 analytic disk, 47, 102 analytic polyhedron, 85 analytic set, 36, 160 analytic tangent, 275 analytically dependent, 320 antiholomorphic form, 303 antiholomorphic gradient, 28 Ascoli - lemma of, 23 atlas, 155 automorphism, 226 ball, Banach algebra, 105 Bergman kernel, 370 Bergman metric, 372 Bergman projection, 370 Bertini - theorem of, 225 Betti number, 190 biholomorphic, 33, 159 blowup, 237 boundary conditions, 357 boundary distance, 54, 58 boundary function, 64 boundary operator, 189 bracket, 332 branch locus, 227 branch point, 128 branched covering, 128, 227 branched domain, 229 bubble method, 283 canonical bundle, 174 CaratModory distance, 371 Cart an-Thullen - theorem of, 77 Cauchy integral, 17 Cauchy-Riemann equations, 29 tech cohomology, 185 center - of a modification, 236 chain rule, 32 Chern class, 256 Chow - theorem of, 217 closed map, 226 closure - of en, 245 - of a manifold, 244 coboundar~ 184, 187 coboundary operator, 183 cochain, 183, 187 cocycle, 183, 187 codimension, 151 cohomologous, 174 cohomology group, 184, 187 commutator, 332 compact exhaustion, 163 compact map, see completely continuous compactification - one-point, 153 compatibility condition, 173 compatible coordinate systems, 154 complete hull, 24 completely continuous, 285 completely singular, 49 complex I-form, 261 complex covariant r-tensor, 261 complex differentiable, 14 complex gradient, 15 complex manifold, 155 complex n-space, complex structure, 1, 155 complexification, cone, 216 conical set, 216 conjugation, connected, 6, 88 continuity of roots, 134 continuity principle, 47 continuity theorem, 43 388 Index contractible space, 189 convergence - compact, 12 - domain of, 13 convergent series - of numbers, 10 convex, 8, 66, 74 - strictly, 67 convex hull, 73 coordinate system, 154 Cousin-I distribution, 253 Cousin-II distribution, 254 covariant tangent vector, 261 critical locus, 227 cross section, 176 cuboid, 276 - distinguished, 276 current, 328 - harmonic, 336 8-Neumann conditions, 357 de Rham group, 311 de Rham isomophism, 312 de Rham sequence, 310 decomposition theorem - of Hodge-Kodaira, 339 defining function, 64 degree - of a hypersurface, 220 derivation, 33, 164 derivative - directional, 16 - of a holomorphic map, 30 - partial, 15 - real, 26 derivatives - Wirtinger, 28 diagonal, 162 differential, 260, 300 differential form, 297, 300 dimension, 161 - local, 150 - of a submanifold, 39 - of an analytic set, 142 - of an irreducible analytic set, 141 Dirichlet's principle, 52 discrete map, 228 discriminant, 127 discriminant set, 128 distinguished boundary, divisor, 198 - of a meromorphic function, 198 - of a meromorphic section, 201 Dolbeault group, 311 Dolbeault sequence, 310 Dolbeault's lemma, 309 Dolbeault's theorem, 312 domain, domain of existence, 97 domain of holomorphy, 49, 99 - weak, 49 dual bundle, 178 duality theorem - of Poincare, 337 - of Serre-Kodaira, 338 effective divisor, see positive divisor effective group operation, 172 elementary symmetric polynomial, 126 elliptic operator, 337 embedded-analytic - irreducible component, 135 - set, 135 embedding, 214 embedding dimension, 152 embedding theorem, 145, 258 energy form, 359 envelope of holomorphy, 87, 97 equivalence - of bundles, 174 - of systems of transition functions, 174· Euclidean coordinates, 322 Euclidean domain, 125 Euler sequence, 223 exact sequence, 223 exceptional set, 236 exhaustion function, 58 exterior algebra, 298 exterior derivative, 301 factorial monoid, 125 faithful, see effective group operation Fefferman - theorem of, 369 fiber bundle, 173 fiber bundle isomorphism, 174 fiber metric, 341 fiber product, 171 filter basis, 100 finite map, 227 finite module, 121 fixed point, 172 form of type (p, q), 298 frame, 177, 313 Frechet space, 285 Frechet topology, 285 Index free group operation, 172 Fubini metric, 319 fundar"nental form, 315 Gauss's lemma, 117 general linear group, 172 genus, 232 - of a Riemann surface, 340 geodesic coordinates, 316 geometric series, 11 germ, 123, 192 globally generated, 176 gluing, 170 - of fiber bundles, 173 - of manifolds, 170 Godeaux surface, 225 good topological space, 189 graph,161 - of a meromorphic function, 242 Grassmannian, 212, 225 greatest common divisor, 117 Griffiths negative, 341 Griffiths positive, 342 harmonic form, 335, 356 harmonic function, 52 Hartogs - theorem of, 307 Hartogs convex, 48, 99 Hartogs figure - Euclidean, 25 - general, 43, 99 Hartogs triangle, 365 Hausdorff space, 153 Hensel's lemma, 120 ,Hermitian form, 3, 261, 321 Hermitian metric, 314 Hessian, 67 H n ,108 Hodge manifold, 346 Hodge metric, 346 Hodge star operator, 323 Holder continuous, 367 holomorphic form, 303 holomorphic function, 13, 96, 257 - on a complex manifold, 156 - on an analytic set, 134 holomorphic gradient, 28 holomorphic map, 30, 158 holomorphically convex, 75, 100, 251 holomorphically convex hull, 75 holomorphically separable, 251 holomorphically spreadable, 251 homogeneous, homogeneous coordinates, 209 Hopf bundle, 238 Hopf manifold, 208, 225 Hopf's a-process, 237 hyperbolic manifold, 368 hyperelliptic involution, 234 hyperelliptic surface, 232 hyperplane at infinity, 210 hyperplane bundle, 220, 221 hyperquadric, 219 hypersurface, 127, 220 - analytic, 160 ideal, 108 identity theorem for analytic sets, 142 for holomorphic functions, 22 for meromorphic functions, 197 for power series, 21 - on manifolds, 156 immersion, 168 implicit function theorem, 34 incidence set, 237 indeterminacy - set of, 243 inner product - Euclidean, - Hermitian, integral class, 346 integral domain, 113 inverse mapping theorem, 33 irreducible analytic set, 141 irreducible component, 131 irreducible element, 117 isomorphism of complex manifolds, 159 - of vector bundles, 177 Jacobian - complex, 30 - real,31 Kahler manifold, 316 Kahler metric, 316 kernel form, 370 Kobayashi metric, 368 Kodaira's embedding theorem, 344 Krull topology, 110 Kugelsatz, 44, 307 Laplacian, 334 Leray covering, see acyclic covering Levi condition, 66 389 390 Index Levi convex, 66 - strictly, 66 Levi form, 57, 263 Levi's extension theorem, 197 Levi's problem, 51 Lie group, 171 lifted bundle, 180 lifting - of a differential form, 304 limit - of a filter, 101 line bundle, 175 - associated to a hypersurface, 200 line element, 314 linear subspace, 218 Liouville - theorem of, 22 locall(>algebra, 192 local homeomorphism, 88 local parametrization, 39, 41 local potential, 318 local uniformization, 231 locally compact, 153 locally finite covering, 153 logarithmically convex, 83 maximal ideal, 108 maximum principle, 22 - for plurisubharmonic functions, 264 - for subharmonic functions, 55 mean value property, 52 meromorphic function, 196 - on a Riemann surface, 158 meromorphic map, 243 meromorphic section, 201 minimal defining function, 194 modification, 236 - generalized, 236 module, 120 Moishezon manifold, 320, 353 monic polynomial, 113 monoidal transformation, 242 monomial,9 Montel - theorem of, 23 Nebenhulle, 364 negative line bundle, 287 neighborhood filter, 100 Neil parabola, 134 Neumann operator, 359 Neumann problem, 357 noetherian, 121 normal bundle, 182 normal exhaustion, 78 normalized polynomial, 113 normally convergent, 11 nowhere dense, 37 Oka - theorem of, 100 Oka's principle, 253 open covering, 153 open map, 88 orbit space, 205 order - of a power series, 119 Osgood - theorem of, 19 Osgood space, 243 paracompact, 153 parameter system, 151 partially differentiable, 16 partition of unity, 163 pathwise connected, 88 period matrix, 348 Picard group, 187 Plucker embedding, 218 pluriharmonic function, 318 plurisubharmonic, 56, 261, 263 - strictly, 59 Poincare map, 301 point of indeterminacy, 196 polar set, 196 pole, 196 polydisk,5 polynomial,9 polynomially convex, 87 positive definite, 261 positive divisor, 198 positive form, 303 positive line bundle, 288 power series - convergent, 108 - formal, 11, 105 power sum, 134 prime element, 117 primitive polynomial, 117 principal divisor, 254 principal ideal domain, 117 projective algebraic - manifold, 217 - set, 217 projective space, 208 projective unitary transformations, 318 proper map, 35, 226 Index properly discontinuous, 205 pseuqoconvex, 60, 103 - strongly, 73 pseudoconvex manifold, 267 pseudodifferential metric, 368 pseudopolynomial, 124 pullback, see lifted bundle pure-dimensional, 143 quadratic transformation, 239 quotient bundle, 181 quotient field, 117 quotient topology, 203 rational function, 223 refinement, 153 refinement map, 153 region, regular closure, 245 regular domain, 360 regular function, 224 regular point, 39, 140, 161 Reinhardt domain, complete, 8, 85 - over en, 95 - proper, Remmert-Stein - extension theorem, 150 removable boundary point, 102 removable singularity, 196 resolution of singularities, 240 Riemann domain - branched, 229 - with distinguished point, 89 Riemann extension theorem - first, 38 - second, 151 Riemann surface - abstract, 157 - concrete, 230 - of.,fi, 88 Ritt's lemma, 151 Riickert basis theorem, 122 Runge domain, 87 saturated set, 203 scalar product, schlicht domain, 90 Schwartz - theorem of, 285 second axiom of count ability, 153 section, see cross section - in a fiber bundle, 186 Segre map, 225 self-intersection number, 241 seminegative line bundle, 352 seminorm, 284 semi positive line bundle, 352 Serre duality, 233 Serre problem, 259 shear, 109 Siegel upper halfplane, 352 singular cochain, 189 singular cohomology group, 189 singular homology group, 189 singular locus, 147 singular point, 39 singular q-chain, 188 singular q-simplex, 188 smooth boundary, 64 smoothing lemma, 59 Sobolev norm - tangential, 358 Soholev space, 358 standard simplex, 188 star, 283 Stein manifold, 251 Stiefel manifold, 212 Stokes's theorem, 305 strict transform, 240 strictly pseudoconvex manifold, 267 strongly pseudoconvex, 267 structure group, 173 StiitzfUi.che, 275 subbundle, 180 subelliptic, 358 subharmonic function, 53 submanifold, 41, 161 submersion, 168 symbol - of a differential operator, 336 symmetric polynomial, 126 tangent bundle, 176 - of projective space, 222 tangent space, 32, 165 - holomorphic, 66 tangent vector, 32, 165 tangential map, 167 tautological bundle, 238 tensor power, 178 tensor product, 179, 261 Theorem A, 252 Theorem B, 253 topological map, 41 topological quotient, 203 toric closure, 246 391 392 Index toric variety, 246 torsion point, 230 torus, 207, 225, 348 transformation group, 172 transition functions,· i 73 transversal, 171 trivial vector bundle, 177 trivialization, 173 - of a vector bundle, 175 tube, 288 tube domain, 87 unbranched point, 133 uniformization, 88, 162 union of Riemann domains, 94 unique factorization domain, 117 vector vector vector vector bundle, 175 bundle chart, 175 bundle homomorphism, 177 field, 176 Veronese map, 224 volume element, 323 weakly holomorphic, 16 wedge product, 297 Weierstrass - theorem of, 19 Weierstrass condition, 113 Weierstrass division formula, 115 Weierstrass formula, 110 Weierstrass function, 226 Weierstrass polynomial, 114 Weierstrass preparation theorem, 113, 116 Whitney sum, 177 worm domain, 366 Zl -regular, 109 Zariski tangential space, 152 Zariski topology, 141 Graduate Texts in Mathematics (contiNud/rom page ii) 64 EDwARDS Fourier Series Vol I 2nd cd 65 WELLS Differential Analysis on Complex Manifolds 2nd cd 66 WA11!RHOUSE Introduction to Affine Group Schemes 67 SERRE Loca1 Fields 68 WEIDMANN Linear Operators in Hilbert Spaces 69 LANG Cyclotomic Fields U 70 MAssEY Singular Homology Theory 71 FARKASIKRA Riemann Surfaces 2nd ed 72 STIlLWEll Classical Topology and Combinatorial Group Theory 2nd ed 73 HUNGERFORD Algebra 74 DAVENPORT Multiplicative Number Theory 3rd ed 75 HOCHSOIlLD Basic Theory of Algebraic Groups and Lie Algebras 76 IrrAXA Algebraic Geometry 77 HEcicE Lectures on the Theory of Algebraic Numbers 78 BURRIsiSANKAPPANAVAR A Course in Universal Algebra 79 WALTERS An Introduction to Ergodic Theory 80 ROBINSON A Course in the Theory of Groups 2nd cd 81 FORSlER Lectures on Riemann Surfaces 82 Barrrru Differential Forms in Algebraic Topology 83 WASHINGTON Introduction to Cyclotomic Fields 2nd cd 84 IRELANIWROSEN A Classical Introduction to Modem Number Theory 2nd ed 85 EDWARDS Fourier Series Vol U 2nd cd 86 VAN LlNT Introduction to Coding Theory 2nded 87 BROWN Cohomology of Groups 88 PIERCE Associative Algebras 89 LANG Introduction to Algebraic and Abelian Functions 2nd ed 90 BR0NDSTED An Introduction to Convex Polytopes 91 BEARDON On the Geometry of Discrete Groups 92 D/ESTEL Sequences and Series in Banach Spaces 93 DUBRovINIFoMENKcVNoVIKOV Modem Geometry-Methods and Applications Part I 2nd cd 94 WARNER Foundations of Differentiable Manifolds and Lie Groups 95 SHIRYAEV Probability 2nd cd 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 CONWAY A Course in Functional Analysis 2nd cd KOBun Introduction to Blliptic Curves and Modular Forms 2nd ed BROCKERIfOM DIECK Representations of Compact Lie Groups GRoVElBENSON Finite Reflection Groups 2nded BERGICHlusTENSENIRBssEL Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions EDWARDs Galois Theory VAR.ADAR.AJAN Lie Groups, Lie Algebras and Their Representations LANG Complex Analysis 3rd ed DUBROVINlFoMENKcVNoVIKOV Modem Geometry-Methods and Applications PartU LANG SLz(R) Sn.VERMAN The Arithmetic of Elliptic Curves OLVER Applications of Lie Groups to Differential Equations 2nd ed RANGE Holomorpbic Functions and Integral Representations in Several Complex Variables 1BITo Univalent Functions and TeicbmU11er Spaces LANG Algebraic Number Theory HuSEMOlJ ER Elliptic Curves LANG Elliptic Functions KARATZASfSHRBVE Brownian Motion and Stochastic Calculus 2nd ed KOBun A Course in Number Theory and Cryptography 2nd ed BERGERlGosnAUX Differential Geometry: Manifolds Curves and Surfaces KEu.l!y/SRlNlVASAN Measure and Integral Vol I SERRE Algebraic Groups and Class Fields PEDERSEN Analysis Now ROTMAN An Introduction to Algebraic Topology ZIEMER Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation LANG Cyclotomic Fields I and U Combined 2nd ed REMMERT Theory of Complex Functions Readings in Mathemlllics BBBINGHAusIHERMEs et al Numbers Readings in MatMmatics 124 DuBRovINIFoMENxolNoVIKOV Modem Geometry-Methods and Applications PartID 125 BERENSTEINIGAY Complex Variables: An Introduction 126 BOREL Linear Algebraic Groups 2nd ed 127 MAsSEY A Basic Course in Algebraic Topology 128 RAuOI Partial Differential Equations 129 PuLTONIHARRIs Representation Theory: A rmt Course Readings in Mathematics 130 DoDSONlPosToN Tensor Geometry 131 LAM A First Course in Noncommutative Rings 2nd ed 132 BBARDON Iteration of Rational Functions 133 HARRIs Algebraic Geometry: A First Course 134 ROMAN Coding and Information Theory 135 ROMAN Advanced Linear Algebra 136 ADKJNslWElNTRAUB Algebra: An Approach via Module Theory 137 AxLBRIBOORDON!RAMBY Harmonic Function Theory 2nd ed 138 COHEN A Course in Computational Algebraic Number Theory 139 BRBOON Topology and Geometry 140 AUBIN Optima and Equilibria An Introduction to Nonlinear Analysis 141 BBCXERIWBlSPF6NNINGlKJumEL GrtIbner Bases A Computational Approach to Commutative Algebra 142 LANG Real and Functional Analysis 3rded 143 DooB Measure Theory 144 DENNISIFARB Noncommutative Algebra 145 VICK Homology Theory An Introduction to Algebraic Topology 2nded 146 BRIDGES Computability: A Mathematical Sketchbook 147 ROSENBERG Algebraic K-Theory and Its Applications 148 ROTMAN An Introduction to the Theory of Groups 4th ed 149 RA~.Foundationsof Hyperbolic Manifolds 150 EISENBUD Commutative Algebra with a View Toward Algebraic Geometry 151 Su VERMAN Advanced Topics in the Arithmetic of Elliptic Curves 152 ZIEGLER Lectures on Polytopes 153 FULTON Algebraic Topology: A rll"StCourse 154 BROWNIPEARCY An Introduction to Analysis 155 KAssEL Quantum Groups 156 KEcHlUs Classical Descriptive Set Theory 157 MALLIAVIN IntegIlltion and Probability 158 ROMAN Field Theory 159 CONWAY Functioos of One Complex Variable n 160 LANG Differential and Riemannian Manifolds 161 BORWElNIERDm.YI Polynomials and Polynomial Inequalities 162 Au>ERlNIBELL Groups and Representations 163 DIXONIMORTIMEIt Permutation Groups 164 NAniANSON Additive Number Theory: The Classical Bases 165 NAnlANSON Additive Number Theory: Inverse Problems and the Geometry of Sumsets 166 SHARPE Differential Geometry: Carlan's Generalization of Klein's Erlangen ProgIam 167 MORANDI Field and Galois Theory 168 EWALD Combinatorial Convexity and Algebraic Geomeuy 169 BHATIA Matrix Analysis 170 BREOON ShcafTheory 2nd ed 171 PE'reRSEN Riemannian Geometry 172 REMMERT Classical Topics in Complex Function Theory 173 DIESTEL Graph Theory 2nd ed 174 BRIDGES Foundations of Real and Abstract Analysis 175 UCKORISH An Introduction to Knot Theory 176 lB! Riemannian Manifolds 177 NEWMAN Analytic Number Theory 178 Cl.ARKEILEDYAEv/STERNIWOLENSKI Nonsmooth Analysis and Control Theory 179 DoUGLAS Banach Algebra Techniques in Operator Theory 2nd cd 180 SRIVASTAVA A Course OD Borel Sets 181 KRESs Numerical Analysis 182 WALlER Ordinary Differential Equations 183 MEGGINSON An Introduction to Banach Space Theory 184 BOllOBAS Modern Graph Theory 185 COXlL1TlUJO'SHEA Using Algebraic Geometry 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 RAMAKRlSHNANN ALENZA Fourier Analysis on Number Fields HARRISIMoRRlSON Moduli of Curves GoLDBLA1T Lectures on the Hyperrea1s: An Introduction to Nonstandard Analysis LAM Lectures on Modules and Rings EsMONDE!MURlY Problems in Algebraic Number Theory LANG Fundamentals of Differential Geometry HIRSCHILACOMBE Elements of Functional Analysis COHEN Advanced Topics in Computational Number Theory ENGEIlNAGEL One-Parameter Semigroups for Linear Evolution Equations NATIIANSON Elementary Methods in Number Theory OSBORNE Basic Homological Algebra EISENBUDIHARRIs The Geometry of Schemes ROBERT A Course in p-adic Analysis HEDENMALMlKORENBLUMIZHu Theory of Bergman Spaces BAOICHERNISHEN An Introduction to Riemann-Finsler Geometry 201 HlNDRY/Sn.VERMAN Diophantine Geometry: An Introduction 202 l I:E Introduction to Topological Manifolds 203 SAGAN The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions 204 EscoFlER Galois Theory 205 FBJxIHAu>ERINlTHOMAS Rational Homotopy Theory 2nd cd 206 MURlY Problems in Analytic Number Theory Readings in Mathematics 207 Goosn.IROYLE Algebraic Graph Theory 208 CHENEY Analysis for Applied Mathematics 209 ARVESON A Short Course on Spectral Theory 210 ROSEN Number Theory in Function Fields 211 LANG Algebra Revised 3rd ed 212 MATOUSEK Lectures on Discrete Geometry 213 FRrrzsCHElGRAUERT From Holomorphic Functions to Complex Manifolds ~*.fr!l&t.fiij ~ (CIP) ~~ Jk~~pj§~ifJ~tliJ~ = From Holomorphic Functions to Complex Manifolds: it -::f~]j(: !Ii:)( / Oto !it ~ tJ] (Fritzsche, K.) tttm-l!I~ifj!l&0~::I~/'R0~, 2009.5 ISBN 978-7-5100-0471-1 I ]A II !it lli )( IV 0174 CD~M~pj§~-!li:)(cz)~t1EJf5-!Ii: 9"~!l&*I!I~i1r CIP ~tJ.ii~¥ (2009) * 11= 4!; : From Holomorphic Functions to Complex Manifolds 1!f: Klaus Fritzsche & Hans Grauert J:j:I ~ 4!;: ]A~~£I§~ifJ~rJiEJf5 ~tEt,fiij~: ~?8 ill !l& 1!f: ijlj 1!f: ~:ri1J ~!Ii: fp jj- :fH~0 ~ Ep ~ ~T: xU~ tftm-l!I~ifj!l&0~::f~]j(0~ tftm-l!I~ifj!l&0~::I~]j(0~ (::f~]j(ljiApq*m 137 010~4021602, ~T1~fii: kjb@ wpcbj com cn EP !l& *: 17.5 IX : 2009 ~ 06 !f}: 100010) 243f ~*: !l&*Jl~i2: % 010-64015659 ~~~iiS: 7f :m 072986 % J1 I!I¥: 01-2009-"1082 -978-7-5100-0471-1/0·686 ffi": 49 00 j[; ... Library of Congress Cataloging-in-Publication Data Fritzsche Klaus From holomorphic functions to complex manifolds Klaus Fritzsche, Hans Grauert p cm - (Graduate texts in mathematics; 213) Includes... of Homotopy Theory 62 KARGAPOwvIMERuJAKOV Fundamentals of the Theory of Groups 63 BOu.DBAS Graph Theory (continued after index) Klaus Fritzsche Hans Grauert From Holomorphic Functions to Complex. .. vector is written as a transposed vector: v t 2 Holomorphic Functions Definition Let E be an n-dimensional real vector space The complexification of E is the real vector space Ee := EffiE, together