Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

538

Gerd Fischer

Complex Analytic Geometry

Springer-Verlag Berlin. Heidelberg. New York 1976

Author Gerd Fischer Mathematisches Institut der Universit~t Mi]nchen TheresienstraBe 39 8000 MLinchen 2 /BRD

Library of Congress Cataloging in Publication Data

Fischer, Gerd, 1939- Complex analytic geometry.

(Lecture notes in mathematics ; 538) 1. Functions of several complex variables.

2. Analytic spaces. 3~ Fiber spaces (Mathematics) I. Title. II. Series: Lecture notes in mathe- matics (Berlin) ; 538. QA3.L28 vol. 538 [CA331] ~lO'.8s [515'.94]

76-27692

AMS Subject Classifications (1970): 32 BXX, 32CXX, 32J10, 32LXX

ISBN 3-540-07857-6 Springer-Verlag Berlin �9 Heidelberg �9 New York ISBN 0-387-07857-6 Springer-Verlag New York �9 Heidelberg �9 Berlin

This .w.ork is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, re- printing, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under w 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.

�9 by Springer-Verlag Berlin �9 Heidelberg 1976 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.

PREFACE

These Lecture Notes arose from courses held at the Universities

of Regensburg, Frankfurt and Munich. The aim of the courses was

to present a survey of the fundamental concepts and results of

"complex analytic geometry" (i.e. the theory of functions of

several complex variables) and to approach the current state of

research in this field.

The choice of the material was governed by the idea that the

presentation should center on (not necessarily reduced.) complex

spaces and their holomorphic maps. Progressing from the funda-

mentals of this theory to problems of current interest is a

particularly long and arduous task. Bearing in mind the reader's

patience, the length of the notes and last but not least the

energy of the author, several basic but voluminous methods and

proofs have only been cited or sketched, particularly if they

are easily accessible in the literature (e.g. []], [3], [11],

[40], [44], [50], [51], [55], [56], [57], [72], [107], [I]I],

[150]). Among these topics are the theory of sheaves, the theory

of local analytic algebras and their dimensions, the theory of

holomorphic functions in ~n including the elementary theory of

analytic sets, and finally the fundamental coherence theorems of

OKA, CARTAN, GRAUERT as well as Theorems A and B of CARTAN-SERRE.

Although these notes are far from being self-contained, the in-

tention was to provide at least the beginnings of a systematic

presentation within the framework of each chapter. Hopefully

they may serve as a partial substitute for a compendium of the

"elements of analytic geometry" which is long overdue.

My sincere thanks go to all who attended the lectures for their

suggested improvements and their endurance, to many colleagues

for their helpful comments, to Inge Pfeilschifter and Ghislaine

Maurer for preparing the camera-ready manuscript, to Camilla

Aman for proof-reading and to Joseph Maurer for sketching the

figures.

Chapter O.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.10 0.11 0.12 0.13 0.14 0 .15 0.16 0.17 0.18 0.19 O.20 0.21 0.22 0.23 0.24 0.25 0.26 0 27 O 28 0 29 0 3O O 31 032 0 33 0 34 O35 O 36 0.37 0.38 0.39 0.40 0.41 0.42 0.43 0.44 0.45 0.46

Chapter I . 1.1 1.2 1.3 1.4 1.5 1.6

CONTENTS

page

BASIC NOTIONS

Ringed spaces I Direct an topological inverse images of sheaves I Morphisms of ringed spaces 2 Monomorphisms and epimorphisms 3 Finite type, finite presentation, coherence 3 " 4 ,,

Germs of homomorphisms 5 Extension of germs 6 Analytic inverse image of sheaves 6 Trivial extension of sheaves 8 Coherence Theorem of OKA 8 Coherence Theorem of CARTAN 8 Complex spaces, holomorphic maps, suhspaces 9 Nilradical 11 Reduction 1 2 Restriction of a holomorphic map 13 Analytic local algebras 14 Holomorphic maps into ~n 14 Holomorphic maps of a reduced complex space 15 Germs of complex spaces 16 Generation of holomorphic maps 17 Immersions and embeddings 19 Gluing of complex spaces 20 Universal property of fibre products and products 21 Direct product of cn and cm 22 Inverse images of subspaces 23 Direct product of subspaces 24 Direct products (general case) 25 The diagonal 26 Universal property of the diagonal 29 Existence of fibre products 29 Kernel of a double arrow 30 Fibre products of reduced complex spaces 30 Immersion of holomorphic maps 31 Stein spaces 32 Theorems A and B 32 Characterization of Stein spaces 34 Privileged neighbourhoods 34 Noetherian properties of coherent sheaves 35 Transporter and annihilator ideals 37 Gap sheaves 37 Analytically rare sets 38 Lemma of RITT 40 Hypersurfaces 42 Constructible sets 43

COHERENT SHEAVES

Complex spaces over S Cones over S Projective varieties over S Linear spaces over S Linear forms Duality theorem

44 44 48 49 51 51

VI

I,.7 1.8 1 . 9

I . 1 0 1 .11

I .12 I .13 1 . 1 4 1 . 1 5 I .16 I . 17 I .18 I .19 I .20 1 .21 I .22 I .23 1.24 1 . 2 5 I .26

Chapter 2.

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2 . 1 0 2 .11 2 . 1 2 2 . 1 3 2 . 1 4 2 . 1 5 2 . 1 6 2 . 1 7 2 . 1 8 2 . 1 9 2.20 2.21 2.22

Appendix.

2.23

2.24 2.25 2.26 2.27

2.28 2.29 2.30 2.31

2.32

Change of base for linear fibre spaces 53 Vector bundles and locally free sheaves 54 Projective variety over S associated to a coherent sheaf 55 Proper and finite maps 56 Algebraic characterization of a finite holo- morphic map 57 Finite Coherence Theorem 58 Finite spaces over a Stein space 58 The analytic spectrum 59 Main theorem on the analytic spectrum 60 Higher image sheaves 63 GRAUERTs proper coherence theorem 64 REMMERTs proper mapping theorem 64 Semiproper mapping theorem 65 Cokernel of a double arrow 66 Analytic equivalence relations 68 Canonical factorization of a holomorphic map 68 Holomorphic maps with connected fibres 70 Stein factorization 70 Petrification 72 Proper equivalence relations 74

DIFFERENTIAL CALCULUS

Tangent space of a complex space at a point 77 Coordinate description of the tangent space 78 Embedding dimension 79 Characterization of immersions 79 Tangent space of a complex space 80 The Jacobian map of a holomorphic map 81 Tangent space of a holomorphic map 83 Tangent space of a direct product 84 Pfaffian forms and vector fields 84 Derivations 85 Restriction of vector fields 89 A theorem of ROSSI 91 Corank of a coherent module 93 Singular locus of a coherent module 95 Regularity criterion for complex spaces 96 The singular locus of a reduced complex space 96 Rank and corank of a holomorphic map 97 Mersions 99 Differential characterization of mersions 1OO Holomorphic retractions 102 Normal space of a holomorphic map 103 Locally trivial holomorphic maps 106

NORMAL AND MAXIMAL COMPLEX SPACES

The Riemann removable singularity theorems for manifolds 108 Weakly holomorphic functions 110 Universal denominators, non-normal locus 111 The Normalization Theorem 112 Removable singularity theorems for normal complex spaces 118 Lifting of a holomorphic map to the normalizations 121 The Maximalization Theorem 122 Lifting of a holomorphic map to the maximalizations 124 Characterizations of maximal complex spaces, Graph Theorem 126 Stein spaces and finite holomorphic maps 127

VII

Chapter 3.

3.1 3.2 3.3 3.4 3.5 3.6 3.7

3.8 3.9 3.10 3.11 3.12 3.13 3.14 315 316 317 318 3 19 3 2O 3 21 3 22

Chapter 4.

4.1 4.2

4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10

DEGENERACIES AND FLATNESS

Dimension of a complex space Finite holomorphic maps Spreading of a holomorphic map Semicontinuity of the fibre dimension Geometric rank and corank Analyticity of the degeneracy sets Image of a holomorphic map with constant fibre dimension Dimension of the image of a holomorphic map Open holomorphic maps and dimension formula iv

Definition of flatness Algebraic consequences Flatness of finite holomorphic maps Flatness criterion of BOURBAKI-GROTHENDIECK Flatness and change of base Flatness of a holomorphic map into Cn Projections are flat Non-flat locus and flatification Flat holomorphic maps are open Flatness of open holomorphic maps Regularity criterion for flat holomorphic maps Bad loci of a holomorphic map

MODIFICATIONS AND MEROMORPHIC FUNCTIONS

o-modifications Proper modifications, purity of branch loci, CHOWs lemma CHOWs Theorem Meromorphic functions Graph of a meromorphic function Meromorphic functions and meromorphic graphs Theorem of HURWITZ Extension of meromorphic functions Meromorphic functions and modifications Theorem of WEIERSTRASS-SIEGEL-THIMM

131 132 134 134 135 137

138 139 142 143 146 148 149 152 152 153 155 155 156 157 159 160

162

169 171 173 177 181 183 185 186 188

0 0 0