Theory of Stein Spaces (Classics in Mathematics)

CLASSI'C'S'tg'MATHEMATICS

Hans Grauert Reinhold Remmert

Theory of Stein Spaces

Springer

Hans l;rauert (b. 1930 in Harem/Ems, Germany) and Reinhold Renunert (h. 1930 in Osnabruck, Germany) met at the University of Munster, where they both studied mathematics and physics from 1949 to 1954. In 1950 they were invited by Heinrich Behnke and Karl Stein to attend their" Oberseminar", which was held on Saturdays, for 2 hours front y a.m. Five years after the tragic events of World War Behnke's old friend Henri tartan visited Munster. His lecture on recent developments in the theory of-Several Complex Variables" was a real eye-opener for the young students and had a strongly formative influence on them: indeed this was to determine the course of their scientific research careers from then on.

In tune 1954 (;rauert and Remmert received their respective doctorates from the University of Munster. In 1957 they both became lecturer I Privatdozent) there. In 1959 resp. 19bo, Grauert and Remmert were appointed full professors at Gottingen resp. Erlangen. The original German edition of-Theory of Stein Spaces"was written at a time when complex spaces, coherent analytic sheaves and the so-called Theorems A and B had already become established notions and theorems. Medicated to Karl Stein, the hook was published in 1977, and the English edition was to follow in 1979.The first announcement of the book, in Springer's promotion, consisted of the picture reproduced above, taken during the boat trip of the annual Bonn Arbeitstagung some time earlier, showing three men on a boat, with the mininlalistic caption "Grundlehren 227"

Classics in Mathematics

Hans Grauert Reinhold Remmert

Theory of Stein Spaces

Springer Berlin Heidelberg New York

Hong Kong London

Milan Paris Tokyo

Hans Grauert Reinhold Remmert

Theory of Stein Spaces Reprint of the 1979 Edition

Springer

Authors

Hans Grauert

Reinhold Remmert

Universitat Munster Mathematisches Institut Einsteinstr. 62 48149 Munster, Germany

Universitat Mtinster Mathematisches Institut Einsteinstr. 62 48149 Munster, Germany

7)'anslator

Alan Huckleberry University of Notre Dame Department of Mathematics Notre Dame, Indiana 46556, U S.A.

Originally published as Vol. 236 of the Grundlehren der mathematischen Wissenschaften

Mathematics Subject Classification (2000): 30A46, 32E10, 32J99,32-02,32A10, 32A20, 32C15

Inside front-cover photography courtesy of Klaus Peters. From The Mathematical Intelligencer Volume 2, Number 2, i98o. Libevy of Caopa. Caulopoj-m-P, bo.d. Data

Orwut Ha

1930.

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Tbevy or Slain W. / H. Grunt R. mornst p. m. - (Clmia in nmthmmds. ISSN 1431-0921) 'Repdot oftheeddion 1979.' Gri9judy publehad as vol. 236 in the utiv: Onaul

L.W. hasop.pbid-th- and htla

dv wall mvi char Wt-b4-fm'

tSHN 3-540-00373-9 (pblc : add-9m ppv)

1. Std. .p c . L mart. RoW,old. EL Title. M. Sale. QA331.068313 2003 515'.94--do2l 2003050517

ISSN 1431-0821

ISBN 3-540-00373-8 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag is a part of Springer Science+Business Media springeronline.com

a Springer-Verlag Berlin Heidelberg 2004 Printed in Germany

The use of general descriptive names, registered names, trademarks etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Printed on acid-free paper

4113142ck-54 3 210

H. Grauert R. Remmert

Theory of Stein Spaces Translated by Alan Huckleberry

Springer-Verlag

Berlin Heidelberg New York

Hans Grauert

Reinhold Remmert

Mathematisches Institut der Universitit Gottingen D-3400 Gottingen Federal Republic of Germany

Mathematisches Institut der Westfilischen Wilhelms-Universitit D-4400 Munster Federal Republic of Germany

Translator:

Alan Huckleberry Department of Mathematics University of Notre Dame Notre Dame, Indiana 46556 USA

AMS Subject Classifications: 30A46, 32E10, 32J99, 32-02, 32A10, 32A20, 32C15

Title of the German Original Edition: Theorie der Steinschen Raume, Springer-Verlag Berlin Heidelberg 1977. With 5 Figures

Library of Congress Cataloging in Publication Data Grauert, Hans, 1930Theory of Stein spaces.

(Grundlehren der mathematischen Wissenschaften; 236)

Translation of Theorie der Steinschen Riume. Includes index. 1. Stein spaces. I. Remmert, Reinhold, joint author. II. Title. III. Series: Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen; 236. QA331.G68313

515'.73

79-1430

All rights reserved.

No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag. © 1979 by Springer-Verlag New York Inc. Printed in the United States of America.

987654321 Berlin Heidelberg New York ISBN 0-387-90388-7 New York Heidelberg Berlin ISBN 3-540-90388-7

Dedicated to Karl Stein

Contents

Introduction

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. XV

Chapter A. Sheaf Theory

§ 0. Sheaves and Presheaves . . . . . . . . . . . 1. Sheaves and Sheaf Mappings . . . . . . 2. Sums of Sheaves, Subsheaves, and Restrictions . . 3. Sections . . . . . . . . . . . . . . 4. Presheaves and the Section Functor r . . 5. Going from Presheaves to Sheaves. The Functor I. 6. The Sheaf Conditions ."1 and .Y2 . . . . . 7. Direct Products . . . . . . . . . . . 8. Image Sheaves . . . . . . . . . . . 9. Gluing Sheaves . . . . . . . . . . . . .

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2 2 3

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§ 1. Sheaves with Algebraic Structure . . . . 1. Sheaves of Groups, Rings, and R-Modules 2. Sheaf Homomorphisms and Subsheaves . 3. Quotient Sheaves . . . . . . . . 4. Sheaves of Local k-Algebras . . . . . 5. Algebraic Reduction . . . . . . . 6. Presheaves with Algebraic Structure . . 7. On the Exactness of r and r .

§ 2. Coherent Sheaves and Coherent Functors 1. Finite Sheaves . . . . . . . . 2. Finite Relation Sheaves . . . . . 3. Coherent Sheaves . . . . . . 4. Coherence of Trivial Extensions . . .

5. The Functors ®° and A'

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4 4

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11

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11

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. . § 3. Complex Spaces . . . . . . . . . . . . . . 1. k-Algebraized Spaces . 2. Differentiable and Complex Manifolds . . 3. Complex Spaces and Holomorphic Maps . 4. Topological Properties of Complex Spaces . . . . 5. Analytic Sets . . . . . . . . . . . 6. Dimension Theory . . . 7. Reduction of Complex Spaces . . . . . 8. Normal Complex Spaces . . . . . . . .

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6. The Functor . Yom and Annihilator Sheaves . 7. Sheaves of Quotients . . . . . . .

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12

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12

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13

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14

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14

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15

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15

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16 18

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18

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19

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20

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21

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Contents

VIII .

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1. Soft Sheaves . . . . . . . . . . . . . . . . . 2. Softness of the Structure Sheaves of Differentiable Manifolds . . . . . . . . . . . 3. Flabby Sheaves . . .

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4. Exactness of the Functor I for Flabby and Soft Sheaves .

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§ 4. Soft and Flabby Sheaves .

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22 22 23 25 25

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Chapter B. Cohomology Theory §

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1. Flabby Cohomology Theory . . 1. Cohomology of Complexes . . 2. Flabby Cohomology Theory . 3. The Formal de Rham Lemma .

§ 2. tech Cohomology . . . . . 1. tech Complexes . . . . 2. Alternating tech Complexes

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3. Refinements and the tech Cohomology Modules fI9(X, S) 4. The Alternating tech Cohomology Modules M.(X, S) . 5. The Vanishing Theorem for Compact Blocks . . . . 6. The Long Exact Cohomology Sequence . . . . . .

§ 3. The Leray Theorem and the Isomorphism Theorems . . . . . 1. The Canonical Resolution of a Sheaf Relative to a Cover . . . . . . . . . . . . . . 2. Acyclic Covers . . . . . . . . . . . . 3. The Leray Theorem . 4. The Isomorphism Theorem fI;(X, .9') = fl9(X, .50) = H4(X, Y) .

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33 34 35 35 37 37 38

40 40 42 43 43

Chapter L Coherence Theory for Finite Holomorphic Maps § I.' Finite Maps and Image Sheaves 1. Closed and Finite Maps . .

2. The Bijection f«(.Y)r -. r[ Y .

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46 47

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3. The Exactness of the Functor f. . . . . . 4. The Isomorphisms HQ(X, .9') = H4(Y, f.(.9'))

5. The 0Y Module Isomorphism f: f.(.9'),, -. n S_ .

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§ 2. The General Weierstrass Division Theorem and the Weierstrass Isomorphism 1. Continuity of Roots . . . . . . . . . . . . . . . . . . . 2. The General Weierstrass Division Theorem . . . . . . . . . . . 3. The Weierstrass Homomorphism OB 4 n.(0,4) . . . . . . . . . . 4. The Coherence of the Direct Image Functor n. . . . . . . . . . . § 3. The Coherence Theorem for Finite Holomorphic Maps 1. The Projection Theorem . . . . . . . . . . 2. Finite Holomorphic Maps (Local Case) . . . . . . . 3. Finite Holomorphic Maps and Coherence

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49 50

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52 52 53 54

Contents

ix

Chapter 11. Differential Forms and Dolbeault Theory §

1. Complex Valued Differential Forms on Differentiable Manifolds 1. Tangent Vectors . . . . . . . . . . 2. Vector Fields . . . . . . . . . . . . 3. Complex r-vectors . . . . . . . . . . . . 4. Lifting r-vectors . . . . . . . . . . . . . . 5. Complex Valued Differential Forms . . . . . . . 6. Exterior Derivative . . . . . . . . . . . . . 7. Lifting Differential Forms . . . . . . . . . . . . 8. The de Rham Cohomology Groups . . . . . . . . .

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§ 2. Differential Forms on Complex Manifolds 1. The Sheaves d' o .r °' and fl' . . . 2. The Sheaves and S1° . . . . . 3. The Derivatives a and d . . . 4. Holomorphic Liftings of (p, q)-forms . .

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64 64 66 67 70 71

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72 73 74 75

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81

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Supplement to Section 4.1. A Theorem of Hartogs

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77 77 79 80

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§ 4. Dolbeault Cohomology Theory . . . . . . . . 1. The Solution of the a-problem for Compact Product Sets 2. The Dolbeault Cohomology Groups . . . . . 3. The Analytic de Rham Theory . . . . . . .

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§ 3. The Lemma of Grothendieck . . . . . . . . . 1. Area Integrals and the Operator T . . . . . . . . . 2. The Commutivity of T with Partial Differentiation . . . . 3. The Cauchy Integral Formula and the Equation (a/dz)(Tf) =f 4. A Lemma of Grothendieck . . . . . . . .

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56 56 58 59 60 60

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Chapter III. Theorems A and B for Compact Blocks C§ 1. The Attaching Lemmas of Cousin and Cartan 1. The Lemma of Cousin . . . . . . . 2. Bounded Holomorphic Matrices . . . . . . 3. The Lemma of Cartan . . . .

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. § 2. Attaching Sheaf Epimorphisms . . . . . 1. An Approximation Theorem of Runge . 2. The Attaching Lemma for Epimorphisms of Sheaves .

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89

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Chapter IV. Stein Spaces . . . . . . . . . § 1. The Vanishing Theorem H5(X, ,9') = 0 . . . . . . 1. Stein Sets and Consequences of Theorem B . . . . . . . . . . . . . . 2. Construction of Compact Stein Sets Using the Coherence Theorem for Finite Maps .

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100 100

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101

x

Contents

3. Exhaustions of Complex Spaces by Compact Stein Sets 4. The Equations H9(H, .') = 0 for q >_ 2 . . 5. Stein Exhaustions and the Equation H1(X, 91) = 0 . .

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108 108 109

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111

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§ 2. Weak Holomorphic Convexity and Stones . . . . . . . . . . 1. The Holomorphically Convex Hull . . . . . . . . . . . . 2. Holomorphically Convex Spaces . . . . . . . . . . 3. Stones . . . . . . . . . . . . . . . . . . . . . . 4. Exhaustions by Stones and Weakly Holomorphically Convex Spaces 5. Holomorphic Convexity and Unbounded Holomorphic Functions . .

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§ 3. Holomorphically Complete Spaces . 1. Analytic Blocks . . . . . . . . 2. Holomorphically Spreadable Spaces . 3. Holomorphically Convex Spaces . . .

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§ 4. Exhaustions by Analytic Blocks are Stein Exhaustions 1. Good Semi-norms . . . . . . . . . . . 2. The Compatibility Theorem . . . . . . 3. The Convergence Theorem . . . . . . 4. The Approximation Theorem . . . . 5. Exhaustions by Analytic Blocks are Stein Exhaustions .

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102 103 104

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112 113 116 116 117 117 118 118 119 120 121 123

Chapter V. Applications of Theorems A and B

§ 1. Examples of Stein Spaces . 1. Standard Constructions . . 2. Stein Coverings . . . 3. Differences of Complex Spaces 4. The Spaces C2\{0} and t:'\(0) . 5. Classical Examples . . . . . 6. r tein Groups . . . . . . .

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136 136

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138

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139 142 144

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4. The Exact Exponential Sequence 0 - 1- O O -. 1 5. Oka's Principle

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§ 3. Divisor Classes and Locally Free Analytic Sheaves of Rank 1 1. Divisors and Locally-Free Sheaves of Rank 1 . . . . 2. The Isomorphism HI(X, O') -+ LF(X) 3. The Group of Divisor Classes on a Stein Space . . . .

§ 4. Sheaf Theoretical Characterization of Stein Spaces 1. Cycles and Global Holomorphic Functions . 2. Equivalent Criteria for a Stein Space . . 3. The Reduction Theorem . . . . . . . 4. Differential Forms on Stein Manifolds . . . 5. Topological Properties of Stein Spaces . . . .

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125 125 127 128 130 134 136

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§ 2. The Cousin Problems and the Poincari Problem 1. The Cousin I Problem . . . . . . . . 2. The Cousin II Problem . . . . . . . . 3. Poincarb Problem . . . . . . . . .

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146 146 147 148

150 150 152 152 154 156

Contents

XI

§ 5. A Sheaf Theoretical Characterization of Stein Domains in C° 1. An Induction Principle . . . . . . . . . 2. The Equations H'(B, OB) = = H'-'(B, OB) = 0 . 3. Representation of 1 . . . . . . . . 4. The Character Theorem . . . . . . . . . .

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§ 6. The Topology on the Module of Sections of a Coherent Sheaf . 0. Frechet Spaces . . . . . . . . . . . . . 1. The Topology of Compact Convergence . . . . . 2. The Uniqueness Theorem . . . . . . . . . . 3. The Existence Theorem . . . . . . . . . 4. Properties of the Canonical Topology . . . . . 5. The topologies for CQ(U, Y) and Z°(U, .9') . . . . . 6. Reduced Complex Spaces and Compact Convergence . . 7. Convergent Series . . . . . . . . . . . . .

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171

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§ 7. Character Theory for Stein Algebras . 1. Characters and Character Ideals . . 2. Finiteness Lemma for Character Ideals 3. The Homeomorphism EE: X -T(T) 4. Complex Analytic Structure on T(T)

163 163

164 165 166 168 170 170

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157 157 159 161 162

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176 176 177 180

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181

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§ 1. Square-integrable Holomorphic Functions

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Chapter VI. The Finiteness Theorem

. . 1. The Space 0,,(B) . . . . . 2. The Bergman Inequality . . . . . . 3. The Hilbert Space 0,',(B) . . . . . . . 4. Saturated Sets and the Minimum Principle 5. The Schwarz Lemma . . . . . . . .

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§ 2. Monotone Orthogonal Bases

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187 187 188 189 190 190

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191

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3. Construction of Monotone Orthogonal Bases by Means of Minimal Functions

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191 192 193

1. Monotonicity 2. The Subdegree

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§ 3. Resolution Atlases . . . 1. Existence . . . . . . . 2. The Hilbert Space CR(U, ,9') 3. The Hilbert Space Zj(U, .9') 4. Refinements . . . . . . .

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§ 4. The Proof of the Finiteness Theorem 1. The Smoothing Lemma . . . . 2. Finiteness Lemma . . . . . . 3. Proof of the Finiteness Theorem .

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194 194 196 197 198

200 200 201 202

Chapter VIL Compact Riemann Surfaces § 1. Divisors and Locally Free Sheaves . 0. Divisors . . . . . . . . . 1. Divisors of Meromorphic Sections

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Contents

XII 2. The Sheaves F(D) . 3. The Sheaves O(D)

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§ 2. The Existence of Global Meromorphic Sections

1. The Sequence 0 -. F(D) -. .F(D') - .F -a 0 . . . . 2. The Characteristic Theorem and the Existence Theorem 3. The Vanishing Theorem . . . . . . . . . . . 4. The Degree Equation . . . . . . . . . . . . § 3. The Riemann-Roch Theorem (Preliminary Version)

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. 206 . 207 . 208 . 208 . 209 . 210 . 210 . 211 . 211 . 212

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Supplement to Section 4. The Riemann-Roch Theorem for Locally Free Sheaves . . . . . . . . . . . . 1. The Chern Function . . . . . . . . . . . . 2. Properties of the Chern Function . . . . . . . . . . . . . . . . . . 3. The Riemann-Roch Theorem . .

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216

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1. The Genus of Riemann-Roch 2. Applications . . . . . .

§ 4. The Structure of Locally Free Sheaves . . 1. Locally Free Subsheaves . . . . . . 2. The Existence of Locally Free Subsheaves 3. The Canonical Divisors . . . . . .

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§ 5. The Equation H'(X, .,lf)

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1. The C-homomorphism O(np)(X) -. Hom(H'(X, O(D)), H'(X, O(D + np))) 2. The Equation H'(X, O(D + np)) = 0 . . . . . . . . . . .

§ 6. The Duality Theorem of Serre

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. 218

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. 225 . 225 . 226 . 227 . 227 . 228 . 229 . 229

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§ 7. The Riemann-Roch Theorem (Final Version) . . . 1. The Equation i(D) = I(K - D) . . . . . . . . 2. The Formula of Riemann-Roch . . . . . . 3. Theorem B for Sheaves O(D) . . . . . . 4. Theorem A for Sheaves O(D) . . . . . . . 5. The Existence of Meromorphic Differential Forms 6. The Gap Theorem . . . . . . . . . . 7. Theorems A and B for Locally Free Sheaves . . . 8. The Hodge Decomposition of H'(X, C) . . . .

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2. Maximal Subsheaves

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3. The Inequality u(g) = u(F) + 2g 4. The Splitting Criterion . . . 5. Grothendieck's Theorem . .

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§ 8. The Splitting of Locally Free Sheaves

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1. The Principal Part Distributions with Respect to a Divisor . 2. The Equation H'(X, O(D)) = I(D) . . . . . . . . 3. Linear Forms . . . . . . . . . . . . . . . . 4. The Inequality Dim. (X) J < 1 . . . . . . . . . . 5. The Residue Calculus . . . . . . . . . . . 6. The Duality Theorem . . . . . . . . . . . . . . .

. 215 . 215

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3. The Equation H'(X, K) = 0

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XIII

Contents 6. Existence of the Splitting 7. Uniqueness of the Splitting

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. 237 . 238

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. 240

Bibliography

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Subject Index

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243

Introduction

1. The classical theorem of Mittag-Letfler was generalized to the case of several complex variables by Cousin in 1895. In its one variable version this says that, if one prescribes the principal parts of a meromorphic function on a domain in the

complex plane C, then there exists a meromorphic function defined on that domain having exactly those principal parts. Cousin and subsequent authors could only prove the analogous theorem in several variables for certain types of domains (e.g. product domains where each factor is a domain in the complex plane). In fact it turned out that this problem can not be solved on an arbitrary domain in C'", m > 2. The best known example for this is a "notched" bicylinder in z2 < -2), from C2. This is obtained by removing the set {(zi, z2) a C2 1z, >_ the unit bicylinder, A :_ ((zl, z2) a C2 I Izl I < 1, 1 z2 < 1). This domain D has the property that every function holomorphic on D continues to a function holomorphic on the entire bicylinder. Such a phenomenon never occurs in the theory of one complex variable. In fact, given a domain G c C, there exist functions holomorphic on G which are singular at every boundary point of G. In several complex variables one calls such domains '(i.e. domains on which there exist holomorphic functions which are singular at every boundary point) domains of holomorphy. H. Cartan observed in 1934 that every domain in C2 where the above "Cousin problem" is always solvable is necessarily a domain of holomorphy. A proof of this was communicated by Behnke and Stein in 1937. Meanwhile it was conjectured that Cousin's theorem should hold on any domain of holomorphy. This was in fact proved by Oka in 1937: For every prescription of principal parts on a domain of holomorphy D c C'", there exists a meromorphic function on D having exactly those principal parts. In the same year, via the example of C'\{0},

H. Cartan showed that it is possible for the Cousin theorem to be valid on domains which are not domains of holomorphy. As the theory of functions of several complex variables developed, it was often the case that, in order to have a chance of carrying over important one variable results, it was necessary to restrict to domains of holomorphy. This was particularly true with respect to the analog of the Weierstrass product theorem. Formulated as a question, it is as follows: Given a domain D in C', can one prescribe the zeros (counting multiplicity) of a holomorphic function on D? It was soon realized that in some cases it is impossible to find even a continuous function which does the job. Conditions for the existence of a continuous solution of this

XVI

Introduction

problem, the so-called "second Cousin problem," were discussed by K. Stein in 1941. In fact he gave a sufficient condition which could actually be checked in particular examples. Nowadays this is stated in terms of the vanishing of the Chern class of the prescribed zero set. Stein, however, stated this in a dual and more intuitively geometric way. His condition is as follows: The "intersection number" of the zero surface (counting multiplicity) with any 2-cycle in D should always be zero. It was similarly necessary to restrict to domains of holomorphy in order to prove the appropriate generalizations of the facts that, on a domain in C, every meromorphic function is the ratio of (globally defined) analytic functions and, if the domain is simply connected, holomorphic functions can be uniformly approximated by polynomials (i.e. the Runge approximation theorem). Poincare first posed the question about meromorphic functions of several variables being quotients of globally defined relatively prime holomorphic functions. He in fact answered this positively in certain interesting cases (e.g. for C"' itself).

2. It is not at all straightforward to generalize the notion of a Mittag-Leffier distribution (i.e. prescriptions of principal parts) to the several variable case. The main difficulty is that the set on which the desired function is to have poles is no

longer discrete. In fact, in the case of domains in C', m > 2, this set is a (2m - 2)-dimensional real (possibly singular) surface. Thus one can no longer just prescribe points and pieces of Laurent series. This can be circumvented as follows: If G is a domain in C'" and U = {Ui}, i e I, is an open covering of G, then the family {Ui, hi} is called an additive Cousin distribution on G, whenever each hi is a meromorphic function on Ui, and on Uiai, = Uio n U,, the difference hio - hi, is holomorphic for all choices of io and i t. In the case of m = 1, this means that hio and hi, have the same principal parts. Thus one obtains a Mittag-Leffler distribution from

the Cousin distribution. A meromorphic function h is said to have the Cousin distribution for its principal parts if h - hi is holomorphic on Ui for all i. Different Cousin distributions can, on the same covering, define the same distribution of principal parts. This difficulty is overcome by introducing an equivalence relation. For this let x e G. Let U be an open neighborhood of x in G and suppose that It is meromorphic on U. Then the pair (U, h) is called a locally meromorphic function at x. Two such pairs (U1, ht) and (U2, h2) are called equiv-

alent if there exists a neighborhood V of x with V c Ut n U2 and ht - h2 holomorphic on V. Each equivalence class is called a germ of a principal part. The set of

all germs of principal parts at x is denoted by jr.. We define W:= U Jrx and SEX

denote by n: ; " -+ G the map which associates to every germ its base point x e G. If U c G is open and It is meromorphic on U then, for every x e U, one has the

associated principal part of h at x, h e Wx. Consequently there exists a map s,,: U - Jr', xi- h,,, such that n - sh = id. It is easy to check that sets of the form sh(U), where U is any open set in G and h is any meromorphic function on U, form

a basis for a topology on W. Further, in this topology, n: Y - G is seen to be continuous and a local homeomorphism. In such a situation one calls lLo a sheaf over G. The fibers of a should be thought of as stalks with the open sets looking

Introduction

XVII

like transversal surfaces given by the maps sh. The map s,,: U .*' is called a local section over U. Every Cousin distribution { U;, hi} defines a global continuous map (section) s: G - Y with it s = id. This is locally defined by s I U, := sh.. The condition that, for all i and j, hi - h, is holomorphic on U; n Uj is equivalent to the fact

that s is well-defined. Two Cousin distributions have the same principal parts if and only if they correspond to the same section in D over G. A meromorphic function h is a "solution" of the Cousin distribution s (i.e. has exactly the same principal parts as were prescribed) exactly when sh = s. It is clear from the above that the sheaf theoretic language is the ideal medium for the statement of the generalization of the Mittag-Leffier problem to the several

variable situation. Of course for domains in C" Oka had solved this without explicit use of sheaves. But even in this case the language of sheaves isolated the

real problems and made the seemingly complicated techniques of Oka more transparent. This was also true in the case of the second Cousin problem, the Poincare problem, etc. Furthermore this language was ideal for formulating new problems and for paving the road toward possible obstructions to their solutions.

Theorems about sheaves themselves later gave rise to numerous interesting applications.

3. The germs of holomorphic functions form a sheaf which is usually denoted by 0. It has already been pointed out that the zero sets of analytic functions are important even in the study of the Cousin problems. Thus it should be expected that analytic sets, which are just sets of simultaneous zeros of finitely many holomorphic functions on domains in the various C'", would play an important role in the early development of the theory. In fact the totality of germs of holomorphic functions which vanish on a particular analytic set form a subsheaf of 0 which frequently comes into play in present day complex analysis. In 1950 Oka himself used the idea of distributions of ideals in rings of local holomorphic functions (ideaux de domaines indetermines). This notion, which at the time of its conception seemed difficult and mysterious, just corresponds to the simple idea of a sheaf of ideals.

The use of germs and the idea of sheaves go back to the work of J. Leray. Sheaves have been systematically applied in the theory of functions of several complex variables ever since 1950/51. The idea of coherence is very important for many considerations in several complex variables. Roughly speaking, a sheaf of 0-modules is coherent if it is locally free except possibly on some small set where it is still finitely generated with the ring of relations again being finitely generated. Even in the early going it was necessary to prove the coherence of many sheaves. This was often quite difficult, because there were really no techniques around and most work had to be done from scratch. The most important coherence theorems

originated with H. Cartan and K. Oka. After the foundations had been laid, coherent sheaves quickly enriched the theory of domains of holomorphy with new important results. In the meantime, in his memorable work "Analytische Functionen mehrerer komplexer Veranderlichen zu vorgegebenen Periodizitatsmoduln and das zweite Cousinsche Problem," Math. Ann. 123(1951), 201-222, K. Stein had discovered complex manifolds which have basic (elementary) properties simi-

XVIII

Introduction

lar to domains of holomorphy. A domain G c C is indeed a domain of holomorphy if and only if it is a Stein manifold. The main point is that many theorems about coherent sheaves on domains of holomorphy can as well be proved for Stein manifolds. Cartan and Serre recognized that the language of sheaf cohomology,

which had been developed only shortly before, is particularly suitable for the formulation of the main results: For every coherent sheaf ." over a Stein manifold X, the following two theorems hold:

Theorem A. The O(X)-module of global sections Y(X) generates every stalk Yx as an Os module for all x e X. Theorem B. Hq(X, .So) = 0 for all q >_ I.

These famous theorems, which were first proved in the Seminaire Cartan 1951/52, contain, among many others, the results pertaining to the Cousin problems.

4. Following the original definition, a paracompact complex manifold is called a Stein manifold if the following three axioms are satisfied: Separation Axiom: Given two distinct points xt, x2 a X, there exists a function f holomorphic on X such that f (xt) # f (x2).

Local Coordinates Axiom: If xo e X then there exists a neighborhood U of xo and functions fl, ..., fm which are holomorphic on X such that the restrictions zi'=f I U, i = 1, ..., m, give local coordinates on U. Holomorphic Convexity Axiom: If {x;} is a sequence which "goes to oo in X" (i.e.

the set {x,} is discrete) then there exists a function f holomorphic on X which is unbounded on {x;}: sup I f (x,) I = oo.

It is clear that a domain in C' is a Stein manifold if and only if it is holomorphically convex. However if one wants to study non-schlicht domains over C' (i.e. ramified covers of domains in C"'), then it is not apriori clear that two points lying

over the same base point can be separated by global holomorphic functions. Likewise it is not obvious that neighborhoods of ramification points have local coordinates which are restrictions of global holomorphic functions. If one allows points which are not locally uniformizable (i.e. points where there is a genuine singularity and the "domain" is not even a manifold, as is the case at the point (0, 0, 0) e V;= {(x, y, z) e C' I x2 = yz}, which is spread over the (y, z)-plane by projection) then the above definition is meaningless, because we assumed that X is a manifold. However, even in the non-locally uniformizable situation above, the following significant weakening of the separation and local coordinate axioms still holds:

Introduction

XIX

Weak Separation Axiom: For every point x0 e X there exist functions f,, ..., f e 0(X) so that x0 is an isolated point in {x e X I fi(x) = =f(x) = 0). Among other things, this allows the consideration of spaces with singularities. Due to the maximum principle, this weak separation implies that every compact analytic subspace of X is finite. It turns out that, without losing the main results, the convexity axiom can also be somewhat weakened: Weak Convexity Axiom: Let K be a compact set in X and W an open neighborhood of K in X. Then k n W is compact, where k denotes the holomorphic hull of K

in X:

k:= {x e X f (x)

sup I f(y) , for all f e 0(X)}. yEK

One way of strengthening the axiom immediately above is to require that k be compact in X. If one does this and further considers only the case where X is a

manifold, then, without the use of deep techniques, one can show that the strengthened axiom is equivalent to the holomorphic convexity axiom (see Theorems IV.2.4 and IV.2.12). For the purposes of this book, a Stein space is a paracompact (not necessarily reduced) complex space for which Theorem B is valid. It is proved that this condition is equivalent to the validity of Theorem A, and is also equivalent to the above

weakened axioms. In particular it follows that if X is a manifold, the weakened axioms imply Stein's original axioms. We will always assume that a complex space has countable topology and is thus paracompact. With a bit of work one can show that any irreducible complex space which satisfies the weak separation axiom is eo ipso paracompact (see 16, 24). 5. We conclude our introductory remarks with a short description of the contents of this book. We begin with two brief preliminary chapters (Chapters A and

B) where we assemble the important information from sheaf theory and the related cohomology theories. The idea of coherence is explained in these chapters. A reader who is really interested in coherence proofs, can find such in our book, "Coherent Analytic Sheaves," which is presently in preparation. Complex spaces are introduced as special C-algebraized spaces. Further we develop cohomology from the point of view of alternating (Cech) cochains as well as via flabby resolu-

tions. Proofs which are easily accessible in the literature (e.g. [SCV], [TF], or [TAG]) are in general not carried out. In Chapter I a short direct proof of the coherence theorem for finite holomorphic maps is given. It is based primarily on the Weierstrass division theorem and Hensel's lemma for convergent power series. The Dolbeault cohomology theory is presented in Chapter II. As a consequence we obtain Theorem B for the structure sheaf 0 over a compact euclidean

block (i.e. an m-fold product of rectangles), K, in C'. In other words, for q > 1, HQ(K, 0) = 0. It should be noted that, although we want to introduce Dolbeault

XX

Introduction

cohomology in any case, this result follows directly and with less difficulty via the

tech cohomology. Chapter III contains the proofs for Theorems A and B for coherent sheaves over euclidean blocks K c C"'. One of the key ingredients for the proofs is the fact that, for every coherent sheaf .9', the cohomology groups, HQ(K, .9'), vanish for all

q large enough. The deciding factor in proving Theorem A is the "Heftungslemma" of Cartan. This is proved quite easily if while solving the Cousin problem, one simultaneously estimates the attaching functions. In Chapter IV Theorems A and B are proved for an arbitrary Stein space, X. A summary of the proof is the following: First it is shown that X is exhausted by analytic blocks. (An analytic block is a compact set in X which can be mapped by a finite, proper, holomorphic map into an euclidean block in some C"'.) The coherence theorem for finite maps along with the results in Chapter III yield the desired theorem free of charge. In order to obtain such theorems in the limit (i.e. for spaces exhausted by analytic blocks), an approximation technique, which is a generalization of the usual Runge idea, is needed.

Applications and illustrations of the main theorems, as well as examples of Stein manifolds, are given in Chapter V. The canonical Frechet topology on the space of global sections 91(X) of a coherent analytic sheaf is described in Section 4. By means of the normalization theorem, which we do not prove in this book, we give a simple proof for the fact that, for a reduced complex space X, the canonical Frechet topology on H°(X, 0) is the topology of compact convergence. Chapter VI is devoted to proving that, for a coherent analytic sheaf So on a compact complex space X, H4(X, .9'), q >_ 0, are finite dimensional C-vector space

(Theoreme de finitude of Cartan and Serre). In this proof we work with the Hilbert space of square-integrable holomorphic functions and make use of the orthonormal basis which was introduced by S. Bergman. The classical Schwarz lemma plays an important role, replacing the lemma of L. Schwartz on linear compact maps between Frechet spaces. In Chapter VII we attempt to entertain the reader with a presentation of the theory of compact Riemann surfaces which results from, among other considerations, the finiteness theorem of Chapter V. The celebrated Riemann-Roch and Serre duality theorems are proved. The flow of the proof is more or less like that in Serre [35], except that, in the analytic case, a real argument for Ht (X, K) = 0 is needed. This is done in a simple way using an idea of R. Kiehl. The book closes with a proof of the Grothendieck theorem on the splitting of vector bundles over CP1.

The reader should be advised that, while the English version is not a word for

word translation of Theorie der Steinschen Raume, there are no significant changes in the mathematics. There are a number of strategies for reading this book, depending on the experience and viewpoint of the reader. Those who are not currently working the field might first browse through the chapter on applications (Chapter V). It gives us great pleasure to be able to dedicate this book to Karl Stein, who initiated the theory as well as collaborated in its development. Various prelimin-

XXI

Introduction

ary versions of our texts were already in existence in the middle 60's. We would like to thank W. Barth for his help at that time. It is our pleasure to express sincere thanks to Professor Dr. Alan Huckleberry

from the University of Notre Dame, South Bend, Indiana, for translating this book into English. Gottingen, Miinster/Westf.

H. Grauert

R. Remmert

Chapter A.

Sheaf Theory

In this chapter we develop sheaf theory only as far as is necessary for later function theoretic applications.

We mention [SCV], [TF], [TAG], and [FAC] as well as [CAS] as standard literature related to the material in this chapter. The symbols X, Y will always denote topological spaces and U, V are open sets. It is frequently the case that V c U. Sheaves are denoted by .9', 501, 9r, ... and for the most part we use S, S1, T, ... for presheaves.

§ 0.

Sheaves and Presheaves

1. Sheaves and Sheaf Mappings. A triple (50, n, X), consisting of topological spaces So and X and a local homeomorphism n:.9' - X from 50 onto X is called a sheaf on X. Instead of (So, it, X) we often write (.", n), .9' or just Y. It follows

that the projection it is open and every stalk Y. := n - '(x), x E X, is a discrete subset of 50.

If (501, n1) and (b"2, n2) are sheaves over X and (p: 9'1 -,f 2 is a continuous map, then q is said to be a sheaf mapping if it respects the stalks (i.e. if n2 = (P = n1).

Since g49'1-) c 592x, every mapping of sheaves tp: 501 - Sot induces the stalk mappings co: xY1x - b2x, x e X. Since n1 and n2 are local homeomorphisms, it follows that a sheaf map q : Y1 - 502 is always a local homeomorphism and is in particular an open map. Let (503, tt3) be another sheaf over X and suppose that 0: 502 -+.9P3 and W: S0i -+ 502 are sheaf mappings. Then tji o ip: 501 - 503 is likewise a sheaf mapping. Since id:." - .9' is a sheaf mapping, this shows that the set of sheaves over X, with sheaf maps as niorphisms, is a category. 2. Sums of Sheaves, Subsheaves, and Restrictions. Let (.9'1i n1) and (502, n2) be sheaves over X. We equip Y1 ®'©2 := ((Pie P2) E Y1 x 502: n1(p1) = 7E2(p2)} = U (` ° xeX

x 6"2x)

Chapter A. Sheaf Theory

2

with the relative topology in So, x .'2. Defining x: Y1 ®"2 -+ X by it(p,, p2) n,(p,}, it follows that (b", ®.2, n) is a sheaf over X. It is called the direct or Whitney sum of .9, and Y2A subset .9" of a sheaf .9', equipped with the relative topology is called a subsheaf of S whenever (.9', n I .9') is a sheaf over X. Thus S9' is a subaheaf of .9' if and only if it is an open subset of 91 and it J Y' is surjective.

Again let ," be a sheaf over X and take Y to be a topological subspace of X.

Then, with the relative topology on V JY := n-' (Y) c ,Y, the triple

(.9' Y,

it I ($" Y), Y) is a sheaf over Y. It is called the restriction of So to Y and is denoted

by .'J YorS"r. 3. Sections. Let .' be a sheaf on X and Y c X be a subspace. A continuous maps: Y -+.9' is called a section over Y if it o s = idr. For x e Y, we denote the "value" of s at x by sx (in the literature the symbol s(x) is also used for this purpose). Certainly s a Y. for all x e Y. The set of all sections over Yin the sheaf 5' is denoted by r(Y, .9). Quite often we use the shorter symbol .9'(Y). A section, s e 9(U), over an open set U c X is a local homeomorphism. The collection {s(U) - lJ s I U (-- X open, s e .9'(U)) forms a basis for the topology of .9'.

:.U

If qi: ,9', - 5`2 is a sheaf mapping then, for every s e .9',(Y), (p O s e .9'2(Y). Hence (p induces a mapping qiY: s --+(p o s. On the other hand, one can easily show the following: A map (p:.9', - 92 is a sheaf map if, for every p e .9',, there exists an open set ,9',(I) with p E s(U) so that the map fD o s: U -- Sot is a section in $"2 (i.e. (p 0 s E .9'2(0))

U c X and a section s E

4. Preebeeves sod the Section Fiattor r. Suppose that for every open set U in X there is associated some set S(U). Further suppose that for every pair of open sets U, Y c X with 0 $ V c U we have a restriction map ry: S(U) -' S(V) g

r'v=id and whenever W c V c U. Then S:= {S(U), rv} is called a presheaf over X. We note that a presheaf on X is just a contravariant functor from the category of open subsets of X to the category of sets. A map of presheaves D: S, Sts where Si = {S,(U), rUn,}, i = 1, 2, is a set of maps '& - {Ov}, OU: S,(U) - S2(U), such that, for all pairs of open sets U, V with V c U,

¢v r,r = r2,, a OU. Thus the presheaves on X form a category. For every sheaf .9' over X we have the canonical presheaf r(,9') := {.9'(U}, rv}, where rr(e) -= s I V. Every sheaf map (p:.9', -+ V2 determines a map of presheaves r(q): r(,9',) - r(y2) where r((p) :_ {(pU). The following is immediate:

r is a covariant functor from the category of sheaves into the category of prerho wes.

3

sheaves and Presheaves

§ 0.

5. Going from Presheaves to Sheaves, The Fanctor r. Every presheaf S = {S(U), rv.} over X determines in a natural way a sheaf .' which is defined as follows: For every x e X the subsystem {S(U), r', x e X} is directed with respect to inclusion of open neighborhoods of x. Thus the direct limit Yx := lim S(U) and xEU

the maps rx : S(U) --+ Y., are defined. We let

= U Sox and define a:.9' --+ X by XEX

a(p) = x when p e Y... Every element s e S(U) determines the set sv'= U r°(s) c Y. The system of subsets of .9', (SI1 U open in X, s e S(U)}, is a xEU

basis for a topology on Y. We equip So with this topology. Then it is easy to verify

that (.9', n) is a sheaf over X. We call r(S)=6P the sheaf associated to the presheaf S.

Let 0: S1 -. S2 be a map of presheaves, where S1 = {S;(U), rj ), i = 1, 2, and 0 _ (&v). Then 0 determines a sheaf map 1"'(4)): (S1) -. t(S2) in the following way: For p a 9' one chooses s e S1(U) with rixs = p and sets `(4,)(p):=r2x4v(s). It is easy to show that this definition is independent of the choice of s and that jc(4) is in fact a sheaf map. Thus 1`'

is a covariant functor from the category of presheaves into the category of

sheaves..

For every sheaf .9' we have the associated sheaf t'(r(So)). One obtains a natural

map gyp:.' -+ )'(I'(.9')) as follows: Let p e Y.. Take U to be an open neighborhood of p so that there exists s e .9'(U) with p = sx. Now define V(p) r. ,(s). Then cp is independent of the choice of U and s, and it is clear that (P is a sheaf map. It is

quite easy to check that 9:.9' -+ t(r(.')) is a sheaf isomorphism and thefunctors PI- and id are naturally isomorphic.

6. The Sheaf Conditions .9'1 and '2. For every presheaf S = {S(U), r'V} we have

the associated presheaf r (Ix(S)). Thete is an 'explicit map between these presheaves: For every s e S(U) the map x -+ rv(s) e Yx, x e U, is a section over U in Y:= I`'(S). This defines a natural map 4)u: S(U) S(U). One has no trouble verifying that

4

{4)v)

is

a presheaf map ¢: S -. 17(f (S)) which induces the identity

t(N): r'(S) -. t`'(S). A presheaf map {4)u} is called a mono-, epi-, or isomorphism whenever al1 of the

maps 4u are respectively injective, surjective, or bijective. The map ¢ above is in general not an isomorphism. It is easy to see that ¢u: S(U) -+ S(U) is injective if and only if the following condition is satisfied:

Sot. If s, t e S(U) are such that there exists an open cover {U,} of U with r°. s = rU, t for all a, then s = t.

In order to guarantee the bijectivity of ¢v, we must require even more: Let 4y: S(V) - S(V) be injective for every open V c U. Then dv is surjecti a (and thus

Chapter. A.

4

Sheaf Theory.

bijective) if and only if the following condition is satisfied:

,"2. Given an open cover (Ua} of U and s, a S(Ua) satisfying rv ,vp sa =

rU; vo s,, for all a and fi, there exists s e S(U) with ru,s = s, for all a. Thus a presheaf is isomorphic to the canonical presheaf associated to its sheaf if

and only if the conditions .9'1 and 612 are satisfied for all open sets. In the

literature, a sheaf is often defined as a presheaf satisfying YI and .9'2 for all open sets in X (e.g. [TFI, p. 109). Remark: One can formulate .9'I and .9'2 in an instructive way by requiring that the sequence S(U)

0

fl S(U,) W fj S(U2 n US), a.$

a

where u, v, w are obtained in the obvious way by restriction, is exact. This means that u maps S(U) bijectively onto the set of x e fl S(Ua) satisfying v(x) = w(x).

Example: For every open U contained in 68, let S(U) be the set of real-valued

continuous functions on U x U. Using the natural restriction maps, S is a presheaf. It is easy to check that S satisfies neither of the above axioms.

7. Direct Products. The interplay between the functors F and r is clarified by the way direct products are defined. If (.";), i e 1, is a family of sheaves on X then one defines S(U) fl .";(U) as the direct product of sets of sections and rV' as the jEI

product of all of the restriction mappings r&: S;(U) - Si(V). Then S - {S(U), r'V', is a pres.'ieaf over X. We set So :_ t(S) and call V the direct product of the sheaves

.9',. It is clear that 9' fulfills conditions Y1 and .92 and thus it is the canonical presheaf of S. We write

=I5". iet

Warning: For every x e X one has a canonical injection Y. -> j 19i.. But, for infinite index sets, it is in general not surjective. The point is that germs (pi), i e 1, of sections in Si., are not necessarily simultaneously realized as the restriction of sections on some fixed open neighborhood of x. Sheaves are frequently constructed using roughly the same procedure as we did for products: One begins with a sheaf, goes to the presheaf level via IF, defines the new presheaf and then returns to the sheaf level by means of t. In the next section we use this principle to introduce image sheaves. Later on, tensor product sheaves (but not Hom sheaves) are obtained in this way as well. 8. Image Sheaves. Let So be a sheaf over X and f: X -+ Y a continuous mapping from X into a topological space Y. To every open set V c Y we associate the set Y(f-'(V)).IfV' c Vthen we let the restriction mapping for sections. Then it is clear that the family {.9'(f -'(V)), p".} is a presheaf over Y which satisfies conditions 911 and .9'2.

5

Sheaves with Algebraic Structure

§ 1.

The associated sheaf r(.9'(f -'(V))) is denoted by fs(.9') and is called the (0-th) image sheaf of .So with respect to f. Due to the natural bijection .9'(j (V)) 99(f-'(V)) (f.(91))(V), we always identify (f*(9))(V) with Every germ a e f,(5")f(x) is represented in a neighborhood V of f (x) by a section s E .9'(f-'(V)). Since f -'(V) is a neighborhood of x in X, s determines a germ s, e Sox which is independent of the choice of the representation and uniquely determined by a. Thus it is clear that,

For every point x e X, there exists a natural map I.: fs('`9'i

-+.9'x, aF--.sx.

If 1p: 5°, .9' is a mapping of sheaves then, for every open set V c Y, one has the map (pf - l(V): (f.(S°1))(V) (f.(So2))(V). The family {(pf_,ty)} is a map of presheaves. We denote the associated sheaf map by fs((p). One sees that fs is a covariant.functor from the category of sheaves over X into the category of sheaves

over Y.

If, along with j, another continuous map g: Y - Z of Y into a topological space Z is given, then one has the sheaves (gf)s(,9') and gs(fk(S')) over Z. For every open set W c Z,

(g.(f.(")))(W) = (f.(9'))(g-'(W)) = S(f - 1(g-'(W))) _.((gf)-'(W)) = ((gf).(.0)(w). Thus

g.(f.(9'))_

(gf).(`').

9. Gluing Sheaves. Let {U.), E, be a covering of X by open subsets, and suppose that on each U. a sheaf .9'i is given. Defining Uij := U, n Uj, we further assume that for each (i, j) we have a sheaf isomorphism O,j:.9'j I Uij .9'i U,,. The family {Y,) is said to be glued together by {Oij} whenever the following "cocycle condition" is satisfied: ®ijt7jk = Dik

on

Ui n Ui n Uk fdr all i, j, k e 1.

From such a family one canonically constructs a new sheaf (see [FAC], p. 201): For every family of sheaves {.9'i} on X which is glued together by {Oi j} there exists a sheaf .9' on X and a family {Bi}i c 1 of sheaf isomorphisms 8i:.50 Ui .9'i so that

Oij = 0i - 0; ' on U. Up to an isomorphism the sheaf 9 and the family {8i) are uniquely determined by {.9'i} and {Oij}.

§ 1.

Sheaves with Algebraic Structure

In most applications the stalks of a sheaf carry additional algebraic structures. Sheaves of local C-algebras are particularly important for us. 1. Sheaves of Groups, Rings, and a-Modules. A sheaf So over X is called a sheaf of abelian groups if, for all x e X, the stalk Yx is an (additively written) abelian

Chapter A.

6

Sheaf Theory

group and "subtraction" .5 e . ° -+ Y, (p, q) --i p - q, is continuous. Note that if (p, q) t .9 a Y, then p, q e '1x with x:= 1r(p) = ir(q). Thus p - q is a well-defined element of Yx.

If fy is such a sheaf of abelian groups and 0x is the identity element in 9%, then the map 0: X ._91, x -+ Ox, is a section in .`P over X and is called the zero section. The set supp Y:= {x e X : 9x # {Ox}} is called the support of Y.

Using the additive structure in the stalks, .9'(U) is in a natural way an abelian

group for all open U c X (e.g. for all s, t c ,9'(U), s - t e Y(U) is given by (s - t)x ,= sx - tx for x e U). In reality, sheaves occur with even more algebraic structure. The further operations are defined analogously, the key point being that they are stalk-wise defined and are continuous. A sheaf of abelian groups .4 over X is called a sheaf of commutative rings, if,

along with the additive structure, there is a further sheaf mapping 9P ®9f , R, (p, q) -+ p q (multiplication), which makes every stalk .4x a commutative ring. If moreover every stalk Ax possesses a multiplicative identity lx, and if the mapping x -i 1x is a section in . (the identity section), then A is called a sheaf of rings with identity. In the following, 9i' denotes a sheaf of rings with identity over X. Obviously 1x # 0z for all x e supp M. A sheaf 9' of abelian groups is called a sheaf of modules over 9, or simply an 9P-sheaf or an 9i'-module, if a sheaf mapping R ® S -+ S is defined in such a way to induce the structure of an 9,,-module on .9x for all x e X. Obviously 9i? is itself an 9Z-module.

As in the case of sheaves of groups, the algebraic structure of a sheaf induces the

same structure on the set of sections via point-wise definitions. Thus 9t(U) is likewise a ring and, if L is a sheaf of .4-modules, the set .91(U) is an R(U)-module. If .9',, ... , .9 are sheaves of A-modules then the Whitney sum So, ® @.91P is an 9P-module with the operations being component-wise defined. In particular, for every natural number p, W° := Me . e : is an Vii-module.

2. Sheaf Homomorphlsms and Subsheaves. We introduce here the relevant notions for sheaves of a-modules. The analogous ideas for sheaves with other algebraic structures go more-or-less along the same lines and will not be discussed. In these considerations .9', and ,t2 are always R-sheaves.

A sheaf mapping .9'2 is called a sheaf homomorphism or an Mhomomorphism if, for every x e X, the induced mapping (p.,: Y,, --- .5r2x is an 9txmodule homomorphism. The sheaves of Ia-modules over X along with the 9P-homomorphism form a category.

In this category .90, and I V2 are isomorphic if and only if there is a sheaf mapping cp: ,51 -+ .92 so that q : 9'1s --+'92x is an 1x-isomorphism for all x e X.

.

§ 1.

7

Sheaves with Algebraic Structure

A subset .9' of 9 is called an a-submodule of .fix if .9' isa subsheaf of 9 and every stalk .9' is an Sax submodule of $ X.

If Y.' is an 9i,,-submodule of Y. for all x e X then 9":= U S; is an .

-

X.x

submodule of .5" if and only if .9' is open in Y.

It follows immediately that if So' and So" are a-submodules of ,", then their star

+ Y" = U (.9' + ,fix) and their intersection .0' n Y"- U (9°= n 9,,) are x.x XEx likewise a-submodules of Y. A sheaf of ideals J, or for short an ideal, is an R-submodule of the a-module A. For every ideal f c a one defines the product

.! $ := U .fX-9xc51, XEx < OD

where

fX - .Sx consists of linear combinations

a,,X

. s,,x, avx E fx, svx C-'9%.

Thus J X .7 X is an R. submodule of YX and, since 5 So is open in .9', .0 - Y is an 9i-submodule of Y. If q : '9' - 92 is an 9i-homomorphism, then the sets

(,p:= U ker q, and XEx

-Ow rp

U im v., are 9i-submodules of .9'1 and 92 respectively. If p: 9, -. 92

xox is

a sheaf homomorphism between sheaves of rings

(i.e.

each mapping

px: A,,, --' azx is a ring homomorphism with p.,(1,) = 1X), then 7t'e3 p is an ideal in 941.

A system of 9i'-sheaves and 9i-homomorphisms -±

r

i E 1,

is called an a-sequence. An. 9i-sequence is called exact at .9'r if S n (pi-, = .*'ex (pl. It is said to be exact if it is exact at every Sor.

3. Quotient Sheaves. Let .P be an a-module and .9' c Y. We set

an .a-submodule of

U 'x/."'z XEx

and define q:.9' - SP/SO' stalkwise via the canonical quotient homomorphism qx:,9% We use the finest topology on [//So' for which q is continuous. Thus a set W c So/5o' is open if and only if q-'(W) is open. Since q($,,) = .9'X/.fix, we have the natural projection ft: Y1,9" -- X so that n q = it. Thus we have the following:

The triple it, X) is a sheaf of .gt'-thodules and q: ," -,1/9" is an Repimorphism with Jtet q = .9'. We call So/.9" the quotient sheaf of So by .°'.

Chapter A

g

Sheaf Theory

Every 9e-homomorphism rp: 99, - b"2 determines the exact R-sequences

0 -- Jt''et rp -- .V, -- .gym (p-0, and 0 -+ .gym cp -- Soz .

b°2 /.gym rp -+ 0,

where 0 denotes the zero sheaf. 4. Sheaves of Local k-slgebras. -Let k be a commutative field and 't-:= X x k the constant sheaf of fields over X (i.e. n: it'' --+ X, (x, a) -. x is the projection). A sheaf of rings 9e is called a sheaf of k-algebras if R is a *'-sheaf with supp 9 = X

such that c(r, r2) = (cr,)r2 for all c e f, and r,, r2 e R.,. In particular, the identity section 1 e 9t(X) is nowhere zero and i:.Jtr -+.*, (x, a) - a - 1X is a sheaf monomorphism (of rings). We identify it' with i(K) c R and k with kUX c RX. A sheaf 9 of k-algebras is called a sheaf of local k-algebras if every stalk RX is a

local ring with maximal ideal m(RX) so that the quotient epimorphism RX -+ RX /m(RX) always maps k onto A. I m(RX).

One identifies RX /m(AX) with k and has a canonical decomposition 9, = k p m(MX) as a k-vector space.

Example: Every topological srace X carries the sheaf 'P of germs of complexvalued continuous functions: The C-algebra '8(U) of continuous functions f: U C is defined for all open sets U c X and' r°: AB(U) -+ W(V), V c U, is the natural restriction. The system {AB(U), r'V) is a presheaf of C-algebras which satisfies .9'l and ."2 and determines the sheaf W. This is a sheaf of local C-algebras such that maximal ideal m(,8,,) consists of the germs f. a W,, which are represented in neighborhoods. of x by continuous function f which vanish at x. If R is a sheaf of local k-algebras and s e R(Y) is a section over a subset Y c X, then s has a value s(x) in k for all x e Y, namely the equivalence class of the germ s,, a A. in k. Thus every section s e R(Y) defines a k-valued function [s]: Y --+ k. The homomorphism s [s] is not in general injective. In other words, a section s is more than the function [s]. A sheaf mapping (p: R, 91.2 between sheaves of k-algebras is called a khomomorphism if every induced map gyp.: Al., -+ 912x is a k-algebra homomorphism. It is blear that k-homomorphisms between sheaves of local rings over k are automatically stalk-wise local (i.e. q, (m(R,X)) c m(R2X4

5. Algebraic Reduction. We let n(RX) denote the nilradical (i.e. the ideal of nilpotent elements) of the stalk RX. Then

4w):. U 11(AX) c R XEX

is open in R and is consequently a sheaf of ideals. We call ii(R) the nilradical of R. The sheaf of rings Red 9i? _= R/tt(R) is called the (algebraic) reduction of R. If

R is a sheaf of local rings over k, then since n(RX) c m(RX), Red R is likewise such a sheaf. We say that R is reduced whenever ti(R) = 0. For example, the sheaf W is reduced.

Sheaves with Algebraic Structure

9

Remark: In the case where .

is a sheaf of local rings the set U nt(. j)is'not

§ 1.

xex

necessarily open. Thus it is not in general a subsheaf of 9i and a construction analogous to the above, replacing u(9tx) with nt(9tx}, does not make sense. 6. Presbeaves with Algebraic Structure. A presheaf S = {S(U), r1V} over X is called a presheaf of abelian groups if S(U) is always an abelian group and rf is always a group homomorphism. A presheaf of rings R = {R(U}, Fvv) is defined analogously. In the following R denotes a fixed presheaf of rings. A presheaf S is called a presheaf of R-modules (an R-presheaf) if every S(U) is an R(U}module and for all a c R(U), s e S(U) it follows that r°(as) = Fuv(a)ru(s). If ." is an 9t-sheaf then F(.9') is an r(9i'}presheaf. On the other hand, if S is an R-presheaf, then t(S) is an t(R}sheaf. One just carries over the algebraic structure via the direct limit map. The continuity of the operations is evident. S2, 0 = (4u) is a presheaf mapping with A presheaf homomorphism ¢: S1 every Ou being a homomorphism of the underlying algebraic structure. The map ping t(4) is thus a sheaf homomorphism. Conversely, every sheaf homomorphism x:.9'1-- b"2 determines a presheaf homomorphism r((p). In the category of R-presheaves, just as in the case of .4-sheaves, we have subpresheaves and quotient presheaves. An R-presheaf S' =.{S'(U), r''C) is called an R-subpresheaf of the R-presheaf S if every S'(U) is an R(U)-submodule of 8(U) and r', is always the restriction of ru to S'(U). If S' is an R-subpresheaf of S, then ,I(U) S(U)/S'(U) is always an R(U}module and, for every open set V c U, the map rV: S(U) -* S(V) induces an R(U)-homotorphism Fu: 9(U) -- S(V). Obviously 9:= { (U , iv} is an R-presheaf. It is called the R-quotient presheaf of S by S' and we write = S/S'. S Every R-presheaf homomorphism 4:.St - S2 determines the R-presheaves Ker ¢ _ {Ker 4,,, pp') and

Im 0 = {Im 4u, v4} where p'v and au are defined in the natural way. We say that an R-sequence St -f- S2 -` . S3 of R-presheaves is exact whenever Im 0 = Ker +(i. Thus every R-presheaf homomorphism 0: S1 - S2 determines two exact sequences of R-presheaves:

and O-.Im4)

0.

7. On the Exactness of Jr and 1. Since the direct limit of exact sequences is again exact, every exact R-presheaf sequence St -m - S2 - S3 induces an exact 1`(R)-sheaf sequence t'(S1) rL f(S2) 12(".' t(S2). In other words l1`' is an exact

functor from the category of R-presheaves to the category of sheaves of r(R)-modules.

In contrast to this, the section functor r is only left-exact. A short exact -4-

.

Chapter A. Sheaf Theory

10

sequence 0 '

So

.9"

0 does induce the exact r(91!)-sequence

o -. r(b,)

-. r(y) -

rp-)

of canonical presheaves. However the last homomorphism is not in general surjec-

tive. Its image is the quotient presheaf s:= r(.9')/r(Y,) and, since 0 - r(b") r(So) -, s 0 is exact and I' is an exact functor, it follows that 1 (S) = Y". Nevertheless S is in general a proper subpresh,;af of r(.9"). The phenomenon of nonexactness of r is the beginning point of cohomology theory.

Coherent Sheaves and Coherent Functors

§ 2.

The notion of coherence of a sheaf of modules is of fundamental importance in the theory of complex spaces. In this section we compile the general properties of coherent sheaves. The symbol 5i< always stands for a sheaf of rings over a topological space X. We use ,90,1,', etc. to denote sheaves of A-modules. 1. Flake Sheaves. Finitely many sections s1, ... , s p E .9'(U) define an 9tu-sheaf homomorphism. au:

(Q ` u, (a1,...., aps)-.a(a1s, ..., aps) ((

P

Y aissi., x e U.

1=1

.

We say that Yu is generated by the section s1, ..., sp if a is surjective. In this case

every stalk .9',i, x e U is a finitely generated (by s1, ..., sp,,) A.module. An A-sheaf So is said to be finite at x e X if there is an open neighborhood U of x and with finitely many sections s1, ... , SP e .9'(U) which generate .9'u. This condition is equivalent to the existence of a neighborhood U of x, a natural number p, and an

exact sequence At -f- .9e ---' 0. The germs of the a-images of the basis

..., 81p) a 9t'(U), i = 1, ..., p, generate Yu as a sheaf of 9Ip-modules. If it is possible to choose U so that a is an isomorphism, then one says that .9' is free at the point x. In this case, the exponent p such that .emu = At is uniquely determined by .9' and is called the rank of .9' at x. An 9e-sheaf So is called finite (resp. locally (811,

free) on X if it is finite (resp. free) at each x e X. It is free if it is globally isomorphic to AP for some p c- F Quotient sheaves of finite sheaves are finite. On the other hand, subsheaves of

finite sheaves are not necessarily finite (even in the case every stalk .9'x is a Noetherian 9i!,,-module). Thus finiteness at x says more than the fact that the stalks are finitely generated in some neighborhood of x. Among the important properties of finite sheaves is the following: If 91 is finite at x and s1, ..., sp E .9'(U) are such that s1ic, ... , s p,, generate Sos as

an A,,-module, then there is a neighborhood V c U of x so that s 1 I V, ..., s,, I V generate the sheaf 91,. In particular, the support supp 9' of a finite sheaf is closed in X.

§ 2.

11

Coherent Sheaves and Coherent Functors

2. Finite Relation Sheaves. If a: ME -* .9'v is an Silo-homomorphism which is determined by sections st, ..., sP e .(U), then the sheaf ofMv-submodules in M{, Re! (st, ... , sP) := Ker a = Y `(a tx, ... , aPx) E

I E aizslx = 0

is called the sheaf of relations of s ..., s,,. One says that .So is a finite relation sheaf at x e X if, for every open neighborhood U of x and f o r arbitrary sections s , ,. .. , SP E Y(U ), the sheaf of relations i et(s 1, ..., s,,) is finite at x. This is the case if and only if, for every sheaf homomorphism a: At -..Sou, Ker or is finite at x. A sheaf of i9-modules .9' is called a finite relation sheaf if it is a finite relation sheaf at all

xEX. 3. Coherent Sheaves. A sheaf of .-modules .9' over X is called coherent if it is finite and a finite relation sheaf. Thus So is coherent if it is coherent at every x c- X (i.e. whenever for every x e X there exists a neighborhood U = U(x) so that Y Iv is coherent). If .5o is coherent then every finite subsheaf of yF-submodules of .9' is likewise coherent. A sheaf of rings 9P is called coherent if gP is coherent as an a-module. This is the case precisely when 3P is a finite relation sheaf. A sheaf of ideals f in yP is said to

be coherent if it is coherent as an A-submodule of A. If -4 is coherent, then the product f t f 2 of coherent ideal sheaves f t, f 2 is likewise coherent (f 1 f 2 is finite!).

If .50 is a coherent a-module then, for every x E X, there exists an open neighborhood U of x with positive integers p and q such that

It - JM -. you -, 0 is exact.

The following is basic for many relations between coherent sheaves: Five Lemma: Suppose that

t m 502

.9 4 .Ys

ms..5.03

is an exact sequence of sheaves of a-modules such that Yt, .5 '2, £14 and .9, are coherent. Then ,5'3 is likewise coherent.

This remark is equivalent to the following: Three Lemma (Serre). If, in the exact £?-sequence

0 ....9'

5.0

50 -- 0,

two sheaves are coherent, then the third is also coherent.

Cr

12

ott & Shelf Theory

We note some important consequences of the Five Lemma: a) L--( cp:.9' - So' be an A-homomorphism between the coherent sheaves .9', Y'.

Then Jire2 cp, fm rp and WoLe2 rp = S'/.gym (p are coherent sheaves of

9P-modules.

b) The Whitney Sum of finitely many coherent sheaves is coherent. c) Let .9" and .9" be coherent . -submodules of a coherent A-module Y. Then

the 91-sheaves ,9' + Y" and .9' n 91" are coherent. d) Let 9P be a coherent sheaf of rings. Then the sheaf of 9P-modules .9' is coherent if and only if, for every x e X, there exist a neighborhood U of x, with positive

integers p and q and an exact sequence

In particular every locally free sheaf of R-modules is coherent.

4. Coherence of Trivial Extensions. If / is an ideal in -4 and 9P// is the associated quotient sheaf of rings over. X, then every sheaf of Alf -modules 91 is in a canonical way a sheaf of 9P-modules. One can study the coherence of .9' as an

9l//-module as well as an 91-module. In this situation we have the following remark:

Let 9t and / be coherent and .9 a coherent R//-sheaf. Then 5o is coherent as an R//-sheaf if and only if it is R -coherent. In particular 11,f is a coherent sheaf of rings.

This implies that, if 9 and the nilradical n(9P) are coherent, then 9PEd A = . / n(:1P) is a coherent sheaf of rings.

The coherence of 9P/f implies that X':= supp(9P//) is a closed subspace of X and that A':= (9P//) I X' is a coherent sheaf of rings on X'. Every 91'-module .I' on X' has a trivial extension .9 on X (i.e..3 = 0 for x e X\X'). In a natural way .9 is an a-sheaf. Denoting the embedding by i: X' -+ X, one can identify . with the image sheaf The following fact (which is a special case of the finiteness theorem, Chapter 1.3) is particularly useful for applications in function theory: Let . and / be coherent, X' '= supp(.W//), and M';= (11,f ) V. Let ' be an A'-sheaf. Then .°l' is coherent if and only if the trivial extension .97 is a coherent sheaf 9t-modules on X.

S. The Functors ®P and N. The system T'_ {.'(U) ®(u) .9"(U), ruv ® r'yu} is

an I ( )-presheaf which satisfies SI and S2 (for the definition of ®, see the standard literature). The associated 91-sheaf 50 ®sr 9°' _ (T) is called the tensor

product of . and 9' (over 9t). It always follows that (. ®ar Y 'XU) = .°(U) ®a(U) 6"(U)

13

Coherent Sheaves and Coherent Functors

§ 2.

and

(v OR 9"),,= Y. ON. Jz'. The tensor product functor is covariant in both entries, additive, and right exact. Moreover, if So and .9' are coherent (resp. locally-free) then 5" OR So' is coherent (resp. locally free).

One defines the p-fold tensor product, ®p, p = 0, 1, 2, ..., inductively by

®p1/,_((& -t ,y)®,9' 1,

with ®° .' .= R. In ((&p Y)., we consider the . ,, submodule ., ll. which is generated by ar,= a, for some pair (p, v) with y#v. Then ..# '= U .ll His an l-subsheaf of ®p Y. The quotient sheaf XEx ApY:_(®".')/.,ll,

p=0, 1,2,...,

is called the p-fold exterior product of.'. Note that A° ." = 9F and nt 5' = So. If

e: ®p .' -- /\" Y is the quotient homomorphism, then e(at ® (gap) =-at n A ap. It follows that

Ap is a covariant functor and, if.S" is coherent (resp. locally-free), then Ap .s' is likewise coherent (resp. locally-free). 6. The Ftmctor Jt°om and Annihilator Sheaves. For -4-sheaves 5o and Y', the set H(U) at°omaw(.S°' U, .9' U) of all 9 ( U-homomorphism .9 j U --+ Y' l U is always an 9r'(U)-module. The restrictions rt: H(U) - H(V) are canonically at hand and the resulting system H:= {H(U), rv) is a r()-presheaf which satisfies

St and S2. The associated 1-sheaf Jt°om.(S', Y')- f (H) is called the sheaf of germs of 9-homomorphisms from .' to .9'. It is always the case that .*' ma(9, .9')(U) = Jroma,v(.9 I U, 59' 1 U).

The functor A'oma is contravariant in the first argument and covariant in the second argument. Additionally, W om, is left exact. Remark: For all x E X there exists a natural 9i'X homomorphism p,,:.ll°oma(9', Y')., - .)romR,,(.',,, Sox) which is in general certainly neither injective nor surjective. The reader should note that the R(U)-module Jt°amatv)(.9'(U), 91(U)) cannot be used for the definition because, among other things, the restrictions ry do not exist.

As in the case of the tensor functors, the .*'om-functor is also coherent:

If .9', 9' are coherent (resp. locally free) then WomR(Y, .5') is also coherent (resp. locally free).

For an 9-sheaf .', we define Woe .',, __ jr, a 5i_ 1). However there exists a largest coherent ideal. This and more is the content of the following famous theorem of H. Cartan and K. Oka.

Coherence Theorem for Ideal Sheaves. Let A be a closed subset of a complex space (X, Ox) such that for every point a e A there is a neighborhood V of a in X and holomorphic functionsfl, ..., f,, e Ox(V) so that

An V = {xeVI f1(x)

=fa(x)=0).

Then A is an analytic set in X. More precisely, the system (5(U)), where U is open

in X and 5(U)'= {f e Ox(U)I f(A n U) = 01, forms a presheaf for the.coherent

ideal sheaf J. It follows that A = supp(Ox/.!), and the pair (A, OA) with OA '= (Ox/JA is a closed reduced complex subspace of (X, CA)'. The sheaf .! is called the nullstellen ideal of A. For every Ox-ideal f,.oi e,4e in4s,

§ 3.

19

Complex Spaces

the radical ideal rad 5 stalkwise by (rad .5)s:_ {fx e &x,s fx e JX for some n e N}. We now formulate the so-called Nullstellensatz of Hilbert and Riickert. The proof of this as well as that for the coherence theorem can be found in [CAS].

It follows that rad f is likewise an

C'x-ideal.

Nulstellensatz: For every coherent ideal 5 c (fix, the radical ideal rad 5 is the nullstellen ideal of A :=supp((r,x/5). The,coherence theorem for ideal sheaves therefore implies that if .f is a coherent ax-ideal then rad f is also coherent. It is well-known that every topological space can be decomposed into its connected components. A much stronger decomposability lemma can be proved for

analytic sets. A non-empty analytic set A in a complex space X is said to be reducible in X if there exist non-empty analytic sets B, C in X with A * B and A * C so that A = B u C. If this cannot be done, then A is called irreducible. Decomposition Lemma. Every non-empty analytic set A in X has a decomposition A = U A; with the following properties: lEl

0) The index set I is at most countable. 1) For every i E I the set Ai is an irreducible analytic set in X, and the family (A,)j E 1 is locally finite in X.

2) For all i, j e I with i

j the intersection A, n A; is nowhere dense in A;. The

family {Al}, E f is uniquely determined by A up to a change of indexing. One calls

the sets Al the irreducible components (branches) of A in X.

If U is a non-empty open subset of X and A is an analytic set in X, then A n U is an analytic set in U. If A, is ar; irreducible component of A in X, then A. n U is not in-general irreducible in U. The .branches of A n U. are the branches of the non-empty intersection A, n U for all irreducible components A; of A in X. Identity Theorem for Analytic Sets. Let A, A' be analytic sets in X, and suppose

that U is an open set in X so that A n U and A' n U have a common branch as analytic subsets of U. Then A and A' have a common branch in"X. 6. Dimensifm Theory, Every complex space X and more generally every analy-

tic set A in X has at every point x E A a well-defined topological dimension, dim tops A e N. This number is always even, and one defines the complex dimension of A at x by dims A

I dim tops A

The number codims A

dims X - dims A

is called the complex codimensitm'st x of A in X.

Chapter A. Sheaf Theory

20

Every stalk O., x e X, is a local C-algebra and thus has an algebraic dimension dim Os (see [AS], Chapter II.4). A non-trivial theorem says that

dims X = dim 0, for all x e X. The following are standard (useful!) results from the dimension theory of analytic sets:

a) The analytic set A is nowhere dense in X if and only if it is at least 1-codimensional.

b) A point p e A is an isolated point in A if and only if dun, A = 0. c) If A is irreducible, then A is pure dimensional (i.e. thefunction dims A, x e A, is constant).

The number dim A= sup(dims A I x e A) is called the complex dimension of A. We note that the case dim A = oo is possible.

d) If

{A,}, E r is the set of irreducible components of A, then dim A = sup{dim A, i e I). If dim A < oo, then there exists j e I with dim A = dim A,. e) If B is likewise analytic in X and B c A, then dim B < dim A.. If dim B = dim A < oo, then A and B have a common branch.

7. Reduction of Complex Spaces. The following is a fundamental theorem-of H. Cartan and K. Oka [see CAS]: The nilradical n(Ox) of a complex space X = (X, Ox) is always a coherent ideal.

Since supp(Ox/n(Ox)= X, it follows that

red X :_ (X, Ored x with

red x == Ox/ n(Ox),

is a complex subspace of X. It is called the reduction of X. The associated holomorphic mapping red X --+ X is likewise denoted by "red" and is called the reduction mapping.

The structure sheaf of red X is in a natural way a subsheaf of the sheaf `Bx of germs of continuous, complex valued functions (i.e. Ored x c `Bx). For all s a O(X s(x) = (red sXx) for all x a X. Every holomorphic map f: X -+ Y between complex spaces canonically determines a map red f red X --*red Y of the reductions such that the following is commutative: red j

red X ------= red Y red

X -- r

One easily verifies that red is a covariant functor.

§ 3.

21

Complex Spaces

A complex space X is said to be reduced at the point x e X if the stalk Ox is reduced (i.e. n(Ox) = 0). The set of non-reduced points of X is the support of n(Ox), and is therefore an analytic set in X. The space X is called reduced whenever it is reduced at all-of its points (i.e. whenever X = red X). The space red X is reduced,

and thus red (red X) = red X.

A point x in a complex space X is called regular or non-singular if the stalk Ox is regular (i.e. isomorphic to a C-algebra of convergent power series). Every regular point is reduced. The non-regular or singular points form an analytic set in X, S, the so-called singularity set of X. The set X\S is a (possibly emp) complex manifold. If X is reduced, then S is nowhere dense in X and is in particular everywhere at least 1-codimensional.

& Normal Complex Spaces. For every complex space (X, (0) the set N c C of elements which do not divide zero is multiplicative (see Section 2.7). Thus the sheaf of quotients .,ll = ON

with

..Kx = (Ox)N.,

x c- X,

is well-defined and is an O-module. One calls .1t1 the sheaf of germs of meromorphic

functions on X. The sections in.* (over X) are the meromorphic functions on X. The reader should note that . K is-not a coherent O-sheaf.

If X is reduced at x, then a germ fx e Ox is in Nx if and only if there is a neighborhood U of x and a representative f e O(U) of fx so that the zero set off is nowhere dense in U. In this case it follows in particular that f -' e ..f!(U). If X is irreducible at x (i.e. Ox is an integral domain), then .lfx is the quotient field of Ox.

A complex space X is said to be normal at x e X whenever X is reduced at x and the ring Ox is integrally closed in &.,. Every regular point is normal. The following is a famous theorem of Oka (see [CAS]): The set of non-normal points of a complex space X is an analytic set in X.

A complex space X is called normal if it is normal at each of its points. The space is everywhere at least

singularity set S of a normal complex 2-codimensional. We will use the following Chapter V :

Riemann Continuation Theorem. Let X be a normal complex space and A an analytic set in.X. Then the following hold:

1) If A is everywhere at least 1-codimensional, then every function which is continuous on X and holomorphic on X \A is holomorphic on X : O(X) = W(X) n (',(X \A) 2) If A is everywhere at least 2-codimensional, then the restriction homomorphism O(X) O(X \A) is bijective.

Chapter A.

22

Sheaf Theory

Normalization Theorem. For every reduced complex space X with singularity set S there is a normal complex space 9 and a finite' surjective holomorphic mapping

t: X

X. The manifold X\l;-'(S) is biholomorphically mapped onto X\S, and

'(S) is nowhere dense in X.

The pair (X, S) is called a normalization of X. Normalizations are uniquely determined up to analytic isomorphisms. If (X, ) is a normalization of X, then the sheaf Ox is an analytic subsheaf of the coherent direct image t9x-sheaf i;,((Og).

The reader can find proofs of these theorems in [CAS].

§ 4.

Soft and Flabby Sheaves

Many important theorems in classical analysis have the following form: Sections of a certain sheaf which are only defined on certain subsets can be extended

to the entire space. The material in this section is devoted to this and related matters.

1. Soft Sheaves. The section continuation problem is satisfactorily solved "in the small" by the following: Theorem 1. Let So be a sheaf (no algebraic structure implied) on a metrizable space X and let t be a section in .9' over a set Y c: X. Then there exists a neighborhood W of Y in X and a section s E S(W) with s I Y = t. The proof uses methods of general topology (see, for example [TF], p. 150). A sheaf .' on X is called soft if, for every closed set A X, the restriction map .9'(X) -+ Y(A) is surjective (i.e. if every section over A is continuable to a section over the entire X). For sheaves of modules one has a handy softness criterion in the form of a separation condition:

Theorem 2. Let X be metrizable and . a sheaf of rings (with identity) over X. Suppose that, for every closed subset A in X and every open neighborhood W of A in

':, there exists a section f e *(X) such that

flA=1 and fjX`W=O. Then every s-module 9 is soft. Proof: Let A c X be closed and t e Y (A). By Theorem 1 there exists an open neighborhood W of A in X and a section s' E Y(W) with s' l A = t. By assumption = Finite holomorphic maps are studied in detail in Chapter 1.

Soft and Flabby Sheaves

f 4.

23

there exists f e M(X) with f f A = 1 and f f X \W = 0. Defines e .V(X) by

s(x) -f (x)s'(x)

for

x e W and

s(x) -= 0

for x e X kW.

Then s is the desired extension of t. 2. Softness of the Structure Sheaves of Differential Manifolds. The sheaf ' of real-valued, continuous function germs on a metrizable space satisfies the separation condition of Theorem 2 (such a space is normal!). Thus on such spaces every

sheaf of W-modules is soft. In the following it will be shown that the sheaf of real-valued infinitely often differentiable functions on a differentiable manifold

also fulfills the separation condition. The proof is based on. the following lemma:

Lemma 3. In R' with coordinates x,, ..., x" let Q and Q' be two open "blocks",

Q:_ {x all"Ia,

Q'__{xead"'I a,<xa 0.

As standard literature, we refer to [EFV], [FAC] and [TF]. We use the same notation as in Chapter A.

§ 1.

Flabby Cohomology Theory

In this paragraph we give a brief report on the basic theorems of flabby cohomology theory. Ii particular we show that an arbitrary acyclic resolution can be used to compute the cohomology groups (the formal de Rham Lemma).

1. Cobomology of Complexes. Let R be a commutative ring (in all of the applications R == 5g(X )). A sequence Ko

'K'-

P

Kgi1 1 ...

of R-modules and R-homomorphisms is called a complex if dg+' dg = 0 for all q.

We write K' = (Kg, dg) for such a complex. The elements of Kg are called qcochains and the maps dg are called coboundary mappings.

If K" = (K'g, d'g) is another complex, then a homomorphism of complexes cp': K" -+ K' is a sequence cp' = of R-homomorphisms Mpg: K'g -+ Kg which are compatible with the coboundary mappings: dgcpg = (pg+, d'g, q >_ 0. With these as morphisms, the complexes form an abelian category.

For every complex K' one defines the R-modules Zg(K') -.= Ker dg

and

Bg(K') _= Im dg-1,

29

Flabby Cohomology Theory

1.

the q-cocycles and q-coboundaries. Since d"' Id" = 0, it follows that Z"(K') c Bq(K'). Thus we may define the q-th cohomology module of a complex K' by q>_ I.

H°(K'):=Z°(K') and Hq(K'):=Zq(K')/Bq(K'), The elements of Hq(K') are called cohomology classes.

If : K" - K' is a homomorphism of complexes, then rpq(Z9(K")) c Zq(K') and cpq(Bq(K")) c Bq(K'). Thus cp' induces homomorphisms Hq(K,.)

H9((q'):

- Hq(K'),

q >_ 0,

of cohomology modules. Thus Hq is a covariant additive functor from the category of complexes with values in the category of R-modules. A sequence K" -# K' -f-' K" of complexes, where (p' and /i' are homomorphisms of complexes, is exact whenever each of the sequences K"q is exact.

Kq -*q

K'q

The following is fundamental for the cohomology theory:

Lemma 1. Let 0 -+ K" K' - - K" -+ 0 be a "short" exact sequence of complexes (0:= zero complex). Then there exists a natural connecting homomorphism M: Hq(K"') -+ Hq+' (K"), q >_ 0, which depends functorially on q' and so that the long cohomology sequence 0

-

H°(K") - ... -

Hq(K") _+ Hq(K')

,

4 Hq+ (K") - .. .

HQ(K'-)

is exact.

Furthermore, the long exact sequence in cohomology preserves commutativity: Let

0 -- K" 0

K'

.

I

I

L"

L'

K"

0

I

--p

0

L"

be a commutative diagram of exact sequences of complexes. Then the diagram of long exact cohomology sequences,

... _, H4(K')

Hq(K-)

1

' Hq+'(K".) _

1

1

Hq(L') -Hq(L"*) is everywhere, commutative.

6'

as

.

1

Chapter B. Cohomology Theory

30

2. Flabby Cobomology Theory. Given an 9i'-resolution

of an 9t-sheaf .9, there exists the associated complex at the section level,

g-°(X)

9"9(X) r-i- ...

g-1(X)

where we always write t94 instead of 1'(t'). Hence one has the cohomology modules

H°(9''(Y)) = Ker t° = .(X), H'(9"'(9y)) = Ker t',,/lm t4*-

q >: 1.

In section 4.3 we embedded every 9t-module .9' in a functorial way in a flabby

9t-sheaf 3r° _= F(b°): 0 -± .y i ' A°. This procedure can be iterated: Let 1,= F(Fo/j(9 )) and let fo: F° A 1 be the composition of the quotient homomorphism 9Fo -, .F°/j(,') with the injection F0/j(So) -+ 9r 1. If 0-.. ,yf .moo ' Jr 1 an exact A-sequence with .4r' flabby for i < q, then we set 99+ 1 =_ A(f'/Jm f" - 1), where f " 1 -j, and we define f' to be the composition of All F'/fm f' with.F9/Jm f q-1- 99+ 1

.-.Jr" is

It follows that every 91-sheaf 9 possesses a canonical flabby £ -resolution

0J+

J° Jro(fy)

1(.0)

...

,,r9(Y) J4 ....

We use F(.9') to denote the associated complex at the section level. Definition (Cohomology Modules): The 9t-modules

H4(X, Y)'= H'(.F'(.y)),

q = 0, 1, ., ...,

are called the cohomology modules of the A-sheaf .50 over X.

Since the flabby funetor.F is covariant and additive, .9'µ+. '(9y) is also such a functor. This has the following consequence:

1) For every 9 Z 0, the functor .y"H9(X, .y) is additive and covariant. The functors 9"-i S(X) and Y--,H°(X, 9') are isomorphic. Since F is an exact functor, Lemma 1 implies the following:

II) Every exact A-sequence 0 - Y'--+ 9 -+ 90" -+ 0 functorially determines a connecting homomorphism induced long cohomology sequence,

0 -. H°(X, Y') -.... is exact.

q ;:0,

that the

H4(X, so) -+ H,(X, ,So") ft H9+ 1(X, ,9') -+ ...

t

31

MebbW Cabomotoa Theory

Remark: The above two statements are frequently used in applications (e.g. in the theory of Stein spaces) in the following manner: Given a sheaf epimorphiem p:.° 5°" for which H'(X, . es p) = 0, then the induced homoroorphism Y(X) - ."(X) between modules of sections is surjective. It should be said here that in this book there appear to be no applications of the higher eohomology functor H', q > 2. However, if in the Stein theory one wants to prove vanishing theorems of the form H'(X, .5/) = 0, q 2! 1, (Theorem B) by a "method de descente," then they are nevertheless indispensible: One first proves

ft claim for every large q and sets it for all q by decreasing induction from q to

q- 1.

Since cohomology preserves commutative diagrams, we have the following:

II') if 0

-

.r

-Y

Y"

iii

0

is a eanm utative diagram of sheaves and sheaf homomorphism whose rows are exact, then all of the induced diagrams

H'(X,

a

.

H4(X, S ") s`

.

H'+'(X,.')

H4+i(X, 9°)

are commwtative.

Since the Exactness Theorem of Section 4.4 says that every flabby resolution of a flabby sheaf determines an exact sequence at the level of sections, we see that, III) for every flabby sheaf So of abelian groups over a topological space X,

H'(X, b) = 0 for all gZ 1. We now prove the following uniqueness theorem for cohomology. Let fi' be a sequence of functors along with connecting homomorphism 3', q,? 0 having dte properties in I), III and III). Then for every q;-> 0, there exists a natural funetor isomorphism F': '(X, S) -+ H'(X, S) which is compatible with the connecting homomorphisms.

Proof: (by induction on q) The existence of F° is clear from I). The exact

a-sequence 0 -. f° - 3 -+ P -+ 0 with the flabby sheaf ' F -.i°(SP) and

Chapter B. Cohomology Theory

32

9-:_.F/.' determines, by II) and III), the following commutative diagram with exact rows (already F° is an isomorphism):

o-' R3°(x, $') -- R°(x, .f) -I1°(x, .f)

10

R1(X, ,Y) ----+ Q j+ t

IFO

Iro

I

o -H°(X, .) -+ H°(x, .f) -+ H°(X, if) - H'(X, s )-O One sees that there is exactly one isomorphism P: 17'(X, ,') H' (X, .') which is compatible with S° and 6°. Let q > 1 and suppose that F" has already been constructed. Since if is flabby, the cohomology sequences give us the following commutative diagram with exact rows:

R4-'(x -'r)

0

as-'

H4(x, .f) -- 0 FS

I F4-1 4

He -' (X, .Y-)

0

ba

,

H4(X, .J')

0

that there exists a unique isomorphism F4: R4(X, 9')- H"(X, 9'), which is compatible with 34-', 61 -'. Again it follows immediately

The Exactness Theorem of Section 4.4 also implies a result which is important for applications in Chapter II.4:

IV) If X is metrizable and So is a soft sheaf of abelian groups on X, then H4(X, So) = 0 for all q >_ 1. For a proof one needs only to observe that in the flabby resolution of 9' all of the sheaves.f 4(.9') are now automatically soft (compare 4.3).

3. The Formal de Rham Lemma. In order to calculate the cohomology modules H4(X, 9'), one does not absolutely need the canonical flabby resolution of Y. Following the standard nomenclature, we call an .*-resolution, 0 So 9-0 . --+ i4 --. , of ,' acyclic if, for all n > O and q >_ 1, H4(X, ._:r") = Q. Theorem (Formal de Rham Lemma). Let

0-,

-'. 370..

...-..r4

be an acyclic resolution of So and if (,") the associated complex at the section level. Then there exists natural A(X)-isomorphisms TP: HP(-_T (.9'))

HP(X, CI),

p = 0, 1, 2, ....

Proof: We construct the ru's inductively. The existence of To is clear. Let _ 7f' ea t' = 9"°/i(.'). In the oohomology sequence associated to_ 0.- ,

33

Cech Cohomology

§ 2.

9-o)_-, ..., we H'(Xi SP) -+H'(X, 1"0(X)--- k(X) an 9i'(X}' induces have k(X) = Ker t; and Im t° = Ker 8°.' Thus 6°

9-0-4

monomorphism TI: Ker t' /Im t° -+ H'(X, 9) which, since H'(X, 9-0) = 0, is also surjective.

For .9 we have the acyclic resolution

C' 1 g ' _LL ... --,

0

where 1 is determined by 9-0

rq 141

. ..

I

9-' modulo 1(50). For the associated section

complex .9 (.9'), we have

H"(9-'01) =

H"+x0+(50)),

n > 1.

Now suppose that the existence of the isomorphisms up to index p for all acyclic resolutions has already been shown. Then; along with the isomorphisms T1,..., TP, we have the isomorphism Since HP(X, 9'°) = HP+'(X, °) = 0, the connecting homomorphism So: HP(X, .y) -+ HP+ (X, .9')

associated to 0 -, So

tP+

-TI

,' -i 0 is bijective. Consequently,

bnt,:

is the desired isomorphism.

.y) O

Remark: The homomorphisms TP, p >_ 0, exist for every resolution of Y. This is easily seen by writing down the double complex behind the above argument. The acyclicity just forces all of them to be bijective.

Due to III), every flabby resolution of.' is acyclic. Thus one can use any flabby resolution of Y to determine its cohomology. We can make the analogous remark (using IV) for soft sheaves. Y"' is a resolution of Y by soft if X is metrizable and 0 .Y° M-sheaves, then there exists natural isomorphisms HP(9 HP(X, .9'), p > 0.

This result will be important in Chapter II.4.

§ 2.

tech Cohomology

The system S = {S(U), rv} always denotes an R-presheaf and U = {Uj}, i e 1, is

reserved for an open cover of X. In this section we introduce the (alternating) Cech cohomology modules H9(U, S) and their limit groups FIQ(X, S). A vanishing theorem for compact "blocks" which is important for later applications, is proved. The theory of the long exact (ech cohomology sequence is discussed in detail. A

Chapter B. Cohomology Theory

34

good readable presentation of tech theory can be found in [TAG] as well as [FAC].

If c e S(U), then we use the suggestive notation c I V for r(c). For every q + I n Ut1. indices io, ..., iq a 1, we write U(io, ..., iq):= U10 r)

1. tech Complexes. For every q z 0, the product C'(U, S) -

fl

S(U(io,

..., i'))

(b..... 4) F 1++,

is an R(X}module. Its elements (the q-cochains) are all functions c which associate

to every (q + 1)-tuple a value c(io,..., iq) a S(U(io,..., iq). One defines the coboundary map d': C'(U, S) -+ C'+ 1(U, S) by

/

q+1

(d'cx , ..., iq+1)'=

4-0

(

(-1 ?c('01 ...,

...,'4+1)) U110, ...,Yk,

,

...,

Obviously d' is an R(X)-homomorphism. One verifies by direct calculations that d'+'d' = 0. Therefore C'(U, S)== (C'(U, S), d') is a complex of R(X}modules, the so-called tech complex with respect to U with values in the presheof S. Every R-presheaf homomorphism op: S' -e S determines R(X)-homomorphisms C'(U, q)): C'(U, S') -+ C'(U, S), q ;?: 0, which are compatible with the coboundary mappings. Consequently C'(U, -) = (C'(U, -)) is a covariant functor from the category of R-presheaves into the category of complexes of R(X}modules. It is clear that the funetor C'(U, -) is additive and exact.

The tech cohomology modules of S with respect to U are defined to be the cohomology modules of the complex C'(U, S): H'(U, S) i= H'(C'(U, S)) = Z'(U, S)/B°(U, S),

q z 0.

Hence we have a sequence H'(U, -), q z 0, of covariant, additive functors from the category of R-presheaves to the category of R(X}modules. Since C'(U, -) is exact, every exact sequence 0 -+ S-* S -, S" -+ 0 of R-presheaves determines (by Lemma 5.1) an exact long cohomology sequence

... -_.. H'(lt, S) - H'(![, S")

°-'

H'+ i /l r' S,) .,...

For every A-sheaf So, we always have the canonical presheaf r(.'). Using this, one sets

C'(U, .9')'= C'(U, r(.9')) = s(u(io,... , iq)) and further

H'(U1 y)'= H'(u, r(s°)),

q Z 0,

§ 2.

35

C ech Cohomology

where H°(U, -) is isomorphic to the section functor .5" ' b"(X) on the category of a?-sheaves. .

2. Alternating tech Complexes. A q-cochain c e C'(U, S) is called alternating if, for every permutation it of {0, 1, ... , q}, it follows that c(i,,(o), ... , i,«q)) = sgn it c(io, ..., iq) and furthermore c(io, ..., iq) = 0 whenever two arguments are the same. The set of all alternating q-cochains forms an R(X }-submodule C;(U, S) of C'(U, S). If c is alternating, then so is d'c. Thus d' induces an R(X)homomorphism d;: C;(U, S) -+ 0.' '(11, S). Hence CQ(U, S) (C;(U, S), d;) is a subcomplex of C'(U, S), called the alternating Cech complex. Most properties of C'(U, S) carry over immediately to C;(U, S). For example, Ca(U, -) is a covariant, additive, exact functor from the category of R-presheaves to the category of R(X}modules. The R(X}modules H;(11, S)- H4(Ca(U, S)),

q

0,

are called the alternating Cech cohomology modules of U with respect to S. In important cases CC(U, S) = 0. For example,

if there exists a natural number d >_ 1 so that, for all pair-wise different indices i°, ..., iq e I, the intersections U(io, ..., i9) are empty, then, for every presheaf S of abelian groups

C;(U, S) = 0 and consequently HH(U, S) = 0 for all q Z d. Proof: Let c E C;(U, S), where q >: d. If the indices are pair-wise different, then ..., iq) = 9 and c(io, ... , iq) = 0. If two indices are the same, then c(io, ... , iq) is zero by the alternating conditions. U(io,

p

The injection of complexes C;(U, S) - C'(U, S) induces an R(X)homomorphism iq(U ): Ha(U, S) -+ H4(U, S), q 0, so that, for every presheaf

homomorphism, (p: S'- S, the diagram H;(U, S')

H'(U, S') ----

-

H:(U, S)

H'(U, S)

is commutative. As a matter of fact one can show that the maps iq(U) are always isomorphisms. We do not know a standard reference for this, but the reader should see [FAC], p. 214. We will never use these isomorphisms anyway.

3. Refinements and the tech Cohomology Modules $'(X, S). Suppose that U = {U,}, i e 1, and 93 = (V), j e J, are covers of X and that 23 is a refinement of U : 13 < U. Every associated refinement mapping tr: J - I determines an 11:(X}

Chapter B. Cohomology Theory

36

q(CO(U, S) - O(93, S), q -> 0, where the homomorphism Cr): iq)} a r[ S(U(i0, ..., iq)) is mapped to the c = {c(i0, ..., c' E fI S(V(jo, ..., jq)) with

q-cochain q-cochain

c'(J0, .. ., jq)_=c(tjo, ..., TJq)IVU0, ...,Jq).

(note that V(j0, ... , jq) c U(Tjo, , Ti,)). One effortlessly verifies that all of the maps Cq(r) are compatible with the coboundary mappings. Thus one sees that if l3 < U, then the refinement map r induces a homomorphism of complexes C(T): C'(U, S) - C'(93, S) and consequently we have R.(X)-homomorphisms hq(t): Hq(U, S)

q > 0.

H41(93, S),

If T': J - I is another refinement map, then q-1

(kgc)(Jo, ...,Jq-I)'

0

(-1) c(rjo, ..., TJ., TL+1. , TJq-1)IVj0, ...,Jq-1)

defines an R(X}homomorphism kq: Cq(U, S) Cq^'(93, S), q >_ 1, which is a socalled "homotopy" operator for the coboundary operator d. In other words, dkq + kq+ Id = CI(T') - Cq(T)

for

q >- 1,

and

kid = C°(T') - C°(T).

It follows that

() *

(C'(T') - C4(T))Zq(U, S) c B9(93, S)

for q > 1,

and

(C°(T') - C°(T))Z°(U, S) = 0.

Thus at the level of cohomology one obtains the same homomorphism for either restriction: h4(T) = hq(i ).. Hence we may write hq(U, 23) in place of hq(T). We note that hq(U, U) = id and, if I1 < D < U, then hq(113, U) = hq(21t, [)ht(il, U). By observing the usual logical precautions, one can consider the "set of all open

covers of X." This set is partially ordered with respect to the relation 93 < U. Every system {HI(U, S), M(93, U)), q = 0, 1, 2, ..., is directed by this ordering. Thus we have the inductive limit Rq(X, S),= Emit Hq(U, S),

q = 0, 1, 2, ....

The R(X}module RR(X, S) is called the q-th eech cohomology module of X with coefficients in the R-presheaf S. We denote the canonical map H9(U, S) FFq(X, S) by hq(U ). Thus h4(%)hq($, U) = hq(U), whenever 93 < U. The functor F7'q(X, -) is again covariant and additive.

§ 2.

37

tech Cohomology

The Cech cohomology modules for A-sheaves ." are defined by

H'(X, Y)R'(X, r(so)),

q > 0.

Acting on the category of 9i'-sequences, ft°(X, -) is isomorphic to the section functor. Hence the functors ." -- H"(X, .") have the property I) of Section 5.2. 4. The Alternating tech Cohomology Modules R;(X, S). The considerations in the preceding paragraph can be repeated mutatis mutandis for alternating tech complexes. Namely, by means of C'(r), alternating cochains are mapped to alter-

nating cochains and thus one obtains a homomorphism of complexes C;(T): C;(U, S) - C'(%, S) as well as the R(X)-homomorphism h;(r): H;(lt, S) H;(23, S), q >- 0. The equations analogous to (*) show that h;(T) is likewise independent of the choice of the refinement map. Correspondingly we write h;(%, U) instead of h.4(-r). Every system {H;(U, S), h;(%, U'.) is directed, the direct

limit R(X}module

H;(X, S).= lir H.(U, S) being called the q-th alternating tech cohomology module with coefficients in the presheaf S.

For sheaves ,°, one sets R;(X, Y):= H;(X, r(.5")). This is again a covariant, additive functor on the category of sheaves which has property I) of 5.2. If 23 < U, then iq(22)hQ(V, U) = h'(13, U)iq(U), where iq(23) and iq(U) are the

natural homomorphisms from Section 2. Thus the limit homomorphism iq: RRg(X, 9") qq(X, .") is induced, having the property that iq h;(U) _ h'(U)iq(U ), q > 0.

5. The Vanishing Theorem for Compact Blocks. For the proofs of Theorems A and B in Chapter 111.3.2 (Theorem 1), we need the following:

Vanishing Theorem. Let B:= (x a R' I a,, < x < b,,, 1 < p < m) be a non-emptycompact block in R' and 9' a sheaf of abelian groups over B. Then

I/;(B,.5")=0 for all q>-3"' The proof will follow from a simple lemma. We denote with norm on R' and associate to every set M c 11'" d(M)'= sup ! x - y I

I

the euclidian diameter,

its

X,yEM

Lemma. Let U = (U;) be a cover of the compact block B. Then there exists.a real number ,1 > 0 (the so-called Lebesgue number of U) so that every open set V in B with d(V) < A lies in some U;.

Proof: Suppose that for every - = 1, 2, ..., there exists an open set V, c B with d(V,) < v-' which lies in no U1. Let p e V,,. Since B is compact, the sequence (p,)

Chapter B. Cohomology Theory

38

has an accumulation point p e B. There exists j e I with p e U,. It is clear that, for v large enough, VV - UU, which is the desired contradiction. We now prove the Vanishing Theorem. I 'r every integer n >- 1 we construct as follows a cover V. of B: Let

41,-- -,1.) =(at+ln (b1-a1),.... a.,+ln (b.-a,.)el8'",10, =0, 1,...,n. B(11,..., I.)'-{xeBIjjxµ-zµ(11,...,1. 11 3'. Since the diameter of every set B(11, ..., 1,) E Z. is smaller than 2m/n max 1 bµ - aµ +, it follows from the Lemma that, for every cover U of B, there exists an index p so that Dv is finer than U. This says that the covers of the form 3,. are cofihal in the direct limiting process which defines H;(X, So). Thus

11.(X, 9) = lim Ho(9% Y) = 0. Remark: It, is clear that the bound 3" can be greatly improved. In fact, a little work yields m + I as the best possible such bound.

6. The Long Exact CobomolW Segaence. For every oxact sequence of presheaves, 0 -+ S' -' S - S" -- 0, and every cover U, we have a ammnutative diagram

0 -. C;(u, S')

-:

0

c:

C.0(111 S)

1

,

l

1

S') -- Co(u, S)'

S")

0

S") - 0

0 (complexes with exact rows By the results in Paragraph 5.1 one has the commutative diagram of long exact eohomology sequences. Since direct limits of exact sequences are exact, it follows that for every exact sequence 0 -+ S'-* S - S" -. 0 of R-presheaves there exists a commutaitlve diagram of long exact colsomology sequences

0 -.

R(X, S') 1

0 - 16 1,

R:(X, S) -, 1

Ha(X S")'

(X S')

1

r) °' , }.+ 1(X, s') _ _, .. .

§ 2.

tech Cohomology

39

For every exact sequence of A-sheaves, 0 -+ So' -+ So -..9" -. 0, one has (see

Paragraph 1.6) the exact f ()ppresheaf sequence, 0 -.

T(.9') -. S--+ 0, .°(U)/So'(U) is a presheaf whose sheaf is in fact .9'". Thus one has a commutative diagram, where 9:= (S'(U), r' '}, with S'(U)

-

Ifi(`,y')-. .(X,.9')-~f (X, T it,

... -174(X, ')

jii+i

1i,

1j,

R9(X, )

a +9+1(X,

with exact rows. The natural presheaf homomorphism 9 -. I'(.9'") induces homomorphisms RR(X, 3') -. 94(x, .9'") and $;(X, 9) -+ I ;(X, .9'"). Thus, if we can prove that these are in fact isomorphisms, we can always replace S' by .9'" in the above diagram. With this in mind, we now show the following:

Let X be paracompact and T be a presheaf (of abelian groups) over X with associated sheaf 9':= (`'(T). Then the natural map a: T -. r(Y) induces genuine isom orphisms

RR(X, T) -. $"(X, I-) and 9;(X, T) -+ LI:(X, .P),

q > 0.

Proof: Assuming the statement in the case f= 0, the general case can be derived as follows: The map a yields 0 --+ Ker a --a. T Im a -+ 0,

o -+ lm a -J

rpm-) ?` 0, where 7` == r(. fl/Im a, as exact sequences of presheaves. The associated long exact cohomology sequences are

-111(X, Ker a)-.$+(X, T)'+R'(X, Im a)-+Itl+i(X, Ker a) -+ and

M (X, Im a) 44 f4Q(X, .P) -. $4(X, t) Since the fimetor 1'.` is exact, '(Ker a) = 0 sheaf; all homomorphi: ,ns

a,: 1 ' ( X , T) -+ RR(X, Im a),

...

Assuming the result for the zero

J*: R IX, IM a)

0,

are bijeotive.

It remains to show that, in the case that if = 0, it follows that I?'(X, -) $,(X, P) - 0. This is obviously contained in the following: Leo>re& 7-L t U s= {U,}, Ii e I, be a locally-finite cover of X. Then, assumbtp r(T) - 0, given a cochain c e C*(U, T), there exists a-refinement ID = {W }, J a J, of

U (with refinement map T: J -. I) so that Cq(T)c = 0.

Proof: Let gf - { t i}, i e I, be a cover of X with P, c U r (Shrinking Theoram Let J ,= X and take T: J --+ I so that x e Vim. Since U is locally finite, every point x

Chapter B. Cohomology Theory

40

possesses a neighborhood W. which has non-empty intersection with only finitely many U;'s. By shrinking if necessary, we may assume that .

1)

Wx c U; for

x e U; and

W. c V for ; E Vi.

Then `I11:_ { Wx}, x e X, is a refinement of both % and tt with refinement map T. Since l; (-_ U,, we may assume that 2)

if

Wx n V, $ O

then

x E U,.

Finally, since f (T) = 0, W,, can be chosen so that 3)

c(io,... , iq) l4 = 0 for all i, .... iq e I and all t E U(io, ..., ii).

For any (q + 1)-points xo, ... , xq with W.. n

n Wxq

Q, we now consider

0(t)c(xo, ..., x,) = c(rxo, ..., txq)I W(xo, ..., xq). Since WXO n W,4 # 0 for

k = 0, ..., q, must have a non-trivial intersection with every V. Thus, by 2), xo E U,,, for all k (i.e. xo E U(rxo, ... , txq)). Now 3) implies that c(rxo, ..., txq) I W;, = 0 and thus c(rxo, ..., rxq) I W(xo, ..., xq) = 0. p Thus, in summary, we have shown that if X is paracompact, then the cohomologyfunctors Hq and H; have the property II) of paragraph 5.2.

We will verify property III) in the next section.

§ 3.

The Leray Theorem and the Isomorphism Theorems

ft.s (X, .q) 4$9(X, ') 4 H9(X ,')

The Leray Theorem is fundamental for later applications in the Stein theory as well as being the key to proving that all of the cohomologies are the same. It states that, for special covers U of X, the groups H'(X, .1) are isomorphic to Hq(U, .9'),

q z 0. By means of a canonical resolution of .9' relative to U, we reduce this

theorem to the formal de Rham Lemma. As an application of the Leray theorem, we show that on paracompact spaces the tech cohomology for flabby sheaves .4r vanislteg. As a consequence the maps iq: H:(X, ,') 4 H'(X, 9) and M (X, .') -24 H'(X, 9) are isomorphisms. 1. '!tie Caaoaical Resolution of a Sheaf Relative to a Corer. If 9' is an M-sheaf over X, then for every open set Y c X, we use S< Y> to denote that 9-sheaf which is the trivial extension of 9'1 Y to X. Thus Se = i,(9 Y), where is Y - X is

the injection. For all open U c X, we have 9'(U) = S(U n Y). Now let U = (U,), i e I, be an open cover of X. For every i-sheaf 9" we form the a-sheaves .9'(io, ..., i,):= .9', io, ... , i, a 4 and further (see

§ 3.

41

The Leray Theorem and the Isomorphism Theorems

paragraph 0.7) we have the direct product sheaf ,cyp :_

(0)

(, 0.....

F1

Y(i°,

.... ip),

P= 0, 1, ....

These product sheaves are obviously 4-sheaves as well. If one introduces the open cover U U = {U;}, U;'= U; n U, i e 1, of U, then Y(U(io, ..., ip) n U) = (.91 U)(U'(i°, ..., i,,)). Thus we may write:

.fp(U)=CP(UIU,.PIU),

(1)

Hence .9"(U) is the p-cochain module of one has the cochain complex

p>_0.

U with respect to the cover U I U, and

C-(U ! U, Sp I U) _ (SeP(U), d$).

If one associates to every section s E 9'(U) the o-cochain c(i) :_ (s I U; n U), then one has an 9i'(U)-homomorphism ju: 5"(U), .5"°(U) such that dgjv = 0. For all U, V with V c U the restrictions S"(U) -+ S"(V) are compatible with d$ and Thus, taking the direct limit of the above complexes, we have an 91-sheaf sequence ,Sp

J

y° da'* yt,

... , Spp dpa ,5pp+t

where d°j = 0 and dp+ tdp = 0. We claim that, for every 9P-sheaf 91 over X and every cover U of X,

(R)

0 --. ,9 --L4

-d0. 6P1 --.. ... -. ,P

Jtpp+ 1 --... .

is an 51r-resolution of Y.

The equations Ker j,, = 0 and Im j = Ker d2, x e X, follow trivially. It thus remains to verify the inclusion Ker d= c Im dz- 1, p >- 1. For that we need the following Lemma:

Lemma. Let Y be a sheaf of abelian groups over a topological space M and let qB = (W), j e J, be an open cover of M. Suppose that Wk = M for some k e J. Then HH(IB, -) = 0 for all q z 1 (i.e. every cocycle c = {e(jo, .. . ,jq)} E Z9(W, T), q >_ 1, is a cobowidary.)

Proof: Let c e Z'(m, 3r). It is always the case that W(jo, ..., jq-,) =

W(k, Jo, ..., j.-1). Thus b'=(b(jo, ..., j,-t)}, where b(jo, .... j.-I)'= c(k,jo, is a (q- 1)-cochain in Cq-'(Ill, T). By the definition of the coboundary map,

(d.-16)(/0, ,%')= .=1

Chapter B. Cohomology Theory

42

However, by assumption,

jj

0 = (dscXk, jo, ... , j,)-= c(Jo, ... , ja) - i (-1)"c(k, jo, ... , IX, ... , r=0

Thus

0

(d'M'b)(jo, ...,j,) _ c(io, ...,j.) Now we can easily prove the claimed inclusions Ker d= c Im dx-1, p >_ 1.

Let sx a Ker d,. Then there is a neighborhood M of x and a representation s e Ker df = ZP(U M, SP I M). We can choose M so small that it is contained in a set Uk. Thus M is a member of the cover U I M and, by the above Lemma (with

9':= So I M), there exists an element t e CP-1(11 I M, 9' 1 M) = 1P- t(M) with df, t = s. Letting t,, e SP,-' be the germ determined by t, it follows that dxP tx=Ss

11

We call the just constructed resolution (R) of .' the canonical resolution of .9' relative to the cover U. We emphasize that it is in general not acyclic. 2. Acyclic Cover. For every flabby sheaf .tom over X, the sheaves W< U> are also

flabby over X. A flabby resolution 0 -, -+,F1 - of [1 induces the flabby resolution 0 - So 0.

The reader can find a simple proof in [TFJ, p. 175. We now assume that the given cover U is locally finite. Thus, for every p z 0, the family {.9'(i0, ..., ii)), io, ..., iP e 1, is locally finite. Hence the Lemma, together with equation (0), implies that

H'(X, lP)

fl

H'(X, [(l, ..., 1,)).

§ 3.

43

The Leray Theorem and the Isomorphism Theorems

Since .9'(io, ..., i) _ Y_ 0, q >_ 1. Thus we can say that,

if U is a locally finite cover of X which is acyclic with respect to So, then the canonical resolution of .' relative to U is an acyclic resolution of Y.

3. The Leray Theorem. The following is now easy to show: Theorem (Leray Theorem). IfU is a locally finite cover of X which is acyclic with respect to the A -sheaf .q', then there exists natural . (X )-isomorphisms H°(11, So) 4 H°(X, .9'),

p = 0, 1, 2, ....

Proof: We now know that the canonical resolution of .9' relative to U,

0 ' _ .q'o -+ 91 , is acyclic. Thus, by the formal de Rham Lemma, there is a canonical 91!(X)-isomorphism of the p-th cohomology module of the section complex (Sot(X), 4) onto HD(X, .9'), p z 1. Now, by equation (1), (.9"(X), dx) _ C'(U, .So). Since H°(C'(U, So)) - HP(U, 9) by definition, the proof is finished.

Remark: The assumption that U is'locally finite is quite important in the above proof (e.g. equation (3) is used). However, with other methods one can show that it is superfluous. In all of the applications in several complex variables one can get along with. locally finite covers anyway (see Chapter 4, 4. The Isomorphi m Tboorem:,114.(X, SP) 119(X, So) H9(X, .9'). If ,F is a flabby sheaf over X, then every covering U is acyclic with respect to F. Since FId(X,'N) = 0 for all'p > 1, the Leray Theorem implies that HP(U, .F) = 0 for all p z I whenever U is a locally finite cover. Since on paracompact spaces every cover has a locally finite refinement, we now have the following:

If.X is par compact

r(X,A):=,0fordlp1.

a flabby sheaf of abelian groups over X, then

Thus on paracotpact spaces, we now know that'the functors f19 (together with the connecting homomorphisms $9) satisfy properties I)-III) of paragraph 5.2. Thus over such spaces the flabby and Cech cohomology theories are isomorphic:

If X is paracompact and . is an A -sheaf, then there are natural isomorphisms Jq4

(X,,")4H'(X,.9"),

q0.

Chapter B. Cohomoibgy Theory

44

In closing we want to make it clear that all of the above remarks for $4 go through for J. The "alternating" A(U)-module .9'(U) Cp(U I U, .So I U) is contained in the 9t(U}module .5o'(U) = CP(U I U, So U). These modules along with the natural restriction mappings form canonical presheaves for the sheaves of A-modules .5" over X. If one "well orders" the index set 1, then (0')

5" _

ri

.So(io,

...,

is 1,

then

Jc, + 5 Jc° J S

+ ab(y,)I. Therefore Ic,1 _ 1, which is exact on U,. We may assume that p, = p and

q, = q independent of i and that the U,'s are pairwise disjoint. Then U is a neighborhood of the fiber n-'(z) and we have the exact sequence

U U, 1

Chapter 1.

52

Coherence Theory for Finite Holomorphic Maps

By Lemma 1.2 we may take U smaller so that U = it -'(V) where Visa neighborhood of z in B. Then the induced map rtu: U -, V is finite and the functor (nru)* carries the sequence (*) over to an exact sequence on V (see Theorem 1.4):

(nu)*(Q) - (nu)*(t'Z) - ('ru)*(`'u) Of course (nu)*(.9'u) = n*(."),,. Moreover, using the polynomial wy which is the restriction of w to V, Theorem 3 yields an Gr v-isomorphism

Thus we have the sequence car -.COP -I' n*(Y)v-40

which is exact on V. This proves that n*(,S")v is coherent. (Here the coherence of

is used in an essential way!) For future reference, we state this result

0'v

explicitely:

Coherence Theorem 4. If .9' is a coherent (OA -sheaf, then n* (Y) is a, coherent CB-sheaf.

§ 3.

The Coherence Theorem for Finite Holomorphic Maps

The purpose of this section is to prove that the direct image of a coherent sheaf by a finite map is coherent. This is proved in three steps. First, Theorem 2.4 is applied in the situation where the sheaf is coherent on some neighborhood of the origin in C x C" and the finite map is the restriction of the projective C x C" -* C" to the support of the sheaf. This is immediately generalized by induction to the case of C' x C" -+ C". Then we show that any finite map can be locally realized as the restriction of such a projection and consequently a local version of the desired coherence theorem is proved. Finally, since coherence is a local notion, the global result (Theorem 3) follows in a few lines.

1. The Projection Theorem. Let w = (w,, ..., z = (zt, ..., z")) be the coordinates in C' (resp. C"). The objective of this paragraph is to prove the following:

Theorem 1. Let .9' be a coherent analytic sheaf on some neighborhood U of the origin (0, 0) e C" x C". Suppose that (0, 0) is isolated in supp ,91 n (C" x t0}). Then there exist open neighborhoods W and Z of 0 c C'", 0 a t" with W x Z c U such that the following hold for the projection (p: IV x Z -+ Z:

1) The restriction map (p supp .9 n (W x Z) is finite. 2) The image sheaf p*(Sw,z) is a coherent sheaf of Oz-modules.

§ 3.

53

The Coherence Theorem for Finite Holomorphic Maps

Proof: a) We may assume that supp 9 intersects the plane C' x (0) at the origin x'= (0, 0). The annihilator sheaf An(Y) c OU is coherent on U and vanishes exactly on supp(."). Thus there is a germ f e An($"),, with f (w, 0):# 0. We may assume that f is distinguished in w1. In order to do this we need only to make a Thus by. the preparation theorem, there linear change of variables in w1, ..., 6

exists a unite e O and a polynomial co., = w; +

a,s wi-' whose coefficients a,,,

are holomorphic germs in w'= (w2, ..., w") and z and which vanishes at

0 e C"'+"-' with f = ew,,. Since a is a unit, co,, is also in An(S),,. Now we can find a neighborhood W1 of the origin in the w1-axes and a neighborhood T c C'"+"-' of the origin in the (w', z)-space with W1 x T c U and such that every germ a,,, has a holomorphic representative a; a O(T). Thus the polynomial co - w, + 6

E a,0,

a O(T)[w1] is a section of An(.9') on W; x T. We let A denote the zero

set of win C x T. Since the equation w(w1, 0, ..., 0) = 0 is solved only by w1 = 0, we can take T to be small enough so that A c W1 x T. Now let 0: W1 x T -+ T be projection on the second factor and define n: A -+ T by n'= 0 A. By Theorem 2.1, if one equips A with the structure sheaf Ci11= O/w0 I A, it is a finite map. Since 9'(w1 x rp,, = 0, we may consider SPW, x r to be a coherent sheaf of O,,-modules and O'.(.'W, x r) = Furthermore, by Theorem 2.4, .9" 1= t(i(.9'1 x r) is a coherent sheaf

of Or-modules. Since supp 9 is a closed analytic subset of A, ' 1 supp So n (W1 x T): supp .9 n (W1 x T) -+ T is likewise finite. We note that all of the above remains valid if T is taken to be a smaller neighborhood of the origin in CM-1 x C".

b) The proof of Theorem 1 is now quickly finished by an argument involving induction on m. If m = 1, then we take W'= W1, Z'= T, cp'= 0 and the above arguments yield the desired result. If m > 11 then we consider the coherent image sheaf of Or-modules, 9', on T. Since {(0, 0)) = supp .9' n (W1 x T), the origin is isolated in the intersection of the plane C" -1 x {0} with supp So' _ /i(supp 9PW, XT). Hence we may apply the induction assumption: There exists a neighborhood W' ofO C"'-' and a neighborhood Z of Oe C" with W' x Z c T so

that the projection x: W' x Z - Z induces a finite map supp Y' n (W' x Z) -+ Z and the image sheaf X,(9',,.. X z) is a coherent sheaf of Oz-modules. We now set W '= W1 x W' and redefine T to be the (possibly) smaller set W' x Z. Thus the pro-

jection rp: W x Z - Z factors through W' x Z: W x Z-*---+ W' x Z x' Z. Since q.(9') = and'.(9wxz) = Y' is coherent, the coherence statement 2) has been proved. Moreover, the restriction of cp to supp So n (W x Z) also factors into two finite maps:

supp 9 n (W X Z) .- supp 9' n (W X Z) - Z. Thus we have the finiteness of 0 ( supp 9 n (W x Z).

2. Finite Holomor hic Maps (Load Case} Let X and Y be complex spaces and P. X -+ Y A holomorphic map. Let 9' be a coherent sheaf of OX-modules.

Chapter 1.

54

Coherence Theory for Finite Holomorphic Maps

Theorem 2. Suppose that xo is an isolated point of supp .' n f - '(f (xo)) in X. Let Uo be any open neighborhood of xa in X. Then there exist neighborhoods U, V of xo e X (resp. f(xo) a Y) with U c Uo and f (U) c V such that the following hold: 1) The restriction of the induced map fu: U - V to supp .9' n U is finite. 2) The image sheaf (fu),(Yu) is a coherent sheaf of Oy-modules.

?roof. a) We first consider the case where Y is a domain in C". We choose a neighborhood V c: Ua of xo in X so that there is a biholomorphic embed-

ding i : U'- G of U' into a domain G c C'". Defining f'-=f I U', the "product map" i x f': U' -+ G x Y is therefore a biholomorphic embedding of U' into G x Y c C'" x C". Now the trivial continuation of the image sheaf to all of G x Y is a coherent sheaf of Oa,, r-modules, ,9''. More(i x f ') 0 over supp .90' n (G x {yo)) = (i x f')(xo), where yo'=f (xo). Thus we are in a position to apply Theorem 1: There exist neighborhoods W and Z of i(xo) e 6 and yo e Y with W x Z c G x Y such that, if gyp: W x Z -+ Z is the projection, the following hold:

1) The restriction of rp to supp So n (W x Z) is finite. 2) The sheaf 9,(9' x z) is a coherent sheaf of OZ-modules. Let V- Z and U '_ (i x f')-' (W x Z). Then, since it is the composition of two finite maps,. fulsupp s" n U: supp b" n U -+ V is finite and the image sheaf (fv) * (Yu) = (ps(5°'w x z) is coherent on V.

b) Now let Y be arbitrary. Since the conclusions of the theorem are local in nature, we may assume that Y is a complex subspace of a domain B in C". Taking j: Y - B, we define the finite map J`: X -+ B by J`:= i c f. Applying part a) to J; there exist neighborhoods U and P of xo e X and yo e B so that ju: U -i induces a

finite map suppY n U

p and such that (J ) (.emu) is a coherent sheaf of

O9-modules. We define V = P n Y as a neighborhood of yo in Y. Since 1m7' c V,

it follows that fu: U - V induces a finite holomorphic map supp .1 n U V. Furthermore (Ju) * (.9'v) is just the trivial continuation of (fu) (.Y) to B. So (fu) s (Yu) is a coherent Or-sheaf.

0

Since supp OX = X, the following case is an immediate application of Theorem 2 in the case where 9' = OX. Corollary, Let f X = Y be a holomorphic map such that xo is an isolated point of

the fiber f -i(f(xo)). Then there exist neighborhoods U and V of xo e X and f (xo) e Y with f (U) c V so that the restriction fu: U -+ V is finite.

3. Finite Holomorphic Maps and Coherence. The theorem below is the main result of this section. It is now just an easy consequence of Theorem 2.

Theorem 3 (The Direct Theorem for Finite Maps). Let f: X -+ Y be a finite holomorphic map and .9' be a coherent sheaf of Ox-modules. Then f,(.9') is a coherent sheaf of Or-modules.

1 3.

55

The Coherence Theorem for Finite Holomorphic Maps

Proof: Let y e Y be arbitrary and x1, ... , x, e X be the different f-preimages of y. By Theorem 2 there are neighborhoods U, and V, of x, e X and y e Y with f (U,) c V so that the induced map fv,: U, - V, is finite and fv (.'v) 1s coherent on Y, 1 5 i 5 t. We may assume that the U,'s are pairwise disjoint. LGt V be a r

neighborhood of y e Y which is contained in U P. Then for 1 S i 5 t we have the finite maps fi: W, -. V, where W,

U, n f -'(V) and f ==f I W,. Furthermore

f,("w,) =fv,.(Yu,)v and thus f,(&w,) is a coherent Ov-sheaf, i = 1, ..., t.

We now choose V so small that U,=f -'(V)

W (this is possible by

Lemma 12) and we consider the restriction off to U, fv: U -+ V. Certainly

fu.(.v)

Since W n Wj = 0 for i * j,

f,(.XV') = n .'(W, n f -'(V')) t

for every open set V' c V. But each f,(Yw,) is coherent. Thus f,(.9'),, is a coherent sheaf of Ov-modules.

0

Chapter H. Differential Forms and Dolbeault Theory

In this chapter Dolbeault cohomology theory is presented. One of the basic tools is the a-integration lemma for closed (p, q)-forms (Theorem 4.1). The proof of this lemma is based on the existence of bounded solutions of the inhomogeneous Cauchy-Riemann differential equation ag/az =f This solution is constructed in Paragraph 3 by means of the classical integral operator

Tf (Z' U)

2Ri 8J

Z)

dC n dZ.

The Dolbeault theory yields among other things the fact that 0) - 0, q z 1, for compact blocks Q c Cm (Theorem 4.4). This vanishing theorem is needed in Chapter III in order to prove Theorem B for compact blocks. In Section 1.2 we collect in more detail than is really necessary the general facts about differential forms on manifolds. We always use X to denote a differentiable manifold of real dimension m and d'R its real structure sheaf. For short we write e for the sheaf IR t- gR + igR. Beginning in Section 2, X has the additional structure of a complex manifold, with complex structure sheaf 0 c O. The symbols U, V are reserved for open sets in X. Real local coordinates on X are denoted by u1, .., u,,.

§ I. Complex Valued Differential Forms on Differentiable Manifolds 1. Tangent Vectors. We need the following representation theorem for germs of functions. Theorem 1. Let u1, ..., u.,, a dc(U) be local coordinates on U c X. Then every germ f= e d'R, x e U, can be written as

.f, =fx(x) + E (u)-x k=1

grx a E.

§ 1.

Complex Valued Differential Forms on Difereafiable Manifolds

57

The values g,x(x) e R of g, at x are uniquely determined by fx: (x},

gwx(x)

N = 1, ..., M.

Proof: Without loss of generality we may take U to be an open set in Rw (the space of real m-tuples (u 1, :.. , uw)) with x the origin. For everyfx e eR there exists

a ball B c U about x = 0 and a representative f e e (B) off,,. In B we have

f(f11,...,uw)-f(0)=

Mat

(f(0,,o,UM,...,Uw)

f(0,...,0'up+1,...,uw))

J'!1 0

ay

If one sets g,,(u1, ..., u,,)'a!o ff/8t (0, ..., 0, tuM, uM+l, ..., uw) dt, then gM E .91(B)

and, by substituting y = tut,

(0,...,O,Y uM+1,..., uw)dY

uMgM=

ay

Thus one has f =f(o) + u1 g1 + - - + uw g on B. Since x = O and u,(x) = 0, this yields the desired equation for fx. The determination of g,x(x) follows trivially by differentiation.

Definition 2. (Tangent Vector). An R-linear mapping i; : d'R -. R is called a tangent vector at x Ecc X if it satisfies the product rule: S(fxgx) =

f gs E SxR

Thus fi(r) = 0 for all constant germs r e R.

The set of all tangent vectors at x e X is an R-vector space which we denote by T(x) and call the tangent space of X at x.

Theorem I Let u1, ..., uw be coordinates on U c X. Then the m partial derivatives

auM

Is:

d'

R,

fx

(x),

P = 1, ..., m,

Chapter II. Differential Forms and Dolbeault Theory

58

form a basis for T(x). For every

e T(x) it follows that

R t = E b(uNx) x

=t

Proof: It is clear that the partial derivative maps are tangent vectors. Since but.

aui

W

these vectors are linearly independent. Thus it remains to verify the equation of By Theorem 1, we have

fs =Is(x) + Lr (FNS fan

gNx E

rR

gN=(x) =

8. (x),

for everyf= a R. Since vanishes on the constant germs, the product rule implies that

V. a 1 4(u x)Ow.(x) =Nit (uNx)'f-

(x).

If in general R is a commutative K-algebra over a field K, and M is an Rmodule, then any K-linear mapping a: R -e M which satisfies the product rule, a(fg) = a(f )p + fa(g). f, g e R. is called a derivation of R with values in M. The set .9(R, M) of all such derivations is itself an R-module. By means of the map R - R,f: -e ff(x), the field R is an fit-module. In this way the tangent space T(x) R). is the module of derivations

2. Vector Fiddle If t is a map which associates to each point x e U c X a tangent vector x a T(X), then one calls l; a vector field on U. If V c U, then associates to each f e SR(V) a real valued function (f ):

(f): V- R, Definition 4. (Differentiable Vector Fields): A vector field 1 on U c X is called differentiable if, for every open V c U and everyf a JR(V), thefunction l;(f) is itself d(fferentiable on V.

The set of all differentiable vector fields on U c X form an 8 (U)-module, V. T(U). If V e U, then one has the natural restriction r : T(U) T(V), Hence e

the system (T(U), rY) is a presheaf on X which satisfies the axioms Sl and S2. The associated 1R-sheaf is denoted by .' and, for every x a X, the stalk Yx is the OX' module of all derivations of d R into itself. The sheaf g is called the sheaf of germs of differentiable vector fields on X.

1.

59

Complex Valued Differential Forms on Dilferentiabte Manifolds

We immediately have the following analog to Theorem 3: Theorem S. If u,, ..., ui,, are coordinates on U c X, then the partial derivatives

auw

form a basis for the 9R(U)-module 5(U). For every to e

y-1

(u,)- µ ,

.'(U

Oul) C- 92(u).

Thus, by means of

,P(U)_, R(Ur,

- ( (ut)+..., ou)

every coordinate system ut, ..., u,,, on U c X determines an fl(U}module isomorphism. Thus -(U) is free of rank m. Hence it follows that the dR-sheaf Y is locally free of rank m. 3. Complex r-vectors. Since C and all of the tangent spaces T(x), x e X, are real vectorrspaces, we can make the following definition for all r z 1:

Definition 6 (r-vector). A (complex) r-vector V at a point x e X is an r -fold, x T(x) -+ C.

f8-linear, alternating map rp: T(x) x .

The r-vectors at x form a complex vector space A'(x). One sets A°(x) _= C. The

Gressmann product n (wedge product) is defined in the direct sum

A(x):= ® A'(x)' r-° as follows: For (p a A'(x) and 0 E A'(x), q) A 0 e Ar+,(x) with q) A 1'/( t, ... ,.+.)

((r + s)!)l(r! s!) Y 6(11, ..., ..., ti)(4r.+i+ ..., ...) for ' C- T(41 It is easy to check that n endows A(x) with the structure of an associative C-algebra with 1 and that

q) A 0 = (-1 ro n 0, for 9 E A(x) and 0 e A'(x) i Here the sum is taken over all i ... , i,+, from I to r + s and d(i representing the sign of the permutation

(l

1t,

a.

2 12

...

4

is Kronecher symbol

Chapter II.

60

Differential Foems and Dolbeault Theory

Now let u1, ..., u,,, be local coordinates on U c X. For a given x e U we let du 1, ... , du,, denote the dual basis of a/au 1 J., ..., a/au. 1, for the dual real vector space T(x)*. (We should really use the notation du1 I,,, ..., du,, I,,, but it is too cumbersome). Then du 1, ..., du,, form a basis for the complex vector space A '(x). Further, as a basis for A'(x), we have {du,, A ... A du,, 11 < t 1
_ 1, on K is ia-exact.

K - Kt x

Proof., We let re denote the C-vector space of all (p, q)-forms on K which, with

respect to the basis dz,, dsj, are free of die,. 1, ..., dam, 0:5 e < m. Note that r, = dP-q(K). We will show by induction on e that every 4p E re satisfying &p = 0 has a D-preimage 0 a `(K). If e = 0 then p = 0 (since q >_ 1) and thus 0 = 0 does the job. Let a >_ 1 and .ake cp E re. Collecting all of ttte terms in cp which contain die, we obtain

p=aAdie+fi, where a e sIP'4-t(K) and fi a S*4P,9(K) are both free of die,

$e re_,. By assumption

0 =$(R =ZaAdie+Zf.

..., dz",. In particular

Differential Forms and Dolbeault Theory

Chapter Il.

78

From this, since die, ... , da," are not found in a, it follows via an easy calculation that, for every coefficient f e d(K) of a, af _ 0 azy

-

for

µ=a+1,..., m.

Hence each such coefficient f is holomorphic in ze+,, ..., z",. We now apply the Lemma of Grothendieck (in the form of the Corollary) to an open neighborhood

B x U of K with B

Ke and U

[I K in which f and all of its deriva-

tives is bounded and holomorphic in ze+,, ..., z,". For the Lemma, z:= ze. Hence, for every coefficient f of at, we obtain a functionTE e(B x U) such that (a}/aae) =f and (aT/azµ) = 0, µ = e + 1, ..., m, on B x U. If one replaces every such f in at with such anI leaving a otherwise unchanged, an easy calculation shows that one has a form & E sP, '(K) such that

Da=andie+y, where yere_,. Now the (p, q)-form 6:= -p& assumption, there exists $ E

y e r, is a-closed. Thus, by the induction = S. Consequently, 0:= & + 3 E d"-q-' is a 6-preimage of 0 on K. Since r. = sVP,9(K), the theorem is sIa.a-'(K) with

p

proved.

Corollary to Theorem 1. For every compact product set K c C", the sequence

0.- OP(K) ±

a , ...?

is exact for all p z 0. Proof: The exactness at the points P-O(K) follows from Theorem 2.6 and the exactness at all other places follows from Theorem 1. p

The proof of Theorem I which we reproduced above and which makes use of the integral operator T is due to Grothendieck, being communicated by Serre on 5/15/1954 ([ENS2], Expose XVIII). Since it is clear that all sequences

O-Oa(K)'.4°.a(K)

ate...

.,0

are exact. Thus the previously announced exactness of the diagram on p. 70 has now been shown. At this point it should be said that Theorem 1 is also valid for open product sets: One can exhaust such domains by compact product sets, solve the a-problem on the compact sets, and take the limit of appropriately chosen solutions as the exhaustion converges to the open set. Such a limiting process is carried out later in the much more general context of Stein spaces (Chapter IV.4). Among other things it will be shown at that time that Theorem 1 is valid for every Stein manifold.

§ 4.

79

Dolbeault Cohomology Theory

2. The Doibeaoh Cohomology Group&40n every m-dimensional complex manifold X and for every natural number p >_ 0, we have the following sequence of sheaves (see 2.3): (*)

0 -p c2'

i.d° a_ydv.t a -

Further, for every closed set M in X, we have the C-complex

'dp.o(M)

a

.

of sections

'%fp.l(M)a ? /p-'"(M)-0.

Definition 2 (Dolbeault Cohomology Groups). The cohomology groups of the complex .d p,'(M) are called the Dolbeault cohomology groups of

M in X. They are denoted by Dolb"(M). Dolby-°(M) = Kerp:

salp.o(M)

-+,Qdp.i(M)

c'pa-'(M) -' 4p'4(M))

Dolbp,4(M) = Ker(D: dp-4(M) -

In particular, if every a-closed (p, q}form on M is s-exact, Dolbp4(M) = 0 for

q> 1. Theorem 3. For every p >_ 0, the complex (*) is a resolution of the sheaf of germs of holomorphic p-forms on X. This resolution is acyclic over every closed set M in X and thus there is the natural C-isomorphism

H'(M, (IP) = Dolbp''(M),

i >_ 0

Proof: Since every paint x e X has a neighborhood basis consisting of relatively compact product domains, the corollary to Theorem 1 shows that (*) is a resolution of CI". Since all sheaves OP-11 are soft on every closed set M c X, the acyclicity follows. The isomorphism is thus a consequence of the formal de Rham Theorem. D The corollary to Theorem 1 says that Dolbp,4(K) = 0

for p >_ 0, q >_ 1,

and K a compact product set in C.

Thus the following is an immediate consequence of Theorem 3:

Theorem 4. IfK is a compact product set in C'", then

H4(K,S2p)=0

for p>0,q> 1.

In particular,

H4(K, 0) = 0 for q z 1.

80

Chapter It.

Differential Forms and Dolbeault Theory

We will make decided use of this last statement in the case of compact blocks (Chapter 111-3.2).

The Dolbeault cohomology groups Dolb°-9(X) vanish as soon as p + q > m = dim X. Thus Theorem 3 implies that, for every m-dimensional complex manifold X, H9(X, UP) = 0 for all

p, q

with p + q > in.

,

In particular H9(X, 0) = 0 for all

q > m.

3. The Analytic de Rham Theory. For every differentiable manifold the differentiable de Rham theory yield the acyclic resolution

0,C

t

'G

°

rat z

of the constant sheaf C by the complex of differential forms. For complex manifolds X, it is appropriate to consider the sequence

d .Ht a .. Hz The Poincare Lemma again implies that (°) is a resolution of C over X. This resolution is in general not acyclic (the sheaves f2° are not soft, rather they are coherent!). However we do want to note the following: Theorem 5. Let X be a complex manifold such that

H9(X,fl')=0

for all

p>0,q>_ I.

Then there exists natural C-isomorphisms

H°(X, C) = Ker(d: 0(X) - 12' (X )) H4(X, C)

Ker(d: 129(X) -+ 129+' (X))/Im(d : 129 - '(X) - Sr(X )).

In particular,

H9(X, C) = 0 for q> m =dim X. Proof: The existence of the isomorphisms follows from the formal de Rham Theorem. The statement about the complex cohomology of X is then trivial since

129=0 forallq>m. The vanishing of the complex cohomology of a real 2m-dimensional manifold X from dimension m + 1 on is a very restrictive topological condition on X. For example, since H2"(X, C) = C, this condition is never satisfied by a compact

81

Supptament to §4i. A Theorem of Hartap

complex manifold X. Thus for compact complex manifolds X, thpi+e are always integers p z 0, q >_ 1 so that H5(X, (F):# 0. The assumptions of Theorem 5 are always satisfied by Stein manifolds. It is not known if there exist non-Stein manifolds of this type (i.e. H'(X, CV) - 0, p >_ 0, q > 1).

Supplement to §4.1.

A Theorem of Hartogs

1, ..., m} denote the polycylinder of Let A,(0) __ ((zt, ... , z",) E C"' Z'. I < s, radius s > 0 about 0 e C'". Using elementary techniques (Laurent developments, see [SCV], p. 31, Theorem 1.3) one can convince himself of the validity of the following:

Theorem 1. Let m' _ 2, s > r > 0. Then every function holontorphic on the "annular region" A,(0)1A,(0) can be continued to a function holomorphic on the entire polycylinder A,(0).

This and Theorem 4.1 imply the following: 0

Theorem 2. Let m >_ 2 and (_ I a da,, e .dO,1(C'") be a closed (0, 1)-form µ=1 M

with compact support (i.e. supp cp

U supp a is compact). Then there exists a

,,=1

function g e 8(C'") with compact support so that Sg = (p.

Pronf. Let A,(0) be a polycylinder which contains supp 9. Take s > r and let v E if (A,(0)) be the function guaranteed by Theorem 4.1 so that Ov = (p I A,(0). In particular

N = 0 on

A,(0)\supp rp.

Thus v e O(A,(0)\A,(0)). By Theorem 1, v I A,(0)\A,(0) is continuable to a function h e O(A,(0)). Define w:= v - h E .?(A,(O)). Then likewise 3w = 9, A,(0). Since v and h agree on A,(0)\A,(0), the trivial extension, g, of w to all of C"' is the desired function.

0

From this we are able to easily derive the following fundamental fact: Theorem 3. (Hartogs' Theorem [SCV], p. 33). Let G be a domain in C', m >_ 2, and K a compact set in G such that G\K is connected. Then the restriction homomorphism O(G) - O(G\K) is surjective.

Proof: We choose a compact set L c G with K c L. Thus if f e O(G\K) is arbitrarily given; f I G\L is always continuable to a differentiable function v E 8(G). This is accomplished by choosing some r e S (G) with r I G\L =_ 1 and r I U(K) __ 0,

where U(K) c L is a neighborhood of K (see Theorem A.4.4). Then one extends f

82

Chapter II.

Differential Forms and Dolbesult Theory

Ov e sxt°''(G). Since by the trivial extension of r f.to all of G. Define trivially to a 0-closed (0, 1)-form t/i Of = 0 on G\L, one can extend 0

rp a sd°-'(C'") with compact support. By Theorem 2 there exists a g e 6(C") with compact support such that Og = rp. Since W vanishes on C'"\L, g I C'\L is holomorphic. Since V vanishes identically outside of its compact support, g I W = 0, where W is the unbounded component of Cm\L Now since every connectivity compon-

ent of C'\L has points of L in its boundary, U W n (G\L) is non-empty. Let h == v - g. Then h e 0(G) and h I U = v I U= f I U. Thus by the identity principle (using the connectivity of G\K), It I G\K =. I G\K.

f In other words, h is the desired holotnorphic continuation off to G.

0

Chapter III. Theorems A and B for Compact Blocks in Cm

In this chapter the main results of the theory of coherent analytic sheaves for compact blocks Q in C" are proved (see Paragraph 3.2). The standard techniques for coherent sheaves and cohomology theory are used, in particular the fact that HQ(Q, So) = 0 for large q (see Chapter B.2.5 and 3.4). Moreover we will bring into play the fact that HQ(Q, C1) = 0 for q > 1. The basic tool which is derived in this chapter is an attaching lemma for analytic sheaf epimorphisms (Theorem 2.3). The proof of this lemma is based on an attaching lemma of H. Cartan for matrices

near the identity (Theorem 1.4) and the Runge approximation theorem (Theorem 2.1).

§ 1.

The Attaching Lemmas of Cousin and Cartan

Unless the reader knows the origins of the problems, reference to the fundamental lemmas of Cousin and Cartan as'attaching" lemmas carries little meaning. Thus we wish to begin this section by remarking that the existence of these lemmas allows us to solve attaching problems. For example, suppose that on a cover U = (U,) of a complex space X one has prescribed meromorphic functions

mi on U, so that mj - m; -'fj e d(U; n Uj) whenever U, n Uj * 9. In other words, one has prescriptions of "principal parts" of meromorphic functions! If

f e 0(U;) can be found so that f j = f - fj on U, n Uj, then the meromorphic function m, which is defined by m; -f on Ui, has the prescribed principal parts. Hence, if one can find such f's, one can "attach" the m,'s to each other. Solving this additive attaching problem (i.e., given the f j's, find the f's) in a very special case is the essence of Cousin's attaching lemma. Cartan's attaching lemma solves the analogous multiplicative problem for holo-

morphic matrices near the identity. This will allow us to attach sheaf epimorphisms.

1. The Lemma of Cousin. We always work in C' with the variable z = (z,, ..., z.). We set x := Re z,, y := Im z, and let

E={z, eCI a<x- 1 be fixed natural numbers and V a non-empty open subset of CV". If a = a(z) = (aa(z))t isp.1

z e v.

is a (p, q)-matrix valued function on V, then we define the norm of a by

Ial'=coax lain,.. 1.

Clearly I a I < oo if and only if each function au(z) is bounded on V. If a (resp. b) is a bounded (p, q4matrix (reap. (q, r}matrix) on V, then a b is a bounded (p, r)matrix and

Ia.bI 0, we define

is valid for

(e+b;,)eB(B')

(e+bi) (e+bo)EB(B")

(e+b;;_

v I D for all n > 0. From the estimates From (-) it follows that a = u I D in (,) and the fact that t < 4, it follows that W

2 Y I b, Ie. = 2KL E t'' = 4Ke < s. =o v=o

(*)

Thus, by Lemma 2, the sequence u converges on B' to an invertible matrix c' c B*(B') such that

V=o

Ib',8.54KIa-eI9.

For the analogous reason, the sequence v converges on B" to a matrix c" E B*(B")

with ic" - eI .. < 4KIa - eID. Furthermore,

e+

converges to e.

Thus,

lima.+, The above lemma only applies to matrices "near the identity." However the same statement holds for arbitrary invertible matrices. Since we don't need this more general version for the proof of Theorems A and B we do not go any further into this matter here.

Sheaf Epimorphisms

§ 2. beenb'y"

We provin sn approximation theorem for holomorphic functions in a very special, geometric situation. This is done by going back to the definition of the Cauchy itgral as a Riemann sum. By means of this approximation theorem and the Cartan Attaching Lemma, we are able to attach epimorphisms of sheaves. Later we will again make decided use of the approximation theorem (see IV.4.4).

Chapter III. Theorems A and B for Compact Blocks in C"

90

In this section we will often write z = x + iy for zr and z'for (z2, ..., z,,). We will always use

R:={zeCIa 1

W Q = R x Q', where R is a compact rectangle and Q' is a compact block in

92

Chapter III. Theorems

B for Compact Blocks in C'

C'"-'. By Theorem 1 there exists a polynomiall= i f zi, with f, e O(Q'), such i=0

that I f - I a < e/2. By the induction hypothesis there exist polynomials f, e C[z2, ..., z",] with

II-

IQ'

where

(n + 1)T

T = max Osisn

I z't I,h

Defining j:= Y_ bit e C[z,, z2,..., z,"], it follows that i=0

13Thus If-1I12e}.

We then set

K- '-R- x K',

K'-R' x K'

and

P==K-nK+=(R-nR+)xK'. We consider analytic sheaves on K. Recall that a sheaf So on a set M in C' or more generally in a complex space X is said to be analytic if ,' is defined on an open neighborhood of M and is analytic there. Correspondingly, sections, homomorphisms, exact sequences, etc. are always defined on open neighborhoods of M.

In the proof of the following theorem we apply both the approximation

theorem and Cartan's Attaching Lemma.

Thmm 2 (Attaching Sections). Let ,9' be an analytic sheaf on K. Suppose that a .'(K -) and ti-, ... , to a .'(K +) are such that their restrictions to P generate the same O(P}submodule of .'(P): O(P)t1- I P = ,E O(P)ti I P.

93

Attaching Sheaf Epimorphisms

$ 2.

Then

an

exists

there

(p, p}matrix

holomorphie

invertible,

on

K",

a- a GL(p, 0(K-)), and sections t1, ..., tP e E(K) so that

(t1 I K-, ..., tPI K-) = (ti-, ..., t; )a-. Proof: By assumption we have equations 9

t; I P = Y to I P ' uai,

P

ti I P =

a=1

p, q,

to I P ' vBi> 9=1

with coefficients us,, vpj a 0(P). We write the sections ti- (resp. t;) as row vectors

t- e $"P(K-) (resp. t+ e 9"9(K+)). Thus, writing the coefficients us; and vaj in matrix form u and v respectively, we have

t+IP=t-IP-v.

(1)

For p > 0 we set

E'=(zeCla-p<x- 1 all of the groups H9(Q, 0°) vanish, the maps 8. are bijective. 3. The Proof of Theorem A for Compact Blocks. We give the proof of Theorem A for compact blocks by induction on d =d(Q). In the case of d = 0 the claim is trivial, because Q is a point z e C' and 50 = Y,, is a finite Os module. Thus we consider the case of d > 1. We let Ad and Be denote the corresponding statements of Theorems A and B for all compact blocks of dirpension d(Q) 0 and an O-epimorphism hj: 0) QJ - b" I Q,.

§ 3.

99

Theorems A and B

01

02

03

0,

05

We first consider h 1 and h2. Since Q 1 n Q2 = Q(e 1) is a (d I)-dimensional block, F,-, implies that the induced homomorphisms of sections,

(0Pt(Q(e1))-''(Q(e1)) and are surjective. Thus, by Theorem 2.3, there exists an analytic epimorphism h1.2: (9P%+P21 Q1 U Q2 -+ °9'1 Q1 U Q2

Repeating this procedure for h1.2 and h3, we obtain in the same way an epimorphism

/

h1,2.3:OPt+P2+P31Q1 U

Q2 U Q3'i (f1Q1 U Q2 U Q3

Continuing on in the obvious way, after (1 - 1) steps we have the desired analytic epimorphism Lr'P"+...,,1

h1.z, ....1:

Q -8 ,V I Q.

Chapter W. Stein Spaces

Stein spaces,are complex spaces for which Theorem B is valid. Theorem A is a consequence of Theorem B and thus is automatically true for such spaces. A complex space is Stein if it possesses a Stein exhaustion. Particular Stein exhaustions are the exhaustions by blocks. Every weakly holomorphically convex space in which every compact analytic subset is finite can be exhausted by blocks and consequently is a Stein space.

§ 1.

The Vanishing Theorem HQ(X, 9) = 0

In this section the central notion of a Stein set is introduced. Compact Stein sets are constructed from compact blocks in C" by means of a lifting process. The main tools for this are the coherence theorem for finite maps and Theorem B for blocks. It is shown that complex spaces X which are exhausted by compact Stein sets

have the property that H9(X, 9) = 0 for all q > 2, where 9' is an arbitrary coherent analytic sheaf. Moreover, whenever such an X possesses a so-called Stein exhaustion, the group H'(X, 5") vanishes.

1. Stein Sets and Consequences of Theorem B. The following language is convenient: Definition 1 (Stein Sets). A closed subset P of a complex space X is called a Stein

set (in X) if Theorem B is valid on P (i.e. for every coherent analytic sheaf 50, H9(P, 9) = 0 for all q >_ I). A complex space which is itself a Stein set is called a Stein space.

It follows that compact blocks in C' are Stein sets. Applying the vanishing of the first cohomology groups, one obtains the following theorem in exactly the same way it was proved for blocks (p. 97): Theorem 1. Let P be a Stein set in X and suppose that h:.5" -+ 3' is an analytic epimorphism between coherent analytic sheaves over P. Then the induced homomor-

phism of sections, hp: ,'(P) -' 9-(P), is surjective.

101

The Vanishing Theorem H'(X, fP) = 0

§ 1.

We say that the module of sections Y(P) venerates the stalk .fix, x c- P, if the image of 91(P) in 9x via restriction 5'(P) sox, s --* s(x), generates 99x as an Ox-module.

Theorem 2 (Theorem A for Stein Sets). Let P be a Stein set in X and So a

coherent analytic sheaf on P. Then 91(P) generates every stalk Yx, x E P.

Proof: Let x e P be fixed. We denote by A( the coherent sheaf of ideals of all germs of holomorphic functions which vanish at x. In other words ,lip = OP for p * x and .,#X = nt(('x) = the maximal ideal of Ox. Defining .4,:= di I P, it also follows that V.9' is coherent over P. By Theorem 1 the sheaf epimorphism, So/.N So(P). Now So -+ So/.4'91, induces an epimorphism of sections, .9'(P) (.f/-4' 9°)p = 0 for

p * x and (So/,N'.y)x =. Yx /nt(Oj9-x.

Thus e is just the restriction map 9'(P) Y X followed by the quotient epimorphism Sox Yx/nt(OJY.,. Now let et, ..., e," be a generating system of the finite dimensional C-vector e,,, space 1x I m(Ox).9x and let sl, ..., s", e ."(P) be sections with I < p< m. By a well-known theorem in the theory of local rings,' the germs slx, ..., s,,,, e '/x generate the Ox module YX. Corollary. Let P be a compact Stein set in X and So a coherent analytic sheaf on X. Then there exists an integer p > 1 and an O I P-sheaf epimorphism,

O°I P-.So such that the associated homomorphism of sections, OP(P) --+ S°(P), is likewise surjective.

Proof: Theorem 2 implies that .°(P) generates '/x for every x e P. Thus the compactness of P allows us to choose a finite open cover

sNS of P and sheaf

p

homomorphisms h,,: O°" I P -- ./ which are surjective on U. Setting p M

v=1

and h:= Y h, it follows that h: Op I P -. .' is surjective on P. µ=1

2. Construction of Compact Stein Sets Using the Coherence Theorem for Finite Maps. From compact blocks in C' we obtain other compact Stein sets by a lifting process. These are very important for the further development of the theory. ' What we need for the above proof is a simple consequence of Nakayama s Lemma (see [AS], p. 213):

Let R be a noetherian ring with maximal ideal m and suppose that M is a finitelygenerated R-module. Then the elements x ... , x e M generate the R-module M if and only if their equivalence classes zt, ... ,

x, a M/mM generate the l'/m-vector space M/mM.

Chapter IV.

102

Stein Spaces

Theorem 3. Let X be a complex space and P c X a set with the following properties:

1) There is an open neighborhood U of P in X, a domain V in C' and a finite holomorphic map t: U --+ V. 2) There exists a compact block Q in C' with

Q c V and t-'(Q) = P. Then P is a compact Stein set in X.

Proof: We note first that P is likewise compact. Let .9' be a coherent analytic sheaf on P. There exists an open neighborhood U' of P which is contained in U such that .9' is analytic and coherent on U'. Since t: U -+ V is finite, there exists a

domain r c V with V = Q and r' 1(V') c U'. The restriction oft to the corresponding map t-1(V') -+ Vis again finite. Thus .9"

t 1(.9' I t -' (V')) is a coherent

sheaf over r. Since P = tthe map t I P: P - Q is also finite. Hence by Theorem 1.1.5 there exist isomorphisms H'(P, .9') = H4(B, (r I P).(Y)), q >_ 0. But

(t I P),(9) = ts(So I P) _ 9'1 Q is coherent. Consequently, applying Theorem B for compact blocks in Cm, Hr(Q, .9-) = 0 for all q > 1. Thus the groups HQ(P, .9') vanish for q Z 1. O 3. Exhavatioos of Complex Spaces by Compact Stein Sets. If X is a topological of compact subsets of X is called an exhaustion of space, then a sequence X whenever the following hold:

1) Every K, is contained in the interior of KY+1: Kv c ICY+1 0

2) The space X is the union of all the K', s: X = U K,. V=1

If X has such an exhaustion then every compact set K C X is contained in some K, and X itself is locally compact. Using an exhaustion, sections in sheaves are frequently constructed by the following simple principle: Let {K,)v21 be an exhaustion of X by compact sets. Let 9 be a sheaf on X and s, e Y(K,) a sequence of sections having the property that sv+ 1 I Kv = sv, v 1. Then there exists a unique section s e .P(X) such that

SI KY=s The proof is trivial.

v> 1.

o

A simple result in general topology states that every locally-compact, second countable topological space possesses an exhaustion. For complex spaces we have the following obvious remark: A complex space X (with countable topology) is exhaustable by a sequence of compact Stein sets if and only if every compact set K c X is contained in a compact

Stein set P e X.

1.

103

The Vanishing Theorem H'1X, y) - 0

Example: Every open block in C" as well as C" itself is exhaustable by compact Stein sets (namely compact blocks).

4. The Equations H'(X, S') = 0 for q >_ 2. If X is a paraoompact space and .9' is Y) can be computed a sheaf of abelian groups on X, then the cohomology via a flabby resolution

as the cohomology of the associated complex of sections (see Chapter B.1). If K c X is compact, then the restriction of (f) is a flabby resolution of So 1 K. One has a commutative diagram,

0.s 9'(X)! 9o(X) 10-.... _, q-z(X) S?, 4-1(X) d , 9' (X) °!.....

0

Y(K) - e(K) _ _ ...

fa-:(K)

-1(K) ._.-, Y9(K)

where the vertical maps are the restrictions. It follows that

0 = H'(K, .") = Ker(d,( K)/lm(dt_, ( K) if and only if the bottom row is exact at the place 94(K). One would expect that if this exactness were the case for all sets of some exhaustion {K,),1 for X, then the top row would also be exact at .94(X). In this direction we prove the following:

Theorem 4. Let X be paracompact and .' a sheaf of abelian groups on X. Let q >_ 2 and suppose that {K,},21 is an exhaustion of X by compact sets K having the property that H4

(Ku, 9) = H'(KV, 9) = 0,

all v.

Then H'(X, .50) = 0.

Proof: For K == Kv we consider the diagram (;) for each v. Since H'(X, Y) Ker dq /I m dq _ ,, in order to prove the vanishing, given a section a e Ker dq, we

must find (l e .''-1(X) with dq_ 1($) = a. Thus it suffices to inductively construct a sequence l3v e Y4-1(K,,) with the following properties: a (K,,,

1) (ppd.-,1 K.)fl,//

2) Y,.1

(Kr = Nr

Then by a remark in Paragraph 3 there exists a unique section i4 E .Soq-1(X) With J11K, = fi,,. Since (;) is commutative, it will then follow that

(d,-1t0)1K,= (dq-IIK,)(fl1K,)= for all v (i.e. dq _

= a).

K,

Stein Spaces

Chapter IV.

104

We now construct the sequence a,. Since aIK, a Ker(dq I K,)

anti H4(K 91) = Ker(dq I K j)/lm(dq- i (K,) = 0,

there exists a sequence.f', e .9q-t(K,) which satisfies (1): (dq _ 1

I K,)P; =aIK, for all v.

We define fit :=f;. Let Y1, ..., P, be already constructed satisfying (1) and (2). Then (d,- 1 I K j , = aIK, and hence (dq - t I K, y+ 1 I K, - fi,) 0. In other I K,). But

words fl',+ t I K, - ft. a Ker(dq

H9-'(K,, 9) = Ker(dq- t I K.)/lm(dq-2I K ) = 0. Therefore there exists y; e Soq2(K,) with dq-2(Yv) _ P,+t K. Now, due to the fact that q > 2, yq- 2 is flabby on X. So there exists y, E 5q- 2(X) which is a continuation to X of y;,. We correct $',+t as follows: Q

+1 - (dq-2Y.)IKv+t e Yq

N,+t

Since dq_t

t

(Kv+t)

d,-2= 0, (dq-IIKv+t)f.+l = (dq-IIKv+t),ffv+1 = aIK,+t

Furthermore (d,,-2y,) I K, = (dq- 2 I K,)y;, = #' + t I K, - f,. Hence

0

( d, , -

The following is now immediate:

Theorem 5. Let X be a complex space which is exhaustable by a sequence {P,),$1 of compact Stein sets. Let Y be a coherent analytic sheaf on X. Then

Ii-,(X, $o)=0 for all q>-2. and the Equation H'(X, So) = 0. We want to analyze 5. Stein under which additional assumptions on the exhaustion used in the proof of Theorem 4 the first cohomology group can be shown to vanish. For this we begi.1 with a complex space X and exhaustion {P,) 1 by compact Stein subsets of X. The commutative diagram

0 -Y(X) t S'°(X)

I0-^-(P.)

'

9'°(P,)

°p

aolp.

"(X) ! _

Jl(p,)

.... .

iIP.

..

§ 1.

105

The Vanishing Theorem HI(X, 9) = 0

is exact at .99°(X) and So°(P,). Thus, using the. injection i, we interpret .9'(X) and .9'(P,) as subgroups of .9'°(X) and .9'°(P,) respectively with

,9'(X) = Ker d°

and .9'(P,) = Ker d° I P,.

In order to prove that H'(X, .50) = 0 one must show that for every a e Ker d1 there exists fi e V°(X) with do f = a. The choice of a sequence IT, e .9'°(P,) with (do I P,));, = a I P, can be carried out as before, because H1(P .9') = Ker(d 1 I P,)/Im(do I P,) = 0. However the construction of a sequence P, e.9'°(P,) which, along with the property (do I P,)f, = a I P additionally satisfies #,,, I P,, = f, is no longer possible. This is due to the fact that P,+ 1 I P, - /3, lies in Ker do I P, = .9'(P,) and, since .9' is not a flabby sheaf, it is not possible to continue it to a section in .9' over all of X. were used in order to obtain In the previous case the equations $,+ 1 I P, ,

by successive continuation a section /3 with dq_ 1 /3 = a. This can be done, however,

in another way: Given a sequence /3, e .9'°(P,), one can determine a "correction sequence" b, e .9'(P, _ 1) which, instead of (2), satisfies the following:

vZ2.

($ +1 +k+ 0 1 P-1 = (P.,+av)I P,-1,

Then, by the remark in Paragraph 3, the sequence of pairs /3 6, determine a If in addition /3 e .9'°(X) with #I P, = (f,+ 1 + b,+ 1) I P,. section (do I P,)/3, = a I P, and 5,+ 1 e .9'(P,) = Ker(do I P,), then do i01 P, = (do I P,)(f v+ 1 I P,) + (do I PJ(a,+ 1) = al P,,

for all v. In other words do # = a. In the following, instead of continuing sections, an approximation of P + 1 I P, /3, e .9'(P,) by sections in Y(X) enters. However for this one needs a good topology on the C-vector space .5o(P,). This is one of the main reasons that the case q = I is significantly more difficult than that of q 2. In the following definition we list the key properties that are needed in order to prove that H1(X, .) = 0.

Definition 6 (Stein Exhaustion). Let X be a complex space and .9' a coherent sheaf on X. An exhaustion {P,),,1 of X by compact Stein sets is called a Stein exhaustion of X relative to Y if the following are satisfied: a) Every C-vector space .5o(P,) possesses a semi-norm space .9'(X) I P, c .9'(P,) is dense in .9'(P,).2

I

I, such that the sub-

b) Every restriction map .5(P,+1) c .9'(P,) is bounded. In other words, there

exists a positive real number M, so that

I s I P,, I, < M, I s

for all

se.9'(P,+1),v>1. 2 A semi-norm has all of the properties of a norm with the one exception that I x I = 0 no longer implies x = 0. Semi-normed vector spaces are topological spaces which are not necessarily Hausdorlt. Thus sequences can have more than one limit.

Chapter IV. Stein Spaces

106

c) If (sj)j. N is a Ceuchy sequence in Y(P,), then the restricted sequence (sl P,-,)j.,, has a limit in .9'(P,_1), v 2.

d) IfsE.f(P,)and IsI,=O, then s+P,_10,vz2.

Maintaining our earlier notation, we now show the following:

Theoran 7. Let X be a complex space and 9 a coherent analytic sheaf on X. Further suppose that there exists a Stein exhaustion (P,) for X relative to 9. Then given a section a e Ker dl there exist two sequences fi, E .9'°(P,) and b, a .9'(P, -1 v Z 3, with the following properties:

/!(dO I P,)P, = a I P,

1)

{Ct+l,*4,+1) P,-1 a (P, +a.)I P,-1 The sequences f,, d, de (j. ection Q 1:..9'°(X) with PIP,-1 = (P, + a.) I P,-1 such that do f - a. In partkuhr II'(X, .9') = 0. 2)

Proof: We have just observed that the existence of two sequences fi 6, having properties (1) and (2) results in the existence of rse Lion ft with do . - a. This obviously implies that H'(X, .9) = 0. The construction of sequences B b, is.carried out in three steps. We may assume that the restriction Y(P,+ 1) -..Y(P,) are contractions (i.e. M, 5 1 for all v).

1) We first construct the sequence f,. As in the proof of Theorem Cone chooses a sequence P. a .Y0(P,) with (do I P,)ff, = at I P,. For this one uses the vanishing of H'(P,, S) - Kttr(d1 I P,)/Im(do I P,). We obtain the fl,'s inductively from the sequence f',. One begins with f 1 ft'1. Let II1i ... , P, be already constructed satisfying (1). Define YY','=TV+1I P, -

I,.

Then (do I P,)y, m a S P. - a l P, = O,

(i.e. y;, 1:.9'(P,))

By a) of Definition 6, y; is approximable by sections in .9'(X). We thus may choose

y, e .9'(X) such that

IY;-y.IP,I, 1. Since 0W is compact, continuity implies that there exist finitely many sections h1, ..., h,, e 0(X) so that

max {IRe 1SKSm

(Im

1

max { I Re hp(p) (, I Im h (p) 1) > I 1:914:5M

for all pEOW. Let Q == {z1i ..., z") E C" I I Re z 15 1, I Im z ( < 1) denote the "unit block" in C". Then the sections h1, ..., It,,, a O(X) determine a holomorphic map

a: X - C",

x -+ (red h1(x), ..., red h(x)).

Chapter IV. Stein Spaces

112

From (_) we see that

n(8W) n Q= ,

Kcn

n

W.

Thus P'= n-' (Q) n W is compact and K c P°. Hence (P, n) is the desired stone. iii) i): Let W c X be the open set associated to the stone (P, it). Since P n W= P and K c P, it follows that k n W c P n W. But P is compact. Thus K n W is compact and W is the desired neighborhood of K.

The following theorem concerning stones is quite important: Theorem 7. Let (P, n) be a stone in X and Q an associated compact block in C'°. Then there exist open neighborhoods U and V of P and Q in X and C' respectively

with n(U) c V and P = n-'(Q) n U such that the induced map n I U: U - V is proper.2

Proof.- Let W c X be the associated (via Definition 5) open set to the stone (P, n). We may assume that W is relatively compact. Thus OW and a(W) are also

compact. Since 0W n n-'(Q) is empty, V'= C"\n(aW) is an open neighborhood of Q. The set U := W n a-'(V) = Win(a(0W)) is open in X, n(U) c V, and it I U: U -' V is proper.-' Furthermore

n- I(Q) n U = 71- 1(Q) n W\n-'(Q) n "WOW)}

The set n-'(Q) n n-'(n(OW)) is empty, because Q n n(W) is empty. Thus n"'(Q) n U = P and, in particular U is a neighborhood of P. 4. Exhaustion by Stones and Weakly Holotnorphically Convex Spaces. Let (P, n) and ('P, 'n) be stones in X with associated maps n: X -. Cm, 'a: X - C and blocks Q, 'Q. Definition 8 (Inclusion of Stones). The stone (P, n) is said to be contained in the stone ('P, 'n), in symbols (P, x) a ('P, 'n), if the following are satisfied:

1) The set P lies in the analytic interior of'P: P c'P°. 2) The space C' is a direct product C° x C' and there exists a point q e C" so that Q x {q} c Q'. 3) There exists a holorrwrphie map qt: X -. C' so that (i.e.

'n = (n, (p)

'n(x) = (n(x) V(x)), x e X).

This inclusion relation is transitive on the set of analytic stones in X. if (', n) c ('P, 'n), then

P C 'P,

P° c 'P°,

and

dim Q 5 dim 'Q.

3 A continuous map f between locally compact topological spaces is called proper if the f-preimages of every compact set is compact. The proof of the following is trivial If f: X -+ Y is a cantfiwout nwp subset of X, then the induced map

-

W°

between topological spaces and W is an open

Y -l(aw) is proper.

ogritpact

Weak Holomorphic Convexity and Stones

§ 2.

113

Definition 9 (Exhaustion by Stones). A sequence {(P,, nj)vz 1 of stones in X is called an exhaustion of X by stones if the following hold:

(P,, r) c (P,+,, nv+,) for all v > 1.

1)

U P° = X.

2)

v=1

Since P, c P°+ 1, every compact set K (-- X is contained in some P.

Theorem 10. The following statements about the complex space X are equivalent:

i) There exists an exhaustion of X, {(P,,, analytic stones. ii) For every compact set K c X there exists an open set W c X so that k (-- W is compact.

Proof: i) ii): Since K c P° for some j, this is contained in Theorem 6. ii) - i): Let {(KY)},, t be an exhaustion of X by compact sets. Using this exhaustion we inductively construct an exhaustion of X by stones. Let (P1_ 1, n1_ 1) with n1_ 1: X -+ C"J-' be already constructed so that K_1 c P°_ 1. Let Q1_ 1 c C'J-' be an associated block. Let K1 u P1_, be the compact set in ii).

Then, by Theorem 6, there exists a stone (P1, r7) with K1 v P1-, c P°. Let nj': X - C", QJ c C" be an associated block, and W c X be an open set such that Pi

W.

We choose a block Q'. c Cm'- with Q1_ c ¢f so that the compact set

7i1_ 1(P1) c C'"%-' is contained in ¢j. We now set 791.= (n1-1, nj ): X .. C'J -' x C" and

Q1

Qj x Q'

.

Certainly

n1 1(Q1) n W = nj lt(Q1) n (nl - 1(Q7) n W) = nl lt(Q/)

P1.

Since n1_ 1(P,) c ¢i, it follows that n;_t1(Qf) c p1. Thus n; '(Q1) n. W = P1 and we have sl& wn that (P1, 7t1) is a stone in X with Q1 as an associated block. From the above construction it is obvious that (P1_ 1, n1_ 1) is contained in 00

(P1, n1). Since U P° = U K, = X, it follows that {(P,, ij)v21 is an exhaustion of X by stones.

1

t

Definition 11. (Weak Holomorphic Convexity). A complex space X is called weakly holomorphically convex if the equivalent conditions of Theorem 10 are

fulfilled.

S" Holomorpbic Convexity and Unbounded Holomorphic Fimctions. In this paragraph we show that the converse-to Theorem 4 is also trut:.

Chapter IV.

L14

Stein Spaces

Theorem 12. Let X be holomorphically convex and D be an infinite discrete set in

X. Suppose that for every p e D there is given a real number ro > 0. Then there exists a holomorphic function h E 0(X) so that

Ih(p)i zrD for all pED. This theorem is proved here only for complex manifolds. In the case of arbitrary complex spaces one is confronted with a convergence problem which can be handled but which requires further considerations (Chapter V.6.7).

In order to prepare for the proof of Theorem 12, we choose an exhaustion of X by compact sets with , = K. Since D is discrete, the sets ,

Do==D n K1,

v>0,

D,,:=D n (K,+1\K,),

are finite and have the property that D. n D, _ 0

when p * v. Obviously

D=UD,. 0

Let q be an arbitrary point in D. Then q e D, for some t and consequently q 0 K,. Thus there exists g. a 0(X) with I gt kx, < 1 and gq(q) I > 1. Let DD(q)'= {p a D, I g.(p) I ? g.(q) O and D; (q) `= D,1 D,(q).

Certainly q F, ,,q). Furthermore, by multiplying g4 by a constant if necessary, we may assume that g9(p) I < 1 for all p e 1;(q).

Now we list the different points in D, as xt1, x,2, ..., x., and write D as a sequence (x j, 0 enumerated as follows: x11,...,x1e,,x2It ...,x2192,...,x1i,...,xh,J,... For every v let t'= t(v) be the index with x, e D,. We claim that

,,x,)

there exists a sequence (h,),, 0, h, e O(X), so that for all v r-1

1)

h,(p) I 2?t ro + 2 +

and

I hi(p) I

for all

I hr(p) 15 nj12-' for all p E D, (x,. Proof (by induction): Suppose h0, , .., h,.1 have been constructed. Since the modulus of is larger than 1 on D;(x,) and less than 1 on K, u D, (x,), h, g,',, does the job ifs is chosen to be sufficiently large. 2)

I h, Ia, 5 nr 12'',

N

We set j =

/Zi

h,, where, h is the function hh which ;is associate to

xk = x,j E D. By 2) above we have I hip k, 5 rte 12-4. Thus 3)

IfI,52-t,

i=1,2,....

Weak Holomorphic Convexity and Stones

§ 2.

115

From these preparations it is clear that the desired function h. should be the 00

OD

ji It is at this

limit of the infinite series Y h, which formally is the same as i=1

o

point where the convergence difficulties arise in the non-reduced case. Thus from

now on we assume that X is reduced. Thus ('(X) c''(X) and 3) implies that Y_ f, converges uniformly and absolutely on compact subsets to a continuous i-1 4 function h with .

Go

OD

d=o

i=1

h= y h,, = Y f and

1

< Y_ n, '2-' < 2` < 1.

h,,(P)

M>1 n+

Thus,, since

+

(ht 1 +

h,,,,

n>1

(`')

Ei

i=1

f'(P)

I

Z

Ii-1t f (P) + i=1

h,j(p)

- 1.

Since p e D,,(x,,), we can apply 1) to show that I- I

,

> r,, + 2 + 1=11=1

hi!(p) I +

h,j(p) I 1=1

Hence

1=t

fi(P) +

Z h,l(P) I -

(t

1=1

1=1 1=1

h,f(p)

h, (p) + i h,,(p) > r, + 2. 1-1

The estimates (+), (ss), and (s*') together yield Ih(p)I z rv.

Chapter IV. Stein Spaces

116

It remains to show that the constructed It e cf(X) is in fact holomorphic on X. If X is a manifold then this follows from the classical theorem that a uniform limit of holomorphic functiors is holomorphic. This statement is also valid on reduced

complex spaces (Theorem 8, Chapter V.6.6). Thus Theorem 12 above is completely proved when X is a manifold and is proved modulo Theorem 8 in V.6.6 for reduced complex spaces. For the non-reduced case, some additional considerations are needed (see V.6.7). For these we need the following remark: If (h ) Z 1 is the sequence constructed above and (m,), 0 is an arbitrary sequence

of positive integers, then the sequence (h;"), 0 has properties 1) and 2) and in particular the series E h'" converges (for reduced spaces) uniformly on compact 0

subsets to a function g e'(X) with I g(p) f > ra for all p e D.

§ 3.

Holomorphically Complete Spaces

In this section the notion of an analytic block is introduced. Complex spaces which possess exhaustions by analytic blocks are called holomorphically complete.

1. Analytic Blocs. For every stone (P, n) in X with an associated block Q c C'", there exist neighborhoods U and V of P and Q respectively such that n(U) c V, P = n-'(Q) n U, and the induced map n I U: U V is proper.. Definition 1 (Analytic Blocks): A stone (P, a) is called an analytic block in X ifU

and V can be chosen so that n' U: U -+. V is finite.' '

It is clear that every block Q c C' has the associated analytic block (Q, id) in

C".

Theorem 2. If (P, n) is an analytic block in X, then P is a compact Stein set in X.

Proof: This follows immediately by applying Theorem 1.3 to the map n'U: U-+ V.

0

There exists an important and easily described class Qf spaces in which every stone is an analytic block. For this we first note the following general fact: Let X be a complex space in which every compact analytic set is finite. Then every

proper holomorphic map f U -- Y of an open subset U in X into an arbitrary complex space Y is finite.

Proof: Every fiber f''(y), y e Y, is a compact analytic subset of U and consequently is a compact subset of X. Thus every f-fiber is finite. p 4 By assumption finite maps are always proper.

.

§ 3.

Holomorphically Complete Spaces

117

The following is now obvious:

Theorem 3. If X is a complex space such that every compact analytic subset is finite, then every stone in X is an analytic block. 2. Holomorphically Spreadable Spaces. We now introduce a classical notion of Stein theory. Definition 4 (Holomorphically Spreadable). A complex space X is called holomorphically spreadable if given- p E X there exist finitely many function f,, ..., f, E 0(X) so that p is an isolated point of the set {x e X I f, (x) = ... =Mx) = 0}. Every domain in C' is obviously holomorphically spreadable. A complex space is called holomorphically separable if, given x,, x2 e X with x, * x2, there exists f e (G(X) such that f (x,) f (x2). It can be shown that every holomorphically separable space is holomorphically spreadable. The proof of this is elementary, but nevertheless uses dimension theory and will therefore not be given here. The following is a simple cdtsequence of the maximum principle. Theorem 5. In a holomorphic separable or holomorphically spreadable complex space X, every compact analytic subset A is finite.

Proof: Let B be a connected component of A. Thus every function f I B, f E O(X), is by the maximum principle constant on B. If X is holomorphically separable and p, p' were distinct points in B, then there would exist h E (0(X) such that h(p) * h(p'). Since this can't happen, B = {p}. If X is holomorphically spreadable, then there exist functions f,, ... , f, e 6(X) such that p is an isolated point of {x e X I f,(x) _ = f,(x) ='O}. Since f) B = 0 for all i, B = (p). Corollary I. In a holomorphically spreadable space X, every stone is an analytic block.

Proof: This is now clear by Theorem 3. Corollary 2. If A is a compact analytic subset of the complex space X and there exists an analytic block (P, rt) in X with .1 c P, then A is finite.

Proof: Let U c X and V c C' with P *a U be such that n U: U - V is finite. Then U is holomorphically spreadable. To see this just note that p E U is an isolated point of (x e U If, (x) = ... = f",(x) = 0), where a(p) = (c,, ..., and

f = (z, -

o a E tr(U).5 Thus by Theorem 5 A is finite in U.

3. Holomorphically Convex Spaces. An exhaustion ((P n,)},,, of X by stones is called an exhaustion of X by blocks whenever every (P n,) is an analytic block. ' In general if f: X -. Y is finite and Y is holomorphically spreadable, then X is holornorphically spreadable.

Chapter IV.

118

Stein Spaces

Theorem 7. The following statements about X are equivalent. i) There exists an exhaustion {(P,,, itj) of X by blocks. ii) X is weakly holomorphically convex and every compact analytic subset of X is finite.

Proof: i) ii): By Theorem 2.10, X is weakly holomorphically convex. Furthermore, every compact analytic subset A of X is contained in some block Pj. Thus, by Corollary 2 to Theorem 5, A is finite. ii) . i): By Theorem 2.10, there exists an exhaustion of X, {(P,, n,)), by stones. By Theorem 3, every stone in X is an analytic block.

Definition 8 (Holomorphically Complete). A complex space X is called holomorphically complete if the equivalent conditions in Theorem 7 are fulfilled.

Every holomorphically convex domain G in C', (in particular, any domain in C) is holomorphically complete. In the next section it is shown that holomorphically complete spaces are Stein spaces.

§ 4.

Exhaustions by Analytic Blocks are Stein Exhaustions

Let .9' be a coherent sheaf on the complex space X and suppose that (P, at) is an

analytic block in X. In this section a procedure is developed to provide the C-vector space .'(P) with a "good semi-norm." The properties of such seminorms are explained in detail with the main motivation for the considerations being conditions a), b) and c) of Definition 1.6 (Stein Exhaustion). The basic result is that an exhaustion {(P,,, n,)), by analytic blocks, where the Y(Pj's are equipped with good semi-norms, is a Stein exhaustion.

I. Good Semi-norms. We need the following remark: If X is a complex manifold and f is a coherent subsheaf of 0 on X, 1 < 1 < ao, then, with respect to the topology of convergence on compact subsets, the module of sections f (X) is a closed vector subspace of 0'(X). Proof. Let f; e f (X) be a sequence having a limit f e &(X). Thus at every point

x E X the sequence of germs f j,, E t,, converges to f, e Ox. Since every 0,, submodule of 0' is closed (see AS, p. 87}, f, E f; for all x e X. Hence

fE f(X). Now let (P, n), n: X -+ C", be an analytic block in a complex space X having associated block Q c C" with Q * $ . Using Theorem 2.7, we choose neighborhoods U and V of P and Q so that n(U) c V, P = n` () (q) r U, and the induced map r: U - V, where T:= it I U is finite. Then P° = r`(Q) is the analytic interior of P.

4 4.

119

Exhaustions by Analytic Blocks are Stein Exhaustions

By the direct image theorem for finite maps, the image sheaf

T `=if(YIU) is coherent on V for every coherent [ on X. Thus by Theorem A for blocks in C"' there exists I 1 and an 0-epimorphism

c: VIQ_9'IQ which induces an C(Q)-epimorphism of modules of sections

eQ: 0'(Q) - 910 Since P = -r- t(Q), there exists a canonical C-vector space isomorphism p: .1(P) - 9-(Q). For every section s e .9'(P), we define I s I by I s I '= inf(I f IQI f e 0'(Q) with cQ(f) = (p(s)).

Theorem 1. The map I I :.1(P) - R is a semi-norm on .1(P). For every section s e .1(P) with I s I = 0, it follows that s I P° = 0. Proof. It is clear that is a semi-norm on ,1(P). In fact it is just the quotient semi-norm on 0'(Q)/Ker EQ = -4-(Q) (using I IQ) transported to [(P) by i. Since Ker EQ is in general not a closed subspace of 0'(Q), the semi-norm is not necessarily a norm. I

I

Let Is I = 0. Then there exists h e 0'(Q) with EQ(h) = i(s) and a sequence hJ a Ker EQ so that lim I h - hJ IQ = 0. In particular lim (hh I Q) = h I Q in the topoJ

J

logy of uniform convergence on compact subsets. Since Jtre4 E is a coherent sub-

sheaf of: O' I Q, the introductory remark above implies that E (h I t) = 0. Thus t(s) I ¢ - ea(h I Q) = 0, and, since P° = r -'(Q), it follows that sf P° = 0. Q In the following we call any semi-norm on [(P) which is obtained as above by a sheaf epimorphism E: 0' I Q -- it. (S I U) I Q a good semi-norm. Z The Compatibility Theorem. Suppose that along with (P, rt) we have another

analytic block ('P, 'it) on X, where 'n: X - C''" and 'Q c C"" is the associated euclidean block. We fix good semi-norms I , 'I I and .1(P), Y('P) as well as I

the associated C-vector space epimorphisms

a: 0'(Q) - .1(P),

'a: O('Q) -* ,1('P).

We will keep this notation for the remainder of this section. In the case P c P' it is important to know if the restriction map,

P: [('P)

Y(P),

s -+ s I P,

Chapter IV. Stein spaces

120

is bounded (i.e. with respect to the semi-norms). When C'N'.= C" x C". and Q - Q x {q} c 'Q, then the C-linear restriction '(Q),

h -' h ' Q,

is obviously a contraction. This fact implies the following basic lemma.

Lemma. Let P c 'P, C'"' = C'" x C", and Q x {q} c 'Q with q e C". Then there exists a bounded C-linear map n: d"('Q) -' (9'(Q) so that Q)

.. ,

.('P) v

n

(*)

dt(Q) a Y(P) is commutative: a n= p o 'a. be the natural basis of 60"('Q). We choose sections Proof: Let (et, ..., g a (9(Q), p = 1, ..., 'l, with a(9v) = p ° 'a(e,.).

Obviously the map rl:

t('Q) - 7t (%

does the job.

(.ft, ... .f t)

M°t

(f I Q)9", F1

The following is now immediate: Theorem 2 (Compatibility Theorem). If P c 'P, C" = CV" x C" and Q x {q} c 'Q with q e C", then the restriction map p: "('P) 9'(P) is bounded.

Proof: Since 'a, a determine the semi-norms the commutativity of (:) implies that whatever bounds exist for I can be used as bounds for p as well.

3. The Convergence Theorem. The main result of this section is based on the following fact.

Let Q, Q* be blocks in C"' with Q c 0*. Suppose that (h;) is a Cauchy sequence in O'(Q*). Then the restricted sequence (h, I Q) has a limit in 0'(Q).

Proof, The sequence h j I Q* converges to some h E (9'(Q*) in the topology of uniform convergence on compact subsets.6 Since Q c 0*, it follows that Ii`m .I h -

hJIQ=0.

O

° The spaces O'(Q) are normed, but not complete. On the other hand, the spaces O'(¢) are complete (with respect to the topology of uniform convergence on compact subsets), but they are not normable (see Chapter V.6.1). This discrepancy makes the introduction of intermediate blocks unavoidable.

§ 4.

Exhaustions

.121

Analytic Blocks are Stein Exhaustions

Now again let (P, n) and ('P, n) be analytic blocks in X. We maintain the notation of the last section, assuming further that the conditions 1) and 2) of the inclusion definition (Definition 2.8) are satisfied:

P c 'P°

C'"' = C' X X C",

and

Q x {q) c 'Q with q e C".

Theorem 3 (Convergence Theorem). If (sj) is a Cauchy sequence in .°('P), then the restricted sequence (sj + P) has a limit s e ."(P). The section s !P° is uniquely determined.

Proof: Since Q x (q) c '¢, there exists a block Q* c C' with Q c 0* and Q* x {q} c 'Q. In a way completely analogous to the Lemma above, one now construct a map q*: 0"(Q)-+ 0'(Q*) so that

is commutative. In this case w denotes the obvious restriction map. Since 'a determines the norm on .9'('P), to every Cauchy sequence (sj) one can find a Cauchy sequence (hj) in 6"('Q) with 'a(hj) = sj. Then (q*(hj)) is a Cauchy sequence in (9'(Q*). Hence, by the fact mentioned at the first of this paragraph, the sequence q(hj) = q*(hj) I Q has a limit in 0'(Q). Consequently s a(h) a ,9'(P) is a limit of aq(h) = p(s j) = s, P. If § E .9'(P) is another such limit, then js - s 1= 0 and therefore, by 0 Theorem 1, s I P° = s `P°. 4. The Approxhn ation Theorem. Again let (P, n) and ('P, 'n) be analytic blocks

in X with C= C" x C" and Q x (q) c 'Q. We choose neighborhoods 'U and 'V of 'P and 'Q respectively so that 'ir induces a finite map 'n J'U: 'U -+'V with 'P = ',r-'('Q) c 'U. We set

Q, :_ (Q x C') n 'Q and P,

:='n-1(Q1)

n 'U.

Obviously P, is compact and 'V is also a neighborhood of Ql. Moreover, (P,, 'ir)

is an analytic block in X with associated block Q, c C" and P, c 'P. Now let .9' be a coherent sheaf on X and 'e: 0 J'Q - 'n*(,9' J'U) l'Q a sheaf epimorphism which induces the map 'a: 0('Q) - .9'('P) and the good semi-norm ,I

I.

By restricting to Q, one gets a sheaf epimorphism t,: 0 1 Q, -+'n*(y 1'U) I Q,. Since 'n*(.9' j'U)(Q,) = . 9'(P1), the map s, determines a C-linear surjection oil: 0'(Q,) -,9(PI) and the associated semi-norm 1 1, on So(P1). By construc-

Chapter IV. Stem Spaces

122

tion the diagram

b'('P) 10,

O1(Q1) a, ,.V(P1 where the vertical maps are natural restrictions, is commutative. Since these maps are all continuous, and since, by the Runge approximation theorem for blocks (see Chapter 111.2.1), V"('Q) is dense in 6'(Q1),

the space .('P)IP1 is dense in.(P1). We now assume that (P, n) is contained in ('P, 'n). Then P c 'P, and there exists a holomorphic map gyp: X -+ C" so that

'n(x) = (a(x), o (x)) a C"' x C" = C", for all x e X.

In this situation the support PI of the analytic block (P1, 'n) can be decomposed in this following way:

There exists a compact set P in X such that 13 n P = q, and

P1=PLIP. Proof: Since 'n = (n, gyp) and Q1 = (Q x C") n 'Q, one can easily verify that P1 = n-1(Q) n 'P.

Since P c 'P and P c n-1(Q), it follows that P c P1. But there exists a neighborhood U of P in X so that P = n-1(Q) n U. Hence P'= P1 \P is compact. 0 The above decomposition of P1 has as a simple (but important) consequence that whenever (P, n) c ('P, 'n) the restriction map r: .(P1) - Y(P) is surjective. Since Q x (q) c Q1, the' assumptions of Theorem 2 are satisfied and or is continuous. These observations allow us to now prove a Runge Theorem for coherent sheaves on analytic blocks. Theorem 4 (Runge Approximation Theorem). If (P, n) and ('P, 'n) are analytic blocks in X with (P, n) c ('P, 'n), then for every coherent sheaf :f on X, the space .9'('P) I P is dense in .5o(P,

Proof: The restriction map .9'('P) - ."(P) is factored into the two other restrictions

.'('P)-°'-

. °(P1)

and Y(P1)-- °-+9'(P).

§ 4.

-123

Exhaustions by Analytic Blocks are Stein Exhaustions

We have already shown that p, (5'('P)) = S'('P) I P, is dense in .9'(P, ). Since or is both surjective and continuous, it therefore follows that op1($"('P)) = .9'('P) I P is 0 dense in .9'(P).

5. Exhau9tions by Analytic Blocks are Stein Exhaustions. It is now relatively easy to prove the following essential result:

Theorem 5. Every exhaustion ((P,, n,)},2 1 of a complex space X by analytic blocks is a Stein exhaustion of X.

Proof. First, by Theorem 3.2, every set P, is a compact Stein set. On each module of sections 5"(P,) we fix a good semi-norm 1

1,. Then-conditions b) and

c) of Definition 1.6 are satisfied. Further we may assume that the restrictions 0 be given. We choose a sequence b; a U8, bi > 0, with

bj < b =t

and inductively determine by the Runge Theorem (Theorem 4) a sequence Si e . 9'(P1) with s, := s_ and

i=1,2 .

I s;+, I Pj - s, ij < bi,

.

Then (s; I P.+, )t> i is a Cauchy sequence in .9'(P,+ ). By the Convergence Theorem (Theorem 3), the restricted sequence (s; I Pi) has a limit t; e .9'(Pi). Since all of the

restriction maps .9'(Pi+ 1) - .9'(Pj) are bounded, tj+ I Pi is also the limit of the sequence (sj I Pj). The uniqueness part of Theorem 3 implies that tj+ I P° = tj I P° But the sets {P°} exhaust X. Thus the tj's determine a global section t e .9'(X) with t I P; = tj, i > 1. Since 1 1, < I 1j, the equation 1

1

tIP1-s=r, -s;IP1 +

J-t j=1

(Sj+1IP1-s1IPI)

yields the estimate

i- t

It IP1-Sit _ 0, whenever there is a Drepresenting Cousin II distribution (Ui, hi) where hi e 0(Ui) for all i. It follows that a meromorphic function h e .Ate*(X) is holomorphic if and only if (h) >_ 0. The classical Cousin II problem, which is also called the multiplicative Cousin problem, asks for a characterization of the principal divisors in .9(X). The exact cohomology sequence immediately gives an answer: A divisor D e 2(X) is the divisor of a meromorphic function h e :;H*(X) if and only if j(D) = 0 e H'(X, 0*). One says that the Cousin II problem is universally solvable on X whenever iy is surjective. The following is analogous to the situation with the Cousin I problem: Theorem 2. The Cousin II problem is universally solvable for a complex space X if and only if the natural homomorphism H'(X, 0*) -+ H'(X, K*) is injective. In particular the Cousin II problem is universally solvable for all spaces X with

H'(X, 0*) = 0. In Section 4 we will look more closely at the group H'(X, 0*) and will show in particular that it vanishes if H'(X, 0) = H2(X, Z) = 0.

We want now to give the divisor group -(X) a more geometric interpretation. For this let X be a reduced space which is irreducible at every point x e X (i.e, locally irreducible). Then every stalk .,l4 is the quotient field of the integral domain Os and ..# = .W., \(0}. A germ in 0 has a nowhere vanishing representation in some neighborhood of x. Every divisor D on X is therefore locally represented by a function f/g where f g 0 are uniquely determined up to nowhere vanishing holomorphic functions. The functions f and g determine well-defined hypersurfaces (i.e. (f = 0}, (g = 0) which may be empty) of positive order. Counting the order of (g = 0) negatively, one can therefore view every divisor D e -9(X) as an analytic hypersurface H in X where (at most countably many) irreducible components are counted with integral multiplicities. The family (Hi) must be locally-finite (i.e. every relatively compact open set U c X intersects only finitely many of the Hi's). If one additionally assumes that every hypersurface in k is locally the first order zero set of some analytic function (this is always true for manifolds), then the divisor group is canonically isomorphic to the additive group of all (even infinite) linear combinations Y_ n, Hi, ni a 1, n, # 0, where (H,) is any

locally-finite family of irreducible analytic hypersurfaces in X with H; i * j. We call the Hi's the prime components of D.

H, for

3. Poiacare Problem. In his work "Sur les fonctions de deux variables" Act. Math. 2, p. 97-113, published in 1883, Poincari had already shown that every

a apser v. Appbcatioi of lbeatemi A and B

140

meromorphic function on C2 is the ratio of two functions which are holomorphic

on C2. Thus the field ..'(C2) is the quotient field of the ring 0(C2} If X is a complex space where .&(X) is the quotient field of O(X) with respect to the elements which are not zero divisors, then one says that Poinc arfi's Theorem holds.

To keep matters simple, we consider here only complex manifolds X. From Paragraph 2 we see that every divisor D a 9(X) is uniquely representable as a linear combination E n, H, of its prime components. For every non-empty open

set U in X . one obtains

the restriction D 1 U e 9(U) as follows: If

H, n U = Y. H,J is the decomposition of H, n U into irreducible components in 0

U,thenDJU= Y. n,JHJwhere n,J:=n,.If D,D'e 1(X)andDIU,D'+Uhavea I'J=1

common prime component, then, by the identity theorem for analytic sets (see Chapter A.3.5), D and D' have a common prime component. The divisor D =

n, H, is positive (i.e. D Z 0) if and only if a1 Z 0 for all i.

Every divisor is uniquely representable as the difference of two positive divisors which have no common prime components:

D = D; - D- with D+ i= Y_ n,H, and D-

- I n,H,. q- 0. By assumption there exists g e .4 (X) with (g) = D" z 0. Thus g e O(X). Furthermore, for f ==gh a &* (X it follows that (f) = (g) + (h) = D+ > 0 and f is likewise in O(X ). Now suppose there is a point x0 e X where and gx, have a non-unit common divisor. Then there exists a neighborhood U of xo with functions p e O(U) so that

fv°P).

gu=PB

The Cousin Problems and the Poincare Problem

§ 2.

141

and

D+ I U = (fu) = (p) + (f),

D I U = (9u) = (p) + (9

where (p) a 2(U) is positive and not the zero divisor. But this implies that D+ I U and D- I U (consequently D+ and D-) have a common prime component. Since 0 this is not the case, we have the desired contradiction. In the next section we will see that the second Cousin problem is universally solvable on every Stein manifold X with H2(X, Z) = 0. From this and Theorem 3 we have the following: The sharp form of Poineare's Theorem holds for every Stein manifold X with H2(X, Z) = 0. The following is a consequence of Theorem A: Theorem 4. The Theorem of Poincare holds for every Stein manifold X.

Proof: Again let X be connected and It e K*(X). The 0-sheaves 0, Oh, and 0 + Oh are coherent subsheaves of ..K.' Thus 0 n Oh is a coherent 0-sheaf (see Chapter A.2.3c). Since J( , is the quotient field of Oz, it follows that (0 n Oh),,

0

for all x e X. The 0-epimorphism 9: 0 - Oh, fx -*fx hx, x e X, determines a coherent ideal I '= cp-' (0 n Oh) with J , $ 0 x E X. By Theorem A there is a global section g # 0 in .0 over X. For every such section we know that g,, 0 0 for all x e X and f '= gh e 49(X). 0

The Poincare problem has had tremendous influence on development of several complex variables. In order to solve the Poincare question, Cousin, in his

1895 work "Sur les fonction de n variables complexes" Act. Math. 19, 1-62, formulated the two problems which are named after him, and solved them in important special cases (e.g. product domains Bt x x B. in C'). Even in the case of product domains, as Gronwall remarked in his 1917 work "On the expressibility of a uniform function of several complex variables as a quotient of two functions of entire character," Trans. Amer. Math. Soc. 18, 50-64, it is necessary to

make the additional assumption that with at most one exception all of the B.'s must be simply connected (i.e. H2(X, 1) = 0, see the next section). In fact Gronwall gave the product domain C* X C* c C2 as an example of a Stein manifold for which the Cousin II problem is not universally solvable and for which Poincare's Theorem in its sharp form does not hold.

' In general if h1,..., h, a ..t(X), then Y -Oh, + + Oh, c M is coherent. To see this note that every point x e X has a neighborhood U so that h,+ U = p,/q with p q E O(U) and q # 0 for all u e U. Multiplication by the common denominator q yields an Ou-monomorphism a:9 Yo and obviously

Ima°OUP, +...+CUP, (-- 0U. Thus Im a (and consequently S/,,) is coherent.

Chapter V.

142

Applications of Theorems A and B

In the following table we summarize the 8 combinations of solvability/unsolvability of our 3 problems and whether or not each can happen

(see [3], p. 192):

Cousin I

Cousin II

Poincare

Possible

1

+

+

+

+

2

+

+

-

-

3

+

+

+

4

+

-

+

+

+

+

+

-

-

+

+

-

+

-

5

6 7

8

-

-

Except for 4), all examples are realizable using domains in C2. Case 4) is ruled out in C2 by Theorem 1.4. As examples, case 5) is demonstrated by the domain 6:= {(Z1, Z2) E

£2

10 < I Zi I < 1, 1 Z21 < 1) V {(Z1, Z2) E C2 I Zi = 0, 1 z21 < 1)

and case 7) is shown by the notched bicylinder D

{(Z1, Z2) E C2 I I Zi I < 1, I Z21 < t}1

{(Zi,Z2)EC2I(IZSI -1)2+(IZ21 -i)2 1. There exists a neighborhood U of p in X with U o supp o = {p) so that (U,-Ou) is isomorphic to a complex subspace of a domain B c C". We identify (U, Ov) with this subspace and let./ be the coherent ideal sheaf in OB so that Ov = (OB/j) (U. Let z,, ..., z," be coordinates for C" which are centered at p. Then the monomials

it+...+im=r, generate the ideal m(OB.j and hence their equivalence classes q,,...,. E eB/f

generate the ideal m(OB//)o _ m(ev,D)' = m;("'. Since the monomials q,, ... j. generate all stalks of OB over B\p, the functions q,, ...,. ( U e O (U) generate every

stalk Ou,,,, x e U\p. Since °(4 = Ov.x for all x E U\p (recall U r supp o = p), the functions q,, ... ,. ( U therefore generate the sheaf 3°(o) ( U. The following is quite useful.

Theorem 2 (Existence Criterion). Let o be a cycle on X so that 0(X) (O/S(o)XX) is surjective and suppose that to every point p e supp o there is arbitrarily assigned a germ gp a OD. Then there exists a function f e e(X) so that fv - 9, e. mp(°) for all

p e supp o.

k

Proof: Let P: O - O/f'(o) be the quotient epimorphism. Since supp(/T(p)) = supp o, a section s e (e/ff(o)XX) is defined by s(p):=go for

14. Sheaf Theoretical Characterization of Stein Spaces

151

p e supp o and s(p) - 0 otherwise. By assumption the induced map p,: 0(X) -

(0/!(o))(X) is an enimnrphism. Consequently there is an f e 0(X) so that mp rl s. Since p(f,) = p(gr), it follows that f, - g, a (Ker p), = 5(o), =

0

for all p e supp o. If every point of supp o is non-singular, then the epiunorphism condition in Theorem 2 guarantees that one can always find a holomorphic function on X whose Taylor series at every point p e supp o with respect to some local coordinate at p is arbitrarily prescribed up to order o(p). We now consider complex spaces X which have at least one of the following properties:

(S) For every coherent ideal f c, 0 it follows that H1(X, 5) = 0. (S') The section functor is exact on the category of coherent analytic sheaves on X

(i.e. every exact sequence 0 -+ Y'--+ 9' -+ 5°" - 0 induces an exact (9(X)sequence 0 - b'(X) -+ .°(X) -. .°"(X) - 0).

Since Z (o) is coherent (Theorem 1), every space having property (S) or (S') satisfies the surjectivity condition of Theorem 2 (i.e. 0(X) --+ (0/.°l"(o)XX) is surjec-

tive). We have two immediate consequences of this: Consequence I. Let X be a complex which has property (S) or (S'). Let (xN)ir20 be a discrete sequence in X and (c.),, 0 an arbitrary sequence of complex numbers. Then there exists a hololitorphic function f e 0(X) with f c", n > 0.

Proof. Let o(x)'= 1 for x = x5, n >_ 0, and o(x) := 0 otherwise. Then, applying Theorem 2 for g., cc e 0.,, there exists f e 0(X) with f - c" e nt,,. In other words f c for all n 0.

Comeque ice 2. Let X be a complex space which satisfies (S) or (S'). Let e == dime mr /MP be the embedding dimension' of X at p. Then there are e holomor-

phie functions f,, ..., ff e 0(X) whose germs f, p, ..., f1, e 0p form a generating system for mr as an Or-module. Proof: Let g1,, ..., g,, e in, be germs which generate mp. Applying Theorem 2 with o(p) s= 2, and o(x) 0 otherwise, there exist functions f e 0(X) with f, - gtp a mp for 15 i < e. Thus the equivalence classes j, p, ... , j1p a mp /mp gen-

erate the C-vector space mp/inp and consequently fl, ..., fp generate ntp as an Op module (see the footnote on p. 101). If p is a non-singular point of X, then the embedding dimension at p is the same as the dimension of X at p. Thus Consequence 2 says that for spaces fulfilling (S) or (S') Stein's global coordinates axiom (see the introduction) holds at all nonsingular points of X.

' The embedding dimension of X at p is the smallest integer e z 0 so that my is generated as an 0e module by e germs in m, (see [AS), Chapter 11.3).

Chapter V. Applications of Theorems A and B

152

2. Equivalent Criteria for a Stele Space. It is now easy to prove that our weakened axioms are equivalent to Stein's original ones. As a matter of fact, the following could be considered the main theorem of this book.

Theorem 3 (Equivalent Criteria for a Stein Space). The following statements about a complex space X (with countable topology) are equivalent: i) X is holomorphically complete (i.e. weakly holomorphically convex, and every compact analytic set in X is finite).

ii) X is Stein. iii) If f is a coherent ideal contained in O, then H'(X, .5) = 0 (property (S)). iv) The section functor is exact on the category of coherent analytic sheaves (property (S')). v) X is holomorphically convex, holomorphically separable, and to every point x0 e X there are a functions ft, ...,f, where e is the embedding dimension of X at x0 and flap, ..., faro generate the maximal ideal m,ro c C.,

Proof: i) = ii): This is the fundamental theorem of Chapter IV. ii)

iii) and

ii) =:. iv): Clear. iii) v) and iv)

v): If D = is a discrete set in X then by Consequence 1 there exists a function h e O(X) with n and in particular I h I D = oo. Therefore (by Theorem IV.2.4) X is holomorphically convex. Given x0, x1 e X with x0 * x1, Consequence I guarantees the existence off e O(X) with f (xo) = 0 and f (x1) = 1. Thus X is holomorphically separable. The last statement in v) follows from Consequence 2. v) . i): Holomorphic convexity implies weak holomorphic convexity and holomorphic separability implies that every compact analytic set in X is finite. Property (S) was first discussed by Serre (see [35], p. 53). It should be remarked that it is enough to require H'(X, .5) = 0 for ideals which coincide with O except on a discrete set. We now mention still another consequence of H'(X, J) = 0:

Theorem 4. Let (Y, Or) be a closed complex subspace of a Stein space (X, Ox). Then every function holomorphic on Y is the restriction of a function which is holomorphic on X.

4Proof: Let 1 c Ox be the coherent ideal associated to Y. Then Y = supp(Ox/J) and Or = (Ox/J)( Y. Since H'(X, ..0) = 0, the homomorphism (t%x(X) -+ Ox /.f(X) is

surjective and the claim follows from the canonical

identification of O,(Y) with (Ox/OM. 3. The Reduction Theorem. Here we consider complex spaces X = (X. Ox) and their reductions red X = (X, Ored x). The kernel of the canonical Ox-homomorphism p: Ox -+ Ortd x is the nilradical .A^ := n(Ox) of Ox. The follow-

133

Sheaf Theoretical Characterization of Stein Spaces

§ 4.

ing is an immediate consequence of the fact that h e Ox(X) has the same complex value at every point in X as the reduced function p,(h) a Ones x(X). If X has any of the following properties, then red X has them too: weak hokimorphic convexity, holomorphic convexity, holomorphically spreadable, Stein.

The converse of this statement is not true in general. In fact Schuster has given examples of complex spaces which are not holomorphically convex (resp. holomorphically separable), but whose reductions are holomorphically convex (reap. iioiomorphically) separable (see [34], p. 285). However, the following is easy to prove.

If P. . Ox(X) -+ ©red x(X) is surjective, then red X will have any of the following Properties if and only if X does: weak holomorphic convexity, holomorphic convexity, holomorphically spreadable, Stein.

We now give a beautiful application of Theorem B: If red X is Stein, then p,: Ox(X) --+ Ond x(X) is surjective.

Proof: We consider the Ox-epimorphisms p.: Ox - .lr'i

with

,lto; %= Ox/.N",

i = 1, 2, ...,

where N-' is the i-fold product of .A with itself. Note that Jr, = red Ox and p1 = p. Since ..ti"' .,fit+1, there are Ox-epimorphisms e;: )f°;+ 1

af°;

with

Ker e; = ,,V'i/Xi+ 1

and

i= 1,2,....

Eipi+1 = Pi,

At the section level the following diagram is commutative.

i+1M

.)ri(x5

-

2(X)--.

1(X)

oridx(X)

Since Y is coherent, all of the products A" are coherent Ox-ideals (Chapter A.2.3). Since N` - Ker e; = V (N + 1) = 0, it follows that let e; is a coherent O,Cd x-sheaf (Chapter A.2.4). Thus, since red X is Stein, H1(X, Jlret et) = 0 and consequently s;,: ,Yi+1(X)-+.*',(X) is surjective. Let h1 a .$"1(X) be given. We successively choose functions hi a .Xoi(X) so that e;(h;) = hi_1, i = 2, 3, .... These will allow us to determine a p,1,-preimage h e Ox(X) of h1.

Applications of Theo. tans A and B

Chapter V.

'-154

The sets x,

{x e X I Y_ = 0} are open in X and X 1 c X2 c

x E X there exists i(x) z 1 with .A'

.

For every

= 0. Thus X = U X. It follows from the

definitions of X1, .)t°, and p, that

Ox, = W,X1,

p,) X, = identity and hiI X E OX(X1).

Since %es e, I X, = 0 and .lt°,+, X, = Ox,, it follows that

ail X,: J°,+, I x, -i

-°,

I X,

is likewise the identity map Ox, - Ox,. Hence hi + 1IXi=hiIX1,

i= 1,2,....

Consequently the family (h1) determines a section

he Ox(X) with hIX, = h1I X1, Since P* = E1. E2.... & i -1 01. and p.(h)IXi =

i = 1, 2, ..

I X, =hIX 1= h, I X1, it follows that

91.x2.... ci-1.(h1)IXi = h, I

and consequently p,(h) = h1.

Remark: We have obviously just shown that if X is a complex space whose .N"1/.K + 1) vanish 1:5 i < oo, then the module of seccohomology groups H'(X, tions Ox(X) is the inverse projective limit lim (Ox /. ')(X), and in this case

p,: Ox(X) O.d x(X) is always srrective., L a result of the above, we now have the following remark: Theorem S (Reduction Theorem). A complex space X is Stein if and only if its reduction red X is Stein. In closing it should be noted that a reduc ed complex space is Stein if and only if

its normalization 9 is Stein. This follows immediately from the fact that the normalization mapping l:: $ - X is finite and except for the singular points in X is biholomorphic.

0

4. Differential Forms on Stein Manifolds. For every complex manifold X, thesheaf L2' of germs of holomorphic p-forms is coherent on X (see Chapter 11.2.2). In the Stein case H4(X, L2") = 0 for all p 0, q ,-ft 1. Thus the following is a simple comequence of Chapter 11.4.2.

Theorem 6. Let X be a Stein manifold and p 0, q Z 1. Then f& every 0 there exist 0 e dva-'(X) with q) R 1y.

(p, q)-form q e dP4(X) with

155

Shed Theoretical Characterization of Stein Spaae

§ 4.

The following is a corollary of the results in Chapter II.4.3: Theorem 7. For every Stein manifold X there are natural C-isomorphisms

H°(X, C) = Ker(d 0(X )), H9(X, C) =

4 >_ 1.

Since by the formal de Rham Theorem (Chapter 1I.1.8) it follows that there are always natural C-isomorphisms

H°(X, C)

Ker(d I I(X )l

H'(X, C)

Ker(d I s11(X ))ldd'-1(X ),

q > 1,

in the Stein case we have the commutative diagram 0

' dI>Z'-'(X) ----

Ker(dIC)'(X)) - H4(X, C)

-

-0

(D) I

with exact rows, where the vertical maps are the natural inclusions. This situation has the following consequence: Theorem 8. Let X be a Stein manifold and a e ,SI(X) a differentiable differential form whose differential da is a holomorphic differential form. Then there exists a differentiable form P e sl(X) so that a - d$ is holomorphie.3

Proof. It is enough to prove the theorem for forms a e 0'(X), 1 5 r < oo. By assumption da a Ker(d I fY+'(X)). Since ar+1(da) = 0 e H'-"'(X, C), the diagram (D) guarantees the existence of a form S e f1(X) with db = da. The form a - S Ker(d i d'(X )) determines a cohomology class x,(a - S) e H'(X, C). We choose a holomorphic r-form s e Ker(d I tY(X)) which determines the same class, Then

a - S - e e Ker(d I.d'(X)) and n,(a - S - a) = 0. Thus there is a differentiable (r - 1)-form f e o0'-' (X) with d ft = a - S - E. Consequently a - d f = S + eeSY(X).

Corollary to 'Theorem & Let X be a Stein manifold and a any d-closed differentiable differential form on X. Then there exists a holomorphic differential form y on X so that a - y is a d-exact (differentiable) differential form.

' For forms a e sf°(X) this theorem says that every differentiable function a with a holomorphic differential da is in fact holomorphic: a e 0(X). This statement is trivially true for any complex manifold (see Chapter 1I.2.6).

Chapter V.

156

Applications of Theorems A and B

This corollary means for example that on a Stein manifold there exist holomorphic differential forms with arbitrarily prescribed periods. The "Stein" assumption in Theorems 6, 7, 8 was made in order to guarantee that Hq(X, OP) = 0. It is not known if a complex manifold with HW(X, i2') = 0 for all p >_ 0 and q z 1 is necessarily Stein.

5. Topological Properties of Stein Spaces. If X is any m-dimensional complex manifold, then C)q = 0 for all q > m. Thus the following is an immediate consequence of Theorem 7: Theorem 9. If X is an m-dimensional Stein manifold, then

H4(X,C)=0 for all q>m. This theorem, given by Serre in 1953, is a purely topological, necessary condition for a complex manifold, in particular for a domain in C^, to be Stein. The following is a homological reformulation of this. Theorem 91. If X is an m-dimensional, Stein manifold, then the integral homology group Hq (X, Z) is a torsion group for all q >_ m.

Proof: By general theorems from algebraic topology Hq(X, C) is isomorphic to

the group Hom(Hq(X, Z), C) for every q. In the case when HQ(X, C) = 0 the group cannot contain a free element, because it would give a non-trivial homomorphism I,(X, Z)-+C. O Until 1958 the problem of whether or not Hq (X, Z) for q > m could contain non-trivial torsion elements remained open. At that time A. Andreotti and T. Frankel [1], using the embedding theorem for Stein manifolds and Morse theory, showed that this is indeed not possible: If X is an m-dimensional Stein manifold, then

H*(X, Z) = 0 for q > m and

H11,(X, Z)

is free.

Again using the methods of Morse theory, J. Milnor (see [28], p. 39) sharpened this result:

Every complex m-dimensional Stein manifold is homotopy equivalent to a real m-dimensional CW-complex.

This statement can still be greatly improved: In every m-dimensional Stein manifold X there is a real m-dimensional, closed CW-complex K which is a "strong deformation retract" of X. In other words, there is

a continuous map f X x [0, 1] -+ X with the following property:

f(x,0)=xforallxeX, f(p,t)

x[0,1),f(Xx{1})=K.

A Sheaf Theoretical Characterization of Stein Domains in C"

§ 5.

157

It is natural to ask which CW-complexes can arise as retracts of Stein manifolds. In this regard one can show the following: (see [17], p. 468ff.): Tube Theorem. Every paracompact, real m-dimensional, real-analytic manfold R has a Stein tubular neighborhood X. In other words, 1) X is a complex m-dimensional Stein manifold and R is a real-analytic submanifold of X ;

2) R is a strong deformation retract of X.

It is also quite reasonable to ask if the lower homology groups Iii (X, Z), 1 5 q < m, of an m-dimensional Stein manifold can be arbitrarily prescribed. In 1959 K. J. Ramspott (Existenz von Holomorphiegebieten zu vorgegebener erster Bettischer Gruppe, Math. Ann. 138, 342-355) showed that quite a bit can in fact be prescribed: For every countable, torsion free, abelian group B there exists a Stein domain X in C2 whose f rst Betti group (i.e. the quotient group of Hl (X, Z) by its torsion group) is isomorphic to B.

Furthermore Narasimhan [29] showed the following: For every countable abelian group G and every natural number q > 1 there is a Stein domain (even a Runge domain) X in C2r+9 so that H (X, Z) is isomorphic to G.

Using complex analytic methods, the theorem of Andreotti-Frankel was generalized in [29]: If X is an m-dimensional Stein space, than

Hq(X, Z) = 0 for q > m and H,"(X, Z) is torsion free.

§ 5.

A Sheaf Theoretical Characterization of Stein

Domains in Cm A domain B in C"', 1 < m < oo, is Stein if and only if it is holomorphically convex. In this section we will show that such domains have a particularly simple sheaf theoretical characteli_=.ation.

1. An Induction Principle. Our beginning point is the following classical result:

Lemma (Simultaneous Coti.tinuability, see [BT], p. 121). Let B CC' be a domain which is not holomorphically convex. Then there is a point p e B and a nolyeylinder A about p x iv' A I- B so that every f e C(B) has a Taylor series on A which converges compactly on A to a function F e 0(A). Let W be the,connected component of B n A which contains p. Then f I W =,F I W.

158

Chapter V.

Applications of Theorems A and B

Proof: Since B is not holomorphically convex, there exists a compact set K c B whose hull ie relative to B is not compact in B. Let A,(c) denote the polydisk

I< of radius t about c in CM. There exists a real number r > 0 so that B'

U A,(a) is 4EX

a relatively compact subset in B. Consequently I f ja, < bo for every f e 0(B). For such a function, the Taylor series off converges on A,(a) for all a and the Cauchy inequalities yield. where

pi!...P.!azi'...a4

e O(B).

Since k is bounded in C', there exists p e f( such that A- A,(p) is not contained in B. We consider the Taylor series off about p (= the origin): fN

...

ZN 1,

ZmN1

0

Since p e 1C, it follows that I L.... I fN, N. JX. Applying (*), we see that the above power series converges on A to F e O(A). The identity principle implies that f and F coincide on the connected cohbpoaent of B n A which contdlQs p. We now use the above lemma to prove a theorem which will in turn allow us to

give an induction argument for a sheaf theoretical characterization of Stein domains. Theorem A domain B in CTM is Stein if and only if at least one of the following conditions is fulfilled:

a) For every (m - 1)-dimensional, analytic plane H c CM Me intersection

B n H H C "'

is Stein and the restriction OB(B) -+ OB H(B n H) is

surjective.

**)'For every complex line E c C", the restriction OB(B) - OB r(B n E) is surjective.

Proof. If B is Stein, then every subspace B n H is Stein and OB(B)

n H) is surjective by Theorem 4.4. Thus (:)holds.

)...): Let E be given and choose a hyperplane H c C'" which contains E. '[hen OB(B) - OB n s(B n E) is the composition of OB(B) -+ OB r, 8(B n H) and

ee,.48 n H) -. OB s(B n E). The first map is surjective by assumption and the seamd is surjective since B n H is Stein. Is remains to show that (s.) implies the holomorphic convexity of B. Suppose this is so the case and choose p, A, and Was in the lemma. Let q e AFB and take

§ S.

159

A Sheaf Theoretical Characterization of Stein Domains in C"

E to be the complex line through p and q. On the real segment pq E n A from p to q there exists a first point y ¢ W. Thus y e a(W n E) n A. We now choose a

function I on E = C' which is holomorphic on Ely and has a pole at Y. By assumption there exists a function f e 0B(B) with f I B n E = I I B n E. By the lemma there exists F e 0,(A) with F I W = f I W. But F is holomorphic at y e A. Consequently Jis bounded along all sequences approaching y e 8(W n E). This is contrary to 7 having a pole at y. Thus B is holomorphically convex and con11 sequently is Stein.

= H'-'(B, OB) = 0. 2.. The Equatiom H'(B, OB) = Theorem 1 in order to prove the following:

We now apply

Theorem 2. Let B be a domain in C'". Then the following are equivalent:

i) B is Stein. H"'-(B, OB) = 0.

ii) H'(B, 08)

Proof: It is enough to prove ii) i). We proceed by induction on the dimension m. For m = 1 it is clear. We will use (a) for the induction step when m > 1. Let

H c C' be an (m - 1)-dimensional analytic plane which intersects B. Thus B' := B n H is a non-empty domain in H ^_- C". We choose a linear function I

on C' which vanishes on H. Thus 08. ^- (08/108) I B' and H4(B', OB.) H4(B, 081108) for all q > 0. Thus we have the exact sequence

0 -08 -" , ire

OB /' 98 -0,

where A is defined by h, -+ 1, h, for h: E 0=, z e B. The associated exact cohomology sequence is OB(B) - OB.(B') --. H'(B, OB) -, .. . - H° (B, OB) -+ H9(B', OB)

H4 +'(B, OB)

... s

From this we read off the fact that 0B(B) - OB.(B') is surective and that H'(B', 08) = = H' -2(B', OB) = 0. This along'with the induction assumption implies that B' is Stein.

0

Remark: The assumption in Theorem 2 that B is a domain in C"I is quite important. There certainly are manifolds X which' are not Stein and all of whose cohomology groups H"(X, 0), q > 1, vanish. For example every projective space P. is such a manifold. Nevertheless in 1966 in "On sheaf cohomology and envelopes of holomorphy," Ann. Math. 84, 102-118, H. B. Laufer proved the following generalization of Theorem 2: _

Chapter V.

160

Applications of Theorems A and B

Every subdomain B of an m-dimensional, Stein manifold whose cohomology groups

H"(B, 0), I S p < m, all vanish is Stein.

The proof uses among other things the fact that every point p e X is the simultaneous zero set of m-functions in O(X ).

The condition (a) in Theorem 1 car be further exploited: Let B be a domain in C" for which the Cousin I problem is universally solvable. Then for every analytic, (m - 1)-dimensional plane H c C* which intersects B, the restriction OB(B) -- 08 y(B n H) is surjective.

,

Proof (also see [3], p. 183-4): Let H = ((zt, ..., -z,,) e C" zt = 0). For every point z E B we choose an open polydisk neighborhood U, c B so that Us n H = q whenever z 4 H. Let g(z2, ..., z") E OB ,,,,(B n H) be given and let

f _(zt, ..., z.):=

g(z2,

.. ., Z.

IU:EIla(U-)

if zeH

and

f'(z,,... , z_) := 0 a 08(U,)

for

z e H.

Sincef' - f'';is holomorphic on U, n Us, for all z, z' e B, the family {U., f'} is a Cousin I distribution on B. Thus by assumption there is a function F which is meromorphic on B satisfying r' _= F I U, -f' E OB(U,). We set G'= zt F and note that G is holomorphic on B\H since F is holomorphic there. Moreover for all

z0H

GI U.=z,r'+gI U, c- 09(U,). Hence G E OB(B) and G 1 B n H = g. Consequently OB(B) -. OB n(B n H) is

p

surjective.

The following is now an immediate corollary:

Theorem 3. The following statements about a domain B c C' are equivalent: i) B is Stein.

ii) The Cousin I problem

is

universally solvable on B, and for every

(m - 1)-dimensional plane H which meets B the intersection B n H is Stein. Proof: i)-* ii): This is adirect consequence of Theorem 1.1b and Theorem 1.16.

ii) co- i): By the remarks directly above, it is clear that B has property (a) of Theorem 1. Thus B is Stein.

0

When m = 2, Theorem 3 says that a domain B c C2 is Stein if and only if the Cousin problem is universally solvable (Cartan [CAR], 1934).

§ 5.

A Sheaf Theoretical Characterization of Stein Domains in C'

161

I Representation of 1. If f,, ..., f,, are holomorphic functions on a complex space X, then in order for there to exist functions g, e O(X), 1 < i < 1, with 1=

r=t

g, f1, the set {f1 =

= f = 0) must be empty. The converse is true for

Stein spaces. For a proof of this, we begin with a preparatory theorem: Theorem 4. Let So be a coherent sheaf on a Stein space X and take .9" to be an 0-subsheaf which is generated by finitely many sections s1, ... , s, in the O(X)-module 9'(X). Then s 1, ..., s1 generate the O(X)-module Y'(X ). In particular if every stalk

.9'r is generated as an 0,, module by s,,,, ..., s,,, a 5",,, then s,, ..., s, generate the O(X)-module .1(X ).

Proof:

f'.) -

Define

on X the 0-homomorphism o: 0'-+.1 by (fl., ...,

f,, s,s. Since X is Stein, the induced O(X)-homomorphism O'(X) -p

r=t 59'(X) is surjective. Ifs,,,, ... , s,,, generate Y. for all x E X, then 59'(X) = 91(X).

The main result of this paragraph follows as an application of Theorem 4: Theorem S (Representation of I by everywhere locally relatively prime functions). Let X be a Stein space and fl, ..., f, e O(X) be holomorphic on X with {x e X I f, (x) fi(x) = 0} = Q. Then there existfunctionsg,, ...,g, e O(X) so that 1

Proof. The ideal J' Of, +

i f g,

1=

1

+ Of, is coherent. Since f,, ..., f have no

common zeros, J. = 0,, for all x e X. It follows from Theorem 4 that J(X) _ O(X),andas aresult The following partial converse shows that the conclusion of Theorem 4 is quite strong: Theorem 6. The following statements about a domain B (-- C.'' are equivalent:

i) B is Stein.

ii) If f,, ..., f E 0(B) are such that {z e B I ft(z) _ . = fj(z) = 0) = 9, then t

there exist g,, ..., g1 E O(B) with 1 Proof: It is enough to show that ii) implies the holomorphic convexity of B. Let

D be a discrete set in B. We may assume that D has an accumulation point c = (c1, ..., cm) e C', because if not, then one of the coordinate functions z,, ... z," would be unbounded on D. Since c 4 ti, it follows that the m functions z - c,,. I < p < m, have no common zeros on B. Thus there exist functions g,, ... ,

Chapter V.

162

Applications of Theorems A and B

m

g. a (9(B) with 1 = I gµ(z - c,,). At least one of the g,,'s must be unbounded on v=t

0

D, as otherwise letting z tend to c would show that 1 = 0.

4. The Character Theorem. We now prove a theorem which is closely related to Theorem 6. A C-algebra homomorphism x: 0(X) -+ C is called a (complex) character. As usual we let z,, ..., z,,, be holomorphic coordinates in C'. The following is due to J. Igusa 123]: Theorem 7 (Character Theorem). The following statements about a domain B in CM are equivalent: .

i) B is Stein. ii) For every character x: C9(B) -+ C, it follows that (x(z1),

..., X(z, )) E B.

iii) For every character x: 0(B) - C there is a point b e B so that for all f e 0(B) it follows that x(f) = f (b). 1

Proof: i) . ii): Suppose (x(zl), ..., x(z,,,)) ¢ B. Then them functions z, -

E

0(B), 1 < p < m, have no common zeros in B. Thus by Theorem 6 there exist functions g,, ..., g,,, a 0(B) with 1= g,,(z,, This yields the following µ=1

contradiction:

M

I = x(I) = p=1

x(gµ) -

x(zµ - x(z0)) = E AM - 0 = 0. µ=1

ii) - i): Spppose B is not holomorphically convex. Then the lemma in Section 1

on simultaneous continuability guarantees the existence of a point p = (p,, ... , pp) e B and a polycylinder A about p with A if- B so that every f e 0(B) has a pOwer expansion at p which converges compactly to F e 0(A). The map 0(b) A), f --+ F, is a C-algebra homomorphism. Thus for every c = (c,, ... , E the map X,: 0(B) C, f -. F(c), is a character. Since p, + (z is the ylof series of z about p, it follows that c and consequently cc- B by ii). This contracts A q` B. Hence B is holomorphically convex. i) A ii)

. iii): Let x be given and b:= (x(z1), ..., x(z,.)) e B. If there oxists f e 0(B)

with' x(f) # f (b), thenf - X(f), z, - X(zl), ... , z,, - X(z.) have ne.common zeros in B. Applying Theorem 6, there exist functions g, g,, ..:, g,, a 0(B) so that

I = g - (f - X(f )) + Y_ gM - (z, - x(zv)) u=1

Since X(h - X(h)) = 0 for all h e 0(B), we have the following contradiction:

I=Al)=x(g)-0+ µ.l iii)

ii): This is trivial, because (x(z,), ..., X(z )) = b.

0

§ 6.

The Topology on the Module of Sections of a Coherent Sheaf

163

The following is a beautiful consequence of Theorem 4:

Theorem 8. If B c C' is Stein and b = (b,, ..., be B, then every f E O(B) can be represented in the form

f=f(b)+

fN'(za-bA

0=t

15µ<m.

where

Proof. Let J1 be the ideal in Oe which is generated by the sections zt - bt, ..., z,, - b,,,'E O(B). Then .F is coherent. Since B is Stein, Theorem 4 implies that

5(B) = (9(B) . (zt - bt) + ... + O(B) . (z,,, - bm). But 5b = m(Ob) and 5= = O, for z # b. Thus f - f (b) e 5(B) for all f e (9(B). O Theorem 8 says that every character ideal Ker Xb, b E B, where Xb: O(B) -. C is

defined by f-.f(b), is generated by the m elements zl - Xb(zi), ..., z,,, - Xb(zm) Generalizations of this as well as of the character theorem can be found in Section 7.

§ 6.

.

The Topology on the Module of Sections of a Coherent Sheaf

The goal here is to make the C-vector space .9'(X) of global sections of a

coherent sheaf S9' into a Frechet space. It will be shown that this Frechet

topology on 9"(X) is uniquely detergiined by certain natural demands. The techniques of construction involve analytic blocks. If X is reduced, then this Frechet topology on O(X) turns out to be the topology of compact convergence. The results of this section have important applications in Chapter VI. In fact some of them were already used in Chapter IV.2.5. 0. Frechet. Spaces. Usually a C-vector space V is called a Frechet space whenever it is locally convex, metrizable, and complete. The following definition is more convenient for function theoretical purposes. Defthitioii 1 (Frechet space). A topological C-vector space V is called a Frechet space if there is a sequence I I,, v = I, 2, ... , Of semi-norms on V so that

(')

m

d(v, w)'=

r:t

defines a complete metric d on

2

v - w I, +Iv-wv,weV,

V A" Induces the given topology.

Chapter V.

164

Applications of Theorem' A and B

Remark: It is always the case that (s) defines a translation invariant psetdometric on V. Thus d is a metric if and only if

Ivt.=O for all vZ 1=ov=0. The following properties are easy to verify: If U is a closed subspace of a Frechet space V, then U with induced topology and V/U with the quotient topology are Frechet spaces. if (V}t E N is a sequence of Frechet spaces, then Fl V with the product topology is

.

to N

a Frechet space.

On a given C-vector space it is impossible to have two Frechet topologies one of which is finer than the other. One can even say more: Banach Open Mapping Theorem. Every C-linear, continuous map 'F: V -+ W of a Frechet space V onto a Frechet space W is open.

A proof of this can be found in any standard functional analysis textbook. 1. The Topology of Compact Coevergence. The following is well-known: Theorem 2. If X is a complex manifold (with countable topology), then the vector

space 0(X) of holomorphic functions on X is a Frechet space (even a Frechet algebra) with respect to the topology of compact convergence. If {U,}Y2 t is a count-

able open cover of X such that every a is compact, then this Frichet topology is given by the sequence I

I, : o(X) -' R,

I f Iv '= I f to, = max I f (x) I,

f e O(X),

X& U.

of sup-norms.

The proof is exactly the same as that for domains in C. In general the topology of compact convergence is not describable by one (or finitely many) semi-norms: Let X be a locally-compact topological space and A an R-subalgebra of the algebra W(X) of all complex-valued continuous functions on X. Suppose that there is an unbounded function u e A. Then there is no R-vector space norm on A which induces the topology of compact convergence.

Proof: Suppose that there exists such a norm I

(. The map A - A. a - ua, is a continuous

R-linear map and is therefore bounded. That is, there exists M e R so that I ua < M a I for all a e A. So for every r > 0 it follows that I ru a 1 5 rM I a 1. We choose r so that z= rM < 1. The function v- rue A is likewise unbounded on X, and thus I to 1 5 e I a I for all a.4. By induction one sees that

lei 5 e"-' I v l for alt n- 1, 2, ... . Since e < I, it follows that e converges to 0. This contradicts the. feet that v is unbounded.

IJ

As a corollary of this theorem we note that the space t'i(X) of all holomorphic functions on a non-compact, holonwrphically convex space X does not carry a norm which induces the topology of

compact convergence.

6.

The Topology on the Module of Sections of a Coherent Sheaf

165

The topology of compact convergence is compatable with the sequence topology in the convergent power series ring (see [AS), p. 58):

Theorem 3 (Compatability Theorem). If X is a complex manifold and 0(X) carries the topology of compact convergence, then every restriction map

0(X) - 0x,

x e X,

f'-'fx,

is continuous whenever 0, is equipped with the (convergent power series) sequence topology.

Proof: Every point x e X lies in a compact polycylinder A c X. The restriction s. The 0(X) -+ O(A) is continuous when we equip O(A) with the sup-norm 0 restriction O(A) -+ Os is likewise continuous (see [AS], p. 58).

For every natural number l > I we furnish the C-vector space O'(X)

0(X )

with the product topology. If / is a coherent subsheaf of 0', then the module of sections /(X) is closed in O'(X) (see Section 4.1) and is consequently a Frechet subspace of 0'(X). Thus the quotient space 0'(X)//(X) is a Frechet space. 2. The Uniqueness Theorem. Now let X be an arbitrary complex space and So a coherent 0-sheaf on X. There is no obvious notion of compact convergence for sequences of sections sY E So(X ). In order to find a Frechet topology for .9'(X) we

think along the lines of Theorem 3. Every stalk S's, x e X, carries the natural sequence topology (see [AS], p. 86ff). This topology is always Hausdorff. This, along with the theorem of Banach (i.e. the "open mapping theorem"), shows that the compatibility property of Theorem 3 is already quite significant: Theorem 4 (Uniqueness Theorem). If there is a Frechet topology on So(X) so that the restriction mappings 6'(X) -+ .9s, s -+ s,, x e X, are continuous, where .Sox carries the sequence topology, then it is unique.

Proof: Let two such topologies on .(X) be given. We denote them by V and W. Furnishing V x W with the product topology, the projections V x W -+ V and V x W -+ W are continuous and the diagonal A c V x W is mapped bijectively in both cases. It is enough to show that A is closed in V x W, because it will then be a

Frechet space with the induced topology and, by the theorem of Banach, will therefore be homeomorphic to both V and W. Since the restrictions V - Sx and W - S. are continuous, it follows that if we equip Yx x Y. with the product topology, then A : V X W - Y. X Sos,

(v, w)'-+ (vx, Wx)

Chapter V.

166

Applications of Theorems A and B

is continuous. Since Y. is Hausdorff, the diagonal Ox is closed in Y s x Y.. Thus for all x e X the preimage As ' (AF) is closed in V x W. But A =n Ax ' (Ax). So A xcx is also closed. 3. The Existence Theorem. In this paragraph we construct a topology on Y(X) for any given coherent sheaf Y on a complex space X. This topology is compatible with the sequence topology on every 5Yx, x e X. The following simple lemma shows that this comes down to a local problem.

Lemma. Let (X,),.z1 be an open cover of X such that every .'(X,) carries a Frechet topology with every restriction 9°(X,.) --+ Y., x e X,,, continuous, v > 1. Define the C-linear map i, by :: Y(X) -+ V:= 11 .9'(X,),

s -+ (s I Xv)Y2 i.

V=1

Then r maps Y(X) bijectively onto a closed subspace of the Frechet space V. The Freehet topology on .9'(X) which is induced by i is such that all of the restrictions So(X) Yx, x c- X, are continuous.

Proof: By results in Paragraph 0, the space V is Frechet with the product topology. The mapping i is C-linear, and, since U X, = X, it is also injective. The

image space Im t c V consists of all sequences (s,),2 t, s, c- b(X), for which s,. = sBx whenever x e X. n X. In order to give a better description of this set, we consider for every point x e X, the C-linear map 1,x: V -+ S' which is defined by composition of the projection V -+ ."(X,) with the restriction 9'(X,.) -+ Sox. It is clear that rl,x is continuous. Setting I v, x) I x e X n X,), it follows that for every triple (µ, v, x) E I the set "lµ, V, x):= {v E V I

Jlvx(v))

is a closed C-subspace of V.°

Now

Imt= 0

L(µ,v,x).

(g.v.x) e I

Thus Im i is closed in V and is consequently a Frechet space. We transport the Frechet topology on Im t back to .9(X) by J - '. Since the restriction bx:.9'(X) --+ Sox is just {x = t1,x o t for x e X it follows that all of the restrictions Y(X) -+ Y x are continuous. If ,,: A -. B, i = 1, 2, are continuous maps between Hausdorff spaces, then the set (a e A 1,1,(a) ry=(a)} is closed.

§ 6.

167

The Topology on the Module of Sections of a Coherent Sheaf

It remains to show that every point p e X has a neighborhood W so that 9(W) carries a Freshet topology which is compatible with the sequence topology on Sox, x e W. Every point p e X possesses an open neighborhood U ' X and a holomorphic embedding n: U -+ V of U into a domain V c C"'. We choose a compact block

n-'(Q). Then (P, n) is an analytic block in U with p e P°, where P° _= n-'(d) is the analytic interior of P. Whenever we have this situation, we call (P, n) a block neighborhood of p.

Q r V with n(p) e ¢ and set P

Lemma. Let So be coherent on X and (P, n) be a block neighborhood. Then there is a Freehet topology on .1(P°) so that all of the restrictions .9'(P°) -+ 9'x, x E P°, are continuous.

Proof: There exists t >_ 0 and a 0-epimorphism e: 01 Q -' x.(bl U) I Q. We .restrict a to 0. Since 0 is Stein and the kernel sheaf f :_ Yet e I ¢ is coherent, we obtain an exact sequence

0 -'f&

eV)

'x,(."IU)(0)=-s (P°)-'0.

Thus So(P°) is algebraically isomorphic to 0'(0)/f(¢) which carries a natural Freshet quotient topology. The isomorphism induces a Freshet topology on

1(P°). We have to show that every restriction Cx: 1(P°) -, .fix, x E P°, is continuous. Let z == n(x). We note that the injectivity of it implies that it 1(z) = x. It is obvious

that the following diagram of natural C-linear maps is commutative:

610) h n,(So I U)(0) ~

Os y X.(991 U)=

9(P°)

._. Y.

Now e, is open and e= is a continuous Os-epimorphism. By the results of Paragraph 1, the restriction wo= it also continuous. Since Ox is isomorphic to a quotient

algebra of 0, the sequence topology of the Os module n,(9' I U)f agrees with the sequence topology of the Ox module Sox. Thus lx is continuous for all x e P°. D The main theorem is now an immediate consequence of the two lemmas:

Theorem 5 (Existence Theorem). Let X be a complex space (with countable topology) and Y a coherent sheaf on X. Then there exists a Freshet topology on 0'(X) so that all of the restrictions 11(X) x e X, are continuous.

Remark: In the proof of the second lemma, Theorem B for open blocks (namely H'($, f) = 0) was used. It should be noted that one can get away with

168

Chapter V. Applications of Theorem A and B

using only Theorem B for compact blocks. As above, one uses the Frbchet space F s= O1(d)/J(0) S(P°)-one doesn't yet know equality! Since e,: O`(Q) - s,(.° I UXQ) .1(P) is surjective, one can construct a C-linear map a: $9(P) -+ F with Ker a c {s a .f(P)Is {P° = 0). Further one can obtain continuous, linear maps q : F -- .fir, x e P, so that rp,, - a is always the restriction. The proof of the existence theorem now runs essentially like that above if one chooses in the

first lemma a sequence of block neighborhoods (P

where U P° = X and .=i

V is the corresponding product of Frbchet spaces F, c .9'(P°).

0

We call the Frechet topology which is characterized by Theorems 4 and 5 the canonical topology on .9'(X), and always think of .9'(X) as equipped with it. The following is a corollary of Theorems 4 and 5: If X is a complex manifold, then the canonical topology on O(X) is the topology of compact convergence.

There is an important generalization of Theorem 8 which is quite useful for studying convergence: Let (s,), 2 ° be a sequence of sections s, E 1(X). Suppose that for every x e X there is a neighborhood U" so'that (s, I U"J a 1 converges in the canonical topology of [1(U") to a section s(') e .9'(U"). Then (s,)v21 converges in the canonical topology of

.'(X)toseY(X)sothat s(U"=s(") for allxeX.

This follows immediately from the first lemma. Since X has countable topology, one can obtain the sequence X, from the Ui's. 4. Properties of the Canonical Topology. We begin with some remarks about arbitrary complex spaces:

Theorem 6. The canonical topology on the module of sections in an arbitrary coherent sheaf has the following properties:

a) If U c X is open, then the restriction map pu:.1(X) --+ 1(U) is continuous. b) If a: So - Jr is an analytic homomorphism between coherent sheaves over X, then the induced map a5: Y(X) -+ 9-(X) is continuous. c) If f: X - Y is a finite holontorphic map, then for every coherent sheaf .5' on X the isomorphism is f.(9'XY) ac .9'(X) is a homeomorphism. Proof: In all cases we procede as in the proof of the unicity theorem (Theorem 3). We set V =_ [1(X) and define in the various cases appropriate sets:

a) W'=.9'(U),

b) W-9-(X), c) W =f,(.9')(Y),

C==Graph po={(s,sjU)}c V X W C- Graph ax = {(s, ax(s))} c V x W C:=Graph i-I _ ((s, f- 1(s))} c V x W.

1 6.

169

The Topology on the Moduk of Sections of a Coherent Sheaf

In each case C is proved to be a closed subspace of the Frechet space V x W by realizing it as the intersection of closed sets:

a) C

n ((v, w) E V x Wlw,=v,) XIU

b) C= n {(v,w)E Vx W+wx=az(v,j} SEX

c) C = n {(v, w) E V X W I J,,(w,,) = v., whenever f (x) = y). )EY

The point is that the restrictions to stalks are continuous! The continuous projection of V x W onto V (on V and on Win case c)) induces a bijective map from the Fri chet space C. By the Banach open mapping theorem it is a homeomorphism.

0

In the situation of Theorem 6b), the ax-image of 9(X) is not in general closed in

i'(X). This is due to the fact that not every global section in a(.9') c ,9r is the image of a section in 50(X).

Closedness Theorem. If 50 is a coherent subsheaf of the coherent sheaf if, then 50(X) is a closed subspace of .T(X).

Proof: All of the restrictions P(X) - Yx are continuous. Since 50X is closed in

Px, it follows that 3r(X) n 50X is closed in if(X ). Thtis n v(X) n YX) is XeX

likewise closed in 3(X).

It is now possible to prove several interesting corollaries. Corollary 1. If X is Stein and a: So -+ 311' is an analytic homomorphism between two coherent sheaves over X, then a4(50(X )) is closed in r(X ).

Proof. Since X is Stein, the induced homomorphism 50(X) - a(50)(X) is surjective. Thus ax(50(X)) = a(.PXX), and the claim follows from the Closedness Theorem.

Corollary 2. Let 50 be a coherent sheaf on a Stein space X, and let st, ..., s, e 50(X) be finitely many sections. Then 0(X)st +

+ 0(X)s, is a closed 0(X )-

submodule of 50(X ).

Proof: By Theorem 5.4 we know that O(X)St +

+ 0(X)s, = .°'(X), where

.' is the coherent 0-subsheaf of 50' which is generated by st, ..., s,. The Closedness Theorem says that 50'(X) is closed in 50(X ).

We see in particular that if X is Stein, then every finitely generated ideal 5 in 0(X) is closed in O(X).

Chapter V. Applications of Theorems A and B

170

In closing we note without proof the following: Approximation Theorems Let X be Stein and (P, a) be an analytic block in X.

Then for every coherent 67-sheaf .9', the image Im p of the restriction map 5°(X)- 5°(P°) is dense in .'(P°). p: S The Topologies for Cq(11, .5") and ZQ(U, .50). Let .9, 9' be coherent on X ad 8

.

U = {U,} be an open (countable) cover of X. Then every C-vector space U,p... ,. = U,o n logy. The C-vector space 50(U,o ...

n U,1, is a Frechet space with the canonical topo-

Cq(U, SO) = fl 5(U10...,

of q-cochains is therefore likewise a Frechet space with the product topology,

0:5 q < oo. We summarize some relevant properties of these spaces in the following:

Theorem 7. Every cochain space C9(U, 50), q = 0, 1, 2,..., is a Frechet space. All coboundary maps 0: 0'(11, 9) -+ Cq(U, 5°) are continuous, and the cocycle spaces Zq(11, ,0) = Ker 0, q = 0, 1, ..., are Frechet subspaces of Cq(U, 9). If 97:.50' -+90 is an Ox-homomorphism, then every induced (C-linear) map CC(U, .9') - C4'(U, .0) is continuous.

Proof: Every coboundary map 8 is a finite sum of restriction maps, and, is therefore continuous by Theorem 6a). Since the space Z'(U, .9') is the kernel of 0, it is closed and is consequently a Frechet subspace of C'(U, .50). The map y(U,., 9p:.°' -. 5° induces continuous maps .9"(U,0 ... (see Theorem 6b)). Thus the induced mapping Cq(11, .9") -+ C4(U, .50) is also continuous.

The following is needed in Chapter VL3.4: Suppose that the cover U'= {U,} is finer than B = {U,} with U, c U, for all i e 1. Then every C-linear restriction map p: Cq(U, .50) -+ 0(11', is continuous.

Proof: By Theorem 6a), all of the restriction maps ./(U,0...,.)- Y(U;o ...,,) are continuous. Thus the claim is immediate. Remark: Every coboundary space Im 8 c C5(U, .9') is a topological C-vector space. Consequently every cohomology module H5(U, .So) (_ Ker Ohm 8) is also a topological vector space. In this way (by taking the limit) one obtains a natural topology on the space Hq(X, Y), q = 1, 2, .... The main problem is that this is in general not a Frechet topology, because as a rule the coboundary spaces Im 8 c C!(U, .50), q = 1, 2, ..., are not closed and are therefore not Frechet spaces.

6. Reduced Complex Spaces and Compact Convergence. In this denotes a reduced complex space. The C-vector space O(X) is thus a subspace of the

§ 6.

The Topology on the Module of Sections of a Coherent Sheaf

171

C-vector space W(X) of continuous, complex valued functions. Hence, along with

the canonical topology of Theorem 5, 0(X) carries the topology of compact convergence. We let V denote 0(X) with the canonical topology and let W be

0(X) with the topology induced from l'(X ). Our goal is to show that V = W. We begin by remarking that the identity map id: V - W is continuous. Proof: Let {f1} be a sequence in V which- converges to f e V (in the canonical topology). Let S denote the analytic set of singular points of X. Now the restric-

tion map 0(X) - 0(X \S) is continuous in the canonical topology, and on the complex manifold X\S the canonical topology and the topology of compact convergence are the same (see the remark at the end of Paragraph 3). Thus f I X \S converges compactly to f I X\S. The singular set S is a proper analytic subset of X which is nowhere dense. Thus compact convergence on X\S implies compact convergence on X. The following is an immediate consequence of Banach's Theorem: if W is a Freshet space, then V = W.

By a well-known theorem of analysis, the space W(X) of all continuous functions on X is a Freshet space. Hence it only remains to prove the following:

Lemma. The space W = 0(X) c Y(X) is closed in(X ). Proof: Let (f) c O(X) be a sequence which converges compactly to f e '(X). Let S be the nowhere dense analytic set of singular points of X. Then, by Theorem 2, f I X\S E O(X\S). Now if X is normal, then the Riemann removability theorem (Chapter A.3.8) implies that f e &(X).

In the general case we take a normalization : X X of X. Hence the lifted converges in the canonical topology of 0(X) to sequence 7 :=A T.=f o e 0(X). Since is finite, Theorem 6c) implies that the section i;,(J) e ,(Og)(X) is the limit of the sequence ,(f) E ,((1 )(X) in the canonical topology on the vector space ,(0x)(X) of global sections of the coherent 0x-sheaf ,(0x). Now, since Ox is an Ox-subsheaf of the space 0(X) is closed in ,(O,r)(X) (by Theorem 6a)). But ,(];) = f e 61(X). Thus (1) E 0(X). Furthermore, since is biholomorphic on X\S, it follows that 1;,(P) =f on X\S. Finally we recall that S is nowhere dense in X. Hencef = ,(J) E O(X). In summary, we have proved the following:

Theorem & if X is a reduced complex space, then the topology of compact convergence on 0(X) yields a Frechet space structure which coincides with the canonical topology on 0(X). 7. Convergent Series. In this paragraph (X, Ox) denotes an arbitrary complex. space and (X, O,ea x) its reduction. If f E Ox(X), is a sequence, then the

Chapter V. Applications of Theorems A and B

172

co

notion of convergence of the associated series Y f, in the canonical topology of v=1 OD

Ox(X) to f e O(X) is well-defined. If Y f, converges to f e Ox(X ), then the =1

"reduced series"

red fY, where red f e O«d x(X ), converges compactly to red f v=1

(by Theorem 6b)). In general it is not the case that the convergence of Y red f. OD

V=1 W

implies the convergence of

fY. As an example we consider the following com-

plex subspace X of the z-plane C:

X:= U

with

n=1

x =n

and ox

= U Ox, with n=1

On X we consider the sequence .f :_ (.f1,.,

..f

.

...) with

v(z -

modulo (z -

M

Since red f,. = 0 for all v, the series

red f,. = 0 converges. Nevertheless

for

W

m = 1, 2, ... the series

does not converge. However

'0

f: has a non-zero

limit in C,(X), because f;: (0, ..., 0. ....) The above example is indicative of the general situation: Theorem 9 (Convergence Theorem). Let X be an arbitrary complex space and 00

(f,.),.Z 1'f c- Ox(X ), be a sequence so that the reduced series c' ed

red f,. converges in

(X). Then there is a sequence (m,.),., of natural numbers so that the series

f converges whenever n, > m,.. V=1

For the proof we construct a sequence of semi-norms (I 1i)!, , on Ox(X ), and for every i a sequence of natural numbers so that the following hold: 1) The semi-norms

I

i;, i > 1, determine the Frechet topology Ox(X).

Q

2) The series v=1

1_f', I; of real numbers converge, whenever k,. > 1;,

Having done this, we define m,, = max{l1v, ..

,

1,)

§ 6.

173

The Topology on the Module of Sections of a Coherent Sheaf

and note that whenever n, >_ m, the series I f converges for all of these semi=t

norms, thus proving the theorem. It remains to carry out the construction. Now every block (P, X) in X (in the sense of Paragraph 3) determines a ' ni-norm I Irz on (!2 (X) as follows: Since n: U V is a holomorphic embedding of a non-empty open set U in X into a domain V in some C", and since P = n-'(Q) is the preimage of a compact block Q c V with Q + 9, one has an e'v-epimorphism £: ev --+ n.(cu)

which induces an epimorphism EQ: (rv(Q)- n. W,

= ("v(P)

at the section level. One sets

.f '=i1 f{IgIQIg E ("(Q), &Q(9) =.f IP},

fe (;x(X).5

A block (P, n) is called distinguished in U if it is relatively compact in X and V

is a Stein domain. Since X has countable topology, there exists a sequence of distinguished blocks (P1, n,) in X so that the relatively compact sets U P°, n = 1, =t

2, ... , exhaust the space X. It is clear from the definition of the canonical topology that, for every such sequence of blocks, the sequence (I I;);,, with I Ii `= I determines the Frbchet topology of C x(X ). Thus the condition 1) above can be fulfilled by a sequence of semi-norms coming from distinguished blocks. The following lemma is now important: I> 0 satisfying the following: For every f e (Ox(U) with If It,, < t there exists a constant M f > 0 so that

I fkln 1. Given this lemma, we can easily find the natural numbers for condition 2): One chooses ly > I big enough so that M f,, q'' < 2-'', r > 1. The convergence of the

red f,. along with the relative compactness of U implies that

reduced series ,=I

there exists an index j = j(U) so that I f It. < t for all r > j. Thus

I f k.° IR < 00 1

whenever k, >_ l_ 5 The reader should note the analogy between this construction and that for "good" semi-norms in Chapter IVA 1.

Chapter V. Applications of Theorems A and B

174

We now come to the proof of the lemma. Using the epimorphism E: O'y -

we identify a.(Ov) with Oy/J where 5 := X ea E. We may further

identify a.(Orcd u) with Oy /rad 5, where rad .f is the nilradical of J. Let p. Oy -+ Oy/rad J. Since V is Stein, s and p induce epimorphisms at the section level: ev: Ov(V) - a.(Ou)(V) and pv: O'(V) a.(Ofed u)(V). Finally, identifying a.(Ou)(V) with Ox(U) and it.(t,,d0(V) with Orrdx(U), one has a commutative triangle

Ox(U)

red

(0red x(U)

of continuous, C-linear epimorphisms, where these vector spaces carry their canoni-

cal topologies. In this case red denotes the homomorphism at the section level which is induced by the reduction epimorphism Ox -' Ored X. Since the domain U c X is Stein, red is surjective. For simplicity we have written E, p instead of Ev, Pv

The following fact is now useful:

Proposition. For every r e R, 0 < r < 1, there exists a t > 0 so that for every f e Ox(U) with I f I U< t there is a g e Ov(V) with

p(g) = red f and

I g IQ 0,

Ej=_(zje CI Izji - no. Since this is true for all H CA it follows from (4) that !I f -f IIB no. In other words, f e 0k(B) and lim11fi -f 11B = 0. Hence 0k(B) is complete. Every bounded set in 0k(B) is a family of functions holomorphic on B which by

Theorem 3 is uniformly bounded on any compact K c B. But the classical theorem of Montel states that such a set is relatively compact in 0(B). O In addition to Theorem 4 one can say that every closed and bounded set in 0k(B) is compact in Ok(B). This is an immediate corollary of Theorem 4 and the following:

(6) If f, e Ok(B) is a 11

II B-bounded sequence which converges compactly to

f e Ok(B) then f c- Ok(B) and (1f IIB _ 1 and denote by M > 0 the constant of the Schwarz lemma. Suppose that k is a Hilbert subspace of (9 (A) equipped with an orthogonal bases {g1, g2, .. .} which is monotone at 0. Then for every e e N there exists d e N such that for all v c- 01,(A) a

v - I (v, g,)g, =1

< Ma'IIvII,. AI

Proof: The monotonicity implies that for every e e N there exists d E -.N such d

that for every v c- H the vector w:= v the origin. The Schwarz lemma implies that lity of {gj} yields IIwli, < IIvlI,

(v, g,)g, vanishes of order at least e at iiwJI,. < MaeIIwlJ, and the orthogona-

2. The Subdegree. In order to construct monotone orthogonal bases we need a

slightly more general function than op. For this we set 1:= (1, ...,

I},

N°'={v=(v1,...,v.)Iv _ ments a = (i, v), a' = (i', v') e A we say that a < a' whenever anyone of the following'three conditions is satisfied: 1)

Ivi < Iv'I

2) I v I = (v' i and there exists j, 1 < j < m, with vj < v; and v,, = v' , fork > j.

3) v=v'andi 0. We set ai '_ (h, g;, i = 1, 2, .... Suppose h e H j. Then, for all i < j, the vectors v:= gi + ch, c e C, are in H*. Since gi is a minimal function,

IIvFI2= 1I9i112+ca,+c,+

Icl2b

_ IIg1II2

for all c e C. Since b > 0, this is impossible unless ai = 0. Consequently every It E Hj is perpendicular to the linear span ((g,, ..., gj_,))c.

Chapter VI.

194

The Finiteness Theorem

On the other hand suppose a, = = aj_, = 0. We set a, _= w(h) and normalize h so that h e H; . Thus for all t e R the vector w = (1 - t)g, + th lies in H40Hence Ilwll2 = (1 - ty11g, i12 + ta, + ta, + t2b Z iig,112

for all t e R. But (1 - t)2Ng,112 + t2b < jig, 02 for small t > 0. Thus a, # 0 and s >- j. In other words h e Hj. If g; e H1 is another minimal function, then u g'j' - gjE H;+,. From the O above remarks, (u, gj) _ (u, gj) = 0. Hence 11u1(2 = 0 and g; = gj. The following important fact is now easy to prove.

Theorem 2. If H is a saturated Hilbert subspace of ((B), then every set Hj contains a unique minimal function gj. The family {g,, g2, ...) is a monotone ortho-

gonal system for H at p e B. If in addition B is connected, then (g,, g2, ...) is a monotone orthogonal basis for H.

Proof: It is only necessary to prove the last statement. If h E H is a vector which

is perpendicular to each of the gJ's, then w(h) > co(g;) for all j. Consequently O oy(h) z lim o,(gj) = oo. Since B is connected, this implies that h = 0. Remark: For k = m = 1 and H = 0,,(B) one finds a presentation of the theory of the minimal functions in the book of H. Behnke and F. Sommer: Theorie der analytischen Funktionen einer komplexen Veranderlichen, Springer Verlag,1962, (2. Aufl.), p. 270.

Furthermore reference should be made to the Ergebnisbericht of H. Behnke and P. Thullen: Theorie der Fupktionen mehrerer komplexer Veranderlichen, Springer-Verlag, 1970, {2. Aufl.), p. 1970.

§ 3.

Resolution Atlases

In this section X always denotes a compact complex space and .P is a coherent analytic sheaf on X. If U = {U;}, sist* is an open cover of X, then the vector space

C4(U, ,') _ fl

of q-cochains is a finite product of Freshet spaces

and is thus a Freshet space in a natural way. In order to specify a subset C}(U, .') which is a Hilbert space we introduce the idea of a resolution atlas. The purpose of this section, in particular the reason for studying these Hilbert space, is to make the necessary preparations for the proof of the finiteness theorem.

1. Existence. A triple (U, 0, P) is called a chart on X whenever U * 9 is open in X and 4': U -, P is closed holomorphic embedding of U into a polycylinder

P={zeC"I jz,I _ 0

Every such family

x P,, is the polycylinder about 0 in C, n U,, is non-empty, then m 1= n(q + 1), with each radius r. If U,0 ... ,,, = U,0 n be given. Then P,0 ... 14- P,0 x :

-. P

x - MOW, ..., 044

is likewise a closed holomorphic embedding.' The image sheaf (01. ... ,,) . (S U.. ... ,, )

is therefore coherent on P,0 ... and

(U-.

'j.

0,.- "M

are equal to the polycylinRemark: Even though all of the polycylinders der of radius r about 0 e C', it is for cohomological reasons advantageous to keep the indices.

Definition 1 (Resolution Atlas). A system 21 = (U,, (D P,, e,o...,.) is called a resolution atlas for ,9' on X if

1) (U (D P,)ls,is an atlas of charts on X and

2) there exists t e N so that the maps o'1 P,o...It -+(D,,,...,,).(`' U,,,...,,)

are analytic sheaf epimorphisms for q = 0, 1, ..., t, - 1 and i, Note that if 21 is a resolution atlas then U = (U,, ..., U,*} is a Stein covering of X.

Theorem 2. For every coherent sheaf 90 on X there exists a resolution atlas.

on X and Proof: We begin with an arbitrary atlas of charts choose concentric polycylinders P, Cc }t, of radius r, r < F, so that U, d5-'(P, t = 1, ... , t still gives a covering of X (the Shrinking Theorem shows that this is possible). The induced maps',: U, - P,, (, dt, I U,, are likewise closed holomorphic embeddings. For all indices to, ..., tq it follows that (4610

... y). (' I U,0 ... ,0I P,0 ... ,, =

... ,.).

(y I U,0 ... ,a).

' We only need lb,,...,. to be holomorphic and finite. If U,, ...,l is empty, we define 0,,...,to be the empty map.

Chapter VI. The Finiteness Theorem

196

is a relatively compact subset of P,o x tees the existence of sheaf epimorphisms Since P,0...

x P,., Theorem A guaran-

P,,...

f.,Q... ,,:

Note that, since for all it e N we have the epimorphism 0"+ 1 - 0", (f1, ..., f.. J- (ft, ..., f"), we may choose I to be independent of to, ..., 14, The system U = (U O P e,o...,,) is the desired resolution atlas for Y.

0

is a reso2. The Hilbert Space CJ(U, Y). Suppose that 91 = (U,, (D P,, lution atlas for .91. Then P,o... , is always the polycylinder A of radius r about O e C', where m:= n(q + 1). Thus C'1(21)°=

r[

L'(P,o...,,),

q = 0, 1, ..., t,,, - 1,

is canonically isomorphic to some 0"(0), and is therefore a Frechet space in a natural way. Every sheaf epimorphism (by Theorem B) map (s,

,)p,0

determines a C-linear, continuous surjective

: MP"...) - (N

)(P

"M-9, I U

0:-_ -V(U),

where ,"(U," ... ,) carries the canonical Frechet topology, The most important map for our considerations is the induced product map E: C9(`21)-.3 0(u, Y).

It is clear that e: C4(2I) -+ C°(tt, ,9') is a C-linear, surjective, continuous map between Frechet spaces.

In the Frechet space 0(2t) we have the Hilbert space C1(21):_

- (rh(A)

that the injection Cg(2I)-, C9( 0 so that, given , ' E Zg(11', .9"), there is a cocycle .' E Zg(11, .S') and a cochain !

rye Cg "'(U", .9') with

4'JU"=41U"+an,

IICIh 0 so that {C' E Zg(u', .9')I IIC'II1. = p} c OW)-

We claim that L '= p -' is the desired constant. To see this let ,' be an arbitrary ($ 0) element of Zg(U'. So). We find c E C so that p. Hence there are elements * E Z9(U*, .9") and w' E C9-'(11', "!') with Ill' + Ow'

Now we

define

C :=

Certainly

cs' lu" _

and Q *, co') E W.

JU e Zf(U, .9')

lu" + aw and

IICII. 0 and L > 0 be the const?nts of the Schwarz Lemma and the Smoothing Lemma respectively. We choose e e N large enough so that t:= LMa` < 1.

By Theorem 2.1 there exists a positive integer d so that for every vector d

v e Zg(21) the vector w:= v - E (v, gi)gi satisfies i=1

twit,. < MaelwIl.

(1)

Combining the projection ZR(221) =- Cf(U, Y) and the injection Z,J(U, So) -+ ZR(U ", .9') we have a continuous map Z9(21) -+ Z1(U ", .") which we denote by v u = e(v) U". For d as above we have the following: Finiteness Lemma.

ZX(U, .')I U" = i C. + oCr"'(11", 9) where 41,....gdEZA(U,3")i=1 v,,

Proof' Let 4 e Zf(U, .") be given. It is enough to construct two sequences vo, .. , and q,, qZ, .. , , where vie Zg(91) and , a Cg-'(U", ,9'), so that with wi d

defined by wi := ri - Y (vi, gt)gt we have i=1

vo=bJU,

(2)

wi=L'i+1+dq1+

.'->0,

and

(3)

Hv

I < tillvo 1.

VgiJIst'

0ILO11,

j> 1.

Assuming we have such sequences, (2) implies (4)

5

U - n+ 1 =

n

(t'i -i=O

1)

_

n

2: 2: (v.f, g i)9i + Z t9gi,

i=1i-O

i=O

n>0.

Chapter VI. The Finiteness 771eorem

202

Since t < I, it is immediate from (3) that the following converges: E

(5)

CF-1(U", )

0

Now a: CQ-'(U", So) -+ C4(U", S") is continuous and

qj+, converges to q in 0

aqj+,. Furthermore, using (3)

the Frechet topology (Theorem 3). Thus aq = 0

and the fact that

I (uj, g.) I - 0 for every x e X then the divisor D is called positive. For divisors D,, D2 e Div X, we write D, < D2 when D2 - D, is positive. The group of divisors is directed with respect to this relation. In other words, given two divisors D. D2 a Div X, there exists D3 a Div X so that D, < D3 and D2 < D3. The set DJ :_ {x e X: n,,(D) * 0), called the support of D. is always finite.

'

1. Divisors of Meromorphic Sections. Let .t be a locally free sheaf. Given a local coordinate t e m every germ sx e . can be uniquely written in the form sx = t'"ix,

where ix e .

x

1mx,f x. The exponent m is uniquely determined by sx. We define G(sx)'= m

to be the order of sx with respect to.F. If e,(sx) > 0 (resp. (sx) < 0) then we call x a

zero (resp. a pole) of sx. The situation o(sx) = oo only occurs for the identically zero germ. It follows that 3x = isx e .f G(sx) > 0) and nTx.'1zx = {s e .FA ; o(sx) > o}.

Chapter VII. Compact Riemann Surfaces

206

Each section s e .9rm(X), s * 0, is called a global meromorphic section of 9F. If s * 0 then it has only finitely many zeros and poles. Thus the following definition makes sense.

DefWd a L (The divisor and the degree of merontorphie sections). Given se.9rm(X), (s) °_ Y ziX

x e Div X

is called the divisor (with respect to F) of s. The integer deg(s) is called the degree (with respect to F) of s.

It follows that (s) is positive if and only if s has no poles, or equivalently, if

s e .flX). Warning: The order functions o and the divisors (s) of sections s e .9=°°(X) depend heavily on the sheaf with which one starts out. For example, with respect to 0, the zero divisor is associated to the section 1 e ar,'(X ). On the other hand, if 0 is replaced by 0(D) (see Paragraph 3) for some D E Div X then (1) = D. In the following it will always be completely clear with respect to which sheaf we are forming divisors.

From the above remarks it follows that every. meromorphic function h e 0"(X)* has an associated divisor (h). Such divisors are called principal divisors. The map .&(X)* -+ Div X, which sends h to its divisor (h), is a group homomorphism. For all h a e)-(X)* and s e F' (X)*, it follows easily that (hs) - (h) + (s). The image group, P(X)==Im(.4'(X)* -' Div X) c Div X, is called the group of principal divisors and the quotient group

Div X/P(X) is called the group of divisor classes on X. Following the classical language of algebraic geometry, one says that two divisors D and D' are linearly equivalent if each is a representative of the'same class of divisors. In other words D and D' are linearly equivalent if D - D' a P(X ). 2. The Sheaves .9r(D). Given a locally free sheaf 3r and a divisor D, we define an

analytic subsheaf 5(D) of F by {sx a Fz : o(s.) z -o=(D)}

and 5(D) = U .F(D)., c F. XeX

Except for points in I D 1, P and 5(D) agree. For x e I D 1, P,, is made smaller (rasp. larger) when oX(D) < 0 (reap. *.(D) > 0). More precisely, it follows that, if

§ 1.

Divisors and Locally Free Sheaves

207

t E inx is a local coordinate at x, then

In One obtains an C -isomorphism of .F(D) onto at x by multiplication by and it is always particular .F(D) is a locally free sheaf with the same rank as .

true that ,F* (D) = F'. In the following lemma we summarize the laws which follow immediately from the general sheaf calculus:

Lemma. Let .F, JF,, and F2 be'locally free sheaves and D, D1, D2 divisors on a compact Riemann surface X. Then

1) Every exact C-sequence 0 -+ F 1 -+ g - F2 - 0 determines in a natural way an exact C%-sequence 0 -+ 91(D) - F(D) -. 92(D) -+ 0.

2) If 9 = 9, + i~ 2 then 9(D) _ 9 1(D) + F2(D). 3) There is a natural 0-isomorphism F(D1)(D2) =;.f(D1 + D2)4) If D1 5 D2 then F(D1) is an analytic subsheaf of F(D2). The reader can easily carry out the proof. We note here that property 4) will play an important role in the next section.

3. The Sheaves 0(D). The above considerations are in particular valid for F = C. All sheaves O(D), D e Div X, are locally free of rank 1. Two sheaves, 0 (D1)

and 0(D2), are analytically isomorphic if and only if D1 and D2 are linearly equivalent.

Proof: The sheaves C(D1) and C(D2) are analytically isomorphic if and only if 0(D1 - D2) = C. Let D:= D1 - D2. Assuming that D1 and D2 are linearly equiv-

alent, D = (h) with h e #(X)*. In this case we obtain an C-isomorphism 0 O(D) byfx "fi h.. Conversely, given an 0-isomorphism C C(D), the image of 1 e C(X) in C(D)(X) is a meromorphic function h e ..Y(X)* with D = (h). O By tensoring .F with C(D), one gets all sheaves JF(D). This is seen from the natural 0-isomorphism , ® C(D) -. .F(D), defined by s ® hs+-. h,, sx. Remark: If ,q' is a coherent sheaf and D e Div X then .5o(D) ;= So ® C (D) is coherent. The reader should note that the statements 1)-3) in the lemma of Paragraph 2 are in fact valid for coherent sheaves as well as locally free sheaves. The tensor product of two sheaves of the type 0(D) is again a sheaf of that type. Given two divisors D1 and D2, there is a natural isomorphism,

C(D1)®0(D2)-+C(D1 +D2), defined stalkwise by s1s s2x. One usually identifies the group H1(X, 0*) with the group of analytic isomor-

phism classes of locally free sheaves of rank 1 over X. In this way, the group operation in H1(X, 0*) corresponds to the tensor product of sheaves. The homo-

Chapter VII. Compact Riemann Surfaces

tog

morphism b in the long exact cohomology sequence,

O-

Div X 6 H'(X, C*),

tt* -.9 -+ 0, is in fact defined by D F-- e (D). The kernel of b is just the group P(X) of principal divisors and therefore one has a natural injection,

related to the short exact sequence of sheaves, 0 - e-

Div X/P(X) C. H'(X, (I*),

of the group of divisor classes on X into H'(X, C").

§ 2.

The Existence of Global Meromorphic Sections

We will show here that every locally free sheaf (not identically zero) on a compact Riemann surface has "many" global meromorphic sections. This follows

from a "characteristic theorem" which will give rise in the next section to a preliminary version of the Riemann-Roch theorem. In particular it is proved that for every p e X there is a non-constant holomorphic function on X\p which has a pole at p. This shows that X\p is Stein, from which it follows that H4(X, .9') = 0, q > 2, for any coherent sheaf ..' on X.

1. The Sequence 0 - .Y(D) -+ .Y(D') 0. Let D, D' be divisors with D < D' and let .Y be a locally free sheaf of rank r. Then there is a natural exact sequence (* )

0-+.F(D)-+flD') T - 0,

where T F(Y)/.(D). Since this sequence plays such an important role in our considerations, we will now write down its basic properties: From the definitions it follows that for every x e X (1)

x = C'"=,

with

nx = cx(D') - r(D).

The support of T is therefore the support of D' - D and (2)

dime .Y(X) = r deg(D' - D).

Since .Y has finite support,

H'(X, .T) = (0). Consequently the first part of the cohomology sequence associated to (*) is the .exact sequence (3)

.Y(D'))-0.

§ 2.

The Existence of Global Meromorphic Sections

209

Thus (4)

if D< D' then dime H1(X, F(D)) Z dim, H1(X, .,F(D')).

2. The Characteristic Theorem and an Existence Theorem. Let So be a coherent sheaf on X. The characteristic of .9', Xo(.9'), is defined by

Xo(Y) = dime H°(X, .9') - dime H'(X, 50). It will be shown (Paragraph 4) that X,(9) is the Euler-Poincare characteristic. In particular it is proved that for every p e X there is a non-constant holomorphic function on X\p which has a pole at p. This shows that X\p is Stein, from which it follows that 134(X, .9') = 0, q >- 2, for any coherent sheaf .9' on X. Lemma 1 (The characteristic theorem). Let be a locally free sheaf of rank r and D a divisor on a compact Riemann surface X. Then Xo(J'(D)) = r deg D +

Proof: We will show that, for arbitrary divisors D and D',

Xo(F(D)) - r deg D = Xo(.F(D')) - r deg D'

(°)

The claim of the lemma will then follow with D'- 0. First we suppose that D < D'. The alternating sum of the dimensions of the vector spaces in the exact cohomology sequence (3) is therefore zero. In other words 0 = Xo(F(D)) - Xo(F(D')) + dime 9"(X ).

If one substitutes r deg(D' - D) for dime .T(X) (see (2) above) then (o) follows immediately.

If D' is arbitrary, then one chooses D" e Div X with D 5 D" and D' S D". Then it follows from the above that Xo(F(D)) - r deg D =

Xo(F(D ))

- r deg

D" a= Xo(.9(D')) - r deg Y.

Theorem 2 (Existence theorem). Let _'F be a locally free sheaf of rank r and D a divisor on ,a compact Riemann surface X. Then

dime -IF(DXX) > r deg D + X,(-,F).

In particular if JF $ 0 and deg D > 0, then

lim dime F(nDXX) = oo. RIM

210

Chapter VII.

Compact Riemann Surfaces

We have therefore established that every locally free sheaf .IF * 0 has nonidentically zero meromorphic sections. This is clear from Theorem 2, since F(D)(X)

is always contained in .$°°(X). 3. The Vanishing Theorem. By Theorem 2,

dime t9(npXX) z n + Xo(r) for any p e X, and all n e Z. Hence there exists no e FJ so that lr(npXX) contains a non-constant function h for all n > no. Since (h) + np > 0, every such function is non-constant, and holomorphic on X \p, and has a pole of order at most n at p. This shows the foilowing:

For every p e X there is a non-constant meromorphic function on X which is holomorphic.on X\p.' Moreover, Xpp is Stein. In particular every compact Riemann surface can be covered by two Stein domains.

Proof: Let h be as above. Then h: X\p - C is a finite holomorphic map. Thus X\p is Stein. If P1, P2 C_ X are different points, then (X \pi, X\p2} is a Stein cover of X. The following is now a consequence of the general theory. Theorem 3 (Vanishing theorem). Let .9' be a coherent. analytic sheaf on X. Then

H'(X,.)=0,

q>2.

For every compact complex space X and every coherent sheaf .9' on X, almost

all of the groups

vanish. Thus the Finiteness Theorem allows us to define the Euler-Poincare Characteristic,

E (-1Y dime H'(X, Y) e Z.

r-o

Hence Theorem 3 shows that for compact Riemann surfaces

AY) = Xo(b) 4. The Degree Equation. An amusing consequence of the characteristic

formula, Xo(C(D)) = deg D + Xo(O), is the degree equation: For linearly equivalent divisors D and D', deg D = deg D'. In particular deg D = O for all principal divisors. ' With a bit more effort one can show at this point that for every p e X there exists no e N so that for every n > no there is h e 4'(X) which is holomorphie on X\p, and which has a pole of order n at p (a forerunner of the Weierstraps pp Theorem)

The Riemann-Roch Theorem (Preliminary Version)

§ 3.

211

Proof: If D and D' are linearly equivalent then, from Paragraph 1.3. C (D) C' (D') and consequently Xo(( (D)) = X,#1 (D')). The characteristic formula yields

degD+Xo(()=degD'+Xo(C`) The reader should note that if X # P,, then not every divisor D with deg D = 0 is a principal divisor. Remark: The degree equation can also be interpreted mapping theoretically, and proved in this way

as well. For this, note that every h e .#(X) defines a branched covering h: X -. P, (the case of )r identically constant is trivial). Certainly, if every point of h-'(0) and h-'(oo) is counted with its branching multiplicity, then (h) = h-'(0) - h-'(oc). For every p e P the sum of the multiplicities of the points of h-'(p) is the sheet number s of the covering h: X -. P,. It follows that deg h = s - s = 0.

§ 3.

The Riemann-Roch Theorem (Preliminary Version)

The classical problem of Riemann-Roch consists of determining the dimension of the C-vector space, H°(X, CA (D)), of all global sections of the sheaf (I (D). The characteristic theorem gives a preliminary solution of this problem.

1. The Genus Theorem of Riemann-Roch. The following notation is standard : I(D):=dimc H°(X, C- (D))

and

i(D):=dime H'(X, ND)).

For linearly equivalent divisors D and D', I(D) = l(D') and i(D) = i(17). We further note that the dimension I(D) > 0 if and only if there is a positive divisor D' which is linearly equivalent to D. In particular I(D) = 0 for every D with deg D < 0.

Proof. The first statement is clear, since the divisors which are linearly equivalent to D are of the form D + (h) for h e . "*(X ). The second statement follows from the first, since linearly equivalent divisors have the same degree.. p

For the zero divisor C' (X) - C and thus l(O) = 1. The natural number g

i(0) = dimc H'(X, 0)

is called the genus of X. From this definition it only follows that g is a complex analytic invariant of X. However in Paragraph 7.1 we will show that g is in fact the topological genus of X (i.e. Ht (X, C):-- C2 ). For every divisor D it follows that Xo(0(D)) = 1(D) - i(0),

and in particular Xo(0) - 1 - g.

Chapter VII.

212

Compact Riemann Surfaces

Thus the characteristic formula X0((' (D)) = deg D + Xo((r) can be restated as follows :

Theorem I (Riemann-Roch, preliminary version). If D is a divisor on a compact Riemann surface X of genus g, then

1(D) - i(D) = deg D + 1 - g. Remark: The Riemann-Roch problem (i.e. the determination of the number 1(D)) is not satisfactorily solved by Theorem 1, because the term i(D) appears as a dimension of a first cohomology group. However, by means of Serre duality it can be interpreted as the dimension of a 0th cohomology group (see Paragraph 6). The final solution of the Riemann-Roch problem is given by Theorem 7.2.

2. Applications. The following Riemann Inequality Theorem 1: 1(D) > deg D + 1

is

a special case of

-g

This inequality yields the first classical existence theorems. For example, since 0(D)(X) contains a non-constant meromorphic function whenever 1(D)> 2, we have the following:

For every divisor D with deg D z g + I thei a exists a non-constant meromorphic function h with (h) + D >- 0.

In particular, given p E X, there always exist non-constant functions which have poles of order at most g + 1 at p, and which are holomorphic on X\p. One can state this as a theorem about coverings of P 1: Every compact Riemann surface X of genus g is realizable as a branched cover with at most g + 1 sheets of the Riemann sphere P,. In particular, if g = 0 then X = PI. Since 1(D) - 0 when deg D < 0, it follows from Theorem 1 that, for every

DEDivXwith degD 0 for every hs E 0(D), 0(D)xsx c F,. Thus 2' is an analytic subsheaf of F. Due to the fact that s -* 0, 2' is isomorphic

to 0(D). Furthermore s e -W(-DXX) c 2"(X) and, when D >_ 0, s e Y(X ). Finally 0(D),, = Ox g;, where o(gx) = --- ox(D). Defining t, g,, sx, it follows that Sex = 0, t,, with o(t,) = 0 for all x E X. Therefore,. by the above remark, F/2' has everywhere. free stalks. .

Chapter VII. Compact Riemann Surfaces

214

Remark: The sheaf ..P = e'(D)s is the only locally free subsheaf of .IF of rank 1

with s E 2°°(X). To see this, let S be such a sheaf. Then .9, = ( v, for some h_,:=m;'. Then v, = h, s,, and, since v, e ,F where s, = m, v and m, a .,#,. Let o(v,)>_ 0, h, E tr(D),. Consequently v, e O(D),s, or 2, c 2,. The O, module Since contains therefore a submodule which is isomorphic to Y, /.9, = 0. Hence %F, /2, is free, and since 2/2', is in any case finite, P, .

0

forallxand2_Y.

2. The Existence of Locally Free Subsheaves. The foundation for the study of the structure of locally free sheaves is the following theorem. Theorem 3 (Subsheaf theorem). Every locally free sheaf .'F * 0 contains a locally free subsheaf which is isomorphic to O(D) for some D e Div X. One can choose D = (s), where s E

Proof: From Paragraph 2.2 it follows that .F has a non-trivial global meromorphic section. Thus the theorem is an immediate consequence of Theorem 2. 0

The following is an immediate corollary of Theorem 3.

Them 4 (Structure theorem for locally free sheaves of rank 1). Every locally free sheaf 2' of rank 1 is isomorphic to a sheaf O(D) with D E Div X. Furthermore one can choose D = (s) with s e Sem(X )'. This theorem says that in the cohomology sequence which is associated to the short exact sequence of sheaves 0 - O 2 0,

... Div X --- H' (X, O') -- H' (X, . K') --- H' (X , 2) - --

,

the homomorphism b is surjective (see Paragraph 1.3). Thus one has a natural group isomorphism,

Div X/P(X) 4 H'(X, O'), of the group of divisor classes on, X onto H'(X,

Remark: Since S is surjective, the map H'(X, ..K') H'(X, 2) is injective. Clearly :a is a soft sheaf and thus H'(X, -9) = (0). Thus Theorem 4 is equivalent to the equation

H'(X,

(0).

3. The Canonical Divisors. The sheaf of germs of holomorphic 1-forms over X is a locally free sheaf of rank I. Thus one can apply Theorem 4: Theorem S. There is a unique divisor class on X so that for every divisor K in this

class, II' = O(K).

Supplement to Section 4: The Riemann-Roch Theorem for Locally Free Sheaves

215

One calls K a canonical divisor and its class the canonical divisor class on X. The significance of canonical divisors appears in Section 6. If X = Pt then every divisor -2x°, x0 E X, is canonical. This follows from the fact that, if z is a coordinate on X\x0, then dz is a differential form on X which is holomorphic and nowhere vanishing on X\xo and has a pole of order 2 at x°. Of course one must use the fact that on P, the degree of a divisor determines its class. In the case of elliptic curves the zero divisor is canonical.

Supplement to Section 4: The Riemann-Roch Theorem for Locally Free Sheaves The generalized Riemann-Rock problem consists of determining the dimension of H°(X, .F(D)) for every locally free sheaf F and every divisor D. In order to do this one needs to carry the idea of degree over to the case of locally free sheaves. We use the fact that Xo coincides with the Euler-Poincare characteristic X. Moreover the additivity of X (i.e. for every exact sequence of coherent sheaves 0 - 9' - 6" -+ .So" -+ 0, one has X(.9') = X(.9') + X(.9")) plays an important role.

1. The Chern Fanction. We denote with LF(X) the set of analytic isomorphism classes of locally free sheaves over X. A function c: LF(X) Z is called a Chern function if

1) For D e Div X, c(O(D)) = deg D. 2) For every exact sequence 0

0 of locally free sheaves, one

has c(F) = c(F') + c(F"). The following is straight forward: Theorem 1. The function c: LF(X)--* Z, defined by 3)

X(.:F) - rank - F X(O)

is a Chern function.

Proof: From Lemma 2.1 we have c(O(D)) = X(O(D)) - X(O) = deg D.

The additivity follows from the additivity of both X and rank.

Remark: The results in Paragraph 4.2 imply that this c is the only Chern function. To see this note first that if y is any such function then, since the locally free sheaves of rank 1 are just, the divisor sheaves, y(6") = c(.P) for every locally free sheaf St' of rank 1. Suppose now that y = c for all locally free sheaves of rank less than r and let .O' of rank r z 2 be given. Then there is an exact sequence of

Chapter VII. Compact Ricmann surfaces

216

locally free sheaves

where 2' and have rank' l and r - 1 respectively. The induction hypothesis and the additivity imply that y(F) = c(F). 2. Properties of the Chern Function. The Chern function behaves nicely with respect to tensor products:

c(.F (9 2) = rank .9r c(2') + c(.), when rank 2' = 1.

4)

Proof: Let .P = O(D). Then F ® .P = Jr(D) and c(2') = deg D. Thus c(.F ®-') = X(-,F(D)) - rank F(D) - X(O)

= rank F deg D + X(F) - rank f X(O)

0

= rank .4 F c(2') + c(.F).

When Y, and 22 are locally free sheaves of rank 1, it follows from 4) that

c(f, (922) = c(2',) + c(2'2). Thus the map H'(X, O') - Z, defined by 2' - c(.'), is a group homomorphism from the analytic isomorphism classes of locally free sheaves of rank I to Z. The reader can check that the map y in the exact cohomology sequence .

...-.H'(X,6)--'H'(X,6*) Y,HZ(X

l)---4

is the Chern function, provided H2(X, Z) is identified with Z in the natural way.

D We remark without proof that, if .F is locally free of rank r greater than 1,

c(F) = c(det .F), where det .F == A F is the locally free determinant sheaf of I

rank 1. Thus, letting F be the vector bundle associated to A, c(F) is just the first Chern class of F, c,(F) a H2(X, Z) = Z.

3. The Riemann-Roch Theorem. Equation 3) in Proposition I above can be rewritten as a Riemann-Roch theorem: Theorem 2 (The Riemann-Roch theorem for locally free sheaves). Let .

be a

locally free sheaf of rank r and D a divisor on a compact Riemann surface X of genus g. Then

dime H°(X, F(D)) - dime H'(X, .F(D)) = r(deg'D + I - g) + e(F). Proof: On the left hand side we have X(.F(D)) which, by the characteristic, theorem, is the same as r deg D + X(F), If one writes r X(0) + c(.W) for X(P), then, since X(0) - 1 - g, the claim follows immediately.

The Equation H'(X,.Il) = 0

§ 5.

§ 5.

217

The Equation H' (X, .elf) = 0

The inequality from Paragraph 3.2, l(np) > n + 1 - g, is already strong enough to show that for every divisor D on X and every point p e X the cohomology groups H'(X, ((D + tip)) vanish for all n > 0 (we already know that the dimension of these vector spaces is constant for n > 0 and is unbounded for n < 0). This

"Theorem B for compact Riemann surfaces," which will be made even more precise in Section 7, has an immediate consequence the fact that H'(X, .,K) = 0. 1. The C-homomorphism C(np)(X) -+ Hom(H'(X, C(D)), H'(X, C(D + tip))).

Let D be a divisor in Div X, p a point in X, and n > 0 a fixed integer. Every function f e C(np)(X) determines the C -homomorphism Of: (I (D) - C(D + tip). hx If f # 0, then Of is injective and the sheaf Of in the exact sequence

0- C(D)

C(D + tip)

has finite support

Jmtpf-0, hntpf = C(D + np)lf- C(D),

(p f)., = 0 for every x 0 D I u {p}, where fX is a unit in C ,, ).

Thus H'(X,Jrtpf)=0and every .function f

0 in C(np)(X) induces a C-rector space epimorphism

Of: H'(X, C(D))

H'(X, ((D + np)).

If f = 0, then we define Of to be the zero mapping. Lemma 1. The map 0: C^(np)(X) - Homc(H'(X, (I (D)). H'(X, C (D + tip))). is C-linear.

Proof: The claim becomes clear when one looks at how Of is defined via the Cech complex. Let It = {U;}, i e 1, be a cover of X with cochain groups C' (t1, ((D)) = fl C (D)(t',Q.. ) iu-il

and

C'(11, C (D + tip)) = 11 C (D + np)(U,,,, ). io.iI

Associated to every C-homomorphism tp f, f e C (tip), are the following homomorphisms at the section level: Of(io, ii): C'(DXU;a1) . C(D + np)(Ur,1),

h-- (f I Ub,t)h,

io, i, a 1.

Chapter VII. Compact Riemann Surfaces

218

The collection'Pf

{cpf(io, i1)} is a homomorphism

'Pf: C'(U,

(r(D + np))

which induces the homomorphism of: H'(X, C(D)) it is clear that the definition of rp f(io, i 1) in

H'(X, e(D + np)). From

4Paf+be(io, it) = acpf(io, it) + btps(io, i1), a, b e C;f, g e ((np)(X ), io, it E 1.

Consequently `Pf+b,, = a'I'f + b'P9 and thus oaf+bo = at/if + bOr.

2. The Equation H'(X, ((D + np)) = 0. The following is an easy corollary to the Riemann inequality and Lemma 1: Theorem 2. Let D e Div X and p e X. Then

H'(X, C'(D + np)) = 0 for all n > no := (dimc H'(X, c(D)))2 + g Proof: Let d °=dimc H'(X, (%(D)). Then, by 2.2(4), it follows that dimc H'(X, Gr(D + np)) < d for all n >: 0. Hence the map >G (in Paragraph I

above) maps C9(np)(X) into a C-vector space having dimension at most d2. But d does not depend on n and, by 3.2, dime &(np)(X) >_ n + 1 - g. Therefore, whene% er n >: no = d' + g, the C-linear map 0 is not injective. Thus for such an n there

exists f 0 in ((np)(X) with (p f = 0. Since pf must be an isomorphism, H'(X, (9(D + np)) = 0. 3. The Equation H'(X, .,!!) = 0. We can now prove the following fundamental theorem :

Theorem 3. H'(X, ..#) = 0.

Proof: Let E H' (X, K) be an arbitrary cohomology class. We choose a finite cover U = {U,) of X, a shrinking of U, R3 = {l;}, with V,Cc Ui, and a cocycle e Z'(U, ..K) which is a representative of We let where ioi, E .1(Uioii). Then (

ioii I Vote) = Ps E Z' (23, -,1)

is likewise a representative of . Since V;C U; for all i, each function ,a , has finitely

many zeros and poles on Vio,,. Thus one can find a divisor D E Div X so that ' a Z'(93, ((D)). Let p E X\IDI. Then, since 0(D) c Cr(D + np) for all n > 0, it follows that ' c Z'(13, 9(D + np)) for all such n. Since S''(D + np) c A' and H'(X, 0(D + np)) = 0 for n > ro, it follows that ' is cohomologous to zero as an element of Z' (D, M). Thus ?; = 0, and consequently H' (X, A) = 0. 0

The Duality Theorem of Serre

§ 6.

219

The most important applications of the equation H'(X,.4') s 0 are found in the next section. In dosing this section we note in passing a rather simple consequence. Let 0m denote the sheaf of germs

of meromorphic 1-forms. The differential d: M (11, which is defined in local coordinates by dh,,-dh,/dz dz, is C-linear. The C-sheaf d.0 c 0' (it is not an 0-sheaf!) consists of all residue free germs of meromorphic 1-forms (i.e. germs of abelian differentials of the second kind; for the idea of a

residue see Paragraph 6.5). Since H'(X, .A') - 0, the short exact C-sequence 0 -. C - A' - d.IY -.0 induces the following exact oohomology sequence:

0- .C- H°(X,..d')-H°(X,d.4')---+H'(X, C)-.0. This means that

H'(X, C) s H°(X, d.&)/dH°(X, M). In words, the cohomology group H'(X, C) is isomorphic to the quotient space of global abelian differentials of the second kind modulo differentials of global meromorphic functions.

§.6.

The Duality Theorem of Serre

In this section we will establish a natural isomorphism between the space of differential forms H°(X, Q(D)) and the dual space of the first cohomology group

H'(X, 0(-D)), where D is a given divisor on X: H°(X, O(D)) 4 H1 (X,

Given to e R(DXX), the associated linear form O(w): H'(X, 0(-D)) -. C will be obtained as a residue map by applying the residue theorem. We reproduce here the algebraic proof of Serre ([GACC], Chapter II), making use of the equation

H'(X, -#) = (0).

1. The Principal Port Distributions with Respect to a Divisor. Defining A° = ,,K/0 as the sheaf of germs of principal parts, we considered in Chapter V.2.1

the exact sequence 0 - 0 -+. ,K - Jr -. 0. In the following we generalize this slightly. Let D e Div X and define .W(D)'=.,K/0(D) as "the sheaf of germs of principal parts with respect to D." Then we have the exact sequence (1)

It follows immediately that A (D), like 2, is soft. In fact, every section over an open set U has discrete support in U. Since every stalk A (D), is the quotient module .It' /0(D) this implies that the C-vector space Jt°(D)(X) of global distributions of principal parts is canonically isomorphic to the direct sum ®. W, 10(D),: zeX

(2)

.;r(DXX) = (D #.10(D). =

:.X

:.x

:.x

O(D),.

Chapter VI!. Compact Riemann Surfaces

220

2. The Equation H'(X, 0(D)) = I(D). Let R be the set of all maps F = (fr) which assign to every x e X a germ fX a J1 so that almost allf: are holomorphic. Clearly R is a C-vector space. Further, given D e Div X, R(D) :_ {F a R: f, e 0(D)X}

is a subspace of R. Every element of the direct sum ® .,11 X (resp. ® 0(D)X) is a XEX

SEX

family { fX} x e X withf5 e

(resp. fX e 0(D)X), where almost all fX vanish. Thus

we have the natural C-linear injection ®.I1X -+ R which maps ® 0(D)X into XeX

XeX

R(D), and which induces a C-isomorphism

yf (D)(X) = e -0X/ ® C(D)X = R/R(D).

(3)

XEX

XEX

Every meromorphic function h determines an element (h5) e R. Thus, identifying. flX) with its image in R, .61(X) n R(D) = C (D)(X). We now set 1(D) := R/(R(D) + . #(X)).

The standard isomorphism theorems of linear algebra yield 1(D)= (R/R(D))/(.,K(X)/(0(D)(X)).

(4)

The following is an easy consequence of the definitions.

Theorem 1. For every D e Div X there is a natural C-isomorphism H1(X, 9(D)) = I(D). Proof- Associated to the short exact sequence (1) we have the exact cohomology sequence

o-((D)(X)-J1(X) -.(D)(X) -;H'(X, C(D))

.H'(X.

By Theorem 5.3, H'(X, .-11) = (0). Thus

H1(X, (' (D)) ; .' (D)(X)/Im E, where Im e

61(X)'C'(D)(X ). The claim now follows from (3) and (4) above.

O

The reader should note that the spaces R, R(D) and .11(X) have very large infinite dimensions, but that the finiteness theorem implies that I(D) is finite dimensional.

§ 6.

221

The Duality Theorem of Serve

and G = (gs) E R, we define the product C-algebra, but it is more FG (f=g,J. Equipped with this product, R is of course a -.K(X) c R. Let a: R -' C be a C-linear importantly also an algebra over the field form ha: R C by ha(F) = form and h E -#(X). Then one defines the C-linear cz(hF). Thus Homc(R, C) becomes an .11(X)-vector space. We explicitly state two simple, but important, properties:

A Linear Forms. For maps F =

a) If . #(X) c ker a then ..#(X) c ker ha. b) If R(D) c ker a then R(D + (h)) c ker ha. Proof. The statement a) is trivial. If F = (fr) e R(D + (h)) thenf, E e%(D + (h))x and therefore o(hx f,,) >_ - ox(D). Hence hF E R(D). This proves b).

We denote by J(D) the dual space of 1(D). It follows from a) and b) that every meromorphic function h E ,,#(X )* determines a natural C-linear mapping,

J(D) -+ J(D + h), defined by AH hA.

Proof: Every A e J(D) is a C-linear form A: R/(R(D) + .#(X)) -+ C and is therefore liftable to a C-linear form x: R -+ C such that a vanishes on R(D) + .tif(X ). From a) and b) it follows that ha: R C vanishes on R,(D + (h)) +,#(X). Clearly ha induces a C-linear form hA E J(D + (h)), which is uniquely determined by A and h. It is obvious that the map A -+ hA is C-linear.

Since hA is no longer in J(D), we go over to the space J °= U J(D), D

the union of all J(D)'s for D e Div X. For two divisors D, < D2, we have R(D,) < R(D2). Thus J(D,) >_ J(DZ) when D, < D2. This immediately implies that every finite subset of J is contained in some J(D). The set J is thus a C-vector space which is filtered by the subspaces J(D), D E Div X. The above defined map, J(D) J(D + (h)), gives us a mapping .

1(X) x J

J. Since Homc(R, C) is an ..H(X }vector space, the following remark

is obvious.

Theorem 2. The set J is, with respect to the operation. N(X) x J . J. a rector space over ff (X).

4. The Inequality Dimx(x, J < I. The critical point of the proof of the duality theorem is the following surprising dimension estimate. It is obtained by taking a limit of the preliminary form of the Riemann Roch formula. Theorem 3. The.,lf(X )-vector space J is at most l-dimensional.

Chapter VII. Compact Riemann Surfaces

222

Proof: Let A., p e J. We choose D E Div X with A, ,U E J(D). Let p c- X be fixed.

For every f e 0(npXX) it follows that, since D - np < D + (f ),f . E J(D + (f )) c J(D - np). Similarly gp E J(D - np) for all g e (i(np)(X ). The C-linear map

-N(X)®.,K(X)-+J,

(°)

(f, g)'-'.1X +gp,

therefore induces by restriction a C-linear mapping (9(np)(X) ® C(np)(X)

(o)

J(D - np)

for all n E Z. If A and p were linearly independent then both maps (o) and (o) would

be injective. This would mean that

(+)

2 dime 0(npXX) < dime J(D - np)

for all n E Z. From the Riemann inequality (Paragraph 3.2) it follows that (+ +)

dime C'(np)(X) = l(np) >- deg(np) + I - g = n + 1 - g.

Furthermore (Paragraph 3.2), as soon as deg(D - np) = deg D - n is negative,

dime J(D - np) = dim, 1(D - np) = i(D - np) = g - 1 - deg(D - np). Thus, for large n, (+++)

dimeJ(D-np)=g- 1 - deg D + n.

From (+ +) and (+++) one infers that, for n large enough,

2 dim, 9(np)(X) > dime J(D - np). This is contrary to (+) and therefore A and p must be linearly dependent over

4(X).

0

Remark: There are divisors D such that H'(X, 0(D)) * (0). In other words, J(DJ# (0). Thus J is a 1-dimensional 4(X)-vector space. S. The Residue CaIcuh . We write A for the sheaf D' of germs of holomorphic 1-forms on X. Note that, since dime X = 1, (Y = 0 for 1 > 1. The sheaf R is locally free of rank 1. Thus, given a local coordinate t c- nt., at x, every germ o) , E f is uniquely written as co., = It,, dt with h. E m.,. The residue, Res., w.,, of w., at x is invariantly defined as the coefficient of r' in the Laurent development of h, with respect to t. In other words Res., o.

=2I J h dt, OR

where H is a small disk about x, and h r= 0(R) is a representative of h.,. If co, e i2, then it is clear that Res., w, = 0.

223

The Duality Theorem of Serre

6.

Now let w E i2°°(X) be a global meromorphic differential form and F = (fx} E R. Then, for almost all x E X, jxwx e Q. Thus the sum

<w, F> = Y Resx(fxwx) E C xcX

is finite. We summarize some properties of this pairing in the following: Theorem 4. The map

:f°(X)xR-.L, defined by (co,

<w, F> is a C-bilinear form. Furthermore

for all h e !((X ),

0)

_ <w, hF>

and

if w e fl(D)(X) and F E R(-D) then = 0.

1)

Proof: The C-bilinearity of ( , >, as well as 0), is clear by definition. Let to E f2(D)(X) and F = (fx) e R(-D). Then °(.fxWx) _ (fx) + °(wx) ? °x(D)

-.(D) = 0.

That is, for all x c- X, f co c- f2x, and Res., (fxco) = 0. This proves 1).

The following theorem is essential for our further considerations.

Theorem S (Residue theorem). If to E i1 (X) and h E .11(X) then <w, h> = 0.

Proof: Since hw e Q'O(X), it is enough to give a proof for h == 1. It must be shown, therefore, that E Res,, wx = 0. Let x,, ..., x" E X be the poles of co and xX

let H,, ..., H. be pairwise disjoint "closed disks" about the xv's. Applying Stokes' theorem, we have Res,, co, XEX

"

1

= v°1 tai eH, I

CO

_

1

r

2ni

I

w=-

1

J

dw = 0,

X%VH,

since co is holomorphic outside of U H,., and thus dw vanishes identically on this set.

6. The Duality Theorem. Every differential form to E fl-(X) determines a C7linear form, CO*: R -,C,

Chapter VII. Compact Riemann Surfaces

224

defined by Fi-+- D. In particular i(0) = g: On a Riemunn surface X with genus there are exactly g linearly independent global holomorphic 1-forms:

, ,1

dime H'(X, f)) = g.

For every positive D, H°(X, Q(-D))

H°(X, 0). Thus using Theorem 3.1. we can estimate i(D) and 1(D) from above: If the divisor class of D contains a positive divisor then i(D) < g and 1(.D):5; deg D + 1.

CAs$St VIL Comps Riemann Swfaca

226

Since 12 = 0(K) (see Theorem 4.5), it follows that 0(- D) = 0(K - D) for all. D e Div X. Thus the above dimension equation can be written in the form

i(D) - I(K - D).

(1).

In the case of D = 0 this leads to I(K) = g and similarly, in the case of D - K, we have i(K) = I(0) = 1. In other words H'(X, £2) = C. Hence it turns out that

X(t2) = I(K) - i(K) = g - 1 = -X((}3

(2)

It follows easily now that g is only a topological invariant: Theorem 1. The genus g of a compact Riemann surface X is a topological invariant. In fact

dims H'(X, C) = 2g. Proof: We consider the exact sequence of sheaves 0

-C.

.0

Q-; 0.

Due to (2) above,

X(X,C)=X(9)-X(f2)=2-2g. Since H°(X, C) and H2(X, C) are both isomorphic to C,

2-2g=X(X,C)='1-dimeH'(X,C)+1.

0

We will make further remarks about the structure of H'(X, C) in Section 7. 2. The Formula of Riemann-Rock. It follows from Theorem 3.1 that, for all D E Div X, 1(D) - i(D) = deg D + 1 - g. If one writes l(K - D) instead of i(D) then one obtains the formula of Riemann-Roch. Theorem 2 (Riemann-Roch, final version). For every divisor D on a compact Riemann surface X of genus g,

I(D)-I(K-D)=degD+1-g. Since 1(K) = g and 1(0) - 1, we find by setting D = K that g - I = deg K + 1 - g. Thus we have the degree equation for diffirrential forms: For every differential form w e

degK=degto-2g--2:

§ 7.

The Riemann-Roch Theorem (Final Version)

227

This equation contains for example the fact C) = (0). One sees this by noting that the differential dz, where z is an inhomogeneous coordinate, has degree

- 2. Thus, since 2g - 2 = - 2, P, has genus 0. From the above it follows that every non-trivial ru E n '(X) has degree -X(X), where X(X)is the topological Euler-Poincarb characteristic of X. Consequently, if a: X -. X' is an s-sheeted ramified covering map between compact Riemann surfaces X and X', and W e Div X is the ramification divisor of a, then x(X) + deg W = s X(X')

Proof: Let w' E 0°(X'), and define co s

A direct calculation shows that deg(w) _

dcg(w') + deg W. Since deg(w) _ -X(X) and deg(w') = -X(X), the claim follows immediately.

0 In particular this shows that deg W is always even, and, when X' = P, with X(X') = 2, it follows that

deg W= 2(s+g-1)

3. Theorem B for Sheaves 0(D). The following important application of the Riemann-Roch formula uses the simple fact that, when deg D < 0, 1(D) vanishes. Theorem 3 (Theorem B). Let D e Div X wit/: deg D > 2g - 1. Then

a) H'(X, 0(D)) = (0) b) 1(D) = ddg D + 1 - g. Proof. a) Since deg K = 2g

- 2, deg(K - D) < 0 and therefore i(D) _

1(K - D) = 0. b) This follows immediately from a) and the Riemann-Roch formula.

0

Theorem 3 is the optimal form of Theorem B in the sense that if the degree of a divisor D is less than 2g - 1, then the cohomology group H'(X,, 0(D)) may not

vanish. For example, by Serre duality, H' (X, 0(K)) = H°(X, 0) = C and deg K = 2g - 2.

4. Theorem A for Sheaves ((D). Let 2 be a locally free sheaf of rank I over X. and let x e X. Then the following are equivalent: i) The module of sections 2(X) generates the stalk Y. as an Os-module. ii) There is a section s e 2(X) with o(s,) = 0. iii) There is a section s E 2(X) with Y, = 0, s,.

M,,,dimc H'(X, 2(-x)) - 2g. Then for every x < X there exists a section s e O(D)(X) with O(D). = Oxsx.

Proof: Let 2' := O(D). If H'(X, V(-x)) = 0, then the above equivalences guarantee the existence of such a section. Now 3'(-x) = O(D - x). But by assumption, deg(D - x) >- 2g - 1. Hence Theorem B implies that H'(X, O(D - x)) = 0. Theorem 4 is the optimal form a Theorem A in the sense that for deg D < 2g the sheaf O(D) may not be generated by its global sections. For example, let D = K + (p). Then O(D) = Sl(p). A section in O(D)(X) is just a differential form co which is holomorphic on X\p, and, if it would generate O(D)P, would have a pole

of order 1 at p. This would contradict the residue theorem. There are divisors with deg D < 2g, and for which Theorem A, however, is valid, namely K: If g * 0 then Theorem A holds for the sheaf D = O(K). In other words, given x e X there exists a holomorphic differential form co E f1(X) which does

not vanish at x.

Proof:

By

the

above equivalences,

we only need to show that

dime H'(X, fl(-x))< dimc H'(X, S2). Since dime H'(X, 0) = 1, it is enough then to prove that i(K - x):5 1. Suppose 1(x) = i(K - x) >- 2. Then there exists h e . K(X) n O(X\x) with a pole of order 1 at x. The map h: X -+ P1 would be

biholomorphic, and, since g * 0, we have reached a contradiction. Thus

i(K-x)- 2. Then every stalk of fl(D) is generated by a global meromorphic differential form co e fl(D)(X ).

Proof: Since Q(D) = O(K + D) and deg(K + D) z 2g, the result follows from Theorem 4. In particular if D- mp, m >- 2, then we have the following: Let p e X and m >- 2 be given. Then there exists a meromorphic differential form

229

The Riemann-Roch Theorem (Final Version)

§ 7.

on X which is holomorphic on X rp and has a pole of order m at p. Furthermore, for every p, and P2 E X With pt # P2, there exists a meromorphic form on X which is holomorphic on XO{p,, p2) and has poles of order I at p, and p2.

Proof: Define D:= p, + p2. Then there exists w c- Q(D)(X) which generates the stalk 0(D)p1. This form must be holomorphic on X1{p,, P2), have a pole of order I at p, and have a pole of at most order I at p2. But the residue theorem requires

that it has a pole of exactly order I at p2. 6. The Gap Theorem. A natural number w >- 1 is called a gap value at p e X if

there is no holomorphic function on X\p which has a pole of order w at p. If X = P, then there are no gap values. But if X P, then g # 0 and w = 1 is always a gap value. We write 1, := l(vp) for each v > 0 and note that 1,
_2g- 1. Then .P is a direct summand of F.

Proof: We first note that ..*'om(W, 2') = (D .)Eoonr(2 i=2

, 2'). Let 2 0(D) and

2, = 0(D,), 2 < i < r. Then .*'o n(ft,, .2) = 0(D - Di). Hence, since deg(D -- Di) = c(f) - c(f,) >_ 2g

-i

for i = 2, ..., r, it follows (by Theorem B) that, for all such i,

H'(X, .1t°om(9i, 2')) = (0). Thus

H'(X, 0om(1, 2')) t=2

H'(X, -*° n(9' , 9')) = (0)

and, by the splitting criterion, .F is a direct summand. In the above proof we used the following fact: For all divisors D, Y E Div X there is a natural 0-isomorphism

0(D' - D) 4 Jrom,(0(D), 0(D')).

0

$ B.

The Splitting of Locally Free Sheaves

237

Proof Let t e nix be a local coordinate at x. Then O(D),, = t- -(°' 0x and C(D')x = t- °xiD't . Ox. Thus every germ hx c- 0(D' - D)x = t" °-(D'-D) 0x determines, by multiplication, an 0X homomorphism (homothety), O(D)x -. O(Y)s,

ox(D' - D) defined by g, t- hx gx. (Observe that o(hx gx) = o(hx) + (gx) ox(D) = - ox(D')). Since 0(D), and 0(D')x are free 0z modules of rank 1, every homomorphism O(D), -p 0(9)x is such a "homothety". Thus the map 0(D' - D) -,. Yoos,(0(D), 0(D')), defined by associating to each germ the homothet i defined above, is surjective. The injectivity of this map is trivial and thus we 0 have established an isomorphism.

5. Grothendieck's Theorem. We fix a point p "at infinity" in P, and set

0(n).= 0(np) for every n e Z. Then n is the Chern number of 0(n) and 0(n) = 0(m) if and only if n = m. Furthermore, if 2' is a locally free sheaf of rank I over P, then 2' = 0(n), where n = c(2).

Proof: Certainly 2' = 0(D) for some D e Div Pt. On P,, however, two divisors are linearly equivalent if and only if they have the same degree. Hence 0(D) is isomorphic to 0(deg D) and, since deg D = c(2'), we have the desired result. O Every sheaf 0(n,) $ 0(n2) ® ... ® O(n,), n1, ..., n, e Z, is locally free of rank r. The splitting theorem of Grothendieck says that one obtains all locally free sheaves over P1 in this way: Theorem S. (Grothendieck, Let F be a locally free sheaf of rank r k 1 over P1: Then there exist integers nt, ..., n, (uniquely determined up to a permutation) such

that 1, U,'a{ze P, 11 i - 1.

The sections #' in JF(-mXPI) generate a non-zero 0-subsheaf .°l of F(-m). Thus

A- .-r(m) + 0 is an invariantly (by .F alone) determined O-subsheaf of .F. With this language we now prove the uniqueness part of Theorem 5:

Uniqueness Lemma 7. Let .4F _ 21 ® ®27, _ .,Pi ® ®2; be two split>_ n tings of iF with 2' = O(ni), 2i - O(ni), 1 < i 5 r. Assume that n1 >- n2 z and ni z n2 z > n;. Then, with d == dime A(-mXP') z 1,

1)Y1®...®Yd =ri®...(D Yd _

2,=O(m)=Y,,for i=1,..., d,

and

2) n,= =nd=nt= Proof.1

Since

=nn=m;n,=nifori=d+1,...,r.

9F = 91® - ® .P

.fit(-m)(P1), where

that .F(-m)(P1) = 0(11) with 1,- nt - m :!g 0 (by the definition it

follows

of m). Now 9(1XP1) = 0 for

1(nd+1)®...ED 0(n:)

Since F/,r has rank r - d < r, and since n,, n; are monotonically decreasing, it follows by induction that n, = n; for i = d + 1, ..., r. Corollary. Let .F = 5r $ .X°, where 4, V are non-zero locally free sheaves over P1. Then µ(-,F) = max{µ(W), µ{l(`')}. P r o o f : Since .F = (9(n1) m

®0(n,), it follows that µ(F) = max{nl, ..

,

n,}.

We see now that every locally free subsheaf 2' of rank 1 in f7 with c(2') is necessarily maximal, because the Splitting Lemma implies that F = 9 ® 2', and the above corollary shows that u(-,',) = max(µ(l), µ(.4°)) = c(2). The following remark is also quite easy to see.

Let .F = I ®r, where 4, 0 are non-zero locally free sheaves with µ(.F) > µ('#). Let 2 be a locally free subsheaf of rank 1 in F so that c(.') > µ(4). Then 2' is a locally free subsheaf of jr.

Proof: Let 2' = Os with s e

and deg(s) = c(2'). Let a:.F

I

denote the natural sheaf projection. Then by the remark in Paragraph 1, either n(s) = 0 or deg(s) < deg(a(s)). The latter is not possible, because deg(a(s)) n>max{n2,.... n,}. However, for every n S max{n2, ..., n,}, there are in the above setting locally free subsheaves 2 = 0(n) in .F. For this, see Prop. 2.4 in "On holomorphic fields of complex line elements with isolated singularities", Ann. Inst. Fourier 14,99-130 (1964), by A van de Van.

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Springer-Verlag, 1976. Cartan, H.: Collected Works (Oeuvres Scientifique), Springer-Verlag, 1979. Cartan, H.: Differential Forms, Boston: H. Mifflin Co., 1970.

DF]

Articles [1] Andreotti, A., Frankel, T.: The Lefschetz theorem on hyperplane sections. Ann. Math. 69, 713-717 (1959).

2 Le Barz, P.: A propos des revitements ramifies d'espaces de Ste' Math. Ann. 222,63-69 (1976). [3 Behnke, H Stein, K.: Analytische'Funktionen mehrerer Veranderlichen zu vorsegebenen Null_ and Polatellenffachen. Jahr. DMV 47, 177-192 (1937),

[4] Behnke, H., Stein, K.: Konvergente Folgen van Regglarititsbereichen and die Merornorphiekonvexitat. Math. Ann. 116, 204-216 (1938}. [5] Behnke, H., Stein, K.: Entwicklung analytischcr Funktionen auf Riemannschen Fiachen. Math. Ann. 120, 430-461 (1948) [6] Behnke, H., Stein, K.: Elementarfunktionen wA Riemannschen Ffichen. Canad. Journ. Math. 2, 152-165 (1950).

[7] Cartan, H.: Les problemes de Poincare et do, Cousin pour Its. Pondiaoa de plusieurs variables complexes. C. R. Aced. Sci. Paris 191, 12841287 (1934]

Bibliography

241

[8] Cartan, H.: Sur k premier probleme de Cousin. C. R. Acad. Sci. Paris 207, 558-560 (1937). [9] Cartan, H.: Varibtbs analytiques complexes et cohomologie. Coll. Plus. Var., Bruxelles 1953, 42-55.

[10] Cartan, H., Serre, J.-P.: Un thborime de finitude concernant les variitbs analytiques compactes. C. R. Acad. Sci. Paris 237, 128-130 (1953). [11] Fornaess, J. E.: An increasing sequence of Stein manifolds whose limit is not Stein. Math..Ann. 223, 275-277 (1976).

[12] Fornaess, J. E.: 2 dimensional counterexamples to generalizations of the Levi problem. Math. Ann. 230, 169-173 (1977), (13) Fornaess, J. E., Stout, E. L.: Polydises in Complex Manifolds. Math. Ann. 227, 145-153 (1977} 14 Forster, 0.: Zur Theorie der Steinschen Algebren and Moduln. Math. Z. 97, 376-405 (1967). Forster, 0.: Topologische Methoden in der Theorie Steinscher Raume. Act. Congr. Int. Math. 1 IS 1970, Bd. 2, 613-618.

[16] Grauert, H.: Charakterisierung der holomorph-vollstiindigen Riiume. Math. Ann. 129, 233-259 (1955).

[17] Grauert, H.: On Levi's problem and the imbedding of real-analytic manifolds. Ann. Math. 68, 460-472(1958} 18 Grauert. H.: Bemerkenswerte pseudokonvexe Mannigfaltigkeiten. Math. Z. 81. 377-391 (1963). 19] Grauert, H., Remmert, R.: Konvexitat in der komplexen Analysis: Nicht holomorph-konvexe Holomorphiegebiete and Anwendungen auf die Abbildungstheorie. Comm. Math. Helv. 31, 152-183 (1956).

[20] Grauert, H., Remmert, R.: Singularititen komplexer Manngifaltigkeiten and Riemannsche Gebiete. Math. Z. 67, 103-128 (1957) [21] Grauert, H., Remmert, R.: Zur Spaltung lokal-freier Garben uber Riemannschen Flichen. Math. Z. 144, 35-43 (1975} [22) Grothendieck, A.: Sur la classification des fibrbs holomorphes sur ]a sphere de Riemann. Amer. Journ. Math. 79, 121-138 (1957} [23] Igusa, J.: On a Property of the Domain of Regularity. Mem. Coll. Sci., Univ. Kyoto, Ser. A, 27, 95-97 (1952).

[24] Jurchescu, M.: On a theorem of Stoilow. Math. Ann. 138, 332-334 (1959). [25] Markoe, A.: Runge Families and Inductive Limits of Stein Spaces. Ann. Inst. Fourier 27, fasc. 3 (1977).

[26] Matsushima, Y.: Espaces Homogines de Stein des Groupes do Lie Complexes. Nagoya Math. Journ. 16, 205-218 (1960).

[27] Matsushima, Y., Morimoto, A.: Sur Certains Espaces Fibres Holomorphes sur une Varied de Stein. Bull. Soc. Math. France g8, 137-155 (1960). 28] Milnor, J.: Morse Theory. Ann. Math. Studies 51, Princeton Univ. Press 1963. 291 Narasitnhan, R.: On the Homology Groups of Steio Spaces. Inv. Math. 2, 377-385 (1967). 30] Oka, K.: Sur les fonctions analytiques de plusieurs variables It. Domaines d'holomorphie. Journ. Sci. Hiroshima Univ., Ser. A, 7,115- IV (1937). [31] Oka, K.: Sur les fonctions analytiques de plusieurs variables III. Deuxiime problime de Cousin. Journ. Sci. Hiroshima Univ., See. A, 9, 7-19 (1939). [321 Oka, K.: Sur lea fonctions analytiquee de plusieurs variables IX. Domains fini sans point critique intirieur. Jap. Journ. Math. 23, 97-155 (1953} [33] Schaja, G.: Riemannsche Hebbarkeitssatze fiir Cohomologieklassen. Math. Ann. 144, 345-360 (1961}

[34] Schuster, H. W.: Infnitesimale Erweiwungen komplexer Raume. Comm. Math. Helv. 45, 265-286 (1970). (35] Serre, J.-P.: Quelques problimes globaux relatifs aux varietis de Stein. Coll. Plus. Var., Bruxelles 1953, 57-68.

[36] Stein, K.: Topologische Bedingungen ftir die Existenz analytischer Funktionen komplexer Verinderlichen zu vorgegebenen Nullstellenfiachen. Math. Ann. 117, 727-757 (1941} [37] Stein, K.: Analytische Funktionen mebrerer komplexer Verinderlichen zu vorgegebenen Periodizitatsmoduln and dps zweite Cousinsche Problem. Math. Ann. 123, 201-222 (1951). [38] Stein, K.: Uberlagerungen holomorph-vollstindiger komplexer Raume. Arch. Math. 7, 354-361 (1936).

Subject Index

acyclic cover

- topology

42

168

algebraic dimension 20

Cartan, Theorem of 179 Cartan's Attaching Lemma 88 Cauchy Integral Formula 74 Cauchy-Riemann differential equations 65 tech cohomology module 35, 37

- reduction

- - -, alternating

32

acyclic resolution

additive Cousin problem additive functor 25

137

176

algebra, Stein

8

- complex

15

algebraized space

alternating tech cohomology module 35, 37

- -, alternating

- - complex

character

- cocycle

35

- ideal

35

analytic blocks

- hypersurface

- class 20

- function 216 closed complex subspace Closedness Theorem 169 coboundary 29

cochain 28, 34

-, alternating 88 84

35

cocycle 29 codimension, complex

19

- - - finite holomorphic maps

92

- - of Oka

block neighborhood

165

16

coherent sheaf 11, 96 cohomology classes 29

- -, Dolbeault

116 91

79

- modules, alternating tech 35, 37

- -, tech

35, 37

- -, (flabby) of an M-sheaf 30

- - of a complex

29

- sequence, long exact completeness relations

214

complex

214

- flabby resolution 30 2

- resolution relative to a cover

52

- groups, deRham 63

167

-, Approximation Theorem for -, exhaustion by 117 branches of an analytic set 19

- presheaf

95

Coherence Theorem for ideal sheaves

Banach's Open Mapping Theorem Bergmann Inequality 189

- - class

17

- map 28

Attaching Lemma for sheaf epimorphisms

canonical divisor

209

chart 194 Chern class 144

90, 122, 170

blocks, analytic

144

- Theorem

annihilator sheaves 13 antiholomorphic p-forms 70 Approximation Theorem of Runge

- sections

162

characteristic 209

129

- interior 111 - set 18 - -, irreducible 19 - -, pure dimensional - sheaves 16, 92 - spectrum 180 - stones 111

- - of Cartan - - - Cousin

35

162, 176 163, 176

- Theorem

116

35, 37

35

42

28

- codimension 19 - dimension 19, 20 - manifold 16

29, 30. 34, 38 188

18

Subject Index

244

dimension, topological 19 direct product of sheaves 4

complex (cont.)

- space

16

- -, holomorphically complete

- (Whitney) sum of sheaves

118

- -, - convex 109 - -, - separable 117 - - - spreadable 117 - -, irreducible

21

- -, normal

-- -, Stein 101

21 reduced 21

- r-vector

- group 113

- value of a holomorphic function - valued differential form 60

17

complexes, homomorphisms of 28 conjugation 62

connecting homomorphism 29 constant sheaf 8 Continuation Theorem of Riemann continuity of roots 48 Convergence Theorem

covering

cycle

121, 172

- - - sheaves

-, Stein

138

102 105

- Theorem 108, 127 exponential homomorphism

- sequence

143

142

exterior derivative 62

- product of sheaves

77

206 section 206

19

finite mapping

- sheaves

13

45

10

Finiteness Lemma for character ideals

- Theorem

-, Formal Lemma 32 -, Theorem of 64

- resolution

30

- sheaf 25

derivation 58 derivative, exterior 62

Formal deRham Lemma 32 Frbchet spaces 163 free sheaves 10 function, holomorphic

62

determinant sheaf 216 8-exact

77 diameter 37

-, meromorphic

17

21

Fundamental Theorem of Stein Theory

differentiable manifolds

16

16

- vector field

Gap Theorem of Weierstrass 229

58

differential forms, complex valued dimension, algebraic 20

-, complex 19, 20 - of a block 96

177

186, 202

Five Lemma 11 flabby functor 25

deRham cohomology groups 63

- maps

8

- Theorem 26, 47 exhaustion

- of a divisor

-, total

152

Exactness Lemma 26

137

Decomposition Lemma for analytic sets degree equation 210

---

151

16

equivalent criteria for a Stein space Euler-Poincari characteristic 210 exact sequence of presheaves 9

21

150

3-closed

138

206

-, holomorphic - Theorem 126

18 42 127

-, acyclic -, Stein

- - section

embedding dimension

Cousin Attaching Lemma 84

- I problem (additive) 136 - Il distribution 139 - II problem (multiplicative)

205

- of a meromorphic function

-, positive 139 Dolbeault cohomology groups 79 domain of holomorphy 134 Duality Theorem of Serre 219, 225

17

59

-I distribution

49

- classes 146

- -, weakly holomorphically convex - subspace, closed 16

- -, open

2

distinguished block 173 Division Theorem, Weierstrass divisor 138 -, canonical 214 - class, canonical 214

60

- value genus

229

211

glued family of sheaves 5 gluing sheaves by cocycles

147

124

Subject Index

245

good semi-norms 118 graph of a holomorphic map Grothendieck, lemma of 75 -, Splitting Theorem of 237 groups, Stein

-, toroid

manifold, complex

136

-, finite 45 -, holomorphic

136

-, proper

- map

- section

17

86 66

holomorphically complete

- convex

- - hull

- set

118

109 108

- separable

14

nilradical 8 normal complex space normalization 22

117

- spreadable

206

minimum principle 190 monotone 192 multiplicative Cousin problem 138

16

- matrix - p-form

16

112

matrix, bounded holomorphic 85 maximal subsheaf 233 meromorphic function 21

Hartogs' Theorem 81 Hodge decomposition 231 holomorphic embedding 16

- function

16

-, differentiable 16 mapping, closed 45 -, differentiable 16

17

117

hull, holomorphically convex hypersurface, analytic 129

- Theorem

113

21

22

nullstellen ideal

18

Identity Theorem for analytic sets 19 image of a presheaf homomorphism 9

Oka Principle

- - - sheaf homomorphism

Oka's Coherence Theorem 16 open complex subspace 17 Open Mapping Theorem of Banach 165 ordering 192, 205 orthogonality relations 188

- sheaves

7

4

inclusion of stones 112 interior, analytic 111 intersection of submodules 7 irreducible analytic sets 19

- complex spaces 21 - component of a divisor

140

Isomorphism Theorem for sheaf cohomology 43

k-algebraized space

15

kernel of a presheaf homomorphism

- - - sheaf homomorphism

9

-, Theorem

145 145

paracompact space 18 Pfaffian form 61 p-forms, antiholomorphic 70

-, holomorphic 66 Poincarb, Lemma of 64 - problem 139 -, Theorem of 140 point character 176 (p, q)-form, differentiable positive divisor 139 presheaf, canonical 2

7

k-homomorphism 8 k-morphism 15 Kronecker symbol 59

- of abelian groups - homomorphism

66

9

9

-, mapping of 2

Lebesque number 37 Lemma of Grothendieck 75

- - modules

9

64 Leray Theorem 43

- - rings

lifting of differential forms 62

-ideal sheaf

- - r-vectors

- part distribution

- - Poincarb

9

principal divisor

60

138 129 137

- - - w.r.t. a divisor

linear equivalence 149 locally free sheaf 10

product of complex spaces

- - subsheaf 213

- - ideal sheaves

long exact cohomology sequence

29, 30, 34, 38

- sheaf

146

7

219 17

Subject Index

246

projection of a sheaf I Projection Theorem (local) 52 proper mapping 112 pure dimensional 20

section functor 2 sections

2

-, attaching 92 -, meromorphic

206

semi-norm, good 118

Separation Theorem 24

quotient presheaves 9

- sheaves

sequence topology 165 Serre Duality Theorem 219, 225 sheaf, analytic 16, 92

7

quotients, ring of

-, sheaf of

14

14

radical ideal 19 reduced complex space 21 reducible analytic sets 19 reduction, algebraic 8

- map 20 - of a complex space 20

- Theorem

154

198

161

- -, relative to a cover -, flabby 30, 33

21

204 61 137

42

divisor 219

- - - - real-valued differentiable functions 16

- - - - 9t-homomorphisms 7

-, locally free

- - ideals

10 1

7

- - k-algebras

211, 226 sheaves 216

R-resheaf 9 9-resolution 26

8

- - local k-algebras

- - modules

--rings 6

9f-module 6

13

-, gluing 5 - homomorphism 6

- mapping 134

- - - -, for locally free

8

6

-,reduced -, Stein -, soft

7

7

7

Runge, Approximation Theorem of 90, 122, 170 r-vector, complex

- - - - meromorphic functions

- - - - - part distributions w.r.t. a

Riemann-Roch, Theorem of

91-submodule

16

- - - - Pfafian forms - - - - principal parts

Riemann Continuation Theorem 21

-, exact

138

- - - - holomorphic functions

- - - - - sections

194

8

66

58 61

-----p-forms 66

-, canonical flabby 30

91-sequence

61

- - - - differentiable (p, q)-forms

- - - - divisors

11

2

- domain 134 - -, unramified - inequality 212

70

16

------- r-forms

resolution, acyclic 32

soft 33 r-form 61 9i'-homomorphism

-, free 10 - of germs of antiholomorphic p-forms

- - - - - vector fields 11

- Theorem 223 restriction map 2

- atlas

11

25

- - - - --- continuous functions

222

-of a sheaf

-,flabby

functions

regular point 21 relation sheaves, finite

residue

10

-, finite relation

------- r-forms

- - - resolution atlas

representation of 1

-, finite

- - - - complex-valued differentiable

refinement map 18 - of a covering 18

relations, sheaf of

- of abelian groups -, coherent 11, 96

59

saturated sets 190 Schwartz Lemma 190, 192

8 229 22

Shrinking Theorem singular points 21

18

Smoothing Lemma 200 soft sheaf 22 spectrum, analytic 180 splitting criterion 235 Splitting Theorem of Grothendieck 237 square integrable 187

Subject Index stalk

247 - - - sheaves 0(D)

1

227

Stein algebra 176

- - (Fundamental Theorem)

- covering 127 - exhaustion 105

Theorem of Cartan

- group

124

179

- - deRham 64

- - Oka

136

145

- set 100

- - Poincar6

- sheaf 229 - space 100 stone 111

- - Riemann-Roch 211, 226 - - -- - for locally free sheaves

structure sheaf

16

Structure Theorem for locally free sheaves of rank 1 214 subdegree

192

submodule

7

140

216

Three Lemma I I topological dimension 19 topology, canonical 168

-,weak

180

toroid groups 136 total derivative 62

subpresheaf 9 subsheaf 9

12

trivial extension of a sheaf

Tube Theorem

157

-, locally free 213 -, maximal 233

- Theorem 214

unramified Riemann domain

subspace, closed complex

-, open complex sum of submodules sums of sheaves I support of sheaves

support of a divisor - - - sheaf

134

16

17 7

value, complex, of a holomorphic function

- of a section

8

Vanishing Theorem 210

1

- - for compact blocks

6

37

vector field 58

6

- -, differentiable

58

tangent space 57 - vector

weak topology

57

180

tensor product of sheaves 13 Theorem A for compact blocks 96

weakly holomorphically convex 113 Weierstrass Division Theorem (general) 49

- - - locally free sheaves

- Gap Theorem 229 - homomorphism 51

229

- - - sheaves 0(D) 227 - - - Stein sets

101

- - (Fundamental Theorem)

124

Theorem B for compact blocks 97

zero section

- - - locally free sheaves

- set of an ideal

229

6 18

17

Table of Symbols

Sox, Wx

)r n (5°, .9') .dn Y 13

1

5°1 ®$"2

1

r(Y, 50), ."(Y)

2

Ox

2

r(Ss)

57

2(R, M) 58

d61, gX, dc, d'X

0

2

51

T(x)

7 15

2

ry

A

A, 14

5°y, 5°1 Y 2 s,,

1 46, 48

13

.I

15

59

A'(x), A(x) 59

16

4A' 59

16

f»,f*

60

r(S)

2 3

Hol(X, Y) X1 x X2

r(D)

3

Gph f 17

.sag

qi 62 d 62 sIv.a 66 S2° 66 a, 0 67, 68

r((p)

17 17

rad f 19

[I Sot 4

f*(b)

5

Jx 5 .f*((p)

dim top., A 19 dim,, A 19 codim., A 19

5

dim A 20 red X 20

tGI

supp 5° 6

5°'n.9" Y'+ Y"

J -Y 7

7 7

.,N

A''e2q 7 5°/5°' 7

Jn 4

7

m(Ax)

8

(e

8

s/s'

supp qp

9

Ker 0 9 IM 0 9 Rel(s1, ..., s,,) 'a,Ee4 q 12 .9' Or Y' 12

Z-(K') 28

d(Q)

Bq(K') 28 Hq(K') 29

Al, 14, 108 P° 111

®°Y 13

n°So

13

.0

30 30

37

. 40

(h)

138

D+, D-

140

expf 143 c(D) 36

R (X, S), $;(X, So) 37 d(M) 37

137

2 138 34

36

$q(X, S), R (X, .So) iq

96

0* 138 .,#* 138

30

iq(U) 34 hq(A3, U), hq(U) 11

81

B(V), B*(V) 86

Cq(U, S) 34 Hq(U, S), Hq(U, .9') C;(°1l, S) 34 H;(U, S) 34

red R 8

69

Tf 72

21

Hq(X,.9')

8

61 61

.F(.9') 25

37q(.9')

s(x) 8 n(om)

CI°

red 20

6

9P°

.sat'

144

(,* 144 DC(X) 146 0(D) 146

G(Ji)

146 146 LF(.,K) 147

g.g'

Table of Symbols

LF(X) 147 supp o 150

C9(21), Cg(21) 196 Cg(U,.9) 196

,(o)

150 176 .T(T) 180 t,, 6, 182, 183 Xp

gl

183

1If IIB Oh(B)

(f, g)B Oh (B) op wp(f)

(f)

IICIIa

197

Z9(21), Zg(21), Zg(U, .t)

210 1(D), i(D) 211 g 211 X(.9°)

197 c(Fl 215

21' < 91 198 J 204

det .F 216

F (X)*, .F (X )* 204

R, R(D) 220

.le(D)

219

187

Div X 205

1(D) . 220

187

deg D 205

J(D), J 221 Res., wx 222

188

IDI

,"(D)

192

193

205

206 206 . °t(D) (s)

189 191

F(a), F(a)*

Hj, Hf

249

193

207

Xo(') 209

<w, F>

223

K 225 µ(r) 233 O(n)

237

Acknowledgement The authors are grateful to D. N. Akhiezer who, while translating "Theory of Stein Spaces" into Russian 1989, made the correction to our proof of Theorem B in this Addendum.

Addendum by D. N. Akhazierl

Our goal here is to make more transparent the exposition in §4 of Chapter IV. We retain the notation and conventions introduced in the preamble to the section. In that preamble, as well as in most other places where no changes are required, we use the pieces of the main text. Some formulations of important theorems are changed, but the numbering is the same as in the main text.

1. Good Semi-norms. Topology in ,9' (P°). We want to enlarge Section 1 of §4. In particular, we assume that Theorem 1 is proven and the definition of a good semi-norm is given. This done, we fix an epimorphism of sheaves c : &'I Q -; rr#(,9' I P) and use good semi-norms to construct a metric on "(P°). We take an exhaustion of Q 00

C...C Q,

Q1

1

where all Q, are (compact) euclidean blocks in cm with the same center. The map c induces the &(Q,,)-epimorphisms

n.('99 IP) (Q0,

EQn :

which define good semi-norms I .I v in .9' (P°). We put 00

d (Sl, S2)

E2 v=1

Ist - S21 v

1 + Isl - 521v

I Prof. Dmitri Akhiezer. Institute for Information Transmission Problems, B. Karetny 19, 101447 Moscow, Russia

Addendum

252

where Si, s2 E S"(P°) (see also Chapter V.6.0), and make the following observation which one should compare with the second lemma in Chapter V.6.3. d is a metric on .9' (P°). Considered with this metric, Y (P°) is a Frechet space. Proof.. The first assertion follows from Theorem 1. To prove the second one, we s E 5° (P°). Then we can find bounded sequences take any Cauchy sequence

fl P. By Montel's theorem, ff,x E O'(Q,,), so that x) = to some function there is a subsequence ff,).), which converges uniformly on

fE

One has

EQ (fv+.)IQ- = Edy-1U014-1 = where s E 9(P°). It follows that

sln-1(Q,-.)

n P,

converges to s in the sense of metric d.

0

The topology on So (P°), induced by d, does not depend on the exhaustion used in the construction. As we will show, this topology does not depend on the epimorphism E as well. Furthermore, we will see that the topology does not change if (P, n) is replaced by another analytic block (P,'n), where're = (n, (p).

Let 'r = (jr, cp) be a holomorphic map from X to C1" = C' x C", n > 0, and let Q* C C" be a euclidean block such that V(P) C Q*. Then P ='rr-1(Q x Q*) fl U. One can find open neighborhoods'U of P in X and'V of Q x Q* in C s" for which the map'nl,u : 'U -->'V is finite. We fix an epimorphism of sheaves ' : O''IQ x Q* ''r* (5 I P), construct an exhaustion { Q,, x Q*,) of the above type for the open set Qx Q*

and denote by 'd the associated metric on Y (P°). Theorem 1'. The topologies on SP(P°) induced by d and 'd coincide. Proof: Let el, ..., ei E t' I (Q) be the standard basis sections. We denote by'e1, ...,'el the preimages of EQ(el), ..., EQ(el) E n*(So I P) (Q) ='n*(. ° I P) (Q x Q*) in O"(Q x Q*) under 'EQ. Further, for f E we denote by 'f E O(Q x Q*,,) the holomorphic extension off to Q, x Qv, constant along each fiber {q} x Q*, q E Q. The norms of C-linear operators

x Qv),

fx ex -

'fx 'ex (v = 1, 2, ..., )

are bounded by a constant which does not depend on v. Therefore the identity map

id: (SP(P°),d) - (5(P°),'d) is continuous. By the Open Mapping Theorem of Banach, this map is a homeomorphism.

0 2. The Compatibility Theorem. Suppose that along with (P, n) we have another analytic block ('P, 'it) in X which is defined by a holomorphic map 'n : X - C'" and the associated euclidean block 'Q C CS"

.

Addendum

253

Theorem 2 (Compatibility Theorem). If (P, n) C ('P,',r) is an inclusion of analytic blocks in X then the restriction map g :.P ('P) -> 9SP (P) is bounded with respect to good semi-norms.

For the proof we need the following lemma.

Lemma. Let (P, n) c ('P, 'n) be an inclusion of analytic blocks in X. Then there exists an analytic block (Pt, n) such that

(P, n) C (Pi, n) C ('P, 'n) Proof: By Definition 8 in IV.2.4, we have C'" = C" x C' and accordingly = 'n (n, rp).

Furthermore, there is a point q E C" such that Q x (q) C 'Q We put

Q' :='Q n (C"' x {q}) C C'° and denote by Q* the image of 'Q under the projection map C " -* C". We can choose the open neighborhoods U C 'P ° of P and V C Q' of Q in such a way that the map nI U : U -* V is finite and u n n-t(Q) = P. We

can also find an euclidean block Q, C C' satisfying Q C Qi C Q, C Q'. Now let Pt := n-1(Q1) n U. Then (Pt, n) is the analytic block which we need. Proof of Theorem 2: According to the lemma one can decompose the restriction map Q : S° (P) -> 5° (P) as follows:

So('P)-->Y(P°)-,9'(P), sHSIP°HSIP.

(*)

The maps Y' ('P) - So (P°) and y (P°) -> So (P) are continuous, if $" (P°) is equipped with the topology defined by '7r and, respectively, by n. But these topologies coincide by Theorem 1'. It follows that o is also continuous and therefore bounded.

3. The Convergence Theorem. The considerations of the preceding section lead to the following result. Theorem 3 (Convergence Theorem). Let (P, jr) C (P, '7r) be an inclusion of analytic blocks in X. For every Cauchy sequence {si } in . ° ('P) the restricted sequence {si I P} has a uniquely determined limit in So (P).

Proof. We choose an analytic block (Pt , ,r) as in the above lemma and consider again the decomposition (*). Since the first map in (*) is continuous, {sj I P?) is a Cauchy sequence in .9' (P°). This sequence converges and has a uniquely determined limit in So (P?). But the second map in (*) is also continuous. Therefore, the sequence obtained by restriction to P is also convergent and has a uniquely determined limit in

Y(P). 4. The Approximation Theorem. We start as in the main text and proceed without changes until the decomposition

Pt=PUP, PnP=e

254

Addendum

is proved. This decomposition has a simple (but important) consequence that whenever

(P, n) C (P, '7r) the restriction map a :. ° (P1) -+ 5 (P) is surjective. The rest of the section is as follows. We extend the euclidean blocks Q, Q1 and'Q to Q, Q1 and, respectively, 'Q. Instead of P, P1 and 'P we get P, P1 and, respectively, 'P. We carry out these modifications in such a way that

P nP=e. By Theorem 1', the spaces .° (P1°) and So (P°) carry Frechet topologies. Furthermore, the restriction map .9' (Pl°) --)- .9' (P°) is a continuous epimorphism. We are now in a position to prove the approximation theorem for coherent sheaves on analytic blocks.

Theorem 4 (Runge Approximation Theorem). If (P, n) and ('P,tr) are analytic blocks in X with (P, n) C ('P, fir), then for avery coherent sheaf .i° on X the space ,° ('P) I P is dense in .9' (P). Proof: Lets E So (P) be a given section. Since or : S° (P1) --)- .So (P) is surjective,

there is a section s1 E .Y'(P1) with o(sl) = si1P = s. Appropriately modifying the blocks as above, we can extend s1 to some 91 E .° (P1). Then, as we know, there exists a sequence {s1"t} in Y(P) such that s(")IPi -> s"1 in .©(P1). Since the restriction map S° (P1°) -> S° (P °) is continuous in Frechet topology, one has s(") I P °

-).

9i 1 P ° in .° (P °). By the definition of a good semi-norm, it follows that

st">IP-s11P=siIP=S.

5. Exhaustions by Analytic Blocks are Stein Exhaustions. The last section is not changed. Here are the main results again. Theorem 5. Every exhaustion (P,,, 7r,)),2:1 of a complex space X by analytic blocks is a Stein exhaustion of X.

Fundamental Theorem. Every holomorphically complete space (X, O) is a Stein space. For every coherent analytic sheaf S° on X the holomorphic completeness of X implies the following:

A) The module of sections 5°(X) generates every stalk K, X E X, as an module.

B) For all q > 1, H9 (X, 91) = {0}.

Errors and Misprints

1) p. 3, line 1: The Functor t 2) p. 3, line 3: x E V (instead of x E X) 3) p. 3, lines 12, 6, and I from below: in each mapping, the second S should be So 4) p. 4, lines 21-22: in the sentence on these lines S and Y should be interchanged 5) p. 14, line 1: instead of "open in 9"' it should be "open in J9 6) p. 38, line 6: the correct formula is this one

B(11,...,1m):={xEB1 lx14 -xµ(lt,...,lm)I