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.
llbwrie dv Staavhm R9urm EoaWh)
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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. . § 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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16 18
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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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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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48 48
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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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§ 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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. . . . . . . . . . . . . . . . . . . § 3. Theorems A and B . . 95 . 1. Coherent Analytic Sheaves on Compact Blocks . . . . . . . . . . . . . 96 2. The Formulations of Theorems A and B and the Reduction of Theorem B to Theorem A 96 . . . 3. The Proof of Theorem A for Compact Blocks . . . . . . . . . . . 98 .
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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
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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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§ 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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1. The Number µ(.r) .
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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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. 240
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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<:{ rs 5",, = 0), and s lo
Chapter A. Sheaf Theory
14
U sin .fix. Then sin S° is open in .? and is consequently a sheaf of R-ideals. xeX
One calls sVw 5° the annihilator of Y.
If . is coherent then the annihilator of every coherent 9t-sheaf is a coherent sheaf of ideals. 7. Sheaves of Quotients. A set .,K c R is called multiplicative if ..K is an open subset of 9t and -Kx is multiplicative in the ring 9tx for all x c- X).' If ..K is such a
set, then, for every non-empty open set U, .#(U):= {r e R(U) I rx a ..Kx for all x e U} is multiplicative in the ring R(U). The ring of quotients is an R(U)-module and, since one has the natural restrictions rv: yt(U)_ (v) --, A(V),,,,(v), the system {9P(U).,,,(u), rvu) is a presheaf of rings. The associated sheaf of rings is denoted by 9i?,,,,, and is called the sheaf of quotients of I with respect to .,K.
It is clear that R.,, is also an A-module. Since .4' is open in 9t, every stalk x e X, is canonically isomorphic to the ring In particular, if 9 is a sheaf of local k-algebras, then is likewise a sheaf of local k-algebras. One frequently identifies 9,,,,, with U (9Px) ,,,,, defining a xeX
basis for the topology as follows; For every open set U c X and every pair f e A(U), g e ..#(U), let [f, g, U] be the set of all germs fx/gx e x e U. The collection of all such [f g, U] is a basis of open sets. For the theory of meromorphic functions (see Chapter V.3.1), the following set, no element of which is a zero-divisor, is an important multiplicative set:
U Xx, A^,, -the elements of 9tx which are not zero-divisors.
xeX
Every X. is multiplicative in.*,. Furthermore, if.* Is coherent, then .K is open and consequently is multiplicative in R.
Proof: Let n e R(U) be such that it., a .A for some x e U. Define the v by rr n, rr, for y e U. Then (X e2 p)x = 0. Since .Yet p is coherent, there exists a neighborhood V of x in U with
-Rv-homomorphism p: atv -- R'
JYea py - 0. This means that n, E X. for all y e V (i.e. ny e _K).
§ 3.
p
Complex Spaces
k ihis section, X, Y, Z always denote Hausdorff spaces equipped with sheaves When there is no confusion we drop the indices which indicate the space. We reserve k for a field.
Ax, gr,
' A subset T of a commutative ring B with.unit 1 is called muttiplicative if 1 e T and, whenever a, b e il', iii e T. Every muttiplicative set T c B determines the ring of quotients BT whose elements are the equivalence classes of the following equivalence relation on B x T: Two pairs (b t,) e B x T we called equivalent if there exists a t e T so that t(t2 b, - tl b2) = p. One writes each equiv tce dais as a "friction" bit sad carries out the- ai*tbm is operatioetin'the1ttlaay Way.
§ 3.
Complex Spaces
15
1. k-algebraized Spaces. A space X together with a sheaf -4 of local k-algebras is called a k-algebraized space. Thus, for example, the pair (X, ie), where (e is the sheaf of germs of complex valued continuous functions on X, is a C-algebraized space.
If (X, 9l) is a k-algebraized space and f: X -, Y is a continuous map, then the image sheaf f,(9) is a sheaf of k-algebras. We emphasize that this is not in general
a sheaf of local k-algebras. For every x E X we have (see 0.8) a canonical map f.: J;(c)f(X) -+ 91X. It is always a k-algebra homomorphism. A k-morphism (fj): (X, .fix) --+ (X, Sly) of k-algebraized spaces is a continuous
map f X -* Y together with a k-algebra homomorphism f: Ay f (9Px) so that every map which arises from the composition
where x is arbitrary in X and y := f (x), is local. In other words, m(. y) is mapped into m(9i'x,.J. The at ove formulation of a k-morphism is due to Grothendieck. An equivalent definition which uses neither image nor preimage sheaves is possible. However it is less elegant.
Example: Every continuous map f X - Y between topological spaces determines a C-morphism (X, 'x)4 (Y, WY). The map}: 'W,, ff((ex) is obtained by lifting continuous functions:./,: c' (V)-+f4(`')(V) = 'Q f -'(V)) by g g o f If (f,3'): (X, 9lx) -- (Y, My) is a k-morphism and Sx is an Sex-module, then
f.(,9x) is an ff(9lx}sheaf and consequently, via 1, an Sly-module. Every Sex-homomorphism ip:.x = Yx uniquely determines an Sly-homomorphism
f.(rp}: Thus f. is a covariant functor from the category of 9lx-modules into the category of Bly-modules.
The identity k-morphism (X, Mx)--' (X, 9lx) is given by the identity maps id: Sex -+ Sex. If (f ): (X, 9i'x) (Y, Sly) and (g, 9): (Y, Mr) - (Z, Az) are k-morphisms, then (h, h-): (X, Mx) -+ (Z, 9!'z), where h'= g o f and h'= g.(]') o g: Rz is also a k-morphism. Thus
id: X - X and
t4e k-algebraized spaces with k-morphisms form a category.
A k-morphism (f,7 (X,-9lx) -+ (Y, Sly) is an isomorphism if and' only iff is a homeomorphism and f.is a sheaf isomorphism.
Remark: If (f l) is a k-morphism, then the map.1 is strongly bound to the
underlying continuous map f. For example,
if the sheaves 91x, Sly are reduced, then for a given continuous map f: X there. exists at most one map f: Sly -+fs(9Zx) so that (f,7) is a k-morphism.
Y,
2. Differentiable and Complex Manifolds. Let U be a non-empty open subset in R', the space of n-tuples of real numbers. Then one has the R-algebra 9'(U) and the C-algebra 8c(U) of R-valued and C-valued infinitely. often d4fTerentiable funs-
Chapter A. Shelf Theory
16
tions. Obviously gc(U) _ "(U) + i9a(U). In both cases one has the natural restrictions ry for V c U. The system (cl'k(U), r'}, where k stands for either R or C, is a presheaf of k-algebras which satisfies S I and S2. The associated sheaves on R", 8R and 1c, are sheaves of local rings over R and C respectively. Thus (R", d") is an R-algebraized space and (R-, 8c) is a C-algebraized space. An R-algebraized space (X, !S) is called differentiable manifold if, for every x e X, there exists a neighborhood U of x in X and a domain D in RW" so that the l -algebraized spaces (U, cl''u) and (D, d') are isomorphic. The sheaf .91 is called the sheaf of germs of real-valued differentiable functions on X. It uniquely determines the sheaf 8X = Sz + i4'X of complex-valued differentiable functions on X. Obviously 49X is a subsheaf of local C-algebras of the sheaf Wx. Further one has a
natural conjugation isomorphism (linear over R), -: 9X .- SX, which fixes SX element-wise. The morphisms between differentiable manifolds are the differentiable maps (i.e.
arbitrary continuous maps which lift differentiable functions to differentiable functions).
In the space C" of m-tuples (zt, ..., z,") of complex numbers, the C-algebra 0(U) of holomorphic functions is defined for every non-empty open set U. Again one has the natural restrictions rvr so that the system {0(U), r} is a presheaf which satisfies SI and S2. The associated sheaf of local C-algebras is denoted by 0. Thus (C", 0) is a C-algebraized space. If one identifies C'" with the space R2M of real 2m-tuples (x1, y,, ..., y :=Im z, then 0 is a C-subsheaf of 9C
W W.
A C-algebraized space (X, Ox) is called a complex manifold if, for every x e X
there exists a neighborhood U of x in X and a domain D in C"` so that the C-algebraized spaces (U, Ov) and (D, OD) are isomorphic. The sheaf Ox is called the sheaf ofgerms of holomorphic functions on X (the structure sheaf of the complex manifold). Since 00 c ec , it is clear that every complex manifold (X, 0) determines a differentiable manifold (X, t>PR) with 0 c 9c c W. A fundamental theorem of K. Oka [see CAS] states that,
for every complex ma ifold (X, 0), the structure sheaf O is coherent. Morphisms between complex manifolds are called holomorphic maps (i.e. arbi-
trary continuous maps which lift holomorphic functions to holomorphic functions), In local coordinates holomorphic maps are represented by systems of holomorphic functions. 3. Cemplex Spaces and Hooomorphic Maps. If D c C" is a domain and f c OD is a coherent ideal, then the support A of the coherent OD-sheaf OD // is a closed set in D. The sheaf 0.,, _= OD // f A is a sheaf of local C-algebras on A which, by the results in Section 2.4 is coherent. The C-algebraized space (A, 0A) is called a closed complex subspace of D. The injection t: A -+ D determines a holomorphic embedding (i,1): (A, 04) - (D, 0D) where 1: OD - i*(0A) = OD If is the quotient map. A complex space X = (X, ex) is a C-algebraized space in which every point has a neighborhood U so that (U, 0v) is isomorphic to a complex subspace (A, 0,t) of a domain in some C". The structure sheaf Ox of a complex space X is coherent. Sheaves of Ox-modules are called analytic sheaves.
§ 3.
17
Complex Spaces
Complex spaces with holomorphic mappings as morphisms form a category. We
let Hol(X, Y) denote the set of all holomorphic maps (f,7): (X, Ox)- (Y, e'r) We often write f: X -+ Y for short. If X is a complex space and X' is a non-empty open subset of X, then (X', is likewise a complex space. One has a natural holomorphic map (X, Ox}, where i id I X' and its the canonical map from Ox to (i, ): (X', i,,(Ox), the latter being the trivial extension of Ox to X. One calls X' together with i an open complex subspace of X. Every section s e O(X) in the structure sheaf is called a holomorphic function on X. The equivalence class of the germ sx a 0x in Ox /ni,, = C is called the (complex)
value s(x) e C of s at x e X. Since t = Ox can contain non-zero nilpotent elements, a holomorphic function s an X is in general more than this C-valued function x -+ s(x).
If (f,, f): X --+C is a holomorphic mapping and Z E O(C) is the identity function, then we obtain a section by lifting z as follows: s:=7(z) E f. (ox)(C) = ox(X );
s(x) = f (x).
It is routine to show that the association Hol(X, C) -+ 0(X ), (f,1) --1. s, is bijective.
Therefore one identifies the holomorphic functions on X with the associated holomorphic maps from X to C. If (X, Ox) is a complex space and 5 c Ox is a coherent ideal then (Y, Or), with Y'=supp(Ox/J) and Or := (Ox/,f) I Y, is again a complex space. The injection 1; Y -. X together with the quotient homomorphisms is Ox -+ t,(Oy) = Ox/.f is a "holomorphie embedding" (Y, Or) - (X, Ox). One calls (Y, lP) the closed complex subspace of (X, Ox) associated to f.
There exists a product in the category of complex spaces: For every two complex spaces X 1, X2 there exists a complex space X and maps ni: X - X,, i = 1, 2, so that if Z is any complex space, the map
Hol(Z, X)-+Hol(Z, X1) x Hol(Z, X2),
f- firs f n2 °f),
is bijective. The space X and the maps nt, i = 1, 2, are uniquely determined up to isomorphism.
Topologically speaking, the above product is just the Cartesian product X1 x X2. Thus we use the suggestive notation X = X 1 x X2.
Holomorphic maps factor through their graphs: Let f: X - Y be a holomorphic map. Then there exists a canonical closed complex
subspace Gph(f) of X x Y, the "graph" off, and a biholomorphic map 1: X -+ Gph(f) so that f = n o ti where it is the restriction of the projection X x Y -. Y to Gph(f }
Chapter A. Sheaf Theory
18
4. Topological Properties of Complex Spaces. Every complex space is locally compact. From here on we will only consider complex spaces and differentiable manifolds which have countable bases of open sets for their topologies. Such space are metrizable. At this point we remind the reader of some of the basic notions and results of 'gent ral topology which are used, for example, in the cohomology theory. A covering 23 = {V% E J (the elements of a covering are always open sets) of a topological space X is called a refinement of a cover U = (Ui); E, of X (symbolized by 23 < U), if every Vj is contained in some U. Thus there exists a map t: J -+ 1 of the index sets so that Vj c Uttn. for all j e J. We call t a refinement map. A Hausdorff space is called paracompact if for every cover U of X there exists a refinement % of U which is locally finite (i.e. every point x e X has a neighborhood which intersects only finitely many of the sets V e 23). Every paracompact space X is normal. Every closed subspace of a paracompact space is paracompact. Every metrizable space, in particular a complex space or a differentiable manifold, is paracompact. A space which is countable at infinity (i.e. a space which is a countable union of compact subsets) is paracompact whenever it is locally compact. The following has numerous applications: Shrinking Theorem. For every locally finite covering {U1} ; E, of a normal space X, there exists a covering (V); E, of X (with the same index set) so that f c U. for all
ie7.
5. Analytic Sets. If (X, Ox) is a complex space and i c Ox is a coherent ideal, then A= supp(Ox/5) is an analytic set in X. One also refers to A as the zero set of the ideal J. Analytic sets are therefore just the supports of closed complex subspaces of. (X, Ox). The support of every coherent Ox-sheaf Y is an analytic set,
because supp .' = supp(Ox A n S). A given analytic set is the zero set of many different coherent ideals (e.g. supp(Ox/3) = supp(Ox/J") for n >_ 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
such that Q c Q' (i.e. a,, < a and b < b,, for all ii). Then there exists a function r e gR(R) with the following properties:
a) 0
Proof: We first consider the case m = 1. For every two real numbers c, d with c < d we associate the real valued function q,S(x)!=exp
11 x_d
_
1
c , fore<x
and
qa(x) - 0 otherwise. By a well-known theorem in calculus, qd is infinitely often differentiable on I8'. By
definition qm(x) z 0 for x e I8' and fd q,(x) dx > 0. Thus pa(n),= (Jx q (x) c
dx)' (f
gce(x) dx)
Furthermore
0
for
xeF,
p,d(x)=0 for x
p,,(x) = 1
for
x > d.
a d°R(I8').
Chapter A. Sheaf Theory
24
for x < b1 and r(x)'= Since a1 < a, and b1 < b1, the function r(x) _= 1 - pb,bl.(x) for x >_ b1 is well defined. Furthermore r has the desired properties. Now let m > 1. Let ru be an infinitely often differentiable function of x,, alone so that a) and b) are fulfilled for the intervals and
(xµ a18'Ia,<x,,
. r,(x.,) is the desired function.
Then r(x1, ... , x.,) := rl (x,)
Theorem 4 (Separation theorem): Let X be a differentiable manifold with (real) structure sheaf 4' = 8" (of course with countable topology). Let A e X be closed and W c X an open neighborhood of A. Then there exists a section f e 8(X) so that
fIA=1,
(IX\W=O, and 0
for all
xEX.
Proof: Let p e A and choose a neighborhood W c W of p which is diffeomorphic to a bounded domain in R'. In W we choose neighborhoods U, U' of p with U C U' C W such that, with respect to the embedding W c. R, they are mapped onto open blocks Q and Q' respectively. Thus Q C Q' and, by lifting the function constructed in Lemma 3 and extending it trivially to all of X, we obtain a function 8(X) with
gI U=1,
glX\U'=0, and 0
Suppose that A is compact. Then there are finitely many points pl, ..., p, e A so that the neighborhoods that we just constructed, U1, ..., U. (U1, ..., U;), cover the set A. Let g. be the function associated to U. and let
f=1-fl(1-g0)E8(X). a=1
Then all the values off lie between 0 and 1. Furthermore m
f(x) = 1
for
x e U U. and f(x)=O for x e X\U UQ, 1
O
1
Y
where U' D U.. Since A c U U. and U Ua C W, f has the desired properties. 1
1
Now let A be an arbitrary closed set in X. We begin with a locally-finite covering (U,)1El of X by relatively compact open sets and (via the Shrinking Theorem) we choose a covering {l;};E, of X with V,c Ub i c- I. Then every set A n V; is compact in X. From what was proved above, there exists a section f E 8(X) with values between 0 and 1 so that
f I A n V, = 1
and f I X\U, = 0 (as sections) for all i c- I.
25
§ 4. Soft and Flabby Sheaves
Since the covering (U,)1., is locally finite, the product
g- n (I -f) iel
defines a differentiable function on X. Namely if x a X is fixed, then there is a neighborhood Z of x and a finite subset T of I so that U, n Z= 0 for all i e 1\T. Since f,, I Z = 0 for i e I \T, g Z is the finite product f (i. - f). Obviously isT
fI - g e 8(x) is the desired section. The following is an immediate consequence of Theorems 2 and 4. Theorem 5. If X is a d fferentiable manifold, then every sheaf of 8s.-modules over X is soft.
3. Flabby Sheaves. A sheaf So over X is called flabby ,for every open set U c X, the restriction mapping 9(X) - .9(U) is surfective. On a metrizable space, every flabby sheaf is soft (Theorem 1!). The converse is not valid. For example the structure sheaf 8s. on a differentiable manifold Wn6t flabby. Every sheaf .P determines a flabby sheaf 9r(.9' As follows: For every open set U c X, one defines .
F(U):= fl 9_ = (s: U -+YuI as = id). xEt/
If ru are the obvious restrictions, then the system {F(U), r°} is a presheaf which satisfies .9'1 and .92. The associated sheaf, denoted by .F(Y), is obviously flabby. Its sections over U c X are all "not necessarily continuous sections" in So over U.
One has the canonical injection j:.9 - ,F(Y). Every sheaf mapping (p: 9" -+ .So determines a sheaf mapping.9r((p): F(.9") F(So) by which .F is a covariant functor (Flabby Functor!). If 9 is an 9?-sheaf, then F(Y) is also an A-sheaf and j is an 9t-monomorphism. It is easy to show that the functor -,F is additive and exact on the category of 9t-sheaves. 3
Flabbiness (and also softness) is not destroyed by continuous mappings: 1f f: X -+ Y is continuous and ,' is a flabby (soft) sheaf on X, then the image sheaf f*(. ') is flabby (soft) on Y.
The proof is trivial. 4. Exactness of the Functor r for Flabby and Soft Sheaves. The section functor r is left exact, but not in general exact. Thus the following is quite important. ' A functor T on a category in which the morphisms a: A -. A' and P: A A' can be added is called additive if, for all such morphisms with T-images Ta: TA --s TA', TO: TA - TA', it always follows that T(a + P) = Ta + Tf.
26
Chapter A. Sheaf Theory
Exactoem Lemma. Let 0 - SP'- Y -+ So" - 0 be an exact sequence over X. Then, in the following cases, the induced sequence 0 - .°'(X) - .9'(X) -* 9(X) -+ 0 is exact: 1) .9'' is flabby. 2) X is paracompact and 9' is soft.
For a proof see [TF] p. 148 and 153. We reproduce here the proof of 2). Let s" e .9'"(X). We can always lift s" locally to a section Y. Since X is paracompact, there exists a locally finite (open) cover (U,),. I of X and sections s, E .9'(U,) which are preimages of s" I U. By the Shrinking Theorem there exists a cover {V,}, e j of X with l7 ,c U, for all i e I. The set E of all pairs (J, t4 where J c I and t e So (U Vj) is a preimage of the section s" I U Vj, is non-empty and partially
Jai
J*
ordered J(J1, t1) 5 (J2, t2) if and only if J1 C: J2 and t2l U V,= t,. By Zorn's
jell
lemma there exists a maximal element (L, s) in E. Since the family {i) is locally finite, the set VL U Vj is closed in X. Suppose there exists i e I\L Then the JeL
sections s e .9'(VL) and s, a .9'(U,) differ by a section t' a .9''(YL n V). Since .9' is soft and VL n V, is closed, we can continue t' to a section s', e .9''(U ). Thus s and s, - s; are equal on VL - 171 and consequently s is continuable to. VL U fl. Unless Y = q, this contradicts the maximality of (L, s). Thus L = I and s represents s" on all of X. Corollary. For every exact R-sequence 0 --+ .9'' -+ .9' -+ .9'" -+ 0 the following are
true: 1) If 9' and Y are flabby, then .9'" is flabby 2) If X is paracompact and .9'' and .5o are soft, then ,9'" is soft.
One usually calls a long exact s-sequence 0-+
9'..+ ...- CY9__+ ...
of 9t-sheafs and a-homomorphisms an (injective) i-resolution of the R-sheaf .9. The following theorem, which is important for cohomology, is a consequence of the above corollary. Exactness Theorem. Let 0
.9' -+ Y -+
-, So9 -
be an a-resolution of . °
over X. If 1) every sheaf 5', .9'9, q. > 0, is flabby, or
2) X is paracompact and all sheaves 5°, Y q, q _> 0, are soft, then the associated sequences of sections, 0 --+ 9(X) -+ .9'°(X) -+ - .9'4(X) -+ - , is exact.
Proof: We define .°l°9 == Jt ez(9'9 - 9'q+ 1) = Jm(99-1 - 9'e) and,, by the assumed exactness of the long sequence, we obtain the exact A-sequence
§ 4.
Soft and Flabby Sheaves
27
.°L''-' -+.So'-' q z 1. Since T' - .i is flabby (soft), it follows inductively from the above corollary that all of the sheaves T' are flabby.(soft). Thus all of the induced sequences 0 -+ -T9-'(X) - So'- '(X) -+ -''(X) -+ 0 are 0
exact
(i.e.
-°l''(X)= Im(Y'-'(X)--.Y (X))). Since TI(X) = Ker(Y'(X)-'
9''+ '(X)), the exactness of the long sequence follows in both cases.
Chapter B. Cohomology Theory
The cohomology groups HQ(X, So) of a topological space X with coefficients in a sheaf of R-modules So are introduced via the canonical flabby resolution of Y.
Moreover the tech and alternating tech coholomology groups, 119(X, 9) and H;(X, .9'), are studied (section 2). By means of the important Leray Theorem (section 3) it is proved, for paracompact spaces, that H3(X, Y) 4 Hg(X, 9')
Hg(X, .'),
q > 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 <2
n
1
23
have a non-empty
Then 93, is a cover of B. The sets B(I1, ..., l.,) and B(1;, ...
intersection if and only if lµ - l 1 < 1 for p = 1, ..., m. For a fixed (11, ..., 1.) e Z', there exists at most 3'" integral lattices points (l1, ..., I.,) which fulfill these conditions. Consequently, the intersection of every 3" 1 different elements of V. is empty and it follows that V) = 0 for all q > 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 -+ of .°
F°
is a
flabby resolution of S' I U over U and since F'
(2)
In
A'(U), we obtain for
U = U c X,
order to compute the cohomology .9'(io, we note. the following:
1P = fl
q Z 0.
groups Hf(X, .IP), where
Lemma. Let (.PJ), j e J, be a locally finite family of (-sheaves (i.e. every point x e X. has a neighborhood V so that all but finitely many 9'j ( V are the zero sheaf over V). 'then, defining if _= 9 J, we have the-canonical isomorphism,
H°(X, 9-) = fl H'(X, f'), J.,
q > 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
H'(X, fP°) = [I H'(U(io, ..., ia), S),
p, q z 0.
It is now easy to give a sufficient condition for the acyclicity of the canonical
resolution (R). One calls an open cover U of X acyclic with respect to 9 if He(U(io, .. ., ip), .9i) = 0 for all p >_ 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 <...
4
where the product is taken over all increasing (p + 1)-tuples in 1°+ t One further verifies that 0(.510) c .9"+1 and that the induced I-sequence, (R')
0 - .So .
.. .
9'O °'-° $01 1-
is as before an A-resolution of Y. In fact the construction in the proof of the Lemma shows that every alternating cocycle is an alternating coboundary. We refer to this as the canonical alternating resolution of .So relative to U. It exists for every open cover. If U is now locally finite, then (from (0')) we obtain (3')
H'(X, 9Pa) =
fl
H4(U(ip,
...,
ip), .9').
10<.
From this one infers that, for every locally finite cover U which is acyclic with respect to .9, the canonical alternating resolution-(R') is acyclic. Consequently in this case (by the formal de Rham Lemma) the cohomology module H'(X, .9') is isomorphic to the p-th cohomology module of the section complex (.9',(X), d;.). But (.9' (X), d;.) = C;(U, .9'). So HP(X, ,9') = H'(Ca(U, .)) = H$(U, .),
p = 0, 1, ....
If .So = F is flabby, then, as at the beginning of this section, it follows that HH(U, F) = 0 for every locally finite covering U of X. Hence, in the paracompact case, the functors 11, likewise have property III). Therefore the flabby and alternating cohomology theories are isomorphic. We summarize now as follows: .
If X is paracompact, then, for every a-sheaf $0, there exists natural isomorphisms
H;(X, .50) 24 f1'(X, .9') 4 H'(X, 9),
q Z 0.
The first isomorphism is as a matter of fact the map i* introduced in Section 2.4. To see this note that the homomorphism from 1R;(X, S) to $'(X, S) is induced by the canonical homomorphism from .So; to .9". Since the sections in .So; (resp..9") are the alternating q-eochains (resp. q-cochains), the-map . -+ 54" is induced by the injection CR(U, .9') - 11(U, .5°). The latter determines the map i.. 0
As a side result we have the Vanishing Theorem cf Paragraph 2.5 for compact blocks B for the flabby cohomology as well:
H'(B, .9') = 0 for all large q.
Chapter I. Coherence Theory for Finite Holomorphic Maps
In this chapter it is shown that for every finite holomorphic map f: X - Y the image functor (* is exact in the category of coherent Ox-sheaves. It is further proved that if . ' is a coherent Cs-sheaf, then J*(S') is coherent. These two theorems are important ingredients in the proof of Theorem B (see Chapter IV, 1.2).
§ 1.
Finite Maps and Image Sheaves
The main result of this section is Theorem 4 which is concerned with the exactness of the image funetor.f*. Several elementary facts about finite maps are needed for its proof. We begin by assembling these. The reader can find a detailed presentation of these foundations in [CAS]. 1. Closed and Finite Maps. All topological spaces which occur. X, Y, ..., are assumed to be Hausdorff.
Definition 1. (Closed finite maps): Let f: X - Y be a map. One says that f is closed if the image by f of every closed set in X is again closed in Y. One calls f finite if it is first continuous and closed and if, for all y e Y, the fiber f (y) is finite.
The following lemma is extremely useful:
Lemma 2. Let f: X - Y be closed and U c X an open neighborhood of a fiber f -'(y) for some y e Y. Then there exists a neighborhood V of y in Y so that
f -'M C U.
Proof., Since U is open, f (X \U) is closed in Y. The set V - Y\ f (X \U) is open
and contains y. Clearly f -'(V) c U.
From here on we consider only finite mappings. The identity id: X -. X is Y and g: Y - Z are finite then the composition gf. X -. Z is likewise finite. Suppose X' is closed in X. Then the induced map f': X'-+ Y, defined by f' f I X', is a finite map. Let V be open in Y and let U '= f -'(V). If we define fu by fu'=f I U, then it is again a finite map. trivially finite. If f.- X
Chapter I. Coherence Theory for Finite Holomorphic Maps
46
2. The Bijection f.(,y), -+ fl 5°x,. Let f: X - Y be finite. Take y to be an arbitrary point in Y and x1, ..., x, E X its different preimages in X. Let So be a sheaf on X. Then for every open set V c X we associated So(f -'(V)). This functor is a canonical presheaf whose sheaf is called the direct image sheaf, f«(5o). Every
germ a, e f,(.9"), is represented in a neighborhood Vo of y by a section the section s deters e 5"(f - '(V0)). Since f (V0) is a neighborhood of every i = 1, ..., t. These germs are determined by r, independent mines germs sx, e of the choice of the representations s. Thus the map F
MY), -iII5'x,, I
U'H(Si,,...,Sxr)
of the stalk f (.9"), into the cartesian product of the stalks 59"x, is well-defined. In
fact s,,, =fx,(u,,), wherelx,: f,(b),-+Yx, is the map considered in Chapter A. Theorem 3. The map I is bijective.
Proof: Injectivity: Let a,, a, e f,(9),. There exists a neighborhood V' of y such that a, and a, have representatives s, s' e .9"(f -'(V)). If1(a,) = J(a;) then, for all i = 1, ... , t, there exists a neighborhood of x1, U; c f -'(V'), so that s U. = s' U;.
Since U U. is a neighborhood off -'(y), Lemma 2 implies that there exists a
neighborhood V of y with V c V and f -'(V) c U U. Thus s I f ' (V) _ s'' f -'(V) and consequently a, = a,.
'-'
Surjectivity: Let (sx,, ..., sx) be an arbitrary point in 11 .9"x,. For each i we choose a neighborhood U; of x; and a section Si e .9"(U) which represents .9"x,. Since X is Hausdorff, one can assume that the Ui's are pairwise disjoint. Define ,
U:= U U;, we thus have a section s e .9"(U) which is defined by s I Ui = s;, t=1
i= 1, ..., t. Again using Lemma 2, since U is a neighborhood off -'(y), there exists a neighborhood V of y such that f -'(V) c U. Certainly s I f -'(V) e So(f -'(V)) and its germ, a, e J,F(S), satisfies fia,) = (sx,, ..., sx).
p
3. The Exactness of the Functor f..-Again let f: X -+ Y be finite and as above let
x1...., x, e X be the distinct preimages of y e Y. Take .9" and Sf' to be sheaves over X and p:.9"' - .9" a sheaf mapping. Thus we have the induced mapping of sheaves f#((p): f#(S°')-+f*(bo). Now 9 gives us maps of the stalks (p.,,: Y' i--+ 9' Hence we have the diagram
(D)
47
Finite Maps and Image Sheaves
¢ 1.
where,' and f are the bijections associated to 5" and 91 respectively (see Theorem 3). It follows immediately that diagram (D) is commutative. Let .1' and 5o be sheaves of abelian groups and suppose that q :.9' -+ .' is a sheaf homomorphism in the category of such sheaves. Then f,(.°') and f,(f') are sheaves of abelian groups, the mappings `'andfare group isomorphisms and f,(9) is a homomorphism of sheaves of abelian groups. This immediately implies that all of the maps in diagram (D) are group homomorphisms. Hence the following is evident. Theorem 4 (Exactness Theorem): Let f- X Y be afnite map and suppose that .9' -..9' -+ . " is an exact sequence of sheaves of abelian groups on X. Then the sequence of image sheaves
MY,) .f.(9') -+f.(."), is likewise exact.
Proof: By the remarks above, the two sequences, t 1
and
fs(Y'), -+f.(`'), -+4(9'"),, are isomorphic. Therefore the exactness of the first implies the exactness of the second.
Q
4. The Isomorphisms H4(X, .9') = H°(Y, f,(.')). The following direct application of Theorem 4 is needed in Chapter IV. Theorem 5. Let f: X -- Y be a finite map. Then for every sheaf of abelian groups (C-vector spaces) on X, 50, there exist natural group (C-vector space) isomorphisms
H°(X .50) = H 9(Y, f,(.)),
q
0.
Proof: Let 0 -+So ' YO-AL-911 ... be the canonical flabby resolution of .50 on X. By Theorem 4 the sequence of image sheaves,
is exact. Furthermore f,(.5ot), i >_ 0, is flabby (see A.4.3). Hence (*) is a flabby resolution of f,(.5') and as a result
H°(Y,f.(S')) = Ker(f,(9°XY)-1f.(S°+')(Y)FIm(f;(b°-i(Y))-'f.(`$)(Y))
Chapter I. Coherence Theory for Finite Holomorphic Maps
48
But the canonical isomorphisms between f;(./Q)(Y) and 9' (X) are compatible with the homomorphisms f (d9) and d°. Consequently
Ker[fs(9')(Y) .f(9'' 'XY)] =
Ker[.9'9(X)-..V9+
(X)],
q > 0.
Since the analogous remark holds for the image groups,
Ker[.9 (X)
H9(Y,
for all q
_
9'
(x) .
`(X
HQ(X, 99)
0.
5. The 0, Module Isomorphism: f5(.'), - fl Yx,. Now let X, Y be complex spaces and f: X -+ Y a finite holomorphic mapping. Thus if So is an Ox-sheaf, then
j (.') is Oy-sheaf. Let y e Y and {x ...... x,} = f t(y). Then we can associate to is <_ i < t. Every Ox; module, the lifting homomorphisms f !: 0, therefore in the obvious way an 0. module. Consequently fl Yx, is an Oy-module.
Likewise the stalk f.(, '), is an 0,-module. The multiplication is defined by taking
hof e
It eOy(V),seP(f'(V))and letting
OX(f -'(V)) is the lift of It by f. This has the following consequence (also see [CAS]):
1f f: X -+ Y is a finite holomorphic map, then every bijection
J:f#(. '),'' 11 xr
y e Y,
{ x 1.
. .
..
.
r) =f `(A
is an c!', module isomorphism.
The General Weierstrass Division Theorem and the Weierstrass Isomorphism § 2.
The key tools of this section are the Weierstrass division theorem and Hensel's lemma for convergent power series. The reader can find these in [AS] (Theorem 1.4.2, p. 34/35, and Theorem 1.5.6, p. 49/50). The main result here is that for certain finite holomorphic maps f, the direct image map f,, preserves coherence. 1. Continuity of Roots. Let B be a domain in C". We use z = (z1, ..., z") for the coordinates in C" and consider a monic polynomial, w = w(w, z)'= wb + a1(z)wb- t + ... + ab(z) a O(BKw), in another variable w e C. The degree of co is b, 1 < b < oo, and the coefficients are
holomorphic on B: a, a O(B) for all j. The zero set of w, A:= ((w, z) e C x BI(D(w, z) = 0),
§ 2.
The General Weierstrass Division Theorem and the Weierstrass Isomorphism
49
is an analytic set in C x B. The projection C x B -+ B induces by restriction a continuous, surjective map a: A -- B. Each fiber, n-'(z), z F B, contains at most b different points. Furthermore we have the following Theorem 1 (Continuity of the Roots). The map n: A - B is closed and therefore finite. Proof.- Let M be closed in A and y e B be an accumulation point of n(M). Thus
there is a sequence (c y,) e M so that the y,'s converge to y. We note that
c; = - (a, (y,c ,' + . + ab(y,)). Jat(y,)I +
Hence, if
+ c, J > 1,
then
Jc, + 5 Jc° J S
+ ab(y,)I. Therefore Ic,1 <-max{1, IaI(yy)I +...+ Iab(y.)1).
(_)
Since (y,) converges to y, each sequence aj(y,), 1 < j < b, converges and is in particular bounded. Thus the sequence {c,} c C is bounded and has a subsequence {c,,} which converges to c e C. Hence {(c,k, y,J} converges to (c, y) e A. By assumption M is closed. Thus (c, y) e M and consequently y = n(c, y) a n(M).
2.'The General Weierstrass Division Theorem. We retain the notation used in the above paragraph. Let y e B be fixed and xi = (w,, y) be preimages of y under the map n. For the sake of brevity we write Os, (resp. 0,) instead of Oc. (resp. Oa,). It follows that O,[w] c 0,<w - w1> = Ox,. Thus every polynomial p e 0,[w] determines at each x; a germ p,,, a Osi.
Theorem 2 (Weierstrass Division Theorem). Let t germs f e Ox;, 1 < i< t, be arbitrarily prescribed. Then there is a polynomial r e 0,[w] with deg r < b and t germs qt a 0,,,, 1 <- i < t, such that fi = gicus, + ri for all i. The polynomial r and the germs q1, ..., q, are uniquely determined by f , ... , f, and w.
Remark: If t = 1 then co,,, is a Weierstrass polynomial in (w - wt) and Theorem 2 is the usual Weierstrass division theorem. Proof: Let y = 0. Then co(w, 0) = (w - w,)b, ... (w - w,)b"
where b, > 1 and
b, = b. By Hensel's lemma there exist monic, pairwise rela-
tively prime polynomials co;(w, z) a O,[w] so that, for the germ cry e Oy[w] determined by cu a O(B)[w], Wy z Wt ... W(,
Chapter I. Coherence Theory for Finite Hobmotphie Maps
50
where w,(w, 0) = (" wit, 1 5 i:5 t. Every induced germ w,,, a Os, 25 O,<w - w,> is distinguished in w - w, and of degree b,. Furthermore each polynomial e,- n wj c- O,,[w] induces a unit e,,,, a 0,,,. This follows because 1#i
e,(x,) _ f wj(w,, 0) = n (w, - wt)b1 ¢ 0. Clearly co, = e, wi, 1 J*t
0
J#i
After the above preparations, the existence and uniqueness statements now follow in a few lines:
Existence: It is enough to prove existence statement for t-tuples of the special form (0, ..., 0,f, 0, ..., 0), f e 0,,,. The more general result will follow immediately by addition. By the usual Weierstrass Theorem, given fie;,; a 0,,,, we have the decomposition fiei-' = qt - w,., + r,,,,
where q,e0r'E0,[w-w,]and degr'
f =9rw:,+rs,. If j # i, then, since w,(xj) = (wj - w,r' # 0, every germ wj,, e is a unit. Thus one can determine q j a Ox j for all j # i. It follows that, for such j, e=i 0 = q j wx, + r).
Uniqueness: It is enough to show that if we have t equations q, w=, + r,,, = 0, where q,,, a d) and r e 0,[w] with deg r < b, i = 1, ..., t, then r = 0 and is distinguished in (w - w,) and since w:, = eu, w,,, it q, _ "' = q, = 0. Since follows that - rx, = (q, is the Weierstrass decomposition of - r;, a 0,[w - w,] in 0,,, = 0,<w - wi> with respect to wt,,,,1 S i S t. Thus q, e,x,
is a polynomial in w. In other words, w, divides r in 0,[wj, i = 1, ..., t. Since w1, ..., w, are pairwise relatively prime, w, = wl w, divides r in 0,[w]. But deg r < deg w. Thus r = 0 and consequently q, = - = q, = 0. 3. The Weiierstrass Homomorphism ObB n.(O,, ). We maintain the above notation. For short we write 0 instead of 0c k B. The polynomial w generates a sheaf of ideals /'= wO. The quotient sheaf 0/,# is "supported" on A. We let OA denote the restriction of O// to A. Thus A is a complex space with structure sheaf OA. The projection n: A -+ B is holomorphic and (by Theorem 1) is finite.
We now construct an OB-sheaf homomorphism ft:O°B- r(&). It will be shown that it is in fact an isomorphism. Let U be open in B and let s = (ro,
b-,
rb_ 1) a ObB be a 'section. The polynomial r =_
-a
.,.,
r0 0 e 0(C x U) determines
The General Weierstrass Division Theorem and the Weierstrass Isomorphism
§ 2.
51
(module f) a section s e (O/f KC x U). By the canonical isomorphisms, (O/f)(C x U) = OA(n-'(U)) = x.(OA)(U). Thus we may think of s as a section s E n.(OAXU). The map nc:Obe(U)- n«(OA)(U),
s
is obviously an OB(U)-module homomorphism. The collection of these homomor-
phism is compatible with the restriction maps. Hence we obtain an OB-sheaf homomorphism n: OB -, n.(CA). We call n the Weierstrass homomorphism with respect to co.
Theorem 3. The
Weierstrass
homomorphism
n: OB
x.(OA)
an
is
OB-isomorphism.
Proof: It is enough to show that every homomorphism of germs, n=: OB.= n.(OA)s, z e B, is bijective. Let x1, ..., x, e A be the preimages of z under the map
it. We identify &.Ifx, with (O/f )x, and bring into play the OB,5-isomorphism
fl (9//)-y = fl
n=: x.{OA)z
1
1
Thus we have the commutative diagram 6
H 1
1,-1
where T=(r02, ... , rb- 1,=)
= (rx, mod 9.., ... , r, mod J1.) and r:= Y rp= wa. The 0=0
bijectivity of irs is equivalent to the bijectivity of T=. As one easily confirms, the latter is exactly the content of Theorem 2. p
It should be mentioned in passing that up to this point we have not used the coherence of the sheaf O. Using Theorem 3 one can carry out by induction an elegant coherence proof for this structure sheaf. The reader can find this and related details in [CAS]. 4. The Coherence of the Direct Image Functor n.. Let z e B be arbitrary and let x1, ..., x, e A be its preimages under it. If .° is a coherent OA-sheaf, then for
each i there exists an open neighborhood Uj of x, and a sequence OV, pU,
u, - 0, p,, q, >_ 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, < m}.
and A'(x) = 0 for r > m,
In particular dim A'(x)
4. Lifting r-vectors. Suppose that X and Y are differentiable manifolds and f: X - Y is a differentiable map. If x c- X and y := f (x) e Y, then every germ g, e 9' is lifted via f to the germ (g, o f ) e 9,'. Thus every tangent vector e T(x) determines the tangent vector fM
: 9" -+ 88,
gr ~ (g' o f )x,
at y e Y. The map fM: T(x) - T(y) is DB-linear. Associated to f,, we have the C-algebra homomorphism
f*: A(y) - A(x) with f *(A'(y)) c A'(x),
r >_ 0,
defined by
(J *(PM,
t') Of (' ), ...,JMlyX
where rp e A(y) and 41, ..., , a T(x). One also uses the suggestive notation of rp o f instead of f *. Obviously M is a contravariant functor and M is covariant.
We now write fM and f in terms of local coordinates. Suppose that n ==dimR Y and that w1, ..., w,; (resp.,u1, ..., u,,) are local coordinates on W c Y (resp. U c X). We assume that f (U) a W. Thus f I U: U - W is represented by n functions w,, = f,,(u1, ..., u,,,) c d"(U), 1 < v a; m. In this language JM
su1
au;
(x) awr and f M (dw1)
af.
au
(x) du,,,
I < i,
j<
m, n.
Since f * is linear, it is therefore determined for all r-vectors:
fo a,, ... , f * (dw,1) A ... o f * (dw,,) t st, <...
5. Complex Valued DiffereMbil Forms. If the map p associates to each point x e U c X an r-vector ip, e A'(x), then one calls tp an r-vector over U. If S 1, ..., a T(V) are differentiable vector fields on V c U and 4L. a T(x) denotes the
1.
61
Complex Valued Differential Forms on Differentiable manifolds
tangent vector of , at x, then, to every r-vector W over U, we have a complex valued function
x - (D(t, ..., ,Xx) - (PA I., ... , .:)
(WI, ..., '): V -+ C,
Definition 7 (Differential forms of degree r). An r-vector 9 on U e X is called a (complex valued) differential r form (of degree r) on U, if for every open set V c U and every system 1, ..., , a T(V), the function cp( 1, ..., g,) is differentiable on V. The set of all r-forms on U is an 8(U)-module A'(U). If V c U, then one has the natural restriction rv: A'(U) -+ A'(V), I V. Thus one sees that the system {A'(U), rY} is a canonical presheaf whose sheaf sad' is a sheaf of c'-modules
called the sheaf of germs of (complex valued) r forms on X. If u1, ..., u,,, are local coordinates on U, then dul, ..., du,, E A' (U). For every
vector field
_
f
f c O (U), and every p = 1, ..., m, it is clear that
du (g) =f, More generally we have the following:
Theorem & If u1, ..., u,, are local coordinates on U, then the family {du,, n - n du,, 11 < 11 5 every r -form (p e A'(U),
V=
S 1,:5 m} is a basis of the 1(U)-module A'(U). For
E
a,1...
du,, n ... A du,,,
where a,, ... ,, _ qp(a/ax,,, ... , a/ax,,) E 1(U).
p
The proof is trivial.
By the above, every local coordinate system u1, ..., u,, on U induces an 1(U)-module isomorphism, (9p ( I I ... I
I ))
Thus A'(U) is free of rank (m) and hence r the 1-sheaf d' is locally free of rank (m).
One should note that sad° =1. The sheaf d' is often called the sheaf of germs of Pfafan forma on X. Another way of looking at d' is sit _ >toomR(I, C), where R and C denote the obvious constant sheaves. The 9-sheaf d' is canonically isomorphic to the r-fold Grassmann product of the S-sheaf sal' with itself over the sheaf of rings d :
0' =A d' = ,&' n
n 0'
(r-times). The 1-sheaf A =_ ® A' of germs of all differential forms on X is locally
free of rank r.
Differential Forms and Dolbauk Theory
Chapter Il.
62
The reader should note that we always consider complex differential forms, while in tho ivory of differentiable manifolds one usually studies only real differ_ ential forms. Conjugation in the field C, zi-41, induces a conjugation : 9 --' d which associates the function 1, defined by x'-+ f (x), to each f e 1(U). Furthermore one has a natural conjugation which preserves the ,,degree: If 9P e A'(U) is given is local a . ,, du,, n A du;,M ten 40= Y, a,, ...,, du,t n A du,, is the coordinates by complex conjugate differential form to op. For a e C and tp, * e A(U), one has the following elementary rules:
a =a,
q, +j -rp+J,
(P ^A
= PA^
.
6. Exterior Derivative. The differential operator d: .d -.d which maps .d' into .d'+' plays a fundamental role in the study of differential forms. The following theorem describes the action of it on modules of sections on open sets U. Theorem 9. There exists a unique C-linear map d: A(U) - A(U) having the following properties:
1) d maps A'(U) Into A't1(U), r,e 0. 2) For every differentiable function f e 1(U) = A°(U) and every vector field 3) do - 99-7 for all -p e A(U).
4) d(g7n fir)-dipnq',+(-IT'cnduff, for 9p a A'(U).
5)dod-0
For a proof at [ARC] as well as jDF]. One calls d the exterior or total derivative. If a 1, ... , u_ are local coordinates in U and f e 1(U), then 2) immediately implies that
df =
du, P
UUr
Thus, as is suggested by the notation, the exterior derivative of ui is du1.,In general if
A(U
1P =1:
dq fr du, ndu,,
ndu,,.
7. Uftigg Ditfeotial Forum. Let X, Y be differentiable manifolds and f X - Y a differentiable map. U W c Y is open, then the differential forms on W
Complex Valued Differential Forms on Differentiable Manifolds
§ 1.
63
can be lifted via f to differential forms on f -'(W): There exists a C-algebra homomorphism
f : A'(W) - A'(f -'(W)),
(p"(D -f
Applying the notation and equations of Section 4, the action of f * is easily computed in local coordinates: Let ul,..., u," (resp. w1,
..., w") be local coordinates on U c f -'(W) (resp. W).
Suppose that the map f I U: U - W is given by w = f,,(ul, ..,, u",) e 9a(U
1
rp =Za,,...,,dw,,A...Adw,,eA'(W),
a,, ..." E &(W).
then
f"((P)IU:=((p°f)fU=F(a,,...,,of)df,,n...Adf.EA'(U where a,, ...,, o f e 8(U) denotes the lift of the function a,, ...
It is clear that
TW(T) =f*(0) Less trivial, but nevertheless true, is
f *(d(p) = d(f *(p) for all rp a A(W). In other words, exterior differentiation commutes with lifting. For proofs of all of these statements we refer the reader. to [DF]. Remark: The lifting of differential forms is never used in this book. 8. The de Rham ( obomology Groups. Using the sheaves .sd' and exterior differentiation d, one forms on X the C-sheaf sequence
g=
a ...-_. a
Since d c d = 0, this is a complex with Ker
a
L.sf-
.o
d d° = C.
Theorem 10. Let i denote the embedding of the constant sheaf C in d°. Then the sequence (s)
o- C - ' +,salo _L d' ° d , d," -mo o
is exact and consequently a resolution of the sheaf C.
This local theorem is an immediate consequence of the famous
Chapter II. Differential Forms and Dolbeault Theory
64
Lemma of Poincere. Let D be a convex domain in R' and rp a A'(D), r - 1. Assume that V is closed (i.e. drp = 0). Then rp is exact (i.e. there exists an (r - 1)-form 0 e A'- I(P) such that d/i = q,). For a proof we refer the reader to [DF]. Since all sheaves d' are soft, the sequence (*) is an acyclic resolution of C over
X. The formal de Rham Lemma (Chapter B.1.3) therefore yields the classical Theorem of de Rham: Let X be a differentiable manifold and d', r z 0, the sheaf of germs of (complex valued) differentiable r -forms on X. Then there exist the following natural isomorphisms:
H°(X, C) = Ker(d ld°(X)) and
H4(X, C) = Ker(d I d'(X ))/dd -' (X),
q > 1.
One usually calls the groups on the right sides of these isomorphisms the de Rham cohomology groups of X (with complex coefficients). Although they are obtained via the differentiable structure of X, they are topological invariants of the manifold X, being isomorphic to "singular" cohomology groups.
The above considerations for the d-operator will be used again for the aoperator in later sections. This will lead to the Dolbeault cohomology groups (see Section 4). At the point where the Poincare Lemma enters for the d-operator, we will use the Lemma of Dolbeault which is proved in Section 3.
§ 2.
Dif Tential Forms on Complex Manifolds
In this section X always denotes a complex manifold of dimension m with structure sheaf 0. The inclusion d c d' together with the conjugation : A -+ A gives rise to a double gradation of A which contains important information about X. L IFbe
Shum0i.o,y1o.iand Q'.If
are complex coordinates in U c X, then, since d is C-linear, dz, = dx + idy, di = d.%, - idyp, 1 < p < m. Furthermore dx _ }(dz,, + di,,),
dyp = Zi (dz - di,,),
p = 1, ..., m.
Thus the family (dzl, dil, ..., dz., dim) is a basis for .cl'(U) as an gf(U)-module. Hen c the following is immediate:
Theorem 1. If z, ..., z,,, are local complex coordinates on U c X, then for every r z 0, an 8(U)-basis of sd' (U) is given by the (2uum) fo rms
Adi1.1 1
1 51t <... <./t S
P+q= r}.
§ 2.
65
Differential Forms on Complex Manifolds
Proof: These 12m forms generate d'(U). Since d'(U) is a free 8(U)-module
of rank (
2m
rr
Ir
O
) , they form a basis.
For every function f e d(U) we have
(Af ax dx + Ay, dy }
df If as usual one sets
- i a f) az.'(aX 2 a
and
ya
a
==
i f ),
2
"
aZ
k = 1, ... , m,
yv
v
`
then one obtains
df = E t a dzA + of
di,)
for all f e f(U).
A function f e d(U) is holomorphic on U if it satisfies the Cauchy-Riemann differential equations:
of
aiw=0,
µ=1,...,m,
or equivalently when
df = E of dz,,. We now introduce three important subsheaves of sl1. We start with the inclusion d0 c s11 and denote by sd 1'0 the 9-subsheaf of 01 generated by W. One calls sd1.0 the sheaf of germs of differentiable (1, 0}forms on X. Explicitly, '0 =
r=1 00
X E X.
gi, dfi.I gix E 9x, fix E Ox J
ydi.o Since d is inter-
Along with .W1,1 we consider its conjugate 8-sheaf changable with conjugation,
x-
!=1
g1x
I gix E d'x, J ix E Ox
,
x E X.
In other words, can be described as the d-subsheaf of sd 1 which is generated by dfi. One calls if" the sheaf of germs of deferentiable (0, 1}forms on X. 0°.1
Chapter If. Differential Forms and Dolbeauft Theory
66
Finally we let Cl' denote the 0-subsheaf of sd' which is generated by W. Thus
c sd'-0 and
Sts={
XEX.
gi.dfi. l9i.,fuEOx
One calls Cl' the sheaf of germs of holomorphic I -forms on X. sd°- ', and Cl' are uniquely determined by the comHence three sheaves
plex structure on X. In local coordinates, they are described as follows: Theorem 2. Let zl, ..., z., be local complex coordinates on U c X. Thus
as an S(U}nodule and a 1) The 1 -forms dzl,..., dz., form a basis of basis of W(U) as on O(Uymodule. 2) The 1 -forms dil, ..., di., form a basis of sd°"(U) as an d'(U)-module.
Proof: 'Certainly dzp a sd'-0(U) and dip e sd"(U), 1 S p < m. If x e U and
f e 0s then df
(i) (dzp. Hence dz,, ..., dzgenerate sd0 as an 8z
F module and thus the sections dz,, ..., dz,, generate the t(U)-module sd'-0(U).
Analogously it follows that sd°"(U) is generated over d(U) by dz,,..., di. and that dz,,..., dzi,, form an 0(U) generating system for il'(U). Since dz,, di,,..., dz.,, di., are linearly independent over 9(U) (see Theorem 1), the claim follows.
2. The Sheaves .d" and W. Using the Grassmann product n which is defined for arbitrary sheaves of rings, we now construct new sheaves from the ir-sheaves 01.o, sd°,t and the 0-sheaf Cl'. For natural numbers p, q we form (over d) the Grassmann product sd D.4 ,= sd l.° n ... A sd t.o n ja(o. t n ... A JVO",
p-times
q-times
and (over 0) the.Grassmann product Op '= 0' A... A 01, p-times
'here we agree that
sd° _ 9 and A°
T. Obviously sd"-4 is an 6-
subsheaf of .td'4' and OP is an d-subsheaf of sd°.
Ddidsa 3. The A-sheqf .d'+ is called the sheaf of germs of differentiable (p, q)-forms on X. The 0-sheaf tl is called the sheaf of germs of holomorphic p--oars on X. The (p, q}forms are rather easily described in local ooordinat:
§ 2.
67
Differential Forms on Complex Manifolds
Theorem 4. Let z1, ..., z,,, be local complex coordinates on U c X. Then 1) The family
2) -d'(U) = ® .arvr(U) as an d(U)-module. r+4= 3) The f a m i l y {dzi, A" A dzl,, 1 5 it < . < i,, 5 m), is an 0(U)-basis for (2'(U}
Proof: The definition of ad" implies that the family in 1) always generates W'4(U) as an .9(U}module. Since the union of such families over all p, q with p + q = r is a basis for d(U) (see Theorem 1), each family is a basis of ad'4(U). The same reasoning proves the validity of equation 2). Statement 3) now follows trivially.
Core". For all r,2: 0
AP- ® .VP-4 (as i-modules). s+t=
In particular ad' ad'-0 therefore coherent. Clearly A'
The 0-sheaf O' is locally free of rank
(rn)
0 for all p > m = dim X.
From 1) of Theorem 4 it also follows that
a and ff. If one composes d : d -+ .rd' with the projections on the first and second summands of the decomposition sd' = j/ "o ®.rdo.', then one has two C-linear maps
a:0-+Jdt.o and 'b:8- jifo,' with d = a + a. If zl, ..., i. are local coordinates on U c X, then
A dz,, 0'_il azO
ff =
.
Thus f is holomorphic if and only if
T I azo
di,,,
for f e d'(U).
0. In other words
These considerations can be immediately generalized: For every (p, q)-form w = E a,,.-- ,, j, ... j, dz,, n ... A dz,, n dzj, n ... A d1 j. e ad'*(U)
Chapter II. Differential Forms and Dolbeault Theory
68
we set M
aw _ Y, (E aa".°,, dz ndzi, n... Adzyndzj, n =1
,.J
az
ndilj a ,qP+t (U)
and
aa,, -JA di n dzf, n (, \i.1 Oz
Adzi, Adze, A... Ada,4) a AP'4+1(U).2
Thus we have defined two C-linear maps
and : fa/P9-+ Since do) =
tj
n da;. and, for functions
dai,... j, n dzt A
as
as
da =
adP9+1.
m
dz
0=1 aZN
I. t
aZa
daM
it follows that
dw=&o+&w for all to e,c/Pq In particular d.riP.a C dP+1.4®dP.4+1
Both maps a,
are in the defined obvious way as C-linear maps
a:s+l-+sf and These operators, along with the total differential operator d, play important roles in the theory of complex manifolds. Their formal properties are summarized in the following:
Theorem 5. For a and defined as above, we have the following identities:
d=a+ Top = ZO
6-a=0,
o,
for all (p a A,
a((p n *) = a(p n O + (-1)°+°(p n aO
for (p a
<< 2 In this can Z means that the sum is taken over all index combinations with 1 < is and
15)l<<ji5nr.
m
69
Differential Forms on Complex Manifolds
§ 2.
and
J((PAO) = &PAO + (-1}p+4V AOO
for q e
saitp,a
Proof: By definition
o=d-d= (a+D)-(a+D)=aoa+D-Z+(a-0+a The equations in the first row now follow, since, for every cp e The equation and (06 + aa)cp E D - Drp e a o i(p E 0q = Dip follows immediately if one writes cp, acp and acp in local coordinates and
notes that for functions f it is always the case that
of 6 azu^aZ.
µ=l, .., M.
Because of this it is also only necessary to verify the remaining equation which involves a and this is again just a calculation in local coordinates. The operators a and are "created equal", However, since the sheaf G is usually given priority over L in complex analysis and since & = Ker 5 19°, the operator 0 is
more in the foreground of interest. Theorem 6. A (p, 0)-form cp a AP-0(U), p >_ 0, is holomorphic if and only if it is a-closed (i.e. a = 0). Thus
for all p> O.
S2p = Ker
Proof: If cp =
a;, ... ,, dz;, A . . A dz1, in local coordinates, then m
µ=t
f
(aa, d 0211
Adz A' Adzi,)c d1(U).
Since the forms da Adz;, A. A dzform a basis for
O q = 0 if and only if
for all p and all i,, ..., ip 8a;,...;p=0. aiN
In other words, a,,...;, a d(U) or, equivalently, cp a S2p(U). Remark: Since acp = $0, the form q is in Ker a if and only if w e Ker 0. Hence, since Ate, we have the analog to Theorem 6: = Ker 01 A°p.
Chapter H.
70
Differential Forms and Dolbeault Theory
One calls OP.the sheaf of germs of antiholomorphic p-forms on X. The results which we jai obtained for 0 and are suggestively illustrated by the following diagram of C-homomorphisms (i always denotes the natural injection): 0
0
0
0
0
d-
d
0 -.--i C
0
1
a.. d0.t
0 ---'O - '--8
aI
0_
61 -.L ,*1.0 a
a
SI0.2
.. `jt.1 a `(2.2 e...,... a .fit," -------+0 a
at
at
0-- --ii"' _ 1-
0
a
a
a
a
0.---(12-_1 ti.2.0 a 2.1 a .. at
a ... a 0.1"
_.2
...
at
,".o a,at.t_? aid".2 a
AM
a
-=0
al
....
a
aS".m
-0
This diagram is anticommutative in every square. Every row and every column is a complex of sheaves. The family (.Wl'9, 0, 0) is thus a double complex. The associated simple complex is {d', d). The main result of the next section is that all of the rows and columns in the above diagram are exact.
Remark. Frequently one sees d' and d' in the literature instead of 0 and a. 4. Holomorphic Map of (p, q)-forms. In Paragraph 1.7 we saw that a differ-
entiable map f: X
Y induces on any open set W c Y a C-algebra
homomorphism
A(W) into A'(f -1(W)) for all r, is interchangable with conjugation, and with the exterior differential operator: T no
f 0 d= d 0
1 3.
71
The Lemma of Grothendie *
In the complex analytic case f has further properties. Theorem 7. Let X, Y be complex manifolds and f: X -. Y a holomorphic map. Then
f'(A'4(W)) c A','(f -'(W))
f'oa=a°f',
f'(n'(W)) c Q'(f -'(W))
f'o =Dof'
Proof: All of these statements are local. So let z1, ..., z,, (resp. w,, ..., wa) be local complex coordinates on U c f (W) (resp. W). Let f I U : U --+ W be given by the functions w, = f,(il, ..., z.) a a)(U), v = 1, ..., n. Every (p, q}form co on W can be written
9) =1: ail ... ,dwa,A Adwa,Adwj,A...Adw1. aj
Then by the result in Section 1.7 °f)dfa,n...Adf'Ad1j,A
f'(.p)I U=L, (a,,...j.
.AdJ+eJlP'O(U1/
a.l
The fact that f (V) e d'4 follows since the holomorphicity of f implies that then f' (qf) = n dfi Since f and ail... a,, are holomorphic, the coefficients
df, a .d'"0 and consequently dj e a&-'. If op is a holomorphic ail ... a, Of dfi, n .
a,,..- j, of are holomorphic. Thus is in the p-fold exterior product of 0(U} module spanned by d0 and is thus in il'(U).
In order to we the fact that f' commutes with 8 and 8, we begin with the equation f o d = d o f . Since d = a + 0 and f' is additive,
f (8q) + f
8f
+ f for every (p,
q}form P.
Now
f'(8,),
8(f'q,) E .d'+ t.f
and f 0( lp), a(f'9P) E .d''4{'.
Thus, since f preserves the double gradation of .d, we have the desired equalities.
§ 3.
° he Lemma of Grothendieck
In this section preparations are made for the proof of the fact that both the rows and columns of the diagram on p. 70 are exact. It is necessary to solve the differential equation 8g/al .= f, where the given function f depends on a parameter as well as z. For this we need theorems from Lebesgue integration-theory as well as the Stokes' theorem in the form of a generalized Cauchy integral formula. The letter z always denotes a complex variable and ul, ..., ud are real variables. We reserve r and z for complex variables of integration.
Chapter II. Differential Forms and Dolbesuk 71nory
72
1. Area Integrals and the Operator T. Let B always be a bounded domain in the z-plane and U an open set in the parameter space Fe of d-tuples (u 1, . , . , u,}. We consider complex valued continuous functions f on B x U for which the integral
(Tfxx,u)-- ff Z-Z
(1)
21Ri
B
exists for all points (z, u) e B x U. From the Lebesgue integration theory -we know that every continuous function from B x U which is bounded on B x {u), u e U, has this property. Note that j j dZ -
{ -zz
exists and f f
By the Lebesgue dominated convergence theorem, (Tf )(z, u) = lira
(2)
1
2n!
ff G/K&
fLu)
C-Z
dC A dt,
B
ddC -, z)2 doesn't?
(S
(z, u) e B x U,
where K, is the closed disk of radius z > 0 about z. The function 7y: B x U -+ C has different behavior with respect to u than it does with respect to z. In order to study the behavior of u in equation (1), it is convenient to free the denominator of the integrand from the variable z by making the substitution ,: Z - z. Thus one obtains
(Tf xz, u) - tai
(3)
f1 B,
f (z
+ 11, u dy n dil,
(z, u) e B x U,
1
where B. - {q e C' z + q e B) is the set B translated by z. One gets rid of the dependency of the domain of integration on z by extending f trivially (by 0) to all of C x U, calling the extended function I In this way we have (31)
(Tf Xz, u) = 2n. JJ
J(z + ry' u) dry n dit,
(z, u) e B x U.
In polar coordinates, q = re`°, (3f can be written as follows:
(Tf)(z,u)=n f f e-'° f(z+re",u)drpndr,
(z, u) e B x U.
a,
In this way we see that if p- sup I n - w I < oo is the diameter of B, then, since o.w
f is bounded on every B x {u), Tf is bounded on B x {u}, u e U, with
ITfIs.(.:52pIf JBxl.l.
73
.13. The Lams a(Grotbendieck
We note here the trivially shown (nevertheless important) fact that T is a C-linear operator.
2. The Cammutivity of T with Partial Differentiation. The situation for the parameters u is relatively simple: Lama 1. Suppose Tf exists on d x U and that (af/Du,,)(X, u) is continuous on B x U and is bounded on B x K for all compact K contained in U. Then
/(Tf)=T. Clu a
I.
Proof: The integrand of T is uniformly bounded by a Lebesgue integrable function. Thus one may interchange integration and differentiation by u,,.
Due to the singularity of the integrand, the situation for z and 1 is more complicated and requires stronger assumptions.
Lemnat 2. Let f be continuous on B x U and suppose that supp f n (B x K) is compact for every compact K in U. Then Tf is continuous on B x U. If in addition 8f/8z, 8f/8z exist and are continuous on B x U, then,
(Tf) - T az'
a M) T 8z'
Proof: We represent Tf by (3'):
Tf
7,
2Ri JJ f (z +q
u) dq n dq.
By the assumption on suppf, the integrand is continuous on C x U, and on C x B x U (where n e C, z e B, u e U) it is uniformly bounded by a Lebesgue integrable function. Thus Tf is continuous in B x V. The additional assumptions imply that the integrand has z andz derivatives which are integrable over C. Thus the uniform boundedness implies that 8/8z and 8/ai commute with the integral operator T.
Applying the above lemmas in the case where f is smooth, we have the following:
Lensnta 3, Let f e 1(B x U) and suppose that supp f n (B x K) is compact for every compact K c U. Then
Tf e il(B x U 8
(Tf)=Tf, az aZ
8 aZ
(Tf)=Tf, ai
au,
(Tf)=T 8f, auM
I<µ
Chapter H. DifferenfW Fame an8 Dolbeauh Theory
74
Proof For any (also higher) partial derivative g of f, the support of g has compact intersection with every B x K. Thus by the above lemmas, we have the desired interchangability of T with the various differential operators. At the same time all of the first partial derivatives of Tf are continuous on B x U. By an iterated application of this, 7y e 6(B x U). 3. The Caochy Integral Fornala and the Equation (8/a XTf) = f. Functions of the form Tf appear as "correction , terms" in the generalized Cauchy integral formula If G is a bounded domain in C and 8G is piece-wise smooth, then, for every complex valued function h which is continuously differentiable on G, Stokes'
theorem is equivalent to the equation
1
!5' (4) d? A
h(4) 6 1 4 _° f f dh(4) Ado
.
Fo- integrands of the form
f (O
z e G,
{-z ,
f continuously differentiable on
this equation is no longer valid. This is due to the fact that the singularity at z yields an additional contribution, which is in fact quite simple to compute. In order to derive a formula for h one puts a closed disk of radius e, K,(z) c G, around z. Since his continuously differentiable on G\Ks z), Stokes' theorem can be applied on G\K.(z) instead of G.
Now ( - z)-1 is holomorphic on G\{z}. So it follows that (8h/81) _ z)-1 and therefore
(8f/8Z) . (a)
if
of
ff d-j
z)- dZ AdZ _
ea
GAS
j-d. -z
The following is just a consequence of letting e -+ 4 in (s):
Theorem 4 (Cauchy Integral Formula). Let f be a continuously differentiable function on 0s. .Then
I Of tai 1!
z)-1 d A dZ = f(z)
(C) '
l
J 6G
()Z
- z = se''',
Proof. In polar coordinates
axA:)
_ 21
AO -)
2.
zds=i f f(z+eeI)do; 0
zeG.
§ 3.
The Lemma of Grothendieck
75
and thus lim
f
f
`-0 ax,(:)
z
d = 2nif (z).
-d
Substituting this in (s) and using
the result follows.
Remark: If f is holomorphic on G, then (af/az) = 0 and the above formula becomes the Cauchy integral formula of classical function theory. We now use Theorem 4 to show that, for functions with compact support, the operator T is a left-inverse of the differential operator (0/02). Lemma 5. Let f be a continuously differentiable function on a bounded domain B and suppose that supp f is compact in B. Then
T=f on
B.
Proof: Let z e B be fixed. Since supp f is compact in B, we may extend f trivially
(smoothly) to all of C and, letting G be a domain containing B,
of
ar
1 T di (z) = 2ni f
I
0
z) -' d AdZ.
If one chooses G with piece-wise smooth boundary, then, using the fact that f =_ 0 on 8G, the Cauchy integral formula yields the desired result. 0
4. A Lemma of Grothendieck. It can now be shown that for all "reasonable" functions, T is a right inverse for a/az. In other words, Tf solves the differential equation (ag/az) =f
Theorem 6 (Grothendieck). Let f e 9(B x U) and suppose that every partial derivative off is bounded on every B x K, where K is compact in U. Then 1) I Tf IB x t., 5 2p I f IB x
2) au (Tf)=T M
3)
u e U, where p = diameter of B,
on Bx U,µ=1,...,d, M
(Tf)=fon B x U,
4)Tfecf(BxU). Proof.. The statement 1) wap.proved in paragraph 1. Statement 2) follows from Lemma 1. In order to prove statements 3) -and 4), let zo e B be fixed and choose a
Chapter 11.
76
Differential Forms and Dolbeault Theory
compact rectangle R c B with zo E R. There exists a function r e $(C) which is
identically I on R and vanishes outside of a neighborhood R' of R which is relatively compact in B (see Chapter A.4.2). We set f1
(1 - r).f,
f2 '= rf
Then f,,f2 e e°(B x U) and f =f, +f2. Since f is bounded on every B x K, so are f, and f2. Thus Tf, and Tf2 exist on B x U and Tf = Tf, + Tf2. Now, by the choice of r, the function f, vanishes identically on R x U. Thus
(Tf1)(z, u) =21ri 1
JJ 81R
fi(, u) dg AdZ,
(z, u) e B x U.
S-z
As a function of z, this integral behaves quite well as long as z varies in R and stays away from the boundary OR. In this case the integrand is infinitely often differenand all derivatives are bounded. It follows that Tf, I R x tiable in z, i and the
U E 4'(R x U) and one can interchange differentiation and integration. Since the integrand is holomorphic in z,
az (Tf,) = 0 on R x U. We consider the integral Tf2 whose integrand has a. singularity at z. By the choice of r, suppf2 n (B x K) is compact for every compact K c U. Thus, by Lemma 3, Tf2 e 9(B x U) and (a/ai)(Tf2) = T(af2 /az). It follows from Lemma 4
that (a/aiXTf2) = f2 on B x U and, since f2 = f on R x U, (a/aiXTf2) = f on R x U. Since Tf = Tf, + Tf2, statements 3) and 4) now follow from the just derived properties of Tf, and Tf2. In the applications of Theorem 6 in the next sections, U is always an open set in
the complex vector space C' with variables w,, ..., wd. The interchangability equations 2) remain valid for the real and imaginary parts of each w,. Thus it
77
Dolbeault Cohomology Theory
§ 4.
follows that
a (TfTef aw;
Hence we can rephrase Theorem 6 as follows: Corollary. Let U be an open set in C' with variables (global coordinates) w1...., wa and suppose f e 8(B x U) satisfies the same conditions as in Theorem 5. Then, if f is holomorphie in the variables wi, it follows that Tf is also holomorphic in the w; s.
§ 4.
Dolbeault Cohomology Theory
Associated to every closed set M in a complex manifold X are its Dolbeault cohomology groups DolbP. (M). By using the results of Section 3, we will show that these groups vanish for every compact product set in C". Among other th ngs H9(M, i2P). this implies the fundamental Dolbeault isomorphism 1. The Solution of the a-problem for Compact Product Sets. If M is a subset of X
and V e A(M) is a differential form, then rp is said to be a-closed (over M) if &p = 0. It is said to be a-exact if there exists 0 e A(M) with 00 _ gyp. Since Z - Z = 0, every a-exact form is a-closed. The a-problem consists of characterizing the subsets M of X on which every a-closed (p, q)-form is a-exact. The following solves this problem for compact product sets:
Theorem I (Dolbeault-Grothendieck). Let K1, ..., K," he compact sets in C and x K," their product in C'". Then every a-closed (p, q) form, q >_ 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
Chapter III. Theorems A and B for Compact Blocks in C"
84
Thus E is an open, non-empty rectangle in the z1-plane. Furthermore let a', b' E R
with a
(z e E I a' < x).
E"
and B:=E x U, Let U # Q open in C' -' (using the variables z2, ..., Then B = B' u B" and B''= E' x U, B" E" x U, D (E' n E") x U. D = B'nB". r1
d
c
a' b'
a
x
b
Theorem 1 (Cousin Attaching Lemma). There exists a real constant K so that for every bounded analytic function f e O(D) one can find bounded analytic functions
f' e 0(B') and f" a O(B") with
1) f=f'ID+f"ID If"IB-<-KIfID-
2)
Proof. We choose a positive real number b < j(b' - a') and a smooth real valued function r(x) with values between 0 and 1 so that
r(x)=0 for x
for x>b'-b.
We then set P(Z) 'P(Z)
r).f (z)
for for
zED
q(z)__
z E B' \D,
((1_ r(x)) f (z) for z e D 0 for z e B"\D.
Certainly p e if(B') and q e if(B") and both functions are holomorphic in Z21 ... , z,,,. Moreover
f=PID+qID,
IPIB'<- IfID,
IgtB-.<
I fID.
The fact that f is holomorphic implies that ap (z)
8i,
= 1 dr (x) ' f (z), 2 dx
Z E B'
and
aq (z)
oil
_ -1 dr
2 TX
(X)
' f (Z),
z E B".
§ I.
85
The Attaching Lemmas of Cousin and Cartan
The function
1 dr 2 dx (x) f(z),
h(z).
ZEB
=B'
V B"
is defined and smooth on B and holomorphic in z2, ..., Furthermore drx
IhIB<MIIID, where M'=supp XER
(x)
f
2
By the corollary to Theorem 2.3.5, there exists a function h e E(B) which is holomorphic in z2, ..., z", so that ah = h,
ai
I
where p := diameter of E.
IB < 2p I h IB,
We now define our desired functions.
f' p-hlB' and f"'=q+hlB". Thus f = f' I D + f " (D. Since af'/ai, = 0 f " E O(B"). Estimating in the obvious way,
If'IB<
and of"/ai, = 0, f' a O(B') and
IPIB'+.IhIB- If ID+2pMIf ID=KIf ID,
where K'=1 + 2pM. In the same way I f" (g. < K I f , and thus the theorem is
0
proved.
2. Bounded Holmnorphic Matrices. Let p, q >- 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
Chapter III. Theorems A and B for Compact Blocks in C"
86
We let e = (ba) denote the q x q identity matrix and we note the following: (2q2)- t > 0. Then, for every (q, q)-matrix a on V with Let s,=q-' a - e 5 s, the inverse matrix a-' exists and is bounded on V with Ja ' 5 2.
Proof: Let h;= e - a. Then I h' j < q'-' I h 1' < q-'(sgy for all i z 0. Since sq < 1, the sum a
Y ht converges on V and f-o
la-'s
2 sq
i=O
by the choice of s.
We note further that, W
for every sequence
of (q, q)-matrices on V such that 2 Y- I a,. - e =o
()
a0a, a.-eI <2 Y- Ia,,-e
for all
n>-0.
v=0
The proof follows immediately by induction on n from the fact that
anal ...
a-
e)
+(a0a, ...
-e).
13
A (p, q)-matrix function a(z) = (aa(z)) is said to be holomorphic on V if each
component aa(z) is a holomorphic function on V. The C-vector space of all (p, -q)-rnatrices which are holomorphic and bounded on V is a Banach space with respect to the norm I I .
In the following p is always equal to q. The bounded holomorphic matrix is not an functions on V form a C-algebra. The reader should note that algebra norm. We let B(V) the I
I
inverse of a holomorphic matrix function is holomorphic, the above remarks show
that
'aeB(V)jIa-eI < s) -z B* (V)' For a later construction (in the proof of Cartan's Lemma) we need the following:
Lemma 2. Let g,, h, e B(V) be sequences with
2y+g,l<s and 2Y_ 0
Ih,1<s.
0
' An a 6 8(V) can have a holoma`phic inverse a-' which is not bounded (Le. 9'(V) is a proper subset of G[{q, 0(V)}).
§ 1.
The Attaching Lemmas of Cousin and Cartan
Then the products
u.:=(e+go)(e+gt)... (e+g.-1)(e+g.)E B(V) (e+hl)(e+h0)eB(V) v converge uniformly on V to the matrices u, v E B*(V) with 00
00
Iu - eI <_2Y Ig,
and
Iv - el < 2YIh,I. v=0
v=0
Proof: It suffices to verify the statement for the sequence u,,. By (Ik) (noting that
s < q-1), it follows that'
(e+g;)...(e+g;)-ej <2IgI, for all j>i>0. v=j
In particular I u - e I < 2 Y I gv I <_ s. Thus the sequence I u I is bounded by 0
1+ s. Hence IuN -- u I = I u,,{(e + g,. t) ... (e + ge,) - e) I < 2q(l + s)
Y_
I go I
v=v+t
for all t > v > 0. Consequently (uv) is a Cauchy sequence in B(V), converging to a
holomorphic matrix function u. Since (u - e 15 s, it is also the case that I u - e I < s. Thus, by the choice of s, u e B*(V). 3. The Lemma of Cartan. In this paragraph we use the notation B = E x U, B',
B" and D which was introduced in Paragraph 1. We will use K to denote the constant of the Cousin Lemma. All matrices which enter into our discussions are (q, q}matrices and we again let s = q - ' - (2q2)-'. The matrix norms over B', B", and D are specified by I IB-, I Ig,, and I ID. The results of the following lemma are basic for the proof of Cartan's Lemma.
Lemma 3. Let E E R be such that 0 < e < sK-' and t = 4g3K2e < 1. Then, for every holomorphic matrix a = e + b e B(D) with I b ID <_ E, there exist holomorphic matrices
a'=a+b'eB(B'}, a"=a+h"EB(B"), and a=a+beB(D) having the following properties:
1) 1b'1B,.
2)a=a'(Daa"ID.
I9 ID<-tIbID,
Chapter M. Theorems A and B for Compact Blocks in C'
88
Proof: Applying Cousin's Lemma to the q2 components of b, there exist matrices b' E B(B'), b" a B(B") and b = Y J D + b" I D with I b'IB 5 K I b ID and I b" IB. 5 K I b ID. Defining a' e + b' and a" e + b", it follows that
a'ID a"ID = a+ (b'ID VI D).
(*)
Since K I b I D < s (recall how E was chosen), a' and a" are invertible on Jr and B" respectively (see Lemma 2) and
Iale,.<2.
Ia-tle,<2,
(**)
For a'= (a' I D)-' a (a" I D)-" e B(D), we therefore have the equation in 2). Note that D.
Thus, by (*), E = - (a' I D)-' - b' I D - b" I D - (a" I D)-'. From this, using the estimates in 1) and (**), it follows that jbID54q3-K2-IbID'
The definition oft and the* fact that I b ID 5 E now imply that 16ID
t I b ID.
We are now in a position to prove Cartan's Lemma.
Theorem 4 (Cartan's Attaching Lemma). For every q z 1, there exists a real
constant e > 0 with the 3ollowing property: Given a matrix, a e B(D) with I a - e ID < e, there exist invertible matrices c' a B*(B'), c" a B*(B") so that
and
Ic-eID54KIa' eID,
(c"- eID54KIa-eID.
Proof: In addition to the restriction placed on E in Lemma 3, we further require
that, 2t 5 1 aid 4Ke 5 s. We set a = e + b and L- I b ID. We now inductively define three sequences
a"=e+b, aB(D
a. = e + b; a B(D)46
a4d
a' = e + b,' a B(D)
satisfying I&Y-,IB 5 KLt°
(°)
IbvID 5 U",
(°°)
av-t = av-tID - a, - a"
ID.
Ib"-tl1., <- KLt"
and
§ 2.
89
Attaching Shear Epimorphisms
a,'_,, and a,, are already constructed, then 1b,11 ):5 For this let ao :=a. If Lt" < e and, applying Lemma 3 to a,,, we obtain matrices a;, a'; and a,,,:= a, Thus, (-) is valid with all indices increased by 1. Further (by 1) of Lemma 3), I b', I,, < K I b, ID,
I h,+ I ID < t I b, I.,
and
I
5 K I b, Jr..
Hence, using these inequalities as well as the induction assumption, b,+,, b', and by. Now for all n > 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<_x<-b,c
K:=Rx K'. 1. An Approximation Theorem of Runge. By carrying over the well-known methods from the theory of one complex variable, we show the following:
Theorem 1 (Rune). Let 6 > 0 be arbitrary. Then for a given f e O(K) there exists a polynomial f in z with coefficients holomorphic on K' such'that
If-1IK
f (w) = 2iri j L (C' iL C-z OE
R I
E
aE
The integrand k(r;, w) := (2ni(C - z)) - r f (C, z') is uniformly continuous on the
compact set 8E x K. Thus there exists p > 0, so that, for all C, 1' e bE with I < 2L,
where L is the circumference of E.
We now decompose 8E into intervals 1 1 < v < n, of length p,, < p and fix points C, e I. Then k(t;,, w) is holomorphic on (C\{lq}) x K' and g(w) __ ± k(C,,, w)p .-1
91
Attaching Sheaf Epimorphisms
§ 2.
is a Riemann sum for the above Cauchy integral. In fact f (w) - g(w) _
I
(k(C, w) - k((,, w)) dC,
w e K,
and thus b
b
If-glx< ,.=12L £ Pv=-. A,. We
For every point {, we find an open disk A, c C with R c A, and C,
choose a Taylor polynomial t, e C[z] from the Taylor development of p,(2i i(C, - z)-') about the center of A, (i.e. the first part of the Taylor series) so that I P"(2ni(c, - z)) ` - t,(z) lit <_ 2nd ' where M, = I
Ix
Y
Then a
J(w)
Y J(,, Z')t,(Z), v=1 YY
Z E C,
Z' E K',
is a polynomial in z with coefficients which are holomorphic on K. Moreover, for w e K,
g(w) -J(w) = i J(C,, z) - [p,(2ni(C, - z))-' - t,(z)] v=t
Therefore
Ig-1IK:5
b
"
,
M,,2nLM,
b Z
Combining this with the estimate above,
If -111':5 6.
0
For later applications we note the following corollary. It is necessary to intro-
duce the notion of a (compact) block in the (z,, ..., z,"}space C'": It is just a compact block in the underlying real space Rz", with respect to the variables
Rez1i...,Rez,",Imzt,...,Imz",. Corolimy (Approximation Theorem for Blocks). Let e > 0 and suppose Q c C' is a compact block. Then for every f e 0(Q) there is a polynomial J E C[z t, ... , z,"] so
that
If -1 1q:5 e Proof: (by induction on m). Thecase of m = 1 is clear by Theorem 1. Form > 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-1I12<-
IQ
tE Ifi -AIQ,
I J'
IQ<<_E.
Remark: The rectangle R in Theorem 1 can be replaced by an arbitrary compact set (The construction of the disk Ay which contains R and does not contain C, still works!). Correspondingly the corollary above is valid for product sets in C"` whose facto are compact and convex.
2. The Attaching Lemma for Epiinorphisms of Sheaves. As in the previous paragraph, K = R x K' is the product of a compact set K' in the z'-space CO-'
with a compact rectangle R={zeCIa<x5b,c
define the subrectangles
R-
{zeRlx<e} and R+'-{zeRlx>e}.
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
and
E
(zeEjx>e-p).
Consequently if U is a bounded open neighborhood of K' in C"i-1, then
D:=(E'nE")xU is a bounded open neighborhood of P. We choose U and p so small that all of the functions us,, vp j and a fortiori the matrices u and v are homomorphic on some
open neighborhood in C' of the compact set D = (E' n E") x U. yt P
E'
E"
d.Pt i
1
d4-
c -c-P t
n'P
a
a-P
a
b
b.P
x
94
Chapter III. Theorems A and B for Compact Blocks in C'
Since E' n E" is a compact rectangle, the approximation theorem implies that, given b > 0, there exist holomorphic functions ua; E O(C x C) which are polynomials in z so that a= 1 ,
uai - uai In < a,
.
..,q,
i= 1,.. ,p.
Using these functions we form the (q, p}matrix function u = (ua;) which is holomorphic on E" x U and, since K+ E" x U, define the section t+ e .9'. (K+) by t+ ,=t4 UIK+. By equation (1) above,
t+IP-t-IP=t+(P If one introduces the (p, p}matrix
a:=a+v(u-u) which is holomorphic on D, then
t+IP=t
(2)
Since u is near u, a is near the identity e. More precisely (see Paragraph 1.2),
la - eln
S.
If S is taken small enough, then I a - e Io will be smaller than the e of the Cartan Attaching Lemma. Thus, for b > 0 small enough, there exist two holomorphic invertible matrices c' e GL(p, !)(E' x U)) and C" E GL(p, 0(E" x U)) such that (3)
We now set
a- -*c'( K- e GL(p, (O(K-)),
C:=C"- I
IK+ E GL(p, (9(K+)),
and
t+ =t+
. c c- SoP(K+).
Then, by (3), a (P = a I P c I P and furthermore, by (2), Hence
r-= is a section in .q2°(K) with
j t-a-, on KIT+, on K+
t I K = t`a-.
Theorems A and B
§ 3.
95
Remark: Since the assumptions of Theorem 2 are invariant under exchanging
+ and -, there also exists a (q, q)-matrix a+ a GL(q, O(K+)) so that (t; , ..., tq )a+ is continuable to an element of _414(K).
The following theorem, which is essentially a straight forward application of Theorem 2, is the main tool for proving Theorems A and B on blocks. Theorem 3 (Extension Lemma for Sheaf Epimorphisms). Let .9' be an analytic sheaf on K and suppose that
h-:O'IK` -.fIK- and h+: ('1K+-.9'IK+ are analytic epimorphisms having the property that, on P = K` n K+ Im((h I P)p: OP(P) _ 9'(P)) = Im{(h+ I P)p: 09(P) - b°(P)}.
Then there exist matrices a- a GL(p, O(K" )) and a+ a GL(q, O(K+)) such that the a- and h+ a+ are extendible to sheaf ho isomorphism s gyp: Op K -+ .9' and 0: V1 K - Y. In particular, the sum rp + +G: Op+"I K - S is surjective. maps l h-
Proof It is enough to prove the existence of C. For this, let tI-, ..., tD a .9'(K-) (reap. t; , ..., t, a So(K+)) be the images under ho (reap. hp) of the canonical generators of Op(K-) (reap. M(K+)). By assumption the restriction of these sections generate the same O(P)-submodule of .9"(P). Hence, by Theorem 2,
there exists a- a GL(p, O(K-)) and there are sections ti, ..., to a O(K) so that (111K-, Thus the map tp: Op I K
..., tplK-)= (ti, ..., tP )a-.
.9', defined by
r=
tax, is an extension
a-. Since a induces an invertible germ a. a GL(p, and since h is a sheaf epimorphism, N I K - is likewise a sheaf epimorphism. Thus the sum to K, of h
(p + 0: Op+* 1 K -+ S is a sheaf epimorphism.
§ 3.
0
Theorems A and B
As in the last section (see 2.1), Q denotes a non-empty compact block in C'". Thus
Q=RxQ', where R = {z e C ja < x < b, c < y < d} is a compact rectangle in the z'= z,-plane and Q' is a compact block in the z' = (z2, ..., C". The
Chapter Ill. Theorems A and B for Compact Blocks in C
96
dimension d(Q) is defined inductively by d(Q),=d(R) + dim Q', where, in the case
ofm=1, d(R) _=
0,
if a=b and c=d
2,
if
1,
otherwise
a < b and c < d
It follows that
05d(Q)<2m, and in fact d(Q) is just the topological dimension of Q. Whenever a < b we define, for every e e [a, b], the set Q(e) == R(e) x Q',
where R(e) __ {z e R I Re(z) = e}.
Then Q(e) is a compact block in ' having dimension exactly 1 less than d(Q). Sets of this type are used in the following to make induction proofs on d(Q) possible.
1. Coherent Analytic Sheaves on Compact Blocks. An analytic sheaf $' on Q is called coherent on Q if there exists an open neighborhood U of Q in ' and a coherent analytic sheaf .9' on U such that 91 Q = Y.
If 9 and 9' are coherent 0-sheaves on Q and h:9 - .f- is an O-homomorphism, then the sheaves Jtrez h, Im h, and'oi(et h are also analytic and coherent on Q.
Proof: There exists an open neighborhood U of Q and coherent analytic ' on U such that 5? l Q = .9' and J" I Q = S. Further h:.9' -+,f' is a
sheaves
section h e F(Q, Oem(S, T)). By Theorem A.4.1 there exists an open neighbor-
hood 0 c U of B and a section h e I'(U,
J)) with 9 IQ = h. Thus
h: So -.f- has been continued to an 0-homomorphism h: 9 1 U -+ Si U. It was shown in Chapter A.2.3 that C et h, fm h, and oj(e2 1^i are all analytic and coherent on U. Therefore the desired result follows, since A '_et h = Jfez hIQ, etc.
Remark: If M is a closed subset of a complex space X, then one has the more general notion of a coherent analytic sheaf on M. The above statements also hold in this more general situation, the proofs carrying over word for word. 2. The Formulations of Theorems A and B and the Reduction of Theorem B to Theorem A. The main results for coherent analytic sheaves on compact blocks were summarized by Cartan and Serre in the form of two theorems:
Theorem A. For every coherent 0-sheaf ,' on a compact block Q c ', there exists a natural number p and an exact O-sequence
o°IQ-'-.0.
§ 3.
97
Theorems A and B
Theorem B. For every coherent 0-sheaf .9' on a compact block Q c Cm, H4(Q, ,92) = 0
for all
q >_ 1.
One can also formulate Theorem A as follows: There exist p sections in .9'(Q) whose germs at each point z c- Q generate the stalk .9's as an C. -module.
Before proceeding, we note an important immediate consequence of Theorem B:
Every exact 0-sequence on Q,
between coherent analytic sheaves induces an exact 0(Q)-sequence
Y(Q)
' g -(Q)
0.
Proof: Associated to the exact sequence 0 -.-f'ei h
S° -
-a 0,
we have the cohomology sequence
Y(Q) -'
H1(Q, Xet h)
...
We proved in Paragraph 1 that .JE'et h was a coherent analytic sheaf on Q. Thu by Theorem B H'(Q, .*'et h) = 0. Since blocks are in particular product sets in C', we already have Theorem B at our disposal in the case of compact blocks and the structure sheaf 0 (Theorem 2.4.4). Since in general H4(X, ,9'D) is isomorphic to the p-fold direct sum of H4(X, .9') with itself, we thus have
H°(Q,0°)=0,
p,q? 1.
This simple case of Theorem B makes it possible to reduce Theorem B to Theorem A in the case of compact blocks and arbitrary coherent sheaves.
Theorem I. In the case of compact blocks, Theorem A implies Theorem B.
Proof: Let 9' be a coherent 0-sheaf on Q. We will show that, given a natural number k 2: 1, there exists a coherent analytic sheaf .9'k on Q and isomorphisms H4(Q, .9')
H4+k(Q, '9k)
for all q> I.
The claim follows immediately from this, since the vanishing theorem for compact blocks implies that all of the groups H4+k(Q, b°) vanish when k >_ 3"'.
98
Chapter 111.
Theorems A and B for Comet Blocks in CO
It is enough to consider the case of k = 1, as the general case follows from this one by repetition. Assuming Theorem A, there exists p z 1 and an epimorphism h: 0° I Q -+ Y. Thus 9' h is a coherent analytic sheaf on Q and the short exact sequence
0-Y, -+ODIQ-+ gives rise to the exact cohomology sequence
-. ga(Q, 0°) -,' H4(Q, 9°)
H"+i(Q, 6i) -' H*+:(Q, OP)-- ...
Since for q >- 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) <- d. Then Ad implies B, (Theorem 1).
Furthermore, as a consequence of Bd, it follows that every exact sequence
be -+ Y .M 0 of coherent 0-sheaves on Q with d(Q) 5 d induces an exact sequence
of sections Y(Q) P(Q) -+ 0. If we denote this consequence by Fe, then it is enough to show that Ad- 1
and Fd-
1
imply
A,.
Without loss of generality we may assume that a < b. Then Q(e) is a
(d - 1)-dimensional compact block and, using the induction assumption Ad_,, there exists a natural number p(e) and an exact 0-sequence Ole)IQ(e)
There exists an open neighborhood Ue c C"' of Q(e a coherent analytic sheaf .Yon Ue withf .Y I Ue n Q = 9' I Ue n Q and a section e e I'(U1, .lt°onr(OV(e1, .')) with 1, I Q(e) = 9e. Since the sheaf Wod'et he is coherent on U. and its support does not intersect Q(e), one can find s > 0, so that the d-dimensional block
a
Q(e), -R(e), x Q', where R(e),={z e R I e - e < x < e + e) is contained in Ue and the homomorphism induced by k" h(e): pp(e) I
Q(e). -i ° I Q(e)1,
is surjective. Since the interval [a, b] is compact, the block Q can be covered by finitely many of these Q(e),'s, say Q1, ..., Q,. We may assume that these blocks come from a decomposition a = eo < e, < e, = b of the interval [a, b] with Qj _ {(z, z') e Q I ej-, < Re z < e;}. For each Qj we have an integer p1 > 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,
Then, as it should be, (doIP,+1)Y,+1 = (doIP,+1)f,+1 - (d07,)IP,+1 = aIP,+1 - 0.
107
The Vanishing Theorem H'(X, 60) - 0
§ 1.
2) We now construct the sequence 8v. The differences 01+ 11 Pv - X = Y: - Yv l P. E `'(P,)
no longer vanish (as in Theorem 4). Nevertheless they are "small":
1 Pv - #J, - qv
(*)
For every v > 1 we consider the sequence S(jv):= 8v+jI
j = 0, 1.... .
Pv - P, E S(Pv),
As one easily verifies, by direct substitution, for all
s 1jv1 - s(jv+111I P, = lv+1 I Pv - fiv
(0)
v and j.
In the third step below we will show that, for all v >- 1, the sequence s; v' I P-, 1 E
Y(P,_ 1) has a limit in ,°(P,_ 1). Let Sv be such a limit, v restrictions are continuous (by b) of Definition 6).
1. Now, all of the
Thus ( +lll lim s;v Pv-, = 6v+1 1 P,- 2,
j- ID
and consequently lim
s
j-,D
I Pv _ 1 = 61-61- 1 I P'-1. 1'1)P_1
Together with (0), this implies that
/
p{j
I ((uv - by+1)I Pv-1) - ((P +1 - Pv)IPv- 1)Iv-1 = 0.
Thus (2) follows directly from d) of Definition 6. 3) We verify here the fact that s v1 I P,_ 1 has a limit in S"(Pv_ -I). By'c) of Definition 6 it is enough to show that for all v, the sequence s1;v1 e .5P(P,.) is a Cauchy sequence in 91(P,).
Since all maps
.9'(P,) are contractions, the estimate (*) implies that
IPv - P.+ji- IIPvIv< I$v+µlPv+,.-I - Nv+µ-1L+p.-1 C
qv+y-1
for all p >- 1. As a consequence, for all i and j with J> i, qv+µ- 1 <
slv) - S(v11
p=i+t
qv
(1,.
-q
This clearly implies that s;` is a Cauchy sequence in Y(P,).
0
Chapter IV.
108
Stein Spaces
of X by compact Stein sets Pv is called a Stein exhaustion An exhaustion of X whenever it is a Stein exhaustion relative to every coherent sheaf Y. Thus in closing this section we have the following clean formulation:
Theorem 8 (Exhaustion Theorem). Every complex space X which has a Stein exhaustion is a Stein space.
§ 2.
Weak Holomorphic Convexity and Stones
In this section (analytic) stones are defined. They are useu in Section 4 for the construction of Stein exhaustions. One is naturally led to the notion of a "stone" while carrying out a careful study of the fundamental idea of holomorphically convex complex spaces. 1. The Holomorphically Convex Hull. Let X be a complex space with structure sheaf e? = Ox. We denote by red: O -, 0red X:= 0/11(0) the reduction map. Since Ored x c `Bx, every section h e O(X) determines a complex-valued continuous func-
tion red h e'(X ). Thus for every point x e X the "value" h(x) :_ (red h)(x) e C as well as the absolute value I h(x) I - I (red h)(x) I> 0 is well-defined (see Chapter A.3.5). As a consequence, for every subset M X, we have
lhIM,=sup Ih(x)I, XCU
0
Ihlk,
Since red It is continuous on X, it follows that I h IM < oo whenever M is compact.
Definition I (Holomorphically Convex Hull). For any set M in X the holorr rphically convex hull of M in X is the set
cI := n {x e X I T h(x) I< I h IM}. keAX)
We often use the more precise notation 14fx instead of M. The elementary
properties of the hull operator " are listed in the following: Theorem 2. For any subset M in X, the following hold:
1) M is closed in X. In particular, for every p e X\1N, there exists a function he0(X)w$h Ih(p)I.
2) McMandM=M.
Weak Holomorphic Convexity and Stones
§ 2.
109
3) IfMcM',then acM'. 4) If tp: Y X is a holomorphie map, then
W t(M)r c (p(Mx) The proof follows directly from the definition of We note that 1) above immediately implies (by taking powers of h) that I h(p) I and I h IM can be chosen to be arbitrarily small and large respectively. A particular consequence of 3) is that,
for arbitrary sets M, M' c X, M n M' c M n M'. Thus
M n M' = M n M', whenever M = M and M' = M'. The statement 4) contains the following:
If U c X is an open subspace of X which contains M, then My c Mx.
The hull operator satisfies a certain product rule: If X x X' is the product of the complex spaces X and X', then
MxM 'cix4' for all sets McX and M' c X'. Proof: Obviously M x X' c M x X' and analogously k x M' c X x M'.
Since M x M' c (M x X') n (X x M'), the claim follows.
In general M is bigger than M. However, in many important cases a = M. For example, for every compact block Q in C"', Q = Q.
Proof: By the above product rule it is enough to consider the case,of m = 1. In this situation, given p e C\Q, there exists a disk
T-=(z eCj Iz - aI 5 r}, such
aeC,
r>0,
that Q c T and p # T. Defining f:=z - a e 0(C),
it follows that
If(p)I > I f 6 (i.e. p 0 Ql. Z Holomorphieally Convex Spaces. The holomorphically convex hull k of a compact set K c X is by no means always compact. Spaces having this property are given special prominence by the following definition. Defpition 3 (Holomorphically Convex). A Complex space X is called holomorphically convex if the holomorphically convex hull K of any compact K e X is itself compact in X. Compact complex spaces are always holomorphically convex. If X is holomorphically convex, then so is its reduction red X = (X, 0,,,+x Furthermore,
Chapter IV. Stein Spaces
110
if X and X' are holomorphically convex, then the product X x X' is also holomorphically convex.
This follows immediately from the product rule for the hull operator since every compact set in X x X' is contained in a product K x K' of compact sets. There exists a simple sufficient condition for holomorphic convexity: Theorem 4. Let X be a complex space and suppose that, given an infinite discrete set D a X, there exists a holomorphic function h e O(X) which is unbounded on D (i.e. I h ID = co). Then X is holomorphically convex.
Proof: If K is compact in X, then I h Jx < oo for all h e O(X ). By assumption the
hull k = n (x e X I h(x) 1 S 1 h 6) doesn't contain a sequence of points which is ti
discrete in X. Since K is always closed, this implies its compactness.
El
At the end of this section we will see that the condition in Theorem 4 is also necessary for the holomorphic convexity of X. The proof of this fact is substantially more demanding. Corollary to Theorem 4. Let G be a domain in C" such that, for every boundary point p e 0G, there exists a holomorphic function f on an open neighborhood U of G with
Gn{xeUjf(x)=0}=O and f(p)=0. Then-G is holomorphically convex.
Proof: We will show that for X '= G the condition of Theorem 4 is fulfilled. For
this let D be an infinite discrete Oet in G. If D is unbounded in C", then some coordinate function of C' is unbounded on D. If D is bounded, then it has an accumulation point p e G. Let f be the function guaranteed by the assumption for the point p. Then h'=f -' I G e 0(G) is unbounded on G. It now follows immediately that every domain G in the plane C t is holomorphically. convex. Consequently every product domain
G, X G2 x
x G. c C",
G '= domain 'in C,
is holomorphically convex. In particular every open block (i.e. the interior of a block) as well as every open polycylinder in C" is holomorphically convex.
Remark: The corollary further implies that every linearly convex domain G in C'"(_ R2') is holomorphically convex. This follows since through every boundary
paint p e 8G of such a domain there exists a "supporting hyperplane" E with G n E = Q. The hyperplane E can be described by an equation l(z) + 1(;) ,= 0, where l is a linear, holomorphic function. Certainly 1(p).-0, but G q,!fz e C"(l(z)=0) _ ¢.
§ 2.
III
Weak Holomorphic Convexity and Stones
3. Stones. It is of interest in Stein theory to introduce a weakening of the notion of holomorphic convexity. We will need a condition like the following: For every compact set K in X there exists a neighborhood W of K in X such that K n W is compact. In order to better understand this condition, we begin by introducing a simplified language. Definition 5 (Stone). A pair (P, n) is called an (analytic) stone in X whenever the following conditions are satisfied:
1) P is a non-empty, compact set in X and n: X - C" is a holomorphic map of X into C'".
2) There exists a (compact) block Q in C' and an open set W in X such that
P=n-1(Q)n W.
Since P c n-'(Q) implies that P c n-1(Q) and since Q = Q, it follows that
J5
r) WcPn W= P. Thus Pc W=P.
The interior ¢ of Q has, with respect to n I W, the open set P° '= n-' (Q) n Was
preimage. We call P° the analytic interior of the stone (P, n). It is clear that all open sets W with n-'(Q) n W = P lead to the same set P. Since P° is open and contained in P, P° is contained in the interior of P, P. However it is not always the case that P° = P. For example, let X be compact and n: X - C' be the constant map x'-40. Then the pair (X, a)is a stone no matter what block Q c {0} is used. In the case dim Q < 2m, however, ¢ = 9 and consequently X° = 9. .
Theorem 6. Let K be a compact set in X. Then the following are equivalent: i) There exists an open neighborhood W of K in X so that k n W is compact. ii) There exists an open relatively compact neighborhood W of K in X such that the boundary OW does not intersect K. iii) There exists a stone (P, n) in X with Kc P.
Proof: i) . ii): This is trivial since one can take W itself to be relatively compact in X. ii)= iii): Since 8W n K - Q, given a point p e OW there exists a holomorphic function h e 0(X) such that I h lit < 1 < I h(p) I (see Theorem 2.2). Thus max(I Re h IK, (im h (K} < 1 and, raising h to a power if necessary, max(I Re h(p) I
(Im h(p) 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
<2-`<1,
Y_ f
t=0,1,....
i>,
We want to estimate the value of h(p) for p e D, say p = x,k a D;(x,k). For such a
point, p e K,+1 and thus Ih(p)I
(P) f JZ
iI f(P)
f (P)I -l.
There exists a largest index I with p e D;(x,,), where 1 < k < 1:!5 n,. For all m satisfying 1 < m 5 n p e D; (x,,,,) and consequently I h,m(p) I < n,`'2-'. Hence
.,>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 <9'(P,+,)-'.9'(P,) do not increase the semi-norms. It remains to show that condition a) is also fulfilled (i.e. for every v, .9'(X) I P, is
dense in .9'(P,)) and it is enough to verify this for v:= 1. Thus let s e .9'(P1) and b e R with b > 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 <_ It, - sP111+ Y bj. j=1
Letting j - oo, I t I P1 - s 1, < b, and thus every section s e .9'(P,) can be approximated by global sections t e S'(X ). 0
One can now combine Theorem 5 with Theorem 1.8 and Definition 3.8 and prove the main theorem of Stein theory:
124
Chapter IV. Stein Spaoes
Fundamental Theorem. Every holomorrhically complete space (X, O) is a Stein space. For every coherent analytic sheaf 9' on X, then holomorphic completeness implies the following:
A) The module of sections ."(X) generates every stalk .'z, x e X, as an Ox-module.
B) For all q 2 1, HQ(X, ,') = 0. Furthermore, the original axioms stated in Section 4 imply both A) and B).
Chapter V. Applications of Theorems A and B
In this chapter we give some of the significant classical applications of Theorems A and B. Along with the Cousin problems and the Poincare problem, we carry out a detailed discussion of Stein algebras. We also become quite involved with the problem of topologizing the module of sections .©(X) of a coherent sheaf. It is necessary for us to use facts from complex analysis which cannot be proved here. For example we need general results from dimension theory, the Riemann continuation theorems for functions holomorphic on normal spaces, and the normalization theorem.
§ 1.
Examples of Stein Spaces
In this section the idea of a Stein space is explained and clarified through recipes for construction, examples, and counterexamples. In doing this, important theorems can, to a certain extent, only be reported.
1. Standard Constructions. If X., a e A, are the connected components of a Stein space X, then for every nonempty A' c AX' = U X. is a Stein space. We !e A'
summarize in the following several other simple recipes for making Stein spaces. Theorem 1. Let (X, Ox) be Stein. Then the following hold: a) Every holomorphically convex open subspace (U, Ov) of X is Stein. b) Every closed complex subspace (Y, Oy) of X is Stein. c) Ifh E Ox(X), h # 0, and H:= {x e X (h(x) 0), then the difference X\H is Stein.
d) If f:, - X is a finite holomorphic map of a complex space Z into X, then Z is Stein.
e) If X' is Stein, then the Cartesian product X x X' is also Stein. Proof: a) This is clear, because X being holomorphically spreadable implies U is as well. b) Since the injection s : Y -. X is finite, this is a special case of d). c) If X
has no singularities, then h-' e O(X\H) is a function which is unbounded on any
Chapter V. Applications of Theorems A and B
126
sequence of points approaching H. If X is arbitrary, then the proof is too difficult to reproduce here. d) Since f is proper, Z is also holomorphically convex. The finiteness off implies Z is holomorphically spreadable as well. e) Since X and X' are holomorphically spreadable, X x X' is likewise holomorphically spreadable.
The product rule Kt x K2 c Kt X K2 implies that X x X' is holomorphically
0
convex.
Remarks: 1) Every C'", 1 < m < oo, is Stein. Thus, by l,b) above, every closed
submanifold of some C' is likewise Stein. One can in fact prove the converse statement (see [CA], p. 122 ff.):
Embedding Theorem. Every n-dimensional Stein manifold can be phically mapped onto a closed complex submanifold of C2"+'
The unit disk is even realizable as a closed complex curve-without signularities in C2.
One can also prove the following generalization of the embedding theorem for Stein spaces: Let X be a finite dimensional Stein space and p e X an arbitrary point. Then there are finitely many functions f1, ..., f,, E 0(X) with the following properties:
1) The holomorphic map f: X C", x -' (f1(x), ...&(x)), is injective and proper. In particular in the topology of C", the subspace f (X) is homeomorphic to X. 2) There exists a neighborhood U of p in X which is mapped biholomorphically by f I U onto a closed complex subspace Y of a polydisk A c C". The property 2) above is always fulfilled as long as fl p, ...J., a trp generate the ideal ttt(Op) (compare Section 4.2). 2) In 1956 K. Stein [38] showed that every unramified covering space of a Stein
space is Stein. An important generalization of this was recently proved by P. LeBarz [2]. By using the solvability of the Levi problem, he showed the following:
Theorem. Let n: Z -> X be a holomorphic map between complex spaces having the following property: Every point x e X has an open neighborhood U so that It -1(I.') is the disjoint union of complex spaces W1, W2, ... such that every induced map n, I Wy: W -+ U is finite. Then, if X is a Stein space, Z is likewise Stein.
In particular it follows that every local-analytically ramified cover of a Stein space is Stein.
3) From the beginning of the subject, the following question has held great interest in Stein theory: Is a complex space X which is exhaustable by a sequence X 1 C X2 C X3 C ... of open Stein subspaces itself a Stein space? Every compact analytic subspace of such a space X is certainly finite. Nevertheless, if a Runge
condition is not imposed on the Xv's, the space X is not necessarily holomorphically convex. In fact E. Fornaess [11] described a 3-dimensional, nonholomorphically
convex,
complex
manifold M having
an
exhaustion
§ 1.
127
Examples of Stein Spaces
by open submanifolds M. each of which is biholomorphically equivalent to the unit ball B (or the unit polydisk) in C3. The maps B 4 Mi are explicitly defined with the deciding fact being that for all s > 0 the polynomial map M1 C M2C M3 C
T: C'- C',
(z, w, t)" (z, zw + et, (zw - 1)w + 2swt),
is injective on the set {(z, w, t) t I < 1/2s}. By a refinement of the construction [13] one can even obtain a limit manifold M on which all holomorphic functions are constant. Fornaess recently found 2-dimensional examples of the same phenomena (see [12]). The Fornaess manifold M is certainly not isomorphic to a domain in C'. On the contrary, in 1938 Behnke and Stein [4] proved the following: Exhaustion Theorem. Every domain B in C' which is exhaustable by a sequence . is itself Stein.
of Stein domains B1 C B2 C
This theorem holds more generally for any unramified Riemann domain B over C"' (for this idea see Section 5 as well as [19], in particular Theorem D", p. 161). In 1956 Stein [38] showed the following: Let X be a reduced complex space and X 1 C X 2 CC... be an exhaustion of X by Stein domains. If every pair (X X, 1) is Runge (i.e. U(Xv+,) is dense in O(X,) in the topology of compact convergence), then X = U X, is Stein.
Recently A. Markoe [25] proved the following: Let. X be reduced and X 1 C X 2 Cc... be an exhaustion of X by Stein domains. Then X is Stein if and only if H1(X, O) = 0.
2. Stein Coverings. The following fact is trivial to prove, but powerful:
Let X be a complex space and U1, U2 holomorphically convex (resp. Stein) subspaces of X. Then U1 n U2 is holomorphically convex (resp. Stein). Proof: Suppose U1 and U2 are holomorphically convex and let K c U1 n U2 be compact. By Theorem IV.2.2 A
.u2cKv,nkv2
Since kv, u2 is closed in any case and kvl, kv2 are compact by assumption, Ku, v= is compact. If U, and U2 are Stein, then, being a holomorphically convex subspace of a Stein space, U1 n U2 is Stein.
The language introduced in the following is now appropriate. Definition 2 (Stein Covering). An open cover U = {U;}, i e I, of X is called Stein if U is locally finite and each Ui is Stein.
128
Chapter V.
Applicati es of Theorems A aced B
The introductory remark above shows that if U is a Stein cover, then all of the n U,, are also Stein. Hence Theorem B intersections U(io, ... , i*) = U,o n implies that such a U is acyclic for every coherent 0-sheaf fi over X. Thus the following is a consequence of the Leray Theorem.
Theorem 3. If U is a Stein cover of an arbitrary complex space X and .' is a coherent sheaf of 0-modules on X, then
H:(U, .Y) = H'(U, .5") = H'(X, `')
for allgz0. Corollary. If X is a compact complex space, then there exists a natural number a = a(X) so that for every coherent sheaf ." on X
H'(X, .") = 0
for all gZa. Proof: There exists a finite Stein cover of X !
Since every point x e X has a neighborhood basis of Stein spaces, there exist arbitrarily fine Stein covers. More precisely, every open cover 90 of X has a refinement U which is Stein.
Proof: Without loss of generality we may assume that to= {Wj}, j e J, is already locally finite and that WWj is always compact in X. By the Shrinking Theorem we may choose a cover V = {V}, j e J, of X so that VjC W for all j e J. Now each compact set 1j has a finite cover {Uji, ..., by Stein open sets where Ujk c Wj. The collection (Ujk) is therefore a Stein cover of X. 3. Differences of Complex Spaces. If A is an analytic set which is everywhere at
least 2-codimensional in a normal complex space X, then the second Riemann Continuation Theorem says that every function f which is holomorphic on X\A is continuable to all of X. If A 0 0, then the difference X \A is not holomorphically convex. This follows because, given a point p e A it is impossible to find a function f e O(X\A) which is unbounded on sequences approaching p. The following is a more general version of this remark.
Theorem 4. Let (X, OX) be an arbitrary complex space and A an analytic set in X which is at least 2-codimensional at some p e A. Then the difference space (Y, OY), where Y = X\A and Oy OX I Y, is not holomorphically convex and therefore not Stein.
Proof: It is enough to show that the reduction (Y, Orcd Y) is not holomorphically convex. Let (9, O g) be a normalization of (X, x) and : Jf - X the
1 1.
129
EumpIss of Stein Spsoa
finite normalization map (see Chapter A.3.8). Then 2-4`(A) is at least 2at every point in Z-'(p) Thta, given a section f e O,.,,(n codimensional in the function f'=f o c- Os(X \A is holomorphically continuabie acrop all points of C- 1(g Since C is finite and f is continuous at all points of C- '(V), the finitely many numbers in the set f (r - '(p)) c C are the only possible accumulation points for f on sequences approaching p. Thus everyf e Od r(Y) is bounded at p and, by 0 Theorem IV2.12, Y is therefore not holomorphimlly convex. The above theorem implies that a difference space Y'= X \A, where A #. (l, can
be Stein only if A is everywhere 1-dimensional. Such analytic sets are called analytic hypersurfaces. In a reduced space X the zero set of a coherent principal
ideal sheaf f # 0, where f, # (0) for all x e X, is an analytic hypersurface. (Recall that f is a principal ideal sheaf when f, is a principal ideal in 0, for all x e X.) In the case of complex manifolds the converse is true. That is, the sheaf of germs of holomorphic functions which vanish on an analytic bypersurface H is a principal ideal sheaf (see Chapter A.3.5). The following is a nice application of Theorem A. Theorem S. Let X be a reduced complex space and H an analytic hypersurface in
X so that the sheaf of germs of holonwrphic functions which vanish on H is a principal ideal sheaf #. Then, if X is a Stein space, it follows that Y'= X \H is also Stein.
Proof: It suffices, given p e H, to find f e O(Y) which is unbounded on sequences approaching p. For this we construct a meromorphic function h e &(X) which is holomorphic on Y and whose germ h, a -9. satisfies by a, = 1 for some a, a m(0,). It is clear that f'= h Y will do the job. We begin by defining an 0-subsheaf of #: Y- U Y. where XEX
Y,'=(f,e-0. f,f,c0,).For all xeYwe have f,= 0.,. Thus .Y,=0,for all x e Y. By assumption f is a principal ideal sheaf. Hence, given a point p e H,
there exists a neighborhood U = U(p) so that fv = g0v for some g e 0(U).
Since f, 0 0,, it follows that g, a nt(0,). Now X is reduced and H is a 1-codimensional analytic set which is nowhere dense in X. Thus no germ g x e U, is a zero divisor in 0,. Consequently g - ' e ..K (U) and Yv = Ov g-' = Ov. Hence the sheaf 5 c -K is locally-free and therefore is coherent. Since X is Stein, given a point p e H, Theorem A guarantees the existence of a
section h e b"(X) which generates the stalk .9',. In other words, there exists VP a1,so that go'=v, h,. But.Yisthesame asOon Y. So h I Y e O(Y) and, since g, a m(O,), it follows that 1 = a, h, where a, '= v, g, a m(0,). It should be remarked that Theorem 5 is also valid for arbitrary complex spaces. However the above proof must be modified. The main point is that, even
though the germs g, can be zero divisors in 0 they are still "active" elements [CAG].
In the case that X is not a complex manifold, the assumptions on H in
Applications of Theorems A and B
Chapter V.
130
Theorem 5 are quite restrictive. For example, even in a normal complex space X, a hypersurface may not be locally defined as the zero set of a single function. An instructive example for this and other phenomena is the affine cone Q3 defined
by the polynomial q:= zi + z2 + z3 + z;. The pair (Q3, OQ,)," where
Q3 ,= {z C C4 q(z) = 0} and
04, :_
j Q3,
is a 3-dimensional, normal Stein space which has only an isolated singularity at 0. The two equations z1 - iz2, z3 = iz4 define a hypersurface H in Q3 which in every neighborhood of 0 is not describable by a single function. The difference, space Q3\H is.not Stein. The reader can find a detailed discussion of this example and the higher dimensional Segre cones in [20]. Without proof we report the following:
If K is a normal Stein space and H is a hypersutface In X, then X{H is Stein in
either of the following situations:
1) dim X = 2 2) At each of its points H is locally the zero set of a single holomorphic function. Case 1)' was proved by R. R. Simka (On the complement of a curve on a Stein space of dimension two, Math. Z. 82, 63=66(1963)) using the desingularization of X. 4. The Spaces C2\{0} and
Let in z 2-and d > 2 be natural numbers with
d S m. In C" with coordinates z1, ..., z," we consider the (m - d)-dimensional analytic plane
G _d0=[(zls..., zm)C C"IZ1 =... Zd=0}. Since d Z 2 the space
C"\E"-4 is
not Stein (Theorem 4). Every set
U, := {(Z1, ..., z",) e C" I zi # 01,
1 < i < d,
is an open Stein subspace of C"\E"-4. Consequently the d sets U1i ..., U4 form an acyclic cover U of C"'\E"-4. Thus, by Theorem 3, H4(C"'\E'"-4, O)
= H;(U, D) for all q = 0, 1, ....
In particular H9(Ct\Ew-4, O) = 0
for all
q> d.
These groups cannot vanish for every q = 1, ..., d - 1, since (e.g. use Theorem C'"\E"-4 would be Stein. In fact it can be shown that
5.2)
H4-1(C"`\"-4,O)*0
and HQ(Cm`E"-4,O)=0 for
g=1,...,d-2.
§ 1.
131
Examples of Stein Spaces
We want to go through the calculations in two special cases. From now on we let d = m and consequently Eni-d = {0}. We consider the first cohomology group H'(Cm\fO}, C) = H'(U, C') = Za(U, (P)/B;(U, O),
where U = {U1, ..., U,,}.
a) Let m = 2. Then Z0(U, O) c C;(U, (0) = O(D), where D = U1 n U2. Since C;(U, C) = 0, it follows that Z;(U, C) = O(D). Furthermore C°(U, O) _ O) is given by NU1)(D C(U2) and the coboundary map da: C;(U, (fl, f2)'-'f2ID - f, I D. In this case ft and f2 have convergent Laurent series on U1 and U2 respectively:
ft = Y a,,,.z; z2
with
aM, = 0
for all
v < 0,
j2 =
with
bM, = 0
for all
t < 0.
,,.ra2
bM,, e,z2 M, v e z
The functions in O(D) are the convergent Laurent series on D = C2`(zt z2 =
f
,
v
CMV z; zz,
where the cM,.'s are not required to satisfy any extra conditions. Since such a series is in B.' (U, C) if and only if it can be written in the form f2 I D - ft I D,
Y cM, zi e2 E )(D) I cM, = 0 for all p, v with u < 0, v < 0 M,v E P
Hence the infinite dimensional C-vector space
V:= h =
dM,z; z2
1 I
E O(D) d,., c- 0 whenever p > 0 or v Z 0
M.v c S
ttt
is complementary to Ba(U, C) in O(D). Thus
Ha(U, O) = v(D)/Ba(u, O) = V. In particular we see that the C-vector space H'(C2\{0}, O) is not finite dimensional.
In the next example we show that the first cohomology groups with coefficients in O for non-Stein domains in C3 may vanish. b) Let m = 3. As earlier we write U;, = U, n U3 and U,,k = U,3 n Uk. Then Ca(u,
3
v) = ® v(U,), c.(u, v) = O(U12)(O.v(U23)® v(U31), 1-1
Chapter V.
132
Applications of Theore s A and B
and
C.'(U, 0) = O(U124 The map d. I: C; (U, 0) -. C; (U, 0) is given by (f12sf23,f31) -'1121 U123 +1231 U123 +1311 0123,
where f12 E 9(U12),123 E O(U23),
and131
C-
MUM). TbUS
Z.(U, 0) = {(f12,f23,131) E CI(U, 01 112 +123 +131 =0 on U123)
The map d;: C;(U, O) - C;(U, O) is given by (11,12,13)~ (112,123,131)
'= (121 U12 -f1I U12,f31 U23 -121 U23,f1I U31 -131 U31),
where f, e 0(U,). We now derive the vanishing of the first cohomology which was originally shown by Cartan in 1937 [8]: H1(C3\(0), 0) = H.(U, 0) = Z1(U, 0)/Im do = 0.
Proof: Given a triple (112, 123, f3s) E C;(U, 0) with 112 + 123 + 131 = 0 on
U123, we must find functions f, e O(U;), 1:!g i:5 3, so that 112 =12 -fi, f23 =13 -f2, andf31 =f1 -f3. We begin by considering the Laurent developments of the f j's:
112=a,,,,ZI2Z3, 123=
bµ, zl Z2 Z3,
Al =L.Cp,,e, ZyZ3. Since112 e O(U12j, every a,,,, vanishes for p < 0. The corresponding coefficients off23 and 131 vanish: (s) a,,,,, = 0
for p<0, b,,,,=0 for p<0, and
c,,,, = 0 for v<0.
Furthei sore f12 +123 +131 = 0. Thus we always have a,,,, + b,,,, + c,,,o = 0 and consequently
a,,,, = 0,
(ss)
b,,,, = 0, co,,. - 0,
p < 0 and v < 0, whenever v < 0 and p < 0, whenever p < 0 and p < 0.
whenever
1.
133
Exampks of stein spaces
g+ + +, g+ +, g + , etc. correspond top z 0, v k 0, and p Z 0, denote the respective subseries of g which notation, we state p z 0, v < 0, and p z 0, p < 0, v z 0, and p < 0, etc. Using this in a convenient way: equations (.) and g,,,o zj z2 e3 w e let
For every Laurent series g =
f12=f12 ++f1+2-+ +f 12 +,
+f23
f23=f 23 ++f 23 +
(°)
+f31++ +f31+-
Al =f31++
Since fl2 +f23 +f31 = 0, we have the following identities:
(oo) f2 1
-(f23++ +f 31 +), -f 23
-f3t++
f 12 ++
-f12
=f23
= f 31+
We now set r1==r31++
+f3--i+
,
f2
23 +f12
f3 =f23
Note that f 31++ E 0(C3) and f31++ E O(U1) Thus fl e O(U1). Analogously f2 e O(U2) and f3 e O(U3). Furthermore (o) and (oo) imply the following:
f2 -fl = -(f23 + +f31++) +f 12 + -f31+± sf 12 + +f 2 + +112 + -f12,
f3-f2=f23
+123 +-f12 +-f23 -+f23
A -f3 =f31 + +131++ -f23
+f23 +-f23,
=f31++ +f31 + +f31+ -f31
Since H'(C3\{O}, O) = 0 and, as was remarked above, the groups H'(C3\{O}, 0) vanish for all q a 3, Theorem 3.7 implies that H2(C3\(0}, 0) # 0.
In fad this space is even infinite dimensional. The fact that H'(C3\(O), O) = 0 is a special case of the following:
Let X be a Stein manifold, A an analytic subset of X which is at least 3codinlensional, and Y == X \A. Then H'(Y, O) = 0.
This in turn is contained in the following cohomological generalization of the second Riemann Continuation Theorem (see [33]):
Let X be a complex manifold and A an analytic subset of X which is at least
r-codimensional. Then for every locally free analytic sheaf b the restriction homomorphisms
H'(X\A, Y) --+ H4(X, Y}, are bijeetive.
q = 0, 1, ..., r - 2,
Chapter V.
134
Applications of Theorems A and B
5. Classical Examples For years there was an outstanding problem in function theory which withstood numerous attacks by many distinguished mathematicians using quite strong techniques. This so called Caratheodory conjecture asserted the existence of a non-constant analytic function on any given non-compact Riemann surface. In 1948 some work of H. Behnke and K. Stein which had been completed in 1943, appeared in the Mathematische Annalen. Among other things this contains the following generalization of the classical Runge approximation theorem ([5], p. 445): Let X be a non-compact Riemann surface and B a domain in X so that X \B has no
compact connectivity components. Then every holomorphic function on B is uniformly approximable on compact subsets by functions holomorphic on X.
From this it follows immediately that every non-compact Riemann surface is Stein.
The proof of the Behnke-Stein approximation theorem rests on a generalized Cauchy integral formula f (z) = 2ni Jf (g)A(g, z)
where the meromorphic differential form A(C, z) dC replaces the classical Cauchy kernel dC/c - z. The difficulty of constructing this kernel is overcome by using techniques from the theory of compact Riemann surfaces (e.g. Schottky Verdopplung). In the meantime proofs using methods from real analysis have been found. For example in [ARC] p. 239 there is a proof which is due to Malgrange. Using the methods and results of their work [5], Behnke and Stein showed in
1948 [6) that the Mittag-Leffier Partial Fraction Theorem and the Weierstrass Product Theorem (i.e. the Cousin Theorems) are valid on non-compact Riemann surfaces. The following lemma appears at the end of their paper: Hilfssatz C: Let D be a discrete set in a non-compact Riemann surface X. For every p e D let zp be a local coordinate at p. Suppose that at all p e D there is s,
prescribed a finite Laurent-Taylor series hp =
a,, zp, 0 < mp, np < oo. Then
there exists a function H which is meromorphic on X, holomorphic on X\D, and whose Laurent development at p with respect to zp agrees with hp up to the nD th term.
The reader can find detailed expositions on questions in this general area in the lecture -notes of A. Huckleberry (Riemann Surfaces: A View Toward Several Complex Variables, Math. Inst. Munster, WS 1974/1975) as well as the Heidelberger Taschenbuch 184 of 0. Forster (Riemann Surfaces, Springer-Verlag, Heidelberg 1977). A domain B in C' is called a domain of holomorphy if there is a function f e 0(B) which is singular at every boundary point p e a B (see [EFVJ, p. 38ff). The classical
theorem of Cartan-Thullen says the following: A domain B in C' is a domain of holomorphy if and only if it is Stein.
§ 1.
135
Examples of Stein Spaces
A reduced complex space X together with an open holomorphic map (p: X -+ Cm is called a Riemann domain over C" if every fiber (p-'(cp(x)), x e X, is
discrete in X. If (p is in addition a local homeomorphism, then X is said to be unramified. Every Riemann domain over C' is holomorphically spreadable. Moreover the following holds: Every Stein Riemann domain X over E' is a domain of holomorphy (i.e. there exists f e O(X) which is not continuable to any "properly larger" Riemann domain).
This result cannot be turned around. In fact the following was shown in [20]: There is a 2-sheeted, ramified domain of holomorphy X over C3 which is not Stein (X can be chosen as manifold).
The question of whether or not there exists such a domain over C2 appears to be still open. In his ninth work [32] Oka showed in 1953 that every unramified domain of holomorphy over cm is Stein.
On the other hand the following was shown in [18]: There exists a noes-compact 2-dimensional complex manifold Y with the following
properties: 1) Y is a ramified, finite-sheeted domain of meromorphy over the complex projective plane P2-
2) Every holomorphic function on Y is constant and in particular Y is neither a domain of holomorphy nor holomorphically convex.
In the proofs of all of these statements the notion of pseudoconvexity plays a deciding roll. We can't go any further into this matter here. The simplest non-Stein domains in E2 are the non-complete prope Reinhardt domains (see [BT], p. 52). The notched bicylinder
C2IIZII <1, 1Z21 <1}\{(Z 1,z2)eC2I(IZII -1)2
+(IZ2I -i)2
surface F. Thus the domain X - P2\F c P2 is homeomorphic to R'. Now every non-constant h E 0(X) must be singular on F. However a theorem of Hartogs states that if the singularity set of an analytic function is a 2-codimensional topological manifold, then that manifold must be an analytic hypersurface. Thus every function analytic on X must be constant.
Chapter V.
136
Applications of Theorem, A and B
6, Stem Groups. A complex lie group G is called a Stein group if the underlying complex manifold is Stein. Every abelian Lie group is complex analytically isomorphic to a product group C" x C'" x T, where T is a so called toroid group (i.e. 0(T) = C). Complex tori are obviously toroid groups. There are however non-compact toroid groups. The following is obvious: x T is holomorphically convex (resp. Stein) if and The lie group A a C" x only if the toroid part T is compact (resp. 0).
Every simply-connected, connected, solvable complex Lie group is isomorphic
as a complex manifold to some C' and is consequently Stein. All of the linear groups GL(m, C) are Stein. Every semi-simple, connected, complex Lie group is complex analytically isomorphic to a closed complex subgroup of some GL(m, C) and is therefore Stein. The Stein groups were intensively studied by Y. Matsushima [26, 27]. In particular he gave a Lie algebraic characterization of such groups. For a complex Lie group G we denote by Z°(G) the connected component of the identity e e G of the center of G.
Theorem. The following statements about complex lie groups are equivalent: i) G is holomorphieally convex (resp. Stein). ii) The toroid part of Z°(G) is compact, equal to a compact torus (resp. 0, i.e.
Z°(G) = C" x Ce').
.,'One sees in particular that G is Stein if and only if it is holomorphically spreadable. The following can also be shown: If G is connected and Stein, then the underlying complex manifold of G is affine algebraic.
It should be noted that the realization of G as an affine algebraic variety may not be related to any algebraic group structure on G.
§ 2.
The Cousin Problems and the Poincare Problem-
The Cousin problems and the Poincare problem are classical problems from 19th century complex analysis. They had a tremendous impact on the development of several complex variables. The reader can find the sheaf theoretical formulation and solution of these problems in the work [9], [35] which was published in 1953. These readings.are emphatically recommended. 1. The Cousin I Problem. For every complex space X, the 0-sheaf of germs of meramorphic functions on X is defined as the sheaf of fractions
..K - 0 where .,K = O,)n,,
x e X,
J 2.
The Cousin Problems and the Poineare Problem
137
and where N is the multiplicative set of elements of 0 which are not zero-divisors (see Chapter A.4.5). It should be noted that ..0 is not a coherent 0-sheaf. The structure sheaf 0 is an 0-subsheaf of .,dl. The quotient sheaf
is called the sheaf of germs of principal parts on X. An element of Jr(X) is called a principal part distribution on X. Associated to the short exact sequence 0 -' 0 -+ .,rtl - ..t° - 0 we have the long exact cohomology sequence
0-.o(X) --
O) -_-.H'(X, .4')
In particular, for any meromorphic function h e 4(X) we have its principal part distribution qr(h) a .l!'(X). For any principal part distribution s c- O(X) there is a cover U = {U,) of X and meromorphic functions h, E .,#(U,) with (h,) = s J U,. On U,j we have g,j hj - h, a O(U,j) and the family (g,,) is an (alternating) 1-cocycle in Z'(U, 0) which represents the cohomology class C(s) a H1(X, 0). Every family {U,, h,), h, E 4(U,), with hi - h, E O(U,f) determines a principal part distribution s e .*'(X). One calls {U,, h,} an s-representing Cousin I distribution. A meromorphic function h E ..&(X) satisfying qs(h) = s is one such that h - hi is holomorphic on U, for all i (see Section 2 of the introduction to this book).
The classical Cousin I problem (also called the additive Cousin problem) amounts to asking for a characterization of the principal part distributions which belong to meromorphic functions. The long exact cohomology sequence gives an immediate answer:
A principal part distribution s e .t!°(X) is the principal part distribution of a meromorphic function h e 4(X) (i.e. s = (p(h)) if and only if C(s) = 0 e HI (X, 67). One says that the first Cousin problem is universally solvable on X whenever (p
is surjective. Since Im to = Ker C, this is the case if and only if C:.W(X) -. H'(X, 0) is the zero map, and this happens exactly when H'(X, 0) - H'(X, ..K) is injective. We summarize these facts as follows:
lbeo 1. The Cousin I problem on a given complex space X is universally solvable if and only if the natural homomorphism H'(X, fl -+ H'(X; ..K) is injeetive. In particular the Cousin I problem is universally solvable for all spaces X with
H'(X, 0) = 0.
Theorem B implies that the additive Cousin problem is universally solvable for all Stein spaces. Oka [30] first proved this for domains of holomorphy in C'". In the case of non-compact Riemann surfaces this is just the Mittag-Leffler Theorem (see [6]).
The sufficient condition H'(X, 0) = 0 for universal solvability of the Cousin I problem is also satisfied on all compact, Kiihler manifolds X whose first Betti number is zero. Thus for example the first Cousin problem is universally solvable
Chapter V. Applications of Theorems A and B
138
for all projective rational manifolds, in particular for all complex projective
spaces P. In Section 1.4 we showed that H'(C3`{0}, 0) = 0. Thus in this non-Stein case the Cousin I problem is universally solvable. On the other hand there are principal part distributions on C2\{0}, where H'(C2\{0}, 0) * 0, which do not belong to a meromorphic function. More generally in the 2-dimensional case we have the following (compare to Theorem 5.2): The first Cousin problem is universally solvable for a domain B in C2 if and only if B is Stein.
With this we see that the sufficient condition H'(B, 0) = 0 (B a domain in C2) is also necessary for the universal solvability of the Cousin I problem.
2. The Cousin II Problem. We let 0x and ./f denote the groups of units in the rings 0X and ..KX respectively. The sets
0* °= U Ox and .ff* = XEX
J #x XEX
are open in 0 and K. They are subsheaves of abelian groups with respect to multiplication and 0* is a subsheaf of K*. The sections in 0*(X) are just the nowhere vanishing analytic functions on X. The quotient sheaf
91 =.,K*X* is called the sheaf of germs of divisors on X, the sections in 2 being called divisors.
We write the operation in the group of divisors 2(X) additively. Associated to the short exact sequence 1-+ 0* 2 -. 0, we have the long exact cohomology sequence: 1
o'(x)
#*(X) ± , (X)
, H'(X, 0*) _._, H'(X, .,![*)
Thus for any h e ,-K*(X) we have its divisor tfr(h),e 2(X). Divisors ti(h) of meromorphic functions are called principal divisors. They are denoted in the classical way by (h). It follows that
(9h) = (g) + (h) for all g, h e .K*(x). For every divisor D E 2(X) there exists a cover U = (U,} of X and meromorphic functions Al a .i(l*(U,) with t/i(h,) = D I U,. On Uy we have g,j =huh, ' E 0(Uu), where the family (g,,) is an (alternating) 1-cocycle in Z;(U, P) which represents the cohomology class q(D) a H'(X, 0*). Every family {U,, h,}, h, a ..K*(U,), with hjh, ' e 0*(U,j) determines a divisor
§ 2.
139
The Cousin Problems and the Poincarb Problem
D e 3(X). One calls (U,, hi) a D-representing Cousin II distribution. A meromor-
phic function h e ..#*(X) has divisor D (i.e. ,li(h) = D) if and only
if
hi h"' a 0*(Ui). for all i. A divisor D is called positive, denoted by D >_ 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
+420
After the above preparations the following is easy to prove.
Tboorem 3. Let X be a complex manifold on which the Cousin II problem is universally solvable. Then the following sharp form of the Poincare Theorem holds on X :
Every meromorphic function h e &(X), h * 0, is the quotient fig of two holomorphic functions f, g e O(X) whose germs f., g a O= in the (unique factorization) ring
O are relatively prime for all x e X. Proof: We may assume that X is connected. Then h e ..&*(X) and (h) a 9(X) is
well-defined. Let (h) = D+ - D- with D+, D- >- 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
again denote an arbitrary complex space. The cohomology group H'(X, C*), which by Theorem 2 plays the deciding role for determining the solvability of the Cousin II problem, is far more difficult to actually compute than is H'(X, (9). We will see that H'(X, 0*) contains topological information about X. The main tool for studying this group is the classical exponential map, the relevant properties of which are contained in the following: Lemma. For every complex space X there is an exponential map 9: 0 -' 0*. The sequence
071--+0 s+0*-+1, where Z denotes the constant sheaf of integers.
§ 2.
143
The Cousin Problems and the Poinearf Problem
Proof.' For every open set U X the algebra 0(U) is Frechet (see Section 4). For everyf e 0(U) the series Z00(f '/v!) converges to an element exp f e 0(U). It is 0
easy to see that exp(f + g) = exp f exp g and in particular exp f e 0*(U). Thus the map 9v: 0(U) -+ 0*(U), is
f -. exp 2nif,
a homomorphism. The family {9v} determines a sheaf homomorphism
9: 0 - 0*. Obviously'ker 9 = Z. In order to show equality it is enough to show that every
f e m with exp f =-1 is the zero germ. As a consequence of exp f - I we have
f=
f!=f2-g where = fJ+' . g' _ .
g:_(1/2!)+(f/3!)+...aOX.Thusf=f2g=fX82=
0 .
.
and consequently f e n mx = {0}. This completes the J=1
verification that ker 9 = Z. Every unit e ==1 + f f e nt., has a logarithm
(-1).+1 fy,
h:=1og(1 +f):= E v=1
v
with exp h = e. Thus 9 is surjective. We will now exploit the exact cohomology sequence associated to the exponential sequence. One first observes that if H'(X, Z) = 0, then the mapping 49(X) 0*(X) induced from 9 is surjective. This is just the classical theorem which states that on a (cohomologically) simply-connected manifold every non-vanishing holomorphic function has a logarithm. More important for us is the following piece of the exact cohomology sequence:
H1(X, 0) .-.-- H'(X, t2) -L H2(X Z) -. H2(X, 0) We see immediately that if H1(X, 0) = 0, then H'(X, t*) is isomorphic to a subgroup of H2(X, Z). Hence the following is an immediate consequence of Theorem 2.
Theorem S. If X is a complex space with HI (X, 0) = 0 and H'(X, Z) = 0, then the Cousin II problem is universally solvable on X.
If X is a non-compact Riemann surface then H2(X, Z) = 0. Consequently the Cousin II problem is universally solvable on every non-compact Riemann surface (this is the generalized Weierstrass product theorem, see [6]).
It is already clear here that H2(X, Z) contains important information about the solvability of Cousin II problems. Even in the 1930's it was still believed that
Chapter V. Apps=sow of Theorem A and B
144
the fundamental group xi(X) would play a great role in these considerations
(based for example on GronwalFs work). For example it is stated in [BTJ p.102 that it is completely open whether or not the Cousin I and Cousin U problems are always solvable for simply connected domains of holomorphy. In 1953 [35] Serre first gave an explicit example of such a domain where the Cousin II problem is not universally solvable. His example is
G'={zeC311zi+zi+zj-1$ <1}cC3. Clearly G is analytically isomorphic to the product of the unit disk and the {z E C3 1zi + z2+Z2 = 1}. Since Q is retractable to the 2-sphere S2, it follows that * (G) = 0 and H2(G, Z) - Z. The Cousin II problem is not universally solvable in G. In fact G n (zt - iz2) has two disjoint components neither of which can be the divisor of a meromorphic function. It is easy to generalize Theorem 5. For this purpose we compose the homomor-
aifne quadric Q
phism b in the above cohomology sequence with the homomorphism from the cohomology sequence associated to 1- 00 !?(X) -" + H;(X, .,K - 2 -+ 0. We thus obtain a. group homomorphism c:.9(X)
. H'(x,
.H2(x, z).
D
-
- a(n(D)
which associates to every divisor D a 2-dimensional integral cohomology class c(D) a H2(X, Z), the so-called characteristir class (Chern class) of D.
If X is an m-dimensional complex manifold, then c(D) is dual to the (2m - 2)-dimensional integral homology class which is determined by D (i.e. the hypersurface along with multiplicities). The following is just a consequence of the basic definitions: Theorem 6. If D is a principal divisor on X, then its Chern class c(D) vanishes. If H'(X, 0) = 0, then every divisor D with c(D) = 0 is principal.
Proof. By Theorem 2 a divisor D is principal whenever q(D) = 0. Thus principal divisors satisfy c(D) = S(n(D)) = 0. If H'(X, OJ= 0, then h is injective. Con0 sequently c(D) = 0 if and only if n(D) = 0.
In Section 3 (Theorem 3) we will see that if X is an irreducible, reduced Stein space, then every cohomology class in H2(X, Z) is in fact the Chern class of a divisor. This implies the following:
If X is an irreducible, reduced Stein space, then the Cousin 11 problem is universally solvable if and only if H2(X, Z) = 0.
5. Oka's Principle. In the case of reduced complex spaces the structure sheaf 0 is a subsheaf of the sheaf r8 of germs of continuous functions. One defines W to be the sheaf of germs of nowhere vanishing continuous functions. Obviously it fol-
1 2.
145
The cousin Problema and the Poinatt Problem
lows that 0* a W*. Again we have an exact exponential sequence as well as the commutative diagram o
z
p
0*
0
0
Z
qj
rB
0.
Associated to this we have the following commutative diagram of exact eohomology sequences:
...
_-;H.(X, p),. H.(X, p*) -:H,+t(X, Z) - H.+t(X, p)_._._....
1
(X,
11
11'
1
W)--.H*(X, qj*) -.H*+t(X, Z) -H-1- 1(X,
W)-:...
Since the sheaf W is soft (Chapter A.4.2 it follows that H'(X, W) = 0 for all i z 1 (Chapter B.1.2). Thus we have the following:
Let X be a reduced complex space such that for some q z 1 H4(X, 0) _ H*+'(X, 0) = 0 (e.g. X a Stein space). Then the injection 0* -*' ' induces an isomorphism H9(X, 0*) ac H*(X, (*).
This is a rudimentary form of the important "Oka Principle" which can be vaguely stated as follows: On a reduced Stein space X, problems which can be cohomologically formulated have only topological obstructions. In other words, such problems are holomorphically solvable if and only if they are continuously solvable.
Oka's famous theorem illustrates this principle: Theorem of Oka ([31],1939). A Cousin II Distribution {U,, h,} on a reduced Stein space X has a holomorphic solution if and only if it has a continuous solution.
Proof: The obstruction to solving the problem is the cocycle (g,j) E Z;(U, 0*), where U = { U,} and such that gj hj h; ' E 0*(U,j) represents the cohomology class of the associated divisor D. If there is a continuous solution s E p1(X), then s, h,(s I U,)-' a 1e*(U,) and since g,j = sjst ', the cocycle (g,j) is cohomologous to zero in Z;(U,'W*). But H'(X, 0*) and H'(X, W*) are canonically isomorphic. Thus q(D) = 0 and a holomorphic solution also exists. 0 The literature on the Oka principle is voluminous. For a more detailed account
we refer the reader to the report of O. Forster [15) and the list of references contained in it.
146
-
Chapter V. Applications of Theorem A and B
Divisor Classes and Locally Free Analytic Sheaves of Rank 1 § 3.
When considering complex spaces X on which the Cousin II problem is not necessarily universally solvable, one is naturally led to consider the quotient group DC(X) ;_ -q(X)/lm 0 of the group of divisors by the subgroup of principal divisors. One calls DC(X) the group of divisor classes of X and the elements are called divisor classes.
The groups DC(X) and H'(X, 0*) turn out to be isomorphic to certain groups of isomorphy classes of analytic, locally-free sheaves of rank 1 over X. These identifications permit us to derive non-trivial statements about such sheaves on one hand and about the group of divisor classes on the other. 1. Divisors and Locally Free Sheaves of Rank 1. Given a divisor D e .9(X) we can associate to it an 0-subsheaf 0(D) of the sheaf of germs of meromorphic functions: If (Uj, hi) is a Cousin II distribution for D on X, then h, h,- ' t 0*(U,j) and consequently (0 1 Uj)Jt ' Ujj = (0 1 Ujj) hi ' Ut,. Thus we have an't7-subsheaf O(D) of M with 0(D) I U. = (0 1 U,)h, '. Since hi e ..dl*(U,), it follows that (0 1 U,)h,-' 0 1 U. Thus O(D) is locally free of rank 1. Obviously O(D) does not depend on the choice of the Cousin 11 distribution for D. The collection G(.,K) of all 0-subsheaves of .t which are locally-free of rank 1
on X is a semi-group: For .9', 2' a G(4') it follows that 2' 2' e G(..A"), where 2.2' is the product sheaf with stalks (_T 2')x := 2' - Y' . The sheaf 2.2' is isomorphic to the tensor product 9 ®e 2'. For every .P e G(J!) we have a covering (U,) of X so that 2' I U, = (0 Uj) fj with fj e M* (U,). Since (O + U, j)(f, I U,,) = 2 I U,, = (01 U.;)(f, I Uf, it follows that f, fj' E 0*(Uj;). Thus .(U,,f; t) is a Cousin II distribution on X which represents a divisor D e 2(X) such that 0(D) = L Consequently G(..K) is a group; the mapping 2(X) - G(.,K), D
0(D), is a group isomorphism:
0(D + D') = 0(D) - 0(D') = 0(D) ®e O(D').
We see that D = (h) for some h e ..#*(X) if -,.id only if 0(D) = Oh' (i.e. whenever h-' is a section of 0(D)). Furthermore .9 a G(.,K) has a section s e 2(X) with Ys = Os,, for all x e X if and only if 2' is isomorphic to 0: An
isomorphism (p: 0 2 determines such a section sgp(1)E 2(X) n #*(X and every such section determines an isomorphism 0 - 2 defined by f, -+ fx sx. Thus a divisor D is principal if and only if 0(D) is isomorphic to 0.
Two sheaves 2, -T' c- G(JI) are analytically isomorphic if and only if .P' 2 -' is isomorphic to 0. Following the classical language, two divisors D, D'
§ 3.
147
Divisor Classes and Locally Free Analytic Sheaves of Rank 1
are called linearly equivalent if D' - D is a principal divisor. Thus the above considerations imply that two divisors D, D' a Q(X) are linearly equivalent if and only if their sheaves 0(D), 0(D') are isomorphic.
Introducing the group LF(..A)
G(.lf)/subsheaves which are isomorphic to C!,
the (analytic) isomorphism classes of locally free subsheaves of rank I on X, we can summarize as follows:
The isomorphism .9(X) - G(.,A"), D - (!'(D), induces an isomorphism DC(X) -+ LF(.,N)
of the group of divisor classes onto the group of isomorphy classes of sheaves in G(..6f ).
2. The Isomorphism H'(X, C"*) LF(X). Using the exact cohomology sequence associated to 1-+ 0* -+ .!!* -+ 0, one has the natural isomorphism DC(X)= 2(X)/Im 0 Im P1 E H'(X, 0*). We identify the groups DC(X) and Im q. From the results in the previous section we see that every divisor class in DC(X) = Im q E H'(X, 0*) determines an isomorphy class of locally-free sheaves of rank 1 over X. This construction can be carried out for all cohomology classes in H'(X, (0*). For this we begin with a cover U = (Ui) of X. For every cocycle (gi,) E Z;(U, (0*) we have
gil E 0*(Ui,) and gi;g;k = gik
on
Ui,k.
We define an analytic sheaf automorphism 0;;: 61 Ui, -+ 0 1 Ui, by fx -+ gi,x fx.
Thus Oi; ° bJk = 0ik on U.{k. By Chapter A.0.9 there exists on X a gluing L = (L, 9i) of the sheaves 0 I U. with respect to the automorphisms 0;;, where yi:. ' I Ui -+ 0 Ui is an isomorphism with 0i, = 9,9; ' (i.e. gi, = 9,9i '(1)). Since the maps 9i can be interpreted as 0 Ui-isomorphisms, .' is a locally-free analytic sheaf of rank 1. We call £° the sheaf glued by the cocycle (gi,).
If D n .9(X) is a divisor which is represented by a Cousin II distribution {Ui, hi}, hi E ..K*(U,), then q(D) is represented by the cocycle (gi,), gi, = h, hi- '. The associated glued sheaf is isomorphic to 0(D), because we chose (0 1 Ui)hi ' for the sheaf 2' I Ui and the isomorphism 9i: 2' I Ui -+ 0 1 U. is given by fx hiz ' f .
If U' is another cover of X and (g3) E Za(U, 0*) is a cocycle which glues together the sheaf 2', then one verifies directly that P and .P' are 0-isomorphic if and only if (gi,) and (gs) represent the same cohomology classes in H'(X, 0*). Letting LF(X) denote the set of all isomorphy classes of analytic, locally free sheaves of rank I over X, we have therefore defined an injection
y: H'(X, 0*)-. LF(X) which associates to a divisor class q(D) a H'(X, 0*) the isomorphy class of 0(D).
Chapter V.
148
Applications of Theorems A and. B
The map y is surjective: For every locally free sheaf 2' of rank 1 over X we have
a cover U = {U,} of X and the isomorphisms 3:.P I U, - 0 1 U,. The family (g+1) defined by g,1 - 319i '(1) a O(U11) is a cocycle which glues together Y. The tensor product 2 ®m 2 of locally-free sheaves of rank 1 is again a locally-
free sheaf of rank 1: If 9,: 2 U,
0 U, and 3j: .P' U, - 0 ( U, are the isomor-
phisms, then
A,`=9,® d:21U1®61u,/'I Ui -+OI U,®elu,01 Ut are isomorphisms for which Al 5T' = 9191 ' ® 919 '. Since 0 ®e 0 = 0.0 = 0, it follows that g,, = g,1' gj, is the associated cocycle. Thus 21 ®a 21' is
isomorphic to the sheaf which is glued by the product cocycle. Since isomorphic sheaves yield isomorphic tensor products, a tensor product is likewise defined in LF(Xj. In fact y(vv') = y(v)® y(v') for all v, v' a H'(X, The following is a summary of the above discussion.
Thegrem 1. The set LF(X) of (analytic) isomorphism classes of locally free sheaves of rank 1 over X is a group with respect to tensor product. The map y: H'(X, LP(X) is a group isomorphism and takes the group of divisor classes DC(X) c H'(X, onto the subgroup LF(.,11') c LF(X) of isomorphism classes which are represented by 0-subsheaves of ii. For every divisor D e 2(X) the sheaf O(D) is in the isomorphism class y(ti(D)).
As a corollary we note that
if X is a complex space with H'(X, 0) = H2(X, Z) = 0, then every locally free sheaf of rank 1 on X is free (i.e. isomorphic to the structure sheaf 0). Proof: The exponential sequence implies that H'(X, consists of one isomorphism class which contains 0.
0 and thus LF(X)
0
3. The Group of Divisor Classes on a Stein Space. The group LF(4') is in general a proper subgroup of LF(X). There is however a simple sufficient condition for an ,isomorphism class in LF(X) to be in LF(Jt ):
Lemma. Let X be reduced and 2 a locally free analytic sheaf of rank I on X which has a section s e 2(X) whose zero set is nowhere dense in X. Then there is a positive divisor D e D(X) so that 2' Is isomorphic to 0(D).
Proof. We may assume that L is glued together by isomorphisms 9,: 211 U, -+ O I U,
with
associated
cocycle
g,1= 3,9 '(1). Thus the zero sets of f
(g,,) a Z'(U, 0') defined
by
3,(s U,) a 0(U,) are nowhere dense in
U,, and consequently fj e .,&*(U,). On U,1
g,1f, =31gi'(f)=31(s)=f1.
§ 3.
Divisor Classes and Locally Free Analytic Sheaves of Rank 1
149
Thus gi1= f; f i ' e 0*(U1) and {U,, is a Cousin 11 distribution on X whose associated divisor D is positive (recall f e 0(U,)). Since 2' and 9(D) both have associated cocycle (g;;), they are isomorphic.
0
The following is now an immediate consequence of Theorem A. Theorem 2. If X is an irreducible, reduced Stein space, then LF(.,K) = LF(X). In other words, every locally free, analytic sheaf of rank 1 on X is isomorphic to 0(D)
for some positive divisor D e 2(X).
Proof: By Theorem A there exists a section s $ 0 in 2'(X ). Since X is irreducible, the zero set of s is nowhere dense in X and the claim follows from the lemma 0 above.
The isomorphism y: H'(X, 0*) -- LF(X) with y(DC(X)) = LM(X) translates the above theorem into the following: Theorem 2'. If X is an irreducible, reduced Stein space, then
DC(X) = H'(X C"*) Every divisor D E 2(X) is linearly equivalent to a positive divisor.
The next result is now an immediate consequence of Theorem 2': Theorem 3. If X is an irreducible, reduced Stein space, then every 2-dimensional, integral cohomology class in H2(X, Z) is the characteristic class c(D) of a positive divisor D e -9(X).
Proof: Let v e H2(X, Z) be given. Since H2(X, &) = 0, the map S: H1(X, tr*) -+ H2(X, Z) is surjective (i.e. there exists u e' H'(X, (n*) with o(u) = v). By Theorem 2' there exists a positive divisor D e -9(X) with tl(D) = u. Consequently c(D) = 5 i(D) = v. O
in his works [36, 37] K. Stein had already dealt (in the language of homology) with the question of which cohomology classes in H2(X, Z) = H2n_ 2(X, Z) on a Stein manifold X can appear as characteristic classes of positive divisors. In his
Habilitationsscrift (published in 1941), which signaled the fruitful entrance of methods from algebraic topology into complex analysis, he solved the problem for
polycylinders. In [37], which was published 10 years later, the question for "infinitely divisible" elements of H2(X, Z) was answered in the posh c for arbitrary Stein manifolds. Theorem 3 was proved by Serre [35] and gives the Stein result a final, optimal form.
150
§ 4.
Chapter V.
Applications of Theorems A and B
Sheaf Theoretical Characterization of Stein Spaces
By definition, every group HQ(X, So), q >_ 1, vanishes when X is a Stein space,
and .P is a coherent sheaf on X. We will now give theorems which show that a space X is Stein if only certain cohomology groups H4(X, S") are zero.
1. Cycles and Global Holomorphic Functions. The following language is convenient :
Definition I (Cycle). A map o: X -s 10 ;_ {0, 1, 2, ...} is called a cycle on X if its "support" supp o {x E X (o(x) * 0} is discrete in X (it would be more precise to call ,a a 0-dimensional, nonnegative cycle).
Every cycle o determines the analytic ideal sheaf
°(o) °= U .°l°(o)x, Xex
It follows that
9H. = m;
(ms = the maximal ideal in O.J.
Os for every x ¢ supp o.
Theorem 1. The ideal sheaf X(o) is coherent.
Proof: The "1 section" 1 E T(,XX\supp o) = O(X\supp o) generates f°(o) over X\supp o. Let p e supp o and r == o(p) > 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><;
Lemma Let q # and let (P, ir) be a distinguished block in X. Then there exists 1= l(n) E 7L and t = t(n) > 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 <_Mf. lk for all k> 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 << r.
Proof: By Banach's theorem p is open. Since W:= (h e Oy(V) I I h IQ < r) is a neighborhood of zero in Oy(V), it follows that p(W) is a neighborhood of zero in
O, 0 so that {v e O,ed x(U) I I v IK < t} c p(W). In particular, for every f e Ox(U) with I f Iu < t there exists g e Ov(V) with I g 6 :5 r and p(g) = red f. Li
Continuing with the proof of the lemma, we note that since Q C V is compact,
there exists l >- 1 so that (rad .gyp I Q c f I Q. Let r:= q/2 and let t > 0 be the number guaranteed by the above proposition. Take f e O4(U) with I f Iv < t. There exists h e Ov(V) with z(h) = f and (by the proposition) there is a g e Oy(V) with p(g) = red f and I g IQ < r. Define v:= h - g e Oy(V). Then p(v) = 0. In other words v e (it''ea pXV) = (rad fRV). Hence v` I Q e 5(Q).
Now let
It
k
(k)?,uEE(v,
1+1
(U
k=1+1,1+2,....
175
The Topology on the Module of Sections of a Coherent Sheaf
§ 6.
Then ykIQ E f(Q). If one sets
,k-.V a 0,(V),
xk,=N=0 i 1k N
k > 1,
then hk = (g + v)k = xk + yk, and fk = e(hk) = s(xk) + e(yk)
for all k > 1.
Since yk Q E f (Q), it follows that e(yk I Q) = 0. Consequently f k I P = e(xk I Q) _ &Q(xk) for k > 1. Hence, by the definition of the semi-norm I f k I,, < I xk IQ
for all
k>1.
Recalling that I g Ic < r, we estimate I xk IQ as follows:
Ixk6 _< µi
(k)
IvlQ = r' NE
Ig1Q
()(IgIhIvIQr.
Since I is fixed, the number
Mfg= max {(IgfQ'IvIQY)E R I SNSI
depends only on f Now
k
N=o P I I
<
(1 k = 2k and q = 2r. Thus ,=o111
xk Ic < rkM f2k = M f qk
for all k > 1.
Thus we have concluded the proof of the Lemma as well as that for Theorem 9. As an application of the Convergence Theorem we now show that Theorem 12 in Chapter IV.2.5 holds for arbitrary, holomorphically convex spaces X. Recalling the proof of that theorem, given an infinite discrete set D c X and a family jr,),. n of real numbers, we constructed a sequence (h,), a 0, h, a 0(X ), so that every series
E (red h,r
V.0 g(p) I
m, >_ 1,
converges compactly to a function g e r'(X) with
r, for all p E D (see the remark at the end of Chapter IV.2.5). By
Theorem 9 (i.e. the Convergence Theorem) one can choose particular m,'s so that 0 the series h," converges in 0(X) to h e 0(X). Since red Cx(X) -+ 0,ed x(X) is .=o continuous, red h = g and thus I h(p) I >_ r, for all p e D.
A second, completely pretentious proof of Theorem 12 of Chapter IV.2.5 should be noted. One uses the following "Reduction Theorem," the proof of which requires further preparation (in particular the "big coherence theorem" [CAS]):
Chapter V. Applications of Theorems A and B
176
For every weakly holomorphically convex complex space X there is a holonwrphic map p: X - X of X onto a Stein spare g so that the following hold: 1) p is proper and all fibers p-'(p(x)), x c- X are connected.
: (X) -- Grx(X ). 2) p induces an isomorphism p*(0y From this, Theorem 12 for weakly holomorphically convex spaces X follows in three lines: If D is discrete and infinite in X, then the properness of p guarantees that p(D) is discrete and infinite in X. Since X is Stein, there exists an h e e,t (X) so rp rp for all p e D. For h:= p*(h) E (1 x(X) it follows that I h(p) that h(p(p))
forallpeD. In particular we see that every weakly holomorphically convex space X is holomorphically convex.
This result also goes back to K.-W. Wiegmann: Structuren auf Quotienten komplexer Raume, Comm. Math. Hely. 44, 93-116 (1969).
§ 7.
Character Theory for Stein Algebras
A C-algebra is called a Stein algebra if it is algebraically isomorphic to a C-algebra O(X) of holomorphic functions on a Stein space X. In this section we will show by means of character theory that finite dimensional Stein spaces are completely dt.termined by their Stein algebras. We will always use X to denote a complex space and T := C (X) the C-algebra of all functions holomorphic on X. 1. Characters and Character Ideals. A C-algebra homomorphism X: T -y C is
called a character (of T), and the ideal Ker X c T is called a character ideal. Character ideals are maximal ideals in T, and a character is uniquely determined by its character ideal. Every point p E X determines the point character Xp: T -+ C, f f (p), whose ideal is Ker yp = i.(E T l .1P e nt((' p)
Every point character xp is continuous (T is assumed to be equipped with the
canonical Frcchet topology), because yp is just the composition of the continuous restriction map ((X)-(,, with the continuous quotient map C'p -+ ep/nt(c'p) = C.
Different points can determine the same point character. For example on a compact space they are all the same. On the other hand Y 'X is holomorphically separable, then XP * yw for all p, q e X with p # q.
It is possible to give characters which are not point characters. In fact, as Theorem 5.7 shows, this is the case for every non-Stein domain in C'".
§ 7.
Character Theory for Stein Algebras
177
The following remark is quite useful:
(') If x is Stein and 10 T is an ideal in T, then every finite set (fl, ..., f) a I has at least one common zero p E X.
Proof: If f1, ... , f had no common zeros, then we could find g,, ... , g, e T with
1 = i g, f (see Theorem 5.5). Since I # T, this is impossible.
0
i=1
As an immediate consequence of (s) we note that if X is Stein, then for every finitely generated, maximal ideal M in T there is a unique point p e X so that M = Ker XP
Proof. Let M = Tf1 + + Tf . By (s) the functions f1, ...,fl have a common zero p e X. Thus M c Ker XP. But the maximality of M implies that M = Ker XPa. Since X is holomorphically separable, p is uniquely determined.
l
We are now able to show the following:
Theorem 1. Let X be a Stein space, p e X, and let I be an ideal in T which is generated by finitely many functions h1, ..., h1 E T. Then the following statements are equivalent:
i) The point p is the only common zero of h 1, ..., h,, and the germs h 1,, ... , h,P E 0, generate the ideal m(OP).
ii) I'=If eTI f,Ent(P, )foralls=1,2,.... iii) I = Ker XP.
Proof: i)= ii): Let s >_ 1 be fixed. Let f be the ideal sheaf generated by the finite many sections h;t h21 h,', v, + + v1= s. Then f, = m(OP)', and
f = 0 for x * p. Thus f (X) = If e T I f, e m(OP)'}. Now the products hvl M y
-
h;', v 1 +
+ v, = s, generate the ideal P. So by Theorem 5.4 it fol-
lows that P= f (X). ii):* iii): This is clear, because Ker X = If e TI f, a m((9P)). iii) = i): For every q e X \p there exists an f e T with f (q) = 1 and f (p) = 0. Thus p is the only common zero of h1, ..., hi. By Theorem 4.3, v) there exist functions g1, ..., g, e T whose germs at p generate nt(0,). Since g1, ..., g, E Ker XP, the germs h1,, ..., h,, also generate m(OP). O
2. Finiteness Lemma for Character Ideals. The goal of this paragraph is to give a proof of the following: Theorem 2 (Finiteness Lemma). Let X be a finite dimensional Stein space. Then every character X: T -'C is a point character XP with finitely generated character ideal Ker X.
We need a dimension theoretical remark for the proof:
Chapter V. Applications of Tboorems A and 8
178
Lemma. Let A be a d-dimensional analytic set in a Stein space X, 1 S d < oo. there exists a holomorphic function g e T so that every set A n (a e X I g(x) - c), c e C, is at most (d - 1)-dimensional. Then
Proof. Let A,, i = 1, 2, ..., be the d-dimensional prime components of A. Choose two points p,, q, e A; which are not in any other prime component of A (see Chapter A.3.5-6). The set D of all such p, and q, is discrete in X. By the results in Paragraph 4.1 there exists g e T which has different values at the different points of D. If the set A< A n {x e X I g(x) = c) were d-dimensional, then A, and A would have a common d-dimensional prime component A,. This would imply that g(pi) = g(q,) - c, and this is impossible.
We now prove Theorem 2. Let X be an arbitrary character of T, and let m
dim X. If m z 1, then let g1 e T be the function guaranteed by the lemma for
A'=X. It follows that ht'=g1 -X(g1)a Ker X, and the ml-dimensional set X, '= {x e X I h, (x) = 0} is non-empty with m1 < m. If ml at 1, then let g2 be the function guaranteed by the lemma where A X 1. With h292 - X(g2) a Ker X, it follows from (s) that X2'_ {x e X I h1(x) = h2(x) = 0) * q. Furthermore, since X 2 = X 1 n (x e X I h2(x) = 0), we have M2:= dim X2 < mi. Continuing on for at most in steps, we obtain a 0-dimensional, non-empty set Xk = {x e X I hi(x) = h2(.T) _
= ha(x) = 0), where
h1, ..., hk a Ker X.
Since Xk is discrete in X, the results in Paragraph 4.1 show that there exists go a 0(T) which maps Xk injectively into C. We set ho '= go - X(go) and note that the set of common zeros of ho, ..., hk a Ker X, which by (a) must be non-empty, consists of one point p. Thus (again by (*)) h(p) = 0 for every h e Ker X. In other
words Ker X e Ker XP. By the maximality of Ker X, it follows that Ker X = Ker Xp and consequently X = Xp. By Theorem 4.3 there are finitely many more functions hk+ 1, ..., h, e T whose germs generate the ideal m(O,). Thus (by Theorem 1) Ker X. is generated by ho, h1i
.., hi a T.
Corollary. If X is Stein and finite dimensional, then every character X is continuous with closed ideal Ker X (with respect to the canonical Frechet topology on T).
This is clear, because X must be a point character. Remark 1. The assumption in Theorem 2 that X is Stein is indispensible. In fact an example of a 3-dimensional manifold Y is constructed in [18] with points p e Y whose character ideals Ker Xp in 0(Y) can not be finitely generated. The algebra 0(Y) is therefore not a Stein algebra There exists an analytic set A c Y so that the
difference Y\A is a ramified holomorphically separable domain over C', and whose function algebra O(Y\A) is isomorphic to 0(Y). In particular this shows that there are holomorphically separable domains G over C3 whose algebras 0(G) are not Stein.
§ 7.
179
Character Theory for Stein Algebras
Remark 2: The assumption in Theorem 2 that X is finite dimensional is dispersible. In fact Theorem 2 can be sharpened as follows: Sharp Version of Theorem 2. If X is a Stein space, then the follqwing statements about a maximal ideal M c T are equivalent:
i) Relative to the canonical Freshet topology M is closed in T. ii) There exists a point p E X so that M = Ker XP. iii) M is finitely generated. We want to say a few words about the proof The main difficulty is the implication i) . ii). For this one needs the following theorem of H. Cartan: Let X be Stein and I * T be a closed ideal in T. Then the ideal sheaf .F which I generates is coherent and I = 5(X),
This theorem is proved in [CAS]. As a corollary one gets the following sharpened version of (s) in Paragraph 1: (i} If X is Stein, then every closed ideal I
T has a non-empty zero set.
If I had an empty zero set, then Jz = O for all x e X (i.e..f = 0). Thus, by Cartan's theorem, I = T. The implication i) =:- ii) is immediate from
In order to verify ii)
iii), given p e X, one first constructs finitely many
functions h1, ..., h,, a Ker X,, which have no common zeros other than p on the union X' of all prime components of X which go through p (X' is Stein and finite dimensional!). Next one can find h4+1 E Ker X, which is nowhere zero on X\X'.
This is done by letting.9 be the union of the prime components of X which don't
contain p and noting that Y =_ {p) u f is a closed subspace of X. Thus the function h e O1(Y) with h(p) = 0 and h 1I = 1 is by Theorem 4.4 continuable to a function h4+t a O(X). If one adds to {h,, ..., k+1} functions g,, ..., g, a Ker XP whose germs at p generate the ideal m(49,), then by Theorem 1 one has a generating system for Ker X,. The implications iii)= ii)= i) are clear. Remark 3: In every Stein algebra T which belongs to a complex space X which contains a discrete infinite set D, there are maximal ideals which can not be finitely
generated: The set L:= (f e T I f (x) = 0 for almost all x e D) generates an ideal 1 T whose zero set is empty. By Zorn's lemma 1 is contained in a maximal ideal M c T. This ideal can not befinitely generated. It is dense in T, because if M * T then M would be an ideal which contains M as a proper subset which is contrary to the maximality of M. The quotient field TIM of any Stein algebra by a maximal ideal M is in a natural way an extension field of C which is in fact C for character ideals. It is amusing to note that TIM is (in, a highly rat-canonical way) always algebraically isomorphic to C.
Chapter V. Applications of Theorems A and B
180
3. The Homeomorphism fi: X --. X(7). For every complex space X with C-algebra T = 0(X) we call the set 1(7) of all characters x: T - C the (analytic) spectrum of X. One has the canonical map pHXp.
The map 8 is injective if and only if X is holomorphically separable. We equip X(T) with the so-called weak topology: The sets
V( ;E):={gpe-1-(T)IIV(fi)-X(fi)I <e,..., I'(f")-X{f")I <E), where e ranges over all positive real numbers and (f . .... f"} is an arbitrary finite subset of T, form a basis of open neighborhoods about the character X. It is easy to we that
X - £" (T)
is continuous.
Proof: Let p e X and V:= V(Xp; fi, ..., f"; E) be arbitrary. One can choose a neighborhood U of p in X so that for all p e U it follows that I f,.(x) - M p) I < E,
v = 1, ..., n. Thus 8(U) c V. The following is proved by applying the generalized embedding theorem of Paragraph 1.1: Theorem 3. The following stgtements about a finite-dimensional complex space X are equivalent:
i) X is Stein.
ii) E: X
1(T) is a homeomorphism.
Proof: i) . ii): Since E is injective and (by Theorem 2) is also surjective, it is enough to show the openness of E. For this let p e X and U be a neighborhood of p in X. In order to give a V:= V(Xp; fi, ...,f.; E) with 8(U) = V, we have to find functions fi, ..., f" e T and E > 0 so that
{XeXIIfi(x)-fi(p)I <E,..., Ifn(x)-f"(p)I <E}CU. Since X is finite dimensional, the generalized embedding theorem guarantees the X C" existence of finitely many functions fi, ...,f, e T so that F== (fl, maps X homeomorphically onto a closed topological subspace of C". Now the polycylinders
{(zi,...,z")FCm IIz;-f(p)I < r, v = 1,...,n} fotm a basis for the topology of C" at the point (fi (p), W i= {x E X 1.1 fr(X) -f,(p) I < r, v a 1,
...,
Thus tbe sets n},
r > 0,
Character Theory for Stein Algebra-
§ 7.
181
form a neighborhood basis at p in X. Consequently there exists an e > 0 with U.
W,
is injective, the space X is holomorphically separable. In order to show that X is holomorphically convex we let K be an arbitrary compact ii)=:* i). T.,st, since
set in X and k its holomorphically convex hull. We wish to show that R is compact in X. Thus it is enough to prove that every ultrafilter a- on R has a limit p e X. For every f e T we know that f (jV) is an ultrafilter. basis on the compact disk (zc- C I
C:Thus limf(jr)eCexists.
I
The map f r- lim f (ar), f e T, is obviously a character. Thus, since 8 is surjective, there is a point p e X so that for all f e T we have f (p) = lim f ($r). Con-
sequently E(jV) is an ultrafilter basis in 1(T) which converges in the (weak) topology of T(T) to Xp = (p). That is, 3(p) = dim 8(Y. Since 8' 1 is continuous, it follows that p = dim In the proof of i) =:- ii) above we actually proved the following: If X is a finite dimensional Stein space, then for every character X e X(T) there
exist finitely many functions h1, ..., h e Ker X so that the system ha; b), forms a neighborhood basis of X in &'(T) (Just set Ua'= V(X; hl, hi `=fi -I(#) I
,
Remark 1: The implication ii) - i) of Theorem 3 w4s proved by R. Iwahashi. It should be noted that for domains X in C" the theorem of Igusa (Theorem 5.7) shows that the bijectivity of 8 implies that X is Stein. Hence for domains if ! is bijectivo, then it is.a homeomorphism. Remark 2: As in the case of Theorem 2, the finite dimensionality assutpption in Theorem 3 is superfluous. In the proof of ii) - i dim X < oo was never used. The proof of i) = u) can be modified as follows: First we note that 3 is injective and (by the sharpened version of Theorem 2) is al$o surjective. Let X' be the union of the prime components of X which go through p. As above, since dim X' < 00, we have functions f1, ..., fa e T so that F =_ (fl, ..., f ): k'-+ Ca is injective and proper. We can also find f., 1 e Ker XD withfa+ l I (X X') = 1. Thus for sufficiently small e > 0 the domain {x e X (f,,(x) - f (p) I < e, 1 < v 5 n + 1} is contained in U.
Remark 3: The use of the embedding theorem in the proof of Theorem 3 seems unavoidable. In the proof of Theorem,2, the much weaker dimension theoretical lemma sufficed.
4. Complex Analytic Structure on X(T). Let p e X be an arbitrary point in the complex space X, and continue the notation T O(X ). The restriction homomorphism t: T -. OP maps I Ker Xo into in == m(O,} Thus for ovary n - 1, 2, ..., we
have an induced C-algebra homomorphism r,: TIP -+ (.) 'nta so that: the
Chapter V.
182
Applications of Theorems A and B
diagram below (where the vertical arrows denote the quotient homomorphisms) is commutative.
T
OP
Lemma. If X is Stein and finite dimensional, then every map r": TIP -+ OP /fit" is bijective.
Proof.- Let n be fixed. We first show that t:" is surjective. Let fp e OP/m" with preimage fP E OP be given. By Theorem 4.2 there exists h E T with hP - f, E nt". Thus the equivalence class k e T/P satisfies r"(h) = f,. In order to prove the injectivity of i", we let g e T/I" with r"(g) = 0. Take g e T to be some preimage of g. Thus r(g) a m". Now Theorem 2 implies that I is finitely
generated. Hence by Theorem 1, ii) it follows that g e I", and consequently
g=0.
Given a character X e X(T) we consider now the completion 1'z of T with
respect to the Ker X-adic topology:
tz = lim T/(Ker X)". UMof OP with respect to the m(OP}adic topology is denoted' by OP. The completion That is, OD OP/{m(Op))". The following is a consequence of the above lemma:
.'lbeorem 4. Let X be a finite dimensional Stein space and X: T - C a character. Then, ;he restriction T -Oa-t(0 induces a C-algebra isomorphism iz: q: ac bs tx-
Proof This is clear, because the maps r": Ti!" - Op/m" are isomorphisms. We now set
T"'= WVIZ--('))
T"
Then tz: Tz -i Ox-,tzr is a C-algebra homomorphism and in particular I^zand Tz are noetherian local rings over C. The natural C-homomorphism j.': T - f'z has
the (deal. ft (Ker X)Y as its kernel and maps Ker X into the maximal ideal m(T1) ,
of Tz. ihu.j (Kei X) is a 1i ,-generating system of the maximal ideal m(T1) of D1. We have the commutative diagram below, where the vertical arrows dente the inchwions
183
Character Theory for Stein Algebras
§ 7.
Tx
rr
f
Os-'(x)
f
Tx ^r
-0g--(x)
The following invariant description of the algebras Tx and fz can now be proved:
Theorem S. Let T be a finite dimensional Stein algebra, take X e 9-(T) and let 0
g
g" E jx(Ker X). Then every formal power series Y_ a,..... gi'
gn
0
a,._. e C, converges in the
topology to a uniquely determined element 00
g e Tx. Moreover g e Tx if and only if the series Yap,
"zi'
..
z"" converges
0
compactly in some neighborhood U of 0 e C". If (h1, ..., h;} c Ker X is a generating system of Ker X, then every f e Tx (resp. T is in fact a formal (resp. convergent) power series in f, := jx(h 1), ... ,. f jx(h1).
Proof: For every analytic local ring OD with completion J 5P
is a C-
epimorphism of an algebra C[[YY, ..., Y,]] of formal power series in finitely many indeterminants Y1, ..., Y, onto Otl which maps the C-subalgebra of convergent power series onto Or. It follows immediately from the elementary theory ofquo-
tient algebras that every formal power series in arbitrary elements vl, ..., v" a m(O,) represents in the m(Opyadic topology an element v e OD, and v e On if and only if the series converges. If (w1, ..., w,) c in (0,) is an ())y-generating system
of m(ao}, then it is clear that every w e Io is a power series in w1...., w,. One now observes that every T-generating system of Ker X carries over by jx to a 1, -generating system of m(Ti). Thus all of the claims of Theorem 5 follow from the above diagram. It is now easy to equip the analytic spectrum 9'(T) of a Stein algebra T with a complex analytic structure sheaf 9. First we put the weak topology on 9'(T). Then let
,p
UT x e f(r)
and let n: 9- -+ 9'(T) denote the map obtained from the "stalk projections" Tx --+-X. We introduce a topology in 9' as follows:
Let X E 9'(T) be fixed and (hl, ..., h,} be a generating system of Ker X.
If f e T, then by Theorem 5 there is a power series P = P(zl, ...,
z1)
184
Chapter V. Applications of Theorems A and B
W
Y- av, ... v, z9 0
z; I <e, 1 < i < 1)
zr', which converges in a polycyfinder A,
whose radius e = c(f) depends on f, so that f = P(fl, ..., f) with J , := j,(h,). By the observation after Theorem 3, the functions h1, ..., h, can be chosen so that the sets
6>0,
U6.={rpe!1(T)I I(p(h,)I <6, 1
form a neighborhood basis of X in X(T). For all 6 < s and rp a U6 the power series P converges in some neighborhood of 0:= (,p(h1), ... , (p(h,)) a A, Thus (p is associated in a unique way to a convergent power series Pq =
a,,.-. ,((p)z I
zV ,
0
where
P,(zt - Oht)+ ..., z, -(p(h,)) = P(z1, ..., z,) for all (z1, ..., z,) near 0. Since h; - rp(h;) E Ker (p, it follows from Theorem 5 that fq'= Po(fr - rp(h 1), ... , f - (p(h,)) a T.. Obviously ff = f We set W6(f) :- {f.I (P E U6),
6 < e = s(f),
and let the topology in 9- be generated by such W6(f )'s as f runs over 3. It is now
routine to show that
(-f-, it, X(T)) is a sheaf of C-algebras over &'(T) with germs S = TX.
Further we define
t''= U t z+ where tx: Tx
O$ _'tx)
Then we have the commutative diagram
and (E, is an isomorphism of ringed spaces (X, 0) - (X(T), ,Y). In summary we see that (X, 0) is completely reconstructable from T:
Theorem 6. Let (X, 0) be a finite dimensional Stein space with Stein algebra T = 0(X). Then the ringed space (X(T), P) is canonically isomorphic to (X, 0)
(and is thus in particular Stein).
It is easy to see that the map (X, t )'"" T
dx(X)
§ 7.
Character Theory for Stein Algebras
185
of finite dimensional Stein spaces to their Stein algebras is a contravariant functor
and the map
T ^"'(X(T), 3) of Stein algebras to their analytic spectra is likewise a contravariant functor. Thus one quickly arrives at the following result: The category of finite dimensional Stein spaces and the category of finite dimensional Stein algebras are anti-equivalent.
It turns out to be irrelevant whether or not one puts the additional condition of Frechet on Stein algebras and whether or not the morphisms are continuous. In fact the following are easy to show: Every Stein algebra T possesses a unique topology so that T is a Frechet algebra.
Every C-algebra homomorphism of a finite dimensional Stein algebra into a second Stein algebra is continuous.
In the above anti-equivalence theorem the assumption of finite dimensionality is again unnecessary. For this we refer the reader to the more detailed literature in [CAS] and [14].
Chapter VI. The Finiteness Theorem
The main purpose of this chapter is to prove the following: Finiteness Theorem (Cartan, Serre): Let X be a compact complex space. Then for every coherent sheaf If on X all of the cohomology modules H4(X, .9'), 05 q 5 oo, are finite dimensional C-vector spaces.
It is easy to sketch the idea of the proof. If 23, 213 are two Stein coverings of X
with ) < 9, then the maps in the bottom row of the following commutative diagram are bijective: Z4('IC3,,°)
Z4(93, 50)
H4(lU, .°) -- H4(93, So) =- H4(X, 9). Since 0 is surjective, it follows that w(Z°(`, Y) 113) = H4(9D, So).
Due to the compactness of X, we will be able to show that B and U3 can be chosen
(dependent on .i) so that for each q there are finitely many cocycles 1, ..., d e Z4(93, .9') with d
Y
Z4(2B, `") 193 C : C4, 0 004-, (9!, ."). 1 I
d
But , Ker tp = 8C4-1(93, So). Thus H4(!B, .9') _ Y Co(l;,) and consequently
dimcH4(X,.°)5d
1 .
The proof of Cartan and Serre makes use of the finiteness lemma of L. Schwartz concerning continuous linear maps between Frechet spaces. The proof which we give here is instead dependent on the Schwarz lemma of classical function theory.
ft.
187
Square-integrabk Holomorphic Functions
We replace the sup-norm by the L2-norm (going back to S. Bergman) on the Hilbert space of square integrable holomorphic functions. This turns out to be much easier to handle. Following the classical pattern we construct monotone orthogonal bases by means of minimal functions. The smoothing of cocycles follows in a simple way via the Banach open mapping theorem. It should be said that this method can be further developed in order to carry out a proof of the general finiteness theorem involved in the coherence of the image sheaf by proper holomorphic maps. The smoothing part is however significantly more complicated.
§ 1.
Square-integrable Holomorphic Functions
Let C' be equipped with coordinates z = x,, + iy,,, 1 S µ < m. We denote its euclidean volume element by d A := dx 1 d y, dx., d y.. For every domain B c C, the space 0,,(B) is defined to be the Hilbert space of holomorphic functions on B which are square-integrable with respect to d1. If 0(B) is given its natural Frbchet topology, then the injection OM(B) -- 0(B) is compact (Theorem 4). A Schwarz lemma is proves for the spaces 0,',(B). -
I. The Space OM(B). For every function f e 0(B) we put
IIIIIB'= j If(z)I' dA < oo. The set of square-integrable holomorphic functions on B, tl'9M(B):={fc- 0(B)I IIf1IB < oo},
is a normed C-vector space (not a C-algebra!) with II case that (1)
IIfIiB <_
Its as norm. It is always the
vol BIf IB,
where vol B denotes the euclidean volume of B. Thus if B is a bounded domain, all of the bounded functions in 0(B) belong to OM(B). The following is an immediate consequence of (1):
(g) If vol B is finite (e.g. B.bounded) and the sequence (f1) c OM(B) converges uniformly to f e 0(B), then f e 0a(B),
lim Ilf -fl(B = o and . lim (If,IIB = IIfNB. J
I
1
The reader should note that convergence on compact subsets been not in
la.oonvergence. For example tihe s sjue ce jig on the unit dsk 1} c C has no 1 Ia limit.
Chapter VI. The Finiteness Theorem
188
The above remark implies in particular that, (3) if B' is a relatively compact subdomain of a domain B, then O(B) C O,,(B') and the "restriction" O(B) -+ 0,(B') is continuous (with the Frechet topology on 0(B) and the H 11B-topology on Oj#r)). A positive-definite hermitian product is defined on (9,%(B) by
f, g e 9,,(B).
(f g)B == J J d.1, B
ha and the Schwarz inequality, I (f, g)a I S i AB uIgIlg, is valid. An explicit calculation of ( , )B is rarely possible. However for monomials on polydisks one obtains the following: The associated norm is H
'Theorem 1 (Orthogonality Relations). Let A= E, x
x E., where
rj> 0,
Ej=_(zje CI Izji
and z" _= z f! ... z",', z' = zi' ... z.; be two monomials, yj Z 0, vj z 0. Then r2r1+2
(e, e),, = 0 for u # v
and
lIzvJIa
Proof.. By Fubini's Theorem, (z", z')s
" n"` .
j;1
1
r2v_+2
..... m
v1+1
v",+1
(z 7J, z)))E,. Using zj = pea', the
claims follow from the fact that . 2x
(d1' zj1)E _
pf+v1+leurf-vd+ dq dp.
00J
2 The Bergmae Inequality. The monotone convergence theorem from integration theory implies that
11th=ais HfIB
(4)
for all f e O(B), where B' runs through all relative compact subdomains of B. The following is an easy consequence of this: Theorem 2 (Completeness Relations). If A is a polydisk centered at .0 e C", then iifAe
(0.12Hz'Js 0
for all
f_azvep(A 0
§ 1.
189
Square-integrable Holomorphic Functions
<m
Proof: On every concentric polydisk A'CA, the Taylor polynomials
a, z' 0
converge uniformly to f I A'. Then by (2) and Theorem 1 if IIA. = lim
a, z'
e
=lim('0Y, 0
la.I IIz"IIs
The claim now follows from (4): Go
If II = sup
sup Ilz'IIs =
00
laYl2IIfIlo =E 0
RICA 0
IavI2IIz"IIe
0
The completeness relations, which among other things imply that yield C"k(A) is
a Hilbert space with {z'} as an orthogonal basis, also imply the "Cauchy inequalities" Ia,.I
! IIJIIe,
v=(V1,...,V.),
0,
IIz IIe
for the II
IIe-norm. The inequality for v1 =
Since f (0) = ao and
= vm = 0 is quite important for us.
r, (see Theorem 1), it can be written as
11 1 IIe = >" r;
follows: (5)
If(0)I 5
(
nrrl ...
Ilf1I,
for all fe(0,,(A).
This implies an analogous inequality for arbitrary domains B: Theorem 3 (Bergman Inequality). Let K c B be compact and d be the euclidean distance from K to the boundary of B. Then
If Ix = dr 11f 11B for all f E (1,,(B). Proof: Let p c- K. Since dA is translation invariant, we may assume that p = 0. The polydisk A centered at p with polyradius (r1, ..., r = d "'-' d lies in B. Since If IIe < IIf IIB, (5) implies that I f (p) I S
(( d
IIfIIe for all points
p E K.
3. The Hilbert Space (9 (B). For every natural number k >_ I we have the k-fold k
direct sum 0"(B) :_ (D 9k(B) equipped with the inner product. k
Chapter VI. The Finiteness Theorem
190
where
I = (fi, ...,fk g = (gt, ..., gk), and f, gi E Ok(B). Theorem 4. For every domain B, the space &k(B) equipped with (, )B is a Hilbert space, 1 < k < oo. The injection 0k(B) -+ Ok(B) is continuous and compact (i.e. every bounded set in Ok(B) is relatively compact in Ok(B)). Proof: It is enough to consider the case k = 1. From the Bergman inequality it follows that 11 ,-convergence implies convergence on compact subsets. Thus 0,,(B) -+ 0(B) is continuous. If f1 E 01,(B) is a Cauchy sequence, then it is also a Cauchy sequence in 0(B) and therefore converges compactly to f e 0(B). Let e > 0 be given and choose no so that 11 f, -fi Ili, < e for all i, j > no. Thus, for every relatively compact subdomain B' of B, (1) implies that II
11f; -fil61 <-11f, - f 11 B' + 11f
-fIIB'
< e + (vol 9)1J21 f -f Ie..
Letting i oo one sees that II fj -f IIB < e for j >- no. Since this is true for all H CA it follows from (4) that !I f -f IIB <- e for j > 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 <- limllfi 118.
4. Saturated Sets and the Minimum Principle. A subset S c &h(B) is said to be saturated if there exists a Frechet closed set T c Ok(B) so that S = T n Ok(B).
Theorem 5 (Minimum Principle). Every non-empty saturated set S c Ok(B) contains an element g with IIg1IB = inf(Ilv0l8I v E S).
Proof: There exists a sequence g1 e S with limllgi IIB = m - inf{IIviiB I v e S}.
Since this sequence is
1
11
IlB-bounded, there is a compactly convergent sub-
sequence gj* with limit g e Ok(B). From (6) it follows that 119111<_ l llg; I1a = M. Since S is saturated, g e S and therefore OgIIB = m by the definition of m.
0
5. The Schwarz Lemma. The classical Schwarz lemma can be stated as follows: Let E, E' be disks centered at the origin in the w-plane with radii 0 < r' < r. Let a'= r'r- 1. Suppose h e 0(E) vanishes of order e at.the origin. Then IhIE'
Ihk
1 2.
Monotone Orthogonal Bases
101
The Schwarz lemma for the sup-norm in polydisks is an easy consequence: Let A {z c- C' I I zp i < r} and A' {z a C'" I I zs I < r'}, 0 < r' < r, be polydisks in CO. If the function f e 0(A) vanishes of order e at the origin, then
If
Ie.
Proof: Let c e A', c * 0. The complex line L,:= (z = cw I w E C) c C" cuts A, A' in concentric disks E _= L. n A, F- L, n A'. Considering E and E as disks in the w-plane, they have radii ry and r y respectively where y = (max I c, I )- 1. The func-
tion f I E'= h is holomorphic on E and vanishes of at least order e at the origin. Since the ratio of the radii is still a, an application of the one variable Schwarz lemma yields
0
If(c)I = Ih(c)I s IhJE.sa`IhiE
A'= {z6C'"IIz,I < r),
0
be polydisks in CO. Let a be any real number with rr'' < a < 1. Then for any k z 1, there exists a constant M > 0 (dependent on r, r' and a) with the following property: ff f e &h(A) vanishes of order e at the origin, then
IIf1I8'5a`MIIf1(8.
Proof: It is enough to consider the case where k = 1. Let r- - a- 'r and a'= {z e CO I I z, I < t}. Hence r' < -r < r and &'C A A. By the Bergman inequality, there exists a constant L so that
IfIa5LafIL for all fEa,(A). Applying the Schwarz lemma for the sup-norm to the disks A' and A, it follows that
If I. a'IfIa. Since II f IIs 5
§ 2.
vol(A') I f k, we see that M'= L Jvo1(A' does the job.
Monotone Orthogonal Bases
Using classical techniques, monotone orthogonal bases are constructed for saturated Hilbert subspaces of 0,'r(B) by means of minimal functions.
1. Moootooicity. Let B be a domain in C' and fix p e B as well as the natural number k. For short we write F := 0'(B). Every vector f = (fx, ..., ft) E F has a well-defined order at p:
°I(f )'=lnln(°o(fI ..., Or(fi)
Chapter VI. 'The Finiteness Theorem
192
A sequence (g j) c F is said to be monotone at p if oa{gj) S ... and
49,(91) S Op(92)1- ...
I'm op(gj) = co. j
For H a Hilbert subspace of 01(B), we will look for orthogonal bases (g1, g2, ...) of H which are monotone at p. Such bases were originally considered by S. Bergman. The monomials in the example of the polydisk A form (when properly indexed) a
monotone orthogonal bases for 0,,(A) at the origin. The key property of monotone orthogonal bases is expressed in the following
theorem. This will be the main ingredient for a convergence argument in Section 4.
Theorem 1. Let A and A' be polydisks centered at 0 e C' with r > r' and take a E t8 such that r'r-' < a < 1. Let k >_ 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
This relation < is a linear ordering of A with each non-empty subset of A having a
uniquely determined smallest element. The subdegree at p of a non-zero f = (fl, ..., ff) E F, wp(f }, is now defined as follows: Let f, = the Taylor development of f at p. Then (0,(f ) = min{(i, v) e A I a,,
If wp(f)
a,v(z - z(p))` be
0).
v*), then op(f) = I v' (. Thus wp(f) < wv(g) implies that op(f) <
op(g). Consequently
every sequence 91, g2, ... with wp(g1) < . . . < wp(gj) <
is monotone in p.
§ 2.
193
Monotone Orthogonal Bans
In that which follows we write w instead of w,. 3. Construction of Monotone Orthogonal Bases by Means of Minimal Functions. Maintaining the notation of the last section, we begin with the following remark : a} u {0} is a Frechet sub-
For every index a e A the set F(a) :_ (f e F I w(f) space of F.
Proof: If c e C*, then w(cf) = cuff). Furthermore w(f + g) >- min(w(f), co(g)). Thus floc) is a linear subspace of F. It is obviously closed. In F(a) we have the set
F(a)*:= If=
a,,,(z - z(p))')
e IEI
where (i*, v*) := a. It is clear that F(a)* is closed in F.
If H is a vector subspace of F, then the set A. 1_ {w(f) I f E H, f * 0} c A is countable (perhaps finite) and consequently An = {a,, a2, ...} where or, < a2 <
. We set H; := H n F(aj) and H' := H n F(aj)*,
j = 1, 2, ....
It follows that H = H,
H2 R . This sequence is strictly decreasing unless Ay is finite. Every g, e H! with Ilgjll = inf [1191l I g e H7} is called a minimal function. Existence Theorem. If H is a saturated Hilbert subspace of 0'(B), then every H' contains a minimal function.
Proof: Since H is saturated, there exists a set T which is closed in F with H = T n (ryt(B). Thus H* = (T n F(aj)*) n ((B). Since F(aj)* is closed in F, this implies that H; is saturated. Hence the minimum principle yields a minimal function in each HT, Lemma. Let H be a Hilbert subspace of Ch(B) such that every H7 contains a minimal function g j. Then H j is the orthogonal complement o f ((g,, ..., g j_ ))c in H.
Furthermore gj is the unique minimal function in H!. P r o o f : Let h E H with b:= 11h 1I 2 > 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
reR,r>0.
195
Resolution Atlases
§ 3.
Since X is compact, it is trivial to show that there exist finitely many charts (U,, 0,, p,), t = 1, ... , t,, on X such that all of the polycylinders P, are the same, and. X = Y U,.
is called an atlas of charts on X. Let q >_ 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(<21) is continuous and compact (Theorem 1.4). We define now the space C1(U, .9') c C4(U, Y) of the "squareintegrable q-cochains (with respect toll[)" by Recall
CR(U, S') := c(CR(121)) = C$(2l)/Ker r n C%(`2t ).
We let
GIs denote the Hilbert space norm on Cg(21) and set IIl,II 'inf{IIvIIAl v E Q2i), e(v) = C),
C E CZ(U, 5°).
§ 3.
Resolution Atlases
197
The following is now easy to prove:
Theorem 3. The space Cg(U,.9') is a Hilbert space with
as norm. The
injection Cf(U, .9') - C9(I1, So) is continuous and compact.
Proof: Since e is continuous, Ker s is closed in C9(21). Furthermore the injecis continuous. Thus Ker e n Cg(21) is a Hilbert subspace of tion Cg(21) and Cg(UI, .So) is a quotient Hilbert space with II 11; as its Hilbert norm. The continuity and compactness of the injection Cg(I1, Y) C9(Il, Y) follows
immediately by considering the diagram Cg(21),
- C9(21)
i
Cip, So)
I.-
- C9(II, .e).
It is obviously commutative, the injection in the first row is continuous and compact, and the projections s are open. 3. The Hilbert Space Zg(U, So). We now define the Hilbert space of "squareintegrable q-cocycles (with. respect to 21)," Zg(I1, So), as a Hilbert subspace of Cg(U, .9'). If 8 is the q-th boundary map in the cochain complex (C*(11, So), 8*), then 8 is a continuous map of Frechet spaces and Z9(11, So) = Ker 8 is a Frechet subspace of C9(I1, Y). Thus, since s is continuous,
Z9(2)- e_ 1(Z°(u, '')) is a Frechet subspace of C4(91). Recalling that the injection Cg(2l) --. CQ(91) is continuous, we therefore have the following:
The space Zg(21):=Z9(2I) n Cg(21) is a Hilbert subspace of Cg(21)= 0;,(A) which is saturated in C4(21) = 9c(A). The injection ZZ(2I) -+ ZQ(21) is continuous.
The space Zg(U, So) along with its norm analogous to that for Cg(I1, 9):
is now defined in a way which is
ZZ(U, 9) := e(Zf (21)) - Zg(21)/Ker s n Zf(2I ),yy IICII,1'-mf{IIvIIAIv E Zg(21), e(ll)= y S)+
b e Zg(U,.).
Theorem 4. The space Zg(U, 9) is a Hilbert space with II
11g, as norm. The
injection Zg(U, Y) - Z9(Ii, .9') is continuous. The injection Zg(U, .o) -. Cg(U, 9) is an isometry.
Chapter VI.
198
The Finiteness Theorem
Proof- Since Ker r. r'. ZW(`?I) is a Hilbert subspace of ZZ(`!1) (see the proof of Theorem 3), the first statement is obvious. The second statement follows (as in the proof of Theorem 3) from the commutative diagram ' Z9(9l )
ZZ(91) r
Zt(U, .9') r
.
Z4(u, Y)
It suffices to note that a is open and that the injection in the first row is continuous. Since per definition IICII. S Ntll, for all C e ZZ(U, .9'), the map ZZ(U, ,9')-. CZ(U, So) is a contraction. But, observing that CZ(91) = ZZ(91) + ZZ(91)1, it is clear that if v e CZ(21) has non-trivial projection on ZZ(9I)1, then 1 v (s > vo I,, where vo a ZZ(91) is the element of minimal length which is mapped to C e ZZ(U, .So). Hence the injection is an isometry.
Every Hilbert subspace of a Hilbert space has an orthogonal complement.
Thus every vector in the quotient space has a preimage of the same length. In the case of ZZ(U, So) this says that
given a vector C e ZZ(U, S") there exists an e-preimage v e ZZ(91) with Ilvlle =
4. Refinements. For every refinement U' of U one can define the groups ZZ(U, .9') I U'. In the following, special refinements of U are introduced. Let 91 = (U 0 and 91' = (U;, 0;, P;, be resolution atlases for ,° on X which have the same index set (1, ..., ts). Furthermore assume that the polycylinders P,, P; are both n-dimensional with 1= r (see Definition 1.2). Definition S (Refinement). The resolution atlas 91' is called a refinement of 91, denoted by 91' < 91, if 1) P; is relatively compact in P, (i.e. 2) and 3)
U,'
r' < r
=m,-'(P;)and0;=0,IU,, 1
e....I (O'I P;o...J for all ta...... tt, where P;o..,, =P;o x . x P; .
Condition 3) above makes sense, because P;0 ... ,qC P,0...,4 by 1) and P,O... "My I U,O... ,') 11';a ... ,4 = (`Y1p
(9 I U;0 ... ,')
by 2). It is easy to construct such refinements. 'Theorem 6. Every resolution atlas 91 for So possesses a refinement.
Proof: One shrinks the polycylinder P, so that the'; preimg$es still cover X. Then it is enough to restrict the maps to the smaller preimage sets.
199
Resolution Atlases
1 3.
If 21, 21* are resolution atlases for .' with associated covers U = {U,} and 11* = (Ut) such that 21 < 21*, then U,C U' for all. Thus u is a refinement of u and one has (see V.6.5) the continuous, C-linear restriction map p: C'(U *, .9') -+ C'(U, So). One also has the C-linear, continuous restriction map
a: C'(9
) - C'(91),
u t-- v l 4
which restricts component-wise the vectors in C4'(91*) Z &(A*) to A. (All objects in 21 * are starred. Thus A* = P,*,... , )It follows from the definitions of p and or (see Definition 5, 3)) that the diagram C'(21*)
C4(RI)
C'(U*, 9')
C'(U, 50)
(* )
is commutative. Writing C'(U*, .9')IU for Im p, this implies the following: 'Theorem 7. If 21 is a refinement of the resolution atlas 21*, then C'(U*, 50) I U C C1(U, Y1,
Z'(U*, )IU c zz(u, 5')
and the induced mappings Ce(U*, 5') -+ CZ(U, So), Z'(U*, 5') -+ ZJ(U, fi) are continuous. Proof: Since A C A*, 0(21*)IA12F IM a C CI(21
Z4(21 *) I A = Zi(21 l.
and the induced maps 0(21*) -+ Cj(9[), Z1(%*)-+ZZ(9I) are continuous (see Section 1.2). The commutativity of (* ) implies that Im p = c(Im a) and CJ(U, SP) e(Cj(91)) by definition. Thus- C'(U*, J)IU a Cj(U, .9). The diagram (#) induces
the commutative diagram below. The map in the first row is continuous. By the definition of the Hilbert quotient topology, e* and a are open maps. Thus the map in the second row is continuous.
C'(21 *) -------- C091)
C'(U*, 50) - C1;(U, 50) The statements about Z'(U *, .9') and Z'(U, .i) are verified analogously. 0
2tx 4.
The lniIcness Theorem
Chapter Vt
The Proof of the Finiteness Theorem
As before, X denotes a compact complex space and .y" is a coherent sheaf on X. 1. The Smoothing Lemma. If 11, '21' are resolutions for .'1 with `11' < `2I then the
cochains in Cg(U,.9') look "smoother" than those in Cg(U', Y J An important problem is to "smooth" a cocycle l;' E Zg(U'..9") by finding a cocycle e Z9(11, ,9') such that ' and determine the same cohomology class in H9(X, .9'). As in the last section, if `21",'21'.... are resolution atlases, then we let Ill, ... and I!. II 11t., ... denote the Hilbert norms on Cg(9t", .5e), Cg(`2(', 9'), ..., and Zg(`21", .c'), Z7,(`21', Y)-. respectively. II
II
II
Smoothing Lemma. Let 121, `21', 91", 9(* denote resolution atlases Jor .'I on X with < 11' < `21 < `21*. Let q c- . Then there exists a real constant L. > 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
and
IjnII
°s
Proof: The map
a: Z9(U*, .9') x C*Sr) Z9(U',
U' + an',
is continuous. Since U*, U' are Stein coverings, a is surjective (see the introduction of this chapter). Consequently Banach's Theorem says that or is an open map. The restrictions fl: Z9(U*, ,5") -+Z%(U, y), C9 '(QI', r) -. Q-'(lt", .9") are
continuous (Theorem 3.7). Thus there exists a neighborhood W of 0 in Z9(U*, Y) x C9_ '(U', 5") so that (I n,(W) and y n2(W) are contained in the unit balls of ZZ(U, So) and. Cf `'(U", f) respectively. The set a(W) is an open neighborhood of0 E Z9(U', .9"). Recalling that the injection Zg(U', J') --+ Z9(U', ;') is continuous (Theorem 3.4), it follows that .(W) n Zg(11', /') is a neighborhood of 0 E ZZ(U', Y). Thus there exists p > 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. <_ I,
and
co := y(w') = cv' I11" E
IIwHI;..
5 1.
The Proof of the Finiteness Theorem
§ 4.
201
For := c-'C e ZR(U, ,f) and q r= c-'m a Cf -'(U", So) we have
' )U" = Since I c ` '
and
S `Ii" + coq
= L 1, the proof is complete.
2. Finiteness Lemma. We continue with the notation of the smoothing lemma. Let q e N be fixed. For the sake of brevity, we write 11 11 for the norm JJ Ne on ZR(9t ). Since ZR(91) is a saturated subset of Oh"(A), it has an orthogonal basis -
P,, g2,
...) which is monotone at the origin. We may assume that +% 11 = 1 for all j.
For the polydisks A', A we have r' < r. Let a e 68 be such that r'r-' < a < 1 and let M > 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 <_ llvj 11 (Schwarz inequality), m
c; = Y (vj, g,) a C,
(6)
1 < i < d.
J-0
By continuity, lim
0. Thus, putting (4), (5) and (6)
0 implies Jim d
together, Z U" _
ci g, + dq.
We now construct the sequences v0, v,, ... and q,, q2, ... using an inductive d
procedure. Suppose vj, tlj have been constructed. Then wj'= vj -
(vj, gj)g;. Let
W'- wj I 0' E ZW(21') and let E : ZZ(2t') -+ ZR(U ", Y) be the projection map which is in fact a contraction. From the smoothing lemma we deduce the existence of E ZJ(U, .50) and qj+i e CC-'(U", .) so that with
(7)
< Llle'(w )II,..
Since Zf (U, Y) carries the e-quotient norm from Zf (U, ,9'), there is a preimage vector vj+t a Z%(2t) of such that 11vj+1 11 = (see Paragraph 2.2). Now e'(w') = s(w j) I U' (see diagram (*) in Section 3.4). Thus v j+, = I U" and wj = e'(w')IU". Since e' is a contraction, 11E (w')11, < 11
Furthermore 11w
Ilwj Ila.. Then (7) implies (2) with the estimates 11vj+1 11 <- Lllwjlls,
and
11gj+t 11:- < Lllwj11A,.
Since l1wj 11, < Ma`Nvj 11 (see (1) above) and t = LMa`, Ilvj+t 11 !5; tllvi11,
Ilgj+1 1110- s tlivJ11.
The induction assumption (3) now yields the desired estimates for vj+1, qj+t 3. Proof of the Finiteness Theorem.
Finiteness Theorem. If X is a compact complex space and .' is a coherent analytic sheaf on X, then every cohomology module HQ(X, .5"), 0 <_ q Sao, is finite dimensional.
§ 4.
203
The Proof of the Finiteness Theorem
Proof: From 3.3 we have four resolution atlases 91" < 1Il' < Ill < 121* for Y. Theorem 3.7 implies that Z9(11*, . ')IU c Zg(U, SI). Now there exists a positive integer d so that we can apply the finiteness lemma: There exist cocycles g,, . . , gd E Z9(tt", .9') so that d
z°(U*
L9++aC9
Hence the cohomology classes determined by g,, ..., 9d generate H9(U", .9'), and, since U" is a Stein cover, dime H9(X, .9') < d < oo (see the introduction to this chapter, where %; corresponds to U", `213 to U* and ;
The finiteness theorem is the foundation for the theory of compact Riemann surfaces which is presented in the next chapter. In this special situation only the groups H°(X, .9') and H'(X, 50) for locally free sheaves play a role. If one is only interested in Riemann surfaces, then the above considerations for resolution atlases are technically simpler. A sketch of a proof of the finiteness theorem for this special case, which goes along the lines of the Cartan-Serre proof, can be found in the book of R. Gunning: Lectures on Riemann Surfaces, Princeton University Press, 1966, p. 59 if. Instead of Frechet spaces, the Hilbert spaces of square integrable cocycles are likewise used. Instead of monotone orthogonal bases, the lemma of L. Schwarz for Hilbert spaces (which is substantially easier than the Frechet version) is used.
Chapter VII. Compact Riemann Surfaces
In the theory of compact Riemann surfaces it is possible to make particularly elegant applications of the finiteness theorem. For such considerations we will always let X denote a connected, compact Riemann surface with structure sheaf (. With script letters like 5) we will denote, as before, coherent analytic sheaves over X. If the support of such a sheaf is finite then ,Y- will usually be written. For such a
sheaf it is easy to see that H1(X, ) = (0). The symbols .IF, Ir are reserved for locally free 0-sheaves. The letter 2' is usual exclusively for locally free sheaves of rank 1. All tensor products are formed over tr. Since X is 1-dimensional, every stalk W. of the sheaf .,# of germs of meromorphic functions on X is a discrete valuation field with respect to the order function o, Recall that, for h e .,4= and t e rn. a local coordinate, o,,(h) = n, where h = t"e and e e V , is a unit. The order of the identically zero germ is defined to be infinity. The valuation ring associated to ox is (1)= with in z as maximal ideal. The goal of this chapter is the derivation of the Riemann-Roch theorem along with the Serre duality theorem. Further a criterion for the splitting of locally free sheaves is proved. An immediate corollary of this is the classification of the locally free sheaves over the Riemann sphere.
§ 1.
Divisors and Locally Free Sheaves
Every locally free sheaf .F is a subsheaf of the 6-sheaf,
of germs of meromorphic sections of F. We identify every stalk .:F f with !lx X = U t".Fx, where t e M., is a local coordinate at x. The sheaf.t ' is not nEz
coherent. However it contains important locally free subsheaves which are related to F. These will be introduced in this section. We write ' (X )* (resp..y (X )*) for
5 (X) minus the identically zero section (resp..F(X) minus the identically zero section).
§ t.
205
Divisors and Locally Free Sheaves
0. Divisors. In Chapter V.2.2, we considered the exact sequence
for arbitrary complex spaces. The sheaf.9:_ M *10* is called the sheaf of germs of divisors. In the case of a Riemann surface, every stalk -9x is isomorphic to Z. and every non-trivial s e -9(U) over an open set U, has discrete support I s I in U. This is the so-called skyscraper property of _Q. For a compact Riemann surface X, the divisor group,
Div X:=9(X), is canonically isomorphic to the free abelian group generated by the points x e X. Consequently every divisor D is of the form
D = xeX I x -Y, where nx e Z and nx = 0 for almost all x. Throughout we write (.,(D) instead of nx. The integer
deg D:= Y f x(D) xeX
is called the degree of D. The mapping Div X - d, which associates deg D to D, is a group epimorphism. If ox(D) >- 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,
i(D) = g - 1 - deg D.
One sees that, with the exception of the case g = 0 and deg D = -1,
H1(X, 0(D)) * (0) for all divisors of negative degree. Furthermore, lim des D-. -,o
dime H'(X, ((D)) = oo.
Remark: The existence of meromorphic functions which have poles (perhaps of high order) only at a prescribed point is guaranteed by the results in Paragraph 2. The improvement here is that the minimum order which can be prescribed can be estimated independent of the point p by the genus.
213
The Structure of Locally Free Sheaves
§ 4.
§ 4.
The Structure of Locally Free Sheaves
We will show that every locally free sheaf, which is not the zero sheaf, contains locally free subsheaves of the form C(D). This theorem is the most important aid in the study of general locally free sheaves (see, for example, the supplement of this section as well as Section 8).
1. Locally Free Subsheaves. The considerations of this section are formal in nature. The following language is useful: Definition 1 (Locally free subsheaves). An analytic subsheaf F' of a locally free sheaf .F is called a locally free subsheaf of .F if
0) F' is itself locally free. 1) The quotient sheaf .9r/.9r' is locally free. The rank equation for a locally free subsheaf .y' of a locally free sheaf .F,
rk .F = rk ,F' + rk is an immediate consequence of the definition. Thus every locally free sheaf 2' of rank 1 contains only 0 and 2' as locally free subsheaves. The requirement 1) in the above definition is quite restrictive. For example a germ t,, e 9x always generates a free submodule t,, (' in J F, but the quotient module .°t,, /tz p'x is in general not free. For an explicit example, take 9 = 0 and t a non-unit. On the other hand if t,, is a unit then there is no problem: If t,, E f, and o(tz) = 0 then 9 /t 0,, is a free Ox-module. Proof.- Let F,,, = 0Y and is = (t,, ..., tl), t, E
Since o(t,,) = 0, some t,, say t1,
is a unit. Let e- t i f and define a: Ox - 0x-1 by
a(fi,...,fr)=1f:-eflt2,...,f,-ef1t.) Thus or is an epimorphism with kernel 0 t,,. It is easy to find locally free subsheaves in
13 .9r':
Theorem 2 Let A be locally free and D the divisor of a meromorphic section
s e A'(X). Assume s * 0. Then the sheaf .', defined by 2 O(D)s = 0(D) is a locally f r e e s u b s h e a f o f F. The section s is always in 2 (X) and, when s e .9r(X
se2'(X).
Proof: Since o(h., s,,) = o(h,,) + o,r(D) > 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-+
Proof: This mapping is obviously additive. Let h e .#(X)' F e R and (.o e f1Q(X) be given. Then by 0) in Theorem 4 and by the definition of hw*,
(hw)*(F) = = <w, hF> = w*(hF) = hw*(F).
From the residue theorem we see that ..11(X) c ker co*. Moreover, if w e f1(D)(X) then R(- D) c ker co* (Theorem 4, 1)). Thus, if co e f1(D)(X ), co* induces a C-linear form, OD((o): R; (R(- D) + . /!(X )) - C.
In other words, if co e Q(D)(X) then Oo(w) e I(-D)* = J(-D). Thus, for every divisor D e Div X. we obtain a C-linear map
0p: Q(D)(X) - J(- D). This mapping extends in a unique way to a C-linear map
0: fZ'(X)-+J. The reader should note that, if w E f1(D) n f2(D')(X ), the elements OD(w) and O0 ((u) agree in J. The following lemma is the preparatory step in the duality theorem: Lemma 6. The
mapping 0
is
. #(X)-linear.
If O(w) e J(- D),
then
w e f1(D)(X ).
Proof : The.11(X)-linearity of 0 follows from the definition of the .1&(X )-vector space structure on J (see Theorem 2) and the ,.f!(X )-linearity of wr-' w* (see the above remarks). It remains to prove the last claim. Let p e X and n:=,,.p(w) + 1. Define FO := (.fx) E R by fx := 0 for x * p and fp = t-", where t e mp is a local coordinate at p. Clearly e,(fp(op) = -1 and thus w*(F0) = Resp(fwp) 0. Now, since O(w) e J(-D), co* vanishes on R(-D).
Thus F. 0 R(-D). Equivalently fP 0 C-(-D)P or,
in
I'P(- D) < 0. This is the same as saying - n -
0. In other words oP(w) +
other words, -.(fp) +
,c. (D) >t 0 for all p. Thus co c- f1(D)(X ).
The following is now immediate. Theorem 7. The maps 0- : f1°°(X) -+ J and O,,: f1(D)(X) - J(- D) are bijectire.
Proof: If O(w) = 0 then, for every D E Div X, O(w) e J(-D). Thus, by Lemma 6, co c- C1(D)(X) for all such D. Since n f1(D)(X) = (0), w = 0 and 0 is D therefore injective.
The Riamann-Roch Theorem (Final Version)
1.
-' ? 1
By Theorem 3, dim_H(%) J S 1. Since 0'(X) # (0). the .!/(X)-monomorphism 0 is automatically surjective. Let D e Div X be given. Then, as the restriction of On to f)(D)(X), 0, is injective. Every A E J(-D) has a pre-image under n, w e 0'(X). Again by Lemma 6. D W e Q(D)(X ). Thus 0p: f)(D)(X) - J( - D) is bijective. Finally the duality theorem is an easy consequence of Theorems 1 and 7. .
Theorem 8 (The duality theorem). Let X he a compact Riemunn surface and D E Div X. Then there exists a natural C-isomorphism H°(X, Q(D))
H' (X, ((- D))*.
Proof: By Theorem 1 there is a natural C-isomorphism between H' (X, (r(- D)) D))*, of the dual and I(-- D). This induces a C-isomorphism, J(- D) H' (X, spaces. Composition with 0D gives us our isomorphism: Ho (X, n(D)) . J (- D) -» H' (X, C (-. D))*.
In this form the duality theorem is a classical theorem in the subject of algebraic curves. The more general form (for complex manifolds of higher dimension which are not necessarily compact) was first formulated and proved by J.-P Scrre in 1954 (Un theoreme de duality, Comm. Math. Helv. 29, 9--26 (1955)).
§ 7.
The Riemann-Roch Theorem (Final Version)
The results of this section are consequences of Theorem 3.1 and the duality theorem. The strength of the Riemann-Roch theorem will he demonstrated by some selected (classical) applications. We will always use K to denote the canonical divisor. The notation 1(D) and i(D), which was introduced in Section 3. will be consistently applied. 1. The Equation 1(D) = l(K - D). Since every finite dimensional vector space
has the same dimension as its dual space, the duality theorem implies the following:
i(D) = dime H'(X, 6(D)) = dime H"(X, f)(-D)). Thus i(D) is the number of linearly nidependent meromorphie differential fornis (
such that (to) >- 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))
. it.): Let t, a Y. be such that Y, = 0, - t,. By i) there exist sections
s e 2(X) and germs f, 0, 1Sµ m, so that t, _ F f,,,s,,,. Thus some s,,,(x) E C must differ from 0. Hence for that section
I
6.
ii)= iii): Every germ t, e 2, with o(t,) = 0 generates 2, as an 0,-module.
228
Ebapter VII.
iii)
Compact Riemann Surfaces
i): Trivial.
ii). iv): The sheaf 9" which is defined by 0 - 2'(-x)- 2- 5- -+0 has support only at x where its stalk is C. Thus we have the following cohomology sequence:
o ---
H'(X, 2'(-x))-H'(X, 2')-'0 Therefore there is a section s e 2'(X) with o(sx) = 0 if and only if
2'(-x)(x) F .T(X), This is equivalent to H'(x, 2'(-x))-+H'(x, .P) being injective.
Theorem 4 (Theorem A). Let D e Div X with deg D >- 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)<1.
5. The Existence of Meromorphic Differential Forms. The following existence theorems are immediate consequences of Theorem A for differential forms: Let D e Div X with deg D >- 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, <
1 and w is a gap value if and only if 1,,, = lw_,.
Proof: In the exact sequence 0 - H°(X, C" (vp)) --, H°(x, Cc ((v + 1)p))
(X) -+ .
the space .T(X) =
v = tP((v + 1)p)p/C(vp)p is 1-dimensional. This implies that lv < 1,., , 5 1, + 1. By definition w is a gap value if and only if every h c- &(wp)(X ) is already in p'((w - 1)p)(X). That is, C(wp)(X) = C'((w - 1)p)(X) or equivalently
Iw = 1.-,-
If v > 2g - I it follows from Theorem 3 that 1, = v + 1 - g. Thus for v > 2g, 1, = I,-, + 1 and there are no gap values greater than 2g - 1. One can improve this remark: Theorem 5 (Weierstrass gap theorem). Let X be a compact Riemann surface with
genus g > 0. Then, for every p E X, there exist exactly g gap values, w,, ... , w9, which we order so that
1=w, <w2
1.
Proof: By Theorem 3, 12,-, = g. Hence 1 < 1° < 1, < < 129 _ , = g. Since li+, - li < 1, there must be exactly g - 1 indices i such that li+, - 1; = 1. Thus, since to = 1 and 12p_ 1 = g, there must beg indices i such that 1;+, = li. That is, there are exactly g gap values.
7. Theorems A and B for Locally Free Sheaves. We call a locally free sheaf "Stein" if Theorems A and B are valid: A) H°(X, F) generates every stalk F. as an Car module.
B) H'(X, F) = (0).
Chapter VII. Compact Riemann Surfaces
230
Using this terminology we have the following:
Lemma 6. Let 0 - Y F - 4 - 0 be a sequence of locally free sheaves. Assume further that 2 and I are Stein. Then .F is also Stein.
Proof: Since H'(X, .P) = H'(X, 9) _ (0), it follows immediately from the exact cohomology sequence that H'(X, f) = (0). Further we consider the commutative diagram 0
. H°(X, 2')
p
H°(X, 9) -' 0
H°(X, .F)
Y.
0
x
1x
x
0
where the rows are exact and the maps in the columns associate to a section its germ at x. By assumption, the images of Ax and yx generate (as Os modules) Px and 4x respectively. It follows therefore that the, image of H°(X, F) under (p. generates .,F,r as an Ox -module.
Theorem 7. Given a locally free sheaf over a Riemann surface X, there exists a natural number n'-so that, for every D E Div X with deg D >- n', ',the sheaf flD) is .
Stein.
Proof: (by induction on the rank r of .f). Since every locally free sheaf of rank 1 is isomorphic to a sheaf of the type G1,(D), the case of r = 1 handled by Theorem 3 and 4. Now we assume that r > 1 and that the statement holds for sheaves of rank at most r - 1. By the "subsheaf theorem" (Theorem 4.3), there
exist locally free sheaves 2 and 4 of rank 1 and r - 1 so that the following sequence is exact for every D e Div X :
0- 2(D)By the induction assumption and Lemma 6 above, there exists n' a 1 so that, for deg D >_ n', F(D) is Stein. The following is H°(X, 0(D)) = (0).
analogous
to the
fact
that
if deg D < 0,
then
Theorem S. Let F.. be a locally free sheaf over a compact Riemann surface X.
Then there exists n- a 71 so that for every D e Div X with deg D < n-, H°(X, F(D)) = (0).
Proof: (by induction on the rank r of F). If the rank of F is I then F = 0(D') for some D' a Div X and n- := -deg D' has the desired property. Ifrank F > 1 then, as in the proof of Theorem 7, we choose locally free sheaves 2' and V of rank
§ 7.
The Riemann Roch Theorem (Final Version)
231
I and r - 7 respectively so that, for every D e Div X, we have the exact sequence 0
Y(D)
.F(D) -+W(D) - 0.
The claim follows immediately from the induction hypothesis and the exact cohomology sequence. 8. The Hodge Decomposition of H'(X, C). We want to develop a better understanding of the vector space H'(X, C) = CO. As earlier, we begin with the resolution of'the constant sheaf C :
Since H°(X, C) = H°(X, (0) = C and H2(X, 0) = (0), the associated cohomology sequence yields
0-fl(X)- a -H'(X, C)--+H'(X, 0)--' H'(X, S2) B H2(X, C)-O. But H2(X, C) = H'(X, 0) = C. Hence fi is bijective and we therefore have the exact sequence
0-!Q(X)-
H'(X,C)-H'(X,&1)-->0.
The map a can be explicitely described in the following way: Let CO e f2(X) and
U = {U,} a covering of X by contractable neighborhoods. Then there exists j e 0(U,) so that df, = co (u, The function a,j:=f - j is therefore constant on U. (we take these intersec(ions to be connected). The family {a,j(w)} forms a 1-cocycle in Z'(U, C). This cocycle represents the cohomology class a(w) e H'(X, C). Since R c C, H'(X, R) is an R-vector subspace of H'(X, C). We now. show
Lemma 9. IM(a) n H'(X, R) = (0). Proof: Suppose that a(w) E H'(X, R). From the above description of a it follows that there is a covering {U;) of X and functions j e (((U,) such that dj = to (v, and every function f; -f is constant and real on U,;. Let g, exp(27r f ). Then ( g, 12 - (g; 12 on U,; and therefore ((g, (2) determines a real valued continuous function, g, on X. By the maximum principle g is identically constant and
thusw=0. There are of course many C-vector spaces V in H'(X, C) so that V ® Im a = H'(X, C). However there is one particular V that is quite natural. For this we consider the "conjugate" resolution of C,
Chapter VIZ.
232
Compact Riemann Surfaces
Here ( 1= n (see 11.2.3, in particular the diagram). For reasons similar to those above, the associated cohomology sequence is
0-.C (X).....H'(X,C)-.HI(X, t Using this we prove
Theorem 10 (The Hodge decomposition). The C-vector space H'(X, C) is the direct sum of the spaces Im a and Im &, where &: C(X)-+H1(X, C) = a(f)(X))
Proof: Since H'(X, C) = C2' and Im at - Im & . C', it is enough to show that Im a n Im & = (0). The conjugation C - C, defined by cr. e, determines an Rlinear involution a: H'(X, C) -4 H'(X, C) which has H'(X, R) as a fixed space. Obviously or leaves every element of Im a n Im & fixed. Thus Im a n Im & c Im at n H1(X, R). Hence, by Lemma 9, Im a n Im & = (0).
§ 8.
D
The Splitting of Locally Free Sheaves
By means of a formal splitting criterion we give a sufficient condition for a 'locally free sheaf .IF over a compact Riemann surface to contain a locally free subsheaf of rank 1 which is a direct summand of ,F (Theorem 4). Every such locally free sheaf contains maximal subsheaves of rank 1. On the Riemann sphere P1, the
maximal subsheaves are direct summands (Splitting Lemma). As a corollary, it follows that every locally free sheaf -F of rank r over P1 is isomorphic to a sheaf b(n;p), where n1, ..., n, e Z are uniquely determined up to order by F (a theorem of Grothendieck). The presentation here follows along the lines of [21].
1. The Ntunber p(F). Let 2' be a locally free sheaf of rank 1 over a compact Riemann surface X. Then, for every s E 2°°(X)*, deg s only depends on 2' (see Paragraph 4.2). For locally free sheaves of rank r > 1 this is not in general true. For example, consider 9 = (9(nt p) ® (D 0(n, p) for some p E X. Then 9 has sections of degree n1, n2, ..., nr. In order to define an integer which can be used in place of the "degree". we make the following observations: Every homomorphism n:9
phism n: F '(X) -+ 9'(X). If
I between locally free sheaves induces a homomor-
233
The Splitting of Locally Free Sheaves
4 B.
is an exact sequence of non-zero locally free sheaves, then, for every section s E ,F°'(Xr, either a) n(s) = 0, in which case s = i(s') with s' e F''O(X)* and deg s'- deg s, or
b) n(s)
0, in which case deg(s) 5 deg(a(s)).
Proof: If .fix is complementary to i(.F') in .F then for every germ tx e ,9rs there exist uniquely determined germs tx e A", is e A"' so that tX = i(t;,) + t=,
o(ts) = min{o(t'), o(t=)},
x E X.
If n(s) = 0, then s'e i(.F'(X )) (i.e. s = i(s) with s' a .V'*(X )*). But o(sX) - ,(s.,) for all x e X. Hence deg s = deg s'.
Now suppose that n(s) * 0, and let sx = i(t;) + t'. Sincenx maps 9rx isomorphically onto W., it follows that o(n(s).) = o(tx). Hence -(s.,) !5; o(n(s)j for all . x e X, and consequently deg(s) < deg(a(s)). For every locally free sheaf F over X, we define µ(F) as follows: p(er) =sup(deg(s)is e -W°°(X)*).
The following direct consequence of the above remarks is useful in showing
that µ(F) < oo: Let 0 -* F' --- 9 -* I -* 0 be an exact sequence of non-zero locally free sheaves. Then
lt( ') < p(F) < max(p(F'). Now it is easy to prove that u(.F) is bounded. Theorem 1. Let F be a locally free sheaf on a compact Riemann surface X. Then the degree function, p(, fl, is finite.
Proof: (by induction on the rank r of JF). For r = I the statement is trivially true. There exists an exact sequence of locally free sheaves 0 £° -+ F -+ I -* 0, where 2 and I have rank I and r - 1 respectively. The proof follows immediately by the induction hypothesis and the above estimates.
2. Maximal Subsheaves. The number µ(.5) can be characterized by the Chern numbers of the locally free sheaves of rank 1: (2)
u(31) = max(c(2): 2' is a locally free subsheaf of .F of rank 1).
This follows from the facts that every section s e .W'(X)* determines a locally free
subsheaf .' of rank 1 with c(.') = deg s, and conversely that a locally free subsheaf determines such a section (see Section 4). We call a locally freest heaf.P of
Chapter VIL Compact.Riemann Surfaces
234
Such sheaves are generated by maximal deg s = u( )). We have seen that sections (i.e. sections s e ,F°°(X)* with
rank I in f maximal if c(am)
every non-zero locally free sheaf .F possesses maximal subsheaves.
Theorem 2. Let -,F be a locally free sheaf on a compact l3iemann surface X. Let & be a maximal subsheaf in F and D E Div X. Theh £(D) is a maximal subsheaf in F(D). In general,
p(F) + deg D _< p(F(D)). Proof: Since !(D) is a locally free subsheaf of JJF(D), c(2(D)) < µ(.F(D)). Of course c(Y(D)) = c(&) + deg D. Thus, since c(&) = µ(.F), (.}
M(,fl + deg D5 p(.flD)).
We now apply (*) with _4r replaced by V (D) and D by -D. Thus
p(JI(D)) + deg(- D) 5 µ(g). Hence there is equality in (*) and, in particular, c(2'(D)) = i(F(D)). 3. The Inequality Riemann inequality:
u(.!F) + 2g. We begin with a consequence of the
Lemma. Let .F be locally free of rank 2 and $ a maximal subsheaf which is c(f) < 2g. isomorphic to (r. Then, defining Proof: Since H'(X, 0) = CB, the cohomology sequence associated to 0 -+ (r .F -. 9 -+ 0 starts out like
iF(X)-+¶(X) "' Co-...-. Now the sheaf I is locally free of rank 1. Thus if c(#) > 2g, then, by the Riemann inequality,
dimc1(X)>-c(g)+ 1 -gg+2. Hence the kernel of V would be at least 2-dimensional, and .f(X) is at least 3-dimensional. However, since the rank of .F is 2, every C-vector space F. /mx F. is 2-dimensional. Thus the restriction map '(X)-+.rx/ms.x has a non-trivial kernel. In other words, there would be a section s * 0 in JF(X) which vanishes at x. This implies that deg s 2- 1. But that is contrary to the assumption that
'4F) e-- c(Y) = 00 = 0.
0
§ S.
235
The Splitting of Lgcally Free Sheaves
We now show
Theorem 3. Let F be a locally free sheaf of rank r -> 2 and let Y be a maximal Then, defining I .f1.4°, subsheaf of µ(W)
µ() + 2g
Proof: As usual we start with the exact sequence
0 -- P __--.
4
0.
At first we consider the case r = 2. There exists a divisor D E Div X with 2 = 0(D) and deg D = c(2) = u(,f). Applying the above Lemma to the sequence we find that c(l(-D))<2g. Since c(T(-D)) = c(s) - deg D, the claim follows by noting that u(T) = c(gr) and deg D = u(-,F). Now consider the case of r > 2. Let 2' be a maximal subsheaf of,#. We have an
induced exact sequence 0 - 2 --s ,f' -+ 2'
x/fix =
0, where 9':=
For all
`fix/
Thus W' is a locally free sheaf of rank 2. Clearly .' is a maximal subsheaf of .F', and therefore u(4) = c(am) = Consequently
0
u(9)<+2g=u(3)+2g. 4. The Splitting Criterion. One says that the short exact (°-sequence,
splits if there exists an C-homomorphism l; :.°2 2 with it l = id. Thus, in this case, .Se = Vl.) Yl ® Y2. There is a simple formal splitting criterion: An exact &-sequence of locally free sheaves 0 -- Y 1 -+ ,y
SQ -s 0 splits when
H'(X, .atom(q, `Pt)) = (0).
Proof: Since I is locally free, the above short exact sequence induces the exact
C -sequence of sheaves,
0 - 3r
('4, 9'l) -, Jtoosn(9, 2) --' Jf°osn(gr, 9) -, 0,
Compact Riemann Surfaces
Chapter VII.
236
which in turn induces an exact cohomology sequence
-.H°(X,
.om(w, #)) H'(X; .lt°om(9f, 9,))
If n, is surjective, then there exists a global section = id. n
e Hom(l, .9') with
Remark: The splitting criterion is valid for any complex space. In the case of Stein spaces the relevant cohomology group is always zero. Thus we have the following observation. Every locally free subsheaf of a locally free sheaf over a Stein space is a direct summand.
For 1-dimensional Stein manifolds (i.e. non-compact Riemann surfaces), it follows that every locally free sheaf of rank r is isomorphic to the sheaf 0. (Recall that H' (X. 0) - H'(X. Z) = 0. Thus H'(X, 0, and thus every locally free sheaf of rank 1 is free.)
The application of the splitting criterion which is relevant to us is the following:
Theorem 4. Let .F be a locally free sheaf of rank r over a compact Riemann surface of genus g. Suppose that £° is a locally free subsheaf of rank 1 in .F haviiig the following properties:
1) The quotient sheaf IF== . f 2' is a direct sum £°2 ® .. (B 2', of locally free sheaves of rank 1.
2) Fori=2,...,r,c(2')-c(2,)>_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
Lett> 1, U,'a{ze P, 11 i 1). Let A e GL(r, t9(U 1,)) be a hole morphic invertibk r x r - matrix. Then there exist holamorphic, invertible matrices P e GL(r, 0(U,)), Q e GL(r, O(U2)) so that the matrix D'= PAQ is a diagonal matrix with t'', ..., z"', n, E Z, as diagonal terms.
6. Existence of the Splitting. Using the following lemma, both the existence and uniqueness are proved by successively "splitting off" maximal subsheaves,:. Splitting Lemma 6. Let A be a non-zero locally free sheaf over P 1, and Jet Y be a locally free subsheaf of rank 1 with . (Se) >_ 1. Then 2' is a direct sign mand of A.
In particular every maximal subsheaf of A is a direct sw ins nd.
Chapter VII. Compact Riemann Surfaces
238
Proof: We proceed by induction on the rank r of F, where the induction hypothesis is the existence part of Theorem 5. If r = 1, then the lemma is trivially true, because 2 = F. If r > 1, then the sheaf W/Y is locally free of rank r - 1,
and by assumption is therefore a direct sum 22 ® ® 2, of locally free sheaves L of rank 1. Now, mace{c(22),
..., C(Itr)} < POW) 5 c(2) + 1.
-1, 2 < i < r. By Theorem 4 it follows that 2' is a direct sum of F. If 2' is a maximal subsheaf of .9r, then from Theorem 3 it follows that c(2) _ El p(.,F) >- p(.F/2). (The genus is zero!).
In particular, c(.') - c(2t)
7. Uniqueness of the Splitting, For every locally free sheaf F on P1 we let
m- p(F). Then .fl-mXP1)= (s e F (P1)I deg(s) = m} v {0} * 0, so
d'=dime #"(-m)(P1) >- 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<0, and &(P1)=C.
Thus there trust be d equations nt = in. Since the n,'s are monotonically decreas-
For every
and 2(-m)=0, 1 < i:s d. i 5 d, 2t(-mXPI) generates the sheaf
211(-m).
Thus
239
t 9. The Splitting of Locally Free Sheaves
® .',(-m)(P1)
f
generates
the
d
sheaf ® 2',(-m). f
Since
Jr(-m)(P1) _ d
d
2',(-m)(P1), it follows that f _ ® .P,(-m). Hence A = 9-(m) _ ®1 2i. i=1
The above can obviously be repeated for the splitting 9 = 2i ® . (D Y;. Thus 1) has been proved as well as n; = = n/; = m. It now follows that =(!>(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)) <- µ(t) and c(2') > µ(l). Thus s e (J(re2 n)(P1) =.)t°(P1 ). Hence .P = Os c A. In particular we have the following: 0(n,) with nl > max{n2, ..., n,} has only one locally A sheaf .F = 0(n1) (D free subsheaf 5 which is isomorphic to 0(nl), and there are no such with 2' = 0(n
n1>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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[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
7) p. 39, lines 1-2 : see Paragraph 1.7 of Chapter A _ 8) p. 77, line 5 (Statement of Corollary): as in Theorem 6 9) p. 81, line 15 (Statement of Thm. 2): instead of "closed" it should be "8-closed" 10) p. 85, line 6: Theorem 2.3.6 11) p. 133, line 2 from below: the correct mapping is this one
.,) - H9(X\A, S°) 12) p. 164, line 18 (Statement of Thm. 2): instead of U it should be U 13) p. 168, lines 5-4 from below: the uniqueness theorem (Theorem 4) 14) p. 189, line 8 from below (Statement of Thm. 3): there should be an inequality (not equality) 15) p. 201, line 2: instead of c-' it should be Icy-1 (twice)
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1).Gilbarg,N.S.Trudinger Elliptic Partial Differential Equations of Second Order ISBN 3-S4o-41100-7
H. l,rauert, R. Remmert 1 hurry of Stein Spaces ISBN 3-540-00373-8 I1. I[a,se Number 1 hcruc ISBN 3-540-42794-X
F. Iliriebruch lirpological Methods in Algebraic Geometry ISBN I.540-58003.0 1..Ilormander The Analysis of Linear Partial Pilterential Operators I - Distribution Ihcor% and FourierAnahsis ISBN 1-540-00002-1 K. Ito, H. P. McKean, Jr. T. Kato
Per l urba11.r11
S. Kobayashi
I hffusion Processes and their Sample Paths ISBN 3-540-60029.7
I heory for Linear operators ISBN 3-540-58061-x
Iran,tormaiion Groups in Differential t,eumetry ISBN 3-540-58059-8
1. Lindenstrauss, L. Tzafriri Classical Banach Spaces I and II ISBN 3-540-00028-9 R.C. Lyndon, P. E Schupp
t
-nibinatorial Group Theory ISBN 3-540-41158-5
S. Mai Lane Honnrlo y 1,11N 3-540-58062-x D. Mumford Algebraic Geometry I - Complex Projective Varieties ISBN 3-540-58057-1 0.1.0'Meara Introduction to Quadratic Forms ISBN 3 540-00504-I G. Polya, G. Szego Problems and Theorems in Analysis I - Series. Integral Calculus. Theory of Functions ISBN 3-540-03040-4 G. Polya, G. Szego Problems and Theorems in Analysis 11 - Theory of Functions. Jeros. Poh-nonuals. Determinants. Number 1 heory. Geometry ISBN 3-540-63080. 2 S. Sakai
I -Algebras and
ISBN 3-540-03033-I
C. L. Siegel, J. K. Moser Lectures on Celestial Mechanics ISBN 3-540-58050-3 T. A. Springer Jordan Algebras and Algebraic Groups ISBN 3-S-10-0012-1
R. R. Switzer Algebraic Topology: Homology and Honwtopy ISBN 1-540-42750-4 A.Weil Basic Number Theory ISBN 3-540.58055-S A. Weil
Elliptic Functions According to Eisenslein and Kronecker ISBN 3-540-05036-9
K.Yosida Functional Analysis ISBN 3-540.58054.7
0. Zariski Algebraic Surfaces ISBN 3-54o-58658-x
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CLASSICS-' IN 'MATHEMATICS
Springer-Verlag began publishing books in higher mathematics in fy2o, when the series Gruntllehreu der motlteutatischen lt'issenschaffen, initially conceived as a series of advanced textbooks, was founded by Richard Courant. A few years later, a new series Ertebnisse tier ;llatheruatik and threr Grenzgebiete, survey reports of recent mathematical research, was added.
01' over 400 books published in these series, many have become recognized classics and remain standard references for their subject. Springer is reissuing a selected few of these highly successful hooks in a new, inexpensive softcover edition, to make them easily accessible to younger generations of students and researchers. Theory of Stein Spaces From the reviews:
Iheorv of Stein spaces provides a rich variety of methods, results, and motivaltons - a hook with masterful mathematical care and judgement. It is a pleasure to have this fundamental material now readily accessible to any serious mathematician."
/. Ee'Ns in 8ulh'tut of the London Alilt henrotrcel .Snore tr (tvtiol
"Written by Ihv(i mathematicians who played a crucial role in the development of the modern theory of several complex variables,
this is in important hook.' /.8. Cooper in
.tlatherrrutrsihe Nachrichtert 1 ry'u)
ISBN 3-540-00373-8
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ISSN I-t;r-o82r ) springeronline.com
