Table Of ContentEncyclopaedia of
Mathematical Sciences
Volume 74
Editor-in-Chief: R. V. Gamkrelidze
Ho Grauert Tho Petemell Ro Remmert (Edso)
0 0
Several
Complex Variables VII
Sheaf-Theoretical Methods
in Complex Analysis
Springer-Verlag Berlin Heidelberg GmbH
Consulting Editors of the Series:
AA Agrachev, AA Gonchar, E.F. Mishchenko,
N. M. Ostianu, V. P. Sakharova, A B. Zhishchenko
Title of the Russian edition:
Itogi nauki i tekhniki, Sovremennye problemy matematiki,
Fundamental'nye napravleniya, VoI. 74,
Publisher VINITI, Moscow (in preparation)
Mathematics Subject Classification (1991):
32-02,32Axx, 32Cxx, 32C35, 32FI0, 32F15,32S20, 32S45,32Jxx,55N30
ISBN 978-3-642-08150-7
Library of Congress Cataloging-in-Publication Data
Kompleksnyl analiz-mnogie peremennye 7. English
Several complex variables VII: sheaf-theoretical methods in complex analysis /
H. Grauert, Th. Petenell, R. Remmert (eds.).
p. cm. - (Encyclopaedia of mathematical sciences; v. 74)
Includes bibliographical references and indexes.
ISBN 978-3-642-08150-7 ISBN 978-3-662-09873-8 (eBook)
DOI 10.1007/978-3-662-09873-8
1. Functions of several complex variables. 2. Sheaf theory. 1. Grauert, Hans, 1930-.
II. Peternell, Th. (Thomas), 1954-. III. Remmert, Reinhold.
IV. Title. V. Title: Several complex variables 7. VI. Series.
QA33I.7.K6813 1994 515'.94-dc20 93-32959
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© Springer-Verlag Berlin Heidelberg 1994
Originally published by Springer-Verlag Berlin Heidelberg New York in 1994
Softcover reprint of the hardcover 1s t edition 1994
Typesetting: Asco Trade Typesetting Ltd., Hong Kong
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List of Editors and Authors
Editor-in-Chief
R. V. Gamkrelidze, Russian Academy of Sciences, Steklov Mathematical Institute,
ul. Vavilova 42, 117966 Moscow, Institute for Scientific Information (VINITI),
ul. Usievicha 20a, 125219 Moscow, Russia
Consulting Editors
H. Grauert, Mathematisches Institut, Universitat Gi:ittingen, Bunsenstr. 3-5,
37073 Gi:ittingen, FRG
Th. Petemell, Mathematisches Institut, Universitat Bayreuth, Universitatsstr. 30,
95440 Bayreuth, FRG
R. Remmert, Mathematisches Institut, Universitat MUnster, Einsteinstr. 62,
48149 MUnster, FRG
Authors
F. Campana, Departement de Mathematiques, Universite Nancy I, BP 239,
54506 Vandoeuvre-Ies-Nancy Cedex, France
G. Dethloff, Mathematisches Institut, Universitat Gi:ittingen, Bunsenstr. 3-5,
37073 Gi:ittingen, FRG
H. Grauert, Mathematisches Institut, Universitat Gottingen, Bunsenstr. 3-5,
37073 Gi:ittingen, FRG
Th. Petemell, Mathematisches Institut, Universitat Bayreuth, Universitatsstr. 30,
95440 Bayreuth, FRG
R. Remmert, Mathematisches Institut, Universitat MUnster, Einsteinstr. 62,
48149 MUnster, FRG
Contents
Introduction
1
Chapter I. Local Theory of Complex Spaces
R. Remmert
7
Chapter II. Differential Calculus, Holomorphic Maps
and Linear Structures on Complex Spaces
Tho Petemell and R. Remmert
97
Chapter III. Cohomology
Tho Petemell
145
Chapter IV. Seminormal Complex Spaces
Go Dethloff and Ho Grauert
183
Chapter V. Pseudoconvexity, the Levi Problem and Vanishing Theorems
Tho Petemell
221
Chapter VI. Theory of q-Convexity and q-Concavity
Ho Grauert
259
Chapter VII. Modifications
Tho Petemell
285
Chapter VIII. Cycle Spaces
F. Campana and Th. Petemell
319
Chapter IX. Extension of Analytic Objects
H. Grauert and R. Remmert
351
Author Index
361
Subject Index
363
Introduction
Of making many books there is no end;
and much study is a weariness of the flesh.
Eccl. 12.12.
1. In the beginning Riemann created the surfaces. The periods of integrals of
abelian differentials on a compact surface of genus 9 immediately attach a g
dimensional complex torus to X. If 9 ~ 2, the moduli space of X depends on
3g - 3 complex parameters. Thus problems in one complex variable lead, from
the very beginning, to studies in several complex variables. Complex tori and
moduli spaces are complex manifolds, i.e. Hausdorff spaces with local complex
coordinates Z 1, ... , Zn; holomorphic functions are, locally, those functions which
are holomorphic in these coordinates.
In the second half of the 19th century, classical algebraic geometry was born
in Italy. The objects are sets of common zeros of polynomials. Such sets are of
finite dimension, but may have singularities forming a closed subset of lower
dimension; outside of the singular locus these zero sets are complex manifolds.
Even if one wants to study complex manifolds only, singularities do occur im
mediately: The fibers of holomorphic maps X -+ Y between complex manifolds
are analytic sets in X, i.e. closed subsets which are, locally, zero sets of finitely
many holomorphic functions. Analytic sets are complex manifolds outside of
their singular locus only: A simple example of a fiber with singularity is the fiber
zi -
through 0 E (C2 of the function f(z 1, Z2) := Z 1 Z 2 resp. z~.
Important classical examples of complex manifolds with singularities are
quotients of complex manifolds, e.g. quotients of (C2 by finite subgroups of
{G (-
~). ~ _~)}
SL2(C)· For the group G:= the orbit space (C2jG is
isomorphic to the affine surface F in (C3 given by z~ - ZlZ2, the orbit projection
(C2 -+ F is a 2-sheeted covering with ramification at 0 E (C2 only. Hence F =
(C2jG is not a topological manifold around 0 E F. This is true whenever the
origin of (C2 is the only fixed point of the acting group.
All these remarks show that complex manifolds cannot be studied success
fully without studying more general objects. They are called reduced complex
spaces and were introduced by H. Cart an and J-P. Serre. This was the state of
the art in the late fifties. But soon reduced spaces turned out to be not general
enough for many reasons. We consider a simple example. Take an analytic set
A in a domain U of (Cn and a holomorphic function f in U such that flA has
J
certain properties. Can one find a holomorphic function in a neighborhood
2 Introduction
v c U of A such that IIA = flA and that the properties of flA are conserved
I?
by This is sometimes possible by the following step-wise construction: Let A
be the zero set of one holomorphic function g which vanishes of first order. Then
we try to construct a convergent sequence f. of holomorphic functions on V,
J
A eVe U, such that f.+l == f.modg·+1, where fo:= f. The limit function
may have the requested properties. This procedure suggests to form all residue
rings oflocal holomorphic functions on U modulo the ideals generated by g.+l.
This family leads to a so-called sheaf of rings over U which is zero outside of A.
We denote the restriction of this sheaf to A by lPA, Sections in lPA, are called
again holomorphic functions on A. For v = 0 these sections are just the ordi
nary holomorphic functions on the reduced complex space A, i.e. lPAo = lpAo For
v > 0 the sections can be considered as power series segments in g with coeffi
cients holomorphic on A in the ordinary sense. This is expressed geometrically
by saying that A, with the new holomorphic functions, is a complex space which
is infinitesimally thicker than A. We call (A, lPAJ the v-th infinitesimal neighbor
hood of A. The sheaf lPA, has, at all points of A, nilpotent germs #0. This
phenomenon cannot occur for reduced complex spaces. Infinitesimal neighbor
hoods are the simplest examples of not reduced complex spaces.
The topic of this book is the theory of complex spaces with nilpotent ele
ments. As indicated we need sheaves already for the definition. Sheaf theory
provides the indispensable language to translate into geometric terms the basic
notions of Commutative Algebra and to globalize them.
2. Sheaves conquered and revolutionized Complex Analysis in the early
fifties. Most important are analytic sheaves, i.e. sheaves f/ of modules over the
structure sheaf lPx of germs of hoiomorphic functions on a complex space X.
Every stalk ~,x E X, is a module over the local algebra lPx,x, the elements of
~ are the germs Sx of sections S in f/ around x. An analytic sheaf f/ is called
locally finite, if every point of X has a neighborhood U with finitely many
sections Sl' ... , sp E f/(U) which generate all stalks ~, x E U. This condition
gives local ties between stalks.
For the calculus of analytic sheaves it is important to know when kernels of
sheaf homomorphisms are again locally finite. This is not true in general but it
certainly holds for locally relationally finite sheaves, i.e. sheaves f/ having the
property that for every finite system of sections Sl' ... , sp E f/(U) the kernel
of the attached sheaf homomorphism lp~ -+ f/u, (fl' ... , fp) 1-+ 'LJjSj, is locally
finite. Locally finite and locally relationally finite analytic sheaves are called co
herent. Such sheaves are, around every point x E X, determined by the stalk ~;
this is, in a weak sense, a substitute for the principle of analytic continuation.
Trivial examples of coherent analytic sheaves are all sheaves f/ on CC", where
~ = 0 for x # 0 and 90 is a finite dimensional CC-vectorspace (skyscraper
sheaves). It is a non-trivial theorem of Oka that all structure sheaves lPC[n are
coherent.
Now a rigorous definition of a complex space is easily obtained: A Hausdorff
space X, equipped with a "structure shear' lPx of local CC-algebras, is called
Introduction 3
a complex space, if (X, @x) is, locally, always isomorphic to a "model space"
(A, @A) of the following kind: A is an analytic set in a domain U of <en, n E N,
and there is a locally finite analytic sheaf of ideals in the sheaf @u, such that
~ = @u on U\A and @A = (@u/~)IA. In the early fifties complex spaces were
defined by Behnke and Stein in the spirit of Riemann: Their model spaces are
analytically branched finite coverings of domains U in <en. In this approach the
structure sheaf @x is given by those continuous functions which are holomor
phic outside of the branching locus in the local coordinates coming from U. It
is known that Behnke-Stein spaces are normal complex spaces.
A complex space X, even if a manifold, may not have a countable topology.
If the topology is countable the space admits a triangulation with its singular
locus as subcomplex. Hence the topological dimension is well defined at every
point: it is always even, half of it is called the complex dimension. Further
more all complex spaces are locally retractible by deformation to a point, in
particular all local homotopy groups vanish and universal coverings always
exist.
Sheaf theory is a powerful tool to pass from local to global properties. The
appropriate language is provided by cohomology. This theory assigns to every
sheaf !f of abelian groups on an arbitrary topological space X so called co
homology groups Hq(X, !f), q E N, which are abelian. There are many coho
mology theories, for our purposes it suffices to use Cech-theory.
For analytic sheaves all cohomology groups are <e-vector spaces. These
spaces are used to obtain important results which, at first glance, have no con
nection with cohomology. E.g. vanishing of first cohomology groups implies, via
the long exact cohomology sequence, the existence of global geometric objects.
For Stein spaces, which are generalizations of domains of holomorphy over <en,
all higher cohomology groups with coefficients in coherent sheaves vanish (The
orem B), this immediately yields the existence of global merom orphic functions
with prescribed poles (Mittag-Lerner, Cousin I). If X is compact all cohomology
groups with coefficients in coherent sheaves are finite dimensional <e-vector
spaces (Theoreme de Finitude).
3. Stein spaces are the most important non compact complex spaces. Histor
ically they were defined by postulating a wealth of holomorphic functions and
are characterized by Theorem B. They can also be characterized by differential
geometric properties of convexity, more precisely by the Levi-form of exhaus
tion functions. Stein spaces are exactly the I-complete complex spaces, i.e. all
eigenvalues of the Levi-form are positive. Natural generalizations are the q
complete and q-convex spaces, where at most q-I eigenvalues of the Levi-form
may be negative or zero. The counterpart of q-convexity is q-concavity. For
all such spaces finiteness and vanishing theorems hold for cohomology groups
in certain ranges, such theorem generalize as well the finiteness theorems for
compact spaces as the Theorem B for Stein spaces. Most important examples
of convex/concave spaces are complements of analytic sets in compact com
plex spaces. If Ad is a d-dimensional connected complex submanifold of the
4 Introduction
n-dimensional projective space lPn then the complement lPn\ Ad is (n - d)-convex
and (d + I)-concave.
The notion of convexity is also basic in the theory of holomorphic vector
bundles on compact spaces. A vector bundle is called q-negative, if its zero
section has arbitrarily small relatively compact q-convex neighborhoods. If q =
I the bundle is just called negative; duals of negative bundles are called positive
or ample. The Andreotti-Grauert Finiteness Theorem can be used to obtain
Vanishing Theorems for cohomology groups with coefficients in negative or
positive vector bundles. As a consequence compact spaces carrying ample vec
tor bundles are projective-algebraic.
For normal compact spaces the notion of a Hodge metric can be defined.
Spaces with such a metric always have negative line bundles, hence normal
Hodge spaces are projective-algebraic. For a complex torus a Hodge metric
exists if and only if Riemann's period relations are fulfilled.
Serre duality holds for q-convex complex manifolds if the cohomology
groups under consideration have finite dimension. For compact spaces, i.e. 0-
convex spaces, duality is true in every dimension. For concave spaces the field
of merom orphic function is always algebraic. For details on all these results see
Chapters V and VI.
4. Whenever there is given a complex space X and an equivalence relation R
on X the quotient space X/R is a well defined ringed space. It is natural to ask
for conditions on R such that X/R is a complex space. To be more precise let X
be normal and of dimension n and assume that R decomposes X into analytic
sets of generic dimension d. Then R is called an analytic decomposition of X if its
graph is an analytic set in the product space X x X. Under certain additional
conditions the quotient X/R is an (n - d)-dimensional normal complex space
and the projection X -+ X/R is holomorphic.
In important cases the analytic graph is a decomposition of X only outside
of a nowhere dense analytic "polar" set. Then the limit fibers of generic fibers are
all still pure d-dimensional and we get a "fibration" t/J in X whose fibers may
intersect. We call t/J a meromorphic decomposition resp. a meromorphic equiva
lence relation of X. A simple regularity condition guarantees that, by replacing
polar points by the fiber points through them, one obtains a proper modifica
tion X of X such that t/J lifts to a true holomorphic fibration J of X with
d-dimensional fibers. The quotient Q := X/J is called the quotient of X by the
meromorphic equivalence relation t/J, this space Q is always normal.
Simple examples are obtained by holomorphic actions of complex Lie groups;
e.g. if <e* acts homothetically on <en, the family t/J consists of all complex lines
through 0 and we have Q = lPn-t.
The theory of analytic decompositions is set-theoretic and not ideal-theoretic.
An ideal-theoretic approach seems to be possible only for "proper" decomposi
tions; then the theorem of coherence of image sheaves can be applied. The
theory of decomposition is discussed in Chapter VI, §§ 2-4, and in Chapter V,
§l.
