Several Complex Variables VII: Sheaf-Theoretical Methods in Complex Analysis PDF
Preview Several Complex Variables VII: Sheaf-Theoretical Methods in Complex Analysis
Encyclopaedia 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 This work is subject to copyright. AII 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 Berlin Heidelberg GmbH. Violations are liable for prosecution under the German Copyright Law. © 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 4113140/SPS -5 4 3 2 I O -Printed on acid-free paper 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.
The list of books you might like
