Table Of ContentAspe~f
Mathematic~
Edited by Klas Diederich
Vol. E 1: G. Hector/U. Hirsch: Introduction to the Geometry of Foliations, Part A
Vol. E 2: M. Knebusch/ M. Kolster: Wittrings
Vol. E3 : G. Hector/ U. Hirsch: Introduction to the Geometry of Foliations, Part B
Vol. E 4: M. Laska: Elliptic Curves over Number Fields with Prescribed Reduction
Type (out of printl
Vol. E5 : P. Stiller: Automorphic Forms and the Picard Number of an Elliptic Surface
Vol. E 6: G. Faltings/G. Wustholz et 01.: Rational Points*
Vol. E 7: W. Stoll: Value Distribution Theory for Meromorphic Maps
Vol. E 8: W. von Wahl: The Equations of Navier-Stokes and Abstract Parabolic
Equations (out of print)
Vol. E 9: A. Howard/P.-M. Wong (Eds.): Contributions to Several Complex Variables
Vol. E 10: A. J. Tromba: Seminar of New Results in Nonlinear Partial Differential
Equations*
Vol. Ell: M. Yoshida: Fuchsian Differential Equations*
Vol. E 12: R. Kulkarni, U. Pinkall (Eds.): Conformal Geometry*
Vol. E 13: Y. Andre: G-Function and Geometry*
Vol. E 14: U. Cegrell: Capacities in Complex Analysis
Vol. E 15: J.-P. Serre: Lectures on the Mordell-Weil Theorem
Vol. E 16: K. Iwasaki/H. Kimura/S. Shimomura/M. Yoshida: From Gauss to Painleve
Vol. E 17: K. Diederich lEd.): Complex Analysis
*A Publication of the Max-Planck-Institut fUr Mathematik, Bonn
Volumes of the German-language subseries 'Aspekte der Mathematik" ore listed on page 342.
Klas Diederich (Ed.)
Complex Analysis
Asped~f
Mathematic~
Klas Diederich (Ed,)
Complex Analysis
Dedicated to H. Grauert
Proceedings of the International Workshop
Wuppertal1990
I I
Vleweg
Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Compl"x analysis: proceedings of the
international workshop
Wupperta119911 Klas Diederich (ed.). -Braunschweig:
Vieweg, 1991
(Aspects of mathematics: E; Vol. 17)
ISBN-13: 978-3-322-86858-9
NE: Diederich, Klas [Hrsg.]; Aspects of mathematics IE
ISBN-13: 978-3-322-86858-9 e-ISBN-13: 978-3-322-86856-5
001: 10.1007/978-3-322-86856-5
Prof. Dr. Klas Diederich
Fachbereich Mathematik
Bergische Universitiit Gesamthochschule Wuppertal
Vieweg is a subsidiary company of the Bertelsmann Publishing Group International.
All rights reserved
© Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig 1991
Softcover reprint of the hardcover 1s t edition 1991
No part of this publication may be reproduced, stored in a retrieval
system or transmitted, mechanical, photocopying or otherwise,
without prior permission of the copyright holder.
Cover design by Wolfgang Nieger, Wiesbaden
Preface
Dedicated to H. Grauert on the occasion of his sixtieth birthday, the International Work
shop Complex Analysis was held in Wuppertal from February 12 to 16, 1991, with 143
mathematicians participating from all over the world. To the great delight of everybody,
an important group of colleagues from the DDR and countries of eastern Europe was able
to participate for the first time. The congress was organized by K. Diederich (Wuppertal),
J .E.Fornaess (Princeton) and 1. Lieb (Bonn).
In complete accordance with the width of the mathematical work of H. Grauert, the
congress was intended to give a broad survey over as many areas of active research in
complex analysis as possible. Its programm was opened by the lecture "Complex Analysis
in the Golden Fifties" by R. Remmert, contained in this volume on page 258. Exposi
tory lectures on the history and recent developments of important areas were given by J.
Dingener, O. Forster, F. Hirzebruch, J. J. Kohn, L. Lempert, T. Ohsawa, S. 1. Pincuk,
O. Riemenschneider and Y. T. Siu. In 82 seminar lectures in 4 parallel groups, recent
research was presented to the congress. It was a particular pleasure to see, how many
young researchers contributed to this activity.
The participants were invited to submit research notes for the Proceedings. \Ve asked the
authors to keep the length of their contributions at 8 to 10 pages in order to be able to give
in the volume an impression of the width and liveliness of modern research in Complex
Analysis. We thank all participants, who contributed an article, for their collaboration.
- The main subjects of the articles are: Cauchy-Riemann-equations and CR-structures,
convexity, deformation theory, envelopes of holomorphy, function algebras, group actions
in the complex domain, Hodge theory, instantons, Kiihler geometry, Lefschetz theorems,
holomorphic mappings, N evanlinna theory, singularities, twistor theory, uniformization.
The congress was financially supported by the Bergische Universitat-Gesamthochschule
Wuppertal, the Land Nordrhein-Westfalen, the Deutsche Forschungsgemeinschaft and the
Max-Planck-Institut fiir Mathematik Bonn. It is a great pleasure to thank all these
institutions for their help. Without it this congress would not have been possible.
Very valuable help in the preparation of this volume was given to me by Gregor Herbort
and Wolfgang Schwarz. And also the publisher Vieweg Verlag always had an open mind
for the special needs of this enterprise. To all of them I return my thanks.
Wuppertal, April 5, Hl91 I(Jas Diederich
Contents
Smooth Proper Modifications of Compact Kahler Manifolds .................. 1
L. Alessandrini and G. Bassanelli
LP-Estimatesfor ainC .................................................. 8
E. Amar
Canonical Resolutions in Sheaves on Schubert and Brieskorn Varieties ......... 14
V. Ancona and G. Ottaviani
La forme hermitienne canonique pour une singularite presque isolee ........... 20
D. Bartlet
A Generalized Cousin Problem for Subvarieties of the Bidisk ................. 29
D. E. Barrett and B. A. Taylor
Scalar Curvature and Twistor Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 33
P de Bartolomeis and L. Migliorini
Some Remarks on Weighted Estimates for a ............................... 40
B. Berndtsson
Circular Models and Normal Forms for Convex Domains. . . . . . . . . . . . . . . . . . . .. 44
f. Bland and T. Duchamp
LP-Estimates with Loss for the Bergman-Projection and the Canonical Solution
a. ................. , ................................................
to 52
A. Bonami
Distortion Function and the Heat Kernel of a Positive Line Bundle ............ 60
T. Bouche
Twistor Spaces and Non-hyperbolicity of Certain Symplectic Kahler Manifolds ., 64
F. Campana
Envelopes of Holomorphy of Domains in Cn .......................................... 70
E. Casadio Tarabusi and S. Trapani
Representing Measures in the Spectrum of Ir(n) ........................... 77
V. Cegrell
Estimees Ck• a pour l'equation a = f dans les convexes ....................... 81
1. Chaumat et A.-M. Chollet
Local Hyperconvexity and Local Hyperconcavity ........................... , 89
M. Col/oiu
a
Domaines estimation maximale ......................................... 92
M. Derridj
VII
Deformation of Compact Rieman Surfaces with Distinguished Points .......... 101
G. Dethloff
Local Extension of Holomorphic L2-Functions with Weights .................. 106
K Diederich and G. Herbort
On the Boundary Behavior of the Caratheodory and Kobayashi Distances in
Strongly Pseudoconvex Domains in Cn ................................................ 111
A. A. Fadlalla
Zur Klassifikation der l-konvexen komplexen Raume ....................... 115
K Fritzsche
Holomorphic Mappings into Convex Domains .............................. 127
1. Graham
On Second Order Hypoelliptic Differential Operators and the a-Neumann
Problem ............................................................... 134
P. Greiner
Equisingularity of Analytically Constructible Sets ........................... 143
Z. Hajto
Fonction de Artin d'un germe d'espace analytique ........................... 148
M. Hickel
Local Peaks Sets and Maximum Modulus Sets in Products of Strictly
Pseudoconvex Domains .................................................. 155
A. Iordan
Hyperbolic C*-Actions on Affine Surfaces .................................. 160
L. Kaup and K-H. Fieseler
On Complex Manifolds Exhausted by Biholomorphic Images of Generalized
Complex Ellipsoids E(n; n1, ... , ns; P1, ... , Ps) ............................... 169
A. Kodama
The Hard Lefschetz Theorem for Concave and Convex Algebraic Manifolds .... 175
S. Kosarew
Sur la resolution des equations de Cauchy-Riemann tangentielles pour
a
les formes support compact dans les varietes .............................. 188
Ch. Laurent-Thiebaut
Imbedding Pseudohermitian Manifolds into a Sphere ........................ 194
L. Lempert
Deformations of Strongly Pseudo-Convex CR Structures and Deformations
of Normal Isolated Singularities ........................................... 200
K Miyajima
The Behavior of Multiplier Ideal Sheaves under Morphisms .................. 205
A. M. Nadel
Local Geometry of Decoupled Pseudo convex Domains ...................... 223
1. D. McNeal
VIII
A Vanishing Theorem on Kahler Manifolds with Certain Stratified Structures
000 231
T. Ohsawa
Hodge-Kohomologic und Steinsche Mannigfaltigkeiten
00000000000000000000 0 0 235
Tho Peternell
The Cauchy-Riemann Equations in Convex Domains
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 247
1. C. Polking
Integral Kernels and Holder Estimates for a on Pseudoconvex Domains of
Finite Type on C2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
254
R. M. Range
Complex Analysis in the Golden Fifties
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 258
Ro Remmert
On q-Convex Exhaustion Functions of Complements of CR-Submanifolds
0 0 0 0 0 0 264
W Schwarz
A Characterization of Homogeneous Bounded Domains
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 268
S. Shimizu
Singular Sets of Separately Analytic Functions
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 278
1. Siciak
Some Recent Results Related to the Uniformization Problem in Several
Complex Variables
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 287
Y-T. Siu
Product Decomposition of Non-reduced Space Germs
0 0 0 0 • 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 299
K. Spallek
Modular Subgerms and the Isomorphism Problem in Deformation Theory
0 0 0 0 0 0 306
H. Stieber
A New Method to lntroduce a-priori Estimates for the a-Neumann Problem
00 0 • 310
K. Takegoshi
Maximal and Semi-Maximal Estimates for a on Pseudoconvex Manifolds
b 000000315
M. Derridj and Do S. Tartakojj
Degenerations of Instantons
0 0 0 0 0 0 0 0 0 0 0 •• 0 0 0 0 0 0 0 0 0 0 0 0 0 • 0 0 0 0 ••••• 0 0 0 0 0 0 0 0 0 0 324
Mo Manuyama and G. Trautmann
Second Main Theorems in Number Theory and Nevanlinna Theory
000000000000331
P. M. Wong
Symplectic Techniques in Holomorphic Group Actions .
0 0 0 0 • 0 •• 0 0 0 0 0 0 0 0 0 0 0 0 0 • 335
T. Wurzbacher
IX
Smooth proper modifications of compact Kahler
manifolds
Lucia Alessandrini and Giovanni Bassanelli
Abstract.
We study the class of compact complex manifolds which are proper modifications of compact
Kahler manifolds. It is shown, by means of new results about positive aa-closed currents, that they
carry a balanced metric. The notion of p-Kahler manifold is introduced in order to attempt a
classification of these modifications.
1. Introduction.
M ,
A classical result of Blanchard [BL] tells us that a compact manifold which is obtained
from a Kahler manifold M by blowing up a smooth submanifold, carries a Kahler metric too.
Varouchas [V A] generalizes this result by showing that the same holds for analytic spaces M and
""
M. But the statement fails to be true if one replaces the blow-up with a generic modification, as the
well-known example of Hironaka shows (it is a modification of P3 along a singular curve, which
turns out to be a non-Kahler threefold).
M
Our aim is to investigate, in the case of a modification f: ~ M between compact complex
manifolds, what kind of geometrical, or metric, properties "come back" from the range manifold M
M . M
to First of all, we prove that if M is Kahler then carries a balanced metric. Using the same
techniques, we prove here that the class of compact balanced manifolds is invariant by smooth proper
M
modifications, i.e. if M is balanced, then can be endowed with a balanced metric too. The
general case of modifications of analytic spaces is still open.
What we use to attain our goal is a characterization of Kahler and balanced manifolds in terms
of positive aa-closed currents (see [HL] and [MI]) and some results on positive aa-closed currents
similar to the support theorem for locally flat currents. As a matter of fact, it seems that this kind of
tool allows a deeper analysis of the geometry of the modification; in fact, these results are true not
only for currents of bidimension (1,1) (the Kahler case) or bidegree (1,1) (the balanced case), but
1
also for (p,p)-currents, 1 $ P $ n-I , which can be used to define the Kahler-degree of a manifold
(see § 3). Thus we take some (rather simple) proper smooth modifications of compact Kahler
manifolds and compute their Kahler-degrees smaller than n-I . It turns out that the Kahler-degrees
depend on the homology of the exceptional set of the modification.
2. Balanced manifolds.
Balanced manifolds have been studied intensively in [MI], to which we refer for this topic; in
[GA] they are called semi-Kahler or of special type. We only recall here the definition and the
characterizations given in [MI] and in [GA].
1 Definition. A balanced manifold M is a compact complex n-dimensional manifold which
satisfies one of the following equivalent conditions:
i) M admits an hermitian metric h such that, if ro is the Kahler form of h , dron-l = o.
ii) M admits an hermitian metric h such that, if ro is the Kahler form of hand S is the
formal adjoint of d in the metric h, it holds Sro = 0 .
iii) M admits an hermitian metric h such that the torsion I-form 'th of the canonical
hermitian connection of h is zero.
iv) there are no non zero positive (n-I,n-I)-currents on M which are (n-I,n-I)-components
of boundaries. (Recall that a real current T of bidimension (n-I,n-I) is the (n-I ,n-I)-component of a
as as ).
boundary if there exist a current S of bidimension (n-I,n) such that T = +
Ai Ai
v) M admits an hermitian metric h with respect to which = Aaf = 1/2 ,"d f E COO(M).
A proper modification of a Kahler manifold is not necessarily Kahler: indeed, Hironaka [HI]
constructed a Moishezon non algebraic (hence non-Kahler) threefold X which is a proper
modification of P3. Nevertheless, X carries a balanced metric, as we proved in [AB2]. From a
geometrical point of view, the obstruction for X to be Kahler is a curve homologous to zero on the
exceptional divisor E: in a dual manner, our proof lied in showing that, since E is not homologous
to zero, X is balanced. This is an example of a more general situation, which we shall consider in
theorems 6 and 7. Let us state our ingredients: first of all, we recall the definition of a smooth proper
modification (see [GRI] page 214).
2. Definition. Let M be a compact complex m-anifold of dimension n. A s-mooth proper
modification of M is a compact complex manifold M with a holomorphic map f: M -t M such
2
