Complex Differential Geometry: Topics in Complex Differential Geometry Function Theory on Noncompact Kähler Manifolds

159 Pages·1983·3.039 MB·English

by  Shoshichi Kobayashi, Camilla Horst, Hung-Hsi Wu (auth.)

DMVSeminar
Band3
Springer Basel AG
Complex Differential
Geometry
Topics in
Complex Differential Geometry
by Shoshichi Kobayashi and
Camilla Horst
Function Theory on
Noncompact Kahler Manifolds
by Hung-Hsi Wu
1983 Springer Basel AG
Authors
S. Kobayashi and H. Wu C. Horst
University of Ca1ifomia Mathematisches Institut der
Department of Mathematics Ludwig Maximilians-Universität
Berke1ey CA 94720 (USA) Theresienstrasse 39
D-8000 München 2 (FRG)
The seminar was made possible through the support of the Stiftung Volkswagenwerlc.
2nd edition 1987
Library of Congress Cataloging in Publication Data
Kobayashi, Shoshichi, 1932-
Comp1ex differential geometry.
(DMV seminar; band 3)
Bib1iography: p.
Inc1udes index.
Contents: Topics in comp1ex differential geometry /
by Shoshichi Kobayashi and Camilla Horst - Function
theory on noncompact Käh1er manifo1ds / by
Hung-hsi Wu.
1. Geometry, Differential. I. Horst, Camilla.
11. Wu, Hung-hsi, 1940- III. Tide. IV. Series.
QA641.K72 1983 516.3'6 83-5995
ISBN 978-3-7643-1494-1
CIP-Kurztitelaufnahme der Deutschen Bibliothek
Comp1ex differential geometry.
(DMV-Seminar; Bd.3)
Enth.: Topics in comp1ex differential geometry
/ by Shoshichi Kobayashi and Camilla Horst. -
Function theory on noncompact Käh1er manifo1ds
/ by Hung-Hsi Wu
ISBN 978-3-7643-1494-1 ISBN 978-3-0348-6566-1 (eBook)
DOI 10.1007/978-3-0348-6566-1
NE: Deutsche Mathematiker-Vereinigung: DMV
Seminar; Kobayashi, Shoshichi: Topics in comp1ex
differential geometry; Horst, Camilla; Topics in
comp1ex differential geometry; Wu, Hung-Hsi:
Function theory on noncompact Kähler manifo1ds
All rights reserved.
No part ofthis publication may be reproduced, stored in a retrieva1 system,
or transmitted in any form or by any means, e1ectronic, mechanica1, photocopying,
recording or otherwise, without the prior permission of the copyright owner.
©1983 Springer Basel AG
Originally published by Birkhäuser Verlag Basel in 1983
ISBN 978-3-7643-1494-1
5
CONTENTS
Part I. Topics in CompleY. Differential Geometry
by S. Kobayashi and C. Horst
Preface • • • 8
0. Notations and conventions 9
1. Basic concepts of compleY. differential geometry 11
2. Affine structures and holomorphic affine connections 30
3. Projective structures and holomorphic projective connections 36
4. Quadric structures and holomorphic conformal structures 44
5. Holomorphic tensor fields 54
References 64
Part II. Function Theory on Noncompact Kahler Manifolds
by H. Wu
Preface 68
0. Background Material 70
1. First Lecture 97
2. Second Lecture 106
3. Third Lecture 119
4. Fourth Lecture 129
5. Fifth Lecture (Open Problems) 139
Bibliography 149
IndeY. • • • • 156
Topics in Complex Differential Geometry
by Shoshichi Kobayashi and Camilla Horst
8
Preface
These notes were written by Camilla Horst on the basis of the
lectures I gave during the week of June 22-26, 1981 at the DMV
Seminar on Complex Differential Geometry in Dusseldorf. My aim
was to make the contents of my survey lecture at the DMV annual
meeting in 1980 (published in Jahresberichte, 1981) accessible to
beginning research students by providing a little more details.
I suggest therefore that the Jahresberichte paper be read as an
introduction to these notes.
I am very grateful to Frau Horst for bearing the burden of
writing the notes. Much of the basic material in Chapter 1 was
added by her. I would like to thank also Professor G. Fischer for
giving me the opportunity to conduct a seminar and to meet a
number of young German mathematicians in pleasant surroundings.
Last but not least, Frau Horst and I would like to express our
thanks to Frau G. Weinert for a beautiful typing job.
s. Kobayashi
in Berkeley
9
o. Notations and conventions
Let M be an n-dimensional complex manifold, p EM,
z = (z 1 , ••• ,z n) =(x 1+ v.-~• y 1 , •. ,xn +Y-Tyn) aholomorphic
coordinate system around p with z (p) = o.
0.1 By
(r,s)
1 C00-(r,s)-forms
M
we denote
s;Jr holomorphic r-forms
M
the sheaf of
holomorphic vector fields,
EJ1
furthermore: .,<t> := .,<r,s) .tM:=.tM(o) (the sheaf of
M r+s=t M '
C00-functions).
0.2 Let E be a C00-vectorbundle of rank r over M. In general the
elements of E a:r will be considered to be column vectors.
pc-:<
However, if we are given a fixed vector bundle E, then for
the dual bundle E* the elements of E~ will be looked upon
as row vectors.
0.3 The sheaf of C00-sections in a differentiable vector bundle
will be denoted by 8M(.,E) or by .tM(E) if there is no
danger of confusion with IM(U) = {C00-functions over UcM}.
If E is a holomorphic vector bundle, then the sheaf of
holomorphic sections will be denoted by OM(.,E) or OM(E).
10
0.4 Since E........,. OM(E) is an equivalence between the isomorphism
classes of holomorphic vector bundles of rank r and the
isomorphism classes of locally free OM-modules of rank r,
we often will not distinguish between E and OM(E).
0.5 The holomorphic tangent bundle of M will be denoted by
T<1•0>M, the antiholomorphic tangent bundle by T(o,1)M,
and Ta:M := T(1 ,o)MEDT(o,1)M. We have T(1 ,o)M=
~'p'
p E9G:
az
T(o,1)M = E9G: ~~ , and eM= 0(.,T<1'0)M).
p az~ p
Similarly we have T*(1,o>M, the holomorphic cotangent
bundle, T*(o,1>M, and T*a:M, where T*(1,o)M=EDG:dzi and
p p
T*(o,1>M = EDG: dzi.
p
The sheaf of holomorphic sections in T*(1,o)M is given
by S21w
The canonical line bundle det T* ( 1 '0 ) M= /\nT* ( 1 '0 ) M is
denoted by KM.
0.6 For a differentiable vector bundle E we let
~~r,s) (.,E)= ~~r,s) (E):= ~~r,s) ®E, the sheaf of E-valued
(r, s) -forms.
r r r
If E is holomorphic, then S2M(. ,E)= S2M(E) := S2M®E.
0.7 Mostly we will omit the subscript Min ~~r,s) ,S2~ and eM.
0.8 Throughout the text, we will adopt the Einstein sum-
mation convention of summing over the same upper and lower