Geometry and analysis on complex manifolds : festschrift for Professor S. Kobayashi's 60th birthday

GEOMETRY AND ANALYSIS ON COMPLEX MANIFOLDS

GEOMETRY AND ANALYSIS ON COMPLEX MANIFOLDS Festschrift for Professor S Kobayashi's 60th Birthday

Editors

T Mabuchi Osaka University

J Noguchi Tokyo Institute ofTechnology

T Ochiai University ofTokyo

World Scientific Sinqapore • New Jersey • London L Singapore • Hong Kong

Published by World Scientific Publishing Co. Pie. Ltd. P O Box 128, Farrer Road, Singapore 9128 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 73 Lynton Mead, Totteridge, London N20 8DH

GEOMETRY AND ANALYSIS ON COMPLEX MANIFOLDS Copyright © 1994 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, includingphotocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 27 Congress Street, Salem, MA 01970, USA. ISBN 981-02-2067-7

Printed in Singapore by UtoPrint

Dedicated to Professor Shoshichi Kobayashi

J

Professor Shoshichi Kobayashi

vii Preface

This volume presents papers dedicated to Professor Shoshichi Kobayashi com­ memorating the occasion of his sixtieth birthday, January 4, 1992. In October of the year, we had an opportunity to celebrate his Kanreki (the sixtieth anniversary of his birth) at the international symposium "Holomorphic Mappings, Diophantine Geometry and Related Topics" held in Research Institute for Mathematical Sci­ ences, Kyoto University. This meeting has naturally led us to organize a committee to publish a book in his honour. It is now a great pleasure that we can publish the present volume, where all included results are very hot and new with most papers either received or revised on or after the latter half of 1993. The principal theme of this volume is "Geometry and Analysis on Complex Manifolds", and the emphasis is put on the wide mathematical influence of Profes­ sor Kobayashi on areas ranging from differential geometry to complex analysis and algebraic geometry. This book therefore covers various materials such as holomor­ phic vector bundles on complex manifolds, Kahler metrics and Einstein-Hermitian metrics, geometric function theory in several complex variables, and symplectic or non-Kahler geometry on complex manifolds. These materials are parts of the sub­ jects on which Professor Kobayashi has made strong impacts and to each of which he is continuing to make many deep invaluable contributions. This volume is edited by using AMS-TEX and PlainTEX. We owe very much to Professor Ichiro Enoki for his beautiful treatment of the style files for this volume.

Toshiki Mabuchi Junjiro Noguchi Takushiro Ochiai

VIM

A List of papers dedicated to Professor Shoshichi Kobayashi on his 60th birthday, but not included in this volume.

K. Cho, K. Matsumoto and M. Yoshida, Combinatorial structure of the config­ uration space of 6 points on the projective plane. H. Pujimoto, Unicity theorems for the Gauss maps of complete minimal

surfaces.

T. Kawai and Y. Takei, On the structure of Painleve transcendents with a large parameter. H. H. Khoai, Height of p-adic holomorphic functions and applications. K. Maehara, Diophantine problem of algebraic varieties and Hodge theory. M. Ohta, On cohomology groups attached to towers of algebraic curves. F. Sakai, On the irregularity of cyclic coverings of the projective plane. T. Sakai, Some geometric inequalities in Riemannian

geometry.

T. Sasaki and T. Uehara, Power series solutions around a singular point of the system of hypergeometric differential equations of type (3,6) by use of special values of sF2W. Stoll, O'Shea's defect relation for slowly moving targets. M. Zaidenberg, Hyperbolicity in projective spaces.

ix

Contents

Preface Toshiki Mabuchi, Junjiro Noguchi and Takushiro Ochiai A list of papers dedicated to Professor Shoshichi Kobayashi on his 60th birthday, but not included in this volume Complex Finsler Metrics Marco Abate and Giorgio Patrizio

vii

viii 1

Stable Sheaves and Einstein-Hermitian Metrics Shigetoshi Bando and Yum-Tong Siu

39

Generalizations of Albanese Mappings for Non-Kahler Manifolds Ichiro Enoki

51

Examples of Compact Holomorphic Symplectic Manifolds which Admit no Kahler Structure Daniel Guan

63

A Torelli-Type Theorem for Stable Curves Yoichi Imayoshi and Toshiki Mabuchi

75

Linear Algebra of Analytic Torsion Hong-Jong Kim

96

KP Equations and Vector Bundles on Curves Yingchen Li

113

Some Topics in Nevanlinna Theory, Hyperbolic Manifolds and Diophantine Geometry Junjiro Noguchi

140

On the Extension of I? Holomorphic Functions IV: A New Density Concept Takeo Ohsawa

157

Symplectic Topology and Complex Surfaces Yongbin Ruan

171

Automorphisms of Tube Domains Satoru Shimizu

198

Tensor Products of Semistables are Semistable Burl Totaro

242

1

COMPLEX FINSLER

METRICS

MARCO ABATE Istituto di Matematiche Applicate, Via Bmum.no 25B, 56126 Pisa, Italy and

GIORGIO PATRIZIO Dipartimento di Matematica, Seconda Universita di Roma, 00133 Roma, Italy

0. Introduction A complex Finsler metric is an upper semicontinuous function F:Tl,0M —► R + defined on the holomorphic tangent bundle of a complex Finsler manifold M, with the property that F(jp;lv) = \(\F(p;v) for any (p;v) € TlfiM and ( e c . Complex Finsler metrics do occur naturally in function theory of several vari­ ables. The Kobayashi metric introduced in 1967 ([15]) and its companion the Caratheodory metric are remarkable examples which have become standard tools for anybody working in complex analysis; we refer the reader to [16, 18], [20], [1] and [14] to get an idea of the amazing developments in this area achieved in the past 25 years. In general, the Kobayashi metric is not at all regular; it may even not be con­ tinuous. But in 1981 Lempert [21] proved that the Kobayashi metric of a bounded strongly convex domain D in C™ is smooth (outside the zero section of T1,0D), thus allowing in principle the use of differential geometric techniques in the study of func­ tion theory over strongly convex domains (see also Pang [24] for other examples of domains with smooth Kobayashi metric). We started dealing with this kind of problems in [2]. In particular, [3] was de­ voted to the search of differential geometric conditions ensuring the existence in a complex Finsler manifold of a foliation in holomorphic disks like the one found by Lempert in strongly convex domains, where the disks were isometric embeddings of the unit disk A C C endowed with the Poincare metric. And indeed (see also [4]) we found necessary and sufficient conditions (see also Pang [23] for closely related results). In that case, because the nature of the problem required the solution of cer­ tain P.D.E.'s, the conditions were mainly expressed in local coordinates somewhat hiding their geometric meaning. The aim of this paper is to present an introduction to complex Finsler geometry in a way suitable to deal with global questions. Roughly speaking, the idea is to isometrically embed a complex Finsler manifold into a hermitian vector bundle, and then apply standard hermitian differential geometry techniques, in the spirit of [17]. A coarse outline of the procedure can be described as follows. Let M be the complement of the zero section in T1,0M. We assume that the complex Finsler metric F is smooth on M, and that F is strongly pseudoconvex, that is that the Levi form of G = F2 is positive definite. Now let V C TxfiM 'M be the verticai bundie,

2 that is the kernel of the differential of the canonical projection 7r: TlfiM —* M. Using the Levi form of G, it is easy to define a hermitian metric on V; moreover, there exists a canonical section t of V giving an isometric embedding of M into V — that is for any v € M the norm of L{V) with respect to the given hermitian metric on V is equal to F{v). Let D be the Chern connection on V associated to the metric, and denote by H the kernel of the bundle map X *-* Vx H. Using 0 , we can transfer both the metric and the connection on H, obtaining a canonical hermitian structure on Tl-°M, and the associated Chern connection preserves the splitting. Finally, the horizontal radial vector Geld \ = ©°t is a canonical isometric embedding of M into H- Then our idea is that the complex Finsler geometry of M should be described by using the differential geometry of the Chern connection D restricted to H, using x as a means of transfering informations from the tangent bundle to the horizontal bundle and back. For instance, the Kahler condition introduced in [3] becomes the vanishing of a suitable contraction of the horizontal part of the torsion of D (here we say that the metric is weakly KahJer); and the necessary and sufficient conditions for the existence of complex geodesic curves (see [3, 4]) are expressed by constant holomorphic curvature and a symmetry property of the horizontal part of the curvature of D\ cf. Lemma 8.3. This approach is in the spirit of the one developed by E. Cartan [10] for real Finsler metrics; see [26], [22], [11], [8], [9] and the forthcoming monograph [5] for an account in modern language. On the other hand, to our surprise we were unable to find in the literature a comparable approach in the complex case. Rund, in [27], described the Chern connection on the horizontal bundle, but only in local coordi­ nates. Fukui in [13] studied the Cartan connection on a complex Finsler manifold, which is in general different from the Chern connection (see [5] for a comparison). Faran [12] studied the local equivalence problem, without dealing with global ques­ tions. Only Kobayashi [17] explicitely used the Chern connection, but he seemed unaware of the relevanceof the horizontal component. It should be mentioned that we choose to work on M instead of the projectivized tangent bundle mainly for keeping more transparent the relationships between global objects and local com­ putations (which are often simplified by consistently using the homogeneity of the function G and its derivatives). However the two approach are completely equiva­ lent. In fact, the role of the canonical sections t and x m o u r context is analogous to the role of the tautological line bundle in [17]. We hope that our work will clarify the subject of complex Finsler geometry, opening the way to new research in the field. The content of this paper is the following. In sections 1 and 2 we describe in detail the construction outlined above of the Chern-Finsler connection. In sections 3 and 4 we define the (2,0)-torsion, the (1,1)-torsion, the curvature of the ChernFinsler connection on the horizontal bundle, we derive the Bianchi identities and we discuss Kahler Finsler metrics. In section 5 we introduce the notion of holomorphic curvature.

3 In sections 6 and 7 we derive the first and second variation formulas for a strongly pseudoconvex Kahler Finsler metric, giving a good example of global computations made using the tools introduced before. As a corollary, we prove the local existence and uniqueness of geodesies for a strongly pseudoconvex weakly Kahler metric, without assuming the strong convexity of the metric. Finally, in section 8 we deal with strongly pseudoconvex Finsler metrics of con­ stant holomorphic curvature, providing a first step toward their classification. As a consequence of results of this section and of [3] we get for example the following: Theorem 0.1. Let F:T1,0M —► 1 + be a complete strongly pseudoconvex metric on a simply connected complex manifold M. Assume that

Finsler

i) F is Kahler; ii) F has constant holomorphic curvature — 4; iii) R(H,K,x,x) = R(x,T R+ satisfying i) G = F2 is smooth on M; ii) F(p; v) > 0 for all p € M and v € M p ; iii) F(p; (v) = |C|F(p; v) for all p € M, v € T}% = GfadG0f

= fg.„ dz» + f |

di>f,

7

where rjj-y = GTOtG0fy and rjj ;/1 = G^Gpfw This is only part of the connection we are looking for: our next goal is to canonically extend D to a (l.O)-connection on TlfiM. Let us consider the bundle map A:T 1 0 M -» V defined by MX) = Vxi, and set H = kerA c TlfiM. We claim that H is a horizontaJ bundJe, i.e., TlfiM = H © V. Indeed, in local coordinates MX) = [Xa + w%(X)v0]da, where X = X^d^ + Xada.

Then a local frame for H is given by {Si,..., 6n}, where

-r%yda

6/1 = dfj, -

— note that f^t;^ = 0 — and the claim is proved. It is not difficult to check (see [5] for a coordinate-free proof) that setting

e(4.) = sa for a = 1,... ,n we get a well-defined global bundle isomorphism 0 : V - » H ; then we can define a (l,0)-connnection D on H just by setting

vxH = e[vx(e-1H)] for any X e TCM and H € X(H). By linearity, this yields a (l,0)-connection on T1,0M, still denoted by D: the Chern-Finsler connection. Using the bundle isomorphism 0 : V - > Wwe can also transfer the hermitian structure (,) onH just by setting VH^eTtv

{H,K)v =

(e-1(H),@-1{K))v,

and then we can define a hermitian structure on T1,0M by requiring H be orthogonal to V. It is easy to check then that D is the Chern connection associated to this hermitian structure, that is X(Y,Z) = {VxY,Z) +

(Y,Vxz)

for any X e T^M and Y, Z e X(T^°M). From now on we shall work only with the frame {6^, da} and its dual co-frame {dz^,^"} given by tPa = dva + r f dz» = dva + GfaGf]ll dz»,

8 where we have set

r« = t%y = GfaGflll.

Writing Writing

w|

we get we get

■pa

—G

= T%„dz»+ T%r,

Ta Gfff-r — -L-,0,

(2.1)

r ^ = G^'S^Gpr) = G™(G0f„ -

G&Jl).

Note that

r£M = a/j(r«)

and r « = r ^ . y ;

(2.2)

in particular, this is exactly the connection introduced by Rund [27]. So we have described a canonical splitting of the holomorphic tangent bundle of M in a vertical and a horizontal bundle, and defined a canonical connection on it, preserving this splitting. In the following subsections we shall begin the study of this connection, introducing torsions and curvatures; here we first describe a few properties of the splitting. First of all, the next lemma shows that the local frames {6\,..., 6n} enjoy some nice and convenient properties: Lemma 2.1. Let D be the Chern-Finsler connection associated to a strongly pseudoconvex Finsler metric F, and let { 6 1 , . . . , Sn} be the corresponding local horizontal frame. Then i) ii) iii) iv)

[6M, Su] = 0 for all 1 < fi,v < n; [