Hyperbolican,Manifolds Holomorphic Mappings An Introduction
Shoshichi
(Second Edition)
yashi
Hyperbolic Manifolds and
Holomorphic Mappings
An Introduction (Second Edition)
Hyperbolic Manifolds
Holomorphic Mappings An Introduction (Second Edition)
Shoshichi Kobayashi University of California, Berkeley, USA
10 World Scientific NEW JERSEY
LONDON
SINGAPORE
BEIJING
SHANGHAI
HONG KONG
TAIPEI
CHENNAI
Published by
World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224
USA ofce: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
The first edition was published in 1970 by Marcel Dekker.
HYPERBOLIC MANIFOLDS AND HOLOMORPHIC MAPPINGS (Second Edition) An Introduction Copyright m 2005 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, including photocopying, recording or any information storage and retrieval system now (mown 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., 222 Rosewood Drive, Danvcrs, MA 01923, USA. In this case permission to photocopy is not required from the publisher.
ISBN 981-256-496-9 ISBN 981-256-589-2(pbk)
Printed in Singapore by World Scientific Printers (S) Pte Ltd
Dedicated to the Memory of
Professor S. S. Chern and Professor K. Yano
Preface to the New Edition
I introduced the intrinsic pseudodistance dx in 1967 and published the first edition of this monograph in 1970 and a survey article in the Bulletin of the American Mathematical Society (1976). In the 35 years since the appearance of the first edition, the subject of hyperbolic complex spaces has seen increasing activities. In 1973 the Mathematical Reviews created two new subsections "invariant metrics and pseudodistances" and "hyperbolic complex manifolds" within the section "analytic mappings" (which is now called "holomorphic mappings"). Since 1980 several books on intrinsic pseudodistances and related topics have appeared, each emphasizing certain aspects of the theory:
T. Franzoni and E. Vesentini, Holomorphic Maps and Invariant Distances, 1980. J. Noguchi and T. Ochiai, Geometric Function Theory in Several Complex Variables, 1984 (English translation in 1990). S. Lang, Introduction to Complex Hyperbolic Spaces, 1987.
M. Abate, Iteration Theory of Holomorphic Maps on Taut Manifolds, 1989.
S. Dinen, The Schwarz Lemma, 1989. M. Jarnicki and P. Pflug, Invariant Distances and Metrics in Complex Analysis, 1993.
In addition, Encyclopaedia of Mathematical Sciences, Vol. 9 (1989), Several Complex Variables, contains the following two chapters: Chapter III Invariant Metrics by E. A. Poletskii and B. V. Shabat,
Chapter IV Finiteness Theorems for Holomorphic Maps by M. G. Zaidenberg and V. Ya. Lin.
A recent undergraduate level book by S. G. Krantz "Complex Analysis: The Geometric Viewpoint" (1990) in the Carus Mathematical Monographs Series of the Mathematical Association of America is an elementary introduction to function theory from the viewpoint of hyperbolic analysis. vii
viii
Preface to the New Edition
In order to give a systematic and comprehensive account of the theory of intrinsic pseudodistances and holomorphic mappings, in 1998 I published Hyperbolic Complex Spaces as Volume 318 of Grundlehren der Nlathematischen Wissenschaften. However; the old book is not subsumed by this new book and continues to provide the easiest access to the theory, especially for students. Incorrect or no longer relevant statements have been deleted from the first edition and, in order to bridge the chasm between the first edition and the voluminous Grundlehren hook of more than 450 pages, the postscript has been added. Shoshichi Kohayashi Berkeley, June 2005
Preface
This book is a development of lectures delivered in Berkeley in the academic year 1968-69. Its object is to give a coherent account of intrinsic pseudodistances on complex manifolds and of their applications to holomorphic mappings. The classical Schwarz-Pick lemma states that every holomorphic mapping from a unit disk into itself is distance-decreasing with respect to the Poincare distance. In Chapter I, we prove Ahlfors' generalization to holo-
morphic mappings from a unit disk into a negatively pinched Riemann surface and present some of its applications in the geometric theory of functions. In Chapters II and III, various higher-dimensional generalizations of the Schwarz-Pick-Ahlfors lemma are proved. The raison d'etre of the first three chapters is to provide interesting examples for the subsequent
chapters. It is therefore possible for the reader to start from Chapter IV and go back to Chapters 1, II, and III only when he must. In Chapter IV, we introduce a certain pseudodistance on every complex manifold in an intrinsic manner. A complex manifold is said to be (completely) hyperbolic if this pseudodistance is a (complete) distance. The classical pseudodist.ance of Caratheodory and this new pseudodistance
share two basic properties: (1) they agree with the Poincare distance on the unit disk, and (2) every holomorphic mapping is distance-decreasing. Among the pseudodistances with these two properties, the Caratheodory pseudodistance is the smallest and the new one is the largest. These pseudo-
distances permit us to obtain many results on complex manifolds by a purely metric space-topological method. They enable us also to gain a geometric insight into function theoretic results. Elementary properties of these pseudodistances and of hyperbolic manifolds are given. In Chapter V, we study holomorphic mappings of a complex manifold
into a hyperbolic manifold. In Chapter VI, which is, to a large extent, based on M. H. Kwack's thesis, we give generalizations of the big Picard theorem to higher-dimensional manifolds. Although there is more than one ix
Preface
x
way to interpret the big Picard theorem geometrically, we consider it as an extension theorem for holomorphic mappings. To avoid technical complications associated with complex spaces, we consider only complex manifolds in Chapters IV, V, and VI. In Chapter VII, we indicate how some of the results in these three chapters could be generalized to complex spaces.
In Chapter VIII, the relationships between hyperbolic manifolds and minimal models are studied. The generalized big Picard theorems are essen-
tially used here. To a large extent this chapter is based on J. Zumbrunn's thesis. Closely following the constructions of the pseudodistances in Chapter IV, we define in Chapter IX two kinds of intermediate dimensional measures on a complex manifold in a intrinsic manner. These measures have been studied more thoroughly by D. Eisenman in his thesis. Our approach is perhaps a little more differential geometric. At the end of Chapter IX, we list a few unsolved problems on hyperbolic manifolds. In preparing my lectures on hyperbolic manifolds, I had numerous useful conversations with H. Wu. By solving some of the problems listed in the
first draft of this book, P. Kiernan has helped me to make a number of improvements. It was through Professor Chern's papers on holomorphic mappings that I was led into this topic. I wish to express my thanks to these mathematicians. Shoshichi Kobayashi Berkeley, January 1970
CONTENTS
Preface to the New Edition
.
.
.
.
.
..
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
Preface ...................................... Chapter I The Schwarz Lemma and Its Generalizations . . .. .. The Schwarz-Pick Lemma . 1 . . . 2 A Generalization by Ahlfors . . . . . 3 The Gaussian Plane Minus Two Points . . . . 4 Schottky's Theorem . . . 5 Compact Riemann Surfaces of Genus >_ 2 6 Holomorphic Mappings from an Annulus into an Annulus .
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Chapter II Volume Elements and the Schwarz Lemma . 1 Volume Element and Associated Hermitian Form 2
Basic Formula .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
5 10
.
.
.
.
.
.
.
.
11
.
.
.
.
.
.
.
.
12
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
16 17
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
19
.
.
.
.
.
.
.
20
.
Chapter Ill Distance and the Schwarz Lemma 1 Hermitian Vector Bundles and Curvatures 2 The Case Where the Domain is a Disk . . 3 The Case Where the Domain is a Polydisk 4 The Case Where D is a Symmetric Bounded Domain
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Affinely Homogeneous Siegel Domains of Second Kind 6 Symmetric Bounded Domains . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
36 37 40 40
.
.
.
.
.
.
.
.
.
41
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.. .. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
. .
xi
25 28 33
.
.
Chapter IV Invariant Distances on Complex Manifolds 1 An Invariant Pseudodistance . . . . . . 2 Caratheodory Distance . . . . . . . . . . 3 Completeness with Respect to the Caratheodory Distance 4 Hyperbolic Manifolds . . . . . . . 5 On Completeness of an Invariant Distance . . . .
3
.
5
.
1
.
.
.
1
.
. Iolomorphic Mappings f : M' M with Compact M' 4 Holomorphic Mappings f : D - M, Where D is a Homogeneous Bounded Domain . . . .. . . .
ix
.
3
.
.. ..
vii
.
.
.
.
.
..
.
.
.
.
.
.
44 45 49 52 56 63
Xii
Contents
Chapter V Holomorphic Mappings into Hyperbolic Manifolds 1 The Little Picard Theorem . 2 The Automorphism Group of a Hyperbolic Manifold 3 Holornorphic Mappings into Hyperbolic Manifolds . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
67 67 67
.
70
. .
. .
Chapter VI The Big Picard Theorem and Extension of Holomorphic Mappings 1
2
3 4 5 6
.. .. .
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
77 77 78 81 84 85 88
.
.
.
.
.
.
.
.
.
.
.
Manifolds
.
.
.
.
.
.
.
Chapter VII Generalization to Complex Spaces .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Complex Spaces . . . . . . . . . . . . . . . 2 Invariant Distances for Complex Spaces . . 3 Extension of Mappings into Hyperbolic Spaces 4 Normalization of Hyperbolic Complex Spaces . 5 Complex V-Manifolds (Now Called Orbitfolds) 1
.
.
.
6
.
.
Statement of the Problem The Invariant Distance on the Punctured Disk . . Mappings from the Punctured Disk into a Hyperbolic Manifold Holornorphic Mappings into Compact Hyperbolic Manifolds Iolomorphic Mappings into Complete Hyperbolic Manifolds . Holomorphic Mappings into Relatively Compact Hyperbolic
Invariant Distances on 11/I'
.
.
.
.
.
.
.
.
.
.
.
.
.
Chapter VIII Hyperbolic Manifolds and Minimal Models Merornorphic Mappings 2 Strong Minimality and Minimal Models 3 Relative Minimality 1
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Puslscripl
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
93 93 95 96 98 100 100 103 103 104 108
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
129
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
115 115 118 125
.
.
.
.
. .
.
Chapter IX Miscellany . Invariant Measures 2 Intermediate Dimensional-Invariant Measures 3 Unsolved Problems 1
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Bibliography .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
135
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
143
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
145
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
147
Summany of Notations
Author Index
.
.
.
.
Subject Index
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
CHAPTER I
The Schwarz Lemma and Its Generalizations
1
The Schwarz-Pick Lemma
Let D be the open unit disk in the complex plane C, i.e.,
D= {zEC;Izj <1}. Let f : D -> D be a holomorphic mapping such that f (0) = 0. Then the classical Schwarz lerrrrna states for z E D
If(z)I < IzI
and If'(0)I <_ 1.
and the equality j f'(0)I = 1 or the equality If (z)I = Izj at a single point z
0 implies f (z) = Ez with jej = 1.
Now we shall drop the assumption f (0) = 0. If f
: D - D is an
arbitrary holornorphic mapping, we fix an arbitrarily chosen point z E D and consider the automorphisms g and h of D defined by for ( E D, f (`) - f (ZX
for C E D.
Then the composed mapping F = h f g is a holornorphic mapping of D into itself which sends 0 into itself. Since F(0) = 0 and F'(0) _ h'(f(z))f'(z)g'(0), we obtain F'(0)
- IzI2
1 - If(z)12f i
,(z )
The Schwarz Lemma and Its Generalizations
2
Hence, 1
I2
1 f(z)I2 lf'(z)I
or
'z 1
If If(z)I2
for z E D.
lIZ12
We may conclude the following:
Theorem 1.1 Let f be a holomorphic mapping of the unit disk D into itself. Then
IdIIfI2
fortED,
1I&I
and the equality at a single point z implies that f is an automorphism of D.
This result, which was derived from the Schwarz lemma, is actually equivalent to the Schwarz lemma. In fact, if f : D -* D is a holomorphic mapping such that f (0) = 0, then by setting z = 0 in the inequality above, we obtain
If'(o)I < 1, and if I f'(0)I = 1, then f is an automorphisni of D. Moreover, pIf(<)I
Idf I
1-If12
Jo
<
/'ICI
Jo
Idzl
1-Iz12'
Hence,
1 + If (()I c to 1 + I(I
log 1 -
if (()I
g 1 - ICI'
which implies
_ 1 + If(()I < I+ I(I
2
if (U It follows that If (()I implies the equality
-1
1-If(()I = 1-ICI
2
1-ICI
-1.
J. The equality If (()I = ICI at a single point C
0
If'(z)I 1 1 - If(z)I2 - 1 - IZI2 for all z lying on the line segment joining 0 and C. By Theorem 1.1, f is an automorphism of D, and hence, f (z) = ez for some a with lel = 1. This proves that Theorem 1.1 implies the classical Schwarz lemma.
2
A Generalization by Ahlfors
3
We shall now consider Theorem 1.1 from a differential geometric viewpoint. If we consider the Kaehler metric dsD on D given by dzdz
2
dsD =
(1
- IzI2)2' then the inequality in Theorem 1.1 may be written as follows: f'(dsD) _< dsD.
The metric dsD is called the Poincare metric or the Poincare-Bergman metric of D. Now, Theorem 1.1 may be stated as follows:
Theorem 1.2 Let D be the open unit disk in C with the Poincare-Bergman metric dsD. Then every holomorphic mapping f : D -+ D is distancedecreasing, i.e., satisfies f'(dS2) < dsD,
and the equality at a single point of D implies that f is an automorphism of D.
If f is an autornorphism of D, then the Schwarz lemma applied to both f and f -1 implies that f is an isometry. We note that the Guassian curvature of the metric dsD is equal to -4 everywhere. (In general, the Gaussian curvature of a metric 2hdzdz is given by -(1/h)(02 log h/0zl82).)
2 A Generalization by Ahlfors Let Da be the open disk of radius a in C, Da = {z E C; Izi < a}. Then the metric 4a2 dz d2
dsa = A(a2 - Iz12)2
(A > 0)
on Da has Gaussian curvature -A. The following theorem of Ahlfors [1] generalizes the Schwarz lemma.
Theorem 2.1 Let M be a one-dimensional Kaehler manifold with metric ds2M whose Gaussian curvature is bounded above by a negative constant -B.
Then every holomorphic map f : D. - M satisfies
f' ds,2y < gd.
The Srhwarz Lemma anal Its Cenc7u1izat1oni,
4
Proof Let. u be the nonne ative function on D defined by
r
((1s7:L1)
J
-
ads7.
e. want to prove that it <= A/B on D. Although u may not attain its maximum in (the interior of) Da in general, we shall show that we have only to consider the case where a attains its maximutin in D,,. Let r he a positive number less than a. Let. zo be an arbitrary point of D. Taking r sufficiently close to a, we may assume that zo is in Dr. We denote by clsr the metric on Dr obtained from ds2 by replacing a by r but keeping the same constant A. From the explicit expression for ds, it is clear that (rl.,')z(, - (d.sa)z, as r -> a. If we define a nonnegative function ur on Dr by f "(ts 2ti1) = u,ds ', then ur(z(i) -y as r -* a. Hence it suffices to prove that 11r .41 B on D,., If we write f *(d.s. ,,) = h dz dE on D, then h is hounded on Dr (the closure of Dr). On the other hand, the coefficient 4r'I.4(r1 - , 2)2 of d,sr approaches infinity at the boundary of Dr. Hence, %
the finiction ur defined on Dr goes to zero at the boundary of Dr. III particular, "r attains its maxirmm'i in Dr. The problem is thus reduced to the case where it attains its maximum in D,,. We shall now impose the additional assumption that u attains its maximum in D, say at zp C D,,.. If u(20) = 0, then u = 0 and there is nothing to prove. Assume that u(za) > 0. Then the mapping f : D,, -+ Al is nondegenerate in a neighborhood of z0 so that f is a biholornorphic map from an open neighborhood U of zc) onto the open set f (U) of Al. Identifying U with f (U) by the map f, we use the coordinate system z of D,1 C C also as a local coordinate system in f (L.1). If we write (Is 4,1 = 2h dz d. on f (U.). 2 = 2y dz dz, then u = h.; y. then f' (ds 12 1) = 2h d-- dz on U. If we write dsZ
The Gaussian curvature k of the metric ds11 = 2h dz d is given by
k=-
1 02 log h. h i)z i):
The Gaussian curvature -A of the metric ds", = 2cjdzd- is given by
-A--g1 02dzlogd,y Since k < -B by our assumption. we have 32 log u
_
i)2 log h
i)'` log g
-
kh - :4y >= BJ(- .4y. jzr> i)zr)z 0-- r:")< Since log u attains its maximum at zu. the left-hand side in the inequality
above is nonpositive at .a and so is the right-hand side. Hence. .4/B
J
7'Fi' Gatosian Plane Mznu TWO 1'o.nn1R
big - It, at zo. Since it attains its niaxinluni at zc, it follows that. A;'B ? ri O
everywhere. 3
The Gaussian Plane Minus Two Points
In view of Theorem 2.1 we are naturally interested in finding a onedimensional Kaehler manifold whose Gaussian curvature is hounded above by a negative constant. As we shall see later (see Sec. 4 of Chapter IV). the Gaussian plane C cannot carry such a. metric. The metric
d.s`'=2(1+1-1'dzd4 on C has curvature k = -1(1 + I: 2)3, which is strictly negative everywhere but is not bounded above by any negative constant. If a one-dimensional complex manifold Al carries a Kaeliler metric whose Gaussian curvature is bounded above by a negative constant, so does any covering manifold of Al. If Af is the Gaussian plane minus a point, say the origin, that is, Al = C {0}, then the universal covering manifold of ;ll is C, the covering projection being given by FC
6`-`z E C
- {0}.
This shows that C - {O} does not admit a Kaehler metric whose Gaussian curvature is bounded above by a negative constant. Consider now the Gaussian plane minus two points, say Al = C - {0, 1}. If we use the modular function ,1(z I, we can show that Al carries a complete Kaehler metric with negative constant curvature. Let H denote the upper
half-plane in C. i.e.. H _ {z - x + iy E C; y > 0}. Then the modular function A gives a covering projection A : H - C - {0.1}. If we digress a little, the group of deck-transformnations is given by
"-zCH.
c z, + d
=H
with
rn
ii
` c
d)
SL(2: Z )l
a
c
b
d) _
1
0
0
1
mod 2.
This group. known as the congruence subgroup mod 2. is a normal subgroup
of index 6 in the modu'_ar group SLi2: Z. and its fundamental domain i given by F in Fig. 1. The boundary of F and the imaginary axis are mapped into the real axis by A. In Fig. one sees roughly how the fundamental domain F is mapped onto C - to. 1 } by A. On the upper half-plane H. the
The Schwarz Lemma and Its Generalizations
6
0
I
0
1
.k(.)
Fig. 1.
a(0) Fig. 2.
metric ds2 = 2dz dz%Ay2 has curvature -A and is invariant by the group of holomorphic transformations of H. It follows immediately that this metric induces a metric of curvature -A on C- {0, 1}. Unfortunately, the modular function A is so complicated that the induced metric on C- {O, 1} cannot be expressed in a simple form in terms of the natural coordinate of C - {0,1 }. (For the definition and basic properties of the. modular function A. see for instance Ahlfors [2' and Ford [1].)
We shall now give a more elementary construction of a metric with Gaussian curvature 5 -4 on C - {0. 1). The construction is due to Grauert and R.eckzicgel [1]. Given a positive C°C function g(z. z) defined in an open
set in C. we define a real-valued function K(g) defined in the same open set as follows: log g fi(gl=-91 c02 Oz Of
The definition is motivated by the fact that K(g) is the Gaussian curvature of the metric ds' = 2y dz dz. We first prove the following:
Proposition 3.1 For positive functions f and g, we have (a) cK(c.q) = K(g) for all positive numbers c:
(:h) f9K(fq) = fK(f) +9K(9): (c) (f + g)2K(f + g) f 2K(f) - 92K(9): (d) If K(f) -k, < 0 and K(g) < -k2 < 0, then
K(.f - 9) S -k1 k2;'(kj + k2).
1lums Two Point4
The (.(aussia.7a Tulle
7
Proof (a) and (b) are immediate from the definition of K(y). (c) follows from the following. which can he easily verified from the definition of K(q). .f9 (.f -1_
If
9)"f2 K(f) + g2K(y) - (f + y)2K (f - y)] Of
dq dzoz
0.
(d) is a consequence of (c) and .. k1k27;(ki - k2).
[f2K(f) +g2K(y)]/(f + y)'` This latter inequality follows from
-(f 2k1 192k2)l (f
g)2 < -kik2/(kt + k2).
We set
p(z. Z) =
2z12a-2(1
+
]22122.),
where n is a constant., 0 < a < 5. Then p is a positive C.'' function on C - {0}. We have
K(p) - -4a2/ (1 +
12:120)3s
We set f (z, =) = P(2, =)P(z
Then f is a positive C'"' function on C - {0, 1}. By (b) of Proposition 3.1, we have
K[f(z, f)] = p(
h [p(z.
)]
1) T K[p(z - 1.
- 1)]
Since
<
K[p(2.-5) p(z - 1,z - 1)
a2
for0<1:1
1
2(1 + 320)
and
K[p(z-1;z-1)]
<
a2
p(z.z)
-
2(1 + 32a)
for0< z 1 <
1
,
- 2
we obtain z
if0<, iz
<= 2
or
0
The Schwarz Lemma and Its Generalizations
8
On the other hand, for Iz - 11 > z, we have 12z - 2I2a-2(1 + 12z - 2120)(1 +I2zl2a)3
12z - 2120-2/5(I2z - 2I-2/5 + 12z - 2120-2/s)
x (12z - 2I-2/5 + I2z12a12z
- 2I-2/5)3
C 1 (1 + 1)(1 + I2zI2a12z - 2I-2a)3
< 2(1 +
32x)3.
Similarly, for IzI > 2, we have 12z120-2(1
It follows that if
IzI
I2z12a)(1
+
+ 12z - 2120)3 < 2(1 +
32a)3.
>_ 2 and Iz -1I ? 2, then
K[f(z,z)]
K[p(z, 4)1
- P(z-1,z-1) + < =
-
2a2
(1 +32-)3
K[p(z -1, z - 1)] P(z,z) 2a2
(1 + 32a)3
4a2 (1 +
32a)3'
Since -4a2/(1 +3 2r)3 < -a.2/2(1 + 32n) for 0 < a < K[f (z, z)] <_
a2
we have finally
for z E C - {O,1}.
2(1 + 320)
We have shown that the metric ds2 = 2f (z, 2) dz dz on C - {0, 1} has Gaussian curvature K(f) <- -a2/2(1 +3 2a) < 0. But this metric is not complete. Since ds = v JIdzl has a singularity of order Iz1a-1 at 0, the point 0 is at a finite distance from any point of C - {0, 11. Similarly, for the point 1. It is also easy to verify that the point at infinity of C is at a finite distance from any point of C - {0, 1} with respect to the metric ds2. We shall make an adjustment to obtain a complete metric whose Gaussian curvature is bounded above by a negative constant. Choose a real-valued C°° function s(z, z) on C such that
s(z,z)=1 for izl<1, 0 < s(z, z) < 1
for
1
4-< IzI <-3
s(z,z) = 0 for IzI >
31
'
3
The Gaussian Plane Minus Two Points
9
We set
q(z, a) = s(z, z)/Iz12(1og Iz12)2
=0
for 0 < I z I <- 3,
forzI> 1
3
We set
h(z,z)=q(z,z)+q(z-1,z-1)+ Since K[q(z, z)] _ -2 for 0 < jz] <
1 q(z,)
we obtain
K[h(z,2)]=-2 on (0
4}. l
JJJ
l
JJJ
We set
g=f+ch, 0
K[g(z,z)]
2k'
2k'
2+ck' <
2+k'
<0
on the domain {0 < jzj < 4} U {0 < Iz - 11 < 4} U {jzj > 4}. In the complementary set which is compact, we use the following estimate (cf. (c) of Proposition 3.1):
K(g) = K[c(f + h) + (1 - c) f ] < c2(f + h)2K[c(f + h)] + (1 - c)2f 2K[(1 - c)f ]
(f +ch)2 _ c(f +h)2K(f + h) + (1 -c)f2K(f) (f + ch)2 < c(f+h)2K(f+h)-(1-c)f2k' (f + ch)2 If we take a sufficiently small c, then K(g) is bounded above by a negative constant on the compact set we are considering. We have thus constructed a complete metric ds2 = 2g dz dz on C - {0, 1 } whose Gaussian curvature K(g) is bounded above by a negative constant,
say k. Multiplying the metric by a suitable constant, we may assume that
k=-4.
The Schwurz Lemma and Its Generalizations
10
4
Schottky's Theorem
One of the best known consequences of the generalized Schwarz lemma is the following theorem of Schottky [1[.
Theorem 4.1 Given a complex number a with a 54 0, 1 and a real number
r with 0 < r < 1, there exists a positive number S = S(a, r) with the following property: If f is a holomorphic map from the open unit disk D = {z e C; IzI < 1} into C - {0, 1} such that f (0) = a, then If (z) I < S(a, r)
forizl
-4; such a metric was constructed in the preceding section. If we set r' _ a log(1 + r)/(1 - r), then the set {z E C; IzI < r} coincides with the set set Al = C - {0,1 } and let dsM be a complete metric with curvature
of points in D whose distances from 0 with respect to dsD do not exceed r'. Let N(a; r') be the set of points in M which are at distance r' or less from a with respect to the metric dsM. Since the metric dsM is complete,
N(a;r') is compact and hence is contained in the set {z E C; IzI < S} for a suitable positive number S. Let f : D M be a holomorphic map such that f (0) = a. Since f is distance-decreasing, i.e., f * (ds 21) Theorem 2.1, it maps the set {z E D; IzI < r} into N(a,r').
dsD by
Since the construction of the metric ds r in the preceding section is fairly explicit, we can give an estimate on S(a,r). Let dsM = 2g dz d2 he the complete metric with curvature < -4 constructed on C - {0, 1} in the preceding section. From our construction, we see that g > A2h on C - {0,1 } for a suitable positive number A, where h was defined in Sec. 3. From the definition of h, we see that, in the domain {z E C; IzI 3}, h coincides with 1
IzI2(log
.
Iz12)2
If we use the polar coordinate system (r, 0), then we have ds ,1 = 2g dz dz > 2A2h dz di A2 dr dr on {z E C; IzI 2r2(logr)2
3}.
Let a and b be two points of C such that 3 < Ial < IbI. Then the distance from a to b with respect to dsM is greater than //'IbI
Adr
= A tog
vrlogr f
log IbI
log al
5
Compact Riemann Surfaces of Genus >_ 2
11
If IaI < 3, then it suffices to replace log IaI by log 3. On the other hand, the distance from 0 E D to z E D with respect to ds2 is equal to
llogl+IzI 2 1-IzI' Let f : D - M = C - 10, 11 be a holomorphic map. Since it is distancedecreasing, we have A
logIf(z)I
1+IzI
1
I log log if (0)1 < 2 1°g 1 - IzI>
provided that I f (0)I >_ 3 and I f (z) I ? 3. If I f (0)I < 3 and I f (z) I >_ 3, it suffices to replace log if (0) I by log 3. In general, we obtain
l(I
+Iz logIf(z)I <Maxog3,B/ 1-IzI
c 111
where B = Max{log 3, log I f (0) I } and C = f /A. If we wish to estimate the constant C, we have to write down the function s(z, z) in Sec. 3 explicitly.
Similar estimates on log f (z) have been obtained by Ostrowski [1], Pfluger [1], and Ahlfors [1]. See also Heins [1]. 5
Compact Riemann Surfaces of Genus ? 2
In this section we prove the following statement:
Theorem 5.1 Every compact Riemann surface M of genus g ? 2 admits a Kaehler metric whose Gaussian curvature is bounded above by a negative constant.
Proof If we make use of the classical result that every compact Riemann surface M of genus g >_ 2 is a quotient of the upper half-plane H by a discontinuous group r of linear fractional transformations acting freely on H, we see easily that M admits a Kaehler metric of constant negative curvature. We shall give here a more elementary proof following Grauert and Reckziegel [1].
Since the genus of M is g, there are g linearly independent holomorphic 1-forms Wi, ... , w9 on M. We set ds2 =
Wi W i + W2GJ2.
In terms of a local coordinate system z of M, we may write
ds2 = (ffl + f2f2) dz dz,
where wi = f1 dz.
If Pi, ... , pk is the set of common zeros of w1 and W2i then ds2 is a Kaehler metric M - {pi, ... , pk}. By a simple calculation we see that the Gaussian
The Schwarz Lemma and Its Generalizations
12
curvature K is given by
K = -2I fif2 - fl f2I2/(flfl + f2f2)3. Since w, and W2 are linearly independent, it follows that f2 0- 0 and fl/f2 0 constant, and consequently K $ 0. Hence there are at most finitely many points Pk+l, ... , pm in M - {pi, ... , pk} where the curvature K vanishes.
For each pi, choose a neighborhood Vi in such a way that vi n l' = 0 for i # j. In each Vi we shall modify the metric so that the Gaussian curvature becomes negative everywhere on M. We fix i, i = 1, ... , m. Choosing a local
coordinate system z in Vi such that z(pi) = 0, let r be a positive number and
B = {z; IzI < r} C= V B' = {z; I z I < r/2}.
Let a(z, z) be a C' function on Vi such that
0<_a(z,z)S1, aIB'-1, aI(I -B)=0. Let c be a constant, 0 < c < 1, and set
g= f + ch,
where f = fi fl +f212
and h= a
(1 + z2).
Then the metric g dz dz coincides with ds2 on Vi - B. Since the curvature of ds2 is bounded above by a negative constant on the compact set. B - B', the metric g dz dz has a strictly negative curvature on B - B' if c is sufficiently
small. From the estimate on K(g) given at the end of Sec. 3, we see that the curvature of g dz dz is strictly negative in B' if c is sufficiently small. 13
Remark It is known (see, for instance, Springer [1, p. 270]) that if the genus g >- 1, there is no point of M where all holomorphic 1-forms vanish. It is therefore possible to choose wl and w2 in the foregoing proof in such a way that wlwl -4- W2,7)2 is positive definite everywhere on M. But this does not simplify the proof. 6
Holomorphic Mappings from an Annulus into an Annulus
Let A be the annulus in C defined by
A={zEC; 0
6
Holomorphic Mappings from an Annulus into an Annulus
13
Let Al and A2 be two annuli with moduli M1 and M2, respectively:
Ak={zEC; 0
For each integer m such that Imril <_ 1112/M1, we define a holornorphic mapA2 with deg f = m as follows: ping fm . At
f,n(z)=zm forzEA1. From topology we know that any mapping f : Al -+ A2 (holornorphic or not) of degree 771 with Irral < M2/M1 is homotopic to fm. In fact, any two mappings Al into A2 with the same degree are homotopic to each other. We know also that for any integer m there is a mapping (not necessarily holomorphic) of Al into A2 with degree m. But we have Theorem 6.1 Let Al and A2 be annuli with moduli M1 and M2 as above. If f : Al -a A2 is a holomorphic mapping, then Ideg f I <- M2/M1. If M2/M1 is an integer, then a holomorphic mapping f : Al -+ A2 with degree m = ±1112/M1 coincides with the mapping fm up to a rotation.
Proof We consider the band B = {z E C; -b < Im z < b} of width 2b. We know that it is conformally equivalent to the open unit disk or equivalently to the upper half-plane H = {w E C; Im w > 0} in C. Indeed, the mapping zEB ie"/2n E H is a biholomorphic mapping. The invariant metric dsH of curvature -1 on H is given by dsH =
dw dw 2 V2
where to = u + iv.
If we induce this metric to the band B by the holornorphic mapping given above, we obtain the following invariant metric dsB of curvature -1: dsB =
72 dz dz
4b2 cost (pry/b)
where z = x + iy.
We consider now a holomorphic mapping p from B onto the annulus A = {w E C; r < 1w < 1/r}, r = e-2"6, defined by p(z) = e2rriz
z E B.
Then p : B -- A is a covering projection. We denote by dsA the metric on A induced by dsB. Then the rectangle {z E B; 0 <- x <- 1} is a fundamental
14
The Schwarz Lemma and Its Generalizations
domain for this covering space (B, A, p). The projection p maps the upper edge, the lower edge, and the two vertical edges of this rectangle onto the inner boundary, the outer boundary of the annulus A, and the segment {w = u + iv E A; u > 0, v = 0}, respectively. It is also clear that p maps
{z=x+iyeB;0<x<_1,y=0)onto the unit circle {wEA;IwI=1}, which is the generator of the fundamental group 7ri (A). Consider a curve w(t), 0 < t <_ 1, in A which represents the generator of irl (A). We may assume that w(0) = w(1) > 0. To compute its length with respect to d$A,
we consider a curve z(t) in the rectangle {z E B;0 < x < 1} such that p[z(t)] = w(t), Re[z(0)] = 0 and Re[z(1)] = 1, and compute the length of z(t) with respect to dsB. From the expression of dsB given above, we see that the curve w(t) has the least length when it is the circle Iw(t)J = 1, i.e., when z(t) is real for all t. This least length is given by
J0
irdx
it
2b
2b*
We have shown that the circle w(t) = e21,it, 0
t 1, represents the generator a of -7r1(A) and has arc-length 7r/2b with respect to ds2 and that
any closed curve in A representing a has arc-length ? it/2b, where the equality holds only when the closed curve coincides with the unit circle up to a parametrization. Similarly, the curve w(t) = e2m7rit 0 < t < 1, representing ma E rri(A) has arc-length im7r/2bl and any closed curve representing ma has arc-length > JTn7r/2bi unless it coincides with w(t) _ e2m1rit up to a parametrization.
Given two annuli Al and A2, we consider the bands B1 and B2 of widths b1 and b2 and covering projections p1 : B1 -* Al and P2 : B2 -+ A2 he a holomorphic mapping. Since B1 is A2 as above. Let f : Al holomorphically equivalent to the unit open disk, we may apply Schwarz-
Pick-Ahlfors lemma (Theorem 2.1) to the map f o pl B1 , A2 and see that f o pl is distance-decreasing, i.e., (f o pi )*ds2 , -<_ dsB Since pl is a local isometry, we see that f itself is also distance-decreasing, i.e.. :
.
f'd 2 , _<- dsA1. We consider the unit circle w1(t) = e2nit, 0 t 1, in Al and its image curve w2(t) = f [wl (t)], 0 _5 t < 1, in A2. Since wi (t) has arclength rr/2b1 and f is distance-decreasing, w2(t) has arc-length <= -7r/2b,. If deg f = m so that w2(t) represents mat E irl (A2), where a2 is the generator 7r/2b1. of ir1(.A2), then w2(t) has arc-length Jmrr/2b2l. Hence, lmrr/2b21
that is, Im; < b2/b1. Since rk = e 2ibk and Mk = log(1/rk) _ -2logrk for k = 1, 2, we have b2/b1 = M2/M1. Hence. Irrtl < M2/M1, which proves the first assertion of the theorem.
6
Holomorphic Mappings from an Annulus into an Annulus
15
Assume that ?112/A11 is an integer and Iml = Ai2/111. Then w2(t) has arc-length lmr/2b21 = 7-,/2b1. Hence, the closed curve w2(t) coincides with
e2mait, p < t 5 1, up to a parametrization. Since f is distance-decreasing and w2(t) has arc-length r/2b1, f maps wi(t) onto w2(t) isometrically. This implies that w2(t) coincides actually with e2mnit up to a rotation. It follows
that f and fin coincide on the unit circle wl(t) of Al up to a rotation and hence they coincide of A, up to a rotation.
Corollary 6.2 Let f be a holomorphic mapping from a annulus A = {z E C; r < jzj < 1/r} into itself. Then either f is homotopic to a constant map or is of the form f (z) = e±2ni(z+a), where a is a real number. Theorem 6.1 is due to Hither [1]. The proof presented here is perhaps a little more differential geometric. In connection with the result of this section, see also Schiffer [1], Jenkins [1], and Landau and Osserman [1, 21.
CHAPTER II
Volume Elements and the Schwarz Lemma
1
Volume Element and Associated Hermitian Form
Let M be a complex manifold of complex dimension n and vhf a volume element of M. By definition vp is a 2n-form which is positive in the following sense. In terms of a local coordinate system z1, ... , zn of M we may write
vM = i'`Kdz1 Adzl A... Adzn Adin, where K is a positive function. To this volume element vM we associate the 2-form pM defined by
oM = 2i E Rap dza A dz0 , where
Rap = -a2 log K/8za 803, and also the Hermitian form hM defined by
hM = 2ERa0 dzad2O. For the reason given below, we shall call hM the Ricci tensor of M (with respect to the volume element vtis).
Example 1 Let M be an n-dimensional Kaehler manifold with metric dsM =
2gapdza d21 . The volume element vM of M is given by
vm =i'CdzI Adzl A
..Adz"Adin.
where G = det(gap).
Then the Ricci tensor (or rather its components) is given by (see, for instance, Kobayashi and Nomizu [1, Vol. II]) RaT3 = -d2 log G/8za 8z3. 17
18
Volume Elements and the Schwarz Lemma
What we have said so far is true also for a Hermitian manifold, provided that we use the Hermitian connection. (For details, see Sec. 1 of Chapter III.)
Example 2 Let Al be an n-dimensional complex manifold and H the Hilbert space of holomorphic n-forms cp such that in
2
ypAy
.11
The inner product in H is defined by (S^,
)= fM tif
z
YA for',
H.
Let pa, 401, P2, ... be an orthonormal basis for H (provided that H is nontrivial). Then 2n-form VM1 defined by VA1 =
i?1
E ;oJ A `t'J J
is independent of the choice of an orthonormal basis. If for every point i. of M there is an element. E H which does not vanish at x, then vM is a volume element of M and is called the Bergman kernel form. The Hermitian form hm associated with v.1 is easily seen to be negative semidefinite. If h,kf is negative definite, -hAf is called the Bergm.an metric of Al. Since the Bergman metric -h.,1.f is a Kachler metric, it gives rise to a volume element in the usual sense, which will be denoted by vi1.1. We shall denote by hM the Hermitian form associated with v'. From these constructions we see that VM, hh1, v'M, and h'M are all invariant by the group of holomorphic transformations of M. If M is homogeneous, i.e., if the group of holomorphic transformations is transitive on M, then v''M = cvM, where c is a constant. It follows that if Al is homogeneous, then hh1 = h,ti1. In other words, if and denote the components of the Bergman metric and of its Ricci tensor, the homogeneity of M implies R«,q = -gap. If M is a bounded domain in C' with Euclidean coordinate system z1, ... , zn, then H may be identified in a natural manner with the space of square integrable holomorphic functions on M with respect to the Lebesgue measure in dz' Adz 1 A .. A dzn A dzn. For a bounded domain M of Cn, v,tif is always positive and -hM is always positive-definite. If we write
A dzn A dzn, we call K(z, z) the Bergman VM = inK(z, z) dz' A dz1 n kernel function of the domain Al. For more details, see Kobayashi [1] and Lichnerowicz [1] as well as Bergman [1]. For homogeneous M, see Koszul [1], Hano and Kohayashi [1]. and Pyatetzki-Shapiro [1].
Basic Formula
2
19
The closed 2-form (1/47T)cpM is known to represent the first Chern class cl(M) of M (see Chern [2] and Kobayashi and Nomizu [1, Vol. 2]).
Basic Formula Let M and M' be two complex manifolds of dimension n with volume 2
elements vM and vm,, respectively. Let hM and hM' be the Ricci tensors associated to vM and vm,, respectively. Let f : M' -* M be a holomorphic mapping and define the ratio u = f*(vM)/vM' by f *(vj) = u vm,. Then u is a nonnegative function on M'. At a point where u is positive, i.e., a point where f is locally biholomorphic, we want to calculate the complex Hessian of log u. Let p E M' be such a point and choose a local coordinate system z1, ... , z" in a neighborhood of p and a local coordinate system wl, ... , w" in a neighborhood of f (p) in such a way that the map f is given by wi = zi, i = 1, ... , n. We write
vM = PKdw' Adwl A ... Adwn Adw", vMw = i"K' dzi A dzl A ... A dz" A dx".
Then
u = f *K/K' and
2E
a,log u
dz`Ydi'? = hM
apaza9z13
- f * It M
We have proved (Kobayashi [2]).
Theorem 2.1 Let M and M' be complex manifolds of dimension n with volume elements vh.1 and vM, and Ricci tensors hM and hM', respectively.
Let f : M' - M be a holomorphic mapping and let u = f * (vM)/vW . Then at a point of M' where u is different from zero, the following formula holds: 82 log u
2
;4,« a>3
p
dza dz = ItM- - f *h,1.
a,p
where zl, ... , z" is a local coordinate system at the point.
In general, for a Hermitian form It on a complex vector space V we denote by dim+ h (respectively, dim- h) the dimension of a maximal dimensional complex subspace of V on which It is positive (negative) definite.
Volume Elements and the Schwarz Lemma
20
Hence h is positive (negative) definite on V if and only if dim+ h = dim V (dire- h = dim V).
Proposition 2.2 Let f : Al' -* AI, h11, hA1,, and u be as in Theorem 2.1. Then we have
(1) If dim+ hh1, > dim+ hAl or dim- hlf, < dim- hm everywhere, the function u attains no nonzero local maximum on M'; (2) If dim+ h,%1- < dim+ h,.f or diva h5,1- > dim- h711 everywhere, the function u attains no nonzero local minimum on Al'.
Proof Assume that u attains a nonzero local maximum at p E Al'. Then the Hermitian matrix (52 log a/dz" az`f) is negative semidefinite at p. By Theorem 2.1, hn1- - f *h w is negative semidefinite at p. Since u(p) -A 0, f * is nondegenerate at p so that dim+ f *h,ti1(p) = dim+ h. j [f (p)and
dim- f *hnf(p) = dim- hAf [f (p)]. Hence,
dim+ h,
' (p) <_ dim+ hM [f (p)],
dim- hnf' (p) > dim hm [f (p)]
.
This proves (1). The proof for (2) is similar.
Remark If Al' is a Hermit.ian manifold with Hermitian metric ds2 =
2 y q dz
dz`3, then the complex Laplacian (logu.) of log u is defined by
(log u)
g'R 92 log a 8z° &0
From Theorem 2.1 we obtain 1
(logu) = R.a1- - 1T ace(f*hm), where R .,, is the scalar curvature of Al' and Trace (f *hM) denotes the trace of f * hh, with respect to the metric ds2 of Al'. This formula was proved by
Chern [2]. If M' is a Kaehler manifold, then (logu) = 20(logu), where A is the ordinary Laplacian.
3
Holomorphic Mappings f : M' -+ M with Compact M'
We shall apply the results of Sec. 2 to the case where M' is a compact complex manifold.
Theorem 3.1 Let Al and Al' he n-dimensional complex manifolds volume elements v,,,1 and v.Nf, and Ricci tensor hm and hAt',
u1th respectively.
3
Holomorphic Mappings f : M' -+ M with Compact M'
21
Let f : M' --* M be a holomorphic mapping. Assume that M' is compact. Then
(1) If dim+ hM, > dim+ hM or dim- hM, < dim- h1,1 everywhere, the mapping f is everywhere degenerate, i.e., f *vM = 0;
(2) If dim+ hm, < dim+ hM or dim- hr11- > dim h,-,l everywhere, the mapping f is degenerate at some point of M', i.e., f*vM vanishes at some point of Al'.
Proof (1) Assuming the contrary, let p be a point of M' where a = f *vM fvM, attains its maximum. Then u(p) # 0. Now, our assertion follows from (1) of Proposition 2.2. (2) Similarly, (2) follows from (2) of Proposition 2.2.
Corollary 3.2 Let f ; M' -> M be as in Theorem 3.1. Then (1) If the Ricci tensor hm, is everywhere positive definite on Al' and if the Ricci tensor hM is nowhere positive definite on Al, the mapping f is degenerate everywhere on M'; (2) If hM is everywhere negative definite on Al and if h.1, is nowhere negative definite on M', f is degenerate everywhere on M'; (3) If hM is everywhere positive definite on M and if h1, is nowhere positive definite on Al', f is degenerate somewhere on Al'; (4) If hM, is everywhere negative definite on Al' and if h.1 is nowhere negative definite on Al, f is degenerate somewhere on M'. Let WWM be the 2-form associated to VA.1 (see Sec. 1). Let 4 h1 be its rzth exterior power and define a function r,11 by 1
2 n (P Al
= rA.,v.41.
SO that, locally,
rM =
det(RQ, )
K
Where vu =inKdz1 Ad!1A
For M', we define rr.1, in the same way.
Tlte°elua
=2iFRajdz°Adz
.
3.3 Let M and M' be n-dimensional complex manifolds with.
Volume elements vM and v11,, respectively. Assume
(a) The Rieci tensors hM and h,1.1, are negative definite;
Volume Elements and the Schwarz Lemma
22
(b) rM(p)/rM,(p') > 1 for p E M and p' E M'; (c) M' is compact. Then every holomorphic mapping f : M' - M is volume-decreasing in the sense that f *vM /vM, < 1.
Proof We set u = f*vM/vm, as before. We have
f*Wti1 =f*rMLi
nM'
rM
vM
=f*rM>= u rM
,
where the last inequality is a consequence of (b). Let p' E M' be a point where u attains its maximum (which may be assumed to be nonzero). To 1 at p'. But this prove that u 1, it suffices to show that f follows from the inequalities:
hM, < f*hM < 0 at p', where the first inequality is a consequence of Theorem 2.1 and the second inequality follows from (a).
Corollary 3.4 Let M and Al" be n: dimensional Hermitian Einstein manifolds with metric dsn and dsM, such that hM = -dsM (i.e., Raq = -g,,,.-3) andhAf- = -dsA1,. If M' is compact, every holornorphicmapping f : M' -+ AI is volume-decreasing.
Theorem 3.5 In Theorem 3.3 or in Corollary 3.4, assume further that Al is also compact. Let V (M) and V (M') denote the total volumes of M and Al', respectively:
V(M) = if vM,
V(M') = f VM'.
a
(1) If V(M') < V(M), then every holomorphic mapping f : Al'
M is
degenerate everywhere on M';
(2) If V(M') = V(M), then every holomorphic mapping f : M'
Al is either degenerate everywhere on Al' or a volume-preserving biholomorphic mapping.
Proof The topological degree deg f of f can be given by deg f = V(M) JM' f vM.
3
Holomorphic Mappings f : M'
M with Compact M'
23
Since f «VM < vM' by Theorem 3.3, we obtain
deg f < V(M')/V(M) Since f is holomorphic, f'vM is nonnegative, i.e., f'"vh.1/vhf' > 0, and deg f ? 0 Moreover, deg f = 0 only when f ivM = 0, i.e., only when f is degenerate everywhere on M'. Since deg f is always an integer, we may con-
clude that deg f is a nonnegative integer such that deg f < V(M')/V(M),
and degf = 0 if and only if f is degenerate everywhere on lvf'. If V(M') < V(M), then deg f = 0. This proves (1). If V(M') = V(M), then f "vM < vN1' and
f"`v,tir < V(M) = 1. V(M) Assume that f is not everywhere degenerate. Then
0 < deg f =
1
V(M)
M,
f*VM = vn.1'.
Being an everywhere nondegenerate mapping of a compact manifold into another compact manifold, f : M' --* M is a covering projection. Since degf = 1, it follows that f is a biholomorphic mapping. Corollary 3.6 Let M be a compact Hermitian Einstein manifold with negative definite Ricci tensor (i.e., R,,,ii = -cy0il with, c > 0). Every holomorphic mapping f : lvi -+ lvi is either degenerate everywhere or biholomorphic and isometric.
Proof By Theorem 3.5, if f is nondegenerate somewhere, then f is hiholomorphic and volume-preserving. If f is volume-preserving, it preserves the Ricci tensor hM and also the metric -(1/c)h1,1. The proof of Theorem 3.5 gives also the following:
Corollary 3.7 Let M and lvi' be compact Hermitian Einstein manifolds such that hM = -cdsh.1 and h1.1' = with c > 0. Then, for every holomorphic mapping f : M' -' M, we have deg f < V(,'11')/V(M).
Remark In certain cases, the total volumes V(M) and V(M') may be expressed in terms of topological invariants of ll1 and M'. Assume hn1 = -cdsM and hti1' = -cdsi1, as in Corollary 3.7. Since the first Chern classes ci(M) and cl(M') are represented by (1/41r)cpM and (1/47r)cp,tif., we see
24
Volume Elements and the Schwarz Lemma
that cl(M)" and cl(M')n are represented by (-1)navhj and (-1)navs.,,, respectively, where a is a positive constant. Hence, the ratio of the volumes may he expressed as the ratio of certain Chern numbers, namely, V(M')/V(M) = c i [M']/ci [Pvl].
Another case that may be of interest is the case where both M and M' are covered by the same universal covering manifold which is a homogeneous Hermitian manifold. Since M and M' are locally homogeneous and isometric, the integrands in the Gauss-Bonnet formulas for M and M' are the same constant. Hence, V(M')/V(M) = X(A1')1X(A ), where X denotes the Euler-Poincare characteristic. If we apply the last remark to Riernann surfaces of genus >_ 2, then we may conclude the following:
Corollary 3.8 (1) If M and Al' are compact Riemann surfaces such that genus of M > genus of Al' >_ 2,
then every holomorphic mapping f : AI' -> Al is a constant m.ap; (2) If M and Al' are compact Riem.ann..surfaces such that
genus of Al = genus of Al' > 2, then every holomorphic mapping f : Al' --> Al is either a constant map or a biholom.orphic mapping.
This corollary may be derived easily from the nonintegrated form of the second main theorem of Nevanlinna theory. As a matter of fact, Nevanlinna theory implies not only the inequality of Corollary 3.7,
degf < X(AP)/X(M) for compact Riemann surfaces Al and Al', but also the equality
degf = X(11')/X(M) + n1(M')/X('11), where n1(M') is a certain nonnegative integer called the stationary index of f. For details, see Wu i1, Corollaries 3.2 and 3.3]. A generalization of the equality above to higher dimension is not known. We conclude this section by a remark related to Corollary 3.6. Let 1I be a compact complex manifold such that a suitable positive power K"' of
4
Holoneorphic Mappings f : D -. M
25
canonical line bundle K admits nontrivial holomorphic cross sections. its Then every holomorphic mapping f of M into itself which is nondegenerate at some point is a covering projection (see Peters 111). This result of Peters is related to Corollary 3.6 as follows. If M is a compact Hermitian manifold with negative definite Ricci tensor [or with negative first Chern class c1(M)1. then the line bundle K71 for a suitable positive integer in admits sufficiently many holomorphic sections to define an imbedding of M into a projective space (Kodaira [1]). It should be noted also that if H1(AI;R) = 0 and
cl(M) = 0, then K itself admits a holomorphic section which vanishes nowhere, i.e., K is a trivial bundle. Other related results of Peters will be discussed in Chapter VIII. 4
M, Where D is a Holomorphic Mappings f: D Homogeneous Bounded Domain
Although we have a homogeneous bounded domain D in mind, we shall treat D as an abstract complex manifold satisfying certain conditions. Lemma 4.1 Let D and Al be n-dimensional complex manifolds with volAl be a holomorphic ume elements VD and v,ti.M respectively. Let f : D mapping. Assume
(a) The Ricci tensors hD and hA, are negative definite; (b) rM(q)/rD(p) >_ I for p E D and q E M; (c) The function u = f*vRf/vD tends to zero at the boundary of D in the
sense that for every positive number a the set {p E D; u(p) > a} is compact.
Then f is volume-decreasing, i.e., u < 1. See Sec. 3 for the definition of rAl and rD.
Proof In the proof of Theorem 3.3, the compactness of M' (which corresponds to D here) was needed only to ensure the existence of a point where u attains its maximum. Since (c) guarantees the existence of such a point, Lemma 4.1 follows from the proof of Theorem 3.3.
Lemma 4.2 In Lemma 4.1, f is volume-decreasing if (c) is replaced by the
following assumptions:
(c') there exists a sequence D1 C D2 C such that
C D of open
Volume Elements and the Schwarz Lemma
26
Uk Dk = D; (c'.2) each Dk carries a volume element vk for which (Dk, M, f IDk) satisfies (a), (b), and (c) of Lemma 4.1 and moreover lira Vk = VD
k-oo
(pointwise on D).
Proof Applying Lemma 4.1 to f : Dk -* M, we obtain f*vAI/vk < 1 on Dk. Hence,
f*VM/VD = lira f*vAf/vk C I. k-.oo
We shall now apply Lemma 4.2 to the case where D is a bounded domain
in C. With respect to the natural coordinate system z1, ... , zn in Cn, the Bergman kernel form bD can be written as (see Example 2 in Sec. 1)
bD=i"K(z,2)dz1Adz' where K(z.
is the Bergman kernel function of the domain D.
Lemma 4.3 Let D be a bounded domain in C" with the Bergman kernel form bD. Assume that (i) the Bergman kernel function K(z, z) of D tends to infinity at the bound-
ary of D; (ii) D is starizke in the sense that, for each 0 5 a < 1, the set
Da = {az E C": z E D}
(z being considered as a vector)
is contained in D.
If ba denotes the Bergman kernel form of Da for 0 < a < 1, then lim ba = bD;
a-1
and for any complex manifold Al of dimension n with volume element cNt M, the function f *v, i/ba on and for any holomorphic mapping f : D Da tends to zero at the boundary of Da.
4
Holomorphic Mappings f : D -+ M
27
proof For each 0 < a < 1, the mapping ga : D -+ Da defined by ga(z) = az is a biholomorphic mapping. Hence, g*(ba) = bD. If we write
bD =i"K(z,z)dzl and
then the invariance ga(ba) = bD implies
K(z, z) = a2nKa(az, az) for z E D. It is now clear that lima.1 ba = bD. If we write
then the function L(z, z) is continuous on D and hence is bounded on the closure D. C D for each a. Since K(z, z) goes to infinity at the boundary of D by assumption, so does ICa(z, z) at the boundary of Da. Hence,
L(z,2)/K0(z,2) tends to zero at the boundary of Da. Since f*vM/b, = L/Ka, this completes the proof.
Theorem 4.4 Let D be a bounded domain in C" with the Bergman kernel form bD and the volume element vD defined by the Bergman metric. Assume
(i) the Bergman kernel function K of D goes to infinity at the boundary of D;
(ii) D is starizke; (iii) the ratio VD/bD is a constant function on D.
Let M be an n-dimensional Hermztian manifold with metric 2 E9ap dw° diba and volume element vM. Assume (iv) the Ricci tensor hM = 2 R, dw°dwI3 is negative definite and the associated 2-form 9M = 2i F_ R°,3 dw° A down satisfies (_1)",,nM/2nVM
1,
1.
Then every holornorphic mapping f : D -- Al is volume-decreasing, i.e.,
f'vM/vD < 1.
Volume Elements and the Schwarz Lemma
28
Proof Let dsD be the Bergman metric of D and hD its Ricci tensor. By (iii) 11D is the Ricci tensor of the volume element bD as well as that of VD and hence is equal to -dsD. If we define rD as in Sec. 3, then we have rD
Since (-l)'?hf ? I by (iv), we have rhf(q)/rD(p) ? 1 for p E D and q E M,
which verifies (b) of Lemma 4.1. Since vD/bD is a constant function on D, va/ba is also a constant function on Da. It is therefore clear from Lemma 4.3 that
lirnva=VD and
f'vhf/Va tends to zero at the boundary of Da, which verifies (c') of Lemma 4.2. Since hD = -dsD, hD is negative definite. On the other hand, hht is negative definite by assumption. This verifies (a) of Lemma 4.1. Now, Theorem 4.4 follows from Lemma 4.2. 0
Remark If 11 is a compact complex manifold with ample canonical lint bundle, Al admits an Hermitian metric satisfying (iv). Let D be a bounded homogeneous domain in C". As we remarked in Example 2 of Sec. 1, the ratio uD/bD is a constant function on D. By a well-known theorem of Vinberg, Gindikin, and Pyatcttki-Shapiro [1], D is biholomorphic with an affinely homogeneous Siegel domain of second kind. In the next section we shall show that every affinely homogeneous Siegel domain of second kind
satisfies (i) and (ii) of Theorem 4.4 as well as (iii). Thus, Theorem 4.4 may he applied to every bounded homogeneous domain D in C" and. in particular, to every Herrnitian symmetric space D of noncompact type.
5
Affinely Homogeneous Siegel Domains of Second Kind
Following Pyateztki-Shapiro [11 we define the Siegel domains of second kind. Let V he a convex cone in R", containing no entire straight lines. A mapping
F : C"' x C"'
C" is said to be V-Herm.itian if
(1) F(u, v) = F(v, u) for u, u e C"', (2) F(aul + bu.2, v) = aF(ul, v) + bF(u2, v) for ul . 'U2, 4! E Cm, a, b E C,
5
Affinely Homogeneous Siegel Domains of Second Kind
29
(3) F(u, U) E V (the closure of V) for u c C'", (4) F(u, u) = 0 only when u = 0.
The subset S of C"+- defined by
S = {(z, u) E C" x C'; Im z - F(u, U) E V) is called the Siegel domain of second kind defined by V and F. Before we prove that S is equivalent to a bounded domain in Cn+m, we consider the Siegel domain of second kind in C1+m defined by
Imz-IuhI2_..._IumI2>0 where z, u1, ... , um are the coordinate functions in C1+m We prove first this domain is biholomorphic with the unit ball 1z012
in
+ ... + IzmI2 < 1
C1++n- We set
z-i
z0
z
z+i'
1
zm = 2um
2u1
z+i
z+i'
Then we have m
1k=O
2(Imz-Iu112-...-luml2),
Izk12= Iz + iI
which proves our assertion.
We shall now prove, following Pyateztki-Shapiro, that the Siegel domain S of second kind defined by V and F is equivalent to a domain contained in a product of balls. By a linear change of coordinate system, we may always assume that V is contained in the cone yl > 0, . . . , y' > 0. If V is the cone yl > 0, ... , y' > 0, then the components F' (u, u), ... , Fn (u, u) are positive semidefinite Hermitian form in ul, ... , u"`. We represent each Fk(u,u) as a sum of squares of linear forms:
Fk(u,u) = ILiI2 +... +
IL;kI2.
We define new Hermitian forms F1, ... , F" as follows. We set
F1(u,u) = F1(u,u) P2 (U, U) = 'I L,2, 12, Where the prime indicates that the summation is restricted to those L2 that are not linear combinations of L .. , L9j . Then we set P3 (U, U)
= E'IL9I2, a
Volume Elements and the Schwarz Lemma
30
where the prime indicates that the summation is restricted to those 0 that are not linear combinations of Li, ... , L91, L?, ... , L92. Similarly, we define F4, ... Fn. Let S be the domain defined by
Imzk-Fk(u,u)>0 k=1,...,n. Since Fk(u, u) > Fk(u, u) for all k, it follows that S contains S. We shall show that S is equivalent to a product of balls. By (4) in the definition of F, the system of linear equations Ls = 0
k = 1,...,n,
s = 1,...,sk
has only one solution, u = 0. It follows that the number of the linear forms Ls which really appeared in the construction of F1, ... , Fk is equal to m. By construction, these forms are linearly independent. If we take them as new variables v1, ... , v', then the domain S is defined by
Imz1-Iv112
-...-
Iv'"' I2
>0,
Imz2_Ivm1+112 _...- Ivm212 >0,
................................. Im zn -
Ivmn-1+112 _...
-
I,Um12
> 0.
It follows that S is equivalent to a product of balls in C1+" C1+m2-m1
The following proposition is due to Hahn and Mitchell [2].
Proposition 5.1 Let S be the Siegel domain of second kind in Cn+" defined by V and F. Let t be a fixed point of V and set t
Sk = {(z, u) E Cn X Cm; Im z - t - F(u, u) E V
for k = 1, 2, 3, .. .
SkCSk+l. For each k, the
Then
translation t (z,u)ES- (z+k,u
ESk
gives a biholomorphic mapping from S onto Sk.
The proof is straightforward and hence is omitted. For the Siegel domain S of second kind defined by V and F, an automorphism of V is a linear tranformation A of V such that, for a suitable
5
Affinely Homogeneous Siegel Domains of Second Kind
31
complex linear tranformation B of C', the following holds:
A F(u, u) = F(Bu, Bu)
for u E C'.
Then, for any x0 E R'7 and uo E Cm, the affine transformation z --- Az + Xo + 2i F(Bu, uo) + i F(uo, uo)
u ->Bu+uo is an automorphism of the Siegel domain S. If V is homogeneous, i.e., the group of automorphisms A of V is transitive, then S is homogeneous under the group of holomorphic transformations (which are affine in C'+'). In fact, if yo c V so that (iyo, 0) E S and if (zl, ul) is an arbitrary point of S
so that yl = Imz1 - F(ul,ut) E V, then the transformation
z -> Az+Rez1 + 2iF(Bu,ul) +iF(ul,ul)
u - Bu+u, maps (iyo,0) into (zj,ul), where A is an automorphism of V sending yo into y1 and B is a linear map of C72 satisfying A F(u, u) = F(Bu, Bu). The Siegel domain S of second kind defined by V and F is said to be affinely homogeneous if V is homogeneous. A theorem of Vinberg, Gindikin, and Pyatetzki-Shapiro [1] says that every homogeneous bounded domain in C" is biholomorphic to an affinely homogeneous Siegel domain of second kind.
Proposition 5.2 Let D be a homogeneous bounded domain in Cn and K(z, z) its Bergman kernel function. Then K(z, z) goes to infinity at the boundary of D.
Proof Assume that the proposition is false. Then there exists a sequence of points z1, z2,... in D which converges to a point on the boundary of D such that K(zk, zk) < a
fork=1,2,3.... ,
where a is a positive constant. Fix a point zo of D. For each k, let fk be an
automorphism of D such that fk(zo) = zk.
Let Jk denote the Jacobian of fk at zo (with respect to the Euclidean coordinate system in C^). Since the Bergman kernel form of D is invariant
Volume Elements and the Schwarz Lemma
32
by fk, we obtain K(zo, zo) = K(zk, 4)IJkI2 for k = 1, 2, 3,.... It follows that IJkj >_ b > 0
for k = 1,2,3,....
Consider { fk} as a family of holomorphic mappings from D into C' which are uniformly bounded. Taking a subsequence if necessary, we may assume that { fk} converges to a holomorphic mapping f : D Cn uniformly on all compact subsets of D. It is well-known that uniform convergence of { fk} entails uniform convergence of all corresponding partial derivatives of all orders of {fk}. In particular, the Jacobian Jf (zo) of f at zo satisfies
IJf(zo)I =limIJkl ?b>0. Hence, f gives a homeomorphism of a neighborhood U C D of zo onto the neighborhood f (U) of f (zo) in Cn. Since f (zo) is a boundary point of D,
f (U) is not contained in D. On the other hand, being the limit of If,,) f maps D into D. This is a contradiction. C ,
Proposition 5.2 shows that an affinely homogeneous Siegel domain of the second kind satisfies (i) of Theorem 4.4. Instead of (ii) in Theorem 4.4. we use Proposition 5.1. Now, from Theorem 4.4, we obtain
Theorem 5.3 Let D be an n-dimensional complex manifold which is biholomorphic with an affinely homogeneous Siegel domain of second kind and let dso be its Bergman metric. Let Al be an n-dimensional EinsteinKaeh.ler manifold with metric dsAf, such that its Ricci tensor h,1 is equal
to -ds2
M.
Then every holomorphic mapping f : D -+ M is volume-
decreasing.
If we assume the result of Vinberg, Gindkin, and Pyatetzki-Shapiro, then in Theorem 5.3, D can be any homogeneous bounded domain in C. Dinghas [1] obtained Theorem 5.3 when D is a unit ball in Cn. A result similar to Lemma 4.2 was proved by Chern [1] under stronger curvature assumptions. Hahn and Mitchell [1] and Kobayashi [2] proved Theorem 5.3 independently when D is a symmetric bounded domain. The generalization to the case where D is a homogeneous bounded domain is due to Hahn and Mitchell [2]. But in both papers Hahn and Mitchell assumed unnecessarily that f is biholomorphic.
Symmetric Bounded Domains
6
6
33
Symmetric Bounded Domains
We shall first summarize known results on the Bergman kernel functions of the so-called Cartan classical domains. For details we refer the reader to Hua [11, Tashiro [1), and Hahn and Mitchell [1[.
According to E. Cartan [11, there exist only six types of irreducible bounded symmetric domains - four classical types and two exceptional types. The four classical domains R1, RII, Rill, and RIV are defined as follows:
Ri = {m x n matrices Z satisfying I,,, - ZZ* > 0}, RII = {symmetric matrices Z of order n satisfying I - ZZ* > 0}, R111 = {skew-symmnetric matrices Z of order n satisfying In - ZZ* > 0},
Riv = {z = (z1i ... , z") E C"; Izz'12 + 1 - 2zz' > 0, Izz'j < 11, where Im denotes the identity matrix of order in, Z* is the complex conjugate of the transposed Z' of Z, and z' is the transposed of the vector z. For the domain R;, (j = I, II, III, IV), we denote by K b j, ds? and vj the Bergman kernel function, the Bergman kernel form, the Bergman met. ric, and the volume element defined by ds2 , respectively. The ratio v, /6j will be denoted by cj. We denote by V(R3) the total volume of R3 with respect to the Euclidean measure of ambient complex Euclidean space. Then
V(RI) = V(RII)
1!2!... (m - 1)!1!2!... (n 2!4! ... (2n - 2)!
1)!mn
n(n+1)/2
n!(n + 1)! ... (2it - 1)! 2!4! ... (2n - 4)! n(n-1)/2 V(Rill) =
(n-1) .n 1
V(Riv)
I
n
1 7f
2n-1n!
K (Z , 2) 1
K11(Z, Z) =
KIII(Z, Z)
1
V(RI) 1
{d e t(Im
V(RI1)
- ZZ* )} -(m+n)
{det(In - ZZ*)}-(n+1)
1
V(Rn1)
{det(I,, -
ZZ*)}-
(n-1) ,
Volume Elements and the Schwarz Lemma
34
KIV(z,z) = V(Rjv (Izz'I2 + 1 1 ds = 2(mrt + n) Trace[(!,,,. -
2zz')-n.
ZZ')-I dZ(In
-
Z*Z)-I dZ'],
ds2 = 2(n + 1) Ttace[(I -
ZZ')-1 dZ(I,,
- Z'Z)-1 dZ']
dsiri = 2(n - 1) Ttace[(1,, -
ZZ')-1 dZ(I
-
dZ'],
dsiv = 4nAdz[A(I,i - z'2) + (In - z'z)z'z(In - z'2)] dz', where
A = Izz'I2 + 1 - 22z, cI = (m. + n)mn V(RI), cii = 2n(n-')/2(71 + 1)n(n+I)/2 V(RII), 1)]n(i-1)/2 V(Riil), CIII = [2(n civ = (2n)n V(Rtv).
If we denote by G; the automorphism group of the domain Rj, then a transformation Z G R3 , W E Rj belonging to G, is given as follows. For RI we have
11' = (AZ+B)(C,Z-D)-1, where A, B, C, D are matrices of dimensions m. x in, in x n, n. x m, and n x n, respectively, satisfying the relations
AA' - BB' = I,,,,
AC' = BD',
CC' - DD' = -In.
For R1I we have
W = (AZ + B)(BZ + A)-1, where
A'B = B'A,
AA' - BB' = In.
For R111 we have
W = (AZ+B)(-BZ+A)-1, where
A'B = -B'A,
A'A-B'B=In.
6
Symmetric Bounded Domains
35
For Riv we have
w={ [((zz' + 1), (zz' - 1)) x
A' + zB']
/ I
l
i)1
((2(zz'+1),2(zz'-1))C'+zD'),
where A, B, C, and D are real matrices of dimensions 2 x 2, 2 x n, n x 2, and n x n, respectively, satisfying the relations
(C
D
1
1
0
jn)
(CC A D)l
=
0In)
,
det
(AC
B
D) =1.
Let Z E \Rj - W E R3 be a holomorphic mapping from the domain R,, into itself and let J(Z) be its Jacobian. By Theorem 4.4 we have the following inequalities:
For j = I,
IJ(Z)I2 s {det(Im - WW')/det(Im - ZZ')}m+n < {det(I11 - ZZ')}-(m+n) For j = II,
IJ(Z)I2 <- {det(In - WW')/det(In - ZZ*)}n+1 ZZ*)}-(n+1) <_ {det(I, For j = III,
IJ(Z)I2 < {det(In - WW')/det(In < {det(In
-
ZZ*)}n-1
ZZ')}-(n-1)
For j = IV, IJ(z)12
{(Iww,l2
+ 1 - 2ww')/(Izz'I2 + 1- 2zz')}n
(Izz'I2 + 1
- 2zz')-n.
CHAPTER III
Distance and the Schwarz Lemma
1
Hermitian Vector Bundles and Curvatures
We shall summarize basic local formulas of Hermitian differential geometry. For a fuller treatment, see Kobayashi and Nomizu [1, Vol. II]. Let E be a holomorphic vector bundle over a complex manifold M with
fiber C'S. We denote by Ep the fiber of E at p E M. A Hermitian fibermetric g assigns to each point p E M a Hermitian inner product gp in Ep. A holomorphic vector bundle E with a Hermitian fiber-metric g is called a Hermitian vector bundle. For a local coordinate system z1, ... , zn of M, we set
Zi = a/azi,
Z7 = Zi = a/azi for i = 1, ... , n.
Let el, ... , er be holomorphic local cross sections of E which are everywhere linearly independent. With respect to this holomorphic local frame field, the components g,,,a of g are given by
for a,/3 = 1,...,r.
gyp =
The components of the Hermitian connection are given by
Y
All the other components are set equal to zero. The covariant derivatives Vz;ep are defined by
r' e,
Vz,ea =
The components Kpi3 of the curvature R are defined by
KpiJeQ = R(Zi, Z?)e,3 = ([Vz;, Vz;] - V[z:.z,])eQ. 37
Distance and the Schwarz Lemma
38
Then Or a
al a2g,7
azJ
az'az
g
ab E7 agpl agEb
+E9g
azi azi
If we set Ka(3i6 = g[R(Zi, ZS)eA, eal, then a2
gaR
Ka013
_
= az'azi
g
E7
7,E
a9al a90 azi azi
The components of the R.icci curvature are given by
Ki; _
a
Kai; _ - > gaaKa(-i.; a,p
If we set G = det(gap),
then
a2c 1 aG aG c aziaz) + c2 azi azi 1
Kip
a2 log c
aziazj Given an element s =
saea of Ep, we consider the Hermitian form
K,
KaQii sasa dzi dzi
a Ai,J
at p E M. If K, is positive (respectively, negative) definite for every nonzero s, the Hermitian vector bundle E is said to have positive (respectively, negative) curvature. If the Hermitian form
2 E Ki; dzi d2' is positive (respectively, negative) definite, then E is said to have positive (respectively, negative) Ricci curvature (or tensor).
Let E' be a holomorphic subbundle of E with fiber C. With respect to the Hermitian fiber-metric induced by g, E' is also a Hermitian vector bundle. The theory of Hermitian vector subbundles is essentially the same as that of R.iemannian submanifolds. We shall give the equation of Gauss which relates the curvature of E' with that of E. In choosing local holomorphic cross sections el,... , e,. of E we may assume that e1,. .. , eq are
1
Hermitian Vector Bundles and Curvatures
39
cross sections of E. If we fix a point p of M, then we may further assume
that el, ... , er are orthonormal at p, i.e., (go)p = (g(e", ep))P = b"p. If we denote the curvature of E' by K'0 then
_(Kr ) _ _(K _) "(3ij P "/3i; P -
(agaeagep) E ` azi ai,j JJ Q=q+1
P
i,j=1,...,n.
for
If s
r
s"e" is an element of EP C EP, then K9 =
KY
-
g" ag"e agog s" 9" d? dzj. azi aV
This implies the inequality K.9' !5
i.e., Ks - K9 is a positive semidefinite Hermitian form. If M is a Hermitian manifold and E is the tangent bundle of M, then the connection considered above is nothing but the classical Hermitian con-
nection. If s, t c EP are unit vectors, then K, (t, t) = -
Kiaktsts-?tkt l is
what is called the holomorphic bisectional curvature determined by s and t in Goldberg and Kobayashi [1]. In particular, K,(s, s) is called the holomorphic sectional curvature determined by s. If M' is a complex sub-manifold
of M, then the tangent bundle E' = T(M') of M' is a Hermitian vector subbundle of ECM'. Then the formula K' < K, above implies a similar inequality for holomorphic bisectional curvature (and hence for holomorphic sectional curvature also). In summary we state
Theorem 1.1 (1) If E is a Hermitian vector bundle over a complex manifold M and E' is a Hermitian vector subbundle of E, then
K',SK3 forsEE'; (2) If M' is a complex submanifold of a Hermitian manifold M, then the holomorphic bisectional (or sectional) curvature of M' does not exceed that of M.
Distance and the Schwarz Lemma
40
2
The Case Where the Domain is a Disk
We prove a generalization of Theorem 2.1 of Chapter I. Let D. be the open disk {z c C; Izi < a) of radius of a and let ds2 be the metric defined by 2= dsa =
4a2 dz dz
A(a - Iz12)2
Theorem 2.1 Let Da be the open disk of radius a with the metric ds2 and let Al be an n-dimensional Hermitian manifold whose holomorphic sectional curvature is bounded above by a negative constant -B. Then every holomnorphic mapping f : Da -> lbl satisfies
f (dsA' < Bdsa. Proof We define a function it on Da by setting u. d82
and want to prove u < A/B everywhere on Da. As we have shown in the proof of Theorem 2.1 of Chapter I, we may assume that u attains its maximum at a point, say z0, of Da. We want to prove that u <- A/B at ze. If u(zo) = 0, then u = 0 and there is nothing to prove. Assume that u(z0) > 0.
Then the mapping f : Da -> M is nondegenerate in a neighborhood of z() so that f gives a holomorphic imbedding of a neighborhood U of zU into M. By Theorem 1.1, the holoniorphic sectional curvature of the onedimensional complex submanifold f (U) of Al is bounded above by -B. Since dim f (U) = 1, the holomorphic sectional curvature of f (U) is nothing but the Gaussian curvature. The rest of the proof is exactly the same as that of Theorem 2.1 of Chapter I.
Theorem 2.1 is essentially equivalent to "Aussage 3" in Grauert and Reckziegel [1), in which they assume that the curvature of every onedimensional complex submanifold of Al is bounded above by -B. 3
The Case Where the Domain is a Polydisk
Let D = DQ = Da x . . . x Da be the direct product of I copies of disk Da of radius a. Let dsD be the product metric dsQ + -t- dsQ in D. Since dsQ has constant curvature -A, the holomorphic sectional curvature of ds2 varies
between -A and -All.
Theorem 3.1 Let D = DQ be a polydisk of dimension I with metric dsD = d.s' +
+ dsn and let M be an n-dimensional Hermitian manifold
4
The Case Where D is a Symmetric Bounded Domain
41
whose holomorphic sectional curvature is bounded above by a negative con-
stant -B. Then every holomorphic mapping f : D M satisfies A
f`(dsnl) < BdsD.
Proof Let (r1i... , rj) be an 1-tuple of complex numbers such that Es =1 IriI2 = 1. Let j : Da D be the imbedding defined by j(z) = ('riz, ... , rjz). From the explicit expression of dsa given in Sec. 2, we see that j is isometric
at the origin of D0. Let X be a tangent vector of D at the origin. For a . . , rj) we can find a tangent vector Y of Da at the origin such that j. (Y) = X. Then, for any holomorphic mapping f : D -> M, we have
suitable (r1, .
IIf.XII2=
AIIYII2 = AIIXII2,
where the inequality in the middle follows from Theorem 2.1 (applied to f o j : Da
111) and the last equality follows from the fact that j is isometric
at the origin of D,. Since D is homogeneous, the inequality
Ii f.X II2
<_
(A/B) II X II2 holds for all tangent vectors X of D.
4 The Case Where D is a Symmetric Bounded Domain Let D be a symmetric bounded domain of rank 1. With respect to a canonical Hermitian metric, its holomorphic sectional curvature lies between -A
and -All for a suitable positive constant A. For every tangent vector X x Da of D, there is a totally geodesic complex submanifold DI = Da x (a polydisk of dimension l) such that X is tangent to D. (If we write D = G/K and g = t + p in the usual manner and denote by a a maximal Abelian subalgebra contained in p, then dim a = rank D = 1. Without loss of generality, we may assume that X is an element of a under the usual identification of p with the tangent space of D at the origin. If J : p -. p is the complex structure tensor, then the manifold generated by a + Ja is the desired totally geodesic submanifold D. To see this we have to use the fact that, for suitable root vectors Xa, and X_,,; with i = 1, ... ,1, a is spanned by Xa, + X_ai, i = 1, ... ,1. But this is rather technical and will not be discussed any further. We shall show later explicitly what Da is for each of
the classical Cartan domains.)
Distance and the Schwarz Lemma
42
From Theorem 3.1 we obtain
Theorem 4.1 Let D be a bounded symmetric domain with a canonical invariant metric dsD whose holomorphic sectional curvature is bounded below by a negative constant -A. Let M be a Hermitian manifold with metric ds2,,,1 whose holomorphic sectional curvature is bounded above by a negative constant -B. Then every holomorphic mapping f : D - M satisfies f*(d52
A
BdsD
Corollary 4.2 Let D be a symmetric bounded domain with a canonical invariant metric dsD whose holomorphic sectional curvature is bounded below by a negative constant -A. Let M be a symmetric bounded domain of rank l so that its holomorphic sectional curvature lies between -lB and -B. Then every holomorphic mapping f D -> M satisfies fk(ds1.f)
BdsD.
Corollary 4.3 Let D be a symmetric bounded domain of rank l so that its holomorphic sectional curvature lies between -lB and -B. Then every holomorphic mapping f : D -> D satisfies f *dsv S l d.5D.
Both Corollaries 4.2 and 4.3 have been obtained by Koranyi 111. We shall now exhibit for each of the Cartan domains, a totally geodesic polydisk of dimension l (= the rank of the domain).
RI = {m x n matrices Z satisfying I - ZZ' > 0}, rank = min(m, n).
D' = {Z = (zip); zip = 0 for i # j}, l = min(m,n). RI, = {symmetric matrices Z of order n satisfying I - ZZ' > 0}, rank = n.
Dn={Z=(zip); zip=0fori 4 j}. Rill = {skew-symmetric matrices Z of order n satisfying I - ZZ' > 0},
rank = D' = {Z = (zip ); zip = 0 except z12 = -z21, z34 = -z43, .. .
l=
L2
]
4
The Case Where D Is a Symmetric Bounded Domain
RIV = {z = (z1, ... , zn) E C'; Izz'12 + 1 - 22z' > 0, Izz'I < 1}, rank = 2.
D2 = {(zj,z2,0,...,0) E RIV}, where the right-hand side is biholomorphically mapped onto D2 by (zl, z2, 0, ... , 0) ---* (zI + iz2, zl - iz2).
43
CHAPTER IV
Invariant Distances on Complex Manifolds
An Invariant Pseudodistance
1
Let D denote the open unit disk in the complex plane C and let g be the distance function on D defined by the Poincare-Bergman metric of D. Let Al be a complex manifold. We define a pseudodistance dm on M as follows. Given two points p, q E M, we choose points p = po, p,.... Pk-1, Pk = q of M, points a,_., ak, b1,. .. , bk of D, and holomorphic map-
pings fl, ... , fk of D into M such that fi(ai) = pi_1 and fi(bi) = pi for i = 1, ... , k. For each choice of points and mappings thus made, we consider the number g(ai,bi) + ,.. + g(ak,bk).
Let dm (p, q) be the infimum of the numbers obtained in this manner for all possible choices. It is an easy matter to verify that dA.f : M x Al -> R is continuous and satisfies the axioms for pseudodistance: dm (p, q) >_ 0,
d..j (p, q) = der (q, p),
dnf(p, q) + d..f (q, r) ? dga (p, r)
The most important property of dp4 is given by the following proposition, the proof of which is trivial. Proposition 1.1 Let 161 and N be two complex manifolds and let f III -> N be a holomorphic mapping. Then :
d,w (p, q) > d,v (f (p), f (q))
for p, q E M.
Corollary 1.2 Every biholomorphic mapping f : M --. N is an isometry. i. e.,
daf (p, q) = dN(f (P), f (q))
for p, q E .M1l.
The pseudodistance d,Af may be considered as a generalization of the Poincare- Bergman metric for a unit disk. We have 45
Invariant Distances on Complex Manifolds
46
Proposition 1.3 For the open unit disk D in C, dD coincides with the distance p defined by the Poincare-Bergman metric.
Proof By the Schwarz lemma (cf. Theorems 1.2 and 2.1 of Chapter I), every holomorphic mapping f : D --. D is distance-decreasing with respect to p. From the very definition of dD we have dD (p, q) ? P(p, q)
for p, q E D.
Considering the identity transformation of D, we obtain the inequality dD(p, q) < e(p, q) -
The following proposition says that dm is the largest pseudodistance on M for which f : D -4M is distance-decreasing. Proposition 1.4 Let M be a complex manifold and d' any pseudodistance on M such that for a, b E D
d'(f (a), f (b)) <_ e(a, b)
for every holomorphic mapping f : D -* M. Then dm (p, q) > d'(p, q)
for p, q E ,tit.
Proof Let po, ... , Pk, al, ... , ak, b i , ... , bk, A ,- .. , fk be as in the definition of dM. Then k
k
d'(pi-1,pi) _
d'(p,q) _
d'(fi(ai),fi(bi))
k
p(ai, bi )i=1
Hence, k
d'(p, q) <_ inf
P(ai, b,) = dq(p, q). i= 1
Proposition 1.5 Let M and l' be two complex manifolds. Then d,kf (p, q) + d z, (p, q') > dm. m, ((p, p'), (q, q') ) > Max[dM (p, q), dm, (p', q')]
for p, q E M and p', q' E M'.
13
I
An Invariant Pseudodistance
47
proof We have dM(p,q) +dM'(p ,q') > dMxM'((p,P ), (q,p )) +dm.m,((q,P ), (q, q')) d ,fxM'((p,P ), (q, q')),
where the first inequality follows from the fact that the mappings f M x M' defined by f (x) = (x, p') and m-+ M x M' and f' : M' :
fr(x') = (q, x') are distance-decreasing and the second inequality is a consequence of the triangular axiom. The inequality dMxM'((p,p'),(q,q')) > Max[dM(p, q), 4'(p', q')} follows from the fact that the projections M x
M' M and M x lei' -> M' are both distance-decreasing.
Example 1 If D is the open unit disk in C, then dDXD((p,P),(q,q'))=Max[dD(p,q),dD(p,q')I for p,q,p,q'ED. To prove this assertion we may assume (because of the homogeneity of D) that p = p' = 0 and dD(0, q) > dD(0, q'), i.e., JqJ > Jq'I. Consider the holomorphic mapping f : D -, D x D defined by f (z) = (z, (q'/q)z). Since f is distance-decreasing, we have dDxn((O,0),(q,q')) = dDxD(f(O),f(q)) <_ dn(O,q). More generally, if Dk denotes the k-dimensional polydisk D xx D, then dDk((pi,...,pk),(gl,...,gk)) =lvlax[dD(pi,gi); i = 1,...,k).
This shows that dDk does not coincides with the distance defined by the Bergman metric of Dk unless k = 1. As we see from the definition of the pseudodistance, dnf is defined in a manner similar to the distance function on a Riemannian manifold. It is therefore quite natural to expect the following result:
Proposition 1.6 Let M be a complex manifold and Al a covering manifold of M with covering projection 7, : Al -+ M. Let p, q E Al and P, q E Al such that 7r(p) = p and 7r(q) = q. Then d,Nf (p, q)
= inf d,ti-1 4
'here the infimum is taken over all 4 E .171 such that q = 7r(q).
Invariant Dzstan.ces on Complex Manifolds
48
Proof Since 7r 111 --> Al is distance-decreasing, we have :
dAf (p, q) < inf d,N (P, 4) q
Assuming the strict inequality, let dM (p, q) + e < inf do-r a
where e is a positive number. By the very definition of dAt there exist points an, ... , ak, bl, ... , bk of the unit disk D and holomorphic mappings
fl,....fk of D into Al such that p = f, (a,),
f1(b1) = f2(a2),..., fk-1(bk-1) = fk(ak),
fk(bk) = q
and k
dn.t (p, q) + e >
Lo(ai, bi) i.=1
Then we can lift, f1, ... , fk to holomorphic mappings f1, ... , fk of D into Al
in such a way that P = f1(a1),
fi(bi) = fi+l(ai+1)
for i=l,...1k-1,
7 O f, = fi
for i = 1,.,.,k.
If we set fk(bk), then 7r(q) = q and d,-1(p, q) __< X:k1 p(ai, b.i). Hence. dAt (P, ii) < d,tit (p, q) + e, which contradicts our assumption.
Remark It is not clear whether the following is true: dAt (p, q) = min d,tir 4
i.e., whether info d,tit (p, q) is really attained by some
Example 2 Let Al be a Riemann surface whose universal covering is a unit disk D, e.g., a compact Riemann surface of genus ?2. From Propositions 1.3 and 1.6 we see that dA..1 is the distance defined by a Kaehler metric
of Al with constant curvature <0.
Example 3 It can be seen easily from the definition of the pseudodistance d_.t that for a complex Euclidean. space C" the pseudodistance d, is trivial.
i. e.. dc (p, q) = 0 for all p, q E C". More generally, if M is a com.plcx manifold on which a complex Lie group G acts transitively, then dti1 i trivial. For each point p E Al, let U be a neighborhood of p such that
2
Caratheodory Distance
49
every point q E U lies on the orbit of a complex 1-parameter subgroup of G through p. Hence, for every point q E U there is a holomorphic mapping
f : C --' M whose image contains both p and q. Since f is distance-
decreasing and do is trivial, we have dnf (p, q) = 0. To prove that dA.f is trivial for any pair of points on M, we connect the pair by a chain of small open sets U and apply the triangular axiom. In particular, for a compact homogeneous complex manifold M or for Al = C - {0}, dnf is trivial. If Af is a compact Riernann surface of genus 0 or 1, then dtir is trivial.
Example 4 LetM={zEC"; r 1, I do not know what dM looks like, although I know that dM is not complete, since M is not holomorphically complete (see Sec. 4 of this chapter).
2
Caratheodory Distance
Let D be the open unit disk in C with the Poincarc-Bergman distance o. Let M be a complex manifold. The Caratheodory pseudodistance cm of Al is defined by cnf (p, q)
= sup g(f (p) ), f (q)) for p, q E M, I
where the supremum is taken with respect to the family of holomorphic
mappings f : Al -* D. We prove that the pseudodistance dtif defined in Sec. 1 is greater than or equal to ctif. That will in particular imply that CM(p,q) is finite.
Proposition 2.1 For any complex manifold Al, we have d..r (p, q) ? c..r (p, q)
for p, q E M.
Proof As in the definition of d,%j(p, q), choose points p = pa, p1, .... Pk-1, pk = q of Al and points a1,.. . , ak, b1,. .. , bk of D and also mapPings fi. .. , fk of D into Af such that f, (a,) = p,_1 and fi(bs) = pi. Let f
Invariant Distances on Complex Manifolds
50
be a holomorphic mapping of M into D. Then k
k
e(f O fi(ai), f O fi(bi))
e(ai, bi) ? i=1
i=1
e(f
fl(ai),f o fk(bk))
= e(f (p), f (q)),
where the first inequality follows from the Schwarz lemma and the second inequality is a consequence of the triangular axiom. Hence, k
dm (p, q) = inf
e(a:, bi) ? sup e(f (p), f (q)) = cm (p, q). i=1
It is an easy matter to verify that cm : M x M
R is continuous and
satisfies the axioms for pseudodistance:
cM(p,q) > 0.
cM(p,q) = cM(q,p),
cM(p,q)+cm(q,r)
cM(p,r)
The Caratheodory pseudodistance cm shares many properties with dAj.
Proposition 2.2 Let M and N be two complex manifolds and let f : M -+ N be a holomorphic mapping. Then CMt (p, q) >_ c,v (f (p), f (q))
for p, q C M.
The proof is trivial.
Corollary 2.3 Every biholomorphic mapping f : M --, N is an isometry. i. C.,
cM (p, q) = cn, (f (p), f (q))
for p, q E M.
The Caratheodory pseudodistance may be also considered as a generalization of the Poincare-Bergrnan metric for D.
Proposition 2.4 For the open unit disk D in C, CD coincides with the distance p defined by the PoincarE-Bergman metric.
Proof Using the Schwarz lemma for a holomorphic mapping f : D - 1) we obtain e(p, q) > CD (p, q)
for p, q E D
from the very definition of CD. Considering the identity transformation of D. 0 we obtain the inequality g(p, q) 5 CD (p, q).
2
CarathEodory Distance
51
The following proposition says that cm is the smallest pseudodistance on M for which f : M -* D is distance-decreasing.
proposition 2.5 Let M be a complex manifold and c' any pseudodistance on M such that p, q E M
c'(p, q) >_ o(.f (p), f (q))
for every holomorphic mapping f : M -+ D. Then cM(p,q) <
(p, q)
forp,q E M
The proof is trivial.
The following proposition can be proved in the same way as proposition 1.5.
proposition 2.6 Let M and M' be two complex manifolds. Then
cM(p,q)+cm,(p',q')
cmtxM'((p,p'),(q,q')) Max[cmt (p, q), cm, (p', q')]
for p, q E M and p', q' E M'. x D, then Example 1 If Dk denotes the k-dimensional polydisk D x CDk((p1,...,pk),(gl,...,gk)) = MaX[CD(pi,gi); i = 1,...,k] for pi, q, c- D.
The proof is similar to the one in Example 1 of Sec. 1. Hence, dDk = CDk.
Example 2 Let M = G/K be a symmetric bounded domain of rank I in C" (in its natural realization). For a suitable subspace C' of C", M n C' is a polydisk Di = D x ... x D and Al can be written as a union of polydisks k(D'), k E K, where k is considered as a unitary transformation of C". In particular, we have
DICMCD". Since the injections D' -i M and M
D" are distance-decreasing, we
have CD1 > CM >_ CD" on D'. From Example 1 we see that CD1 = CD" On
Di. Hence, CDc = CM
on D'.
Using the result in Example 1 of Sec. 1, we obtain a similar result for dD,
44d dM. Given any two points p, q E M, there is an element of C which
52
Invariant Distances on Complex Manifolds
sends p and q into Dl. We may conclude therefore that, for a symmetric bounded domain M, the two distances cm and dtif coincide.
Example 3 Since cm < daf in general, we see from Example 3 of Sec. 1 that if M is a complex manifold on which a complex Lie group G acts transitively, then cm is trivial.
Example 4 If M is a compact complex manifold, then cm is trivial. This is clear, since every holomorphic function on M must be a constant function. This shows that there is no analogue of Proposition 1.6 for cm.
Example 5 Let M = C - A, where A is a finite set of points, or more generally, let M = Cn - A, where A is an analytic subset of dimension < n - 1. Then cm is trivial. A holomorphic mapping f : M -+ D may be considered as a bounded holomorphic function on M. By the socalled Riemann extension theorem, f can be extended to a holomorphic mapping of C" into D. By Liouville's theorem (or by Example 3), f must be a constant function and hence cM is trivial.
Example 6 Let Al = {z e C"; r < IzI < R} and B = {z E C"; IzI < R}. If n ? 2, then c,.f = CB I M. This follows from the fact that the envelop of holomorphy of Al is B, so that every holomorphic mapping from Af into the unit disk D can be extended to a holomorphic mapping from B into D. On the other hand, I do not know what the Caratheodory distance car
looks like if n = 1. This is in contrast to the situation in Example 4 of Sec. 1, where df.f is known for n = 1 but not for it >_ 2. The Caratheodory distance was introduced in Caratheodory [1, 2]. For recent results on the Caratheodory distance, see Reiffen [1, 2]. 3
Completeness with Respect to the Caratheodory Distance
Let Al he a complex manifold. Throughout this section we are interested in the case where the Caratheodory pseudodistance cm is a distance, i.e., the case where the family of holomorphic mappings f : Al D separates the points of Al. Since every bounded holomorphic function on M, multiplied by a suitable constant, yields a holomorphic mapping of Al into the unit disk D. the necessary and sufficient condition that cm be a distance is the
following: For any two distinct points p and q of M, there is a bounded holomorphic function f on M such that f (p) f (q). It is very close to
3
Completeness with Respect to the Carathkodoiy Distance
53
assuming that M is a bounded domain in a Stein manifold. In particular, aU compact complex manifolds are excluded. In general, we say that a metric space Al is complete if, for each point p of M and each positive number r, the closed hall of radius r around p is a compact subset of M. If Al is complete in this sense, then every Cauchy Sequence of Al converges. The converse is true for a Riemannian metric, but not in general.
proposition 3.1 If Al and M' are complex manifolds with complete Caratheodo7y distance, so is Al x Al'.
Proof This is immediate from Proposition 2.6. Proposition 3.2 A closed complex subrnanifold M' of a complex manifold M with complete Caratheodory distance is also complete with respect to its Caratheodory distance.
Proof This is immediate from the fact that the injection M' - Al is distance-decreasing.
Proposition 3.3 Let Al and Ali, i E I, be complex submanifolds of a complex manifold N such that Al = ni M. If each Ali is complete with. respect to its Caratheodory distance, so is Al. Proof Since each injection Al Ali. is distance-decreasing, the proposition follows from the following trivial lemma.
Lemma Let Al and Al i E I, be subsets of a topological space N such that M = f i Ali. Let d and di be distances on. Al and Mi such that d(p, q) >_ di(p, q) for p, q E Al. If each Ali. is complete with respect to d;, then Al is complete with respect to d.
We shall now give a large class of bounded domains which are complete
with respect to their Caratheodory distances. Let G be a domain in C" and fl, ... , fk holomorphic functions defined in G. Let P be a connected component of the open subset of G defined by fi(z)I < 1,...,lfk(z)I < 1.
Assume that the closure of P is compact and is contained in G. Then P is called an analytic polyhedron.
Invariant Distances on Complex Manifolds
54
Theorem 3.4 An analytic polyhedron P is complete with respect to its Caratheodory distance cp.
Proof Let F be the set of holomorphic mappings of P into the unit disk D. Let o be a point of P. Given a positive number a, choose a positive number b, 0 < b < 1, such that
{z ED; o(fi(o),z)<_afor i=l,...,k}C{zED; IzI<_b} Then
{p E P; cp(o, p) < a} = (p E P; o(f(o), f (p)) a for f E F} C {p E P; o(fi(o), fi(p)) a for i = 1, ... , k} C {p E P; I ff(p)l <_ b for i = 1, ... , k}.
Since {p c P; I fi(p)l _< b for i = 1, ... , k} is compact, {p E P; cp(o, p) S a} is compact.
In the definition of an analytic polyhedron, we used only a finitely many holomorphic functions fl, ... , fk. If we take an arbitrary family of holomorphic functions f,Y, a E A, on G and assume that the set defined by I fa(z)I < 1, a E A, is open, then we call each connected component of this open subset of G a generalized analytic polyhedron. The proof of Theorem 3.4 is valid for the following:
Theorem 3.5 A generalized analytic polyhedron P is complete with respect to its Caratheodory distance cp.
Let hj be real analytic functions on G of the form 00
hj=
Ifjm1
2 ,
j=1,...,k,
m=1
is holomorphic in G. Let S be the set of sequences {am} of complex numbers such that Em'=1 laml2 = 1. Then where each
{zEG:Ihj(z)I<1}= n {a,,,}ES
{z;
T, a,, fjm(z) m=1
This shows that a connected component of the open subset of C defined by
{zEG;Ihj(z)j <1 for j=1,...,k} is a generalized analytic polyhedron. To such a manifold. Theorem 3.5 is therefore applicable.
3
Completeness with Respect to the Caratheodory Distance
55
So far, we have given sufficient conditions for a manifold to be complete with respect to its Carathcodory distance. Wire shall give now some necessary conditions.
Horstmann [1] has shown that a domain in C' which is complete with respect to its Caratheodory distance is a domain of holomorphy. We shall give a generalization of Horstmann's result. Given a family F of holomorphic functions on a complex manifold M and a subset K of M, we set KF = {p E M; I f (p)I < supl f (K)I for all f E F}, where supl f (K)I denotes the supremum of If (q)I for q E K. Then KF is a closed subset of M containing K and is called the convex hull of K with respect to F. The convex hull of KF with respect to F coincides with KF itself. If KF is compact for every compact subset K of M, then M is said to be convex with respect to F. If F C F, then KF' D KF. Hence, if M is
convex with respect to F' and if F' C F, then M is convex with respect to F. If M is convex with respect to the family of all holornorphic functions on M, then M is said to be holomorphically convex.
Theorem 3.6 Let M be a complex manifold with Caratheodory distance. Fix a point o of M and let F be the set of bounded holomorphic functions f on M such that f (o) = 0. If M is complete with respect to cM, then M is convex with respect to F and hence is holomorphically convex.
Proof Let a he a positive number and B the closed ball of radius a around o E M, i.e.,
B = {peAl ;c,til(o,p)
suplf (B)I for f E F}
= { p E M; g(0, f (p)) < sup o(0, f (q)) for f e F } I.
C
lp
qEB
E Al; g(0, f (p)) < sup cnt (o, q) qEB
_ {p E Al; e(0. f (p)) < a} = B.
Since BF contains B, we conclude BF = B.
1J1
ll
56
Invariant Distances on Complex Manifolds
If f is a bounded holomorphic function on M, then, for a suitable pos-
itive constant c, cf is a holomorphic mapping of M into the open unit disk D. This means that, in Theorem 3.6, we may choose F to be the D such that f (o) = 0. family of holomorphic mappings f : Al It is not clear if the converse to Theorem 3.6 holds. As we shall see, shortly, Al need not he complete with respect to cm even if M is holomorphically convex. It seems, however, very reasonable to expect that Al is complete with respect to cm if it is convex with respect to the family of bounded holomorphic functions. Let Al be a complex manifold of complex dimension n and A an analytic subset of dimension <--_ it - 1. Since every bounded holomorphic functions
on Al - A can be extended to a bounded holomorphic function on Al by Riemann's extension theorem, it follows that Al - A cannot be convex with respect to the family of bounded holomorphic functions if A is nonempty and that clif coincides with CM -A on M - A. In this way we obtain many examples of holomorphically convex manifolds which are nor convex with respect to the family of bounded holomorphic functions and not complete with respect to their Carathdodory distances. The punctured disk D - {0} is the simplest example. The question of completeness with respect to the Caratheodory distance may he quite possibly related to that of completeness with respect to the Bergman metric (see Bremermann !1 and Kohayashi [1] on the latter question). Since D is the completion of D - {0} with respect to the Carathcodory distance CD_{o} and every holomorphic mapping f : D - {0} --* Al is distance-decreasing with respect to cD_ (0) and cm, we have Proposition 3.7 If Al is a complex manifold with complete Carath.eodor.q distance, then every holomorphic mopping f from. the punctured disk D - { 0) into Al can be extended to a holomorphic mapping of D into Af. 4
Hyperbolic Manifolds
Let Al be a complex manifold and dtif the pseudodistance defined in Sec. I. If dm is a distance, i.e., d,tif (p. q) > 0 for p # q, then Al is called a hyperbolic manifold. A hyperbolic manifold Al is said to he complete if it is complete with respect to dif. As in Sec. 3, Al is complete with respect to dtif if. for each point p of M and each positive number r, the closed ball of radius 7' around p is a compact subset of Al. As we shall see later, for a hyperbolic manifold Al, this definition is equivalent to the usual definition in terms of Cauchy sequences.
4
Hyperbolic Manifolds
57
The proofs of the following three propositions are almost identical to those of Propositions 3.1, 3.2, and 3.3.
proposition 4.1 If M and M' are (complete) hyperbolic manifolds, so is MxM'.
proposition 4.2 A (closed) complex submanifold M' of a (complete) hyperbolic manifold M is a (complete) hyperbolic manifold.
proposition 4.3 Let M and Mi, i c I, be closed complex submanifolds of a complex manifolds N such that Al = I '7 M. If each Mi is a (complete) hyperbolic manifold, so is M.
From Proposition 2.1 we obtain
Proposition 4.4 A complex manifold Al is (complete) hyperbolic if its Caratheodory pseudodistance cm is a (complete) distance.
Corollary 4.5 Every bounded domain in C" is hyperbolic. Every generalized analytic polyhedron is complete hyperbolic.
The second statement follows from Theorem 3.5. The following proposition is immediate from Proposition 1.4.
Proposition 4.6 If a complex manifolds If admits a (complete) distance d for which every mapping f of the open unit disk D into Al is distance-decreasing, i.. e., d'(f (a), f (h)) _< o(a, b) for a, b E D, then it is (complete) hyperbolic.
The results so far have their counterparts for Caratheodory distance. We shall now give results which are proper to the invariant distance d f,.1.
Theorem 4.7 Let Al be a complex manifold and Ai a covering manifold of M. The Al is (complete) hyperbolic if and only if Al is (complete) hyperbolic.
Proof Assume that Af is hyperbolic. Let p,q E Af and dm (p, q) = 0. Let P be a point of Nf such that. ; (p) = p, where it Al Al is the :
covering projection. By Proposition 1.6, there exists a sequence of points
4i,...,yj,... of ,f such that 7r(gi) = q and lirnidcf(p, qi) = 0. Then the Sequence {Q7} converges to P. Hence 7r(4i) converges to p. Since 7r(4i) = q,
we obtain p = q. This proves that Al is hyperbolic.
Assume that Al is complete hyperbolic. Let B,. be the closed ball of
radius r around p E ll'I, i.e., I 3 r . _ {4 E Al; d ,
-. 1
<_ r}. Similarly, let 13,.
Invariant Distances on Complex Manifolds
58
be the closed ball of radius r around p = ir(p) E M. By Proposition 1.6, we have
Br C 7r(Br+h)
for b > 0.
Since Br+a is compact by assumption and B, is closed, B,. must be compact. Hence, M is complete.
Assume that lhl is hyperbolic. Let p, q E Al and dM (p, q) = 0. Since the projection 7r is distance-decreasing, we have dy1(ir(p),7r(q)) = 0, which
implies 7r(p) = ir(q). Let U be a neighborhood of p in M such that 7r U ir(U) is a diffeomorphism and 7r(U) is an e-neighborhood of ir(p) with respect to dA.1. In particular, U does not contain q unless p = cj.
:
Since dti1(p, q") = 0 by assumption, there exist points at, .... ak, bl, ... , bk of D and holomorphic mappings f i .... , f k of D into M such that p =
fi(ai),ff(bi) = fi+l(ai+1) for i = 1,...,k - 1 and fk(bk) = q and that. k
ei=1 g(ai, bi) < E. Let aibi denote the geodesic arc from ai to bi in D.
Joining the curves f1(aibi), ... , fk(akbk) in M, we obtain a curve from p to q in M, which will he denoted by C. Since it o fl, ... , -7r o fk are distancedecreasing mappings of D into M and a1b1.... , akbk are geodesics in D. every point of the curve ir(C) remains in the c-neighborhood 1r(U) of 7r(p). Hence, the endpoint q must coincide with p. Assume that Al is complete hyperbolic. We shall prove here that Al is
complete in the sense that every Cauchy sequence with respect to is convergent. In the following section, we shall prove that the completeness in this usual sense implies the completeness defined at the beginning of this section. Let {pi} be a Cauchy sequence in Al. Since the projection 7r is distance-decreasing, {7r(pi)} is a Cauchy sequence in M and hence converges to a point p E Al. Let E be a positive number and U the 2e-neighborhood of p in Al. Taking e small we may assume that 7r induces a homeomorphism of each connected component of 7r-1(U) onto U. Let N be a large integer such that 7r(pi) is within the E-neighborhood of p for i > N. Then every point outside U is at least E away from 7r(pi). Let Ui be the connected component of 7r-1 (U) containing pi. We shall show that the 6-neighborhood of pi lies
in Ui for i > N. Let q be a point of iff with d1(pi, q) < E. We choose points a1, ... , ak, b1, .... bk of D and holomorphic mappings fl_., fk of D into Al in the usual manner so that Ek p(aj,b)) < E. Denoting by ash, the geodesic are from a) to b;, let. C be the curve from pi to q obtained hr joining fi(aibi)..... fk(akbk) in M. Let C = ir(C). From the construction of C, it is clear that C is contained in the e-neighborhood of 7r(pi) and 1
4
Hyperbolic Manifolds
59
hence in U. It follows that C lies in Ui. Let P be the point of Ui defined by p = ir(q). Then {pj } converges to P. 0
A complex manifold M is called a spread (domaine etale in French) over a complex manifold M with projection -7r if every point p E M has a neighborhood U such that 7r is a holomorphic diffeomorphism of U onto the open set 7r(U) of M. This is a concept more general than that of covering manifold. From the proof of Theorem 4.7 above, we obtain the following: Proposition 4.8 A spread k over a hyperbolic manifold M is hyperbolic. A similar reasoning yields also the following:
Proposition 4.9 If a complex manifold M' is holomorphically immersed in a hyperbolic manifold M, then M' is also hyperbolic. In Sec. 3 we saw that the punctured disk D - {0} is not complete with respect to its Caratheodory distance. But it is complete hyperbolic. In fact, the half-plane {z = x + iy E C; y > 0} is the universal covering space of D - {0}, the projection being given by z -' ezz. We may also consider the unit disk D as the universal covering space of D - {0}. Our assertion follows from Theorem 4.7. More generally, we have Theorem 4.10 Let M be a complete hyperbolic manifold and f a bounded holomorphic function on M. Then the open submanifold M' = {p E M) f (p) # 0} of M is also complete hyperbolic.
Proof Multiplying f by a suitable constant, we may assume that f is a holomorphic mapping of M into the open unit disk D. We denote by D' the punctured disk D - {0}. Let o be a point of Al' and let a and b be positive numbers. Since D' is complete hyperbolic, for a given positive number a we can choose a small positive number b such that {z E D; IzI > b} D {z c D*; do. (f (o), z) : a}. We set
A = (p E Al; dM(o, p) < a},
A' = {p E M'; dm, (o, p) _< a},
B={pEM;If (p)I ?b}.
B'={pEM';If(p)I ?b}.
Since dMM' (o, p) ? dti1(o, p) by Proposition 1.1, we have
ADA'.
Invariant Distances on Complex Manifolds
60
Since b is positive and Al' = {p E Al; f (p) # 01, we have
B = B'. Since f : M'
D* is distance-decreasing, we have
A' C
{pE M';lf(p)I >b}=B'=B.
Since A is compact subset of M by the completeness of M and B is closed in M, the intersection A n B is compact. Since B = B', it follows that A n B
is in M'. Since A n B is a compact subset of M' and A' is closed in Al', the intersection A' n (A n B) is a compact subset of W. Since both A and B contain A', it follows that A' coincides with A' n (A n B) and hence is a compact subset of M', thus proving that Al' is complete hyperbolic. It should be observed that M' above can never be complete with respect to its Caratheodory distance cM' unless Al = M'. In fact. cw = CM 101' (see Sec. 3).
Theorem 4.11 A (complete) Hermitian, manifold M whose holomorphic sectional curvature is bounded above by a negative constant is (complete) hyperbolic.
Proof Let ds' denote the Poincare-Bergman metric on the open unit disk D. Let ds 1.1 be the Hermitian metric of Al. If we multiply ds ti.l bya
suitable positive constant, we have by Theorem 2.1 of Chapter III the following inequality for every holomorphic mapping f : D --> M.
I*(ds11) 5 dsD,
If we denote by d' and o the distance functions on M and D defined by and dsn respectively, then this inequality implies that f is distancedecreasing with respect to d' and o, i.e., d'(f (a), f (b)) S o(a, b) for a, b E D. Now the theorem follows from Proposition 4.6.
Corollary 4.12 The Gaussian plane minus two points C - {a, b} is e complete hyperbolic manifold.
This follows from the result in Sec. 3 of Chapter I and Theorem 4.11. Similarly, the following corollary follows from Theorem 5.1 of Chapter I. Corollary 4.13 Every compact Riem.ann surface of genus g >_ 2 is a con7pact hyperbolic manifold.
4
Hyperbolic Manifolds
61
As a consequence of and also as a generalization of Corollary 4.12 we have the following example (due to P. J. Kiernan).
Example 1 In P2(C), let L1, L2, L3, and L4 be four (complex) lines in general position. Let a = L1 fl L2 and b = L3 fl L4. Let Lo be the lines through a and b. Then M = P2(C) - UQ=o Li is complete hyperbolic. In fact, M is biholomorphic to the direct product of two copies of C - {0, 1). To see this, we consider Lo as the line at infinity so that P2(C) - Lo = C2. Then L1 is parallel to L2, since L1 fl L2 = a E Lo. Similarly, L3 is parallel to L4. But L1 and L3 are not parallel, since L1, L2, L3, and L4 are in general position in P2(C). It is now clear that M = C2 - U4=1 Li is affinely equivalent to the direct product of two copies of C - {0, 11. On the other hand, P2(C) - U =1 Li is not hyperbolic, since it contains Lo {a, b} = P1(C) - {a, b}, which is not hyperbolic. Let pj, j = 1, . . , 4, be four points in general position in P2(C). Connecting every pair of these points, .
we obtain six lines L1, j = 0,1, ... , 5. Since P2(C) - Ut o Lj is contained in M = P2(C) - Ui=o L3, it is also hyperbolic. I do not know whether it is also complete. For higher-dimensional analogues of this example, see Kiernan [1].
The following result is also due to Kiernan [4].
Theorem 4.14 Let E be a holomorphic fiber bundle over M with fibre F and projection ir. Then E is (complete) hyperbolic if M and F are (complete) hyperbolic.
Proof Assume that M and F are hyperbolic. Let p, q E E. If 7r(p) # Rr(q), then dE(p,q) > d,Ns(7r(p),7r(q)) > 0. Assume 7r(p) = 7r(q). Choose a neighborhood U of 7r(p) in M such that 7r-1(U) = U x F. Let B,, be the ball in M centered at (p) and of radius s with respect to dM. Denote by
Dr the disk {z E C; Jzj < r}. Choose s > 0 and r > 0 in such a way that B2a C U and dD(z, 0) < s for z E Dr (where D = D1). Thus, if f : D E is holomorphic and f (0) E ir-1(Be), then f (Dr) C U x F. Choose c > 0 such that dD (0, a) > c dDr (0, a) for all a E Dr/2. Let fi : D - E be holomorphic mappings and let a; , bi he points of D such that p = fl (a,), fi(b1) = f2(a2).... , fk(bk) = q. By homogeneity of D, we may assume that ai = 0 for all i. By inserting extra terms in this chain if necessary, we may assume also that bi E Dr/2 for all i. We set po = p, p1 = fl (bl ), ... , pk = fk(bk) = q. We have two cases to consider. Consider first the case where at
Invariant Distances on Complex Manifolds
62
least one of the pi's is not contained in 7r-1(B3). Then it is easy to see k
k
k
dD(0,bi) ?
dE(fi(0),fi(bi)) _
dE(Pi-1, Pi) i=1
k
dM(7r(pi-1),ir(pi)) > s. i=1
Consider next the case where all pi's are in -7r-1(B,,). Then k
dD (0, bi) > c 1: dD, (0, bi ) i=1
i=1 k C
W(Pi)) ? c dF (p, q),
dF i=1
where p : U x F -; F is the projection. This shows that dE(p,q) > min[s,cdp(P,q)] > 0. Thus E is hyperbolic. Assume that M and F are complete hyperbolic. Let {p , } be a Cauchy sequence in E. Then {7r(pn)} is a Cauchy sequence in M and therefore
1r(pn) - xo for some xo E Al. Choose a neighborhood U of xc in M such that 7r-1(U) = U x F. Choose s > 0, r > 0, and c > 0 as above. Given E > 0 with 2e < s, choose an integer N such that pn E it-1(B,) and dE(pn, pm) < E for n, rn > N, We shall show that dF(cp(pn), co(pra)) <
e/c for n, in > N. We fix ri, in > N. We choose holomorphic mappings D Pn = fl (0), fl(b1) = f2(0),..., fk-1(bk-1) = fk(0), fk(bk) = Pm, We may again assume without loss of generality and yk 1 dD(0, bi) < that bi E Dr/2 for all i. Since k
k
k
E > E dD(0, bi) E dE(fi(0), fi(b)) > E dM (irO fi(0), 7r o fi(b)), ,=1
i=1
i=1
it follows that fi(0) E -7r-1(B22) C 7r-1(B3). Hence, k
k
> >dD(0,bi) > c>dDr(0,bi) k
C E dF(W 0 fi(0), v o fi(bi)) ,=1
> cdF(o(Pn),w(Pm))
5
On Completeness of an Invariant Distance
63
This shows that {cp(pn)} is a Cauchy sequence in F, and therefore
v(pn) - yo for some yo E F. Clearly, pn -; (xo, yo) E U x F. Thus E is complete.
Remark Since F may be considered as a closed complex submanifold of E, it follows that if E is (complete) hyperbolic, then F is also (complete hyperbolic. On the other hand, Al need not be hyperbolic even if both
E and F are complete hyperbolic. Let B* = {(z,w) E C2; 0 < z12 + ]w[2 < 1}. Then B* is a holomorphic fiber bundle over P1(C) with fiber D* = {z E C; 0 < jzj < 1}. The bundle space B* is hyperbolic and the fiber D* is complete hyperbolic, but the base space Pi (C) is not hyperbolic.
If we set E = {(z, w) E C2; z # 0 and IzI2 + I wI2 < 1), the E is a fiber bundle over C with fiber D*. This furnishes an example where E is even complete.
By a reasoning similar to the proof of Theorem 4.14, Kiernan [4] proves that if E is a Hermitian vector bundle over a hyperbolic manifold M, then the open unit-ball bundle {X E E; 1IX11 < 1} is hyperbolic. We conclude this section with another example of a complete hyperbolic manifold.
Theorem 4.15 A Siegel domain of the second kind is complete hyperbolic. Proof In Sec. 5 of Chapter II we proved that a Siegel domain of the second kind is equivalent to a domain contained in a product of balls. From that proof it is not difficult to see that a Siegel domain of the second kind can be written as the intersection of (possibly uncountably many) domains, each of which is biholomorphic to a product of balls. But a product of balls is complete hyperbolic. Our assertion follows now from Proposition 4.3.
5 On Completeness of an Invariant Distance We saw in Theorem 4.7 a similarity between the pseudodistance dm and
a Riemannian metric on M. We shall point out here another important similarity.
Given any subset A of a complex manifold M and a positive number r, let U(A; r) be the open set defined by U(A; r) = {p E 141; dM(p, a) < r for some point a c A}.
64
Invariant Distances on Complex Manifolds
With this notation, we have
Proposition 5.1 Let o be a point of a complex manifold Al and let r and r' be positive numbers. Then U[U(o; r); r'] = U(o; r + r').
Proof The inclusion U[U(o; r); r'] C U(o;r + r') is true for any pseudodistance and makes use of the triangular axiom only. In order to prove
the inclusion in the opposite direction, let p E U(o; r + r') and set dM(o, p) = r + r' - 3e. Then there are points ai, bi E D and holomorphic mappings f i : D -t M, i = 1, ... , k, such that f, 1(a1) = o,
fi(bi) = fi+1(at.+1)
for i = 1,...,k - 1,
fk(bk) = p, k
E P (ai, bi) < r + T - 2e. t=1
Let j be the largest integer, 1 5 j 5 k, such that i-1 `o(at,bi)
If we set
q=fj(cj), then dn.I (o, q) < r and dAj (q, p) < r' so that p E U(q; r') C U[U(o; r): r'].
In the proof of Theorem 4.7 we promised to show that if a hyperbolic manifold Al is complete in the sense that every Cauchy sequence converges, then the closure U(o: r) of U(o; r) is compact for all o E M and all positive
numbers r. We shall show that this assertion is true for a larger class of metric spaces.
Theorem 5.2 Let Al be a locally compact metric space with distance function d satisfying the equality U[U(o; r); r'] = U(o; r + r')
5
On Completeness of an Invariant Distance
65
for all o E M and all positive numbers r and r'. Then M is complete in the sense that every Cauchy sequence converges if and only if the closure U(o; r) of U(o; r) is compact for all o E M and all positive numbers r. Proof We have only to prove that if M is (Cauchy-) complete, then U(o; r) is compact; the implication in the opposite direction is trivial.
Lemma U(o; r) is compact if there exists a positive number b such that U(p; b) is compact for every p E U(o; r).
Proof Since M is locally compact, there is a positive number s < r such that U(o; s) is compact. It suffices to show that if U(o; s) is compact, so is U[o; s+ (b/2)]. Let pl, P2.... be points of U[o; s + (b/2)]. Let ql, q2, -be points of U(o; s) such that d(pl, q,) < 4 b. Since U(o; s) is compact, we may assume (by choosing a subsequence if necessary) that ql, q2, ... converges to some point, say q, of U(o; s). Then U(q; b) contains all pi for large i. Since U(q; b) is compact by assumption, a suitable subsequence of Pi, P2.... converges to a point p of U(q; b). Since p1, p2, ... is a sequence in U[o; s + (b/2)] and U[o; s + (b/2)] is a closed set, the limit point p lies in U[o; s + (b/2)]. This completes the proof of Lemma.
The proof of Theorem 5.2 is now reduced to showing that there exists a positive number b such that U(p; b) is compact for all p E M. Assume the contrary. Then there exists a point p1 e 1L7 such that U(pj; is non2) compact. Applying Lemma to U(pl; 2) we see that there exists a point
P2 E U(pl; 2) such that U[p2; (1/22)] is noncompact. In this way we obtain a Cauchy sequence pl, p2, p;l, ... such that Pk E U[pk_1i (1/2k-' )] and U[pk; (1/2k)] is noncompact. Let p he the limit point of the Cauchy sequence p1, p2, .... Since Al is locally compact, for a suitable positive number c, U(p; c) is compact. For a sufficiently large k, U[pk; (1/2k)] is a closed set contained in U(p; c) and hence must be compact. This is a contradiction.
Remark Let lvi he a hyperbolic manifold and M' its completion with respect to the distance dAf. Unfortunately, 1I" need not be locally compact and hence it need not be complete in the strong sense that every closed ball of radius r in 111' is compact. An example of a hyperbolic manifold ,11 such that M' is not locally compact has been found by Kiernan [4]. A similar example of a Riemannian manifold has been found by D. B. A. Epstein (see Kobayashi [4]).
CHAPTER V
Holomorphic Mappings into Hyperbolic Manifolds
1
The Little Picard Theorem
The classical theorem of Liouville states that every bounded holomorphic function on the entire complex plane C is a constant function. We may prove this using the distance-decreasing property of the Caratheodory distance. Let D be a bounded domain and f : C -> D a holomorphic mapping. Since the Caratheodory pseudodistance cc for C is trivial and the Caratheodory pseudodistance CD for D is a distance, the distance-decreasing property of f implies that f is a constant mapping. The same reasoning applies to the invariant pseudodistance dM. Delete two points a and b from C. We know (see Corollary 4.12 of Chapter IV) that C - {a, b} is hyperbolic. On the other hand, do is trivial. Hence, every holomorphic mapping f : C C- {a, b} is a constant mapping; in other words, every entire function with two lacunary values must be a constant function. This is the so-called little Picard theorem. We may now state Theorem 1.1 Let M' be a complex manifold on which a complex Lie group acts transitively. Let M be a hyperbolic manifold. Then every holomorphic mapping f : M' --+ M is a constant mapping.
The fact that the pseudodistance dM, for such a manifold M' is trivial was established in Sec. 1 of Chapter IV (see Example 3).
2 The Automorphism Group of a Hyperbolic Manifold The following lemma is due to van Dantzig and van der Waerden [1]. For the proof, see also Kobayashi and Nomizu [1, pp. 46-50].
Lemma The group I(M) of isometries of a connected, locally compact metric space M is locally compact with respect to the compact-open topology, and
its isotropy subgroup Ip(M) is compact for each p E M. If M is moreover compact, then I(M) is compact. 67
Holomorphic Mappings into Hyperbolic Manifolds
68
From this lemma, we derive the following
Theorem 2.1 The group H(M) of holomorphic transformations of a hyperbolic manifold M is a Lie transformation group, and its isotropy subgroup H7,(M) at p c ,'f1 is compact. If Al is moreover compact, then H(M) is finite.
Proof Let I(M) the group of isometrics of M with respect to the invariant distance dM. Since H(AI) is a closed subgroup of I(M), it follows from
Lemma that H(M) is locally compact with respect to the compact-open topology and H,(M) is compact for p E M. By a theorem of Bochner and Montgomery [1], a locally compact group of differentiable transformations of a manifold is a Lie transformation group. Hence, H(M) is a Lie transformation group, thus proving the first assertion. Assume that M is compact. Another theorem of Bochner and Montgomery [2] states that the group of holomorphic transformations of a compact complex manifold is a complex Lie transformation group. The second assertion follows from the following theorem.
Theorem 2.2 A connected complex Lie group of holomorphic transformations acting effectively on a hyperbolic manifold M reduces to the identity element only.
Proof Assume the contrary. Then a complex one-parameter subgroup acts effectively and holomorphically on M. Its universal covering group C acts essentially effectively on M. For each point p E M, this action defines a holomorphic mapping z E C z(p) E M. By Theorem 1.1, this holomorphic mapping C Al must he a constant mapping. Since the identity element 0 E C maps p into p, every element z of C maps p into p. Since p is an arbitrary point of Al, this shows that the action of C on Al is trivial, 0 contradicting the assumption.
Corollary 2.3 Let Al be a Hermitian manifold whose holomorphic sectional curvature is bounded above by a negative constant. Then the group H(A1) of holomorphic transformations of M is a Lie transformation group. and its isotropy subgroup Hp(M) is compact for every p E M. No complex Lie transformation group of positive dimension acts nontrivially on Al. If M is moreover compact, then H(M) is a finite group.
Proof This is immediate from Theorem 4.11 of Chapter IV and from Theorems 2.1 and 2.2.
O
2
The Automorphism Group of a Hyperbolic Manifold
69
We shall explain Theorem 2.2 from a slightly different viewpoint. Let X be a holomorphic vector field on a hyperbolic manifold M. Although it generates a local one-parameter group of local holomorphic transformations of M, it may in general not generate a global one-parameter group of holomorphic transformations. If it does, we call X a complete vector field. If X is holomorphic, so is JX, where J denotes the complex structure of M. Theorem 2.2 means that if X is complete, then JX cannot be complete. For a bounded domain in C', Theorems 2.1 and 2.2 have been proved by H. Cartan [1]. (Of course, the second assertion of Theorem 2.1 is meaningless for a bounded domain.) We note that for a bounded domain these results of H. Cartan may be obtained by means of its Caratheodory distance. For a bounded domain, the Bergman metric may be also used to prove these results. Unlike the Caratheodory distance, the Bergman metric may be constructed even on some compact complex manifolds, e.g., any algebraic hypersurface of degree d > n + 2 in For more details, see Kobayashi [1]. It would he of some interest to note here that one does not know if a generic algebraic hypersurface of degree d > n+2 in Pn+1(C) is hyperbolic or not (cf. Problem 4 in Sec. 3 of Chapter IX). Theorem 2.2 is essentially equivalent to Theorem D (or Theorem D') in Wu (2]. For a systematic account of the holomorphic transformation group of a complex manifold, see Kaup [1].
From Corollary 4.13 of Chapter IV and Theorem 2.2 we obtain the following result of Schwarz and Klein (see Schwarz [1] and Poincare [1] ).
Corollary 2.4 If M is a compact Riemann surface of genus g ? 2, then the group H(M) of holomorphic transformations of M is finite.
Hurwitz [1] proved that the order of H(M) does not exceed 84(g - 1). For similar results on algebraic surfaces, see Andreotti [1]. The following result is originally due to Bochner [1], Hawley [1], and Sampson [1]. Corollary 2.5 Let M be a compact complex manifold that has a bounded domain D C C" as a covering manifold so that M = DIF, where I is a Properly discontinuous group of holomorphic transformations acting freely on D. Then the group H(M) of holomorpphic transformations of M is finite.
Proof Since D is hyperbolic (Corollary 4.5 of Chapter IV), M is also hyperbolic by Theorem 4.7 of Chapter IV. Our assertion now follows from Theorem 2.2. 0 It is possible to obtain Corollaries 2.4 and 2.5 from the following result (Kobayashi [5]): If M is a compact complex manifold whose first Chern class
70
Holomorphic Mappings into Hyperbolic Manifolds
cl(M) is negative, then the group H(M) of holomorphic transformations of
M is finite. A compact Kaehler manifold M with negative Ricci tensor satisfies cl (M) < 0.
Theorem 2.6 Let M be a hyperbolic manifold of complex dimension n. Then the group H(M) of holomorphic transformations of M has dimension S 2n + n2 and the linear isotropy representation of the isotropy subgroup Hp (M), p E M, is faithful and is contained in U(n).
Proof Let o : Hp(M)
GL(n; C) be the linear isotropy representation. Since Hp(M) is compact by Theorem 2.2, its image e[Hp(M)] is compact and is contained in U(n) [after a suitable change of basis in the tangent space Tp(M)]. Since Hp(M) is compact, we can find a Riemannian metric
on M invariant by Hp(M) so that Hp(M) is contained in the group of isometries of M with respect to that Riemannian metric. It follows then that g is faithful. Hence, dim Hp(M) < dim U(n) = n2. Finally, dim H(M) dim M + dim Hp(M) < 27t + n2.
Remark It is not difficult to see that if dim H(M) attains its maximum 2n + n2 in Theorem 2.6, then M is biholomorphic to the open unit ball in C". We shall only outline its proof. The assumption dim H(M) = 2n + n` implies that HH,(?lf) = U(n) for each p E M and H(M) is transitive on Al. Then there is a Hermitian metric on M which is invariant by H(M). Since
the isotropy subgroup Hp(M) = U(n) contains the element -I, there is no nonzero tensor of odd degree (i.e., odd number of indices) at p, which is invariant by Hp(M). In particular, the torsion tensor field of the invariant Hermitian metric must vanish. Hence, the metric is Kachlerian. Since U(n) is transitive on the unit sphere, the holomorphic sectional curvature of this invariant Kaehler metric is a constant. Since M is hyperbolic, this constant must be negative. Since a homogeneous Riemannian manifold of negative curvature is simply connected (see Kobayashi and Nomizu [1, Vol. II, p. 105]), M is simply connected. Being a simply connected complete Kaehler manifold of negative constant holomorphic sectional curvature, M is biholomorphic with the open unit ball in C", (see Kobayashi and Nomizu (1, Vol. II, pp. 169-170]). 3
Holomorphic Mappings into Hyperbolic Manifolds
The following theorem is similar to the result of Van Dantzig and Van der Waerden (see Sec. 2), but it is easier to prove.
3
Holomorphic Mappings into Hyperbolic Manifolds
71
Theorem 3.1 Let M be a connected, locally compact, separable space with pseudodistance dM and N a connected, locally compact, complete met-
ric space with distance dN. The set F of distance-decreasing mappings f : M N is locally compact with respect to the compact-open topology. In fact, if p is a point of M and K is a compact subset of N, then the subset F(p, K) = If E F; f (p) E K} of F is compact. proof Let If, I he a sequence of mappings belonging to F(p, K). We shall show that a suitable subsequence converges to an element of F(p, K). We
take a countable set {pi} of points which is dense in M. We set Ki = {q E N; dN (q, K) < dM(p, pi)}. Then Ki is a closed dM (p, pi)-neighborhood
of K and hence is compact. Since each f, is distance-decreasing so that fn(pi) is in the compact set Ki for every n. By the standard argument of taking the diagonal subsequence, we can choose a subsequence {fnk } such that {fnk (pi)} converges to some point of Ki for each pi as k tends to infinity. By changing the notation, we denote this subsequence by { fn} so that {fn (pi) I converges for each pi. We now want to show that If. (q) } converges for each q c Al. We have
dN(fn(q), fm(q)) < dN(.fn(q),.fn(pi)) +dN(fn(Pi)7 L. (Pi)) + dN (fm (pi), fm (q))
< 2dM(q,pi) + dN(fn(Pi), fm(Pi))Given any positive number e, we choose pi such that 2dM(q, pi) < e/2. We choose also an integer no such that dN(fn(pi), f, (Pi)) < e/2 for n, m > no. Then dN(fn(q), f. (q)) < e for n, m > no. This shows that { fn(q)} is a Cauchy sequence for each q. Since N is complete, we may define a mapping
f:M --+Nby f (q) = n-x lim f, (q). Since each fn is distance-decreasing, so is f. Since fn(p) E K for each n, it follows that f maps p into K. We shall complete the proof by showing that the convergence fn(q) --* f (q) is uniform on every compact subset C of M. Let b > 0 be given. For each q E C, choose an integer nq such that dN(fn(q), f (q)) < 6/4 for n > nq. Let Uq be the open 5/4-neighborhood of q in M. Then for any x c Uq and n > nq, we have
dN(fn(x),1(x))
dN(fn(x),fn(q)) +dN(fn(q),f(q)) + dN (f (q), f (x))
<2dM(x,q)+4 <S.
Holomorphic Mappings into Hyperbolic Manifolds
72
Now C can he covered by a finite number of Uq's, say U2 = Uq, , i = 1.... , s. It follows that if n > maxi {rtq, }, then dN [f. (x), f (x)] < b
for each x E C.
Comparing this proof with that of Lemma for Theorem 2.1, we find that Theorem 3.1 is easier for the following reasons: In Theorerrr 3.1, N is assumed to be complete and there is no need to prove the existence of f As an application of Theorem 3.1 we have Theorem 3.2 Let M be a complex manifold and N a complete hyperbolic manifold. Then the set F of holomorphic mappings f : M -- N is locally compact with respect to the compact-open topology. For a point p of M and a compact subset K of N, the subset F(p, K) = If E F; f (p) E K} of F is compact.
Proof This follows from Theorem 3.1 and from the fact that if a sequence of holomorphic mappings f, E F converges to a continuous mapping f in the compact-open topology, then f is holomorphic.
Theorem 3.3 Let M be a hyperbolic manifold and o a point of M. Lct f : M - M be a holomorphic mapping such that f (o) = o. We denote bp dfo : To(M) - T<,(M) the differential off at o. Then (1) The eigenvalues of dfo have absolute value <_ 1; (2) If dfo is the identity linear transformation, then f is the identity transformation of M; (3) If Idetf0I = 1, then f is biholomorphic mapping.
Proof Let r be a positive number such that the closed r-neighborhood B,. = (p E M; dn1(o, p) r} of o is compact. Let Fo denote the set of all (holomorphic or not) mappings of Br into itself which leave o fixed and are distance-decreasing with respect to dh1. By Theorem 3.1, F0 is compact.
Let f : M - M he a holomorphic mapping such that f (o) = o. Let A he and eigenvalue of dfo. For each positive integer k, the mapping f k, restricted to Br, belongs to Fo and its differential at o has eigenvalue ak. If Jai > 1, then IAkI goes to infinity as k goes to infinity, in contradiction to the fact that Fo is compact. This proves (1). For the sake of simplicity, we denote by d'0 fo all partial derivatives of order m at o. We want to show that if dfo is the identity transformation of T0(M), then d"' f0 = 0 for m > 2. Let m be the least integer >_ 2 such that
3
Holomorphzc Mappings into Hyperbolic Manifolds
73
0. Then d"(f k ) = kd- f o for all positive integers k. As k goes to infinity, d"t (f k )o also goes to infinity, in contradiction to the fact that Fo is compact. This proves (2). Assume Idet dfol = 1. From (1), it follows that all eigenvalues of dfo have absolute value 1. Put dfo in Jordan's canonical form. We claim that dfo is then in diagonal form. If it is not, it must have a diagonal block of this form: dm fo
IA
0
1
A
JAI=1. 0
A
The corresponding diagonal block of (dfo)' is then of the form f Ak
kAk-1
*
I
kAk-1
0
Ak
It follows that the entries kA'-1 immediately above the diagonal of (dfo)k diverge to infinity as k goes to infinity, contradicting the compactness of Fo. Since dfo is a diagonal matrix whose diagonal entries have absolute value 1, there is a subsequence {(dfo)k} that (dfo)kconverges to the identity matrix as i goes to infinity. We denote now by Fa the set of all holoinorphic mappings from M into itself leaving the point o fixed. Assume that M is complete. By Theorem 3.2, Fo is compact. Then there is a subsequence of {f ',} which converges to a holomorphic mapping h E Fo. By change of notations, we may assume that If k; } is this subsequence. Then dho = lim d(f ki)o is the identity transformation of To(M). By (2) of the present theorem, h must be the identity transformation of M. Since Fo is compact, there is a convergent subsequence of {fk;-1 }. By change of notations, we may assume that {f ki -I} converges to a holomorPb.ic mapping g E Fo as i goes to infinity. Then
f o g = f o (lim f k,-1) = lim fk, = identity transformation. Similarly, g o f is the identity transformation of M. This shows that f is a
biholomorphic mapping with inverse g.
If M is not complete, we argue as follows. Let r he a small positive number such that the closed r-neighborhood Br of o is compact. We denote
74
Holornorphic Mappings into Hyperbolic Manifolds
again by Fr, the set of all mappings of B, into itself which leave the point o fixed and are distance-decreasing with respect to dM. By Theorem 3.1, F,., is compact. We obtain a sequence If k, } which converges to an element h of Fo. Since the convergence is uniform, h is holomorphic in the interior of B, and dho = lim d(f k, )o is the identity transformation of To(M). n-on, the proof of (2) above, we see that h is the identity transformation of B,.. Let W be the largest open subset of lvl with the property that some subsequence of If k; } converges to the identity transformation on W. (To prove the existence of W, consider the union W = UWj of all open subsets It'., of M such that on each W- some subsequence of { f k; } converges to the identity transformation. A countable number of Wj's already cover W. We consider the corresponding countable number of subsequences of { f k, } and can extract a desired subsequence of f k= for W by the standard argument, using the diagonal subsequence.) Without loss of generality, we may assume that fk converges to the identity transformation on W. Since W contains
the interior of B,., it is nonempty. Let p E W and U a neighborhood of p with compact closure U. Since lim f k+ (p) = p and each f k; is distancedecreasing, there is a neighborhood V of p such that f k; (V) C U for i >_ i;,.
Let F be the set of all distance-decreasing mappings from V into U. By Theorem 3.1, F is compact. We extract a subsequence from { f k; } which is convergent on V. Since it has a converge to the identity transformation on V n W, it converges to the identity transformation on V. This proves that W is closed and hence W = M. The remainder of the proof is the same as the case where Al is complete. D
For a bounded domain, Theorem 3.3 is due to H. Cartan [2, 3] (and also to Caratheodory [3]). Theorem 3.3 has been proved by Kaup [1] under weaker conditions. It has been also proved independently by Wu [2] under slightly stronger conditions. Let M be a domain in C" and D the closed unit disk in C. According to Oka [1], Al is said to be pseudoconvex if every continuous mapping f
D x [0, 1] -> C' such that (1) for each t E [0, 1] the mapping ft : D -4 Cn defined by ft(z) = f (z, t) is holomorphic and (2) f (z, t) E M unless I z I < 1 and t = 1, maps D x [0, 1] necessarily into M.
For other definitions of pseudoconvexity and their equivalence with this definition, see Bremermann [2] and Lelong [1]. As an application of Theorem 3.2 we shall prove the following theorem.
3
Holomorphic Mappings into Hyperbolic Manifolds
75
Theorem 3.4 If a domain ltii in C" is complete hyperbolic, then it is pseudoconvex.
proof Let dD denote the boundary of D, i.e., dD = D - D. Since f (OD x [0, 1]) is a compact subset of M, there exists a compact neighborhood K of f (OD x [0, 1]) in M. Then we can find a point zo E D such that f (zo, t) E K for 0 < t < 1. By Theorem 3.2, the family F(zo, K) of M which send zo into K is compact. Since holomorphic mappings D each ft is in F(zo, K) for 0 < t < 1, the limit mapping fl must be also in F(zo, K). In particular, fl maps D into 1b7, i.e., f (z, 1) E M for z E D.
Let B be the unit open ball in C'. From Theorem 3.4 we see that Bn - {0} (where 0 denotes the origin) is not complete hyperbolic for n >- 2.
CHAPTER VI
The Big Picard Theorem and Extension of Holomorphic Mappings
1
Statement of the Problem
The classical big Picard theorem is usually stated as follows:
If a function f (z) holomorphic in the punctured disk 0 < Izl < R has an essential singularity at z = 0, then there is at most one value a ( oc) such that the equation f (z) = a has only a finite number of solutions in the disk. We may rephrase the statement above as follows. If f (z) = a has only a finite number of solutions in the disk, then it has no solutions in a smaller
disk 0 < jz[ < R', R' < R. Hence, the big Picard theorem says that if a function f (z) holomotphic in the punctured disk 0 < Izi < R misses two values a, b ( x%) , then it has a removable singularity or a pole at z = 0, i.e., it can be extended to a meromorphic function on the disk jzI < R. In other words, if f is a holomorphic mapping from the punctured disk
0 < Izj < R into M = P1(C) - {3 points}, then f can be extended to a holomorphic mapping from, the disk I z I < R. into P, (C). We consider the following problem in this section.
Let Y be a complex manifold and -4f a submanifold which is hyperbolic and relatively compact. Given a holomorphic mapping f from the punctured disk0 < 1z1 < R into Al, is it possible to extend it to a holomorphic mapping from the disk I z I < R into Y?
We shall give an affirmative answer in some special cases. We shall consider also the case where Y and Al are complex spaces and the domain is of higher dimension. The following example by Kiernan [41 shows that the answer to the question above is in general negative. (It is slightly simpler than the original example of Kiernan.) Let Y = P2(C) with homogeneous coordinate system v, w). Let
M = {(1, v, w) E P2(C); 0 < wI < 1, jwj < !e1/' I}. Then the mapping 77
78
The Big Picard Theorem and Extension of Holomorphic Mappings
(1, v, w) -+ (v, wel/") defines a biholomorphic equivalence between M and
D* x D, (where D* = {z E C; 0 < JzJ < 1} and D = {z E C; IzI < 1}). Hence, M is hyperbolic. Let f/: D* -4M be the mapping defined by f (z) = 11, z, I e1/2)
z E D*.
Then f cannot be extended to a holomorphic mapping from D into P2(C 2
The Invariant Distance on the Punctured Disk
Let D be the open unit disk in C and D* the punctured disk D - {0},
i.e., D* = {z E C; 0 < IzI < 1}. Let H be the upper half-plane {w = u + iv E C; v > 0) in C. The invariant metric dsH of curvature -1 on H is given by 2
dsH =
dw dw v2
Let p : H --+ D* be the covering projection defined by
z=p(w)=e2"iw forwEH. Let dsD. be the metric on D* defined by p (dS2
- dsH.
Since dz = 27riz din and zz = e-4"", we obtain easily 2
dsD
-
4dzdz zz[log(1/zz)]2
For each positive number r < 1, let L(r) denote the arc-length of the circle JzJ = r with respect to dsD.. Then 27r
log(1/r)' In the definition of the pseudodistance dh1 in Sec. 1 of Chapter IV, the distance e on D is the one defined by the Poincare-Bergman metric dsD of D. Without loss of generality (i.e., by multiplying a suitable positive con-
stant to the metric), we may assume that dsD has curvature -1 so that H is not only biholomorphic but also isometric to D. Then the invariant distance dH of H is the one defined by d.sH. By Proposition 1.6 of Chapter IV the invariant distance dD coincides with the one defined by dsH.. We state the result of this section in the form convenient for later uses.
2
The Invariant Distance on the Punctured Disk
79
proposition 2.1 Let L(r) be the arc-length of the circle IzI = r < 1 with respect to the invariant distance dD of the punctured unit disk D*. Then L(r) = a/ log(1/r), where a is a constant. In particular, line L(r) = 0.
r-+e
Since we want to consider not only complex manifolds but also complex
spaces, we state the following proposition for a metric space M, not just for a hyperbolic manifold M.
proposition 2.2 Let M be a locally simply connected metric space with distance function d.,1. Let D* be the punctured unit disk with the invariant distance dD.. Let f : D* - M be a distance-decreasing mapping. Assume that there is a sequence of points {zk} in D* such that limk-,o zk = 0 and f(zk) converges to a point PO E M. Then for each positive r < 1, f maps the circle IzI = r into a closed curve which is homotopic to zero.
Proof Set rk = 141- Since the closed curves
yk(t) =
f(rke2,rit)
0 < t < 1, k = 1,2,...
are all homotopic to each other, it suffices to prove that, for a sufficiently
large integer k, the closed curve P10 is homotopic to zero. Let U he a simply connected neighborhood of po in M and take a smaller neighborhood
V such that V c U. Let N be an integer such that f (z,) E V for n > N. Since f is distance-decreasing, Proposition 2.1 implies that. the arc-length of the closed curve f (yk) approaches zero as k goes to infinity. Hence, if k > N is sufficiently large, f (yk) is contained in U. Since U is simply connected, f (yk) is hornotopic to zero.
Corollary 2.3 Let M be a locally simply connected, compact metric space with distance function dAf . Let D* be the punctured unit disk with the invariant
M is a distance-decreasing mapping, then f maps distance dD.. If f : D* each circle izI = r < 1 into a closed curve which is homotopic to zero. Let M, D*, and f be as in Proposition 2.2 or Corollary 2.3. Let M,f be a covering space of M with projection rr. Then f can he lifted to a mapping
f:D*--> hl such that f =7ro Theorem 2.4 Let .11 be a complex manifold which has a covering manifold M with Carath.eodory distance ctil. Let D* be the punctured disk. Let f : D*
M be a holom.orphic mapping such that, for a suitable sequence
80
The Big Picard Theorem and Extension of Holomorphic Mappings
of points zk E D* converging to the origin, f (zk) converges to a point po E Al. Then f extends to a holom.orphic mapping of the (complete) disk D into Al.
Proof Since
is assumed to be a distance, dti1 is also a distance by Proposition 2.1 of Chapter IV and hence Al is hyperbolic. By Theorem 4.; of Chapter IV, Al is also hyperbolic. Since f D* M is distance:
decreasing with respect to dD. and d,.f, we can lift f to a holomorphic mapping f : D* , Al as we have seen above. Then f (zk) converges to a point po E Al such that ir(po) = po, where it is the projection Al -+ Al. The map-
ping f : D* - Al is distance-decreasing with respect to the Caratheodorv distances cD and cA;1. Since the disk D is the completion of D* with respect
to the Caratheodorv distance CD' (see Sec. 3 of Chapter IV), f can be extended to a holomorphic mapping f : D -> Al such that f (0) = po. It follows that f can be extended to a holomorphic mapping f : D -+ Al such that f (0) = po. Corollary 2.5 Let Al be a compact complex manifold which has a covering manifold Al with Caratheodory distance c,ti1. Then every holo-morphic mapping from the punctured disk D* into Al can be extended to a holomorphic mapping from the disk D into Al.
Remark Theorem 2.4 can be applied to a complex manifold Al of the form Af/I , where Al is a bounded domain in C't and r is a properly discontinuous group of holomorphic transformations acting freely on Al. If Al = Al/r is compact, Corollary 2.5 applies. From Theorem 2.4 we obtain also the following:
Corollary 2.6 Let Al be a complex submanifold of a complex manifold Y such that Al is compact. Assume that Al has a covering manifold Al with Caratheodory distance c.A1. Then every holom.orphic mapping f of the punctured disk D* into Al satisfies one of the following two conditions:
(1) f can, be extended to a holomnorphic mapping of the disk D into Al:
(2) For every neighborhood N of the boundary 8A1 = 11 - Al of Al in Al, there exists a neighborhood U of the origin in the disk D such thrt
f(U-{0})cN. If we set Y = Pl (C) and M = Y - {3 points}, then Corollary 2.6 yields the classical big Picard theorem immediately.
3
Mappings from the Punctured Disk into a Hyperbolic Manifold
81
The results of this section are due essentially to Huber [2], who obtained & generalization of the big Picard theorem in the following form: Let Y be a Riemann surface and M a domain of hyperbolic type. Then every holomorphic mapping f : D* -* M can be extended to a holomorphic mapping f : D -+ Y. This follows from Corollary 2.6 and from the fact that the boundary of M is contained in another subdomain of Y which is of hyperbolic type. But this last fact is known only for Riemann surfaces. 3
Mappings from the Punctured Disk into a Hyperbolic Manifold
In this section we shall prove the following theorem of Kwack [1], which generalizes Theorem 2.4.
Theorem 3.1 Let Al be a hyperbolic manifold and D* the punctured unit disk. Let f : D* --+ Al be a holomorphic mapping such that, for a suitable sequence of points zk E D* converging to the origin, f (zk) converges to a point po E Al. Then f extends to a holomorphic mapping of the unit disk D into M.
Corollary 3.2 If Al is a compact hyperbolic manifold, then every holoAl extends to a holomorphic mapping of D morphic mapping f : D* into M. Proof As in Sec. 2, we set rk = IzkI, 'Yk(t) =
In other words,
f(rk(2ait)
0
is the image of the circle I z I = rk by f .
Let U be a neighborhood of po in Al with local coordinate system W1.... , W. We may assume that po is at the origin of this coordinate system. Let E be a small positive number and let V be the open neighborhood of PO defined by
V:1 0 1< E
i=1....,n.
Taking e sufficiently small, we may assume that V C U. Let Lt' be the neighborhood of po defined by TV : I& I < E/2
i = 1.... , n.
The problem is to show that, for a suitable positive number 6, the small punctured disk {z E D*; Jzj < b} is mapped into U by f.
82
The Big Picard Theorem and Extension of Holomorphic Mappings
Since the diameter of yk approaches zero as k goes to infinity by Proposition 2.1, all but a finite number of yk's are contained in W. Without loss of generality we may assume that all yk's are in W. By taking a subsequence of {zk} if necessary, we may assume also that the sequence Irk) is monotone decreasing. Consider the set of integers k such that the image of the annulus rk+I < Izj < rk by f is not entirely contained in W. It this set of integers is finite, then f maps a small punctured disk 0 < jzj < 5 into W. Assuming that this set of integers is infinite, we shall obtain a contradiction. By taking a subsequence, we may assume that, for every k, the image
of the annulus rk+I < Izj < rk by f is not entirely contained in W. For each k, let Rk = {z E D*; ak < Izl < bk}
be the largest open annulus such that (1) ak < rk < bk and (2) f maps Rk into W. We set o'k(t) = ake2rrit Tk(t) = bke2ait.
0 < t < 1, 0 < t < 1.
In other words, ok is the inner boundary of the annulus Rk and Tk is the outer boundary of the annulus Rk, From the definition of ak and bk, :t is clear that both f (ok) and f (Tk) are contained in TV but not in W. By Proposition 2.1, the diameters of f (ok) and f (Tk) approach zero as k goes to infinity. By taking a subsequence if necessary, we may assume that the sequences { f (ok)} and { f (Tk)} converge to points q and q' of IV - TI', respectively. Since po is in W and both q and q' are on the boundary of 11'. the points q and q' are distinct from po. (The point q might coincide with the point q'.) By taking a new coordinate system around po if necessary. we may assume that u'1(q) 54- 'u'1(po) = 0,
w1(q')
u,,1(po) = 0.
Let f (z) = [f 1(z), ... , f" (z)] he the local expression off on D. Then
khillx f1(uk) = "u'l(q) lim fl (Tk) = a'l(q) k-x kliIll -.z f 1(Zk) = il'l (PO) = 0-
f I(L) -
Mappings from the Punctured Disk into a Hyperbolic Manifold
3
It follows that if k is
83
then f1(zk)
f1(ak) U f1(Tk)
K
If k is sufficiently large, we can find a simply connected open neighborhood
Gk of w1(q) in C such that f 1(Qk) C Gk and f 1(zk) V Gk. We apply Cauchy's theorem to the holomorphic function 1/[w1 - f' (zk)] and the 1
closed curve f 1(ok) in Gk. Then dw1
f
w1 - fl(zk)
_ - 0.
This may be rewritten as follows:
f1i(z)
Jak f1(z)-f1(zk)
dz = 0,
where f 1'(z) = df 1(z)/dz. Similarly, if k is sufficiently large,
dz=0.
f11(z) JT, f1(z) - f1(zk)
On the other hand, the principle of the argument applied to the function f 1(z) - f 1(zk) defined in a neighborhood of the annulus Rk which is bounded by the curve Tk - ak yields the following equality:
f
1' z k
f 1(z)
-'
1'
(zk)dz - f k
z
f1(z) -(fi(zk)dz
= 27ri(N - P)>
where N and P denote the numbers of zeros and poles of f1(z) - f1(zk) in Rk. In the present situation, P = 0 and N 1. We have arrived finally at a contradiction.
U
From Theorem 3.1 we obtain also the following:
Corollary 3.3 Let Al be a hyperbolic submanifold of a complex manifold Y such that N is compact. Then every holomorphic mapping f of the punctured disk D* into Al satisfies one of the following two conditions: (1) f can be extended to a holomorphic mapping of the disk D into M; (2) For every neighborhood of N of the boundary 8M = M - Al of M in M, there exists a neighborhood U of the origin in the disk D such that
f(U - {0}) c N. This generalizes Corollary 2.6.
The Big Pirard Theorem and Extension of Holomorphic Mappings
84
4
Holomorphic Mappings into Compact Hyperbolic Manifolds
As an application of Theorem 3.1 we shall prove the following result. of Kwack [1].
Theorem 4.1 Let M be a compact hyperbolic manifold. Let X be a complex
manifold of dimension m and A an analytic subset of X of dimension < m - 1. Then every holomorphic mapping f from X - A into M can be extended to a holomorphic mapping from X into M.
Proof We shall first show that it suffices to prove the theorem when A is a nonsingular complex submanifold of X. Let S be the set of singular points of A. Then S is an analytic subset of X and dim S < dim A (see, for instance, Narasimhan [1, pp. 56-58]). Since A - S is a non-singular complex submanifold of X - S, we extend first f to a holomorphic mapping from X - S into M. Since dim S < dim A, we obtain the theorem by induction on the dimension of A. We shall now assume that A is nonsingular. For each point a of A we
want to find a neighborhood U in X such that f I U fl (X - A) can be extended to a holomorphic mapping from U into M. We may therefore assume that X is a polydisk
Dx
Dm-1 =
{z,t1,...,tm-1)
E
Cm; lz l < 1, jt11 < 1,..., Itm-1l < 1}.
and that A is contained in the subset defined by z = 0. For the sake of simplicity, we denote (t', .... ttn-1) and (z, t1 ... , tm-1) by t and (z, t). respectively.
For each fixed t E D»i-1, we have a holomorphic mapping ft from the punctured disk D" into .41 defined by ft(z) = f (z, t). Applying Corollary 3.2 to each ft, we extend ft to a holomorphic mapping ft : D . Al and we set
f (0, t) = ft(0). We have to prove that this extended mapping f : X -» ?1 is holomorphic. By the Riemann extension theorem, it suffices to show that f : X --+ .11 is continuous at every point of A. To prove that f is continuous
at a E A, we may assume without loss of generality that a is the origin (z, t) = (0, 0). We set p = f (0, 0) E M. Let e' be any positive number. Let V he the --neighborhood of 0 in D with respect to the Poincare distance v = dD. Let IV be the E-neighborhood of 0 in D'-1 with respect to the distance
do,.'-l. W 'e shall show that f maps V x W into the 3e-neighborhood of p in Al with respect to the distance d,tit. Let (z, t) E V x W. Since the restriction of f to D x {0} is holomorphic and since f : D x {0} --+ Al i'
5
Holomorphic Mappings into Complete Hyperbolic Manifolds
85
distance-decreasing, we have dAs (f (0, 0), f (z, 0)) < dD (0, z) < E.
We shall first consider the case z # 0. Since f : D" x D"-1 -, M is holomorphic and hence distance-decreasing, we have
dM(f(z,0),f(z,t)) < dD,..-, (0, t) < 6,
where the equality is a consequence of Proposition 1.5 of Chapter IV. We have now
dm (f (0, 0), f (z, t)) < dm (f (0, 0), f (z, 0)) + dm (f (z, 0), f (z, t)) < 2E,
provided that z # 0. We shall now consider a point (0, t) E V x W. Choose any z E V different from zero. Since f : D x {t} --, M is holomorphic and hence distance-decreasing, we have
dM(f(z,t), f(0,t)) < dn(z,0) < E. Hence, dm (f (0, 0), f (0. t))
dm (f (0, 0), f (z, t)) + dm (f (z, t), f (0, t))
<2e+F=3E.
0
This proves that f is continuous at (0, 0). 5
Holomorphic Mappings into Complete Hyperbolic Manifolds
In Theorem 4.1, if A is smaller, it suffices to assume that Al is complete hyperbolic. Before we make an exact definition, we prove
Proposition 5.1 Let D"' = {(z1,.. . , zm) E Cm; 1, ... , m} and let A be a subset of Dm = D x
Izi l
< 1 for j =
Dm-1 of the forrn A = {0} x A',
where A' is nowhere dense in DI-1. Then the distance dD,,,_A is the restriction of the distance dD.., to Dm - A.
Proof Let p and q be two points of Dm - A. Since the injection D"' - A --+ D"' is holomorphic and hence distance-decreasing, we have dDm_A(p, q) ! dDm (p, q). To prove the proposition, it suffices to show the
opposite inequality for every pair of points (p, q) belonging to a dense subset of (Dm - A) x (Dm - A).
The Big Picard Theorem and Extension of Holomorphic Mappings
86
Let S be the subset of (Dm - A) x (Dm - A) consisting of pairs (p, q) for which there exist points a, b E D and a holomorphic mapping f : D Dm -A such that do..' (p, q) = dD (a, b), f (a) = p, and f (b) = q. If (p, q) E S, then dom-A(p, q) = dDm-A(f (a), f (b)) < dD(a, b) = dDm (p, q)
It suffices therefore to prove that S is a dense subset. Let p = (al, ... , am) and q = (b',... , bm) be arbitrary points of D1- A. To show that every neighborhood of (p, q) in (Dm - A) x (Dm - A) contains a point of S, we may assume without loss of generality that a', b' and 0 are mutually distinct, for the set of such pairs is dense in (Dm - A) x (D"' - A). The distance dDm (p, q) is equal to the maximum of dD(a) , b3), j = 1, ... , 71,
say dD(ak, bk) (see Example 1 in Sec. 1 of Chapter IV). We set a = ak and b = bk so that dDm (p, q) = dD (a, b). Since dD (a', b') < dD (a, b) for there exist holomorphic mappings f j : D -> D such that j= f j (a) = aj and f j (b) = bj for j = 1, .... in. Since a1 bl, we may impose the additional condition that f, be injective. Then f i ' (0) is either empt'y' or a single point c E D. If f i '(0) is empty, then the mapping f : D -. D"' defined by f(z) = (f1(z),..., f,, (z)) sends D into Dm - A, since f,(z) never vanishes. In this case, (p, q) belongs to S. Assume c = fl-'(O). Then the mapping f D D"' defined above maps D into Dm - A if and only if , fm (c)] is not in A. If f (D) C Dm - A, then (p, q) belongs to S. (f2(c)....
So, we have only to consider the case [f2 (c), ... , f,.,., (c)] E A', We assert that given a positive number E there exists a positive number h
such that for any points c E D (j = 2, ... , rn) with dD (cj, f j (c)) < b there exist automorphisms h j : D . D satisfying
hj(fj(c)) =c', dD(a',hj(a')) <e, and dD(b', hj(b')) < e. We shall first complete the proof of the proposition and then come back to the proof of this assertion. Given E > 0, let 0 > 0 be as above. Since A' is nowhere dense in D"I
there exists a point (c2, ... , c') E D"'-' - A' such that dDW, fj(c)) < D D be as above. We consider the for j = 2... . , m.. Let hj : D [a',112(a2)..... hm(am)] and q' _ [b'.12(b2),...,hm(b`)] of points p' = Dm defined by f'(z) _ Dm and the holomorphic mapping f : D (f1 (z), h2(f2(z)).... , h,(fm (z))). Since a1, b', and 0 are mutually distinct both p' and q' are in Dm - A. Since It, : D D is distance-preserving, We
5
t!olomorphic Mappings into Complete Hyperbolic Manifolds
87
have dD(hi (a3), hj(bi)) = dD(ai,bi) and hence dDm(h, q') = Max{dD(a2, bi); j = 1, ... , m} = dD(p, q) = dD(a, b)
Clearly, f'(a) = p' and f'(b) = q'. This shows that (p', q') belongs to S. Since dD (a2, hj (a4)) < E, we have dDnn (p, p') < e. Similarly, dDm (q, q') < e.
Thus, the E-neighborhood of (p, q) in (Dm- A) x (Dm - A) with respect to dDm x dD,n contains a point (p', q') of S. This completes the proof of the proposition except for the proof of the assertion made above. To simplify the notations in the assertion above, we denote aj, bi, f!(c), c1, and h j by a, b, c, c', and h respectively. Then the assertion we have to prove reads as follows:
Lemma Let a, b, c E D be given. Then for any e > 0 there is a 6 > 0 such that for any c' E D with dD (c, c') < 6 there exists an automorphism h : D -+ D satisfying
h(c) = c',
dD(a, h(a)) < e,
and
dD(b, h(b)) < E.
In order to prove the lemma, it is more convenient to replace D by the upper half-plane H in C. Given e > 0, choose a1 > 0 and b2 > 0 such that
dH(a,r(a+t)) <e and d11(b,r(b+t)) <E for any real numbers t and r such that Ill < D1 and Ir - 11 < S2. The set {r(c + t); I t I < S1i Ir - 11 < b2} contains the 6-neighborhood of c for some 6 > 0. Given c' in the b-neighborhood of c, we can find an automorphism
h : H -* H of the form h(z) = r(z +t) such that h(c) = c', dH(a, h(a)) < E
0
and dH(b, h(b)) < e.
As an application of Proposition 5.1 we prove
Theorem 5.2 Let Al be a complete hyperbolic manifold. Let X be a complex manifold of dimension in, and let A be a subset which is nowhere dense in an analytic subset, say B, of X with dim B < 7n-1. Then every holomorphic mapping f : X - A M can be extended to a holomorphic mapping
X -+AI.
Proof As in the proof of Theorem 4.1, we can reduce the proof to the special case where X = D' _ {(z',.. , z`n) E C"a; Iz21 < 1} and B is the subset defined by z1 = 0 so that A is of the form A = {0} x A', where A' is nowhere dense in D"'. Since f : D"` - A M is distance-decreasing, f can be extended to a continuous mapping from the completion of the .
88
The Big Picard Theorem and Extension of Holomorphic Mappings
metric space D"' - A into M. By Proposition 5.1, D"' is the completion (if D' - A with respect to the distance dnm_A. By the Riemann extension Al is necessarily theorem, the extended continuous mapping f : D"' holomorphic.
Theorem 5.2 contains the following result of Kwack [1], which was proved by a different method.
Corollary 5.3 Let M be a complete hyperbolic manifold. Let X be complex manifold of dimension m, and let A be an analytic subset of dimension S m - 2. Then. every holomorphic mapping f : X - A 11 ct
can be extended to a holomorphic mapping X
M.
For extension of a holomorphic mapping X - A M where Al has a covering space which is a Stein manifold, see Andreotti and Stoll [1]. 6
Holomorphic Mappings into Relatively Compact Hyperbolic Manifolds
The following result is the best solution we can give to the problem stated in Sec. 1 at the moment. Theorem 6.1 Let. Y he a complex manifold and Al a complex sub-manifold of Y satisfying the following three conditions: (1) Al is hyperbolic; (2) the closure. of Al in Y is compact; (3) given a point p of AI - Al and neighborhood U of p in Y, there exists a neighborhood V of p in Y such that V C U and the distance between Al n (Y - U) and Al n V with respect to dtit is positive.
Then every holomorphic mapping f from the punctured disk D' into 11 can be extended to a holom.orphic mapping from the disk D into Y.
Proof Let {rk}, 0 < rk < 1, be a monotone decreasing sequence with lim rk = 0. We consider {rk} as a sequence of points in D' converging 10 the origin. Since Al is compact, we may assume (by taking a subsequence if necessary) that { f (rk)} converges to a point po of M. The case where po is in Al has been already considered (see Theorem 3.1). We assume therefore that po E A1 - Af. Let U he a neighborhood of po in Y. Let -;,k denote the
circle :z! = rk in D. We claim that there exists an integer N such that
f (-yk) C U fork > N.
6
Holomorphic Mappings into Relatively Compact Hyperbolic Manifolds
89
To prove our claim, assume the contrary. By taking a subsequence if necessary, we may assume that each circle yk has a point zk such that f (zk) is not in U. Taking again a subsequence if necessary, we may assume that the sequence If (zk)} converges to a point qo of Al. In view of Theorem 3.1, we have only to consider the case where qo is in Al - M. Since f (zk) is outside U, the limit point qo is also outside U. Taking U sufficiently small, we may assume that qo is outside U so that Y - U is an open neighborhood of qo. By (3), there is a smaller neighborhood V of qo
such that V fl U = 0 and the distance between M n U and M fl V with respect to dm is a positive number 6. Then
d,.f(f(rk),f(zk)) ? J for large k, because f (rk) E U and f (zk) E V. On the other hand, the arc-length L(yk) of yk tends to zero as k goes to infinity (see Proposition 2.1). We have
dar(f(rk),f(zk)) N.
The rest of the proof goes as in the proof of Theorem 3.1. (In order to conform with notations in the proof of Theorem 3.1, take a neighborhood W of po defined by wl2 I < e/2 in terms of a local coordinate system wr,... , iv" in U as in the proof of Theorem 3.1. By taking a subsequence, we may also
assume that f (yk) C IV for all k. We then proceed as in the proof of Theorem 3.1.)
If we set Y = Pi(C) and Al = P1(C) - {3 points}, then conditions (1), (2), and (3) are met and we obtain the classical big Picard theorem,
Example 1 Let Y = P2(C) and pj, j = 1, 2, 3, 4, be four points in general position in P2(C). Drawing a complex line through each pair of these four points, we obtain a complete quadrilateral Q = U; =o Li as in Example 1 of Sec. 4 of Chapter IV. Let Al = P2(C)-Q. Then conditions (1), (2), and (3) are satisfied. While (1) was verified in Sec. 4 of Chapter IV, (2) is evident. To verify (3), let p he a point of Q = 11 - Al. Without loss of generality, we may assume that p is not on the line Lo, which we shall consider as the line at infinity so that P2(C) - Lo = C2. By changing indices if necessary, We may assume that Li and L2 meet at infinity, i.e., are parallel in C2 and
90
The Big Picard Theorem and Extension of Holumorphic Mappings
that L3 and L4 are parallel in C2 = P2(C) - Lo. After a suitable linear transformation, we may assume that in C2 L1 = {(0, w) E C2; w E C},
L2 = {(1, w) E C2; W E C},
L3 = {(z, 0) E C2; z E C},
L4 = {(z, 1) E C2; z E C},
so that N = C2 - (L1 U L2 U L3 U L4) _ (C - {0,1}) x (C - {0,1}). Let U be a neighborhood of p in P2(C). We may assume that U does not meet Lo, i.e., U is contained in C2 = P2(C) - Lo. It is now clear that there exists a neighborhood V of p in C2 such that V C U and the distance between Nn (C2 - U) and N n V is positive with respect to dN . Since M = N - L;,
the distance between M n [P2(C) - Uj = M n (C2 - U) and M n V is positive with respect to d11. This proves our assertion. Rom Theorem 6.1 we may conclude that every holomorphic mapping f from the punctured disk D* into M = P2(C) - Q can be extended to a holomorphic mapping from the disk D into P2(C), where Q is a complete quadrilateral. In Example 1 of Sec. 4 of Chapter IV, we constructed a complete hyperbolic manifold M by deleting a certain five lines from P2(C). It is not clear whether Theorem 6.1 is valid for such a manifold M.
Theorem 6.2 Assume M and Y satisfy conditions (1), (2), and (3) of Theorem 6.1. Let X be a complex manifold of dimension in and A a locally closed complex subm.anifold of dimension <- m. - 1. Then every holomorphic
mapping f : X - A -> lvi can be extended to a holomorphic mapping from X into Y. I do not know whether the theorem is valid when A is an analytic subset with singular points.
Proof We may assume that X = D' = D x D` and A = {0} x D"i-1 so that X - A = D' x D"`-1. We denote a point (z, t', ... , t"'-1) E D"' by (z, t). Given a holomorphic mapping f : D" x D'-I - 111, the restriction of f to the punctured disk D* x {t} can be extended to a holomorphic mapping from the disk D x {t j into Y for each fixed t (by Theorem 6.1) Y is continuous We have to show that this extended mapping f : D"' at every point (0, t) of A. It suffices of course to show that f is continuous
at the origin (0,0) E D x D'n-1. Let p = f(010) E Y. If p is in
.11,
then the proof is the same as in the proof of Theorem 4.1. We assume therefore that p E bf - Al. Let U be a neighborhood of p in Y defined by w1 I < a, i = 1 , ... , n, with respect to a local coordinate system w , ,
. . .
"
6
Holomorphic Mappings into Relatively Compact Hyperbolic Manifolds
91
around p. Let V be a neighborhood of p described in condition (3). Let b be the distance between Al n (Y - U) and M n V with respect to dM. Let r be a positive number such that the disk {(z,0); JzJ < r} is mapped into V by f. Let r' be a positive number such that dDm-1 (0, t) < b if
Itil
If0
dD x Din-1 ((z, 0), (z, t)) <_ dD,,,-1(0, t) < r',
where the first inequality follows from the distance-decreasing property of M and the second inequality follows from the distancef : D* x D"i-1
decreasing property of the injection t c Drri-1 - (z,t) E D* X D"1. Since f (z.0) is in V and f (z, t) is less than r' away from f (z, 0), it follows that f (z, t) is in U. By the Ricmann extension theorem f is holomorphic in {(z, t); I z I < r and JtzI < r'}.
Example 2 Let Al he the quotient of a symmetric bounded domain by an arithmetically defined discrete group r. Let Y be the Satake compactification of Af. If the action of r is free, then _11 and Y satisfy conditions (1), (2), (3) of Theorem 6.1. Even if the action is not free. Al and Y satisfy similar conditions (see Sec. 6 of Chapter VII). Theorem 6.2 applied to this example yields a simple proof of a result of Borel (see Kobayashi and Ochiai [1]). Although Y is a complex space with singularities, we shall see in Chapter VII that all the results in this section hold when 11f and Y are complex spaces.
CHAPTER VII
Generalization to Complex Spaces
1
Complex Spaces
We shall review quickly the definition and basic properties of complex spaces. For details and proofs, we refer the reader to Gunning and Rossi [11 and Narasimhan [1].
Let Il be an open set in C". A subset U of St is called an analytic set in Il if every point a E SE has a neighborhood Na such that U n Na is given as the common zeros of functions f i, ... , ff holomorphic in Na, i.e.,
UnNa={xENa;fi(x)=...= fp(x)=0}. If follows that U is closed in S2, that Il - U is dense in S2 if U # S2 and that S2 - U is connected if 0 is connected. Let S2 and Sl' be two open neighborhoods of a point. a E C's. Let U and
U' be analytic sets in 0 and S2', respectively. We write (0, U) - (i', U') at a if there exists an open neighborhood SZ" C SZ n S2' of the point a such that U n S2" = U' n f2". Clearly, - is an equivalence relation. We call an equivalence class an analytic germ at a. We denote the analytic germ defined by (SZ, U) at a by Ua. For each point a E Cn. let O",a denote the ring of germs of holomorphic functions at a. We denote by O" the sheaf of germs of holomorphic functions on Cn. Let U be an analytic set in S2 C Cn. At each point a E S2, we denote by
Ja = ,7(Ua) the set of germs of holomorphic functions in St vanishing on the germ Ua. Then ,7a is an ideal of On,a. If a U, then ,7a = On,a. We denote by ,7 = ,7(U) the sheaf of ideals J, , a E Q. of the analytic set U. i.e., ,7 = Uacn ,1a. It is subsheaf of O(S2) = O X52. Consider the quotient sheaf O(c)1O(U). Its stalk at a E 52 is given by On,aIJa; in particular, it is zero for a ¢ U. The restriction of the quotient sheaf O(c)I,7(U) to U, denoted by Ou, is called the sheaf of germs of holomorphic functions on U. A section of the sheaf OU over an open subset V 93
Generalization to Complex Spaces
94
of U is called a holomorphic function on V. This is equivalent to the following more elementary definition. A continuous, complex-valued function on V is said to be holomorphic if it is locally the restriction of a function holomorphic is Q.
An analytic germ Ua at a E S2 is said to be reducible if it is a union of two analytic germs at a, each of which is different from Ua. Then an analytic germ Ua is irreducible if and only if J(Ua) is a prime ideal of On,a. Every analytic germ Ua can be uniquely written (up to order) as a finite union Ua = Uk=l U,,,a of irreducible analytic germs U,,,a such that, for each v, U,,,a ¢ Up76 v U,,,a.
For an analytic set U in 52 C Cn, the local dimension dims U at a E i2 is defined as follows. If a ¢ U, then dim,, U = -1. If a E U, then dim,, U = n - s, where s = codima U is the dimension of a maximal dimensional linear subspace L C Cn through a such that a is an isolated point of L n U. WWe define dim U = MaxaEu dims U. If dima U = k for all a c U, then U is said to be pure k-dimensional. If U is irreducible, then U is pure k-dimensional for some k. Let U be an analytic subset in S2 C C'. A point a E U is called a regular point of U of dimension k if a has neighborhood N in S2 such that U n N is a k-dimensional submanifold of N. A point. a E U is said to be singular if it is not regular. The set of singular points of U forms an analytic subset
SofcsuchthatdimS<-- dim U - 1. A local geometric description of an analytic set is given as follows. Let U be an irreducible analytic set of dimension k in an open set S2 C C12.
Let a E U. Then there exists a local coordinate system z1, ... , z' in a neighborhood Dn = { I z` l < 1;i = 1,. .. , n} of a in i with the following properties:
(1) The point a is at the origin (0, . , 0) and is an isolated point of the subset ((0,..., 0, xk+l,... , zn) E U n D'} of U n Dn; . .
(2) If we set Dk =
{(xi.... ,xk);Iz:l < 1
for i = 1,...,k}, the neap
7r: U n Dn -* Dk defined by 1
-7rz ,
.
. ., z k, x k + 1
,
.
. ., x n) = (z 1 ,
.
.
.
, zk)
is surjective and proper; (3) There exists an analytic subset S of dimension -< k - 1 of U n D' such that S contains all singular points of U n Dn and 1r : U n Dn - S Dk - -7r(S) is a finitely sheeted covering space. [Moreover, 7r(S) is an analytic subset of Dk of dimension < k - 1.]
2
Invariant Distances for Complex Spaces
95
Let X be a Hausdorff topological space and 0 = Ox a subsheaf of the sheaf of germs of continuous functions on X. The pair (K, O) is called a complex space if every point a E X has an open neighborhood U such that U is an analytic set in an open set Q C C" and OI U is the sheaf of germs of holomorphic functions on the analytic set U. The sheaf 0 is called the structure sheaf of X. For any open subset U of X, the continuous sections of 0 over U are by definition the holomorphic functions on U. For the sake of simplicity, we often denote (X, 0) by X. A continuous map f : X --y Y of one complex space X into another, Y, is said to be holomorphic if f`(Oyf(a)) C Ox,a for every a E X, i.e., if h o f is a function holomorphic in a neighborhood of a e X whenever h is a function holomorphic in a neighborhood of f (a) E Y. Since a complex space is locally an analytic space, such local concepts as "regular point", "singular point", and "local dimension" can he defined in an obvious manner. A complex space X is reducible if it can be written as a union of two complex spaces, each of which is different from X. For a complex space X, its dimension dire X is defined to be the maximum of its local dimension. If its local dimension is k everywhere, X is said to be pure k-dimensional. If X is irreducible, its local dimension is a constant. 2
Invariant Distances for Complex Spaces
Let X he a connected complex space. Then we can define the Caratheodory pseudodistance cx and the invariant pseudodistance dx as in Chapter IV.
In defining dx, we have to use the fact that any two points p and q of X can be connected by a chain of analytic disks. To establish this fact, it suffices to prove that, given a (singular) point a of an irreducible analytic set U, there is a holomorphic mapping f of the unit disk D into U such that f (D) contains a and also a regular point of U. Assuming that U is a k-dimensional analytic subset in an open set S2 C C", we take a local coordinate system zl,.. , z" in 0 satisfying the three conditions (1), (2), and (3) of Sec. 1. Since T(S) is an analytic subset of dimension <- k - 1 in Dk, we may assume that the coordinate system satisfies the following conditions:
7r(S) n D' = {0),
where D1 _ (z', 0, ... , 0) C Dk.
By (1) we may assume, by taking a smaller neighborhood of a if necessary, that a is the only point of U which projects upon the origin 0 of Dk. By (3), a : it-l(D1) - {a} -' D1 - {0} is an s-sheeted covering projection, where s
Generalization to Complex Spaces
96
is a positive integer. Then it is not hard to see that there is a holomorphic mapping f : D -4 7r-'(D1) C U such that
ir[f(u')] = (w',0,...,0) E D' C Dk. Then f (0) = a and f (w) is a regular point for u; E D - {0}. A complex space X is said to be hyperbolic if dX is a distance. A hyperbolic space X is said to be complete if it is complete with respect to dX All the results in Chapter IV can be immediately generalized to complex spaces except those which make sense only for nonsingular complex manifolds (e.g., Theorems 3.4, 3.5, and 4.11). Similarly, the results in Chapter V can be also generalized to complex spaces. For some results in Secs. 2 and 3 of Chapter V, the following fact (see, for instance, Gunning and Rossi [1, p. 158]) plays an essential role. The space of holomorphic functions on a complex space is complete in the topology of uniform convergence on compact subsets. .
For a complex space X, the tangent space TT(X) at x E X can be defined as the space of derivations on the ring of germs of holomorphic functions at x (see, for example, Gunning and Rossi [1, p. 152]). It is then not difficult to generalize Theorem 3.3 of Chapter V to a complex space. 3
Extension of Mappings into Hyperbolic Spaces
Theorem 3.1 of Chapter VI may be generalized as follows:
Theorem 3.1 Let Y be a hyperbolic complex space and D* the punctured Y be a holomorphic mapping such that, for a unit disk. Let f : D* suitable sequence of points zk E D* converging to the origin, f (zk) converges
to a point yo E Y. Then f extends to a holomorphic mapping of the unit disk D into Y. Corollary 3.2 If Y is a compact hyperbolic complex space, then every holo-
morphic mapping f : D* - Y extends to a holomorphic mapping of D into Y.
Proof Let U be a neighborhood of yo in Y which is equivalent to an w1....,-wn be the coordinate analytic subset in an open set S2 of C. Let system in Cn. We may assume that yo is the origin of this coordinate system. Then the rest of the proof goes in the same way as in Theorem 3.1 of 11 Chapter VI.
3
Extension of Mappings into Hyperbolic Spaces
97
Theorems 4.1 and 5.2 of Chapter VI may be generalized as follows without any change in their proofs.
Theorem 3.3 Let Y be a compact hyperbolic complex space. Let X be a complex manifold of dimension in and A an analytic subset of X of dimen-
sion < in - 1. Then every holomorphic mapping f : X - A -> Y can be extended to a holomorphic mappings from X into Y.
Theorem 3.4 Let Y be a complete hyperbolic complex space. Let X be a complex manifold of dimension in, and let A be a subset which is nowhere dense in an analytic subset, say B, of X with dim B < in - 1. Then every holomorphic mappings f : X - A -, Y can be extended to a holomorphic mapping of X into Y. Corollary 3.5 Let Y be a complete hyperbolic complex space. Let X be a complex manifold of dimension in, and let A be a analytic subset of dimen-
sion < in - 2. Then every holomorphic mapping f : X - A -+ Y can be extended to a holomorphic of X into Y.
Similarly, Theorems 6.1 and 6.2 of Chapter VI may be generalized as follows:
Theorem 3.6 Let Z be a complex space and Y a complex subspace of Z satisfying the following three conditions:
(1) Y is hyperbolic; (2) The closure of Y in Z is compact; (3) Given a point p of Y - Y and a neighborhood U of p in Z, there exists a neighborhood V of p in Z such that V C U and the distance between Y fl (Z - U) and Y fl V with respect to dy is positive.
Then every holomorphic mapping f from the punctured disk D' into Y can be extended to a holomorphic mapping from the whole disk D into Z.
Corollary 3.7 Let Y and Z be as above. Let X be a complex manifold of dimension in and A a locally closed complex submanifold of dimension <-Y can be extended to a holomorphic mapping from X into Z.
to - 1. Then every holomorphic mapping f : X - A
We conclude this section by showing that, in theorems of the kind discussed here, the assumption that X is a nonsingular manifold is essential. Let Y be a projective algebraic variety in Pn(C). Let C(Y) be the affine cone of Y, i.e., the union of all complex lines through the origin of Cn+t
Generalization to Complex Spaces
98
representing the points of Y. We recall that Y is said to be projectivel\ normal if C(V) is a normal complex space. We make use of the following theorem (see Lang [1, p. 143]). If Yis a normal projective algebraic varicq. then Y can be imbedded into some PP(C) in such a way that Y is projectively normal, i.e., C(Y) is normal. In particular, let Y be a nonsingular projective algebraic manifold which is hyperbolic, e.g., a compact Riemann
surface of genus > 1. Then Y is projectively normal in some Pn(C) and C(Y) is nonsingular except at the origin. Let 7r : C(Y) - {0} -> Y be the restriction of the natural projection C"+1 - {O} -' PP(C). It is clear that -7r cannot be extended to a holoinorphic mapping from C(Y) into Y. This shows that Theorem 3.3 does not hold X = C(Y) and A = {0} and hence that the cone C(Y) is singular at the origin. We give another example due to D. Eisenman. Let M be a compact complex manifold and L a negative line bundle over M. According to Grauert [2], the space L/ obtained from L by collapsing the zero section to a point is a complex space. Let f be the projection from L - {zero section) onto Al. It is clear that f cannot be extended to a (continuous) mapping from L/M into M. To obtain a counter-example, all we have to do is to take any compact hyperbolic manifold, e.g., a compact Riemann surface of genus _? 2 as Al. This example shows also that if L is a negative line bundle over a com-
pact hyperbolic manifold, then the point of L/1l corresponding to the zero section of L is a singular point. For if it were a nonsingular point, f would be extended to a mapping from L/M into Al by Theorem 4.1 of Chapter VI. More generally, we may take a complex vector bundle E which is negative in a certain sense in place of a negative line bundle (see Grauert [2]). A similar reasoning shows that if we obtain a complex space by collapsing a complex subspace of a hyperbolic manifold to a point, then the resulting space has a singular point. This fact will be taken up in Chapter VIII. 4
Normalization of Hyperbolic Complex Spaces
A complex space X is said to be normal at a point a E X if the ring 0.x,,, of germs of holomorphic functions at a is integrally closed in its ring of quo-
tients. If X is normal at every point of X, then X is said to be normal. A normalization of a complex space X is a pair (Y, 7r) consisting of a normal complex space Y and a surjective holomorphic mapping r : Y -+ X such that (i) 7 , -: Y
X is proper and 7r-1(a) is finite for every a E X ;
(ii) If S is the set of singular points of X, then Y - it-1(S) is dense in Y and 7r : Y - it-1 (S) -, X - S is biholomorphic.
4
Normalization of Hyperbolic Complex Spaces
99
The normalization theorem of Oka (see Oka [2], Narasimhan [1, p. 118])
says that every complex space X has a unique (up to an isomorphism) normalization (Y, rr). The following result is due to Kwack [1].
Theorem 4.1 Let X and Y be complex spaces and let f : Y --+ X be a proper holomorphic mapping such that f-1(a) is finite for every a E X. If X is hyperbolic, so is Y. Corollary 4.2 If (Y, rr) is a normalization of a hyperbolic complex space X, then Y is also hyperbolic. Proof Since F is distance-decreasing, we have dr (p, q) > 0 if f (p)
f (q)
Let p and q be two distinct points of Y such that f (p) = f (q). Let V and W be disjoint open neighborhoods of p and q respectively such that
f-1[f(p)] = (V UW) n f-1[f(p)] Assuming dy(p,q) = 0, we shall obtain a contradiction. Let {yt} he a sequence of curves in Y joining p and q such that their lengths L}-(-y,1) measured in terms of dy satisfy
LY(?n) < 1n Let Un be the closed ball of radius 1/n around the point f (p) in X with respect to dx. Since f is distance-decreasing, f (yn) is contained in Uk for k S n. In other words, -yn C f-1(Uk)
if k < n.
Since V and W are disjoint and y,, is a curve from p E V to q E W, there exists a point pn on yn which is not in V U W. Since Uk is compact for sufficiently large k, so is f-1(Uk). Since pn E f-1(Uk) for k <_ n, it follows that the sequence {pn} has a subsequence which converges to a point, say po, of Y. Since pn ¢ (V UW) and VUW is open, the point pe is not in V UW .
On the other hand, f (pn) E Un and n Un = { f (p)}. Hence, f (po) = f (p). This implies po E f _'[f (p)] = (V U W) n f -1 [f (p)] C V U W, contradicting the statement above that po V U W.
100
5
Generalization to Complex Spaces
Complex V-Manifolds (Now Called Orbifolds)
Let G be a finite group of linear transformations of C'. Then the quotient space C"/G is, algebraically, an affine algebraic variety and, analytically, a, normal complex space (see H. Cartan [4]). A complex space X is called an n-dimensional complex V-manifold if every point p of X has a neighborhood U which is biholomorphic to a neighborhood of the origin in C"/G, where Gp is a finite group of linear transformations of C" (which depends on p It follows from the result of Cartan that a complex V-irlanifold is a normal complex space. The notion of V-manifold was introduced by Satake [1. and has been extensively investigated by Baily [1, 2]. .
Let Al he a complex manifold and G a finite group of holornorphic transformations of Al leaving a point o fixed. With respect to a suitable local coordinate system with origin at o, the action of C is linear. In fact, let B be the open ball of radius r around o with respect to a Riemannian metric invariant by G, where r is chosen so small that B may he considered as a hounded domain in C". Then C may be considered as a group c,f holornorphic transformations of a bounded domain B leaving a point o fixec:.
By a classical theorem of H. Cartan '[3] (see also Bochner and Martin [I the action of C is linear with respect to a suitable local coordinate system. Let Al he a complex manifold and I' a properly discontinuous grou:, holornorphic transformations of M. Since the isotropy subgroup of F at each point of Al is finite, the quotient space ,ll/I is a complex V-manifold. In particular, if Al is a hyperbolic manifold, e.g.. a hounded domain in C", all:l IF is a discrete subgroup of the group H(1.1) of holomorphic transformations of Al, then ,11/I is a complex V-manifold. In Secs. 4 and 5 of Chapter VI, we discussed the problem of extendin., a holornorphic mapping f : X - A Al to a holornorphic mapping front X into 11, where X is a complex manifold, A is an analytic subset of X. and Al is a hyperbolic manifold. As I have indicated in Sec. 3, the results of Secs. 4 and 5 of Chapter VI can be generalized to the case where .11 is hyperbolic complex space. On the other hand, the assumption that X is nonsingular complex manifold seems to be rather essential. It is clear t}ia' the results can be generalized to the case where X is a complex 11-manifolu 6
Invariant Distances on M/T
Let M be a complex manifold (or more generally, a complex space) and Ia properly discontinuous group of holomorphic transformations of Al. 11"e set Y = In addition to the pseudodistance dye defined earlier, we can
6
Invariant Distances on Al/I'
101
define another pseudodistance d' as follows. Let IT : Al -+ Y = Al/I' be the natural projection. A holomorphic mapping f from a complex space X into
y is said to he liftable if there exists a holomorphic mapping f : X Al such that f = IT o f. It is said to be locally liftable if every point of X has a neighborhood U such that the restriction of f to U is liftable. If X is simply connected, then a locally liftable mapping f : X Y is liftable. If the group r acts freely on Al so that Al is a covering space of Y, then every holomorphic mapping f : X , Y is locally liftable. In the definition of dy-. we used chains of holomorphic mappings from the unit disk D into Y. If we use only locally liftable holomorphic mappings from D into Y, we obtain
a new pseudodistance which will be denoted by dy. Since D is simply connected, to define dj, it suffices to consider only liftable holomorphic mappings from D into Y. In general, we have d' (p, q) ? dy (p, q)
for p, q E Y.
If r acts freely on Al so that IT : Al -* Y is a covering projection, then d'y = dy. The following proposition is trivial.
Proposition 6.1 Let Al be a complex space and r a properly discontinuous group of holom.orph.ic transformations of Al. Let f be a locally liftable holomorphic mapping from, a, complex space X into Y = A-11117. Then
dx(p,q)
for p,q E X.
The following proposition is a generalization of Proposition 1.6 of Chapter IV. The proof is similar. Proposition 6.2 Let Al be a complex. space and r a properly discontinuous group of holommorphic transformations of Al. Let IT : Al - Y = AI/1' be the natural projection. Let p, q c Y an d p, q E Al such. that i (q) = p and rr(p) = q. Then. dy.(p.q) q
where the infim:am. is taken over all q E .11 such that 7r(q) = q.
The proof of the following proposition is similar to that of Theorem 4.7 of Chapter IV.
Proposition 6.3 Let Al. F and Y be as above. Then d'. is a (complete.) distance if and only if Al is a (complete) hyperbolic space, i.e., if and only if dA.t is a (complete) distance.
102
Generalization to Complex Spaces
Many of the results in Chapters IV, V, and VI may be generalized to M/t and locally liftable holomorphic mappings. We mention one. Theorem 6.4 Let M be a hyperbolic complex space and r a properly discontinuous group of holom.orphic transformations. Let f be a locally liftable holomorphic mapping from the punctured disk D* into Y = M/1' such that, for a suitable sequence of points zk E D* converging to the origin, f (zk j
converges to a point po E Y. Then f extends to a holomorphic mapping from the unit disk D into Y.
CHAPTER VIII
Hyperbolic Manifolds and Minimal Models
1
Meromorphic Mappings
There are several definitions of a meromorphic mapping from a complex space into another complex space. Remmert [1] and Stoll [2] have made systematic studies on meromorphic mappings. A m.eromozphic mapping f from a complex space X into a complex space Y in the sense of R.emmert is a correspondence satisfying the following conditions:
(1) For each point x of X, f (x) is a nonempty compact subset of Y; (2) The graph F f = { (x, y) E X x Y; y E f (x) } off is a connected complex subspace of X x Y with dim F f = dim X ; (3) There exists a dense subset X ` of X such that f (x) is a single point for each x in X*. Denote by it the projection from X x Y onto X. Denote by 7r the restriction of it to the graph F f. Since f (x) = 7r 1(x) = F f f1 it(x), it follows that f (x) is a complex subspace of Y. Let E be the set of points of F f where 7r is degenerate, i.e.,
E={(.x,y)E['f;dimf(x)>0}. Let
N=7r(E)={xEX;dim f(x)>0}. Then E is a closed complex subspace of codimension > 1 of F f and N is a closed complex subspace of codimension > 2 of X (see Remmert [1]). It is then clear that f : X - N Y is a holomorphic mapping. From Corollary 3.5 of Chapter VII we obtain
Theorem 1.1 Let f be a meromorphic mapping from a complex manifold X into a complete hyperbolic space Y. Then f is holomorphic. Observe that X is assumed to be a nonsingular manifold. 103
Hyperbolic Manifolds and Minimal Models
104
If we crake use of Theorem 3.1 of Chapter VI and the property of a meromorphic mapping that. f (x) is nonempty, then we can actually show that every meromorphic mapping f from a complex manifold into a hyperbolic space (complete or not) is holomorphic.
Remark Let X and N be as above, and set X' = X - N. We indicate the proof that any holomorphic map f': X' , PrC extends to a meromorphic map f of X into P,,,C. In order to show that. every holomorphic line bundle 1;' on X' extend,
to a line bundle
on X, we first prove that every point p E X has a
neighborhood U such that l;'' is trivial on U' = U fl X'. Let U be a small coordinate ball neighborhood of p. Since N is of at least real codimension 4, we have H2(U', Z) = 0. We claim also that H'(U', St) = 0, where S2 is the sheaf of germs of holomorphic functions. Let w' be a a-closed (0, 1)-form on U'. Since codim N > 2, w' extends to a a-closed (0, 1)-form w on U.
Since H1(U,Il) = 0, we have w = af, which implies w' = af. From the exact sequence
H1(U', si) - H1(U', W) -, H2(U', Z)
...
we conclude H1(U', R*) = 0, showing that U has the desired property.
We cover X by such neighborhoods U;, and set U, = U; n X'. The transition functions g'ij: Ul n U,1 -+ C of l;' extends to g,j: U, fl U, - C, which become a transition function for i;. Let wo, w1, ... , w` be a homogeneous coordinate system for P7,,C. Let Va be the open set in PrC defined by w° 0. We set hal3 = wr3/w' on Va fl Va. Let rl he the line bundle over PmC defined by transition functions This bundle admits m + 1 linearly independent sections to, t1.... , t,,, .
Given f : X' PmC, set ' = f *77, and so, si, ...... s, be the sections of induced by to, tl, ... , t,n. The map f is given by p [so(p), si (p), ... , s;,, (p) p E X'. We extend l;' to a bundle l; over X and the sections so, si, ... , s;,, to sections so, s1..... s,,, of e. Then the map p F-+ 180 (p), sl (p), ... , s,n (p)], p E X, sends X meromorphically to P,,, C.
2
Strong Minimality and Minimal Models
We say that a complex space X is strongly minimal if every meromorphic mapping from a complex space U into X is holomorphic at every simple (i.e., nonsingular) point of U. This is the definition used by Weil [1, p. 2 V in showing that every Abelian variety is minimal.
2
Strong Minimality and Minimal Models
105
Let X and X' be complex spaces. We say that X and X' are bimeromorphic to each other if there exist meromorphic mappings f : X -+ X' and g : X' -+ X such that x E g1 f (x)]
for x E X and x' E f [g(x')] for x' E X'.
We consider the case where f : X -+ X' is holomorphic and hence single-
valued. Let N' = {x' E X'; dimg(x') > 01 and N = f-1(N'). Then, codim N > 1 and codim N' ? 2 (see Sec. 1). The restricted mapping f : X - N --+ X- N' is biholomorphic. We call g : (X', N') -+ (X, N) a monoidal transformation with center N' and its inverse
f:(X,N)-+(X',N') a contraction. For details on monoidal transformations and contractions we refer the reader to Moisezon [I]. Given a class of bimeromorphically equivalent complex spaces, a space Xo is called a minimal model of the class if, for every space X in the class, there is a contraction f : (X, N) --+ (Xo, No). It is clear that in the given class there is at most one minimal model.
It is clear that if X is a strongly minimal complex manifold, then it is the minimal model in its class of bimeromorphically equivalent complex manifolds.
Theorem 1.2 may be restated as follows:
Theorem 2.1 Every complete hyperbolic space is strongly minimal.
As we remarked at the end of Sec. 1, we may drop the assumption of completeness. But we are primarily interested in compact spaces. This theorem generalizes a result of Igusa [1] (every compact Kaehler manifold with negative constant holomorphic sectional curvature is strongly minimal) and a result of Shioda [1] (a complex manifold which has a bounded domain of C' as a covering manifold is strongly minimal). The following result implies that every complex torus is also strongly minimal.
Theorem 2.2 If X is a complex space which has a closed complex subspace X of CN as a covering space, then X is strongly minimal.
Proof Let U be a complex manifold of dimension m. Let f be a meromorphic mapping of U into X. We want to prove that f is a holomorphic
106
Hyperbolic Manifolds and Minimal Models
mapping from U into X. Since the problem is local with respect to the domain U, we may assume that U is a ball in C"' and that f is holomorphic on U - A where A is an analytic subset of dimension S m - 2. Then U - A is simply connected so that the holomorphic mapping f : U - A
X
can be lifted to a holomorphic mapping f : U - A -+ X C C''. Since f is given by N holorrrorphic functions, it can be extended to a holomorphic mapping f : U -+ C^' by Hartogs' theorem. Since X is closed, f maps U X, then irof is a holomorphic mapping of U into X which coincides with f on U - A. p into X. If we denote by zr the projection X
Theorem 2.1 implies that. a compact Kachler manifold with negative holomorphic sectional curvature is strongly minimal and hence is the minimal model in its class of bimeromorphically equivalent complex manifolds. We shall now show that a compact Kaehler manifold with negative R.icci tensor is the minimal model in its class of bimeromorphically equivalent complex manifolds. Let E be a holomorphic vector bundle over a compact complex mani-
fold Al, Let r(E) denote the space of holomorphic cross sections of E. If the restriction mapping a E r(E) --+ a(x) E Ez is surjective for each point, x of Al, then E is said to have no base points. Assuming that E has no base points, set r = dim ET,
k + r = dire r(E).
To each point .i of Al, we assign the kernel of the restriction mapping F(E) Ex, which is a k-dimensional subspace of r(E). In this way we obtain a holomorphic mapping of M into the complex Grassmann manifold
Gk,r(C) of k-planes in the (k + r)-dimensional vector space r (E). If this mapping M -+ Gk.r(C) is an imbedding, then the bundle E is said to be very ample.
If a (local) holomorphic section of E is a covariant holomorphic tensor field of M. then E is called a covariant holomorphic vector bundle over M. The bundle of complex (p, 0)-forms on Al is a covariant holomorphic vector bundle. In particular, the canonical line bundle, i.e., the bundle of (n.0'forms on M (where n = dim M), is a covariant holomorphic vector bundle. We prove first the following theorem.
Theorem 2.3 Let E be a very ample covariant holomorphic vector bundle over a compact complex manifold M. If f : M -+ Al is a meromorphic mapping which is nondegenerate at some point, then f is biholomorphic. In particular, every bimeromorphic mapping f of Al onto M is biholomorphic.
2
Strong Minimality and Minimal Models
107
Proof Let w E r(E). Since w is a covariant holomorphic tensor field on M, f induces a covariant holomorphic tensor field f *(w) on Al - N, where N is an analytic subset of codimension >_ 2 in M. Since f * (w) is given, locally, by a system of holomorphic functions, f * (w) extends to a holomorphic section
of E over M by Hartogs' theorem. We shall show that the linear mapping f* : I'(E) --* r(E) is injective. Let U and U' be open sets in M such that
f maps U biholomorphically onto U'. If w E F(E) and f*(w) = 0, then f * (w) vanishes identically on U and, consequently, w vanishes identically on U'. Since w is holomorphic, it vanishes identically on M. This proves our assertion. Since dim r(E) < oo by compactness of M, it follows that
f* : r(E) -> r(E) is a linear autoinorphism. This linear isomorphism f* induces an automorphism p of the Grassmann manifold Gk,, (C) of k-planes in r(E), where r = dim Ex and k + r = dim F(E) as above. We consider M as a complex subinanifold of Gk,r(C). Then f : M --+ M is the restriction Gk,r(C) to M and hence is biholomorphic. of cp-1 : Gk, (C)
Corollary 2.4 Let E be a very ample covariant holomorphic vector bundle over a compact complex manifold M. Then M is the minimal model in its class of bimeromorphically equivalent complex spaces.
Proof Let X be a complex space bimeromorphic to Al, Let f : X and g : M - X be meromorphic mappings such that x E g[f (x)]
for
Al
x E X and p E f [g(p)] for p c M.
We want to show that. f is holomorphic, i.e., single-valued. By Theorem 2.3,
f o g M -- LVI is biholomorphic and f [q(p)] = p for p E M. Since g : M X is surjective by x E g[f (x)], it follows that f is single-valued. 0 Example 1 Let M be a compact Kaehler manifold with negative Ricci tensor or, more generally, with negative first Chern class cl(M). From a result of Kodaira [1] it follows that a suitable positive power Kr" of the canonical line bundle K of Al is very ample. (The converse is a triviality.) Theorem 2.3 implies that a meromorphic mapping of M into itself which is nondegenerate at some point is a biholomorphic mapping of Al onto itself. This is a result of Peters [1]. By Corollary 2.4, Al is the minimal model in its class of meromorphically equivalent complex spaces. This is due to P. Kiernan, who has pointed out to me that Theorem 2.3 implies Corollary 2.4 immediately.
Hyperbolic Manifolds and Minimal Models
108
Example 2 Consider r nonsingular hypersurfaces V1,.. . , Vr of degree d1i ... , d, in Pn+r(C) respectively. Suppose that they are in general position so that the intersection M = V1 fl fl Vr is a non-singular manifold of dimension n. If we denote by h the second cohomology class of M corresponding to a hyperplane section of M, then a formula on Chern classes
(see Hirzebruch [1, p. 159]) implies c1(M) = (n + r + 1 - Ei=l d;)h. If > di > n + r + 1, then c1(M) < 0 and Theorem 2.3 and Corollary 2.4 apply to M. In particular, a nonsingular hypersurface M of degree d in Pn+i(C)
with d > n + 2 has c1(M) < 0 and is the minimal model in its class of bimeromorphically equivalent complex spaces.
3
Relative Minimality
Let X and X' be complex spaces which are bilneromorphic to each other with meromorphic mappings f : X --> X' and g : X', X such that
xEg[f(x)] for xEX and x'Ef[g(x')] for.x'EX'. In Sec. 2, we defined the notions of monoidal transformation and contraction. If f is holomorphic, i.e., if f : (X, N) --> (X', N') is a contraction, then we write
x>x'. The minimal model X0 in a class of himeromorphically equivalent complex
spaces satisfies, by definition, the inequality X > Xo for every complex space X in the class. On the other hand, a space X in a class of bimeromorphically equivalent complex spaces is said to be relatively minimal if
the class contains no space X' (other than X itself) such that X > X'. The minimal model may or may not exist but is unique (if it exists) in each class. On the other hand, relatively minimal models always exist but are not necessarily unique in each class of birneroniorphically equivalent complex manifolds (see Corollary 3.5 below). The following statement is obvious.
Theorem 3.1 The minimal model is relatively minimal in its class
of
bim.eromorphically equivalent complex spaces.
We shall find other sufficient conditions for a complex manifold to be relatively minimal. For this purpose, we review quickly results of Kodaira and Spencer [1] on divisor class groups. A divisor of an n-dimensional compact complex manifold M is a finite sum of the form E niVi, where ni E Z
3
Relative Minimality
109
and Vi is an (n - 1)-dimensional closed complex subspace of M. The set of divisors on M forms an additive group G(M), called the group of divisors. Every meromorphic function f on M defines a divisor (f) = (f )o - W001 where (f )o is the variety of zeros of f and (f),,,, is the variety of poles of f. The subgroup of G(M) consisting of divisors (f) of meromorphic functions f is denoted by G,(M). The factor group A(M) = G(M)/G1(M) is called the group of linear equivalence on M. Each divisor > ni Vi of M gives rise to a complex line bundle over M as follows. Let {Ua} be an open cover of M, where each Ua is sufficiently small. Let fai be a holomorphic function on Ua such that Ua n Vi is defined by fai = 0. If Ua fl Up # 0, we set
gaol = [J(fai/fsi)n` We associate to > niVi the line bundle defined by the transition functions {gap} and obtain a homomorphism of the group G(M) into the group F of line bundles over M, whose kernel is exactly Gl(M), and hence an isomorphism of A(M) = G(M)/G,(M) into F. It is known that if M is a closed complex submanifold of a complex projective space (i.e., if M is projective algebraic), then A(M) F. Given a complex line bundle l; E F with transition functions {g,,O} we define a 2-cocycle {ca,3y} of Al by log g
+ log 9i?-r + log 9tia = 2-7rical37,
caR7 E Z.
This induces a homomorphism F -* H2(M; Z), denoted by cl. The class e1 (C) E H2(M; Z) is called the first Chern class of C. Let P be the kernel of c1 : F -+ H2(M; Z); it is called the Picard variety attached to Al: In other words, P is the subgroup of F consisting of line bundles with trivial Chern class.
Let S2 denote the sheaf of germs of holomorphic functions on M. We define a homomorphism from H'(M; S2) onto P as follows. Given a 1-cocycle
{hap} representing an element of H1(M;c), we set gaj? = exp haf3.
Then {ga13} defines an element of P. We have now an exact sequence
0 --+ H1(M; Z) - H1(M; S2) -4P --+ 0,
where H1(M;Z) -* H1(M;c) is induced by m c Z --> 27rim E Q. Assume that Al is a compact Kaehler manifold. Then the homomorphism F --+ H2(M; Z) maps F onto H1,1(M; Z), where H1.1(M; Z) is
110
Hyperbolic Manifolds and Minimal Models
the subgroup of H2(M; Z) consisting of elements which are mapped into H1,1(M; C) under the natural map H2(M; Z) -+ H2(M; C). Thus we have
F/P Since H1(M;St)
H1"1(M; Z).
H°"1(M;C), we have
P
H°'1(M;C)/J
with
J= ( where He denotes the harmonic form representing the class c and FLT denotes the projection onto the space of (0,1)-forms.
1
Theorem 3.2 Let Al and Al' be compact complex manifolds of dimen-
sion n, F(M) and F(M') the groups of line bundles over Al and Al' respectively, and P(M) and P(M') the Picard varieties attached to Al and Al' respectively. Let f : (M, N) f* : F(M')
(Al', N') be a contraction. Then
F(M) is injective and maps P(M') into P(M), but the
image f*F(M') does not contain the line bundle over Al defined by the divisor mN for any nonzero integer in. Proof We first note that N is given as the set of zeros of the Jacobian of f so that codim N = 1. As we stated in Sec. 1, a result of Remmert implies that codim N' > 2. Let t;' be a line bundle over M' and let i; = f *t;' be the induced line bundle over Al. Assume that is a trivial line bundle. Then t; admits a holomorphic cross section a which never vanishes on Al. Since f : 1L1- N -* M'-N' is biholomorphic, f * induces an isomorphism between l;'I(A1' - N') and t;I(AM - N). Let a' be the holomorphic cross section of ;' over M' - N' corresponding to the section al (M - N). Since codim N' ? 2. we can extend a' to a holomorphic section, denoted again by a', over Al" by Hartogs' theorem. Since the set of zeros of a' on Al' is either an empty set or an analytic subset of codimension 1 and must be contained in N'. it must he empty. This shows that a' never vanishes on Al' and hence ;' is trivial. The colnmutativity of the diagram
0 --' P(M)
F(M) --f H2(Al; Z)
0 -+ P(M')
F(Al) - H2(Al'; Z)
T
implies that f* maps P(Al') into P(M).
3
Relative Minimality
111
To prove the last assertion of the theorem, let t; be the line bundle associated to the divisor mN and let a be a holomorphic cross section of whose zero set is precisely mN (multiplicity counted). Assume that e = f"i;', where i' is a line bundle over M'. Let a' be the holomorphic cross section of l;' over Al' - N' corresponding to cI(M - N). We extend a' to a holomorphic cross section over M. Again looking at the zeros of a' as above, we obtain a contradiction unless m = 0. Corollary 3.3 If M is a compact complex manifold with its Picard variety P(M) = 0 and its second Betti number b2(M) = 0, then M is relatively minimal in the class of compact complex manifolds bimeromorphic to M.
Proof Assuming the contrary, let (M, N) -+ (Al', N') be a contraction. Since P(M) = 0, we have P(M') = 0. From rank F(M) = rank[F(M)/P(M)] < b2(M) = 0, we obtain
rank F(M') = rank F(M) = 0. On the other hand, the bundles associated to the divisors ntN, m
0, are
nontrivial.
Corollary 3.4 If M is a compact Kaehler manifold with dimH1'1(M; C) = 1, then Al is relatively minimal in the class of compact Kaehler manifolds bimeromorphic to M.
Proof The proof is similar to that of Corollary 3.3. Assuming the contrary, (M', N') be a contraction, where Al' is a compact Kaehler let (Al, N) manifold. The following two facts imply the corollary immediately:
(1) rank[F(M')/P(M')] = rank H1'1(M'; Z) ? 1; C) = dim H1,0 (M; C) (2) dim P(Af) = dim = dim H" (11'i C) dim H01 I (M'; C) = dim P(M'). Since (1) is a well-known property of a compact Kaehler manifold Al', we prove only (2). Since M and Al' are compact Kaehler manifolds, we have H1°0(M; C) = the space of holomorphic 1-forms on Al, H1 "0(Af'; C) = the space of holomorphic 1-forms on M'.
112
Hyperbolic Manifolds and Minimal Models
It suffices to show that the map f' : C) --i H1.°(M; C) is an isomorphism, where f is the contraction (M, N) , (M', N'). Let w' he a holomorphic 1-form on M'. If f'w' = 0 on M, then w' = 0 on Al'- N' and hence w' = 0 on M' since codim N' ? 2. Let w be a holomorphic 1-forin on M. Consider the holomorphic 1-form on M- N' which corresponds to wI(:bf - N) under f and then extend it to a holomorphic 1-form w' on M'. Then f'w' = w on M - N and hence f'w' = w on M. Corollary 3.5 Let M be a compact (Kaehler) manifold. Then in the class of compact (Kaehler) manifolds bimeromorphic to M, there is one which is relatively minimal.
Proof If Al is not relatively minimal, let (M, N) -. (M', N') be a con-
traction. By Theorem 3.2, rank F(M) > rank F(M'). If Al' is not relatively minimal, let (M', Ni) (Al", N") he a contraction so that rank F(M') > rank F(Al"). Since rank F(M) is finite, this process must stop eventually. As we shall see shortly, a relative minimal model in the given class need not he unique (see Example 2 below).
Example 1 Let Al be a compact Kaehler manifold with positive sectiinui,l curvature (or more generally, with positive holomorphic bisectional curvature). It is known (Bishop and Goldberg [1], Goldberg and Kobayashi [1_) that dim H1 1(M; C) = 1. By Corollary 3.4, Al is relatively minimal in the class of compact Kaehler manifolds which are bimeromorphic to M.
Example 2 Let Al be a compact homogeneous Kaehler manifold of the form G/H, where G is a connected compact semisimple Lie group and H is a closed subgroup with 1-dimensional center. Every irreducible coinpact Hermitian symmetric space is such a manifold. Rom the exact homotopy sequence of the bundle G --+ G/H, it follows that b2(M) = 1. By Corollary 3.4, Al is relatively minimal in the class of compact Kaehler manifolds which are bimeromorphic to Al. According to Goto [1], Al = G/H is a rational algebraic manifold and hence is bimeromorphic to a projectiVP space.
(C) with n >- 3. More generally, let 141 = Vl'1 .. f 1;. be a nonsingular complete inter(C) with n > 3 as, in Example 2 section of closed hypersurfaces in
Example 3 Let Al he a nonsingular closed hypersurface in
3
Relative Minimality
113
of Sec. 2. Then b2(A1) = 1, (see, for example, Hirzehruch 11, p. 1611). By Corollary 3.4, M is relatively minimal in the class of compact Kaehler manifolds which are himerolnorphic to M. In Example 1 of See. 2 we saw that if M is a compact complex manifold
such that a suitable positive power K"` of its canonical line bundle K is very ample, then -1f is the minimal model. We prove now the following:
Theorem 3.6 Let M be a compact complex manifold such that a suitable positive power K"" of its canonical line bundle K has no base points (i.e., for each point p of M, there is a holomorphic cross section of K'' which does not vanish at p). Then M is relatively minimal in its class of bimeroMorphically equivalent complex manifolds.
Proof Let f : (M, N) -> (M', N') be a contraction, where M' is a complex manifold which is bimeromorphic to Al. As in the proof of Theorem 2.3, we
see that f induces a linear isomorphism from I'(K") onto I'(K'"), where K' denotes the canonical line bundle of M', I'(K") the space of holonlorphic cross sections of K" over M', and I'(K'") the space of holomorphic cross sections of K'" over Al. We claim that f is nondegenerate everywhere on M. In fact, if f is degenerate at a point p of Al, then f'(w') vanishes at
p for every w' E I'(K'"'). Since f' : I'(K") -+ I'(K'") is an isomorphism, this means that every w E I'(K"') vanishes at p, in contradiction to the assumption that K"' has no base points. Since f is nondegenerate everywhere, it is a covering projection from M onto Al'. On the other hand, f is bimeromorphic and hence must be hiholomorphic.
Example 4 Let 14 be a compact complex manifold with cl(11) = 0. If H1(M, S2) = 0 (where S1 is the sheaf of germs of holomorphic functions), in particular if Al .is Kaehlerian and bl(AI) = 0, then Al is relatively minimal in its class of himeromorphically equivalent compact complex manifolds. To prove our assertion, consider the exact sequence
0-3Z-->fi-,9'- O. where
* is the sheaf of germs of non vanishing holomorphic functions and Then this yields the 52' is given by following exact sequence:
the homomorphisln 9
H'(111.52) -, H1(M,SI') - H2('11,Z)
Since the homolnorphism H1(M, S2') -> I1.2(M, Z) maps a complex line bundle into its Chern class, a complex line bundle l; E H1(,11,c') with
114
Hyperbolic Manifolds and Minimal Models
ci (1;) = 0 comes from H' (?b1, Q). If we assume that H1 (M, l) = 0, then cl (C) = 0 implies that t; is a trivial bundle. Applying this result to the canonical line bundle K of Al, we see that cl(Al) = -cl(K) = 0 implies that K is trivial provided H'(M, S2) = 0. If K is trivial, then M admits a nonvanishing holomorphic section. Our assertion follows from Theorem 3.6.
Example 5 Let M = V, n
fl V, be a complete intersection of non_ singular hypersurfaces V ,7... , Vr in as in Example 2 of Sec. 2. Assume n ? 2 so that b1(ML1) = 0. If dl,... , dr are the degrees of V1,. .. , V,. respectively and satisfy the equality n+r+ 1 = dl + +dr, then M is relatively minimal in its class of bimeromorphically equivalent compact complex manifolds (see Example 2 of Sec. 2 and Example 4 above). In particular, a nonsingular hypersurfa.ce M of degree n + 2 in Pn+1 (C) with n >_ 2 is relatively minimal. Compare this with Example 2 of Sec. 2.
CHAPTER IX
Miscellany
1
Invariant Measures
Let M be a topological space with pseudodistance d. If p is a positive real number, then the p-dimensional Hausdorff measure p, is defined as follows. For a subset E of M, we set
pp(E) = sup inf E(8(Ei))r; E = U Et, 6(Ei) <
where 8(E) denotes the diameter of E. If M is a complex manifold, the invariant pseudodistances cm and dM defined in Chapter III induce Hausdorff measures on M. Since these pseudodistances do not increase under holomorphic mappings, the Hausdorff measures they define do not increase under holomorphic mappings. There are other invariant ways to construct intermediate dimensional measures on complex manifolds. For a systematic study of various invariant measures on complex manifolds, we refer the reader to Eisenman (1l. In this section, we shall briefly discuss invariant measures which may be considered as direct generalizations of ctif and dM. Let D he the open unit ball in C". The volume element defined by the Bergman metric of D,1 induces a measure on D,,, which will be denoted by p. Theorem 4.4 of Chapter II implies that every holomorphic mapping f : D D,, is measure-decreasing with respect to p, i.e.,
ti[ f (E)] < p(E) for every Borel set E in D.
Let M be an n-dimensional complex manifold. Given a Borel set B in M, choose holomorphic mappings fi : D,, Al and Borel sets E, in D, for i = 1.2.... such that B C U, f,(Ei). Then the measure /1tif(B) of B is defined by
knt(B) = inf
115
(Ei),
Miscellany
116
where the infimum is taken over all possible choices for fi and E. In analogy to CM we define another measure -yM as follows. For each Borel set B in Al, we set 'YA1(B) = sup li[f (B)]+
I where the supremunl is taken with respect to the family of holomorphic mappings f : M -> Dt1. From the definitions of these measures, the following proposition is evident.
Proposition 1.1 If f is a holomorphic mapping of a complex manifold ill into another complex manifold N of the same dimension, then /LN[f(B)] < µA1 (B)
and 'YN[f(B)] < ni(B)
for every Borel set B in M.
Corollary 1.2 If f : M N is biholomorphic, then l-LN[f(B)] = /M(B)
and -tN[f(B)] ='YM(B)
for every Borel set B in M. The following two propositions follow from the fact that a holomorphic mapping of Dn into itself is measure-decreasing with respect to p.
Proposition 1.3 For every Borel set B in a complex manifold Al, we have 'YA1(B) < IA1(B).
Proposition 1.4 For the unit ball Dn itself, both IUD,, and 'YD,, coincide with the measure p defined by the Bergman metric. The following proposition is trivial.
Proposition 1.5 Let M be an n-dimensional complex manifold.
(1) If p' is a measure on M such that every holomorphic mapping f : Dn
Al satisfies
p'[f (E)] !5 p(E) for every Borel set E in Dn, then µ'
(2) If p' is a measure on Al such that every holomorphic mapping
f : Al D satisfies p'(B) >_ p f (B)]
then p..'>iA1
for every Borel set B in Al.
I
Invariant Measures
117
We call a complex manifold M measure-hyperbolic if pM(B) is positive for every nonempty open set B in M. In contrast to Theorem 4.7 of Chapter IV, the following proposition is easy.
Proposition 1.6 Let M be a complex manifold and If f a covering manifold of M. Then 11 is measure-hyperbolic if and only if M is.
If M is a hounded domain in Cn, then yti1(B) is positive whenever B is a nonempty open set in lbf. From Propositions 1.3 and 1.6, we obtain Corollary 1.7 Let M be a complex manifold which has a bounded domain of C" as a covering manifold. Then lbf is measurre-hyperbolic. In analogy to Theorem 4.11 of Chapter IV, we have
Theorem 1.8 Let M be an n-dimensional Hermitian manifold with and Ricci tensor Rid such that Hermitian metric ds211 = 2 E is negative semi(i.e., the Hermitian matrix (Rid (Rid) 5 definite) for some positive constant c. Then M is measure-hyperbolic.
Proof By normalizing the metric, we may assume that, c = 1. If ds2 = 2 E h 1dz'd I denotes the Bergman metric in D,,, its Ricci tensor is given by -hid. By Theorem 4.4 of Chapter II, every holonlorphic mapping f Dn -* M is volume-decreasing with respect to the volumes defined ds2 and dsh1. If we denote by p' the measure on Al defined by ds 11, then Proposition 1.5 implies that p'(B) < p!.1(B) for every Borel set B in M. Our assertion is now clear.
In particular, every compact Hermitian manifold with negative definite Ricci tensor is measure-hyperbolic. Theorem 1.9 A compact complex manifold with negative first Chern class is measure-hyperbolic.
Proof Since such a manifold admits a volume element whose Ricci tensor is negative definite, the theorem follows from Lemma 4.1 of Chapter II.
Example 1 Let M = V, n . . ^ VV, be a complete intersection of nonsingular hypersurfaces V1,. .. , VV, of degrees dl , ... , d,. in Pn+r+i (C) such that
Miscellany
118
d, + - + d,. >_ n + r + 2. Then M is measure-hyperbolic (see Example 2 ill Sec. 2 of Chapter VIII).
Theorem 1.10 If a complex manifold M is hyperbolic, it is also measurehyperbolic.
Proof Let µ2n denote the 2n-dimensional Hausdorff measure defined by the distance dm. From Proposition 1.5 we obtain iM(B) R_! µ2.(B).
On the other hand, µ2n(B) is positive for every nonempty open set B (see Hurewicz and Wallman (l, Chapter VIIJ). Cl
Example 2 Let M' be a measure-hyperbolic complex manifold and let g (M', N') -+ (M, N) be a monoidal transformation (see Sec. 2 of :
Chapter VIII). Since the contraction f : (M, N) -> (lli', N') is measuredecreasing and, for every nonempty open subset U of M, f (U) contains a nonempty open subset of M', it follows that M is also measure-hyperbolic. This furnishes an example of a compact Kaehler manifold which is measurehyperbolic but is not relatively minimal. It shows also that the converse to Theorem 1.9 does not hold (see Example 1 in Sec. 2 of Chapter VIII).
Example 3 In connection with Theorem 1.9 and Example 2, we mention that an algebraic manifold of general type is measure hyperbolic (see Kobayashi and Ochiai [2]). 2
Intermediate Dimensional-Invariant Measures
Let M be a complex manifold of dimension n. Let B be a real k-dimensional differentiable manifold (with or without boundary) together with a differentiable mapping tp : B -+ M. For each positive integer m, we define a k-dimensional measure p(B, p),n of (B, ;p) as follows. Choose a countable open cover {Bi} of B, a differentiable mapping hi Bi -+ D, (the open
unit ball in C"'), and a holomorphic mapping fi such that
D, -+ M for each i
fiohi =;i!B, We denote by ds2 the Bergman metric of Dm. Then hi ds2 is a positive semidefinite quadratic differential form on Bi and hence induces a (possibly degenerate) volume element, i.e., a nonnegative k-form, on Bi, which will be
2 Intermediate Dimensional-Invariant Measures
119
denoted by vi. (The construction of the volume element from a Riemannian
metric can be applied to h; ds2 even when hi ds2 is degenerate. At the points where h* ds2 is degenerate, the volume element vi vanishes.) We set
µ(B, (p)m = inf E i
v=,
jai
where the infimum is taken with respect to all possible choices for {Bi, hi, fi}. The measure µ(B, p)m can be infinite. If we cannot find {Bi, hi, f} satisfying the condition fi o hi = cplai, then we set µ(B, (p),, = oo. We have clearly p(B, p)1 'e p(B, (P)2
{L(B, (p)3
.. .
If p(B) is contained in a compact subset of M and if m
n, then µ(B, co)m
is finite.
In analogy to the measure -ym defined in Sec. 1, we define another k-dimensional measure y(M, (p)m as follows. Choose a holomorphic mapping f : M Dm. Let v f be the volume element defined by cp* (f * ds2); it is a k-form vanishing at the points where W* (f * ds2) is degenerate. We set y(B, cp)m = sup f
of , B
where the supremum is taken with respect to all possible holomorphic mappings f : M -+ Dm. Clearly, we have y(B, (p)1 s y(B, IP)2 < y(B, 0)3 < ...
.
We shall see (Proposition 2.3) that y(B, cp)m is finite if W(B) is contained in a compact subset of M. If B is a real submanifold of M and cP is the injection of B into M, then we write y(B)m and y(B)m for µ(B, (p)m and y(B, p)m, respectively. The following proposition is evident.
Proposition 2.1 If f is a holomorphic mapping of a complex manifold M into another complex manifold N and if B is a real k-dimensional manifold with a mapping : B M, then lt(B, f o ip)m < µ(B, cp)m
and y(B, .f o p)m < y(B;,p)m.
Corollary 2.2 If f : M -+ N is biholomorphic, then p(B, f o o)m = IL(B, ip)m
and y(B, f o ip)m < y(B, cp)m.
Miscellany
120
Proposition 2.3 If M is a complex manifold and B is a real manifold with a mapping cp : B --+ M, then
'y(B, V), < -y(B, G)2 < ... < p(B,
1.
Proof It suffices to prove the following inequality for every pair of positive integers p and q: ?'(B, p)p < A(B, co),.
If we make use of the notations {Bi, hi, fi} and f used in the definitions of (p)q and -y(B, (p)p respectively, then we have the following commutative diagram:
Bi-"'+M4Dp.
hi \ 1fi Dq
Denote the metrics on D7, and D. by dsp and dsq, respectively. Their . . . 2 . 2 2 . * 2 (f dsp,) = hi fi (f dsp) = hi (.fi f dsp) hi (dsq); where the last inequality follows from the Schwarz lemma for f o fi : Dq Dp (see Corollary 4.2 of Chapter III). On the other hand, the volume elements vi and v f are induced, respectively, from h.i (dsq) and y ' (f' (Isp). Hence; v f < vi on Bi so that
Lvi
v. 113
0
The following proposition follows also from the Schwarz lemma for holo-
morphic mappings D,,, -* D and D --+ D,,,. Proposition 2.4 Let B be a real manifold with a mapping cp : B -j Dr, Then, for ni n, both p(B,'p),,, and -y(B, cp),,, coincide with the integral over B of the volume element defined by cp' ds2, where ds2 denotes the Bergman metric of D. So far, we have been using the expression "k-dimensional measure rather loosely. If, for each real k-dimensional manifold B with a mapping p : B - M. v(B, p) is a nonnegative real number (including the infinit,) and if v(B, yo) = inf E. v(Bi, yo), where the infimum is taken with respect to all countable open covers {Bi} of B, then we shall call v a k-dimensional measure on Al.
2
Intermediate Dimensional-Invariant Measures
121
Proposition 2.5 Let v be a k-dimensional measure on a complex manifold M of dimension n. Assume m > n.
(1) If v(B, f o 0) < p(B,V))m for every real k-dimensional manifold B with ip : B Dn and for every holomorphic mapping f : Dn --+ M, then v(B, cp) <= a(B, cp),n for every real k-dimensional manifold B with
cp:B->M; (2) If v(B, cp) > y(B, f o tp),n for every real k-dimensional manifold B with co : B then v(B, cp)
-+ M and for every holomorphic mapping f : M - Dn, y(B, eo)m for every real k-dimensional manifold B with
W:B --+ M.
Proof (1). Let {Ui} be a countable open cover of M with biholomorphic U. We set Bi = cp'1(Ui) and i = fti 1 o;p1 Bi. From mappings fi : Dn the assumption on v and Proposition 2.4 we obtain v(Bi,,p) = v(Bi, fi o 7'i) C p(Bi, ,bi)m =
L.
vi,
where vi is the volume element defined by 0i ds2. Hence
v(Bi, o) C i
Vi. d
JB,
Taking the infimums of the both sides with respect to all possible choices for {Ui}, we obtain easily the desired inequality. The proof of (2) is similar. 0 We shall say that a complex manifold M is (k, rrt)-hyperbolic if p(B, cp)n,
is different from zero for every real k-dimensional manifold B with an imbedding 5o: B M. The following proposition is easy to prove.
Proposition 2.6 Let M be a complex manifold and M a covering manifold of M. Then M is (k, m)-hyperbolic if and only if Al is. The proof of the following corollary is similar to that of Corollary 1.7.
Corollary 2.7 Let M be a complex manifold which has a bounded domain of Cn as a covering space. Then M is (k, m)-hyperbolic.
More generally, we have the following theorem, the proof of which is essentially contained in the proof of Theorem 1.9. Theorem 2.8 Every hyperbolic complex manifold is (k, in)-hyperbolic.
Miscellany
122
We shall now define norms in the kth homotopy group 7rk(M. XO) of a 'If complex manifold M. Let a be an element of 7rk(M, xo) and let cp : Sk
be a mapping representing a. We set M(a). = inf u(Sk, co)n1,
y(a)m = inf -y(5 k' P)""
where the infimums are taken with respect to all (p representing a. Since Sk is compact, µ(a),,, is finite if m is not less than the complex dimension of M. It is easy to verify
u(a + a')m < µ(a). + u(a')m, y(a + a')m < y(a)m + y(a )m for a, a' E 7rk(M, xo). If k = 1, then a + a' should be replaced by au', since 7rl (M, xo) need not be Abelian.
It is easy to see that
p(a)m > y(a),n
for a E 7rk(M, xo)
and
µ(f*a)m < u(a),,,
and y(f*a)m < 'Y(a)m
for every a E Irk(M, xo) and for every holomorphic mapping f of M into another complex manifold V.
Example 1 Let M = {z E C7s; 0 < r < jIzII < 1}. Then 7r2i_1(M, x0) = Z. Let a be the generator of 7r2r_1(M, xo). We claim that y(a)n is the volume of the hypersphere {z E C"; IIzII = r} with respect to the Bergman
metric of the unit ball Dn provided that n > 2. Let c : S2r-1 ---* M he a mapping representing a. Let f : M -* D be a holomorphic mapping and v f the volume element of Stn-1 defined by Ip* f * ds2, where ds2 is the Bergman
metric of D. Since n ? 2, f can be extended to a holomorphic mapping
of D into D. Since the extended mapping f : D D is distancedecreasing by the Schwarz lemma, it follows that co* f * ds2 S cp* ds2. If we denote by v the volume element of S2':-1 defined by Ip* ds2, then
y(S2n-1,
of = v. P)n =sup f IS2r -1 JStn-I
S2n-1(r) denote the sphere {z E Cn; IIzII = r}. Considering the mapLet S2n-1 -, rcp(x)/II:p(x)II E S2n-1(r), we see easily that the integral ping x E
fs,,,-, v is greater than the volume of Stn-1(r). On the other hand, we can find yo such that this integral is arbitrarily close to the volume of S2i-1(r). This completes the proof of our claim. This example shows that y(a),n is nontrivial sometimes.
2
Intermediate Dimensional-Invariant Measures
123
Similarly, we can define norms (or rather, pseudonorms) in the homology groups of a complex manifold. Both µ(B, W)m and -y(B, cp),,, can be defined when B is piecewise differentiable, e.g., a simplex. We can therefore define v(s)m and 'y(s)m when s is a differentiable singular simplex of M. If c = Fi nisi is a differentiable singular chain, we define
µ(c)m =
7lC)m = i
lnil7(si)m.
i
If a E Hk(M, Z), then we define
µ(a)7 = inf µ(c) c
-y(a)m = inf 7(c)m, c
where the infimums are taken with respect to all cycles c representing the homology class a. If B is a complex manifold and cp : B M is a holomorphic mapping in the definition of µ(B, cp)m given at the beginning of this section, it is natural to modify the definition of µ(B, cp)m by considering only holomorphic hi.
We obtain a slightly different measure µ(B, co);,, in this way. If B is a complex space, we can still define µ(B, cp);,, by ignoring the singular locus of B. The singular locus has a lower dimension anyway. In particular, if V is a complex subspace of M, then we can define µ(V);,,. If B is of complex
dimension r and if cp(B) is contained in a compact subset of M, then µ(B, p);,L is finite for m > r. Assume that M is a Hermitian manifold with metric dsh1 whose holomorphic sectional curvature is bounded above by a negative ' constant. We shall normalize ds,2L1 in such a way that every holomorphic mapping f Dm M is distance-decreasing, i.e., f' dsM= ds2. This is possible by Theorem 4.1 of Chapter III. In defining µ(B, w),,,, we choose a countDm and able open cover {Bi} of B and holomorphic mappings hi : Bi fi : Dm -i M such that fi o hi = '1 B,. Then
dsM = hi (f;
hi ds2
on Bi.
It follows that the volume element vi on Bi constructed from h* ds2 is bounded below by the volume element obtained from cp' dsM. This implies that µ(B, ;c')f71 is bounded below by the volume of B with respect to the volume element constructed from a positive semidefinite form cp' dsM. In particular, if V is a complex subspace of M, then µ(V ),,, is bounded below
Miscellany
124
by the volume of V with respect to the metric induced by dsM. Hence,
Proposition 2.9 Let M be a Hermitian manifold with holomorphic sectional curvature bounded above by a negative constant. Then µ(V);,, > 0 for every complex subspace V of positive dimension.
Let M be a compact complex manifold. Every r-dimensional closed complex subspace V of M is a 2r-dimensional cycle. A cycle of the form c = Eni[,, where each `, is a closed complex subspace of dimension r, is called an analytic r-cycle. We say that > niVi is positive if all ni's are positive. By considering only those elements of H2r(M; Z) which can be represented by analytic r-cycles, we obtain a subgroup H,(M; Z) of H2r(M; Z). For a E H,:(M; Z), we set
µ(a)n = inf
nilu(Vi)'M
where the infimums are taken with respect to all analytic r-cycles c E niVi representing a. Theorem 2.10 Let M be a compact Kaehler manifold with negative holom.orphic sectional curvature. If a E HH(M; Z) can be represented by a positive analytic r-cycle, then p(a)', is positive.
Proof Let c = E niVi be any analytic cycle representing a. Let w he the Kaehler 2-form of M. The volume of V with respect to d$1 is given by the integral fv w'. This integral does not exceed µ(t2);,, (if dsM is normalized as in the proof of Proposition 2.9). Hence, µ(c)m
-
Inilµ(Vi)m
Inil
r v,
ni
wr
wr.
v,
Since wr is closed, the integral f: wr depends only on a and not on c. Since a can he represented by a positive analytic cycle, this integral is positive. It follows that µ(c);,, is bounded below by a positive constant which depends oillV oil (.X.
It seems difficult to find criteria for p(a), to be positive. It would be of interest to investigate relationships between the pseudonorms defined above and the pseudonorm defined by Chern, Levine, and Nirenberg (see Nirenberg [1], Chern, Levine, and Nirenberg I1]). I suspect that their pseudonorm for Hk(AI; Z) lies between the two pseudonorms I introduced (see Sec. 3). here and is close to
3
3
Unsolved Problems
125
Unsolved Problems
In order to state some of the unsolved problems on hyperbolic manifolds, we introduce notions closely related to that of hyperbolic manifold. Let M and N be complex spaces and denote by Hol(N, M) the set of holomorphic mappings from N into M. A sequence fi E Hol(N, Al) is said to be compactly divergent if given any compact K in N and compact K' in 111, there exists j such that fj(K) fl K' = 0 for all i > j. Fix a metric o on M which induces its topology. Hol(N, 111) is said to be normal if every sequence in Hol(N, M) contains a subsequence which is either uniformly convergent on compact sets or compactly divergent. According to `Vu [2], M is said to be taut if Hol(N, lvi) is normal for every N. If Hol(N, Al) is equicontinuous for every N with respect to some metric 0 on M, then M is said to be tight. We shall say that a complex space Al is Caratheodory-hyperbolic or C-hyperbolic for short (respectively, complete C-hyperbolic) if there is a covering space 1%1 of Al whose Caratheodory pseudodistance cti1 is a distance (complete distance).
We say that a complex space Al admits no complex line if there is no holomorphic mapping from C into Al other than the constant maps. These various concepts are related in the following manner: C-hyperbolic
complete C-hyperbolic 2
complete hyperbolic 31
taut
i'
2' It
i,., 'Clbol1c tt4 no colnp 1ex l Ine 3'
1"
tight
The implications (3), (3'), and (1") have been proved by Kiernan [2] and Eisemnan [2] independently. The other implications are either trivial or have been proved in Chapter IV. A bounded domain which is not a domain of holomorphy provides an example to show that the converses to (1), (1'), and (1") are not true. D. Eisenman and L. Taylor have shown that the converse to (4) is not true by the following example. Let
111={(2,w)EC2;IzI<1,Izw1<1}-{(O,w);IwI1}. The mapping h (z, w) -4 (z, zw) maps Al into the unit bidisk, and is one-to-one except on the set z = 0. Let f : C lvi be a holomorphic :
mapping. Then ho of is a constant map. It follows that either f is a constant
map or f maps C into the set {(0, w) E M}. But this set is equivalent
126
Miscellany
to {w E C; Jwj < 1} which is hyperbolic. Hence f is a constant map in
either case. To see that M is not hyperbolic, let p = (0, b) E M with b # 0. We shall show that the pseudodistance dm (O, p) between the origin
0 and p is zero. Set pn = [(1/n), b]. Then dM(0, p) = limdM(0, pn). Let an = min[n, (n/Ibl)1/2]. Then the mapping t E D -- (ant/n, a,,bt)M maps 1 /an into pn. This shows that lim dm (0, pn) < lim dD (0, 1 /an) = 0. The following example by Kaup shows that the converse to (2') is not
true. Consider a principal bundle C2 - {0} over P1(C) and let L be the associated line bundle. L may be also obtained from C2 by blowing up the origin. Let B be the set of elements of L of length less than 1 with respect
to the natural Hermitian fiber metric. B may be obtained from the unit ball in C2 by blowing up the origin. It is a unit disk bundle over P1(C). Choose three points in P1 (C) and remove the closed disk of radius 2 from
the fibers over these three points. Then the resulting space is a simply connected hyperbolic manifold which is not C-hyperbolic. It would be natural to ask the following questions.
Problem 1 Is every taut manifold complete hyperbolic? (I believe that the answer is probably affirmative.) For each positive integer n, let 2
C
2 + 1) + 1
if n is even,
Cn+11 Jt\rn+31 2 J+1 ifnisodd. 2
Let M be the manifold obtained from Pn (C) by deletion of o(n) hyperplanes in general position. Wu [3] proved that M admits no complex line. The first
few values of a are. e(2) = 5, e(3) = 7, e(4) = 10, e(5) = 13. On the other hand, Kiernan [1] proved that if M is the manifold obtained from Fn(C) by deletion of only 2n hyperplanes in general position, then M admits a complex line and hence is not hyperbolic. Hence, in Wu's result, e(n) is the best possible for n = 2, 3. But it may not be the best possible for n > 4. Problem 2 Delete from Pn(C) e(n) hyperplanes in general position. Is the resulting manifold M hyperbolic? Is it possible to extend every holomorphic mapping from the punctured disk D* into M to a holomorphic mapping from D into Pn(C)? (For this question, see Theorem 6.1 of Chapter VI.) Is it also possible to lower e(n) to 2n + 1?
+ (zn)'1 + 1 = 0. Let M be Cn minus the hypersurface (z1)' + As Kiernan [1] pointed out, the holomorphic mapping f : C -4 M
3
Unsolved Problems
127
defined by f (t) = (t, (-1)1Idt, 0, ... , 0) is nonconstant. This implies that, if z0, Z1, ... , z denotes the homogeneous coordinate system for Pn (C), then minus the hypersurface (zl)d + - + (zn)d = 0 admits a complex line for any degree d. But this hypersurface is rather special. -
Problem 3 Let M be P,(C) minus a "generic" hypersurface of high degree d. Is Lvi hyperbolic? We may further ask questions similar to those in Problem 2. Problem 4 Is a generic hypersurface of large degree in P,,(C) hyperbolic? Is it at least without complex line? The surface (zo)d+(zl)d = (z2)d+(z3)d contains a rational curve Z0 = z2 = U,
Zl = Z3 = 'v.
Let X be a compact complex space and M a complex space admitting an analytic subset of a bounded domain in C" as a covering space. Let a E X. If f and g are holomorphic mappings from X into M such that f (a) = g(a) and f.(u) = g.(u) for all u E 7rl(X, a), then f = g (Borel and Narasimhan [1]). Actually, Borel and Narasimhan proved a little more.
Problem 5 It is probably possible to extend the result of Borel and Narasimhan to the case where M is C-hyperbolic. Will it be also possible to extend it to the case where M is hyperbolic? Problem 6 Is a compact C-hyperbolic or hyperbolic manifold projectivealgebraic?
In connection with the problem above, we pose
Problem 7 Classify the two-dimensional compact hyperbolic manifolds.
Problem 8 Investigate the structures of a homogeneous hyperbolic manifold. Is there any other than the homogeneous bounded domains in C"? If M = G/H is a homogeneous hyperbolic manifold, then M is noncompact by Theorem 2.1' of Chapter V and G has no nondiscrete center by Theorem 2.2 of Chapter V. (If a holomorphic vector field Z on M generates a one-parameter group belonging to the center of G, then iZ will also generate a global one-parameter group of transformations of M in contradiction to Theorem 2.2 of Chapter V unless Z = 0.)
Problem 9 Extend Theorem 6.2 of Chapter VI to the case where A is an analytic subset with singular points. We shall now elaborate on the statement made concerning the intrinsic seminorms of Chern, Levine, and Nirenberg in Sec. 2. Let u be a real C2
Miscellany
128
function on a complex manifold M. We define a real k-form, 1 < k <- 2n = 2 dim M, wk(u) as follows: w2p-1(u) = dcu A
(dd`u)P-1
W2p(u) = du A dcu A (dd`u)P-
where dC = i (a - a). We denote by Fk = Fk (M) the set of real C2 functions u satisfying the following three conditions:
(i) 0
Let B be a real k-dimensional differentiable manifold and (P : B -+ ?l1 a differentiable mapping. We define vk by
7k(B,yo) = sup J cp*[wk(u)] uEFk B
Then Vk is a k-dimensional measure on M in the sense defined in Sec. 2. If f is a holomorphic mapping from M into another complex manifold, then Vk(B, f o yo) !-5 vk(B, ip)
Problem 10 Does there exist a (universal) constant c so that -y(B, So).
C Vk(B,
)
la(B, ti )"
for k-dimensional B and mappings ;o: B -. M? In view of Proposition 2.5 it suffices to prove these inequalities in the special case where M is the open unit ball Dn in C'. To prove the second inequality, it would be necessary to obtain an estimate similar to the one proved by Chern, Levine, and Nirenberg [1]. Their basic inequality says that if K is an open subset with compact closure in Dn, then
fK2
+ any minor of {uk}I)dV <_ C
with C a fixed constant independent of u. Here uj = au/azi, u_,k = a2u/azj a2k, and dV represents the usual volume element in C".
Postscript
For the assertions made in this postscript without reference, the reader is advised to look up the Grundlehren book, Hyperbolic Complex Spaces. 1
dM and cm
I took it for granted that when a complex manifold Al is hyperbolic, the distance function dM defines the manifold topology of M. However, this is not completely trivial. For a Riemannian manifold it takes also a little argument to see that the Riemannian distance defines the manifold topology. Both dm and the Riemannian distance belong to a class of distance functions called "inner distance". It is the property of an inner distance that it defines the manifold topology. On the other hand, the Caratheodory distance cm may not define the manifold topology. In Sec. 2 of Chapter IV, we showed that for a polydisk M = Dn, CM coincides with dM. It is a deep result of Lempert that for any geometrically convex domain M in C", cm coincides with dM. As for Example 1 in Sec. 1 of Chapter IV, we have generally dx xx'((p, p ), (q, 4 )) = max{dx (p, q), dx, (P , q')},
for (p, p'), (q, q') E X x X'. Although a similar formula holds also for the Caratheodory pseudodistance, its proof by Jarnicki and Pflug is much harder. 2
Hyperbolicity with Partial Degeneracy
In this book, we mainly considered two extreme cases with respect to the intrinsic pseudodistance dx of a complex manifold X, namely the hyperbolic case and a few examples of X such as C and C` for which dx is identically zero. However, there are important non-hyperbolic complex manifolds X with non-trivial dx. 129
Postscript
130
The complement of 2n + 1 hyperplanes in general position in the complex projective space is known to he hyperbolic. Let X be the complement of only n + 2 hyperplanes in general position in PnC. Then X is not hyperbolic. However, d x (p, q) vanishes only when p and q belong to the so-called diagonal hyperplanes. I elaborate on this when n = 2. The reader is advised to draw the configuration while reading the following explanation. The complement of 5 lines in general position in the projective plane P2C is hyperbolic. (A line P2C is a projective line P1 C.) Let X be the complement of 4 lines Li, i = 1, 2, 3.4 in general position in P2C. The configuration has 6 vertices, pi, = Li n L;, 1 < i < j < 4. Consider, for examples, the line L12 through the vertices p12 and p34. As a line in X, L12 has two points missing, namely P12 and p34 since they are on the deleted lines. Since the projective line L12 minus two points is biholomorphic to C' = C - {0}, the distance dX (p, q) = 0 for any two points p, q E X n L12, (see Example 3 in Sec. 1 of Chapter IV). The same holds for the line L13 connecting P13, P24 and the line L1.1 connecting p14, P23 The three lines L12, L13 and L14 are called the diagonal lines of the given configuration. It is now clear that if p, q E L12 U L13 U L 14,
then dx (p, q) = 0. It is a non-trivial fact that dx (p, q) = 0 only for such a pair p, q. This result can he generalized to all n. The example above leads us naturally to the following definition. A complex space X is said to be hyperbolic modulo a closed subset A if dx (p, q) > 0 for distinct points p, q unless both p and q are in A. Although it is generally believed that a generic compact complex space is hyperbolic, concretely given compact complex spaces are often hyperbolic modulo a proper closed complex subspace A. 3
Hyperbolicity Criteria
In Theorem 4.11 of Chapter IV, we gave a differential geometric criterion for hyperbolicity in terms of negative holomorphic sectional curvature. Although this helps us to understand the geometric meaning of hyperbolicity, it is not practical to use in concrete examples.
The most powerful criterion for hyperbolicity is Brody's criterion. A complex space is said to be Brody-hyperbolic if it admits no nonconstant holomorphic maps from C. Every hyperbolic complex space is trivially Brody-hyperbolic (see Theorem 1.1 of Chapter V). An example of a domain in C2 Sec. 3 of Chapter IX shows that the converse is not true in general. The theorem of Brody says that for a compact complex space the converse holds.
4
Hyperbolic Imbedding
131
Brody's theorem, combined with Nevanlinna-Cartan theory of value distributions, seems to offer the best approach to the hyperbolicity question.
I conjectured that a generic hypersurface of PnC of large degree (> 2n + 1) is hyperbolic and its complement is hyperbolically imbedded in PnC. There have been a number of papers written in the past 30 years on this conjecture. I mention here only the paper of H. Fujimoto, from which the interested reader should be able to trace back other papers; Some con-
structions of hyperbolic hypersurfaces in Pn(C) in Advanced Study Pure Math. 42, Complex Analysis in Several Variables, 109-114, Math. Soc. Japan. It is an open problem to extend the theorem of Brody to the situation where the space X is hyperbolic modulo a subspace. 4
Hyperbolic Imbedding
The big Picard theorem is considered the pinnacle of the classical function theory. Theorem 6.1 of Chapter VI generalizes the big Picard theorem to higher dimension as extension theorems for holomorphic maps. The terminology hyperbolic imbedding did not exist at the time of writing of the first edition. However, the three conditions in Theorem 6.1 are precisely the conditions for hyperbolic imbeddedness. We define hyperbolic imbedding in the way currently stated.
Let Z be a complex space and Y a complex subspace with compact closure Y. We say that Y is hyperbolically imbedded in Z if for any two distinct points p, q e Y there exist neighborhoods Up and UQ of p and q such that dy (Up n Y, U. n Y) > 0. In applications, Z is usually compact, Y is the complement of a closed complex subspace of Z, (so that the boundary of Y is precisely this closed complex subspace). For example,
in the case of the classical big Picard theorem, Z = P1 C and Y is the complement of three points, say oc, 0 and 1. Then Theorem 6.1 reads as follows:
If Y is hyperbolically imbedded in Z, every holomorphic mapping f from the punctured disk D* into Y extends to a holomorphic mapping f from the disk D to Z. There are many other ways of defining hyperbolic imbedding, including the one which relates it Montel's theory of normal families. Here, we give a definition in terms of the relative pseudodistance dy,z. Let Fy,z c Hol(D, Z) be the family of holomorphic maps f : D -+ Z such
that f -1(Z - Y) is either empty or a singleton. Thus, f E Fy,z maps all of D, with the exception of possibly one point, into Y. The exceptional point
Postscript
132
is of course mapped into the boundary 8Y. Making use of holomorphic disks belonging to .Fy,z, we define a pseudodistance dy,z on Y. Since
Hol(D, Y) C Fy,z C Hol(D, Z), we have
dz < dy,z < dy,
where the first inequality holds on Y and the second inequality is valid on Y. Then Y is hyperbolically imbedded in Z if and only if dy,z(p, q) > 0 for all distinct pairs p, q E Y. Brody's criterion has been extended to a criterion for hyperbolic imbeddedness by Green, Zaidenherg and others. 5
Intrinsic Infinitesimal Pseudometrics
In Sec. 1 of Chapter I. we defined the Poincare metric d4 of the unit disk D. The distance function p on D is the integrated form of ds2 and, conversely.
dsv is the infinitesimal form of p. The intrinsic pseudodistance d x is a generalization of p. Then there is naturally a corresponding generalization 6D21 of i.e., the infinitesimal form of d y . We denote it by F,V; actually, FD is the generalization of dsD. The advantage of da over Fv is that d.y can be defined for singular complex spaces and infinite dimensional spaces while there are some technical problems in defining Fa for spaces other than non-singular complex manifolds.
It is not a trivial matter to rigorously prove that dx is the integrated form of Fk, and this was done by Royden. He used it also prove that on a Teichmiiller space T the Teichmuller distance coincides with dT. It is an important result with many applications. 6
Self-Mappings
Although we discussed automorphisms of a hyperbolic manifold in Chapter V, we have not considered more general self-mappings. Abate's book mentioned in Preface presents a comprehensive account on iterates of holomorphic self-maps in hyperbolic complex spaces. We mention here also J. E. Fornaess' Dynamics in. Several Complex Variables, CBMS 87 (1996). Amer. Math. Soc., which gives applications of the intrinsic distance and hyperbolicity to complex dynamics.
8
Principle for the Construction of dM
133
Finiteness Theorems In Chapter V, we proved that the automorphism group of a compact 7
hyperbolic complex manifold is finite. This is a beginning of more general finiteness theorems on Hol(X, Y) based on distance-decreasing property of holomorphic mappings. Various finiteness theorems for holomorphic maps and sections, have been obtained by Noguchi and others. Some of the finiteness theorems are motivated by the Mordell conjecture over function fields, proved by Grauert and Manin, and also by Lang's conjectures in Diophantine geometry. 8
Principle for the Construction of dM
The construction of dM is based on a very general and simple principle, which may be used to construct other quantities. Let C be the category of complex manifolds with holomorphic mappings as morphisms. The category C contains a "model object" D with Poincare distance p. For any object of C, i.e., any complex manifold M there are many niorphisms from D to M to connect any pair of points of M. That is all we needed to construct dM. So the same construction works for the category of complex spaces with singularities and also the category of almost complex manifolds. We may even consider the category of infinite dimensional complex spaces. In Chapter IX we considered the intrinsic measures or volume elements.
This is also based on the same principle with the unit ball with invariant volume element as model object. Compact measure hyperbolic manifolds are closely related to compact complex manifolds of general type. We can apply the same principle to real projective structures. Let P be the category of real manifolds with projective connections and P the subcategory of manifolds with (flat) projective structure, the morphisms being projective maps. As model object, we take the open interval I = 1). We define the distance function p in I by p(a, b) = I log(-1, 1; a, b)j,
where (-1, 1; a, b) denotes the cross ratio of -1.1, a, b. This is an analog of the Poincare distance. The interested reader is referred to my paper Projectively invariant distances for affine and projective structures. Banach Center Publ. 12, Warsaw 1984, 127-152. So far, we have not found such a construction in the conformal differential geometry.
Bibliography
Ahlfors, L. V.
[1] An extension of Schwarz's lemma, Trans. Am. Math. Soc., 43 (1938), 359-364. [2] Complex Analysis, McGraw-Hill; New York, 1953. Andreotti, A. [1] Sopra it problems dell'uniformizzazione per alcune classi di superficie algebriche, Rend. Accad. Nazi, Pisa, XI(4) 2 (1941), 111-127.
Andreotti, A., and Frankel, T. T. [1] The Lefschetz theorem on hyperplane sections, Ann. Math., 69 (1959), 713-717.
Andreotti, A., and Stoll, W. [1] Extension of holomorphic maps, Ann. Math., 72 (1960), 312-349. Baily, W. L., Jr. [1] The decomposition theorem for V-manifolds, Am. J. Math., 78 (1956), 862-888.
[2] On the imbedding of V-manifolds in projective space, Am. J. Math., 79 (1957), 403-430.
[3] On Satake's compactification of Vn, Am. J. Math., 80 (1958), 348-364. Baily, W. L., Jr., and Borel, A. [1] Compactification of arithmetic quotients of bounded symmetric domains, Ann. Math., 84 (1966), 442 528. Bergman, S.
[1] Uber die Kernfunktion eines Bereiches and ihr Verhalten am Rande, J. Reine Angew. Math., 169 (1933), 1-42; 172 (1935), 89-128. Bishop, R. L., and Goldberg, S. I. [11 On the second cohomology group of a Kaehler manifold of positive curvature, Proc. Am. Math. Soc., 16 (1965), 119-122. Bochner, S. [1] On compact complex manifolds, J. Indian Math. Soc., 11 (1947), 1-21. Bochner, S., and Martin. W. T. [1] Several Complex Variables, Princeton Univ. Press, Princeton, N.J., 1948.
135
136
Bibliography
Bochner, S., and Montgomery, D. [1] Locally compact groups of differentiable transformations, Ann. Math., 47 (1946), 639-653. [2] Groups on analytic manifolds, Ann Math., 48, 659 (1947). Borel, A., and Narasimhan, R. [1] Uniqueness conditions for certain holomorphic mappings, Inventiones Math., 2 (1967), 247-255. Bremermann, H. J. [1] Holomorphic continuation of the kernel function and the Bergman metric in several complex variables, in Lectures on Functions of a Complex Variable, Univ. of Michigan Press, Ann Arbor, 1955, pp. 349-383. [2] Die Characterisierung von Regularitatsgebieten durch pseudokonvexe Funktionen, Schriftenreihe Math. Inst. Univ. Munster, No. 5 (1951), pp. 1-92. Caratheodory, C. [1] Uber das Schwarzsche Lemma bei analytischen Funktionen von zwei kornplexen Veranderlichen, Math. Ann., 97 (1926), 76-98. [2] Uber die Geornetrie der analytischen Abbildungen, die durch analytische 1lrnktionen von zwei Veranderlichen vermittelt werden, Abhandl. Math. Sera. Univ, Hamburg, 6 (1928), 97-145. [3] Uber die Abbildungen, die durch Systeme von analytische Funktionen von mehreren Veranderlicher erzeugt werden, Math. Z., 34 (1932), 758-792. Cartan, E. [1] Sur les domaines bornes de 1'espace de n variables complexes, Abhandl. Math. Sem. Univ. Hamburg, 11 (1935), 116-162. Cartan, H.
(1] Sur les grouper de transformations analytiques, Actualites Sci. Ind., 9, (1935), Herrnann, Paris. [2] Les fonctions de deux variables complexes et le problerne de la
representation analytique, J. Math. Pures Appl., 10 (1931), 1- 114. [3] Sur les fonctions de plusieurs variables complexes, l'iteration des transformations interieures d'un domaine borne, Math. Z., 35 (1932), 760-773. [4] Quotient d'un espace analytique par un groupe d'autornorphismes, algebraic Geometry and Topology, Symp. in Honor of Lefschetz, Princeton Univ. Press, Princeton, N.J., 1957, pp. 90-102. [5] Quotients of complex analytic spaces, in Contributions to Function Theory. Tata Institute, Bombay, 1960, 1-15. Chern, S. S. [1]
Characteristic classes of Hermitian manifolds, Ann. Math., 47 (1946), 85-121.
[2]
[3]
On holomorphic mappings of Hermitian manifolds of the same dimension, Proc. Symp. Pure Math., Vol. 11 (1968), Entire Functions and Related Parts Analysis, pp. 157 170, Amer. Math. Soc., Providence, R. I. The integrated form of the first main theorem for complex analytic mappings in several complex variables, Ann. Math., 71 (1960), 536-551.
Bibliography
137
[4] Complex analytic mappings of Riemann surfaces I, Am. J. Math., 82 (1960), 323-337. [5] Holomorphic mappings of complex manifolds, Enseigment Math., 7 (1961), 179-187.
Chern, S. S., Levine, H. I., and Nirenberg, L. [1] Intrinsic norms on a complex manifold, in Collected Papers in Honor of K. Kodaira, Springer, Berlin, 1969. Dinghas, A.
[1] Ein n-dimensionales Analogon des Schwarz-Pickschen Flachensatzes fur holomorphe Abbildungen der komplexen Einheitskugel in eine Kahler-Mannigfaltigkeit, Arbeitsgemeinschaft fur Forschung des Landes Nordrhein-Westfalen, 33 (1965), 477-494.
[2] Uber das Schwarzsche Lemma and verwandte Satze, Israel J. Math., 5 (1967), 157-169. [3] Verzerrungssatze bei holomorphen Abbildungen von Hauptbereichen auto-
morpher Gruppen mehrerer komplexer Veranderlicher in eine KahlerMannigfaltigkeit., Sitzber. Heidelberg. Akad. Wiss. (1968), 1-21. Eisenman, D. [1] Intrinsic measures on complex manifolds and holomorphic mappings, Mem. Am. Math. Soc., No. 96 (1970).
[2] Holomorphic mappings into tight manifolds, Bull. Am. Math. Soc., 76 (1970), 46-48. Ford, L. R.
[1] Automorphic Functions, McGraw-Hill, New York, 1929. Cindikin, S. C., Pjateckii-Sapiro, 1. I., and Vinberg, E. B. [1] Homogeneous Kaehler manifolds, in Geometry of Homogeneous Bounded Domains, C.I.M.E., 3° Ciclo, Urbino, Italy, 1967, pp. 3-87. Goldberg, S. I., and Kobayashi, S. [1] On holomorphic bisectional curvature, J. Diff. Geometry, 1 (1967), 225-233. Coto, M. [1] On algebraic homogeneous spaces, Am. J. Math., 76 (1954), 811-818. Grauert, H. [1] Characterisierung der Holomorphiegebiete durch die volLstandige Kahlersche Metrik, Math. Ann., 131 (1956), 38-75. [2] Uber Modifikationen and exzeptionelle analytische Mengen, Math. Ann., 146 (1962), 331-368. Grauert, H., and Reckziegel, H.
[1] Hermitesche Metriken and normale Familien holomorpher Abbildungen, Math. Z., 89 (1965), 108-125. Grauert, H., and Remmert, R. [1] Komplexe Raume, Math. Ann., 136 (1958), 245-318.
138
Bibliography
Gunning, R. C., and Rossi, H. [1] Analytic functions of
several Englewood Cliffs, N.J., 1965.
complex
variables,
Prentice-Hall,
Hahn, K. T., and Mitchell, J. [1] Generalization of Schwarz-Pick lemma to invariant volume in Kahler manifolds, Trans. Am. Math. Soc., 128 (1967), 221-231. [2] The same title, II, Can. J. Math., 21 (1969), 669-674. Hano, J., and Kobayashi, S. [1] A fibering of a class of homogeneous spaces, Trans. Am. Math. Soc., 94 (1960), 233-243. Hawley, N. S. [1] A theorem on compact complex manifolds, Ann. Math., 52 (1950), 637-641. Heins, M. [1] Selected Topics in the Classical Theory of Functions of a Complex Variable, Holt, Rinehart & Winston, New York, 1962. Hirzebruch, F. [1] Topological methods in algebraic geometry, Springer, Berlin, 1966.
Horstmann, H. [1] Caratheodorysche Metrik and R.egularitatshullen, Math. Ann., 108 (1933). 208-217.
Hua, L. K. [1] Harmonic analysis of functions of several complex variables in the classical domains, in Translations of Mathematical Monographs, Vol. 6, Americait Mathematical Society, Providence, R. I. (1963). Huber, H. [11 Uber analytische Abbildungen von Ringgebieten in Ringgebiete, Compos. Math., 9 (1951), 161-168. [2] Uber analytische Abbildungen Riemannscher Flachen in sich, Comment. Math. Helv., 27 (1953), 1-72. Hurewicz, W., and Wallman, H. [1] Dimension Theory, Princeton Univ. Press, Princeton, N.J., 1942. Hurwitz, A. [1] Uber algebraische Gebilde mit eindeutigen Transformationen in sich, Matfi. Ann., 41 (1893), 403-442. Igusa, J.
[1] On the structure of a certain class of Kahler manifolds, Am. J. Math., 76 (1954), 669-673.
Jenkins, J. A. [1] Some results related to extremal length, Ann. Math. Studies, No. 30 (1953), pp. 87-94.
Bibliography
139
Kaup, W.
[1] Reele T ansformationsgruppe and invariante Metriken auf komplexen Raumen, Investiones Math., 3 (1967), 43-70. [2] Holomorphe Abbildungen in hyperbolische Rdume, Centro Internazionale Mat. Estivo, 1967, pp. 111-123. [3] Hperbolische komplexe Raume, Ann. Inst. Fourier (Grenoble), 18 (1968), 303-330.
Kiernan, P. J. [1] Hyperbolic submanifolds of complex projective space, Proc. Am. Math. Soc., 22 (1969), 603-606. [2] On the relations between taut, tight and hyperbolic manifolds, Bull. Am. Math. Soc., 76 (1970), 49-51. [3] Quasiconformal mappings and Schwarz's lemma, Trans. Am. Math. Soc., 147 (1970). [4] Some remarks on hyperbolic manifolds, Proc. Am. Math. Soc., 25 (1970). Kobayashi, S. [1] Geometry of bounded domains, Trans. Am. Math. Soc., 92 (1959), 267-290. [2] Volume elements, holomorphic mappings and the Schwarz lemma, in Proc. Symp. Pure Math., Vol. 11 (1968), Entire Functions and Related Parts Analysis, pp. 253-260. Amer. Math. Soc., Providence, R. I.
[3] Distance, holomorphic mappings and the Schwarz lemma, J. Math. Soc. Japan, 19 (1967), 481-485. (4] Invariant distances on complex manifolds and holomorphic mappings, J. Math. Soc. Japan, 19 (1967), 460-480. [5] On the automorphism group of a certain class of algebraic manifolds, Tohoku Math. J., 11 (1959), 184-190. Kobayashi, S., and Nomizu, K. [1] Foundations of differential geometry, in Interscience Tracts, No. 15, Vol. I (1963), Vol. 11 (1968). Wiley, New York.
Kobayashi, S., and Ochiai, T. [1] Satake compactification and the great Picard theorem, J. Math. Soc. Japan, to appear. [2] Mappings into compact complex manifolds with negative first Chern class, J. Math. Soc. Japan, to appear. Kodaira, K. [1] On Kahler varieties of restricted type, Ann. Math., 60 (1954), 28-48.
Kodaira, K., and Spencer, D. C. [1] Groups of complex line bundles over compact Ki hler varieties. Divisor class groups on algebraic varieties, Proc. Natl. Acad. Sci. U.S.A., 39 (1953), 868-877.
Koranyi, A. [1] A Schwarz lemma for bounded symmetric domains, Proc. Am. Math. Soc., 17 (1966), 210-213.
Bibliography
140
Koszul, J. L. [1] Sur la forme Hermitienne canonique des espaces homogenes complexes, Can. J. Math., 7 (1955), 562-576. Kwack, M. H. [1] Generalization of the big Picard theorem, Ann. Math., 90 (1969), 9-22.
Landau, H. J., and Osserman, R. [1] Some distortion theorems for multivalent mappings, Proc. Am. Math. Soc., 10 (1959), 87-91. [2] On analytic mappings of Riemann surfaces, J. Anal. Math., 7 (1959/60), 249-279.
Lang, S.
-
[1] Introduction to algebraic geometry, in Interscience Tracts, No. 5 (1958). Wiley, New York. Lelong, P.
[1] Domaines convexes par rapport aux fonctions plurisousharmoniques, J. Anal. Math., 2 (1952), 178-208. [2] Fonctions plurisousharmoniques et fonctions analytiques de variables reeles, Ann. Inst. Fourier (Grenoble), 11 (1961), 515 562. Lichnerowicz, A. [1] Varietes complexes et tenseur de Bergmann, Ann. Inst. Fourier (Grenoble), 15 (1965), 345-407. Moisezon, B. G.
[1] On N-dimensional compact complex varieties with N algebraically independent meromorphic functions, Am. Math. Soc. T1 ansl., Ser. 2, 63 (1967), 51-177. Narasimhan, R. [1] Introduction to the theory of analytic spaces, Lecture Notes in Mathematics, No. 25 (1966), Springer, Berlin, Nirenberg, L. [1] Intrinsic norms on complex analytic manifolds, Ist. Nazl. Alta Mat. Symp. Mat., II, 227 (1968). Oka, K. (1) Domaines pseudoconvexes, Tohoku Math. J., 49 (1942), 15-52. [21 Sur les fonctions analytiques de plusieurs variables, VII. Lemme fondamental, J. Math. Soc. Japan. 3 (1951), 204-214; 259-278. Ostrowski, A. [1] Asymptotische Abschatzung des absoluten Betrage einer Funktion, die die Werte 0 and 1 nicht annimt. Comment. Helv. Math., 5 (1933), 55-87. Peters, K. [1] Uber holomorphe and meromorphe Abbildungen gewisser kompakter komplexer Mannigfaltigkeiten, Arch. Math., 15 (1964), 222-231.
Bibliography
141
Pfluger, A. [1] Uber numerische Shranken im Schottky'schen Satz, Comment. Math. Helv., 7 (1935), 159-170. Poincare, H. [1] Sur un theoreme de M. Fuchs, Acta Math., 7 (1885), 1-32.
Pyatetzki-Shapiro, I. I. [1] Geometric des domaines classiques et theorie des fonctions automorphes, Dunod, Paris, 1966. Rieffen, H. J. [1] Die diflerentialgeometrischen Eigenschaften der invarianten Distanzfunktion von Caratheodory, Schrift Math. Inst. Univ. Munster, No. 26 (1963). [2] Die Caratheodorysche Distanz and ihre zugehorige Differentialmetrik, Math. Ann., 161 (1965), 315-324. Remmert, R. [1] Holomorphe and meromorphe Abbildungen komplexer Raume, Math. Ann., 133 (1957), 328-370. Sampson, J. H. [1] A note on automorphic varieties, Proc. Natl, Acad. Sci. U.S.A., 38 (1952), 895-898.
Satake, I.
[1] On a generalization of the notion of manifolds, Proc. Natl. Acad. Sci. U.S.A., 42 (1956), 359- 363. [2] On the compact ification of the Siegel space, J. Indian. Math. Soc., 20 (1956), 259-281. Schiffer, M. [1] On the modulus of doubly connected domains, Quart. J. Math., 17 (1946), 197-213. Schottky, F.
[1] Uber den Picardschen Satz and Borelschen Ungleichungen, Sitz. Ber. Preuss. Akad. Wiss. (1904), 1244-1263. Schwarz, H. A. [1] Uber diejenigen algebraischen Gleichungen zwischen zwei veranderlichen Grossen, welche ein Schar rationaler eindeutige umkehrbarer Transformat.ionen in sich selbst zulassen, Crelle's J., 87 (1879), 139-145. Shioda, T.
[1] On algebraic varieties uniformizable by bounded domains, Proc. Japan Acad., 36 (1963), 617-619. Springer, G. ,1] Introduction to Riemann Surfaces, Addison Wesley, Reading, Mass., 1957. Stoll, W. [11 About the universal covering of the complement of a complete quadrilateral, Proc. Am. Math. Soc.. 22 (1969), 326-327.
Bibliography
142
[2] Uber meromorphe Abbildungen komplexer Raume I & 11, Math. Ann., 136 (1958), 201-239; 272-316. Tashiro, Y. [1] The curvature of the Bergman metric, Sci. Rept. Tokyo Kyoiku Daigaku, 8 (1963), 280-296. van Dantzig, D., and van der Waerden, B. L. (1] Uber metrisch homogenen Raume, Abhandl. Math. Sem. Univ. Hamburg, 6 (1928), 374-376. Vinberg, E. B., and Gindikin, S. G. [1] Kaehlerian manifolds admitting a transitive solvable automorphism group, Mat. Sb. (English Transl.), 74(116) (1967), 333-351. Vinberg, E. B., Gindikin, S. G., and Pyatetzki-Shapiro, I. I. [1] Classification and canonical realization of complex bounded homogeneous domains, Trans. Moscow Math. Soc., 12 (1963), 404-437. Weil, A.
(1] Varietes abeliennes et courbes algebriques, Actualites Sci. Ind. (1984). Hermann, Paris. Wu, H.
[1] Mapping of Riemann surfaces (Nevanlinna theory), Proc. Symp. Pure Math., Vol. 11 (1968), Entire Functions and Related Parts Analysis, pp. 480-532.
[2] Normal families of holornorphic mappings, Acta Math., 119 (1967), 193-233.
(3] An n-dimensional extension of Picard's theorem, Bull. Am. Math. Soc., 75 (1969),1357-1361.
[4] The Equidistribution Theory of Holomorphic Curves, Princeton Univ. Press, Princeton, N.J., 1970.
Summary of Notations
We summarize only those notations that are used most frequently throughout this monograph.
C Cn D D"
Dn Dn PP(C) P d11
CM
The field of complex numbers. Vector space of n-tuples of complex numbers (z1, ... , zn). Open unit disk in C, i.e., {z E C; IzI < 1}.
Punctured disk D - {0}. Polydisk D x ... x D (n times). Open unit ball in Cn, i.e., { ( z ' , . .. , zn); E1zjJ2 < 1}. Complex projective space of dimension n. Poincare distance (non-Euclidean distance) in D. the pseudodistance of M defined in Sec. 1 of Chapter IV. the Caratheodory pseudodistance of M defined in Sec. 2 of Chapter IV.
143
Author Index
Numbers after names show the pages on which their papers are quoted.
A Ahlfors, L. V., 3, 6, 11 Andreotti, A., 69, 88
Grauert, H., 6, 11, 40, 98 Gunning, R. C., 93, 96
B Baily, W. L., Jr., 100 Bergman, S., 18 Bishop, R. L., 112 Bochner, S., 68, 69, 100 Borel, A., 127 Bremermann, 11. J., 56, 74
Hahn, K. T., 30, 32, 33 IIano, J., 18 Hawley, N. S., 69
H
Heins, M., 11 Hirzebruch, F., 108, 115 Horstmann, H., 55
IIua, L. K., 33 Huber, H., 15, 81 Hurewicz, W., 118 Hurwitz, A., 69
C Carathedory, C., 52, 74 Cartan, E., 33 Cartan, H., 69, 74, 100
I
Chern, S. S., 19, 20, 32, 124, 128
Igusa, J., 105
D
J
Dinghas, A., 32
Jenkins, J. A., 15
E
K
Eisenman, D.. 115, 125
Kaup, W., 69, 74 Kiernan, P. J., 61, 63, 65, 77,
F
125, 126
Ford, L., R., 6
Kobayashi, S., 17, 18, 19, 32, 37, 39, 56, 65, 67, 69, 70, 91, 112, 118 Kodaira, K., 25, 107, 108 Koranyi, A., 42 Koszul, J. L., 18 Kwack, M. H., 81, 84, 88, 99
G Gindikin, S. C., 28, 31 Goldberg, S. I., 39, 112 Coto, M., 115 145
Author Index
146
R
L
Landau, H. J., 15, Lang, S., 98
Reckziegel, H., 6, 11, 40, Reiffen, H. J., 52
Lelong, P., 74 Levine, H., 124, 128 Lichnerowicz., A., 18
Remmert, R., 103 Rossi, H., 93, 96
M Martin, W. T., 100 Mitchell, J., 30, 32, 33
Sampson, J. 11., 69 Satake, I., 100 Schiffer, M., 15 Schottky, F., 10 Schwarz, H. A., 69 Shioda, T., 105 Spencer, D., C., 108 Springer, G., 12 Stoll, W., 88, 103
S
Moisezon, B. G., 105 Montgomery, D., 68
N Narasimhan, R., 84, 93, 99, 127 Nirenberg, L., 124, 128 Nomizu, K., 17, 19, 37, 70
0 Ochiai, T., 91, 118 Oka, K., 74, 99 Osserman, R., 15 Ostrowski, A., 11
T Tashiro, Y., 33
V van Danzig, D., 67 van der Waerden, B. L., 67 Vinberg, E. B., 28, 31
P Peters, K., 25, 107, Pfluger, A., 11 Poincar6, H., 69 Pyatetzki-Shapiro, 1. 1., 18, 28, 31
W Wallman, H., 118 Weil, A., 104 Wu, H., 24, 69, 74, 125, 126
Subject Index
A
G
ample, 106 analytic germ, 93 analytic polyhedron, 53 generalized, 54 analytic set, 93 reducible, 94
generalized analytic polyhedron, 54
group of divisors, 109 group of linear equivalence, 109
H Hausdorff measure, 115 Hermitian fiber metric, 37 Hermitian vector bundle, 37 holomorphic bisectional curvature,
B base point, 106 Bergman kernel form, 18 Bergman kernel function, 18 Bergman metric, 18 big Picard theorem, 77 bimeromorphic, 105
39
holomorphic sectional curvature, 39
holomorphically convex, 55 hyperbolic, 56
C
C-, 125
C-hyperbolic, 125 Caratheodory hyperbolic, 125 Caratheodory pseudodistance,
(k, m)-, 121 measure-, 117
49
K
compactly convergent, 125 complete metric space, 53 complex space, 95 complex V-manifold, 100 contraction, 105 convex hull, 55 convex with respect to F, 55 covariant vector bundle, 106
k-dimensional measure, 120 (k, m)-hyperbolic, 121 L
liftable (mapping), 101 locally, 101 linear equivalence, 109 group of, 109
D
little Picard theorem, 67
dimension of a complex space, 95 divisor, 108
local dimension, 94 locally liftable, 101
group of -s, 109
147
Subject Index
148
M measure hyperbolic, 117 meromorphic mapping, 103 minimal, 104, 108 relatively, 108 strongly, 104 minimal model, 105 monoidal transformation, 105
R reducible, 94, 95 analytic set, 94 complex space, 95 regular point, 94 relatively minimal, 108 Ricci tensor, 17 S
N negative curvature, 38 negative Ricci curvature, 38 no complex line, 125 normal complex space, 98 normal family, 129 normal point, 98 normalization, 98
P Picard theorem, 67, 77 big, 77 little, 67
Schottky's theorem, 10 Schwarz lemma, 1 sheaf of germs of holomorphic functions, 93 Siegel domain of second kind, 28 affinely homogeneous, 31 automorphism of, 31 singular point, 94 spread, 59 strongly minimal, 104 structure sheaf, 95
T
Poincare (-Bergman) metric, 3 positive curvature, 38 positive Ricci curvature, 38 projectively normal, 98
taut, 125 tight, 125
pseudoconvex, 74 pseudodistance d,1.1, 45 pure k-dimensional, 94
V-manifold, 100 volume elements, 17
V
