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

GEOMETRY AND ANALYSIS ON COMPLEX MANIFOLDS

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

Editors

T Mabuchi Osaka University

J Noguchi Tokyo Institute ofTechnology

T Ochiai University ofTokyo

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

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

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

Printed in Singapore by UtoPrint

Dedicated to Professor Shoshichi Kobayashi

J

Professor Shoshichi Kobayashi

vii Preface

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

Toshiki Mabuchi Junjiro Noguchi Takushiro Ochiai

VIM

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

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

surfaces.

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

geometry.

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

ix

Contents

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

vii

viii 1

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

39

Generalizations of Albanese Mappings for Non-Kahler Manifolds Ichiro Enoki

51

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

63

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

75

Linear Algebra of Analytic Torsion Hong-Jong Kim

96

KP Equations and Vector Bundles on Curves Yingchen Li

113

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

140

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

157

Symplectic Topology and Complex Surfaces Yongbin Ruan

171

Automorphisms of Tube Domains Satoru Shimizu

198

Tensor Products of Semistables are Semistable Burl Totaro

242

1

COMPLEX FINSLER

METRICS

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

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

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

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

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

Finsler

i) F is Kahler; ii) F has constant holomorphic curvature — 4; iii) R(H,K,x,x) = R(x,T<°M —> R+ satisfying i) G = F2 is smooth on M; ii) F(p; v) > 0 for all p € M and v € M p ; iii) F(p; (v) = |C|F(p; v) for all p € M, v € T}<°M and < € C. We shall sistematically denote by G the function G = F2. Note that it is important to ask for the smoothness of G only on M: in fact, it is easy to see that G is smooth on the whole of Tl,0M iff F is the norm associated to a hermitian metric. In this case, we shall say that F comes from a hermitian metric. To start, we need a few notations and general formulas. In local coordinates, a

4

vector v € T*'0M is written as

—"I where we adopt the Einstein convention. In particular, the function G is locally expressed in terms of the coordinates {z1,... ,zn,v1,...,vn}. We shall denote by indices like a, ft and so on the derivatives with respect to the ^-coordinates; for instance, d2

G

a0

°-

dvadv0' On the other hand, the derivatives with respect to the z-coordinates will be denoted by indices after a semicolon; for instance, d2G

n

_

d2G

av "" ~~ dz»dzv ' ~ d!?dva' For our aims, we ought to focus on a smaller class of Finsler metrics. A complex Finsler metric F will be said strongly pseudoconvex if

iv) the Levi matrix (G a ^) is positive definite on M. This is equivalent to requiring that all the F-indicatrices IF(p) = {ve T^M

I F(v) < 1}

are strongly pseudoconvexes. As we shall see in section 2, this hypothesis will allow us to define a hermitian metric on a suitable vector bundle. The main (actually, almost the unique) property of the function G is its (1,1)homogeneity: we have G(p-Xv)=S
(1.2)

Thus differentiating with respect to £ or ( and then setting ( = l w e get G

a

^

= Ga,

Ga0vl3 = O,

(1.3)

and G a / 3 7 v1 = -Ga0,

Ga0y W = Ga/3,

Gafa v*1 = 0,

(1.4)

5 where everything is evaluated at (p;v). On the other hand, differentiating directly (1.1) with respect to £ or £ and putting eventually £ = 1 we get Ga va = G,

GaP vav0 = 0,

Ga0 vavP = G.

(1.5)

It is clear that we may get other formulas applying any differential operator acting only on the ^-coordinates, or just by conjugation. For instance, we get Gaa.w 5 = G^,

(1.6)

and so on. Assuming from now on (unless explicitely noted otherwise) F strongly pseudoconvex, we get another bunch of formulas. As usual in hermitian geometry, we shall denote by (G0a) the inverse matrix of {Gap), and we shall use it to raise indices. First of all, applying G" a to the first equation in (1.3) we get G0aGa=vP, = u".

(1.7)

and thus, applying (1.6), G/3;/JG

Ga = G:li.

(1.8)

a

Recalling that (G^ ) is the inverse matrix of {Gap), we may also compute derivatives of G^a: DG0a = -GBaG^{DG^), (1.9) DG^), where D denotes any first order linear differential operator. As a consequence of (1.4) and (1.9) we get G§a W = -G^G^G^s

W = 0,

(1.10)

and recalling also (1.7) we obtain GnG*a = -GpG^G^G^

= -G^G^v"

= 0.

(1.11)

2. T h e Chern-Finsler connection To any hermitian metric is associated a unique (l,0)-connection such that the metric tensor is parallel: the Chern connection. The main goal of this section is to define the analogue for strongly pseudoconvex Finsler metrics. Let 7r: M —> M denote the restriction of the canonical projection of T^'^M onto M. The vertical bundle V C TlfiM is, by definition, the kernel of the differ­ ential dn: TlfiM —► Tl'°M. It is easy to check that V is a complex vector bundle of rank n over M\ a local frame for V is given by {di,..., dn}, where we set

d

da = -— dv<*

and

Ou.

=

M

d "^

1

a*"'

6 for a, \i = l , . . . , n . We shall denote by X(V) the space of smooth sections of V; more generally, X(E) will denote the space of smooth sections of any vector bundle p: E —* B. Let h 1 °M «—» T 1 lfi '°M be the inclusion and, for v € M p , let Let jp: T*'°M <-* T M be the inclusion and, for v € Mp, let lfi -» T , 1 ' 0 ^ Ky Tp M kv:T}'0M->T^°(Tpl'°M) denote the usual identification. Then we get a natural isomorphism denote the usual identification. Then we get a natural isomorphism fcjj = d(j„(v))v °kv T^M - v», h, = d(J*(v))v ° K- Tl'°v)M -* vv,

r-

and, by restriction, the all-important natural section t:M--» V given by and, by restriction, the all-important natural section i: M —» V given by t(«) = tv(v) € V„. t(v) = LV(V) e V„. In local coordinates, In local coordinates, d C'alu! ) 1 —=d a\v;

(«)/

in particular, if v = va(d/dza)

then t(n) =-- vada\v. i{v)--

i is called the radial vertical vector field. The first observation is that a strongly pseudoconvex Finsler metric F defines a hermitian metric on the vertical bundle V. Indeed, if v € M and W\, W2 € V„, with Wj = Wfda, we set {Wx,W2)v

=

GaB{v)W?wl.

Being F strongly pseudoconvex, {,) is a hermitian metric. Note that the third equation in (1.5) says that G == (* i ) ; so i is an isometric embedding of M into V. Following Kobayashi [17], we now consider the Chern connection D on the vector bundle V: it is the unique (l,0)-connection on V such that the hermitian structure previously defined is parallel. In other words, D: X(V) —> X(T^M® V) is such that X{V, W) = {VXV: W) + {V, VyW), for any X e Tl'°M and V, W e X(V). In local coordinates, the connection matrix (wf) is given by u>% = GfadG0f

= fg.„ dz» + f |

di>f,

7

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

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

-r%yda

6/1 = dfj, -

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

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

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

{H,K)v =

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

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

(Y,Vxz)

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

8 where we have set

r« = t%y = GfaGflll.

Writing Writing

w|

we get we get

■pa

—G

= T%„dz»+ T%r,

Ta Gfff-r — -L-,0,

(2.1)

r ^ = G^'S^Gpr) = G™(G0f„ -

G&Jl).

Note that

r£M = a/j(r«)

and r « = r ^ . y ;

(2.2)

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

[6M, Su] = 0 for all 1 < fi,v < n; [
Proof: i) If suffices to compute. First of all, Proof: i) If suffices to compute. First of all,

i W -r« M r»)a a i

where r™

= 9|/(I?,) and so on. Now

where T?^ = (^(T^) and so on. Now,

r!)!»-« a

ra

;l/M

(~tfai/~1

/~i

p
V*f;nv — Wf;lA ;MJ. r^Ta(/~i s~i p<7 \ - « l ^ r ^ / l - <-*
i V - = G^(Gff?i„IY„ -

"'OTp*

pa

p
*ni

1

1

f-1T0t(/-1

p
;i/I ; M ) l n n p ( j\

_

£r;v ;^ — « l^ff;" ;M " " T p l ; W I ; M J , p a pff /~iTa(s~< p<7 ^» p o r*a \ 1 *ff;^ ;./ ~ *-* l W f ; ( « l ; l , ~ C r ^ p i ^ 1 ; | / ) ,

and the assertion follows. Note that we have actually proved that MI?,,) = ff„CT£).

W).

ii) Indeed,

[*„.*«] = PM - r^AA] = d«(i?M)d„ = rs.„d„.

(2.3)

9 iii) In fact, using (1.8) we get 5M(G) = G;M — r°pGa = G;M — GTaGf:liGa = G;M — Gitl = 0. iv) Finally, °s(G a ) = Ga-ji — T.pGaf = Gatji — Ga-ji = 0, where TfA =T%.

D

The philosophical idea behind our work is that to study the geometry of a com­ plex Finsler metric one should transfer everything (or most of it) in the horizontal bundle, and then apply the usual techniques of hermitian geometry there. We shall better substantiate this idea later, for instance in sections 6 and 7 discussing vari­ ation formulas; here we begin to show how to lift objects (e.g., vector fields) from the tangent bundle up to H. The main tool is provided by the horizontal analogues of the isomorphisms uv. live. M, we set Xv = ©u o h,- Tff(„)M -+ Hv. The horizontal radial vector Geld x € X{H) is then defined by X = 6ot; a

a

in local coordinates, if v — v (d/dz )\p

we have

X(v) =vaSa\v. Using the isomorphisms Xv we can induce an embedding of M into H which respects the Lie algebra structure. To be precise, a vector field £ e X(Tli0M) may be lifted in two different ways to vector fields in T1,0M: via the horizontal lift

£*(») = *,(«*(*))), and via the vertical lift

. € * » = *.(«*(«))) ■

A consequence of Lemma 2.1 is that the horizontal lift is a Lie algebra homomorphism: Proposition 2.2. Let D be the Chern-Finsler connection associated to a strongly pseudoconvex Finsler metric F on a complex manifold M. Then: i) [X(H),X(H)} C X{H) and [X(V),X(V)] C X(V); ii) ifiu & € X(M) then [g,tf] = & , & ] " , [&,$} = 0 and [ff,&] € X(V). Proof: i) Take HUH2&

X{H). Locally, H3 = H^S^; hence

[Hi,Ha] = («?M
(2.4)

10 (where we used Lemma 2.1) is horizontal. Analogously, if Vi, V% € X(V) Vj = Vfda, we get [Vl,V2] = (vfdp(V2a) which is vertical. ii) Locally, fc = $(d/dz»)

V20d0(V?))da,

-

with

(2.5)

and tf = ( # o TT)^; SO (2.4) yields

KF, *?] = ((# °*)Mtf°*)-- fe"»«^))«» Now « „ ( $ o TT) = {dtf/dz")

Kf.tf] =

o TT; therefore

l^1 8z" ^2dzv )

*M = K I . & ] " .

On the other hand, (f = (£? o 7r)da and ^ ( ^ o TT) = 0 yield

Kiv.&v]=o. Finally,

Kf.tf] = [(tffj) °T + ((^) o7r)r^j da, D

again by Lemma 2.1.

Note that, as a consequence of ii), the obvious map of X(V) into X(H) induced by the complex horizontal map 0 : V —► X is not an isomorphism of Lie algebras; it suffices to remark that G(£v) = SH for all £ 6 X{M). 3. Torsions and kahlerianity As it may be expected, the next step is the study of the Chern-Finsler connection is to describe its torsion(s) and clarify their geometrical meaning. The tangent bundle TlfiM (and hence M too) is naturally equipped with a T^Af-valued global (l,0)-form, the canonical form n = dz>i®dli + dva ®da€

#(A1,0M ®

Tl'°M).

It is easy to see that as soon as we have a strongly pseudoconvex Finsler metric — and hence the canonical splitting T1,0M = H®V — one has r} = dzli®S^

+ ipa® da-

Extending as usual the Chern-Finsler connection D to an exterior differential (still denoted by D) on T 1 '°M-valued differential forms, it is very natural to consider

11 the torsion Dr\ of the connection. Since r\ is a (l,0)-form, Drj splits in the sum of a (2,0)-form 8 and a (l,l)-form r. We shall call B the (2,0)-torsion of the ChernFinsler connection, and r the (1, l)-torsion of the Chern-Finsler connection. Locally, we may write O = 6f,®6ll + 0a®da

and

r =

Ta®da,

where, setting rj} ;j| = % ( r ° ) ,

r a = av a = -foCrj) d^ A dzv - rg.M «u* A y^;
(3.1)

6a = drpa -- / A w ^

= i[Mr«) - Mi?J]
(3.2;

by (2.3), (2.1) and (2.2). One may wonder whether these torsions are the right generalizations of the usual torsion in the hermitian case. The answer is a double yes. First of all, a standard argument using the definitions shows that torsions and covariant derivative are related as usual: VXY - VYX = [X, Y] + 9(X, y ) , V x F - VYX lfi for any X, Y e X{T-.1,0 M),

= [X, Y] + T(X, Y) + T(X,

Y),

(3.3)

where, by definition, VXY = V X F .

Furthermore, the vanishing of (part of) the (2,0)-torsion can be again interpreted as a Kahler condition — but with some care, because 6 is composed by a horizontal part and a mixed part. To be precise, we shall say that a differential form 7 on M is horizontal if it vanishes contracted with any V e X(V). The decomposition T1,0M = H®V induces a projection p*H of the differential forms onto the horizontal forms; the horizontal part of a form 7 is then p*H(7). There is a corresponding projection on the vertical forms, of course, but we shall not need it now because the vertical part of both torsions 6 and r is zero. For this reason, the form 0 — p*H{0) will be called the mixed part of 0. In local coordinates, p*H{0) = {K^dztihdzv)®6a

and

d - p*H{6) = (T^^

hdz")®8a.

The next proposition discusses the meaning of the vanishing of the (2,0)-torsion 6 or of one of its parts.

12 Proposition 3.1. Let F be a strongly pseudoconvex Finsler metric on a complex manifold M. Then: i) the mixed part of the (2,0)-torsion vanishes iff F comes from a hermitian metric; ii) 9 vanishes iff F comes from a hermitian Kahler metric. Proof: i) The mixed part of the torsion vanishes iff G ^ 7 = 0 for all P, fi and 7. Conjugating, this is equivalent to having dj{Gpp.) = d^Gpp) = 0, that is Gpp,(v) depends only on n(v) — and this happens iff F comes from a hermitian metric. ii) It follows from i) and the fact that when F comes from a hermitian metric 9 = (9ap) o n e h a s

a.

=

,dgggy 90f 9fc

9zf '

a For this reason we say that a strongly pseudoconvex Finsler metric F is strongly Kahler if the horizontal part of the (2,0)-torsion vanishes, that is iff MH,K eH

6(H, K) = 0.

This is exactly the notion of kahlerianity introduced by Rund [27]. However, as we shall see later on (see sections 6 and 7), studying the geometry of a strongly pseudoconvex Finsler metric it turns out that this assumption is too strong and not quite natural. So it is appropriate to introduce two more notions of kahlerianity. We shall say that F is Kahler if 0(H,x)=0, = 0, e(H, x) =

\/HeH VHeH and that F is weakly Kahler if VHeH H

= 0. (0(H, ),X)=O. im,x),x)-X

In local coordinates, F is strongly Kahler iff F Q pa

z/

_

pa a . p

M'

it is Kahler iff 1

/*;*«" = pa ,«*;

it is weakly Kahler iff t = 0, Go[r^ ; v -- r j > " = o,

that is iff

0 = [GWW - GKft + e„,r* y = K W - Gufw + G„a?r?jv»w.

13 In particular, if F comes from a hermitian metric then these three conditions are all equivalent to the usual Kahler condition, because Gvaf = 0 for a Finsler metric coming from a hermitian metric. There are other characterizations of strongly Kahler Finsler metrics. To F we may associate the fundamental form $ = iGap dza A dlP, which is a well-defined real (l,l)-form on M. Then the strong Kahler condition is equivalent to the vanishing of the horizontal part of d $ . To express it more clearly, set dti =P*H°d, dH =p*H°d and dH=p*Ho 5, so that again dH = dn + BHTheorem 3.2. Let F be a strongly pseudoconvex Finsler metric on a complex manifold M- Tien the following assertions axe equivalent: i) F is a strongly Kahler Finsler metric; ii) VHK - VKH = [H, K] for all H, K e X{H); hi) d / / $ = 0; iv) dn$ = 0; v) for any po € M there is a neighborhood U ofpo in M and a real-valued function 4> e C00^-1^)) such that $ = idHdH on it~\U). Proof: i) -*=>- ii) follows from (3.3). iii) -$=$• iv) holds simply because $ is a real (l,l)-form. iv) <=> i). Indeed, (2.1) yields d*(X,Y,Z)

=

i(8(X,Y),Z)

for all X, Y, Z 6 TlfiM\ hence d # $ vanishes iff pH o 6 vanishes, that is iff F is strongly Kahler. v) => iv) follows from Lemma 2.1.i). iii) =$. v). Let 7 be any horizontal form. In local coordinates, defined on a coordinate neighborhood of the form 7r -1 ({7), one has 7l(pi«) = 7 A S ( P ; V) dzA

A

dzB

>

for suitable multi-indices A and B. On U we may then consider the family of forms iv\P = 7 A B ( P ; V) d*A

where here {dzj} is the dual frame of {d/dzj}; the v-coordinates just as parameters.

A

dzB

>

in other words, we are considering

14

The gist is that the following formula holds: {dH-y)v = d(7«)Then we may now apply the Dolbeault and Serre theorems (with parameters) to $„ in a possibly smaller neighborhood of po — still denoted by U — to get a function 4>v € C°°(t/,M) depending smoothly on v such that 4>„ = idd<j>v. Then setting

4>(p;v) = v(p) DD

we get $ = idndH, as required. From this point of view, a strongly pseudoconvex Finsler metric is Kahler iff d/f*(- ,X> ■)

= EO,

and it is weakly Kahler iff dH$(-,X,X) =0. We end this section pointing out that also the vanishing of the (l,l)-torsion r has a nice geometric meaning: Proposition 3.3. The (1, l)-torsion r vanishes iff the frame {6^, da} is holomorhic. Proof: Indeed the frame {6^, da) is holomorphic iff its dual coframe {dz^, rpa} is, which happens iff the forms ipa are holomorphic, that is iff ra = 3ijja = 0 for a = 1,... ,n. D 4. The curvature tensor The curvature tensor R: X{TlfiM) -> X(tf(T£M) Finsler connection is given by R = D o D, that is VX e XiT^M)

Rx =

<8> T^M)

D(DX).

Analogously we have the curvature operator Q e X(tf(T£M) defined by (cf. also [17]) Q(X,Y)Z Z = R. RZ(X,Y). Locally, f2 is given by n = n£ ® [dz0 ®8a + tP0® da], where

of the Chern-

ng = du$ - u j A u "

Decomposing fi into types, we get

n = n' + n",

g> /^'°M

® T1%aM)

15 where SI' is a (2,0)-form and SI" a (l,l)-form. Locally,

(fl')g = 6u% - wj A w?,

(«")? = &4.

f2 has no (0,2)-components because the connection forms are (l,0)-forms. Actually, even SI' vanishes: indeed, by definition Gr0dGaf.

w£ = So

3w£ = dCT" A S G a f = -CT^G^dG^ = (Gf»dGaf)

A (Gp0dG^)

A dGaf

(4.1)

= < A ^ .

So n = fi" and

Sl% = Suf, exactly as in the hermitian case. The relation between curvature and covariant derivatives is the usual one: V x W - VyVx = V [ x ,y] V * V F - VrVx V

X

VF-VFV

= V[xF] + X

il(X,Y);

= V[XF],

for any X, Y e ^ ( T 1 ' 0 ^ ) . We can also recover the Bianchi identities in this setting: Proposition 4.1. Let D:X{TX>°M) -» -V(rc*M ® T ^ M ) be the compiex Jinear connection on M induced by a good complex vertical connection. Then

DO = rf A fi, DT = T]V

A fi,

7X2 = 0, where T/H = dz^ ® 5M and 77v = VQ ® 4
+ r 0 A w« = aava + ^

A^a = -aav a + &/*" A ^ = / A &$,

by (3.2), and so DT = nv A ft. Finally, 3ft£ = 0 and --u ^ A d u ^ + & ^ A w « = -d[du%-uZAue) asia0 -u} ASI^+SI} AW? ==aawg dn%-u}*n?;+n}Au>% dduj%-u;}Aduj%+du}Au,° = -d(du^-ui^)= o, by (4.1).

□

=o

16

In local coordinates, the curvature operator is given by ft£ = R%^ dz» A dz" + R%S.B ^ A dzv + R%^ dz» A1JP + R°m ^ A W>, where

p«

_

-6p(r^)-r^5P(r^),

R

%6;B = ~Sp(^ffs)

Rr

0T,li

^3«7

=

_

= _

~ R?0;P<

r

r

,

,

(4.2)

r

^7( ?;/i) "" / 3 o - ^ '

< M r ^ ) = Rf/3^y-

In particular, since {DT)a = {r]v A ft)Q = ip" A ft? = fl°Mi7 t/>" A dzM A dz" + R%B.g i>° A i>s A dz" + J#, ; / t Vff A dz^ A V^ + R^ i>° A ^ A iP = - i ? « M P dz^ A f A dz" - J J ^ dz" A r

A ?=»,

the vanishing of r implies the vanishing of most of the curvature. Another consequence of (4.2) is an unexpected relation between ft and T: Lemma 4.2. Let D be the Chern-Finsler connection associated to a strongly pseudoconvex Finsler metric F on a complex manifold M. Then T = ft(-, •)(•■

Proof: Recalling (4.2), (1.3), (1.4) Proof: (1.4) and and p<* „0 — r a

r%y = o,

we have R2 ..„«"0 = = --S,-JTa..). R0;„uV R

06

y

%\n fi=

R

l

R

06

y

-fctrs.), = o,

-r= 0,

and the assertion follows from (3.1).

□ □

5. Holomorphic curvature One of the most useful concept in hermitian geometry is the notion of holo­ morphic sectional curvature. To find the correct, analogue in our setting, we first need a closer look to the horizontal part of the curvature operator. We define the horizontal curvature tensor R by RV{H,T<,L:IJ)

=

(Q(H,K)L,M)V

for all H, K, L, M e ~HV and v e M. In local coordinates,

R(H,K,L,M)

=

The symmetries of R are easily described:

G^Rl^H^W^W.

17

Proposition 5.1. Take v e M and H, K, L, M e Hy. Then R{K,H,L,M) R(K,H,L,M)

= -R(H,K,L,U); -R(H,K,L,M);

(5.1)

R(K, Jt\H, i t , Z/. M). R{K, H, if, M, L) -=- RJH^WJ^M).

(5.2)

Furthermore, if 8H0 = 0 we also have == R(H,T(,L,Jl) fl(ff, # , £ , M) === R(H,M,L,K). *(JT, M , L , * ) . «R(L,T(,H,Tl) ( £ , ff.ff, M) =

(5.3)

Proof: (5.1) follows immediately from the observation Q${K, H) = -Q%(H,7(). To prove (5.2), we start from fig = 8LJ% Bwl ==B{G 8(Gfafa8G 8G0f) ==-G^G^BG^ -G^G^dG^

fa AAddG G ^0f ++ GfaGdBG d8G0f0f; ;

in particular, Ga^% = -Gf»dG^

A BG0f + BdGm.

On the other hand, pTGa& = G GVT A SBG^ G ^ D--- GGaT&T B8G TY, fif? = G »BG^ "8GT^ A 8dG T^

n

hence Gasfii0 = — GpcSlzjIn our case, this means that 0 = -G^Q?.(K.H)M Li 0Tn = -G0&^{Kjl)M

l3 R(K. H. M,L) M. L) ■G^Q'S(K.H)M Ty R(K,H, == G a^Q$(K,W)M^r

=

Ga0G-n°(H,K)L-yM0 a^(H,K)L-rW

= R{H,~K,L,~M), and (5.2) is proved. Now, (5.3). First of all, SHO = p*H(D6), because we saw that the (2,0)-part of D6 vanishes. Proposition 4.1 says that DO = r)H A fi; in local coordinates,

(VH A n)a = dza A fi? =Kw

dz A

"

dz>x A

&? + Kt-,9dz"

A

i>6

A

^"

(5.4)

+ i?^ ; / x dz" A dz" A in + R%^ dz" A i>6 A ^F; in particular, d//(9 = 0 iff fl?.^ = R%p. Then G rR^ ^K-HaM- =

fl(L, tf, ff, M) =

a 7pa fl(L,F, F , M ) = GarR^pL"l0 H M^

GafR^p^K^H"Mr =

= GafR%aPL»WH<'W

R(H,K,L,M).

= R(H,K,

L,H).

Finally, R(H, M, L,

=

R{M,H,K,L)--= R(K, H, M, L) --=

R{H,K,L,M).

K)-. R{H,~M, L,T<) = R{M,~H,K,~L) = R{K,~H,M,T) = R{H%T(,L,"M). D

18

We remark that (5.2) is equivalent to {fl(H, K)L, M) = {L, tl{K, H)M)

(5.5)

for all H, K, I, M G H. Now, one possible approach to the holomorphic sectional curvature is to con­ sider the (horizontal) holomorphic flag curvature KF(H) of F along a horizontal vector H efiv:

KFF(H) (H) =

9

jM^R(H,H,H,H).

(H,H)lR(H'H'H^h

Exactly as in the hermitian case, if dud = 0 then the holomorphic flag curvature completely determines the horizontal curvature tensor: Proposition 5.2. Let R, S: Hv x Hv x Hv x Hv —* C be two quadrilinear maps satisfying (5.2) and (5.3). Assume that VH € Hv

R(H, H, H, H) = S(H, ~H, H, H).

Then R = S. The proof is very similar to the traditional one for hermitian metrics; see [19] and [5] for the details. We do not discuss it here because, from a certain point of view, the holomorphic flag curvature is not the right generalization of the holo­ morphic sectional curvature. In fact, roughly speaking, it contains too many infor­ mations. Requiring, for instance, that the holomorphic flag curvature is constant means imposing very strong constraints on the behavior of the complex Finsler metric, constraints that are somewhat beyond the geometry of the metric which lives naturally on the tangent bundle of the manifold. Of course, one may study such requirements, but in this case the theory seems to be a standard consequence of the hermitian geometry of vector bundles without significant application to the function theory of the manifold. A different notion appears to be a more appropriate tool for the applications in complex geometry (see [17], [2], [3], [4], and sections 6 and 7 where we discuss variational formulas; cf. also [11] and [8] for similar arguments in the real case). Namely, let F: T1,0M —► E + be a strongly pseudoconvex Finsler metric on a complex manifold M, and take v e M. Then the holomorphic curvature KF(V) of F along v is given by KP(v) = KF(X{v))

= ^jafl(x(«),X(w),x(«),X(«)).

Clearly, KF((v)

= KF{v)

for all C £ C*; so this is the holomorphic curvature discussed by Kobayashi [17]. Note that, by Proposition 5.1, the holomorphic curvature is necessarily real-valued.

19

In local coordinates we get

2

Kp KF == -^G )v»W a80{T^ -£3G«*p(r°)vV

(5.6)

If F comes from a hermitian metric, (5.6) gives exactly the classical holomorphic sectional curvature. Furthermore, our definition recovers another important ge­ ometrical characterization of the holomorphic sectional curvature, and provides a firm link with the theory of invariant metrics on complex manifolds (cf. [3]). Wu [30] has shown that for a hermitian metric g on a complex manifold M, the holomor­ phic sectional curvature of g along v G Tp , 0 M is the maximum value attained by the Gaussian curvature at the origin of the pull-back metric ip*g when : A —► M with ip(0) = p and m+ just by setting Vw £ TtM

F°{u) =

F(u0),

where u >-* u0 — (u — iJu)/2 is the standard isomorphism between T%M and TlfiM (J is the complex structure on TtM). F° satisfies all the properties denning a real Finsler metric, but perhaps the indicatrices are not necessarily strongly convex. Nevertheless, we may use it to measure the length of curves, and so to define geodesies; and one of the main results of this section is a theorem ensuring the local existence and uniqueness of geodesies for weakly Kahler Finsler metrics only under

20 the strong pseudoconvexity hypothesis — a striking by-product of the complex structure. Let us fix the notations needed to study variations of the length integral in this setting. The idea is, as usual, to pull back the connection along a curve; but since our connection lives on the tangent-tangent bundle, the details are a bit delicate. A regular curve a: [a, b] —* M is a C 1 curve with never vanishing tangent vector. Here, we mean the tangent vector in r 1 , 0 M , obtained via the canonical isomorphism with TtM: so we set

*>-^®-£w

dza

where d1,0 is the composition of the differential with the projection of TCM onto Tl-°M associated to the splitting TCM = TlfiM 0 T°^M. The length of a regular curve a with respect to the strongly pseudoconvex Finsler metric F is given by

L(
F(&(t))dt,

Ja

exactly as in the hermitian case. A geodesic for F is a curve which is a critical point of the length functional. To be more precise, let ao- [a, b] —> M be a regular curve with F(&o) = CQ > 0. A regular variation of M such that (a) 0 for every s € (—e, e). A regular variation E is fixed if it moreover satisfies (d)
£<»>=

=0

for all fixed regular variations E of CTQ. Our first goal is to write the first variation of the length functional; we shall then find the differential equation satisfied by the geodesies (see also [2]). Let E: (-£, e) x [a, b] -» M be a regular variation of a regular curve a0: [a, b] —» M. Let p:Y,*(TlfiM) — ( - e , e ) x [a,6] be the pull-back bundle, and 7:E*(T 1 -°M) -► T1,0M be the bundle map such that the diagram 7 Ti,oM E*(Ta-°M)

-I

(-£,£) x [a,fr]

E

I"

M

21

commutes. Two particularly important sections of E*(T1'°M) are

r == 7

1

(d 1 ' 0 : ' ( * ) ) ■

as a a at az«'

as° a » ( * ) ) - a« az«' the restriction of 1/ to s = 0 is the variation vector of the variation E. Note that setting £*M = 7" 1 (M), we have T e #(£*M) and [/ = =

7- 1 (d 1 '°

T(* 1 *)=7- 1 (ff.(t)). Now we pull-back Tl,0M over £*M by using 7, obtaining the commutative diagram 7*(T1'°M)

-U

I

TlfiM

I

E*M

-U

1

M

;

■ I

(—£,e) x [a, 6] —> M note that 7*(T1,0M) is a complex vector bundle over a real manifold. The bundle map 7 induces a hermitian structure on 7*(T1,0M) by

vx,y e Y(TlfiM)v{x,Y)v = mx),*/(Y))l(v). Analogously, the Chern connection D gives rise to a (l,0)-connection D*: ^(7*(T 1 '°M)) -» *(TC*(E*M) 7*(T1'°M)), where TC*(E*M) = T,*(E*M) ® C, by setting VXY = r

1

(V d ..o 7 ( x ) 7(^)), 1

V^y = 7 - ( v 5 r ^ f y 7 ( n ) , for all A" € Ta(E*M) and Y e X(-y*(Th°M)). In particular we have X{Y,Z) = X(( 7 (y), 7(^)7) = W ( ( 7 ( y ) , f f l ) ) = (d1'°7(X) + di.o 7 W)((7(n>7(^)»

= {VXY, z) + (Y: v*Tz) + (v^y, z) +
(6-1)

22

for all X € T,(E*M) and Y, Z e ^_(7*(T1-0M)). We may also decompose TB(E*M) = H* © V*, where as usual a local real frame for V* is given by {da, ida}, and a local frame for H* is given by

6t = dt- (rfa o 7 )^-d M ,

«s = ds - (r?a o 7 ) ^ - » M .

where dt = d/dt and ds = d/ds. Therefore, setting TH = d1,07(*t) and UH = d1,07(55), we have 9E^M <9E TH(v) = -£r(s,t)6uLtv) dt and

= Xi(v)(&s{t)) 6 Wj(v)

ayM

X77(»)(7(^(«,«))) (»)(#(M)))e UH(v) = -^-(«,*)**.l7(») == X €W H r7 W W, for all v € E*M(S t)j they are the horizontal lifts of 7(T) and 7(f) respectively. In particular, TH(TH*S)) = x(*.)(6.2) If we take v e (E*Af)(s(), then d^iUV'M)) d ^ T ^ E ' M ) ) C 2T^,W $j#

and

7 (7*(T

1

'°M)„) = ! $ , £ .

Therefore we also have a bundle map E:TB(E*M) —► 7*(T1,0M) such that the diagram TB(E"M) ^ * 7*(T1'°M) d107 \

| T r

i,

°M

commutes. Using 2 we may prove three final formulas: 7(V£2(y) - V^S(A-)) = V d ,,o 7(x) dl-°-y(Y) - V„..o7(y) dl>°7(X) = [d1-°7(X), ^.° 7 {y)] + fl(d1'07(x)1 d 1 ' 0 ^ ) ) ,

( 6(6.3) 3)

-

for all X, Y € <*,(T1(E*M)); 7 o (V^ Vy - Vy Vj^) = (Vdi.o7(X) Vdi;o7(y) - Vdi,o7(K) V d i.o 7(x) ) 07 V7 = V[di.o7(x),di.o7(y)] ° 7 ,

(6-4)

and

7 ° (7Jvf - vfyj) = (v*dl.„7(x)iarKRFJ] + n o ^ x ) , d M ^ ) o 7, (6.5) for all X, r 6 TB(E*Af). We are now able to prove the first variation formula for weakly Kahler Finsler metrics:

23

Theorem 6.1. Let F: TlfiM -+ K+ be a wealdy Kahlej- Finsler metric on a complex manifold M. Take a regular curve o<j\ [a, 6] — ► M with F(do) = CQ > 0, and a regular variation E: (-e,e) x [a, b] ~* M ofcr0- Then H dgg =■ i ; |■R *e {"f 7 HT,"T\ .%1 [ - R Ree ^f <[/", (UH,>vVT,T „ + w TT "\.) * ,dt\. 0 0dtj. 1 ■* /(TO la >ds (0) Ja

In particular, if E is a fixed variation, that is E(-,a) = cr0(a) and E(-,6) =
(6.6)

Proof: By definition,

fc(«) = [ (G(as))1/2dt; Ja

therefore din ds

1 " 2caJ

f6

1

Ja

BW*. ) ] * =" 2cs Ja

- c S(*), S(«i)>T*,

where cs = F(a s ) and we used

a/

G(*.) = (x(*,),x(*.)>*. = (H(fie),H(5t))T, by (6.2). Now, using (6.1) and (6.3), we get ~ < S ( * « ) , S ( « , ) ) r = ±M3(«.),S(* t )) T = i{(Vj.H(5 t ),H(« t )) T + (S(«,), V^S(fi t )) T + (Vi-H(5t),E(«t)>T + (S(«,), Vj.S(«t)>r} v H(« 3(*t), = Re Re{(V (V|. + <(V| S(«,»T B ( *f)>T ))T + 4*.E(6 t),H(6 t ),H(5 t ))T} ir7 3(«,), H H H ,T"), = Re{(VJ,lS(«.), + <[t/« .r^i <0(t/" 2(«,)>T ,T">*. = Re{
Since F is weakly Kahler, (6.2) yields H H (9(UHH,TH,T )&m = 0. (9(U ),TH),T )&,=0.

Furthermore,

,rrH „a, [tf" .T»] Vu

v raE \W

35% H-1

fdX"\

IP" c

"(^r)*"5

dE" dE"

at

/aE"\\ «-(£)} v

24 since

E" dE" (dZ»\ dT," fd^\ _ d 22sM fd^\ a _E^ (&P\ 5 + + 6e 6o 6v dsdt -ds"{-W) ^sVW)~Ji \df) ~dT \ at ) ~ dadt _ d 2 !^ _ dE" / d E " \ dW (dY,»\ "\ ds )+ dt bo\ds

-~dtds~~~dT

(6.7)

)'

we get [U«,T*] + VmT» = ^

fc(^)

"V ds J

at

SM = VWU».

(6.8)

Then

i_a_,„,^ „ (E(6 ),E(6 )) t

2ds

t

T

= Re{T + (V^-S(tf.},H(*«))r}

= Re{5 t T}

= Re{|(t/",r»},3-~

( " jyrpH+TH

(6.9)

'*» | '

and the assertion follows.

□

As a corollary we get the equation of geodesies: Corollary 6.2. Let F: T1,0M —* IR+ be a weakly Kahler Finsler metric on a com­ plex manifold M, and let a: [a, b] —> M be a regular curve with F(a) = CQ > 0. Then cr is a geodesic for F iff VTH+¥WTH where T H (u) = X» (*(<)) € Hv for alive

= 0,

(6.10)

M„(t)-

Proof: It follows immediately from (6.6). X0

□ D

+

Corollary 6.3. Let F: T ' M - » R be a weakly Kahler Finsler metric on a complex manifold M. Then for any p e M and v e Mp with F(v) = 1 there exists a unique geodesic a: {—e,e) —> M such that
= [ ( < T % + ^ ^ ) ( ( T Q ) + r° M (*)^o»]««= [aa + r5,(*)^]* a .

So (6.10) is a quasi-linear O.D.E. system, and the assertion follows.

□

Thus the standard O.D.E. arguments apply in this case too, and we may recover for weakly Kahler Finsler metrics the usual theory of geodesies. In particular, if the metric F is complete we can define the exponential map exp : T^°M —* M for any p e M. See [5] for details.

25

7. Second variation of the length integral Our next goal is the second variation formula, which holds for Kahler Finsler metrics. To express it correctly, we need two further ingredients. The first one is the horizontal (l,l)-torsion TH, simply defined by TH(X,Y)

=

@(T(X,Y))

=

n(X,Y)X.

The second one is the symmetric product ((,}}: H x H —> C locally given by Vif, KeHv

{(H, K))v = Ga0(v)

HaK0.

It is clearly globally well-defined, and it satisfies Vtfew

«ff,x)> = o.

Theorem 7.1. Let F: T^'°M — ► M+ be a Kahler Finsler metric on a complex mani­ fold M. Take a geodesic a0: [a, b] —» M with F(&0) = 1, &nd let E: (-e, e) x [a, &] —>M be a reguiar variation of ao- Then dip

.b

^ ( 0 ) = Re, o | a rb

+ Jo

v

^

T

H

+ T

HI/

«||2

^Re^,^)^

Re[(n(T", [/")£/", T"),>0 - { 0 ( ^ , ^ ) ^ , 1 " ) ^ + «T"([/",T^*X

-

((TH(TH,W),UH))&o]}dt.

In particular, if E is a fixed variation such that Re{£/ , T ),j0 is constant we have

>

>

■

b H UH \\l TH+T>rU == [ {\\V £{WT» +W \ o H H H H HH) -Re[{n(T ,U")UH,TH)i0 ff > 44 -- (Sl(U Re (n(T^|y*)tf^T (WH,TH)U ,T )&0 :T»)U ,T ia H H H H UHo)) - {(THH(T (TH ,W), ))\jdt (uHH,W), ,W),u»)). - &o -((r ,U»),UHU))&o &o] } dt. + '{Tr (U

(7.1)

Proof: During the proof of the first variation formula — in (6.9) — we saw that fb (VJ,S(*.),H(*t)>T+ (V| r H(6 s ),S(5 t )) T (s) = Re / dt. ds Ja ({E(6t),E(St)h)1/2

d£*

26

So we need to compute d r (VJ t E(6 s ),E(5 t ))T+ T

-dsf I"

({E(St),E(6t))T)1/2 J _ MV; t 5(g,), 5 ( ^ ) ) T + M V ^ E f e ) , 5(g t )) T ((E(St),E(5t))T)l/2 lT + (V|-H(*.).H(*,)>T ~ 2

//-V*\ - / c « \ s / 2

*

, ,

%

*,(=(«i), =(«t))T-

((i(fi«),a(*t)>r)

(7.2) Since, when s = 0, the denominator of the first term is equal to 1, and the denom­ inator of the second term is equal to 2, we may forget them. Let us call (I) the numerator of the first term, and (I) the numerator of the second term. First of all, (6.9) yields 2

^Re(I) = Re > » , T » > , -
(7.3)

1

because OQ is a geodesic. The computation of (I) is quite longer. First of all, using (6.3), (6.4) and (6.5) we get (I) = T ),V^E(St))T S(*«), + (v;,(Vl 6E(6 t))T tE(6sVf

m))T

+ (V^Vf r S(fi s ),E(* t )) T + (V| 7 V^S(^),E(5t)>r + r

= T - t ))TT--■ (V ( v [THJJW] lT«iR H 4- T THi7" H + {VTHUH{VTHU»,9(U + 0(U« THH,T))a, ))&,

H (Q(U»,W)U»,T (WH,T»)UH,THU )&S,

U E/",r")^ TH + T (V^VJ.S(*.),S(«,)> T --- (V , + [WuH]"> w + (V£V|-S(« s ),E(S t )) T - ( v ^ — , £ / " , r ) * a

+ T + (VWUH, [UH,TH] +

+

H

Ii

H

VjprT"),,

(v¥Bu ,e(u ,T ))i,.

27

Recalling (6.8), (6.2) and that F is Kahler we get (I) = . . ,W)U ,T )A. (il(THH,W)U + (Q(U(Q{U ,THH)U T"h. H W - \V[T",U"]MT"j7»]+[T»,U"]+[T"JJ \"\T».U "1+IT" U" + \T".U"]+\T».V ]^

T U",,T" ' >*. *•

+ T + (v^s(«.), vj,s(«.))T + (VJ,S(«.),V| r 2(fi,)) r + (V^S(ff.),V| r H(*.)) r . Now \T»

Tl"\-

dE* / 3 E " \ 5E" . / 3 E " Y 6 , at H~37,r~arH~§rJJ "'V

™-f?w^-«^+f«.(f)«.-£*(3> [TH

uH\ =

tit

ds

»p(rj)a0 - ^(rf P )^] + 2L- ^ ^

j «P.

1

dl>dY,

3E" . / d l > \ .

[W >,(4)^ - ^(r- jdj + ^ 5 ( ^ )fi,P^-Kw{*-«a*| ~~arH~arr/l tf*(D*-|*(f)* UH]--

[T",[/"] =

A

tit. ds

dE" . (dY,"\

~dT8*\-arj

dE" . / d E " \ l .

~ ~dT 6*VdTJi*»

and so (6.7) and (3.1) yield \T", U") + [TH,U»] + [ r " , UH] + [W,W] H HH = ,U,W) ) = T(UHT(U ,W)H,TH)--T(T - T(T

H + T(UH,TH)--T(T",U + T(U",W) T(T",W). )

Furthermore, if V e V we have (VVU»,T»)&.

=Gai,(a,)\n

[ * , ( § £ ) +r? 7 (a 8 ) ( § £ ) ] « ?

= G a (a,)r? 7 (a.) ( ^ ) V7 = G „ ( * . ) ( — ) V"

= «e(n u"K> and

(VFtf», I*)*. &yU",T") a.

= G^(^)F7 ^ ( ^ ) ] (A -WJ*M%)U = 0.

Therefore ^[T",t/"]+(T",f/"]+[TH,y"l+[T»,t7ff]^ H HH H

' - ^ )*»

H H H H ))it, H))ai, = ({r iT (U(V,Wy,U ,T»XU»)) ))&£iM --((rH(T {(rH,U«),U (T ,W),U

28 and thus

(Vl+TE(6s),v;t+TtZ(6t))T (I) == St(Vl+7;E(6s),E(St))T H H H H HH H H H - (fi(T
H

H

HH

H

(7.4)

H

- ((r ,W),U « T " ((U C / ",W),U , T ^ ) , [ / )) >% ((r"(T ,W),U ))&a ))&a + |||V^B(*.)||3.. |V^B(*.)||3.. as S + ((r"(T Recalling that for s = 0 we have V* ■g-S(St) = 0 because a0 is a geodesic, (7.2), (7.3) and (7.4) evaluated at s = 0 yield the assertion. □ So we have obtained the second variation formula for strongly pseudoconvex Kahler Finsler metrics. Besides its own intrinsic interest, (7.1) is useful for com­ paring the real Finsler metric F° and our original complex Finsler metric F. The idea is that measuring the length of curves both using F° and using F we end up with the same function (.■£,'■, therefore writing the second variation formula either in terms of F or in terms of F° (assuming the convexity of the latter, of course) we get equal objects. The second variation formula for real Finsler metrics has been computed by Auslander [6] (see also Chern [11], Bao and Chern [8] and [5]), in a setting similar to ours and in terms of the so-called horizontal flag curvature of the Cartan connection. Unexpectedly, it turns out that for general Kahler Finsler metrics the curvature terms appearing in our second variation formula (7.1) do not yield the real horizontal flag curvature; there are pieces of the latter vanishing after the integration. The comparison is carried out in detail in [5], where we prove for instance the following: P r o p o s i t i o n 7.2. Let F: T1,0M —+M+ be a convex (i.e., with strongly convex indicatrices) Kahler Finsler metric on a complex manifold M such that (V x r)(x, V) = 0 For all V e V. Then the horizontal Bag curvature of the Cartan connection associ­ ated to F° is given by

)v Re[(n(X,H)H,X)vX)v- --(n(H,x)H, (n(H,x)H,X)v Re[{n(x,H)H, for all He

H H H + ((r((T (H,x),H)) (H,x),H)) {(TH(X,H),H)) v---{(T (X,H),H)) V] V] v

H.

Luckily, here we shall not have to compute the real horizontal flag curvature. Indeed, we shall only need Auslander's version [7] of the classical Cartan-Hadamard theorem. Looking at the proof, one sees that the main point is the positivedefiniteness of the Morse index form — which is a consequence of but it is not equivalent to the negative-semidefiniteness of the curvature. Since the Morse in­ dex form is computable via the second variation formula, conditions ensuring that the integrand in our formula (7.1) is positive suffice to yield the Cartan-Hadamard theorem for Kahler Finsler metrics. Again, see [5] for details. 8. Manifolds w i t h c o n s t a n t holomorphic c u r v a t u r e

29 A very natural problem now is the classification of Kahler Finsler manifolds of constant holomorphic curvature. In this respect, the Finsler situation is much richer than the hermitian one; for instance, Lempert's work [21] and [3] imply that all strongly convex domains of C n endowed with the Kobayashi metric are weakly Kahler Finsler manifolds with constant holomorphic curvature —4. The last theorem of this paper is a step toward this classification; roughly speak­ ing, we shall prove that a simply connected Kahler Finsler manifold of nonpositive constant holomorphic curvature is diffeomorphic to an euclidean space. Further­ more, in the case of constant negative holomorphic curvature our results show that the Finsler geometry of the manifold is pretty much the same of the one of strongly convex domains endowed with the Kobayashi metric. The idea is to apply the Cartan-Hadamard theorem; to do so, we need to esti­ mate the curvature terms appearing in the second variation formula. Let F:TltQM -> 1 + be a strongly pseudoconvex Finsler metric on a complex manifold M. We say that F has constant holomorphic curvature 2c € 1 if 2 (n( ,= cG\ X,x)x,x)=cG =

(8.1)

that is iff Kp = 2c. The idea is to differentiate (8.1) in such a smart way to get all the informations we need. We start with a couple of computational lemmas. L e m m a 8 . 1 . Let F: T1,0M - » 1 + be a strongly pseudoconvex Finsler metric on a complex manifold M. Then ((VwwQ)(H,K)x,x) {(V n)(H,K)X,x)--

=--(T»(H,0(K,W)),X) (TH(H,6(K,W)),X)

In particular,

forallWeV eH. for all W € V and H, H, K Ken.

((vwn)(H,x)x,x) = o for all W £ V and H € H. Proof: Since we are interested only in the horizontal part, we may replace fi by QH = Q% ® dz13 ® 6a.

Since Vw dz0 = 0 and V - ^ 6 a = 0, we have Vwn"

= (VWQ%) 9 dz0 ® Sa.

Again, we only need the horizontal part, that is 1 v P jr(Vw«|) = R%^(W) dz» A dz". llP^{W)dz' Adz P'HCVW"*) = WWiR.%^) (R%»*)dz"dz» AAdrdz - --R%

30 Recalling (4.2), taking H, K, LeHwe

get

1 H <(V K)L, = Ga[W{R%^) - R ^ {(V {H,K)L, R l i^ ^- ^ mW wm^ L^ * wn"(ff, wn X) x) =

[d^(r^) ^ i t , + a,(r^^(r^)) = -Ga[d^s &p{v%.^n^d^v% p(r^) -- W abD{Y^)) ra0a6p(r^)T^]H^K^L0m

-

a = = -GalSfidjiTp.^) -Ga[8Dd^r%j -- r; r ^;,9f 9 f (r (r%j + a,(rg„)« a,(rg„)«PP(rf (rfM +W feOUiX M)) + ;M) + ;M)r^

p a p f po

'■0a'- */;PL 7;.M

- «P-(rQ;M)r^ -

T%a6p{T"j%\mWL^m

(where r? ;M 3^(r^) and we used Lemma 2.1.ii)), ; M = Sy(r^) r^GadgiFf.^) G0a8i,{F^.^) -- G G ^^rrr ^. -rr"? ^ = -[5p(G -MGada9/9(r^ ;£;G0a5(r".M) + G^S^T^) 0(r^));/i)) --- r^ f -- SfiGJ^jTif 6p(Garp.fl)r%lj -- GeaG 6-p{Tl)Y^\mWL^n 60a m p{T°JT^]H»WL<

(where we used G GaF^ ■■ = G0
= -[-SoiGvpTU +G 5 (r%.J- - 6pS(G aaF^)F^ = -[-S F^)F^ p(Ga0r^)+G0a6e(T^) 0a p p(G G0a -GSp(F^)F^}mK^L^Wy. 6p(Tl)T^}H"K^L^W^. 0a Hence ((VwQ)(H,K)x,x) = -[-*p(G*/3r";M) + ^ ^ ^ ( r ^ . ^ ) - *p(GQrg;M)rg? -

G ^ r g r ^ K t f ^ ^ ,

= 6p-(GQrfM)r^^^M^, (where we used (1.3) and v0T^ = T^), = GQsp-(r« ) r ^ f f ^ w ^ = (T"(ff,0(*:,wo), X ), because 0(tf, W) = -V^ K"W^6P, by (3.1). Finally,. = GC {(V Q)(H,x)x,x) ((VwwQ)(H,x)x,x) = = 0,

"H^Wi GaSp^^WH^Wi •Wi)iV

because Y^W = 0.

□ lfi

Lemma 8.2. Let F: T M —» E+ be a strongiy pseudoconvex Finsler metric on a complex manifold M. Then ((V ((Vvn)(H,K)x,x) = < Hr (e(H,V),K), " ( 0 ( t f , n t f )X,)X > vn)(H,K)X,x)----(T

31

for all V € V and H, K e H- In particular, ((vvn)(x,K)x,x)--= 0 for all V € V and K e H. Proof: Again it suffices to consider QH =fijg® dz@ ®Sa; so Vvfi" = (v^n^) ® dz13 ®sa-n°®

ui}(v) dz0 ®sa + n}® dz0 ® u>°(v)6a.

We are interested only in the horizontal part. Taking H, K € ft we get Ga(Vvna0)(H,W) = Ga[V(R%^) - R%.^V{V)\H*W = -Ga[dxs-v{T%j - sB(r%p)r^ + d^Mr^))

- Ta0j,(r>p)r^}vxH»K*

= -Ga[8Mn,») - n-,Mn*) - W ^ + SxTOMiT,.) a M rr p r ^ + r| «p(rx ) r^r? r^. r ^ / / ^ a iM iM p ■ a w , p + rgr«>(ri.Jr 0M wKx\vhH,iKV (where we used [^,9^] = r^.pdf), = -MGJeir^)) - r%BGad0(r?j - 6p(Garg.p)r£A + Gadx(r^)6e(r^) + G0a6,{TlJ

- GpaT%.^Tlfi - G0<J6-v{T^)Tlx]VxmW

(where we used 5p(Ga) = 0 and GaY%a = G0a), = -{-6p(Ga0T%ltt) + 6,{dp6^Gx)) + r\.PGa0T^ - 6*{GaT%p)Tlx + Gadx(Ta0a)8P(r^) + G0aS,(T^) - G0aT%;iiT%P - G0a8*{T°p)Tlx)VxH»Kt (where we used Garf;/X = 0 and G a r£ ; / i = 6„(GX)), = -l-SoiGapTl^) + G0asp(r^)

+ 6D{d06^Gx)) - Su(GaTa0.p)T^x + Gadx(T%)Sp(T^) - G0a6p(r<>p)rZx]vxH»Kt.

Furthermore, GaO^(H,K)u}(V) = -G^iT^

+

= -[So(S^G^)) +

F^M^WJx^^W Gla6,(ri)]r}xVxH>*Kf;

X Gaau*(V)Sl}(H,K) -GJ°x[6+0{TlJ+T}MTlW H»To; u>5(V)H}(H,K)==-GaT°xM?lJ T}MT»WXH»K»

= -G^oiTU

+

r}M^Wx^K^.

32 Summing up we find <(Vyfi)(ff,tf)X,X> <(Vyfi)(ff,i?)x,X> = -[-6*(G -hMG ^ r ^ ) + SptfoS^Gx)) SptfoS^Gx)) - 6 5p(G + + Gad Gxa(T% dx(rar0cr )6)S a0r%J 0{GaT% a r^p)Yl ; p )r^ x P(Tf p(T^) ll) L + Gfi,60
= 6pp(G (Gaavvf,f,T%. T%. + p)1* = -- [[ M M «« % % ** M M (( G GA A )) )) -- 6 p)1* x x + xx i + Gix5 >{TlW H»K* + G^SS>{Vj ll)]V H' K^

0 a 0 GG avavdx{T dx{Ta0a0a)6 )6ff{V^) {V^)

X -[-^(G; Gxa^W CxM^lW^^ P )r^ A - Gx = -[S GxM^lW ^W B(G;PWU aSe(T^)+ + 0 0 0), (where we used v^d^Gx) = v Gx ^ v^^fix, ^GXa0 --== 0), (where we used t ^ « M ( G x ) = v^Gx^0 - v^V^Gxa - vv0T^G Xa0 = S^G.^T^H^W = SoiGaY^Vl^H^W = S^G.^T^H^W = SoiGaY^Vl^H^W A = G ff ff 5 = G 5 55 (( rr ff pp )) rr ££ AA V V \ f/ff ^^ ==

H (TH(r(d(H,V),K), x)(e(H,V),K),x).

The final assertion follows from 6(x, V) = 0.

□

In the following computations we shall need some symmetries of the curvature operator, summarized in L e m m a 8.3. Let F: T1,0M —> E + be a strongly pseudoconvex Finsler metric on a complex manifold M. Then i) (Cl(H, x)x, X) =
(dHe(H,x,x),x)=o

(8.2)

for all H eeH; ft; ii) (Q(H, K)XX> , x) ) for (n(XX,, K)H, Xx) for allH,KeH allH, K eH iff iff (n(H,K) x)--==
(8.3)

eH. for all H,K H, Ken. Proof: It follows immediately from (5.4) and Proposition 4.1.

□

Now we can start. The first step is: Proposition 8.4. Let F: Tl'°M —> K+ be a strongly pseudoconvex Finsler metric on a complex manifold M, with constant holomorphic curvature 1c e ffi. Then

(5He(H,x,x),x)=o

(8.4)

cGX. T rHH{X,X) {x,X)- == CGX-

(8.5) [8.5)

for all H e H iff

33 furthermore, they both imply (Q(X,K)X,X)=CG( X,K)X,K) (tl(x,K)x,x)----cG(

(8.6)

for all K € H. Proof: Take W g V a n d let K = ®(W) 6 H\ note that VWX = 0 and VWX = @(W) = K. Then W(cG W(cG22)=2cG{ ) --= 2cG( X,K); X,K);

w(U(x, x)x, x) = <(vwn)(x, x)x, x) + (8.7) = (fi(x,tf)x,x} + = cG 2 and hence (8.7) yields

(Q{ (n(X,K)x,x) X,K)x,x)

= 2cG(x,K)---{T(THH(X,X),K). = 2cG(x,*:> (X,X),K).

(8.8)

//

Subtracting ( r ( x , x). K) = ( ^ ( x . x ) X i ^ ) to both sides, we find that (8.5) holds if and only if

{nH(x,K)x,x) = (nH(x,x)x,K)

for all K € H, that is, recalling (5.5), iff (tt(K,x)x,x) == (n(x,x)K,x), (Sl(x,X)K,X), MK,X)X,X)-and thus, by Lemma 8.3, iff (8.4) holds. Finally, if (8.5) holds, (8.8) yields (8.6).

□

The second step requires (8.3): Proposition 8.5. Let F: Tl'°M —> 1 + be a strongly pseudoconvex Finsler metric on a complex manifold M with constant holomorphic curvature 1c € M. Assume that (8.2) holds. Then (n(H,K)X,x) ++(n(x,K)H, =-c{(H,x)(x,K) c{(H,X){x,K) {n(H,K)x,x) (n(x,K)H,x}X)

+ (X,x)(H,K)} (x,x)(H,K)},, +

(8.9)

for all H, K e H. In particular, if (8.3) holds then (H,-K)}

(8.10)

for allH, K G H. Proof: Take V, W € V such that 9 ( F ) = # and B{W) = if and extend them in any way to sections of V (and thus extend H and K as sections of H via Q). We have V(cG(X, K)) = c[{H, X)(X, K) + G(H, K) + G(X, VVK)],

34 and

V(Q(X,K)X,X) = ((vvmx,K)x,x) = (Sl(H,K)x,x) (U(H,K)x,x)

+ {ii{H,K)x,x) + (n(xyvK)x,x)

+ («(x,v(tl{x,VvK)X,X) 7K)X,x)

+

+ (n(x,K)H,x)

(a{x,K)H,x),
thanks to Lemma 8.2. Since (8.2) holds, we can use Lemma 8.4 (that is, (8.6) applied both to K and to VyK) to get exactly (8.9). Finally, (8.10) follows from Lemma 8.3. □ So we have obtained one of the hermitian product terms. This immediately yields one of the symmetric product terms: Proposition 8.6. Let F: r 1 , 0 M —> E + be a strongly pseudoconvex Finsler metric on a complex manifold M with constant holomorphic curvature 2c € ffi. Assume that (8.3) holds. Then rH(K,x)- =-\{(K,X)X T»(K,X) ^{(K,X)X+(X,X)K} + (X,X)K)

(8.11)

for all K € H. In particular, ({H,r (K,x))) =--(X,X)((H,K)) | <x,x) ((#,#)> IB,'. H^(K,m-

(8.12)

oH H K (= X for allH,Ke H.

Proof: We get H

(H,i ' (K,x)) ==--(H,Q(K,x)x)-(H,TH(K,X)) (H,n(K,x)x) == {Q(X,K)H, (n(x,K)H,Xx)) for all H, K € H, thanks to Lemma 4.2 and (5.5). Then (8.10) yields (8.11), and (8.12) follows immediately. □ For the other symmetric product term we need the weak Kahler condition: Proposition 8.7. Let F:T1,0M M —► m+ be a weaJdy Kahler-Finsler metric on a complex manifold M such that (8.3) hoids. Then

w-

H ({H,rrH (x,K)))=0 (x,K))) == 0

for allH, Ke H. Proof: The Proof: The weak Kahler condition {0(H,x),x) e H implies (8(B,X),X)-- == 0 for all H € W H" K <= -W VH,KeH

(((Vj?e)(H,x),x) ( V ^ ) ( H , X ) , X ) === 0 a

(8.13) a

because V^X = 0 = VKX. Now, writing 6 = 6 ® 6a, we have V ^ = ( V ^ 5 ) Sa and a V-^0 / f « ff(r^) (r» „) dz» dz" A dz" dzv + Ws KT6 y A dz". dz' f(T^) f(TZy) ^ v^ a == K-S

35

Therefore (8.13) implies

Ga\8rVZJ - Sr(T%ff)]H't1P^ = 0

(8.14)

for all H, K € H. Writing the curvature in local coordinates we find : (n(x,A:)/f,x> = -G a [5 f (r° „) + T%6r(V>„ )]H»Krv" ,i 7

(Q(X,K)H,X) = -Ga[Brp%¥) + mm, #)*,*> = -GJSAT"...) + So (8.14) yields So (8.14) yields (CI(X,K)H,X)- - (n(H,K)X,x)

=

T$m6r(P$,)]R i Z 'r, T°6AT"„)]H' Wvv.

-Gar^6f(r^)H^Krv-

(n(x,K)H,x) - (n(H,K)x,x) = -Gar*Ha6r{r°v)H»Wvv = {{H,Ta {X,K))), = ((H,r (X,K)h

a□ ana tne assertion assertion follows rouows from trom (8.3). (p.i and the We are left with the last term: We are left with the last term: Proposition 8.8. Let F: TlfiM — ► M+ be a strongly pseudoconvex Finsler metric on a complex manifold M with constant holomorphic curvature 2c G E. Assume that (8.2) hoJds. Then (Sl{H,x)K,x) = c{{H,x)(K>X) for aJlH.K

+ (X,X)((H,K))}

6 H.

Proof: First of all, we have

(n(H,x)x,x) = (xMx,H)x) = = CG(H,X,),

(8.15)

by (5.5) and (8.6). Now take W &V such that Q(W) = K; then W{cG(H,X)) ==--c{{K,x)(H, W(cG(tf,x)) c{(K,xW,x)X)

+ G(V W-H,x)}, +G{V wH,x)},

w(ti(H,x)x,x) = {(vwn)(H,x)x,x) + (n(VwH,x)x,x) + {n(H,x)K,x), and so (8.15) yields (Q(H, x)K,X) = c(K,X)(H,x)

~ <(V w n)(ff,x)x,x)-

Now Lemma 8.2 gives H

((vWwQ)(H,x)x,x) w),x),x) == {n(o(H,w), {Sl(0(H,W),x)x,x) {(V Q)(H,X)X,X)-- = = (r (e(H, (TH(0(H,W),X),X)-x)x,x) = cG(6(H.W).v). = cG{6(H,W),x), again by (8.15). But -G^taK^H" {0(H,W) ) = = -G^K'H" (9{H,W),<X X) and we are done.

= #.*». = - «-((H,K)), D

36 We can finally collect all our computations in Corollary 8.9. Let F: TlfiM —> M+ be a weakly Kahler-Finsler metric on a com­ plex manifold M. Assume F has constant holomorphic curvature 2c € K and that (8.3) holds. Then Re[{fi(x,K)H,X) X)- --{Q(H,x)K, (n(H,x)K,XX) ) Re[(n(x,K)H,

H + ({H,T + ({H,TH(K,X))) (K,X)))-

H

(x, H(K)))\ --((H,r {(H,T X,K)))]

»1

^Re\G{(H,K) -{(H,K))} -2{K, == %RB[G{{H,K) - {(H,K))} + + (H,(H,x){(x,K)-2(K, X] X)}] X){(X,K) (8.16) for 6 H. forallH,K all H,KeH. Proof: It follows from Propositions 8.5, 8.6, 8.7, 8.8 and Corollary 7.2.

□

We are then able to prove the announced T h e o r e m 8.10. Let F:TlfiM —» K+ be a complete Finsler metric on a simply connected complex manifold M. Assume that: i) F is Kahler; ii) Fhas nonpositive constant holomorphic curvature 2c < 0; iii) (dH6(H, x,K), x) = 0 for all H, K € H; iv) the indicatrices of F are strongly convex. Then exp p : T^°M —> M is a homeomorphism, and a smooth diffeomorphism outside the origin, for any p e M. Furthermore, M is foliated by isometric totally geodesic holomorphic embeddings of the unit disk A endowed with a suitable multiple of the Poincare metric if c < 0, or by isometric totally geodesic holomorphic embeddings ofC endowed with the euclidean metric ifc = 0. In particular, if 2c = —4 then F is the Kobayashi metric of M, and if c = 0 then the Kobayashi metric of M vanishes identically. Proof: Let R(H, K) denote the left-hand side in (8.16) — i.e., the curvature terms of (7.1). Corollary 8.9 yields R(H, H) = | Re{G[(H, H) - ((H, H))] + (H, x) {(x, H) - 2(H,

X >]}.

In particular, if H = \ we get

= o, R(x,x)R{X,X)=Q, and if (H, x) = 0 we get R(H, H) = ~ Re[(H, H) - ((if, H))] = ~ Re[(iH, iH) + ((iH, iff»]. Now, in local coordinates the quadratic form H H-Re[(ff,ff) + ((ff,ff})]

(8.17)

37 is represented by the Hessian of G; by iv), it is positive definite. So (H,x) = 0 implies R(H,H)
= 0. ((H,;r"(x,K)})((H,TH(X,K)))=0-

References 1. M. Abate, Iteration theory of holomorphic maps on taut manifolds, (Mediterra­ nean Press, Cosenza, 1989). 2. M. Abate and G. Patrizio, Uniqueness of complex geodesies and characterization of circular domains, Man. Math. 74 (1992), 277-297. 3. M. Abate and G. Patrizio, Holomorphic curvature of Finsler metrics and com­ plex geodesies, to appear in J. Geom. Anal. 4. M. Abate and G. Patrizio, Complex geodesies and Finsler metrics, to appear in Proceedings of the Semester in Complex Analysis, Banach Center, Warsaw, 1992 5. M. Abate and G. Patrizio, A global approach to Finsler metrics, in preparation. 6. L. Auslander, On the use of forms in the variational calculations, Pac. J. Math. 5 (1955), 853-859. 7. L. Auslander, On curvature in Finsler geometry, Trans. Am. Math. Soc. 79 (1955), 378-388. 8. D. Bao and S. S. Chern, On a notable connection in Finsler geometry, Houston J. Math. 19 (1993), 138-180. 9. A. Bejancu, Finsler geometry and applications, (Ellis Horwood Limited, Chichester, 1990). 10. E. Cartan, Les espaces de Finsler, (Hermann, Paris, 1934). 11. S. S. Chern, On Finsler geometry, C.R. Acad. Sc. Paris 314 (1992), 757-761. 12. J. J. Faran, Hermitian Finsler metrics and the Kobayashi metric, J. Diff. Geom. 31 (1990), 601-625. 13. M. Fukui, Complex Finsler manifolds, J. Math. Kyoto Univ. 29 (1989), 609624.

38 14. M. Jarnicki and P. Pflug, Invariant distances and metrics in complex analysis, (de Gruyter, Berlin, 1993). 15. S. Kobayashi, Invariant distances on complex manifolds and holomorphic map­ pings, J. Math. Soc. Japan 19 (1967), 460-480. 16. S. Kobayashi, Hyperbolic manifolds and holomorphic mappings, (Dekker, New York, 1970). 17. S. Kobayashi, Negative vector bundles and complex Finsler structures, Nagoya Math. J. 57 (1975), 153-166. 18. S. Kobayashi, Intrinsic distances, measures and geometric function theory, Bull. Am. Math. Soc. 82 (1976), 357-416. 19. S. Kobayashi and K. Nomizu, Foundations of differential geometry, vol. I! (Wi­ ley, New York, 1969). 20. S. Lang, An introduction to complex hyperbolic spaces, (Springer, New York, 1987). 21. L. Lempert, La metrique de Kobayashi et la representation des domaines sur la boule, Bull. Soc. Math. France 109 (1981), 427-474. 22. M. Matsumoto, Foundations of Finsler geometry and special Finsler spaces, (Kaiseisha Press, Otsu Japan, 1966) 23. M. Y. Pang, Finsler metrics with the properties of the Kobayashi metric on convex domains, Publications Mathematiques 36 (1992), 131-155. 24. M. Y. Pang, Smoothness of the Kobayashi metric of non-convex domains, Int. J. Math. 4 (1993), 953-987. 25. H. L. Royden, Complex Finsler metrics, in "Proceedings of Summer Research Conference," Contemporary Mathematics (AMS, Providence, 1984), pp.119124. 26. H. Rund, The differential geometry of Finsler spaces! Springer, (Berlin, 1959). 27. H. Rund, Generalized metrics on complex manifolds, Math. Nach. 34 (1967), 55-77. 28. M. Suzuki, The intrinsic metrics on the domains in Cn!, Math. Rep. Toyama Univ. 6 (1983), 143-177. 29. B. Wong, On the holomorphic sectional curvature of some intrinsic metrics, Proc. Am. Math. Soc. 65 (1977), 57-61. 30. H. Wu, A remark on holomorphic sectional curvature, Indiana Math. J. 22 (1973), 1103-1108.

39 STABLE SHEAVES AND EINSTEIN-HERMITIAN METRICS SHIGETOSHI BANDO Mathematical Institute, Tohoku University, Sendai, 980-77, Japan and

YUM-TONG SIU Department of Mathematics, Harvard University, Cambridge, MA021S8, USA

After the pioneering work of Kobayashi (cf. [8]), the relation between stable holomorphic vector bundles and Einstein-Hermitian connections is now quite wellunderstood by works of Donaldson, Narasimhan, Seshadri, Simpson, Uhlenbeck, Yau and others (cf. [3,4, 5,9,12,13,15,16,17,18,19,20]). The purpose of this paper is to generalize the results for vector bundles to the case of reflexive sheaves by extending the notion of "Einstein-Hermitian metrics". We recall the definition of stable sheaves and introduce the notion of admissible Hermitian metrics on torsion free sheaves. Definition. Let £ be a torsion free coherent analytic sheaf on an n-dimensional compact Kahler manifold (X, ui) with the fundamental form w. We define the slope n(£) of £ by H{£) = (d(£) U [w]n_1)[A-]/rank£:. 1) £ is called (semi-)stable, if for any subsheaf S such that 0 < rank<S < rank£ we have fi(S) < fi(£) (n(S) < (i(£)), respectively. 2) £ is called poly-stable, if it is a direct sum of stable sheaves £* with the same slope (i(£i) = n(£). 3) A Hermitian metric h of £ defined on the locally free part of £ is called admissible, if its curvature tensor F is square integrable and its trace AF with respect to the base metric w is uniformly bounded. The admissible Hermitian metrics play the role of the ordinary Hermitian metrics for vector bundles. Theorem 1. Any reflexive sheaf £ on an n-dimensional compact Kahler manifold (X, w) admits an admissible Hermitian metric. The meaning of admissibleness of a Hermitian metric is understood by the following Theorem 2. Let (E, h) be a holomorphic vector bundle with a Hermitian metric h defined on a Kahler manifold (Y, w) (not necessary compact nor complete) outside a closed subset S with locally finite Hausdorffmeasure of real co-dimension 4. Assume that its curvature tensor F is locally square integrable on Y, then a) E extends to the whole space Y as a reflexive sheaf £, and for any local section s € T{U, £), log+ h{s, s) belongs to H^.

40 b) If AF is locally bounded, then h(s, s) is locally bounded, and h belong? to L\Xoc for any finite p where £ is locally free. c) If (E, h) is Einstein-Hermitian, then h smoothly extends as an Einstein-Hermitian metric over the place where £ is locally free. This is a combination of the removable singularity theorem [1] and the slicing theorem [14]. Theorem 2 and the compactness theorem for Yang-Mills connections [10] yield the following Corollary which generalizes the results of Uhlenbeck [16], [17] to the higher dimensional case. Corollary 2. Let (Ei, hi) be a sequence of Einstein-Hermitian holomorphic vector bundles with fixed first and second Chern classes on a compact Kahler manifold (X, uj). Then there exist a subsequence, which we still call (Ei, ht), and a reflexive sheaf £ with an admissible Einstein-Hermitian metric h such that under suitable gauge change (Ei,hi) smoothly converges to (£,h) outside a closed subset with Unite Hausdorff measure of real co-dimension 4. For examples of convergence to sheaves in higher dimension, see [11]. Our main result is Theorem 3. A reflexive sheaf £ on an n-dimensional compact Kahler manifold (X, u) admits an admissible Einstein-Hermitian metric, if and only if £ is polystable. Corollary 3. For a poly-stable reflexive sheaf £, we have the Bogomolov (2rc2(£) - ( r - l)c,(£)2)

inequality

U M " " 2 [ X ] > 0,

where r = rank£. And t i e equality holds if and only if £ is locally free and its Einstein-Hermitian metric gives a projectively flat connection. For non-stable ones we have the following result. Theorem 4. Let £ be a reflexive sheaf on (X, ui). If£ is not stable, then it "breaks up" into a direct sum of Einstein-Hermitian sheaves via the heat equation considered below. It is an interesting question whether the sum is isomorphic to the reflexization of Gr£ or not. Remark. We can work with torsion free sheaves. But since it would result in some complication of the statement and since metrics essentially deal with the reflexization, we restrict ourselves to reflexive sheaves. The idea of the proof of Theorem 1, 3 and 4 is as follows. First let us assume that X is projective algebraic. Then there exits a resolution • • ■ —► E^ —> E% —► £v —► 0 of the dual sheaf £ v of a reflexive sheaf £ on (X, w) by holomorphic vector bundles. Taking its dual 0 —► £ —► E0 —► Ex, we get an embedding of £ into a holomorphic vector bundle E0 as a holomorphic subbundle outside S where £ fails

41 to be locally free. We fix an arbitrary Hermitian metric ho on EQ. It induces a Hermitian metric h on £ outside S. For the general case, we perform the above construction locally, and patch the locally defined Hermitian metrics to get the global h. We deform h by the heat equation

dft _ 1 = =-(V=1AF -{V-IAF - - K£)i), \(£)I), dt

^h-1h

where X(£) = 2irnn{£)/[u\n[X] and I stands the identity endomorphism of the fibre. We show that the solution for t > 0 is an admissible one. If £ is stable, we obtain an admissible Einstein-Hermitian metric as the limit as t tends to infinity. 1. Removable singularity theorem Here we consider Theorem 2. As the theorem is of local nature, we can assume Y to be a domain in C n , S of finite Hausdorff measure of real co-dimension 4 and F square integrable. The following is a direct consequence of the removable singularity theorem [1] and the slicing theorem [14]. Lemma 1. Let (E, h) be a Hermitian holomorphic vector bundle defined on the complement of a closed subset S of a product of balls B 2 x B™ -2 C C 2 x C™ -2 Assume that S has finite Hausdorff measure of real co-dimension 4 and there exists a compact subset K of B 2 such that S is contained in K x B " ~ 2 If the curvature tensor of (E, h) is square integrable, then E extends to the whole space B 2 x B " - 2 as a reflexive sheaf. Let us first consider a) of Theorem 2. Fix an arbitrary point, say 0, in Y. Then by assumption, for a generic projection p : Y C C™ —► C™ -2 the set S n p - 1 ( 0 ) consists of a countable number of points which may accumulate only at 0. Shrinking the domain Y, we may assume we are in the situation of Lemma 1. Thus E extends as a reflexive sheaf £. Fix an arbitrary section s e T(Y,£). We put Yt = p _ 1 ( t ) , St = SnYt and Ut = log + h(s, s)\Y . Let

Atut > -41*11, where A t is the Laplacian on Yt. Then we get for any e > 0 / JYt

|V(M)|2 < /

4\Ft\4>2ut + f

JYt

|V0| 2 « 2

JYt

< e f ( M ) 2 + 4£- x / JYt

** Vt

^2|Ft|2+ / JYt

\V<j>\\l

42

We apply the Poincare inequality and get /

\V(ut)\2<2 f \V2\Ftt\

JYt

JYt

JYt

Here and hereafter we denote a general positive constant by C which may differ in the different appearance. We integrate it in t and get for a compact subset K' C B n - 2

I Jp-l(K')

\V'(
[

\V<j>\2u2 + C f

Jp-\(K')

2\F( tI 2 .

Jp-1(.K')

where V stands the derivation along the fibres of the projection p. Since we can use any generic projection p, it shows the desired result u € Hloc. Once log + h(s,s) € H\oc is known, it is easy to see that A F e L^c log h(s, s) £ Lj^c in view of

implies

+

A l o g + h{s,s) > - 2 | A F | . c) is a consequence of b) and the standard regularity theorem of linear elliptic partial differential equations. The last half of b) follows from Proposition 1. Let h be a ranic r Hermitian matrix valued function defined on an n-dimensional Kahler manifold (Y, u) which belongs to H1 Assume that h and hr1 are uniformly bounded and it satisfies A^a/i/i" 1 ) = /

(1)

in a weak sense with a uniformly bounded function f, then h belongs to C}'a for anyO < a < 1 and admits an estimate depending only on ||/i||£,°°, | | ^ _ 1 | U ~ , | | / | | L ° ° and the geometry of (Y, UJ). One can show Proposition 1 by modifying the argument which was used to show the corresponding result for harmonic mappings in [7]. Since the proof is essentially the same, we restrict ourselves only to pointing out the places where change is necessary. Again we work locally. We denote the complex Laplacian by D = A-v/^499 and its Green's function by G(x, y). Step 1: h is continuous. The equation (1) means that for any matrix valued function k € L°° n H1 with compact support we have 71 n1_1 '1dhJ Iftidhhtidhh^dku - - + ftTfkw ftrfkun n:== 0. 0.

43

Let 0 be a cut off function such that = 1 on B(xo,p), = 0 outside B(xo,2p) and |V| < 2/»_1. We substitute k —
f

< C\\h\\L~(l + ||/|U
JB(xo,p)

as in [7]. Thus for any 0 < p « 1 there exists po such that p < po < yfp and * / ivftl2 < -£—. Po JB(X0,P0) -log/9 Let ho be the average of h on the ball B(xo,po). Then there exists a positive constant C such that n(tr h /i 0 + %Thah - 2r) > - C . Hence applying the Poincare inequality we obtain sup \h - h0]2 < C sup (tr/,/i0 + trfto/i - 2r) B(ao,2 _1 «))

B(i 0 ,2-'p)

< C(po2n [

JEB{x0,po)


f

\h-h0\2

+ p20)

\Vh\2 + p2)

1 JB(xo,po) + P). -l0g/9

/9 Step 2: h is Holder continuous. Fix- la0 gsufficiently small positive number p and Step 2: /i is Holder continuous. Fix a sufficiently small positive number p and let ho be the average of h on B(xo,2p)\B(xo,p)- Since h is continuous, there exist positive constants a and C such that in a neighborhood of xo it holds a\h-h0\2 >o|V/i|2-C. Multiplying it by <j>G(-,xo), integrating by parts and again applying the Poincare inequality, we obtain

p2~2n

f JB(XO,P)

\Vh\2
G(;xQ)\Vh\2 JB{XO,P)


Ih-htf + p*)

J B(x0,2p)\B(x0,p)


f

\Vh\2 + p2),

JB(x0,2p)\B{xo,p)

which implies the Holder continuity. Step 3: h belongs to Cx'a Instead of the formula (6.32) in the page 84 in [7], we apply the following formula. For k e H1 with compact support f trdhdku"-1 + / trdhih^ho - I)Bku)n-1 + f tifh0kun with h0 = h(x0)- Then the argument in [7] shows the desired result.

= 0,

44

2. Heat kernel estimate For later use, we show the uniform boundedness of heat kernels for a certain type of degeneration of metrics. Proposition 2. Let (X, w) be an n-dimensional compact Kahler manifold and n : Y —► X a Wowing up with non-singular center. Fix an arbitrary Kahler metric 6 onY and set uit = it*u> + e6 for 0 < e < 1. Let He be the heat kernel with respect to the metric we, then we have a uniform estimate 0 < Hc < C{t~n + 1) with a positive constant C. For the proof we start with the following lemma. Lemma 2. If a real n-dimensional Riemannian manifold (M,g) satisfies the Sobolev inequality (J\\^)~

< SJ\V4>\

for e C&M),

then the product manifold M x R also satisfies the Sobolev inequality with the Sobolev constant S n / n + 1 Proof. For an arbitrary point ( i , ( ) i n M x R and a function 6 CQ{M x R), we have \
and (/

\
VM

<S [ \V{y,t)\dy.

'

JM

Thus /

W^

= I dt [ \<j>(y,t)\*?dy

JMxR

JR

JM

<j^dt[Jj<j>(y,t)\^dy)~^J

\
<S ( dt [ \V<j>\(y,t)dy([ dy [ \V
= S([

JM

VM

JR

I

|V0|)^

Lemma 3. Let (X, ui) be an n-dimensional compact Kahler manifold and ir : Y —► X a blowing up with non-singular center. Fix an arbitrary Kahler metric 6 onY and set ue = 7r*w + eO for 0 < e < 1. Then (Y, we) satisfies the Sobolev inequality (j \*\A)~

<S I \V\

for
C\Y\

45

with S independent

ofe.

Proof. We denote the blowing up of Cfc at the origin by Cfc and its restriction to a ball B fc by B fc . We fix an arbitrary point p in X. Then in a neighborhood of p the blowing up is given by n : B fc x B ' —► B fc x B ' for some k, I. We first show the uniform Sobolev inequality for the functions <j> whose supports are contained in Bk x B ' : 2n-l

( / |0|^T)

^

2

" <S j\V\.

For two metrics go, Si such that C\g$ < gi < C2|2), where (z,w) is the coordinate of B fc x B ' . Changing the coordinate by (z, w) —► v ^ ( z ! u ')i w e reduce the problem to the verification of Sobolev inequality on (Cfc x Cl, \/^idd(\z\2 + log|z| 2 + H 2 ) ) . Since that for (C fc , v ^ T d d ( | z | 2 + log|z| 2 )) is trivial, Lemma 2 implies the uniform Sobolev inequality on B fc x B ' . Let p, be a partition of unity on X corresponding to the covering given by B fc x B''s considered above. Then, since it holds sup|V7r*j0i|We < sup|Vpj| w , summing up the Sobolev inequalities for {■K*Pi)4> we get the desired uniform Sobolev inequality on Y The following result obtained by Cheng and Li [2] concludes the proof of Propo­ sition 2. Lemma 4. If a reai n-dimensional compact Riemannian manifold (M, g) satisfies the Sobolev inequality

(J\4>\^)~

<sj\V4>\ + \eC\M),

then the heat kernel H satisfies the estimate 0 < H < C(t~nl2 constant C which depends only on n and S.

+ 1) with a positive

3. Existence of admissible Hermitian metrics Let £ be a rank r reflexive sheaf on an n-dimensional compact Kahler manifold (X, u). We take a finite cover { Ua } and local resolutions • • • —► £ ^ a —► E^a —► £ v | r / —> 0 of the dual sheaf £v by holomorphic vector bundles, which admits the following commutative diagram.

•■• ^

Ka\uanu0

1 ■•• ^

EUuanu0

— Ka\uanu0

w ^

1* El0\uanu0

^

v

£v\uanu0

— o

II v

- £ £ \uanu0

— o

46 Taking its dual we get the following commutative diagram. 0

y p

► £\u -*=♦ Eo, E0,aa\U \uaar\Ua nuff 0 — —» £\u„nu,. anu0

-^ ^

E E1
K 0o

V0

-- £\u — Eoji\u £\uanu 0 0 ■***Eo,0\u ar\u anu anu 0 $-^ ^

E\u Elt0 nuQ0nu0 liPa\u

This allows us to see £ locally as a subsheaf of locally denned holomorphic vector bundles Eoa. The relfexive sheaf £ is locally free outside a subvariety S of codimension at least 3 and outside S the embedding £ —> Eoa realizes £ as a holomorphic subbundle. We take arbitrary Hermitian metrics ha on Eo,a and a partition of unity pa subject to the covering { Ua }. Set ha = YlPp^c^eThen, the Hermitian metrics { ha } on { £ o , a } give the same pull back metric
= ~(V=1AF

- \{£)I),

(2)

where X(£) = 2im(j.(£)/[u!]n[X] and / stands the identity endomorphism of the fibre. (To be precise, the heat equation is to be satisfied outside 5.) Since the equation has singularity on 5 , we make a regularization to solve it. We take blowing up with non-singular center finite times iti : X{ —► -Xi-i, (t = 1 , . . . k), XQ = X such that n = ■Kkirk-i ■ ■ - f i : -Xfc —* X is biholomorphic outside 5 and the kernel of 7r*<£0,a : n*Eo
(2)fc

Here A.k,e means the trace taken with respect to the metric wjt>e. Since every thing is now in the smooth category, the result of Donaldson [4] says that (2)fc has a solution untill infinite time. The curvature tensor satisfies the following. dAk€F —^-=Dt,eAfc,£F,

(3.i)t

47

d\Ak,(F\2 ^ ^ = aafcfc,£|A, |A fc*,*F\ , £ F| 2 2-- iVAfc,, |VA f c , tF| F|22, , dt d|A , £ F| ^ ^fcS ^
iL^

|Afc,£F|2 = - /

[f

(3.3)fc

|VAfcfc ,<,£F\\ |VA F|2,

(3.4)fc

|Afcfc,,££F|(t,y) F|(t,y)< f

|Afc,£F|(0,y),

(3.5)fc

|A F|(t,x) \Ak, I fc , £fF\(t,x)<

^F\(0,y] fffc,e,x,y)\Ai (t,x,y)|A fc,eF|(0,y), HUt

(3.6)fc

±[

Jy£Xk

(3.2)fc

Jxk

~ Jy£Xh ~ Jy€Xk Jy€Xk

where Qt>£ and Hk,e(t,x,y) and its heat kernel.

are the complex (crude) Laplacian with respect to u>fc,£

L e m m a 5. Let F be the curvature tensor of the initial metric h which is induced by the metric ha on Fo.a- Then its trace A i i £ F on (Xi, u>i]£), 0 < c, < 1, is uniformly integrable. Proof. Let Fo, a be the curvature tensor of the holomorphic Hermit ian vector bundle (Fo,a, h t t ) and pa the orthogonal projection onto £. We take a subcovering { Va } of { Ua } consisting of compact subdomains Va C Ua. Then on each domain Va we have the inequality y/^lF — V—^PaFo,aPa + y/—^dpaf\dpa < V—lpaFo,aPa < CLJ with some positive constant C, which shows that y/^lF is bounded from above on X^ It holds | A i | £ F | a £ £ < ( | A < > £ ( ^ F - CUJI)\ + | A i i £ C W | X £

< ntr(2Cw/ - - / ^ I F ) ufc1 < n t r ( 2 C W ---V-y/^lF)u>£~*, < ntr(2CW -^)"C;. with £i = ( 1 , 1 , . . . , 1). And since F gives the curvature of the holomorphic vector bundle E on Xk,

L

xk

t r ( 2 C W - v ^ T F ) ^ - 1 = ( 2 C r M - 2nc1(E)) U [ u * ^ ] " - 1 ^ * ! -

A direct calculation shows (cf. [8], [15]) L e m m a 6. Let (E, h) be a holomorphic Hermitian vector bundle of rank r on an n-dimensional compact Kahler manifold (X,u>). Then we have (2c 2 (F) -

Cl(F)

2

) U H " - 2 m = c / (|F| 2 - |AF| 2 )a/*, Jx

(2c2(E) - — C l ( £ ) 2 ) U M"- 2 [A-] = c / (|F°| 2 - |AF°| 2 ) W ", r Jx

48 where cn = (4n2n(n — 1))

1

and F° stands the trace free part.

Proposition 2 says that for a fixed e' = (ei, € 2 , . . . , «fc-i) the heat kernel Hk,t has a uniform bound for 0 < ek < 1. It is easy to see that Hk 0 and a uniform L°° bound for t > to > 0 or on a compact set disjoint from the exceptional set. Proposition 1 gives C 1 ' " estimate on h. Lemma 6 gives the uniform L2 bound on F. Thus we can substruct a subsequence which converges to a solution of the heat equation on Xk-i as ek tends to 0, ^-h'1 = -(^TA - ( / = l A Jffce__11((£t ,F ,F ~ - A(£)/), \{S)I), (2)fc_, dt at such that F has a uniform L2 bound for t > 0 and Ak-ir(L,F has a uniform L 1 bound for t > 0 and a uniform L°° bound for t > to > 0. It holds

iU^-L | /

I/

|Afc_^F|2=-/

\A \Ak-i,e'F\(t,y) k^c,F\(t,y) < [1

|VA |VAfcf c__MM
(3.4)*.

\Akk.^F\{Q,y), \A -i>e,F\(0,y),

(3.5)fc_i

Hk^c,(t,x,y)\Ak_lti,F\{0,y). ffe-M'(*,iB)3/)|Afc_ u'Fmy.

(3.6) fc _!

i

\A ,F\(t,x) < [f |Akfe-it£_i^F|(t,x)< JyeXk-i

Again taking the limit ek~i —► 0, we have a solution of the heat equation on Xk-2Repeating the argument, we obtain a solution of the heat equation (2) on X which gives the desired admissible Hermitian metric h for the positive time t > 0. Since £ and E is isomorphic outside 5 which has at least co-dimension 3, by Fubini's theorem we get

—c^u c jf (|(|F°| F°I2--- | A|AF°| U Hn "- i [ Xu ] >c F ° | ) w »" .

[2cC22(£) (2 (f) -- ^±Cl(£)2) r

And it holds

2

2

n

22

w

(4)

Jx

n r

2 |VAF| 2 (i f \VAF\ (t,y)< I |AF| \AF\22(t(t00,2/),y). )2/)

Jto

JX

Jto

JX

<-L

(5)

JX

4. Einstein-Hermitian metrics Let £ be a reflexive sheaf on an n-dimensional compact Kahler manifold (X, w). Then as shown in the previous section, we can solve the heat equation (2) with an admissible solution h(t) untill infinite time. By Simpson [13], the stablity condition on £ implies the existence of a subsequence h(ti), tt —► 00 which converges to an admissible Einstein-Hermitian metric. The inequality (4) implies (2c2(£)

n ~22[X] ^ Crr l-ci(enu ( f ) 2 ) UM M"\F°\\ 2u" " [X] >>ccnn Jx[ \F°\

L

49 for the Einstein-Hermitian metric, which shows Corollary 3. For a general £, the inequality (5) enables us to take a subsequence h(ti) such that / |VAF|a(t<,y)—0. Jx By the results in [18], [19], taking suitable gauge change we can further take a subsequence so that the sequence of the corresponding holomorphic connections converges to a weak solution of V A F = 0 outside a closed subset S" C X\S of locally finite Hausdorff measure of real co-dimension 4. According to the eigenspaces of AF, it splits into a sum of holomorphic Einstein-Hermitian vector bundles with I? curvature defined on X\(S U 5"), which extend to reflexive sheaves. The following proposition shows the poly-stability of an Einstein-Hermitian sheaf as in the holomorphic vector bundle case. Proposition 3. Let (£, h) be a reflexive sheaf with an admissible Einstein-Herm­ itian metric on a compact Kahler manifold (X,ui). According as n{£) < 0 or n{£) = 0, the sheaf £ admits only zero section or parallel sections s € T(X, £). Proof. If s is a global section of £, then by Theorem 2 b), \s\ is bounded on X and satisfies a|s|2 = |Vs|2-A(£)|s|2>0. Remark. We get the admissible Einstein-Hermitian metric solving the heat equa­ tion. It is also possible to get it as a limit of Einstein-Hermitan metrics of the holomorphic vector bundle E on Xk which is /i-stable with respect to the metrics Wfc,e for sufficiently small £, by taking e —> 0. Acknowledgements The first named author would like to express his thanks to Professors Itoh, Mabuchi, Noguchi and Ohsawa for helpful discussions and Professor Maruyama for pointing out an error in the first draft. He also would like to acknowledge his gratitude to Max-Planck-Institut fur Mathematik for hospitality. The main part of this work was done during his stay in Bonn. The second named author would like to acknowledge support from the National Science Foundation. References 1. S. Bando, Removable singularities of holomorphic vector bundles, Tohoku Math. J. 4 3 (1991), 61-67. 2. S. Y. Cheng and P. Li, Heat kernel estimates and lower bound of eigenvalues, Comment. Math. Helv. 56 (1981), 327-338. 3. S. K. Donaldson, A new proof of a theorem of Narasimhan and Seshadri, J. Diff. Geom. 18 (1983), 269-277.

50 4. S. K. Donaldson, Anti self-dual Yang-Mills connections over comlpex algebraic surfaces and stable vector bundles, Proc. London Math. Soc. 50 (1985), 1-26. 5. S. K. Donaldson, Infinite determinants, stable bundles and curvature, Duke Math. J. 54 (1987), 231-247. 6. D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, second edition, (Springer, Berlin and New York, 1983). 7. S. Hildebrant, Harmonic mappings of Riemannian manifolds, in "Harmonic mappings and minimal immersions", Lect. Notes in Math. 1161, (Springer, Berlin and New York, 1985), 1-117. 8. S. Kobayashi, Differential geometry of holomorphic vector bundles, Publ. Math. Soc. Japan, (Iwanami Shoten and Princeton Univ. Press, 1987). 9. J. Li and S.-T. Yau, Hermitian-Yang-Mills connection on non-Kahler manifolds, in "Mathematical aspects of string theory", Adv. Ser. Math. Phy. (World Scientific, Singapore, 1987), 560-573. 10. H. Nakajima, Compactness of the moduli space of Yang-Mills connections in higher dimensions, J. Math. Soc. Japan 40 (1988), 383-392. 11. T. Nitta, Compactfication of moduli spaces of Einstein-Hermitian connections for null-correlation bundles, in "Recent topics in differential and analytic geom­ etry" , Adv. Studies in Pure Math. 18, (Kinokuniya, Tokyo and Academic Press Inc., Boston, New York, 1990), 397-416. 12. M. S. Narasimhan and C. S. Seshadri, Stable and unitary vector bundles on a compact Riemann surface, Ann. Math. 82 (1965), 540-564. 13. C. Simpson, Constructing variation of Hodge structure using Yang-Mills theorey and applications to uniformization, J. Amer. Math. Soc. 1 (1988), 867-918. 14. Y.-T. Siu, A Hartogs type extension theorem for coherent analytic sheaves, Ann. of Math. 93 (1971), 166-188. 15. Y.-T. Siu, Lectures on Hermitian-Einstein metrics for stable bundles and KahlerEinstein metrics, (Birkhauser, Basel-Boston, 1987). 16. K. Uhlenbeck, Removable singularities in Yang-Mills fields, Comm. Math. Phys. 83 (1982), 11-30. 17. K. Uhlenbeck, Connections with IP-bounds on curvature, Comm. Math. Phys. 83 (1982), 31-42. 18. K. Uhlenbeck, A priori estimates for Yang-Mills fields, preprint. 19. K. Uhlenbeck and S.-T. Yau, On the existence of Hermitian- Yang-Mills connec­ tions in stable vector bundles, Comm. Pure Appl. Math. 39 (1986), S257-S293. 20. K. Uhlenbeck and S.-T. Yau, A note on our previous paper. On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Comm. Pure Appl. Math. 42 (1989), 703-707.

51

G E N E R A L I Z A T I O N S OF A L B A N E S E M A P P I N G S FOR NON-KAHLER MANIFOLDS

ICHIRO ENOKI Department of Mathematics, Faculty of Science, Osaka University Toyonaka, Osaka 560 Japan

0. Introduction Holomorphic forms on a compact, complex manifold M are not d-closed in gen­ eral. However, the ordinary Albanese mapping of M is related only with d-closed ones [1]. In this paper, in order to study the relations between the structuer of M and the spaces of holomorphic l-forms, including non-d-closed ones, we shall gen­ eralize the notion of Albanese mappings in two ways. The first one is a reduction with respect to holomorphic l-forms. The second concerns the Albanese torus. Firstly, we give the generalization as reductions. Definition. We say a (reduced) complex analytic space V has ample l-forms if the sheaf Qy of the Kahler differentials on V is genearted by global sections on V. The ordinary Albanese mapping can be regarded as an universal mapping to a Kahler space with ample l-forms. We remove this restriction to the target spaces. Definition. Let M be a compact complex manifold. A surjective holomorphic map­ ping p: M —► V to a complex space V with ample l-forms is called a Albanese reduction of M if for any surjective holomorphic mapping p': M —> V' to a complex space V' with ample l-forms there is an unique holomorphic map a : V —» V such that p' = a o p. T h e o r e m A . Every compact complex manifold has the Albanese

reduction.

Secondly, the Albanese torus can be generalized to complex parallelizable mani­ folds, which are quotient spaces G/T of complex (non-abelin in general) Lie groups by discrete subgroups T. Definition. A holomorphic mappnig A: M —> L of a compact complex manifold M to a compact complex parallelizable manifold L is is called a generalized Albanese mapping if for any holomorphic mapping A': M —* V to a compact complex paral­ lelizable manifold L' there exists an unique holomorphic map r: L —* V such that A' = r o A. We call L the generalized Albanese manifold of M. Compact complex parallelizable manifolds are always quotient spaces of complex Lie groups by discrete subgroups by Wong's theorem [2]. Moreover, we note that every holomorphic map, T: L —» L', between compact comlex parallelizable manifolds, L = G/T and L' = G'/V, is covered by a composition of a complex Lie group

52 homomorphism h: G —> G' and a left translation Lg by g e G', where G and G' are simply connected, T and T' are discrete. Thus, in the definiton above, we have the commutative diagram:

where 7r, n' are the projections to the quotient spaces. Not every campact complex manifold has the generalized Albanese manifold. We shall give a sufficient condition to exist: Theorem B. Let M be a compact complex manifold and a : M —► A the (ordinary) Albanese mapping of M. Let /\pa*: /\p H°(A,n\) -> H°(M,SfM) denote the linear map induced by a. Assume one of the follwing conditions: a) dim A > 3 and / \ a* is injective; b) dim A = 2 and /\ a* is injective; or c) dim A < 1. Then M has the generalized Albanese mapping A: M —» L. Corollary. If a compact complex manifold M has no nonconstant furnctions, then M has the generalized Albanese mapping.

meromorphic

In Section 1 of this paper we recall the relation between Lie groups and Lie-Hopf algebras, which is dual to the popular relation between Lie groups and Lie algebras. In Section 2 we consider the well-known mappings to a Lie group characterized by pull-backs of invariant 1-forms. Section 3 is devoted to study the structure of complex spaces with ample 1-forms. The proofs of Theorems A and B will be given in Section 4. We note that results in Sections 1,2 and 3 are needed only for the proof of Theorem B; the proof of Theorem A requires no preliminaries. 1. Lie groups a n d Lie-Hopf algebras We begin by recalling the relation between Lie groups and Lie-Hopf algebras. Definition 1.1. A (complex) Hopf algebra $ with comultiplication \i: $ —► $ g > $ is called a (complex) Lie-Hopf algebra if a) /z($) c A 2 * C $ 8 $ and b) (n <8> id - id ®n) o /x = 0.

.. y>n n e * form a basis, then there are detremined constants c3 so that l£tp\. 1 If ip ,... tp € * form a basis, then there are detremined constants cf so that

PW) = 2 Y* ^ ■i k

A

^

and C

^ = ~°iki'

53 We call cf-k structure constants of the Lie-Hopf algebra ($, ft) with respect to the basis <^1,...e$4>e$. )(X,Y) (Condition b in Definition 1.1 corresponds to the Jacobi identity on g.) Taking a dual basis X\,..., Xn e g of g, ip'(Xj) = 6ij, we have

lxj,xk] = Yti

For each complex Lie group G let Q(G) denote the space of all right invariant holomorphic 1-forms on G. Identifying /\p ft(G) with the space of all right invariant holomorphic p-forms on G, we can regard the exterior derivative d as comultiplication d: ft(G) —> A Cl(G). Then (Q(G), d) is a complex Lie-Hopf algebra and its dual is the Lie algebra of G (consisting of right invariant holomorphic vector fields). Thus the fundamental theorem on the relation between Lie groups and Lie algebras can be stated as follows: E a c t l . 2 . The category of finite dimensional complex Lie-Hopf algebras, ($, u) 's, is equivalent to the category of simply connected complex Lie groups via a contravariant functor ($,/i) -> G* with ($,/*) =* (fi(G$),d). 2. Mappings t o Lie g r o u p s Let M be a complex manifold. For a linear subspace $ C ff^M.ft]^), $ A p C H°(M, fi^) denote the image of A P $ under the natural linear map l

ep: f\H°{M,n /\H°(M,nlM) M)^ -

let

'H°(M,tf H°{M,tf> ).). MM

We call a linear subspace $ C H°(M, il]^) involutive if d $ C $ A 2 Moreover we call a pair ( $ , u) of a subspace <£ C H°(M, fl]^) and a comultiplication u.: $ —» /\ $ a Lie-Hopf subspace if 1) $ is involutive with d = e% o n on $ and 2) ($,/i) is a Lie-Hopf algebra. Lemma 2 . 1 . Let $ C H°{M,Q}M) be an involutive subspace. Ife-i\ /\2$ —» $ A 2 is injective, then the structure of Lie-Hopf subspace on $ is unique, if exists. If e^: A 3 * —* ^ A 3 J8 injective, then $ C H°(M,O,]^) has an unique structure of Lie-Hopf subspace ($,/*)■ Proof. If e2 is injective on A 2 *i t n e n t n e m a P ^ : * —» A 2 * with d = e 2 o u is unique. The injectivity of e$ on A 3 $ implies the injectivity of e2 on A 2 *• Hence

54 we have an unique linear map /i: $ —► / \ 3 > C $ ® 3 > such that d = e2 o /x on $. The relation d o d = 0 on $ can be written as d o d = e3 o {fj. ® id — id ®//) o /z = 0

on $.

This implies condition b) in Def 1.1 since ej, is injective on /\ $.

D

Let 7r: M —» M be the universal covering space of a complex manifold M. Fixing reference points o € M and o G M with 7r(o) = o, we identify the fundamental group 7Tj(M) = Ti(M, o) with the covering transformation group of M over M by the usual manner. For a Lie group G let R(g) denote the right translation of G by g e G. Let e 6 G be the identity element of G. Proposition 2.2. Let G be a simply connected complex Lie group. Let ( $ , / J ) be a finite dimensional Lie-Hopf subspace ofH°(M, 0.]^). Then, for any complex LieHopf algebra bomomorphism r\: (ft(G), d) —> ( $ , /z), there is an unique holomorphic mapping A: M —* G of the universal covering TT: M —► M to G such that ir'r) = A* and A(5) = e. Proof. Taking a basis, w 1 , . . . , ui", for J7(G) and the dual basis, Xi,..., Xn, for the Lie algebra of right invariant holomorphic vector fields on G, set w = ^ w1®Xi and ip = J^jJ?(w')®Xj. Then u; and

G and M x G —> M respectively. The Maurer-Cartan equation, dw = [w,w], means that the curvature of w vanishes. It follows that the curvature of ip also vanishes since r) is a Lie-Hopf algebra homomorphism. Therefore we have an unique horizontal lift M —> M x G through the point (o, e) € M x G, which induces, via the projection M x G —» G, an unique map A: M —» G with A(o) = e. Condition "horizontal" is nothing but A* = TT'T] on H(G); this also implies that A is holomorphic since so is tp.

O

In view of Fact 1.2 we have three corollaries. Corollary 2.3. For each finite dimensional Lie-Hopf subspace ( $ , /*) ofH°(M, il\f) there exist a triple (G, A, p) consisting of a simply connected complex Lie group G, a holomorphic mapping A: M —> G and an anti-homomorphism p: iti(M,o) —* G such that a ) ( T * ) _ 1 A * : (fi(G),d) -> ($,n) is a Lie-Hopf algebra isomorphism and A(o) = e, b) A o 7 = il(p(7)) o A for 7 e 7ri(M, O). Moreover G is unique up to isomorphisms, A is unique up to automorphisms of G. Proof. By Fact 1.2 the Lie-Hopf subspace ($, /z) corresponds to a simply connected complex Lie group G with (fi(G), d) = ( $ , /*), which is unique up to isomorphisms. Fix a complex Lie-Hopf algebra isomorphism 77: fl(G) —* $ . Let A: M —> G be the holomorphic map given by Proposition 2.2 with A* = TT'T} on ft(G). Let M: M —*

55 G be another holomorphic mapping with property a). Then by Fact 1.2 a LieHopf algebra automorphism (A*) - 1 o A'* of S7(G) induces a complex Lie group isomorphism h such that h* - (A*) - 1 o A'* and hence -K*T\ = (ho A')* on fl(G). Therefore A = h o A' by the uniqueness of A. Define p : 7Ti(M) -> G by p{*i) = A(7(o)) for 7 e 7Ti(M). Then (■R(p(7))_1 o A o 7)(o) = e

for 7 € 7ri(M).

On the other hand we have (i?(p(7))" 1 o A o 7 )*w = ir*tj(w)

for w 6 fi(G),

since w e Q(G) is right invarinat and 7r*7}(w) is 7ri(M)-invarinat. Thus by the uniqueness of A we obtain A o 7 = i?(p(7)) o A. It follows, for 7,6 € n\(M) iJ(p(7 o 5))" 1 o A o (7 o 6) = i?(p(fi)) _1 o {i?(p( 7 ))- 1 0 A o 7} o 6. Therefore R(p("/ o 6)) = i?(p(7)) o R(p(6)), i.e., p is an anti-homomorphism.

D

Corollary 2.4. Let f: M\ —> Mi be a, holomorphic mapping between complex manifolds. Let ($j,p^) be a finite dimensional Lie-Hopf subspace of H°(Mi,ill) and let (Gi, Aj,pi) be the associated triple given by 2.2. Assume /*$2 C $1 and / * : ($2, Pi) —* (3>i,/-»i) is a Lie-Hopf algebra homomorphism. Then there is an unique complex Lie group homomorphism h: G\ —► G2 which makes the following diagram commutative. Mi

-^->

'1 M2

Gi

<-^~

4 -^->

G2

TTXCMI)

'■{ <-^-

TTI(M 2 )

where / : Mi —> M2 is a lift of / to the universal covering spaces of of Mi with references points 5i € Mi, /(5i) = en, and /„ is the homomorphism induced by f. Moreover, if f*: ($2,^2) -» ( * i i ^ i ) is injective, then h is surjective. Proof. By a) of Corollary 2.2 and the assumption on / , (AJ) _1 o/*oA2: (fi(G2), d) -> (fi(Gi), d) is a Lie-Hopf algebra homomorrphism; and by Fact 1.2 this induces an unique complex Lie group homomorphism h: G\ —* G2 such that A\oh* = /* ° A£ on n(G2). Therefore, by the uniqueness in Proposition 2.2 we have h o Ai = A2 o / . It follows that the right half of the diagram is also commutative since Aj determine Pi by b) in Corollay 2.3. If /* is injective on $2, then (AJ) - 1 o / ' o A^ is injective on fi(G2) since / covers / . Hence h is surjective by Fact 1.2. D As a special case of the above we have

56

Corollary 2.5. Every holomorphic map f: Mi -* M% between compact complex parallelizable manifolds, Mi = Gi/Ti, is covered by a composition of a complex Lie group homomorphism h: Gi —* G2 and a left translation Lg by g e G2 so that the following diagram is commutative:

where I \ C G, are discrete and 7Tj: Gi —► Mi are the projections to the quotient spaces. 3. Spaces with ample 1-forms In this section we shall examine the structure of complex spaces with ample 1-forms. We use the following well-known lemmas. Lemma 3.1. Let E be a holomorphic vector bundle of rank r over a compact complex manifold M which has no non-constant meromorphic functions. Then for any linear subspace V of H0(M,O(E)), k

dimV ==--max{A; max{fc I| / f\\ VV / f0 0 in Moreover, if dim V = r, then /\k V = H°(M, 0(/\k

k

H0(M,O(/\E))}. H0(M,O(/\E))}. E))

forl
Proof. Let s\,..., sm be a basis of V, m = dim V. Assume S\ A • • ■ A S/t / O o n M. We shall claim that, if s A sj A ■ ■ • A s^ = 0 on M, then s, s\,..., Sfc are linearly dependent over C. In fact, outside the zoro locus of s\ A ■ • • A s j there defined holomorphic functions cl uniquely so that s = Ylic%si- I n terms of a holomorphic local trivialization e\,..., er of E, express s = J^ a?tj and Si = J2 ■ trfej with a?, t^ holomorphic. By an elementary linear algebra, each c' is a rational functions of a J , b\ with constant coefficients. Thus all cl extends to meromorphic functions on M, which are all constants by assumption. Assume now dim V = r. Then si(x),..., sT(x) is a basis for the fiber of E over x provided si(x) A • ■ ■ A sT(x) =f= 0. Therefore, for any ip e H°(M, 0(/\k E)) there are determined holomorphic functions c1 defined outside the zero locus of si A • • • As r so that ip = J^j c'si, where / = {u, ...,ifc} and s; = S i , A - A s i t . By the same reason above all c1 extend to meromorphic functions on M, and hence constants. D

57 L e m m a 3.2. Let M be a compact complex manifold of dimension n. Then every holomorphic (n — l)-/orm on M is d-closed. Proof. Let w be a holomorphic (n— l)-form on M. By Stokes' theorem JMduiAdw = 0, which implies dw = 0. □ We need two more lemmas. Lemma 3.3. Let p : M —> N be a surjective holomorphic mapping between com­ pact complex manifolds M and N. Let S C N be the set of all critical values of p. Assume every fiber ofp is connected and N is Kahler. Then

H°(M, ni,) n H°(M

- p-^sj.p'nj,) c P*H°(N, nlN).

Proof. Let

Jv, Jv,

JdVj JdVj

71 2 tpAdipAu tpAdrpAujn~-2..

Set Uj = p~l{Vj). Since the volumes of fibers of p are bounded from below, there is a positive constant C independent of j such that / ipAdipA uf1-2 A uin~2 A hm-n JdVj JdUj for all j , where m = d i m M . Moreover the right hand side tends to zero as j* —► oo since p*i> =

Adi>A w"" 2 = 0. JN-S i^°° JdV, This implies dip = 0 and hence (fy? = 0 on M . By integration

fx (p which is constant along each fiber of p. Since every fiber is connected, there defined on any simply connected open subset of N a holomorphic function / so that (p*/)(x) = Jx (p, i.e., ip = p*df. Consequently ip extends to a holomorphic 1-form on N so that (p = p*ip.

D

58 L e m m a 3.4. Let p: M -» N be the algebraic reduction of a compact manifold M onto a projective manifold N with d = d i m M — dimiV. Suppose that the homomorphism / \ d H°(M, SlxM) —> H°(M, ft^/iv) induced by the naturaJ projection n it -* °it/N fe nontrivial. Then dimH0(M,Q1M)/p*H0{N,il\f) = d. Proof. Let r : fi^ —«• ^ / ^ be the natural projection. By assumption we have rpi e H°(M, Q}M) with r ( ^ A - • • A rj)d) ^ 0 on M. Let ip e H°(M, Q]^). Then there are meromorphic functions /* on M such that r

i- By Lemma 3.3 it suffices to show dfr = 0 on M ' = p~l(C) with sufficiently general curve C on iV. We may assume H°{C, fij,) ^ 0. Let n € ff°(C, fij,) be nonzero and set Vo = P*V for simphcity. Since rip = 52±=i /**"&> there is a meromorphic function / ° on M' such that V = E?=o f V»»- Set w = Vl A- - • Arpd and w' = ( - l ) i _ V l A- ■ • A & - l At/'i+i A- ■ -A^j for 1 < i < d. Then V be a desingulalization of the complex space V. We map assume M has the algebraic reduction it: M —» TV. Let a^f: M —> .A(M) and a w : iV —> A(N), respectively, be the Albanese mappnigs of M and N. By the uni­ versality of a there is a holomorphicr map r : A(V) —> J4(JV) such that ajy07r = aojj on M. Let d = dim M—dim AT. Then the natural linear map ep: / \ p H°(M, n\f) —> H°(M, SfM) is notrivial for p < n = dim M . Therefore, by Lemma 3.4 there is a ddimensional subspace * C f l ° ( V , il^) such that ^9+ntH°(N, &„) = # ° ( M , ftj^). It follows that ir has no multiple fibers and that /J(7T _ 1 (J/)) = (T O Q) _ 1 (QJV(J/)) for y e N. Moreover * spans ^L^-ir.,) at each point of /i(w _1 (i/)) and hence the dimension of the Zarisky tangent space of A*(w_1(v)) at each point is at most d. Since d = dim/i(ir _1 (3/)) = d i m $ , every M(7T _1 (J/)) is smooth. Consequently ev­ ery a _ 1 ( v ) , v e V, is smooth and there is a subspace * C i?°(V, fiy) which spans ft a-'(») a t e a c n P o i n t o f a _ 1 (*0- In particular each a - 1 (w) is a complex parallelizable manifold. Let ip1,..., V m € * be a basis. Let 1/ c Q ( V ) be a small open simply connected neighborhood of a point in a(V) Let F be the universal covering space of a _ 1 ({7)

59 and Q : F —* U the map induced by a. Since * gives a trivialization of ^ - ' M Q is smooth, we have

an<

^

dp = ^2 4k^ A v> f c +Yl Q * e j A ^ + a ' & i > A*

i

where c V s are constants, ). Thus the monodromy group of A„ is independent of v, say T, and hence A induces a holomorphic local trivialization oCl(U) ^Ux G/T. a Let V be a reduced and irreducible analytic space and y.: M —» V a desingulalization of V. We call a subspace $ C H°(V, Q}v) involutive if /**$ C # ° ( M , 0]^ ) is an involutive subspace. Proposition 3.6. Let V be an irreducible normal complex space with ample 1forms. Let a: V —» A(V) be the AJbanese mapping of V and

Apa*: AP#V(n«VP -* #°(W) t i e linear map induced by a. Assume that H°(V, Qv) is involutive and one of the follwing conditions: a) dim A(V) > 3 and fa a* is injective; b) dimA(V) = 2 and /\ a* is injective; or c)dimA(V)
n\{v))

and let d = dim K - d i m a(V). Then by Lemma

3.4 there is a d-dimensional subspace * C H°(V,nv)

such that * A d ^ 0 and

60 H°(V, Qv) = * e *. By Prop 3.5 * gives a trivialization of the cotangent bundle of each fiber of a. Since A P ( * © * ) = T,s+t=P A" * © A* * . it follows that either (i) the natural map A 3 ( $ + * ) - ► ( * + * ) A 3 is injective, (u) dim($ + * ) = 2 and the natural map A 2 ( * + *)—►($ + * ) A 2 is injective, or (Hi) dim($ + * ) = 1. Thus by Lemma 2.1 H°(V, ilv) has the unique structure of Lie-Hopf space such that the inclusion $ —► H°(V,QV) is a Lie-Hopf algebra homomorphism. Let p: M —> V be a desingulaization of V. Let A —> A(V) and M —> M be the universal covering spaces of A(V) and M respectively. Let (G, A, p) be the triple given in Lemma 2.2 associated to the the Lie-Hopf space p*H°(V, Qv). The triple associoted to the Lie-Hopf space /i*$ is determined by the lift &M '■ M —> A of the Albanese map a>M'- M —> A(M) = A(V). Moreover, since the inclusion $ —» H°(V, Qv) is a Lie-Hopf algebra homomorphism we have by Cor. 2.4 a complex Lie group surjective homomorphism h: G —> A and a translation T : A —> A such that a = T o h o A. Since .ff0^, f2^) = * © $, the dimension of fibers of h is d. In particular, A maps each fiber of CUM onto a certain fiber of h. In view of Prop. 3.5 it follows that p{w\(M)) is discrete in G. Since the simply connected complex Lie group G is a complex linear group and hence G is a submanifold of a Stein manifold, Thus A factors through a holomorphic map of the universal covering space of V, induces a holomorphic map i:V—*L = G/p(iri(M)). Then i*: H°(L, il\) = H°(V, Q.v) is an isomorphism by the definition of A. It follows that i is an immersion since V has ample 1-forms. Moreover Toh induces a holomorphic map 77: L —* A(V) so that ay = i) o i. This imlpies that 77: L —* A(V) is the Albanese map by the universality of a. Finally, the uniqueness of L and i follows from the uniqueness of the triple (G, A, p). D 4. Existence of the generalized Albanese mappings Let M be a compact complex manifold. Lemma 4.1. Let f: M —> X and g: M —► F be surjective holomorphic mappings onto complex spaces with ample 1-forms, X and Y respectively. Then there are a surjective holomorphic map h: M —> Z onto a complex space Z with ample 1-forms, holomorphic maps a: Z —> X and r : Z —> Y such that f = a o h and g = T oh. Moreover: 1) IfrH°(X, Q^) C g*H°(Y, 0\r), then we can take Z to be a finite unramified covering ofY. 2) If both f*H°{X, n]f) and g*H°(Y, nj,) are invoiutive, then we can choose Z so that h*H°(Z, illz) is also invoiutive. Proof. Let f x g: M -> X x Y denote the direct product of / and g, namely (/ x g)(x) = (f{x),g(x)) for x e M. Set Z = (/ x g)(M) and h = f x g: M -> Z.

61

Let a: Z —> X and r : Z - » K b e the map induced by the projections X x Y —► X and X x Y —► V respectively. Then f = cr o h and 5 = r o / i . Since X, Y are with ample 1-forms, so are their direct product X xY, and its subvariety Z. Moreover, if both f'^iX,^) and g*H0(Y,Q\,) are involutive, then h*H°{Z,Qz) is also involutive. Suppsose /*if°(X,n3 f ) C s*ff°(y,n^). Then /i*fr 0 (Z,n^) = g*H°{Y,Q\r) by construction, and hence T*H°(Y, fiy) = /f°(Z, fi^) s i n c e ^* i s injective on H°(Z,0,z). Since Y and Z are complex spaces with ample 1-forms, this implies that r is an unramified covering. □ We call a subspace $ C H°(M, Q.lM) geoemetrically realizable if there is a surjective holomorphic map / : M —► V onto a certain complex space V with ample 1-forms such that $ = f*H°{V, Qv). Proof of Theorem A. By Lemma 4.1 there is an unique maximal geometrically realizable subspace * C H°(M, il]^), and we have a surjective holomrphic map p: M —» V onto a certain complex space V with ample 1-forms such that \P = h*H°(V, £ly). Taking a finite unramified covering if necessary, we way assume the induced homomorphism h,: TTI(M) —► iti(V) is surjective so that, if p can be lifted, p': M —► V , to a finite unramified covering V of V, then V = V. Therefore p: M —> V is the Albanese reduction of M by 1) of Lemma 4.1. D Proof of Theorem B. Note that, if * , * ' C H°(M, Q]^) are involutive, then * + * ' is also involutive. Therefore by Lemma 4.1 there is an unique maximal involu­ tive geometrically realizable subspace $ C H°(M, illM), and we have a surjective holomrphic map / : M —* V onto a complex sapce V with ample 1-forms such that $ = f*H°{V, Q\,). Taking a finite unramified covering if necessary, we way assume the induced homomorphism / « : TTI(M) —► 7ri(V) is surjective so that, if / can be lifted, / ' : M —» V, to a finite unramified covering V' of V, then V' = V. By the maximality of $, the ordinary Albanese mapping a: M ^ A of M factors via / through the Albanese mapping a y : V —► A so that a = ay ° /• By Prop 3.6 the assumption of Theorem B implies there is an immersion t: V —> L of V into a compact complex parallelizable manifold L such that i*: TT\(V) —> ni(L) is an iso­ morphism. Let X = 10 f: M —> L. We claim that this is the generalized Albanese mappnig. Let Ai: M —> L\ be a holomorphic mappnig to a compact complex parallelizable manifold L\. For simplicity, set LQ = L and Ao = A. Let p: M —* M and pi\ Gi —► L,, respectivel, be the universal covering spaces of M and Li. Fixing a reference point o € M of the universal covering space M of M, choose a lift A*: M —► Gi of Aj so that Xi(o) is the identity element of Gi for i = 0,1. Then AJ/f°(Li,n^ i ) and $ are involutive and hence AJif°(Li,n^ ) C $ by the maximality of $ . Since e 2 : A 2 $ -» $ A 2 i s injective by Prop 3.6, AJ: pJ//°(Li,n}, ] ) — p*$ is a Lie-Hopf algebra homomorphism and hence induces a complex Lie group homomorphism h: Go —► Gi so that AJ = XQ O /I* on pJ/f°(Li,ft^ ). By the uniqueness in 2.2 we

62 have Ai = hoXo. Moreover h induces a m a p r: LQ —» L\ since Ao*(7Ti(M)) = 7Ti(Lo). Thus we obtain the desired commutative diagram:

References 1. Blanchard, A., Sur la varietes analytiques complexes, Ann. Sci. Ecole Nomr. Sup. 73 (1958), 157-202. 2. Wang, H. C , Complex parallelizable manifolds, Proc. Amer. Math. Soc. 5 (1954), 771-776.

63 E X A M P L E S OF C O M P A C T H O L O M O R P H I C S Y M P L E C T I C MANIFOLDS WHICH ADMIT NO KAHLER STRUCTURE DANIEL (ZHUANG-DAN) GUAN* Department of Mathematics, Princeton University, Princeton, NJ 08544 USA

1. Introduction Let M be a complex manifold of dimension 2n. A holomorphic symplectic structure or form is a closed holomorphic 2-form w on M with maximal rank (see [15, p.74]). In [19] Todorov claimed that certain compact holomorphic symplectic manifolds are Kahler and asked if every irreducible compact holomorphic symplectic manifold of dimension more than 4 is Kahlerian. Some counter-examples are given in this note. Although we are primarily interested in holomorphic symplectic structures, we consider also real symplectic structures as well. We note that both the real and imaginary parts of a holomorphic symplectic form are symplectic forms. Starting from homogeneous manifolds three methods could be employed to con­ struct examples: (1) Consider an open or compact homogeneous manifold and an invariant struc­ ture. (2) Construct a homogeneous manifold with an invariant structure and a quo­ tient, which may not be homogeneous, in particular, a compact quotient of an open manifold. (3) Examine a structure (not necessarily invariant) on an open or compact ho­ mogeneous manifold. We call a 2n-dimensional compact manifold M with a closed 2-form ui almost sym­ plectic if the cohomology class [u>]n is nonzero, i.e., JM u>n ^ 0. In this note examples of compact almost symplectic complex manifolds given by the constructions above are examined. For the method (1), one has the following theorem of Huckleberry (see [12]): Proposition 1. Every compact homogeneous symplectic complex manifold is a product of a torus and a rational homogeneous space, both with standard symplectic structures. Remark. It seems that this proposition is still true even if we do not assume the existence of an invariant complex structure. We know that it is true if the group is reductive (see [7]) or the isotropy subgroup is discrete (see [12]). • e-mail: [email protected]

64 So every compact homogeneous holomorphic symplectic manifold must be a torus, which is Kahlerian. By the method (2), we can obtain many examples of real (or holomorphic) symplectic or pseudo-Kahler manifolds. The two examples in [2] are of the form G/T where G is a real Lie group with a right-invariant complex sructure and Y is a uniform discrete subgroup; they are not complex homogeneous. In [10] there are examples of nilpotent real Lie groups G with right-invariant pseudo-Kahler struc­ ture but with no Kahler structure. By considering compact quotient of these exam­ ples, we obtain many such examples. Every closed 2-form on such a space is, by an averaging procedure, cohomologous to a form which comes from a right-invariant 2-form of the universal covering. Examining the examples which come from the method (3), we have BorelRemmert's Theorem (see [3]): Proposition 2. Every compact complex homogeneous manifold with a Kahler metric is a product of a torus and a rational homogeneous space. Because of this Proposition we can easily contruct examples of compact holo­ morphic symplectic manifolds which admit no Kahler structure. So we discuss more about the method (3) here. In this note the following theorems are obtained by a hamiltonian argument (cf. [12]): Theorem A. Every compact homogeneous complex manifold with an almost sym­ plectic structure is a product of a rational homogeneous space and a complex parallelizable solv-manifold with a symplectic structure which is right-invariant in its universal covering. Theorem B. If the structure in Theorem A is holomorphic, then the manifold is a complex parallelizable solv-manifold with a holomorphic symplectic structure which is right-invariant in its universal covering.

■{(:

X

x ;

D(

hen G 1 ^ 1 x ( Q ™ ) f' tthen with the the c/G r z+iz with Gc/Gz+iz 0 right-invariant closed 2-form dx A dw + dy A dz is a compact holomorphic symplectic nil-manifold which cannot be Kahlerian by Proposition 2. We also get:

let G For example, let G-= < I 0

-)}•

Theorem C. If the structure in Theorem A is given by a (1, l)-form, manifold is a product of a rational homogeneous space and a torus.

then the

This can be regarded as a generalization of Proposition 2 as well as the main result in [7]. From this theorem, we see that the examples in [2] cannot be homo­ geneous. The more precise structure of the solv-manifolds in Theorems A and B might be discussed in a forthcoming paper.

65 As a tool we prove also that the cohomology of a compact complex homoge­ neous manifold G/H can be calculated from a subcomplex of the de Rham complex consisting of forms that are invariant under the left action of a maximal compact subgroup of G and "invariant under the right action of N/H°" in a sense (for detail see the proof of Lemma 1). Three examples of compact holomorphic symplectic manifolds without Kahler structure will be given in Section 4. We notice that all the examples in this paper are not simply connected and real homogeneous. Some examples of nonhomogeneous or simply connected compact holomorphic symplectic manifolds which are not Kahlerian have been found in [11]. 2. P r e l i m i n a r i e s 1. Throughout this note, (M, w) denotes a connected symplectic manifold or almost symplectic manifold. We call (M, u>) a symplectic manifold if M is a 2ndimensional connected differentiable manifold, and w is a closed real 2-form with nowhere vanishing w". We call (M, w) an almost symplectic manifold if M is a 2ndimensional connected compact differentiable manifold with nonzero cohomology class [ai]n. Note that an almost symplectic structure induces a symplectic structure on an open set. In this note we assume that M is a compact complex manifold with complex structure J and that a Lie group G acts on M transitively and holomorphically. Since M is compact we can assume that G is a complex Lie group. We call a symplectic structure (M, w) (1) a pseudo-Kdhler structure if ui(JX, JY) = u>(X, Y), and a Kahler structure if moreover u>(X, JX) is positive for all nonzero X. (2) a holomorphic symplectic structure if ui(JX, JY) = —u>(X,Y). Similarly we can also define the almost pseudo-Kdhler structure, almost Kahler structure and almost holomorphic symplectic structure in the compact case. Remark. Our definition of "holomorphic symplectic'' is equivalent to the usual one that there exists a closed holomorphic (2,0) form 6, called a holomorphic symplectic form which is nondegenerate (or [6%] / 0 in the almost symplectic case). The symplectic form in (2) can be the real or imaginary part of the usual holomorphic symplectic form. Conversely, we can get a usual holomorphic symplectic form from the w in (2) by S(X,Y) = \[u{X,Y) - iu(X,JY)). Since 2w = S + 6 and S is a (2,0) form, 6 is closed if and only if u> is closed. 2. A rational homogeneous manifold Q is a compact complex manifold which can be realized as a closed orbit of a linear algebraic group in some projective space. Equivalently, Q = S/P where 5 is a complex semisimple Lie group and P a parabolic subroup, i.e., a subgroup of S which contains a maximal connected

66 solvable subgroup (Borel group). Every homogeneous rational manifold is simplyconnected and is therefore an orbit of a compact group. In general, a quotient K/L with K compact and semisimple carries a if-invariant complex structure which is projective algebraic if and only if L is the centralizer C(T) of a torus T c K. A parallelizable complex manifold is the quotient of a complex Lie group by a discrete subgroup. It is a solv-manifold or nil-manifold according as the complex Lie group is solvable or nilpotent. 3. If G is a complex Lie group and H c G is a closed complex subgroup, then we have the normalizer fibration G/H —» G/N, where N = Nc{H°). Let Q and H denote the Lie algebras of G and H, respectively. The base space G/N is realized as the Ad(G)-orbit of the subspace H in the Grassmann manifold of subspaces of Q that have the same dimension as that of H. If G/H is compact, then G/N is a rational homogeneous manifold and N/H is a compact parallelizable homogeneous manifold. Remark. The fibration above is due to J. Tits [20]. In the case G is only a real Lie group, there is a generalization of this fibration which is defined by all the anticanonical sections coming from /\n Q, where we regards Q as the corresponding holomorphic vector fields on G/H and n — dimG/H. Equivalently, we have a fibation G/H -> G/NR where 1 1 NRR = {geG\gH^g{geG\gH°g= = H°, H°, jAd(g) jAd(g) == Ad(g)j Ad(g)j mod mod H} H)

and j is a linear map in Q which induces the complex strcture on G/H (see [14]). We call this fibration the Q-anticanonical fibration. If there is an invariant volume on G/H, there is another fibration, the global canonical fibration G/H —► G/I with / = { g € G | t r s / w ( a d ( j M(g)X)

-jo

ad(Ad( 5 )X)) = 0 for alLY e G }

(see [13, p.236]). Note that x and y in G/H define the same point in G/L if and only if fx(x) = fx(v) for all X G Q, where fx is the divergence of the vector field JX. This fibration is slightly different from the one in [13] (which we call the local canonical fibration). As in the proof in [7] we always have N^ C / and the equality ATR = / if G/H is compact. 4. Let (M, w) be a symplectic manifold and G a Lie group of symplectic diffeomorphisms of M, i.e., a smooth action G x M —> M such that g*w = UJ for all g e G. Let Ham/ oc (M) be the set of smooth vector fields X on M such that Lxu> = 0. In this situation we have the following sequence: 0 ^ R -U C°°(M)

s

^ Ham i0C (M) ^ - Q

where i realizes the real numbers as constant functions, the skew-gradient sgrad(/) is a vector field Xf such that iX/u = u(Xj, ) = df, and a is the natural Lie algebra

67

homomorphism arising from the G-action. The associated Lie algebra structure { , } on C°°(M) is defined by {f,9) = w(sgrad(/),sgrad(p)). It follows that sgrad: C°°(M) — ► Ham(0C(M) is a Lie algebra homomorphism, and we are confronted with a lifting question: Does there exist a Lie morphism A: Q —> C°°(M) such that sgradoA = a ? If such a lifting exists, we refer to the G-action as a Poisson action (with regard to the lifting). In this case the G-equivariant dual map

$:M^5*,

*(*)(£) = A(fl(aO

is called the moment map. If every (/-field is the skew-gradient of some function, i.e., if for every £ € C the associated vector field is of the form £M = sgrad(/j), then the G-action is called a hamiltonian action (our definition is different from the one in [9]). The following is a list of elementary observations in the above setting (see [9]): (1) Let G' be the commutator of G, then the G'-action is hamiltonian. And if G is a semisimple group, then it induces a Poisson action. (2) Suppose that £ € Q can be lifted. Then {xeM\df {x e M i(x) 0} == {x { i egJ!f M | {|Cifcf(x) 0}. | dft(x) = --= 0} M ( i ) == 0}. (3) If the G-action is Poisson with moment map $ : M —> Q*, then ker(d$x) = { v € TXM\ux(v,w)

= 0 for all w € TxG{x) } = (TxG(x))L,

where (TxG(x))L is the skew-orthogonal complement to the tangent space of the G-obit G(x). ( (1), (2) and (3) come from the properties of the Poisson bracket.) (4) If G is as in (3) and G(x) = G/H is a generic orbit with moment fibering *IG(X) : G/H -» G/J = G(*(x))

x € hi,

then H° < J° and J°/H° is abelian (see [9, p.190]).

68 3. The Proof of Results Lemma 1. Let (M,u>) be a compact complex homogeneous space with an almost symplectic structure. Let G be the identity component of the group of holomorphic automorphisms of M, H the isotropy subgroup, and N = Nc(H°) the normalizer ofH° in G. Then we can choose a cohomologous symplectic structure which is leftinvariant by a maximal connected compact subgroup of G such that the induced symplectic structure on the covering space G/H° is invariant under the right action ofN/H°. Poof. First we apply the averaging method to the left action of a maximal compact subgroup of G. Noting that u>n is nonzeo in the cohomology group and any 1parameter group fixes H2(M, Z) (the latter is discrete), we see that the resulting 2-form is in the same cohomology class as w, so it is an almost symplectic structure. Secondly, we apply the averaging method to the right action. Since gH°n = gn(n~1H°n) = gnH° for all g £ G and n E N, we see that N induces a right N/H"-action on G/H°. Let w be the 2-form on G/H° induced by w. Then w is (right) iy/H°-invariant. Now consider the averaging map A defined by A{Q){x) = f(N/H0)/(H/H°)n*('*'(xn^dn, where dn is the Haar measure of (N/H°)/(H/H°) induced by the left invariant Haar measure of group N/H° and we asume that J(N/H°)/(H/H°) d"n = 1- Here we notice that n —> n*(u;(xn)) induces a map from {N/H°)/{H/H°) to EpA P r x *(Ar/H°), A is wel defined. But n*(u>) usually does not induce a form on (N/H°)/(H/H°), the argument in the first paragraph of this proof does not work here, so we need a different argument. A(u) induces a closed 2-form A(LJ) on M. If we can prove that A(UJ) is in the same cohomology class as u>, then we are done. Actually we can define the averaging operator sending the de Rham complex to the right iV-invariant de Rham complex and prove that it induces an isomorphism of the corresponding cohomology groups. The proof of the isomorphism reduces to the case M = N/H by an argument which is similar to the argument in the proof of the existence of the Leray spectral sequence (see [8, p.201]); we outline the proof as follows: (1) we apply Theorem 4.6.1 in [8] to the base Q and the right TV-invariant de Rham sheaf complex which is regarded as a sheaf complex on Q, (2) the right AT-invariant sheaves have a partition of the unity on Q so that they are acyclic for q > 1, (3) we get a spectral sequence for the right iV-invariant sheaf complex similar to the Leray spectral sequence for the de Rham complex. In the case in which M = N/H, we can assume that G = N is a complex Lie group and H is a uniform discrete subgroup of G, i.e., G/H is compact. Now we define a right-invariant Riemannian metric g on G/H by a positive bilinear form of the Lie algebra Q of G. Then the proof of Proposition 7.23 in [18] works in our situation; we outline the proof here: (1) the argument in p.114 of [18] reduces the question to that of the cohomology of the Lie algebra Q with coefficient ker(jl), (2) the

69 ker(.A) consists of all functions perpendicular to the A-harmonic functions, (3) by the existence of Green's operator we see that the Laplacian J2 g^XiXj, which is an element of the universal enveloping algebra U (Q) of Q, induces an automorphism on the cohomology of ker(A), (4) by definition HP(Q, ker(A)) = E x t j ^ J R , kei(A)) is a (7(£)-module and the Laplacian A vanishes on the real number field R, we see that the cohomology of ker(A) is zero so that A induces an isomorphism of the cohomology group of the de Rham compex to the N-invariant de Rham cohomology group (see [18, p.119]). Q.E.D. Remark. By the same argument we generalize Nomizu's Theorem in [17] and so Benson-Gordon's Theorem in [5] to arbitrary Lie groups. L e m m a 2. (cf. [12]) For a manifold M as in Lemma 1, the Q-anticanonical fcion is a product.

fibra-

Proof. Let M = G/H —► G/N = Q be the £-anticanonical fibration. Since the base Q is a rational homogeneous space, there is a connected compact semisimple Lie subgroup K in G which acts on Q almost effectively and transitively. By Lemma 1 we may assume that the symplectic structure is left-invariant under the action of K. Let q € Q and L = Iso^lq} be the isotropy subgroup of K at q. Since K is semisimple, the K-action on M is Poisson. In particular, if £ € K. (the Lie algebra of K) and if T is the closure of the 1-parameter group expt£ in K, then

Fix M (T) =

{X\ZM(X)

= 0 } = {xldf^x)

= 0} ± 0.

Now let T be a maximal torus in L. Recall that the fixed point set FixQ(T) of the T action on Q is finite. Furthermore, since H/H° is a discrete subgroup of N/H°, it follows that if x € FIXM(T) and if F is the (/-anticanonical fiber through x, then F C Fixjtf(T'). For g € L let T9 = gTg-1, and let g(t) € L be a continuous curve with g(0) = e (the identity) and g(l) = g. Since Fixjif (T3^) is continuous in t and since q is an isolated point in F i x g ( r 9 ' ' ' ) for all t, it follows that F C F i x M ^ 9 ) . Since L = \JgeLTg ( s e e t 1 ' P-89l)> t h i s implies that F c FixM(L). Thus for all x e F, ISOK{X} = Iso#{<7} = L, i.e., the A"-orbits in M are sections of the Qanticanonical map. Let S be the minimal complex subgroup of G that contains K, and Lc the smallest complex subgroup of S that contains L. It follows that Lc acts trivially on F. Let P = lsos{q). Then P = Lc ■ Ru(P), where Ru(P) is the unipotent radical of P. Let / be the group of ineffectivity of the P-action on F. Now / is a normal subgroup of P and Lc C / . Further Ru(P) is generated by one dimensional subgroups which are normalized by a maximal torus of Lc Since the maximal torus action is non-trivial on each of these root groups, it follows that I = P. Thus the S-orbits in M are also sections. But they are holomorphic sections, and consequently the £-anticanonical bundle yields a product structure M = F x Q. We have the desired result. Q.E.D.

70

L e m m a 3. (cf. [12]) The G-anticanonical map and the K-moment map coincide. The symplectic form in Lemma 1 is nondegenerate on both F and Q. Proof. Let XQ e M and put m = dimR.ftr(xo). Since all K-orbits in M are of dimension m, it follows that ranks$ = m for all x e M. Furthermore, the fibration $ : K(x) -» K{$(x)) is a torus bundle (Property (4) in 4 of section 2). Since L = ISOK{X) already contains a maximal torus (due to the fact that K/L = Q is a homogeneous rational manifold), this map is finite. But the base space Q is simply-connected. Hence $ : K(x) —> K($(x)) is injective for all x € M. In other words, if F$ is a $-fiber through a point XQ e M with ISOA-{X} = L, then F$ n if (xo) = {xo} and F$ = { x € M | Isojf (x) = L }. Since all K-orbit in M are sections of the S-anticanonical bundle, this is also a description of the S-anticanonical bundle fiber F through xo- Moreover, u = u\p + U\Q + uipQ with WFQ a form representing a class in i f 1 ( i r ) x Hl(Q), which must be zero by Hl{Q) = 0, hence u\p and U\Q are nondegenerate (by Property (3) in 4 of section 2). Q.E.D. Proof of Theorem A. By Lemmas 2 and 3 we get that M = Q x F with a symplectic structure on Q which is left-invariant by a connected maximal compact subgroup of the automorphism group of Q and with a symplectic structure on F which is right-invariant by N/H° By setting N\ = N/H° and T = H/H°, we have F = Ni/T. Let M be the Lie algebra of Ni and A/" = M © R - b e t h e central extension Lie algebra of Lie algebra Ni with Lie bracket {x © o, yffi6} = [x, y] ®U>F(X, y) for x, y € M\, a, b € R. Let i be a linear form of M which is zero on Mi and identity on R, then dl(x © a, y © b) = —u)p{x, y). Let V be a complement of R in Af which is invariant under S a semisimple part of M, so M = R © V. Then AT* = R* © V*. We let 2 = l0 + h with ! 0 £ R*, d £ V* and dl0(S, M) = 0. Then wi = w - <«i induces a closed 2-form in [w]. But u>i(«S, y ) = 0 for all Y e A/i, i.e., [u]n = 0 if 5 ^ 0 . Hence Ni must be a solvable group. Q.E.D. Proof of Theorems B and C. If the original structure is holomorphic, then Q must be a point because H2(Q, O) = 0. If the original structure is pseudo-Kahler, then F is a torus because every complex pseudo-Kahler homogeneous space is a quotient of a Cm for some m e N (see [7], 3.3). Q.E.D. 4. E x a m p l e s We will give three different types of compact holomorphic symplectic manifolds that are not Kahlerian: (a) a complex homogeneou nil-manifold, (b) a complex solv-manifold which is not a nil-manifold, (c) a nonhomogeneous real parallelizable manifold. We obtin (a) and (b) by method (3), while (c) is obtained by the method (2).

71

(a) Let

1 a;

°-{6i!)-(ST)}M = Gc/Gz+iz and ui = dx A dw + dy A dz as a, right-invariant form. Theorem C shows that there is no pseudo-Kahler form on M. Remark. After we had finished this paper, we found that this example is the complexification of the Kodaira-Thurston surface. In general, if G R / G Z is a compact (real) symplectic nil-manifold, then Gc/Gz+iZ is a compact holomorphic symplectic nil-manifold. (b) Let

«-(S ?)»(J ?)■ where T is the complex 1-parameter subgroup of SX(4, C) that contains

M (o with A = ( j

0^

:-)

i I • ^ being the 2 x 2 identity matrix, and R being a 4 x 2 matrix

of the form fi=

(C2

(o

cc22)-)-

0

Let M = G/Gz, where

*- (? V)«(i *V) and

•{(?

Tz =

0 A-

■)l

n€z|

with a right-invariant symplectic form which is dt A dc + dx\ A di2 + dy\ A dj/2 at the identity point of G. Here t corresponds to the coordinate of T and c to that of C in the expression of G, while (xi.j/i), t = 1,2, are linear coordinate systems (different from the original ones) with bases {e\, e 2 }, i = 1,2, of the upper-left C 2 and lowerright C 2 in the definition of R, where that e\, j = 1,2, are eigenvectors of A with eigenvalue 2±& and e^ are eigenvectors of A with eigenvalue 3~^. Theorem C shows that there is no pseudo-Kahler form on M. (c) Now we consider a real nilpotent group G with a right-invriant almost complex structure J. We choose a C-basis {wi,...,w„} for the complex forms of type

72

(1,0) on Q ® C; then {wi,.. . , w „ , w i , . . . , w„} is a real basis of Q*. The structure equations have the form dui = ^2

A

ijk Uj A Uk + £

}
^ijfc WJ A wfc + £

j,k j,k

^Jfc

W

J

A UJk

j
for 1 < i < n. Then J is a complex structure if and only if Cjjjt = 0. Moreover (see [6], [4] and [16]), we have Proposition 3. The struture equations for the Lie algebra Q of a nilpotent Lie group G with a right-invariant complex structure have the form dUi =

^ijfc

w

B

j AWk

ijk

Uj

Auk

( l < t < n).

j,k
j
Conversely, the structure equations above define a niipotent Lie group G with rightinvariant complex structure. Moreover, if Atjk, B^ are integers, then there is a uniform discrete subgroup ofT of G such that J induces a complex structure on G/T. In this case the quotient manifold admits a KaMer structure if and only if the group is abelian. Now we construct (cf. [6]) an example of holomorphic symplectic manifold of complex dimension 4 with structure equations: dwi = 0 du)2 = 0 dW3 = Wi A (^2 + ^ 2 ) duii = W\ A u>2

and wi A W3 + W2 A 014 as a right-invariant 2-form. A direct calculation shows that there are some pseudo-Kahler forms but there is no Kahler form on G/T. Moreover we have the following procedure to find new examples from old exam­ ples which is suggested by Professor S. Kobayashi (see [21]): Proposition 4. Let (Q, H, j , UJ) be a complex symplectic, or holomorphic symplec­ tic, or pseudo-Kahler algebra (i.e., it comes from a complex symplectic, or holomor­ phic symplectic, or pseudo-Kahler homogeneous space, see [7]), A an~R-algebra, and I an R-h'near function on A such that the bilinear form k(x, y) = l(xy), (x, y 6 A), is nondegenerate. Then (A®G, A®H, id®j, fc®w) is also a complex symplectic, or holomorphic symplectic, or pseudo-Kahler algebra. In particular, if Q is nilpotent, H = 0 and A has integral structure constants and if the constants A^, B^ in Proposition 3 are all integers, then (A ®G,id® j , k<S>u) induces a complex sym­ plectic, or holomorphic symplectic, or pseudo-Kahler structure on some compact quotients of a nilpotent group G& ofQ®A. So we have many examples of irreducible compact holomorphic symplectic man­ ifolds of dimension more than 4 which admit no Kahler structure. As an example

73 of dimension 2 we have the well-known Kodaira-Thurston manifold (see [6, p.62]). One can easily see that all these examples are not simply connected. Acknowledgements Here I like to take this chance to express my thanks to Professors S. Kobayashi and A. Todorov for their interests and encouragement in this work. I also thank Professor A. Wolf, for showing me [18] which has led to a break-through for this work, and Professor Huckleberry, for giving a simpler proof for the main result in [7]References 1. J. F. Adams, Lectures on Lie Groups (Benjamin, 1969). 2. L. de Andre, M. Fernandez, A.Gray & J. Mencia, Complex Manifolds with indefinite Kdhler Metrics, Proc. Sixth Intern. Colloq. Diff. Geom., Santiago de Compostela 1988, Cursos. Congr. Univ. Santiago de Compostela 61 (1989), 25-50. 3. A. Borel k. R. Remmert, Uber Kompakte Homogene Kahlersche Mannigfaltigkeiten, Math. Ann. 145 (1962), 429-439. 4. C. Benson k C. Gordon, Kdhler and Symplectic Structures on Nilmanifolds, Topology 27 (1988), 513-518. 5. C. Benson k C. Gordon, Kdhler Structures on Compact Solvmanifolds, Proc. Amer. Math. Soc. 108 (1990), 971-980. 6. L. Cordero, M. Fernandez k. A. Gray, The Frolicher Spectral Sequence for Com­ pact Nilmanifolds, Illnois. J. Math. 35 (1991), 56-67. 7. J. Dorfmeister k Z. Guan, Classifications of Compact Homogeneous PseudoKdhler Manifolds, Comm. Math. Helv. 67 (1992), 499-513. 8. R. Godement, Topologie Algebrique et Theorie des Faisceaux, Publ. Inst. Math. Univ. Strasbourg 13 (1954). 9. V. Guillemin & S. Sternerg, Symplectic Techniques in Physics (Cambridge Univ. Press, 1984). 10. Z. Guan, Classification of Pseudo-Kdhler Surfaes (1990). 11. D. Guan, Examples of compact holomorphic symplectic manifolds which are not Kahlerian II, in preparation. 12. A. T. Huckleberry, Homogeneous Pseudo-Kahlerian Manifolds: A Hamiltonian Viewpoint, preprint (1990). 13. J. Hano k S. Kobayashi, A Fibering of a Class of Homogeneous Complex Man­ ifolds, Trans. Amer. Math. Soc. 94 (1960), 233-243. 14. A.T. Huckleberry k E. Oeljeklaus, Classification Theorems for Almost Homo­ geneous Spaces (Publ. Inst. Elie Cartan, Nancy, 1984). 15. S. Kobayashi, Differential Geometry of Complex Vector Bundles (Iwanami,

74

Tokyo and Princeton Univ. Press, Princeton, 1987). 16. Mal'cer, A Class of Homogeneous Spaces, Amer. Math. Soc. Transl., 39 (1951). 17. K. Nomizu, On the Cohomology f Homogeneous Spaces of Nilpotent Lie Groups, Ann. Math. 59 (1954), 531-538. 18. M. S. Raghunathan, Discrete Subgroups of Lie Groups, Ergebnisse der Math. 68 (Springer, 1972). 19. A. Todorov, Every Holmorphic Symplectic Manifold Admits Kdhler Metrics, preprint (1991). 20. J. Tits, Comm. Math. Helv. 37 (1962), 111-120. 21. K. Yano & S. Kobayashi, Prolongations of Tensor Fields and Connections to Tangent Bundles 1, —General Theory—, 3. Math. Soc. Japan. 18 (1966), 194210.

75 A T O R E L L I - T Y P E T H E O R E M F O R STABLE C U R V E S YOICHI IMAYOSHI Faculty of Science, Osaka City University, Sumiyoshi-ku, Osaka, 558 Japan and

TOSHIKI MABUCHI Department of Mathematics, Osaka University, Toyonaka, Osaka, 560 Japan

Introduction Let C be a stable curve of genus g > 2, i.e., C is a projectve algebraic reduced curve with dime H1 (C, Oc) — g such that the following conditions are satisfied: (1) Singular points of C are, if any, ordinary double points. (2) No smooth rational components of C meet the remainder of the curve in fewer than three points. For the desingularization v: C—>CofC, writing C as a union \Sa=lCa of irreducible components, we can express C as a disjoint union \Jra=lCa of nonsingular irreducible curves Ca of genus ga such that u(Ca) = Ca. Let m and na be respectively the numbers of the singular points in CSjng and v~l (Csing) n Ca, where C S j ng denotes the set of singular points in C. We can then write CSing = {pi,P2, ■ ■ ■ ,pm }■ Let Cteg be the set of nonsingular points in C. We obviously have 1m = n\-\-n% + V nr. Moreover, by dime H 1 ( C , O c ) = 5> gHra=1"(g- 1a-l). m + E^==l(0a 9- - 11 == m ). The condition (1) above shows that each point pi of CSing has an open neighborhood Ui in C written in the form Ui = { (z, w) e C 2 ; zw = 0, \z\ < 1, \w\ < 1 } , for some holomorphic functions z, w on Ui- Fix once for all a positive integer /i, and let wc denote the dualizing sheaf of C. Then w®'1 has a holomorphic local base dz®i*

6 ==

,

+ (-l) M

s„dw^

over the open neighborhood Ui. Let Resj: if°(C,u;®' i ) - > C b e the residue map sending each T € F°(C,w® M ) to Resi(r) 6 C defined by Resi(r) = ( r / ^ ) ) p . .

76 Consider the mapping H°(C,u$") 3 f H A(T) := (Resi(T),Res 2 (T),...,Res m (r)) e C m We then define Gr{H°(C, wgM)} := {H°(C, wg M )/Ker A} © Ker A = C m ffi KerA. Put ||y|| := ( E £ 1 | t t | a / " ) ' , ' a for y = ( W l l f t l . . . ,ym) G C m . To each a € Gr{if°(C,w®' J )}, we associate a nonnegative real number ||Q1')} onto C m and KerA respectively. Note that ||A:<7||c = k\\a\\c for all k e C, and that ||<J||C = 0 if and only if a = 0. For stable curves C" and C", the vector spaces Gi{H°(C,uc?)} and Gr{H°(C", w®,^)} are called C-linearly isometric if there exists a C-linear iso­ morphism t: Gi{H°(C,u^)} = Gi{H°(C",uj®!t)} which is an isometry, i.e., ||t((r)||c" = \W\\c for all a e Gi{H°{C ,u®?)}. Moreover, such an i is called a C-Zinear isometry. On the other hand, note that i' _1 (C s i n g) is a finite set consisting of 2m elements. Consider a partition II of ^ _1 (C s ing) = {q1,q2,- ■ ■ i<72m} m t ° a family of m sets of cardinality 2, where we observe that the set P of all such partitions is a finite set consisting of (2m)!/(2 m m!) elements. Given such a partition II, we define Cn as the stable curve obtained from C by identifying all 2g+l. Then C' is isomorphic to Cn for some He P. Note that the extension data for 0 -» Ker A -> H°(C, wg") -» H°(C, w®M)/ Ker A —> 0 are essentially given by the choice of II in P. 1. A key lemma characterizing the integral

f~ | r | 2 ^

Let ip = -n + 1 and £ > -fj. + 1. Put *kj j '■ : = An{e » ( <++3-j)/(ji-3 3 - j ) / ( M - 3 ++ jj), ),

ije€{{il,,22}},,

77 where fi is an integer satisfying fj, > 2. Moreover, ki denotes +00 if /z = 2. We now define a real-valued function f(t) on { t e R; |t| < e } (0 < e « : 1) by

-L

2/ dz A dz. [ \

ltrPf^^^dzAdz. ff{t):= it): "

2

If k < kj with j e {1,2}, we can also define real numbers a.j by
a2 :

-IL** -H*1 4(! (Re

2-Ai)(Ri

iz A dz, i(lu

dz A dz. 2 ■*)'}■——

For functions u = u(t) and v = v(t) in t with v(t) # 0 for t ^ 0, we write u as o(v) or O(u) according as lim t _o u(t)/v(t) = 0 or lim t _o \u(t)/v(t)\ < +00. As in the case /z = 2 (see [6], [1]), we can prove the following key lemma for all integers H > 2: Theorem 1.1. Let 2 < /i e Z. We put v := 2(n~1k + \)/{k - £), where v denotes +00 if k = £. Then we have: (1) (2) (3) (4) (5)

Ifk> fci, i.e., 2/n < v < 1, then f(t) = /(0) + C 0 |t|" + 0(1*1"). Ifk = ki, i.e., v = l, then f(t) = / ( 0 ) + Cit + C2\t\ + o(\t\). Ifk2
where Co, C2, C3, C4 are positive real constants independent constant independent oft.

oft, and C\ is a real

This theorem will be proved later in Section 6. Remark 1.2. Let u > 3. In this case, use the inequality v > 2/fi or 1 > according as k > k\ or fc < k\. It then follows that

2/fi,

/ ( t ) = /(0) + o(|t| 2 /"). Definition 1.3. Let A € (0,2] U {+00}. Then concerning the smoothness at t = 0, a real-valued function h(t) on { t € R; \t\ < e } (0 < e < 1) is said to be of order A, denoted by d(/i) = A, if we have the following: (1) For A = +00, both h'(Q) and /i"(0) exist, and h(t) === fc(0) h(t) /»'(0)t + o(\t\2).)h(0) ++/»'(0)t + ^ ^f'tt2 2 ++ o(\t\

78

(2) For A = 2, the function h(t) is differentiable at t = 0, and there exists a real constant &o > 0 such that h(t) = ft(0) + fc'(0)t + 6o|t| 2 log(l/|t|) + o(|t| 2 log(l/|t|)). (3) For 0 < A < 2, there exist real constants &i, 62 with 62 > 0 such that h(t) = h(0) + bit + b2\t\x + o(\t\x), where we can set 61 = 0 for A < 1, and if A > 1, the function h(t) is differentiable at t = 0 with b\ = h'(0). Let Vk,t denote 2(/x_1fc + l)/(fc - 1 ) or +00, according as k > k
u^.

2. E x t r e m a l sets In this section, assume /i > 2. For C as in the introduction, we put V+(C)ll := C = {y = (j/i, 2/2, • • ■, Vm); Vi € C for alii } and V~ (C)M := Ker A. Consider the vector space m

ViO,,

:= Gr{H°(C,

W®")}

= V+(C) M 8

V-{C)».

For a S { 1 , 2 , . . . , r}, let Qa denote the canonical sheaf of Ca. Moreover, for each i 6 { 1 , 2 , . . . , m}, we define vector spaces V+(C) : = { mj ;y / ej C m ; y J == Oiij^i} O i f i / i } = {{0,...,0,y { ( 0 , . . . ,i,0,...,0);y 0 , 2 / i , 0 ,leC} . . . , 0 ) ; 2 / t e C } = C, tl:={yeC v; + (c%: V„-(C%::= {a {^ WO*

6 Ker A; Supp(a) C C a } »« H^C^Q^dfi KerA; fl°(Ca,wf (0« -

l)Da)),

_1

where Da denotes the reduced effective divisor ^ (C S i n g ) n Ca on Ca. Note that deg.D a = na. Since C is a stable curve, na > 3 - 2g a for each a. Then by Riemann-Roch's theorem, dim c V a -(C) M = d l m c f l 0 ( C a i O f ' , ( ( M - l ) J D a ) ) r 0 if (M,s Q ,n Q ) = (2,0,3), > J

1 I. 2

if (fi,ga,na) otherwise.

= ( 2 M - l ) ( S o - 1) + fo- l ) n a

= (2,0,4) or (2,1,1) or (3,0,3),

Note also that V+(C) M = e M + ( C ) „ and V"(C) M = © ; = 1 K - ( C ) M . Elements a' and a" in V(C) M are called ^-related (see also [4]) if ||(1 - S)a> + B<J»f^

=

(1

_ 5)2/^1^11^

+ aa//.||0A||»/Mi

0 < 5 < 1.

Furthermore, vector subspaces V" and V" of V(C)^ are called ^-related if every
79 Fact 2.1. V + ( C V V2+(C)M> . . . , V,+ ( C V Kf (C)„, V 2 -(C)„, . . . , V~{C)^ are mutually fi-related subspaces ofV^)^. This //-relatedness enables us to define the concept of extremality as follows. An element a in V(C) M is said to be extremal if a cannot be written as a sum a' + a" of two nonzero C-linearly independent /x-related elements a', a" in V(C)^. Then the set £ ( € % of all extremal elements in V(C) M is described as follows: T h e o r e m 2.2. £ ( C ) „ = ( U & V / t C ) , , ) U (U r a = 1 V-(C%). Proo/. By Fact 2.1 above, we clearly have E ( C % C (U^ 1 ^+(C) / i )U(U^ = 1 V r a -(C) M ). Since dim c V r . + (C) M = 1 for all i, Fact 2.1 also implies £ ( C % 3 U ^ V ^ C V Therefore, it suffices to show EiC)^ 3 V£~ (C)M for all a. For contradiction, assume that an element a in V~ (C) M with some fixed a is written as a sum a' + a" of Clinearly independent /^-related elements a', a" in V(C) M . Then the proof is reduced to showing that it is absurd. We can now write a' = a'a + T and a" = a"a — r for some<^, a£ € V~{C)^ and r € ^ + ( C ) ^ e ( e ^ a V ^ ( C ) M ) . Put u(s) := ( l - s ) 2 / " + 2 1 _1 s for s € R with 0 < s < 1. Since a' and a" S 2/ M - ( i - 2s) /' and s : = (1 - s ) are /^-related, we have

(i - s?'n KBc'" + s2/MIKHcM - K + ^allc/M} + "MIMIc"* = (1 - S)2/"Kllc/M + -""Kile" 1 - 11(1 - »Wa + KfJ" + " W r f " = (1 - s)2/"H<x'||^ + s^W'fJ" _ ||(1 _ sy + sa»fJ> = o.

The following two cases can occur: Case 1: n = 2. Then u{s) = 2s, hence 0 = (1 - s){ \\a'a\\c + \\sa'^\\c - \\a'a + « j * | j c } + 2S\\T\\^ > 2 s | | r | | ^ > 0. By setting s = 1/2, we have K | | c + | | < | | c = \\a'a + o^tlc and r = 0. In view of /i = 2, 0 = \\a'\\r+

a'atflC

^a + ^lb = / (KI + KI - K + ^D-

Since volume forms \a'a\ + \crHa\ and |o-^ + \cr'a + o^\, it follows that \a'a\ + \a'a\ = Wa + v'al everywhere on Ca. Hence, the meromorphic function a'al^a o n C<* i s real-valued, and is a real constant. Therefore, a' (= a'a) and a" (= a^) are C-linearly dependent, in contradiction. Case 2: n > 3. Then for 0 < s < 1, we have u(s) = s1^ + oCs 2 ^). Since IKIIc" _ I K + ^ H c "

=

°(* 2/ ' J ) = o(s 2 ^) by Remark 1.2, it now follows that

(1_s)2/,s-2/nKn2/, + u(s)||T||2/M o(s (l-s)^s^\Kf^ + u(s)\\rf^ DCS2/")h= =

2

= ^ S^(\K\\ = ( I K2^+\\T\\ | | ^ +2J») Hr||^) ++ 0O(S ( ^2)H.

80 In particular, we obtain a"a = r = 0, and it therefore follows that a" = 0. This is a contradiction, as required. □ Put £ + ( C % := U^V+iC),, a n d £ - ( 6 % := lfa=1V-(C)». For each (a, b) € I?, define (C;a,b) := {a € * ( C ) ; ( p a , n a ) = (a,6)}, where $(C) := { 1 , 2 , . . . ,r}. Put further E

(a,b)(C)»

'■-

U

a£${C;a,b) Va (C%i

K(P)„

:=

U

a6$,(C) Va (C)f»>

# $ ( C ; a, 6) := cardinality of $ ( C ; a, 6), where $„(C) := $(C) \ ($(C; 0,3) U $ ( C ; 0,4) U $ ( C ; 1,1)). Note the following fact which is straightforward from the definition of extremality: Lemma 2.3. Let C' and C" be stable curves with a C-linear isometry u: V^C')^ = V(C") M . Then i maps £ ( C % onto £(C*'%. Remark 2.4. (1) Let n = 2. Then V-(C)„ = {0} holds for a 6 $(C;0,3), and dime V-iC)^ > 2 holds for a e *»(C), while dim c V+iC),, = 1 = dim c V7(C% for a e $ ( C ; 0,4) U # ( C ; 1,1) and i € { 1 , 2 , . . . , m}. Therefore, in Lemma 2.3, the isometry i maps £ „ " ( C % and £ + ( C % U £ ( - j4) (C") M U E^^C')^ onto ^ ( C " ) M and E+(C")li U E7f.iAC'')tl U £?," j J C " ) ^ , respectively. In particular, there exists a bijection $»(C") 9 a *-> a 6 $»(C") such that i maps each V~(C) C-linearly isometrically onto V^(C"), and moreover m' + # $ ( C ' ; 0 , 4 ) + ##(C";1,1) = m" + # $ ( C " ; 0 , 4 ) + # $ ( C " ; 1,1), where m' and m" are the numbers of singular points of C" and C" respectively. (2) Let/x = 3. Then dim c Vj+(C)M = 1 = dim c V^(C) M for a e $(C;0,3) and all i, while dim c VQ-(C)M > 2 for a e $ ( C ; 0,4) U $ ( C ; 1,1) U $»(C). Hence, in Lemma 2.3, we see that i maps E+iC^^UE^iC'),, and £ ~ 4 ) (C")pU£ ( - 1 ) (C")^U£'-(C')^ onto £ + ( C " ) M U S - 3 ) ( C " ) M and B ( - i 4 ) (C") ( 1 UB^ 1 ) (C") p U£»-(C")^, respectively. In particular, m ' + #*(C";0,3) = m " + # $ ( C " ; 0 , 3 ) . (3) Let n > 4. Then dim c V;.+(C% = 1 < dim c Va~(C)M for a e $(C*) and all i. Hence, in Lemma 2.3, we obtain a stronger statement that L maps E+(C')a and E-iC')^ onto E+{C")y, and £ r ( C " % , respectively. 3. W e i e r s t r a s s points In this section, we consider a pair (X, £>), called a punctured Teichmuller pair, of a nonsingular irreducible projective curve X of genus g and an effective reduced divisor D of degree n on X such that n > 3 — 2g. Let ^ > 3 and put W :=

81

H°{X,w^((n ~ !)£>)) and TV := dim c W = (2/x - l)(g - 1) + (fJ, - l)n, where OJX denotes the canonical sheaf of X. (For the case /i = 2, see [6] and [1].) Then aeg{w%"((t*-l)D)}-29

= M2<7-2)+(p-l)n-2S ^ { ^ _ 3 > J

if n > 3 - 2g, otherwise.

Therefore, the invertible sheaf w ^ ( ( ^ — 1)D) on X is generated by global sections, and if n > 3 — 2g, it is further very ample. For each p e X, there exists a C-basis {<7i, (72,..., (7jv} for W such that 0 = ordpffi < ordp(T2 < . . . < ord i — 1 for some i. Note that the set X° of all non-Weierstrass points is Zariski-open and dense in X (see for instance [2]). For each p & X, writing the aboveCT/Vas r/p, we have ordp?7p > i V - 1 .

(3.1.1)

where the equality holds if and only if p € X° This 7?p, called a Weiersirass section at p on X , is uniquely determined by p up to a constant multiple, and the complex line CT)P in W depends holomorphically on p e X. Now, suppose (M,3>") i {(3,1,1), (3,0,3), (4,0,3)}. Then N > 3, and either N > \ig + 1 or (ffin) £ { (li 1)> (2,0) }. Moreover, given a point p on X, we see from the inequality d e g K " ( ( M - 1)L>)} - ( i V - l ) =

5


(3.1.2)

that any nonzero element 6 in W with ord p 6 > N — 1 has its unique highest zero at p on X. Here, every r\v with p € X is a typical example of such a 6. Consider real constants e

:= I/JV- M ,- M + I = 2 M " 1 {l + ( i V - l ) - 1 } , f 2(/x- 1 + fc-1), if k > 2,

ejt : = i/fc 0 = <

(_ +00,

otherwise.

Definition 3.2. To each pair (c, T) e W x TV with a / 0, we associate a real-valued function h0tT{t) in £ € R by hatT(t) := / x \a + tr\2^. In view of Corollary 1.4, define SJ £ (0, 2] U {+00} by 8] := mind(/i ff(T ),

j€{0,l},

82 where W0 := W and Wi := {T G W ; d ( V r ) > <5o }. Note that d(h„iT) = <5J for general elements r in Wj. Put k! := max p€l) ordp<7 and k" := max peX ^ £ ,ord p
(3.2.1)

where if either k! is zero or D = 0, then A;'-1 denotes +oo. Furthermore, put E' := { o r d p < r ; p e D } and S" := { o r d p < r ; p e X\D}. Then min{i/A(_A(+ii_/i+2> f/t",i} > «f e AoUAi

ifn>3-2p,

(3.2.2)

where A0 := {f^_M+1,_M+2, vjt»,i} and Ai := { ^ 1 _ M + 1 , _ M + i , i/fc2i0 ; h e S', fc2 € S"}. In view of the inequalities \x > 3 and 2/p. < e < ejv +9 _i < e^-i now show the following:

< 2, we shall

L e m m a 3.3. Suppose (jJ,,g,n) <£ {(3,1,1), (3,0,3), (4,0,3)} and 0 / a e W. Then (1) 6Q < e if and only if oidp a > N — 1 for some p e D; (2) SQ = e if and only if ordp a = N — 1 for some p € D, where in each of (1) and (2), t i e point p is uniquely determined by a. Lemma 3.4. Suppose {n,g,n) $ {(3,1,1), (3,0,3), (4,0,3)} and 0 # a € W In view of the fact that ord 9 a < N + g — 1 for every q 6 X, let us take an integer k such that N-l 6 or one of the following occurs: (a) N = 3, and (p, g, n) = (3,0,4) or (4,1,1) or (5,0,3). (b) N = 4, and {p,g,n) = (3,1,2) or (5,1,1) or (6,0,3). (c) N = 5, and (p, 9, n) = (3, 2,0) or (3,0,5) or (4,0,4) or (6,1,1) or (7,0, 3). Proof of Lemma 3.3. We first assume k' > N - 1 . Then by (3.1.2), we have k" < k'. Hence, by (3.2.1), 8% = 2fi-1{l + k'"1) < e, where 6% = e just in the case k' = N-l. Conversely, assume <5g < e. Since k" < N + g - 1, we have e < eN+g-i < ejt«.

83 Therefore, by (3.2.1), 6% = 2/i- x (l + A;'-1) < e. Hence k' > N - 1, where the equality k' = N - 1 holds just in the case 6% = e. D Proof of Lemma 3.4. (1) Consider the case where k is not divisible by fi. By (3.2.1), 5Q = min { e ^ , , ek„}. Since fc is not divisible by n, the equality 6$ = e^ holds if and only if we have ek = ekn, i.e., A;" = k. (2) We next consider the case where /i | k and AT < 5. Since 7 V - l < f c < i V + p - l , one of the following occurs: 1 (a) AT = 4, and (fi,g,n,k,ek,k(k- (*- - 1I)" ) - **) = (3,1,2,3,4/3,2). 1 (b) AT = 5, and (»,g,n,k,ek,k(k- (k- - 1l)) - e f c ) = (4,0,4,4,1,4/3). _1 (k- - 1l) (c) AT = 5, and (/i, fff n,fc,e*,*(* ) - e f c ) = (3,2,0,6,1,6/5).

,51-.. „ Then by (3.2.1) and (3.2.2), it is easily seen that the equality (*ft,£f) = (ek,k(k l) - 1 efc) holds if and only if k" = k.

(3) By n = 0, we have k" := maXpg^ ord p cr and 6g = vk»fi = ek"SQ = ek if and only if k" = k, as required. □

-

Therefore,

Remark 3.5. Assume that n> g + n and n > 0. Let fc be the integer as in Lemma 3.4, so that A T - l < f c < A T + g - l . Then by AT = (2/x - \){g - 1) + {(j. - l)n, it follows that 0 < n < -k + (2g - 2 + n)/i 2, we consider the spaces W := H°(X', w®^((/x— l)D')), W" := H°{X',J%t,{{n - VjD")). Regard each a in W as a meromorphic /z-ple l-form on X with poles only at D of orders less than (i. Recall that, for genera g1, g" of X', X", we have n' > 3 — 2(J and n" > 3 — 2p" respectively, where n' := deg.D' and n" := deg£>". We further set ||a||' := {fx, \a\2^}^2 2 2 and ||r||" := {/ x „ \T\ ^}^ for all a e W and r e r . Note that dim c W = ( 2 ^ - l ) ( f l / - l ) + ( A t - l ) n ' and dim c W" = ( 2 / x - l ) ( 5 " - l ) + ( A t - l ) n " . For p = 2, recall the following results of [6] and [1]: Fact 4 . 1 . Let \i = 2, and assume that neither (g1\n') nor (g",n") belongs to { (2,0), (1,2), (1,1), (0,4), (0,3) }. Suppose moreover that there exists a C-iinear isometry between (W, || ||') and (W", \\ \\"). Then g' = g" and n' = n". The same method as used for Fact 4.1 allows us to prove the following: T h e o r e m 4.2. Let (J, > 3. Suppose that there exists a C-iinear isometry 7: W' = W" between (W, || ||') and (W", || ||"). Then g1 = g" andri = n".

84 Proof. By setting N := dim c W (= dim c W"), we have N - p + 1 = (2p - \){g' 1) + (A« - l)(n' - 1) = (2p - l)(p" - 1) + (At - 1)(»" - !)■ The following two cases are possible: Case 1. N - p < 1. Note that, in this case, both (g',n',N) belong to one of the following sets: {(0,3,/x-2), ( 1 , 1 , / i - l ) }

if M > 5;

{(0,3,2), (0,4,5), (1,1,3)}, {(0,3,1), (0,4,3), (1,1,2), (1,2,4)}

if M = 4; if ju = 3.

and

(g",n",N)

Hence, since (g1, n', N) and ( 2. Since 7 is an isometry, we have \\a + tr\\' = \\j{cr) + tl(T)\\" f° r a u o-, T e W and ( € R . Hence, at t ~ 0, the order of smoothness in t for the function {|| 2. Lemma 4.3. Let (X, D) be a punctured Teichmiiller pair, and let W, N be as in Section 3 with fx>2. Suppose a, r € W andN-fj. > 2. Consider hatT(t) := Jx \a+ tr\2^ as in Definition 3.2. Then r e tf°pf,wfM) if and only if d(haiT) > ew+g-„ for all 0 ^ c e W. Proof. By AT - At > 2, we have ejv+9-M < 2. The proof is divided into two parts. Step 1. Let r e /f°(X,w® M ). We observe that the highest possible order \ a of a zero of a section a e W for the invertible sheaf w®M((At - l)D) satisfies the inequality \ a < deg{wfM((At - l)D)} = N + g - 1. Therefore, by Corollary 1.4, we obtain d(h^T) > min { vki; \ a - fj, + 1 < k e Z, 0<£ eZ} > ejv+g-^Step 2. Conversely, let us assume r ^ H°(X, w®M). Then r regarded as a section of W®M((A« - 1)1?) has a zero of order A, with A < p - 1, at some puncture pe D. Let T]p eW be asm Section 3. Then the highest order, denoted by A, of a zero of the section np for W®M((AI - 1)D) is taken just at the point p and satisfies A > N — 1.

85 We now put a = r\p. Then v in Theorem 1.1 with k = X - fj, + 1 and £ = X - fi + 1 satisfies V v

1 1 1 1 1 2(»-1k+l)/{k-e) k+l)/(k- -e) < < 2{p 2{fj,-~ + + 1 ) " 1 (l))} = 20*~ + {k (k + 1 - {l-n^ - 1 )} 1 l 11 <2{»+ + {N-n {N-n + l r ^ l A*" ) } < e ^, N+g < 2{/x" + 1)- * ( i - A*" )} < ^N+g-n,

since we have k> N - fi>2, £ <-1, and N + g > ng. Therefore by Theorem 1.1, it follows that d{hatT) < ejv +g _ M , as required. □ Let t: V ( C % = V(C") /1 be a C-linear isometry, where C and C" are stable curves. Recall that, for C in the introduction, we have numbers r, m, n\, n%, ... , fh> 3ii 92, • • • , ffr- For stable curves C" and C", we similarly associate numbers r', rri, n'v n'2, ... , n'r,, g[, g'2, ... , g'r, and r", m", n'/, n^, . . . , < „ , ^ ' , gf, . . . , g£, respectively. Moreover, write C" and C" respectively as unions U„ = 1 C^ and U^ = 1 C^ of irreducible components. Then u is called type-preserving if, with {C{, C j , . . . , C'r,} renumbered suitably, the following conditions are satisfied: (1) r' = r" and m! = m"; (2) ( < £ , < ) = ( s £ X ) and t(V~(C%) = V-(C'% for all a. For such an t, by setting m := m' = m", we write ^ + ( ( 7 % = ©JljV^C"),, and V+(C"% = ©™! V+(C") M . Then renumbering { V + ( C % K + ( C % , . . . , V+(C\} if necessary, we may further assume t(V^+(C")/j) = V^iC")^ for all i, since Theorem 2.2 and Lemma 2.3 allow us to obtain i ( V + ( C % ) = V + (C*'% from (1) and (2) above. T h e o r e m 4.4. (1) Assume pi>4. Let i: V ( C % = V ( C % be a C-iinear isometry for stable curves C and C" Tnen t is type-preserving. (2) Assume that /i = 3 and that m' + #$(C";0,4) + #$(C"; 1,1) = m" + # $ ( C " ; 0,4) + # $ ( C " ; 1,1) for stable curves C and C". Given a C-linear isome­ try t: V^C')^ = V(C") M , we can find a type-preserving C-iinear isometry between V ( C % and V(C"%. Proo/. (1) By Lemma 2.3 and (3) of Remark 2.4, we have r' = r" and m' = m", and may assume i(V+{C')^) = ^ + ( C " % and t(V"(C) M ) = ^ - ( C ' % for all i and a, where { V+(C%, V+{C')^ ..., V+,(C% } and { V f ( C ) M , V 2 " ( C % , . . . , V ? ( C % } are renumbered appropriately. Then by Theorem 4.2, t is type-preserving. (2) In view of Lemma 2.3 together with (2) of Remark 2.4, m'+

#$(C";0,3) = m" + # $ ( C " ; 0 , 3 ) ,

t(£+(C')/1U£(-3)(C%)

+

= E (C")^U£(-i3)(C")M,

(4.4.1) (4.4.2)

and t (£ ( - 0 , 4 ) (C% U £ ^ , ( 0 % U « T ( C % ) = ^ 0 , 4 ) ( C " ) M U ^ ^ ( C * ) * . U £ . - ( C % . This last equality and Theorem 4.2 allow us to obtain a bijective correspondence $(C"; 0,4) U $(C"; 1,1) U * , ( C ) ~ $(C"; 0,4) U 3>(C"; 1,1) U * . ( C ) ,

a «-» a

86 such that the sets { a; a € $(C*'; 0,4) }, { d; a € $(C"; 1,1)}, { d ; a 6 $»(C") } coincide with $(C";0,4), *(C";1,1), $»(C") respectively, and that the following holds: <-(Va(C')) = Vr(C"),

a€*(C;0,4)U*(C';l,l)U*.(C).

(4.4.3)

In particular, #$(C"; 0 , 4 ) = # $ ( C " ; 0 , 4 ) and # $ ( C " ; 1,1) = # $ ( C " ; 1,1). Hence, m' = m" and by (4.4.1), we have #$(C";0,3) = # $ ( C " ; 0 , 3 ) . This together with (4.4.2) and (4.4.3) gives us a type-preserving C-linear isometry between ^(C")^ and V(C")p, as required. D 5. Proof of Main Theorem To prove Main Theorem, we consider punctured Teichmuller pairs (X',DI), (X", D") as in the previous section. Retain the notation in the previous section. For fi = 2, recall the following result in [6] and [1]: Fact 5.1. Suppose p, = 2. We further assume that neither (g',n') nor (g",n") be­ longs to {(2,0), (1,2), (1,1), (0,4), (0,3)}. L e t 7 : W =• W" be a C-hnear isometry between (W, \\ ||') and (W", || ||"). Then there exist an isomorphism u: X' S* X" and a constant £ 6 C with |£| = 1 such that u*(D") = D' and that 7 _1 (3 and that either \x > g' + n' or n' = 0. Suppose also that (u-,g',n') / (3,1,1). Let 7 : W £ W" be a C-Jinear isometry between (W, || ||') and (W", || ||"). Then there exists an isomorphism u: X' = X" such that u*{D") = D'. Moreover, if (/z, g', n') =£ (4,0,3), then we can find a constant C € C with ICI = 1 such that 7 ~ V ) = < > V for all a e W". Proof of Theorem 5.2. We first observe that (fj,,g',n') ^ (3,0,3). Next by The­ orem 4.2, g' = g" and n' = n", so that we put (g,n) := (g1 ,n') = (g",n"). If (fi,g,n) = (4,0,3), then there obviously exists an isomorphism u: X' = X" such that u*(D") = D'. Therefore, without loss of generality, we may assume that (|i,p,n) f {(3,1,1), (3,0,3), (4,0,3)}. Now, to each p e X', we can associate the Weierstrass section r\p e W at p on X' as in (3.1.1). By Remark 3.5, either we have 7i = 0 or every integer k with N — l N - 1, we see from Lemmas 3.3 and 3.4 applied to {X', D') that (a) p e D' if and only if 6l{Vp) = 6$ < e; (b) p € X' \ D' if and only if 6^Vp} = 6$ = ek for some k as above.

87 Then by Lemmas 3.3 and 3.4 applied to (X",D"), a point u{p) € X" is uniquely determined by p in such a way that 7(??p) coincides with r}uu,\ up to a constant multiple, where 7?u(p) € W" denotes a Weierstrass section at the point u(p) on X". Thus, to the C-linear isometry 7 : W = W", we can associate a holomorphic map u: X' -> X" such that 7(C r/p)

= CTJ U ( P ) ,

pel'.

Similarly, to the C-linear isometry 7 _ 1 : W" = W', we can associate the holomorphic map v: X" —► X ' such that 7-

1

( C T 7 , ) = City,),

? eX"

It then follows that v ou = idx< and uov = idx". Therefore, u is an isomorphism of X' onto A"" Now by (a) above, f w = ^ ( " p ) < e if and only if p e D' Then by Lemmas 3.3 and 3.4 applied to (X",D"), we see that u(p) € D" if and only if p € D'. Hence, u*(D") = £>'. Since I " 9 g n 7 " 1 (r]q)/(u*r)q) € C is a holomorphic function, and since ||w*»?9||' = ||»?9||" = ||7-1(»7g)||' for all q e X", there exists a constant ( € C with |£| = 1 such that 7 _1 (r/ g ) = C^Vq f° r all g € X". By the following Lemma 5.3, we now conclude that 7 -1 ( ord„ 7j„ > N — 1, which is a contradiction, as required.

Q

Proof of Main Theorem. In terms of the notation in Section 4, let us write C and C' respectively as unions U Q = 1 C a and U^ = 1 C^ of irreducible components. Let us further write V ( C ) , = ( C i ^ f W ® ( » L i ' ' . " W ■""» V ( C % = ( e ™ > + ( C " ) M ) 0 (©^' =1 V^"(C')^), where m and m' are the numbers of singular points in C and C" respectively. Note that we have a C-linear isometry t: 1^(C% = V ( C % . Then by (1) of Theorem 4.4, 1 is type-preserving. Hence, we may assume (1) r = r' and m = m'\ (2) (ffa,Ba) = (5a. O and i(V£-(C)„) = V£-(C% for all a; (3) t (V+(C)) M = V+(C") M for all i. Let va: Ca —» C a and v'a- C'a —> C"a be the desingularizations. As in Section 2, on CQ and (?„, w e n a v e reduced effective divisors £>Q := v~l(Ca n C s i ng ) and •^o : = ^a~l{C'a fl C ^ ) respectively. For the canonical sheaves Qa and w a of

88 Ca and C'a respectively, let us identify V-(C)„ V-{C')» = H*{&a,riZ»Un

= H°(Ca,u>^{(n

- l)Da))

and

- l)D'a)). Now,

/ / > 2 p + l = 2m + 3 + T,ra=12{ga - 1) = 3 + E£ = 1 {n Q + 2{ga - 1)}, hence ga + na < n for all a. Therefore, Theorem 5.2 together with (2) above allows us to obtain isomorphisms ua: Ca = C'a, a = 1,2,... ,r, such that u^(D'a) = Da for all a. It is now easy to see that C" = Cu for some II e P □ 6. P r o o f of T h e o r e m 1.1 Let ip = ip(z) ^ 0, V = i>(z), n>2, f(t), k, £, is, kj, a,j be as in Section 1. The purpose of this section is to prove Theorem 1.1 by studying the behavior of f(t) as t -* 0. We may assume that tp{z) = zk, ip(z) = /h(z), where h is a non-vanishing holomorphic function on A satisfying \h\ < C5 for some positive constant C5 > 1. If k < £, then f(t) is a real analytic function in t with 01 = /'(0) and 02 = /"(0)/2. In fact, for fc < £,

-b

/(*) = fit)--

2k 2 / \z\ ^ |1 \l + +
:

7A

^ 2

dz, dz A dz,

and we see that \\ + tzi-kh{z)\2li1 has a power series expansion in t, which is uniformly convergent on A for |£| £, i.e., 2//U < v < + 0 0 .

For fi = 2, Theorem 1.1 is known by Royden [6] (see also [1] and [2]). We deal with the case \i > 2 by a method slightly different from Royden's, though our approach is valid also for fj, = 2. We first observe that there are complex numbers cap with a power series expansion OO 00

|1 ++ Zz|| 22 //"M== |l

£ ca0zazP, a,/3=0

(6.1)

convergent uniformly and absolutely on compact subsets of A. Here c 00 = 1, c 10 = c

oi = VA*. C 20 = c02 = (1 ~ AO/( 2 A« 2 ). c n = 1/M2, and we have caa

> 0 for all a.

Rewrite (6.1) in the form |l + z| 2 /" = l + - R e z + ^ { ( 2 - M ) ( R e z ) 2 + //(Imz) 2 }

(6.2)

+ higher order terms in z and z. Introduce a new real parameter s by sk~e = t with 0 ^ \t\
C^^:=C_1.

89

where AJ(t) := { w e C; \s\ < \w\ < 7T 1 } and Aj(t) := { C 6 C; R < |C| < M " 1 } . Note that A(i) := { f e C; |C| < |s| _ 1 } is just A0 U Ai(t). (6.3) Proof of (1) of Theorem 1.1: e /(*) - /(0) = |[f {\* /(t) {\zkk + tz h(z)\2^2 / / i --■\zkk\V» \k2|^2/M} }^ ^O dzdzAAdzdz /(*)-/(0) {|^ tz^)| ■ /

= = W 1*1" /

■/Aft)

fc f c fc 22// {IC fc--'■t++/.(-KsQ?/*C)l 2/ " -- l ClC " 1-'l ""}^dCAdc }^ dC A d~c ICI M/ " {ic

= iti'JbM + WJiW, where we define Jo(s) and Ji(s) by JbW := j ^ ICI^" {IC fc - f + M » 0 I V " - I C ^ I 2 ' " } ^ rfC A df, * ( »W) := := /

2 lC|2fc/ Hs C) C"*^ Ck+t |\2/»" --i}^-dCAdf. ~ 1j ~ ^ A df. ICI *^" { 11 + A(« l

JAM

Now we consider the asymptotic behaviour of JQ(S) as s —> 0 (i.e., 4 —» 0). Then Jo := lims_,o Jo(s) is easily computed as follows: J0=

f

\(\"fr

VAo =

{ |f*-< + A(0)|a/" - |C fc - £ | 2/ " } ~± l

>

2

d< A dC

1 9 2 |2/M _ ij ^ J -Jfc+£i ++r r-fc-W-ifc^ fRri+WM if*" ( |11 r i+ ! / " - - l)d6»Udr.

Jo

For any fixed constant r > 0, the function C 9 2 H-> 11 + r -fc+ * _1 /i(0) z \2^ e C is subharmonic. Therefore,

f( | 1 + ' r

-fc+f- ■xft(0)fr.gV^C ( 11 + r-fc+^MO) e ^ 1 * |2//* | 2 ^ _-- l) l ) d0 > 0

for all r > 0. Moreover, by A(0) / 0, we have \Ck~e + /i(0)| 2//i r < 1, where we set r = |f|. Hence, Jo > 0 and Jo(s) = Jo + o(l),

as s ~» 0.

(6.3.1) |£*-*| 2 / M

> 0 for (6.3.2)

Next we consider the asymptotic behaviour of Ji(s) as s —> 0. Note that AJ(0) = {w € C; |u>| < i? _ 1 }. Let 1A;(«) = lA;(t)(w) denote the characteristic function of AJ(£). Then we can write J\(s) in the form x

{" ++ /i(su; ' )iu*-

Ms) llA;(t)(^) {w)\\lL /s..r /i(«) = [ 1 t) 7J&1(0) A ; (o)

M) V - l dwAdw. h(aw'P-l)\w\+Wri?^dw/\diB. - i 1J})v^Kt\2-4-(2fc/ 22

-< |2/*i .

90 k t 2k -t\V»< CC6&\w\ -1 < OnA*l{t),by\h{sw-1)wk-1e)w \ k~e\ < 1/2, we have \l+h(svj\l+h(sw~1)w1k-)w \ ^-l \w\k-~-el OnAl(t),by\h(swfor a positive constant Cfe Ce independent of s. Therefore, Therefore,

1A!(*}CW) III + M - "

-1

) w*-*!^*1 - 11 M - 4 - W " ) < C e k r 4 - ^ / ^ ^

on Af(0). Since k > ku i.e., ~ 4 - ( 2 f c / / * ) + k - £ > - 2 , the function |w|-4--(3*/W+*-* is integrable on AJ(0). Hence, by Lebesgue's dominated convergence theorem, J\ := lims_,o >M*) is written as follows:

j j == /j Ji ■■

JAKO) JAUO)

2fc/ri dw A dw. M+ M h(0)w H - 44--((2fc/M) (\\l |i + 0 ) ^ k^~le\2^-/ -M1 ->i — }— ^—dwAdw. l l

J

2

Further by (6.1), we have a power series expansion |1 + \2^ -.- 11 = \l + h{0)w h{0)wk-k'£-t\V*

J2 c^fc(O)aAfOJ0u/<*-'>o'u;(fc-W, (<*,/3)#«l,0)

which converges uniformly and absolutely on compact subsets of A. Here, the summation is taken over all nonnegative integers a, P such that (a,/?) ^ (0,0). Since caa > 0 for all a, it follows that °° a a = Y] J2^C2nc aa\h(0)\ Ji = aa\h(0)\

//

rR'1

r-3-(2k/ li)+2(k-e)a dr r-3-(2k/ )i)+2(k-e)adr

OO

= 2nR^kM+2J2{~2-(2k/n)

+ 2{k-e)a}-1\h(0)\acaaR-^k-e'>a

> 0.

Hence, we see that J\ > 0 and that Ji(s) = J\ + o(l) as s —> 0. This together with (6.3.2) completes the proof of (1) of Theorem 1.1, since f(t) - / ( 0 ) = \t\vJ0{s) + \t\vJi(s). a (6.4) Proof of (2) of Theorem 1.1: Write h(z) as a power series E^L 0 ft Q z a on A with coefficients ha e C. In the proof, we shall also show that the constant C\ in (2) of Theorem 1.1 is given by Cx = 2TT (A; + tyl

Re{hk_e}.

(6.4.1)

We here use the same notation as in (6.3). First, by the same argument as in (6.3), we see that Jo > 0 and that Ms)

= Jo + o(l),

as s -> 0.

(6.4.2)

91 Next, we consider the asymptotic behaviour of Ji(s). Then J\(s) is written as a sum K\(s) + K2{s), where we set

-L

JifiW Ki(s) := [

4 a k / f n + /.(^-^Hp/, |w|-4-(2fc/„) |u)|-( ")(|l + fc(a«;-1)u;fc-<|2/'1

- 1 - -

fj, U

M k e R e { Re{h(sw/ i ( s t o - Vl)w '\ ->}\^--^dwAdw, ——- dw A dio,

)

2 1 2

i 1c k e K K22(s) {s) := := /I -|w|-4-W -\w\^ R e~^^'Re{h(sw{ h ( s w - 1 ) w f )w ^ }-^}^-dwAdw. dwAdtD. M;(t) M 2

We shall first study Ki{s). Note that fe = fcj. Then - 4 - (2fc//i) + 2(fc - £) > - 2 . As in (6.3), Lebesgue's dominated convergence theorem implies that Ki(s) = Ki+ o(l),

as s -> 0,

(6.4.3)

where K\ is such that H- 4 -( 2 */"> / |1 + fc(0)iuw|2/'i - 1 - - Re{h{0)wk-£}\

#! = / JA;(0)

l

M

^dwAdw. J

2

In view of the power series expansion a a k |1 + h(Q)w /i(0)w fck--ef | 2 /" - 1 - - RelMO)^-'} Re{/i(0)u;' £ - f } === Y, E C«Pcaflh{0) h(0)ahWw^-^ h(0)f>w(k-e>ffl<MW, wl -W, a+0>2 ^M Q+/3>2

the same argument as in the proof of J\ > 0 allows us to obtain # i > 0. We finally compute K2(s) as follows:

aejf>* r ""0

a R K (s) = = -* ]Re I f ] s ha (j K22(s) /* I»I lc*=0

47T<

I

1

3 - -(2*/M)dr = = ^ R■Re{ft e { ^k_/} } / * \-3. --Wridr =

N

fJ2\v=T(k-t-<-<•>«a)9 d de * \\ )|

'*) Of"™

- 3 - ( 2 * : / M ) + * : - ^ -d r . rr-3-(2W+^-a

2?rt

j

)| fi2+(2fc/kM Re{/i f c ^}{|<|-"^Re{hk_e}{\tr-R^ M}.

Since f(t) - / ( 0 ) = |t|"{Jb(s) + # i ( « ) + #2(s)}, the above expression of K2(s) together with (6.4.2) and (6.4.3) implies (6.4.1) and completes the proof of (2) of Theorem 1.1. D

92 (6.5) Proof of (3) of Theorem 1.1: In view of the definition of a\, we have fit) - /(0) - oit o 2 7 2 _t z\2k h(z)ze-ek-t\-k0n = / M *^l\l" { |1 + h(z)z ^ _-- 11--- - Re{ft(*)** Re{h(z)zt~ }t 1 ^ A*

II

H . } ^ - ddz AAdzdz Z

2 / \t\ v»\\i+h(soe \(\ 4\i+h(so^ \ 7JA(t) JA{t) I 2k/

= \t\L = i*r

k 2/i1 -k\2/ii

_ i _ 1 Re{h{s C) ^-<=} J V^ZI ^

A df

= \t\"Io(s) + \t\"h(s), where functions 7o(«) and Ii(s) are such that

Us) = J I C P ^ j l i + A ^ O C ^ P - 1 - H R e {/ l ( s C) C***} } ~ -
Ji(a) = / 7A;(*)

ft(fltu-1)ti;fc-'|a/'*

I

-1- 1

2

W '>}-' J ~22

Re{/i(suf Re{h(sw~1)wk'e}>^——

A H*

_

dw AAdw. du;.

Since Io(s) is expressible as 2

M/

/..

'' I

/

/ dr, r 1 + ( 2 k / " ) + * - ' R e { f t ( a r e ^ r T"' ) e ^ = T « ( f c -"■•■do )},

and since 1 + (2k /n) +£Jo > 0 in (6.3) gives us h ■= lirailo(s) = / = ^

k> — 1 by k < k\, the same argument as in the proof of

|C| 2fc/ ' i {|l + / i ( 0 ) ^ - f c | 2 / ' i - l - - R e { / i ( 0 ) ^ " ' : } } ^ d < A d C

Kl2fc//1{|i + MO)c*-*!27*1 - i } ^ l d c A d c > o.

Now by (6.2), we have | |1 + h(sur1 2

Cj\w\ ^

a/ i /"} ^ d c Adc \dC -*i "}^U>

w a

/ A ICIM/""{IC*{ic*-'+Hsoi --ic*ic -< + ft(*OI/"2/"-

l

> on A\(t)

V ~ ' l 2 / m " - 1 - (2/fi) Re{h{s w~1)wk-i}

for some positive constant CV independent of s.

I <

93 T h e n b y h ( a u r 1 ) t o f c - < = E ~ = 0 \ha\ sa rk-e-oceV^i'(«.+(*-<-<*)»)>-")»)

we obtain

(Re{fc(-«;- 1 )^-'}) 2 = f:» Q r a *- 2 '-«»{ £ My^(fl)}, <*=0 *■ /?+ 7 =a ' OO

-1)wk~ -*})*..

(lm{h(sw

= E*<* 2k--21r

a=0 a=0

i

E *■ 3+7=Q /3+

7=a

where 4 ^ ( 0 ) := (-1)'cos{8 0 + 81 + {2k-2£-/3--y)0 for any integer z. Therefore, £2(5) is OO

E* a=0

Q

r

J

kr1-"!

E ^+7=a

L

W

a=0

l\s\

2ir(l - M) +2 ^ t2

3+-V-Q

X!

c o s

M N

{(2-

, /Jo 0

^

^ +^

A4 TT(l 2

} + c o s { $ 0 - 0 7 + ( 7 - 0)0 }

-/OAJJ^H-«A [ 1 ] , ( e ) } c w l d r J

r-Hk-i)-

I

r-^-^dr

•'I 8 '

0+- r=2(fc-f) 2«

(*)},J

r

IMIM 2/j2

A [l1 y3,7

2

OO

r-^-'dr J\s\ •//l»l s ■

a = 0n

-■ w

cos + 6 \hg\ \hJcos(6a T = 5AJ..on.(fci ^- £)( 1(1 --- '2t*R2(t(~'~)>h B+~t=2{k-l) E IMIM ^ + *r) 2

2 k e

0+J--=2(k-l)

+

27T|fc0|2 V-

2

{^

= C 9 log — + 0 ( 1 ) ,

7T

+ log ^ \ t \

00

|/la| a * } + aE =l

2 2 2a ( 1 - - i ? % .>l )

ass^O,

for some positive constant Cg independent of s. Now by k = k2, we have v = 2. Hence, we see that f(t) - /(O) - axt = |*| 2 {/ 0 (s) + h(s) } = \t\2{I0(s) + Li{s) + L2(s) }, where I0(s) + Li(s) + L2(s) = I0 + Z-i + C 9 log(l/|i|) + O(l) as i -> 0. This completes the proof of (4) of Theorem 1.1. D (6.7) Proof of (5) of Theorem 1.1: By setting A := := \l |1 + + 1 h(z)zi-k\2/l' t{z) M*) i|(2 -

e k 2

k

-th(z)z 1 - -- \ ^-l--Re{h{z)/R e { / i ( z ) / - f c }}tt A*

M )(Re{/ l ( 2 )/-'=})

2

+

n{lm{h{z)J-k})2y,

5 , ( 0 := |i + A(«C)<'- f c l 2 / ' , -i-?Rfi{M*C)C'- f c } A4

- ^ { ( 2 - A0(Re{/i(SOC"})2 + ^M/i^OC* - *}) 2 },

94 k > k2, i.e., - 4 - (2k/fi) + 2(k - i) > - 2 , Lebesgue's dominated convergence theorem together with an argument as in the proof of K\ > 0 in (6.4) shows that fc-f|2/M | t t-4-(2fc/ / | - * - WMr t)l|/ |1i ++ /l(0)w /,(o)iu*-'|V»i |w|

h ! ■= : = l ilim/i(s) m / i ( s ) == //

Z

kl

fc -- 11- - - Re{^(0)w Re {h(0) w Adw dw>>C0. -' }\^—^dw- dw A M

<>}^

Hence, f(t) - /(0) - axt = |i|*(Io + h) + o(\t\") with 70 + ^l > 0, as required. D (6.6) Proof of (4) of Theorem 1.1: We use the same notation as in (6.5). Then by k (= fo) < fci, we have IQ := lim,,_,o IQ{S) > 0. Now we put o2 Re{/i(suT 5i(«. if) == - R gi(s,w) e l / i t s u r 1 ) ^ " ^~% },

"W

1 fc l k w) == i { ( 2 - /-i ) fi)(Re{h(sw~ 4- Mfi(lm{h(sw~ ( R e { M s »- )w - V k- -f }V) 2 + ( M M s ^- )w V W- }-e}) ) 22\.} -

952(s 2 ( S :, U )

■?K

Using these, we can write Ji(s) as a sum Li(s) + L2(s), where Li(s) Li(«) = /

1

[Iwr^Wri w r 4 ~ ( U / " ) {\l (|l +

k

2

_1 h(sw)w -'\2/^'' ft(sw )w*^|

- l1- ~9i(s,w) g i ( s , w ) --5 292(s,w) ( s , i o ) >—— \——dw/\dw, dw - /[ L22(s) (s) =

2 /rt M--44-( M - ( 2*/") * 9»(«, 10) ^\ *P dw dw A A AB. dw. 2 { S , w) 22

J&t(t) ./A?m

Note, by (6.2), | |1 + h{s w'l)wk-l\2^ - 1 - gi(s,w) - g2(s,w) | < CsM 8 ** - 0 on A\(t) for some positive constant C% independent of s. Since k = fo, i.e., —4 (2fc//x) + 3(fc - £) = - 2 + (k - £) > - 1 , Lebesgue's dominated convergence theorem allows us to express L\ := lim5_,o L\ ($) as a bounded integral /

JA;

(0)

M - ^ W l l + h(0)wk-e\2'» - 1 - g,(0,w) - g2(0,w)\ ^ I J 2

dw A dw.

Next we compute I/2(s)- In the expression of h(z) as a power series E£L0 haza, we have ho ^ 0. We write w and ha in the form (f w = re^6re^ , 6, \

ha=

> 0, 9 e R , rr > 9

l/ialev^ ",

0aeR.

95 we have the equalities 2k

2k f(t) '»A^A ^dzhdz a\t -a -- 2at2t --== f( \z\\z\ A dz t(z)t(z) —-— dz ( O ) ■-- oit / ( * ) -- //(O) 2* JA

7

= 1*1 / = |t|"

l£|2fc/p 2fc/f, B {0

C == |t|"{JW \t\"{M0(8) + M1(8)}, Bss(C) ^^ d C V A Ad d< 0(») + M1(s) } ,

lC|

JAW

where Mo(s) and Mi(s) are such that M AibW 0(s) = /

2fc/ e / / J |C| l C l 2 f"B ^d p dCCA A ddC, C, s-(BC s ()0^

JAo

4 /[ H' H " -"■W« * "JB,(iuB , ( i u )= JA-M

Mi(a) = Mi{a)

4

la

/, ,

1

1

) . ^ d■i du; u A A( t du>. e. 2

By /c < fca, we have (2k/n) + 2x(t - k) > -2 for x < 2. Therefore, M 0 := lims_>o MQ(S) exists. We finally study the behaviour of M\(s) as s —> 0. By (6.2), | B s ( w _ 1 ) | < Cio|«;| 3 ^ fc_ ^ on Aj(t) for a positive constant Cio independent of s. Hence, Mi(s) = O ( l ) + 0(\t\3~v). Thus, / ( t ) - /(0) - ait - a2t2 = \t\u{M0 + 3 2 0(1) } + 0(\t\ ) = o(\t\ ), as required. D Remark 6.8. In (5) of Theorem 1.1, put v := min{3, v}. In (6.7) above, we actually proved a stronger statement that the equality f(t) — f(0) — a\t — a2t = 0(\t\") holds for k < k2. References 1. C. J. Earle and I. Kra, On isometries between Teichmuller spaces, Duke Math. J. 41 (1974), 583-591. 2. F. P. Gardiner, Teichmuller theory and quadratic differentials (Wiley-Interscience, New York, 1983). 3. T. Mabuchi, Compactification of the moduli spaces of Einstein-Kahler orbifolds, Adv. Stud. Pure Math. 18-1, (Kinokuniya and Academic Press, Tokyo and Boston, 1990), 359-384. 4. T. Mabuchi, Orthogonality in the geometry of IP-spaces, in preparation. 5. T. Mabuchi, Precompactness for a family of multiplicative systems of If -spaces, in preparation. 6. H. L. Hoyden, Automorphisms and isometries of Theichmiiller space, Ann. Math. Stud. 66 (1971), 369-384.

96 L I N E A R A L G E B R A OF A N A L Y T I C T O R S I O N HONG-JONG KIM* Department of Mathematics, Seoul University, Seoul 151-742, Korea ABSTRACT Analytic torsion was first introduced by Ray and Singer [33, 34, 35, 32, 42, 6, 29, 30] in the study of Reidemeis ter-Pranz torsion [7, 10, 13, 26, 27, 28, 36]. Recently, it has become more and more interesting [4, 11, 12, 15, 14, 17, 31, 38, 46]. In this elementary article we study finite dimensional linear algebra of analytic torsions and its analogy for elliptic operators. The original definition of analytic torsion has concerned only flat hermitian connections but we consider its natural extension to an arbitrary elliptic complex of hermitian vector bundles.

1. Linear Algebra 1.1. Regularized determinant In this section we fix a real or complex field as a scalar field W. For an endo­ morphism P of a vector space V of finite dimension n, the regularized or renormalized determinant1 of P , denoted by det P , is the product of all nonzero (complex) eigenvalues of P counted with multiplicity. For a nilpotent endomorphism, the regularized determinant is equal to 1. One can see easily that det(P') = (det PY

Vi = 0,1, 2 , . . . .

The regularized determinant is nonzero and equal, up to sign, to the coefficien t of the lowest degree monomial in the characteristic polynomial. Proposition 1.1.1. The map |det| : E n d y - ^ F x -> M. is upper semi-continuous. Proof. Let det(t - P) = tn - O n ^ t " " 1 + • • • + (-l)n-kaktk be the characteristic polynomial of P , where ak ^ 0. Then it should be obvious that det P = ak and det(t + P) = tn + On-it"-1 + ■ ■ ■ + aktk Thus

detP = l i m d e t ^ + P ) • e-mail: [email protected]; Partially supported by GARC, KOSEF (1993). 'The notation det x P is also suggestive.

(LO)

97 where k = k(P) depends on P. Observe that k(P) = limi_ 100 dimker(P i ) = dimker(P n ). Thus the "stable nullity" k(P) is an upper semi continuous function of P Now the result follows from (1.1.2). Note that det is continuous when restricted to each subvariety of End V with fixed "stable nullity". Proposition 1.1.3. For any nonzero scaJar c, det (cP) = cl det P, where I = l i m i _ 0 0 d i m P t ( V ) = d i m P n ( V ) denotes the "stable rank" of P Proof. Note that the stable nullities k of P and cP are equal and hence their stable ranks I = n — k are also equal. Now * w ™ ,. det(t + cP) ,. det(ci + cP) cn ,. det(f + P) det(cP) = hm *—r = hm . ,.k = ~rk hm , v k ' «-0 t <-o (ct) c *->o tk = c'detP. This completes the proof. We will see later in (2.2.4) that the stable rank of some elliptic operator Q is equal to the value of the zeta function (Q(S) at s = 0. Proposition 1.1.4. If Pi is an endomorphism ofV\ and P% is an ofVi, then det(Pi © P 2 ) = det Pi • det P 2 .

endomorphism

Proposition 1.1.5. If two endomorphisms P\ : V\ —» V\ and Pi : V2 —» V2 are equivaient in the sense that there exists an isomorphism h : V\ —f V2 such that hP\ — Pih, then the regularized determinant of Pi and Pi are the same, i.e., det(/i o Pj o /T 1 ) = det Pi. 1.2. Zeta function and the regularized determinant For an endomorphism P of a finite dimensional vector space V over C, define the zeta function (p of P by CP(S):=£A-S,

VseC,

A

where A runs through all nonzero eigenvalues of P counted with multiplicities and A _ s = | A | ~ s e _ i s a r g A (we fix an argument argA for each A). Then CF(0) i s equal to the stable rank of P and d e t P = exp(-Cp(0)).

98 1.3. The analytic torsion of a linear map Suppose that we axe given a linear map P : V0 -> Vi

between inner product spaces Vo and Vi of finite dimension. Then we have the adjoint P* : Vi - Vo of P Since the nonzero eigenvalues of P*P and PP* are the same with the same multiplicity, they have the same determinant, which is a positive real number. We now define the analytic torsion of P : Vo —>■ V\ between inner product spaces to be T(P) := v/det(P*F) = x/det(PP*) € R+. It is obvious that r{P)=r{P*). Remark 1.3.1. Suppose hi is an automorphism of V. for i = 0,1. Then it defines a new inner product (•, ■}" := (hi(-), /ij(-)) on Vi and hence we have a new adjoint P* of P . Then P* = (/iS/i 0 )- 1 J P*(M^i)In particular, if hi is a nonzero scalar Hi, then the new torsion f ( P ) is equal to jjM T(P), where I is the stable rank of P*P, which is equal to the rank of P We list some properties of the analytic torsion. Proposition 1.3.2. For two linear maps P : VQ —► Vi and Q : WQ —> W\, T{P®Q) T{P®Q)

==- r(P)T{P)-T{Q). ■ r(Q).

-1 If there exist isometries ho :■VQw0 and Vo —+ > WQ and hi h\ ■Vi : V% ~->Wi —» such W\ such Qx =oP<3 /hioPoho , that that Q ---h 10"1 then oP<03/lQ T(h1oPoh -1) • 1 ) === T(P). r(fci T(P).

Proposition 1.3.3. If P is an endomorphism of an inner product space V such that P*P = PP*, then T(P) = | d e t P | . In this case we have a geometric interpretation of T(P). Note that if P is normal, i.e., PP* = P*P, then P:(kerP)-L-+(kerF)-L is an isomorphism and r(P) is equal to the volume change ratio of this isomorphism.

99

1-4- Analytic torsion of a chain complex Now suppose that we are given a chain complex P : 0 -► Vb -^ V! A ... - ^ H Vi — 0

(1.4.1)

of finite dimensional inner product spaces. Then the analytic torsion of this chain complex is defined as the alternating product

=n(^)(-i)'-

rP^H(rPq)^)\ TP:

?>o 9>0

Let P* be the adjoint of Pq and let Uq;^P*qPq +

Pq^_v

Then by the Hodge theory, Vq = im(P9_i) © im(P*) © ker(D9) is a direct sum of Dg-invariant subspaces. Since a,|im(P,_i) = Pq^\P*_^ and Dq\im(Pq) = PqPq, we have, by the proposition (1.1.4) Proposition 1.4.2. deta, =

(T(P,_I)

• r(Pq))2.

Thus we have r P = n(detD,)(- 1 ) , + I «/ 2 .

(1.4.3)

9>0

This identity will serve as a definition for the infinite dimensional case. 1.4-4- K V+ a n d V- denote the direct sum of all Vq for even q and for odd q, respectively, then we have maps P+ V+ v_, P-:V-^ P_ : K. -> F+ p+ :■V+v+-»- V-, where P± = (P + P*)\V±. Now P_ = (P+)* and (P+)*P+ = Do © Q2 © • • •,

(P-)*P- = Di © D3 © ■ ■ •,

and hence r(P + ) = n r ( P , ) = r(P_). 9>0

1.4.5

More generally, one may be interested in

rt:=\[r{P n: qf 9>0 9>0

for t e R.

W

100 1.4-6. From the above proposition, one can see easily that

n^eta,)'-1)' = 1 in the complex (1.4.1). This phenomenon is also true for infinite dimensional cases (2.2.5). p

p

1.4-7. If we only consider the torsions of exact sequences P : 0 —> V& —> Vy —» • ■ • —> Vi —> 0, then P+ : V+ —> V_ is an isomorphism and hence T(P+) = Ylq>o T(Pq) is a continuous function of P. Since Eleven T (^9) ali^ Ilq:oddT(pq) a r e u P P e r semicontinuous, T(P) = T(P+)/(Y\q:oddT(Pq))2 = (Ylg:eveaT(Pq))2/T(P+) is a lower and upper semi-continuous function of P and hence is a continuous function of P. I.4.8. If we use the zeta function (1.2), then the torsion of the chain complex (1.4.1) is given by rP = n ^ e t D ^ -

1

^ = exp {\ £ ( - l ) » < ( 0 ) ) .

q>0

We now show that

<J>0

E(-1)*^(o)

is equal to the alternating sum of the ranks of Pq's. Let zq = dim ker Pq, bq = dim im Pq-\ and vq = dim Vq. Then Vq = ker P 9 ©im P* and hence dimimP* = vq — zq = bq+\. Now CD,(0) is the dimension of imD, = im Pq-\ © im P* and hence

E t - ^ w o ) = Et-1)'^+6«+i) = E t - 1 ) ^ as claimed. 1.5. Torsion vector and analytic torsion For a finite dimensional vector space V, let det V be the highest exterior power of V and let V~l be the dual space of V If V. = {VQ, ..., Vj} is a sequence of finite dimensional vector spaces, then det V. := det V0 ® (det V i ) - 1 ® . . . ® (det Vj) ( _ 1 ) '. Suppose we are given a chain complex

p : 0 -» Vb A Vi A ... J H y; _+ 0 of finite dimensional vector spaces. For the moment we do not assume inner prod­ ucts. Let H*(P) = {H°{P),..., Hl(P)} be the cohomology spaces of the complex. Then

101

Proposition 1.5.1. There is a canonical isomorphism [2, 3, 19, 30] detV.~

det

H'(P).

We may regard this isomorphism as an element f(P) of (det V.)~l ® det H'(P), which will be called the torsion vector of the chain complex P If we assume inner products on each vector spaces Vjt, then det V, inherits a canonical inner product, and by the Hodge theory, det H'(P) also inherits an inner product. Thus one can measure the length of an element in (det V . ) - 1 ® det H'(P). One can see easily that in [30]. Proposition 1.5.2. |f(P)| = T(P). 2. Analytic Torsion for Elliptic Complexes Suppose we have an elliptic complex2 P : 0 -> C°°{E0) A C°°{Ei) - ^ . . . - ^ H C°°(Ei) -+ 0

(2.1)

for hermitian vector bundles EQ, EI, ..., Ei over a compact Riemannian manifold M, where C°° denotes the space of smooth sections. Then we have the formal adjoints P* and the 'Laplacians' dq. Motivated by (1.4.3), the analytic torsion of the elliptic c omplex P is defined by i 1 +1 2 r(P):=l[(deta
9>0 9>0

where the meaning of det □ is explained in the following. 2.2. Zeta function and the regularized determinant of a self-adjoint, positive semidefinite elliptic operator Let Q : C°°(E) -> C°°(E) be a self-adjoint, positive semi-definite elliptic (pseudo differential linear) operator of order m > 0 on a hermitian vector bundle E over a compact Riemannian manifold (M, g) of dimension n. Let A^ denote the fc-th non zero eigenvalue of Q counted with multiplicity. Then it is well known [41, p. 124] that the limit \n/m n/m Xk

lim l i m ^ kr k—KX> k—*oo

K

exists as a finite positive real number. Thus the series oo

CQ(B)

:

oo t=1

!

A

fc

fc=lAfc

We assume the order of the linear differential operators P) 's are all the same and positive.

102 converges and holomorphic for complex numbers s with large real part. We now show that (Q{S) has a meromorphic extension for all s € C and regular at s = 0. For this, consider the heat operator of Q; x -tQ ■c00-> (E)^C' {E). ee-tQ : C°°(£;) C°°(£).

It is an integral operator with kernel Kt(x, y):E iC^a;, y) :E y->E y -> x, Ex,

x,y € x,yeM,t>0 M, i > 0

so that for any section s of E, tQ ((ee - ^S){x)= s ) ^ ) = = I/

Kt(x,y)s(y)Sg(y)

JM

satisfies the heat equation ( ^ + Q)st = 0, l i m ^ o st = s, where 6g denotes the canonical density on (M,g). Then for each x € M, there exists an asymptotic expansion (cf. [41, p. 114], [16], [37]) as t \ 0 OO

Kt{x,x)

~ ^^(x)^-"'/"1,

0i € C°°(End£)

j=0

uniformly in x € M. Corollary 2.2.1. Let OO

hQ(t) : = ^ e _ A ' ' = Tr(e-"5) - dim ker Q. hQ(t) :=J2e~Xkt = Tr(e- t Q ) - dim ker Q. Then ast\0 there exists an asymptotic expansion Then as t \ 0 there exists an asymptotic expansion OO

M*) ~X> t ( i - " ) / m > as t \ 0, where a, = = J/ M tr(0i) t r ( ^ ) <5p Sg ifi^n

ceR

tr e and oann = JM Jtr(6 6g -- dim ker Q. M (n)n)6g-

2.2.2. Now by the Mellin transform, we have 1 f°° CQ( S ) = T ^ T / t s_1 /iQ(t)rft, 1 s \ ) Jo

Res»0.

The zeta function is regular at s = 0 and CQ(0) = a„ = JMtt{en)6g and CQ(0) is a reaZ number.

- dim ker Q

103

Now the regularized determinant3

of Q is defined by

d e t Q = exp(-<£(0)). Thus the torsion of the elliptic complex (2.1) is 1 \ TP = e x p ( - £ ( - l ) « g C D , ( 0 ) ) .

(2.2.3)

Proposition 2.2.4. Let E be a hermitian vector bundle over a compact Riemannian manifold M. Let Q : C°°{E) —> C°°(E) be a positive semi-definite self-adjoint elliptic operator. Then for any positive real number c, CcQ(s) = C-$CQ{S),

Se C

and det(cQ) = c ^ ^ d e t ( Q ) . Proof. Let A denote the eigenvalues of Q so that cA's are the eigenvalues of cQ. Then for R e s > > 0,

t.w-E w

= c-aC<3(s)

and hence it is true for all s € C. Now C'CQ(S) = -c- s (logc)CQ(s) +

c-%(s)

at regular point s and hence C^>(0) = -(logc)Cq(O) + CQ(0). Finally, det(cQ) = exp(-C^(0)) = exp((log C )C Q (0)) exp(-Cg(0)) = <*><°>det(Q). This completes the proof. This proposition says that (Q(0) is the 'stable rank' of Q (cf. Proposition (1.1.3)). Note that / tr(0„) Sg = dim ker Q + (Q(0). JM

Proposition 2.2.5. In the elliptic complex (2.1), £ ( - l ) ? C a , ( s ) = 0, q>0 3

Also called a functional

determinant.

VseC

104 and

I[(
1.

9=0

Proof. For A > 0, let T\(Eq) denote the A-eigenspace of □,. Then we have an exact sequence

o-,r A (£ 0 ) - ^ r A ( £ 0 The spaces r\(Eq) spaces, then

**...->rx(Et)-o.

are finite dimensional and if c\,q denote the dimension of these

£(-!)«<*., = 0. «>o Now for s e C with large real part,

£(-!)«&,« = E(-D ? E cjf = E E ( - D ^ = °9>0

A>0

9

A

q

This shows the first assertion. Now we have E(-1)?CD,(0)=0 9>0

and the second assertion is clear. This completes the proof. Proposition 2.2.6. Let Q : C°°(E) — C°°(E) and Q' : C°°{E') — C°°(E') be positive semi-definite self-adjoint elliptic operators on M. Suppose there exists an isometric bundle homomorphism h : E —» E' such that Q'h = hQ. Then AetQ = detQ'. Corollary 2.2.7. Suppose that an elliptic complex (2.1) is equivalent to another elliptic complex P': 0 -» C°°(E'0) - ^ C°°(£i) - ^ . . . - ^ i * C°°(EJ) -v 0 in the sense that there exist isometric bundle homomorphisms that /i 9 +iP 9 = P,/i 9 , then r(P)=r(P'). Proof. hq+x Proo/. Note that P' P gq = = hq+l PqPhqqh-1q

1

hq : Eq —» £ ' such

and {P' (P^)* ^ ^qh+q+lj ~\ - 1 . Thus qY = /hqi P*

a'g =

hqaqhq-1

and the result follows from the above proposition.

105 2.3.

Duality Let P : 0 -» C°°(£ 0 ) - ^ C°°(£i) - ^ . . . - ? H C°°(JSi) -> 0

be an elliptic complex for hermitian vector bundles EQ,E\,... Riemannian manifold M. Then we have the adjoint P* : 0 -> C 0 0 ^ ) - ^

,E\ over a compact

. . . -^> C°°{E0) -> 0.

Proposition 2.3.1. r ( P ) = T(P*)(-1>'+1 Proo/. Note that ,+ ( i),+i,/a( -1)detD V2 = =( J](5etn,)(= fI(detD(. - 9 ) =n( " «) "i),a ",+l(i",)/2 ■if T(n=n(*t°*-,) o>0

T
9>0

-,+!((. -9)/2

q

and hence

r(p*ri),+i = ( n ^ e f 9)(-inV2(n(detD9)("1),+1,/2). 9>0

q>0

which is equal to T(P) by the Proposition (2.2.5). This completes the proof. 2.4- Change of the metric Let P : C°°(Eo) —» C°°(£i) be a differential operator between hermitian vector bundles (£i, ( , ),-), i = 0,1, over a compact Riemannian manifold (M, p). Let ( , )t be a new hermitian structure on Ei. Then {•, -)i = (Hi-, -)i for some self-adjoint positive definite endomorphism Hi of (Ei, ( , )j). Let g be a new Riemannian metric on M so that the new Riemannian density satisfies Sg = XSg for some positive function A on M. Then the new adjoint of P is given by (cf. (1.3.1)) P* = i f l o ^ P ' F i A . In particular, if A is constant and Hi = pi id for some constant pi > 0, then p~* = Pip* P0 Now in the elliptic complex (2.1), if we use the new Riemannian density Sg = XSg (A = const > 0) and new hermitian structures ( , )j = ft( , )i, then

Ai = £l±ipi*p + -^p,_ 1 p;_ 1 . Pi

Pi-\

106 Corollary 2.4.1. Suppose we have a homothetic change of the Riemannian 5g = X6g on M and a homothetic change ofhermitian structures ( , ){ — p on Ei for some constants p, po > 0, then CA;(S)=P_SCA.(S)

Ai=pAi, CA; (0)

= CA, (0),

Ck; (0) = - (log p)^

CA

(0) + CAi (0)

,

det At = P <<°> det Ai,

In particular, f ( P ) is independent

density po(, )i

f ( P ) = p~* E ( - I ) « C A , ( 0 ) T ( P )

of\.

3. Flat Bundles Let E —» M be a hermitian vector bundle with a flat connection D . Then we have the associated elliptic complex dD : 0 - A°(E) *B+A\E)1*+

> An{E) -* 0.

The analytic torsion of this complex will be denoted by T(D). Proposition 3.1. Let D be a flat iermitian connection over an even dimensional oriented M. Then r(D) = 1. Proof. The Hodge star gives a commutative diagram

Thus from Corollary (2.2.7) and Proposition (2.3.1), T(D)-T(D) ==

1 r(ZT)T(D*)=T(D)= r(2>)-». .

Thus r = 1. This completes the proof. Remark 3.2. If D\ is another flat hermitian connection on E equivalent to D, then there exists an isometry h : E —> E such that dDi = h~ o do ° h. Thus T(DI) = T(D) by Corollary (2.2.7). Thus r is a function on the moduli space of flat hermitian connections.

107 3.3. Homothetic change of the metric Suppose D is a flat connection on a hermitan vector bundle E over a Riemann ian manifold (M, g) . If g = fi2g for some constant /i > 0, then (,)", = fi-2^ , ), on A"(E) and hence A, = /i _ 2 A,. Thus (cf. 2.4.1) f (£>) =

ME(-I)'«CA,(0)T(JD)

3.4. Example For example, consider the Laplacian A = - (J5) acting on the space of smooth functions on the circle S J (r) = {re' 9 : B 6 R} = R/2nrZ of radius r > 0. Then the spectrum of A is {£y : k e Z} and hence 00

25

1

C A « = 2 £ ^ = 2r2sCR(2s),

Res > -, 2

where 0? denotes the Riemann zeta function. Since [44] CR(0) = - ~ ,

Cfi(0) = - l o g v ^ ,

we have CA(0) = - 1 , CA(0) = -21og27rr, det A = (27rr)2. Thus the analytic torsion of the de Rham complex on the circle is T(Sx(r), d) = VdetA = 2nr. If we consider a complex line bundle L over S 1 (r), then L must be trivial and hence we may assume that L — S 1 (r) x C. We also assume the standard Hermitian structure on L. Then a Hermitian connection D on L is given by D = d + iui for some (real valued) 1-form u on S*(r). One can see easily4 that any Hermitian 4

If ^ / s , , . a = k + u for k € Z and 0 < u < 1, then — I l

u + igiT^dgt - udS = 0,

where s*(re") = e"*. Thus u + ig*~ dgk - ud0 = df = -ie-"d(eif) g = gte" 6 C°°(S\r), 1/(1)) so that

for some / e C ^ S ' M . I R ) . Now let

g" D = d + iw + g~ldg = d + iudB. In general, if D is a flat connection on a hermitian line bundle L over a compact Riemannian manifold M, then any flat connection on L is equivalent to D + iuj for some harmonic real 1-form u on M, unique modulo ff'(Af, Z). For, if D = D + iui is a flat connection, then ui is closed and hence there exists a smooth function / : M —> H such that u> := u + df = u — ie~',d(e'1) is harmonic.

108 connection on L is equivalent to Du := d+ iudB for some constant u e M, unique modulo additive Z. We now fix u G M such that 0 < u < 1. Then the Laplacian is given by

Au=

-M^ + ")

and hence Spec Au = {('" ' " ) 2 : k 6 Thus the corresponding zeta function is, for Res > j ,

CM = £ ( ~ r

2 s

= r"2s ( f > + u ) " 2 s + X > + d - w))"2s)

itez

Vit=o

ife=o

= r- 2 s (C(2s,u) + C ( 2 s , l - u ) ) ,

/

where C( s . u ) : = 53itlo(k + w ) _ s ' s t n e Hurwitz zeta function for Res > 1 [44, p.36]. Since the analytic continuation of C,{s,u) is given by, for R e s < 0, ,.

sCIS, v

,

U); =

2 r ( l — s) I . 1 j - .,

(27T) 1 - 8 v

'

^

cos2m7ru

^

m1"8

< Sin -7TS >

I

2

l.

-.

1

^ ^ sin27rmu I

2

^

1- COS -7TS

m=l

>

m=l

=

ro1-*

>,

[

,1

we have, for Re s < 0, . . . _ 2 . 2 r ( l — 2s) „ . ^-^ cos2m7ru c (s) = r 2 s m 7 r •s ^ (2TT)1-2S m1-25

"

•72^r-

S^^

771=1

Thus Cu(0) = 0 and

c4(o) - 2 f; ^ ^

=

_2l0g n _ e w , = _2l0g |e-

7(1

m=l

= — 2 log(2 sin 7ru) and d e t A u = (2sin7ru) 2 ,

T(DU) =

2sinnu. 1

Note that for 0 < u < 1 the cohomology spaces HpJS ^)) is independent of r.

are trivial and T(DU)

109 4. Instantons on 4-Manifolds To an instanton D o n a compact oriented 4-manifold, there associates an AtiyahHitchin-Singer complex [1] and hence one has an analytic torsion T(D). Then r is gauge invariant and hence is a function on the moduli space of instantons. 5. Holomorphic Vector Bundles Let £ - » J t f b e a hermitian holomorphic vector bundle over a compact (complex) hermitian manifold M of complex dimension n, and let d

p . #rf"p : n0 _^ -> AAPflfp\ <°{£) H'. /tp,lcc^ Ap'\S) -*

.

►AP,ntc\ Ap'n{£) -» 0

be the Dolbeault complex. Then we have the p-th analytic torsion TP(£). there is a commutative diagram 0^

AP'°(£)

- £ - »►

A*l{S)

Since

n - > • • • - *► Ar> -> {£) -*^ 0 AP
'1 0 -^ n - p . r c ^ . ) we have

• -* ^"-P'^f*)-* 0 Jjn+l

{( 1)n+1 r Pp (£) Tn( £ : )== T „ P_{£*) . p ( n~ -

5.i. Einstein-Hermitian Connections Let £ -» M be a C°° hermitian vector bundle over a compact Kahler manifold M with the Kahlr form $ . Then a hermitian connection D for £ —> M is called an Einstein-Hermitian connection5 if the curvature ifo is of type (1,1) and the mean curvature6 KD is a constant multiple of the identity endomorphism of E ([22]). Then there exists an elliptic complex [18] 0 -» A°(EndE)

- ^ A 1 ( E n d S ) - ^ ^ ( E n d £ ) A,A0'3

(End E) &>

...-^4°-n(End£)^0 5 This connection was called Hermitian-Einstein [5, 20, 8, 9, 43], Hermite-Einstein [18, 24], EinsteinHermitian [21, 22, 25], or Hermitian Yang-Mills [23, 45], since A ^ is proportional to the hermitian metric hjj and D is a critical point of a Hermitan Yang-Mills functional. Since it was first recognized and studied by Professor S. Kobayashi as a differential geometric concept of stability, it might be better to call it a Kobayashi connection. 6 I f e i , . . . , e n i s a n orthonormal basis for the holomorphic tangent space of M at a point m, then the mean curvature is a self-adjoint endomorphism Km '■= J2Z=i ^e0*<» °^ the fiber Em- A hermitian connection D = D' + D" induces Laplacians A = A ' + A " Then

A' - A " = v ^ I I A , RD] : A"(E) - . where A is the adjoint of the exterior multiplication e x t $ : A'(E) A")U°(£)-

A"(E), —* A*(E).

In particular, Ko = (A' —

110

and hence we have an analytic torsion T(D) for each Einstein-Hermitian connection. The group of the bundle isometries of E acts on the space of Einstein-Hermitian connections and the analytic torsion is invariant under the action. Thus the analytic torsion is a function defined on the moduli space of Einstein-Hermitian connections. This moduli space is closely related to the moduli space of stable bundles. References 1. M. F. Atiyah, N. J. Hitchin, and I. M. Singer, Self-duality in four dimensional Riemannian geometry, Proc. Roy. Soc. London A 362 (1978), 425-461. 2. J.-M. Bismut, H. Gillet and C. Soule, Analytic torsion and holomorphic deter­ minant bundles I. Bott-Chern forms and analytic torsion, II. Direct images and Bott-Chern forms, III. Quillen metrics on holomorphic determinants, Commun. Math. Phys. 115 (1988), 49-78, 79-126, 301-351. 3. J.-M. Bismut and W. Zhang, An extension of a theorem by Cheeger and Muller, Asterisque 205 (Soc. Math, de France, 1992). 4. P. Braam, First steps in Jones- Witten theory, preprint, Univ. of Utah. 5. N. P. Buchdahl, Hermitian-Einstein connections and Stable Vector Bundles Over Compact Complex Surfaces, Math. Ann. 280 (1988), 625-648. 6. J. Cheeger, Analytic torsion and the heat equation, Annals of Math. 109 (1979), 259-322. 7. G. de Rham, Complexes a automorphism.es et homeomorphie differentiables, Ann. Inst. Fourier, Grenoble 2 (1950), 51-67. 8. S. K. Donaldson, Anti-self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. Lond. Math. Soc. (3) 50 (1985), 1-26. 9. S. K. Donaldson, Infinite Determinant, Stable Bundles and Curvature, Duke Mathematical J. 54 (1987), 231-247. 10. B. Dubrovin, A. Fomenko and S. Novikov, Modern Geometry Methods and Applications, Part I, II, III, GTM 9 3 , 104, 124 (Springer, 1985). 11. G. Faltings, Lectures on the Arithmetic Riemann-Roch Theorem, Annals of Math. Studies 127 (Princeton Univ. Press, 1992). 12. J. Fay, Kernel Functions, Analytic Torsion, and Moduli Spaces, Memoirs 464 (AMS, 1992). 13. W. Franz, Uber die Torsion einer Uberdeckung, J. Reine Angew. Math. 173 (1935), 245-254. 14. D. Freed, On determinant bundles, in "Mathematical aspects of string theory", ed. S.-T. Yau (World Scientific, 1987), 189-238. 15. D. S. Freed and R. E. Gompf, Computer Calculation of Witten's 3-Manifold Invariant, Commun. Math. Phys. 141 (1991), 79-117. 16. P. Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem (Publish or Perish, Inc., 1984). 17. D. Johnson, A Geometric Form of Casson's Invariant and its Connection to

111

Reidemeister Torsion, Unpublished lecture notes. 18. H.-J. Kim, Moduli of Hermite-Einstein vector bundles, Math. Z. 195 (1987), 143-150. 19. H.-J. Kim, Analytic torsion, Proc. of 1st GARC SYMP. on Pure and Appl. Math., Part II (Global Analysis Research Center, Seoul National Univ., 1993), 11-30. 20. S. Kobayashi, First Chern class and holomorphic tensor fields, Nagoya Math. J. 77 (1980), 5-11. 21. S. Kobayashi, Einstein-Hermitian vector bundles and stability, in Global Riemannian Geometry, ed. T. J. Willmore and N. Hitchin (Ellis Horwood Limited, 1984), 60-64. 22. S. Kobayashi, Differential geometry of complex vector bundles 15 (Iwanami Shoten and Princeton Univ. Press, Tokyo, 1987), Publ. Math. Soc. Japan. 23. J. Li and S. T. Yau, Hermitian-Yang-Mills connections on Non-Kahler Mani­ folds, in "Mathematical Aspects of String Theory", ed. S. T. Yau (World Sci­ entific, 1987). 24. M. Lubke, Chernklassen von Hermite-Einstein-Vectorbundeln, Math. Ann. 260 (1982), 133-141. 25. M. Lubke, Stability of Einstein-Hermitian Vector Bundles, Manuscripta Math. 42 (1983), 245-257. 26. J. Milnor, Two complexes which are homeomorphic but combinatorially distinct, Ann. of Math. 74 (1961), 575-590. 27. J. Milnor, A duality theorem for Reidemeister torsion, Ann. of Math. 76 (1962), 137-147. 28. J. Milnor, Whitehead Torsion, Bull. Amer. Math. Soc. 72 (1966), 358-426. 29. W. Muller, Analytic torsion and R-torsion of Riemannian manifolds, Advances in Math. 28 (1978), 233-305. 30. W. Muller, Analytic torsion and R-torsion for unimodular representation, Pre­ print, MPI/91-50. 31. D. Quillen, Determinant of Cauchy-Riemann operators on a Riemann surface, Func. Anal, and its Appl. 19 (1985), 31-34. 32. D. B. Ray, Reidemeister Torsion and the Laplacian on Lens Spaces, Advances in Math. 4 (1970), 109-126. 33. D. B. Ray and I. M. Singer, R-torsion and the Laplacian on Riemannian Man­ ifolds, Advances in Math. 7 (1971), 145-210. 34. D. B. Ray and I. M. Singer, Analytic torsion for complex manifolds, Ann. of Math. 98 (1973), 154-177. 35. D. B. Ray and I. M. Singer, Analytic torsion, Proc. Symp. in Pure Math., AMS 23 (1973), 167-181. 36. K. Reidemeister, Homotopieringe und Linsenraume, Hamburger Abhandl 11 (1935), 102-109.

112 37. J. Roe, Elliptic operators, topology, and asymptotic methods, Longman Scientific and Technical, 1988. 38. S. Rosenberg, Spectral Geometry and the Witten Laplacian, Surveys in Geom­ etry (1992), Kanazawa Univ.. 39. M. Rothenberg, Analytic and Combinatorial Torsion, Contemporary math. 105 (1990), 213-244. 40. M. Rothenberg, Torsion in Topology and Geometry, Proc. Workshop in Pure Math. 11 (1992), 232-266. 41. M. A. Shubin, Pseudodifferential operators and spectral theory (Springer, 1987). 42. I. M. Singer, Eigenvalues of the Laplacian and Invariants of manifolds, Proc. Intern. Congr. Math., Vancouver (1974), 187-200. 43. Y.-T. Siu, Lectures on Hermitian-Einstein metrics for stable bundles and Kahler-Einstein metrics (Birkhauser, 1987). 44. E. C. Titchmarsh, The theory of the Riemann Zeta-function, Oxford Science Publication, 1986. 45. K. Uhlenbeck and S. T. Yau, On the existence of Hermitian-Yang-Mills con­ nections in Stable Vector Bundles, Comm. Pure and Applied Math. 39 (1986), 257-293. 46. E. Witten, On Quantum Gauge Theories in Two Dimension, Commun. Math. Phys. 141 (1991), 153-209.

113 K P EQUATIONS A N D VECTOR BUNDLES ON CURVES YINGCHEN LI Department of Mathematics, University of California, Davis, CA 95616, USA

Introduction Moduli spaces of vector bundles on curves have been studied from various view­ points in the past 30 years. In this article we focus on their connections with inte­ g r a t e equations of KdV type. Historically, this connection between vector bundles on curves and KdV equations was first discovered by Krichever [15]. Afterwards, several major developments took place. Among them are the solutions of the Schottky problem [20, 2, 29], characterization of Prym varieties [18] and classification of commutative rings of ordinary differential operators [21]. In this article we give a survey of these aspects of the theory as well as some recent progress that we have made on the relation between the Hamiltonian structures of the moduli spaces of Higgs bundles on Riemann surfaces and the Heisenberg KP systems. In §1 we follow the treatment of [22] to derive the Lax equations of the KdV and the KP systems from the viewpoint of isospectral deformations of ordinary differential operators with formal power series coefficients. After that we formulate the KP system as dynamical motions on certain infinite dimensional Grassmannian following Sato [26]. A generalization of this to matrix-valued differential operators will be given. In §2 we formulate two categories and prove they are equivalent. In §3 we prove the characterization theorem of Prym varieties as appeared in [18]. In §4 we consider relations between the Heisenberg KP systems and Hitchin's integrable systems of cotangent bundles of moduli spaces, as well as moduli spaces of Higgs bundles. The main new results are contained in 4.3. 1. W h a t is t h e K P system? The Kadomtsev-Petviashvili equation is the following nonlinear partial differen­ tial equation 3 1 (1) T % — (tit - jUxxx - 3UU ) = 0 (1) X X

for the unknown function u(x, y, t). The well-known KdV equation is obtained from (1) by dropping the y-dependence of u: ut - -,uXxx - 3M«I = 0

(2)

4 Of course, it is impossible to see the connection of these equations with algebraic geometry of vector bundles on curves from the above expressions. We will mainly focus on equation (1) since it is more general than the KdV equation. A geometric

114 way to understand (1) is to introduce a large system of nonlinear equations which contain the KP equation (1) as the first equation. This system of equations, called the KP system or the KP hierarchy, are the equations that describe the universal isospectral deformations of ordinary differential operators. 1.1. Lax equations First of all, let usfixthe coefficient ring R = C[[x]]. We consider ordinary differential operator with coefficients in R of the form P = dn + ai(x)dn-1

+ ■■■ + o„(i),

where a{(x) G R and d = g|. The eigenvalue problem Pf(x)

= Xf(x)

(3)

has n linearly independent solutions in the function space R for any given A € C. Thus one can say that the spectrum of the operator P is the whole complex plane C. However since for each complex number A, the A-eigenspace is of dimension n, one should really view the spectrum of P as an n-sheeted covering of C. If one considers the point at infinity also as an eigenvalue, one gets a compact Riemann surface which is an n-sheeted covering of the Riemann sphere. Definition 1. Let {P(t)\t € M} be a family of ordinary differential operators param­ eterized by a space M. It is said to be an isospectral deformation of P = P(0) if there exist differential operators Qi(t),Q2@), • ■ • ,Qjv(t) analytic in t such that the following system of differential equations ( P(t)f(x,t) \wJ(x,t)

=

Xf(x,t)

= Qi(t)f(x,t),

i = 1,2,---

has nontrivial solutions for every complex number A. The crucial thing is that A is independent of the parameter t so that the first equation in the above system says that the family of operators {P{t)\t e M} indeed have the same spectrum as P(0). The rest of the system that involve Qi(t) are the boundary conditions imposed on the solutions f(x,t). The compatibility condition of the above system reads as follows: 0 = ^-(P(t)f(x, t)- -Xf{x,t)) = -^(P(t)f(x,t)-Xf(x,t)) ati = =

~(P(t)f(x, t)) -X—f(x,t) -t(P(t)f(x,t))-\~f(x,t) ati

P(t)^-f(x,t)t) - XQi(t)f(x,t) XQi(t)f(x, = (-^P(t))f(x, (£-P{t))fb:,t)t) + P(t)^f(x, ati

t) ati (~P(t))f(2■,t) + P(t)Qi(t)f(x,t) P(t)Qi(t)f(x,t) -- Qi(t)P(t)f(x,t). = {^P{t))f(x,t) Qi(t)P(t)f(x ,t). ati

(4)

115 Or equivalently -^-P(t)

= [Qi(t),P(t)],

i = l,2,---,N,

(5)

This is the Lax equation of isospectral deformations of the operator F(0). Of course there are other compatibility conditions. For example, starting from dt.dt f(x>t) — d^dt f(x't) an< ^ making a similar calculation we can derive the socalled Zakhrov-Shabat equation:

^-t-«*■*'■

™

To derive the K P system, let us introduce more notations. Let D denote the set of all formal ordinary differential operators with coefficients in R, this is a ring again. Let E denote the set of all formal pseudo-differential operators with coefficients in R, where by a pseudo-differential operator we mean an expression of the form am(x)dm + ■■■ + ai(x)d + ao(x) + a-i^d'1 + a_ 2 (a:)d~ 2 + • • • - If a m (x) ^ 0, we say that this pseudo-differential operator has order m. Such a pseudo-differential operator is called monk if Om(x) = 1 and normalized if ao(x) = 0. Using elementary algebra we can show easily the following proposition Proposition 1. (a) M P 6 E is a monic element, then there is unique monic element QeE such that PoQ = QoP=h (b) If P e E is a monic eiement of order m, then there is a unique monic eiement QeE of order one such that Qm = P; Moreover if P is normalized, so is Q. (c) For every normalized monic first order pseudo-differential operator L = d + U2(x)d~1 + U3(x)d~2 + ■ ■ ■, there is a monic zero-th order operator S such that 5 _ 1 L 5 = d; Such an S is unique up to a pseudo-differential operator with constant coefficients. So suppose the family {P(t)\t € M} are monic differential operators and nor­ malized of order n, taking the n-th root we get a corresponding family of pseudodifferential operators {L(t)\t € M} of order 1. Analogous to the Lax equation (5), we consider the following equations for L(t): ^rL(t)

= [Q,(t),L(t)}

(7)

Clearly by taking the n-th power of a solution of (7) we can get a solution of (5). The converse is also true but less obvious. To see it, we rewrite the equation ^ = [Q i ) L»]a S dL 1 oL „ o r, idL ^^ I -L£ ^ 1 ++ L^L»-> L—Ln~2 ++ ... ■■•+ + L L«-^^T oti oti oti Oti Oti dti ^[QuL]^-1 + L[QU L]Ln~2 + • • • + L^lQi,

L]

(8) (8)

116 Let A = %^- [Qu L\. Then (8) is equivalent to AU1-1 + LALn~2 + • ■ • + Ln~1A = 0 Multiplying both sides of (9) by S'1 satisfies S~XLS = d, we get {S-1AS)dn~1 Let B = S^AS.

(9)(9)

from the left and 5 from the right, where S

+ d(S'1AS)dn~2

+ ■■■ + dn-1{S'1AS)

=0

(10) (10)

Then equation (8) becomes Bd"-1 + dBdn~2 + ■■■ + dn~xB = 0

(11) (11)

Now note that dB = Bd + B', by comparing leading terms on both sides of the equation we conclude that B = 0. This proves our claim. Goin back to (7), note that since the left hand side of (7) is a pseudo-differential operator of order at most —1, the right hand side [Qi(t),L(t)] is forced to be a pseudo-differential operator of order at most —1 also. Let us use E(-1) to denote the set of all pseudo-differential operators of order at most —1, Theorem 1 ([8]). Let L = d+u2(x)d~1+U3(x)d''2-{ be an arbitrary normalized monic pseudo-differential operator, then the set {Q 6 D\[Q, L] € E'1} is a vector space generated by (Lm)+,m = 0,1,2,•••, where £(£)+ denotes the differential operator part of L(t)m Using this theorem we can rewrite equations (5) as -^P(t) ati

=- [P(t)f,P(t)} [p(o! ,p(t)\

(12)

This system is called the n-th KdV hierarchy. It describes universal deformations of an arbitrary normalized monic n-th order differential operators. The unknown functions are the coefficients of the operator P{t). If one makes some constraints on the coefficients, the system specializes to a relatively smaller system. For example, set P = d2 + 2u, then the system reduces to the KdV equation (2). A general question arises here as whether one can find a universal system which makes all KdV hierarchies to be its specializations. The simple minded idea of embedding the space of lower order differential operators into the space of higher order operators would not work because of our special requirements on the first two coefficients (monic and normalized). The Key idea of Sato is to identify the roots of differential operators. So, we write down the following system of infinitely many nonlinear differential equations with infinitely many variables: ^-L(t) ati

i ==[(L (t))+,L(t)}, [0(t))+ ,L(t)l, i -== l1, 2, 2, -, - - ,,

(13)

117 where L{t) = d + u2(x,t)d-1 + u3{x,t)d-2 + ■ ■ ■ , t = (*i,*a»-- •) andu j (a;,t) ) j = 2 , 3 , ■ • • are unknown functions. (13) is called the KP hierarchy. It is the system that governs the universal isospectral deformations of ordinary differential operators, because by making con­ straints on the unknown functions v.j(x, t)'s one can recover the deformation equa­ tions of any given differential operators. For example, if one makes assumption that the n-th power of L(t) is a differential operator, then the KP hierarchy specializes to the n-th KdV hierarchy. 1.2. Infinite

Grassrnannians

One has seen that the KP system is rather simple when written in Lax form. In order to see the connection with algebraic geometry more directly, we start from the Lax formalism and introduce Sato's infinite Grassmannian. This is a gigantic creature, but the price will be paid back when we interpret the KP system in a rather trivial way using this infinite Grassmannian. The whole point in the following process is to use a new symbol z to denote the operator d. We let C((z)) denote the formal Laurent series ring in the variable z. As a vector space over C, it has a splitting C((z)) = C[z _1 ] © C[[z]]z. Basically the infinite Grassmannian will be the set of vector subspaces of C((z)) that is comparable with C [ z - 1 ] . To make this more precise, consider the quotient C((;z))/C[[z]].z. For any vector subspace W C C((z)) we will use jw '■ W —* C((z))/C[[z]]z to denote the natural map. Recall a linear map between vector spaces is said to be Fredholm if both the kernel and the cokernel are finite dimensional. Definition 2. We define the infinite Grassmannian of index \i as Gr(fi) = {W C C((z))|7v^ is Fredholm of index fi}. The topology on Gr(fi) is induced by the filtration given by the pole orders of formal Laurent series. The tangent space of Gr(n) at a point W is given by Tw(Gr{fi))

= HomcontiW, C((z))/W),

(14)

where ifom COTl< (WiC((z))/W) denotes the set of all continuous homomorphisms of vector spaces W and C((z))/W Now we claim that each element in C((z)) produces a vector field on Gr(iu,). In fact, given £ G C((z)), for each W € Gr(fi) we have W <-» C((z)) - ^ C((z))/W

(15)

This is a tangent vector at W Hence we get a vector field on Gr(fj,). It is clear that the Lie bracket of these vector fields are induced by the trivial brackets in

118 the commutative algebra C((z)). In other words, we have obtained a commuting set of vector fields on the infinite Grassmannian. Note that actually elements of C[[JZ]].Z produce zero vector fields, so the meaningful ones are produced by elements in C[z - 1 ]. Remarkably enough, this set of commuting vector fields together is the KP system written in Grassmannian language. We would like to point out that the vector field given by z~n corresponds exactly to the n-th Lax equation in the differential operator formalism. Briefly, the correspondence (which we will call Sato' identification) is as follows. First consider the ring of pseudo-differential operators E as a left ^-module. Let E ■ x be the left maximal ideal generated by x and consider the quotient E/E ■ x. We identify this vector space with the space of formal Laurent series C((z)), where d is identified with z _ 1 In this way E acts on C((z)) and hence on Gr(y) for any H. On the other hand, by proposition 1, for the operator L(t) in the system (13) we can find a degree zero monic operator S(t) such that 5(<) _ 1 • L(t) ■ S(t) = d. Using this we can rewrite system (13) in terms of S(t) as follows: d(S(t)) 1 -(SM&Sit)-1)--S{t). )-■ S(t). ^ l = -(S(t)&S(t)dti

(16)

Now given a W £ Gr(n), let W(t) = S{t) ■ W. The KP system (16) then gives the dynamical motion of W. Clearly the i-th equation in (16) corresponds exactly to the vector field on Gr([i) given by z~l. For more details, see [26]. Remark. There are several other formulations of the KP system. The most impor­ tant one is the r function formulation. But we will not use them. For these aspects of the theory, see [6]. The Grassmannian formulation can be generalized easily to the vector-valued case. This corresponds to isospectral deformations of matrix ordinary differential operators. Let V = © n C((z)) and V*"1) = {£f=1 ajz3\oj € C®"}. We define the index fj, infinite Grassmannian of vector valued functions of size n as Grn(fi) = {W C Vl'yw is Fredholm of index /x}, where 7 : W -» V/V(-~1'> is the natural map. As before we have TwiGrJji))

&.

Hmncont{W,VIW).

We claim that each element of the formal loop algebra gl(n, C((z))) produces a vector field on Grn(^i) as follows. Let ip e gl(n,C((z))). We use ^wW to denote the composed homomorphism IV-tV-^V-,

v/W

(17)

119 $w{ip) is a tangent vector at W. Since \p produces a tangent vector at each point W, we get a vector field on Grn(n) which will be denoted by $ ( ^ ) . It is easy to verify that the Lie bracket of these vector fields corresponds exactly to the bracket of the general Lie algebra. So we get a huge set (in fact, a Lie algebra) $(gl(n, C((z)))) of vector fields on the infinite Grassmannian. It is from this set that we will create systematically various integrable systems. In this setting we take it for granted that a completely integrable system with the underlying infinite Grassmannian means a commutative Lie subalgebra of vector fields from the above set. In particular, any commutative subalgebra of the loop algebra gl(n, C((z))) gives rise to a completely integrable system. The true reason that they are indeed "integrable" can be found in [18] where we use the formalism of matrix pseudo-differential operators to translate a commutative Lie subalgebra of gl(n, C((z))) into a system of nonlinear partial differential equations written in the Lax form. Now we would like to specialize to a particular type of algebra. These are the types of algebras depending on the choice of an ordered partition of n and a monic element y of the form y = zr + E m = i z r + m Here we remark that r is always a negative integer and we call that — r is the order of y, the reason is that after identifying z = 9 _ 1 , — r will be the order of of the differential operator part of y. Given an element y as above , we use hk{y) to denote the k x k matrix whose off-diagonal elements are 1 and up-right entry is y and zero elsewhere.

( °1 hk(y)

0

0 0 1

0 0 0

0 0 0

0

0

0

1 0 0 0 1 0 /

Vo o o

0 0

y \ 0 1 0

This matrix has the nice property that hk(y)k = yhxk- We use H^(y) to denote the algebra generated by hk(y) over the field C((j/)). As abstract algebras we have the following isomorphism % ) ( y ) = C((y))[x}/(xk

- y) S Cfty 1 /*)),

where x is an indeterminant. Now for any ordered partition n = («i, nz, ■■ ■ , n;) of the positive integer n and a monic element y as above, we define the following subalgebra of the loop algebra gl(n,C«z))):

HD(y) = ®lj=1H(ns)(y) £ eJ =1 C((» 1 /"0). Let us explain the notation here. Given an element from each H^iy), we get I elements. We then put these I elements along the main diagonal to form a blockwise diagonal matrix. So Hn{y) is the set of all these block-wise diagonal matrices

120 and we call it the maximal commutative subalgebra of type n associated with the element y. The set of commuting vector fields *(H„(y)) given by Hn(y) is called the Heisenberg KP flows of type n associated with the element y. For an explanation of the this terminology, see [18]. A special case that is worth emphasizing is when y — z and the partition is (1,1, • • ■ ,1). In this case we call the Heisenberg flows the n-component KP flows. The importance of the n-component KP flows lies in the fact that its traceless part can characterize Prym varieties. We will see this in §3. 2. Vector bundles on curves and the Krichiver functor 2.1. Moduli of vector bundles on curves The theory of holomorphic line bundles on a compact Riemann surface is equiv­ alent to the classical Abel-Jacobi theory of divisors. It is well-understood. The theory of higher rank vector bundles on curves have been studied intensively since early 60's. First of all we note that the only topological invariants of a holomorphic vector bundle on a compact Riemann surface is its rank and degree. For rank one case, the set of isomorphism classes of line bundles on a curve X is given by the Picard group Pic(X). Pic(X) splits into a disjoint union according to the degrees of the line bundles.

Pic(X) = | J Jd(X),

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deZ

where Jd(X) denotes the set of isomorphism classes of line bundles of degree d. All of the Jd(X)'s are isomorphic to a p-dimensional complex torus. In higher rank case, the situation is more complicated. Fix rank n and degree d, we do not have a good moduli space for all vector bundles of degree d and rank n. To get a good moduli space, we have to restrict to those stable ones. By definition, a vector bundle E on X is stable (semistable) if for every subbundle F c E, we have deg(F) degiF} deg[E)_ (19) uK ; U9j rk(F) rfe(F) ~' rk(E) rk{E) ' Let Ux (n, d) denote the moduli space of stable bundles of rank n and degree d on the curve X. It is known that Ux{n,d) is a smooth quasi-projective algebraic variety, its tangent space at a point E € Ux{n, d) is H^{X, E®E*). Using Riemann-Roch one can calculate easily that the dimension of Ux(n, d) is n2(g - 1) + 1. There is a natural compactification of Ux(n, d) which parametrizes the so-called s-equivalence classes of semistable bundles of rank n and degree d, see [27] for the precise definition of s-equivalence. We only want to point out that in each of these s-equivalence classes there is a unique representative of the form ©JR. where each

121

Ei is a stable bundle of the same slope n = &. We use Ux(n,d) to denote this compactification. It is known that Ux{n, d) is a normal projective algebraic variety. 2.2. The Krichever functor The relation between algebraic geometry of vector bundles on curves and the KP system can be best seen from the Grassmannian picture. The basic process is the Krichever functor. Prom a given vector bundle on a curve together with some local data we can construct a point in the Grassmannian; conversely, starting from a point of certain type in the Grassmannian, we can con­ struct a curve and a bundle on it with auxiliary local data. Recall that a point in the infinite Grassmannian corresponds to a solution of the KP system, so the Krichever functor means that there is a good correspondence between vector bundles on curves and solutions of the KP equations. Since the KP system describes deformations of differential operators, we can say that deformations of vector bundles correspond to deformations of differential operators. Historically, Krichever [15] was the first to discover that fine bundles on curves with auxiliary data give solutions to the KdV and KP equations. Afterwards, Segal and Wilson [31] reformulated Krichever's results as a map from the set of algebro-geometric data to the infinite Grassmannian, hence connecting the works of Krichever and Sato. Segal and Wilson's original paper only gave a map, the functoriality of the construction in both directions were discovered in [21]. In [18], we extended the Krichever functor of [21] to a more general situation. We describe below the construction for the KP system first and then indicate how to generalize to the vector-valued situation. Let A" be a projective curve of genus g, p € X a smooth point on X. Let z be a local parameter around p and Up = 5j>ecC[[z]] its formal neighborhood. The advantage of using a formal neighborhood instead of an open neighborhood in the usual topology or Zariski topology is its uniqueness. There is no ambiguity. Let A be the ring of regular functions on the affine open curve X — {p}. Of course, we have X = Spec(A) U Up and A C C((z)). Let L be a line bundle of index p. = h°(X,L) - ^{X, L) on X and : L\Up ^ 0Ur{-\) a local trivialization of L on Up. The reason we choose Our(-l) instead of Ou„ is because in the definition of Grassmannian we formed C((z))/C[[z]].z. It will also be very clear from our cal culation of cohomology of L below. With the data (X,p, L, ) at hand we will construct a point in the Grassmannian Gr(ti). We take W = H°(X\{p},L). Then WcH°(UpP\{p},L) WCH°(U \{P},L) 0 a^ HH°(u (Up\{p},O (-(-i)) p\{P},oUp Up -1)) = = C((z)) C((z))

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122 So W is a subspace of C((z)). To see it is indeed a point in Gr(fi), we calculate the cohomology groups of the line bundle using the covering of X by X\{p} and Up. l

H°(X\{p},L)nH°(U ,L) r\H \uP p,L) H°(X\{p},L)

H°(X,L)^ H°(X,L)^

£= w WnC[[z]]z n c[[z]]z Kerinw) es Kerh w)

H \X,L) =

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H°(Up\{p},L) ifO(X\{p},L) + /f 0 ([/ p: i )

C(W) W ++ C[[z]]z C[[z]]z S cofcer(7w) cofcer(7Vv) C±L

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This shows W € Gr(p). Summing up we get the following: P r o p o s i t i o n 2. Given a set of geometric data (X, L,p, 4>), where X is a projective curve, p is a smooth point, L is a line bundle of index p. and <j>: L\ur — 0[/p{— 1), we can assign natural ly a point W e Gr(fi). This is essentially the result of Krichever in the setting of Segal and Wilson [31]. The meaning is clear: from the above set of algebro-geometric data, we can construct a solution of the KP system! A natural question arises as to whether there is an inverse construction, i.e., given a point on the Grassmannian Gr(p), can one construct some algebro-geometric data? This problem lies at the heart of the solution of Schottky problem using the KP equations. Let us analyze the situation. Given a point W € Gr(p). We may consider its stabilizers Aw = {a € C((z))|aW C W} and then take the spectrum of this ring to get at least an affine curve. Unfortunately, for a generic point W € Gr{p), its stabilizer turns out to be the scalars C. So in this case, the solution does not carry any algebro-geometric information in our context. However as Witten's conjecture implies, some point of this kind contains surprising amount of information about intersection theory on the moduli space of curves (see [33]). Now, assuming the stabilizer Aw # C, we can actually prove that Aw is a ring of Krull dimension 1. Intuitively this is clear since Aw is a subalgebra of C((z)) (which is not too large). For a detailed argument of this, see [21]. So taking the spectrum we get an affine algebraic curve Spec(Aw) and then W, as an A^-module, gives a vector bundle on Spec(Aw)In general, this is not necessarily a line bundle. The rank r can be determined as follows: recall that each element of Aw is of the form a = a-mz~m + a_( m _ 1 )Z _ ( m _ 1 ) -) (ao + a\z + aiz2 + ■ ■ ■, where m > 0. The element is said to have order m if

123 o—m j1 0. We use ord(a) to denote this number. With this understood, we have r = g.c.d.{ord(a)\a 6 Aw}- The reason is that W contains an element of the form a = z~m + a_^m_i)Z~^m~^ + ■ ■ • + oo + a\z + a2z2 + ■■■ for every sufficiently large m > 0 and on the other hand Aw contains an element of the form a = z~rm + a_ ( r m _ 1 ) 2H'' m ~ 1 ) H h a 0 + a\z + o 2 z 2 H for every large m. Actually all of the above argument works equally well if we consider any inter­ mediate ring between C and Aw, the reason for doing this is to obtain a categorical equivalence instead of merely a two way maps. It is introduced in [21] the notion of a Schur pair (A, W), where W is a point of Gr(/j.) and A stabilizes W and A # C. From such a pair we can repeat essentially the above construction to obtain a vector bundle on the affine curve Spec(A). Now we would like to explain the compactification process to obtain a projective curve and a vector bundle on that curve. Intuitively, one simply thinks of z as a local coordinate around the point at oo. Practically, we take in consideration of the filtration in F = C((z)). Let Fm = {a-mzz~mm

+ ■ ■- •++ OQ oo++ aiz aiz ++ ■■ > ■-■m■7 |a_ ^0m } . f 0}.

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We have: n n .•. •. •QCf~F~ (2CFF~ "( n( n_ 1_ )1 ) c C■ ■ •■ ■C CF°F°c CF1F1c •C• • •C • F•mC C FmFm+1 C Fm+1 • C • ■C •• ■ ;24)

Let Am = A n F " \ Wm = W n Fm. Then gr(A) = ®™=0Am is a graded ring of Krull dimension 2 and gr(W) is a graded module over gr(A). We take X = Proj(gr(A)). This is a projective curve which compactifies Spec(A) by adding in one more smooth point. The graded ring structure on gr(W) allows us to get a vector bundle on X. For this kind of construction in standard algebraic geometry, see Hartshorne's book [10]. All the local data can be presented through standard algebro-geometric means. It can be checked that the two processes above are inverse to each other and both constructions are functorial. To give a precise meaning of this last statement, we introduce two categories. Definition 3. We define the category S(n) of Schur pairs as follows: (1) The objects of this category are the pairs (A, W), where W € Gr(/j.) and A C Aw, Ay^C; (2) A morphism (t, a) from a pair (A\, Wi) to another pair (A2, W2) consists of two inclusions 1: A\ C A2 and a : W% D W%. Now we would like to define the category of geometric data. Note that previously when we dealt with the case of a line bundle, we only had a quartet. But in higher rank case, as has appeared in our inverse construction of the Krichever map, we need to specify a local trivialization of the vector bundle.

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124 Definition 4. We define the category £(n) of geometric data as follows: (1) The objects of this category are the set of quintets (X,p,E,ir, Up is a locally cyclic covering, i.e. if we use the same parameter for the ring Oup, then n is given by z —» zT; Finally (p '■ E\up = ■7T„(0[/ (—1)) is a local trivialization of E in the formal local neighborhood of p. Two quintets (X, p, E, n, {X2,P2,E2,n2,
„l "4

[V2 h /3

Op, -

* UP2

commutes, where /3 is the morphism of formal schemes determined by /?; and E2\uP2

j,

>

*{ *2*G = T2*0£/„(-l) : Ua{-\)

4>0*Ei\uP1

J/3.W>i) : & if3 rw bUo i ((-l) -i) tnO uO

where ^ is the homomorphism of sheaves on C/P2 determined by ip. T h e o r e m 2 ([21]). There is a contravariant functor from C(fi) toS(^i) these two categories anti-equivalent.

whichmakes

2.3. A generalization Now we would like to indicate how to generalize the above theory to a more general setting. On the geometric side, we will choose several points on a curve and consider morphisms between curves and push-forward of vector bundles; on the other hand, we consider vector-valued Grassmannian. So let n = (ni,ri2, ■ •• ,nj) be an ordered partition of n. Definition 5. A set of geometric data of a covering morphism of algebraic curves of type n, index \s, and rank r is the collection (/: (C„,A,n,^,$) -

(C0,p,7T,f*F,
of the following objects: 1. n = {ni,n2, ■■■ ,n{) is an integral vector of positive integers rij such that n = ni+n2-\ \-ni-

125

2. C„ is a reduced projective algebraic curve defined over k, and A = {p\,p2, • ■ ■ ,pi} is a set of I smooth points of C„. 3. II = (71-1, • • • ,7fj) consists of a cyclic covering i nor phis m itj : U0j —► Uj of degree r which maps the formal completion U0j of the affine fine A.\ along the origin onto the formal completion Uj of the curve C n along pj. 4. T is a torsion-free sheaf of rank r defined over C„ satisfying that H = dim H°{Cn,T) - dim Hl(Cn, T). 5. # = (j :TVi ^U

6. 7. 8. 9.

*A°uJ- -D).

where Ty} is the formal completion of T along pj. We identify (pj and Cj ■ <j>j for every nonzero constant Cj e C*. Co is a projective integral curve with a marked smooth point p. f : Cn —► Co is a finite morphism of degree n of C n onto Co such that / _ 1 (p) — {Pii • •" iPi} with ramification index nj at each point pj. 7r: U0 —► Up is a cyclic covering morphism of degree r which maps the formal completion U0 of the affine line A£ at the origin onto the formal completion Up of the curve Co along p. -Kj : U0j —» Uj and the formal completion fj : Uj —> Up of the morphism / at pj satisfy the commutativity of the diagram U0j

—'—>■

[h

*'[ U0

Uj

-^-+

Up,

where tpj : U0j —► U0 is a cyclic covering of degree nj. 10. 4> : (f*F)Up -^ T . ( © i = i V > j * ( ^ ( - l ) ) ) is an (/»Oc„)^-module isomor­ phism of the sheaves on the formal scheme Up which is compatible with the datum $ upstairs. Definition 6. A triple {AQ, A B , W) is said to be a set of algebraic data of type n, index fi, and rank r if the following conditions are satisfied: 1. W is a point of the Grassmannian Grn(fi) of index p. of the vector valued functions of size n. 2. The type n is an integral vector (m, ■ ■■ ,n{) consisting of positive integers such that n = n\-\ 1- nj. 3. There is a monic element y 6 L = k{(z)) of order - r such that AQ is a subalgebra of k((y)) containing the field k.

126 4. The cokernel of the projection 7 ^ : A0 —> k((y))/'k[[y]\ has finite dimension. 5. An is a subalgebra of the maximal commutative algebra Hn(y) C gl(n, k((y))) of type n such that the projection 1A

"-

n

Hn(y)ngl(n,k[[y}\)

has a finite-dimensional cokernel. 6. There is an embedding AQ C An as the scalar diagonal matrices, and as an Ao-module (which is automatically torsion-free), An has rank n over AQ. 7. The algebra An C gl(n,k{(y))) stabilizes W C V, i.e. An ■ W C W. Proposition 3. For every set of geometric data in definition 5, there is is a unique algebraic data having the same type, index and rank. Conversely, for each set of algebraic data in definition 6, we can construct a unique set of geometric data of the same type, rank and index. Briefly, in the correspondence, we take W —
127 T h e o r e m 3 ([20]). Every finite dimensional integral manifold of the KP Rows in X((j) is isomorphic to a Jacobian variety of some algebraic curve and the KP Bows restrict to linear flows on the Jacobian; Conversely every Jacobian variety of an algebraic curve appears as an integral manifold of the KP Sows. Remark. Since no other abelian variety appear in this context, this theorem provides a characterization of Jacobians. Remark. Note that in the statement of the theorem, we used all equations in the KP system. But since the global vector fields on a Jacobian J(X) is a g-dimensional vector space, where g is the genus of X, only finitely many independent equations in KP system are sufficient to generate all flows in J(X). In other words, for a given curve X and a line bundle L on it, we only need finitely many equations from the K P system to generate all deformations of L. Shiota [29] went further in this direction to give a affirmative solution of Novikov conjecture which says that the single K P equation (1) is enough to characterize Jacobian varieties. Remark. It is known that deformations of line bundles on a hyperelliptic curve correspond to the standard KdV hierarchy (2-reduction of KP hierarchy). In other words, KdV hierarchy characterizes Jacobians of hyperelliptic curve. Similarly we can say that the n-th KdV hierarchy characterizes Jacobians of curves which allow a degree n morphism to P 1 with a ramification point of index n. Let us give a sketch of the proof of theorem 3 using Krichever functor in 2.2. Let M be a finite dimensional integral manifold of the KP flows. Let IT(W) € M, where W € Gr(fx). Let Aw = {a £ C((z))\aW C W} be the maximal stabilizer of W. We note that Aw is completely determined by W. So from W we get a pair (Aw, W). So the subspace of Tn(W)X(n) generated by the KP flows at U(W) is C((z))/(Aw + C[[z]]). By assumption we have Tn(w)M

= C((z))/(AW

+ C[[z]])

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But Tn(w)M is finite dimensional, so C ^ Aw C C((z)). This shows (Aw,W) is a Schur pair. Actually the rank of this Schur pair must be 1 because of the finite dimensionality of C((z))/(Aw + C[[z]]). By the Krichever functor in theorem 2, we get a quartet (X,p, n, L), where X is projective curve with X\{p} = Spec(Aw)- A similar calculation as in the Krichever correspondence shows that C((z))/{AwW C((z))/(A

(X,Ox) x) + C[[z}})^H c[M]) = lH\X,O

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Thus we have l M Si H^X.Ox) TU{W) =H (X,Ox) U(W)M

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This shows that the KP flows produces all infinitesimal deformations of the line bundle L on X.

128 If we use Sato's identification we can write all other points in II (M) in the form S{t)W, where S{t) = 1 + S_i(*)3 _ 1 + S_ 2 (t)d - 2 H . Correspondingly we get Schur pairs ( S ( 0 4 w S ( i ) _ 1 , S(r)W) = (Aw, S(t)W). By applying the Krichever functor we obtain quartets (X,p, <j>(t),L(t)). The difference between L(t) and L is a line bundle of degree 0 which glues together Aw and C[[.z]]z along Spec(C[[z]])\{p} with the transformation S(t). The action of T c will wipe out the deformation on the part of Co of curves. Let Nf : J(C) -f J(Cb)

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be the norm homomorphism which is defined as follows. First given a divisor J2 npp of degree 0 on C, we send it to a degree 0 divisor Ylp^pfip) o n Co- It then can be checked that this map respects linear equivalence relations among divisors. So it induces a map Nf : J(C)

—> J{CQ).

Another elegant and equivalent way of defining the norm homomorphism is by using the direct image. Given a line bundle L on C, let f„L be its direct image which is a vector bundle on Co- Then Nf(L)

= det(f.L)

® det(f.Oc(R)),

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where R is the ramification divisor of / . So up to a translation by det(f*Oc(R)) the norm homomorphism is simply the map obtained by taking the determinant of the direct image. Now the Prym variety of the morphism / is defined by Prym(f) = Ker(Nj). This is a subgroup of J{C). It might have several components. But it can be shown that if the morphism is generic, then Prym(f) is connected. So it is a complex torus. The characterization theorem for Prym varieties is based on the following ques­ tion and observation. Starting with a point W in the vector-valued Grassmannian (See 2.3) Grn(n), let the Heisenberg KP flows Hn(y) of type n associated with y act on W. The

129 natural question we would like to answer here is, what is the geometric structure of the orbit of W under the Heisenberg flows Hn(y)? The lesson we learned from the characterization of Jacobian varieties tells us that we should really project this orbit to certain quotient space of Grn(ii). Supposedly this quotient process will wipe out all the unnecessary local data when we want to see the geometric structure of the orbit. Such quotient is the following: Let H+ = Hn(y) l~l gl(n, C[[y]]y). Let r n (j/) = {In + -<4|^4 € H+(y)}. This is a group. Its natural action on the vector space V = ©"C((z)) induces a free action on Grn(y). The quotient space we will use is Zn(fj,,y) = Grn(fj,)/Tn(y). This quotient depends on the choice of y and the partition n. We use again n to denote the projection from Grn(n) to Zn(u.,y). Now the first important observation we made is the following: Proposition 4. Suppose the orbit of the Heisenberg KP Hows Hn (y) starting from W projects down to a Unite dimensional manifold in Zn(fi, y). Then the projection image of the orbit is almost the Jacobian variety of some algebraic curve. If y is of the form y = z + OQ + a _ i z _ 1 + a_2-z -2 + ■ • •, then the projection image is a Jacobian variety. The proof of this proposition uses the more general Krichever functor. Let Xn be the orbit. Let A„ = {a e Hn(y)\aW c W} be the maximal stabilizer of W inside Hn(y) and AQ = {a € ((y))\aW C W} the maximal stabilizer inside C((y))). So we get a set of algebraic data (J4O!^I», W). The Kricever functor then provides us with a set of geometric data of the form / : ( C „ , A , n , F , * ) -►

(C0,p,n,ftF,4>)

By the assumption the tangent space of U.(Xn) at n ( W ) is finite dimensional and is given by

TnwX

" ~ Anl%y)

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But this coincides exactly with /f 1 (C„, Oc„) if we use formal neighborhoods in the cohomology calculation. This suggests that the integration will give the Jacobian of C„. But technically, it is slightly different since the initial W we started with may not give a line bundle on C n . We know from the Krichever functor that if y is of the form y = zr + aT-\zT~1 H ao + a _ i z _ 1 H , then actually W gives rise to a rank r vector bundle F and consequently the Heisen­ berg K P flows Hn(y) produce the family {F® C\L € J(Cn)}. As a variety this is a quotient of J{C„) by a finite subgroup. In the special case when r = 1, it is J{Ca). Now we consider the system of flows given by C((y)). If y is of the form y = z1 + ar-izT~1 + 1- ao + a _ i z _ 1 + • • •, then we call this system an r-reduced KP system. Under the assumption of the above proposition, similarly we can show that

130 the projection image of the orbit of W under the action of the r-reduc ed KP flows C((j/)) is given by the family of vector bundles {f*F®N\N G J(Co)}- This is clear, beacuse W, as an Ao-module, gives f,F when going through Krichever functor and the r-reduced KP system C((j/)) produces the Jacobian of Co- A result in [17] says that the map from J(C0) to {f„F ®N\Af e J(Co)} is finite for any F and is an isomorphism for generic F. The tangent space calculation shows

T (U(X0)) =: T n{w)U{X0)

-^

Hl{X 0 E(Xo xJ ^A^yfy^ -^ A„ec[M]r '°

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How does the Prym variety Prym(f) creep into this picture? Let us consider the case r = 1. As we have seen above, beginning with W the Heisenberg KP flows Hn(y) produce a family of points W(t)'s. The projection image in Za(fi,y) is isomorphic to the Jacobian J(Cn). If we view W(tys as modules over AQ, then we get a family of vector bundles {f*(F ® L)\L G J(Cn)}. In order to see the Prym variety, we take determinant det(f*(F L)). If we view it as a deformation of det(f*F), then the infinitesimal version is given by the trace of an element in Hn(y). In particular this means that traceless part of the Heisenberg flows Hn(y) would fix the determinant of ft(F ® L). In light of the equivalent definition, we see that the orbit of W under the traceless Heisenberg flows projects down to the Prym variety Prym(f) in Za(fj.,y). This is particularly true for the n-component KP system with y = z. On the other hand, given a morphism / : C —» Co of degree n, we can always find a point p e CQ such t h a t / _ 1 ( p ) consists of n distinct points. By supplying local data and then applying the Krichever functor we can realize the Prym Prym(f) as a projection image of an orbit under the traceless n-component KP flows #(i,i,... ,i)iz)Of course there are infinitely many ways to do this. Finally we can state the characterization theorem: Theorem 4 ([18]). A variety is a Prym variety if and only if it can be realized as the projection image in some Z(i i,... ,I)(A*I Z) of an orbit of the n-component KP flows #(l,l,...,l)(z).

4. Moduli of Higgs bundles and the Heisenberg K P s y s t e m s In [9] Hitchin studied the holomorphic symplectic geometry of the cotangent bundle T*Uj({n,d) of the moduli space of stable bundles on a compact Riemann surface. He proved that this is a completely algebraically integrable Hamiltonian system. E. Markman [19] extended this result to the moduli space of Higgs bundles. In this section we explore the relations between Hitchin-Markman systems with the Heisenberg KP systems. 4-1. Hamiltonian systems of finite

dimension

Let us review the basic concepts of a Hamiltonian system in symplectic geometry.

131 Since we will consider the holomorphic symplectic geometry of the moduli space of Higgs bundles, all objects in this section are holomorphic. So Let (M,w) be a holomorphic symplectic manifold of complex dimension 2n, where w is a closed nondegenerate holomorphic 2-form. u> can also be viewed as an isomorphism, called the Hamiltonian mapping from the cotangent bundle to the tangent bundle H : T*M -> TM A holomorphic vector field £ is called a Hamiltonian vector field if the Lie derivative L^w = 0. This means 0 = L/M = (di(0 + id(£))w = d(i(Z)u)

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where t is the contraction. So there exists a local function H(x) satisfying d(H(x)) = - t ( 0 w

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Conversely, given a function f(x) (local or global), we can get a vector field Xf = W(df(x)). This vector field is Hamiltonian because £«(d/(x))" = d(L(H(df(x))u)

= ~d(df) = 0.

The differential equation -£ = £(g{t)) corresponding to a Hamiltonian vector field £ is called a Hamiltonian equation, where g(t) is the unknown function. Now we define Poisson bracket on a symplectic manifold (M, w). This is an operation in the set of smooth functions on M. Given two functions / and g on M, we define their Poisson bracket by

-Xgg(f) {f)=u{X {/, 9} = = */(
(35)

Under this bracket operation, the space of holomorphic functions H°(M, OM) be­ comes a Lie algebra, moreover, the mapping / t-» Xf is a Lie algebra homomorphism from H°(M, OM) to the Lie algebra of vector fields on M. Using the Poisson bracket, one can rewrite the Hamiltonian equation for a given function f(x) as follows:

f = {/,
(36)

Actually Hamiltonian structure can be defined on a more general class of manifolds called holomorphic Poisson manifolds (or Poisson varieties). Such a manifold has a Poisson structure, i.e., a bilinear form { , } on the space of holomorphic functions satisfying anticommutativity, Jacobi identity and Leibnitz rule (1) anticommutativity: {f,g} = - { # , / } ; (2) Jacobi identity: {/, {g, h}} + {g, {ft, / } } + {h, {/, g}} = 0; (3) Leibnitz rule: {ft, fg} = {ft, f}g + /{ft, g}.

132 The Poisson bracket makes H°(M, OM) into a Lie algebra, and moreover the natural map />-»•{/, } is a Lie algebra homomorphism to the space of vector fields. An element in the image is called a Hamiltonian vector field. Two functions are said to be Poisson commuting if their Poisson bracket is zero. Given a Poisson structure on M, we can also define a bundle homomorphism (Hamiltonian mapping) H :T*M —> TM as follows: write each small local section of T*M in the form df, where / is a locally defined function. Then for any other locally defined function g on the same open set, we use Poisson bracket to obtain a new (locally defined) function {/, g}. Because of the Leibnitz rule, this process is independent of all the local choices and gives a globally defined Hamiltonian mapping. The rank of the Poisson structure at a point p € M is by definition the rank of the Hamiltonian mapping at the point p, Tip : T*M —» TpM. It is known that a Poisson manifold is a foliation of symplectic submanifolds. 4-2. Moduli of Higgs bundles Now we review the results of Hitchin, Beauville and Markman about finite di­ mensional integrable systems in the moduli theory of vector bundles on compact Riemann surfaces. Let X be a compact Riemann surface of genus g > 2 and Ux(n, d) the moduli space of stable vector bundles of rank n and degree d. Let T*Ux{n, d) be its cotangent bundle. It has a tautological holomorphic symplectic structure. Recall that the tangent space of Ux(n, d) at a point E € Ux(n, d) is H^{X, £nd(E)), so by Serre duality, the cotangent space at E is H 1(X,£nd(E))* = H°(X, £nd(E)Kx), where Kx is the canonical line bundle of X. Thus each point in T*Ux(n, d) is a pair {E,4>), where

Kx)- We can interpret <j> as a homomorphism of vector bundles 4>:E^E®KX. (37) Taking the characteristic coefficients of this homomorphism we get an element ch((/>) € W = ) = (chi(4>),ch2(4>), ■ ■ ■ ,chn{4>)), where each chi(4>) is the trace of the composed homomorphism E —► E ® Kx —► E ® K^2 -»•<.—>£ x K^. Thus we obtain a holomorphic mapping ch

w

T*U« T*Usx(n,d)^W x{n,) (E,)^ch(ci>) (E,4>)

(38)

This can be considered as a set of dim(W) functions. From the viewpoint of inte­ grable system, ideally we expect this to be just half of the dimension of T*Ux(n, d).

133 In fact recall that dim(Ux(n, d)) is equal to the dimension of its tangent space at an arbitrary point E. And in turn this can be calculated by using the Riemann-Roch formula which turns out to be n2(g - 1) + 1. In particular dim{T*Ux{n,d)) = 2{n2{g - 1) + 1). On the other hand n

dim{W) = J2dim(H°(X,Kf) i=l n

= S + £ > ( 2 < 7 - 2 ) + l-<7) i=2

= n2(g - 1) + 1

(39)

It is proved by Hitchin using the method of gauge theory that ch gives n2(g — 1) + 1 generically independent holomorphic functions. These functions are Poisson commuting with respect to the natural symplectic structure on T*Usx{n,d). This makes T*Ux(n,d) into a completely algebraically integrable Hamiltonian system. Given a generic point s = (si,S2, ■■■ ,sn) € ®"=1H°(X,KX), the fibre of the Hitchin map is isomorphic to a large Zariski open subset of the Jacobian variety of the spectral curve Xs = Spec{Sym{Kxl)/Xs), where Sym(Kxl) is the sheaf of symmetric algebras generated by Kxl and I s is the sheaf of ideals generated by the image of the homomorphism given as the direct sum of the homomorphisms Kxn -^+ Kx~%\ i = 0,1,2,..., n, SQ = 1. Intuitively, Xs is the curve sitting in the total space of cotangent bundle K\ (as an open algebraic surface) which admits a map of degree n to the base curve X. The fibre at p 6 X is given by the solutions of the polynomial equation yn — s \(p)ynl + • ■ • + (—l) n s n (p), where y should be viewed as the variable along the fibre direction of \Kx]■ Since an element (E, has characteristic coefficients [s\, ■ ■ ■ ,sn), one can view the whole level set as a sort of "isospectral" deformations of the pair (E, (j>). In this way, vector bundle E on X can be viewed as a line bundle on the spectral curve Xs, hence a point in the Jacobian of Xs. Formally, if one denotes the map from Xs to X by n and the line bundle on Xs by L, then E = TT,(L). This is Hitchin's remarkable discovery in [9]. It has many beautiful applications to the moduli space of vector bundles. If one examines the above process, essentially everything works if one replace the canonical bundle Kx by a line bundle L assuming deg(L) is large. In fact there is a notion of stability for a pair {E,) is a Higgs pair

134 (F, ip), where F is a subbundle of E invariant under <j> and ip is the restriction of <j> to F. The Higgs pair (E, ) is said to be stable if for every Higgs subbundle (F, ip), the inequality deg(F) „ deg(E) K(40) rk(F) " rk(E) ' rk(E) holds. If one replaces the strict inequality < by the relaxed <, one obtains the notion of semistable Higgs pairs. In [28] C. S. Simpson proves that the set of isomorphism classes of stable Higgs pairs is parametrized by a quasi-projective smooth algebraic variety. Given L, we use MmggS(L) to denote this moduli space. The construction of spectral curves in Hitchin's theory was generalized by Beauville, Narasimhan and Ramanan to the case with arbitrary twisted endomorphism. Given a triple (L, E, 4>), where E is a vector bundle of rank n, L is a line bundle of positive degree and

E® L is a, twisted endomorphism, one can construct a curve in the total space of L as before. Let s = (si, Sg, • - ■ , sn) be the characteristic coefficients of
1 l)/l ), Spec(Sym(L Spec(Sym(L)A Ss),

where I s is the sheaf of ideals generated by the image of the homomorphism given by the direct sum of the homomorphisms L~n -£-* L~(n~'\ For fixed s and L, it is proved in [4] that there is a one one correspondence between the set of pairs (E, <(>) with ch{) = s and the degree d + n(n — \){g — 1) line bundles on Xs. Subsequently, E. Markman [19] proved that for given line bundle L the set Mmggs(L) of all pairs {E,<j>) together with the map 1 (L)(L) - ©?=i Hef^H^X,^) MHiggs \X,U) HLL :■MHiggs H-» ch{) {E,4>) ch() (E,4>) H->

is also an algebraically completely integrable Hamiltonian system (see [9] and [19]). To state Markman's result more precisely, we have to introduce some notations. Let 4> € H°(X, Snd(E) ® L) be a Higgs field. Consider £nd{E) -> £nd(E) ® L tp \—>) is given by Hl{C(E^)* By Grothendieck's Duality Theorem for hypercohomology, we have Hl(C^E^)* S H 1 (C* ( B ^)),

135

where C*^E^ is the complex obtained by dualizing C^Erf) i.e., £ndE V ® ATX -» £nd£ ® if*

tnen

tensoring by Kx,

Now given a section ?? e H°(X,L ® jRfj^), we get a a homomorphism from the complex C* to the complex C, henceforth a homomorphism T?E .-, —> T(E,$)- This gives a Poisson structure on Mmggs(L). It is with respect to this Poisson structure that the functions given by characteristic coefficients are Poisson commuting. The generic fibre of the Hitchin map again is the Jacobian of the spectral curve Xs given by sections s e ®?=1H°(X, V). Remark. We should remark that Markman's construction of the Poisson structure depends on the choice of a section of L ® Kx, so it is rather surprising that the same set of Hamiltonian functions are commuting with respect to a family of Poisson structures. Similar phenomenon also occurs in the Hamiltonian formalism of the KP system (see [5]). In a forthcoming paper we will describe the relation of these two phenomena. Remark. In the case when L = Kx Markman's construction reduces to Hitchin's. 4-3. Hamiltonian systems on Higgs moduli spaces and Krichever functors We would like to show that the Hamiltonian equations of the Hitchin-Markman and the Heisenberg KP systems are related by the Krichever functor. Basically, us­ ing Krichever functor, we can interpret the Hamiltonian functions on Higgs moduli space as a subset of certain algebra of the kind Hn(y) as in §2. Hence the Hamil­ tonian equations in Hitchin-Markman's systems can be viewed (at least generically) as reductions of Heisenberg systems. First of all, let us look at the system 1\X,V) (L) ®UH\X,U) (L) -- ®U H Higgs HLL :• MMHiggs ch{<j>) (E,4>) -\—¥ch(4>) (E,
An independent set of functions can be obtained by taking a basis in the dual space of 0™=1-ff°(A', U). Now using Serre duality and the choice n e H°(X, L Kxl) we made earlier in 4.1, we have

(©?=1ff°(*,L8)r = ®UHl{X,L~i®Kx) - ®UH\X,L~i+l) = ®^Hl{X,L-i)

(41)

Instead of taking an explicit basis of this space ®?~QH1(X, L~'), we simply consider the flows generated by this set of functions on M H , 9 9 S ( L ) . We will relate this set

136

of flows with the Heisenberg KP flows on the quotient Grassmannian Z^^... ^(p.) through Krichever functor. To do this, Let us fix a point p G X and z a local parameter around the point p. Given a point (E, <j>) G M H J 9 S S ( L ) , we take the characteristic coefficients s = ch(4>) G ®?=1H°(X, Ll). This in turn gives a spectral curve Xs = Spec(sym(L~1)) /ls. Let / : Xs —> X be the natural morphism. Let A = / _ 1 ( p ) . If p is not a branch point, then A is a set of n distinct points and the local parameter z can also be used as local parameter at every point of A. Similarly any local covering ■K around p will automatically induce local coverings around each point in A. However since we will be only interested in the rank one bundles on Xs, these local coverings are always canonical. Finally, E is given by the direct image of some line bundle, say CE- We use this notation since CE is uniquely determined by E and . Anyway, we have obtained the following data: (/ : (XS,A,I1,CB)

- (X,p,w,E~f*(CE)))

(42)

The only difference between this set of data and the geometric data in definition 5 is that we do not have local trivializations of the bundles involved. However as is pointed out in the characterization theorem of Prym varieties, when we pass from the Grassmannian Grn(fi) to the quotient Grassmannian Zn(p,,z), the trivialization data will be wiped out. So we do not need them. However an addi­ tional difficulty appears here, namely, when (E, ) changes, the ramification type in (/ : (Xs, A , n , CE) —> (X,p,ir,E = / » ( £ B ) ) > will also change. So we can not put Mfjiggs{L) into a single quotient Grassmannian (recall that the quotient Grass­ mannian depends on the choices of a coordinate y and a type n). To overcome this difficulty, first we should be satisfied if we can embed a large part of Mniggs {L) (say a Zariski open subset) into the quotient Grassmannian and secondly, we should prove that for generic s G ®f=iH°(X, Ll), p is not a branch point so that - ^ x ... ^(/x, z) will be a good ambient space. P r o p o s i t i o n 5. There is a point p G X such that for generic s G ®™=1H°(X, U), p is not a branch point of f : Xs —> X. Proof. Given s = (si, s2, ■ ■ ■ , sn) G ©f =1 if°(X, L'), consider the equation n 1 xnn- - - Sxx"" s i x " "11 ++ S2X s2xn-2n~2- -■■■ X (-l^Sn-iX ■■■ + (-l) - sn-1x

+

n

n (-l) Ssnn (-l)

(43) (43)

defined in the total space of the line bundle L. Here we consider a; to be a variable along the fibre direction of L. Its derivative is n l nx n x "~--1 --(n-(n - l)Slxn-2

n n 3 1 (n- 2)s ~* - - ••• + + {n2)s„_ n-2x2 a; + ( (-- ll )) ""-"\ > ^- i

(44)

From this one can see that the resultant is an element in H°(X, Z/ W a -1> ). We use , S2, • • ■ , Sn ) to denote this resultant. It is clear that p G X is a branch point of

137

/ : Xs —> X if and only if the section R(s\ , S2,•■ • ,sn) vanishes at p. In other words, if and only if R(sus2,--- ,«n) € ff°(X,Ll!1^(-p)). Let R : ®t^H°(X, U) -» ff^L**^) denote the map which sends s to its resultant. Now since M = H»{X,L^) H0(X,Li *) and UpeX H°(X,L^ = and fn W ^ tf fl 0^ ^L! ^^ -*p( - -P)) ) ) === {0}, {0}, pzxH«{X,L^{v))(- -P)) =

we conclude that there i s a p e X such that Im(R) <£ H°(X,Lisi:TJ1(-p). This implies that a generic s gives a spectral curve Xs whose map to X is not branched at p. This proves the proposition. Now finally we can state the generic embedding theorem. Theorem 5. Let U C ®f=1H°(X, I>) be the Zariski open subset such that for each s eU, f : Xs — ► X is not branched at p. Let M = ch~l(U) be the inverse image of U under the Hitchin map. Then M can be embedded in the quotient Grassmannian 2(1,1,-,1)0*1 *)■ Now let (E,
S H^X,®?-^-*) "') l = ®?-*H e ^ H ^ J(X,Lf.i-1)

(45)

This shows that it coincides with the linearization of Hitchin-Markman system (41). More details of this relationship between Hitchin-Markman system and the n-component KP system will be given elsewhere. References 1. M. R. Adams and M. J. Bergvelt, The Krichever map, vector bundles over algebraic curves and Heisenberg algebras, preprint (1991). 2. E. Arbarello and C. De Concini, On a set of equations characterizing the Riemann matrices, Ann. Math. 120 (1984), 119-140. 3. A. Beauville, Jacobiennes des courbes spectrales et systemes Hamiltoniens com­ plement integrates, Acta Math. 164 (1990), 211-235. 4. A. Beauville, M. S. Narasimhan and S. Ramanan, Spectral curves and the gen­ eralized theta divisor, J. reine angew. Math. 398 (1989), 169-179. 5. L. A. Dickey, Soliton equations and Hamiltonian systems (World Scientific, 1991).

138 6. E. Date, M. Jimbo, M. Kashiwara and T. Miwa, Transformation groups for soliton equations, in Nonlinear integrable systems-Classical theory and quantum theory, ed. by M. Jimbo and T. Miwa (World Scientific, 1983), 39-120. 7. J. M. Drezet and M. S. Narasimhan, Groupes de Picard des varietes de modules de fibres semistables sur les courbes algebriques, Invent. Math. 97 (1989), 53-94. 8. I. M. Gel'fand and L. A. Dickey, Russ. Math. Surv. 30 (1975), 77-113; Frac­ tional powers of operators and hamiltonian systems, Punc. Anal. Appl. 10 (1976), 259-273. 9. N. Hitchin, Stable bundles and integrable sytems, Duke J. Math. 54 (1987), 91-114. 10. R. Hartshorne, Algebraic Geometry, GTM 52 (Springer, 1977). 11. V. G. Kac, Infinite-dimensional Lie algebras (Cambridge Univ. Press, 1985). 12. T.Katsura, Y. Shimizu and K. Ueno, Formal groups and conformal field theory over Z, Adv. Stud. Pure Math. 19 (1989), 1001-1020. 13. N. Kawamoto, Y. Namikawa, A. Tsuchiyaand Y.Yamada, Geometric realization of conformal field theory on Riemann surfaces, Commun. Math. Phys. 116 (1988), 247-308. 14. M.Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, preprint (Max-Planck-Institut, 1991). 15. I. M. Krichever, Methods of algebraic geometry in the theory of nonlinear equa­ tions, Russ. Math. Surv. 32 (1977), 185-214. 16. S. Kobayashi, Differential Geometry of Complex Vector Bundles (Iwanami, Tokyo and Princeton Univ. Press, Princeton, 1987). 17. Y. Li, Spectral curves, theta divisors and Picard bundles, Intern. J. Math. 2 (1991), 525-550. 18. Y. Li and M. Mulase, Category of morphisms of algebraic curves and a charac­ terization of Prym varieties, preprint (Max-Planck Institut, 1992). 19. E. Markman, Spectral curves and integrable systems, Thesis (University of Penn­ sylvania, 1992). 20. M. Mulase, Cohomological structure in soliton equations and Jacobian varieties, J. Diff. Geom. 19 (1984), 403-430. 21. M. Mulase, Category of vector bundles on algebraic curves and infinite dimen­ sional Grassmannians, Intern. J. Math. 1 (1990), 293-342. 22. M. Mulase, Algebraic theory of the KP equations, to appear. 23. D. Mumford, Prym varieties I, in "Contributions to analysis," ed. L. Ahlfors, et al. (Academic Press, New York, 1974), 325-350. 24. D. Mumford, An algebro-geometric constructions of commuting operators and of solutions to the Toda lattice equations, Korteweg de Vries equations and related nonlinear equations, in "Proceedings of the international symposium on algebraic geometry, Kyoto 1977," Kinokuniya Publ. (1978), 115-153. 25. E. Previato and G. Wilson, Vector bundles over curves and solutions of KP equations, Proc. Symp. Pure Math. 49 (1989), 553-569.

139 26. M. Sato, Soliton equations as dynamical systems on an infinite dimensional Grassmannian manifold, Kokyuroku, RIMS, Kyoto Univ. 439 (1981), 30-46. 27. C. S. Seshadri, Fibres vectoriels sur les courbes algebrique,, Asterisque 96 (1982). 28. C. S. Simpson, Moduli of representations of the fundamental group of a smooth projective variety, preprint (Princeton University, 1989). 29. T. Shiota, Characterization of jacobian varieties in terms of soliton equations, Invent. Math. 8 3 (1986), 333-382. 30. T. Shiota, The KP equation and the Schottky problem, Sugaku Expositions 3 (1990), 183-211. 31. G. B. Segal and G. Wilson, Loop groups and equations of KdV type, Publ. Math. I.H.E.S. 6 1 (1985), 5-65. 32. E. Witten, Quantum field theory, Grassmannians, and algebraic curves, Commun. Math. Phys. 113 (1988), 529-600. 33. E. Witten, Two-dimensional Gravity and intersection theory on moduli spaces, Surveys in Diff. Geom. 1 (1991), 243-310.

140 S O M E T O P I C S I N N E V A N L I N N A THEORY, H Y P E R B O L I C MANIFOLDS A N D DIOPHANTINE

GEOMETRY

JUNJIRO NOGUCHI* Department of Mathematics, Tokyo Institute of Technology Ohokayama, Meguro, Tokyo 152, Japan

We will discuss open problems in the Nevanlinna theory in several complex variables, the theory of hyperbolic manifolds, and Diophantine geometry. Some of them are already posed ones and known, and the others may be new. In the course, we will give a survey of the affirmative solution of S. Lang's conjecture on rational points of hyperbolic spaces over function fields ([51]). We will also give new examples of hyperbolic hypersurfaces of large degree of P n ( C ) , and those of which complements are hyperbolic and hyperbolically imbedded into P n ( C ) . They affirmatively support some problems posed by S. Kobayashi in his famous book [26]. (These are joint works with K. Masuda [36].) In the end we will propose algebraic and arithmetic analogues of hyperbolic pseudodistances, and discuss some problems. For the general reference of this article, see [26], [32], and [54]. Remark. This is an expanded version of [53]. Contents §1. Nevanlinna Theory §2. Hyperbolic Manifolds §3. Algebraic and Arithmetic Hyperbolic Pseudodistances 1. Nevanlinna T h e o r y 1.1. Transcendental Bezout problem The transcendental Bezout problem, say, on C n asks if it is possible to estimate the growth of the intersection of two analytic (effective) cycles, X\ and X% by the growths of Xi,i = 1,2. In general, the answer is negative; M. Cornalba and B. Shiffman [11] constructed an example of Xit i = 1,2 in C 2 such that the orders of Xi,i= 1,2 are 0, but that of X\ O X2 can be arbitrarily large. On the other hand, W. Stoll [57] established an average Bezout theorem as follows. Let Xi, i= 1,..., q be effective divisors defined by entire functions Fi(z), i = 1,..., q on C n with Fi(0) = 1. One says that Xi or FJ(JZ), i - 1, ...,q define a complete intersection Y = n? = 1 Xi if Y is of pure dimension n — q, or empty, and that Fi(z),i = l,...,q define a stable complete intersection if Fiti(z) = Fi(tiz1,..., Uzn), i = 1,..., q define complete intersections for all t = (h,...,tq) with 0 < tt < 1. Put Yt = C\qi=1{Fiu{z) = 0} • Research partially supported by Grant-in-Aid for Co-operative Research(A) 04302006 represented by F. Maeda (Hiroshima University), and by Grant-in-Aid for Scientific Research 92087.

141 (with multiplicities). Let N(r;Y) set

denote the ordinary counting function of Y and

N(r,Y) N(r,Y) = fI ■■■■ •• / I Jo Jo Jo Let M(r;Fi) proved

N{r;Y N{r;Y t)dt 1---dt t)dt 1---dt q. q.

denote the maximum modulus function of Fi(z). Then W. Stoll [57]

T h e o r e m 1.1.1. For any 0 > 1 there is a positive constant Cg such that
N{r,Y) < Cg Y[logM(6r;Fi). i=l

In the proof the following type of estimate plays an essential role: f1 log 11 dT 1 for an entire function F(z) and ||z|| < r. Any probabilistic measure /i in the unit disk would give a similar result if the above type of estimate holds. Therefore it is interesting to ask P r o b l e m 1.1.2. Characterize what kind of measures can be applied to get an average Bezout estimate? 1.2. Nevanlinna's inverse problem For a meromorphic function F on C we have Nevanlinna's defect relation:

53 6F(a) < 2. o€P>(C)

The defect 6F{O) has a property such that 0 < 6F(O) <1 and 8F(O) = 1 if F omits the value a. As a consequence, there are at most countably many a e P*(C) such that SF((I) > 0; such a is called Nevanlinna's exceptional value. Conversely, for a given (at most) countably many numbers 0 < Si < 1 with X) 5j < 2 and points Oj e P 1 ( C ) . D. Drasin [13] proved the existence of a meromorphic function F such that 5 F (aj) = Si. It is known that the defect relation holds for a meromorphic mapping / : C m —* P n ( C ) with respect to hyperplanes in general position (H. Cartan, L. Ahlfors, W. Stoll, E.I. Nochka [40], and W. Chen [10]), and for a dominant meromorphic mapping / : C m —» V into a projective manifold V with respect to a union of hypersurfaces with simple normal crossings (P. Griffiths et al.). W. Stoll asked P r o b l e m 1.2.1. Does Nevanlinna's inverse problem hold for f : Cm —> P n ( C ) with respect to hyperplanes in general position, or for a dominant meromorphic mapping f : C m —> V into a projective manifold with respect to a union of hypersurfaces with simple normal crossings? This may be a hard problem, but the following will be easier.

142 1.3. Order of convergence of Nevanlinna's defects Given a divergent sequence {zi}^, we classically defines its order by the infimum of p > 0 such that J ^ E i \zi\~p < °°. Thus for a sequence { w i } ^ converging to 0 we may define its order of convergence by the supremum of a > 0 such that 12Zi \wi\a < °°- As seen in 1.2, there are at most countably many Nevanlinna's defects values a, of a meromorphic function F on C. W.K. Hayman [20] proved that the order of convergence of {6p(aj)} does not exceed 1/3 for F of finite lower order A; i.e., EMai)a
(1.3.1)

for a > 1/3. Moreover, A. Weitsman [61] proved the above bound 1.3.1 for a = 1/3. Let / denote a linearly non-degenerate meromorphic mapping / : C m —» P"(C). Then V.I. Krutin' [29] proved that Theorem 1.3.2. Let f be of finite lower order A and a > 1/3. Then there is a constant A(a, A) > 0 such that Y^6f(Di)a

a


n

for any family of hyperplanes Dj of P (C)

in general position.

Conjecture 1.3.3. The above estimate still holds for a = 1/3. Now let / be a dominant meromorphic mapping / : C m —> V as in 1.1, and UDj a union of hypersurfaces of V with simple normal crossings. Problem 1.3.4. Does the estimate J26f(Di)a

a


hold for a > 1/3 and for f of finite lower order X? 1.4- Order of a meromorphic mapping into a protective manifold We first recall the definition of the order of a meromorphic mapping / : C " —► M into a compact complex manifold of dimension m. Let w be the associated (1,1)form of a hermitian metric ft on M. The order functions of / are defined by

*/( r ) = r2m-2 J

f

i'W^iwi2)"11'

flUIKH I*

V2T

'-Ma*. ■f JO

I>(r) =

Pf =

s

in

Then the order of / is defined by

IE l^IM =

r—oo

logr

s

m

^1M,

r-nx,

Note that pj is independent of the choice of u> (ft).

log r

/

143

Theorem 1.4.1. Let / : C" —> M be a meromorphic mapping. Assume that the rank of df at generai points is m, and that pf < 2. Then any global holomorphic section of symmetric tensor Slflk{M) of the bundle Q.k{M) of holomorphic k-forms, k = 1,..., n over M, must identically vanish. Proof. Since there is no available reference showing this fact, we here give a proof. Let n be a non-zero global holomorphic section of S'fifc(M) over M. Put

/v =

^2r}a(z)ea, a

where {e a } is the canonical base of holomorphic sections of SlQk(Cm) associated with a base of the vector space C m . Then there is a constant C > 0 depending only on m, k and I such that

(Efei2)1/fc'(^IWI 2 )" 1 * c±ru

A

(^aiNp)m_1 •

We may assume that f*T](0) ^ 0. Then the submean property implies

± l,- E i-<°»ia)"" - « > •■ Therefore we have tf(r)

>c-

Cir ,

so that p / > 2. Q.E.D. Let / : C" —» V be a dominant meromorphic mapping into a projective manifold V of dimension n. Conjecture 1.4.2. If the order of f is less than 2, V is unirational. By Theorem 1.4.1 V is rational for n < 2, and for n = 3 V is rationally connected by J. Kollar, Y. Miyaoka and S. Mori [35]. Here one also should remark that any non-constant holomorphic mapping of C m into a complex torus has order > 2, and that by Brody's method any non-constant holomorphic curve / : C —> M into a compact complex manifold M yields a nonconstant holomorphic curve g : C —> M of order at most 2. Thus it is interesting to ask Conjecture 1.4.3. If V admits a non-constant holomorphic curve f : C — ► V of order less than 2, then V contains a rational curve. In other words, can one construct a non-constant holomorphic curve g : C —> V from / such that Tg{r) = O(logr)?

144 1.5. Holomorphic curves Let / : C —> V be an algebraically non-degenerate holomorphic curve into a projective manifold V, and U? =1 A a union of hypersurfaces of V with simple normal crossings such that the first Chern classes of the line bundles [Di\ determined by Di are the same w > 0. P. Griffiths [18] posed C o n j e c t u r e 1.5.1. The following defect relation holds: holds

£*/(A)< i=i

rCl(-Av)i L

)

J

; t u + cx(K (AV) > 0}. where N [ s "i =/£^k i] ] = inf{t €e RR;tuj v) Assume that V is an Abelian variety A. Then CI{KA) = 0. By making use of the solution of Bloch's conjecture (A. Bloch [5], T. Ochiai [55], M. Green and P. Griffiths [17], and Y. Kawamata [24]), the above conjecture implies that any non-constant holomorphic curve into A can not miss a smooth ample divisor of A. This follows from the following conjecture due to Griffiths [18]: Conjecture 1.5.2. Any holomorphic curve into A missing an ample divisor D of A must be algebraically degenerate. Ax [3] confirmed this when / is a one-parameter subgroup, answering a question raised by S. Lang. M. Green [16] proved that A\D is complete hyperbolic if D contains no translation of an Abelian subgroup. J. Noguchi [41] proved Conjecture 1.5.2 in the case where D contains two distinct irreducible components which are ample and homologous to each other. His arguments were based on an inequality of the second main theorem type ([41,43,44,46]): T h e o r e m 1.5.3. Let V be an n-dimensional complex projective manifold, D a complex hypersurface of V and a : V\D —» A the quasi-Albanese mapping. Let f : C —» V be a holomorphic curve. Assume that the closure of a(V\D) in A is of dimension n and of log-general type, and that / ( C ) is non-degenerate with respect to the linear system ofH°(V, f2n(logL>)). Then we have the following inequality of the second main theorem type: KTf{r)

< N(r, f*D) + small order term,

where N(r, f*D) stands for the counting function of f*D without counting multi­ plicities, and K is a positive constant independent of f. If K > 1 for an Abelian variety, then this implies Conjecture 1.5.2. Thus it is interesting to investigate K. Problem 1.5.4 ([42]). Compute the above positive constant

K.

See [50] for a new type application of the Nevanlinna calculus to a moduli prob­ lem. Cf. [17,48,34,33] to see how the methods used in the Nevanlinna theory are related to the topics discussed in the next section.

145 2. Hyperbolic Manifolds 2.1. Finiteness and rigidity theorems Let X and Y be compact complex spaces. Assume that Y is hyperbolic. In 1974 S. Lang [30] posed a conjecture to claim the finiteness of the number of surjective holomorphic mappings from X onto Y (cf. also Kobayashi [27]). This has motivated many works. See Zaidenberg-Lin [64], and [49]. The first result in this direction was given by S. Kobayashi and T. Ochiai [28]: Theorem 2.1.1. There are only finiteiy many surjective meromorphic mappings from a compact complex space onto a compact complex space of general type. At the Taniguchi Symposium, Katata 1978, T. Sunada asked the following prob­ lem: Problem 2.1.2. Let f,g:M^>Nbe two holomorphic mappings from a compact complex manifold M onto another N of general type. If f and g are topologically homotopic, then f = g. This is proved for Kahler N with non-positive curvature and negative Ricci curvature (Hartman [19] and Lichnerovich's theorem), but still open for N with KN > 0. The above Lang's finiteness conjecture was affirmatively solved by Noguchi [51] in 1992: T h e o r e m 2.1.3. Let X and Y be as above. Then the set Mer sur j(X, Y) of surjec­ tive meromorphic mappings from X onto Y is finite. It is interesting to recall the following conjecture also by T. Sunada [58]: Conjecture 2.1.4. Let f,g : X —> Y be two topologically homotopic holomorphic mappings. Then f = g.

surjective

In the case of C-hyperbolic manifolds there are works by A. Borel and R. Narasimhan [6] and Y. Imayoshi [21,22,23]. H. Nakamura [38] recently gave a partial answer to this conjecture for varieties of a special type, too. Lately, Makoto Suzuki [59] proved the following non-compact version of Theorem 2.1.3: T h e o r e m 2.1.5. Let X' be a Zariski open subset of a compact complex space, and Y' be a complete hyperbolic complex space which is a Zariski open subset of a compact complex space, and hyperbolically imbedded into it. Then the set Merdom(^i Y) of dominant meromorphic mappings from X' into Y' is finite. In view of this result we may ask Conjecture 2.1.6. i) Let Y be a complete hyperbolic complex space with finite hy­ perbolic volume. Then the holomorphic automorphism group Aut(F) of Y is finite.

146 ii) Let X be also a complete hyperbolic complex space with finite hyperbolic volume. Then MeTdom(X, Y) is finite. G. Arerous and S. Kobayashi [2] proved that if M is a complete Riemannian manifold of non-positive curvature with finite volume, and if M admits no non­ zero parallel vector field, then there are only finitely many isometries. The proof based firstly on the fact that the group Is(M) of isometries of M is compact. In Conjecture 2.1.6 i), Aut(F) is compact. In Conjecture 2.1.6 ii), it suffices to show that the space Holdom(^, Y) of dominant holomorphic mappings from X into Y is finite, and then we see that Hold 0 m(^, Y) is compact, too. In the case of dimension 1, Theorem 2.1.3 is de Franchis' theorem, and we know a stronger theorem called Seven's theorem. One may ask for a similar statement for compact hyperbolic complex spaces. Conjecture 2.1.7. We fix a compact complex space X and set Sev(X) = {(/, Y); Y is hyperbolic and f : X -* Y is surjective and holomorphic}. Then Sev{X) is finite. Making use of the idea of the proof of Mordell's conjecture over function fields for hyperbolic spaces proved by Noguchi [51], Theorem B (see 2.2), we see that any element of Sev(X) is rigid ([51]). Let (/, Y) e Sev(X). Then the diameter and the volume of Y are bounded by those of A". In light of these facts, it is interesting to ask Problem 2.1.8. There is a positive constant v(n) such that the hyperbolic volume Vol(y) > v(n) for every hyperbolic irreducible complex space Y of dimension n. 2.2. Hyperbolic fiber spaces and extension problems In Noguchi [51] (cf. also [45]) the analogue of Mordell's conjecture over function fields for hyperbolic space which was conjectured by S. Lang [30] was affirmatively solved: Theorem 2.2.1. Let R be a non-singular Zariski open subset of R with boundary dR and (VV, ir, R) a hyperbolic fiber space such that (VV, w, R) is hyperbolically imbedded into (VV, it, R) along dR.

(2.2.2)

Then (VV, n, R) contains only finitely many meromorphically trivial fiber subspaces with positive dimensional fibers, and there are only finitely many holomorphic sec­ tions except for constant ones in those meromorphically trivial fiber subspaces. It is a question if Condition 2.2.2 is really necessary. In the case of 1-dimensional base and fibers, this is automatically satisfied by a suitable compactification (see J. Noguchi [47]). On the other hand, we know an example of hyperbolic fiber space (VV, 7r, A*) over the punctured disk A* such that even after a finite base change it has no compactification at the origin into which (VV, 7r,A*) is hyperbolically imbedded along (over) the origin (see Noguchi [52]). Condition 2.2.2 was essentially used in the proof to claim the extension and convergence of holomorphic sections, so that the space of holomorphic sections forms a compact complex space.

147

Question 2.2.3. is there any exampie of a hyperbolic fiber space (W, TT,R) of which holomorphic sections do not form a compact space. From the viewpoint of holomorphic extension problem it is interesting to ask Problem 2.2.4. Let (W, IT, A") be a hyperbolic fiber space which does not satisfy 2.2.2 at the origin. Is there a holomorphic section having an essential singularity at the origin? The following is based on the same thought. Problem 2.2.5. Let Y be a hyperbolic complex space (or a hyperbolic Zariski open subset of a compact complex space) which does not admit any relatively compact imbedding into another complex space so that Y is hyperbolically imbedded into it. Then, is there a holomorphic mapping f : A* —> Y with essential singularity at the origin. In the case of a compact Riemann surface M, T. Nishino [39] generalized the one point singular set to the set of capacity zero: Theorem 2.2.6. Let E C A be a closed subset of capacity zero and f : A\E —> M a holomorphic mapping. If the genus of M is greater than 1, then / has a holomorphic extension over A. Later, Masakazu Suzuki [60] extended this to the higher dimensional case: Theorem 2.2.7. Let M be a complex manifold whose universal covering is a polynomially convex bounded domain of C m Let D be a domain of C™ and E C D a pluripolar closed subset. Let f : D\E — ► M be a holomorphic mapping. If the image f(D\E) is relatively compact in M, in particular if M is compact, then f extends holomorphically over D. Thus one may ask Conjecture 2.2.8. Let N be a compact Kahler manifold with negative holomor­ phic sectional curvature, or more generally a compact hyperbolic complex space. If E is a pluripolar closed subset of a domain D C C", then any holomorphic mapping f : D\E —> N extends holomorphically over D. 2.3. Hypersurfaces of Pn(C) S. Kobayashi [26] claimed Conjecture 2.3.1. A generic hypersurface of large degree d of P"(C) is hyperbolic. Conjecture 2.3.2. The complements of generic hypersurfaces of large degree of P"(C) are hyperbolic. We lately proved the following existence theorem ([36]):

148 T h e o r e m 2.3.3. There exists a smooth hyperbolic hypersurface of every degree d > d(n) of P " ( C ) , such that its complement is complete hyperbolic and hyperbolically imbedded into P n ( C ) , where d(n) is a positive integer depending only on n. In the above theorem d{n) may be very large. In the case of P 3 ( C ) the possible smallest degree of hyperbolic hypersurface is 5, since the Fermat quartic of P n ( C ) is a Kummer K3 surface. Thus we ask C o n j e c t u r e 2.3.4. A generic hypersurface of degree 5 of P 3 ( C ) is hyperbohc. The followings are examples for Conjecture 2.3.1. Examples 2.3.5. a) (R. Brody and M. Green [8]) The hypersurface of P 3 ( C ) defined by d2 d/2 zx)d/2 ' ++ s(z s(z00zz22))d/2 =0. 0. = 44 ++ -• •■• ++4zi ++t(zt(z00Z!) is hyperbolic for even degree d > 50 and for generic t,s e C*. b ) ( A . Nadel [37]) The hypersurface b) (A. Nadel [37]) The hypersurface

3 zz^(z60e(z3+tz ) + 4e+3 + 4e(4+tzl) + 4e+3 = 0 + tz\) + z
3

3

3

is hyperbolic for ('■ > 3 (6e + 3 > 21) and for t € C* except several numbers. is hyperbolic for e > 3 (6e + 3 > 21) and for t e C * except several numbers. c) (J. Noguchi) There are simpler ones (see [36] for the proofs) : d

d

d/3

, dd==34e e >> 2 428, , ■ + zd + + t(z0zlZ2 t(z) Qz1z2z3= )dl0i=Q, 4z ++----+z

t iee C C*..

d ) ( K . Masuda and ' '(C) hyperzd + • • ■ + zJ.d Noguchi + t{z0z1z2[36J) z3)d'i I n P= 0, dwe = have 4 e > 2the 8 , the i efollowing C*. 4 bolic hypersurtaces: d) (K. Masuda and J. Noguchi [36]) In P ( C ) we have the the following hyper­ bolic hypersurfaces: d 3 2 d 3 3 4zd ++ •-■■■ ■+ +Z«z+d +h(zlzh(z2) 2'z2)d/3 + +t2(zt2Z(z3) 2' z3)df3+ +t a t3^(z2n3Zi)d/3 d 3 +U(z2Zl1)d/3 ' +U{zjz

= 0, tj €e C*, C*, d = - 3e > 192.

This is hyperbolic for (tj) in a non-empty Zariski open subset of (C*) 4 In fact, it is so for (tj) = (—1, - 1 , 1 , 1 ) . We call this case a generic case. d 2 d i z4d + ■ ■++4 z+ + t1(z t^zszjz^ 3z z5) l + ■-■■

++ t2(tZl2(zZl2z52)zd5^)d^

tjEC*,

++ t3(tZl3(zZl2z22)zd2?)d4^

d = 4e>196.

This is a generic case, and one may put t\ = t2 = t3 = 1. 2 2 d 3d 3 • • ++zd4++h(z h(z z2z)2)' / 44 + -• ■■ d3 +U(zlz +U(zl^)5)d/3'

3 + t5(z2z1)d< '3

d d3 3 d 3d 3 ++t2(z\z t2(z\z ++ t3(zlz t3(ztz 3)3)l ' 4)4)' '

= 0,

tj-GC*, tj G C*,

d= 3e> 43. = 3e >2243.

== 0,0,

149 This is also a generic case, and one may put all U = 1. 4

d d/i + +zi z+ h(zl zd + ■ ■ ■ + h{zl^) Z2)V

4+-

d/2 d/2 +U{z = 0, +U{ziZ 4zl)1)

3 dd + +t2(z t2(ziz hizizrf'* 2z33)) ^'* + + Hzlztf"

%GC*, tj€C*,

d == 4e 4 e > 2256. 56.

This is hyperbohc for all (tj). These are the first examples to be found in P 4 ( C ) . See [36] for examples in P 5 ( C ) . A. Nadel used Siu's theorem ([56]) to get his Example b). In the case of the other examples, H. Cartan's second main theorem with truncated counting functions and the ramification theorem ([9]) were used. But you should not try to prove Example d) "by hand", since you have to check possibly about 2,000 4 x 4-matrices to be of full rank. We have checked them by computer....Thanks to "MATHEMATICA". G. Xu [62] recently proved an interesting theorem answering to a conjecture of J. Harris: T h e o r e m 2.3.6. On a generic hypersurface of degree d> 5 in P 3 ( C ) , there is no curve with geometric genus g < d(d - 3)/2 — 3. This bound is sharp. Moreover, if d>6, this sharp bound can be achieved only by a tritangent hyperplane section. Now it is of interest to recall Bloch's conjecture [5]: Conjecture 2.3.7. Let f be a holomorphic curve from C into a smooth face of P 3 ( C ) of degree 5. Tien / is algebraically degenerate.

hypersur­

Note that Conjecture 1.5.1 implies this. Now we discuss the hyperbolicity of the complements of hypersurfaces of P n ( C ) . In the simplest case of P 2 ( C ) , it is known that d must be greater than 4 (M. Green [15]), and that the complement of 5 lines in general position is hyperbolic and hyperbolically imbedded into P 2 ( C ) . Conjecture 2.3.8. The complement of a generic smooth curve of degree 5 of P 2 ( C ) is hyperbolic and hyperbolically imbedded into P 2 ( C ) . We know at least the existence of such a curve by M.G. Zaidenberg [63]. Theorem 2.3.9. For each d > 5 there exists an irreducible smooth curve of de­ gree d of P 2 ( C ) whose complement is hyperbolic and hyperbolically imbedded into P2(C). If we allow d to be larger, we may give some concrete examples. Examples 2.3.10. a) (K. Azukawa and Masaaki Suzuki [4]) The complement of the following curve 4 + 4 + 4 + t(z0zi)d/2 + t(z0z2)d/2 =0 in P 2 ( C ) is hyperbolic and hyperbolically imbedded into P 2 ( C ) for even degree d > 30 and for t / 2,4. Remark that the curve in this case is smooth, and this

150 provided the first example of a smooth curve in P 2 ( C ) of which complement is hyperbolic. b) (A. Nadel [37]) The curve in P 2 ( C ) 6(

e+3 : + 3 z%*{z\ zz tz\)= 0 z0 (z\ 0 + tz\) tz\) + zt + zt*(zs ++tzl) l 6 e 33 &el z7 0 z

has the same property for e > 3 and and for t € C* except several numbers. c) (J. Noguchi) We also have d d/3 = z4d + + z*t ++zi t(z02z1)zd/3 =00 z2d + t(z 0 ziz 2) =

for d = 3e > 21. d) (K. Masuda and J. Noguchi [36]) M.G. Zaidenberg [63] constructed an irre­ ducible hyperbolic hypersurface of degree 350 of P 3 ( C ) , of which complement is also hyperbolic. Here we give another one of lower degree. The complement of the hypersurface of P 3 ( C ) defined by d3

d3

z{ + • •■• + + 4zi + H{z\z t 1 ( 2 ?z)2') d/3 + t 2{zlz (z22z)3)' d^ 4+2

2

3

2 md 3 + th{zl^) 3(z 3z4) /

d13 3 + tUizlzrf =0 A{z\zi) f

with generic (t,) € (C*) 4 and d = 3e > 171 is hyperbolic and hyperbolically imbedded into P 3 ( C ) . Here, one may put t% = t2 = — £3 = —£4 = —1See [36] for more examples in P 4 ( C ) and P 5 ( C ) . 2.4- Non-algebraic hyperbolic manifold So far, all known compact hyperbolic manifolds or complex spaces are algebraic. In conversations with I. Graham and also lately with Y. Kawamata, the followig problem has been raised. Problem 2.4.1. Is there any non-algebraic compact hyperbolic

manifold?

3. Algebraic and Arithmetic Hyperbolic Pseudodistances 3.1. Algebraic hyperbolic pseudodistance Let V be an algebraic variety defined over a number field K, and a : K <—► C be an imbedding. Then V naturally carries a structure of a complex manifold denoted by V7. The following conjecture is due to S. Lang [30]. Conjecture 3.1.1. If V is hyperbolic for a a : K ^-» C, then there are only finitely many rational points on V. Note that its function field analogue was proved as seen in Theorem 2.2.1. G. Faltings [14] proved Conjecture 3.1.1 for V contained in an Abelian variety defined over K. The hyperbolic hypersurfaces of P n ( C ) given in Example 2.3.3 have a very different feature to hyperbolic subspaces contained in Abelian varieties in the sense that those hypersurfaces have no Abelian differentials. As a test case of Conjecture 3.1.1 in dimension > 2, it would be interesting to ask the following question because of the simplicity of the defining equation:

151 Question 3.1.2. Let X be the hyperbolic hypersurface of P 3 (C) defined by 8 88 *o tizoztzizs)7 7=0. = 0. *o28 + •-••• •++z!4 ++t(zozi.Z2Z3)

with an algebraic number t ^ 0, or a rational t / 0 for simplicity. contain only finitely many rational points?

Then, does X

Related to Conjecture 3.1.1 he also posed ([31]) P r o b l e m 3.1.3. Let V be an algebraic variety defined over a number field K. If V" is hyperbolic, is VT hyperbolic for another r : K <-» C ? Remark. The hyperbolic spaces given in Examples 2.3.3 have such property as stated above. Through a discussion on this problem at M.P.I., Bonn, S. Bando mentioned an idea to use chains of algebraic curves to connect two points on an algebraic variety instead of chains of holomorphic mappings from the unit disk. Here we explore this idea. Let M be a complex algebraic variety, and P,Q 6 M. Let {{fi,Ci,pi,qi)}ei=0 be a chain of smooth algebraic curves with algebraic morphisms / , : C* —► M and points Pi,qi e Ci so that P=fo(po), P = /O(PO),

= fi(Pi),l
fe{qe)=Qft(qt) = Q-

Let dc,(Pti9i) denote the hyperbolic pseudodistance of the 1-dimensional complex space Ci, and set

DM(P,Q) = inf {£ ^ M ii) DM{P, Q) is a continuous function deBning a pseudodistance on M. iii) If DM (P,Q) isa distance, then it is an inner distance and its topology is the same as the underlying differential topology. iv) (Distance decreasing principle) For an algebraic morphism f : M —► N, we have DMM(P,Q)>D N(f(P),f(Q)). (P,Q) >D N(f(P),f(Q)). So, we call DM{P, Q) the algebraic hyperbolic pseudodistance of M. For instance, DM{P,Q) = 0 for the complex projective space, the complex affine space, and Abelian varieties. During the preparation of this paper the author lately learned that J.-P. Demailly, L. Lempert and B. Shiffman [12] proved an approximation theorem for the Kobayashi-Royden infinitesimal form on a complex quasi-projective manifold, and

152 Theorem 3.1.5. DM(P, Q) = dM(P, Q) for non-singular quasi-projective

M.

Moreover, K. Kitayama [25] remarked that the same holds in general: Remark 3.1.6. DM(P,Q) = dM(P,Q) for any complex algebraic variety M which may be singular or non-(quasi-)projective: In fact, the analogous statement holds for a Zariski open subset of a Moishezon space. 3.2. Arithmetic hyperbolic pseudodistance Let V be an algebraic variety defined over a number field K, and P,Q 6 V(K). The arguments in 3.1 naturally leads to the following definition of the arithmetic hyperbolic pseudodistance. Set

I >J V^( P ' , Q,Q"),* ) ,

j

Dv (P,Q) == p[K: DvKK(P,Q) ^ QQ] ]£

where a runs over all possible imbedding of K into C. We call DyK(P,Q) the arithmetic hyperbolic pseudodistance, which also satisfies the distance decreasing principle for if-morphisms. If L D K is a field extension, then DVK (P,Q)-VK(P,Q)

==

DDVL (P,Q). VL(P,Q).

There is another way to define the arithmetic hyperbolic pseudodistance DyK {P, Q). We make use of only chains {(fi, Ci,Pi, gi)}f=0, connecting P and Q such that all fi : Ci —> V axe defined over K and p%,qi G Ci(K), and set

DVK(P,Q) = ini\j2DciK(j>u DyK(P,Q), and DvK(P,Q) satisfies the distance decreasing principle not only for K-morphisms but also for field extension L/K. DVK{P,Q)>D >DVL(P,Q). VL{P,Q). Dv K (P,Q)

In what follows, dVl((P,Q) stands for DVK(P,Q) or DVK(P,Q). If V is a curve of higher genus, dvK (P, Q) is a distance. For this moment we do not know any substantial implication from this pseudodistance, but may consider many many problems!!! Some of them are Problem 3.2.1. i) What is the lower bound ofdVl((P, Q) in the case where dvK(P, Q) is a distance. ii) What is the relation between dyK{P, Q) and the heights hK(P) of rational points P e V(K). For instance, is there a relation between l/'mi{dVK(P,Q);P,Q e V(K)} and swp{hK(P); P e V(K)}? Here it is of some interest to recall the following result due to J. Kollar, Y. Miyaoka and S. Mori [35]:

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157 O N T H E E X T E N S I O N OF L 2 H O L O M O R P H I C F U N C T I O N S IV: A N E W D E N S I T Y

CONCEPT

TAKEO OHSAWA Deapartment of Mathematics, Nagoya University, Chikusa-ku, Furo-cho, Nagoya, 464-01 JAPAN

Introduction In a series of previous works [14], [16], [17], [18], we have studied extension prob­ lems for L2 holomorphic functions from complex submanifolds of Stein manifolds. In [17], we arrived at a conclusion that an extendability condition is given by a geometric condition in terms of the curvature as well as in many other extension problems in several complex variables (cf. [5], [6], [11]). On the other hand, in func­ tion theory of one complex variable, although extension (or interpolation) problems have been discussed already for a long time (cf. [9], [3], [19], [23], etc.), the curva­ ture concept does not seem to have entered into the picture. Usually the conditions of interpolatability have been described in terms of highly analytic terms, like the convergence of a sequence associated to the Weierstrass canonical product (cf. [9]) or that of the Blaschke product (cf. [3]). Generalizations to the several variables seem to have been done in the same vein (cf. [1], [10], [12], [13]). Nevertheless, the recent works of Seip and Wallsten ([20], [21], [22]) show a remarkable similarity with our previous results. Namely, they have exploited a more geometric concept of density to solve certain interpolation problems, and the density is nothing but the curvature in many situations, not to mention Einstein's theory of gravity. Inspired by their work as well as their motivation arising from interest in signal analysis and quantum mechanics, we would like to extend our previous works to generalize their 1? interpolation results to several variables. To state our result, let M be a Stein manifold of dimension n equipped with a positive Radon measure d^M- By A2(M, d/ij^) we denote the set of those holomor­ phic functions f on M satisfying ll/ll 2 :■= ■= //

JM

2 \f?dnM < oo. oo. \f\ dm <

With respect to the norm ||/||, A2(M,d^M) will be considered as a Hilbert space (cf. Hormander [8]). Let S C M be a closed complex submanifold and let d/is be a positive Radon measure on S. We say (S, d/j,s) is a set of interpolation for A2(M,dfiM) if there exists a bounded linear operator 2 I: A (S,dfi {S,dns) s)^

2 2 M) M) ^ ■A {M,dfi A {M,dn

158 satisfying I{f)\s = f for all / € A2(S, dps)- Our aim is to give a sufficient condition for (S, dfis) to be a set of interpolation in terms of the curvature it induces on M\S. To describe that condition, it is convenient to introduce the notion of polarization function of S. Let dfc(z, S) denote the distance from a point z € M to the union of fc-dimensional components of S, measured by any fixed Hermitian metric on M. A continuous function g: M —► [—00,00) will be said to be a polarization function of S if g is of class C 2 outside <7-1(—00) and the function n-l

g(z)-2^2(n-k)logd -k)k{z,S) log dk{z,S) g(z)-

- 2 £(n-

is bounded on the compact subset of M. Clearly the concept of the polarization function does not depend on the metric. The set of polarization function of S will be denoted by 11(5). Moreover we put Ub{S) :=

{9 e n(5)

s u p 3 < 00 > M J

For any g £ U(S) and any volume form dVjtf on M, let dV^ [p] be the minimal element of the set positive Radon measure d\i on S satisfying 9 9 /f fdn fdfi>> lim / /e"~fe~ X{-txi-t-i
for all nonnegative continuous functions / with compact support on M. Here XA denotes the characteristic function of the set A. Roughly speaking, dVj^ [g] amounts to a generalization of the logarithmic ca­ pacity in one complex variable. In fact, let M be a Riemann surface admitting the Green function G: M x M —» (0,00], let S consist of one point p and let z be any local coordinate around p such that

v^T dz —-— az A az dVM dz dV === —^— M

2 at p. Then g := —2G(p, ■) is a polarization function of S and dVM[g] = 2irc/g(0)25p. Here cp{z) denotes the logarithmic capacity of M with respect to z and 6P denotes the Dirac measure supported on p. If S consists of an infinite number of points of M, there exist in general no bounded subharmonic polarization function of S, so that the quantity d+(S) := sup

jinf V=lddg/dVM I g g Ub(S) J €

[0,oo]

is an invariant of (M, 6VM , S) which effectively measures the density of S at infinity. In the present situation, one may suspect that d+(S) has something to do with the interpolation property of S with respect to some canonical measure on S. If M = C this is the case, indeed (see Section 3). However, in general \/^lddg must compete with the curvature form of dV^. Namely, as a fundamental result we shall establish the following.

159 Main T h e o r e m . Let M be a Stein manifold equipped with a volume form 4VM of class C2 and let S be a closed complex submanifold of M equipped with a positive Radon measure dfj,. Then {S,du-) is a set of interpolation for A2(M, dVM) if there exists age Ii.b(S) such that dVM[g\ < d/j, and { e | v / z l ( 9 a ( l + e)g + KM) > 0 on M \ S } D [0, e0] for some e0 > 0. Here KM denotes the curvature form ofdViuIn view of the above result, we would like to introduce a new density concept. For any plurisubharmonic function <j> on M of class C 2 , for any volume form 6VM of class C2 on M and for any positive Radon measure d/i on S we put II*(S, dfj.) = {g G Ub(S) | V=l(dd(4> + g) + KM) > 0 and sup e~e*dVJvf [g}/d/j, < oo for any e > 0 }. s Then we define the upper density of (S, dfi) with respect to

*(5, d\i) := h u > | rT*(S, dfj.) / 0}. Roughly speaking, D^(S, dju) corresponds to the degree of the bundle KM ® [S] in case codimS = 1. It will turn out that the quantity D^(S, dfj.) is closely related to Beurling-Malliavin-Landau's upper uniform density. To explain this relation, let us recall first that the upper uniform density of a discrete subset T = {z,}^j of C is defined as

^(r):=H5sup#^l|z-^l 0. Seip-Wallsten [22] has shown the following: Theorem. Let T be a uniformly discrete subset of C and Jet <$r be the Dirac measure supported on F. Then (r,e~a^ 6?) is a set of interpolation for A2(C, y/^Te-a^dz

A dz) if a >

TTD+{T).

In [20] it had been shown that the condition a > wD+(T) is also necessary for the interpolatability of T in the above sense. Combining these results with Main Theorem and an estimate for the generalized Weierstrass' cr-function, it will be shown that D++(T) (T) ==- -D^(r,6 -D+(r,6r) r)

(1)

7T

holds for <j>(z) = \z\2 and for all uniformly discrete subsets T of C. Here the right hand side of (1) is defined with respect to the Lebesgue measure of C. Hence the following is a generalization of Seip-Wallsten's theorem.

160 Theorem 0.1. Let M be a Stein manifold equipped with a volume form dV^ of class C2, let S C M be a closed complex submanifold equipped with a positive Radon measure da, and let be a plurisubharmonic function of class C2 on M. Then (5,e" ai ^d/i) is a set of interpolation for A2{M,e~°"t'dVM) whenever a > Z?^(5,d/i). Obviously (1) is a generalization of Gauss-Bonnet's theorem on elliptic curves. Therefore it is interesting to know purely geometric interpretations of D^(r, Sr) in different situations. For instance, let M = C n , let T C C™ be a lattice and let 5 be a T-invariant divisor equipped with a T-invariant volume form dy,. Then D^(5, dfi) = 1 if (27r)_1v/—Tdd

retraction

The reader is refered to [7] for a proof. We shall fix a neighborhood U D S with a holomorphic retraction r y : U —> S throughout. Let ^4(5) and A(U) denote the set of holomorphic functions on S and U, respectively, and let us define a map

Iu-.A(S)^A(U) MS)- -MU) Iv: by Iuif) = f°ruWhat we shall do for the proof of Main Theorem is to consider an increasing family {Mi}^ of Stein open subsets of M such that oo

M = {JMit i=\ 2

a family of bounded operators / , : A {S, dy) -> A2(Mi,dVM) obtained by modifying /[/ by utilizing certain solution operators to the d-equation on Mi, and then a weak limit of the sequence {h}^. For that, we shall appeal to the following existence theorem.

161 Theorem 1.2. (cf. [18] Theorem 1.7 and remark after Theorem 1.6). Let X be a complex manifold of dimension n admitting a complete Kahler metric, let E —» X be a holomorphic line bundle with a C°° fiber metric h, let Kh be the curvature form ofh, and let 8 be a C°° d-closed positive (1,1) form on X. Suppose that there exist a bounded C°° function T): X —» (0, oo) and a continuous function c: (0, oo) —> (0, oo) such that V—l(vKh — ddr] — c(r))~ldr\ A Brj) > 9 on X. Then, for any d-closed E-valued (n, 1) form v with finite I? norm with respect to h and 0, say ||V||A^, there exists an E-valued (n,0) form u satisfying dWc(v) + *}u) = v and | | « | | M < | | v | | M . With notations as above, let IJ''q(X, E, h, 6) denote the Hilbert space of .E-valued (p, 5) forms w satisfying | | w | | ^ < c>o. Then we have the following. Corollary 1.3. Under the assumption of Theorem 1.2, there exists a C-linear map T: L^iX^^^) n K e r a -^ Ln<°(X,E,h,0) satisfying S o T = Id and ||(C(T?) + X/2 V)- T\\ < 1. 1

Proof. One has only to put T=(B\ T =--(BUKerd)^(Kerd)1. )-1. 2. Proof of Main Theorem Let ip be any C°° strictly plurisubharmonic exhaustion function of M and let Mc = { z 6 M | i>{z) < c } for any e e l . Given any g and d\i as in the hypothesis, we shall show that for each c G K there exists a bounded linear map 2

IIcc: A2(S,d») (S,dp)- ^ - A2(M A2{M c,dVMM)) C ,dV such that Ic(f) = f on Mc n S for all / € A2(S, du). Clearly we may assume that g and dV^ are of class C°° for that purpose. Let p: K —► R be any C°° function satisfying p(s) = 1 for s < — 1 and p(s) = 0 for s > 0. We put tc = sup{-g(.z) I z e 8U n Mc+1} and for each t > tc define a map ec,t from A2(S, dp,) to the set of C°° functions on Mcby o n SU I f\ _ / P<<9 + Wu(f) PP P(9 + *) e-cAI) - I otherwise Q and consider the equation du = 8ec,t(f)In order to apply Theorem 1.2 we put X = =

MCC\S \S

and E = K*x,

162 where Kx denotes the dual of the canonical bundle Kx of X. As a fiber metric of E we put h = e~9~6^dVM\X, where 8 is an arbitrary positive number. X admits a complete Kahler metric since M c is Stein and 5 is complex analytic. Moreover we put gt = log(e 9 + e"') - suplog(e 9 + e~') - 1 M

and r] = r)t = 2{-gt +

log(-gt)+2eo1).

Then we have sTlddgt /-Taa, =

-V^lddri and

r—(

n (

eSddg e9dBg

s.

e^dg

hdg\

^9_^i\

= 27=T( (1 - gtl)ddgt + ft-2^* A Bgt)

%/=Ta7j A BJ] < 8 N / = I ( 1 + g t _2 )dg t A 8gt.

Therefore V / ^ T ^ K / , - ddr, - r,-3dr, A 5??) > 2V^T(2e^ 1 « ft + (1 >2V=i(eolK

g;l)ddgt)

+ ddgt)

9

>*n(f '*

do

-t\2

T

c

'

^Sd&A > 0.

Hence the conditions of Theorem 1.2 are fulfilled if we put 0 = 6t= 2v^l(e 0 Kh + ddgt) 3

and c(s) = s . On the other hand, it is easy to see that for any / e A2(S, dfi) l|3e C i t (/)|| M < oo if we regard e C;t (/) as a K^-valued (n, 0) form, and that Hm | | 9 c , , t ( / ) | | M < 2 s u p | p ' ( S ) | | | / | | from the assumption on dy,. By applying Corollary 1.3 for each t and <5, we obtain a C-linear map nx T === TT«.«: (x,x,h,0) tr. L '\X,K K*x

h,e) ^L ^n

n

L '°(X,K ,h,6) >°{x, Kx xM)

(f)

163 such that 8 o T = Id and

IK^ + ^ - ^ r j i < 1. Then we put h,6 = et,i - Ttj o 3 o etiS. Clearly Itj(f) extends to a holomorphic function on Mc that agrees with / on Mc D S, so that 7t,6 can be naturally identified with a bounded linear map from A2{S,dp) to A2{Mc,e-s,t'dVM). Therefore, letting J - » 0 and t -» oc, and choosing a weakly convergent subse­ quence of Itjg, we obtain an operator Ic from A2(S, dp) to A2(MC, dV^) s u c n that Ic(f)\S = f for all / in A2(S,dp). From (t), the norm of Ic has a bound which is independent of c. Hence, by letting c —» oo, we obtain the desired interpolation operator as a limit of a weakly convergent subsequence of Ic. Let us mention again a corollary of Main Theorem (cf. Theorem 0.1 in the introduction). Corollary 2.1. Let M be a Stein manifold equipped with a volume form dVM of class C2, let S C M be a closed complex submanifold equipped with a positive Radon measure dp, and let D^(5, dp). Remark. As we shall see in Section 3, the condition a > D^(S, dp) is not necessary in general for (S,e~°"^dp) to be a set of interpolation for A2(M,e~a^dVM)The necessity holds if M = C, dV^ = (the Lebesgue measure), S is uniformly discrete, dp. = 8s and (z) = |z| 2 . In this para­ graph, we shall show the equality + 7T"- ^ ( r . f r ) . D- F(T) D (r) = w-Wpfy).

(*)

First we recall the following: T h e o r e m 3 . 1 . (cf. [20] and [22]). ( r , 5 r ) is a set of interpolation for A2{C, y/^le-^dz A dz) if and only ifD+(T) < n~lp. An obvious consequence of Theorem 3.1 and Main Theorem is that D+(T) < T r ^ C T . f i r ) .

164 To show that D+{T) > 7T -1 £>*(r,5r), we first note that for any a > D+(T) there exists a discrete set f = {z m n e C\m,n e %} D F and a positive number Q such that the square lattice A = { A mn | Am„ = ^JVfa.{m + y/^ln)

for m, n 6 Z }

satisfies \zmn

Amn\ < Q

for all m and n (cf. Beurling [2], p.356), zoo being the closest point to 0. Since D+{f)

= a and D^(T, Sr) < D^(f, Sf), it suffices to show that + D+(f) D (f) > >

1

t>

1 t ■Kn- D' D' '(f,6 (f,6t).

For that we put

G{Z) = (Z- ,oo)n' f1 - r - ) exp ( r - + * j£-) where the prime denotes omission of the term corresponding to m = n = 0. Lemma 3.2. [22], Lemma 2.2) There exist constants C\, C2 and c in (0,00), de­ pending onJy on Q and inf{ \zmn — zmin'\ | (m,n) ^ (m1, n') } such that for every z we have Cie-cWl°zWd{z,T)

< |e-?W 2 G(z)|

|e-?l 2 l 2 G(z)| < C2ee^°s^

(1) (2)

and for every zmn 6 T we have | e -f|z™| 2 G"( Z m n )| > c i e -cIw|log|iw|_

(3)

Here d(z, T) := inf w € r \z — w\. Therefore for any 6 > 0, g(z) := 21ogG(z) - a\z\2 - S(\z\2 + I) 3 / 4 g n 6 ( r ) by (1) and (2), and for any e > 0 there exists a constant Ce > 0 such that dVM[g] <

Cee^28T

by (3). If 0 > a, there exists a 6 > 0 such that g(z) + p\z\2 is plurisubhaimonic. This means that D+(t,6f.) < a which was what we wanted to show.

165 O p e n Question. Find an elementary proof of (*). From the sub-mean value property of subharmonic functions, it is clear that £>^(r, fir) = d+(T) if <j>(z) = \z\2 and T is uniformly discrete. Therefore, the equality (*) can be written more explicitly as D+(T)

= - inf { 7 TT

— A > - 7 on C \ T for some g e Ub{T) \ ,

(

OZOZ

)

which implies in particular that 7T = inf I 7

—-JL > - 7 on C \ (Z + v ^ Z ) for some g e n 6 ( Z + V ^ I Z ) > .

This observation leads us to the following alternative proof of the equality (*) for the special case. Proof of (*) for the lattice: Let T C C be any lattice, let q: C —► C/T be the canonical projection and let p £ C/T be any point. Then the line bundle associated to the divisor p admits a fiber metric whose curvature form u> is proportional to dz A dz. Since

—— / 2ir 27r

vc/r Jc/r

u == 1,

one has q*, y(

r-

u) I == cc 1xdz dz A A dz dz.

■

Here c denotes the area of any fundamental domain of T. This implies that there exists a g € II;, (p) such that —

q*ddg = y/^l c~ldz/\

dz

on C / r - {p}. Therefore

ir"*(I\fr) 9 -q*g so that i>*(r, 5 r ) < c, but clearly £> + (r) = c. Thus the equality (*) would follow if Uc"t'(T, 6r) = 0 for all a < C7r, but this is a straightforward consequence of GaussBonnet's theorem. Remark. More generally, if n > 1 and T C C n is a lattice, our theory has a com­ pletely new consequence with respect to the interpolation from T-invariant nonsingular divisor S C C™. Namely, if the first Chern class of [S/T] is represented n

22

£

2_] aa-vdzy A dz„ dzv vdzv A v=l

166 after a suitable unitary coordinate change, one can see similarly as above that, with respect to any T-invariant volume form du- on S, D^{S, du.) = 1 for 4>{z) (z)

n n

= Yl aW v\z"\I22

Therefore

av\zv\2\

I S, exp(-^2

dfij

is a set of interpolation for !

n

n

n A2J U(c A ,

n dz^ AAdz,/dzA1

J2 a ^I^I A 2) f{

n , (V=l) exJexpf(V=I)-

u

if av > 1 for all v. The condition av > 1 {v = 1, . . . , n) seems to be necessary for the interpolatability, since the growth of the volume of S can be estimated by using Jensen's formula. However the author has no idea of rigorous proof for that. 4. Effect of bounded plurisubharmonic functions Let the notations be as before. As we have said in the introduction, we shall investigate the case where (S,dfi) is a set of interpolation for the spaces A2(M, e~a^dVM) with a < D^(S, dfi). For that, we shall restrict ourselves to the plurisubharmonic functions <> / of the form — log(—a). Here a is any negative plurisubharmonic function on M. Once for all we fix a g € Hb(S) and put Uc = g~l([—oo,c)). For simplicity we assume g < — 1. Our first goal is to prove the following. Lemma 4 . 1 . If there exists a positive number e such that V^ldd(eg

- log(-a)) > 0

(4)

and V^l(ddg

+ nM - (1 - e)ddlog(-a))

> 0,

(5)

then for any e e l there exists a bounded linear map Jc: A2(Uc,dVM)

-»

A2{M,dVM)

such that Jc{f)\S = f\S for all f e A2(UC, dVM). Proof of the special case a— -I'm essentially done in Section 2. For the general case, we need another existence theorem for the d-operator which we prepare below.

167

Proposition 4.2. Let X be a complete Kahler manifold of dimension n with Kahler form u), and let (E, h) be a holomorphic Hermitian line bundle over X. Suppose that there exists a C°° plurisubharmonic function on X such that u) = y/^ldd(j)

(6)

and M < IThen, for any e > 0 and any compactly supported E-valued (n, 1) form u, 2(l + r1)(\\du\\l+\\dlu\\l)

+ (l+eWu\\l

(7)

> ||u|I

holds. Here b\ denotes the adjoint of 8 with respect to u and h, and \\ ||/, denotes the L2 norm. Proof. Since u is of type (n, 1), by (6) and (7) we have Donnelly-Fefferman's esti­ mate lldullh + \\d*hu\\h + \\d*u\\h > \\u\\h (9) (9) (cf. [4] and [15]). Clearly (8) is a direct consequence of (9). Combining (8) with the Bochner-Nakano equality for the 9-Laplacian, one can deduce an d priori estimate for the 3-operator. In fact, under the conditions (4) and (5) in Lemma 4.1, we set X = MC\S E = D*X h = e-gdVM\X and w = V^ldd( - c i log(-0) - (1 - c2) log(-a) - 8 log(c - i>)). Here c\, ci and 6 are so chosen that |d( C l log(- f f ) + ( l - c 2 ) l o g ( - a ) + 61og(c-V))| < 1. Prom (8) and the Bochner-Nakano equality we obtain C{\\8u\\l+\\8lu\\l)>

\\g-x8g'u\?h

(10)

for any K^-valued (n, 1) form u of class C°° and with compact support. Here 8g* denotes the adjoint of exterior multiplication by 8g, and C is a constant independent of 6. In view of the existence theorem following from (10), it is easy to see that Lemma 4.1 follows in the same vein as in the proof of Main Theorem. From Lemma 4.1 we infer the following.

168 T h e o r e m 4 . 3 . Let 7 € ( - 1 . oc), g 6 IhJS) and U,: = g _ 1 ([—x. $)■ IfVzri(d&9+ KM) > 0 and there exists a positive number e such that y/^\dd(eg — log(— 0, then (5, efy/) is a set of interpolation for A2(M, {—aydVu) if and only if so is (S, dfi) for A2(UC, (-(J^dVu) for some e e l Proof. Apply Lemma 4.1 for the volume form

(—a)''dV\f.

If M is the open unit disc A = { z 6 C | |z| < 1 } . application of Theorem 4.3 for
—

Ei^^Kr

p(z,Zi)

=

1 0

?^^)

£> (rj = l i m sSup up—-—;— ■ D-[T) — 11IQ , T .r/\ 2 c A log(l - T) wiere Z-Zi ZiZ — 1

Then T , (1 - \z\2)'l+1&r) is a set of interpolation for A2(A, i/^Hl if'i/2 > D~(T) and inf^y p(z,. Zj) > 0.

- |z| 2 ) 7 _ 1 dz / dz)

Proof. One has only to construct a 5 e IIj,(r) such that 0

for some 7 < 7

and Uc is a tubular neighborhood of T such that the area of its component containing Zi decreases as i —> oc not more slowly than (1 — iz,|) 2 for some c. This simple procedure may well be left to the reader. Remark. According to Seip [21j. it is also true that (T. (1 — \z^)'i"xbr) Is a set of interpolation for A2(A, V ^ ' l - \z\2y~ldz A dz) only if 7/2 > D+(T) and infi^jp(z,. Zj) > 0. Therefore, similarly as in the case of C one can prove that D+
D~]°d-a)(Y,(-<j)-26T)

for any uniformly discrete subset T C A with respect to the Poincare metric. Thus. as a further investigation, generalization to higher dimensional domains of these Gauss-Bonnet type equalities seems to be an interesting subject. Acknowledgements The author thanks Professors S. Nakano, S. S. Y. Lu. J. Jost. A. Todorov and J. Noguchi for stimulating conversations. He is also grateful to the department of mathematics of Harvard University for the hospitality and support during the preparation of the manuscript.

169 References 1. E. Amar, Suites d'interpolation pour les classes de Bergman de la boule et du polydisque de C", Can. J. Math. 30 (1978), 711-737. 2. A. Beurling, The Collected Works of Arne Beurling, vol. 2, Harmonic Analysis (Boston, 1989). 3. L. Carleson, An interpolation problem for bounded analytic functions, Amer. J. Math. 80 (1958), 921-930. 4. F. Docquier and H. Grauert, Levisches Problem und Rungescher Satz fur Teilgebiete Steinschen Mannigfaltigkeiten, Math. Ann. 140 (1960), 94-123. 5. H. Grauert, Uber Modifikationen und exzeptionelle analytische Mengen, Math. Ann. 146 (1962), 331-368. 6. P. A. Griffiths, The extension problem in complex analysis, II. Embeddings with positive normal bundle, Amer. J. Math. 88 (1966), 366-446. 7. R. C. Gunning and H. Rossi, Analytic functions of several complex variables, (Prentice-Hall, Inc., Englewood Cliffs. N. J., 1965). 8. L. Hormander, An introduction to complex analysis in several variables. 9. A. F. Leontev, On the interpolation of the class of entire functions of finite order, Doklady Akademii Nauk S.S.S.R 56 (1948), 785-787. 10. Y. Nishimura, Probleme d'extension dans la thiorie des fonctions entieres d'ordrefini, J. Math. Kyoto Univ. 20 (1980), 635-650. 11. S. Nakano, On the inverse of monoidal transformation, Publ. RIMS 6 (1970), 483-502. 12. S. Nakano, Extension of holomorphic functions with growth conditions, Publ. RIMS 22 (1986), 247-258. 13. S. Nakano, Extension of holomorphic functions with growth conditions. II., Math. Japon. 35 (1990), 769-772. 14. T. Ohsawa and K. Takegoshi, On the extension of L2 holomorphic functions, Math. Z. 195 (1987), 197-204. 15. T. Ohsawa and K. Takegoshi, Hodge spectral sequence on pseudoconvex domains, Math. Z. 197, 1-12. 16. T. Ohsawa, On the extension of holomorphic functions II, Publ. RIMS 24 (1988), 265-275. 17. T. Ohsawa, The existence of right inverses of residue homomorphisms, Complex analysis and Geometry (Plenum Press, New York and London, 1993), pp. 285191. 18. T. Ohsawa, On the extension of I? holomorphic functions III: negligible weights, preprint. 19. R. Rochberg, Interpolation by functions in Bergman spaces, Michigan Math. J. 29 (1982), 229-236. 20. K. Seip, Density theorems for sampling and interpolation in the Bargmann-Fock space I, J. reine angew. Math. 429 (1992), 91-106.

170 21. K. Seip, Beurling type density theorems in the unit disk, Invent. Math. 113 (1993), 21-39. 22. K. Seip and R. Wallsten, Density theorems for sampling and interpolation in the Bargman-Fock space II, J. reine angew. Math. 429 (1992), 107-113. 23. H. Shapiro and A. L. Shields, On some interpolation problems for analytic functions, Amer. J. Math. 83 (1961), 513-532.

171 SYMPLECTIC TOPOLOGY A N D COMPLEX SURFACES YONGBIN RUAN* Department of Mathematics, University of Utah, Salt Lake city, Utah 84102, USA

1. Introduction In the theory of smooth complex surfaces, an essential goal is to classify complex or algebraic surfaces by families. A weaker classification, advocated by Friedman and Morgan [7], is to classify complex surfaces by the deformation equivalence. Two complex surfaces are (algebraic, complex) deformation equivalent iff they are in the same connected component of the moduli space of (algebraic, complex) surfaces. A connected component is called a deformation type. By Enriques-Kodaira classi­ fication, one can divide complex surfaces into the sub-categories by their Kodaira dimensions. The most of surfaces fall into the category of so called the surfaces of general type. Ironically, those are the surfaces we understand the least. What Enriques-Kodaira classification is really about is a classification for the surfaces of non-general type. Among them, the largest class is elliptic surfaces. Other two im­ portant classes are rational surfaces and ruled surfaces. Their symplectic analogues have been studied extensively by McDuff [17]. Algebraic deformation type is sub­ tler. We shall focus on the complex deformation type. The complex deformation types of non-general type complex surfaces are completely understood. Rational and ruled surfaces are relatively easy. For the elliptic surfaces, the results are more complicated. In particular, the multiple fiber plays a particular important role. We highly recommend Friedman-Morgan's book [6] for a comprehensive reference on the complex deformation equivalence of elliptic surfaces. Another important general theorem about deformation equivalence is the fol­ lowing classical theorem [1]: Classical Theorem. The number of (—1) exceptional curves on an irrational sur­ face is a complex deformation invariant. For the surfaces of general type, there are many examples. But the method tends to be ad hoc. There is no general theorem. One won't get the whole picture of complex surfaces theory without talking about celebrated Donaldson gauge theory. Donaldson gauge theory concerns about the diffeomorphism classification of smooth 4-manifolds, hence complex surfaces. For the relation between the complex deformation type and the diffeomorphism type, there is a well-known conjecture by Friedman and Morgan: •Partially supported by NSF grant DMS 9102522.

172 F r i e d m a n - M o r g a n conjecture. The deformation equivalence of complex sur­ faces is equivalent to the diffeomorphism equivalence of its underlying smooth 4manifolds. One should point out that Friedman-Morgan conjecture has been solved recently for simply connected complex surfaces of non-general type [8, 9, 19, 2, 20]. On the contrary, for the surfaces of general type, there is no evidence to indicate one way or other. In theory, one can try to calculate Donaldson invariant and use it to distinguish the diffeomorphism types of algebraic surfaces, hence, their deformation types. Donaldson invariant is very effective to distinguish the diffeomorphism type. But as an effective tool to distinguish the complex deformation type, it remains to be tested. In fact, the situation is opposite. Often our knowledge about the defor­ mation types of complex surfaces guided gauge theorists to look for homeomorphic non-diffeomorphic 4-manifolds. This leads to a third category-symplectic category. A symplectic manifold is a pair (V,w), where V is a smooth even dimensional manifold and w is a closed nondegenerate two form. In particular, every Kahler manifold is symplectic with its Kahler form as a symplectic form. The recent work of Gompf shows that sym­ plectic 4-manifolds are beyond just Kahler complex surfaces [12]. In this article, we shall focus on the interpolation of complex and symplectic category. Hence, we will only talk about the complex surfaces and the symplectic form is always a Kahler form unless we mention otherwise. First of all, Siu's solution of Kodaira conjec­ ture showed that the existence of a Kahler form on a complex surface is purely a topological condition, namely, the evenness of b\. We can also define the notion of symplectic deformation equivalence and symplectic deformation type [24] in the same way as that of complex deformation equivalence and complex deformation type. Since the space of Kahler forms with respect to a fixed complex structure is convex, the different Kahler forms with respect to a fixed complex structure are always deformation equivalent to each other. Hence, the choice of Kahler forms doesn't matter for the question of deformation equivalence. In fact, one can prove a stronger theorem: T h e o r e m A. For the complex surfaces of even first betti number, the complex deformation equivalence implies symplectic deformation equivalence. Therefore, if we can distinguish the symplectic deformation types of complex surfaces, we distinguish their complex deformation types. One may ask what is the advantage to study symplectic deformation type instead of complex deforma­ tion type? The reason reflects the fundamental character of symplectic geometry. Symplectic geometry is a topology-like theory instead of a geometry-like theory and particular suitable for studying the deformation equivalence. For examples, an algebraic family of algebraic surfaces appears as a single point in the moduli space of symplectic structures, no matter how complicated this family could be. The symplectic structure ignores the delicate properties inside the moduli space of

173 complex structures and concentrates on the property of the connected component, namely, deformation type. To fully utilize this property, we have to establish some symplectic deformation invariants. They are provided by Gromov theory of pseudoholomorphic curves. In 1985, Gromov observed that much of theory of holomorphic curves works on almost complex manifolds with a symplectic structure. Very first application is his classification of symplectic structures on C P 2 . Since then, tremen­ dous progress has been made on symplectic 4-manifolds by the work of Dusa McDuff [16, 17]. She classified all the symplectic structures on rational and ruled surfaces up to deformation equivalence. A central ingredient in Gromov and McDuff's work is a symplectic invariant based on pseudo-holomorphic spheres. Using Gromov-McDuff invariants, the author was able to generalize classical theorem to the symplectic case. The proof is essentially contained in [24]. Here, we go one step further to introduce Gromov invariant for higher genus pseudo-holomorphic curves. We shall prove that T h e o r e m B . For each A e H2(V4,Z), genus g, we can define a symplectic defor­ mation invariant #(4]o>w), roughly as the number of J-holomorphic genus g curves C representing A and intersecting k many generic points x\,...,Xk e V4, where k = c1(Vi)(A) + (g-l). As an example, we will calculate this invariant for the multiple fibers of elliptic surfaces. Using this calculation, we can prove following theorem: Let {pi> P2- ■■ > Pk} be the multiplicity of multiple fibers of elliptic surface S. Assume that m is the least common multiple. Let Sm = {^ < ^ < • • ■ < - } . S' = {^ < *j£ < • • • < jjr-} is another such order sequence. We define Sm < S'm iff k' > k and jj* = 5- for i < k. Then Theorem C. Suppose that 5, S' are minimal elliptic surfaces of even first betti number. If S, S' are symplectic deformation equivalent, either Sm < S'm or S'm < Remark. It is false to strength the theorem to Sm = S'm. For example, a simply connected rational elliptic surface with one multiple fiber is complex deformation equivalent to a simply connected rational elliptic surface without multiple fiber. Hence, they are symplectic deformation equivalent provided that the first betti number is even by theorem A. But for other simply connected elliptic surfaces, Sm = S'm if we assume that S, S' are complex deformation equivalent. To generalize it to symplectic category, one have to calculate $A for the class A represented by general fiber, where the complex structure is not generic. For reader's information, let's briefly review the present situation about the diffeomorphism classification of elliptic surfaces, which has been a foci of gauge theory for many years. The central problem here is Frieman-Morgan conjecture we mentioned before. First of all, we would like to reduce the problem to minimal complex surfaces. Then we need to generalize Classical Theorem to differentiable

174 category. Namely, we would like to prove a "theorem" that The number of (—1) exceptional curves on an irrational surface is a differentiable invariant. Up to now, this statement is still a conjecture! As we mentioned before, it is a theorem if we replace "differential" by symplectic deformational equivalent. But forr ellip­ tic surfaces, it was proved by Friedman and Morgan. So we reduce the problem to minimal elliptic surfaces. For diffeomorphism classification, it is convienent to divided minimal elliptic surfaces into three category according to its fundamental group: simply connected, finite cyclic, not finite cyclic. For the minimal elliptic surfaces whose fundamental group is not finite cyclic. FM conjecture follows from Ue's work [27]. The most of attention forcused on simply connected case, where there can be at most two multiple fibers with coprime multiplicities. When geo­ metric genus pg = 0, it is called Dolgachev surfaces. The partial solutions of FM conjecture for Dolgachev surfaces were obtained by [9, 21]. For general minimal elliptic surfaces, a partial solution of FM conjecture was obtained by [9]. For the case of pg = 1 (homotopy K3-surfaces), a complete solution was obtained by [19, 2]. Finally, a complete solution of FM conjecture was proved by [20]. For the case of cyclic fundamental group, a partial solution for the case of pg = 0 is obtained by [15]. As far as author's knowledge, the general case of FM conjecture for elliptic surfaces with finite cyclic fundamental group is still open. I should mention that Ue's result is not gauge theoritic: But all the results about simply connected elliptic surfaces was proved by calculating Donaldson invariants. We also should mention another exciting development related to elliptic surfaces. Several years ago, Gompf and Mrowka constructed many smooth four manifolds which are homeomorphic to some elliptic surfaces but not diffeomorphic to connected sum of complex surfaces. In particular, they didn't admit any complex structure. Due to the recent Gompf's work, many Gompf-Mrowka examples admit symplectic structures. It seems to be a natural problem to calculate those invariants for Gompf's examples. The primary goal of this invariant is studying the surfaces of general type. But this method encounters significant difficulty for this purpose. The reason that it works so far is due to the fact that the ordinary complex structure is generic for the cases we study so far. Hence we can calculate invariants directly using the complex structure. But it is not the case for the surfaces of general type. In fact, Donaldson [3] showed that the cases we just discussed are the only cases where the complex structure is generic. This approach is still very much in its infancy. A lot of work needs to be done in order to work for the surfaces of general type. We shall leave it for the future research. So far, we have talked a lot about applying symplectic methods to complex ge­ ometry. It is again true here that one can also apply complex geometry to symplectic topology. In this article, we make a simple observation of such kind. It is actually related to the question of symplectic deformation. The symplectic object we are interested in is 7ro(%m w ), where Symu is the group of symplectomorphisms. The author showed [26] that iro(Symu) is highly nontrivial for P?, the blow-up of the

175 complex protective plane P 2 at fc-points with k < 8. Here, we introduce a much more general method of the monodromy representation of complex deformation. Then we prove that a lot of more complex surfaces have highly nontrivial ^(SymJ). In gauge theory, the notion that a 4-manifold has a big diffeomorphism group plays a special role. We can also define the notion of "big symplectomorphism" in the symplectic category. Furthermore, we shall show that T h e o r e m D . Elliptic surfaces and complete intersections have big symplectomor­ phism groups. The precise definition and proof will be in section 3. An interesting question for future research is what kind of role the big symplectomorphism plays in symplectic geometry, compared to that of gauge theory. The paper is organized as follows: Theorem A is proven in the section 2. Theo­ rem D will be discussed in the section 3. Section 4 to 6 are devoted to establishing the invariant. Theorem C will be shown in the last section. 2. Symplectic deformation equivalence The deformation equivalence is a classical notion. For complex manifolds, we can form the moduli space of complex structures over a topological manifold or for a fixed set of data, which usually is a set of Chern numbers. Then two complex manifolds are complex deformation equivalent if they are in the same connected component of moduli space. A connected component is called a deformation type. We can also form the moduli space with some addition data, for example, the moduli space of complex structures with a Kahler form, the moduli space of complex structures with an ample line bundle. They give Kahler deformation equivalence and algebraic deformation equivalence. We can also form the moduli space of symplectic structures as follows: Every symplectic manifold has a natural orientation. Let VV be the space of all symplectic forms on V inducing the same orientation. VV is an open set in the space of closed 2-forms on V. The orientation preserving diffeomorphism group Diff+(V) acts on VV. Define the moduli space of symplectic structures VV = W/Diff(V). Two symplectic manifolds are called symplectic deformation equivalent if they are in the same connected component of VV. A connected component of VV is called a symplectic deformation type. Obviously, algebraic deformation equivalence implies Kahler deformation equiv­ alence and Kahler deformation equivalence implies symplectic deformation equiv­ alence. What we are interested in is more precise relation between Kahler and symplectic deformation equivalence for Kahler manifolds. Proposition 2.1. Suppose that Kahler manifolds (V,UJ) and (V,a/) are complex deformation equivalent. Furthermore, we assume that there is a path of complex manifolds Vt such VQ = V, V\ = f*V and each Vt admits a Kahler form ut (not

176 necessarily continuous with respect to t), where f : V —> V is a Then (V, ui) and (V, ui') are symplectic deformation equivalent.

diffeomorphism.

Proof. We need to introduce Gromov's notion of a tamed almost complex structure for a symplectic manifold. Let (V, ui) be a symplectic manifold. An almost com­ plex structure J is called w-tamed if u(X, JX) > 0 for any nonzero tangent vector X e TpV. By the definition, the tamed almost complex structure is an open condi­ tion. Hence if J is a tamed almost complex structure and J' is an almost complex structure close to J in C°-norm, J' is tamed as well. On the other hand, fixed an almost complex structure J , we can also talk about the tamed symplectic structure in the exactly same way. One observation is that the space of tamed symplectic structures for a fixed almost complex structure is a convex set. If w and <J are two tamed symplectic forms, tu> + (1 — t)ui' for 0 < t < 1 is obviously a closed 2-form. Furthermore, (tw + (1 - t)u')(X, JX) = tu(X, JX) + (1 - t)uj'(X, JX) > 0 for any nonzero X e TPV. This condition also implies that tui + (1 — t)ui' is nondegenerate and hence a tamed symplectic structure. Now suppose that (Vt,ut) is a path of complex manifolds satisfying the hypothesis. Without the loss of generality, we can assume / = Id. For every t e [0,1], there is a e< > 0 such that if \t' — t\ < et, the complex structure on Vf is wt-tamed. By the compactness of [0,1], there are finitely many 0 = to < U < ■ ■ ■ < tk = 1 such that the complex structure of Vti+l is wtj-tamed. Since both u>ti and w^+1 are tamed symplectic structures on Vji+1, by our observation, we can join them by a path of tamed symplectic structures on VJl+1. In this way, we construct a path of symplectic structures joining UQ and ui\ = u'. Therefore, (V,w) and (V',w') are symplectic deformation equivalent. This proposition is particular interesting because Kodaira conjecture asserts that the existence of Kahler structures on a complex surface is completely decided by the evenness of first betti number b\, which is purely a topological condition. Therefore, Sui's solution of Kodaira conjecture implies: Corollary 2.2. For the complex surfaces of even first betti number, complex de­ formation equivalence implies the symplectic deformation equivalence. Thus, for complex surfaces if one can distinguish symplectic deformation type, one can distinguish complex deformation type. Any symplectic deformation in­ variants are also complex deformation invariants. For the higher dimension, the relationship is much less clear. 3.

iro(Symw)

First, let's recall some notations [26]. Let Diffjf be the group of orientation preserving diffeomorphisms inducing the identity on the cohomology. Then D(V) = j^fjjr- Let-<£ : ir0{Diff+) -> D(V). The latter can be identified as a subgroup of Autr(H*{V,Z)) where Autr(H*) is the automorphism group of the cohomology

177 ring H* preserving the real characteristic classes, i.e., Steifel Whitney classes and Pontrjagan classes. For Sym, the analogue of D{V) is SU(V) = 5 mnDiff+ lt is a subgroup of both D{V) and Autc^, where Autc^ c Autr is the subgroup preserving Chern classes and the symplectic class [w]. We can also define another group. Let Sym,i be the group of diffeomorphisms preserving the deformation type of [u], i,e g e Symd iff there is a family of symplectic forms ut such that wo = w,u>i = g*u>. Let us call g a deformed symplectomorphism. Note that [g*J\ could be different from [a;]. So Sym^ is a subgroup of Diffc -the group of diffeomorphisms preserving Chern classes. Let

sd(v) =

Sym

Symd n

*

Diff+

which is a subgroup of of D(V) and Autc. So far, there is no general method to construct the elements of ^(Syrn^) or iro(Symd)- In this section, we introduce one such method. In [26], the author showed that S^ and Sd are highly nontrivial for del Pezzo surfaces. Using the method to be introduced, we will show that it is the case for a much larger class of algebraic surfaces. 7ro(5ymw(Vr)) has strong relation with the deformation of symplectic structures. Let me introduce a way to construct symplectomorphism from the monodromy representation in complex geometry. First, let me generalize the notion of monodromy representation to symplectic category. Definition 3.1. A family of symplectic deformations of (V, u>) is a fiber bundle 7r : X —* U over a connected manifold U (could be open) such that V = 7r _1 (6), and a closed two form il on X such that fi^-i^) is nondegenerate (hence symplectic) and f2|„.-i((,) = w. L e m m a 3.2. Let (X, Q) be a family of symplectic deformations of (V, w) and V = 7r_1(6) for b 6 U. Then there is a homomorphism p : ivi(U,b) —> iro(Symi(V)). If f2T-i(x) aiways represents a rational class, then p : ITI(U, b) —► TTQ(Symu(V)). Proof. Let 7 C U be a loop at b. 7 induces a map 7 : [0,1] —* U with 7(0) = 7(1) = b. The pull-back by 7 gives a fiber bundle over 7*7r : ^*X —► [0,1] and a closed 2-form 7*fi over 7*X. Itself is clearly a family of symplectic deformations with (7*7r- 1 (0),7*n| 7 ^-i ( 0 )) = (7*7r- 1 (l),7*0| 7 . w -i) = ( V » . But this bundle is trivial. Therefore, there is a diffeomorphism / : V x [0,1] -» y*X. such that /o = f\vx{o} = Id- L e t ft = f\vx{t)- T h e n h i s a diffeomorphism on V and /j'w is connected to w b y w t = f?Q. Hence, it is a deformed symplectomorphism. It may not be a symplectomorphism. But we can modify it to be a symplectomor­ phism by following procedure, provided [u>t] = [wj. This condition is always satisfied

178 if [u)t\ is a rational class. Suppose that [uit] = [«]. By Moser's theorem, there is a family of diffeomorphisms gt : V —> V such that g^uit = w and go = Id. Note that n L - i ( i ) = w. Hence, /i 7 = pi/i :V-^V and h*w = gj/j"w = g{ui\ = w. Therefore, h~f e 52/mu,(V). This process is not unique. It depends on the choice of / and gt. One can easily check that a different choice of / and gt gives the same element of no{Symu(V)). Let's use [/i7] e ^o{Symw) to denote this element. Furthermore, if 7 and Y are homotopic, essentially the same argument will show that [/i7] = [Ay]. So this defines a map p : -n\{U) —» ~Ko{Symw{V)). One can also use the same argument to show that p is a homomorphism. Definition 3.3. We define p : ni(U) —> Trc^Ss/m^K)) as the monodromy represen­ tation of the family of symplectic deformations (X, fi). The families of symplectic deformations exist in abundance for Kahler manifolds. Recall that a family of complex deformation of a complex manifold V is a proper complex analytic map p : S —> U such that p _ 1 (6) = V for some b e U. p may have singular fibers. But the singular locus S(p) C U is of at least complex codimension 1. Hence p : S — p _ 1 (5(p)) —> U — S(p) is a holomorphic fibration. If p give an algebraic family (there is a relative ample bundle over S), then there is a map 7r : S —» P™ such that 7r is an embedding for each fiber. Hence, the pull-back of Pubini-study form of P™ will make p : S — p~1(S(p)) —> U — S(p) to be a family of symplectic deformations. Example 3.4. Consider the blowing up f\ of P 2 at fc-points {x\,..., Xk)- Varying the points (x\,..., x^ gives a projective algebraic family. Therefore we get a family of symplectic deformations p : S —> Sk(T2) - A, where Sk(P2) is the fc-fold symmetric product of P 2 and A is the diagonal. By the proposition, we have the monodromy representation p : 7ri(£*(P 2 ) - A) -» 7T0(Symd(Vl)). If we start from a Kahler form fl of P™ with a rational Kahler class, p maps into no(SymQi 2 ) . We note that 7ri(Sfc(P2) - A) = Sk, the symmetric group on fc-letters. Therefore we construct a subgroup of Sj/m^P 2 .). Its image in Sd(V2k) are just symmetric group on the exceptional classes of /c-many exceptional curves. Hence, S^ C ^ ( P ? ) and for an approper choice of Kahler form u>, Sk C S U (P|). Therefore, both groups are nontrivial for this examples. Using monodromy representation and quadric transformation of P 2 , the author proved a much stronger theorem: T h e o r e m ([26]). For del Pezzo surfaces, Su = Autc^],

Sd(V) = Autc.

Actually, there is a lot of more examples having the samiliar properties as del Pezzo surfaces does. There is a notion of big monodromy group in algebraic ge­ ometry, which incidently is very important in gauge theory. Let 5 be an algebraic surface and qs is its intersection from and ks is the canonical class. Let 0(qs) (resp., SO(qs)) be the algebraic subgroup of automorphisms of H2{S) which preserve qs

179 (resp., and which have determinant 1). Let Stab°{ks) = {a 6 SO(qs);a{ks) = ks}In our old notation, Stab°(ks) is just Autc(S). Clearly, all these are algebraic group schemes defined over Z. We denote by SO%(qs) and SOc(qs) the groups of integral and complex points, and similarly for 0(qs) and stab°(ks). If ir: X —> U is a family of deformations of S, then monodromy representation p : wi (U, b) —> iro{Diff+ (S)). Recall that

U has a big monodromy if Im(<pop)C\Stab\{ks) is Zariski dense in Stab^ks). The surface has a big monodromy if it sits as a fiber in a family with big monodromy. We can obviously generalize this notion to the symplectic case. Definition 3.6. Let (S,UJ) be a symplectic 4-manifold. A family p : X —> U of symplectic deformations of (S, ui) has big monodromy if Im((p o p) n Stab^(ks) is Zariski dense in Stab^,k Jks). (S,ui) has a big symplectic monodromy group if it sits as a fiber in a family with big monodromy. To abuse the notation, we often just simply call that (S, ui) has the big symplectomorphism group. By all previous argument, if a family of big monodromy also happens to give an algebraic family, then it gives a symplectic family of big monodromy. By the results in [10], following two classes of examples have this phenomenon: Proposition 3.7. Suppose that S is a smooth complete intersection or a minimal simply connected elliptic surface. Then S has the big symplectomorphism group for some choice of Kahler form. In all the examples we just talk about, S^ is quite close to Autc. One may think that this is a general phenomenon. In fact, this is not true. There are obstructions for the existence of symplectomorphisms, which are the symplectic extremal rays the author introduced in [26]. 4. Gromov compactness The most important part of Gromov theory is Gromov compactness of the mod­ uli space of pseudo-holomorphic curves. Gromov first outlined it in his paper [14]. His proof is very geometric. Pansu [23] has given more detail along the same line as Gromov's method. On the another hand, there have been successful attempts to put Gromov's compactness into the context of Uhlenbeck bubbling off [22, 18, 29]. This P.D.E approach actually gives more information at the case of higher genus. There is a famous Deligne-Mumford compactification of the moduli space of abstract Riemann surfaces by so called stable Riemann surfaces. It is transpar­ ent from this P.D.E approach how do we describe the Gromov compactness from Deligne-Mumford compactification. Gromov compactness (as well as Uhlenbeck bubbling off) is tied with elliptic regularity. We will not give any more analytic background here. Instead, we will concentrate on its geometric outcome.

180 Let fn : (E 9 ,j n ) —► {V, J ) be a sequence of J-holomorphic maps representing homology class A e #2(V,Z) (in general, we only need that the energy e(/„) = <*/([/„]) is bounded from above). Let's first consider the case j n = j , namely, fixing complex structure on the domain E 3 . This case was proven by Parker and Wolfsen [22]. Up to the automorphisms of (E s , j ) , the only non-compactness in this case is bubbling off. Namely, in the limit there may be some holomorphic sphere bubbling off from a genus g curve. Picture is more complicated actually because there may be bubbles over a bubble. In the end, we get a geometric picture like a bubble tree over (E 9 , j). The limit / is a J-holomorphic map from this bubble tree to V Here, / is J-holomorphic iff / is J-holomorphic on each component of the bubble tree.

Next, let's consider the case of J-holomorphic torus. We know that the moduli space of abstract holomorphic torus is complex one dimensional. If /„ : (Ei, jn) —* (V, J) be a sequence of J-holomorphic maps, we can also let j n degenerate. This case has been considered by McDuff and Salamon [18]. Of course, we would still get a bubble tree in the limit as previous case. In addition to it, there is extra degeneration due to the degeneration of j n . This extra degeneration can be described as collapsing a disjoint set of simple close curves in the same homotopic class. This can also be understood by just taking a cylinder. In the limit, we will collapse a set of parallel circles on the tube. It can be illustrated in following diagram:

181 Now, let's consider J-holomorphic curves of genus> 2. In this case, we have Deligne-Mumford compactification of the moduli space of abstract Riemann sur­ faces. Deligne-Mumford compactification can also be understood differential geo­ metrically. We know that for every Riemann surface of genus > 2, there is a unique hyperbolic metric. The degeneration of hyperbolic metric will give the degeneration of complex structures. Let /„ : (Si, jn) —> (V, J) be a sequence of J-holomorphic maps. The degeneration of j n can be described by collapsing a set of disjoint min­ imal geodesies. The resulting singular Riemann surface j is the Deligne-Mumford limit of stable curve. But the limit of / „ is not a J-holomorphic map on (E s , j). First of all, we could always have bubbles. Furthermore, at the circle 7 we collapse, there may be some additional degeneration. If we concentrate locally on the collar of 7, it is like a cylinder. Then what happen before for J-holomorphic torus will happen here as well. Namely, we may actually collapse several parallel circles. This case has been proven by Ye [29].

In the end, we get a "cusp" curve which is a connected union of finitely many J-homomorphic curves / = Uhi as we described satisfying following properties: (a) (b) (c) (d)

The The The The

image of / is connected. homology class [/] = £[/i,] = A ( in general, lim„_ 0 0 [/ n ] = [/]). energy is preserved. image of fn converges to the image of / .

But one should aware several "bad" situations. (1) Some component hi could be a multiple cover curves, i,e, a multiple cover of another J-holomorphic curves of lower degree; (2) Some component hi could be a constant J-holomorphic curves, i,e, the image is a point; (3) There could be two or several components ftj,,..., hik such that they have the same image. All those bad cases are degenerate case in Fredholm theory and need special attention in Gromov theory. 5. Invariant $A,g,u f ° r * n e most of 4-manifolds In this section, we will establish Donaldson type Gromov invariants over the most of 4-manifolds for any genus. The exceptional case is the case that c\(V)(A) = 0

182 and g = 1, which will be the topic in the next section. The reason we single out the 4-dimension is a technical one. The space of multiple cover curves has the smaller dimension in the dimension 4. It is possible to use a simple compactification to show that the compactification of moduli space carries a fundamental class. In higher dimension, the situation is completely different. Multiple cover curves behave badly. We have to use more complicated compactification by considering the intersection property of the components of cusp curves. Even so, we can only manage to prove the existence of the fundamental class of the compactification of the moduli space of J-rational curves for semi-positive symplectic manifolds. This has been worked out in detail in [25]. For the general case, there are some serious problems. We will leave for the future research. The case g = 0 has been worked out by McDuff [17]. 5.1. Fredlhom Theory Another basic part of Gromov theory is its Fredlhom theory. Namely, various moduli spaces Gromov theory considers are smooth manifolds for a generic choice of tamed almost complex structure. The method is really nothing new. It follows from a quite general procedure called Sards-Smale infinitely dimensional transversality theory. Because of this, it is also often referred as transversality. Nevertheless, it is just a infinitely dimension version of usual finite dimensional transversality theory. The essential new ingredient is that all the maps involved have to be Fredlhom. This is only true because the equation we considered here is elliptic. The idea to apply Sards-Smale theory to the moduli problem is due to Freed-Uhlenbeck [11]. To show that the moduli space of nonmultiple covered J-holomorphic curves is smooth for generic J, we follow the line of that of Freed-Uhlenbeck. The case of J-sphere has been done by McDuff [17]. There are two problems we need to handle in our case for Fredholm theory. The first is the variation of complex structures on the Riemann surface of higher genus which doesn't exist for genus 0. This can be quite easily done through a routine argument except that the moduli space Mg of Riemann surfaces of higher genus is singular. But Mg has a stratification parameterized by the automorphism group of Riemann surfaces. Namely, one can write Mg

= E T ".s ael

where each smooth strata T aiS consists of the Riemann surfaces of a fixed automor­ phism group a. For g > 2, the top strata is the space of Riemann surfaces of trivial automorphism group / and has dimension 6g - 6. For g = 1, top strata has an automorphism group K0 which is the extension of S1 x S1 by Z 2 and has dimension 2. In addition to it, there are two lower stratus consisting of two points. They have automorphism groups « which are the extension of «o by Z 2 , Z 3 respectively. For g = 0, Mo is a point and the automorphism group is PSL2C. In each case, all the lower stratus have at least codimension 2. Basically, one can use the same

183

argument to establish Fredholm theory for each strata Ta,9. It is a rather general argument. The precise structure of T<,i9 is not needed. One only has to know that TQi9 is smooth. Another more serious problem is the multiple cover curves as far as Fredholm theory concerned. Fortunately, this problem is not too bad on dimension 4 except the case that an elliptic curve is multiple cover of another elliptic curve and C\(V)(A) — 0. This case will be discussed separately in the next section. Let E 9 be a surface of genus g. Our basic topological object is A* C A = Map(E 9 , V) the space of non-multiple cover maps. To specify a sobolev norm on -4*, we need a metric on E g . When g > 1, there is an unique hyperbolic metric for an element of Tag = Map(Ea,g,V). Then Aa,g is a Banach manifold. Clearly, there is a liberation Aa,g — ► TQi9 whose fiber over p~l(b) is Map((E,a), V). Let AagC Aa,g denote the map which is non-multiple covering restricted over any fiber

p-\b). There is an evaluation map e : -K,g xTa,s Ea,g -> V, where A*a g xx„ s Ea,g is fiber product over Ta,9. We can pull back the tangent bundle ofV e*TV^A*a,gxJa,gJe°°. We can also form the vertical cotangent bundle over Ea>g and take partial sections to form a Hilbert bundle Q1(e*TV) -> A*ag x Je°°. For each J € J , we can view e*TV as a complex vector bundle. Therefore we can decompose Q1(e*TV) = n1'°(e*TV) © QP'l(e*TV). Both fi1-0,00'1 are Hilbert bundles over A*a
184 where i the inclusion ThTa,g c n ° ' 1 ( r S 9 ) -^-> Q^ifTV). D is elliptic and i is a finitely dimensional operator. Hence D © i is Fredlhom. Furthermore, index(D © i) = 2(ci (A) + n ( l - g)) + dim T a , g . For any J e J, let M*,A ™ = 5j 1 (0) be the moduli space of non-multiple covered, J-holomorphic map / : (£ 9 , a) —> (V, J ) for a e T a , g . The same argument as that of [16] will show that 8 is transverse to the zero section and hence 3 _ 1 (0) is a Hilbert submanifold of T a , 9 x A* x J™. Consider map

§- l(0) -^J?. Sards-Smale theory implies that for a generic J, M*,A n = 8j (0) is a smooth manifold of the dimension predicated by the index formula. Group a acts freely on M*,A aJy Define the moduli space of unparameterized curves M*,A ~ = ■Ml. T\ la. From now on, we take the convention to call an element of Ml. „ „ n a J-holomorphic map and an element of M*,A n = M*,A _„ n / a a J-holomorphic curve. Obviously, if a = 1, two are the same. So far we can conclude that T h e o r e m 5.1.1. For generic J', M*,A « is a smooth manifold of the dimension predicated by the index formula. For generic J, J', there is a generic path Jt e J£° such that (J t M*,A qaj\ x {t} is a smooth cobordism. Corollary 5.1.2. When dimV = 6 and ci(A) < 0, for a generic J there is no J-holomorphic curve to represent A. When d i m f > 8 and c\(A) < 0, there is no J-holomorphic curves other than rational curves or a multiple cover of rational curves to represent A. Proof. index(D ® i) < 2c\(A) if dimK > 6 and g > 1. It is equal to 2c\(A) for g = 1 and 2c\(A) + 2n for g = 0. But there is an automorphism group G for g = 0,1. Hence if M*,A 0 n is not empty, it must have at least dimension 6 or 2 for cases of g — 0,1. Thus under the assumption of theorem the corresponding moduli spaces of unparameterized, non-multiple cover curves have negative dimensions and hence must be empty for a generic J . Fix a such J. Now if A is represented by a J-holomorphic curves / , then / must be a multiple cover one. But we can always choose a h such that / covers h but h is a non-multiple cover one. Since c\(A) < 0, then Ci([h]) < 0. This is impossible if J is generic unless h is a rational curve in the case of dim V > 8. 5.2.

Orientation

The same argument as in [25] will show that Ml. M\A a.a.J),, and UUtM*,. ,, x Itl tA4?■ a .n x{t}

carry a natural orientation.

185 5.3. Fundamental class First of all, by Gromov compactness, we have following easy lemma: L e m m a 5.3.1. Fix a positive number M and an positive integer g. There are only finitely many homology classes in B e Hi{V, Z) represented by a J-holomorphic curve of genus < g and w(B) < M. Proof. Suppose that it is false. Then there are infinitely many such classes Ai € H2(V,Z). Obviously, Ai has no limit in Hiiy,%). Suppose that Ai is represented by a genus go J-holomorphic curve Cj, where go < g and u>(Ci) < M. By Gromov compactness, up to choosing a subsequence, Ci is weakly convergent to a cusp curve C. Then Ai converges to [C]. This is a contradiction. Definition 5.3.2. Define

* W / ) = {{Bi,...,Bt};

(J M{Bitg,tJ) ?4 0 , £ Bi = A}. 9'<9

By the lemma, "D{A,O,J) is a finite set. Remark 5.3.3. We can also allow J to vary in a compact set K. Then Vi^gK\ lijeK v(A,g,J) i s a1*50 finite.

=

To abuse the notation, let M*,A u n = M*tA, n. Now we can write down a compactification of M*,A j jy Let M*,A n = U o - ^ U o a T ) * n e s P a c e °^ n o n " multiple cover curves. Intuitively, the space of J-holomorphic curves M(A,g,j) = M*,A n U {multiple cover curves}. Here, we only care about the image of J-holomorphic curves. Henc, we would like to replace multiple cover curves by their image. This process will change the fundamental class of a J-holormophic curve. Hopefully, it will reduce the dimension of moduli space as well. Let H(k, g, g1) be the space of degree k holomorphic maps from a genus g Riemann surface to a fixed genus g' Riemann surface. We will replace multiple cover curves by

Be eHH22(v,z),H(k,. \J{M{hg,jy,j (V,Z),H(k,g,g') ^V\JJ) fc>i

T

Define M(A,g,J) = M\AtgtJ)

Vk>1 {A< ( f , s V ) ; |

€ H2(V,Z),H(k,g,g')

£ 0}

186 Theorem 5.3.4.

'(A, 9,J) M*(A,g,J)c M'

(J

C

I]

A BM , S(B. ^jy ,J)-

{BU...,B,} gi+-+9t<9

The proof follows from Gromov compactness. Suppose that H{k,g,g') ^ 0. Clearly, g' < g and there are only finitely many choices of such pairs (fc, g'). For generic J, dim M\BgaJ)

< 2Cl(V)(B)

+ 2(g - 1).

If H{k,g,g') ^ 0, by Riemann-Hurwitz theorem, 2{g - 1) = 2k{g' - 1) + 6, where 6 > 0. Hence, O

^M = c,(,)(f,+2tf htJ) dimMl fg,J) 2 = 2cl{V)(-) and equal iff g = l,a{V){B) t>2, dim

J]

_ 2ci(V)(B)+2(s-l) " m ;fc" ' - 1 1 + 2{g'--1) < ~~

= 0. Furthermore, if {Bi,...,Bt}

E V(g,A,J) and

^tBj,.fl<53(2ci(V)(Bj) + 2(ft-l))<2ci(V)(A) + 2 ( f f - l ) - 2 ,

SiH—Hgt<9

where dimX ( j 4 > g ) l i j) = 2ci(V)(A) + 2( 5 - 1). For {^},

d i m X ^ 9 a J ) < ^ ^ g ) / i J ) - -2 for any nontrivial a. HH{k,g,g')

^ 0, by the previous argument,

d i m M \ B g,A

< dimMlAgItJ]

- 2

unless g = l,ci(V)(.A) = 0. Therefore, we show Theorem 5.3.5. Let (V, w) be a symplectic 4-manifold. For a generic tamed almost complex structure J, MA gj j carries a fundamental class unless g = 1, ci(V)(A) = 0. Remark. Here we don't need to use the property that cusp curve is connected. In the dimension 4, the dimension of M*,A „ is 2ci{V)(A) + 2(g - 1), which is increasing as g increases. For higher dimension, dimM*{Aig>J) = 2Cl(V)(A)

+ (6 - 2n)(g - 1),

which is opposite to the dimension 4. Then, we have to use the connectedness of cusp curves to put further constraint to reduce the dimension of the space of cusp curves [25].

187 5.4-

Invariant

To define Donaldson type Gromov invariant, we have to study /i-classes. For 4-manifolds, the only nontrivial class is JJ([V]) [25]. Hence, the only nontrivial invariant is

*<*,) = Mfe(M)pW(A9,^)l for k = ci(V)(A) + (g-l). Because of the multiplicity, it is rather difficult to prove that M([V]) extends over M^„l,J}We take the approach of intersection theory. [V] is dual to a point x e V. Consider the evaluation map

* ? < * * r,J) x G (£)*

e

vk,

where G = / i f g > 2 a n d G = PSL2(C) or KQ if g - 0, or 1. Note that dimA4* jn x G (E)fc = 2c\{V)(A) + 2(g - 1) + 2k = 4k. Hence for a generic element ( x i , . . . , xjt), e _ 1 ( ( x i , . . . , x^)) is a zero dimensional smooth oriented man­ ifold. Now we want to show that e _ 1 ( ( x i , . . . .x^)) is compact and hence finite. Then we can define ®(A,g,J) =

\e~1{{xu...,xk))\,

counted by the orientation. Notice that e _ 1 ( ( x i , . . . , xjt)) is the set of J-curves passing through ( x j , . . . , xj.) together with the number of intersection points. Hence, this definition agrees with what we expected for yr ([V]) [Mu gJt,J)] • To show that e - 1 ( ( x i , . . . , xjt)) is compact, we have to choose {x\,... ,xk) care­ fully to avoid the image of the lower stratus in Mi A a J\- To see what we need to do, let's suppose that e _ 1 ( ( x i , . . . , x^)) is not compact. In particular, there are infinitely many elements in e - 1 ( ( x i , . . . ,xjt)). Since (£ s ) fc itself is compact, then the image of 7r : e _ 1 ( ( x i , . . . , x/t)) —* M*,A t « has infinitely many elements. Choose a sequence / „ € 7r(e _ 1 ((xi,... ,xjt))). It implies that Imfn passes through Xi,..., Xfc. Therefore, we can assume that / „ converges to an element h in M^,g,i,J) such that Im h also passes through x\,..., x^. Because h is not a manifold point, h 0 M*,A XJy We can divide it as three situations: (i) h e Ylgi+...+gt 2 where {Bx, ...,Bt} e V{A,g,J)l (») / € M*AglJ) where M(k,g,g') ^ 0. (iii) h 6 M*, A has n for some nontrivial a. For the first case, h = (h\, ...,ht) several components. Of course, some component could be a multiple cover curve. Suppose that hi is a degree d, multiple cover curve of a genus g[ curve. When hi is a non-multiple cover curve, we just choose dj = 1 and g[ = gi. We need an additional data to describe "h passes through x i , . . . , x t " . This can be done by decomposing { 1 , . . -, k] = Uljlj, where i e Ij means that Im hj passes through x*. (warning: some Ij could be empty). For the simplicity, we also impose the condition 7,-n/g = 0 if j i=- s. Then the condition "/i passes through x i , . . . ,Xk" could produce

188 several such decompositions. What we need here is that there is one. Once we have a decomposition, we can consider I I ( ^ x*°-> (EGJhh/G -i* Vk gi+-+gt
'

Condition "h passes through x i , . . . , x j t " can be interpretered as ( x i , . . . ,xk) € e ( n 9 l + . . . + 9 t < s ( ^ * | i ] 9 , J ) XT 0 . 9 (EG,g>)M)/G). For the second case, if h €

M*Ujy

we need to consider

(MIMXTc"

<9'),G " ^ Ffc

EG

The condition "/i passes through x j , . . . , xkn can be interpreted as {x\,..., xk) £ e( (JV(*A j , xYogEGtg')/G). For the last case, if /i 6 A l ^ g f l ^ , then ( x i , . . . , xk) 6 e ((-^(A o 7) XT « a EG,g)/G). By our calculations in the last section,

dim

n

(-^(f^.j)

Ji+-+Jt<S gi+—+gt
and

J}

XT

XJGS XJGS

^JIAy°$4k-2, -2'

«»

{EGJ,))IG {EGJ,))IG

< 4fc - 2 < 4fc - 2

dim(A^ ga 7) x T i 8 Ea,g < 4fc - 2,

d i m ( A ^ g a 7 ) x Ta , a £ Q , 9 < 4A: - 2, provided that either c\(V)(A) ^ 0 or g / 1. We shall assume this condition through out the rest of this section. Let U be the union of the images of II

X

M

(%,g',J)

G (Erf)1*1,

9i+-+9t<9

-^(4, S ,J) >
^,«7,a,J)

X

« E9

under the evaluation map e. We just show that Vk —U is path connected and of Bansecond category. In particular, if we choose that ( x i , . . . , x^) & U. e _ 1 ( ( x i , . . . , x^)) is a compact, zero dimensional smooth manifold. Thus, it is also finite. Definition 5.4.1. Define $(Aigj){xi,-■ orientation.

■ ,xk) = | e _ 1 ( ( x i , . . . ,xk))\,

counted by the

L e m m a 5.4.2. *(/i, 9 i j)(xi,..., xk) is independent of the choices of ( x i , . . . , x^). Proof. Let (x[,..., x'k) e Vk-U be another generic choice. Since Vk-U is path con­ nected, there is a generic path 7 e Vk — U connecting (x\,..., xk) and (x[,...,x'k) such that e - 1 (7) is a smooth, oriented 1-manifold. Since 7 e Vk - U, e _ 1 (7) is compact. The boundary de~\i)

= e-\{xx,..

.,xk))

U - e - 1 ( ( x ' l l . . . ,x'k)),

where — means the opposite orientation. Hence, $(A,g,J)(xi,

■ • • , Xk) = $(A,g,J)(x[,.

. . , x'k).

We shall omit ( * i , . . . ,xk) and only denote as $(A„tn.

189 L e m m a 5.4.3. ®(A,g,j) is independent of J. Hence, we can denote it by t is a path of symplectic structures, then $(A,g,u0) = $(/i,g,wi)-> i e the invariant depends only on the deformation type of symplectic structuresProof. Let J be a generic almost complex structure tamed with wo and let J be another generic almost complex structure tamed with u>\. Then there is a generic path (Jt) connecting J to J' tamed with ut. Consider r,j«) Ut)

*tt

e J (EG,g)k)/G ( t) yfc

*TC, ,

x{t] X

d i m \ J { M * { A M ) x {*} x T c s {EG^/G

= 2d{V)(A)

+ 2fc+1 = 4k + 1 .

+ 2(g-1)

Hence for a generic ( i j , . . . , x^), e7 ; 1 ,((a;i,..., xjt)) is a 1-dimensional smooth ori­ ented manifold. The argument for compactness is similar. We consider

uU nII

(^(ijjjy ^ , 9 : Jt)

(

*t gi-\ 91-t ■+9n, <9 hgm
xx{t}

e

IAI

(Jt)

G T . ,( iEa* <*> x^.» W ',:)' ) /)/G ^ X

C S

v\ V\

'

U(^(4(9V,)

x

to xT<w (EG,g>))/G ^ Vfc

t

and tjt)

tt

x{t}

X

e

T„, ,Ea

,g)la

Ut)

Vk

for {Bu ■~,B,} e p(A,< ! . « ) ) ■ Let Wt be the image of these space in Vk k. Now for { S i , . . . , B s } 6 ^(A.s.CJ,))Let W( be the image of these space in V . Now

dimu

n

[

dimlj ] t gi-\ J \-g

m

t SiH

{t}X T

(^W'•)/G << 4fc4fc- -L1,

M (f,^) x* XT E IA G <9 ( \^,g'M W «.»; ( cJ )l

\-gm
dim\J(MlUjt)

G.3;

x {*} x T c y (EG>g,)k)/G

< 4fc - 1

and d i m ( J ( A < ^ s a > A ) x {t} x T a j £ ^ ) / a < 4 f c - 1 t

when either ci(V)(^4) / 0 or
(jt)((xi>

■■■'xk)) = e / H f o , ■ • ■ -»fc)) u - e j / ( ( ^ i , • • • ,£*))•

Hence, $(A, S ,J) = $ (As,./')- S o w e h a v e s n o w n Theorem B . $ is an invariant of the symplectic structure, and furthermore it is unchanged by symplectic deformations, i.e., it is an invariant of symplectic defor­ mation type.

190 6. R e m a i n i n g case Now we discuss the remaining case that ci(V)(A) = 0,g = 1, where the con­ struction of previous section fails. The multiple fiber of an elliptic surface falls into this category. As we discussed in the last section, the only problem is the multiple cover curves. Recall that in section 5.3 we deal with multiple cover curves by simply replacing it by its iamge. To be able to use this simple process, we have to show that such process will reduce the dimension of moduli space. We prove it in the end of section 5.3 for all the cases other than c\{V){A) = 0, g = 1. One consequence is that multiple cover curves will not give any contributions to invariant $A- But for the case that c\(V)(A) = 0, g = 1. The process replacing multiple cover curves by their images will not reduce the dimension of moduli space. Therefore, we have to study multiple cover curves in the different way. We shall see in the next section, multiple cover curves does give contributions to invariant. Suppose that A is a primitive class. There is no multiple cover curves. Then invariant $(A,I,V) i s well-defined. Note that dim M*,A t « = 2c\ {A) = 0. Therefore, = for a generic tamed almost complex structure J, M(A,I,K0,J) ^\A l K. J) *S °^ finitely many points. And *(A,l,w) =

\M(A,1,J)\,

counted by the orientation. Now we define $(nA,i,v) by perturbing the pseudoholomorphic equation. First of all, let's see what we can get by just varying the almost complex structure J. Recall that the moduli space Mi of complex structures on torus is homeomorphic to C, but not diffeomorphic. In fact, they are two singular points. For a generic complex structure a € Mi, the automorphism group is KQ. There are two singular points a.i,ai where the automorphism groups are the extensions of K,Q by Z2 and Z3 respectively. Since dim Ml . , > = 2ci(nA) = 0. For a generic J, there is no J-holomorphic torus with complex structures » i , a 2 , representing nA. Furthermore, there is no cusp curves as well. Moreover, by the induction, we can assume that the space of multiple cover curves is also compact. We assume that J satisfies these conditions through out this section. Then, Ml. n is just M*,A j

* and M(A,I,J)

*S compact.

For the Fredholm theory, we only have problem at multiple cover curves. A torus could be either a multiple cover of a torus or a sphere. Since ci (nA) = 0, there is no ./-sphere for a generic J . We only have to worry about the first case. Suppose that it is a degree k cover. Then H(k, 1,1) ^ 0. By Riemann-Hurwitz theorem, there is no ramification point. Hence, it is a free covering. Then, H{k, 1,1) is a finite set. We also know that M^^j) is compact. The space of multiple cover curves is also compact. In this situation, it is a particularly nice case where one can add an abstract perturbation supported in the local neighborhood of multiple cover curves. Incidently, this was the method that Donaldson used in [4]. There are certain rules for doing this. We will refer to [5, p. 155] for a general description. Here, we follow

191

their method. Let's first recall our general set-up. Let M\ = M\ - {ai, 02}- Fix a generic C°°tamed almost complex structure. For each a € M[, there is an unique flat metric ga of area 1 compactible with a. ga defines sobolev norm L\ on A = M a p ( £ i , V) which gives L^(A) a Banach manifold structure. Here, it is important to have a Hilbert manifold (see [5]). Hence, we take p = 2 and k as a large integer. M[ is finitely dimensional. Then, L\(A\^\) is a Hilbert manifold, KQ acts freely on £fc(-4i,i) and quotient B\j. = L t ^ ' ^ is also a Hilbert manifold. Then tangent space Taj(Ll(Ai,x)) = F a ( S i ) © LftSlP(TfV)), where / € p'^a). The tangent space of an orbit of K 0 action is # ° ( T E ) as a subspace of L2k(Q,°(TjV)). Hence, the tangent space Ta ^ = (if°(TE))- L -the orthogonal complement of i7°(TE). As we discussed in the last section, we can form Hilbert bundle Z#2_j(fi0,1(e*TV)) over .4.1,1. «o acts as an isometry of ga and hence an isometry on L^_ 1 (n 0 , 1 (T/V)). Therefore, we can divide by KQ to get a Hilbert bundle over B\ti (still denote it by L^_ 1 (n°' 1 (e*TV)). The pseudo-holomorphic equation dj is a canonical section of L|_ 1 (n°' 1 (e*TV)). Now we would like to construct a perturbation term a of a section of L^_ 1 (n°' 1 (e*TV)) such that the perturbed equation Sj + a is still Fredholm and has the same index as that of Sj. Furthermore, we would like to have a supported in a local neighborhood of multiple cover curves. To do it, we usually need to use a smooth cut-off function which doesn't exist for general Banach manifolds. This is why we need to use a Hilbert manifold instead. To ensure first condition, we can choose a to be a section of L^il0'1 (e*TV)). Then the derivative

Aa A<7

x 1 1 TV)) ::{H (H\TE,a)) -^Ll(Q°'He L2k{^\e*TV)) . Ll.li&He LU^e'TV)) (T^,: a)) TV)) - +

is a compact operator. Hence, A(Bj + cr) is Fredlhom and has the same index as that of Adj. We give such a C\ topology in the sense of uniform convergence of Hall^ and convergence of ACT in operator norm, uniformly on compact sets. To construct the perturbation specifically, we need the following theorem of a local model of the Fredlhom map. L e m m a ([4, Lemma 7]). Let E -i-* F, /(0) = 0, be a smooth map between Banach spaces with the property that L = (L>/)o is a Fredholm operator of index k. If we choose decompositions E = kerL ®E',F = ImL © F', so that dim KerL = d i m F ' + k, then there is a local diffeomorpbism if ofF and a local smooth map : E —> F' such that f o y~l(x)

=Lx + <j>(x) e ImL ®F' = F

192 locally. Note that under this new model, L{x) + {x) = 0. Hence, the solution space is the zero of finite dimensional map 4>\KerL -* $* — CokerL. Applying this lemma to our situation, at a neighborhood of (a, [/]) € M[ x B we can decompose (H0(T'£))-L = KerL®E' for L = D®i and write dj = L + : ( ^ ( T S ) ) 1 —► CokerL. In particular, it induces a map 0' : KerL -> CokerL. By the construction, (D')o = 0. To construct specific perturbation, one only has to perturb <£'. For instance, we can pick up a linear epimorphism A : KerL —» CokerL and for sufficiently small r] > 0 and extend ^4 over £? by a = T?/?(—)4(Proj L (x)) where j3 is a smooth cut off function and ProJKerL ■ (H 0 (TE)) X —► A'erL is the projection. Then we can just perturb 3 to 5 + a. Using this construction, we can prove follow lemma: L e m m a 6 . 1 . If V CC U C 0j,i are open sets with U covered by finitely many slices Tjyj t x W where W c M[ is an open subset such that W is compact, then the set G of "good perturbation" a, supported in U and such that the zero set V D Z(d + a)(zero section) is cut out transversally (i.e., A(§ + a) is onto for all (a, [/]) e V n Z(B + 0 such that for any (a, [/]) e MR(a) \\dJ{[f])\\L2(a) < R'. Since W is compact, we can assume that an converges to OIQ € M[. By the continuity of the norm,

I I I W D I k K ) - \\d?(lf])\\Ll{ao)\ < C\\8?([f])\\Ll{an), where C depends on W, J. Therefore, ||S7([/n])||i|( O 0 ) is bounded. By the elliptic estimate, ||r/nl|[z|+1(ao) i s bounded and hence is compact in L | ( Q O ) . Hence, there is a convergent subsequence. So MR is compact.

193 Now, let's prove that G is open. For any ao € G, by the assumption for any (Q, [/]) € V n Z(dJ +_aQ), A(dJ + ff0) is surjective. Since the space of surjective operator is open and VC[Z(dJ+a0) is compact, there is R > 0 such that &(dj + a0) is surjective for any (a, [/]) e Vr\MR(a0). Since V Ci MR{a0) is compact, for any a close to cro, A(9^ + a) is surjective for any (a, [/]) eVn MR{OQ). On the other hand, if a is close to a0 enough, V7 n Z{dJ + a) C V n Mfl(. We can apply usual Sards theorem to <j> : KerL —» CokerL to find /3n —> 0 such that L + 4> + f3n is transverse. Then we can extend /?„ to Bi?i by a cut off function. Then a + /3 n 6 G and converges to
<7

)li counted by the orientation.

T h e o r e m 6 . 3 . $(nA,i,j,tZ(8j ) such that and UfZ(dj + at) x {t} is an oriented, smooth compact cobordism. d\JZ{b^t

+ at) x {t} = Z{85 + a)\J-Z(S$,

a generic path can show that (at) joins a, a1 The boundary

+ a'),

t

Then, $(nA,l,J,a) = * M , i , J V ) Remark. For the reader who is not familiar with this method, the author wants to emphysis that this method is very general. For example, in gauge theory, there is no need to introduce Freed-Uhlenbeck tranversality theorem if we adopt this method of abstract perturbation. One drawback is that we lost the geomtric meaning of moduli space. One also should be careful to use abstract perturbations at the boundry of moduli space. It could destroy the compactness.

194 7. Elliptic surfaces In this section, we will prove theorem C listed in the introduction. To ensure the existence of Kahler form, we always assume that the first betti number is even. An elliptic surface S admits a holomorphic map n : S —> P 1 such that the general fiber is a smooth elliptic curve F. One can choose a generic complex structure such that singular fibers are nodal elliptic curves and the multiple fibers Fp of the multiplicity p are again smooth elliptic curves. But the homology class [Fp] = L2. Therefore, for every elliptic surface, we obtain a set of number {pi > • • • > pk) representing the multiplicity of the multiple fibers. Assume that m is the least common multiple. Let Sm = {^ < g < • • ■ < £ } . S'm = {% < % < ■ ■ ■ < g } is another such ordered sequence. We define Sm < S'm iff fc' > k and |* = y- for i < k. We shall prove that Theorem C. Suppose that S, S' are minimal elliptic surfaces of even first betti number. Then if S, S' are symplectic deformation equivalent, either Sm < S'm or Sm < Sm. We shall prove the theorem C by using the invariant we defined in last section. Before we state the calculation, we need to relax the genericity condition of J, a. Definition 7.1. (J, a) is yl-good if (i) for any (a, [/]) € Z(3j + cr), i®D is surjective. (ii) There is no cusp J-curve representing class A. It is easy to check that it is enough to use yl-good (J, a) to define $(,4 1 wj with ci(^)=0. Proof of Theorem C. Suppose the complex structure we choose has the property that the singular fibers are nodal elliptic curves and the multiple fibers are smooth elliptic curves. To prove the proposition, we shall calculate $(,4,1,^) for A with c\(A) = 0. The invariant we need is the one defined in the last section. At the first, we will show that $(A,I,U) = ° unless A is a multiple of [F]. It is classical that c\ = /3[F]. If 0 = 0, then c\ = 0 and S, S' are K3-surface and Sm = S'm = 0. Theorem is obvious. Suppose that 0 > 0. If C is a holomorphic curve, C is either a mulitsection or a fiber by our assumption of singular fibers. In particular, either ci(C) > 0 or a(C) = 0. Let C = $laici be an effective curve {CH > 0). If ci(C) = 0, each d has to be a fiber and hence [C] is a multiple of fiber. Therefore, for any A with ci(A) = 0 but other than a multiple of fiber, there is no holomorphic A-curve and no cusp holomorphic A curve. Hence, the complex structure is A-good and $(A,I,U) = °- It is the same argument if 0 < 0. By [6, Corollary 2.9], [F] = mA for a primitive class A, where m is the least common multiple of p i , . . . ,pk. Now we calculate $(nA,i,u) for any n < m. First we check that (J,0) is nA-good, where J is a generic complex structure we choose. Clearly, M\nAXJ) is the set of all multiple fibers of the multipUcity f. Let C be any

195 multiple fiber of the multiplicity EJ. The normal bundle Nc is a torsion element of order E . Then H°(NC) = ^(Nc) = 0. Furthermore, we have an exact sequence 0 --»+ TC -> T TS\ 5 C| c -» Afc Nc - 0. It induces a long exact sequence of cohomology l 0 --► ^ (HC {C)) -»fl*(T5| -* Hl{TS\c)c ) — — 0. 0.

Then, Coker{i ® D) ^ Hl{TS\c)/ImHl{TC) = 0. Now we check the multiple cover curves. Clearly, there is no rational hA curves for h < n. Thus, the only possibility is a multiple cover of another elliptic curve which has to be another multiple fiber Fq such that q = dq>n^, where dpin is the degree of multiple cover. Let sq}n = \H(dqtn, 1,1)|. Then, there are s 9 i n many multiple cover curves of genus 1 covering Fq. Let v : C —> Fq be such a multiple cover curve. Then we have a short exact sequence 0-> 0 — v*TF u*TFqq -^T -f T » 00.. CS^ CS -> v*NFq --> By Riemann-Hurwitz theorem, v : C —» Fq has to be a unramified cover. Therefore, v*TFq = TC. Furthermore, v*NFq is a torsion element of order *■* and hence IP-{u*Fq} = 0. This implies that i © D is surjective at C. It is clear that there is no cusp nA curve. This shows that (J, 0) is n.A-good and there is no need to perturb to an almost complex structure or add a local perturbation around multiple cover curves. We can just use our complex structure to calculate invariant $( n ^i ) W ). A direct counting argument shows that $

(nAl,w) = kT +

5Z

5

9.n*9>

where kq is the number of the multiple fibers of the multiplicity q. Now suppose that S,S' are symplectic deformation equivalent. If Sm = 0 or S'm = 0, there is nothing to prove it. Hence, we can assume that Sm j= 0 / S'm. By the previous argument, there is diffeomorphism / : S —► 5" such that $(nA,i)(S) = $(n/.(.A),i)- Choose ra = ®. $(nA,i) = kpi ¥" 0. Hence, ^ / * ( J 4 ) has to be a multiple of [F1], where F' is the general fiber of IT' : S' —* E. Let A' be the primitive class such that [F1] = m'A'. Since, *(„^',i)(5") = 0 for n < y-- This implies that Zp- < —. The same argument will show that g < 5 - and hence equal. Furthermore, fepj = kpi. The rest of proof follows from the induction. Acknowledgements My special thanks go to Professor Kobayashi who taught me most of my differ­ ential geometry when I was a student in Berkeley. It is my pleasure to be able to contribute to this Proceeding for celebrating his 60th birthday. My work is building on the work of Dusa McDuff. My special thanks also go to her.

196

References 1. W. Barth, C. Peters and A. Van de Ven, Compact complex surfaces (Springer, 1984). 2. S. Bauer, Diffeomorphism types of elliptic surfaces with pg = 1, Warwick preprint (1992). 3. S. K. Donaldson, Yang-Mills Invariants of Four-manifolds, Geometry of Lowdimensional Manifolds: 1, LMS Lecture Notes Series 150 (1989), pp. 5-40. 4. S. K. Donaldson, An application of gauge theory to four dimensional topology, Jour. Diff. Geom. 18, 279-315. 5. S. k. Donaldson and P. Kronheimer, The geometry of four-manifolds, Oxford math. Mono.. 6. R. Friedman and J. Morgan, Smooth four-manifolds and complex surfaces, Springer-Verlag (to appear). 7. R. Friedman and J. Morgan, Complex versus differentiable classification of al­ gebraic surfaces, Topo. Appl. 32 (1989), 135-139. 8. R. Friedman and J. Morgan, On the diffeomorphism type of certain algebraic surfaces I, J. Diff Geom. 27 (1988), 297-369. 9. R. Friedman and J. Morgan, On the diffeomorphism type of certain algebraic surfaces II, J. Diff. Geom. 27 (1988), 371-398. 10. R. Friedman, B. Moishezon and J. Morgan, On the C°°-invariance of the canon­ ical classes of certain algebraic surfaces, Bull. Amer. Math. Soc. 17 (1987). 11. D. Freed and K. Uhlenbeck, Instantons and four manifolds, MSRI Publ. 1 (Springer). 12. R. Gompf, A new construction of symplectic manifolds, preprint. 13. R. Gompf and T. Mrowka, Irreducible ^manifolds need not be complex,, Ann. Math. 138 (1993), 61-111. 14. M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307-347. 15. M. Liibke and C. Okonek, Differentiable structure on elliptic surfaces with finite fundamental group, Compositio Math. 63 (1987), 217-222. 16. D. McDuff, Elliptic methods in symplectic geometry, Bull. AMS. 23 (1990). 17. D. McDuff, The structure of ational and ruled symplectic ^-manifolds, J. Amer. Math. Soc. 3 (1990), 679-712. 18. D. McDuff and D. Salamon, . . . (book). 19. J. Morgan and K O'Grady, The smooth classification of fake KS's and similar surfaces, preprint. 20. J. Morgan and T. Mrowka, On the diffeomorphism classfication of regular el­ liptic surfaces, preprint. 21. C. Okonek and A. Van de Ven, Stable vector bundles and differentiable structures on certain elliptic surfacses, Invent. Math. 86 (1986), 357-370. 22. T. Parker and J. Wolfson, Pseudo-holomorphic maps and bubble trees, J. Geom. Anal. 3, no.l (1993).

197

23. P. Pansu, Pseudo-holomorphic curves in symplectic manifolds, preprint (Ecole Poly, 1986). 24. Y. Ruan, Symplectic topology and algebraic 3-folds, to appear in Jour. Diff. Geom.. 25. Y. Ruan, Topological Sigma model and Donaldson type invariants in Gromov theory, preprint. 26. Y. Ruan, Symplectic topology and extremal rays, GAFA 3 (1993), 395-430. 27. M. Ue, On the diffeomorphism types of elliptic surfaces with multiple fibres, Invent. Math. 84 (1986), 633-643. 28. G. Xiao, Surfaces fibrees en courbes de genre deux, Springer Lect. Notes in Math. 1137 (1985). 29. R. Ye, Gromov's compactness theorem for pseudo-holomorphic curves, preprint.

198

A U T O M O R P H I S M S OF T U B E D O M A I N S SATORU SHIMIZU Mathematical Institute, Faculty of Science, Tohoku University

Introduction There is a key observation in the study of the holomorphic automorphisms of a bounded domain in C n . Let D be a bounded domain in C n By a well-known theorem of H. Cartan [1], the group Aut(Z?) of all holomorphic automorphisms of D has the structure of a Lie group with respect to the compact-open topology, and its Lie algebra g{D) can be identified canonically with the finite-dimensional real Lie algebra consisting of all complete holomorphic vector fields on D.li z\, ■ ■ ■ ,zn are the complex coordinate functions, then every element X of g(D) can be written in the form X--

=t,m-6 a ?=i

are are holomorphic functions on D. Sup­ where z = {z\, ■ ■ ■ , zn) and fi(z), ■■, ■fn(z) • , fn(z) pose that g{D) / {0} and take a non-zero element XQ of g(D). Since g(D) is finite-dimensional, the minimal polynomial P of the endomorphism ad XQ of g (D) is defined. If X is any element of g{D), then X satisfies = 0,

P{adX0)X

(*)

which implies that the coefficients of X are given as a solution of the system of certain analytic partial differential equations. To explain this, for n-tuple v = ("1. ■ • ■ i vn) of non-negative integers, we set d" = (d/dzi)"1 ■ ■ ■ (d/dzn)"" and \u\ = £?=!"«• Write X0 = £ " = i fj(z){d/dZj) and X = T%=1gj{z)(d/dzj), where fj{z), gj(z), j = 1, • • • , n, are holomorphic functions on D. Then we have (adX0)X

= [Xo,X]

■ttwf " g fc=l \j=l

3

a,, sM)a,k-

By a successive use of this relation, we see from (*) that there exist differential operators

J2hu(z)d\ = £*

Pij{z,d)=

3 = I . " - ,n,

M
with the coefficients hv(z) holomorphic functions on D such that 9i{z)r- ■ ,9n{z) satisfy 7),

£>„■(*, £>)#(*) =0,

i ■= ! , ■ • • , n ,

199 where d denotes the degree of the minimal polynomial P Therefore, when there is assumed the existence of many holomorphic automorphisms of D, the functions 9iiz),• • • ,9n(z) have rather restricted form. This observation is extremely useful in the investigation of the structure of B(Z?) when D is a Reinhardt domain (cf. Shimizu [3]) or D is a tube domain which is an object of our study in this paper. If a domain fi in R n has the convex hull containing no complete straight lines, then the tube domain To. = R n + ■/—T^ over fl is biholomorphically equivalent to a bounded domain in C n . When Q is a convex cone, the beutiful result on the structure of B(TQ) is known (see Matsushima [2]). We try to clarify the structure of B{TQ) when Cl is an arbitrary domain in R " whose convex hull contains no complete straight lines. In this paper, we establish the most fundamental result on the structure of S(TQ) (Section 2, Theorem). A more detailed analysis of the structure of g(7h) and some interesting applications of our result will be given in the subsequent papers [4], [5]. This paper is organized as follows. In Section 1, we discuss basic concepts and results on tube domains. Our theorem on the structure of B{TQ) is given in Section 2 together with some important observations. In Section 3, we collect several basic lemmas used in the proof of the theorem. Sections 4, 5, and 6 are devoted to the proof of the theorem. 1. Basic concepts on t u b e domains We first collect some notation and terminologies needed later. The group of all holomorphic automorphisms of a complex manifold M is denoted by Aut(M). we denote by GL(n, R ) K C™ the subgroup of Aut(C n ) consisting of all transformations of the form Cn B z t—► Az + P € C n , where A 6 GL(n, R ) and /? G C n - Two complex manifolds are said to be holomorphically equivalent if there is a biholomorphic mapping between them. If E = {• • • } is a subset of a vector space V over a field F, the linear subspace of V spanned by E is denoted by Ep = {■ - • }pWe discuss basic concepts on tube domains. For each £ € R n , we define a tanslation o^ € Aut(C n ) by <7?(z) =Z + £. Write E = R71 The additive group E acts as a group of holomorphic automorphisms on Cn by i-z

=

CT?(Z)

for i £ E and z e

Cn.

Definition. A tube domain in Cn is a domain D in C™ which is invariant under E, that is, such that £ • D c D for every £ € E. It is immediate that D is a tube domain in C™ if and only if D is a domain in

200 C n given by

D = Rn + y/^in = {x+y/^ly\xeRn,yeQ},

(1.1)

where Q is a domain in R n - When D is given by (1.1), we write D = TQ, and D is called the tube domain over Q. Also, fi is called the base of D. Let TQ be a tube domain in C n The group E then acts as a group of holomorphic automorphisms on TQ. The subgroup of Aut(7fo) induced by E is denoted by Ey n . It is readily verified that if ip e GL(n, R) K C™, then v?(^h) is a tube domain in C n , and we have ^ E i ^ y - 1 = Er n ,, where TQ> = ¥>(Tp.). The converse assertion also holds. Proposition (cf. [3, Section 2, Proposition 1]). Let ip : T^ —> Toy be a biholomorphic mapping between two tube domains TQ and TQ> in C". IfipYsraV-1 = E T ^ I then

for all z € T n and all £ 6 E.

(1.2)

Fix a point ZQ of TQ and, for ( e S , write /—ll m ¥>(zo+£), where Rey>(z0 + £) e R " and Im^(z 0 + 0 e fi' C R " Then, by (1.2), the mapping E = R n 3 £ i—► Rev?(z0 + f ) e R n is affine, while the mapping E = R n B £ i—► Imy>(zo + 0 € R n is constant. This implies that if we write f(zi, ■ ■ ■ , zn) = (
dx~dx"k{z)'

»".J.* = 1 . - , " ,

vanish identically on the subset Eo of TQ given by So = {zo + t 6 C " | £ e R " } . Since Eo is a totally real submanifold of TQ of dimension n, it follows that dz~dz-k{z)'

*^* =

1

.-.».

201 vanish identically on To,, and hence that

for all z €

TQ

and all £ € E.

Combining this relation with (1.2), we see that A£ = A'f for all £ e E, so that A = J4', which concludes that

equivalent,

Let us consider this problem in the case where the bases fi and fi' of TQ and To/ are domains in R n whose convex hulls contain no complete straight lines. When fi and fi' are convex cones in R n , an affirmative answer is given (see Matsushima [2]). On the other hand, when fi and fi' are arbitrary domains in R n whose convex hulls contain no complete straight lines, there is a simple counter example. In fact, consider the upper half plane 7(o,oo) = {x + y/^ly

€ C | x e R, y > 0}

202 and the strip %«■) = {x+ V^ly e C | x <E R, 00oj and T^o.ir) m C are holomorphically equivalent, but not affinely equivalent. However an application of our result in this paper yields that if two tube domains TQ and TQI in C n whose bases ft and ft' are bounded domains in R™ are holomorphically equivalent, then they are affinely equivalent. The proof of this result will be given in the subsequent paper [4]. 2. S t r u c t u r e o f g ( T n ) We begin with some preparations. Let D be a domain in C™ and let z\,- ■ ■ ,Zn be the complex coordinate functions. For j — 1, • • ■ , n, we write dj = d/dzj. Then every holomorphic vector field X on D can be written in the form n

* = £/*<*)«,•. where z = (z\, ■ ■ ■ ,zn) and fi{z), ■ ■ ■ , fn(z) are holomorphic functions on D. The vector field X is called a polynomial vector field if fi(z), • • • , fn(z) are polynomials in £!,••• , zn. Suppose that D is a tube domain in C™ and write D = TQ. Then Y^Tn a c t s as a Lie transformation group on TQ. For every j = 1, • • ■ , n, the holomor­ phic vector field dj on TQ is the infinitesimal transformation of the one-parameter subgroup 3 (zi, • • • ,Zj,■ ■ ■ ,Zn) i—► (zi, • • • ,Zj +1, ■ ■ ■ ,zn) 6 TQ 11 G R }

{TQ

of Aut(Tn). Hence, in the Lie algebra consisting of all holomorphic vector fields on TQ, the subalgebra corresponding to Er n is given by {d\, ■ ■ ■ ,3„}R,. Note that if the base Q of TQ has the convex hull containing no complete straight lines, then TQ is holomorphically equivalent to a bounded domain in C™ and {d\, • • • , 3 n }R is a subalgebra of the finite-dimensional real Lie algebra g (7b) consisting of all complete holomorphic vector fields on TQ. We now state our result on the structure of 0(Tn). Theorem. To each tube domain TQ in C n whose base ft has the convex hull containing no complete straight lines, there is associated a tube domain T^ which is affinely equivalent to TQ such that g (Tn) has the direct sum decomposition B(Tn) = p+z for which p = {X e B{TQ) IX is a polynomial vector Geld},

r f

/ c

+

« = lE{ *h /=r+l£ [ \ I=

\

/

/

\

\ )

v=i«&. e - - U - £ >/=Ia& j=r+l

/

-

JH

203 where r is an integer between 0 and n and aj, i = 1, ■ ■ ■ ,r, j = r + 1, ■ ■ • , n, are real constants. In the rest of this section, we give a description of holomorphic automorphisms of one-dimensional tube domains and some important observations related to it, which not only present a good illustration of our theorem but also are useful in the proof of the theorem. Every tube domain in C is affinely equivalent to one of the following: (i) The whole complex plane 2(_ a a ^ 0 j; (ii) The upper half plane T( 0oo ); (iii) A strip T^a^, where - o o < a < 6 < o o . Write d = d/dz.

In the case (i), Aut(T(_ 00 ^ j ) is a Lie group, and we have fl(T(-oo,co)) = {0.*3}c.

(2.1)

In the case (ii), we have fl(r(o,oo)) = id, zd,z2d}K.

(2.2)

We present the case (iii) as an example of our theorem. Example. Consider a strip Tutf in the complex plane, where - o o < a < b < oo. Then we have flCW = {d, Cec*d, C - 1 e - c z 9 } R , (2.3) where c = ir/(b — a) and C = e " v ^ m / ( M , i n particular, we have 9{T[0tir)) = {d,e'd,e-'d}K.

(2.4)

(2.4) follows from (2.2), because Tmn) and T(o,oo) a r e holomorphically equivalent under the mapping T

(0,n)

3

z

'

> e*

€

r

(0,oo)-

Since 2>o , ) and T( a ^ are affinely equivalent under the mapping %») 3

^

—^-z

+ v / ^ a G r (a , fc) ,

(2.3) is an immediate consequence of (2.4). Combining (2.2) with (2.4), we know that the assertion of our theorem is true when n = 1. Therefore, in proving the theorem, we may assume that n > 2. As a consequence of (2.1), (2.2), and (2.3), we see that if T(a,b) i s a t u r j e domain in C and if there exists a non-zero element of B ( I ( « M ) which is not a polynomial

204 vector field, then T(a,b) is a strip, that is, we have — oo < a < b < oo. This fact will be useful later. The completeness of holomorphic vector fields on a strip is closely related to the existence of a solution of the ordinary differential equation given by ^ = C e * « , (2.5) at where C is a non-zero complex constant. We observe that a solution z(t) of (2.5) is defined for all t € R if and only if e~z^ £ { C } R and that z{t) is not defined for t = e-*(°)/C if e- z (°) 6 { C } R . Indeed, it follows from (2.5) that e-*(') =

_Ci + e - z (°),

and our assertion is an immediate consequence of this relation. 3. Basic l e m m a s In this section, we collect several basic lemmas needed later. We begin with a simple formula which will be frequently used in the proof of our theorem. Lemma 1. If f(z) is a holomorphic function on D and if A, wi, • • ■ ,uin are complex constants, then, for each i — 1, • • • , n, + + (dii-\){f(z)e^ (d - \){f(z)e^-+-+^} »"*"}

= = {(wi {(ui--- X)f(z) X)f(z) + +

lZl+ +u+ + dif(z)}e^ ^, dif(z)}e" - '"- z»,

where di — X is viewed as a differential operator. The proof of this lemma is straightforward, and is omitted. The following lemma gives a solution of the differential equation which appears in the study of automor­ phisms of tube domains. Lemma complex consider D in C n

2. For each i = 1, • • • , n, Let Pi be a polynomial of one variable with coefficients whose mutually distinct roots are given by CJH,--- ,Wjrj) and the differential operator Pi (6\). / / a holomorphic function f(z) on a domain satisfies the system of the linear differential equations on D given by

K{di)f{z) Pl(di)f(z)=0,= 0,

£t == l!,.. , . . •. , n ,n,,

then it is of the form T\

r„

+i+ U Vfcfc /(*) = £E■ ■ ■ £ /*-*.w^»* . . j „ ( z ) e W l J i 2 :™ " ' + w " ' " Z n ,, /w = ■■•E/*.

ii=i

where fjv..jn{z), j% = at most YZ=i ( d e § pi When n = 1, the ordinary differential need a lemma.

J»=I

1, • • ■ ,n, i = 1, • • • ,n, are polynomials inz\,--- ,zn of degree ~ !) ■ assertion of this lemma is nothing but a basic result on linear equations. To prove Lemma 2 in the case where n > 2, we

205

Lemma 3. Let ui\,--- ,wro be mutually distinct complex constants, let fj(z) = Ylk=o ajkZ ' . 3 — li • • • i r o, be polynomials of one complex variable, and consider the holomorphic function f{z) given by

/to' =£/ito<^*. Then, for each j = !,••• , ro, there exists a polynomial Pj of one variable with complex coefficients such that Pj(d)f(z) =

aj0e^,

where d = d/dz. Proof. We observe first that a repeated application of Lemma 1 to the function fj(z)eu'z yields that UfYifjWtP"} z } =={&fj{z)}e»>* ((d3 ---^O'i/ito^ {^/itoK'* if j/ '=== jj and that J

l u^ifjiz^n WjZ} = {( u>.) + f?\z)}e">* ((dd ---^)'{/^toe {K azi0d' ^+/fto}c^ Uj --- Wi r) ajQ

if / j'+3 *j,

where /■ \z) is a polynomial of degree at most dj — 1. Now, for each j = 1, ■ • • ,ro, we define a polynomial Qj of one variable with complex coefficients by

QJ(t) = (t-uj)d>

J ] (t-uyOVM. l
Then we have

Qj(d)f(z) = Q V , where Cj=d3\

n l<j'
{uj-ujl)d>'+\

206 Indeed, by the observation above, we have

(a - ^ ' ) d / + i (( a - «i)*{/iW^">)

QiWiM*)**'} = n l
[1 ( 8 - ^ ) ^ + 1 ({8*/i(z)K>*) l<j'
J]

(8- w ,-.)"i'+i(d > !oj 0 ^)

l
= [[ K-^)^' + 1 rfi!^ 0 e^, l
and we have, for every / = 1, • • • , ro for which f ^ j ,

Qi(d){f?W

= {d--"i)di '*} =

n

(8- - u j . ) * ' + 1 ( ( 8 - - ^ ) d ' '+1{/i'{^)^ '•>)

l
= (8 - Wj)d> J ] (a - ^") dj " +1 ({^' +1 /,<MK''*) 1<Ji
0. Since the constant Cj is not zero by assumption, the desired polynomial Pj is given by CfQj. O Corollary. Let f(z) be as in Lemma 3. If f(z) is identically zero, then, for every j = 1, • • • , To, the coefficients ct^, k = 0 , 1 , ■ • • , dj, of the polynomial fj(z) are all equal to 0. Proof. Suppose the contrary. Then we may assume without loss of generality that, for some j , the coefficients OJQ is not equal to 0. Since, by Lemma 3, there exists a polynomial Pj such that Pj{d)f{z) = ajoeUjZ, it follows that the function Pj(d)f(z) is not identically zero. This contradicts to the assumption that f(z) is identically zero, and our assertion is proved. D We can now prove Lemma 2 by using the induction on the dimension n. As previously remarked, when n = 1, the assertion of Lemma 2 is a well-known result. Let f(z) be as in Lemma 2, and suppose that Lemma 2 holds when the dimension is less than n. In view of the principle of analytic continuation, we may assume that D is a polydisk Ai x • • ■ x A„, where A* = {\zt - Q\ < £ } , i = 1, • ■ • , n. For

207 any point c' = (z\. •• • . ;„ i) of A ( \ ■ • • x A,,_i, consider a holomorphic function fy(:,)) on A„ given by f,'{:n) = / ( ; ! , ■ • • .2»-i. 3»). Since, by assumption, P „ ( $ , ) M : „ ) = PntA,)f{:w

■ ■ • M.- i. =..) = 0.

it follows from the induction hypothesis that /.-■(;„) is of the form •-„

MM,) = V

/<<>»

\

l>

?L M''''" ~" «"' *"•

(3-D

where, for each j n = 1.- . r „ , the integer dj, + 1 denotes the multiplicity of the root uf,,^ of the polynomial P„ and a^. j „ = 1,- ■ , r „ , A- = 0.1, ••• .
0>,W/««(- ») = and consequently o,„oU)

can

o ,M

;V

n t „ i»t

^° written in the form

«V.o(?'l = < "'"•'"•
M = J - ( v o M ( ^ - * ) f*».* would yWki that o,„iU'1 is a holomorphk function on Ai \ • • • x A , , ^ . Thus, by the induction on k. our .assortion is prowd. Since, by assumption 0

= P.$ ) A : ' . M , ) = P»(3

>{x:(i>.

^■l--')^

M £{P,(^
>. = ! \*=° *. = > \fr=0

H \

V

-*

'j/

f,-x»«

^

/

i = !.••■ , n - 1.

208 it follows from the corollary to Lemma 3 that every holomorphic function <Xjnk(z!) satisfies the system of the differential equations given by Pi{di)ajnk{z')

= 0,

i-1,---

,n-l.

Therefore, by the induction hypothesis, the functions 2 a>fc( '). <Xjnk(z')>

1, ' • '■! •djn jn jn ~ = 1, • •' *) .Trnn,, Kk — - U, 0,1, , dj, n,

are of the form

a^co = £]■■■ E fh-^1J,k(z,)^zi+-+^-^-^-\ ^•jt(z') = E - A-

— 1

= U, 0,l, Jn == 1] K= 1, ••■• '■! ", j dj jn 1, '' ' •"• ,,Tr n„,,fc n !n, where fj-i-jn_1j„k{z'), ji = h ■ • •
nn Z1

Zz

d

z * k)dj+j+ EE / M*l> j ( z i ' - -'• i■ •n)Oj, X==EE/-if(r*( i . -' "-" ■<>>=) i Zn)9}'

j=k+l j=k+l

j=i J=l

where f\, ■ ■ , fn are holomorphic functions on D, then where / i , • • • ,f„ are holomorphic functions on D, then k

^'=E^ 1 '"' 'z^ 3=1

defines a complete holomorphic vector field on the domain D' in Ck given as the image of the domain D under the projection Cn B {z\, ■ • • ,Zn) i—> {z\, • ■ • ,Zk) & Ck

The proof of this lemma is straightforward, and is omitted. In the notation of Lemma 4, we say that X' is induced by X. In Lemmas 5, 6, 7 and 8 below, we shall assume that Tn is a tube domain in C n whose base has the convex hull containing no complete straight lines. We observe t h a t g ( T n ) n v c : T g ( T n ) = {0}.

209 L e m m a 5. If a complete holomorphic vector field X on TQ is of the form nn

X

ajdj, i=i

where « i , ■ • ■ , Q „ are complex constants, then atj 6 R for all j = 1, • • • , n. Proof. The element X can be written in the form X = Y + V^IZ, {
where Y,Z e

z = V = I ( y - X) e g(Tfi) n 7 = 1 0 ( T n ) = {0}, and therefore that X = Y e {b\, ■ ■ ■ , 9 n }a- This proves our assertion. D L e m m a 6. If a complete holomorphic vector field X on TQ is of the form X ■-

n

■ ,Zk)dj, j=k+l

then fj{z\,■■■ ,zk), j — k + 1,• -• ,n, are reai constants. Proof. In view of the assumption that fj(z\,- ■■ ,Zk),j = k + 1, • • • , n, are holo­ morphic on TQ, it is enough to show that they take only real values on TQ. TO see this, let £ = (&, •• • , Cn) be any point of TQ and write £' = (Ci, • • ■ , 00- Consider a tube domain TQ« in Cn~k given as a connected component of the open subset {z" €. (jn-k I (£/( z //) g T n } Qf Cn-k S i n c e t h e v e c t o r fidd X = Y™=k+r fj(zi, ■■■ , Zk)dj is tangent to {£'} x TQ« C TQ at each point of {£'} x TQ« and since it is complete, it follows that the restriction of X to {£'} x TQ» induces a complete holomorphic vector field X" on TQ« of the form n

*"= E

/i(Ci,- • ,Ct)a,-.

j=k+i

On the other hand, since the base fi of TQ has the convex hull containing no complete straight lines, so does the base Q" of TQ». Thus, by Lemma 5, we have /)■(£) =

/jCCi, • • •, Ck) e Rforall j = k +1, • • • ,n. □ The following Lemmas 7 and 8 contain basic technics in our study. L e m m a 7. If a complete holomorphic vector field X on TQ is of the form

X = £ (aie**l+JfS,» + /? j e A z i -^ 2 2 ) a,, 3=1

(3.2)

210 where OLJ,fij,j — 1, ■ - ■ , n, are complex constants and A is a non-zero real constant, and if oti = 0:2 = 0 or Pi = 02 = 0, then X = 0. Proof. It is enough to prove the lemma in the case where 0\ = 02 = 0. In fact, if ce\ = £*2 = 0, then, by a change of coordinates C " 3 (zi, Z 2 , Z3, ■ ■ ■ , Zn) I

► (Zl, - 2 2 , Z 3 , ■ • - , Zn) 6 C " ,

we can reduce our problem to the case where 0\= 02 = 0. Now, to prove that X = 0 in the case where 0\ = 02 = 0, we show first that /fit,- = 0 for all j = 3, • - • , n. Suppose contrarily that 0j> ^ 0 for some j ' > 3 and write Y = [X,[ch,X]]. Since X € g(T fi ) and b\ € g(TpJ, we have F G g(T n ). On the other hand, a straightforward computation yields that Y is of the form 2 £ 5 / J , ( - 2 A a i + V=Ia 2 )e 2 A z i a^.

Y = £ J=3

^

Thus, by Lemma 6, the coefficients ~10j(-2\cn 2

+ V=la2)e2Xz\

j = 3, ■ ■ - , n,

of Y must be constant, so that ^ / ^ ( _ 2 A a 1 + v/=Ta2)=0,

j = 3,---,n.

Since 0j> ^ 0, this implies that —2Aaj + \/—1<*2 = 0, or AQI = ( V ^ - T / 2 ) Q 2 - AS a consequence, we have a i a 2 ^ 0. Indeed, otherwise, we see from the relation Aai = (%/—I/2)a2 that a\ = a-i = 0. Again by Lemma 6, the coefficients

e ^^ ^ + ^ ^ + /^^ee ^ '--^^F2 *^,, a^je■Xzi+

j = 3.• 3. • • •• ,«, ,n,

of X must be constant, so that a3 = Pj=0,

j = 3,--,n.

This contradicts to the assumption /fy ^ 0, and therefore 0:10:2 ^ 0. If TQI is a tube domain in C 2 given as the image of the domain TQ under the projection C" 3 (zi, 2 2 , • - • , 2„) >—* (21, 2 2 ) € C 2 ,

then, by Lemma 4, Jf induces a complete holomorphic vector field X' on TQ> of the form 2 X'-X' == aa i^e ^' '++ ^^ a*2!3i ++ c&s**' a 2 e A z +i + ^^ l ;2 " 9^2 .

211 Consider an integral curve z'{t) = (zi(t), 22(f)) of the holomorphic vector field X'. Since X' is complete, z'(t) is defined for all t 6 R, and its components z\ (t) and 22(f) satisfy the system of the ordinary differential equations on R given by • dzi{t) = a I e A *«W +J f I 2!iW ! dt < dz2(t) =: a2eAa^e^^+^^l *'« \ dt

(3 3)

Hence, if we write
(3.4)

on R, where C = 2AQI ^ 0. This implies that, for every point (Cii C2) of Th<, the equation (3.4) has a solution ip(t) defined on R with the initial condition <^(0) = ACi + (V—T/2)C2- Since Tfj- is a tube domain, that is, Th< is stable under real translations, the value of Aft + (■>/—T/2K2 may take all complex numbers. But, as observed in Section 2, if ip(t) is a solution of (3.4) with e - 1 ^ 0 ' g { C ^ R , then it is not defined for t = tT*®)fC. Thus we have a contradiction, and our assertion is proved. Since, by assumption and what we have shown above, 0j = 0 for all j = 1, • • • , n, lation it follows from the relation

[^X] =

2

^\±(aie^+^-PJe^- 4

1:

(;

1

*w

that [^2, X] = (y/=l/2)X, or X = - 2 V ^ I [ 3 2 , X]. Therefore we have X € g(T n ) n v / ^ T g ^ n ) . The relation g(Tn) n v / z To(Tn) = {0} implies that X = 0. Q L e m m a 8. If a complete holomorphic vector Geld X on TQ is of the form (3.2), and ifai = 0\ = 0 or ai = 02 = 0, then X = 0. Proof. Suppose contrarily that X / 0 . Consider first the case where 02 = 02 = 0. Then, by Lemma 7, we have oq/Ji / 0. Applying, if necessary, a change of coordinates C " B (21, Z2, 23, • • ■ , Zn) I

► (21, 22 + C, 23, • • ■ , Zn) £ C n

for some e G C, we may assume that ot\ = 0\. Now, if TQI is a tube domain in C 2 given as the image of the domain TQ under the projection C " 3 ( z l t 2 2 , . - • , * ) > — («i.22) e c 2 ,

212 then, by Lemma 4, X induces a complete holomorphic vector field X' on TQI of the form

X' = ai (e A z i + V ? Z 2 + e A z i -*r^) ftTake a point (Ci i C2) of TQI with C2 £ R and, for each £ e R, write

£>£ = {*! eC|(2i,<2 + 0ern'}. By the definition of tube domains, we have D'c = D'Q for every £ € R- Hence, if T(aj,) is a tube domain in C given as a connected component of the open subset D'0 of C, then it is also a connected component of D'c for every £ G R. Since the vector field X' = a i ^e A z i + ( v / = T / 2 ) 2 2 + ^ - ( v ^ T / a ) ^ 5 l i s t a n g e n t t 0 y ( a 6 ) x {£2 + £} c 2Q< at each point of TUM X {C2 + £} and since it is complete, it follows that the restriction of X' to T(a j) x {£2 + £} induces a complete holomorphic vector field Y on T(0it) of the form y = Q l (e^+^F^+f) + eA*-^(&+«)) 9 l =

cn(j+-]eXzid1,

where 7 = e ( ^ / 2 X 6 > + 0 . Note that Y must be complete on T(ab) for all 7 6 C with I7I = e-'1/2)1™1^ j4 1. As a consequence, T L M is a strip in the complex plane, that is, we have — 00 < a < 6 < +00, because 7 + (I/7) ^ 0. Recalling the example in Section 2, we see that 7 + (1/7) £ { C } R for some non-zero complex constant C. Since this contradicts to the fact that 7 may take all values of absolute value e -(i/2)im< 2 ^ 1 ; w e c o n c l u d e that if a2 = fo = 0, then X = 0. Consider next the case where a i = 0i = 0. An application to X of an argument similar to the preceding case yields that a vector field Y of the form Y=(a'2e^z> + /32e-^=)02 must be complete on a strip T(ab) in the complex plane, where a'2,P'2 are non-zero complex constants. But the example in Section 2 asserts that if a vector field Z of the form k

I \ z = ET^U \i=l

1

is complete on T(a(,), where 7;,uit, i = 1,• ■ ■ , k, are complex constants and Wj, i = 1, • • ■ , k, are mutually distinct, and if 7* ^ 0, then w, e R . Thus we have a contradiction, and this proves that X = 0 in the case where ot\ = /3\ = 0. D

213

4. Proof of Theorem As in Theorem, let Tn be a tube domain in C n whose base Q has the convex hull containing no complete straight lines. We begin with two fundamental lemmas. A set E of complex numbers is said to be closed under complex conjugation if u> e E, then l i g f i . Lemma A. There exist a positive integer d and n sets Ri = {wii,--- ,w iri },

i = !,■•■ ,n,

of mutually distinct complex numbers which are closed under complex conjugation such that every element X ofg(Tn) can be written in the form n

/

ri

r„

>\ W l 2 : iWWI + + u nI+ ; s+ .. „(2)e » '" ' -'" " J E f E ■' ' •Z■^ Efjji-fni- •■*.M« * "• ^" A8-)j . jn=l 3=1 \V i=l j=l j„=i )/' 3l=\

* == E Ex

where fjj,~j„(z), j = l,--- ,n, ji = !,■•■ ,ri,i z\, ■ ■ ■ ,Zn of degree at most d—l. Lemma B. InLemmaA,iffjjl...jn(z)

^ 0 , thenu^,-

a*.

(4.1)c4-1)

= l , - - ,n, are polynomials in ■ ■ ,wnjn are all real numbers.

In this section, we give a proof of Lemma A. The proof of Lemma B is rather long, and will be given in Sections 5 and 6. Proof of Lemma A. For i = 1, ■•• ,n, let Pi be the minimal polynomial of the endomorphism addi of B(TQ). Put d = £™=i(deg Pj - 1) + 1 and, for i = 1, • • • , n, let Ri be the set of mutually distinct roots of Pj. Since Pj is a real polynomial, the set Ri is closed under complex conjugation. Now, let X be any element of fl(Tn). Then X can be written in the form n

X = Ytfj(z)dj, i=l where fi{z), ■ • • , fn{z) are holomorphic functions on TQ. Since n

0 = Pi(addi)X = J2 Pi(di)fj(z)dj, i=l

i = 1, ■ ■ ■ , n,

every function fj{z) satisfies the system of the linear differential equations P «(9i)/j(z)=0, Pi(di)fj(z)=0,

= i» =

!,■■ • ,n. l,---,n.

214 By Lemma 2 of Section 3, we see that /i(z), ■ ■ • , fn{z) are of the form

/*(*) = £ ■ • £ /«i"•i„(2)eUJ"1*1+"'+^*"> Jl=l

i = 1. • • • . n.

Jn=l

where fjji-j„(z), j% = 1, • • ■ , r*i, t = 1, • • • , n, are polynomials in zi, • • ■ , z n of degree at most d— 1. □ Now, Lemma B asserts that we may assume that R% C R, i — 1, • ■ ■ , n. We define a subset R of R n by XX =

XXI X ■ • • X x l ^

and denote by S the subset of R consisting of those elements (wij,, ■ • ■ , wnj„) e i? for which there exists an element X of Q(TQ) such that if X is written in the form (4.1), then fjjv-jn(z) # 0 for some j with 1 < j < n. Note that, since {Si, • • • , dn} C fl(Tn), we have O e S . L e m m a 1. If ui = ( w i j , , - " !wn?„) £ 5 , then there exists a non-zero element Xu e S(TQ) of the form n

Xu = e«u.*+"-+*w.*. £

o %

(4.2)

3=1

where a1, • • • , a n are complex

constants.

Proof. By assumption, there exists an element X € £j(7h) of the form Tl

* =£

/

Ti

£

Tn

• ■ •£

(s=i yjt1=i

\

Afe-• .kn(z)eUlk*Zl+"+""*»*' dk, fe»=j

y

where / ^ . . . ^ ( z ) , k = 1 > ' ' ' i n > fc = 1»'" • i r ii • = h " ■ ■ i n i a r e polynomials in zi, • • ■ , z n of degree at most d - 1 , such that / j j , ...jn (z) ^ 0 for some j with 1 < j < n. We first show that there exists an element X^ € ${TQ) of the form n xW

= £ fk(z)e^+~+""^dk,

(4.3)

fc=i where fk(z), k = 1, • • ■ , n, are polynomials in zi, ■ • • , z„, such that /j(z) / 0. For i = 1, • • • , n, we define a real polynomial Qi by

Qt(z) =

JJ (x - wife), l
215 and set n

x{1)

= J]Q i (oda i ) d x.

Then we have xW(1)

n

/ n

»■„

n

\

+ + k +w d d(fkkl-kn(z)e^^ E n^(^ UQi(di) - ^"" ^) "*"z")) 8*■ k. = E ( E •■■■ E (/^-fc.(2)e"UiZl+

* = fc=i E \JbE ijfc=l \Jfci=X 1=

Jkn=\ f c „ = lti== l

/

Since the degrees of the polynomials fkki--k„(z) are at most d — 1, we see that if km^ jm, then

n^(^)d(/fcfc1-fcm-fc~wea"fci2i+"+wmt" n n

iZm-\

hWnlfc

d ((dmm - ujumk = L ((d -kn(z)eu^ km...kn(z)e^^+--+"^ mkj„i(f / kkl...\Jkki-"km-

= =

I

L

-kfnc(z))e^ (^(/^,.. f c m ... „(^))e (3|(.(/**l-*m

•*") Zl + " +Um k m 2*"+-+<*.» m + " ■+W„fc,-*•))

-))

jZm W l Zl-\ 1 + hWrnt,, +u t z ++-+UI, + w "fcn t z2" ]

*^ - "" - "' -

" " ")

= 0, where L is a differential operator given as a polynomial of d\, di, ■ ■ • ,d,dnn. - This implies that the element X^ of g(Ta) g(!7h) has the form (4.3). The assertion that fdz) fj(z) ^ j= 0 0 follows from the relation fjji—j„(z) / 0 and the fact that if g(z) is a polynomial, then, for a complex constant A with A ^ Wy4, 1 2l+ +tt J z Wl Zi+ +w z ) (s(z)e'" (g(z)eUl**+•••+<-»*.*.) )c<«Wi»i+-+w-M*(di -- AA) '' " '" " ") = fc(zh(z)e h - ™ ",i (di-

where h(z) is a polynomial with deg g(z) = deg h(z). We now show that there exists a non-zero element Xu 6 0(Tn) of the form (4.2). For this, it is enough to show that, for i = 1, ■ ■ ■ , n, if there exists an element Yeg(Tn) of the form

^ = E^Wewl-zl+-+u'"'"2"9fc, Jfc=i

where (^(z),fc= 1, • • ■ , n, are polynomials in Zj, ■ • • , z„, such that some polynomial gk(z) is not zero, then there exists an element Z 6 g(7h) of the form n

2 =

zi+-

•■+W„j„

*"%

k=\

where /i fc (z),fc= 1, ■ • • , n, are polynomials in z,+i, • ■ • , z„, such that some polyno­ mial hk(z) is not zero. Indeed, combined with the result of the preceding paragraph, a repeated application of this fact yields the desired result.

216 Let Y be as in the preceding paragraph and we define a real polynomial Q by Q(x) = x — Wyj.

Let s denote the maximum value of the degrees with respect to z\ of the polynomials gk(z), k = 1, • * • , n, and set Z = Q(addi)sY. Then we have

n

i*i+-

'sT0(diY(ai.(z)euv:

Z--=

■+W-j„

M&

k=l

and

zi+Zn s w ,«!+■ "+»*>/«>M z ■,(z))e^ + -+»»^) == (a? fli (d?gk(z))e»^+-+»^ By element of given by byZ. fl(Tn) isis given By the the definition definition of of the the integer integer s. s, the the desired desired element of g(Tp.) Z. □ a '+W„j„

We write S = {wi,--- ,uT,uT+i,- ■■ ,uiT+q}, where w\, ■ ■ ■ ,u)T are linearly independent over R and, for every t = 1, • • ■ ,q, the vector u>r+t is a linear combination of u}\,*• • , t*v. For i = 1,• ■ ■ ,n, let «j be the element of R™ whose z-th component is equal to 1 and whose components except it are all equal to 0, and take an element A of GL(n, R ) such that t

Aui = Uj,

where lA denotes the transpose of A. element of GL(n, R) K C n given by

i = l,--- ,r, After a change of coordinates under the

C n 9 2 1 ► «; € C", C n 3 z — w € C",

we may assume that

we may assume that S-

< ui,• ■ • ,UT,^2^im,■••

, y ] \ m

I

!=1

t=l

>,

J

where At;, £ = !,••• ,q, i - !,■■■ ,r, are real constants. By Lemma 1, for i = 1,•*• , r, there exists a non-zero element Xu, e g(Tp.) of the form

■X-Ui

= <**£«&. j=i

where of, j = 1, ■ ■ • , n, are complex constants. For brevity, we write Xt = XUi.

217 4-1. The vector fields Xi In this subsection, we study the form of the vector fields Xi. We first determine the constants a\. Lemma 2. For every i = 1, • • • , r, the constant a\ is not zero. Proof. Suppose contrarily that a ' = 0. Then, by Lemma 6 of Section 3, the coeffi­ cients oP{eZi, j = l,--- , t - l , t + l , ••■ ,n, of X, must be constant, so that af=0,

j = 1, ••• , i - l , » + l, ••• , n .

This contradicts the relation Xi=£0. D For i = 1, • ■ • , n, we denote by zz>i the projection of C™ onto C given by T&i{Zi, • • • , Zi, • • • , ZJI) = Zi-

Note that W^TQ) is a tube domain in C. By Lemma 4 of Section 3, for every i = 1, ■ • • ,r, the vector field Xi induces a complete holomorphic vector field on roj(Tn) of the form a\ez'di. Since a\ / 0 by Lemma 2, tUj(Tn) is a strip in the complex plane, and the example in Section 2 implies that roj(Tn) = T(a.:ai+n) f° r some a,i € R. After a change of coordinates Cn3{zi,--,Zn)>—►(toi,--- , t % ) e C , ( Wi = Zi- V-lau i = 1, • • • ,t-, \ Wi = z,, i = r + 1, ••• ,n, we may assume that ro^To.) = T(0i7r) and a ' e R for all i = 1, ■ • ■ , r. By writing Xj as ( 1 / Q | ) X J , we have, for i = 1, • • • , r,

n Xi = e ^ 4 ^

«! = !■

J'=i

We now determine the constants oPt, j j= i. Lemma 3 . It 1 < i, j < r and i ^ j , then a? = 0. Proof. Suppose contrarily that a?{' ^ 0 for i, j with 1 < i,j < r and i / j . We denote by Wij the projection of C n onto C 2 given by w«(iti,--- ,zn) = (zi,Zj).

218 Then w»j(TpJ is a tube domain in C 2 . Since wy(Tn) C Wj(ToJ x Wj{Tn), we see that wy(Tfi) C T (0 ^) x T(0jW). By Lemma 4 of Section 3, X, and Xj induce, respectively, complete holomorphic vector fields X[ and X'- on vaij(Tn) of the form

X'i^e^idi

+ ^dj)

and Xj = ez'{dj + aft).

We divide into the two cases of a{ £ R and of G R. Suppose first that oPi £ R. Consider an integral curve z'(t) = (zj(t),2:j(i)) of the holomorphic vector field X[. Since X[ is complete, z!(t) is defined for all t € R, and its components Zi(t) and 0j(t) satisfy the system of the ordinary differential equations on R given by dzjjt) Ml

dt dt dzj(t) ^ dt

== „*(*), e-M,

(4.4)

= 4e*W. 4e»®.

(4.5)

It follows from (4.4) that C -*M

= _ t + e - z '(°)

for alH G R.

Hence, if we write Zj(£) = Xj(t) + \/—Ti/i(£), where Xj(t),j/i(t) are real-valued func­ tions, then (0,7r) = {&{£) \ t € R } and, for all t € R, Xi(t) = \ogsmyi(t)

+ Xi(0) - l o g s i n ^ O ) .

(4.6)

On the other hand, substituting (4.4) into (4.5), we have the differential equation d(zj(t) - qfc(t)) _ dt

Q

on R, and consequently, Zj(t) is given by Zj{t) = afziit) - ofzi(0) + Zj(0).

(4.7)

Let j/j(t) be the function given as the imaginary part of Zj(t) and write a{ = € + V^T'7, where £,r/ € R. Then, using (4.6) and (4.7), we see that, for all * € R, Vj(t) = j/logsin j/i(t) + tyi(t) + C,

(4.8)

where C is a constant. Since n7y(7h) c r (0)W ) x T(0)7r), so that 0 < %■(*) < TT, and since logsinj/i(t) -> - o o as j/j(t) tends to 0 or ir, (4.8) implies that ^ = 0. This contradicts to the assumption (x{ ^ R.

219 Suppose next that a - G R . After a change of coordinates $ : C 2 3 (zit ZJ) i—► («/,-, Wj) € C 2 , J 2i = Wi, \ Zj =

O^Wi+Wj,

X[ and X'j are complete holomorphic vector fields on $(n7y(7h)) of the form X'i =

e^du

^ = e ^

+

^ ( a ^ + ( l - a ^ ) .

(4.9)

Since o^ € R and since ro^To.) C T(0 T) x T ^ j , it follows that (mij(Tfi)) is a tube domain in C 2 contained in T(07r) x C. We show that $(roy(Tn)) = T(0|ff\ x T(a 6) for some a, b with —oo < a < 6 < +oo. Let Q' be the domain in R 2 given as the base of the tube domain $ ( w y (Xp.)). It is enough to show that Q' = (0, n) x (a, 6) for some a, 6 with - o o < a < b < +oo. Consider an integral curve w'(t) = (wi(t),Wj(t)) of X[. Then u>i(t) and t«j(i) satisfy the system of the ordinary differential equations on R given by dw i(t) _ ewM dt (4.10) dwj(t) 0. dt In view of the observation in the preceding paragraph, it follows from (4.10) that if Vi(t) and Vj(t) are the functions given as the imaginary parts of u>i(t) and Wj(t), respectively, then (0,7r) = {vi(t)\t € R } and Vjft) = Vj(Q) for all t € R. This implies that if (vi,Vj) £ ft', then

{(?,»,-) l€€(o,>r)} e n * . Hence we have ft' = (0,7r) x (a, b) for some a, 6 with — oo < a < 6 < +oo. Since the base of the tube domain vDij{To) is contained in (0,7r) x (0,7r), so that Q' is bounded in R 2 , we obtain —oo < o < b < +oo, and our assertion is proved. It is known that g(T(0,ff) x T(a,b)) l s given as the direct sum of g{T(0„)) and 0(X(a,&))- In view of the example in Section 2, we see that B(*(w«Crn))) =

fl(%,,xrW))

= {dit dj,eWidi, e-Widi, Ce^'dj,

C r V ^ S ^ ,

where c = ir/(b - a) and C = ei-V^i^/ib-a) element of g($(mj S i n c e x>, i s m (Tn))) of the form (4.9) and since of / 0 by assumption, we must have X'j = 0. This is a contradiction, and the lemma is proved. Q

220 By Lemma 3, for i = 1, • • • , r, we have n

/

x2 =

= eZi

\

hi


Write c*j = £f + V^af, change of coordinates

where £f, of e R, i = 1, • • • , r, j = r + 1, ■ • • , n. After a

C* 9 <*!,■••,*»)'—»(«*>••• , w „ ) e C n , j Wi = Zi, i = l,--- ,r, I

w

i = E j = i ( - Q ) ^ + z''

i = r + 1,-■ ■ , n ,

Xj, i = 1, ■ ■ • , r, have the form (

Xl = eZi\di+ \

n

J2 V-lefa j=r+l

\

<

i = l , . . • ,»".

J

^.2. The vector fields In this subsection, we study the form of the vector fields Xu for which w ^ 0,«j, • • •

,ur.

L e m m a 4. If u 6 5 and w ^ {0,ui,- ■ • , u,-}, then w G {-ui, • • • , — fr}. Proof. We denote by WQ the projection of C n onto C given by O7o(zi,--- , 2 n ) = (^l,--- , Z r ) .

Then G7o(Th) is a tube domain in C Since ^ ( T h ) C roi(Tn) x ■■• xTO,.(To.), we see that VSJ0(TQ) C (T(0i7r))r We show that w0{Tn) = (T( 0 l r ) ) r - Let Q0 be the domain in R r given as the base of the tube domain WO(TQ). It is enough to show that fio = (0,7r)r By Lemma 4 of Section 3, for every i = 1, • • • , r, the vector field Xi induces a complete holomorphic vector field onTOO(To,) of the form e^di. A consideration of integral curves of these vector fields eZid\, ■ ■ ■ , e*rdr yields that, for every i = I, ■ ■ ■ ,r, if (yi, ■ ■ ■ ,yr) € Sl0, then {(yi,---

,yi-i,t,Vi+i,---

, i f r ) | { £ (O,TT)} c $V

Hence, if we fix a point (yi, ■ • • , yr) of fio, then we obtain inductively that (0,7r)' x {(j/i+i,• • - ,yr)} c O 0 and our assertion is proved.

fori = l,--- ,r,

221

Now, suppose that w € S and w $ {0,«i,--- ,Ur) and write u = £[=i"^u*> where A* € R, i = 1, • • • , r. Lemma 1 asserts that there exists a non-zero element Xu £ fl(Tn) of the form n n

XiZi+ XlZi++x+x

-A^j

Xul = 1

ee

- - ^Y^ 3' ^Y/°0ec'd3' i=i

n

where a , • • • , a are complex constants. By Lemma 4 of Section 3, X^ induces a complete holomorphic vector field on roo(Th) of the form r

e A l * l + - + A " z -£V0j. On the other hand, we know from the result of the preceding paragraph that s(wo(rn))=fl((T (0llr )) r ) = {dlr--,dr,

e z 'di, e-^dt, • • • , e^dr,

e~^dr}K.

Therefore we must have (Ai,--- ,Ar) € {0,ui,-ui,---

,u'r,-u'r},

where, for i = 1,- •• ,r, we denote by uj the element of R r whose i-th compo­ nent is equal to 1 and whose components except it are all equal to 0. Since u) = (Ai,■•• ,A r ,0,••• ,0) and since ui ^ {0,ui,--- ,u r }, we conclude that w €

{-ui, • ■ •, -ur}. a

To see that S = {0,u\, —u\, ■ ■ ■ ,Ur, —«r}, we need the following lemma. Lemma 5. For every i = 1,■-■ ,r, if (zi, • ■ • , Zj,- ■ ■ , z„) € Tn, then (zi,--- , — Zj + V^TTT,--- ,z„)6Tn. Proof. TO see that (zi, • ■ ■ , - z , + v 7 - ^ , • • • , Zn) € Tn, it is enough to show that (l/i.--- ,-S/i+w, ••• ,j/n) € fi, where j/i,--- , yn are the imaginary parts of z\, ■ ■■ ,z„, respectively. Consider the integral curve (zi(t), ■•■ ,zn(t)) of Xj with (zi(0), • ■ ■ , Zn(Q)) = (*!,■•• ,ZT,). Then zi (£),••■ , z„(t) satisfy the system of the ordinary differential equations on R given by rfZj(t) dzj (t)

~dT~ dzjjt) dt

e*»,

(4.11)

0,

(4.12)

j = 1,--- , i - 1,2-1-1,-■ ,r,

v/^Tof'e^W,

j = r + l , - - - ,n.

(4.13)

222 For j = 1, • • • ,n, write zj(t) = Xj(t) + ^/ ^ T2/j(t), where Xj{t), yj(t) are real-valued functions. Since, by (4.11), e-*W _I

= - t + e- 2<(0)

for all t e R,

0)

we see that if to = 2e <( cosyi(O), then zi(t0) = xi(0) + yf^l(ir-yi(0)).

(4.14)

Also, by (4.12), we have, for j = 1, ■ • ■ ,i - 1,i + 1, • ■ • ,r, Zj(t)

= ZJ{0) for all t 6 R,

(4.15)

while it follows from (4.11) and (4.13) that, for j = r + 1, • •• ,n,

2j(t) = V^Io^C*) - V-\4zi{°) + zi(°)

for a11 e R

'

>

and hence that, for j — r + 1, ■ • • , n, 2 / J W = a f ( x t ( t ) - x i ( 0 ) ) + 2/J(0)

forallieR.

(4.16)

Now, (4.14) implies that Xi(t0) = xi(0),

(4.17)

2/i(to) = -J/i(0)+7r.

(4.18)

By (4.15), we have yj(k)=yj{0),

j = l,---,i-l,i+l,---,r.

(4.19)

Substituting (4.17) into (4.16), we see that ty(*o) = W(0),

j = r + 1, • • • , n.

(4.20)

Since (zi(to), • • - , z„(tQ)) e T n , so that (j/i(t 0 ), • • • , yn(*o)) € ^ , and since (yi, ■ • • , 2/n) = (yi(0),--- >Vn(0)), we conclude from (4.18), (4.19) and (4.20) that (j/i,--- , -Vi + K,"' ,Vn) e H. D For i = 1, • ■ • , r, we define an automorphism TJ of C " by

n : C n B (*i, ■•• ,z„) •—> (wh ■■■ , w n ) e Cn,

{

Wi = -z% + \ / - 1 T , WJ = ZJ,

j = !,-■■ ,t — l , i + 1, - - - ,n.

Note that r " 1 = r,-. Lemma 5 asserts that T,(TOJ C TO. for every i = l , - - - , r . Therefore n , • ■ • , r r gives automorphisms of To.. For i = 1, • ■ ■ ,r, set Yi = (ri)*Xu where ( n ) , denotes the differential of Tj. Then Yi, z = 1, • • • , r, are elements of g(Tp,), and have the form

Yi = e-*L-

J2 S=laidA, i = l,-..,r.

223 Lemma 6. For every i = 1, ■ • ■ , r, if X is an element of 3 (To.) of the form X = ztZx Z)j=i o^dj, where e = ± 1 and a 1 , • ■ ■ , a " are complex constants, then a1 € R. Moreover, X = a'Xi when e = 1, white A" = a'Yj when e = - 1 . Proof. By Lemma 4 of Section 3, X induces a complete holomorphic vector field on t*7j(Tfi) of the form a ' e " ' ^ . Since OTi(Tp.) = T(0)„), the example in Section 2 implies that a1 € R. If e = 1, then X - a ' X t is an element of g(T n ) of the form e 2 ' £ ? = i £ j d.j, where / ? \ ■ • • , / T are complex constants and /?' = 0. By Lemma 6 of Section 3, the coefficients /JV',

; = 1,... , » - i ^ + i,... , n ,

of X — a ' X ; must be constant, so that 0>=O,

j = l, ••• , 4 - 1 , 1 + 1 , . . . , n .

Therefore we have X — a'Xj = 0, or X = alXi. X = a'Yi when e = - 1 . D 4.3. Vector fields in Q(TQ) with non-polynomial

A similar argument shows that

coefficients

For i = 1, • • ■ , r, consider an element X e g(To.) of the form n

X = e<*Y,fj(z)dj,

(4.21)

j=l

where e = ± 1 and / i ( z ) , • ■ ■ , / n ( z ) are polynomials in z\, ■ ■ ■ ,zn. In this subsection, we show that fi{z), ■ ■ ■ , f„{z) must be constants. We begin with the case where they are affine functions. L e m m a 7. If X is an element of Q(TQ) of the form (4.21) and ifdegfj(z) every j = 1, • • • ,n, then fi{z), ■ ■ ■ , fn(z) o,re constants.

< 1 for

Proof. Our assertion for the case of e = —1 follows from that for the case of e = 1 by applying the change of coordinates by 7-;. If i 7^ 1, then, by permuting the coor­ dinates z\ and Zj, we can reduce our problem to the case of i = 1. Hence we may assume without loss of generality that e = 1 and i = 1. For the convenience of com­ putation, we write X\ = ez' YTj=\ <**dj, where a1, ■ ■ ■ ,an are complex constants. Note that a 1 = 1 # 0. Since deg f\ (z) < 1 by assumption, we have n

h(z) = ^2ikZk +P\ k=i k=\

224 where 71, • • ■ , 7 B , / ? 1 are complex constants. We show that 71, • • • , "in are all real numbers and that /j(z), j = 2, - • ■ , n, are given by

(!H

fj{z) = cP rf$>*** \Y^lkzk\ />(*)

fP, + ^>

i = ]=%■■,n, 2,- • ,n,

for some complex constants /3 2 ,- ■ ■ ,/3". Write Z\ = [d\,X] — X. 2, • • • , n, write 2j = [di, X]. Then we have

Also, for t =

n

d d d}' Zi=e*>J2 ifjW Zi=e*>J2difjW }'

» == 1! ., — " .»• ■

?=i

,

«

(4.22) (4-22)

•

.7=1

Since, by assumption, deg/j(z) < 1 for every j = 1, • ■ • , n, the polynomials difj(z), i,j = 1, ■ ■ ■ , n, are complex constants. Therefore, noting that difi(z) = 7i, i = 1, • • • , n, we see by Lemma 6 that 7, e R, i = 1, • - • , n, and n n

Zi = TiXi = eZl J2 W % .

i = 1, ■ •• -,n. , n. *=!,-•

(4.23) (4.23)

J =I

It follows from (4.22) and (4.23) that, for every j = 2, • ■ • , n, d«/j(z) = ot~fr,

i = 1, • • • , n.

This implies that

( "

\

\fc=i

/

/,-(*) =oJ : X I W +^'. J = 2 --•

,«,

for some complex constants 02, ■ ■ ■ , /?" By the result of the preceding paragraph, to see that f\{z), ■■■ , fn(z) are con­ stants, it is enough to prove that 7t = 0, k = 1, - • • , n. We first show that n

'%2<xilj = 0,

(4.24)

J=I

//(*) = ^ r / i ( 2 ) -

i = 2, • ■ ■ , n.

(4.25)

A straightforward computation yields that n n

[X [ X1;1,X] , X ]== e 2 z ' ^ 9 > ( 2 ) a j > 3=1

(4.26)

225

where gj{z), j — 1, • • • , n, are the polynomials given by att

9i(=)

£

otklk,

fc=l n

§M = ^fj{z)

- a>Mz) + o> £ "Sit-

3 = 2, • • • , n.

*=i

If gj(z) ^ 0 for some j with 1 < j < n, then we see from (4.26) that 2«i e 5. This contradicts to Lemma 4. Therefore we have gj(z) = 0 for all j = 1, • • • ,n. This implies (4.24) and (4.25). We now prove that 7^ = 0, k = 1, ■ • • , n. By (4.25), X has the form

X=e-fl{z)£^dj. Consider an integral curve z(t) = (21 (<),-•• , z„(t)) of X. Then 2i(t),••• , Zn(t) satisfy the system of the ordinary differential equations on R given by

^=f-('ViWt)),

(4.27)

*SM = Hje*«AM«)>. j=2,...,n-

(4.28)

or a' Substituting (4.27) into (4.28). we have the differential equations < ( , ,(0

-A

-_a t * i(0)i =-0n,

j = 2,.-,n,

- u>

<«

on R, and consequently, for j = 2, • • • , n, the function Zj(t) is given by ZjLt) = ^ ( ^ i ( 0 - ai<0» + *j(0).

(4.29)

Using (4.29) together with the obvious relation

Sl(t)«.4l=»W-5l
AW*))

n

1

£T*Y^JtW + Z? Jt=l n

£">»

nn

H(=i(0-*J :(0)) 4-2>k*fc(o)+^

- * ■ Q1

(IH \^

it=i

1

(-1(0 - = i ( 0 ) ) + ^7fc2fc(0) + /3 ,

226 and hence, by (4.24), n

1 h(z(t))- = S>k*fc(O)+0 . n

(4.30)

k=l

k=\ (4.27) and (4.30) imply that Zl(t) satisfies

(4.27) and (4.30) imply that z\(t) satisfies dzijt) = (E7^fc(0)+/? X dt \fc=l

e*W.

>

It follows from this relation that e

- z ' « = -Ct + e~ZlW

for all t 6 R,

(4.31)

where C = £ £ = 1 7**fc(0)+/?1. We show that 7^ = 0, fc = 2, • • ■ , n. Suppose contrarily that 7 ^ / 0 for some fco with 2
{C e CI (*i(0), ■ • • ,vi(0).C,%+i(0), • ■ ■ ,zn(0)) e r n } of C. Since 7 ^ is a non-zero real constant, 1

' ' 7fc„C 7fco'

' l1

+ 52 7***(0)+/?3 C C « T(a,b) 6) \
'

is a tube domain in C, which we denote by Tjy^j, and (4.31) holds for all C e Tiai m. Note that, in (4.31), z\(0) e T(0|?r), and hence we have e~* l(0) = £ + V ^ b ? , where 77 < 0. Take an element \T^\v of T(0/ (,-) for which v ^ 0 and let Co be the element of Tfaip) given by Co = (£/??)« + V ^ l « . If to = rj/v, then - C 0 t o + e - * 1 ^ = 0, so that z\(t) is not defined for t = to- This is a contradiction, and our assertion is proved. We show that 71 = 0. The result of the preceding paragraph implies that X has the form

= e-(7i^1 + / ? 1 ) E ^ r By Lemma 4 of Section 3, X induces a complete holomorphic vector field X' on T(0]7r) of the form

X' = (l\z1 + p1)ez*d1.

227

Since the example in Section 2 shows that X' e{di, ez'di,e- "2iai}R, it follows that 71 = 0, and this completes the proof of the lemma. □ We now prove that if X is an element of g(TnJ of the form (4.21), then /i(z), • • * > fn(z) are constants. Suppose the contrary, or that deg fj0(z) > 1 for some jo with 1 < jo < n. It is verified that there exists a polynomial n

P(xi,---,a: n ) = J ] x ™ , i=l

where mi, ■ • ■ ,m„ are non-negative integers, such that if 9j{z) = P(di, ■■■ ,dn)fj{z),

3 = 1,• • ■ , n ,

then deggj(z) < 1 for every j = !,■■• ,n and degjte(2) = 1 for some j'0 with 1 < Jo — n- We define a real polynomial Q by

Q(x1,---,x„) = ( a ; i -ir i n a : rThen we have n n

ZlZl P d Q(add\,Q(adc\, ■■■ , add > ((di
= e 2l Eft (*)»»• i=i

Since Q(addi, ■ ■ ■ ,addn)X € g(Tn) and since deg9^(2) < 1 for every j = 1, ■ • • , n, it follows from Lemma 7 that gi(z), ■ • ■ ,gn{z) must be constants. This contradicts to the relation deg (^(z) = 1. We thus conclude that /i(z), ■ • • , fn(z) are constants. 44. Structure of Q(TQ.) We have shown that, after a change of coordinates by an element of GL(n, R) K C n , we have r

£{Xi,**}RCfl(Ta). Write p = {X € g(Tn) IX is a polynomial vector field}, rr

c

= £ { * ; , FJR. t=i

228 It is obvious that p n e = {0}. To complete the proof of our theorem, it re­ mains to prove that Q{TQ) = p + e. Let X be any element of S(TQ). Since 5 = {0, «i, — «ii • ■ ■ i "r, —Wr}. the vector field X can be written in the form n

x

/ T

r

\

e

= j=iE \i=i (E/*(*>«*+Y,9fiW ~* + >id/ dj> i=i

where fji(z), gji(z),hj(z),

h

j = 1, • • • , n, i = 1, • • ■ , r, are polynomials.

L e m m a 8. The polynomials fji(z), gji{z), j — 1, • ■ • ,n, i = 1, • • • , r, are constants. Proof. We show that fji(z),j = l , - - ,n, i = l,--- ,r, are constants. A similar argument shows that 1 for some JQ,IQ with 1 < jo < «, 1 < io < r. We define a real polynomial P by P(x) = xd, where d is the integer given in Lemma A, and set A"W =

P(addio)X.

Then we have XW

= E (/fWe*0 +»f We"*0) % .7=1

J

where /■ (z),
= deg^z),

j = 1, • ■ ■ ,n.

Furthermore, we define a real polynomial Q by Q(x) = (x + l ) d and set X& =

Q(addl0)xW.

Then we have

*(2) = E/f(*)eM, J'=I ;

where / j (z), j = 1, • • • , n, are polynomials with deg/f(z) = deg/«(z),

j =

l,--,n.

Since A"(2> e g(Tn.) and since d e g ^ ( z ) = d e g / « ( z ) = deg/ J o i o (z) > 1, this contradicts the result of the preceding subsection, and the lemma is proved.

□

229 By Lemma 8, X has the form nn

xX -=E

/ r

r

\

e z* + h*>(*)]$

E ^ '++E£/%««^ ~ ' + >W a,-. a

j=l \ l = l

t=l

/

where a.ji,Pji,j = l,--- ,n, i = 1, ■■• ,r, are complex constants and hj(z), j = 1, ■ • • , n, are polynomials. For i = 1, •■ • , r, write x f = e* £ ? = 1 o^d, and Y? = €~--Z,

E?=■4 0*dj-

Lemma 9. The vector fields X*, Y{ , i = 1, • ■ • , r, are in g (To.). Proo/. We show that X? e fl(Tn), i = 1,--- ,r. A similar argument shows that Y? € g(Tn), i = 1, • • • ,r. We define real polynomials P and Q by P(x) = x + 1 and Q(a;) = xd, where d is the integer given in Lemma A. For every i = 1, • • • , r, we have P{addi)Q{add,)X P(addi)Q(addt)X

2 d === P(add P{add j ]E T (o^e 0j )] t) l) 1 (-- l ) e-^^ e - « ) % (ajie2-' ++ (-l)%

n n

-a,. = 2 2 ^ ocjt "a,. 1=1 i=i

Since P(addi)Q(addi)X

€ s(7h), we see that x f e fl(Tn). D

It follows from Lemmas 6 and 9 that an, Pa, i = 1, • • ■ , r, are real constants and A J^ — PiiY%, Xfj — = QfjjAj, auXu Yf = PuYu

This implies that ^ [ = 1 X- + E i = i ^ n

= *X -$>{*)% j ( * ) d j == E»

ii = = ll ,, -. .. •. , ,r. r.

£ e. As a consequence, we have

( E ^ l + E ^ G€fl(T„). fl(Th)\i=l

i=l

/

Since h\{z),-- ■ ,hn(z) are polynomials, we see that ^™ = 1 hj(z)dj € p. We thus conclude that X is given as the sum of an element of p and an element of e, and the proof of our theorem is completed. 5. Proof of Lemma B Let X be an element of fl(Th) of the form (4.1). Suppose that fjji-jn{z) / 0 and some element among WIJJ , • • ■ , uinjn is not a real number. Under this hypothesis, we shall derive a contradiction. For each index ji, i = 1, • • • , n, we define an index ji € {1, • • • , n } as follows. Consider the element w ^ of Ri. Since Ri is closed under complex conjugation, we have Uijl e Ri. Since the elements of Ri are mutually distinct, there is a unique element u^ of Ri with wij~ = wa,. We set ji = k.

230 Lemma 1. There exists an element X « e s ( T h ) of the form

*«=£( *(1) = E ( E

+ + ■• E^tu*)^ ''- ^**W ■■ Ej^i^^-^A^

it=i (w == i i j■ i

<">

/

kn=jnj*

where / ^ . . . ^ ( z ) , & = 1, ■ • ■ , " , h = ju'ju i = 1,•-- , n , are polynomials in *i, ■•■ , «a, such that fV...jn{z)

^ 0.

Proo/. For i = 1, ■ ■ ■ , n, we define a real polynomial Q, by

Qi(x)=

n

(z--^ifc),

i
n

J O : = nJjQi(adai)dX. j=l

Then we have

(1)

T" n \ » / n a dz ewiii2i+ a Wtti 1+ +w + ,inz z = E 1 E ■■ E E[ftW) ) |a* ••• E n*( ') (/»fc»*.w« ^ 1 -fc»( ) * " ' "^ " ") *--

** = Jfc=l £ \ f a£= l• (1)

fc=i

\Aj«i

n

fc>=l i = l k„=li=l

/)

Since the degrees of the polynomials fkki-kSz) "-m ? Jmi Jmi then

f[Qi(di)d(fkkl- * . . ^ y ^ - H -

■+w m *„ , z m

axe a

* most d — 1, we see that if

+ - ■■+w„t„

ii=l =l

*•)

d z 2 + . uf c.„W . . te(Wzl k)' 2el +w- +"w.— '™ + -+ -^+ ^w "'=» - ^ z^")) )) =L i ((d ((9m "-w Um tmm )i dJ(A(f c/l ... m t t lf c.m....

Zr „ ++-- + w „ t . - ) Wl ( 2 i + - l+•+'»>mk (a^(/ tifcl ...fc ...ik.(z))c^*i* "-- h *-»«» -"-^*-*•) =L i (^(/fcit,. " "fern m.fc„(^))e m

= 0, where L is a differential operator given as a polynomial of d\, ■ ■ ■ ,dn. This implies that the element A"(1) of g(Tn) has the form (5.1). The assertion that ft} , (z) ^ 0 JJ\"'Jn » '

'

follows from the relation fjj^-^iz) / 0 and the fact that iffc*e {ji,ji}< * = 1» ■ • ■ ,«i and if
where h{z) is a polynomial with deg g{z) = deg h{z).

a

231 Lemma 2. There exists an element X^ €fl(Tp.)of the form

* ( 2 ) = E ( E _ •■■ E fc=l

\kl=ji,jl

•ftfa"*.^*+~***"*■) aftl fc,=jn,jn

/

where 'ykk1-k„, k = 1, ■•• ,n, ki = ji,jj, i = 1, ■ ■ ■ ,n, are complex constants, such that J??t...& / 0 for some indices j ' , j{, • • • , j'n with 1 < f < n and j[ € {jiji}, i = !,-•• ,n. Proof. It is enough to show that, for i = 1, ■ • ■ , n, if there exists an element Y € fl(Tn) of the form n

yy =

f

E[ "£ fc=l

lt 2l+ 1 +w 1

z

N1

E ••• E to1-^W^ ' *- *"^"'"'" "'" ")aUfc, £ £**,..*„ We""' fc, ^ /

\*i=iiji

J/

kn=jn'jn

where Sk*,...^^),fc= 1,• -■ ,n, fc = j , , ji, i = 1, • • ■ ,n, are polynomials in 2;^ ■ • • , 2^,, such that some polynomial gkki-kn(z) *s n o * zero, then there exists an element Zeg(Tn) of the form n

z = E( *-SI

/

\

1 42l+ +u t z ^fc,..fen(^)ewit ' '"^*" - " " " ]J *d,k, E _ -■• £E_ W-U*)^ £■

fc=i \ * i = j i , i i

fc.=Jnj'n

/

where hkkl...kn(z), k = 1, • • • ,n, k, = jit]i, i = 1, • •■ ,n, are polynomials in z i+1 , •• • ,z„, such that some polynomial hkki-kn(z) is n °t z e r o - Indeed, combined with Lemma 1, a repeated application of this fact yields the desired result. Let Y be as in the preceding paragraph and we define a real polynomial Q as follows: If uiijt £ R, then Q{x) = (x - uijt) (x - uiiji) and if Wjjs € R, then Q(x)

= X - Wy,.

Let s denote the maximum value of the degrees with respect to Z{ of the polynpmials 9kki-k„(z), k = 1,■■■ ,n, ki = ji,ji, i = 1,••• ,n, and set Z = Q(addi)sY. Then we have

*-E|

V • £ ora)^.-^)^^""^*))*

Jt=i '\\fci=ii,ii k=l fel=il,il

fcn = Jn»Jn

fcn=Jnj'n

//

232 and Q(^)*(fft* 1 ...fc.Wc^*'*' + "-^^^) = ( C ^ f c . . . * . ^ ) ) ^ ' * ' ^ " 4 ^ ' *-, where C is the non-zero constant given by ( C = ±(wy, — uHjl)s when u;^ ^ R, \ C=1

when uHji € R.

By the definition of the integer s, the desired element of 0(TQ)

is given by Z.

□

In Lemma 2, we may assume that 7jj,..-j„ # 0. Lemma 3. There exists a non-zero element X^

€ Q(TQ) of the form

n

X^

= J2{akeUlhZl+'"+UJ''i"Zn

+ /3ke^Zl+'"+^z")dk,

(5.2)

k=\

where ak, Pk, k = 1, • • • , n, are complex

constants.

Proof. We may assume that wi^, • • • , wyt are not real numbers and uit+ijt+1, • • ■ , unj„ are real numbers. By assumption, we have t > 1. If t — 1, then there is nothing to prove. Suppose that t > 2. For i = 2, ■ • ■ , t, we define a real polynomial Qi by Qi{x, y) = (xy - uijJU^l) (xy - Wi^WyJ . and set t

X^

^HQi{addi,addi)X^. i=2

Then we have

*(3) = E ( E ••• E 7u,..fc„IlQI(ai,al)e-1-+-+— U . ft=l \fei =Ji Ji

A„=J„J„

!=2

ForTO= 2, • • • , t, if fci = ji and fcm = j m ,

/

or if fcx = ]i andfc™= j m ,

then

t

n < ?i( 9 i>^)e WltlZl+ "' +Wmtm2m+ " +u '" l " Zn i=2

= n Oi(ft: = n c?i(9i>5o((wifcIwmfcn = II

u + + Qi(di,di)(Qrn(o\,d di){Q m)e ^ --+»">^ ---+»^) m(i

2
-

2
w

ljiW m j m )(wifc,W m jfe m - "

WlfcjZH

e

UljiUmjm)

hWmkm ZmH

|-U>n fcn^n

)=0.

233 This implies that the element X^ and since

of fl(Tfi) has the form (5.2). Since 7#,...j» =£ 0

t

i~[Qi{di,di)eu,1^Zl+-+Uni"Zn

= ce"u1^+-+^~z-i

!=2

where C is the non-zero constant given by C =

Wl

II

* (W'J* ~~ ^ ) ( w i j i - wIiTViji>

i=2

it follows that X^

^ 0.

a

For simplicity, we write X = X^\u>i = wy,,- • • , w„ = unjn. We consider the linear subspace {a>i, • • ■ , w„ } R spanned by w\, ■ ■ ■ , w„ of the field C of the complex numbers viewed as a real vector space. To proceed further, it is necessary to divide into the following two cases: Case (i) dim R {wi, • ■ • , W „ } R = 1; Case (ii) dim R {wi, • • • , W „ } R = 2. We first show that Case (i) cannot occur. In this case, we may assume that u>i ^ 0. Then we have uii = AjOii,

i = 2, ■ ■ • , n ,

where \ , i = 2, ■ ■ ■ , n, are real constants. Note that a>i ^ R. After a change of coordinates under the element of GL(n, R ) x Cn given by C n 3 (zi, • • • , % ) • — (wi, ■ ■ • ,w») G C", ((wi=Zi wi = zi + X2Z z22 -\ + ■ • +1- An^n, \nZn, \wi=zu \wi=Zi, X has the form

= 2, 2, ■ ■ •• •, «, ,n, ii =

x == £»*"*+

^zi)aj,

j =n l

where a ' , /?'•, .7 = 1, • ■ • , n, are complex constants. If a'j= Pi = 0, then, by Lemma 3=1 6 of Section 3, the functions where a ' , /?', j = 1, • ■ • , n, are complex constants. If a'j = f3[ = 0, then, by Lemma a ^ e W I Z l + ^ e w l Z l , j = 2,.. ■ ,n, 6 of Section 3, the functions a'jeUJ1Zl+^"1, j = 2,---,n, must be constants. Since u)\ # 5JT, it follows that Q/. = 0J = 0, j = 2, ■ • ■, n. This contradicts to the relation X ^0. Suppose that oix # 0 or f}[ # 0. If T(„it) is a tube domain in C given as the image of the domain TQ under the projection C " 9 ( zZ i , - - - ■

, Zn) ,z'—► zi 61eC, C, n)>—*z

234 then, by Lemma 4 of Section 3, X induces a complete holomorphic vector field X' on T(a,A) of the form X' = (Qie W l Z l + /3ie' J l 2 l )di Since a[ ^ 0 or 0X ^ 0, we see that T/a,b) *s a strip in the complex plane. But the example in Section 3 asserts that if a vector field Y of the form / 1=

y =

>

1*

is complete on T( a6 ), where %,ui'i, i = 1,- •- ,fe, are complex constants and c*^, i — 1, • • • , k, are mutually distinct, and if % ^ 0, then w[ € R. Thus we have a contradiction. We next consider Case (ii). We may assume that wi, u>2 are linearly independent over R. Then we have w, = A;wi + Wj = A;Wi + /iiO>2, /iiW2,

3,•• ••• •, ,"n, «i = 3, ,

where A,, in, i = 3, • • • , n, are real constants. After a change of coordinates under the element of GL(n, R) K C™ given by ■ ■• ••,i%) ,Oe6 C "C, n , C °n 93 (*i,• ( * ! , • ■■• •, Z,Zn) „ ) 1 t—» > (tUi, (tui,• '' Wi W\=Zl = 1i

zi + + A A3Z3 + 3z3 H

w = z zu wt = t

X has the form

n

X ==

• •' +1- AnZn, A n z„,

W2 == 22 Z2 ++ M3Z3 M3Z3 +"I • • • +1" MnZn, (J-nZn, ™2

n 3=1

u

3, ■ ii == 3, ■•■• ■,n, ,n,

j2(a'jeWlzl+"2Z2 +

^eWl0I+W2Z2)aj,

3=1 where c/, /5jJ, j = 1, • • • , n, are complex constants. Furthermore, after a change of C n 9 (zi coordinates ,%)eC",

C n 9 ( z i ,z\- -=- ,Aiwi z n ) ^+ (Aw2W2, i,...,Oecn, < zi Aiwi + z2 = = y! + A/J.' 2W2, xw\ 2W2, < Z Zi=wi, i — // 23, U>■2 , • •,», 2 = /i'lttll + , Zi = W{, i = 3 , • •• ,n,

where A',/i-, i = 1,2, are the real constants which satisfy f., , 1 A'jwi + Ml^2 = A i w i + ^ i w 2 = 2-' ,

V-

A'2wi + M2W2 = A2Wi +^' 2 W 2 = —2 — ,'

235 X has the form n

X

jZl +

■E(<

^ Z a

+ ^'e5z'-

2 ■

*)%.

where a",/3", j = 1, ■ ■ ■ ,n, axe complex constants. In the next section, we shall show that there can not exist a non-zero element of fl(Tn) of this form. 6. Proof of Lemma B (continued) Suppose that there exists a non-zero element X € g(To.) of the form n n

X=

;52l + ' 2

22

jj=i =i

1

+/3,-e**1- -4 ") %,

where ay, 0j, j = 1, ■•• , n , are complex constants. To complete the proof of Lemma B, it is enough to derive a contradiction under this hypothesis. We write Y = [82, X] and Z = [X,Y]. A straightforward computation yields that 73 V - (a (' Azr *. J b i - ^ d. e^'++^z* ^ 2 2 - Pje^-^*) Y = V ! J2 2

and

j

fi* 1 n a„ zZ = V=I V- e*zi'dj,

22

,•=1 J=l

where 7 j , j = 1, ■ • ■ , n, are the complex constants given by 71 = V -l(«l/?2 + «2/?l), 72 = -ai/?2 + a2/?i + 2 v ^ l a 2 ^ 2 ,

7j = ajfc - aify + y/^iaj^

+ a2Pj),

j = 3, • • •, n.

L e m m a 1. The constant 71 is not zero. Proof. Suppose contrarily that 7i

v^

= ai/% + a 2 /?i = 0.

By Lemma 6 of Section 3, the coefficients ^ 1 * v^T 7je 21 , 2

'■

j = 2,---,n,

(6.1)

236 of Z must be constant, so that 72 = -<*i#2 + a 2 f t + 2v/ = Ta 2 ft = 0.

(6.2)

It follows from (6.1) and (6.2) that

(ai - V-la^Pt

= 0,

or that (a) P2 = 0 or (b) a i - \f-la2 = 0. In the case (a), by (6.1), we have a2 = 0 or 0i = 0. Since X =£ 0 by assumption, this contradicts to Lemmas 7 and 8 of Section 3 applied to the complete holomorphic vector field X on To.. In the case (b), if a2 = 0, then a i = 0, and this contradicts to Lemma 7 of Section 3 applied to X. Therefore we have a 2 ^ 0. Substituting the relation a\ = V / -Ta2 into (6.1) and noting that a2 # 0, we see that ft = — v ' - l f t . It follows from the relations a\ = >/—Ta2 and ft = — ■
v ^ xX === V^l

1

((aa2e^+^ ^ 2 ' ^^ 2

2

foe^-^'Ad,. -- f32e^-^)di

+ z

2

+ ((aa 22e^ + fte* * + e * *2 l *+ J ^ 2 2 + fte*'"2'"^)

22

)& 82

n

+ ]£ (a^ 2 1 "^ 2 2 + fte*21 * 22J $. If Tn' is a tube domain in C 2 given as the image of the domain TQ under the projection C n 3 (JZI, z2, ■ ■ ■ , 2 „ ) H - t ( Z l , 2 2 ) 6 C 2 ,

then, by Lemma 4 of Section 3, X induces a complete holomorphic vector field X' on TQI of the form

X' = ^J=l ((aa22ee^^1' ++ VV FF2222 -- ftei fte*22'-*?* '"^22)2) ftft + + ( aa22ee^^21l ++ VV ? 2 2 + +

22 fte*2'"^ '"^22 )) fte*

ft. ft

Consider an integral curve z'(t) = (zi(t),z2{t)) of the holomorphic vector field X'. Since X' is complete, z'(t) is defined for all t e R, and its components .zi(t) and ,z2(t) satisfy the system of the ordinary differential equations on R given by ' dzi(t) = ^ ( a 2 e ^ 2 ' W + y ? 2 2 « - ftef *»(*>-*T^C*A , dt 1 dz dz2 2{t) ^ = Q W + /3 i ( t ) - vvffII^^(t)_ (t)_ a aj ce^^' W W++^^^ W ^i(*)/ 3 22e e^

\

(fi 1)

dt

Hence, if we write (i) = CO® Ce^W ^ =

(6.4) (6.4)

237 on R, where C = 1/^102 # 0. This implies that, for every point (Ci, C2) of TQI, the equation (6.4) has a solution ip{t) denned on R with the initial condition
Z = eZlb\ + -reZid2 + X ^ e * 1 ^ , 3=3

where 7 and 7J, j = 3, • • • , n, are complex constants. A straightforward computa­ tion yields the following lemma. L e m m a 2. If W is an eJement of g(7fi) of the form n n

w=

Y^(<*'je"lzl+W2Z2 +

P'je^+^^dj,

W = J'=I Y^ia'je"1*1*"2*2 +

fye^+^dj,

where uii,u>2 and a'j, ftp 3 = 1, ■■•, n, are complex constants, then where ui\, u>2 and a'-, /?'-, 3 = 1, ■ • ■ , n, are complex constants, then +1 2 +1D222 ,{ui+l)zl+UI2Z2\Q. n

[z,wq«= f](a;'e^

)

+ $'<

= 1i i, =

where a'L 0L 3 = 1, • ■ • ,n, are the complex constants given by a'{ = ( w i + w 2 7 - l)«ii a 2 = (wi + W27)a'2 - 7<*i> a" = (wi + W27) a j - 7j«i!

i = 3, ■ ■ • , n,

(W1+W27#01-== (aTT + ^ 7 - l)/?i, 1)#, /32' = (wT + W27)/?2-7/3i^ ' = (051 + ^07^ )7^) -^ 7- 7^ ^1 !, ,

Jj = 3, 3 ,•-■-•- ,, 7n1..

A repeated application of this lemma yields that, for k = 0,1,2, • • •, we have (odZ)kX--

=£(«fe(

^+fc)*a + * 3 *

+ /5fe< 1+*)*- - ^

■)fy

238 ',/3J , j j==1,2, af\pf\ , n,arearecomplex complexconstants constantsand andokaf\pf\ 1,2,satisfy satisfy where aj ',/?' , j = 1, • -•■■,n, a (0) f - c y . / pf=P s f - f ch ,

j = l,2, l,2,

fc+1 > == Q + aa<(W)

(6-5)

v5l -l)aW

fc+ 7 (i+'+^-O'f+1) /f = . ++ / 3 (W)=Q fc_vEl7-l)/3p,

G *-^v>K'

(6.6) (6.7)

at1] =(i+*+^)<.r-^*»,

(6.8)

Q+*-^)tf'-rf'

(6.9)

/# +1 > =

It follows from (6.5), (6.6) and (6.7) that, for k = 0,1,2, ■ k

)

«r =«inG+ m+2 T l7 - 1 )' m=0 ^

(6.10)

*

fc+i)

M - A m=0 n (2+m

7

2

V

l).

(6.11)

'

m-0 ^

Since the functions e{i+k)zl+^z^

eC-+k)Zl-^z^

fc

=

o, 1,2, • ■ ■ ,

are linearly independent over C and since the real vector space fl(Tn) is finitedimensional, there exists a non-negative integerfcosuch that (adZ) f c o + 1 X = 0. Then we have a[ ' = p[ + — 0. To proceed further, we divide into the two cases of a\j3\ / 0 and a\fi\ — 0. Suppose first that a i f t / 0. Since a[ fco+1) = ^ f c o + 1 ) = 0, it follows from (6.10) and (6.11) that there exist non-negative integers k'0 and 1'0 such that ^ +^ 0 + ^ 7 - 1 = 0 ,

(6.12)

l + l'o- ^ 7 - 1 = 0.

(6.13)

By adding the left side and the right side of the equations (6.12) and (6.13), respec­ tively, we have k^ +1'0 = 1, so that (a) %, 1'0) = (1,0) or (b) (*:(,,/(,) = (0,1). In the case (a), by (6.12), we see that 7 =
239 that aw = - 0 1 ,

(6.14)

/3$ 1} = 0,

(6.15)

««

2)

=M =

/f> =

0,

^

(6.16) 1

V^i '.

(6.17)

By applying Lemma 8 of Section 3 to the complete holomorphic vector field (adZfX on T n , (6.16) implies that (adZ)2X = 0, and hence that $] = 0. Com­ bining this with (6.15) and (6.17), we have pf> = 0.

(6.18)

By applying Lemma 7 of Section 3 to (adZ)X, (6.15) and (6.18) imply that (adZ)X = 0, and hence that a[' = 0. But, by (6.14), a[ ' is not equal to zero. This is a contradiction. A similar argument shows that the case (b) cannot occur as well. Suppose next that a\f5i = 0, or that (a) a i = 0 or (b) /?i = 0. In the case (a), first note that P\ ^ 0 and Q2 / 0. Indeed, if 0\ = 0 or 0:2 = 0, then, by Lemmas 7 and 8 of Section 3 applied to X, we have X = 0, which is a contradiction. Now, since p(ko+ ) _ Q^ w m | e p1 j . Q^ i t f 0 n o w s from (6.11) that there exists a non-negative integer k'0 such that

pSri^o, « m , #o,

+1, #/ 9 *W+D = =00,i

m = = O,l,...,k' 0 , 1 , ■ ■0i,"-0 m

1.1 + ^ o - ^ ? 7 - l = 0. 2

(6.19) (6.20)

We show that k^ also satisfies

l ++ *0 k'0 ++ V^T ^ 2 7 = ==0.0. 2 1

(6-21)

By (6.10), we have a[k+1) = 0

for all k = 0,1,2, • • •

(6.22)

Therefore, for k = 0,1,2, •■ ■, an application of Lemmas 7 and 8 of Section 3 to (adZ)k+1X yields that p[k+1) = 0 if and only if a 2 fc+1) = 0. Using this fact, we see from (6.19) that Q^

+ 1 )

=0,

4m)^0,

m = 0,1, • " , * £ .

(6.23)

240 On the other hand, for k = 0,1,2, • • •, the constant a i

—a«;2 n(Vm

"2

is given by

+^7),

(6-24)

m=0

because, for m = 0,1,2, • • •, (m+l)

"2

(/ Ii +, m ,+V^- )! 4\ "(m) 7

by (6.8) and (6.22). Combining (6.23) with (6.24), we obtain (6.21). By adding the left side and the right side of the equations (6.20) and (6.21), respectively, we have k'0 = 0, and hence, by (6.20), 7 = V^T. It follows from (6.9) that

= ft - ^ l f t . fl^-A On the other hand, since a ^ = @[1} = 0 by (6.19), (6.22) and the relation k'0 = 0, an application of Lemma 8 of Section 3 to (ad Z)X yields that f}\ * = 0. Therefore we see that ft = \/—Tft. It follows from the relations a i = 0 and ft = v^-Tft that X has the form X = 01e^'-^lz'd1

ft + ( a 2 e ^ 1 + * ? 2 2 + v ^ f t e * 2 ' - ^ 2 2 ) ft

+ £ ( ^ 3 * + ^ + ^ e **- , f s *) ft. If Tn' is a tube domain in C 2 given as the image of the domain TQ under the projection C™ 3 (zi, 22,- • • ,Zn) •—> (21, 22) <= C 2 , then, by Lemma 4 of Section 3, X induces a complete holomorphic vector field X' on To/ of the form X' =

Iz2 + v ^ T f t e * 2 ' - ^ ) ft. A« fte^-^ft Z 2 d i + ( Q 2 e ^ ' + ^

Consider an integral curve z'(f) = (21(f), z2(*)) of the holomorphic vector field X'. Since X' is complete, z'(t) is defined for all t e R, and its components zi(t) and 22 (t) satisfy the system of the ordinary differential equations on R given by

r dzi{t) < dt dz2(t) dza(0 \ dt

= fal« i ( * ) = OLie

*«

•• yzi^e i*iW■ -- ~ -MD+^Mt) + -*!*»»(*>,

(6 25)

-

241 Hence, if we write ip(t) = (l/2)zi(t) + (y/^1/2) z2(t), then
(6.26)

on R, where C = ( is stable under real translations, the value of (l/2)Ci + (\Z = T/2)f2 may take all complex numbers. But, as observed in Section 2, if ip{t) is a solution of (6.26) with e" v ( 0 ) 6 {C}R, then it is not defined for t = e ~ ^ ° ) / C Thus we have a contradiction. A similar argument shows that the case (b) can not occur as well, and the proof of Lemma B is completed. References 1. H. Cartan, Sur les groupes de transformations analytiques, Act. Sci. Ind. (Her­ mann, Paris, 1935). 2. Y. Matsushima, On tube domains, in "Symmetric Spaces,'' Pure and Appl. Math.8, (Dekker, New York, 1972), 255-270. 3. S. Shimizu, Automorphisms of bounded Reinhardt domains, Japan. J. Math. 15 (1989), 385-414. 4. S. Shimizu, Automorphisms and equivalence of tube domains with bounded base, preprint. 5. S. Shimizu, A classification of two-dimensional tube domains, in preparation.

242 T E N S O R P R O D U C T S O F SEMISTABLES A R E SEMISTABLE BURT TOTARO* Department of Mathematics, University of Chicago, 5734 S- University Ave., Chicago, IL 60637, USA

Introduction This paper is devoted to a problem in linear algebra which Faltings formulated and solved using the theory of semistable vector bundles [2]. I will give a somewhat more elementary proof. Here is the notation we need. Let V be a finite-dimensional vector space over C. A filtration of V will mean a decreasing sequence of subspaces Vp of V for p e Z, • • ■ D V-1 3 V° D V1 D ■ ■ • , such that UV P = V and n V p = {0}. An N-filtered vector space, for an integer TV > 1, will mean a vector space V equipped with TV nitrations, denoted Vf for 1 < z/ < TV. Define the slope p(V) by

Kv) = ^(y) =

\

" nrHin [rTDCl'"l

dm7K^ ",P

p d i m g r

'

( n

where gr£(\0 = Vv/V£+ . For any subspace W C V, we have the induced nitra­ tions on W, Wp = W n Vj. We say that an AT-filtered vector space V is semistable if for all W C V, W # {0}, we have fi(W) < fi(V). V is stable if we have strict inequality fi{W) < fj,(V) for W C V, W ^ {0}, W ^ V; and V is polystable if it is a direct sum of TV-filtered vector spaces, all of the same slope. For TV-filtered vector spaces, we can form ©, Horn, ® as TV-filtered vector spaces. For example: (V'®CV")P

-

= E iy')i®
Theorem 1. V, V" semistable =s- V (g> V" semistable. Also, V, V" polystable => V ® V" polystable. Faltings's proof in [2] associates to every TV-filtered vector space V a vector bundle Ey over a suitable algebraic curve, with the property that V is semistable • Supported by NSF grant DMS-9007259. e-mail: totaroQmath.uchicago.edu

243

<=> Ey is semistable. Then one uses Narasimhan-Seshadri's theorem that a vector bundle on a complex curve is polystable «*■ it admits a projectively flat metric. Since the tensor product metric on E <8> F is projectively flat if E and F are, it follows that the tensor product of polystable bundles is polystable. The same statement for semistable bundles is a corollary. Then at last one returns from vector bundles to TV-filtered vector spaces to prove the theorem. This paper gives another proof, essentially just using linear algebra over the complex numbers. We find a simple notion of a "good" hermitian metric on an ./V-filtered vector space V over C such that V is polystable <=>• it admits a good metric. (The proof uses the Kempf-Ness theorem from geometric invariant theory.) It is easy to check that the tensor product of good metrics is good, and the result follows. Althogh this proof is more algebraic in flavor, it still works only over C. (Of course, semistable TV-filtered vector spaces make sense over any field, and those of fixed slope form an abelian category.) Faltings has recently generalized to any field the theorem that the tensor product of semistable ./V-filtered vector spaces is semistable [1], and the method of [6] gives another proof of this generalization. This is a little surprising, given that the tensor product of semistable bundles over a curve in characteristic p need not be semistable (see for example [5]). 1. Statements Convension. In what follows we will assume that all TV-filtered vector spaces V satisfy Vj = V, v = 1,,.. ,N. This can always be arranged by shifting the indexing of the nitrations. In this case, the slope n(V) has the simple definition:

Kvo = ^ £ E d h » e p(V)-

V

p>l

Definition. Let V be an TV-filtered vector space over C. We say that a hermitian metric on V (that is, on the underlying vector space) is good if the sum of all the orthogonal projections V —► V$ for v = 1,..., TV and p > 1 is equal to /*(V) times the identity endomorphism of V. Theorem 2. An TV-filtered vector space V over C is polystable <& V admits a good metric. Moreover this good metric is unique up to automorphisms of the N-Bltered vector space V Theorem 3. The tensor product of good metrics is good. Here Theorem 3 is easy linear algebra, to be proved at the end of the paper. And Theorems 2 and 3 imply the theorem of Faltings.

244 2. Proof of Theorem 2: G o o d metric =>■ polystable We begin by proving the simpler direction: an JV-filtered vector space which admits a good metric is polystable. So suppose that V has a good metric and let W C V be a linear subspace, W / {0}, W ^V. Let iw denote the inclusion W •-» V and {iwT its Hilbert-space adjoint, the orthogonal projection V —» W Similarly let t£ denote the inclusion V$ <-* V, for any 1 < v < N, p € Z. Consider the composition

ft = fwZWYiwThat is, fl is the endomorphism of W C V defined by orthogonally projecting to Vy C V and then orthogonally projecting back to W. Since

fS = Hw%){ihilY, fv has the form AA* and so is a self-adjoint endomorphism of W with nonnegative eigenvalues. Moreover, tf is the identity on W n Vg = W$. So tr/P>dimW?, and

*£E# > ^ ^ d i m ^ v V

pp>>ll

v

p>l

r = )dimW = ^(W n{W)A\mW.

On the other hand, since the metric on V is good, we know that

£ £ « ( « * = MV)-ivi/

p>i

Thus

£ £ ^ = £ £ ^ ( 0 * % = ^(M^) ■ v ) ^ = i*{v) ■ iw, V p>\

V

p>\

and so t r

£ £ ^ = ti{V)A\mW. V p>l

Comparing this with the previous paragraph, we find that fi(W) < n(V). That is, we have proved that an JV-filtered vector space V with a good metric is at least semistable.

245 But we can say more. Suppose that V is not stable, so that there is a subspace W with n(W) = fi(V). By the above inequalities, this can only happen if f$'s nonnegative eigenvalues on {WS)1- c W are actually 0, for all p and v. That is, fl restricted to {Wg)x C W is 0, which implies (using the definition of f£ as orthogonal projection from W to V$ and back again) that (WS)1 C W is contained in (V£)L c V, for all v and p. This means that V = W © W 1 expresses the ATfiltered vector space V as a direct sum of AT-filtered vector spaces. Moreover, since fj,(W) = n(V), we also have ^(W1) = n(V) ( because the slope of W © WL is an "average" of the slopes of W and WL, namely u(w v

ffi

w x , _ g W dim ^ + M (W^) dim M/^ ; dimPy + dim^- 1 ';

And both W and W1 have good metrics. By applying the same argument to W and W1, we eventually write any V with a good metric as a direct sum of stable AT-filtered vector spaces, all of slope = n(V). That is, we have proved that if V has a good metric then it is polystable. 3. Proof of Theorem 2: Polystable => good metric To show that a direct sum of stable TV-filtered vector spaces of the same slope has a good metric, it suffices to show that a stable AT-filtered vector space has a good metric. We begin by explaining the relation of stability for A^-filtered vector spaces to Mumford's notion of stable points in a group representation. Definition. Let G be a reductive algebraic group over C and W a complex repre­ sentation of G. A point x e W, x ^ 0, is called: stable

if the orbit Gx C W is closed and dim Gx = dim G,

polystable semistable

if Gx C W is closed, if 0 is not in the closure of Gx.

Let V be an AT-filtered vector space. To V we can associate a point in N

nn Gr dta^). v=\p>\

just by V i-> {Vv ■ v = 1 , . . . ,AT;p > 1). (Here Gr^(V^) is the Grassmannian of fc-dimensional subspaces of V.) Moreover, we can embed this product of Grassmannians in the projective space

p(®JU®p>iA dimV7 n

246 so that V determines a line in the vector space ®„®P>iAdimV?V.

W := Clearly the group SL(Vr) acts on W.

Choose a nonzero point x in the line in W associated to V. Lemma 1. An N-filtered vector space V is stable if and only if the point x is stable for the action of SL(V) on W The same goes for "semistable." Proof, We have to show that the orbit Gx C W is closed and of maximal dimension -» for all linear subspaces U cV one has

up) < Mn that is, 1 :££dim(t/r dim U v p>\

1 EEdimV?. \vs) < dimV V v

p >l p>\

If all of the subspaces V$ c V have the same dimension (or are 0), then this is proved in Mumford-Fogarty [4], p. 88, using the Hilbert-Mumford theorem. The proof in general is the same. This proves the lemma for stable points. Mumford likewise checks that the point x is semistable ■»• (in our notation) the ./V-filtered vector space V is semistable. QED. Lemma 1 is also true for ''polystable" in place of "stable,'' as will be clear once we prove Theorem 2, but it is inconvenient to prove this now because Mumford and Fogarty do not formulate a polystable version of the Hilbert-Mumford theorem. Next we give the Kempf-Ness characterization of polystable points in a complex representation. Theorem 4. (Kempf-Ness [3]). Let G be a reductive algebraic group over C, W a representation of G, x € W a nonzero vector. Choose a maximal compact subgroup H c G and an H-invariant hermitian metric on W. Then the following are equivalent: (1) x is polystable: that is, the orbit Gx CW is closed. (2) The function dx: G -> R defined by dx{9) = |flx|2 (using the metric on W) attains its minimum on G. (3) The function dx has a critical point on G.

247 More precisely, any critical point of dx gives actually a global minimum, and the set of critical points (if nonempty) is just one double coset in H\G/GX, where Gx is the isotropy group ofx. Of course (1) =$• (2) =*• (3) is trivial. The converse is an elegant application of the Cartan decomposition of G, which in turn is just classical linear algebra in the c a s e G = SL(V). For G = SL(V), one can restate the Kempf-Ness theorem as follows, using the fact that SL(V) acts transitively on the space MU(V) of hermitian metrics on V with a given volume form w. Corollary 1. Let V be a vector space over C with a voiume form u>. Let W C ® r V be a representation of SL(V), x e W a nonzero vector. Then the following are equivalent: (1) x is polystable, that is, the orbit SL(V) ■ x C W is closed. (2) There is a hermitian metric TTIQ on V with volume form w which minimizes the function dx(m) = \x\2m on M w . (Here \x\m denotes the length of x in the metric on W C ®rV associated to a metric on V.) (3) There is a hermitian metric mo on V which is a critical point for the function dx : MU(V) -> R. Moreover, if there is a metric on V as in (3), it is unique up to the action of the subgroup of SL(V) which fixes the point x € W. We want to use this criterion when V is an iV-filtered vector space and x e ® ^ j <3>p>i AdJmVr"''V is a point in the line associated to V. That is, if we choose a basis o ^ i € V, 1 < i < dim V,?, for each subspace V?, then we can take x = (a\tl A • • • A <*}|dimVii) ® («2,1 A • • ■ A o4 j d i m V ? )
M™ = I I I I V0l mK,l A • • ' "Mlm V?)V

p>l

More simply, if we have a pair of metrics mo and m on V, we can say that \Am

1*1*-

T-T y j r volm(V?) s

,pii

vol

™(V2T'

Here both metrics give volume forms on all linear subspaces V$', and the ratio of the volume forms on each V,? is a positive real number.

248

mo,

By Lemma 1 and Corollary 1, every stable iV-filtered vector space V has a metric with associated volume form w, such that mo is a critical point for the function

= n n voi™Ki A ■ ■ • ^.dim v?)

dx(m) =\x\2m v

p>l

on Mu(V). We can assume here that o?vX,... ,oPv&inVP is an orthonormal basis for y$ with respect to the metric mo- The tangent space to MW(V) at mo can be identified with the space of self-adjoint endomorphisms b of (V, mo) with trace 0. In these terms, d(vol m (c^ 1 A---Ac^ d i m V ; ; ,) 2 )| m o (6) = d ( d e t < o { i i ) Q j J > m ) | m o ( 6 ) . Using the fact that < o?vi,oPv, > m „ = % , we can evaluate the derivative of the determinant as a trace: 1

dimVZ dimV,?

= E d(< < i ><* >m)|mo(&) t=i

= tr«o&), where 7r£ denotes the orthogonal projection with respect to the metric mo from V onto Vu. We view 7r£ as an endomorphism of V. (In the previous section's notation, we could write 7rS = iS(iJ)*.) Since all the volumes vol m (a£ j A- • •) are equal to 1 when m = mo, the derivative at mo of the product of these volumes is just the sum of the derivatives. That is:

d(dx)\mo(b) = d(n n vowl

= ££tr(7r£o6) v

p>l

,6). = tr(£5>£' V p>l " p>i

So the metric mo is a critical point for the function dx ■ MW{V) —► K <=$■ >b)-tr(EE^« v pp>\ >l

= 0

249 for all self-adjoint endomorphisms b of (V, mo) with trace 0. Since tr(AB) is a nondegenerate bilinear form on the space of self-adjoint endomorphisms, and J£„ J2P>i 1$ is self-adjoint, this condition is equivalent to

££< € R

\y.

V pp>\ >l

By computing the trace of this endomorphism (namely, £ „ Y^p>\ ^m at last that mo is a critical point for the function dx if and only if

E£*s =MimV £ £ d i m ^ v p>\ p>l

V$),

one sees

lv

v p>l V p>\

= M^)-iv

That is, mo is a critical point for dx : M^V) - » 1 o mo is a good metric in the sense defined before Theorem 2. It follows that every stable .TV-filtered vector space has a good metric. We deduce that every polystable iV-filtered vector space, being a direct sum of stable spaces of the same slope, has a good metric. The uniqueness part of the Kempf-Ness theorem implies that a good metric, when it exists, is unique up to automorphisms of the ^-filtered vector space V. QED, Theorem 2. 4. Proof of Theorem 3 Let V, W be iV-filtered vector spaces with good metrics. We want to show that the induced metric on the iV-filtered vector space V g> W is also good. Let 7r|J and fi, be the orthogonal projections V —> V,f and W —> W$, respectively. For each v and p, the orthogonal projection

o»:V®W^{V®Wfv=

E

V?&Wf

P'+V"=P

is given by

*S = 2>i®pT- £ < ® P S + 1 - ! -

ieZ iez (For example, one can check this for each v by choosing orthonormal bases for V and W which are compatible with the nitrations Vv and Wv.) We have 7rJ, = 0 and pi = 0 for i sufficiently large. It follows that

E£"S = E E * i ® ^ = E E « i « ^ + E E < ® ^ = EE<®w + E E ^ ® ^ ^

t>l

f

i<0

= /*(V) ■ lv ® l w + M(W0 • l v ® l w = M( F ® W) • lv®wThat is, the tensor product of good metrics is good. QED, Theorem 3.

250 References 1. G. Faltings., Mumford-Stabilitat in der algebraischen Geometric, To appear in the 1994 ICM proceedings. 2. G. Faltings and G. Wustholz, Diophantine approximations in protective spaces, Invent. Math. 116 (1994), 109-138. 3. G. Kempf and L. Ness, The length of vectors in representation spaces, in Al­ gebraic Geometry, Copenhagen 1978, Springer Lecture Notes in Math. 732 (1979). 4. D. Mumford and J. Forgaty, Geometric Invariant Theory (Springer, Heidelberg 1965). 5. S. Ramanan and A. Ramanathan, Some remarks on the instability flag, Tohoku Math. J. 36 (1984), 269-291. 6. B. Totaro, Tensor products of weakly admissible filtered isocrystals (as used in p-adic Hodge theory), preprint.