Contributions to Several Complex Variables: In Honour of Wilhelm Stoll

Aspects of Mathematics

Aspelde der Mathematik Editor: Klas Diederich

Vol. El: G. Hector/U. Hirsch, Introduction to the Geometry of Foliations, Part A Vol. E2: M. Knebusch/M. Kolster, Wittrings Vol. E3: G. Hector/U. Hirsch, Introduction to the Geometry of Foliations, Part B Vol. E4: M. Laska, Elliptic Curves over Number Fields with Prescribed Reduction Type Vol. E5: P. Stiller, Automorphic Forms and the Picard Number of an Elliptic Surface Vol. E6: G. Faltings/G. Wi.istholz et. al., Rational Points (A Publication of the Max-Pianck-lnstitut fiir Mathematik, Bonn)

Vol. E7: W. Stoll, Value Distribution Theory for Meromcirphic Maps Vol. E8:., w.:von Wahl, The Equations of Navier-Stokes and Abstract Parabo.IJc'Equations Vol. E9: A. Howard, P.-M. Wong (Eds.), Contributions to Several Complex Variables Band Dl: H. Kraft, Geometrische Methoden in der I nvariantentheorie

The texts published in this series are intended for graduate students and all mathematicians who wish to broaden their research horizons or who simply want to get a better idea of what is going on in a given field. They are introductions to areas close to modern research at a high level and prepare the reader for a better understanding of research papers. Many of the books can also be used to supplement graduate course programs. The series comprises two sub-series, one with English texts only and the other in German.

Alan Howard, Pit-Mann Wong (Eds.)

Contributions to Several Complex Variables In Honour of Wilhelm Stoll

Friedr. Vieweg & Sohn

Braunschweig I Wiesbaden

Professors Alan Howard and Pit-Mann Wong, Department of Mathematics, University of Notre Dame, Post Office Box 398, Notre Dame, Indiana 46556, USA.

AMS Subject Classification: 32 06

All rights reserved © Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig 1986

No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, mechanical, photocopying, recording or otherwise, without prior permission of the copyright holder.

Produced by W. Langeliiddecke, Braunschweig Printed in Germany

ISSN ISBN

0179-2156 3-528-08964-4

Contents Picture of Wilhelm Stoll . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

VI

Foreword . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

VII

Group Picture of the Conference Participants and Explanations . . . . . . . . • . . . . . . . . . . . VIII Arithmetic Hilbert Modular Functions Ill Walter L. Bailey, Jr. The Heat Equation for the a-Neumann Problem on Strictly Pseudoconvex Domains Richard Beals and Nancy K. Stanton

41

Some Examples of the Twistor Construction Daniel Burns

51

Complete Kahler Domains. A Survey of some Recent Results Klas Diederich

69

On the Minimality of Hyperplane Sections of Gorenstein Threefolds Maria L. Fania and Andrew J. Sommese

89

On Meromorphic Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hans Grauert

115

Recent Developments in Homogeneous CR·Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . Alan T. Huckleberry and Wolfgang Richthofer

149

Problems of Value Distribution in Complex Analysis for Several Variables Pierre Lelong

.. . .. .. .. . ..

179

On the Boundary Behavior of Holomorphic Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . L/Jszl6 Lempert

193

Integral Geometry of the Monge·Ampere Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Robert Molzon Logarithmic Jet Spaces and Extensions of de Franchis' Theorem Junjiro Noguchi

227

Remarks on the Nakano Vanishing Theorem Bernard Shiffmann

251

Curvature of the Weii·Petersson Metric in the Moduli Space of Compact Kiihler·Einstein Manifolds of Negative First Chern Class . . . . . . . . . . . . . . . . . . . . . . . . 261 Yum·TongSiu Extension Problems and Positive Currents in Complex Analysis Henri Skoda

. . . . . . . . . . . . . . . . . . . 299

On the Uniformization of Parabolic Manifolds . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . 329 Pit-Mann Wong

Wilhelm Stoll October 1984

VI

Foreword

In 1960 Wilhelm Stoll joined the University of Notre Dame faculty as Professor of Mathematics, and in October, 1984 the university acknowledged his many years of distinguished service by holding a conference in complex analysis in his honour. This volume is the proceedings of that conference. It was our priviledge to serve, along with Nancy K. Stanton, as conference organizers. We are grateful to the College of Science of the University of Notre Dame and to the National Science Foundation for their support. In the course of a career that has included the publication of over sixty research articles and the supervision of eighteen doctoral students, Wilhelm Stoll has won the affection and respect of his colleagues for his diligence, integrity and humaneness. The influence of his ideas and insights and the subsequent investigations they have inspired is attested to by several of the articles in the volume. On behalf of the conference partipants and contributors to this volume, we wish Wilhelm Stoll many more years of happy and devoted service to mathematics.

Alan Howard Pit-Mann Wong

VII

VII I

"'

Cll

l6' "'Cll c.

E

CQ

z

l!!

....::I . !::!

Cl..

c.

::I

e

(.!) Cll

.s::.

....

.....0 Cll

E

Cll

.s::.

~

IX

Participants on the Group Picture 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

X

Qi-keng LU, Professor, Chinese Academy of Science, Peking, China. Pierre LELONG, Professor emeritus, University of Paris VI, France. Wilhelm STOLL, Professor, University of Notre Dame. Marilyn STOLL, Timothy O'MEARA, Provost and Professor of Mathematics, Univ. of Notre Dame. Mike SPURR, Professor, Rice University, Houston, Texas. B. A. TAYLOR, Professor, University of Michigan, Ann Arbor, Mich. Yi-Chuan PAN, Professor, Jackson State University, Jackson, Miss. David BARRETT, Professor, Princeton University, Princeton, N.J. Robert FOOTE, Professor, Texas Tech University, Lubbock, Texas. Weiqi GAO, Student, University of Notre Dame. Alan HOWARD, Professor, University of Notre Dame. Dennis SNOW, Professor, University of Notre Dame. Joanne SNOW, Professor, St. Mary's College, Notre Dame, Ind. Robert MOLZON, Professor, University of Kentucky, Lexington, Ky. Nancy STANTON, Professor, University of Notre Dame. Mary Jo KR EUZMAN, Student, University of Notre Dame. Paula A. RUSSO, Professor, Michigan State Univ., East Lansing, Mich. Eric BEDFORD, Professor, Indiana University, Bloomington, Ind. Zbigniew SLODKOWSKI, Visiting Professor, Univ. of California, Los Angeles. Hans GRAUERT, Professor, University of Gottingen, F.R.G. Giorgio PATRIZIO, Professor, University of Rome II, Italy. Alan HUCKLEBERRY, Professor, University of Bochum, F.R.G. J. RAMANATHAN, Professor, University of Chicago, Illinois. Henri SKODA, Professor, University of Paris VI, France. Wanxi CHEN,Student,Univ.of Notre Dame (from Univ.of Science and Techn.of China). Harry d'SOUZA, Professor, University of Michigan, Flint, Mich. Chong-Kyu HAN, Professor, University of Alabama-Tuscaloosa, Univ. Alabama. Stephen BELL, Professor, Princeton University, Princeton, N.J. Yum-Tong SIU, Professor, Harvard University, Cambridge, Mass. Klas DIEDERICH, Professor, University of Wuppertal, F.R.G. Y. KIM, Student, University of Michigan, Ann Arbor, Mich. Junjiro NOGUCHI, Professor, Tokyo Institute of Technology, Tokyo, Japan. Daniel BURNS, Professor, University of Michigan, Ann Arbor, Mich. Leonard SMILEY, Professor, University of Alaska, Anchorage, Alaska. Yong In KIM, Professor, Michigan State Univ., East Lansing, Mich. Pankaj TOPIWALA, Student, University of Michigan, Ann Arbor, Mich. Walter BAILEY, Professor, University nf Chicago, Chicago, Illinois. Herbert ALEXANDER, Professor, University of Illinois at Chicago, Chicago, Illinois. Pit-Mann WONG, Professor, University of Notre Dame. Kam Wing LEUNG, Professor, Chinese University of Hong Kong. Brian SMYTH, Professor, University of Notre Dame.

43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Andrew SOMMESE, Professor, University of Notre Dame. Michael BUCHNER, Professor, University of New Mexico. Laszlo LEMPERT, Professor, Eotvos Lorimd University, Budapest, Hungary. Norman LEVENBERG, Professor, University of Kentucky, Lexington. Ky. Chia-Chi TUNG, Professor, Mankato State University, Mankato, Minn. Dan GROSS, Professor, Seton Hall University, South Orange, N.J. Steven ZUCKER, Professor, Johns Hopkins University, Baltimore, Md. Bernard SHIFFMAN, Professor, Johns Hopkins University, Baltimore, Md. John WERMER, Professor, Brown University, Providence, R.I. Gary HARRIS, Professor, Texas Tech University, Lubbock, Texas. James FARAN, Professor, Johns Hopkins University, Baltimore, Md. Thomas BLOOM, Professor, University of Toronto, Ontario, Canada. lan R. GRAHAM, Professor, University of Toronto, Ontario, Canada. Mark ANEMA, Student, University of Michigan, Ann Arbor, Mich. Patrick SMITH, Professor, University of Toronto, Ontario, Canada. Harold BOAS, Professor, Texas A. & M. Univ., College Station, Texas. Jay WOOD, Visiting Professor, University of Notre Dame. Bruce GILLIGAN, Professor, University of Regina, Saskatchewan, Canada.

XI

Arithmetic Hilbert Modular Functions Ill Walter L. Baily, Jr.* Department of Mathematics, University of Chicago 5734 S. University Ave., Chicago, IL 60637 U.S.A. INTRODUCTION The purpose of this paper, which is a continuation of [2,3,4,5], is to prove that the special values of Hilbert modular

functions of level 7L generate abelian extensions of

certain

CM fields, by using an essentially elementary theory

of arithmetic Hilbert modular functions, based on the theory of congruence Eisenstein series. The main results are generalizations of the main results of Heeke's thesis [10]. They are also subsumed in more far-reaching results of Shimura and Taniyama [15,13,14]. But our methods are quite different from the latter's and stem directly from Heeke's original ideas. We start from a totally real number field n

over

Q

k

of degree

and consider a family of arithmetic subgroups of

the projective group PGL 2+(k) acting on the product of n upper half planes. Letting K be a purely imaginary quadratic extension of

k , the various immersions of the multiplica-

tive group Kx in PGL 2 +(k) determine fixed points in certain copies of Hn , which are called special points, each of which is associated to a certain order in of integers of

k .

image of an element

K

containing the ring

~

Such a special point is realized as the T

of

K - k

under those immersions of

K in C for which the image of T has positive imaginary parts. This collection of n immersions of K in c as socia ted to T also determines a CM-type for K and a reflex field K*(T) Defining arithmetic Hilbert modular functions as in [3,4], we show here that i f f is an arithmetic Hilbert modular function of level TV and if (T) is a special point associated to the principal order of K and if K* (T) is the corresponding reflex field, then f((T)) generates an abelian extension of K*(T) ~moreover, if Ln. is the abelian extension of K*(T) generated by all such special values of arithmetic Hilbert modular functions of level ~ , then the Galois group of Ln(K*(T) is isomorphic to a sub*Support from NSF Grant No. DMS-8401708

2

group of a certain class group of K. Cf. [101, pp. 55-57 • The point of our developments here is to show much can be done without using the theory of families of abelian varieties and their moduli. Our treatment is based on [10, 3 , 4 , 5 1 and on a paper of Karel [111, in which the classical one variable case is similarly treated. contribution of [ 7 1

Of course, we must emphasize the

in characterizing the more basic geo-

metric and topological properties of the adelic space. We postpone to a subsequent publication consideration of Shimura's reciprocity law for the abelian extension Ln(K*{f) obtained from Kronecker type congruences for the special values.

0. NOTATION AND PRELIMINARY REMARKS. In this article we use the notation of [51.

As a con-

venience to the reader we summarize some commonly used notation.

Denote by

k

a totally real number field with

n > 1 , having ring of integers imaginary quadratic extension of

~

.

Let k . If

[k:Q1

be a purely a E k , a >> 0 K

will mean that a is totally positive. Let L: = {al, ... ,an) be the set of all immersions of k as a subfield of ffi , arranged in a definite order; in general,

0.1 L

=

'V

'V

{a 1 , ... ,an)

will be a "lifting" or set of extensions of

a 1 , ... ,an to immersions of K in C For a given lifting, d~fine [151 the reflex field K* { 't) by K* { 't) = Q { { L:6' E 't 11 I 11 E K}) Let A , A{k) , A{K) , I , I{k) and I(K) denote respectively the rings of adeles of

Q , of

k , of

K , and their groups

of units, the ideles, each supplied with its customary topology. The subscripts oo and f will denote the projections of an adelic object to its archimedean and non-archimedean components respectively, and the subscript + will indicate adelic objects with non-negative archimedean components. ~

resp. ~ will be the maximal compact subrings of Af and of A(k)f respectively. On the other hand, Q , k , and K are naturally imbedded in their rings of adeles and if a belongs to one of these fields, we use a to denote af , A

the non-archimedean projection of the image of

a

in the

3

adeles. Thus, if a E ~ , a E & , and the image of o under this mapping. We denote by by

Hn

its

(i, ... ,i)

H

&

the upper half complex plane

n-th cartesian power, and by

E

is the closure of

Hn, i = /-1

Moreover,

G1

Im z

> 0 ,

i.e , the point is the algebraic

group GL 2 defined over k and G = Rk/QG 1 • We have a cannonical isomorphism ¢ of G 1 (A (k)) onto G (A) such that with respect to sui table integral structures we have = G(~) G 1 and

P

If

~

f(~)

¢(G

1

(&) )

is the group of upper triangular matrices in R = k/Q P 1 . is an open compact subgroup of G(Af) , denote by P1

the corresponding arithmetic subgroup

G(Q)nG+(R)JK

of G+(~) , or its projection into G+(R) , according to the context. W~ denote by ~~ the isotropy group of i.e E Hn in

G+(R) For most other notation we refer to [ 5]

However,

some modifications of our earlier conventions are needed. The center of G is Z = Rk/QZ 1 center of G 1 • In [ 4] , the symbol

,

where Z 1 is the CA,CIK ,w) was used to

denote the space of modular forms of weight to

w

with respect

, and that space was defined as the complex vector

~

space of functions @ on G+(A) satisfying three conditions (i), (ii), and (iii) . In fact it turns out that (ii) is undesirable and unsuited for our purposes, and instead of replacing it with the condition that

@ transform according

to some finite character on the center Z(A) of G+(A) , we drop the condition (ii) altogether. (In fact, condition (i) alone implies that all

z

@(zg)

=~(g)

for all

g

E

G+(A)

belonging to a subgroup of finite index in

and for Z(A).

Indeed, the Eisenstein series (3), p. 596 of [ 3] does not satisfy (ii) unless the function s on double cosets does. Nevertheless, the further discussion of Eisenstein series, and in particular of their convergence, is not affected by this.) Secondly, while most of the open compact subgroups ~ considered in [ 4] were principal congruence subgroups of G 1 (o) of the form ~(n) = {k E G 1 (o) 1 k = 1 2 mod7L}, we want to consider here, as far as possible, an arbitrary open

4 ~

compact subgroup

of

G(Af)

In later sections we return

to the principal congruence subgroups as a practical consequence of their relationship with ray class groups. Returning to the purely imaginary quadratic extension K of k , let B = (6 1 , B2 ) be a k-basis of K . If a E K , there is a unique two by two matrix q B (a l

The mapping

~--~~

a

(

~ ~)

qB(a)

= (:

qB(a)

E

M2 (k)

such that

~ ~)

is called a regular imbedding of

K in M2 Ck) and may be extended in a natural way to an injection of A(k)-algebras of A(K) into M2 (A(k)) If B is replaced by another k-basis B' of K , qB is replaced -1

by qB' SqBS for some S E GL 2 (k) If T E K - k , we write qT in place of qB

B

=

(T

I

l)

1. SPECIAL POINTS AND RAY CLASS POLYNOMIALS 1.1 DEFINITIONS

be a set of reWe refer to §2 of [ 5] . Let @ C G (Af) presentatives for the following decomposition of G+(A) into double cosets:

= VBE@G+(Q)8G+(R)G(~)

G+(A) 'V

Let K and z be as in the introduction and T E K - k be 'V such that ~(T) = ~ , i.e., Im(T 0 j) > 0 , j = l, ... ,n . Then 'V

'V

I(k)f

and let

(T) = (Tol, ... ,Ton) E Hn and T defines an imbedding qT of K in M2 (k) which exin M2(A(k)) We tends naturally to an imbedding of A(K) also associate to any T E K- k its reflex field K*(T) K* (~h) ) , We may assume that each 8 c @ is of the form 8

=

(g'

ry

rJv~-r

~)

I

8'

be the

E

a-module

VT + o

v in

v8 K

id. (8')-l

Let

and let

If now we fix in K an order ~ conbe its order in K taining the integers ~ of k 1 we may define the subset :0: (~) = {T E K - k I!Rv h ) = 1 and a =v(~) of K by v

'V

subset

=vc~~z l

of

£ }

=v

::: (l)

is a zero-cycle of

v(l) = v~(l) rational over K*('f) 0 • Now :::~(~ 'f> its pre-image under the natural (coset) mapping ~~ of onto V(l) A(~ ( 1) , w)

Let over

a3 (l)

= (b 0 , ••• ,bN(l)) be a basis of for some large w , and complement

(!

is v~

/3 (1 )

djiK

= (b 0 , ••• , bN (K)) of A (IK , w) over (! • If w is large enough, g,(l) resp. Q31K provides a set of projective coordinates for a projective imbedding of v(l) resp. of VIK as a reducible projective variety (of several componto a basis

ents), rational over

(!

as a cycle in projective space.

Then ~IK is merely the projection on the first N(l) + 1 coordinates and therefore is defined over (! • Hence,

:::K(~, 'f> is a zero cycle rational over K*('f) 0 also. (The components Vw of VIK are rational over Qm , and permutes them in a manner compatible with translation by the ele-

,£t

ments

]J(o) e: G(Af) for o A(~,w) , then by definition,

e:;if

"'ilxc I(k)f.

R(]J(O)-l)!fl 0

What we wish to show is that

~(cr,

=if>

If if> e: for all

oEJ/t .)

B , as a subcycle of

'f>

is rational over K*('f) Moreover, we can then see at the same time that for each of the narrow ideal class representatives

v , the set

of the components of

Bv =

V(l)

~IK

-1

(Vv)n B , where

, is also rational over

Vv

is one

K*('f) .

1.7 IK-LEVEL MODULAR CORRESPONDENCES AND THEIR FIXED POINTS Let

CIK(s 0 ) =

1Ks 0 ~

be the double coset determining a

modular correspondence eiK(SO) C: V~ X V~ , where SO belongs to G(Af) , and ~ is an n.o.c. subgroup of G(Af) • The projective coordinate ring of VK has a ~-structure A(K) which was described in the last section. Let us assume that s 0 e: M 2 (~) (within our usual identifications). Let if> be a homogeneous element of degree zero in the quotient algebra; that is, if> = a/b , where a , b e: A(R) and are homogeneous of the same weight. Then we form the transformation polynomial of

if>

with respect th

T<J>,s 0 ,1

By its choice,

N

~(s 0 )

llj=1 (xif>

:

ifl(gsj»

satisfies

S (a) if>

N

, SK<s 0 > =~=lsjlK. if>

for every

11

a £ ~ = Gal(~ab/0) . Moreover, l, ••• ,N • Therefore, if

(3(cr)R(S.)

J

then each coefficient is a ~m-arithmetic modular function with respect to ~ and belongs to M(~) • Moreover, ~ is also an element of M(~) • Let

FP(SK(S 0 ))

be the set of fixed or self-correspond-

ing points of V~ with respect to C~(s 0 ) • Essentially as remarked in § 2. 5 of [ 4 ] , FP implies that

q(a)

is

a scalar multiple of the identity, the essential terms on the right hand side are those for

a

K - k •

E

We have seen

that each of the terms on the right hand side is a cycle rational over

0 •

If

pal

prime ideal

then

S!rl FP 00 (q

will follow that 'V

B

=

::IK (CJ", 'f)

n

1

(rr))

, and thus it

priK (FP 00 (q I (rr)) )

I

and the right hand side is a zero cycle rational over K*('f>o· 'V 'f (In fact this will show that B = VI< q I f"'l .::IK (~I ) • ) But for a given such z I we may, by Prop. 1, write A

z = yt;q(a)Bn , y E G+(Q) , 6 £@ , a£ I(K)f, n E AG(?l) IK 00 and where !;(i.e) = (T) . Write n = n 1 u 1 , n 1 £ G(?l) ul £ IKOO . We now show that ene- 1 belongs either to q(I(K)f)IK 00 or to q(tfi(K)f)IK00 , and this says that z £ Yq(c)!;6(K 00 , where c is either in I(K)f or in I(K)f~f

, so that

z £ B~l*B

=

'V

B

, which is what we want.

,

18

In fact we have z = yE;q(a)en £ FP 00 (TI) c FPIK, for every n.o.c. subgroup ~~ of G(Af) Hence, zq 1 (n) £ G+(Q)Z~00 IK 1 and priK 1 (zq 1 (n)) = pr~ 1 (z) ; therefore, by Prop. 3 (with a= 1), there exists b~ 1 ,f £ ~x such that q 1 Cn>

£

n

-1 -1

e

q(b~~ ,f>en~~

-1 -1

so that q 1 (n) = n e q(b~ 1 ,f>enK 1 for some K 1 £ IK 1 • For the rest of this proof, we may assume ~~ = IK(~) for some integral ideal TL of defined over k) . Let en = w = w 1 u 1 , so that w 1 = en 1 £ G(Af) , and put w" -1 -1 en 1 s Then q 1 (n) = w 1 q(b 0 )w 1 for some b 0 in the closure of Kx in I(K)f , or -1

-1

w 1 q 1 (n)w 1 = w"q(n)w" = q(b 0 ) Now K = k(A) , n = Af , and q = qT . We want to prove that AX b 0 £ K , too. For this purpose we use 2.3.1 A LEMMA BASED ON A RESULT OF E. ARTIN [1] Lemma 1. Let q be a regular imbedding of K in M2 (k) , extended to an imbedding of A(K) in M2 (A(k)) . Suppose that b 0 belonas to the closure of Kx in I(K) A AX X that c belongs to K , c £ K ,. such that K = k(c) , -1

and that for some w £ G(Af) we have w q(b 0 )w = q (~) • Then w normalizes the subalgebra q(A(K)) of M2 (A(k)) and b 0 = b for some b £ Kx. Proof. According to [18: p.62] , if 81 , 8 2 are a k-basis of K , there is a canonical identification of A(K) with A(k)8 1 + A(k)8 2 Since K is a eM-extension of the totally real field k , there exists o £ K such that K = k( 0 ) , o2 £ k , and o2 n , then

20 b = E b , where E rn,n belongs to a small congruence subrn rn,n n group of the units of K • Since K is a pure imaginary quadratic extension of

k , the totally positive units of

are a subgroup of finite index in the units of Chevalley's theorem [ 6]

k

K , so by

on the units, we may assume each

Ern,n is a totally positive unit of k. Let n 1 , ... , nt be an independent set of generators of the totally positive units

k

of

(t = [k:Q] - 1)

w

Since

A

q + q(~~>q<s> = q + q(S>q<s> we have in the limit

~

By discarding the early terms in the 1\

A

rn,no

(a

no

(since 13 -F 0

because 1 k: cf. (2)), where

over in

I(K)f , and each

of

k

Hence, if

•

+ 13

and

no

o)

o

Ern = £ is a Cauchy sequence x rn,no E ~ ~ + , the totally positive ~nits A rn 13m = Ern'~n ~ , we have 13m ~ B = E~ , 0

in the idelic topology.

Since

E

~~ ~

A

A-lA

E

E , Brn E k), a unit of

ern = urneO , where

e0 E

X

~

~ ~

13m

= + 1

13m = ern , where we may assume

K, hence (since Say

-F 0 '

are linearly independent

Therefore,

~-11'\

n0

A

in the idelic topology. place,

= ~ n + sn6 ,

q(a) + q(l3)q(S)

sequence, we have for some large enough

brn

, we have

bn

, and i f

q(;n) + q(Bn)q(;5) in the idelic topology.

A

q(~o)

q(o) =

and

=

(~)

,

for every finite A

ern

is a unit of

k

urn E

for all X

~

A xl rn A xt rn urn= n 1 ' .···.nt ' xj,rn E Z1: We may take a subsequence such that for each

+

rn

for all

rn ,

and

x.

J 'rn

of

Then

Z1: ) • ~

Since

r, 2

since

eo

that = Z1: •

ix.J

lim (ern) =

j = l, ..• , t ,

(the standard cornpactification

converges to

eolirn ~rn

1 , this gives

2

X

By a lemma of Artin [ 1 ] , this implies E 0 + = a. j = 1 , •.. , t , h ence -xj = aj /2 E .;; ,..., .!..., 2~ J

That is,

'

=• •

21

X.

J

~

so that

72:

is in fact equal to A

n

=l,,,.,t

j

I

n = + 1

for a unit

A

b

E K

Since a = y , we obtain + E E k , which proves the lemma.

2.3.2

COMPLETION OF THE PROOF OF PROPOSITION

8

B

Thus y"'

=

E

"'"'

~ Eo

En =

"'X

0

E

k .

=

Returning to the proof of Proposition 8, this shows that But w"q(n)w" thus

Int(w")

and since

-1

=

q(b 0 l

gives an automorphism of

=

A(K)

A(k) + A(k)A

automorphism extends to one of have

b0

=

n

or

b0

tralizer in GL 2 (k~) normalizer of T(k5 ) the identity or we have either

=

n

E

the case described in

,,

q(k)

,

, it follows that this A(K)

Clearly we must

=

T(kJ)

X

q(~)

is its own cen-

and N(~) = T(~)~1T(kj) is the It follows that either Int(w") is

Int(lw") w"

over

qCil

is the identity, and therefore

q(A(K)x)

or

lJ.J"

[11:~4.4.4].

£

Thus

q(A(K)x~ , as in

z =

y~q(a)w'u

1

belongs to G+ (Q) E, [q (I (K) f) V

"'B

the full pre-image of

which shows that 3

"-• B

.

Hence

is a zero-cycle rational over

THE RATIONALITY OF

Proposition 9.

C: ~K

q ( 1 fi (K) f)] 611< 00

B

The orbit

OVER B

of

K*cr) 0 .

K*Cr) CIK ( C})

is rational over the reflex field

in

~ ( &, r)

K*Crl

In the statement and proof of this proposition, the notation retains the same meanings as in the statement and proof of Proposition 8. Proof.

We already know that

~ = B\/1*B

is rational

over the totally real subfield K*Crl 0 of K*Crl B and t*B are equal or they are disjoint; for if

Either a*B(T)

1*a'*8(T) a'*t*B(T) for some a , a' t: I(K)f, then clearly B = t*B , so that B = ~ , and B is rational even over

22 'U

K*(t:) 0 • Now suppose

B

and

l*B

are disjoint.

We want to in the composi tum M(IK, K*(I:')) of K*(I:') and M(IK) , such that ci> is constant cf>(B) 'i' cf>(uB) • on each of B and l*B and such that construct an arithmetic modular function

ci>

3.1 STABILIZER SUBGROUPS AND FIXED POINTS OF CORRESPONDENCES priK(~8)

We start from the chosen point

of the orbit

B , where 8 is an element of ~ and ~ £ ~ 8 ,oo n fK • • Kx

let

IK },

(z) =

J

n xq,

YIK' (z) resp. Y;, (z) denote ~· (z) IK' resp. x~, (z)IK' ZIK, ( z) resp. Z;(, ( z) denote the images of these in

z;,

VIK, Clearly Y;, (z)C YIK, (z) and (z) C ~· (z) for all n.o.c. subgroups IK' of IK We claim that z~, (z) is always finite and that for sufficiently small JK' , we have

23

z;, (

z~,

z)

{z)

•

First of all, trivially,

Int IK (q' (b)) C

IKq' (b) IK

which is compact and open in contained in

=

cl< (q' (b))

G(Af) ; since

IK , this implies that

xq'

IK'

contains represen-

tatives of only finitely many left cosets of hence

~·

(z)

modulo

empty); therefore, ~·

(z)

IK'

for each k 'IK,

e:

IK'

IK'

G(Af)

IK'

of

are finite.

IK'

,

Suppose

that is

~ ¢

IK

where

'

w

= e-1 n Then

K

IK ,

there exist

A

IK'

in

such that -1 w q' (elK, )ul

(*) Let

z;, (z)

and

= x ,IK'nw- 1 q'(Kx)w q

for all n.o.c. subgroups

IK'

on the right is finite (possibly

ZIK, {z)

is non-empty for all such

xq ,IK•n..IJ

is open and

=

"shrink to {1}"

A

~,q' (b)~,

-1

k'IK'

through a linearly ordered sequence

of n.o.c. subgroups constituting a neighborhood basis for the identity in G(Af) , and for each ~· , let ~· , ~· , and k'IK' be chosen to satisfy (*) • Automatically k'IK' tends to the identity in the usual topology on G(Af) , and since IK is compact, by going to a subsequence we may assume limiK'

~

{1}~·

ko £ IK

lim IK'

~

A {1}'11 ,

proving the result. 3.2 MULTIPLIER POLYNOMIALS The coset space

~·

(z}JK'/lK'

is finite, because [IK :IK']

XIK, (z)/~tK' (z) , where IK' , by means of the mapping of cosets

is finite, and may be identified with

~K'

(z) =

~(z)()

x.JIK' (z)

x, y

for if

/K

I

nli(z) Let

~K

,

pr IK

~·

E:

=

~IK'

priK'/IK

Then

and if

0.

YWJK' (z) C

xlK'

, then

x

-1

y

(z)

be the canonical projection of

be the canonical projection of

G+(Q)\G+(A)/lK~

vlK •

(z)

xiK'

onto

prlK(~,

VIK , and (z)) =

is a(finite)zero cycle on

priK

onto

V00

be that of

P:tjK'/lK(~,

VIK, G+(A)

onto

(z)MIK' (z))

VIK , as is also

pr(K(~,

(z)) which

E:

25 i t contains. Fix

1jJ E A (IK , w)

for some

zero at every point of for any

::A

=

ljJM(g) where

such that

If

M E G(Af)

c&, 'f>

t-

ljJ(g)

such that

g E G+(A)

w > 0

nomial in the indeterminate

t,

~

=

Pl)J,IK' ,b(t; z)

, define

'

Cl)J!IM) (g)/l)J (g)

is defined as in §6.1 of

l)JjjM

0

is non-

1jJ

[ 5 ]

Define a polyz E ::A(~,~), by

for given

E XIK' (z)IK'/IK'

( t - ljJM(z))

Similarly define P, 1, b(t.; z) with X co' (z) in place of X __ , (z) 'I'' co' q "11< For the time being, we fix b E K , such that K k (b) , and 1jJ , and omit the indices

M

runs over

XII

=IK'

e~ 1 (q I

v

=IK'

.::. t

i

1

1

X

~£ •

I

X

I

E

s

=

Now if

££

x' , then

~

M , M'

=

{u

pr~,

3A(~, E'> ( z) : If

,

t

~

f 1 , ••• ,

S

Denote by x

E

VIK'

for some

u

, and

z

E

E

JK'

T(z;M,M')

or

,

u E

G+(A) , let

M'w

is unchanged if

for any

w

E ~·

•

M or

K

E ~oo

,

then

M'

Hence, there

z , and for each

the family of such sets depends only on w

o

is equal to and that

S

F (x') A

(b))

lj!M(z) + Fu (zM) - Fu (z) ~ lj!M' (z) } •

C£

JK'C IK, the set

Mw

U

are only finitely many such sets for each E

I

and let

xq' = Int(IK) (q' (b))

E E

F (x)

be

S

e~ 1 (q I

M(IK')

u 1 f 1 (x) + • • • + u£f£(x)

X

is replaced by z

(b))

generated by

~-algebra

a variable point of

I

T(z; M,M') With

be such that

separate the distinct points of

(u 1 , •.. ,u£) Fu(x)

If

e;_, (q'

We may assume that each is regular on

f 1 , ... ,f£

=

(ef, E'>.

Let

fOr SOme SUitably large

f 1 , ... ,f£ E M(IK')

the quotient algebra of the f£ •

associated

iS the Union Of all iterateS

(b))

eJKI (ql (b))

Let

V~~

by this correspondence.

A

the finite set

Of

v~,

denote the set of points of

(b))

in the usual way to

Where

* ,

is any subset of

A

as in [ 4 ] ,

,

~, (q' (b)) C. let

4, determines a modular correspondence

§ 2. v~,

u

MOO t - 1/!M

Cz"l

t-1

0

is also of type M(~) [t] on ~ In other words, 1/!M coincides on ~ with the restriction of some function We now calculate 1/!M ( z) and 1/!M ( Z I) in M(ll

has adelic integral entries and hence that

-lA

~

a- 1 6

el7, while the divisor of a S has the same norm 'f to k as -lA the divisor -:(2 of TI Since q' (a S) E IKq' (n)~ and -lA

=

A

, it follows that a S TT mod 7ZCJ Then c E ( id. (c) 7t) = (1) , 1 1

8-

and

-f , c

=

=

id. (NK/k (c)) we have (c)

(n, n) =

since

~X

r...J

(1)

,

,

and

c

1.

So i f Suppose (c)

TT ( E - 1)

we have

Another choice of, say, same conditions as

=

f? .

-=f?

E€

En , where

id. (NK/k (~)) or

c1

E

=1

=

c1

=

::'

ElTT

Let c = TT mod 7Z (9

h

7l. fY' ; hence ,

c - TI € c

1

= id. (lT) = -:f = (TI) , (TI)

mod 1l and

in place of

with

• C

q' (E) E K (fl.,).

satisfying the

also gives

q' (E 1 ) E

-1

K{n) hence, i f M1 £ q' (c 1 )K(7Z-) , we h~~ M ~ 1 e K(n,) If (c) = = (n) , c = E'i with £' ~ ~, then since 'if

= TI

K (1Z.)

with

1?

=1

so that if

mod

-1

M Ml e. K (n,..) H 8 ~(t) and of

common roots of ((En) -1 *z)

we have

c

is defined in the same way with respect to

M1

c 1 , then ~

n..- ,

E ' _ 1 mod 7l.- and q ' ( E ' ) E and if in the same set-up we have c 1 in p·lace of (ii') c 1 = TiE" , then E I E II - l E K (1Z ) (c 1 ) = 1?

_ c

and

~ (

T TT,'I'"' ( t ; z)

are

Pl (z) =

(E 'iT)- 1 *z)

, and each of these is HB~ , but also a simple root of p 1 (z) -1 p 2 (z) . We have p2

(z) =

Therefore, the only possible

not only a simple root of TTT,~

(t; z) if p 1 ( z)

because

= ~ (TT-l* z)

q' (E)

and

q I (E I)

-?

and

p 2 ( z)

both belong to

~ (TI

-1

*Z)

K(n.)

.

"--{? and are in the same ray class modulo n. X AEC/,TT , a :: 1 modxn , a£K and hence 1T (al72 ' to ~x, belongs which imply (since .. 1 mod 7l) TT = 1T E Hence, E - 1 mod Suppose

'12 =

Then where

n.

34 71- 1 u

=

=

-1

q(a)q(a)oo ~q(71b)8Kn

where

K00 E IK00

since

t;q(nb)e

=

--1

(TI

= q(~lt;q(nb)eq'(E)

t;q(71b)e

*z)

=

= q(a)t;q(nb)8K 00 K~

Kn.. E lK (7l-)

and, of course,

7T -l *Z , this implies p 2 (z)

two polynomials

=

=

(71 -l*z)



Therefore, the monic g.c.d. of the

HB(t)

t - ((E71)-l*Z) which is of type

•

p 1 (z)

q (a) E G+ (Q)

=

T71 ,(t ~ z)

and

is

t - (71-l*Z)

K*(~)MUK) [t]

on

B

since both of the

first two polynomials are, on account of the Weber-Perron theorem. This proves Lemma 3 in case ::{? and -;2 are in the same ray class modulo ~ . Suppose on the other hand that :f! and --:;2 lie in distinct ray classes modulo ~ ~ then for every z F B we have p 1 (z) ~ p 2 (z) and the g.c.d.of HB (t) and Trr,(t ~ z) , which is of type K*(~)M(K) [t] , is equal to (t - pl (z)) (t - p 2 (z)) = L (t ~ 'l'v (z)) , say, provided q' (n) in fact belongs to IKq' (71)1K . In fact, this is automatically so, for det(q' (71)) is an idele of k generating a prime ideal

1

of first degree in

k , which

by elementary divisor theory over k~ implies that (* ) q ' (iT) = K1q ' (71 ) K2 , K1 , K2 E IK ( 1 )

A

=

G {Z:l )

and since (*) is a system of equations over the different finite places of

k , and

71

and

71

are units

=1

mod~~

at every place cti7L {where 7lr;- is the power of Cf dividing 71.,), we may assume for each such ff that Kl and K 2 ~ both belong to IK {7l)} , and for all other Cf- , betong to K {l)f" which is the same as K {7Z)'}- if_ 'f.../'Tl- • therefore, we may take Kl , K2 E K(n) = ~ and q' (rr) E IKq' {71)1K . Now each coefficient of L(t, '!'v{z)) is in K*{~)M{~) , and in particular we have {A)

(

71 -l*Z) + (n -l

for every z E N variables of coeffients in for every z E

*Z)

= R0 {'!'l (z) '• • • ''I'N (z))

B , where R0 is a fixed rational function in which the numerator and denominator D have K*{~) and such that D{'!' 1 {z) , •.. ,'!'N(z)) ~ 0 B •

35 [1~

We may now continue the argument of §5.2 of

to its

end in essentially the same way as there, but now for the case of Hilbert modular functions, as follows: Let

be a positive integer such that ~e

e

-:pe

principal ray mod7l...., i.e., Then we have

71

e E IT.

U n,k.1) ~

=II.{)'

1

II _ 1 modJl.

~ IT. '7J v..TL (v)

-e 71 E

and

lies in the

II E Kx

, where

"" = { E E ~I "x E ~(e) 1 mod7L}. Let ~ be an arithmetic modular form in A (IK , w) of weight w > 0 , which is nowhere

=

S

vanishing on the set the ideal where

For each

th..-=

and

M.

I(K)f , we define

are defined for

QUM)

in §6.1 of [5 ], the ideal of the entries of

E

~q'(a)-1)

'lJta

'U<M)

a

~(M)

M E G (Af)

as

being essentially the g.c.d.

Then for each

z E ~A(~,~)

put

¢a (z) = N'lltaw ¢ (a*z)/¢ (z) If

bE I(K)f , then

Setting

¢a (b*z) = N'Ua w ¢ ( (ab) *Z) /¢ (bH) b = ai , i = O, ••. ,e-1 in turn, we get, analogous-

ly to the classical case ¢ (z) ¢ (ad).···.¢ (ae-l*Z) = N?J ew ¢ (ae*Z) /¢ (z) a a a v'a. ae = a.E: with a £ Kx and E e U1?.. v(M 0 )

-1

-:f!

e

M0 has

= NK/krr ,

= NK/Qnaw ¢(au)/¢ (z) if we take

a

=

71

-1

--1

resp. 71

and

II a =

since, for reasons explained earlier,

¢(E:*(~*Z)) = ¢(;*Z)

II

¢((~E:)*z) =

But now if we put

then

¢(~q(~-lb)6) = ¢(q(a) 00 ~q(b)6) = = while

¢(~kooq(b)6)

=

resp. II

xw

1 '

Ri = {Ri 1 , ... ,RiN) and Rij is a rational function 2N variables with coefficients in K* cr-> , i, j = 1, ... ,

where in

1

II

such that the denominator of each

when the

2N

RiJ' (~

variables are replaced by

z £:::A(~ r-)

, and one obtains the same

in place of

'1T

result with

n

•

On the other hand, for each struct the polynomial in

l, ... ,N, we may con-

i

t

~

Pi{t; z)

is non-zero "' {z) , ~ v ('IT -l *z)) ,

= LM

Tn,¢(t;z) E:

c__(q'(TI))/IK -IK

t

-

(

zM)

-~i(zM)

which is in manner similar to that in a previous case seen to be of type

MID
boundary" [8] and immediately gives

c0•

McKean and Singer worked in the more general context of Riemannian manifolds with boundary.

Their work was generalized to elliptic boundary

value problems by Greiner [6] and Seeley [11].

2.

STRICTLY PSEUDOCONVEX DOMAINS

One analogous problem in several complex variables is the heat equation for the a-Neumann problem.

~n+l, n ~

domain in

(0,1)

2,

A form

forms.

Let

Q

be a bounded strictly pseudoconvex

with smooth boundary. For simplicity, we work on u E c 1 (A 0 ' 1 (Q)) satisfies a-Neumann boundary condi-

tions if u

au

0

norm

norm

on

(2.1)

M.

Let AO,q

u

b

o}

norm

(2.2)

and let (2.3)

denote contraction with the vector field is the inward unit normal vector field on plex structure on

~n+l

N"

= .!_(N v2

M and

+ iJ 0 N)

J0

where

N

is the almost com-

Then we can rewrite the a-Neumann boundary con-

ditions (2.1) as Vu

=

vau.

0 =

(2.4)

The a-Laplacian

D

is defined by vu

--* aa

0

If

u

+

-i = :2:u.dz l. ou

-

o} (2.5)

-*a a E Dom o,

then

2

a ui -i 2 :2:. (:2: --.--. dz ). azJa~J i j

(2.6)

44 1 n+l (z , ••• , z )

Here A

form

(0,1)

are the complex coordinates on ,.n+l. ~ 2 0 1 F(z,t) E c (A ' (Q x R+)) solves the heat equation for

the a-Neumann problem if for fixed

t,

F(·,t) E Dom o (2. 7)

a

+

(at

O)F = 0.

The initial value problem for the heat equation for the a-Neumann problem is the following. Given f E c 0 (A 0 ' 1 (Q)), find a solution F of the heat equation (2.7) satisfying (2.8)

lim F ( ·, t) = f. ~

The solution to the initial value problem is given by applying the semigroup generated by

-o,

to the initial value.

given by integration against a smooth kernel

c= 0, if v 1 "I 0,

where v 1 ,v 2 are sections in H0 (Cx,O(Nx ® n* H*)). We remark that these conditions are formally the same as those encountered in the case n=l (X a kahler surface) except that in higher dimensions one has to check that ¢ is closed on n- 1 (t) in (v), and that ¢(v 1 ,acv 1 )) is (positive) definite in (vi).

These last facts are automatic for n=l.

Conversely, one has the following inversion theorem, due originally to Penrose, in a Minkowski form, for n=l, [4]. Theorem: Let z be a complex 2n+l manifold admitting maps n, p, a and differential ¢ as above, verifying (i) to (vi). Then there exists a unique hyperkahler metric on X such that Z=Z(Xl, the twister space for that metric.

54 Proof: By a standard deformation theory argument (cf. [7], for example), one can identify X with a component of the oreal points in the space of all sections of the map n. ~

this identification, we can identify T (X) H0 (C ,O(N )) , where C =p X

X

X

-1

¢

Under

with

X

(x), and T (X) with the o-real X

vectors in H0 (Cx,O(Nx)). For x € X, let Ex=H 0 (Cx,O(Nx ~ n* H*)). By condition (iv), dimE =n, and E=U E is a smooth vector bundle. Denote X x€X X H0 (IP 1 ,0(H)), by V and identify V with the sections in H0 (Z,O(n* H)) pulled-up from IP 1 • above, T (X)~¢

E

®

By condition (iv), and the

Let w0 denote the skew-form on V such The form¢ in (v) defines a non-degenerate V.

that w0 (z,w)=l. skew-form on E, which we continue to call just ¢, and is a non-degenerate symmetric form on T(X)~.

G.9

w0 By condition ¢

(vi), ¢~ w0 is a real symmetric form on T(Xl~¢, i.e., is real on T(X), and is positive definite on T(X). Denote this metric by g. Using this metric, we identify T(X) with T*(X). Then ~ 2 (El&s 2 (V) + s 2 (E)~~ 2 (V). Under these identifications, we have a distinguished three dimensional space of

~ 2 T*(Xl®¢ =

2

2-forms on X, of the form ¢ ® S, for S € S (V). Such a 2-form is real if and only if S is real with respect to o 0 . LetS be real € s 2 (v), with jsj 2 =2: such an S defines an almost complex structure on X, via the 2-form ¢@ S and the metric constructed on X.

More explicitly, fort € IP 1 , let

st be a section in V such that st(t)=O, and 2

jstj =w 0 (st,o 0 (st))=l. uniquely as s = st

-i(st de f.

Then every such Scan be written

® o 0 (stl

+ o 0 (stl

® stl

for some t e IP 1 . Writing¢~ st in terms of the metric g and a skew-symmetric transformation Jt on T(X)g¢, one checks that Jt is +ion the subspace € ~¢·o 0 (st)} and-ion E

® {¢ •St}.

55

Lemma:

The map p, restricted to Zt=n

-1

(t), is holomorphic to

X with the almost complex structure Jt. In particular, Jt is integrable, and ~ ~· st is its kahler form for the metric g. We postpone briefly the proof of the lemma. ~®(o 0 (st)~o 0 (st))

The complex 2-form represented by

is of

type (2,0) for the structure Jt, and its pull-back via p to Zt agrees with the restriction of

~

on Z to Zt.

Since

~

is closed

on zt by condition (v), ~t is closed on X. Since the 2-forms ~t' t€IP 1 , on X span the same three-dimensional space as the forms ~ (iP St, we conclude that ~ ~ St is closed. metric g is kahler for each Jt, and

~t

Hence, our

is covariant constant,

as a linear combination of kahler forms for various of the 1 structures Jt'' t' e P Thus, g is hyperkahler. Finally, the twister space of g is the manifold Z(X)=X x IP 1 , where the complex structure on X x {t} is given by Jt. By the lemma, Z ~ Z(X) by p x cluding the proof of the theorem.

TI

is biholomorphic, con-

Proof of the Lemma: We have only to make explicit the identification of T(X) and the a-invariants in r(c ,N ) to evaluate X

X

X

the differential of p. Fix q € Z such that p(q)=x, n(q)=t. We identify geometrically T(Zt) and the real normal vectors IR IR q N to C at q, and then N with the complex vector space N q

at q.

q

X

X

The o-real holomorphic sections of Nx are nowhere zero

on ex' explicitly, they are of the form s = v®o 0 (st)- o(v) for v € Ex.

®

st

The differential of p sends s(t) to

s € H0 (Cx,O(Nx)) = T(X) ®¢.

So, dp*(v(t)®J 0 (st) (t)) ~ 0 (st) - o(v)~t' and dp*(i v(t)~ 0 (st) (t)) = iv ® a 0 (st)+io (v)~t· On the other hand, since 2

wo(st,o 0 (st)l=lstl =1, we calculate

56

and

Thus, p restricted to zt is holomorphic. §2.

The Geometric Construction Let us fix some notation for describing the generalized

flag manifolds.

G is a complex semi-simple lie group,

e

an

anti-holomorphic Cartan involution fixing u, a maximal compact subgroup of G. algebras.

Gothic letters denote corresponding Lie

Tis a maximal torus in U,

respect to T, and Xa. €

'J- a

~the

roots of 'with

non-zero root vector for a..

For

z 0 € it. c ,, ~ + ( z 0 ) = { a.€~ I .:!:_a. ( z 0 ) > 0 } , and fl.+= (j) ¢ • X+a.. - a.e~+(z 0 J Let C:(z 0 ) be the centralizer of z 0 in1f,1'+=()t(z 0 )tti'tL+' and P + the normalizer off+ in G under the adjoint representation. Our flag manifold M is-the quotient space G/P+.

Since

9(z 0 )=-z 0 , S(P+)=P_, and G/P+ is conjugate biholomorphic to G/P_ via e. The holomorphic tangent bundle to M is the quotient of -1

G x(11f+l by the action of P+' p• (g,X modl"+)=(gp ,Ad(p)X modtJtl• Let B be the Killing form of,. Under B, the dual of~,+ is ~+'

and the cotangent bundle of M is G

of P+, p• (g,Y)=(gp- 1 , Ad(p)Y). P+.)

x~+

modulo the action

(Note that~+ is normalized by

The tangent space to T*(M) at (g,Y)mod P+

is~xft,+

mod

vectors of the form (X,-[X,Y]), X €'!"+ (these are the tangents to the P+-orbit through (g,Y)).

Thus, each tangent vector to

T*(M) at (g,Y) has a unique representation as (X_,X+), with X+ € 11. +.

The canonical symplectic form w on T* (M) is given at

(g,Y) by w( (X_,X+), (Y_,Y+) )=B(X_,Y+)-B(Y_,X+)-B( [X_,Y_] ,Y). Finally, let 1>lct®¢ be the center of ct(z 0 ). The u-invariant kahler metrics on M are given, at e mod P+ € M, by (X,X)=-B(X,ad(z)S(X)), where z € &Lsatisfies a.(z)>O, all

a € ~+ (zo).

Here,

X

e Vr+·

57 Note that the same constructions, mutatis mutandis, work for the homogeneous space G/P_ and its cotangent bundle. We want to build a twistor space in two patches over IP 1 Let u 0 =IP 1 -{oo}, Ua=IP 1 -{0}, t, the affine parameter on u 0 , r; the affine parameter on U00 , r;=l/t on Ua f\ Uoo . We want fiber spaces over u 0 , U00 such that the fibers over 0 and oo will be T*(M) and its conjugate respectively. We view T*(M) as T*(G/P_).

The problem will be to patch these two spaces over

uo" uoo. Let us fix z in the center of

a

e

6+(z 0 ).

Consider the space

P+ acts as above. (g,tz +

X+l~t

e

~(z 0 )

z0 =G

x {¢•z +~+}/P+, where

Since z is central in

which factors through

~(z 0 ),

z0 :

¢=u 0 is well-defined on

map rr is a submersion.

such that a(z)>O, all the map

call it rr.

The

We also have another map

z0 .

We have:

q

ZO

-+lx¢

+rr

+p2

uo

¢

On rr -1 (t), tiO, q I rr -1 (t) is biholomorphic.

The map q gives a

resolution of singularities of the variety in ~x ¢ given as

z

the Zariski closure

x ¢*1 X=Ad(g) (tz), 3ome 0 of {(X,t) € (Note that qlrr- 1 (0) is the moment map of the action

g € G.}.

of G on M.) rr:

Over Ua' we construct Z00 =G x {¢z +~_}/P_, and define (g, r;z + X_) ~ r; € U00 , and q :

(g

r; z + X-)

I

The image q(Z ) is ·ra-x

~

'~-*

•

00

-+

Z. 00

(Ad (g) ( r; z +

X-)

I

r; ) € ~

X

¢.

Consider the transformation f of -2 -1 X, t ) . This takes z 0 to Z00

send1ng (X,t) to (t

over ¢*=u 0 ~ U00 ,

if we set r;=t- 1 .

obtained by gluing

Denote by

z 0 ,zoo in this way.

z

the manifold

The map q is globally

58 defined on Z to

'fJ- ® H® 2

over IP 1 .

Note that Z has a G-action

induced from g• (g' ,X) = (gg' ,X) on G x,., and q is equivariant G acting on '0-0, ~;( [X,I;]

I

-B(I;, [X,Y]).

[Y,I;])

B([X,I;],Y) = -B(X,[Y,I;]). It is clear that 4>0,I; is well-defined, i.e., the right hand side is 0 if [X,I;]=O, that 4>0,I; varies holomorphically with 1; in~, is G invariant of type

(2,0) on Zt' and is readily

checked to be closed on each zt. To represent this on z 0 , take (g,tz + Y0 ) e z 0 , when Y0 € ~+' g € G, tiO, and consider Lie alge~ra elements X_, X' __ in.,._, Y+' Y~ in"+" Let Y+ €f1+ verify [Y+, tz + Y0 ] = Y+: Y+ exists, since tiO and ad(z) is invertible on~+" Under q, the tangent vectors (X_,O) and (O,Y+) to z 0 are repre~ented by [Ad(g) (X_),Ad(g) (tz + Y0 )] and Ad(g) Y+ = [Ad(g) Y+' Ad(g) (tz + Y0 )J at I; = Ad(g) (tz + Y0 ) in similarly for X~, Y~. Hence, on z 0 , we have: 1)

4>0,I;((O,Y+),

(O,Y~))

-B (Ad (g) (Y+) 2)

4>0,I;((X_,O),

4>0,I;((X_,O),

Ad (g)

I

(Y~))

0.

(O,Y+)

= -B(Ad(g) (X_) 3)

=

I

Ad(g) (Y+))

(X~,O))

=

-B(Ad(g)(tz + Y0 ), -B(tz + Y0 ,

[X_,

[Ad(g)(X_),

Ad(g)(X~)])

X~])

From these formulas it is clear that 4> 0 extends holomorphically across t=O in z 0 , and agrees with the canonical 2-form -1 on n- 1 (0) = T*(M). One constructs 4> 00 on Z00 -TI (oo), and extends across Z00 ,

in a completely similar fashion.

Under the patching

map f, we have f*(j> 00 so that 4> 0 , 4> 00 patch to give a section of ~ 2 ®n*(~2 ) Z/Pl

60 globally.

The reality condition (vi) of §1 follows directly

from the fact B(8(X),9(Y)) = B(X,Y), X,Y € Next we calculate the normal bundles of the real curves

e u

(u, tz), u

z0 ),

(represented in

tivity condition in (vi), §1. assume u=e.

and to verify the posi-

Without loss of generality,

By a theorem of Grothendieck, the normal bundle

to our section, call it C, splits as a sum of line bundles:

~

n* H@ii = N.

i=l By the non-degeneracy of Nc

~

n*(H*) =

N~

~

constructed above,

® n*(H).

Since the integers d. are uniquely determined, we see that 1

each di is 0, 1 or 2, and the number of O's is equal to the number of 2's.

To show that all di=l, it suffices to show

that for every vector v in the fiber of N at t=O, there exists a global section V of N over C such that V(O)=v, and V(oo)=O. (This will guarantee that no di=O, and hence no di=2.) X

€,,

let VX denote the vector field on Z, the derivative of

the action of exp(sX) € G on Nc.

For

z.

VX along C gives a section of

If X=X_ €1,_, VX_(O) = (X_,O) €11._

x~+

= T(T*(M)) at

(e,O) = N at (e,O). On the other hand, if X=X+ €~+' c 1 VX (0) = (X+,O) = 0 inN at (e,O). Thus, -t VX defines a + c + holomorphic section of Nc' which vanishes at t=oo, and whose value at t=O can be computed at lim t+O - lim t+O

1 d

t ds(exp(s X+), t z)

Is=O

1 d

t ds(e, Ad(s X+) (t z)) ls=O

(O,[X+' z]) Since adz is invertible on~+' this is an arbitrary vector in 0 Xfl_ c. Nc at (e,O).

Finally, for X_ € '1.-C~-' VX- is 0

61

.

in N at t=co, since in zoo' G X fl_ is divided by p c The preceeding argument shows that the (nowhere vanishing)

® TI* (H*)

holomorphic sections of Nc

over u 0 , equivalent to

1)

t1 VX

2)

are spanned by 1

~

vx_ over 0

over u 0 , equivalent to VX

+

00 ,

over U00 ,

+

The action of a on global sections is given by

for X+ int\.+. 1

tve(x_l'

over u 0 • Thus, ~ 0 (vx , a(VX )) is a constant, which we evaluate pointwise over u 0 as

- t1 B(t z, [X_, 8(X_)]) B(X_, 8([z,x_])) > 0, if X Similarly,

'/ 0.

~ 0 (~ VX ,a(~ VX )) +

+

-B(X+' 8([z,X+])) > 0, if X+'/ 0. Finally, ~ 0 (~ VX ,a(VX ))=0.

Since~ is everywhere non-

+

degenerate, this suffices to verify the positivity of ~(v,a(v))

· §3.

everywhere, concluding the proof of the main theorem.

Global Calculations The metrics constructed in the preceeding

algebraic in nature.

§

are rather

Thus, although the existence proof is

local, one expects these metrics to have global extensions to T*(M), to be complete there, perhaps asymptotically locally

62 flat at infinity, and unique subject to these conditions.

In

this § we work out the global extension in the case where M is a compact hermitian symmetric space.

Without loss of general-

ity, we assume M is irreducible, and G is Sl(n,¢), Sp(n,¢) or SO(n,¢).

1)

We fix the notation briefly.

G = Sl(n,¢).

P+ is given by block upper triangular

matrices p

(: I :}

q

p+q=n

z 0 is of the form~ :pb

:q ') , a,

b real, where

pa + qb = 0 and a-b>O. 2)

G = Sp(n,¢) preserves the skew-form represented by

P+ are as in 1), if p=q, restricted to the subgroup Sp (n, ¢l . 3)

G = SO(n,¢) preserves the symmetric form represen ted by

Q

G

I

n-2

0

p+ is of the form

0

a I 0

n }1

0

n-2

}n-2

}1

-:')

I

where a' denotes transpose.

a

e

zo

¢n,

is of the form

63

0

' ->..)

, A.>O.

For all three cases 8(A)=-A'.

For symmetric spaces

-1\.+ is abelian. We will describe the real curves as in §2 above, but take advantage of the following ansatz, which is immediately valid only in the case of symmetric M: given Y0 € rL +, we seek tC C=C(Y 0 ) e ~- so that (e , tz 0 + Y0 ) mod P+' or equivalently

describes a real section of Z passing through (e,Y 0 ) at t=O. We, of course, want C to vary real analytically with Y0 , etc., all of which will be clear by the construction. Note that (adc) 3 =0 in our cases, so that Ad (etc) e

(tz 0 + Y0 )

t adC

(tz 0 + Yol

= tz 0 + Yo + t[C,Y 0 ] + t

2

t2 [c,z 0 J + - (ad C)2 t > 0

( 1 )

71

t

(z) : = cr (z

z)

is subhannonic and continuous on the unit disc hamOIJ.ic and

Coo

the Kahler IlEtric details see

[s]).

on

ds 2 = d d ct on After changing

subharnonic function

a:• . f

m

6* : = 6' {0}

1Jr

on

0::

( 2 )

6 c 0:: , strictly sub-

and a sinple calculation shCMS that is canplete at

f..*

t

slightly near

which is

Coo

a6

0

(Fbr rrore it extends to a

and strictly subharrronic on

Finally, in the situation of thrn. 2 we choose functions

E ll>(M)

A = {x E M : f 1 (x) = · • • d cr2 on M and r;ut on D

with

:Kahler !1Etric

= f m(x)

f 1 , ... ,

= 0} , a conplete

This metric has the desired properties. Remark.

Notice that

1Jr

is continuous across

A .

'lheorem 2 shows that the property of being canplete Kahler does not inply holanorphic convexity . .Additional assunptions have to be made. They can go into two different directions: A)

Curvature assunptions on the canplete Kahler IlEtric.

B)

Regularity assunptions on the boundary of

D .

we will discuss sane recent results concerning A) . Section 3 then deals with B) . In section 4 we will study the weak 1-canpleteness

In section 2

of certain canplete Kahler danains

2.

D cc M where M is not Stein.

CUrvature conditions

Let M be a canplex manifold and

D cc M an (open) danain. We can

ask the following questions: a)

Which curvature conditions on a canplete Kahler metric

guarantee or even characterize that

D is locally Stein in M

ds 2 (and,

on D

72 therefore, globally Stein if M is Stein)? b)

Let M be non-compact. Which curvature conditions on a corrplete ds 2

Kahler metric

on M guarantee or even characterize that M is Stein~

Let us first consider the question a). It has been clarified very much

[3]

by the work of Cheng and Yau

0![1

In

Theorem 3.

and Mok and Yau

it is shown: Let

D cc M be a (non-ccmpact) danain on a ccrnplex manifold M.

D such that

Suppose there is a complete hermitian metric on - C then

0 ![! .

~

Ricci curvature

~

0 ,

is locally Stein (and, hence, Stein, if M

D

The same statement also holds, if complex manifold M, such that

rr : D

-+

is Stein or M

=

lPn).

M is a Riemann dorrain over a

rr(D) cc M and there is a metric on

D

as above. Remark.

Since a Stein Dorrain

D always can be .imbedded into some

~ ,

it always carries a corrplete Kahler metric with non-positive Ricci curvature (see f. i. Kobayashi, Nanizu

0 ?J ,

prop. 9. 4) .

In generalizing previous work of Griffiths also was proved in

0 ![!

0~

and Shiffman [?~ it

that local Steinness follows fran a

condition on

the holomorphic sectional curvature: Theorem 4. M with on

Let

rr : D

M be a Riemann dorrain over a complex manifold

-+

rr(D) cc M . Suppose, there exists a ccrnplete Kahler metric

ds 2

D which satisfies the following condition : There is a function

A : lR

-+

lR

with

sectional curvature

J-~

A(t) = 0

K of K( T)

for all

ds 2 ~

A (d(q,'1a))

q E D , T E T~ 0 M , where

distance with respect to locally Stein.

such that one has for the holarorphic

ds 2

d ( · , '1al

denotes the geodesic

from a fixed point

'1a

E D . Then

D is

73 Remark.

Again, since a Stein domain

D can be imbedded, it always carries

a complete Kahler metric with non-positive holamorphic sectional curvature. The proofs of both theorems 3 and Kontinuitatssatz of Hartogs holds on

4 are given by showing that the D as a consequence of a Schwarz

lemna for volume fonns (in the case of the Ricci oondition) due to Yau. The existence statement in the remark after theorem 3 can considerably by sharpened for Riemann domains over Stein manifolds. Cheng, Mok and Yau

proved in Theorem 5. such that

[3] Let

and Q~ together:

11

D

--+

M be a Riemann danain over a Stein manifold :vi

rr(D) cc M . Then

D carries a complete Kahler-Einstein metric.

(It automatically has negative Ricci curvature.) Fbr the highly conplicated proof of this theorem, which starts with the solution of the Monge-Ampere equation det (

a2 u

l

with boundary values

~

az. · az.

=

e(n+1)u

J

on strictly pseudoconvex danains in

af1 ,

the

reader is refered to the original articles. VE now

c<J.Ire

to the discussion of question

b) from above. We only can

nention a few results in this direction. Elencwaig considered in situation of a pseudoconvex domain Theorem 6.

If

D.:?] the

D in a Kahler manifold M and showed:

D cc M is a pseudoconvex domain in a Kahler manifold M

with positive holornorphic bisectional curvature, then

D is holomorphically

convex. ~·

Of course,

R.E. Greene and

M need not be holornorphically convex. H.H. Wu [}5~ gave already in 1974 the following list

of different properties which imply the property of being Stein:

74 Theorem 7.

let M be a corcplete Kahler manifold. Then M is Stein, if

it satisfies one of the following properties: is sinply connected and its sectional curvature is

1)

l\1

2)

M is noncc.upact and its sectional curvature is

;;;; 0

~

0 ;

and strictly

positive outside a cc.upact set; 3)

M is noncorcpact, its sectional curvature is

,;; 0

and its holonorphic

bisectia1al curvature is strictly positive. 4)

M is noncc.upact, its Ricci curvature is strictly positive, its sectional curvature is

~

0

and its canonical bundle is trivial.

In all these cases the hypotheses are essentially made such that it is

possible to construct strictly plurisubharrronic exhaustion functions on M using the geodesic distance with respect to the given Kahler metric.

3.

ve

The boundary of a::mplete Kahler domains

now care back to the approach of the thesis of H. Grauert

considering case

[1-[]

by

B) of the introduction. More precisely:

let M be an open carplex manifold and D cc M a carpleteKahler danain. We want

to investigate the nature of the obstructions in

aD

that might prevent

D from being locally Stein. H. Grauert proved in [}-[] in this direction: Theorem 8.

If

D as above has real-analytic srrooth boundary, then

D is

locally Stein. This shows that the obstruction can, indeed, be ruled out by regularity assurrptions on

aD • In

fact, the nature of the counterexanq:>les of

theorem 2 might be considered as a I!Otivation for the question whether all obstructions are located on the so-called "thin carplement" of

D ,

namely the set A

( 1 )

75 This is indeed the case. one has Theorem 9.

D cc M be complete Kahler and suppose that

[4] Let

locally Stein near all points

x E A defined as in ( 1 ) • Then

D is is

D

locally Stein everywhere. Remark. D cc M

As a consequence one obtains that 0 any complete Kahler danain

which is topologically fat, i.e.

i5

= D , is everywhere locally

Stein. This result was first proved by T. Ohsawa

aD

assl.liTption that

is

the same line as T. Ohsawa as the manifolds is the main tool. In

[?'[! (

of H. Skoda fran Proposition.

Let

D cc

[?fl

under the additional

c 1-srrooth. The proof of [4] follows in so- far L 2 -theor:y of

[4]

a

on canplete Kahler

this is combined with the techniques

8~ might also be used) in order to prove:

afl

Then there are holorrorphic functions

n l:

o

hJ. (z) (zJ. -zJ.)

j =1

h 1 , ••• , hn

=1

on

on

D

f i5

z0

be a complete Kahler danain and

such that

D.

Theorem 9 is then a simple consequence of this proposition. The next question which now, obviously, has to be asked is: Question:

What is the nature of thin canplerrents

Kahler domains

2. A =- D 'D

of complete

D cc M ?

In particular, because of theorem 2 one might ask: Are all thin complements

A as al:ove complex analytic? I t turns out that this is

indeed the case under certain additional regularity assumptions on

At first, T. Ohsawa proved 1980 in Theorem 10.

Let

and suppose that

A

[?:[]:

A be a thin complement of a complete K4}ler domain A is a

codinension 2. Then

c

1 -smooth

real submanifold of

i5

of real

A is a complex-analytic hypersurface in

0

i5 •

Ohsawa's method of proof, which, again, makes essential use of the L2 -theory of

a

for solving

a-closed (n, 1) -fonns on complete Kahler

D

76 domains, does not seem to be generalisible to the cases of higher (real) codirnension of

A . Therefore, new methods had to be used in order to show:

[s]

Theorem 11 .

let

D and suppose that codimension

0

A is a real-analytic subvariety of

3 . Then

~

1)

Remarks.

A be a thin corrplement of a canplete Kahler danain

i5

of (real)

A is camplex-analytic.

Notioe, that srroothness of

A is not assumed. This does,

however, not cause essential new difficulties, since it is not difficult to prove (see [5]) that real-analytic subvarieties which are complexanalytic at all regular points, are necessarily corrplex-analytic everywhere. 2)

Theorem 11 says, in particular, that real-analytic subvarieties of

odd codimension

1

>

never occur as thin canplements of complete Kahler

domains.

Because of remark 1 one may assume in the proof of theorem 11 that is srrooth. Then the idea of the proof is as follows: complex-analytic at a point

If

A

A is not

z 0 E A , a small neighborhood

U can be

chosen and a closed 2-dimensional canplex submanifold X c U such that y : =X

nA

is a real-analytic curve. Then the restriction of the given ds 2

Kahler metric

to X ' y

in such a way that there is a plurisubharmonic function is

C~

on

!;,

fact that

n

t;,r

dcr 2 • Furthenrore,

y • The corrplexification of

regular disc t;,r \ y

and induces

X, y

corrplete along

y

da2

~

y

on X which

can be kept

now contains a complex

!;, n X = !;,r n y • let 1Jr : = r is at infinite distance with respect to

c X with

r

dcr2

is changed into a new Kahler metric

I

!;,

•

r d de 1Jr

Then the on

gives an :immediate contradiction to the other fact, that the

Laplacian of

1Jr

on

t;,r

is locally integrable.

This proof is very simple. But canpared to Ohsawa' s theorem 11 the result also seems to be much weaker because of the assumption that

A is

Cw . One has to ask whether Ohsawa's result also holds in higher codirnension for real

c1

subrnanifolds. Surprisingly enough this is not

the case. Namely, one has: Theorem 12. [s] Fbr each integer k ,;;; 3 c~-submanifold A of pure codimension k

there is a closed real in a ball B which is nowhere

77 complex-analytic and such that

B' A is, nevertheless, complete Kahler.

The construction of such manifolds

d;)

technical, since they may not be

Ac B

is naturally rather

anywhere because of theorem 11 . Let

us indicate sorre basic ideas of it for the case of a complex Here

Coo-submanifold

A c B c ~3

A is constructed as a graph over the

coo-function

The function

polynomials

2-dirrensional non-

(i.e. the case of real codimension 4. (x 1 ,x 2 )-plane of a

G:- valued

f :

f

is, of course, constructed by an approximation by canplex

Fk on

~ 2 • But in order to avoid that

A has a kind of

"camplexification"

which 111'0\lld, indeed, prevent make sure that the sequence

but that

B' A Fk I

from being complete Kahler, one has to

JR2

ronverges in the

Coo- sense near

0 ,

diverges for all (z 1 , z 2 ) E C2 ' lR 2 near 0 . The canplete Kahler rretric ds2 on B ' A is obtained by constructing a Fk (z 1 , z 2 )

continuous plurisubharrronic function that

ds 2 :

= d de - 1 satisfies:

The set 1jr (r e

P : = {8 E @,2rr)

has linear measure By this

curves

y

~rethod

~

ie

) < - 1000

for sone

0 < r

~

r0

}

e •

we have ensured that there are many real disjoint

running in

A'

5"1 '

towards

A'

on which

-

1CXXJ

Ic I

every-

where. In fact, the union

W of these curves has positive ~reasure. The

plurisubhannonic function

A math.~

[14] GRAUERI', H.:

f~s.

(1984), 1-45

Charakterisierung der Holonorphiegebiete durch die

vollstandige Kahlersche Metrik. [15] GRIFFITHS, P.A.:

Math.Ann . .111_ (1956), 3.3-75

TwO theorems on extension of holanorphic mappings.

Invent. Math. 14 (1971), 27-62

[}5
Rem:Jvable singularities for positive currents.

J\m.J.

Math. 96 (1974), 67-73 [17] KOBAYASHI, S., NCMIZU, K.:

Foundations of differential geanetry II.

New York 1969. Interscience Publishers [18] MJK, N., YAU, S.-T.:

Corrpleteness of the Kahler-Einstein metric on

bounded danains and the characterization of danains of holaTPrphy by curvature conditions. [19] NAKANO,

s.:

Proc.Syrrp,Pure Math. 39 (1983), 41-59

Vanishing theorems for weakly 1-conplete manifolds.

Number theory, algebraic geanetry and ccmnutative algebra, Kinokuniya, Tokyo ( 1973) [20]

NARASIMHAt~,

R.:

The Levi problem in the theory of functions of

several canplex variables.

Proc. Int. Congr. Math. (Stockholm 1962) ,

385-388 [21

J

OHSAWA, T. : RIMS, Kyoto

[22] OOSAWA, T.:

On canplete Kahler danains with ~'

C1-boundary.

Publ.

929-940 (1980)

Analyticity of canplements of canplete Kahler danains.

Proc. Japan Acad. 56, Ser. A, 484-487 (1980)

87

8fl

RIOffiERG, R.:

Stetige streng pswdokonvexe Funktionen.

.Math.Arm.

175 (1968), 257-286

8-!1

SAOOLLAEV, A. :

A l:x>undary uniqueness theorem in

tfl .

Math. USSR

Sb. 30 (1976), 501-514 8~

SHIFFMAN, B.: manifolds.

8~

Extension of holOIIPrphic maps into hermitian

Math.Arm.

S:rBC:NY, N. :

SKODA, H. :

(1971), 249-258

Quelques problerres de prolongement de courants en

analyse corrplexe.

~'[I

12i

D.lke .Math.J. 52 (1985), 157-197

Application des techniques

L2

a la

theorie des ideaux

d'une algebre de fonctions holamorphes avec poids. Nonm. Sup. ~![!

SKODA, H.:

~

Arm.Sci.Ecole

(1972), 545-580

MJrphismes surjectifs de fibres vectoriels semi-positifs.

Arm. Sci. Ecole Norm. Sup.

11

(1978), 577-611

Klas Diederich Bergische Universitat GHS Wuppertal Mathematik GauBstr. 20 D-5600 Wuppertal Fffi

On the Minimality of Hyperplane Sections of Gorenstein Threefolds Maria Lucia Fania and Andrew John Sommese To Wilhelm Stoll on his Sixtieth Birthday Let

X

be a normal irreducible three dimensional projec-

tive variety whose local rings are Cohen Macaulay and whose dualizing sheaf,

KX

is invertible (see §0 for more details).

We will call such a variety a Gorenstein threefold throughout this article. We say that a pair

(X,L)

with L an ample line bundle on

a Gorenstein threefold has non negative logarithmic Kodaira dimension [I] if there is some integer 0 h (x, (~ ~ Llnl > o . Assume that

n > 0

such that

(X,L)

is such a pair and that there is at This conleast one smooth element of the linear system ILl dition of course implies that X has at most isolated singularities. The logarithmic Kodaira dimension condition on (X,L) implies that all smooth elements of Kodaira dimension (see §2).

ILl have non negative

The main theorem of this paper (which generalizes [So4], [SoS J,

[So6 J) is the following.

Main Theorem.

Let (X,L) be as above.

There exists a pair

(X',L') with L' an ample line bundle on a Gorenstein threefold X

such that there is a holomorphic surjection n: X---+ X', X as X' with a finite set F c X' reg blown up, Which satisfies the following conditions: expressing

a) given a smooth n(S) is the map of S

s E ILl I TI s : s---+ S' onto its minimal model,

b) L' = [n(S)J for smooth S E ILl and there is a one to one correspondence between smooth S' E IL'-FI and smooth S E ILl gotten by sending such S' to their proper transforms in X , c)

Kx,~

L' is numerically effective.

90 ~11

the corollaries of the smooth version of this result

from [So4J, [SoSJ, [So6], and [So7J carry over with little work; we will discuss these results in another place. ~

remarked earlier, the smooth version of this result was

proved by the second author [So4], [SoSJ. classified all the pairs

(X~)

In [So6], he further

with L an ample line bundle on

a smooth threefold X and the logarithmic Kodaira dimension of (X

~)

negative.

These last results were never published be-

cause N. Shepherd-Barron showed that they were easy Gonsequences of Mori's Theory of extremal rays (cf. [Mo], [Ka 2 J); for explicit details on the use of Mori's Theory see [Be+Pa].

We needed (e.g.

[Fa]) the results of [So4], [SoSJ, and [So6] for local complete intersections with isolated singularities.

In this case Mori's

theory does not apply but the methods of [So6J combined with the results of [L+So]

(see (0.9)) work.

We use these methods to prove the theorem stated above. substantial part of this paper is identical with [So6]. a sequel we will deal with the pairs (X

~)

A

In

where the logarithmic

Kodaira dimension is negative. Let us give a detailed description of this paper. In §0 we give background material and results for which we don't know a good reference.

In §1 we recall the basic results

on the Fano-Morin 3 dimensional adjunction process.

We work in

more generality than needed since we will use the results in a sequel to classify pairs with log

(X~)

< 0.

In §2 we prove the main theorem. The authors would like to thank the Max Planck Institut fur Mathematik for its support.

The second author would like to thank

the University of Notre Dame, the National Science Foundation [NSF Grants #M 0. for all effective divisors lowing important lemma. (0.5.1)

Lemma.

Let

S

C

on

This implies that K8 • C ~ 0 S . This implies the fol-

be a connected smooth projective sur-

face of nDn=negative Kodaira dimension. for some t > 0 or (Kg + L)

(Kg + L)

~

Then either

K;

IDs

L · L + 1 •

L = 0 then since some power K~ with t > 0 of Ks t Ks has a non-trivial section, Ks = IDs· Therefore it can be assumed without loss of generality that Ks L > 0 Proof.

If

.

Let n:

S

Then Kg

----+

S' be the map of

S

.

onto its minimal model.

n*Ks• + P where Pis an effective divisor satisfying

-k = P • P =Kg Ks- Kg, · K8 ,. Since S is obtained from S' by a sequence of blowups, we see that k' = c 2 (s) - c 2 (S') equals the number of reduced and irreducible components of P. By the invariance transformations k' Therefore:

of K5 , · Kg, + c 2 (S') under birational k . By the ampleness of L, L · P > k'.

(Kg+L)

(Kg+L) = (n*Ks, + P+L)

(n*Ks, + P+L)

Kg'· Kg, + P • P + L · L + 2 (n*Kg,) > -k+L · L + 2(n*Ks,)

• L + L · P + k'

L

L + 2 (n*Ks,

· L)

with equality in the last inequality only L + L • P > L · L if L • P 0 and (n*K8 , • L) = 0 The former equality implies that S S' and the latter inequality combined with this implies that lemma.

K5 • L = 0 •

This contradiction proves the

0

(0.6) The Hirzebruch Surfaces [Ha 2 , pg. 369ff; Nag]: By Fr with r ~ 0 we denote the rth Hirzebruch surface. Fr is the unique holomorphic lP ~ bundle over lP ~ with a section E satisfying E • E = -r. Let n: Fr--+ lP~ denote the bundle projection. In the case r = 0, F 0 is simply lP ~ x lP ~ • In the cases r ~ 1, E is the unique irreducible curve on Fr with negative self intersection. By Fr for r ~ 1, we denote the normal surface obtained from Fr by blowing down E. In case 2 r = 1, F 1 is lP~. A basis for the second integral homology of Fr is given by E and f, a fibre of n; of course f · f 0 and f • E = 1. The line bundles on F are given by [E]a ® [f]b r and the latter i; ample if and only if it is very ample, and it

94 a > 0

is very ample if and only if [E]a ~ [f]b and

b > ar

to F

a > 0 •

Given a line bundle L on F , the pullback

is of the form {[EJ

r

b > ar + 1 •

and

is spanned by global sections if a~d only if

r A

~

[fJ }

r

for some integer

A

we will use the following generalization of a result of Kobayashi-Ochiai [K+OJ many times.

{0.6.1}

Theorem.

Let

X

be an n dimensional connected normal

irreducible Gorenstein projective variety with isolated singularities.

Let

L

be an ample line bundle on

X

~

Assume that:

K~ ~ Lb :: (!)X

where

b < a < 0 •

Then there is an ample line bundle M on X such that Mt = K-Xl , and Mq = L wh ere b q = t a. In par t'~cu 1 ar t > q > 1 n n, {X, L} "' {JP 5. It is a straightforward check that the second part of Van de Ven's argument works for L ample, spanned by global sections, and satisfying D

L • L > 5.

(0.8.1)

Lemma.

Let L be an ample line bundle on a smooth

connected projective surfaceS. Assume that f(K~ €1 LN) spans A N K5 €1 L where A and N are positive integers.

Let~: S ~ lPa: be the map associated to f(K~ a)

If dim

~(S)

€1

LN).

= 2, then any connected component, E,

of a positive dimensional fibre of

~

is a smooth

rational curve satisfying E · E = -1 on S and: A = N(L E) • *) In particular for large enough n, the map~·: S ~ ~· (S) associated to f(K~n €1 LNn) expresses S as a smooth surface,¢' (S), with a finite set blown up. b)

If dim ~(S) = 1, then a connected component F ~ general fibre of ¢ is a smooth rational curve and:

99 **)

2A = N (L · F)

For large enough n, the maps

1 :

s -

1

(S) associated to

(K~n ® LNn) has connected fibres and maps S onto a smooth curve,

1

(S).

If dim <j>(S) = 0, i.e. K~ ~ LN ~ @S, then either A~ N, or

c)

(S,L) is as in

a) of (0.8), or S is biholomorphic to a

smooth quadratic

in-;~

and

L~s

isomorphic to the restric-

tion of @ 3 (1).

lPa: ~·

In case

a), the assertions about care immediate con-

sequences of lemma (2.3.3) of [So 3 J; the reader can check that the proof of (2.3.3) of [So 3 J still holds under the hypotheses of (0.8.2) above. In case fibre of <j>.

The rest of part

a) is standard.

b), let F be a connected component of general Since KS,F

KF it follows that:

z

A deg(KF) + N deg(LF)

=

0 •

Since LF is ample and {A,N} are positive, this implies that -1 1 KF is ample. Therefore F z lPa: and deg KF = -2 giving**) of b).

The rest of part b) is standard.

Assume that K~ ® LN

z

@s.

If N >A, then by (0.6.1),

we are done. The following is a very slight modification of [L

+ So, (2.3) ].

(0.9)

Theorem (Lipman and Sommese).

Let V be a three dimen-

sional irreducible normal Gorenstein variety. ample line bundles in P

~

Rv , P

v.

V which is biholomorphic tow; "' @ 2 (-2). lP

Let L and H be

Assume that there is a subvariety and assume that*

Assume that there exists a map p: V -

A where:

a)

A is affine and V* is a Zariski neighborhood of P,

b)

P: V* - P

--7

A - p(P) is a biholomorphism.

Assume that there is a smoothS < ILl and H is spanned by glo~al sections in a neighborhood of s u P and Hp ~ @ 2 ( 1) • Assume that s + n E !HI where D ~ P. Then P does Wot meet the Lingular set of

v.

100

Proof.

This will follow from the proof of [L +So,

for any x

E

with isolated singularities such that x Let IH-xl denote the set of D x

P n Sing(V).

E

meet

s.

(2.3)] if

P n Sing(V) we can find an irreducible D E

E

!HI

D.

!HI that contains

E

Note that the base locus of IH-xl does not

Indeed if it did then since H is spanned in a neigh-

borhood of S, it would follow that IH-xl = IH-yi where y Since S + D

l

y but not x by D

Bertini's theorem there is

D

P this is absurd.

c

(0.10)

Lemma.

~(A))+l

~

Let

< dim A.

~(A)

Let a= (dim

~(A)

0

Let A be an effective ample divisor on a con-

nected projective manifold X.

dim

Thus D is irreducible and

V-S which implies that Sing(D) is finite.

phic map with dim Proof.

Thus by

IH-xl with D meetingS trans-

E

versely in a smooth ample curve. Sing(D)

S.

E

: X ---+- IP a: be a holomor-

Then

~(A)

=~(X).

and assume that

~(A)

~~(X).

Then

< a.and:

*)

L •

•

0 in homology

• L • A

La timesJ where L L

~*@

(1). L is spanned by global sections and thus IPII: L can be represented by an effective cycle D that meets

A in a cycle representing D · A. Further D is a non-trivial union of dim X- a > 1 dimensional analytic sets since dim

~(X)

+ 2.

~dim ~(A)

Therefore Lemma.

(0.1)

+ 1 =a

and dim X= dim A+ 1 >dim

*) contradicts the ampleness of A.

~(A)

0

Let A be an ample divisor on an irreducible pro-

jective local complete intersection X.

Assume that there is a

continuous map r: X ---+-A such that rA:A ---+-A is a homotopy equivalence. Proof.

Then dim X < 2.

Assume that dim X > 3.

Then by the first Lefschetz

theorem:

*) Since rA is a homotopy equivalence: **)

r~: H2 (A,II:)

--+

H2 (A,II:) is an isomorphism.

Combining*) and**), it follows that:

101

r

-

*

H 2 (x,~) is

an isomorphism.

Therefore a Kaehler class won X can be written r*n where 2 a+l a+l n E H (A,~). This implies that w r*(n ) = 0 where a

= dim

A.

This is absurd since w raised to the dimension of

X must be non-trivial.

0

The Adjunction Process

§1

Throughout this section L is an ample line bundle on

(1.0)

an irreducible three dimensional normal Gorenstein projective variety, X. smooth S

E

It is further assumed that there is at least one

IL I .

The adjunction process that we use in this paper is a modification of the adjunction process Morin [Ro, pg. 66] used to reprove Fano 1 s classification of threefolds with rational hyperplane sections.

This process used by Morin was based on

the Castelnuovo-Enriques adjunction process for surfaces [C+E]. The following lemma is at the heart of the process. Lemma.

(1.0.1)

Let £ be a holomorphic line bundle on a smooth,

connected, projective threefold, X. divisor on X.

Let S be a smooth ample

K~ ® [S]d ® !.

Let !(d) denote

Assume that:

(1.0.1.1)

£ is spanned by g~_~.e_~~--~!::~!:i:_ons,

(1.0.1.2)

£(d) S is spanned by g_lobal s-~~-"!:_io:t_:l_~-~C?E_ __~ < d < N.

Then there is an integer N 1 > 0 such that either:

a)

(!(N))

Nl

is spanned by global sections, or,

b)

(!(d 1 ))N 1 is spanned by global sections for some nonnegative d

and the map associated to r((C(d 1 ))N 1 )

< N,

1

has an image of less than 3 dimensions. ~·

Let d

1

be the largest integer less than or equal to N

such that there is an N 1 > 0 such that: r (( Since £ is spanned d

1

c< d

> 0.

I )

)

NI

If d

spans ( ( ( d

)

1

Therefore it can be assumed that d

= 1

)

NI

N there is nothing to prove.
N(L ·E)

•

Next assume that S has non-negative Kodaira dimension. We have the following lemma. (1.1.1)

Lemma.

Let £ be an ample line bundle on s, a smooth

connected projective surface on non-negative Kodaira dimension. Assume that ( i s spanned by global sections, h 0 £ • £ > 7 .

Then h (K8

~

£)

~

4 and (K 8 +£)

(()

~ 4, and

· (K 8 +£) > 7.

By (0.5.1), (K 8 +£) · (K 8 +£) ~ 7. By the Kodaira vanishing theorem and the Riemann-Roch theorem for K8 ~ £:

Proof.

h (Ks

~

' ' = xC@s) + i(Ks+£)

. '·

Since sis of non-negative Kodaira dimension, x(@ 8 ) Thus h 0 (K 8 ~ £) > 4.

(0.5) K8 · £ > 0.

~

0 and by

0

Choose N large enough so that LN is very ample, 2

N

o

N L • L ~ 7, and h (L) ~ 4. Theorem (0.8) and lemma (1.1.1) immediately yield that either f(!(n)) spans !(n) for all n > 0 or there is a finite smallest non-negative integer, A, such that r(!(A+l)) doesn't span !(A+l), and !(A) is not ample. By

*) of

a) of lemma (0.8.1), it follos that there is a smooth

rational curve E on

s

such that:

E · E = -1 on S and A= N(L • T:). Therefore by the first equality S is not a minimal model and 'by the second A~ N. Therefore the theorem is proved if S has non-negative Kodaira dimension. Next assume that S is rational.

Choose for N an even

104

number that is large enough so that LN is very ample.

We have

the following lemma. ( 1.1. 2)

Lemma.

Let !. be a very ample line bundle on a smooth connected surfaceS surface satisfying h 1 ' 0 (S) = 0. Assume that K 8 ® !. is ample and spanned by global sections. K 8 ® !. is very ample unless: a)

....~ = K-2 s ,

Then

. K2 l.e. s ""~ ....~ ~ or

b) Proof.

In [So 3 , §3], the second author studied the mapping associated to f(K 8 ® !.) . He showed that under the hypotheses of the lemma, K 8 ® !. is very ample except in the two cases given by (2.5.1) and (2.5.2) of [So 3 J. In the first case S is a two sheeted branched cover of A direct computation using the description in (2.5.1) of -2 shows that K 8 ~ !.. In the second case S is a two sheeted branched cover of a singular quadratic.

A direct computation using the description -3 in (2.5.2) of [so 3 J shows that K 8 ~ !.. D Choose the smallest non-negative integer A such that !(A)

is not very ample; by the first paragraph of this proof such a finite A exists. (1.1.3)

We claim that either the theorem

is true or:

!(A) is spanned by global sections.

To see this note that since !.(A-1) is very ample, it follows from theorem (0.8) that (1.1.3) can fail only if: a.)

s

~

lP

~ and !.(A-1) ~

Q)

2 Ce) for e

1 or 2,

lPa:

or, 8)

1

S is a lP C bundle r: S !.(A-l)F

~

---+-

lP

1

a: and

ID 1 (1) where F is a fibre of r. lPa:

If case a.) occurs then (S,L) is as in case c) of the conclusions of the theorem. If case S) occurs then since

105

K S,F

~

KF

~

@ l (-2) : lPC

-2(A-l) + N(L · F)

=

l

which contradicts the fact that N is even. We claim that either the theorem is true or: (1.1.4)

((A) is ample.

To see this assume that ((A) is not ample. Let~: S -+lP~ be the map associated to r (((A)). Since according to (1.1.3),

r (((A) ) spans ((A) , we can use ( 0. 8 .1) . If dim ~(S) = 2, then exactly as in the case when Shad non-negative Kodaira dimension, we can use (0.8.1) to conclude that A > Nand therefore (S,L) is as in part sions of the theorem that we are proving. conclude from part

a) of the conclu-

If dim

~(S)

= l, we

b) of (0.8.1) that (S,L) is as in part

of the conclusions of the theorem that we are proving.

b)

If

dim ~(S) = 0, the we conclude from part c) of (0.8.1), that {S,L} is as in part c) of the conclusion of the theorem we are proving.

Therefore without loss of generality, we can

assume that {1.1.4} is true. But by lemma {1.1.2), we conclude that either (S,L) is as in part c) of the conclusions of the theorem we are proving or ({A} is very ample.

Since by

the choice of A, ({A) is not very ample, it follows that the theorem is proven if S is rational. Finally assume that S is birationally ruled and .h 1 ' 0 {s}

> 0.

2

Choose N = {12!} · 4 • d · N' where d and N' is chosen large enough so that: {1.1.5}

n.

LN is very ample, N2 (L · L) > 5, and h {LN) :::_ 4 0

By the first paragraph h 0 (((n)) = 0 for all large enough Choose the largest non-negative integer A such that: f(((A-1)) spans

(A-1),

(A-1)

(A-1) > 5,

(1.1.6)

h 0 (((A-l) :::_ 4, and C(A-1} is ample. By the same argument as in the case of rational

s, it can

106

be assumed that: (1.1.7)

r(!(A)) spans !(A) and !(A) is ample.

Since S is birationally ruled: h 2 ' 0 (S) = 0.

(1.1.8)

By (1.1.6) and the Kodaira vanishing theorem for !(A-1): hi(!(A)) = 0 fori> 0.

(1.1.9)

Using (1.1.8),

c

E

(1.1.9) and the residue sequence for smooth

I!(A-1) I: 0

---+

KS

--+

!(A)

--+

KC

--+

0 .

We conclude that: (1.1.10) where g(C) is the genus of C. Since S is birationally ruled and satisfies h 1 ' 0 (s) > 0 it follows that: (1.1.11)

KS · KS ~ 8- 8hl,O(S) < 0 •

Combining (1.1.11) with (1.1.10) we get: !(A)

·!(A) = (KS+!(A-1)) (KS+!(A-1)) = KS · KS + 4g(C) - 4

- !(A-1)

· !(A-1) ~ 8- 8h 1 ' 0 (s) + 4g(C) - 4- !(A-1)

· !(A-1)

4 + 4h 0 (!(A))- 4h1 ' 0 ( s ) - !(A-1) · !(A-1) By (1.1.5) and the fact that h 1 ' 0 (s) > 1, the last inequality becomes:

j.c£ (A)

• !(A)) + 1 < h o (!(A))

Therefore if I show that: (1.1.12)

! (A)

• ! (A)

> 13.

We will conclude that h 0 (!(A)) ~ 4. This combined with (1.1.7) would show the contradiction that (1.1.6) is true with A+l in the place of A.

Therefore to prove the theorem, we must

only show that (1.1.12) is true. Therefore assume that !(A) · !(A) < 12. Let: x =!(A) · !(A) = (AKS+NL) • (AK 8 +NL) =-A 2 K8 · K8 +NT for some integer T. If A = 0 or K . K = O, then N divides x

s

s

107

and therefore x > N > 13.

Therefore we can assume that:

A · KS · KS :} 0 Recalling that Ks · KS

~

0, and using the definition of N, we

conclude that: y = -A 2 + N • T

*) where x = YIKs

.

Ksl, and N = N

IKs

Ks I.

I f Y_~ 12, then from the form of *) and the fact that (12!) 2 2 divides N,

we conclude that y is of the form z

write

.

We can re-

*) :

**)

N'T

where zA =A, and

z 2 ~•

N.

Since 4 divides N', we get a contradiction from**). Therefore (1.1.12) is true. §2

0

Proof of the Main Theorem

Assume that L is an ample line bundle on a normal irreducible Gorenstein projective threefold, X. a smoothS

E

Assume that there is

ILl and that log (X,L) ~ 0

It is easy to see that such S have non-negative Kodaira dimension by means of the argument used in [So4]. through it.

D

E

Let us go

Since log (X,L) > 0, there is an effective

IN(KX+L) I for sure N > 0

If S is not a component of D

we are done by the adjunction formula.

Therefore we can assume

without loss of generality that D r S + ~ where r > 0 and S . N r ~s not a component of E. Thus Ks = Ls ® [f. n S]. Since some power of KS is a product of an ample divisor and an effective divisor, it follows that s is of non-negative Kodaira dimension. By theorem (1.1) and the fact that s has non-negative Kodaira dimension, it is true for all sufficiently large N > 0 that (Kn ® LN) spans Kn ® LN for:

s

s

s

a)

s

all n if S is minimal or

b)

for 0 < n·< A where A> Nand KA ® LN s s is not ample.

108

Since log (X,L) ~ 0 it follows that if r(K~ ® Lb) spans

K~ ® Lb with b

> a then the map associated to

K~ ® Lb has a 3

dimensional image. Therefore by lemma (1.0.1) there is for each n > 0 an integer N' such that r(KnN'® L(N+n)N'> span X KnN'® L(N+n)N' X

In case

a) we see that KX ® L is numerically effective.

Indeed let C be an effective curve.

Letting n go to

oo

We have

· C + ~ n L · C > 0 gives the result.

Therefore the main theorem is proven if all smooth S E ILl are minimal and therefore we can assume without loss of generality that we are in case b). By (0.8.1),

{

*)

(0.7.5) and the fact that A> N we conclude:

For each n > 0 and < N there exists an N' such that r(K~N'® L(n+N)N') s;ans K~N'® L(n+N)N' and the assorted

map ¢ n, N' has a 3 dimensional image. When n map has some positive dimensional fibres.

= N,

the

We can choose the N' large enough so that the maps ¢n,N' have normal images and connected fibres.

Denote this map

from X to ¢ n, N' (X) by ¢ n . We claim that ¢ : X X' = ¢N(X) is the reduction n with the properties of the main theorem. We proceed by ana~

lyzing the positive dimensional fibres of ¢N. ( 2 .1) Lemma. Proof.

By

-

.

Hl (Ka ® Lb) = 0 i f b > 2a 2 > 0 X *) there is an N" > 0 such that K~a-l)N" ® LbN"

spanned by global sections and has a 3 dimensional image. Therefore by (0.2.1) the lemma is proven.

D

From this we see that r(Ka ® Lb) X

for b > 2a > 0.

--+

r(Ka ® Lb-a) ~ 0 S

S

Therefore by choosing the N' large enough it

can be assumed that

109 ~n(S)

has connected fibres with

normal.

Further

~

n,s

is an

embedding for 0 < n < N . By (0.7.5) there are Weil divisors u1 , ••• ,Dk on X such that each Di meets S transversely in a smooth rational curve E.1 such that E.1 · E.1

-1 on S and these E1. are precisely the fibre of ~N,S" By the last paragraph ~N(Eil ~ ~N(Ejl for i ~ j implies that the n. are disjoint. 1

Note K~-l ® LN-l ® LN is spanned in a neighborhood of S. This implies that r(K~-l ® L 2N-l) is spanned in a neighborhood 2

of each ni. Indeed let pi: W ---+D. be the map from the normalization of Di to Di. Since p~(K~-l ® L 2N-ll ~@ 2 (1)

w

1

and since K~-l ® L2N-l is spanned by global sections in a neighborhood of S, it suffices to find a global section of KN-1 ® L2N-l whose restriction to Di vanishes only on X

S n Di = Ei. Choose any section t of K~-l ® L2N- 2 which does not vanish identically on S n Di and let s be the tautological section of S vaishing on S. Then t ® s is the desired section. . N-1 S1nce KX borhood

2N-l . 1s spanned by global sections in a neighof Di and PfCK~-l® L2 N-ll is @ 2 Cll we easily see that ~

L

w

the pi are biholomorphisms. (2.2) Lemma.

There exist large t such that K~ ® L~-l® ~ [Ii]-l

is spanned by global sections. ~Consider ~N,S: S--+ S' ~N,S(S~. Note that there is an ample line bundle £on S' such that ~N,S'£ = K8 ® L8 . Thus by (0.8.1) and since:

Kt ® Lt-1® ® [E.]-1 ~ ~* S

S

i

N,S'

1

the lemma is clear.

(K

S'

® £t-l) D

We claim that ~N has only the Di as positive dimensional fibres. If we show this then by theorem (0.9) ~N: X --+ X' = ~N(X) is a reduction. If there was any other irreducible positive dimensional Variety V such that ~N(V) is a point, then V n s is non-trivial. We claim V n S belongs to uE 1.• Indeed this will follow from i

110 d~N

being of rank 3 on T

d~N

is of rank 2 in TS,x for x

X,x

for x E

F

S-uE.. i

By construction

1

Therefore d~N will be

S-uii.

if we can produce a section of KNN' ® L 2 NN' ,x X for some N' of the form t ® s where t is a section of NN' 2NN'-l KX ® L with t(x) ~ 0 and s is the tautological section of rank 3 on TX

of [S] = L·

This is clear by lemma (2.2) and lemma (2.1).

Further by lemma (2.2) and (2.1) given x E Ert can be chosen so that ts vanishes only to the first order on Ei in a neighborhood of x on s.

Therefore by the implicit function theorem

t vanishes only on a manifold in a neighborhood of x on X. Assume x was chosen so that V n

s 7 x.

Since s ® t vanishes

on V it follows that V c S u Di which implies that the only positive dimensional fibres of ~N are the Di. Let L' =

[~N(S)]

¢N(~Di).

and F =

l

We claim that S' is minimal for all smooth S'

E

IL'-FI

Indeed if not then there is by repeating the whole argument we find a smooth rational curve E in S' E · E

=

-1 on S', L'E

~ ~

lP

=

~N(S)

1 (1) and ¢N(Ii)

This is absurd since L{

~

~i

E

such that E for some E.

where E is the proper trans-

form of S'. This proves that the smoothS' E IL'-FI are minimal. The argument at the beginning of the proof shows KX, ® L' is numerically effective. 0 (2.3)

Corollary.

Let L be an ample line bundle on an irredu-

cible projective Gorenstein threefold, X. Assume there is a smooth s ILl· There is a polynomial p(n) such that:

"

Proof.

h o (Kn ® Ln) = p(n) for n > 0 . X If log (X,L) < 0 the corollary is trivial with p(n) = 0.

If log (X,L) > 0 let 7T : X ---+ X' be the reduction of the main theorem. Since KX' ® L' is numerically effective it foln-1 n l~ws that KX ® L is ample for n > 0. By (0.2.1) H1 (Kn ® Ln) = 0 for i > 0 and therefore by the Riemann-Roch X

theorem the above is true.

0

In a sequel we will give a detailed structure theorem for (X' ,L') in terms of the degree of p(n). The reader can consult [So5], [So6] for a description of these results in the case

111

References [Bo]

E. Bombieri, canonical models of surfaces of general type, Pub!. Math. I.H.E.S. 42 {1973), 171-219.

[Be+Pa]

M. Beltrametti, M. Palleshi, On threefolds with low sectional genus, preprint.

[B+H]

E. Bombieri and D. Husemoller, Classification and embeddings of surfaces, Proc. Symp. Pure Math. 29, ed. by R. Hartshorne, {1975), 329-420.

[ C+E]

G. castelnuovo and F. Enriques, Sur quelques resultats nouveaux dans la theorie des surfaces algebrique, Note V in P+S below. T. Fujita, On the hyperplane section principle of Lefschetz, J. Math. Soc. Japan 32 {1980), 153-169. T. Fujita, On the structure on polarized manifolds of total deficiency, I , J. Math. Soc. Japan 32 {1980), 709-725. R. Hartshorne, Pmple Subvarieties of ~lgebraic Varieties, Lecture Notes in Math. 156 {1970), SpringerVerlag, New York {1977). R. Hartshorne, Plgebraic Geometry, New York {1977).

[Hi]

Springer~Verlag,

F. Hirzebruch, New Topological Methods in Plgebraic Geometry, 3rd Edition, Springer-Verlag, Berlin {1966). S. Iitaka, On D dimensions of algebraic varieties, J. Math. Soc. Japan 23 {1971), 356-373. S. Iitaka, On logarithmic Kodaira dimension of algebraic varieties, Cbmplex ffialysis and ~lg·ebraic Geometry, ed. by W.L. Baily, Jr. and T. Shioda, (1977), 175-189, Iwanami Shoten. Iskovskih and v.v. Sokurov, Biregular theory of Fano 3-folds, Proceedings lllgebraic Geometry Cbnference, Cbpenhagen, 1978, edited by K. Lonsted, Lect. Notes in Math. 732 {1979), 171-182.

[I+SJ

v.~~

[Ka]

Y. Kawamata, A generalization of Kodaira-Ramanujam's vanishing theorem, Math. Jnn. 261 {1982}, 43-46. Y. Kawamata, Elementary contractions of algebraic 3-folds, Jnn. of Math 119 {1984) 95-110.

[Kl]

S. Kleiman, Towards a numerical theory of arnpleness, Jnn. of Math. 84 {1966), 293-344.

[K+OJ

S. Kobayashi and T. Ochiai, Characterizations of complex projective spaces and hyperquadrics, J. Math Kyoto Univ. 13 {1972), 31-47.

112

[Ko]

K. Kodaira, Pluricanonical systems on algebraic surfaces of general type, J. Math. Soc. Japan, 20 (1968)' 170-192.

[L+So]

J. Lipman and A.J. Sommese, On the contraction of projective spaces on singular varieties, preprint.

[Mo]

S. Mori, Threefolds whose canonical bundles are not numerically effective, Ann. Math 116 (1982), 133176.

[J.I.1u]

D. Mumford, The canonical rings of an algebraic surface, Appendix in z2 below.

[Nag]

M. Nagata, On rational srufaces I, Mem. Call. Sci. Kyoto (A) 32 (1960), 351-370.

[Nak]

s.

[P+S]

E. Picard and G. Simart, Theories des Fonctions Algebriques de Deux Variables Independantes, Chelsea Publ. Co., Bronx, New York (1971).

[RaJ

C.P. Ramanujam, Remarks on the Kodaira Vanishing theorem, Jour. of the Indian Math. Soc. 36 (1972), 41-51; Supplement to the article "Remarks on the Kodaira vanishing theorem," J. Indian Math. Soc., 38 (1974), 121-124.

[Ro]

L. Roth, Algebraic Threefolds, Springer-Verlag, Heidelberg, (1953).

Nakano, On the inverse of monoidal transformation, Publ. R.I.M.S. Kyoto Univ. 6 (1971), 483502.

F. Sakai, Semi-stable curves on algebraic surfaces and logarithmic pluricanonical maps, Math. Ann. 254, 89-120 (1980). F. Sakai, D-dimensions of algebraic surfaces and numerically effective divisors, preprint. [Sh+So]

B. Shiffman and A.J. Sommese, Vanishing Theorems on complex manifolds, to appear. A.J. Sommese, On manifolds that cannot be ample divisors, Math. Ann. 221 (1976), 55-72. A.J. Sommese, Non-smoothable varieties, Com. Math. Helv. 54 (1979), 140-146. A.J. Sommese, Hyperplane sections of projective surfaces: !-the adjunction mapping, Duke Math. J. 46 (1979)' 377-401.

[So 4 J

A.J. Sommese, On the minimality of hyperplane sections of projective threefolds, Journal fur die reine und angewandte Mathematik, 329 (1981), 16-41.

113

A.J. Sommese, Ample divisors on 3-folds, Algebraic Threefolds, Springer Lecture Notes in Math 947 (1982) I 229-240. [So 6 J

A.J. Sommese, On the birational theory of hyperplane sections of projective threefolds, unpublished 1981 manuscript. A.J. Sommese, Configurations of -2 rational curves on hyperplane sections of projective threefolds, Classification of Algebraic and Analytic Manifolds, ed. by K. Ueno, Progress in Mathematics 39 (1983) Birkhauser Boston.

[VdV]

A. Van De Ven, On the 2 connectedness of very ample divisors on a surface, Duke Math. J. 46 (1979), 403-407.

[Vi]

E. Viehweg, Vanishing theorems J. reine angew. Math. 335 (1982), 1-8. 0. Zariski, Introduction to the Problem of Minimal Models in the Theory of Algebraic Surfaces, Pub. Math. Soc. of Japan 4 (1958). 0. Zariski, The theorem of Riemann-Roch for high multiples of an effective divisor of an algebraic surface, Ann. of Math. 76 (1962), 560615.

Department of Mathematics University of L'Aquila L'Aquila, Italy Department of Mathematics University of Notre Dame Notre Dame, Indiana 46556

On Meromorphlc Equivalence Relations Hans Grauert Mathematisches Institut, Universitat Gottingen, Bunsenstr. 3-5, D-34oo Gottingen, West Germany Introduction. 1. We denote by

X

a weakly normal (see§ 2.3.) complex

space with countable topology and by

R

c

x

X,

X

X

x

an analytic

set with the following two properties: 1)

R

contains the diagonal

2)

R

is mapped by the reflexion

X x X~ X x X

D

X

c

(x 1 ,x 2 )

through

x €. X

is defined as

denotes the projection (as XX

p2

(x 2 ,x 1 l

onto itself.

Such an analytic set defines a fibration in Xx

~

X

X

x

~

X

p 1 (R !"'\ (X

X. x

The fibre x))

where

onto the first component

will denote the projection onto the second).

Here,

always is considered as a set, not as a complex subspace

with a nilpotent structure.

Definition: i~

R

i~

a

no~mat

an equivalenee

~elation

1)

R

2)

the eodimen~ion o6 the equal :to c ;;: o.

3)

the

p~ojeetion~

pi

eomple~

6ib."Ct..~

R

-+

X

equivalenee 4elation i6: in i~

a~e

X, eon~tant

eve~ywhe~e,

open.

We shall prove in § 5 that under this assumption the .Quotient space dimension

c •

X/R

is a weakly normal complex space of pure

116

2. But the main purpose of this paper is to prove something for meromorphic equivalence relations in normal complex spaces:

~odime~~io~

~et),

a

~owhe~e

that

~uQh

RIX' P

2)

me~omo~phi~

equivate~~e ~etatio~ i~

o6

X

i6:

c

the~e i~

1)

a

i~

Definition: R

=

de~~e a~atyti~ i~

R n (XxP)

~et

P c

~owhe~e

x

(pota~

de~~e ~~

R,

R n ((X,P) x (X'~)) i~ a ~o~mal Qomptex equio6 QOdime~~io~ c i~ X' t.

vate~Qe ~etatio~

We denote by the holomorphic

n :

R~

map

~

has pure codimension

c

R

the normalization of

R

p2 ~ R ~ X.

R

We have

The analytic set

The normal complex space

R

X

decom-

poses into connected components Xi of dimension ni. For xEXi the generic fibre ~- 1 (x) has codimension c+n. We look at the degeneration set E := {(x1,x2) E ~

-1

(x2) < c + ni'

'R

set of

E

X•

J.

~· = ~IR'. The set

'E =

R' = 'R ' E

We put ~

-1

:

codim( x ,x ) 1 2 which is a nowhere dense analytic

Xi},

(see [Re]).

'R

~(E)

-

~-1 (P)

is not analytic in

and

'R,

in

general. But it is a countable union of local nowhere dense analytic subsets like (x 1 ,x 2 ) E

If ~

D ~

R

R

in

~(E)

~(0)

2)

~- 1 (E)

3)

~(D' {0}) c

is.

is a point there are many holomorphic maps D c ~

of the unit disc

1)

X

around

0 E ~

(x 1 ,x 2 ),

is countable,

R'

We consider the (set theoretic) fibred product

Rx

D

with:

and take the union

z

Rx

X D

c

of all irreducible components

'l
D , completely. All

117

zt := z n (R x {t})

fibres

points over

Xj

~- 1 (~(t)).

x

Xi.

If

have dimension t

So we call the

R

fibres in

of the fibres of

J

is generic we have

R:

6-ib'l..if..¢ ...

A.= I, .. ,1

0 Q>... ~

-1

We denote by

(G),

by

128 A

the part of

tv

...

R*

Proposition:

an.d

over

Q

G

and prove:

,.

/1

= R\Q*

RIY*, R*

=

a~e compl~x,

holomo~phic

.6emip~ope~.

Proof: If

S E

~~

is a fibre over

n ~*, B* := ~ n ~- 1 (G)

.. ,zr E p(S)

G,

we take points

on the p-image of the var-

ious images sets of those irreducible components of are completely contained in local complex subspaces

E .

z 1 , ..

S

which

We take pure n-dimensional

Ai c Z, i=1, ... ,r

with the following

properties: 1)

Ai cUi cc Z

are n-dimensional complete intersections

(relatively to U), Ai n p(S) = {zi} .

2)

A neighbourhood

s1

E

over

~~

S n B*

G

= 11

~

is defined as the set of all

~

s1

such that

n B*

can be connected with

by a chain of holomorphic 1-parameter families

Sx(t) n B* , I tl < 1 X

c

-V(S)

•

•

•

over

G

with

p Sx(t) n

i)

A.

1

*¢

and

,k •

We denote by Clearly,

A

v

"' v

v

the inverse image of

is open and its inverse image

again. By projection we obtain subsets

A

Q

in

u" c

"

y

l

A.= I

Q;v

is open

1\

u c Y*/'R*, v

0

:=

c:

II

Q*/R*

which are open in the quotient topology. There is a multiplicity for the irreducible components of the fibres bourhood of

S E S.

~~

If

coming from the generic fibres of a neighS

is generic this multiplicity always is

one. Otherwise it can be a higher integer. It might happen

129

that a settheoretic

S

has various multiplicities coming from

different neighbourhoods. Then these have to be considered as different fibres. Thus, a fibre is the set

S

equipped with

multiplicity of the irreducible components. -The multiplicity carries over to p

s1, s1

y

E

p(S).

We take this multiplicity! Then each

has the same intersection number with

multipli~ity

We use the multiplicity to define the

s 1 n Ai, s 1

intersection points function on

Ai'

y.

If

g

for the

is a holomorphic

we take the elementary symmetric_polynomials

of the values of functions on

E

Ai.

g

s 1 n Ai.

in

~

v .

These all lead to holomorphic

We may assume that an embedding of

in

a complex number space is given by finitely many holomorphic functions. We take these for g. A

We define

V

so small that it is over an open subset of

X,

which is isomorphic to an analytic subset of a domain in the complex number space. We also take the holomorphic functions on ~

V coming from the finite number of coordinate functions on this open subset.

H

Altogether, we obtain a finite set A

v

tions on A

(t,T) E V

The differences A

x

V

I

p(S)

n

~*

n

p E,

E

a neighbourhood of

on

A

V

~

x

V.

which is somewhere dense on each p-

We take the S

I

f E H,

Firstly, we take an infinite set of

image of any irreducible component of contained in

for

maximal.

We proceed as follows: on

- f(T)

define a coherent ideal sheaf

We wish to have the

zi

f(t)

of holomorphic func-

Ai c z

S ,

which is completely

such that

V

is still

This is possible! We just have to use

130

the parametrization of fibres given by Lemma (n) and that the ln-

s

verse image of the "point" make sheaf

e I

smaller. The set

H

a

is closed in

is infinite now. But the ideal

is still coherent. The functions of

different fibres of

irreducible component of

Q

separate the t

):

each

and

this is a well-known statement. smaller we find finitely many

V

points among our infinite set

{zi}

such that the ideal

is spanned by the functions to these become finite and the functions

H

zi

already. Now

So

Qi cc QA..

-1 -

-1

nA. : qA. (Qil n nA. (B*)

follows immediately that

R*

and

H

R*, R*

holomorphic equivalence relations. - We may replace relatively compact open subset

I

has

separate the fibres out of V. R( H,nA. ) .

pact and the maps

t0

~

converges against some irredu-

qA.(t 0

After having made

H

We have to use that for

V :

cible components of

and that we can

R*

Since ~

B*

Qi

QA.

are by a

is com-

are proper, it

are semiproper. So the

Proposition is proved. 2.

Hence

~* = Y*/R*, X*

have holomorphic maps

0•/R*

are complex spaces.

We

~* : Y* ~X*, ~*

such that the diagram Y*

l

·Q>*

,..., X*

~*

B*

1~

rt*

G

is commutative. All fibres of They are mapped by By

,.,

R*

tt*

~*

are of pure codimension

finite open onto an

S n B*, S € N~

this map is bijective. Hence, it is topological. The

n.

131

composition maps

p o ~*

n , "*

~~ ~ p(~~ n B*}.

gives a bijection

The

are proper. Since the inverse image of a generic

point consists of one point only, they are proper modifications. 3. X*

is defined as a set independently of the choice of B.

Since every irreducible component

G with pS,

S' c E

S E Jltcp

enters in

over

G

B*

S'

of a fibre

the set

X*

S €

By making

B

o :

X* ~ G ,

X** ~ X*

over

,..

X*

might

larger it might be-

come finer. We would obtain a proper modification stead of

cp

is just the set of

But already the topology of

depend on the choice of B.

~

X** ~ G

in-

but also a bijective holomorphic map

such that

~**

X

61·~

~ ~

G

commutes.

X*

In general, they are

·x*, X**

qua¢i no4mal:

will not be normal complex spaces. But any local continuous complex function

which is holomorphic outside a nowhere dense analytic set, is holomorphic. When we pass over to the normalization of 6

the map

is still bijective and holomorphic: This follows from the

modification properties. By a well-known theorem (see [CAS)) it is biholomorphic then. This implies that already the old is biholomorphic. Thus, the proper modification only on

X* ~ G

depends

G .

The same holds true for have a commutative diagram

"' Y*

~

B*.

If

B**

6

is larger we

132

B*

Y:•• ---•

6

where of

B**

is a biholomorphic map of

onto an open subset

·"'Y**· 4.

Now we exhaust

open subsets Bv cc

El.'+1

=

Y.

Y* -+Y:+ 1 1B~ : "' ~""

l

X

Y

by a similar sequence

We construct the commutative diagrams

and get isomorphisms

rv

and

G*y

X*v

·"" y

by a sequence of relatively compact

r

l

fV

" v cations

X

GV cc Gv+l cc X

"'Y*...

~

Y*

A

n

I

.

o"

=

x;::: x;+ 1 1G~,

We glue together and obtain proper modifi-

Y-Y, n : X

-.x

such that the diagram

y

l

X

is commutative. The properties 2) and 3) of Proposition (n) are satisfied. Its proof is completed!

133

§ 3.

The Proof of the Main Theorem

1.

Assume now that

X

is a normal complex space and that R

is a regular meromorphic equivalence relation in sion

We denote by

c

R~

n :

ization

R

P c X

is the degeneration set of X.

connected components

~

"" R

:

~X

X

and

X

Xi

with X

The set

E

decomposes into

The codimension c + ni.

is

p2 : X

X

~

X

The space

over

equal

4.1

.

(c + ni)

We apply Proposition

.....

: R

p2 o n

of dimension

1

of the generic fibre of

n

~

of codimen-

a polar set. We take the normal-

~

and put

X

Y

~

~

X. We obtain a com-

X

Z

equal

mutative diagram 1\

n

N

y

"" ~

1 X

image of

1 ~

n

X

,...

R

....,

1"-

where

1'-

X

is a proper modification of .'V

in

R

X

X

X.

"' X under the map

We take the inverse ,..-

~

X

ducible components which are completely in with fibre

'P = n- 1 (P*) S E

~~

,P*

=

p

u

({I {E) •

is contained in E

n

and omit all irre-

....., p

X

"' X

W .

is compact and we

This proves the statement.

We get immediately , that

R

is semiproper.

by lifting a semiproper equivalence relation Y .

B=

We have

R* u A,

where

A

components of dimension less than Q x Q

under

less than

Y x Y

Q x Q

~

The dimension of R'

B

X

.

all fibres of

Since

.....

c . The image

~

Q

"' X

-+

Q

'

Q

to

B

of

A

in

is an analytic covering.

c

=

We have

dim Q

:

R

is mapped by

Q

X~

Pz

from

nowhere dense.

is a sutured complex space.

The generic fibre of fibre of

Y

is less than

(Q,R')

R'

is obtained

is the union of irreducible

is discrete andJi·:p 2 (B) c Q

That proves that

R

is an analytic set of dimension

c , since locally

R' = D u B,

W of the

~

X

N

X

-+

X

onto

is open, this holds true for

hence it is true for the fibres

but these are analytic subsets of

a

X

So the map

stricted to fibres is biholomorphic. Outside

B

X*

n*:X*~x

the map

-+

Q,

ren*

is bijective. Since

n*

is finite it is topological there. If

local holomorphic function there,

f

o n*

-1

f

is a

is a local con-

tinuous function which is holomorphic outside a nowhere dense analytic set. Because So

X

is weakly normal, it is holomorphic.

n* : X*IO'~ ~ n*(X*IQ,~))

is a biholomorphic map. The

rest is a nowhere dense analytic set. That means that a proper modification.

n*

is

146

We call 6.

n* : X*

~

X

In the case of the Main Theorem each fibre in

X

(there denoted by

·x)

the

no4mal~at~on

in

Q

of

R

has a well defined multiplicity on its

irreducible components. We can use like in§ 2.1 complete intersections Q/R'

Ai

to construct local ho·lomorphic functions in

which separate the fibres of

is holomorphic and that

X .

This proves that

R'

Q/R 1 is a weakly normal complex space.

So, in our case we can divide by

R'

spaces are not necessary: The quotient

and the sutured complex X/R

is a weakly nor-

mal complex space. But this is also like this in general. It can be seen that a multiplicity can be locally fined in all of the cases of our Theorem. So

resp. X/R X/R

de-

always is

a weakly normal complex space. This proves our Theorem.

147

Bibliography

[Fu]

Fujiki,A.: On Automorphism Groups of Compact Manifolds. Invent.Math.i!, 225-258 (1978).

K~hler

[STCER] Grauert,H.: Set Theoretic Complex Equivalence Relations. Math.Annalen 265, 137-148 (1983) [CAS]

Grauert,H. and R.Remmert: Coherent Analytic Sheaves. Springer Heidelberg 1984.

[Hi]

Hironaka,H.: Flattening Theorem in Complex-Analytic Geometry Amer.J.Math.97, 503-547 (1975).

[Li]

Lieberman: Compactness of the Chow Scheme: Fonctions de Plusieurs Variables Complexes III (Seminaire Norguet). Lecture Notes in Mathematics 670 (1978).

[Re]

Remmert,R.: Holomorphe und meromorphe Abbildungen komplexer R~ume. Math.Ann.133, 328-370 (1957) .•

[St]

Stein,K.: Maximale holomorphe und meromorphe Abbildungen I,II. Ann.J.85, 298-315 (1963); 86, 823-869 (1964).

Recent Developments in Homogeneous CR-Hypersurfaces A. Huckleberry and W. Richthofer Dedicated to Wilhelm Stoll*

1. INTRODUCTION

Let G be a connected complex Lie group and X

=

G/H

a com-

plex manifold homogeneous under a holomorphic G-action. In order to understand X, e.g. how it fits into a fine classification, details of its function theory, etc., one should use as much

Lie theoretic information about

G as

is possible. In

particular it is often useful to study the orbit structure of real subgroups of

G.

Such orbits are usually not complex sub-

manifolds of X. Conversely, if G is a connected real Lie group and one wishes to understand G and its representations, then one naturally looks for orbits M

G/H where strong analytic or

algebraic tools are at hand. A very interesting setting is where X is a complex manifold where G is acting as a group of holomorphic transformations and M

= G/H

is a G-orbit in X. If

1is the Lie algebra of G, then the complex Lie algebra ~

=~

+ i

AJ

(not necessarily a direct sum) is represented as

an algebra of holomorphic vector fields on X. Ideally the vector fields in

0 are

integrable so that we have an induced

holomorphic action of the associated complex Lie group

e.

In

this situation we have the inclusion of orbits G/H ~G/H and it is possible to derive information about M from the complex homogeneous space If M a

~

=

G/H.

G/H is an orbit in X as above, then we refer to X as

-complexification of M. If in addition the Lie algebra

of vector fields induces a G-action, then we refer to X as a G-complexification of M. In either case M inherits a G-invariant Cauchy-Riemann structure from X (see [G],[AHR] for generalities). With this in mind we can refine our line of * Stoll's beautiful results on parabolic spaces were one of the strong motivating forces for the research described in the present paper.

150

questioning: Given an orbit M

=

G/H of a real Lie group, what

are the possible G-invariant CR-structures on M? What are the possible~

-complexifications X so that M - x i s CR-embedded?

What are the possible G-complexifications? Is it possible to find X

= G/H

complex homogeneous so that M - x is CR-embedded

as a G-orbit? If we reach the ideal situation, i.e. X

G/H

and M

=

G/H

is CR-embedded as a G-orbit, then there is so much structure at hand that it is often possible to make very strong statements. These statements can go both ways: The structure of M yields information about X or vice versa. Example 1: Let V := F 1 x Fn be embedded via the Segre embedding inFm' m = 2m-1. This is equivariant with respect to the usual

=

SL 2 (C) x SLn+ 1 (C) action on V. Note that G acts transitively on the complement F 'V =: X = G/H. Let K = su 2 x SU 1 . · m n+ It turns out that the CR-structure of the minimal K-orbit M in G

X reflects the fact that V is not a complete intersection in Fm (see [BFS]).

D

1\

Example 2: Let G be a complex semi-simple group and A 1\

Ha

para-

bolic subgroup, i.e. G/H is a homogeneous rational manifold. Let 1\

G be a non-compact real form of G. Then the generic orbit of G is open and G has a unique orbit M of minimal dimension ([WO]). Theorems from complex analysis lead one to consider certain CR-cohomology spaces on M.

They

are interesting G-moduls,

because they have stable Hilbert subspaces which realize interesting representation theory for G (see [RP] for a special case).

D

We began studying the interplay between CR-orbits and homogeneous spaces from the point of view of the homogeneous space, i.e. given X

1\

1\

= G/H,

what influence do the natural CR-

orbits M4 X have on the classification theory for X? A most striking application arose in the classification of homogeneous surfaces, where Tomilieri's classification of invariant CRstructures on Heisenberg groups ([TO]), the Andreotti-Fredricks embedding theorem ( [AF]) , and •ranaka' s extension theorem for CR-maps of real quadrics ([TA]) provided a fundamental step in the case of solvable groups (see [OR], [H]).

151 On the other

hand, while working in the complex

homogeneous situation, we realized that the CR-manifolds M

= G/H

are important in their own right and therefore began

an organized study of the subject. The purpose of the present paper is to outline the recent developments in this direction so that the reader has a guide to the somewhat technical details in [AHR and R]. Up to this point most of the fine classification results have been proved for compact homogeneous CR-hypersurfaces. However, the basic tools are available for the higher codimensional case. As far as we know, the first results in the classification theory for compact homogeneous CR-hypersurfaces are due to Morimoto and Nagano ([MN]). They considered the situation where M

= G/H

is CR-embedded in a Stein manifold X. In this case one

immediately sees that M

= an,

where Q is a strongly pseudo-

convex open subset of X. In particular the Levi-form of M is everywhere positive dfinite. Since

n

can be recovered as

the envelope of holomorphy of the CR-functions on M, it follows that G acts as a group of holomorphic transformations on

n.

In order to go ahead with their classification theory, Morimoto and Nagano needed the theory of compact groups. Thus they assumed that rr 1 (M) is finite and as a result any maximal compact subgroup K of G acts transitively on M ([S]). Under these assumptions they proved that either n ~ mn' the ball in Cn, and M is CR-equivalent to s 2 n- 1 with its induced structure, or M is a sphere bundle in the tangent bundle of a compact symmetric space of rank 1 and

n

is the tube around

the 0-section with this sphere bundle as boundary. Since the symmetric spaces of rank 1 are completely classified, i.e. spheres,projective spaces over

m,

C, orR, and the Cayley

projective plane, this is certainly a fine classification theorem. Rossi ([R1]) extended the work of Morimoto and Nagano in ···two ways. First, he used the theory of "filling in holes" ([R2]) to show that if M

= G/H

is an abstract strongly pseudo-

convex compact CR-hypersurface with di~M ~ 5, then M is G-equivariantly the boundary of a domain n in a Stein space X. For the same reason as above G acts as a group of holomorphic

152

transformations

on~-

Rossi's methods also require that a

maximal compact subgroup K of G acts transitively on M. Thus he also assumed rr 1 (M) to be finite. His classification goes as follows. If

~is

smooth, then the results of Morimoto and

Nagano may be applied. If not, then it is easy to see that it has exactly one singular point x 0 . Taking a K-equivariant ~ -1 minimal desingularization rr: ~ ~ ~, it follows that Q:=rr (x 0 ) is a K-orbit and Q can be identified with a K-invariant tube neighborhood of Q in this normal bundle. Ampleness criteria show that Q is homogeneous rational and the normal bundle is very ample. Thus the singular cases in the classification arise as follows: Let G be a semi-simple complex Lie group and P a parabolic subgroup. Let Q := G/P and recall that any very ample principal ~*-bundle

over Q is G-homogeneous, i.e. the bundle can be

described by a homogeneous fibration G/H

~*G/P.

Attaching

thew-section and blowing it down to a point x 0 , we obtain A A an affine cone C with vertex x 0 so that C'{x 0 }= G/H. Let K be a maximal compact subgroup of G. Then any K-orbit M

=

K/L

in C'{x 0 } is a strongly pseudoconvex CR-hypersurface. Any two K-orbits are equivalent under the right C*-action and any two maximal compact subgroups are conjugate. Thus the construction only depends on the ample bundle. The only possibility for a non-singular vertex is when the bundle is the hyperplane section bundle overFn, i.e. C = Cn+ 1 ([HO]). Using non-trivial analytic methods and a complicated check of cases in the spherical case, Burns and Shnider ([BS]) showed that rr 1 (M) is finite if M is a strongly pseudoconvex compact homogeneous CR-hypersurface with di~M ~ 5. Thus, except for possibilities in the 3-dimensional case, the fine classification of Morimoto-Nagano-Rossi is indeed complete.

o

In [AHR] we began a study of abstract homogeneous CRmanifolds and applied our methods to the classification problem for compact homogeneous CR-hypersurfaces. One should note that if X is an arbitrary compact complex homogeneous manifold, then M = s 1 x X is a homogeneous CR-hypersurface. Thus a classification of homogeneous CR-hypersurfaces would naturally contain a classification of homogeneous compact complex

153

manifolds. This is of course not possible at the present time, and thus it is necessary to make restrictions. The general theory in fact shows that problems with Levi-flatness, e.g. S

1

x

X, are critical. Thus we began by considering homogeneous

compact CR-hypersurfaces with non-degenrate Levi-form. Of course this contains the strongly pseudoconvex case discussed above. A fine classification is proved in ([AHR]) and described in § 4 of the present paper. There are no restrictions on dimension. Since one can't always "fill in the holes", there are certain cases where there are no G-complexifications (see

§ 3 for examples). However we give an exact description of these cases. The fine classification of compact homogeneous CR-hypersurfaces M with non-degenerate Levi-form is given in terms of a root theoretic description of a canonical fibration of M which has a strongly pseudoconvex fiber and a compact complex homogeneous rational base. The simplest case of this fibration is the s 1 -fibration of M induced by the cone fibration 1\

1\

G/H

~

1\

1\

G/P in the case discussed above. The Levi-form of M,

C* which is itself an interesting invariant of the group theory, can be explicitely calculated ([AZ]). In fact the resulting characteristic polynomials have a very simple form. In § 5 of the present paper we give a complete description of the compact homogeneous CR-hypersurfaces M

=

G/H which

possess a Kahler structure, i.e. there is a CR-embedding M

'"-X in a Kahler manifold. On the one hand, this may be

thought of as a continuation of the classification theory of Borel-Remmert ([BR]) and Matsushima ([M]) in the complex Kahler case. On the other hand, we hope that this is the beginning of a project which will significantly aid in understanding complex homogeneous manifolds. For example, if X

=

e1~ is Kahler, e.g. Stein or quasi-projective and M

=

G/H

is a compact orbit of some real subgroup of ~, then M inherits this Kahler structure. In fact one only needs for some neighborhood of M to be Kahler. For example this is guaranteed if there are sufficiently many holomorphic or meromorphic

154

functions on X. The results in the hypersurface case indicate that the existence of a Kahler structure on M is very restrictive. This then imposes strong funtion-theoretic conditions on the ambient space.

c

2. SOME BASIC METHODS Let M

= G/H

be a homogeneous real analytic generic CR-

submanifold of a complex manifold X with complex structure J. If R is the maximal J-invariant subbundle of TM, then we refer to (R,J) as the induced CR-structure on M. We wish to study how properties of X affect M and vice versa. The "vice versa" needs to be clarified, because only the germ of X along M can be determined by M. However, in many situations involving group actions there is a global relationship, e.g. if X is complex homogeneous and G is a subgroup of

= ~~~

a having compact

hypersurface orbits in X. If X is not complex homogeneous as above, it is often possible to construct an embedding M

=

G/H

'-+

X

1\

'-+

1\

G/H

=

1\

M ,

1\

1\

where M ~M is equivariant. In this case M is called a Gcomplexification of M. There are at least two different ways of approaching the classification of (M,X): 1) Use complex analytic methods and concentrate on X(extrinsic); 2) Forget X and study Mas a real manifold with some additional structure (intrinsic) . One of the basic ingredients of the following is the interaction between intrinsic and extrinsic methods. We begin by giving an intrinsic characterization of a homogeneous CR-manifold. Proposition 1:

Let M

=

G/H

be~

real-homogneneous manifold.

The G-invariant CR-structures (R,J) dence with the pairs

~Mare

in 1-1 correspon-

(R,J) which satisfy the following condi-

tions: (0)

is ~ subspace of 11 with ~1 c an endomorphism;

R

Rc

·'J

and

J

R -+ R

is

155

( 1) JX = 0 if and only i f X E ,_; ~2

~

l..

~

(2) J X + X E ·L for all X E R; (3) Ad X E R and JAd X-Ad JX E ~ for all g E H, X E R; g g g (4) [X,Y] - [JX,JY] E Rand J ([X, Y] - [JX,JY]) - [JX, Y] - [X ,JY] E It for all X, Y E R. Two pairs (R,J) and (R' ,J') are equivalent if and only if

1

R = R' and JX - J'X E for all X E R then (3) may be replaced !?.Y_ (3) • [X, Y] E R and J[X, Y] for all X E

1,

R'. If H is connected,

[X,JY] E ~

Y E R.

c

The proof of Prop. 1 is a straightforward consequence of the definitions (see [R], p. 17). Corollary 2: Every G-invariant CR-structure on G/H is analytic. The following results of Andreotti-Fredricks provide the first steps toward connecting the intrinsic and extrinsic points of view. ~~~~~~~=~

([AF]). Let M be an analytic CR-manifold of type A Then M admits ~generic complexification (M,T). Given a generic complexification (M,T) of M, there are open neigh(m,~).

borhoods U of T(M)

U such

map f : U -+

in~

and

U of

that f · T

T(M) in Manda biholomorphic

= T.

A

Remark: If M is a generic complexification of M and M is of type (m,~) them dim~~

=m

- ~.

~~~~~~~=~ ~

([AF]). Let f ; M-+ M' be an analytic CR-map between A A analytic generic CR-manifolds M c M and M' c M'. Then

~are open neighborhoods U of M in~ and U' of M'~~· and~ holomorphic map ~ : U-+ U' suc~that ~IM =-;. As a result we have the following simple but useful

Corollary 5: Let M be ~analytic generic CR-submanifold of A complex manifold M. Then for every analytic CR-vector field

~

156 1\

X E rCR(M,TM) there is an open neighborhood U of M in M and ~

holomorphic

vector field Z on U such that

=

X

Z)

(Z +

IM.

Given a homogeneous CR-manifold M

=

G/H, Prop.1-Cor.5

show that M is an analytic generic CR-submanifold of a complex 1\

manifold M. For every g E G there are open neighborhoods U , g

1\

vg of M in M and a biholomorphic map f f

g

=

IM

: U ~ V such that g g g g. Taking M to be smaller if necessary, we may assume 1\

that every

~-vector

field on M is the "restriction" of a holo/\

morphic vector field on Mas in Cor. 5. If G acts almost effectively on M, i.e. the ineffectivity of the G-action is discrete, we have an of~

embedding~~

1\

1\

r 0 (M,TM). The complex hull

with r;spect to this embedding is denoted by

called the M-complexification 1\

M is refered to a

a~

of~.

1\

~and

is

Any such complex manifold

-complexification of M

=

G/H.

One of the main methods for studying M is to find G-equivariant EibrationsM

= G/H

~

G/I such that G/I and I/H are

known. We begin by considering the normalizer fibration G/H

~

G/NG(H), where NG(H)

:= {g E GlgHg

-1

= H}.

In some sense

this fibration factors out the group manifolds which are equivariantly contained in G/H. Of course we must take the CRstructure into consideration. This is done at the Lie algebra level and therefore the fiber is only locally a group. With Prop.1 in mind it is natural to define NCR(H), the CR-normalizer of H, as NCR(H) := {g E GlAd X E R and Ad JX- JAd X g g g

c't

for all X E R}.

Using Prop.1, a direct calculation shows that NCR(H) consists of the g E NG(H 0 G/H 0

~

G/H 0

,

)

such that the right-translation aH 0

~

agH 0

,

are CR-mappings with respect to the CR-structure

on G/H 0 which is induced by the covering map G/H 0

~

G/H.

We are now in a position to give intrinsically defined fibrations of G/H which are reasonable in the category of CRmanifolds.

157 ~Q~~~~~=g·

Let M

=

G/H be a homogeneous CR-manifold and let

L c NCR(H) be a closed subgroup such that H c L and

J ( R, n R)

c R,

n R. Then there is ~ unique G-invariant ~

CR-structure on G/L such that G/H

G/L is a CR-submersion. []

J R and

Using the properties of

J ('1! CR n R)

that

1icR n

c

Corollary 7. There is G/NCR(H) such that G/H

described in Prop.1, it follows thus we have

~unique

~

G-invariant CR-structure on ~

G/NCR(H) is

CR-submersion. []

If we look at NCR(H)/H as a submanifold of G/H with the induced CR-structure, then NCR (H) "normaJ.izes" the CR-structure. As a result we have Proposition 8.

The fibers of the fibration G/H

~

G/NCR(H) are

Levi-flat. Moreover the distribution generated £y the maximal complex subspaces tangent to the fibers of G/H

~

G/NCR(H) is

contained in the Levi kernel of M. []

·'1

We now assume M is embedded in a Let

n : G

~

1\

-complexification M.

G/H denote the projection and 0 : = n (e) . It is

useful to note that the isotropy Lie algebra 1\

~ :=

{Z E

1\

1

I Z(O) = 0} is directly connected with the CR-

structure of G/H. To see this ;:ecall that J

1\

~~

= A•J

+

Jx1,

where

is the complex structure on M and l'J is regarded as a real 1\

subalgebra of

JX(O)

=

1\

r 0 (M,TM). Hence Z =X+ JY E

(\

1

if and only if

Y(O). This means that, considered as elements in

TGe, X and Y are contained in Rand JX

=

Y (mod :. ) . Elementary

calculations then show that NCR(H) where Ad : G ~

= {g/\E Aut(~·:)

1\

1\

G I Adg h( f,( }, 1\ denotes the adjoint action of G on '''

which is induced from the usual adjoint action of G on tj. This (extrinsic) argument shows that G/NCR(H) naturally inherits a G-invariant CR-structure from the Grassmannian 1\ defined by the complex subspaces of ,.. which have the 1\

I

same dimension as :.1 : NCR (H) is just the isotropy N at \ the "point" i'l· Using PlUcker-coordinates we therefore have a G-equivariant map

158

G/H -+ G/ N-+ lPk («::) . This map is also given by the holomorphic sections of the A

anti-canonical bundle of M which are generated by the

A

~-sections

and is therefore a CR-map. We refer to it as the o:-anticanonical fibration of G/H. Since N

= NCR(H)

and G/H-+ G/HCR(H) is a

CR-submersion, the map G/NCR(H) -+ G/N is CR, but in general its inverse is not. However, if G/H is a CR-hypersurface, is totally real, or is a complex manifold, the map G/NCR(H) -+ G/N is a CR-isomorphism. The equivariant embedding G/N

~

lPk(«::) yields a representation

of Gin PSLk+ 1 (C). Let G denote the smallest complex Lie subgroup which contains the image of G (Recall that G is connected!), and let N be the G-isotropy group at the point N. Consequently we have G/H -+ G/N

'-+

G/N

'-+

lPk ( «::) ,

where G/N is a generic CR-submanifold of the complex homogeneous manifold G/N. Since N c: NG (H 0 ) , the fiber N/H 0 of the ';'f -anticanonical fibration can be written as the quotient A/f, where A = N/H 0 and r := H/H 0 is a discrete subgroup in A. Moreover the ~-anti­ canonical fibration of N/H is degenerate, i.e. the base is a point. We call such a manifold flat. Indeed flat implies Levi-flat, but not vice versa. From now on we focus our attention on homogeneous compact CR-hypersurfaces. In this case either G/N is projective rational and N/H is

~

compact flat CR-hypersurface or G/N

is ~ compact hypersurface in

G/N

and N/H is ~ compact

parallelizable complex manifold. In the latter case there is not always a G-complexification of G/H (see

§ 3). However,

in the first case we have ~~~~~~~=~·

Let G/H be

~compact

homogeneous CR-hypersurface

where G is acting effectively. If the base G/N of the

tj-

anticanonical fibration of G/H is projective rational, then there is

~

""

G-complexification G/H of G/H and

equivariant diagram of CR-maps

~

commutative

159

-

G/H

l

1\

1 1\

G/N 1\

where N

1\Q

= N&(H

--+

1\

G/H

1\

G/N

).

c

For details of the results in this section see [R].

3. THE BASE OF THE 1-ANTICANONICAL FIBRATION OF A COMPACT HOMOGENEOUS CR-HYPERSURFACE. Recall that if M

=

G/H is a homogeneous compact CR-hyper-

surface, then we have the

~-anticanonical

fibration

G/H -+ G/ N -+ lPJC In this section we make some detailed remarks about the base. Thus we assume that G is a real group of linear transformations 1\

acting onlPk' G is the smallest complex Lie group in PSLk+ 1 (C) which contains G, and that the G-orbit M = G(x) = G/H is a 1\

real hypersurface in the G-orbit X

1\

= G(x) =

1\

1\

G/H. Without loss

of generality we may assume that G is acting almost effectively on M. Note that the base of the

.lJ -anticanonical

fibration

either satisfies these conditions or is a compact homogeneous rational complex manifold. The latter situation is well-understood. The following is the fundamental for the classification of the linear situation described above.

E~~g~g~~~~~=l· ~

Either M is the equivariant product s 1 X Q of circle and a complex homogeneous rational manifold or the

~-simple

part Kss of any maximal compact subgroup K of G

~transitively

on M. In the latter case rr 1 (M) is finite.

The proof of this fact goes by induction on dimension and uses fibrations of the complex homogeneous space X

1\

1\

G/H.

We assume that the Levi-flat case M = s 1 x Q is wellunderstood and only consider the latter case in the above proposition. Hence we assume that G = K is a semi-simple compact linear group. Of course AutCR(M) may be much larger

160

= alBn,

e.g. M

but going to the possibly smaller compact group

has distinct advantages. In the usual way we have M = K/L

4

S/H =: X

4

]pk

I

where M is a hypersurface in X. Since linear semi-simple groups are algebraic, H is an algebraic subgroup of S, and X is Zariski open in its closure inlPk. It is therefore enough to understand algebraic homogeneous spaces S/H of complex semi-simple linear groups where the general orbit of is

~

~

maximal compact subgroup

real hypersurface.

If X

= S/H

is Stein, then we are in the case handled by

Morimoto and Nagano which was described in§ 1. In fact X is then the tangent bundle of a symmetric space of rank 1. If X is not Stein, i.e. H is not reductive, then H is contained in a proper parabolic subgroup of S. Let P be a minimal such subgroup. Since KP has hypersurface orbits in the fiber P/H of S/H

~

S/P, we are in a good inductive situation and it is

relatively easy to prove E~~g~g~~~~~=~·

Let S be a linear semi-simple· complex group and

H an algebraic subgroup. Suppose that the generic orbit of

~

maximal compact subgroup K of S in the homogeneous space X

=

S/H is

~

real hypersurface. Let P be

subgroup of S containing H (P

=

~

minimal parabolic

S is allowed) . Then P/H is

Stein. This result had already been used in the classification of certain almost homogeneous spaces

([HS],

[A]). In [AHR] it is

pointed out that the "Stein-Rational" fibration S/H

~

S/P is

essentially unique and exactly reflects the structure of

M

=

K/L.

The only cases where P is not unique arise from building the following type of example inside of

s.

Therefore all possible

such minimal parabolic groups can be described (see [AHR]). -t ~~giDgl~· Let S = SLn+l (~) act on 1Pn x 1Pn by A(p,q) = (Ap , Aq). There are exactly twoS-orbits inlPn X

=

x

1Pn' i.e. an open one

S/H which is the complement of a divisor orbit defined

by { (v,w)

I vt

· w

=

0}. There are exactly two minimal parabolic

161

groups, P 1 and P 2 , which contain H. They just arise by projecting on the respective factors of lPn x lPn: en S/H -+ S/Pi = lPn , i = 1, 2 . Given that M

=

c

K/L is described by its complexification

S/H and that S/H breaks into Stein and rational parts, i.e. P/H and S/P, it only remains to describe the possible pairs (H;P) coming from a given S. The situation is so concrete that in principle one could calculate virtually anything one needs, e.g. analytic cohomology as in [BFS]. So far calculations have been carried out in two directions: 1) A detailed root-

,

theoretic description of all pairs (H,P)

([AHR]); 2) An

explicit calculation of the Levi form of M ([AZ]). Since these results are a bit technical, we only discuss here an important case of 1). Consider the class of algebraic manifolds X

S/H as above.

Assume further that a Stein-rational fibration has the form S/H-+ S/P with P/H

=

en, i.e. a homogeneous affine bundle

where the generic Kp-orbit in en is a hypersurface.·~nder these assumptions it is easy to see that P is

repre~ented

in

the complex affine group, that Ru(P) is represented as the full group of translations en, and the reductive part of Pis represented as GLn(C), SLn(C) or Spn in the standard_ way. The latter case can only occur when n is even. Since Kp has a fixed point in P/H, we may take the reductive parts of P and H in a Levi-decomposition to be the same, i.e. P

=

L

~

Ru(P) and H

=

L

~

Ru(H), and Ru(H) is obtained

from Ru(P) by removing n 1-dimensional root groups: Now L is determined by a set TI of simple roots, i.e. the simple factors in L are determined by orthogonal connected chains in the Dynkin diagram for S. We write n as a disjoint union n 1 U n 2 , where the simple factors coming from n 1 yields the factor which is either represented as SLn or Spn. It is not difficult to show that P contains exactly one simple root group which is not in H. Let a be the root corresponding to this group. Then Ru (P) and

=

Ru (PTI U {a}) Ru (Pn 2 U {a})

162

Ru(H) = Ru(Pll U {a}) Ru(Pll U {a})', where P 0 is the parabolic group c~rresponding to the set a of simple roots. Of course n 1 must be orthogonal to n 2 U {a}. Given the above information one can write down the possible diagram. Since n 1 is orthogonal to the connected chain n 2 U {a}, it is enough to describe the latter. The following is a list of all possibilities where the "white" circles represent the roots in n 2 . 0

--

---

0

0- -

a

0

0

a o- --o_:( o

• ••

-

0

>-. a

--e

a

~ ~~g~g~~· Here we discuss G-invariant CR-structures on s 3 . The first such structure which comes to mind is M = am 2 , the c2 with the induced structure. Let

boundary of the ball in K

= su 2

act linearly on ~ 2 as usual. In this way K acts freely

and transitively on M and therefore inherits a left-invariant A

CR-structure. Since K

=

SL 2 (c), one easily finds a G-complexi-

fication, S

3

A A 2 = M = K/L ~ K/L = C' {(0,0)}

,

and the k-anticanonical fibration is just the restriction of the Hopf fibration,

£2 '

A

{ (o,o)}

A

K/L

c1

A

K/B

= lP 1 ,

to M. Now K is not the full group of CR-automorphisms of M. However, this group G is easy to describe: By analytic continuation G = AutCR(am 2 ) ~ Aut 0 (1B 2 ) = PSU(3,1). It should be remarked that the induced Hopf fibration of M is not G-equivariant. The •1-anticanonical fibration is just the embedding inlP 2 induced by the standard inclusion lli 2 ~ lP 2 . In fact G = PSL 3 (~) and we again have a G-complexification, am 2 = M = G/H

A A

0} of su 2 -invariant CR-structures on s 3 which are pairwise inequivalent and which are not equivalent to the above structure M. For this let X:= Q( 2 ) be the affine quadric which is

so 3 (~)-homogeneous

163

via the standard representation of 8L 2 (~)

80 3 (~). Recall that the universal cover of 80 3 (~). Hence we can

G is

=:

" " where H " is the group of diagonal matrices write X = Q( 2 ) = G/H,

G.

in

1 r

~

Let gr := ( 0 1 ), r

" -1 in G/H. " " grHgr Then {Nr

orbits of K

=

8U 2 . N0 manifold, and Nr ~ JP 3 r

>

0, and Nr be the K-orbit of the point

Ir

=

~ 0} is a parameterization of the

8 2 is embedded as totally real sub-

(IR)

as a real analytic manifold. For

0 let Mr be the universal cover of Nr. Equip Mr with the

CR-structure corning from the covering Mr

~

Nr and lift the

K-action. It follows that M is K-hornogeneous, is ~-analytically 3 r equivalent to 8 , and K acts freely and transitively. Thus, for each r > 0 we have a left-invariant CR-structure on K. Identifying Mr with K ~-analytic

~~wmg=l·

= 8U 2 , for every g

automorphism int(g), h

Int(g)

: Mr

~

is~

Ms

~

ghg

-1

E K we have the

, of Mr.

CR-rnap if and only if r

If int(g) is~ CR-rnap and g is not the identity, then r 1 0 -1 0 1 1 -1 -1 and g E { ( 0 1 ) , ( 0 _ 1 ) , ( _ 1 1 ) ' ( 1 -1 ) } ·

= s. = s

a -5 = {(b a:l, Ia 12 + Ib 12 = 1}, and q> := Int(g). Then CR-rnap if and only if q>* : TCR(M )~ TCR(M ) and q> is a e r e s \P* o Jr = Js o \P*. In this case q>* = Ad(g), Proof. Let g

TCR(M) = (( e r and Jr(v) = wr.

(0-1), 1 0

(-ir-1 i ) i ir-1

))JR.=:

The condition q>* : TCR(M )~~ TCR(M ) can be formulated as e r e s follows: ( 1) 2s Irn{ab} = -Irn{b 2 + 2 }

a

so that q>*(v) = Re{z} · v 2 -2 where z = b + a ,

+

Irn{z}

ws'

and

-2 2 -1Re{a -b -2r ab} so that where w

q>*(wr) Irn{w} •V + Re{w} ws' = a2 - b 2 - 2r- 1 ab.

164

The condition that

~*

commutes with the J-operators can be

spelled out as follows: - Im{z} · v + Re{z}·w

=- Im{w} · v + Re{w} · ws. s · r = - ab. Hence b = 0 or rb = - a. If b = 0, then (1) implies that Im{a 2 } 0. Since lal 2 +lbl 2 1,

Equivalently, w = z or b

2

it follows that a = ± 1 • Consequently g

±I and r = s. -2

If rb = - a, then ( 1 ) again implies that Im{a } = 0. Thus a, b E lR. Applying ( 1 ) and (2)' we see that IP*(v) = v and ~*(wr)

= ws.

Now~*=

Ad(g) and therefore application of IP* a -:6 amounts to conjugation with the matrix g = (b a). In particular,

det(wr) = det(ws)' i.e. r = s. Finally, a concrete calculation of centralizers shows that unless Int(g) = I we have r = 1 and g is in the group of order 4 in the statement of the proposition. ~:

Since a CR-isomorphism

Mr

~

[]

Ms induces an automorphism

of AutCR(Mr)' it is useful to note ~~miDg=~·

Let SU 2 act~ Mr as above. Then AutCR(Mr)

Proof. Set G := AutCR(Mr)

0

and consider

0

=

su 2 •

the~-anticanonical

fibration M = G/H

~

G/N.

The fiber is infinitesimally defined by the isotropy algebra 1\ 'II.. :

=

'YI-1

1\

1\

('t,) • Since ~ => sl 2 (C)

and the anticanonical fibration

for the affine quadric is finite G/H

~

(in fact 2-1), the fibration 1\

1\

G/N is likewise finite. Thus G/N is a homogeneous

surface on which SL 2 (c) acts and has an open orbit. Fer the same reason as above, the sl 2 (C)-anticanonical bundle of this orbit is finite. The classification of homogeneous surfaces 1\

then shows that G = SL 2 (c) (see [HL], [OR], [H]). Thus, since 1\ G => su 2 and G is a real form of G (Mr is strongly pseudoconvex!), [] it follows that G = SU 2 . Remarks (1) Since

su 2

acts transitively on Mr' no other Lie

group of CR-transformations acts transitively. (2) Even though Mr is strongly pseudoconvex, one can not "fill in the hole"

(see [AHR]).

(3) There is no G-complexification of Mr' because Mr G/H 1\ 1\ 1\ 1\ would be contained in G/H, G = SL 2 (c) and dimcH = 1. A simple

165

"

check shows that, no matter which 1-parameter group H is chosen, the generic G-orbit is not simply-connected. However, 0

~~QQQ§itiQD_J.

;=~~~i;~~~~lly N = M = am 2 and

Let N be

CR-hypersurface which is 3 the 3-sphere s • Let G = AutCR{N) 0 . Then either ~homogeneous

G = PSL(3,1) or N = Mr for some r > 0. In the

latter case G = su 2 . The CR-manifolds M, Mr' r

>

0, are pair-

wise different. Proof. Since the su 2 -anticanonical fibration of M has positive

*

dimensional fiber, M Mr' r > 0. Analytic continuation arguments show that AutCR(M) 0 = Aut 0 0B 2 ). Thus it remains to discuss the manifolds Mr. Suppose that

~

: Mr

~

Ms is a CR-isomorphism. We may assume

that ~(e) = e, where Mr and Ms are E-analytically identified with K = su 2 and e is the identity inK. Define~ E Aut(K) by g ~ ~g~-1. Now Aut(su 2 ) = Int(SU 2 ) ~ ~ 2 , where ~ 2 is generated by complex conjugation b. Thus ~ = int(h) or ~ = int(h) o b for CR some h E K. Note that b stabilizes Te (Mr) for all r. Since ~(g)· f = ~(g · ~- 1 (f)) for all f E K and ~(e) = e, it follows that ~(g) = ~(g) · e = ~(g) for all g E K. We have already shown that~= int(h) implies that r s (Lemma 1). Thus it remains to handle the case where

~(g)

= int(h) (g). We do

this by carrying out essentially the same calculations as in the proof of Lemma 1. Using the fact that ~*: T;R(Mr)~~ T;R(Ms) we obtain exactly the same equations as in Lemma 1, i.e.

(1) and {2). Further, ~*Jr = Js~* yields z = -w. Thus in this case ra 2 = ab and arguing along the same lines as in Lemma 1 yields r = s. o

166

4. THE FIBER OF THE1-ANTICANONICAL FIBRATION; CLASSIFICATION AND THE CASE OF NON-DEGENERATE LEVI FORM. The aim of this section is to understand the fiber N/H of the 2r -anticanonical fibration M

=

G/H

-+

G/N. Without imposing

further conditions it is impossible to say more than we already know, i.e. N/H is either complex parallelizable or a flat hypersurface. Thus we will impose two kinds of further conditions which we believe to be quite natural from the complex analytic viewpoint: We first consider the case where the Levi form of M

=

G/H is non-degenerate. Secondly, without

any Levi condition, we assume that G/H admits a Kahlerian complexification, i.e. a Kahler manifold X which contains G/H as a real analytic compact hypersurface. In this case we refer to M

=

G/H as a Kahlerian homogeneous CR-hypersurface.

Kahlerian structures on CR-manifolds have been primarily studied by methods

coming from Riemannian geometry (see [KY]) .

It is however our aim to handle this situation by purely group theoretic and complex analytic techniques. The first part of this section is devoted to the study of a compact homogeneous CR-hypersurface M

=

G/H where the Levi

form of M is non-degenerate, i.e. no zero eigenvalues. The basic tools for this are Prop. 2. 8, Thm. 2. 9 and the discussion just prior to Thm. 2. 9. So let us have a look at the )-J-anticanonical fibration G/H

-+

G/N. If G/N is a CR-hypersurface, then N/H is a complex

manifold and 3. 8 (the "moreover" part) implies that N/H is finite. If G/N is a complex manifold, then there is a 1\

1\

G-complexification G/H of G/H as described in 2. 9. In this case N/H is a flat CR-hypersurface. Again by 2. 8 it follows that N/H

= s1

(Note that N is connected, because G/N is simply-

connected.). From the proof of 2. 9 (see [AHR] for details) '

1\

1\

it is clear that G,H can be chosen such that N/H

= C*.

Summing up we have the following ~Q~~~~W=l· Let G/H be ~compact homogeneous CR-hypersurface with non-degenerate Levi form. Then either G/N is projective

167

rational and G/H ~ G/N is an ~

equivariantly embedded in ~

pal bundle over G/N,

s 1 -cR-principal ~

G/H

bundle which is ~*-princi­

homogeneous non-trivial G/N is finite.

o

Remark. Together with the results in 3., Theorem

gives a

complete fine classification of homogeneous compact CRhypersurfaces with non-degenerate Levi form. Now assume that M

=

o

G/H is a Kahlerian homogeneous CR-

hypersurface. If the base G/N of

the~

-anticanonical

Fibration of G/H is a hypersurface, then N/H is obviously a compact complex homogeneous parallelizable manifold, i.e. every connected component of N/H is a compact torus [W]. If the base G/N is not a CR-hypersurface, then it is projective rational. In this case by 2. 9 we have a G-complexification G/H so that G/H

·--~

A A

G/H

1 C5 l

G/N

--->

e~~ A A

where N/H is a flat CR-hypersurface in N/H. Since G/H lives in a Kahler tube X and since the germ of a generic complexification A A

is unique, there is an open neighborhood U of G/H in G/H which is Kahlerian. Thus the same is true for N/H A A

=

N/H

A A

A

A/r , where A

=

A AQ

N/H

A

and r

stand the following situation:

Gbe

Let

a complex Lie group,

~

A A

N/H. Now

A AQ

H/H . Hence we must under-

A

r

A

< G a discrete subgroup

A

A

and G a Lie subgroup of G such that the G-orbit G/r of r A A

in G/r is a compact hypersurface admitting a Kahlerian A A

neighborhood in G/r. Before stating the result in this setting, we want to A A

*

discuss two special cases. First, assume that O(G/rl Let

11:

Now :::

:

= 'Y n i ''I

A ' · L

< G, the G -orbit of ''>t

r is contained in the compact hypersurface G/r . Let A A f E 0 (G/H) be non-constant and note that f 1 G/r is bounded. Consequently the restriction of f to a G~,L-orbit in G/r is constant, and, since the fibers of f and the G,,-orbi ts are

168

1-codimensional, it follows that the G_-orbits are closed.Hence 1\

1\

*

if O(G/H)

"'-

~,

then we have 1\

1\

G/f----+ G/f

l

J

(J

1\

G/G,nf- G/G r -~ 1\ /\"'" It follows that G/G_f = 81 ~ G/G'l'ltf = C*. Thus G/f is a torus ··o bundle over 8 1 which is equivariantly embedded in a torus 1\

bundle over C*. In particular G is abelian or 1-step solvable. The second special case is where G is simply-connected and 1\

abelian. In this case G

=

n 1\ C and r is a discrete subgroup of

rank 2n or 2n-1 and r has rank 2n-1. We then have

= E2n-1;r2n-1 ~

G/f

Cn/r2n-1

=

G/r

1\

1\

and there is always a fibration 1\

G/f

-+

1

G/f

/\t ,

G/L ..-.:'--+ G/L where 8 1

= L/f ~ ~/r

G/L G/L is a torus (see and e.g. [V]). Thus if G is abelian, we always have an 8 1 -CR1\

/\

fibration of G/f onto a compact torus. Note that if r is of 1\

1\

rank 2n and G/f is an irreducible torus, then this fibration 1\

1\

is not induced by a fibration of G/f. 1\

1\

1\

If G is abelian and simply-connected and O(G/f) 1\

rank r

=

*

C, then

2n - 1 and the above shows that

=

G/f 1\

I f 0 (G/f)

=

8

1

X

~

G/L

C*

1\

X

1\

G/L

1\

1\

G/f.

1\

C, then G/f is called a Cousin group.

With these preparations we can.now state the main results in the flat Kahler case. For this it is convenient to refer to 1\

1\

(G, G, r, f), G/r

1\

-+

1\

G/r as above, as the data of a "flat Kahler

CR-hypersurface, FKH". 1\

~g~~~~m=~·

1\

1\

Let (G, G, f, f) be the data of FKH. Then G is A 1\ solvable and there is a closed normal complex subgroup I c G 1\

such that I

1\

0

is contained in G, is abelian, with I

a•~ c~If G~ admits -1\ 1\ = G r.

can choose I

a

non-con~ant CR-fun~n,

=

1\

I 0 f and

then one

"ltl_.

We now state the main structure theorem for flat Kahler CR-hypersurfaces.

0

169 ~Q~~~~~=~·

~ fol~owing

statements are equivalent: (G, G, f, f) is the data of FKH;

i)

G~~

ii) GnLis abelian and either a-is abelian or the

------_, ,

orbits in G/f are closed and the image of the

representation of p : f-> Aut (G_,J, p (y) (g):= ygy is finite. A

Remarks:

A

(1) If (G, G, f, f) is the data of a FKH and

A

O(G/f)=~

A

then Theorem 3 shows that G is abelian. A

A

(2) If G is non-abelian, then 0 (G/r) *C. The holomorphic reA

A

duction G/f -> G/G,.,J is a torus bundle over C* which admits a finite covering which is trivial. Conversely, given a homogeneous torus bundle which is trivial after going to a finite cover, the preimage of s 1 is a FKH. D

Sketch of the Proof of Theorem 3. Using Theorem 2 and the fact that there is a G-invariant measure on G/f, standard integration arguments yield a neighborhood of G in

e which

is right

G-invariant and admits a right G-invariant Kahler metric. Explicit calculations show that nilradical of

e and

e is

abelian or Gm is the

G. In the latter case the Gm-orbits in G/f

are closed ([MOS]). Furthermore, direct calculations show that ad(~)

has purely imaginary spectrum and, since p(f) stabilizes

a full lattice, it consists of torsion elements and is therefore finite.

D

The proof of Theorem 2 is carried out by induction on di~G/H

(see [R]). To give a taste of what is done, we give a

sketch of the proof of one step in the solvable case. Proposition 4. Let

e, G ,~and

f be given as in Theorem 2.

Assume the following: A

1)

G and G are solvable and simply-connected;

2)

O(e/~)

=

C;

3) The center of

e is

discrete;

4) There is an abelian connected complex normal subgroup ~ of

e such

that ~~ is closed in A

A

e and e•c ~A

A

Then there is a closed complex subgroup I c G such that f c I,

170

A A A I/f is connected and I 0 is

~

non-trivial complex Lie subgroup

of G'~~',.: Sketch of the Proof of Proposition 4. Since~ is an abelian normal subgroup of a simply-connected solvable group, it can be identified with its Lie algebra. Consider the map A A A -1 A int : G ~ Aut(N) ~ GL(~), int(g) (n) = gng • Since N is abelian thisfactors through the abelian quotient e;~ and induces arepresentation A A p : G/N

~

A Aut(N).

A

If N c G, then there is nothing to prove. Thus we assume that A

A

N ¢ G. In this case N = N n G is an abelian normal subgroup of A A A A G (Note that G' c G~~c G!) and G/N = G/N. Let H := Nf/N. Since H is a lattice in G/N

= e;~,

it follows from 3) that

A

p (H) c Aut (N) is no·i:: unipotent. Hence there exists h E: H such that adh

=

p(h) -

id~

is not nilpotent.

Now consider the following chains of subspaces in ~: A A Ak Ak+1 No N, N adh(N ), k ;;: 0 k No N, Nk+1 adh (N ) , k ;;: 0 No

.....

N"'l.

=

N n G ' Nk+1 •n

.,_

k adh (N~, k ;;: 0.

It follows that k c Nk c •.•.. Nk+1 c NAk+1 c N.,_ Ak Now adh is not nilpotent. Thus the chain (N ) kEN becomes · f or some k 0 . Th erefore Mko ~s · a complex L~e · sub group stat~onary A ko A of G""'1. and one can show that I = N · r does the job. D

The proof of Prop. 4 shows one of the main difficulties in studying homogeneous CR-manifolds: One can rather easily find fibrations by real groups, but it is much more difficult to find the appropriate complex subgroups. D

171 5. CLASSIFICATION OF

K~HLERIAN

CR-HYPERSURFACES

In this section we describe a fine classification of homogeneous compact Kahlerian CR-hypersurfaces M 1\

a Kahlerian tube with M

->

A

= G/H.

Let M be

M and look at the 1-anticanonical

fibration G/H ... G/N of G/H. We first consider the case where G/N is projective rational. In this case it follows from 1\

1\

Thm. 2.9 that there is a G-complexification G/H with an open 1\

1\

Kahlerian neighborhood U of G/H in G/H and we have the diagram 1\

1\

G/H -+ G/H

+

1\

G/N 1\

where N

+1\

G/N

1\

1\

NG(H 0

).

1\

By Thm. 4.2 we know that N/H 0 is solvable

and that we have fibrations 1\

N/H

--+

+ N/I +S1

1\

-->

1\

N/H

+1\

N/I

N/L where N/I

=

1\

1\

1\

1\

N/I is a torus, I/H is an abelian group, and N/L

is a torus. Thus we have the following picture: 1\

G/H + G/I +S1

1\

-->

G/H

-

+ 1\ 1\ G/I

G/L + G/N

-

1\

1\

G/N.

The most striking question at this point is whether or not G/L is Kahler. A theorem of Blanchard ([BL]) implies that G/I 1\

1\

1

is a Kahlerian hypersurface in G/I. So we have an S -CR-principal fibration of G/I onto G/L which extends to a holomorphic 1\

1\

submersion on some Kahlerian open neighborhood U of G/I in G/I. This does not in general imply that the base of the s 1 -CRprincipal bundle is Kahler.

172

Example (F. Lescure). Take a E ~*with lal< 1 and let

r :=[(:nan

a~)

fibrations:

E GL 3 (C) In E

3

c . . . {0}

J.

'l

Consider the following

1\

GL 3 (~)/H

+C*

T 1\

GL 3 (C)/N

JP2 (C)

where

and 1\

N 1\

1\

={(~

aAb)

\ E ~*, a,b E: C, A E GL 2 1\

(c)} ~

1\

fH/H ~ 7L and T N/fH = ~*I 7L is a compact complex torus. Now'let E = c 3 ..... { o} xJP (~)x denote the fiber product of 2

Obviously

the bundles ~ 3 ..... {0} ~JP 2 (~) and X ~JP 2 (~). Thus we have the following diagram of holomorphic fiber bundles:

=0, the bundle E

~ c 3,{0}

is

~*

triviaL Thus E is Kahlerian. The restriction of E ~ T to T x s 5 c: E is an s'-CR-principal bundle over X·. But X is not Kahler.

a

The critical point in the above example is that the s 1 is fibered out in the wrong direction. In order to make this precise we need to introduce some notation. Let X be a Kahlerian complex manifold containing a compact hypersurface M. Let J be the complex structure on X, g a J-invariant (i.e. Hermitian) Riemannian metric, and w(X,Y) =g(X,JY) its associated Kahler-form. For p E M Ep : = {X E ™p I w (X, Y) = 0

v

y E ™p}

173

=

is 1-dimensional and transversal to RM , i.e. TM p p Ele~entary

RM e E . p p

differential geometric techniques lead to the

following result. Proposition 1. Let M ~

be~

in~

real hypersurface

Kahler mani-

holomorphic map onto ~ complex manifold Y, and assume that rrlM is an s 1 -CR-principal bundle. fold X. Let rr : X

Y be

~

If one of the following conditions hold, then X is Kahler: 1) M is Levi-flat; 2) ker(rriM)*

=

E

for all pEM.

p-

[J

The above shows that in order to show that G/L is Kahler s 1 -fibration in the "E direction". By

we need to find an

p

1\

1\

induction this is reduced to the case where N/H 0 is abelian. 1\

In this situation one has

th~

N/H-CR(resp. N/H-holomorphic)1\

principal bundle G/H

~

1\

1\

G/N(resp. G/H

1\

~

1\

G/N). Since G/N is

projective rational, a maximal semi-simple compact subgroup K of G acts transitively on G/N. Thus the group A

=

N/H

x

K acts

transitively as a group of CR-transformations on G/H and we 1\

may write G/H

1\

A/D. Since A is compact and also acts on G/H,

we may assume that the Kahlerian neighborhood U and the Kahler-metric w are A-invariant. Let rr :

A~

A/D denote the projection and let 0 := rr(e). -1

~

For a subspace v c T (A/D) 0 let V denote the space rr* (V) c Since the Kahler-farro w is A-invariant, it determines an element F c H2 (.ct, m) given by F = rr* (w JA/D) JN'(. Note that C

= N/H

is the connected component of the center of A, and

= -t

~

e ,1;0 • Let p 1 :a~ ,c and it. denote the projections. Now [.k, ,l] = k, H2 (./i,m) = 0,

=

0 for

thus the Lie algebra splits, P2 :/.'If, and d~

/.Jt..

/./t

all~ EC 2 (~,m). Hence, elementary calculations

Pi

2

show that F = p1(F 1 )+ (dF 2 ), where F 1 E H (..c,m) and F 2 E C 1 (k,m) . By the non-degeneracy of the Killing form B on k, we may write F 2 (X) F(X,Y)

=

= B(X,W)

for some WE.fv. Thus we have

F 1 (p 1 X,p 1 Y) + B([p 2 X,p 2 Y],W).

It follows that E0 = p 1 (E 0 ) + p 2 (E 0 ). Furthermore, since the C-orbits in A/D carry a CR-hypersurface structure we obtain E'0 = p 1 (E0 ) + d . The above formula also implies that E'0 n k is

174

the centralizer of W E ~ and D is contained in the centralizer of the torus in K which is generated by W. Now, excluding the Levi-flat case in which we show A/D CR N/H

x

G/N and in which G/L is Kahler (Prop.1), we show

E0

= p 1 (d-) + rL. From this and Prop.1 it follows that the fibration with the group < exp (E 0 n k) > • D yields an S 1 -CR-principal fibration G/H ~ G/L where G/L is Kahlerian. that

It is now possible to state the first result in the classification of G/H. ~Q~~~~ID=~·

Let G/H be

wh~

surface,

~

homogeneous compact Kahlerian CR-hyper-

'/1

G is connected and the base G/N of the

-anti-

canonical fibration of G/H is projective rational. Then there

Gwith

exists a complex Lie group one has ---

G/H -1-

G/N

-

G~

N/H 0 is solvable, and

a;~ -1-

a;~

as in Thm. 2.9. Moreover there is a closed complex Lie group

7\ A

A

I c N such that I

~compact

Hc G n

~

/\--/\0

c N · H0

,

7\

-A

A

A

--

1\7\

N'•H c I, I 0 /H 0 is abelian, I/H is

torus, and there is a closed subgroup L c G such that ~ ~ G/L-is an S 1 -cR-principal b~e over

c L c N and G/G n

$

the homogeneous compact Kahler manifold G/L. If

c

1\1\1\1\-

a is

a

7\7\

closed complex subgroup such that IcJ, J/HcG/H, and J=J 0 1\

1\

1\

1\

•

I',

1\

then J c N and J has the same properties as I. c

The remaining case is that in which G/N is a CR-hypersurface. s 1 xQ (Q projective

In this situation we know that either G/N =

rational) or a maximal compact semi-simple subgroup K of G acts transitively on G/N (see Prop.3.1). In the latter case rr 1 (G/N) = 1, ~ 2 (see [AHR]). In both cases the components of N/H are compact complex tori. The fine classification in this case is handled by methods similar to those in the proof of Thm. 2. For this consider the bundle G/N° n H torus principal bundle. Again A:= N°/N° n H

x K

K

-+

G/N°, i.e. a

acts transitively on G/N° and

acts transitively on G/N° n H. Using F as in

the proof of Thm. 2 and observing that the orbits of

175

c

N° /N° n H are complex, one shows that

We close by stating the main result on the bundle structure of a compact homogeneous Kahlerian CR-hypersurface. ~g~~~~ID=~·

Let M be a compact homogeneous connected Kahlerian CR-hypersurface. Then either 1) M is~ torus bundle over an s 1 -CR-principal bundle over

~

homogeneous complex Kahler manifold T

x

Q

(T a complex torus, Q projective rational), or 2) either M or a 2-1 covering of M is T

x

is

~

CR-product

M' , where T is ~ compact compact torus and M1 ~

simply-connected compact homogeneous Kahlerian

CR-hypersurface which either lies 2-1 covering

of~

CR-hypersurface

in~

-

in~

-

n n

(~)

or is a

-

(~). []

Remarks.

(1) The hypersurfaces M1 in (2) above are classified

via the Stein-Rational fibration (see § 3). (2) The reader should note that by using Thm.4.3 one can say much more about the structure of Min case (1) above. In particular the fiber of the ~ -anticanonical fibration M is either an s 1 -bundle over a torus or vice versa.

~

Q

For detailed proofs of the results in this section see [R].

References [AHR]

Azad,H., Huckleberry,A., Richthofer,W.: Homogeneous CR-Manifolds, Crelles J. (to appear)

[AF]

Andreotti,A.,Fredricks,G.A.: Embeddability of real analytic Cauchy-Riemann Manifolds, Ann. Scuola Norm. Pisa 6, 285-304 (1979)

[Az]

Azad,H.: Levi-Curvature of Manifolds with a SteinRational Fibration, Manuscripta Math. (1985)

[BFS]

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[BL]

Blanchard,A.: Surles varietes analytiques complexes, Ann. Sci. Ec. Norm. Sup. 73, 157-202 (1956)

176

[BR]

Borel,A., Remmert,R.: Uber kompakte homogene Kahlersche Mannigfaltigkeiten, Math. Ann. 145, 429-439 (1962)

[BS]

Burns,D., Shnider,S.: Spherical hypersurfaces in complex manifolds, Inv. Math. 33, 223-246 (1976)

[G]

Greenfield, S.J.: Cauchy-Riemann equations in several complex variables, Ann. Scuola Norm. Pisa, 257-314 (1968)

[H]

Huckleberry, A.: Homogeneous Surfaces,

[HL]

Huckleberry, A.T., Livorni, E.L.: A classification of homogeneous surfaces, Can. J. Math., Vol.XXXIII, No.5, 1097-1110 (1981)

(to appear)

[HO]

Huckleberry, A.T., Oeljeklaus, E.: A characterization of complex homogeneous cones, Math. Z. 170, 181-194 (1980)

[HS]

Huckleberry, A.T., Snow, D.: Almost-homogeneous Kahler manifolds with hypersurface orbits, Osaka J. Math. 19, 763-786 (1982)

[KY]

Kon, Masahiro, Yano, Kentaro: CR Submanifolds of Kahlerian and Sasakian Manifolds, Progress in Math. v.30, Birkhauser, (1983)

[M]

Mats~shima,

[MN]

Morimoto. Y., Nagano, T.: On pseudo-conformal transformation of hypersurfaces, J. Math. Soc. Japan 15, 289-300 ( 1963)

Y.: Surles espaces homogenes kahleriens d'un groupe de Lie reductif, Nagoya Math. J. 11, 53-60 ( 1957)

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A. Huckleberry and \v. Richthofer Fakultat und Institut fur !1athematik Ruhr-Universitat Bochum UniversitatsstraBe 150 D-4630 Bochum 1, FRG

Problems of Value Distribution in Complex Analysis for Several Variables P. Ielong Universite de Paris VI Ma.thematiques 4, Place Jussieu F-75005 Paris dedicated to Wilhelm Stoll

INTROOOCTICN

Value distribution theory for holarorphic functions and holarorphic mappings is an irrportant part of CC!lplex analysis theory was established only for mappings

a:

For a long tilre, the

ii: , i.e. ,

+

for

entire or merarorphic functions defined in the CC!lplex plane ranges in

a:

or

ii:

the

a: ,

values of with

(identified with the Riemann sphere).

We are indebted to Wilhelm Stoll for giving beautiful extensions of the theory to CC!lplex manifolds, opening ways to new problems of value distrihltion theory, and giving to the theory a larger danain of research. Haw does one generalize to

ly for holarorphic mappings the image

f(C)

en

a:

f :

-+

gives a covering of

the general situation of

f : X

the classical results obtained fonre-

0: ? ii:

is not a rational function, of unboW1ded degree. 'Ib consider If

f

Y , we must assume that an exhaustion

+

_ {X:i"~)o of X is given , Xr=> Xr•' for r > r', and r~ Xr =X. We suppose Xr is a relatively CC!lpact danain in X , with image f (Xr) c Y of bounded degree

n (r) , defined by

n(r,a) =card [Xr n f and

n(r) = sup n(r,a)

p * (x)

(III) Iv (x,u)

~

s

zeros

0

G of the

E'

E~

G , and

variety Ms c: G such

x

U E

co

q>v

and there exists

q> :

r >0

u

E~

exists

ex> 0

such

that

u

llx-x0

11 < ex ; the ball

for

aurn

E~

> s isolated zeros , Iu I < r

lui = r

IF(x,u) I ~ ~

n(x,r) =.f-.. for

s'

has

such that the disk

zeros of lP arxi IF(x0 ,u) I ~ c > 0

n

affiF = - (x,u)

~

is open. Let x 0 E G , x 0 f/.

F(x0 ,u)

+

is defined in

is a polynanial with constant coeffi-

cients of a finite number of the variables

the entire function

(with multiplicity). Let

= G or there exists an analytic sub-

= 0 , where

G'

u

x E G , such

such u + F(x,u) ;;; 0 • 'fuen

E~ c: Ms • Moreover Ms = Ms x c

First we prove that

a: •

be of finite order with respect to

F E H (r!)

the analytic set in

is a closed set in

(x,u) , xEG , u E

contains

s'

• By continuity, there

arxi

~F (x,u)F- 1 (x,u)du = s'

f lal=r

B(x0 ,cx)

u

does not belong to

E~ ; E~

is

closed. Now we prove that E~ is contained in an analytic subvariety Min a acmpact ball B c: G , or (equivalently) that E' = E x C is contained s s in an analytic subvariety M of 'B. We give first the equations of M

188 ~

~c:B

in a danain ~

n w =

cl>

~

1

Let be

•

xo

(x01 u0

F(x,u) =

(9)

G (x1 u)

with

lPx (u)

=

G(x,u) 1

= aq(xluo) = q!

s

n

-1

n [II X

= ~

~c:B

1

such that

such that 1

= q, •

~nw

F(xoluo) "f 0

Then

(x 1 u) - xo

II


x (u)

s n [llx - x 0

x E E (12)

Rx (u)

=

is a polyncmial of

s

I: b. (u-u ) J 0

J

G(x 1 u)

1

(11)

of degree at I!Dst p+s-1 •

1

with

bJ.

depending on

x

for

0

I n

• As

a (x,u) by aq' (x,u) , we obtain q D (x,u) = F- 0 (x,u)D' (x,u) v \) (s+1) (v+1) and D~ (x,y) is obtained writing a' . instead a .

t:;. ·• If we replace

v-J

D~

in (13) • Then of

0

~ [F- 1 aF]

= - 1-

(x,u) is holarorphic of (x,u) En

.- - F(x u) "o ' ' · ••

v-J

and is a polynanial

."v+s+ 1

aF (x u) with constant coefficients. v+s+ 1 ' au a/ We suppose that the equations D'\) (x, u) = 0 , for v 'l s+p , define an analytic sUbvariety

Ms

in

B x C • Cbviously,

Ms

is invariant by the

(x,u) + (x,u + v) for v E a: • Moreover Ms contains a: ; E's in contained in B in the analytic sUbvariety Ms n [u=O].

translations E.. X

E' s

'!ben

b/ If

is contained in an analytic sUbvariety of

=0

D'\) (x,u)

in

Q

for v 'l s+p , and i f

E.. x C are analytic sUbvarieties in a neighboorhood

II

/:;. : llx- x0

then there exists solutions ~ (u)

= b0

+ b 1 (u-u0 )

and fran (14)

,

•• •

b0

,

(E

co

X

F(x0 ,u0 ) C)

u (Es

X

for

F(x,u) ¢ 0 , both W and (x0 ,u0 ) f/. W , we construct

lu - u 0 l < r~ , such that /:;. n W = cp ~ bs , not all zeros, and

••• ,

G(x0 ,u)

is a polynan of degree at rrost is a rational function of

poles (with Im.lltiplicity). Then

zeros. The conclusion holds for all with

~

+ bs (u-u0 ) s

,

we deduce that

with at rrost s

re

is restricted to camplex curves, the images of these curves



cannot have arbitrarily large order of contact with the level hypersurfaces of

r . This part of the v.ork is based on results by K. Diederich

and J.E. Fornaess, and J. D'Angelo. A quantitative statement about the

contact between holooorphic curves and level hypersurfaces will then prepare the ground for the final step of the proof, which is the estimation of

q,' • The estimate we thus obtain. is equivalent to the

Holder-continuity of

q, •

Finally a 'lt.Ord about the notation. IbweVer, the occurrence.

c' s

c,c',...

will stand for positive

D1 , D2 and q, , unless stated otherwise). will not necessarily rrean the sarre constants on each

constants (depending only on

198

2. Estimating the boundary distance

Let

o 1 , o2

and

c dist(w,ao 2 )Tl

We shall first derive Lamia 2.2 and then show its equivalence with

LEmna 2.1.

Proof of Lernna 2.2: is,

P.

1

Let

Pi

denote the Bergman projection in Di • That L2 (D. ) to its subspace

is the orthogonal projection of

consisting of holorrorphic functions. In a canpactly supported

;

according to which on

snooth strictly starshaped dana.ins the Bergman projection maps the Sobolev space w112 intoitself.Since (det¢ 1 )(q>o¢) isin CQ(D 1 ) , P 1 maps it into w112 (o 1) • In particular, there is an e > 0 such that det ¢ 1 = P 1 ((det ¢ 1 )q>O¢) E L2 +e, Hence "">

I

/det ¢ 1 12 +e

=

01

I

1-e

/det \jl 1

02

Applying now the sul::Iooan value property to the subharnonic function /det \jl 1 1-e

on balls centered at w E o2 and of radius dist(w,ao2 ) , the second inequality in I.ernna 2.2 is obtained. Since the role of o 1 and

o2 is synmetric, the first inequality holds as well. Proof of the equivalence of I.ernnas 2.1 and 2.2:

Suppose first that (2.1)

oolds. Then using the cauchy estimates and the boundedness of

4>

and

t ,

we have /det q, I (z)

I=

/det o/ 1 (¢ (z))

r1

>

Q)nversely, suppose that (2.2) and (2.3) l:x>ld.

the length of

q, 1 (z)v , where v

we

first want to estimate

is an arbitrary unit vector in

«fl .

Of course we have an upper estimate

(2.4)

/¢ 1 (z)v/ < c'dist(z,ao1 )-1

b.lt we shall need an estimate from below. TO this end, let v1

= v, v 2 , .•• , v n

be an ortl:x>nonnal basis in

«fl .

Then by cramer 1 s rule

and (2.4) (applied to v = v 2 ,v3 , ••. ) c dist(z,aD 1l'TJ < ldet ¢ 1 (z)

I
c dist(z,ao 1 )M

let now to

cf>

(z)

w and

=

v E <J:I1

w . Chcx:>se a unit vector

direction as the vector 0 < t < dist(z,ao 1 )/2

(2.6)

WW 1

the point which lies the nearest

E ao2

W1

•

so that

f (t) =

Put

I!t 0

f (0) + f 1 (0) t = w +

at distance

> ct dist(z,ao1 )M

distance be

2lw-w 1

(2.7)

t < 2c f (O) + f 1 (O) t

and

I=

distance of

-1

cf>

cf>

1

have the scure

(z)v

(z +tv) • Then for

we have

lf(t) - f(O) - f 1 (Oltl

The point

•

cf>

1

(z) vt

2

f"(u) (t-u)dul < C 1 t dist(z,ao1 ) lies on the half line

from w • Choose

t > 0

WW 1

-2

,

so that this

2 dist(w,ao2 ) ; then

lw-w 1 ldist(z,ao1 )

-M

, o 2 . M:>reover, the lw-w 1 I (if z , and

lies in the C<XIq?lement of

f (O) + f 1 (0)

to

o2

will be

therefore w , is sufficiently near to the boundary

ao 1

Now there are two possibilities. Either the value of

res. t

aP 2 ) .

selected

t < dist(z,ao 1)/2 , or t ~ dist(z,ao 1 )/2 . In the first case (2.6) shows that the distance of f(O) + f 1 (O)t to 2 -2 f(t) = cf>(z+tv) E o 2 is less than c 1 t dist(z,ao 1 ) , hence in view of above satisfies

(2.7) lw-w 1

I
act real analytic variety in

does not contain any nontrivial genns of canple.x varieties.

Using the tools developed. by J. D1Angelo in [p'Q and [p~ , this theorem can be given a quantitative fonn. We are now going to review some of the concepts he introduced. and theorems he proved.. Let

~*

stand for the set of nonconstant genns of holcrnorphic

(0::,0) ~ (crfl,p) ,

mappings

p

being a fixed point in

(eventually vector valued.) soooth function near order in

0, i.e., the largest

v

such that

crfl . If

0 E G:: , v (g) !: ..

g(~) - g(O)

is an

g

is its

= O(~v)

(~~0).

Let nOW' of

r

be a real analytic function definerl in a neighborhood

p E crfl , dr (p) t-

0 . We put

ll(r,p)

sup

v (r o f) /v (f) !: ..

f E ~*

and call it the maximal order of contact of the hypersurface

{ z : r (z)

=r

(p)}

with holorrorphic curves.

The order of contact of a proper ideal

I

the ring of genns of holcrnorphic functions at

contained in

~

p

=

(!} ,

p , is an analogous concept,

defined by sup

f E ~*

of

-.llf (I)

By

the Nullstellensatz

I

consists of the single point

inf v (h of) /v (f) h E I

< ..

is finite i f and only if the variety

p , and in this case

I

contains

203 some power of the maximal ideal

(!) •

One also has the estimate

(3. 1)

(see

@i]

In

1

Theorem 2. 7).

@;U

D'Angelo shows how to reduce the carputation of

ll(r 1 p)

to

the carputation of 1:*(I) for certain ideals I c (!) • The way he p achieves that is the following. For the sake of simplicity we shall

assurre

r (p) = 0 •

First D'Angelo shows that there exists a separable Hilbert-space which we are going to identify with

£2 -

"if, -

and three genns of holorro:rphic

mappings H :

(!~flip) ~ (4:10)

I

(ttfllpl ~ (illo)

--+ ('ie

(ttfl 1 p)

1

0)

such that r(z) = 2 Re H(z)

+

=2ReH(z)

IIF(z) 11 2 - IIG(z) 11 2 =

+

Z(IF.(zli 2 -IG.(z)l 2 l

J

Next for any unitary transformation the ideals i.e.

I (U 1 p) c

F. - Z u .. G. J i Jl l Then he proves 1

(3.2)

(!)

p

1

U :

1l ---+ til

he introduces

generated by H and the components of u.. Jl

F - UG

being the entries of (the matrix of)

sup1:* (I(U 1 p)) ~ ll(r 1 p) ~ 2 sup 1:*(I(U 1 p))

u Where

J

U runs over all

u unitary mappings

ie. --+ -df

Theoran 10) • He also proves the following important theoran:

(see @~ 1

U .

1

204

Theorem 3.2:

Suppose

bounded for

p

near

ll(r,p0 ) is finite for some p0 . Then p0 • (See Theorem 4. 11 • )

ll(r,p)

is

@TI ,

Using these tools, it will be easy to prove the following theorem, which essentially says that analytically bounded domains are of finite type (in the sense of D'Angelo) • This theorem has also been found by D'Angelo and possibly by others, too, but it does not seem to have ever been published.

Theorem 3.3: r :

Gfl

~

Let

Dc

afl

be an analytically bourrled domain,

a real analytic defining function of

D • Then there is

a neighborhood

0

such that for

p E0

ll(r,p)

JR.

we have

Proof:

of

aD

and a positive integer

By Theorem 3.2 and tne canpactness of

ll(r,p)


aD

ll(r,p) =

it suffices to prove ®

•

Then by (3.2) there

of unitary matrices such that (v ._._.

(3.3) It can be assumed that for each

.

Then the operator nom of uji need not be unitary.

i,j

I

®

v uji has a limit s 1 ; however, u

the sequence

U= (uji)

will be

Let us now define the ideal J= (H,F.- 6 u .. G. J i Jl l We claim that the variety of

Indeed, let

z

u*

(uij)

I

the point

J

j

1 12, • • •)

consists of the single point

F(z) = UG(z)

is the adjoint of

no:rm at rrost 1, this implies variety of

I

. p .

be a point in this variety. Then

H(z) = 0 where

J

G.- 6 \.1."':' F. J i lJ l

is contained in

I

G(z) = u*F(z)

U • Since both

u

IIF(z) II = IIG(z) II , whence

I

and

u* are of

z E aD • Thus the

a D , so that by Theorem 3.1 it reduces to

p • By the Nullstellensatz we have then

205 d~ I!J/J =

Na.v by the Noether property of

d
p

00

there is an

N such that 1, .•. ,N) •

j

Let (H,FJ.- 1: u~ .G. i J1 1

j

1, ••• ,N) •

By one half of the Banach-Steinhaus theorem (the trivial half), there

is a neighborhood of

p

on which

F.- 1: u~.G.--+ F. J i ]1 1 J

--+

uniformly, as

v

continuity of

dime 0/I

oo

•

1: u .. G. i ]1 1

Apply.ing the theorem about the upper semias a function of

Proposition 5. 3) , we deduce that for

v

I

( !Jo], Chap. 11,

large

01. the other hand

J

since

v

(Uv)*

c (H,Fj

=

1: u~.G. i J1 1

G

(Uv)- 1 . Therefore

~

v

j - ~ uij

&;'1.

"1

dim ID/I(Uv,p)

(3.3). This contradiction proves Theorem 3.3.

1,2, ... )

j

s

I(Uv,p) ,

d , contradicting

206 4. More on the relative position of a real analytic hypersurface and a holarorphic curve

Theorem (3.3) implies in particular that if

v (f) = 1)

(afl, p)

is a holOIIOrphic gem with

f' (0) t- 0

Taylor polynomial of

(about 0) will contain at least one non-

r of

(i.e.,

f : (0::,0) ---7

then the

k'th

vanishing nonconstant tem. Sane general properties of real polynomials and analytic functions pemit us to estimate this nonconstant tem from

below. Lamna 4.1:

let

W: G x

for all

Let

JFil

K be a CC~Tpact subset of G , and

Gc

Iff

be open, let

~

JR

be a positive real analytic function such that

x E G the function

y

~

W(x,y)

at most. Then there are positive numbers

a , b

d

such that

2 -b

W(x,y) > a(1 + IYI )

(4. 1)

Proof:

is a polynomial of degree

Let

y 1 , .•• ,yM denote the cxx:mlinates of

to prove (4.1) for

y's

1 ~ IY 1 1 =max IYil • In the sequel

such that

i

we shall ass\.Ulle that this is the case. Introduce the new variables

y • It will be enough

TJ 1

1/y1 ,

TJi = Y/Y 1

(i=2, ••• M) .

Then

y~(x,y) =

V(X,T])

will be a polynanial in TJ , analytic in

x, TJ,

TJ 1 t- 0 • Hence the

f.ojasiewicz inequality (see

with same positive

a, b,

for

x E K,

and positive when

[!.o]

yields

ITJ 1 1, •.• ,1TJMI ~ 1 •

This gives

(4.1) in the case considered.

we r of

are now ready to estimate a nonvanishing Taylor coefficient of from below. We shall use the notation of Theorem 3 • 3.

207

Lemma 4.2: a , b with

There are a neighborhood

such that whenever

p EL

am

/f' (O)

I

L of

ao

f : (1!:,0) ~ (~,p) = 1

and positive numbers

is a holcm:>rphic gem

then

lf{ll.) (0) 1)-b

L

a (

(4.2)

O<j.J. :::k

~:

Let

L

be any c::arpa.ct neighborhood of

Theorem 3. 3. For L

X

x = (x' x") E

s2n-1 Ct.: ....n x q ...n; ,

1f1

x

~

ao

contained in

of

0

in a neighborhood of

y= (y 2 ,y3' .•• ,yk) E ... "- (k-1) n

(YIJ. E ...n 2 ••• , k) q; ,!J.=,

define

W(x,y)

r(x' +x"~ +

W is then a real analytic function, polynanial in

kL y ~IJ./J..L!) 2

12

IJ.

y , and Theorem

3.3 implies that it vanishes nowhere. Hence Lernna 4.1 applies and we

obtain (4.2).

208 5. Proof of Theorem 0. 1

Lemna 5.1:

Given a positive integer q(t,t) =

following property. If

polynomial of degree at rrost

k , there is a

c > 0

with the

L

2q .. ti tj/i!j! O:::i+j :::k 1 J

k2

such that

jq(t)

1- min Itis 1

is a

(5.1)

then (5.2)

max ltl:::1

jq(t)

I>

c

Proof:

It will be enough to prove (5.2) for polynomials q satisfying q(O) = 0 and 6 lq .. 12 = 1 • The set of such polynanials is, however, 1]

a ccrnpact set, and (5.2) follows by Weierstrass' theorem.

Proof of Theorem 0.1:

Let

Choose

so that (4.2) of Lerrma 4.2 be satisfied with o 2

a< 1 , b , k

substituted for z E o1

(5. 1)

where

r

be a real analytic defining function of

sufficiently near

II cp'

(z)

II

< dist(z,ao 1 )

such that for

o-1

,

stands for operator norm.

> 0 • Then there is a unit vector

(5.2)

> 0

ao 1

Suppose that (5. 1) does not hold for sorre

o

o

o . We claim that there is a

o2.

Icp' (z)vj

~

v E ~

dist(z,ao 1 )

z

near

ao 1

and some

o

is small

such that

o-1

We are going to show that this cannot be the case if enough. For brevity,

'We

introduce positive numbers

left hand side of (5.2) is

;>, ,

fj,

1/}, and the right hand side is

(5.2), together with the cauchy estimate, yields

so that the 6°- 1 . Then

209 (5.3)

Fbr

I; E

a:, Is/


c

210

max

(5.5)

/1;

I:::

A1 + a/X

Q(l;) ~ c Ak(1 +a +be) x-k

min

Q(U

~

J!;J::: A1 +a/X

> c Ak(bo +o +a) = c Aa(k+1/2)

we

rJ.CM

wish to show that a similar

replaced by

r of • To see this, for

If(!;) - F(l;) I :::

Since

f

max II; I I ::: A1 + a/X

is bounded on the disc

J~;l

estimate holds with Q = R o F J!; J:O A1 +a/X 'lfie estimate

If

+ 1) (1;)(A1+a/X)k+1 I I (k+ 1)!

(k

rphic mappings between a:: 2 • Math. Ann. 245 (1979)

certain real analytic domains in 255-272.

215

@-Ffl

K. Diederich, J.E. Fornaess:

BiholO!!Orphic mappings between tY.U-

dimensional Hartogs domains with real-analytic boundaries. Recent Developrents in Several Corplex Variables, Princeton University Press, 1981.

[p-'fl

M. Derridj, D. Tartakoff: solutions to the

On the global real analyticity of

a-Neumann problem.

Contn. Partial Diff. Equ.

(1976) 435-601. [G]

G.M. Golusin: variable.

[! 2 is imbedded into the Jacobian variety, Theorem

228

(2.5) implies that there are only finitely many non-constant rational mappings from an algebraic variety into C (de Franchis' Theorem). H. Fujimoto [Fl] obtained a finiteness theorem for a family of linearly non-degenerate meromorphic mappings from the m-dimensional complex vector space ~m into the n-dimensional projective space Fn(~).

We will give a remark on the

relationship between Fujimoto's result and Theorem (2.5)

(see

Remark (2 .15)). In section 3, we will prove another extension of de Franchis' Theorem, which generalizes the result of [N-Sl , Main ~.

Theorem (1.2)] in the case of the complex number field

Let

V be a complete smooth al~ebraic variety, D a hypersurface of V with normal crossings and V

=

v-o.

Let T(V;log D) be the

vector bundle of logarithmic vector fields alon3 D and Tq(V; log D) denote the q-th exterior powerAqT(V; log D). say that the vector bundle Tq(V; lo~ D) --+ tive over V if there is a proper

V is

morphism~:

We

quasi-nega-

q-

T (V; log

D)~~

N

into the complex affine N-space ~N such that the restriction of ~ over Tq(V; log D)

lv

phism onto its image.

minus the zero section is an isomor-

Let W be another algebraic variety and

Fq(W,V) the set of proper rational mappings f: W

~

V with

rank f .:_ q. (3.1)

Theorem.

If Tq(V; log D) is quasi-negative over V,

then F (W,V) is finite. q

Acknowledgement.

The main part of this paper was written

during the author's visit to the University of Notre Dame, 1984/85.

He expresses his sincere gratitude to the Department

of Mathematics of the University of Notre Dame for the hospitality, and especially to Professor

w.

Stoll for numerous dis-

cussions on the subjects of this paper and related topics. §1 a)

Jet bundle.

LOGARITHMIC JET

f,g:

{~ 1 0)

-+

~

Let

standard coordinate z.

SP~CES

be the Gaussian plane with the

Let M be a complex manifold and

(M;x) germs of holomorphic mappings from nei9h-

borhoods of the origin 0 For a positive integer k

E

~

E

~

into* with f(O) = 9(0) = x E M. we write f ~ g if f and g have

229 the same Taylor expansions in z up to order k for some holomorphic local coordinate system around x. "~"

checked that the relation,

Then it is easily

is independent of the choice

of the holomorphic local coordinate system around x and defines an equivalence relation on the set {f:

(~;0)

~

(M;x)}.

Let jk (f) denote the equivalence class of f. and set Jk(M)x = {jk(f); f:

(~;0)

~

(M;x)},

(1.1)

Then Jk(M) naturally carries the structure of a holomorphic fibre bundle over M with the canonical projection ~=

Jk(M)

--+

M (cf.

[01] and [G-Gl]). The bundle (Jk(M),

is called the jet bundle of order k and jk(f)

~,

M)

Jk(M) is

E

It is noted that J 1 (M) is isomorphic to the holomorphic tangent bundle T(M) over M, and that Jk(H)

called a jet of order k.

has a structure of flag with the natural

~rejection

Jk(M) ~ Jk-l(M) (1.2) Jk (M) -----+ Jk-l (H) --- • • • - - Jl (M) :. T(M) ___, M. such that for a holomorphic sections E f(U,Jk_ 1 (M)) over an

open subset U

M, the restriction Jk(M) ls(U) of

c

Jk(M)--+ Jk_ 1 (M) over s(U) is isomorphic to J 1 (H) lu; T(M) lu. If M is a complex algebraic manifold, then (Jk(M), ~, M) is also a complex algebraic fibre bundle over M. open subset of

~

and G: W

-+

Let W be an

M a holomorphic mapping.

Then G naturally induces the lifting (1. 3)

such that Let over M.

~oJk(G)=

G (cf. [01 , p. 86]).

1

~M

denote the sheaf of germs of holomorphic 1-forms Take a holomorphic section wE f(U, ~ 1 ) over an open

subset U of M.

M

For jk(f)

f*w

Jk(M) lu, put

E

=

A(z)dz .

Then the derivatives djA/dzj (0), 0 ~ j < k - 1, are welldefined, independently of the representative Hence we have a mapping

f for jk(f).

230 Let w1 , ... ,wm with m = dim ~1 be holomorphic 1-forms on U such that w1 A ••• AWm does not vanish any-

which is holomorphic. where.

Then we have a biholomorphic mapping 1Tx (w 1 , .•• ,wm): Jk(Mllu~ux

(1.5)

(~klm

which we call the trivialization associated with {w 1 , ••• ,w m} .

=~ denote the sheaf of germs of meromorphic 1-forms and take~ E r(u, =~). Then as in (1.4), ~induces a meromorphic

Let

vector function

~: Jk (Ml lu

(1.6) b)

a:k •

-+

Jet spaces.

Let X be a complex space with structure

sheaf @X, which is, in this paper, always assumed to be irreducible and reduced unless otherwise mentioned.

We assume for

a while that X is biholomorphically imbedded into a complex manifold M.

Let J(X) denote the ideal sheaf of X.

that a jet jk (f)

E

we say

Jk(M)x with x e: X is tangent to X if Pof have

zero of order > kat 0 for all p set of all jets jk(f)

E

J(X)

E

X

.

Let Jk(X)

X

be the

Jk(M)x which are tangent to X and set Jk(X)

= xWx

Jk(X)x'

1T: Jk(X) - X , where 1T denotes the natural projection. It follows from the coherence of the ideal sheaf J(X)

([Cl]) that Jk(X) is a com-

plex subspace of Jk(M) and 1T: Jk(X) --+X is a holomorphic .fibre space.

Let X

complex manifold M'.

--+

M' be another imbedding of X into a

Then,in the same way as above, we have

another holomorphic fibre space 1T': Jk(X)'--+ X, which is however isomorphic to 1T: Jk(X) --+X as fibre space.

Hence

for general X, we define the holomorphic fibre space 1T: Jk(X) --..X by making use of local imbeddings of X into open subsets of complex vector spaces. the jet space of order k over X.

We call (Jk(X) ,1T,X)

The jet space J 1 (X) of order

1 is isomorphic to the Zariski tangent space

8(X) •

As in

(1.2), Jk(X) carries a sequence of fibrations. (1. 7)

Jk (X) --+ Jk-l (X)

~

•.. --+ J l (X)

5!

8 (X) -----+" X.

such that for a holomorphic sections s e: r(u,Jk-l(X)) on an

231

open subset U of X Jk (X)

Is (U)

;

H

(X)

Iu

.

Let G: W --+ X be a holomorphic mapping from an open sub~

set W of

Then as in (1.3), we have the lifting of G

into X.

(1. 8)

Jk(G): W - Jk(X)

such that

c) folds.

-1--

7ToJk(G) =F. Sheaf and space of logarithmic jet fields over mani-

Let D be a hypersurface of a complex manifold M and

fJM(log D) the sheaf of germs of logarithmic 1-forms alon9 D (cf. [Dl], [Il] and [I2]). For convenience, we recall the 1

1

For x e M-D, the stalk fJM(log D)

definition of fJM(log D).

1

1

is identical to the stalk fJM,x of fJM at x.

For x

X

E

D, take a

neighborhood U of x and irred11cible holomorphic functions a 1 , ... ,a 1 on U such that

u n D Then we define 1 (1. 9) fJM(log D) x

0}

1

L j=l

dcr. C>M

•

1

_ J + fJM

,x crj

,x

1

If D has only normal crossings, then fJM(log D) is locally free.

If M is bimeromorphic to a compact Kahler manifold, the 1

global sections of

fJM(log D) are d-closed (cf. [Dl]).

Let N

be another complex manifold, E a hyFersurface of N and

~: N--+ M a holomorphic mapping such that ~-l(D) ~

c

E.

Then

naturally induces a sheaf morphism

~*:n~(log D)

(1.10)

--+

n~(log E).

If D and E have only normal crossings, then for a mer6morphic

mapping~: N--* M with ~-l(D) lfi*: r (M, ~~(log D))

(1.11) Lets

E

r(u,

c

Ewe have

--+

r (N ,n~ (log E)).

Jk(M)) be a holomorphic section on an open

subset U of M. (1.12)

Definition.

along D if ~

1

wos J"u :

ruE r(U', fJM(log D)),

We say that s is a logarithmic jet field U --+ ~k are holomorphic for all

where U' is an arbitrary open subset of u (see(l.6) for the definition of w).

232 The sets of logarithmic jet fields along D over open subsets of M form a complete presheaf which defines a sheaf

Jk

We call /k (M~ log D) the sheaf of germs

log D) over M.

(H;

of logarithmic jet fields along D over M. Assume that D has only normal crossings. x

E

0

Take a point

D and a holomorphic local coordinate neighborhood

1

m

U(x , ... ,x) around x 0 so that xo = D n

u

(0, ... ,0), 1 R, = {x , ••• ,x = O}

(1 < R, :_ m).

Then any losarithmic 1-form w along D on an open subset U' of U is written as dx 1

w = al-l-+ X

where a. are holomorphic functions on U'. Jk(M) luJ: U x {dx 1 , ... ,dxm}.

(~k)m

be the trivialization associated with

Then a sections

s (x) = (x,Z (x)): U

---+

E

r(U, Jk(M)) is given by

(~k)m ,.,ith

U x

( zi - n(iiJifa} x Y)

R'

Since stant.

is compact and H is an affine space, y must be conThis contradicts (2.11).

Now we show the (2.12)

Lemma.

fi; Y--+ are

e: H.

x,

followin~

Q.E.D.

lemma used in the above proof:

Let Y be an alsebraic variety and

i = 1,2 ••. ,distinct holomorphic mappin5s which

non-de~enerate

with respect to A(X).

Then there is an

irreducible alsebraic curve c in Y such that f. I C: C - x ~

are all distinct and non-degenerate with respect to A(x). ?roof.

Put Z. . ~J

{y

=

Y; f. (y) ~ .

E

=

for i < j.

f. (y)} J

Then Z .. are proper algebraic subsets of Y. ~J

Z =

u

i<j

Z .. ~J

u

u

i

f~ 1 s (X) ~

Put

uS (Y).

Then the subset Z of Y does not contain a non-empty open set of Y.

Take a point y 0

hood U

c

R(Y)

E

Y-Z and a local coordinate neighbor-

such that u is biholomorphic to the polydisc

lim '"' { (;l' ... 'ym) ; Y0

.,.

(O, .. ,O)

Iyj I

=o,

< l} c

a:m,

242 where m =dim Y.

By [N4, Remark (2.3)] we have an analytically

non-degenerate holomorphic mapping ¢: ~ --+ ~m with ¢(0) = 0. Here ¢ is said to be analytically non-degenerate if the imase ¢(~)

is not contained in any proper analytic subset of ~m.

Let {cr 1 , ... ,cr} be . the maximal linearly independent system of . q {'* ( Jn · · ~ N} . h f i are 1R(X) w A ••• A wJn) ; 1 ~ Jl < ••• < Jn Ten non-degenerate with respect to A(X) if and only if -1 -1 -1 fi cr 1 , ... ,fi crq E r(u, £i KR(X)) are linearly independent. Therefore f. are non-degenerate with respect to A(X) if and 1 -1 -1 -1 only if (fio¢) crl, ... ,(fio¢) aq_ E ret:., (fio¢) KR(X)) are linearly independent. for every i. Tij on U.

f~ 1 KR(X)

Fix a trivialization

-1

Then f.

=~

x

~

a. are given by holomorphic functions

1 J Put Tfj = Tij(¢(z)).

Then it follows that

Ti 1 , ... ,Tiq are linearly independent if and only if the Wronskians

d. (z) J

Tfl (z)

T~

Tfi (z)

T~

"t' iiq-1) (z)

T

1q

( z)

1 (

1q_

Z)

;k(q-1) (z) 1q

* 0;

Note that ¢- 1 ( )

that is, the sets {d. (z) = O} are discrete. J

is the countable union of discrete subsets. We take a point z 0 E ll-(uj {dj(z) = O}u ¢- 1 (Z)) and put y~ = ¢(z 0 ). Set ¢ (z) = (¢

1

m (z), ... ,¢ (z)), 00

L

k=O

y' = 0

k

\)

ak (z - z 0

)

,

243 Now we may assume th a t y l

1 ••• 1

. t'1.ons o.f y Ill are t he r e s t r1c

affine coordinate functions over an affine open subset w of R (Y) with U c W c a:£. Let y 1 1 • • • 1 Yf/., be the affine coordinate functions of

a:f/.,.

Consider the affine equations: a

(2.13) y

Ill

a

1

1

. + al

0

Ill

0

1 + a q-1

z +

...

Ill + al z +

Z

Ill

+ a q-1

Z

q-1 I

q-1 I

where z is considered as the affine coordinate function over

a:.

Let V be the intersection of W x

a:£

set of

x a: defined by (2.13)

1

a:

and the algebraic sub-

and V' the irreducible com-

ponent of V containing the set of points defined by (2.13) with z E ~and (y 1 1 .• 1yrn) E ~rn ~ U. Let p 1 : V' ~ (y 1 Z) ~ y E Wand p 2 : V' 3 (y 1 z) ~ z E a: be the projections. Then p 1 and p 2 are locally biholornorphic around (y~ 1 0). Let C be the Zariski closure of p 1 (V') in Y. Then Cis smooth at y'0 1 . around which z gives the local coordinate. of

T1.. 1

di(z 0 T. l.q

1c

lc

n U1

... 1 T.l.q

lc

n

u at

by the construction.

)

n

u are

y'0

E

C with respect. to z are

Since di(z 0

linearly independent.

The Wronskians

)

'I

01 ti 1 1Cnul•••l

Therefore f. l

lc:

C--+ X

are distinct and non-degenerate with respect to A(X). (2.14)

Example.

Q.E.D.

Let C be a compact smooth curve with genus

3 and a: C-+ A the Albanese mapping.

Put

X = a.(C) + a(C) c A. Then X is an algebraic surface of general type.

For y

E C1

we have rnorphisrns: a

y

C 3 x r-+ a(x) + a(y)

EX.

Moving y E C 1 we have infinitely many rnorphisrns ay which are degenerate with respect to A(X). (2.15) position

C- X

Remark.

ox

Let H11 ••. 1Hn+ 2 be hyperplanes in general the n-dirnensional complex projective space Fn(a:)

z1 1 ... 1 Zn+ 2 divisors on a:rn which may be transcendental. Then FujiMoto [Fl] proved that there are only finitel~ many

and

linearly non-desenerate rnerornorphic rnappin~s f:
is not a-exact in general.

However, if we have chosen

then the (1,0)-component of v-v' is globally defined on M 0 and

the ~obtained by using

vdiffers from

The infinitesimal deformation



by the a of the global vector field v-v'.

can also be described by the Cech cohomology class

266

is the limit as t in

0.

I

+

0 of ..!t times the discrepancy of going from M0 to M along w~ = constant t I

and going along w~ = constant in I

0.. I

In this paper we will not use the description of

the infinitesimal deformation by Cech cohomology.

Now assume that M 0 carries a Hermitian metric.

(1.3)

+ there is a unique harmonic representative 1(1. The difference +-1(1 is given

represented by by the

a of a global (1,0) vector field u on M

vector field

von M

0

0

a

a

1

dz~I

u.

(J

von M

O•

Write u =E u~ I

the canonical lifting of

it•

a ·

We now compute the Lie derivative of the volume form of the Kahler-Einstein

metric. Let 1 a b a B w=Adw =g-dw Adw 2 wbdw a aB with wab skew-symmetric in a and b. Then wai= gai waB = -gBa

w =0 aB

was= o. Using the suymmetry of +a B in a and B and using (Ly8) aB = 0, we conclude that Lvw vanishes identically on M0 • The volume form is, up to a constant, the exterior product of n copies of w. Hence the Lie derivative with respect to v of the volume form of the Kahler-Einstein metric is zero.

§3. Kahler Condition.

(3.1) Though the Kahler property of the Weii-Petersson metric was proved already by Kosio [6), to compute the curvature of the Weii-Petersson metric we have to

first differentiate the Weii-Petersson metric once as an intermediate step. differentiation immediately yields the Kahler property.

This

We are going to do this

272 intermediate step of first-order differentiation of the Weii-Petersson metric here in this section. Now assume that ll is an open neighborhood of 0 in coordinates t = (t 1 , ... ,tN).

~ with holomorphic

We now modify the situation in §1 as follows.

replace the unit open interval I by ll so that

11:

We

M + ll is a holomorphic family of

compact Kahler-Einstein manifolds with negative Ricci curvature.

We keep, with

obvious modifications, the notations of §1. Let t' k and t' k be respectively the real part and imaginary part of tk. liftings of

-fir,k ata' k•

Let

Let v'k and v' k denote respectively the canonical

'(F),'( a:• )denote respectively the tangent-bundlek k

valued harmonic (0,1 )-forms representing the infinitesimal deformation in the direction of

a a at'• at"'' k

Define

k

Note that in general the Lie bracket [vk',vk'] is not zero so that corresponding to vk', vk' there is no smooth trivialization of the family M over a two-realdimensional plane inn with t'k, t' k as variables. Let

Since

it follows that

273

(L

a

a

a

J) =(L J) =(L Jl-=0. vk a vk a vk a

The tensor L- J is the complex conjugate of L J. vk vk complex conjugate of

The tensor

a

~ (at.)

is the

k

~(a: ). k

(3.2)

Let dV denote the volume form of the Kahler-Einstein metric gai"

We

N

define the Weii-Petersson metric }_ hil dti & d~ on n by i,j=1

In order for the metric to be positive definite we assume from now on the element of H 1 (Mt,T M/ defined by any nonzero vector of ll of type (1,0) at t is nonzero. Because of (2.3) we have

licity we let T- = (L J)- and v = v.. a vi a 1

trivialization wa

=wa(z,t)

for the family

Choose a smooth

a

M + g such that ar.

11:

= -vi

at M 0 and

J

a

We use the dot • in the superscript position to denote at: of a j

scalar function or a component of a tensor.

As before, a a' aa mean respectively

a

-a· --==a· az az Since on each Mt the g

a

ai

Y T_) Y II

--(~

awa

Applying

T~:0 -valued

(0,1)-form

T~

is harmonic, we have

=0 on Mt for every t, that IS.

a~ and setting t = 0, we get on Mo i

Using ). ti(a a)• 0 = (L ~ )ai= (8 ail. -8 a"0(-a 0 w 11 - g tw

280

y

Ye

y--.

t

ye

(LT)_=(T_) +T_(a wal -T_(a w ), v a a a a a t we get

+g

aa a

a

((-~

t

ut

--a • Y

•

(a w l - g- (a w l )T-l y

ay

a

0

Multiplying out and using normal coordinates at the point under consideration and

a/· •

the fact that (--) t=O awa by

applying

= -(a aw ). )•t=O a

respectively

al •

and ( 0 ) t=O aw

=-cac;w ). )•t=O (obtamed 0

and

to

at.

I

o=

). - az). a a wa + - - a--w ) we obtam 0

az

awa

g

aa

- g

ai

ai

,

y

a T_ + g

uY a

aca

t

a

uY "

a

w l

ut a

-- • Y

g- a .

-

(L-L ll- (L- L j)!:8 gaS g,-dV v.t vk "\1 v. v. a o J I

>.

( V (L -

a

);a

JMo (S a L)) >.

L

v J. vk

-

11 8

J)- g

t~B g-

a s

>.11

) dV

a a

(21-1t(at) t(a-tl-) dV J. a k 8

284 (4.7)

Our final conclusion in the first part of the computation of the curvature

tensor of the Weii-Petersson metric is

R(WP)-.-:-= Jf R.k 1 J

-4

JM

(l M0

l J)p(l l J)q gabg dV v. v. a v v. b pq .. I k J

a6aYaaaa tyt 87 a,r_+ g It:" a T_ uy a v a a uy 1\ a ay a 6 a

= g

ai -a--A Y aS --a-a Y +<ar.>- ~ ai"L + g It:" ->L j a y a ay j a a

a b which vanishes because the only nonzero components of + (at:) a are of the form I

a i

-

+ ( ar-.) a. Here when we use v = v. instead of v. the eleventh term g I

Lj

I

ai - - . a (a wa) a

a

on the left-hand side of the equation in (4.5) vanishes not because of the

-..

a a a -closedness of • (at>- but because j a

a a a and+<ar.>cJ=O. Thus,a"(L T)=O. Thatis,(L L J)_isa"-closed. I vi vi vi a §5. Second Part of the Curvature Computation.

(5.1)

We continue with the notations of §4 and set out to transform the first

287 integral in the expression for R(WP)t.kiTin (4.7). We will do it first for the special case i

= j = k = I.

and then get the general case by polarization. As a first step in

the transformation we compute

a of Lv. Lv.J. y I

Since T

position denote

is a-closed,

I

a-T_ is a II

syiTVTletric

in

a

and

11.

That is,

a~ of a scalar function or a component of a tensor. Then i

is symmetric in a and 11. Using y (L T)_ vi II we conclude that at t

y

y-1: y wa )• - TJ (w )•, a II II -.:

= (T_)• + T_(a II

=0

Ae Y --;~,• Y Y y--. -.: Y• -(a-w) a,T--(a w) a,T_+a-((L T>--T- +TJ (w)) a " II a " II a vi II a II II -.: is symmetric in a and 11. Hence

1-=1" ). y -- • y y y -- • - 2-TJ,T_-(a wA)a,T_+a-:-(L T)_-(a-T-Ha wa> a " II a " II a v. II a a II y --- • 1: • 1: 1-=1" y -L.(a a wa> +a-L..(a wy) +TJ (---T....:.) a all all 1: 111: 2 a

is symmetric in a and 11.

Since the sum of the second and the fourth term is

symmetric in a and II and also both the fifth term and the sixth term are symmetric in a and II, by taking the skew-symmetrization we obtain 1: y 1: y y y 3-(L T)-- ao(L T)- = r-f(TJ L- TJ L). a vi II " vi a II 1: a a 1: II (Note that one can also obtain this result by differentiating with respect to aat. the I

288

integrability condition at(t} = i[t(t},t(t}] and using t• = - 1-T, where t(t} is the

2r-'i

T~; 0 -valued (0,1}-form on M0 such that a local function f is holomorphic on Mt if as a function on M0 through the smooth trivialization f satisfies

(a- t(t}"a,_}f = 0

t = 0 due to the on M0 .} We now use normal coordinates. Since a T_ a- -closedness t

a

of T, it follows that y y a-(L T}-- a-:-(L T}- = a vi a a vi a

M

ty ty a (LT-- LL}. t a a a a

Let a----= T--T--- T--T-aa,ty at ay at ay and let aaa__ be obtained from a.,--- by raising the first two indices. Then the ty pa;y last equation can be rewritten as

when (i}

01

>. means applying a to the first index of (L T}-;;- (which is g, :-:{l T}-}, i.e. v.

regarding ( l

vi

T}2:_ as a TM 1 ,O -valued (0,1}-form, a

o

PY

~pplying a and

"Y

v.

a

1

lowering the index

y, (ii} 0 2 • means applying

a'

to aaa'Ty as a A2 TM 01 •0 -valued (0,2}-form and then

lowering the indices a and a, or in other words

a* is applied to the second set of

indices of 6----. aa,ty

(5.2}

Let X denote the space of all tensors :!-;;--if satisfying the following three aP,Y

symmetry relations: (i} :!----=-:!----(skew-symmetry in the first two indices} aa,yll aa,yll (ii} :!0 -:.= :!-F0 (symmetry in the two sets of double indices} a.,,yu yu,ap (iii}

:!---- + :!---- + :!---- = 0 (vanishing of the sum from the cyclic aa,yll ay,lla all,ay

permutation of the last three indices}.

289 Simple direct verification shows that 9- 0 -"i" belongs to X. a.,,yu For s = 1,2, the operator D applied to a covariant tensor with two sets of s . skew-symmetric indices of antiholomorphic type (e.g, an element of X) means the operator

a applied

to the sth set of skew-symmetric indices,

Let

o• s denote the

adjoint operator of Ds which is the same as applying a" to the sth pair of indices. Let 0 = s

o·sDs + Dso·s and let Hs denote the projection operator onto the kernel

of 0 • Let G denote the Green's operator which is zero on Ker 0 aad equals the

s

s

s

inverse of (the identity operator minus H ) on the orthogonal complement of Ker 0 •

s

s

For !! in X we have the following properties

(d)

(e) [] 1 !! belongs to X

(gl if

o,:: = o, then

ID 2 •,0 1 1:: = e

o• .::, where !'D. •,0

1

1 means the commutator

Properties (a) and (c) follow from simple straightforward computations. To prove property (b), by definition we have

A

where

a

\1

means that the index

a

\1

is omitted and

290

(j)

2 = ~ (-1) v+1 gcn R - "--- ~ aa v -ta,a ••••av···az v=O 2

@

= ~ (-1 )v+1 R- t_P !!--- ~ _ L a a tP,a ••• .a ••• az v=O v v

2 G)= }_ (-1 )v+1

l:

2

v=O

11=0

R- ' - P " - - -

f3

V

a

-

D

-ta,a •••• (p) ••• a ••• az II V

II

II¢V

the subscript

(p)

II

The term

meaning that the subscript i

II

is replaced by

p.

(j)

vanishes because R - = e g and because of the aav aav symmetry property (iii) of E. The term '2' vanishes because R- t_P is \!:1 a a -

-

symmetric in t and p whereas !!--.,

/" 0-

a

'tP,Po•••P\)••• 2

is skew-symmetric in

v

Y" and p.

The term G) is the sum of six terms which can be grouped in three pairs so that the two terms in each pair cancel out because of the symmetry of R-8- 6 in a and a y y and of the skew-symmetry of E-;;--i" in y and 6. Property (d) is obtained from a.,,yu property (b) because of the symmetry property (ii) of !!. To prove properties (e), (f) and (g) we compute explicitly 0 1!! and obtain (01!!)----=- ga'v V-!!----- gat[v-v ]!!---aa,AII a t aa,A11 a' a ta,AII

=- ga'v V-!!----- 2e E---o ' afl,AII afl,AII

+ R-t_P !!---- + R-t_p !!---a II ta,pA a A ta,PII where R "Q = e g "Q is used. The skew-symmetry of (0 1!! }-;;-,-in a and a and the a., a., a .. ,"ll symmetry of (0 1 !! }- 0 , - in (a,a) and (A,II) are clear from this expression. To get a.,,,.ll

291 the symmetry property (iii) for 0 1 E, we cyclically permute I!,A,Il and take the sum from the last four terms of the above expression and get - R-'t_P "----- R-'t_P "----+ R-'t_P "---+ R-'t_P "---a A -'tll,Pil I! II -'ta,pA a II -'t8,PA I! A -'ta,Pil

where the two terms in each of the following six pairs cancel out: the first and the eleventh terms, the second and the twelfth terms, the third and the fifth terms, the fourth and the sixth terms, the seventh and the ninth terms, the eighth and the tenth terms. Property (f) follows from property (e) because applying 0 2 to E is equivalent to switching the two pairs of indices, applying 0 1 , and then switcllng the two pairs of indices again. perty (d), we have

D2.0 1 E = oto 1•D 1 E and 0 1D 2•E = D 1D1•o2•E.

Direct

computations from definitions yield - ·D -(D D . E)---= g pA g a't ( V V-V 2 1 1

p a a

all,ll

E----- V V-V E---;:j 't8,A11 p 8 a 'ta,Ail

-- . E)---=g pA g a't (-V-V V E----+ V-V V E---j. (D D ·D 2 1 1 a 8, 11 a p a 't 8, All 8 P a 't a, All

Hence - • 0 ]::)---= ([D g pA [V-V ](g a't V ::---::i 2 ' 1 a8,1.1 I!' p a 'ta,A11

- (the expression obtained by switching a and I!) A p = [R-I! a

(g

a'[

V

-

a

::----) 'tP,Ail

p

A p a't A a-:t + R-, (g V E---::i + R-- (g V E----)] I! " a 'ta,Pil 8 II a 'ta,Ap

- [the expression obtained by switching a and 8]

292

- (the expression obtained by switching a and a), is

symmetric

in

the symmetry of R-A_p

a

II

a

and

a

and

in A,p and the

skew-symmetry of ::---,in A,p, From symmetry property (iii) of:: we obtain ta,p,.

-.

([0 2

Let

X0

'D/01- 0 1Dz•

be

_

,Od ::)as,-;=- e g the set of

= e oz• on

.

x,

0"[

all

-.

V o"as,'Til= e(Dz Elas,-;·

::

in

X

such that D 1 :: = O.

by property (gl, it follows that

.

Smce 0 1 = 0 2 on X, 1t follows that (0 1 +e)

'D/01

=

Since

. 11 >. 11

a

is that for

;,

A= - 1 - 1 E A- - (dz p.q. ap .. ap,a, ••• aq one has

a

" ••• "dz P) ~ (dz

a, " •••lldz il Q),

294 and in the formulae we have been using for correspond to A-

-

-

-

al•••a.p 8 1••• 8 q

with p

0 1 ,0 1 *,'5 2 ,02 •,

the components

eA a 8 , II

= q = 2. Thus, when we choose local coor-

dinates so that

-.. a a T--=2.1-11!1(-}:-=U-16 1!1(-}::aS at. aS aS at. aa I

I

at the point under consideration, we have

(6,6} M 0 = 8

2 - 8 j l. 1•-=-1-· jMo at. aa at. aa Mo a I

(WP) Thus R .-,- . ..,-at t II II

I

= 0 becomes

+ ((G 1

-

(0 1 +e)

-1- •

-.

)Da 6,0 2 6)Mo

+ (HL L J,HL L J)M - (H 1 6,H 1 6)M 0 vi vi vi vi o

Here we drop the subscript 1 for H in HL L J because we regard l l J as a v. v. v. v. 10 I I I I TMo ' -valued (0,1 )-form on M 0 and there is no need to distinguish between two different kinds of covariant indices.

We will drop subscripts in similar circum-

stances later without further explicit mention, We now further simplify the second and the third terms on the right-hand side of the above expression for ((G1- (Ul +e)

-1 -

R(WP>.-,-.~ II II

•

- • )D2 6,0 2 6)M

o

=

((G 1

-

·-

(U1 +e)

-1 -

)D 1 l

vi

T,D 1 l

vi

T)M

o

(with L l J regarded as TM 1 •0 -valued (0,1)-form) vi vi o

295

(because L L

vi vi

(because HL L

v. v. I

J has

Let

v. v. I

II II

£•

J as

with HL L

v. v. I

J)

I

R(WP~..,... . ..,- to I Ill

get the expression for

We Introduce the following notations to make the expression simpler.

. ..,-.-:-at t

(5.4)

J is a" -closed

I

)> denote the function on M 0 which is the pointwise inner product

296 10 a a of the TM 0 ' -valued (0,1)-fonns +.11 1 a >. a 11 .

a 11

a

>.

a >.

a

11

+< at/a" at/a

a 11

+ •< a\ >a• i- •< atk >i• a1•

Then at t = 0

+4e ((01 +

+e

e)-1(a~k),4>(a~t·)Mo

(OJ 1 + e)-\

L

J,L L

J)M

vi vk vi v R. o a a a a - (H(+(ati)A ''a\)),H(+(at/,.. +»Mo'

Here we have used the fact that L[ ]J is a a-exact and L l J is 3"-closed _ vj'v R. vi vk and the fact that OJ 1 + e) 1 is self-adjoint. Let '1': H 1 (M 0,TM 0 ) x H1 (M 0 ,TM 0 )

+

H2 (M 0,A 2 TM 0 ) be defined by taking the

skew-symmmetric part of the tensor product.

T~:0 -valued

That is, if t, n are a-closed

(0,1)-fonns on M0 defining the classes [tl.(n] in H1 (M 0 ,TM 0 ), then

'I'([ t],[ n)) is defined by the a-closed A2 TM1: 0 -valued (0,2)-fonn t

1\

n with compo-

nents

(t

A

A\1

n)_aB

=-12

>. II

>. II

>. II

>. II

a B

B a

a B

B a

( f;_n_- f;_n_ + n_f;_- n_E;__),

(Straightforward verification shows that E;

1\

n is always a-closed.)

We can now state our result on the negativity of the bisectional curvature of the Weil-Petersson metric,

(5.5) Theorem. let

11:

M + 1'1 be a holomorphic family of compact complex mani-

folds parameterized by an open neighborhood 1'1 ~

0!!!

ct,

Assume ·that each fiber

297 M

t

= 11- 1 (t)

carries a Kahler-Einstein metric whose Ricci curvature is equal to the

metric times a fixed negative constant.

Assume that the element of H 1 (M ,TM ) t

defined by any nonzero tangent vector of 0 ~ (0, 1) !! t is nonzero.

t

Let

X 1,X 2 be two tangent vectors of type (0,1) ~ 0!! 0 and 111 1 ,111 2 be the elements of

H 2 (M 0 ,A 2 TM ) defined by taking the skew-symmetrization of the product of 111 1 and D

--

1112 is zero, then the holomorphic bisectional curvature of the Weii-Petersson metric in the directions of X 1 ,X 2 is negative. To prove this theorem, without loss of generality X1

= a~, i

X2

= aat,

where t 1 , . . .,tN are the coordinates of

we can assume that

~.

Then the vanishing

j

of '1'(111 1 ,111 2 ) implies that

H(cJI(a~.>" cJI(a~.>> = 0

and

J

I

( WP) . -1 a a a a R .-,-.-,-= 8e ((] 1 +e) (cJI(at),cJI(at)>,)M I I J J i i j j D + e (OJ!+ e)-1Lv.Lv.J'Lv.Lv.J)Mo•

J

I

Since the operator {]

1

the right-hand side is positive.

J

First we show the following:

valued smooth function f on M 0 , Suppose the contrary.

I

+ e) - 1 is positive, it suffices to prove that the first term on

Let g =

0

{] 1

1

+ e) - 1 f is also a nonnegative-valued function.

+ e) - 1 f. Then the infimum -A of g is a negative

number and it is achieved at some point P of M 0 (note that that g is real-valued). From OJg)(P• -eA

= eg(P)

for any nonnegative-

S, 1 we ~(4]

1

0

is a real operator so

conclude that

+ e)g)(P)

= f(P),

contradicting the fact that f(P) ~ 0. We now apply it to the function f

= . 1

Since a Kahler-Ein-

I

stein metric is real-analytic, it follows from the harmonicity of cJI(

a~.)

that f is

I

real-analytic and QJ 1 + e)

_,f

is also real-analytic.

From the nonnegativity of the

298 two non-identically-zero real-analytic ftnctions

4J 1

+ e)

-1

f and (t(

a a at.>,+< at.>>, J

we conclude that((]

1

+ e)- 1 f,

) tj tj M

J

is positive. 0

REFERENCES

1.

L. Ahlfors, Some remarks on Teichmiiller's space of Riemann surfaces, Ann. of Math. 74 (1961 ), 171-191.

2.

L. Ahlfors, Curvature properties of Teichmiiller space, ). Analyse Math. 9

( 1961 ), 161-176. 3.

N. Hitchin, Compact four-dimensional Einstein manifolds, ). Diff. Geom. 9

(1974), 435-441. 4.

K·. Kodaira and). Morrow, Complex Manifolds, New York: Holt, Reinhardt and Winston, 1971.

5.

K. Kodaira, L. Nirenberg, and D.C. Spencer, On the existence of deformation of complex analytic structures, Ann. of Math. 68 (1958), 450-459.

6.

N. Koiso, Einstein metrics and comp!ex structures, Invent. Math. 73 (1983),

71-106. 7.

H. Maass, lber eine neue Art von nichtanalytischen automorphen Funktionen, Math. Ann. 121 (1949), 141-183.

8.

H. Royden, Intrinsic metrics on Teichmiiller spaces, Proceedings International Congress Math. 2 (1974 ), 217-221.

9.

10.

H. Royden, Oral communication, detailed paper in preparation. G. Schumacher, On the geometry of moduli spaces. Preprint 1984.

11.

Y.-T. Siu, Complex-analyticity of harmonic maps, vanishing and Lefschetz theorems, ). Diff. Geom. 17 (1982), 55-138.

12.

S. Wolpert, Chern forms and the Riemann tensor for the moduli space of curves. Preprint 1984.

13.

s.-T. Yau, On the Ricci curvature of a compact Kahler manifold and the complex Monge-Ampere equation, 1. Comm. Pure Applied Math. 31 (1978),

339-411.

Extension Problems and Positive Cunents in Complex Analysis H. SKODA Universit~ de Paris VI 4 Place Jussieu 75230 PARIS CEDEX 05

SUMMARY

This paper is a survey of recent developments in the theory of the extension of analytic sets and closed, positive currents.

INTRODUCTION

Since the fundamental works of P. Lelong ([ 17] and

and [19])

Y.T. Siu [ 22], it is well known that positive closed

currents are a vecy

good generalization of the notion of

analytic set because the analytic sets are exactly the sets of density of positive closed currents and because many properties of the analytic set

X are in fact properties of the associated current

of integration [ X] and also because the Lelong-Poincare equation of currents function F

~ aa Log If I = [ X]

am

giVE5 a simple fundamental relation

the analytic hypersurface X

= f-1 (0)

bet~

in the case of codilnen-

sioo one [ 18] . It is natural to ask if the basic theorems al::x:Jut the analytic sets are really theorems of analytic geanetry involving the hard structure of analytic set or if these theorems are in fact

theorems of carplex analysis only involving the soft structure of positive closed current.

the

300 Since the Ramert-Stein extension theorem for currents of Y. T. Si u

f 22]

and R. Harvey [ 12] it became clear that the extension of

closed, positive current was a natural generalization of the extension of analytic set. In this paper, we shall make a sw:vey of recent theorems of extension of analytic objects, essentialy of closed, positive currents.Our main reference will be the very interesting paper of N. Sibony [ 21].

1. EJcr'ENSICN OF ANALYTIC SEI'S AND OF CLOSED POSITIVE CURRENTS.

Perhaps the first IOOtivation of these problems of extension of closed, positive currents across an exceptional set was to obtain a better understanding of the following classical result of E. Bishop (1964,[4]) :

Theorem 1 : Let A be a subvariety of the c:::arplex hermitian manifold

n and X be a subvariety of n \ A, of pure dimension p,

such that for all cx:npact set

K of

n :

(whare vol 2 p means the euclidean volune of real d:imension

X

is a subvariety of

2p)

then

n (of ~ dimension p) .

Of course the main interest of the theorem is to give a necessary and

sufficient condition of extension of dim

X~

X across A in the case

dim A which is not covered by the Remnert Stein theorem. The

special case where dim X W. Stoll [ 26]

= dim

A, is due tow. Stoll [ 25]. In 1964,

gave the following beautiful characterization of

301

algebraic sets which is closely connected with the Bishop's theorem.

Theoran 2 : A subvariety

of!,

X in

of pure d:imension p , is alge-

braic if and only if there exists a constant

c

>

0

such that for

r »0 :

all

vo1 2 (X n B(O,r)) ~ er 2 P p

(where B(O,r)

is the euclidean ball of ce1ter

The proof of

0, radius

r).

W. Stoll was independent of Bishop's theorem. He

used the value distribution theo:ry of holanorphic maps. cne can now

of!

give the following short proof : ('there

lPn_ 1

is isanorphic to

is the hyperplane to the infinite), the assumption

about the growth of the volune of

X in

finite volume for the Fubini metric on theoran (with X in that

lPn

X

Q

= lP n

, A = lP

n-1

is a subvariety of and

lPn\ lP n- 1

)

of!

means that

X has

lPn , therefore the Bishop's

claims that the closure

X

of

lPn. Then Cb::>w' s theorem proves

X are algebraic.

The Bishop's theorem has a nice generalization to the case of closed,

positive currents.

Theoran 3 : Let

A be a closed canplete pluripolar subset of the

oorrplex hennitian manifold on

Q \

A and if T

trivial extension

en

fl.

:f

fl. If

T

is a closed, positive current

has locally finite mass near A, then the of

T

to

Q

is a ~, positive current

302 (A

is cx:mplete pluripolar in

where

u E PSH(n), i.e.

u

n

neans that

A

is plurisubharm:>nic in

and J. Polking proved the theorem in 1974 [ 15] bidillensioo

(p,p)

n

and T

E

n).

when

T

n, u (z)

when

= - ""}

R. Harvey is of

and A is an analytic subset of d:imansion

I proved the theorem in 1981 [ 24] of

= {z

p.

A is an analytic subset

is an arbitrary positive, closed current. My proof

was qUite different fran that of

R. Harvey and J. Polking. I

directly worked with the current

T

potential associated to

T

instead of the plurisubhanronic

by R. Harvey and J. Polking. In 1982,

H. El Mir [ 8] [ 9] generalized my method to the case \\here

A is only

closed, pluripolar. Nevertheless, these two proofs used technical estilnates of the distributioo of the mass of

T

in rome neighborhood

of A. Quite recently, N. Sibony [ 21] found a very nice fonnalization of these proofs. I shall now explain the Sibony' s proof. For the sake of sinplicity,

suppose, T

\\1e

(it is the decisive case) and T

is of biclimension (1 , 1)

has finite mass in

if necessazy, the problan is local relatively to

n

n \ A (\\le shrink A) • The two

following lannas give the basic infonnation about the distribution of the mass of the current

Lemna 1 :

T

near

(Essentially a variation of a Chern-I.evine-Nirenberg estimate)

(Q1e only needs to suppose that subset

u

E

of

€."" (n)

A.

A is closed). let

n. There exists a oonstante

n PSH (n)

such that

u

is

sane neighborhood of A (depending on ! K

TA

i

K be a oanpact

C(K,n) such that for all

;;;. 0, is bounded and vanishes in u), we have :

aau.;;; C(K,Q).IJTII

(n \

A).IJull ..

303

11'111

where

is the mass-measure of the current T.

Proof :

I.et be lete, because

almost all a in

p-pluripolar subset of

o \ A of bi.dilrension (p+l ; p+l)

is a closed,

and i f the mass

is locally finite in a neighborhood of A, then the trivial

extension

1.C

of

T

a?.

exanples

1) 0-pluripolar carplete sets are the usual

Theorem 4 :

a!? ,

is the volun fonn on

-

Notice that

then

Z0

0 ) and such that the differential fonns II~ Bp J J the space of cx:nplex fonns of bidegree (p,p) where

generate

=

,

n-:- 1 (Z

manifold V n

Bp

+ ctP

v n n-:- 1 (zo)

II . {V) , A n

in

a?

is a closed positive current on

o.

309 Proof :

The problem is local. We suppose that

t~ 1

is open in

of

V

on

of

t~ 2

and

locally flat current en if

= v = •\

{y}

A2 • The currents

x

v. If

is a fonn of degree 1

st all



11- 1 (y) n A

proves that

Therefore we have for all

cii'

are

y.

is the trivial extension of the current

y>

(1,1) , closed and because 11- 1 (y) , the theorem 3

d:f

which is defined for alllost all

11- 1 (y) \ A. Because

en

and

(y) n A

"' y> A i *y .p]w, [ J 1, then the mass of

T

is

locally finite in a neighborhood of A. 2) If

T

~

is pluripositif and p

2 , then the mass of

T

is also

locally finite in a neighborhood of A.

Prcof :

We on-ly prove the

with

1)

p

=1

(the important ca.se) , It is a

local result, therefore we SUP.[X)se that the euclidean ball B(0,1) is ocntained in

n and that in B we have A= p- 1 (0)

positive function of class c 2

in

ll

where

p

is a

such that

ddc p ;> CB ,

in

B(0, 1) ,

cfl.

We choose any function

where

C

is a oonstant

h

h< 0

h E

too (B)

=o

in

>

and

B is the Kahler fonn en

such that

B(O,t> ,

in the set

h .;;; 0

0

{z ; ~ k+2, T has a unique closed positive exten-

d~ has a locally

is pluripositive and p ;> k+2, then

flat extensicn to

Q

dd~

Notice that

and cne has :

has bidimension

(p-1,p-1)

and that

p-1 ;> k+l ,

~

therefore

has finite mass and

difT

dd~

is a no:r:mal current using

assertion 1) • Easy counterexanples prove that the results are shiup.

The fundamental tool of the Sibcny' s proof is still a Chern-LevineNirenberg estimate. T A dp Ad~

f

8 2 \ M(z 0 ) 8 2 xM(z 0 ) 9

bcnmded by a constante

C(z 0 ). The details of the

proves that

C (z0 )

(with n

=

Applicaticn of theorem

2)

proves that the last malllber is

is ba.mded by a constante

c

proo~

of this theorEm

for all

z

o

E

n-2 8 .

321

Fubini 1 s t:.heorEm allow to estimate the volutre of

=

vol (r)

J

vol(r (z 0 )) dA. (z 0 )

An-2 Because

r

(!1 x N) \

is

< cJ

dA. (z 0 )

r




Ck if k

~

j

(by maximum principle).

C (M) =lim C. > 0 oo

j+oo

J -

Thus the (1.5)

exists and is called the capacity of M at infinity. Remark.

In many applications we often encounter the situation

where instead of a Kahler form, we have only a closed nonneaative form

w.

We can simply replace (1.1) by (1.3), which

still makes sense, then the concept of capacity can still be defined, provided that the Dirichlet problem is solvable. Proposition 1.1 .

A connected Kahler manifold M with zero

capacity at infinity does not admit any non-constant bounded holomorphic function. Proof.

Let w be the fundamental Kahler form and suppose there

332 is a non-constant bounded holomorphic function f, say lfl < 1. Then by (1.2) we have,

0 < A

/1

From (1.1),

m-1

w

(1.2) and Stoke's theorem, the last term above is

equal to log(l+ I f 12 )d c U.IIW m-1 < - J log(l+ I f !2 )d c U.IIW m-1 ~c.log 2 J - 0 of class c 2 on H, with the property,

exhaustion) T

(2.1) Let u

=

log T then (2.1) together with the assumption that T

is strictly plurisubharmonic imply that the hermitian form (uaS) is positive semi-definite and is of constant rank m-1. Thus it has exactly one zero eigenvalue everywhere. By a direct calculation it is easily seen that (2.1) is equivalent to the identity TaST T- = T a S The complex qradient vector field Z is given by a

Z = T

a/,a =

11az11 2 +

TbTacllazll 2 >,ab"= Jlazll 2 - - + IIAII 2 + 2 IIBII 2 From these we get, Tb"Ta(logllazJI 2 >

,a -b

( II az II 2 II A II

~I I 2 +2 II az II 2 II B II 2 ) Iii az 114 > 2lJBII 2 ;11azll 2 .

a

Observe that llaz II 2

-R -T T

8

Ta\!B

aB

a\! = - trace of B and

since (trace of B) 2 < m liB II 2 , we conclude that b a T T (log II azll 2> > ~ II az II 2 ,ab -m proving Corollary 2.1. A well-known theorem of Ahlfor's asserts that there is no pseudo-hermitian metric on

~

satisfying the conclusion of

Corollary 2.1, hence we have Theorem 2.2 .

If the universal cover of each leaf of the Mange-Ampere foliation is ~ then lla z II 2 0, that is the Monqe-

=

Ampere foliation is holomorphic. The above theorem is due to Burns [6], our computation is perhaps more systematic. §3

Uniformization Theorems

Let M be a complex manifold with a parabolic exhaustion T on M.

Namely T: M--+ [O,c) is proper and continuous (3.1)

and satisfies properties (i), on M* = M- {T=O}.

(ii) and (iii) of definition 1.1

The exhaustion is unbounded or bounded

340 or
0. By property (iii) of a stationary maP, the function a

Rei:(fa(~)-qa(~))h(~)¢a(f(~)) =Rei: a

a

(~~-q (~)~h(~)¢a(f(~))

f

a

extends holomorphically to all

~

By strict convexity the

E ~.

left-handed side above is > 0 (with eauality iff f

= a).

Thus we have , at the oriqin 8fa Re L(l-A)~ (o) fa(o) ~ o with eauality iff f

=

a. In particular, takina a to be a con8fa stant map, we aet Re L(l-) ~ (o) fa (o) ~ 0, and so in

general we have A·< 1. Thus a stationary map is extremal. It can be shown easily that a stationary map is a biholomorphic map of T onto its image. tion T

o

We conclude that the exhaus-

f is strictly subharmonic on

is harmonic on~ and soT

f(~) = 1~1

o

~

and u

o

f = loa T

2 by theorem 3.1

o

f

(actu-

ally for m=l, it is not difficult to see this). In other words T is obtained by first constructina the stationary maps (= extremal discs) f and then pullina back the absolute value function on

~

via f.

By examinina the situation of the strictly convex domain carefully, one realizes that the special relationship between T and the Kobayashi metric remains valid if the Monqe-Ampere exhaustion T satisfies,

345 T T

-1

{~}

(o) consists of exactly one point

(4. 3)

is Coo everywhere after blowing up the point {o}

and u = log

T

(4. 4)

satisfies (4.1) on a neighborhood of o.

Let (r!,T) be a complex manifold with a bounded Theorem 4.4 Monae-Arnpere exhaustion satisfyina (4.1), (4.3) and (4.4) then the leaves of the Honae-Amoere foliation of .M* extends across ~

and are extremal discs of the Kobayashi metric of M.

Remark.

The exhaustion

T

restricted to a leaf is then the ~

oull-back of the absolute value function on disc throuqh o.

In particular

T

via an extremal

is real analytic on each leaf

throuah o. We also remark that the conditions on

T

do not characterize strictly convex domains.

in theorem 4.1 However, if we

know in addition that the Monqe-Ampere foliation is holomorphic then we have, Theorem 4.5 .

Let (M,T) be as in theorem 4.4 and assume in

addition that the Monae-Amoere foliation is holomorphic then M is biholomorphic to a bounded strictly pseudo-convex circular domain in ~m. A domain D in ~m is circular if z complex numbers

A with !AI

~

1.

E

D then

AZ

E

D for all

For details of theorems 4.4

and 4 . 5 see [ 18 J , [ 19 J , [ 2 0 J , and [ 3 7 J • Returnina now to the proof of theorem 3.1 in the previous section where we had established the holomorphicity of the Honqe-Ampere foliation in the case of bounded exhaustion under the additional assumotion that under the C

00

T

is real analytic.

However,

assumption, what we had shown in §3, clearly

implv that the hypothesis of thereom 4.4 is satisfied, thus the leaves of the foliation extend across o as extremal discs and T is real analytic (hence so is !1-;J Z II 2 ) when restricted to each leaf. Since 1!8zl! 2 vanishes up to infinite order at~, it vanishes identically on each leaf.

Thus Z is holomorphic,

completing the proof of theorem 3.1. Notice that in theorem 4.5 (unlike theorem 3.1) the exhaustion T is not assumed to be of class C at the origin, 00

346

therefore holomorPhicity of the foliation has to be assumed. A bounded circular domain D in ~m can always be defined by D = {z E ~mjT(z) < 1} where T(z) = ea jzj 2 and the function q is constant alona each complex line through the oriain, i.e., n-1 ~P • Thus T is not smooth at the oriain

q is a function on

unless D is biholomorphic to the ball. §5

Intrinsic metrics in the bounded case

From the results of the Previous section, we know that the Honcre-Ampere exhaustion is intrinsically related to the Kobayashi metric, we shall further exploit this relationship in this section. Let T be a non-negative

~~onqe-Ampere

sarily an exhaustion) which is bounded. erality assumed that supT

=

1.

function (not necesWithout loss of aen-

In section 2, we have studied

the geometry of the Kahler metric h = Tasdz a dz B•

Consider now

also the hermitian metric g = (1-T)

-2

h

(5 .1)

The followina is a consequence of theorem 2.1, Theorem 5.1 .

tenor of cr and h.

Let K

a

ture of the metric a. (i)

Sq = Sh

(ii)

K

cr

(Z,Z)

and Sh respectively the Ricci

be the holomorPhic sectional curva-

Then on M* we have 2m (1-T)q = Sh

l!azl! 2

-

s0

Denote by

2mT/l-T < -

2mh l-T

1

in particular

o,

-1.

By a direct computation we have

~-Sy]..!V where ively.

~

=

(l-T)- 2 R - - + 2(1-T)- 4 T -[-T- + T 2 u -] By]..!V

Sy

]..!V

]..!V

- - and R - - are the curvature of a and h respect-

SyJ.lV

BYJ.lV

From this we qet

4

(1-T) -

- - 1.

Specialize to the case where M is a strictly convex domain with sMooth boundary in ~m or, more crenerally where M

347

satisfies the hypothesis of theorem 4.4, then the leaves of the foliation extend through the fixed point and are in fact extremal discs of the Kobayashi metric. The metric g = iaaT/(l-T) 2 pull back to the unit discs in ~ via the extremal maps is the Poincare metric

on~.

Thus it can be ~e

thouaht of as a generalization of the Poincare metric. have the

followin~

theorem concernina isometries of this

metric. Theorem 5.1 .

Let (M,T) and (M,T) be complex

fyina the hypothesis of theorem 4.4.

satis-

Let <j>: H-+ M be an

isometry of the correspondina metrics a and defined by (5.1).

manif~lds

q

respectively

If ¢ preserves the correspondina Monoe-

Ampere foliations then ¢ is biholomorphic or antibiholomorphic. The above theorem was proved in [16] for strictly convex domains with the exhaustion obtained in theorem 4.1.

The

?roof in the more general situation above is analoaous.

Basi-

cally the assumptions guaranteed that ¢ maps leaves onto leaves and is biholomorphic (or anti-biholomorphic) on each leaf.

Thus X¢

=

0 if X is tanoent to a leaf.

direction at the point

~

Since for each

there is a leaf through

direction, we have a¢ ·= o at ~·

~

in that

By an argument similar to the

aroument at the end of section 4 we see that 8¢ actually vanishes to infinite order at o; But lla¢11 2 is real analytic on each leaf because both¢ and the metrics are.

Hence IIY¢11 2

vanishes identically. For manifolds (M,T) and (M,T) in theorem 5.1, let Mr and Mr be respectively the Kobayashi balls of radius r from o and o. (cf.

The following is an easy conseauence of theorem 5.1. [16]).

Corollary 5.1

With the notations as above, then every biho-

lomorphic map ¢: Mr

~

Mr is the restriction of a biholomor-

phic map from M onto M. The corollary above was first observed by Bland, Duchamp and Kalka [5] where M and M are strictly convex domains.

This

theorem can be thouaht of as a "weak form" of uniaue analytic continuation for the complex homoaeneous Monae-Arnpere eauation.

348 For applications of this result we refer to [5] and [16]. Consider now a bounded strictly convex domain D with smooth boundary in ~m.

Denote by oD and CD respectively the

Kobayashi and Caratheodory distances from a fixed point p c D. Let T be the unique Honc:re-Arnpere exhaustion centered at p.

X

Take an extremal disc f: throuqh p. f(8D)

c

D of

~

the Kobayashi metric

Construct a complex vector bundle E over

aD by assigning to each point z 0 c f(8D) the complex m

tanoent space to D at z 0

,

i.e., E

zo

{zJ Iu (z ) (za-za) a=la o o

=

O}

where u

(log T)a. Since an extremal disc is a stationary a map, the vector bundle E extends to a holomorphic vector bundle TI:E

~

f(~)

over f

(~).

By the convexity of 8D, the

domain D lies entirely on one side of each tanoent space to the boundary, thus D

c

E. The restriction TijD: D ~ f(~) is

holomorphic and surjective onto the extremal disc. bounded holomorphic function on

f(~)

Thus every

extends to a holomorphic

function on D with the same bound, i.e.,

f(~)

is also an

extremal disc for the Caratheodory metric, therefore CD

= oD.

More generally for any convex domain (bounded or unbounded and not necessarily with smooth boundary), bv takinq an exhaustion with strictly convex domains with smooth boundaries, a limit argument shows that the same is true. Theorem 5.2 .

Let D be a convex doMain in ~m then the Cara-

theodory metric and the Kobayashi metric of D are identically. The above theorem was proved by Lempert [15] and independently by Royden-T•Tonq [21].

For the more aeneral situation of

manifolds satisfyino the hypothesis of theorem 4.4, one expects to have a bounded A > 0 so that it is not clear how A deoends on

~~.

CD~

AoD on D, however,

A clear understandina of

this constant will be extremely helpful on the problem of existence of bounded holomorphic functions.

349 Appendix ~e

collect here, for the convenience of the reader, some

commutation formulas that were used in the computation in §2. (I)

For 1-tensors ¢.l, k..,.. J

{II)

s Rik] ¢s

- ¢ l, . ..,..k J

For 2-tensors ¢i],k'f

¢i],k"f

s RiH ¢SJ

-

t Rjtk ¢it

s Rik"f ¢SJ

-

s Rik] ¢s"f

¢k-.'0 J,lx, ¢ l. ] ..,.., k" )(,

s s Rkj"f ¢si + Rik"f ¢ sj (III)

For 3-tensors

Rs. _ lpq

,~, '¥

s]k -

Rs ikq

,~, '¥

SJP

¢ - +rs ¢ - - r~lp ¢ s]kq + r~lp r~ JO stk io sika

+

rl~k

r!Jq ¢ stp

¢i]p,qk

¢i]k,qp

+ R!]qp ¢itk

- r~]q

- r~lp ¢s]kq + r sik ¢s]pq

-

t r~ Rjqk ¢itp + r~]q lp ¢stk

s rik ¢stp

For the special case in §2 where is the metric, we have

T

is a function and

T

-

pa

350

-

where

T~lp

RI]qp

T

~-

SJkq

+ TS T ~- + ik SJpq

f~Jq

TS T ip Stk

Titk

s £ s = -TJSqpTik + Tsp TJ£q Tik

t s £ s Rjqk Titp = TJSqk Tip- Tsk TJ£q Tip t s Tjq Tip Tstk t s s £ -Tjq Tik Tstp= -TJ£qTikTsp• from which we get the following formula,

351

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