Geometry of Homogeneous Bounded Domains (C.I.M.E. Summer Schools, 45)

E. Vesentini ( E d.) Geometry of Homogeneous Bounded Domains Lectures given at a Summer School of the Centro Internazi...

Author: E. Vesentini

37 downloads 763 Views 14MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

E. Vesentini ( E d.)

Geometry of Homogeneous Bounded Domains Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Urbino (Pesaro), Italy, July 3-13, 1967

C.I.M.E. Foundation c/o Dipartimento di Matematica “U. Dini” Viale Morgagni n. 67/a 50134 Firenze Italy [email protected]

ISBN 978-3-642-11059-7 e-ISBN: 978-3-642-11060-3 DOI:10.1007/978-3-642-11060-3 Springer Heidelberg Dordrecht London New York

©Springer-Verlag Berlin Heidelberg 2011 st Reprint of the 1 Ed. C.I.M.E., Ed. Cremonese, Roma 1968 With kind permission of C.I.M.E.

Printed on acid-free paper

Springer.com

CENTRO INTERNAZIONALE MA TEMA TICO ESTIVO (C.1.M.E.) 3" Ciclo - Urbino 5-13 Luglio 1967

GEOMETRY OF HOMOGENEOUS BOUNDED DOMAINS

Coordinatore

EdDardo Vesentini

. " v GINDIKIN,S. G.-, PJATECCKII-SAPIRO 1. 1., VINBERG E. B. :Homogeneous Kilhler manifolds pag. 1

GREENFIELD S. J. KAUP W. KORANYI A. KOSZUL J. L. MURAKAMI S. STEIN E. M.

: Extel1dibility properties of real Submanifolds of (n . pag.

89

: Holomorphie Abbildungen in Hyperbolische RauIl)e. pag.

109

: Holomorphic and harmonic functions on bounded :3ymmetric domains. pag.

125

Formes harmoniques vectorielles sur les espaces localement symetriques

pag.

197

Plongements holomorphes de domaines symetriques. pag.

261

The analogues of Fatous's theorem and estimates for maximal functions pag.

287

CENTRO INTERNAzIONALE MA TEMATICO ESTIVO (C.l.M.E.)

v ..

S.G. GINDIKIN,LI. PJATECCKII-SAPIRO, E.B.VINBERG

If

HOMOGENEOUS Kl{HLER MANIFOLDS"

Corso tenuto ad Urbino dal 5 al 13 luglio 1967

HOMOGENEOUS KAHLER MANIFOLDS (1)

by

.,

.. S, G, GINDIKIN, 1. 1. PJATECCKII-SAPIRO, E, B, YINBERG

Introduction, R e c all 0 f c e r t a i n res u It s --------------------------------~---

,

I, Definition of homogeneous K!Ihler manifolds,

Let h

=g +

i

be a positive definite Hermitian differential form on

the complex manifold M, Then differential form and

g

is a positive definite symmetric

is a non-degenerate skew-symmetric differen-

."

tial form of type (1,1), and (1)

g(x, y) = ." (Ix, y)

where

I

is the complex structure operator, The complex manifold

with the pos itive definite hermitian differenti:al form K:hlerian if one of

h

M

is called

the following equivalent conditions is satisfied

(Kl)

o

d

(K2) The parallel translation with respect to the riemannian metric g

(K3) form

preserves the complex structure of the

In local coordinates h

tang~nt

space,

i, e,

hll~

Zll, Zll the coe·fficients

of the

can be represented in the form

o2

(2)

logtp

ozeL oz~

where tp is a positive real function. The prof of the equivalence of conditions for example in

[13,27],

An automorphism ( 1 )English

(Kl) - (K3) can be found

of the K:hler manifold

Translation by Adam Koranyi,

M

is

an invertible

- 4 -

holomorphic map preserving the form

h. We shall denote the group of

Il

all automorphisms of the Kahler manifold M by G(M); its connected o component by G (M) • We stall also consider the group GA (M) of all M and the

invertible holomorphic transformations of the manifold ,group

GR (M)

of all isometries of

GR (M) the connected components of the respectively. In

and GR(M)

GA(M)

as a riemannian manifold. Then

0

0

We denote by GA (M) and groups

M

conditions are given in order that

G~(M)

[8, 12]

= GO(M).

some sufficient We are not going

to dlscuss these conditions here. However the connection between the groups

GA (M)

and

G(M)

Ii

The Kahler manifold

G(M)

acts

will be considered in certain cases. M

is called

homoge~~

if the group

transitively on it,

Often the homogeneity of a K~hler manifold is defined by the transitivit)' of the group [3]

and of Tits

GA (M). From the results of A. Borel - R. Remmert [22]

it follows that, if a compact KYhler manifold is

homogeneous in this sense, then there exists on it a K~hlerian structure (compatible with the given complex structure) with respect to which it is a homogeneous in our sense, In the non,.compact case, it is unlikely that the consideration of homogeneous complex manifolds carrying Kghlerian structures will lead to a significative classification. The simplest examples of homogeneous K~hler manifolds are the hermitian space

Il,

the complex torus

and.the unit disc

Tn, the complex projec-

tive space

pn

K in the complex plane.

In the follo-

wing three

paragraphs we shall describe three fundamental types of hoII

mogeneous Kahler manifolds which have an extremely important significance for the theory. n

In the following we shall abbreviate the words "homogeneous Kahler manifolds II by "h.K. m,

IV

- 5 -

These are nian metric

h.K. m.

IS

which have zero curvature in the Rieman-

g. The,) are easy to classify. First of all, every homogene-

ous locally flat

h.K. m. is isomorphic with. the hermitian space

Hn

In fact, by a known theorem of E. Cartan it is isomorphic to an Euclidean space as a riemannian manifold; from the rlhler condition (K2) it follows that

the complex structure is invariant under parallel transla-

tions. Any locally flat

Hn

h.K. m. can be obtained by factoring

by

some lattice. The group

o GA (M)

hull of the groups

for a locally flat h.K. m.

M is the complex

GO (M) • A maximal complex subgroup of GO (M) is the

group of parallel translations. It is transitive on

M.

These h.K. m. 's have been studied by several authors and have been com~letelJ classified (Lichnerowicz [11] [26] ). We note that Wang [26]

, Borel [2]

I

, Wang

found all simply connected complex n

homogeneous manifolds. Some of these do not admit any Kahlerian structure. We formulate the fundamental result concerning this type of h.K. m.

Let

M

be a simply connected compact

h.K. m. • Then the group

is a compact semi-simple Lie group with trivial center, its isotropy subgroup is connected and is the centralizer of a torus. Conversely, if

G is

lizer of a torus in

a connected compact Lie group and

K

is the centra-

G, then there exists an invariant Klthler structure

on the homogeneous space nitely many sub-groups

G/K. Every, complex Lie group has only fi-

(up to conjugation) that are centralizers of tori.

- 6 -

They can all be easily found. Every simply connected compact an algebraic manifold

h.K. m. M

can be realized as

in a complex projective space

way that the automorphisms of

M will be

pn

in such a

the restrictions of unitary

II

projective transformations. However the Kahler structure of M will in general be different from the K~hler structure induced by

pn.

As in the 1 )cally flat case, the group G~ (M) of the simply connected compact this case

h.K. m. M is the complex hull of GO(M)

subgroup

KA

subgroup

GO (M) • However in

has no non"trivial complex subgroups. The isotropy

G~(M)

of the group

(the group

is

KA

is connected and contains a Borel

not the complex hull of the group K

).

Let us look at a typical example. Let

G

be the group

L ni = n

K = K(n l , .•. , n s ) '

of n xn

unitary matrices. Let

be the subgroups consisting of all diagonal

block matrices of order

n l , ., ., ns • We call an (n l , .•. , ns) flag a sequence of subspaces of the hermitian space Hn of dimensions

+ n 2, .•. ,

n, n l

ns

+ ... + ns" l'

homogeneous manifold

G/K

contained successively in each other. The can be realized as the manifold of

(n l , ..• ,n s ) flags. In a natural way it is contained in a complex projective space ; the KRhler structure induced by this inclusion is invariant under the group

G . We should mention that the group effect i vely on group

G,

in this example acts non

The kernel of the action is the center of the

G/ K which

G

is

contained in

K. This is in agreement with the

gener"l theory, since the automorphisms group of a simply connected compact

h. K. m.

The group

always has a trivial center as we remarked before. GA

of all non-singular complex

is the complex hull of the group

G

n Xn

matrices

and acts analyt-ically on

G/ K'

but it does not preserve the Jd.hler structure. The isotropy subgroup of

GA is the the group

KA = KA(n 1, ... ,n s ) which consists of all

- 7-

triangular block matrices with blocks of order

n l , ••. , ns on the diago-

nal. This is how one describes (up to t'1e choice of the K~hlerian structure)

all simply connected complex

h.K. m.

which are connected with

the unitary group. For the other compact Lie groups there is an analogous construction. Matsushima

[14] proved that every compact h.K. m.

rect product of a simply connected compact

Let ce

D

be a bounded domain in

<en. The Bergman metric

n

Kahlerian

the

[1,5,27]

is a di-

h.K. m. ana a complex torus.

n-dimensional complex spa-

defines in

D

a canonical

structure. This structure is invariant under all analytic auto-

morphisms of the domain The domain

D

D

, that

is, now

GA(D)

= G(D) •

is said to be homogeneous if the group

GA (D)

is transitive on it. In the case of a homogeneous domains the coefficients of the Bergman ,metric can

be found on the basis of (2) where for

l(J

one has

to take the density bf the invariant measure. Beside the canonical KMhlerian structure there may exist other K8hlerian structures in a homogeneous domain which are invariant with respect to respect to some transitive subgroup of it. of the other fundamental types of D

the group GA (D)

GA (D)

,or with

Differently from the case

h.K. m. , for the bounded domains

does not contains any non-trivials complex

subgroup. In the following we shall abbreviate the word IIhomogeneous bounded domains n In

as

Dh. h. d. n

>

0 for

y( V , y

all

=1=

x for which the inner product 0 ) • We are not going to prove this

result here • ~xample

3. In order to construct non-symmetric homogeneous Siegel

domains of type

I, one has to construct homogeneous non-self-adjoint

cones (with respect to any scalar product) • Such cones appear first in

1R 5

0

Consider the cone in 1R 5

(6)

Yll

Y33 -

Y22

Y33 -

2 Y13 2 Y23 Y33

Its

> 0 > 0

.

0

adjoint cone is the cone of symmetric positive definite matrices

of the form

This cone is not linearly equivalent with the cone (6) • Correspondingly there exist homogeneous non-symmetric Siegel domains of type

I

in

the following paragraph) n

~

4

0

(Cn

for n ~ 5 • Let

us recall (cf. also

that non-symmetric h. b o d. 's exist in (Cn for

Let us mention also that there exist an analytic continuum of

non-isomorphic Siegel domains of type of all h. b o d. 's

I

in cen

, for

n ~ 11 (in the class

there is a continuum of non isomorphic ones for

n~

7) •

- 17 -

2. Siegel domains of type II, From the concluding remarks of the previous paragraph one can infer that not all h, b, d, IS

are isomorphic with Siegel domains of type 1.

One can get to the same conclusion from simplex considerations too, For the complex ball

with

n .::: 2 , there exists no realization as a Siegel domain of type I ,

This fact will in full

be a consequence of the results of the following

part, but let us show right now that the complex ball cannot be mapped onto a Siegel domain of type

I by a mapping which is holomorphic on

the closed ball, For the proof it is sufficient to remark that the skeleton of a Siegel domain of type

I has real dimension

n

=dim<e D,

while

the skeleton of the ball coincides' with its boundary, i. e. has real dimens ion function

2n-l 1 1

This follows from the fact that the modulus of the reaches its maximum only at the point (1,0,.",0)

zl-2 in the closed ball, and the group of unitary linear transformations acts transitively on the boundary of the ball. The ball can be mapped onto an affine homogeneous domain by setting

z = 2

"

•• 0

z

J

n

=

We obtain as image the domain 2

1m z-

(7)

>0 ,

We describe now a transitive group of affine transformations of the domain

(7), We consider the maps z .... z

(8)

u .... u

+a + +c

2i

I

Uk

~k + i I

ck 2 ,

- 18 -

where

aElR, CE

0) •

(8), (9) generate a transitive group. In fact any

(z, u) satisfying (7) can be mapped by (8) onto a point

> 0)

(iy, 0)

, and this point can be mapped by (9) onto (i, 0) • For the

proof of the transitivity of the group of automorphisms it is enough to prove that an arbitrary point of the domain can be carried onto some given point.

(7) admits the following generalization.

The construction Let

V be a convex cone in m m n

• " > ~ of the space ([: q , It is clear that one can

([;q

choose a basis in

so that the subspace

([;

rna

consists

of step matrices of some type. (The first elements in each row vanish, and the number of these elements does not decrease when we go from one row to the following ) • It turns out that the domains

D(V, F) associated to the form (23)

are non-symmetric unless D(V, F) is a Siegel domain of type 1. We prove this in the simplest case, Let

e=

2, q = 1, let

Then we obtain the domain in

u be matrices with the second row equal to zero. ([:4 given by the following conditions:

> O'Y11-lul

(25)

2

> 0 (y .. =Im IJ

z .. ). IJ

This is the first example of a non-symmetric homogeneous bounded domain constructed by 1. I. Pjatecckii-S~piro in [16] • We give a proof that it is not symmetric.

First of all !:emIE~

we prove the following lemma:

3, The symmetry of .the Siegel domain of type II D(V, F)

- 25 -

at the point

(zo' 0) , if it exists, (z, u)

where

z

-+

-+ (

is of the form

tp (z),

t/J

(z)u)

tp (z) is the symmetry of the Siegel domain

at the point z6 ' and

t/J

depending analytically

on

(z)

is

z

a

of type

linear transformation of

I D(V) {:m

•

We mention first. that the symmetry is unique at every point

~,

(it must be the reflection in the geodesies with respect to the Bergman metric). Let the symmetry at the point (zo' 0) be

(z, u) .... ( tp(z, u), t/J (z, u»

(26)

•

It must commute with every automorphism of

D(V, F) which pre-

serves the point (zo' 0) (because of uniqueness), in particular with (z, u) .... (s, e

i8

u).

Hence tp(z, e

i8

t/J(z, e Because

i8

u) = tp (z, u) ,

u) = e

i8

t/J (z, u)

of the analyticity of tp and t/J in

a

neighborhood

of

0

with respect to u we obtain that the symmetry has the form (26). Setting

u = 0,

domain

we obtain that

D(V)

Lemma 4, ~.

at the point

z Zo

The domain

-+

tp (z)

is the symmetry of the

• (25) is non-symmetric.

The symmetry at the point

(z

= iE,

form

(27)

( z, u) .... (- z

-1

,t/J (z)u)

,

u

= 0)

must be of the

- 26 -

Under an analytic automorphism a point of the skeleton go into another point of the skeleton or to infinity, i

However under the map (27) the point of the skeleton z = ( 1 u = 1 goes into the point z = '12

(1+' _1_1i

must

S1 D

1) 1

'

-I-i) _1+iwhich does not belong to the

skeleton, Therefore there exists no symmetry at the point

(i E, 0) •

Remark, It would be possible to compute the volume element for the Bergman metric of the domain (25), and check that it is not invariant under maps of the form (27) • Example. 5 • Let V be the cone of hermitian positive definite mam , trices of orfier • We realize the space

(2)

(complex~

matrix of order

e,

V

g(y)

we ma k e correspond the map o'f

u

t Y t*

mm \L-

(29) Condition (22) is satisfied , The corresponding domains are symmetric if one of the number q, r We denote by

uk the pair

rows of the matrices chain of subspaces a;mo

of pairs

sl

is equal

to zero,

(U~I) ,u~2)

uk E "41

u

This

determines

Proposition 2 : an element

s

with

by the formula

[. JU,ul.

F

in terms of the algebra

Let

ws that

Ad T is a maximal

triangular sub-group in Ad GO(M) • So it remains to prove that acting on the space

5

(M) is triangular.

We do this only for the case when

M is

a h. b. d.

neral case the proof can be reduced to the case of uses theorem If

M is

Ad T

a h. b. d.

In the ge;' if one

2, § 7 • a

h. b. d.

then the group

Ad GO (M) is the connnected

- 48 -

component

of the identity of some algebraic linear group, This is pro-

ved in § 3

of our article [25] ,

From this it is easy to deduce the algebraicity of the group ad T , In fact

.

ad T

is a maximal splittable solvable

0

Ad G (M).

(Ad T)

(i. e, the smallest algebraic group conta~ a is also a splittable solvable group . Consequently (A~ T)a =

But its algebraic hull ining Ad T)

in

sub~group

= Ad T • Like every connected solvable algebraic linear group Ad T

can

be factored into a semi-direct product Ad T = (Ad T)

where (Ad T)R

group containing all

tive eigenvalues on

In other

r

(Ad T)r

is a commutative

sub~

semi-simple elements whose eigenvalues have modu-

1 , Since all linear

sformations

• (Ad T)

R

is a normal sub-group containing all elements of Ad T

which have positive eigenvalues and

lus

the group

'6

transformations contained in

(by the fact that

contained in (Ad T)I words

all

Let us find the center of

t E. T

have posi-

is splittable) the linear tran-

are equal to the identity on

elements

belong to the center of the group

T

Ad T

for which

G.

Ad iE-(Ad T)I

T,

T

Since

T

is splitt able

it follows

that it is connected, On the other hand the center of the algebra is trivial, since

for any IV (

and hence [jx, x]

f

x

E.

G,

x

f

0,

[jx, xl) = !? (jx, x) > 0 0, Therefore

(Ad T)r = {e f,

and

Ad T = (Ad T)R'

which we had to prove , Corollary 1: If the h.K, m, ble group of automorphisms n

Kahler

algebras,

M

admits a transitive splittable solva-

, then all such groups determine isomorphic

- 49 -

In fact all

~ (M) •

these

By proposition K, which

Ad(M) morphic

as

K~hler subMalgebras of

K:hler algebras are 3

they are conjugate with respect to the group

contains

j

and ."l ,

and

consequently they are iso-

n

Kahler algebras.

Corollary 2: Let Biegel domains of type 1) The domains

D= D(V, F)

and

D1 =D(V 1,F 1) be homogeneous

n.

Then the following conditions are equivalent:

D

and

D1 are affine isomorphic.

and

D1 are analytically isomorphic.

2) The domains D n

3) The Kahler algebras constructed using transitive triangular groups of automorphisms for The implication lary

1,

and

D and

1)~2)

3) ~ 1)

is

D1

are isomorphic.

obvious. 2)

follows from

~3)

proposition

follows

from corol-

1.

Now we introduce the following definition: A K%.hler manifold will be called

if it

~l

admits a

automorphisms. The K:hler algebra

transitive splittable solvable group of

{y )(,

,

"l } will be called normal

j,

is

a

splittable solvable

h.I

0

=0

j uE

since

J1.

1-

5' (condi-

1-IS+a>1

= ju

,

y

=v

0

J:r, IS
0

IS

a

,then

-IS

< I_a

- 64 -

(;;; , 71~) = (j 0 ,j J(cr) Lemma 4. Proof. Since [ that

A

[J(o ,J] 0

=

0 •

We consider the operator

Xl' do J = 0

commutes

o •

A = ad;;o

h

where hE:".

,using the integrability condition it follows

with

j . By the closure of the form

g

and the

3 ,

commutativity of

s>

(Ax, y)

This shows that

+

s>

(x,Ay) =,. 0

the operator

A

(x,YE.

dJo '

is skew-symmetric with respect

to the canonical scalar product. Since its eigenvalues must- be real we have

A=0 ,

Lemma

5;

.1+,u+ y =0,

Let

hE '}{ ~commutes with

If the element so

JJ;

with

Proof.

Let

a 6 0 • Then

[h, a 1 , x) = 0 for all

s> (

Since,

, we have

and therefore

j [h, a]

h,a

0

J

Lemma 6. The ideal for which

g= x

s>

coincides with the set of all

[jg, g] = 0

+h

where

x

E;r,

o = [jg, gJ

hE:

U.

[jh, h]

[ jg, g 1 Then

(mod

and

s>

x E.

~.

([h, a] , j Ih, a]) = 0

.

Proof. It is enough to prove that if Let

it commutes al-

by the closure of the form

by the condition of the lemma, by lemma 2,

'J; then

J) .

o then

- 65 -

Since which

;Jt(

is

a

non-degenerate

was to be shown Lemma 7, The center

and is

z-;;;o

Lemma

From lemma that

Z

Z

1

of the algebra

h

is contained in

0,

do

j - invariant

ProoL

that

K!!hler algebra, we have

shows that

6

4

and the fact

Z:) J , Furthermore

it is clear

, it follows

that

coincides with the centralizer of

Let

-

2

From the integrability condition we find that Cja, jh] = j [ja, h]

for

a11

h E..

J( 2

so that S'(j [ja, h] , tja, h] ) :: S(Cja, jh], tja, hJ ) , Using the condition account

that

(K A 7) of the closure of

[ja, [ja, h} ]

:: 0

and

[jh, fja, h1 ]

(j [Ja, h] , [ja, hJ

This proves that

Ua,h]

s:

and

taking into

0 we obtain

) = 0

o and

ja E.

Z

The purpose of the remaining paragraphs of this part is the proof of the fundamental theorem, Here is the logical scheme of this proof.

- 66 -

ILemmas

8,9 (punto 2,

2)

§

Lemmas 10,11 (punti 1,2,

~ Lemma 12

I

(punto 3

§

3)

I

!!)]

Lemmas 13,18 (punti 4,6

§3)

---.------~.

/

I

/r---------'+-Proposition 3(punti 1, ;--~~)l

________________~J

//

Fundamental theorem 2nd its consequences - lemmas 2 - 7 ( §1, punto 2

§4)

---~-.-------~---~

The theorem will be proved by induction with respect to the dimension of the Ktthler algebra. The dotted arrow indicates that in the proof of lemmas

13 - 18 we use the induction hypothesis.

If it is known in advance that the algebra is non-degenerate, then

the proof becomes considerably simpler. In this case the whole right-hand side of the scheme ( § 3 lemmas

2 - 7 from

in the text) turns out to be unnecessary and so do

§ 1 •

1

1. The fundamental theorem will s ion of the algebra Proposition 1.

be proved by induction on the dimen-

together with the following assertion:

In every normal Id.hleralgebra there is either

- 67 -

an elementary or a commutative non-zero id.hler ideal. Assuming that the fundamental theorem holds for agebras of dimenn ,in

sion less than paragraph 2.

§

§ §

2 and 3 we prove proposition 1 , and in

4 we prove the theorem for algebras of dimension

f

Since the Lie algebra

n

•

is solvable and splittable, there

is a one-dimensional ideal in it. In this ideal we choose a non-zero element

r

so that

[jr, rJ

= r

or

we show that in the first case the element mentary KRhler ideal, and in

§

=0

[jr, rJ r

• In this paragraph

is contained in some ele-

3 we show that in the second case it is

contained in a commutative K'!hler ideal. To start we prove two lemmas connected with these two situations.

g

II:

f

Lemma 8.

Let

gu be the sub-space conSisting of those elements

for which

[g, r ] = [jg, r] = 0 •

(6)

tJD is

Then rator

invariant under

ad gP jr Proof.

and

ad

jr

moreover the ope-

commutes with

The invariance

of!fJ with respect

to

j. is

immediate

from the definitions. From the integrability condition we have [jr, jpJ = j [jr, p]

(7)

for

all

p

E.

£p.

Using the Jacobi identity and

(7)

we find

[[jr,p] ,r] = 0, [j Ijr,p1,r]= L[jr,jPJ ,r]

so that

[jr,p] E

follows.

ffJ.

From (7)

0

the commutativity of

ad jr

and

- 68 -

Lemma 9, For all

u, v

E.

f

d~ ~(etadjr u, etadjr u, etadjr v)

(8)

= S'(jr,etadjr

[u,v]) .

Proof. d ill - 0 -

J (

(tadjr

r"Jr"

e

e

u, e

tadjr)

v

=

tadjr u J, .etadjr v) + O( tadjr J e

- 0 (" -.) Jr, [tadjr e u, e tadjr v

J) -_ 0J ( Jr, "

U"

["Jr e tadjr J

v] ) =

etadjr [ u ,J) v.

(In the second step we used the fact that the form 3, In this point we prove proposition

follows in the case where

[jr, rJ

=

r

2

q

is closed).

from which proposition 1

•

For this we will not use the induction hypothesis concerning the fundamental theorem. Proposition 2. If in the normal dimensional ideal

(r) (1) such that

n

Kahler algebra [jr,r J

=r

there is a one-

, then the algebra

~

can

be decomposed into a direct sum of sub-spaces (9)

where

vf = (jr) + (r) + U is an elementary KlIhler ideal and " 1 [Jr,u] ="2 u for u U; 2) ~ is a KMhler sub-algebra orthogonal to fand commuting 1)

E:

I

with

jr

and

r

.

(1) By (r) we denote the one-dimensional ideal spanned by the vector

r .

- 69 -

Let

Proof, the relation a

and

b

[jr, r in

if be the sub-space constructed in lemma 8,

J =

r

1

g - ajr

Using lemma If in

9

- br

f1J,

E.

decomposes into a direct sum of sub-spaces

~ = (jr)

(10)

Using

gE. ~ numbers

one can find for any element

a unique way, so that

This means that

a)

By

tP,

+ (r) +

we shall study the eigenvalues of

(8) we set

fJ0 then

u;: r , vE

ad

flJ.

jr on

the right hand side

is

that tadjr e r

(11 )

t

=er

we obtain

o (r etadjr v)

( 12)

),

According to lemma on

!JO.

=

a e -t

for

the operator

8

9 (jr,

e

Finally' for any

tadjr ) v gE

= ~

Jr, e tadjr) g

0('

(14)

J

a e

-t

q (e tadjr u,

(15 )

u EO

fP,

product using that

commutes with

by

Let

p E.

E:

(10) and (II)

a e

e tadjr) v

v

-t

+

=

e

b

u, v e a e- t

C7U d , we obtain

t

~

+ bet + c

then in this equality we can change over to a scalar ad.,? jr

tadjr r (e tad "u, e -I) b)

ad j r

for all

Formula (8) now gives for all

( 16)

flO

v E.

Therefore

( 13)

If

all

= ae -t +

(1Y be

commute, We obtain

and

be

t

+ c

for

all

an eigenvector for ad jr

pE.

[f,

VE

1

corresponding to

0,

- 70 -

eigen-value A

Then

By formula e Since

2

e

A(t)

(p, p) = ae

Uj ). Let

-b

can

(p,p)fo,

[ jr,p ] 0Ap,

p

=e

,\ t

p.

(16)

check that the operator also on

tadjr

+ be

t

+ c

and

[jr,z]=f\q+p. Then

.

We

therefore

:;u such that

q be vectors

1

0, :. '2

g:> (and

ad jr is semi-simple on

p

1

have only the values

e

in tadjr

q=e

At + >.t q te p

and

by formula (16) (e tadjrp, etadjr q) = e 2 ~ t (p, q)+(te 2 At (p, p) =ae

-t

t

+be +c.

Being (p, p) = 0, this equality is impossible. c) Since the operator a =0

adg>jr

has no eigen-value -1, we have

in (12) . Looking at the formulas that were deduced from (12), we

see that in each of them lue of

a = 0 , and therefore -

'12

is not an eigen-va-

ad;? jr. We denote by

the eigen-space of

to the eigen-value

• The algebra

~

:;

ad ff-' jr

corresponding

decomposes into a direct

sum of subspaces

UJ =

(17)

(jr)

{I

ad~

Since invariant

under

orthogonal to Since

jr

+ (r) + !fl. + 1 2

commutes with

go 0

the subspaces

j

J?

j • Formulas (12) • (13) and (16) show that

(jr) + (r)

[Po C jJ

+~

2 we have

eigen-values shows that

N

[ r,

= (jr) +

J ]=0 . 0

(r) +

~

2

are

[j0 o

Consideration of the is an ideal.

is

- 71 -

~o,

The sub-space is a sub-algebra (see

§

being the orthogonal complement of

6, part 2) setting

U/?1 ()(}JI = fP

0

N

,

, we obtain

2 the decomposition (9), which has the required properties,

§ 3, Commutative K!lhler ideal. I,

In this paragraph we consider the case where

[jr, r] = O.

Under the hypothesis that the fundamental theorem holds for algebras of lower dimension then

7J

we prove

Proposition 3 : If the normal K1!hler algebra dimensional ideal

(r) such that

ned in a commutative K!lhler ideal

!.J

contains

a one-

[jr, r] = 0, then this ideal is contai-

K.

In all the lemmas of this paragraph the hypothesis of proposition 3

is assumed. By

{fJ we

denote the sub-space constructed in lemma

8

Lemma 10 Proof,

For any

g E

1 from the Jacobi identity and from

the

integrability condition we have [rjr,g], rJ = [jr, [g,rJ] = 0 [j [jr,g] ,rJ = [rjr,jg] ,r]= 0 i. e.

[jr, g]

E~

2, Lemma 11:

Proof: to

(jr)

(ad jr)2 = 0 ,

a) We show that the sub-space

+ (r) • Let p

E:.

r.

[jr,

d1 is orthogonal

By the closure of the form if we have

- 72 -

g (r,

= S'(jr, [r

[jr,p] )

o.

,pJ)

Using lemma 8 we also obtain S>(jr, [jr,p] ) = -

~(jr,j

b) From lemma 10 and

[jr,jp1 ) = - g(r, [jr,jp]) = 0 . a) it follows that

(18)

Formula (8) ·shows then that for any

u, v

rg

£

d 3 O( tadjr tadjr) d 2 D(· tadjr 1) = --3- .) e u, e v =.) Jr, e . [u, v 2 dt dt

=g(jr,

[jr, Ur, etadjr [u, vJJJ )

=

0

I

i. e.

(e

(19)

Whence for

p

(e

eigenvalues on

=p [JJ

tadjr

ZE d'such

tadjr

2 v) = at +bt+c

P, e

tadjr

2 v) = at +bt +c •

we obtain that the operator adjr (and therefore on

c) We assume that x, y,

u, e

EtJ

(20) Setting v

tadjr

(adjr)2

has no non-zero

f! ).

F o. Then tl).ere exist elements

that: [jr,xl

[jr,y] =

=0 ,

0

o.

[jr, zl

,

It is clear that tadjr

e

x

=x

I

By lemma 10 We get

e

tadjr

y

=y+tx,

e

tadjr

t2 z = z+ty+ - x

Y E. &Pa.nd we can substitute

2

p

=y,

v

= z in (20) •

- 73 -

t2

(y+tx, z+ty + -2- x) Since

(x, x)

f

:=

at

2

+ bt + c

0 , this is impossible.

3, With the aid of the operator ad jr

filtration of the Lie

f!

algebra

be the natural projection of ;) We denote by

A

we construct a

U = r;/ (jr) + (r) onto U . 0

Let

the operator induced

the integrability condition it follows that

n

and let

by adjr on commutes

A

j-invariant

U . From

with

j on

U

We set

~ (-1) ()J (0)

:=

1'

If

:=

n

-1

~ (1)

:=

n

-1

~ (2)

(Ker A) , (1m A) ,

(jr)

+ (r)

Lemma 12 : The sub-spaces of the Lie algebra

form a .i-invariant filtration

~(i)

, Furthermore

(21 )

o Proof:

a)

From lemma 11 it follows that

A 2 := 0

Therefore

Ker A :::J 1m A , and we have the inclusions ;; (-1) :::J ~(o) :::J J(l) :::J fj(2)

ud (i)

The invariance of the sub-spaces from the commutativity of

A

with

J'

u 2 = [jr, g2

J ,Since

follows

j.

b) We prove the commutativity of u 1 := [jr, gl

with respect to

Y

(1)

0

Let

(adjr)2:= 0 ,

gl' g2 c

~

and

- 74 -

Therefore

[[jr, GjJ ,[jr,

V]] = 0

Since

~(1) =[jr,~] + it remains to prove

(jr)t(r) ,

that [Or)

+ (r) , [jr, U/J] = 0 (}

but this follows from lemmas 10 and and 11, c)

It follows immediately from the definitions that

[~(-1)

r2) ] c

,

~(1) ,[~(o)

,

~(2)] c ~ (2)

d) We show that [

~(o)

kE.~(O),

Let

,

~(1) ]

u E~(I)

c

r~(O) , ~(2)J c ~(I) , it

. Since

is enough to consider the case where ( jr, [k,g]]

~(1)

= [k, uJ

u = jr, g

and

glC.:r

We have

+ [[jr, k] ,gJ

Since

r· [k, gJ] ,[[jr, kJ , pr,

gJE: ~1)

we also have

[k, uJ e)

We prove

g£r:! '

~(1) •

that

[1 (-1) Let

E.

u

6.

C;; (1)

,zr(I)] , Since

c 1(0) (jr, uJ

=0

and [jr,g]E:1 I )

,

- 75 -

we obtain using the Jacobi identity and the already proved commutativity of

~(1) f)

gl' g2 E

that

r jr,

f;j (0)

It remains to prove that

:! (0)

[g,UJE(:;t).

[g,uJ1= O. Therefore

• Since

[?(2) ,

1

(O)J

11 =[[jr, gJ

[jr, [gl' g2

J

Consequ~ntly

(0)

[gl' g2~?J

The proof of lemma 12

is

c r;;(2) we have

+ [gl' [jr, g2 J]E: ~2) •

,g2l

•

is finished.

4. If in the filtration constructed above

is a commutative

a sub-algebra. Let

Kahler ideal in

V and

CJ (o) = ;; then f proposition

Therefore in the following we shall assume that

fJ (

0)

= ;/(2)

3 is proved.

t

fj .

The induction hypothesis may be applied to the K1ihler algebra Cj(O) • Let

(22) be the decomposition correspondipg to this theorem. Lemma 13 : Proof: lows that

g

Let

[jg, g]

Lemma 14 : Proof: ma 14

r;(1)

E.

C;;(l) . Then

= 0 • By lemma 6

[fJ ,J'J

Let

[[jr, g1

cd .

,0

g E.

jg E

~ (1)

• From (21) it fo1-

this means that

g

E.!;J.

c ~(O)

Cf· ,

x£

J . Then. [jr, gJE:Jl)

and by lem-

= 0 • Therefore [jr, [g,xJ] = [g,[jr,xJ]

o.

Thereby the lemma is proved. From lemma 14

~(or

it follows in particular that if the sub-algebra

is commutative, then it is an ideal in;;

so that in this case

- 76 -

proposition 3 is proved,

71

In the following we assume that

(0) is not commutative, i.e,

o,

tjA.

We denote by

s

the principal Idem potent of H, We denote by

the subspace formed by the vectors which ar{J annihilated by so-

me power of the operator

Lemma 15 ; If

ad

A + p.

js - ;\ , It is clear that

> 0 or A = p. = 0,

I[~, J;l :LJ Proof,

Let

g E:.

[rg,x],J.()+J4+ II)J = Let

= 0 ,

Cjv' XE..(h , Then by lemma

According to lemma 5

then

it is sufficient to

14

prove that

0 ,

YEU.() +,.u + y) , By the commutativity of [rg,X] ,y

J = [[g,y] ,x]

=

d

0 ,

We have

[g,Y]e0'(O) 1

d

_(A+f 0 then [g, yJEJ since the operator adxjs has only non-negative eigen-~alues, Consequently in this case _ [g, yJ ,x J = 0, If If A + P.

~ ma

=).J. = 0 then [g,y}E..

J( +d, x£~

r

and

4, 5. We

consider the graded Lie algebra

[[g,y], x]

=0

by lem·

- 77 -

1

~=

(23)

+ 1(0) + ~(1) + r(2)

(-I)

which i~ associated to the filtered Lie algebra g

61

(1)

g

we denote by

:J . For every, element

the corresponding element

Of:! (1)

(I) .

From (21) it follows that

o ,

(24)

We define

fj (-1)

on

a trilinear operation

the formula

(25)

abc =

nar,

a] ' b

(a, b, c)

~a

b c

by

J ' c1 .

We established some properties of this operation, Lemma

1) The operation (25) is commutative.

16:

pr, abc] = [(Ifr, a]

2)

Proof:

1)

= [[Jr, [ 01(-1) rJ

analogously that 2)

1]

By the Jacobi identity

[[fIr,a] ,bl ,c] since

' b] , Ifr, c

, 0(-1) ] __

If

ra, bJ] , c]

-[[Ifr,b] =

,aJ ' cl

0

0 • This shows

that abc

= bac. One proves

abc = acb

We take the commutator of both sides of equation (25) with

jr

and use the Jacobi identity. From the properties of the graduation and from (24) it follows that

[Ir,

[Jr,

of the three terms on the right

IJr, 1 cJ

which

we

aJ]= 0,

[(jr,al,

[Jr,

bJJ= 0 , so that,

hand side, there remains only [[

B;:~ aJ,

b]

have to prove

~}If we regard the same element g E. ~(i) as an element of ~(i-1) then g = 0 , However in the following it will always be clear which of the subspaces ;; (i) we have in mind ,

- 78 -

The following lemma is fundamental for ·this paragraph Lemma 17 : Proof.

abc = 0

-(-1) a,b,cE.;j

for all

.

Tq the decomposition (22) of the algebra

1

(0)

,

there

corresponds the decomposition

(26)

!(O)

of the algebra

[1(1)

(27) and

from lemma 14

J

[ s, r = jr

so

1J

Jl =

0

that

r e.

or

±

a=O

E;t,

d . Since

t'

where

ad jr

maps

~ (-1)

a= 0

has eigenvalues

(30)

aE

-12

or

it is clear that the

isomorphically onto

operator ad js is semi-simple on 1 + -2 on . it. Relation (29) shows that it is

Let

2

d_ cV • Going over to the al-

then. jr E

From the definition of the sub-spaces fJ{i) operator

(r) is an ideal 1 then and if a= _ -

we obtain the following relation

[ 3S, Jr] = - ajr

(29)

,

that this case is impossible.

Furthermore, if r gebra

,

if follows

.By lemma 2

t (r),

0

if follows that

From lemma 13 [js,r] =ar

,JJ

+ 1(2)

[~(-1)

(28)

1~

. By lemma

fj (

1)

:f (1)

and has eigenvalues

semi-simple also on

1 + - + a there. - 2

a,

~ (-1) A

J

abc

b"" ~ (-1) ~

J

c

E

• The

:fi(-1) , so that V

,

= [fUr,a1 ' bJ c] 1=

o.

~ (-1)

0

and

- 79 -

Then also

[fr, abc l

(31 )

[rUr,al ,b1 ,[fr,c]]

=

f

0

•

From what it has been said above it is clear that (32)

1

+ - +a

or

ct

-

2

Furthermore

1 or 2

1

then

which is imps sible in view of (28)

and (30) •

Using the symmetry of abc ,

we obtain now

A+,u,,u +v, v +

(33)

[ [1(-1) -

-(1)

[jr, aJE$A

'

sfied only in the case where

=

jr, c

or ~

9(1)J

,

[_

I

a , 1 + a or -2

g +b >0

Lemma 15 shows that if

Since

A-

]

,~

E;;; (1) 1'_Cl'

A+v

=

b

(1)J

+a

o,

.

then

= o.

,condition (31) can be sati-

- 2 a < O! A - a ,v

- a are not simul

taneously zero • Using the symmetry of abc

A+,u,,u +v,v +A
Therefore the operator

j •

jz£ Z(jO)) • From the integrability con-

ditions it follows that the operator and from the closure of

[Z(~(l)), ~] CZ(~(l)

i.e

We show' now that the ideal Let

'

1 (1)

j,

it follows

•

is skew-symmetric with respect to the

canonical scalar product and since it has real eigen-values we have A

=0 • 7.

This means that

jz

E

Z(ff(l)) • The .lemma is proved.

Now we can prove proposition 3.

We denote by

N the center of the ideal

a commutative ideal in

3'

Z (~(1))

• This will be

and by lemma 7 applied to the K'!hler algebra

- 82 -

Z( {:f(1)) NF

o.

and ideal

N is

j-oinvariant. Since

N~

~(1)

we have

So proposition is proved under the induction hypothesis of the

fundamental theorem • From propositions

2 and 3 , which we have proved

in

§ §

2

and 3 respectively, proposition 1 follows.

§ 4. Proof of the fundamental theorem. Let

r; be a normal

KAhler algebra and assume that the statement

of the theorem is true for all normal KAhler algebras of dimension lower than that

ff . Then. we can apply propositions

of

Let

N be

of these propositions and let Then ~ t

is

Cj

a Klhler ideal in

!! '

(47)

c-r' }(l ;f" =dl + v

(cf.

§

6, part 2) . By the

can be decomposed into a semi-direct sum

is a commutative

Klhler ideal and

sub-algebra having a principal idem potent

Jt,

is

a

N+

d'

KBhler

s'

Applying the lemma on symplectic representations (cf. 2) to the KBhler algebra

fJ .

satisfying the conditions of one

Cf

a KlIhler sub-algebra of

l'

J'

3 to

be the orthogonal complement of N.

induction hypothesis

where

2 and

§

6 ,part

we see that

(48)

We consider separately the two cases corresponding to the two possible types of the ideal a)

N

N is an elementary Klhler algebra . We set

J =J' , )( =

N

+)(' ,

s = r + s'

,

- 83 -

where

r EN

~ = d +Jt

We show that the decomposition theorem and From

2.

is an element fulfilling the conditions of proposition

s

(48)

commutes with

is

a

principal idem potent for the algebra

d

it follows that jr

is of the kind needed for the

and

r

is an ideal in:t

}t

• Since

fI'

we have

[js,s] =[jr,r]

+

r + s' = s

[js',s']

By the same considerations ad js'

on

ad jr

on

ad js = (jr)

It follows now that the operator ad js

+ (r)

.

is semi-simple on the

sub-space

W= to check how

js

acts on

Consideration of that

~

0, 1,

and has eigen-values

(jr)

+

(r)

+)(, eX

there, and

the sub-space

the eigen-values

of

U

j

ad jr

by s'

on ; ; shows

of the two-dimensional

The representation and

js'

It remains

(ef. proposition 2) .

[elf' ,U]e U .

algebra generated

W= U{?c .

K§hler

will be sumplectic.

By the lemma about the symplectic representations of such algebra,;; ( §

6

,part

2)

u=z(

1

-y

where

ad JS' =

jUre .1

~

U'.,1

onU,;l

- 84 -

Since

ad U jr

setting

2

1 we obtain 2

UJ1+U+U 011 ad js

2

=)

on

This finishes the proof in the case b)

N is

a commutative

U). a)

KHhler algebra.

We set

From (48) . it follows that the ideal decomposition

1 =(f+ Jt

J

is commutative so that the

satisfies the requirements of the theorem

S'Jme problem In

§

5

of the introduction to our lectures

about the structure of the homogeneous

the Ilmain conjecture II

KHhlerian manifolds (h.K. m. )

was formulated, We give some corollaries of it here, It is possible that some of them are valid

for a wider class of manifolds. It should be very

interesting to find direct proofs for these statements.

1.

Each

h. K m, whose points are separated by bounded holompr-

phic functions is a bounded domain. 2.

If the points of a

h. K. m.

are separated by holomorphic

tions but all bounded holomorphic functions all locally flat,

func-

it are constant then it is

- 85 -

4.

Each simply connected h. K. m. is holomorphically convex.

5.

If a h. K. m.

is a

Stein manifold then it may be holomorphical-

ly fibered in locally flat h. K. m. and the base is a bounded domain. For arbitrary hOlomorphically convex complex manifolds the Remmert fibration is known, its fibres being defined as maximal sets on which all holomorphic functions are· constant. The base of this fibration is a Stein manifold and the fibres are compact. This result could be applied for the proof of the main conjecture if the propositions

4

and

5

were proved.

The similar construction for bounded holomorphic functions, about which we do not know any general results, gives immediately the fibration mentioned in the main conjecture. 6.

Each h. K. m.

with

negative Ricci curvature is a bounded do-

main. 7.

Each

h. K. m. with zero Ricci curvature is locally flat.

Hano and Kobayashi considered one canonical fibering of an arbitrary homogeneous complex manifold with invariant measure. It seems to be not difficult to p1'ove that for h. K. m. Ricci curvature and the base

is a

the fibres of this fibering have zero h. K. m. with non-degenerate

Ricci

curvature. It is possible that a further investigation of the base and a direct proof of the statement 7 will lead to a differential-geometrical proof of the main conjecture. 8.

Each compact group of automorphisms of a

h. K. m. has an

orbit which is a complex submanifold. In particular, each one-parametric compact group of automorphisms has a fixed point.

- 86 -

REFERENCES 1.

BERGMANN S. , RUber die Kernfunctionen eines Bereiches und ihr Verhalten am Randell, J. reine und angew, Math., 1933, 169, 1934, 172, 89-128 •

2.

BOREL A., nK'!thlerian coset spaces of semisimple Lie groups" , Proc. Nat. Acad. Sci. U.S.A. , 1954, 40 , 12, 1147-1151.

3.

BOREL A., REMMERT R. , RUber kompakte homogene KMhlersche Mannigfaltlgkeiten ll , Math. Ann. , 1962, 145, 5, 429-439 .

4.

CARTAN E. , IISur les domaines bornes homogenes de llespace de n variables complexes II I' Abh. Math. Sem. Hamb. Univ., 1935, II, 116-162.

5.

FUKS B. A. , II pecial chapters in the theory of analytic functions of several complex variables", Moscow, 1963 (Russian), (English Translation published by the American Mathematical Society in 1965) •

6.

GRAUERT H., nAnalytischen Faserungen tiber holomorf-vollstandigen Raumen ll , Math. Ann. , 135, 3, 263-273.

7.

HANO J. , nOn Kahlerian homogeneous spaces of unimodular Lie groups", Amer. J. Math., 1957 , 79, 4, 885-900.

8.

KOBAYASHI S. , NOMIZU K. , "On automorphisms of a KHhlerian structure II , Nagoya Math. J., 1957 , II, 115-124.

9.

KOSZUL J. L. , II Sur la forme hermitienne canonique des espaces homogenes complexes", Canad. Journ. Math. 7,4, 1955, 562-576.

10.

KOSZUL J. L. , "Ouverts convexes des espaces affines ll , Math. Zeitschr. 79, 1962, 254-259.

11.

LICHNEROWICZ A. , "Espaces homogenes kHhleriennes", Colloque de geometrie differentielle", Strasbourg, 1953, 171-184.

12.

LICHNEROWICZ A. , "Sur les groupes d'automorfismes de certaines variHes KHhleriennes II , C. r. Acad. Sci. Paris, 1954, 239, 21 , 1344-1346.

13.

LICHNEROWICZ A. , IITheorie globale des connexions et des groups d'holonomie", Rome, 1955.

14.

MATSUSHIMA Y. Sur les espaces homogenes KHhleriens dlun groupe de Lie reductif" , Nagoya Math. J. , 1957, 53-60.

- 87 -

15.

16.

POINCARE H. , nLes fonctions analytiques de deux variables et la representation conformeD, Rend. Circolo Mat. di Palermo, 1907, 23, 185.220.

, " PJATECCKII,SAPIRO I. I. , nOn a problem proposed by E. Cartan", Dokl. Akad. Nauk. SSSR. 113 (1957) , 980-983 (Russian) • v

·v

17.

PJATECCKII, SAPIRO I•• , IIGeometry of classical domains and theory of automorphic functionsl!, Moscow 1961 {Russian~ (an enlarged english edition is being prepared for print·) .

18.

PJATECCKII, SAPIRO 1. I., DClassification of bounded homogeneous domains in n-dimensional complex space n, aokl. Akad. Nauk. SSSR, 141, 2 (1961) , 316-319 (Russian) = Soviet Math. Dokl. 2(1961), 1460-1463.

19.

PJATECCKII, SAPIRO 1.1. ,liOn bounded homogeneous domains in an n-dimensional complex space", Izv. Akad. Nauk SSSR, Ser. Mat. 26 (1962), 107 - 124 (Russian) .

20.

PJATECCKII, SAPIRO 1. I. , The structure of 26 (1962) , 453-484.

21.

PJATECCKII," SAPIRO 1. I, nThe geometry and classification of bounded homogeneous domains n, Uspehi Mat~ Nauk. 20 (1965), no. 2, 3 - 51 (Russian) = Russian Math. Surveys, 20 (1965) , no. 2,1 -48.

22.

TITS J. , nEspaces homogenes complexes compacts n , Comm. Math. Relv., .1962, 37, 2, 111 - 120 •

23.

VINBERG E. B., 'The Morozov - Borel theorem for real Lie groupsn , Dokl. Akad. Nauk. SSR 141, 2 (1961), 270-273 (Russian - Soviet Math. Dokl. ~ (1961), 1416 - 1419 •

24.

VINBERG E. B., liThe theory of convex homogeneous cones n , Trady Moskov. Math. Obshch. 12 (1963), 303-358 (Russian) = Trans. Moscow Math. Soc. 13 (1964), 340-403.

25.

VINBERGE.B., GINDIKINS.G., PJATECCKII, SAPIROI.I.,"Onthe classification and canonical realization of complex homogeneous bounded domains', ibd., 359-388 (Russian) = ibd. , 404-437.

26.

WANG H. C. ,nClosed manifolds with homogeneous complex structure" Amer. J. Math. , 1954, 76, I, 1-32.

27.

WElL A. , "Introduction 1958.

v

.. ,

v

~

v'

j algebras",

ibd.

"

a l'etude

des

varieh~s

kllhlei'iennes n , Paris,

CENTRO INTERNAZIONALE MA TEMA TICO ESTIVO (C.L M.E.)

Stephen J. GREENFIELD

EXTENDIBILITY PROPER TIES OF REAL SUB MANIFOLDS OF ([n


EXTENDIBILITY PROPER TIES OF REAL SUB MANIFOLDS OF en by Stephen J. Greenfield (Massachusetts Institute of Technology Cambridge, Mass) A

History In 1906 F. Hartogs [4] discovered that

,,2

a neighborhood of the bicyc1inder in

\I..

a function analytic in

could always be extended to an

analytic function defined in a neighborhood of all the bicyclinder. Not much later h: (

(1910) E. E.Levi [6] 2

found a local analogue of Hartogs' result: let

--;.. R be differentiable and suppose that

M

=h

-1

(0) is a submanifold

of (2 . If h (1)

det

h h

at

then

p €M

-

h

-

h

zl z l zl z 2 z

h 1

-

z2 z 1

hzl

z2 z 2

hz2

f

0

0

z2

functions analytic in any neighborhood of M extend to

be analytic in a fixed open set on one side of M . The boundary of the fixed open set includes a neighborhood of

p

in

M

In the early 1940' s, Bochner and Martinelli

that if

11

M is

of

.

[21

[9] showed

a compact differentiable hypersurface bounding an open set

([n, then any function analytic in a neithborhood of M has an

extension analytic in

71..

(Their result was actually better, for they re~

quired only that the function be defined on

M and satisfy certain appro-

priate partial differential equations.) In 1960, H. Lewy nal manifold

in

[3

[8]

published an example of a four-dimensio-

having the property that functions analytic in a

neighborhood of it extend to be analytic in preceding, Lewy just needed the function satisfying

" T (see

to require - if

N is a compact

D) cannot he. globally imbedded.)

REFERENCES

1. Bishop, E. , "Differentiable manifolds in complex Euclidean space", Duke Math. J. n (1965), 1-22 2. Bochner, S., "Analutic and meromorphic continuation by means of Green's formula", Ann. Math. 39 (1938), 14-19 . 3.

Greenfield, S., Cauchy-Riemann Equations in Several Variables (Bradeis Univ. thesis, 19S7).

4.

Hartogs, F. , "Einlge Folgerungen aus Cauchyschen Intergra1formel bei Funktionen mehrer Ver'anderlichen"Silzb Munchener Akad. , 36 (1906) , 223.

5. Kohn, J.J., "Boundaries of complex manifolds", Proceedings of the Conference on Complex Analysis (Springer-Verlag New York Inc. , 1965). 6. Levi, E. E. , "Studii sui punti singolari essenziali delle funzioni di due o pili variabili complesse", Annali di Mat. Pura ed appl., 3(1910) 61-87 . 7. Lewy, H., "On the local character of the solution of an atypical linear differential equation in three variables and a related theorem for regular functions of two complex variables", Ann. Math., 64(1956), 514-522. 8. ------, "On h,ulls of holomorphy", Comm. Pure Appl. Math. , 13(1960), 587-591.

9. Martinelli, E., "Alcuni teoremi integrali per Ie funzioni analitiche di pili variabili complesse" , Rend. Accad. Italia, 9(1939), 269-300; 10. Newlander, A., and Nirenberg, L. , "Complex analytic coordinates in almost complex manifolds", Ann. Math. , 65(1957), 391-404 . 11. Niremberg, L., "A complex Frobenius theorem", Seminars on Analytic Functions (Institute for Advanced Study- United States Air Force Office of Scientific Research, 1957) . 12. Rossi, H. , report to appear in the Proceedings of the international Congress of Mathematicians (Moscow, 1966). 13. Weinstock, B. , On Holomorphic Extension from Real Submanifolds of Complex Euclidean Space (M.1. T. thesis, 1966). 14. Wells, R. O. , "On the local holomorphic hull of a real submanifold in several complex variables", Comm. Pure Appl. Math. , 19(1966, 145- 165.

- 108 -

15. ----------, "Holomorphic approximation on real-analytic submanifolds of a complex manifold", Proc. A. M. S. , 17(1966), 1272-1275. 16. ----------, "Holomorphic hulls and holomorphic convexity of differentiable submanifolds'~, to appear Trans. A. M. S.

CENTRO INTERNAZIONALE MA TEMA TICO ESTIVO

(C. 1. M. E.)

W. KAUP

HOLOMORPHE ABBILDUNGEN IN HYPERBOLISCHE RAUME


HOLOMORPHE ABBILDUNGEN IN HYPERBOLISCHE RAUME. W. Kaup ( ERLANGEN) Es sei (vergl.

7 0 , so dass

alle

der

Z€ (x) relativ-kompakt in X liegt und y

f

erzeugt, Es sei nun

ZARISK1-offen, Wir du'rfen annehmen, dass

x gilt, Daraus folgt , dass ~X X

normal und Y C X

Y jede Komponente von

X

sehneidet, d. h, jede besehrankte holomorphe Funktion auf Y

ist eindeutig

auf

X

holomorph fortsetzbar, Wegen Bemerkung 3

- 114 -

V, Kaup

gilt somit

cXIY" c Y (was iibrigens mit dem Maximumprinzip auch direkt

folgt) , Andererseits ist fur jedes offene zusammenliimgende U nY

die Menge

Es seien Y C X komplexe

R'aume und f: Y

Abbildung, Notwendig daftir, dass f

T:

X -+ Z gestattet, ist

9y

I

~X Y "

zusammenh'angend, d, h,

~

U C X auch

,

Z eine holomorphe

eine holomorphe Fortsetzung

- falls Y dicht

in

X liegt

- die folgende Be-

dingung: ( l\'t)

Zu jedem

x e:: X gibt

(Yn ) in Y,

so

dass

es eine gegen x konvergente Punktfolge

die Bildfolge (fy ) in Z einen Ifaufungspunkt 11

besitzt, 1: allgemeinen ist Satz 2 : Es sei

diese Bedingung nicht hinreichend, es gilt

X ein normaler komplexer Raum und

Kl- offene Teilmenge, 1st und

f: Y ---. Z

so existiert fur Beweis : Sei eine Folge Sei

einer

Y C X eine ZAR1S-

Zein beschrankt K-vollst'andiger komplexer Raum

eine holomorphe Abbildung, die die Bedingung (.) erIullt, f

genau eine holomorphe Fortsetzung

x ein

beliebiger Punkt in X, Wir

(y) in Y mit

lim

X"

n

V C Z eine

pakt ist

jedoch:

(-Kugel

y

V

von

X ~ Z,

wahlen entsprechend (_)

und q:" lim f(y) Eo Z ,

n

n

(bez, f Z) um

und eine Umgebung

T:

q

so

klein,

biholomorph

dass

V kom-

aquivalent ist zu

beschr'ankten lokal-analytischen Menge in einem Cm, Wegen Satz 1

existiert eine Umgebung U von

x

mit

Riemannschen Fortsetzungssatz ist f bar ,Diese Fortsetzung Die Bedingung ne Iftille fry) C Z

ist

in (*) ist wegen Satz 1 (4) wegen Satz bezuglich

1 (1) ist dann ~

nY) C

in den Punkt

V, d, h, nach dem x holomorph fortsetz-

wegen ( ~) naturlich eindeutig,

(lit) ist

vollsfandig

f(U

z, B, dann eritillt, wenn beziiglich

die abgeschlosse-

S' Z ist, Denn die Folge (y n)

eine CAUCHY-Folge bezuglich auch

die

Q,e,d,

Bildfolge

~Y'

und

(fy) eine CAUCHY-Folge n

Z' Insgesamt erhalten wir daraus

Satz 3: Es seien

Y C X wie in Satz 2 und Zein komplexer Raum, fUr den

- 115 -

W. Kaup

'?z

eine vollst'andige Metrikist

und

homogen

§lZ

T:

Mannigfaltigkeit) ist, ist

~Z

Z ; dann

kann

Radius

z

folgt sodann, Z

liegt,

d. h. Es

und

o

o dass

wenn

r

in

Z

jede Kugel yom

Z

erzeugt. Es

liegt. Aus der

Radius

Romogeniiat

< r relativ-kompkt in

jetzt

jedem

komplexen

Raum

'r:X ~ X

von

X

und

~X

setzen -1

X

wir

eine weitere ste-

die universelle

fur

alle

x, y ~ X

-1

('t' (x), 't' (y))

dann

Satz 4 : (1)' rT ist eine invariante Pseudometrik 1st die universelle Uberlagerung so ist (3)

Automorphismen von

ist insbesondere vollstandig.

~Z

(jX(x,y) :"

(2)

O"x

eine

auf ~ ;

X von X beschr'ankt K-vollsiandig,

Metrik, die die

Topologie von X erzeugt;

1st jede beschranlde holomorphe Funktion ·auf

X

konstant (z. B. wenn

X eine zusammenh'angende komplexe Liegruppe ist) , so ist (4)

K-voll-

> 0 , so daBs die Kugel mit

tige Pseudometrik zugeordnet. Dazu betrachten

dafUr gilt

beschrankt

Topologie von

ein

relativ-kompakt

werde

Uberlagerung

Z

ist Z automatisch eine komplexe

eine Metrik, die die Punkt z E: Z

urn

zu

folgendermassen eingesehen werden: Wegen

gibt also einen r

ist,

(d.h, die Gruppe Aut (Z) aller

operiert transitiv auf

Satz 1

beschr'anktK-vollst'andig

X -+ Z fortgesetzt werden.

eine vollstandige Metrik

standig und homogen Z

Z

ist). Dann kann jede holomorphe Abbilpung f: Y -+ Z

einer holomorphen Abbildung Dass

(z.B. wenn

CTX :;"X :)

(5)

Fur X" E

(6)

1st

C

x gilt

X normal

und

vX~

"S~X

konvergiert .

eine

=

~

;

(l"E" S'E " c E " D ; und

Y

C

X

endlicher Fundamentalgruppe reits dann

G"'x

eine

ZARISKl-offene Teilmenge mit

71'1 (Y), so ist eine

CA DCRY -Folge beziiglich

Folge (y ) in Y be-

n (j y' wenn sie in

X

- 116 -

W. Kaup

Beweis : Jede Decktransformation (d, i, ein Automorphismus g mit

gilt

X

t

xt

wobei

-1

le('l

-1

die

Topologie von

entsprechenden Eigenschaft Abbildung

von

~

von (j

und der

folgt

Tatsache,

f: X -+ Y eine holomorphe

X beschrankt

einfach

dass

aus der

zu jeder ho-

Abbildung

Y mit kommutativen Diagramm

~

~

y

X~

Y

X

~

1;".j, f eine

X

existiert ,

zusammenh'angende komplexe Liegruppe, so ist

eine komplexe Liegruppe, und wegen rametriger Untergruppen Funktion auf

C --., X

X konstant, (5) ist

schen Lemmas,

der

w:

einer

(i. a, verzweigten) analytischen

setzt

werden, wobei

"X

ein

der

und C' A. auf Y

X

X, so kann

gefunden

werden

Y iiberein,

7f

Uberlagerung komplexer

ZARISK1-offene Teilmenge enth'alt (vergl. '"

des Schwarz-

Nachweis von (6) : Die

kann wegen

normaler

X

beschr'ankte holomorphe

eine einfache Konsequenz

Y~Y

auch

Existenz geniigend vieler einpa-

ist jede

und es fehlt nur noch

universelle Uberlagerung

Y als

X erzeugt, wenn

Die Invarianz

K-vollstandig ist;

f: X

(x, ty) ,

(x), y ~ 't' (y) zwei belie big gew'ahltePunkte sind, und aus

(i) folgt, dass

lomorphen

S X'"

= inf

(5' (x, y)

1st

X, Deshalb

operiert eigentlich diskontinuierlich auf

auch

(i)

€.

X

'f= 't g) ist eine Isometrie bezuglich ~ X ' und die Gruppe", r aller

Dec]{transformationen

(j

von

(Y) endlich zu 1 1\ 'f: X ~ X fortgeRaum

(9] ) , Also

ist,

der

stimmen

1st nun (y n) eine Folge in Y mit lim "

Folge (Y ) in mit 'f y = y und lim y €' X ",n n n n Nun ist yn eine CA U CRY - Folge beziiglich 0' und

Y

eine

X

damit auch bezuglich IJ '" , d, h, die Bildfolge y ist eine CA UCRY -Folge

Y

bez'tiglich Vy .

Yn

n

Q.e,d,

-117 -

W,Kaup

Als Anwendung l'asst sich zeigen Satz,5.:Es sei X ein normaler kOIIlplex.e.r Raum und N ex eine analytische Teilmenge, die

folgender Bedingung genugt :

(* *) Zu jedem

x € N gibt es eine zusammenhangende Umgebung

x

in X, so dass die' Fundamentalgruppe 71'1 (U - N) er:v;ilich ist,

Dann gilt

fUrY: = X - N und jeden komplexen Raum

Ie Uberlagerung beschrankt K-vollstandig ist: f: Y

~

U von

Z, die die Bedingung ( .... ) von Seite

Z,

dessen universel-

Jede holomorphe Abbildung erfilllt, ist zu einer holo-

morphen Abbildung X ~Z fortsetzbar , Beweis :Wir d'Ltrfen

w;. (Y)

eine Punktfolge in

Y mit

CAUCHY-Folge beziiglich lich der Metrik

O"z .

endlich annehmen , Sei dann lim

y

n

= x, Wegen Satz

cry, d,h, (f(Yn)) ist

x

~

N und (y n)

4 (6) ist (y) eine n

eine CAUCHY-Folge bezug-

Wegen ( .. ) durfen wir annehmen, dass die Folge

fry ) wenigstens einen Ha'ufungspunkt besitzt, d, h, die Folge f(y) ist bereits n n in Z konvergent,. Q,e.d, 1st itir

X eine komplexe Mannigfaltigkeit, so ist die Bedingung (!II. *)

N z, B, dann

erfUllt,

wenn N eine Codimension

> 1 in X

besitzt, Nen'nen wir jetzt einen komplexen Raum X K-hyperbolisch, wenn

crX

eine vollst'andige

Satz 5

Metrik auf

X ist, so liefert

der Beweis von

speziell

Satz 5': Es seien X, Y und N

wie in Satz 5, 1st Zein K-hyperbolischer

komplexer Raum, so ist jede holomorphe Abbildung f: Y holomorphen Abbildung Bevor solI

--+

Z zu einer

X -+ Z fortsetzbar ,

wir auf die Notwendigkeit der Bedingung (fit lit) eingehen,

der Begriff "K-hyper;bolisch"

n'aher untersucht werden: Da jeder

lokal-kompakte metrische' Raum vollst'andig ist, auf dem eine Gruppe von lsometrien transitiv operiert (vergl. Satz 3), gilt zun'achst : Bemerkung 4:

Jede homogene komplexe Mannigfaltigkeit, deren universel-

Ie Dberlagerung beschr'ankt K-vollstandig

ist, ist K-hyperbolisch.

- 118 -

W. Kaup

Durch einfache Rechnung folgt weiter Bemerkung 5 : Ein komplexer Raum

-

X 1st genau dann K~hyperbolisc\1 wenn

---------~---~--.---......-------.---

die universelle tJberlagerung K~hyperbolisch ist. Speziell ist also ein komplexer Raum K~hyperbolisch, wenn er Uberlagerung eines kompakten Raumes ist und eine beschrankt K~vollsr:1ndige Dberlagerung bes itzt, Da die hyperbolischen Riemannschen Fl'achen gerade den homogenen Einheitskreis als universelle Uberlagerung haben, sind also speziell alle hyperbolischen Riemannschen FEichen K-hyperbolisch, und der Begriff

1I~.::hyperbolisc_h~1

erfahrt dadurch eine gewisse Rechtfertigung, Weiter

gilt Beme£kung 6 : Sind X ~

XY

und

Y K-hyperbolisch, so auch das direkte Produkt

und jede analytische Teilmenge A eX,

Bemerkung 7 :Ist X

ein

K-hyperbo~ischer

komplexer Raum und N C X

die Nullstellenmenge ein..:'E.2-2lf

X

so

X - N K-hyperbolisch,

ist auch

das Komplement

Beweis : Es sei (jy'

giert

Dann

ist

beschr'ankten holomorphen Funktion,

Y: = X - N und (y ) auch n

somit gegen ein

x

eine

(y ) eine CAUCHY-Folge in Y bezliglich n

CAUCHY -Folge bez'ciglich

E X , Nach

O"x

und konver-

Voraussetzung existiert eine holo-

morphe Abbildung f : X

-+ Emit

f -1 (0) = N ,E * : = {tEE : t

f

ist K-hvperbolisch . f

bildet

in

CAUCHY-

Folge beziiglich

'J" E

* ' d, h,

Y

f(x) C

E

.. ab,

E*

d. h, f(y ) ist eine n

und somit

x

e

y, Also ist U"'y

vollst'a'ndig,

Q,e. d,

Jedem komplexen Raum X lisierung

X .... zusammen

Abbildung

0}

~-: X lI

komplexer Raum,

-

mit

ist

in eindeutiger Weise die Norma-

einer diskreten eigentlichen holomorphen

X zugeordnet

(vergl. l~1), X· ist ein normaler

und man zeigt leicht

Bemerkung~: Mit X ist auch die Normalisierung Xi( K-hyperbolisch. Betrachten

wir nun fur

jedes

n >0

das

folgende

((lJ)

- 119 -

W,Kaup

:Es

Bei~pi~l

faltigkeit

sei

der

A

eine kompakte

Dimension

projektiven Raum

komplexe Mannig-

n, die singularit'atenfrei in einen komplex-

P N eingebettet

von kompakten hyp,erbolischen netes. N stets

K-hyperbolische

sei

(ist z, B, A

Riemannschen

erreichbar), A bestimmt im

ein direktes Produkt

Frachen, so ist das f'ur geeig_ N+1 Vektorraum C einen, N+1

analytischen Kegel X, dessen einzige Singularit'at der Nullpunkt 0 C C ist.

{o}

Y: = X -

ist also

eine komplexe Mannigfaltigkeit, und man hat

eine holomorphe Abbildung f : Y -+ A, die nicht holomorph fortsetzbar ist, da f(U Bedingung

n Y)

(* ,) ist also

=A in

auf

f'ur jede Umgebung U von

Satz

5 wie in

Satz

X

0 gilt, Die

51 notwendig und kann

nicht durch eine Bedingung an die Codimension von N

ersetzt werden,

Fur komplexe Mannigfaltigkeiten ist die Bedingung (tHi) jedoch vermutlich

(10J ) .

'uberfliissig (vergl. Es sei

X

ein

..

komplexer Raum, dessen universelle Uberlagerung

X beschr'ankt K-vollstandig ist, Dann ist die Gruppe morphismen von trik

O"x

(1 il

).

X als Gruppe aller biholomorphen Isometrien der Me-

eine reelle Liegruppe,

die eigentlich auf

X

operiert (vergl.

eX) die Gruppe aller Decktransformationen, so Aut(X) ~ N( l")/ r, wobei N(,,) = {g E Aut(X): gr= r g}

1st'

bekanntlich

r

Aut (X) aller Auto-

Normalisator

C

Aut

von

Automorphismen

r mit

in

Aut(X) ist. Versehen wir

ist der

jede Gruppe von

der KO-Topologie (=Kompakt-Offen-Topologie), so

gilt Satz 6 : Es

sei

komplexer Raum menge

X ein und

Y

zusammenhangender normaler K-hyperbolischer C

X das Komplement einer analytischen Teil-

N ex, die die Bedingung

(iA,4jl') aus Satz

5 erfullt, Dann liefert

die Zuordnung g ..... g \ Y einen topologischen Isomorphismus der pe {g 6 Aut (X) : g(N) = N~ Y nicht

auf die Gruppe Aut(Y) . 1st

homogen sein, und ist N zus'atzlich

kompakt, 1st

Grup-

N 1= ~, so kann

kompakt, so ist auch Aut(Y)

N die Singularitatenmenge von X, so gilt Aut(X) = Aut(Y) .

- 120 -

W,Kaup

Be~eis

: Die Beschr'ankungsabbildung

'fJ

von G:=

h

e: Aut(X): g(N)

=NJ

in Aut(Y) ist injektiv und stetig , Wegen Satz 5 I ist If- bijektiv , Da G und Aut (Y)

Liegruppen

mit abz'ahlbarer Topologie sind, ist If' ein topolo-

gischer

Isomorphismus, Die folgenden Aussagen ergeben sich unmittelbar

aus der Tatsache, dass X

operiert (

unter

[I:D)

G = Aut (Y) als lsometriengruppe eigentlich auf

bzw, dass

die Singularifatenmenge von X invariant

Aut(X) ist ,

Daraus ergibt sich nun Satz 7 : Fur jeden kompakten K-hyperbolischen komplexen Raum X ist die

==e --

Automorphismengruppe Aut(X) endlich, Beweis: fiir

Nach

einem

Satz, der von BOCHNER und MONTGOMERY (

kompakte komplexe Mannigfaltigke iten und von KERNER ([14)) Ilir

kompakte komplexe R'aume bewiesen worden ist, ist

Aut(X) eine komple-

xe Liegruppe, die holomorph auf X operiert. 1st G die

1-Komponente

von Aut(X), so ist also speziell fur jedes

durch

finierte Abbildung so

(3J )

ist

G=

peebenfalls Da auf

i II

G -+ X

holomorph und wegen Satz

,Da X

kompakt

x E X die

([13]

kompakt ist, ist

Aut(X)

g~

gx de-

4 (3) konstant, Alals

lsometriengrup-

) und deshalb endlich,

einem kompakten komplexen Raum jedes holomorphe Vektorfeld

integrierbar ist

([121), ergibt sich insbesondere die

Folgerung:

X

1st

ein

so existiert ausser D

kompakter

o kein

K-hyperbolischer komplexer Raum,

holomorphes Vektorfeld auf

X ,

Wir wollen den Begriff "K-hyperbolisch" noch etwas erweitern und setzen Definition: wenn auf

Ein zusammenh'angender komplexer Raum X heisst hyperbolisch, X

eine vollst'andige stetige d(fw,fz)

Iur

alle w, z £ E

heisst

und

hyperbolisch,

Metl'ik ~

d

existiert

mit

D(w,z)

fe Ho 1 (E, X), Ein beliebiger komplexer Raum wenn jede Zusammenhangskomponente hypel'boli.sch ist

Offensichtlich nangt

diese Definition nicht ab von del' speziellen

- 121 -

W.Kaup

Wahl der

invarianten

Metrik

D auf

zusammenh'angenden komplexen Raum so dass

E. Betrachten wir nun auf Y

die

jedem

gr'osste Pseudometrik ky'

stets ky(fw, fz) ~ D(w, z)

gilt, Diese

existiert;

man setze n'amlich 11 k (x, y) : = inf Y

wobei inf

tiber alle endlichen Teilmengen

strecken ist, ren mit ky auch

Ek=1

fur die holomorphe

fl (z) ::: x, fn(zn)::: y

und

wir

nun mit

70 , so dass die Iterierte g = f eine Retraktion ist (d. h, g = g) . Insbesondere ist also jede surjektive holomorphe Abbildung f: Y --t Y ein Automorphismus endlicher Ordnun,s von Y ,

- 123 -

W.Kaup LITERATUR ~11 ANDREOTTI, A. and W. STOLL: Extension of holomorphic maps, Ann. of Math. (2) 7~, 312 - 349 (1960) .

BEHNKE, H. u. P. THULLEN : Theorie del' Funktionen mehrerer komplexer Ver·anderlichen. Erg. d . Math.·~, Berlin: Springer 1934 BOCHNER; S. a. D. MONTGOMERY: Groups on analytic manifolds. Ann. of Math. ~, 659-669 (1947). BOREL, A. a. R. NARASIMHAN: Uniqueness Conditions for Certain Holomorphic Mappings. Inventiones math. ~ , 247-255(1967) . CARATHEODORY, C. : UBER das Schwarzsche Lemma bei analytischen Funktionen von zwei komplexen Ver·anderlichen. Math. Ann. ~,

[6J

76-98 (1927) .

DOUADY, A. : Le probleme des modules pour les sous-espaces analytiques compacts d lun espace analytique donne . Ann. Inst. Fourier 16

,1 - 95 (1!'66) .

GRA UER T, H. : Ein Theorem der analytischen Gacbentheorie. Publ. Math. ~ , 233-292 (1960).

[8] -

u. H. RECKZIEGEL : Hermitesche Metriken und normale Familien holomorpher Abbildungen . Math. Zeitschr. 89, 108 - 125 (1965)

[9J - , u. R. REMMERT: Komplexe Riwme. Math. Ann .

..!1~, 245-318 (1958)

'[10 HUBER,

H. : Uber analytische Abbildungen Riemannscher Frachen in sleh. Comment. Math. Helv. ~, 1-72 (1953)

[1 UKA UP, W : Endlichkeitss'atze

fur Systeme holomorpher Abbildungen in hyperbolische R"aume. In Vorbereitung

[r ~ KA UP,

W. : Infinitesimale Transformationsgruppen komplexer Raume. Math. Ann. ~, 72 - 92 (1965) .

[131 KA UP,

W.: Reelle Transformationsgruppen und invariante Metriken auf komplexen Raumen. Inventiones math. ~, 43 - 70 (1967) .

[r~ KERNER, H.: Uber die Automorphismengruppen kompakter komplexer R'aume. Arch. Math.

Q,

282 - 288 (1960).

[1~ KOBAYASHI, S. : Intrinsic metrics on complex manifolds. Bull. Amer. Math. Soc.

E '

347-349 (1967) .

(16) REIFFEN, H. J. : Die Caratheodorysche Distanz und ihre zugehorige Differentialmetrik. Math. Ann. !..~, 315-324 (1965)

(1 ~ REMMERT,

R. : Holomorphe und meromorphe Abbildungen komplexer R"aume. Math. Ann. ~, 328-370 (1957)

CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C.I.M.E.)

A. KORANYI

"HOLOMORPHIC AND-HARMONIC FUNCTIONS ON BOUNDED SYMMETRIC DOMAINS"


- 127 -

HOLOMORPHIC AND HARMONIC FUNCTIONS ON BOUNDED SYMMETRIC DOMAINS by

Yeshiva University (New York)

The main purpose of these lectures is to study questions of elementary analysis on bounded symmetric domains, namely the realization of these domains as generalizations of the unit disc and the upper halfplane, the study of the structure of their

boundary and

the boundary behaviour of holomorphic functions. This is done in sections

3 to

6 which contain material otherwise available only

in journals (mainly

~.4]

, [161

and [24] ). Some slight simplifications

and improvements have been made here; it will, by the way, be apparent that the subject still has plenty of open problems. A second purpose is to make all this more accessible to the analyst who is not an expert in Riemannian geometry. Our study uses parts of the theory of Riemannian symmetric spaces. For this standard treatise is that of Helgason

[8J ; another

very clear and con-

cise introduction can be found in chapters 1, 2, 8 of the book of Wolf

02J.

These books contain all that is needed here, but they also

contain much more. In our section I we outline how the part of the theory actually needed can be built up with minimum effort. In particular, we do not need locally symmetric spaces and we can avoid most of the topological difficulties.

In this section most proofs are

omitted, but they should be easy to fill in with the help of the references. Section

2 is a slight rearrangement of material contained in

[8J.

- 128 -

A. Koranyi

Because it is the basis of all that comes later, we present this material with proofs.

(This does not mean all proofs; here and also

later some the proofs, preferably the lengthy and uninstructive ones, will be omitted. In these cases easily traceable references will be given). The prerequisites for (i)

the reading of these notes are the following.

The fundamental facts about Lie groups and their homo-

[51

Ch. IV, or , in a nutshell,

[22J

geneous spaces

([8] Ch. II, 'or

Ch. I sec. 5).

For this, in turn, . one needs the basic definitions

about differentiable manifolds (same references) and some facts about covering spaces and covering groups (

[5]

[22J

Ch. I sec. 8, or

Ch. II; a good reference for all the above is also

[ID ).

The fundamentals of the theory of semisimple Lie groups

(ii)

and Lie algebras. (Best exposition' for our purpose in Also to be found in (iii)

Li 1] , [13],

[8]

Ch. III.

etc.)

Some of the elementary facts of Riemanpian geometry:

The existence of geodesics, normal neighborhoods, the notion of completeness.

[8]

and

b2].).

(A few pages in any book on the subject; also in

- 129 -

A. Koranyi

1.

Preliminaries.

Generalities on complex domains. mean a connected open subset main and

f: D-+- D'

we say that

f

G(D}.

of

C.

By a domain we

D'

If

is another do-

is holomorphic with holomorphic inverse,

is an isomorphism.

automorphism of denoted

D

n

D

D' = D ,f

If

The automorphisms

0:

D is said to be homogeneous

is called an

D form a group, if

G(D}

is tran-

sitive on D . If main, then

D is bounded, or isomorphic with a bounded do-

D has an invariant Hermitian metric, the Bergman

metric. More precisely, to everyz E. D there is attached a po-

i gjk(Z}}

sitive definite Hermitian matrix

(j, k = 1, ••. , n) ; the

length of a tangent vector

v

~

)'

= L.J (a

c)

j~ J

is defined by

L

)v/ ~.

- J

+a.---..,J (j z.

J

gjk(z) a j a k

vector can be written in this form);

if

f: D --;,. D'

e. g.

[8] p.293-300).

We denote by D

is an isomor-

I(df}v ID' = Iv ID

phism, then it is also an isometry, i. e. (For the proofs see

(every real tangent

I(D}

the group of all isometries of

as a Riemannian space, and give

By the Mayers-Steenrod theorem

I(D}

I(D}

the compact-open topology.

is a Lie group (a proof in

- 130 -

A. Koranyi

a special case, still sufficiently general for r 1 p. L8 j

170-172).

closed subgroup of

our later use, is in

By the theorem of Weierstrass,

I(D);

G(D)

in particular it is a Lie group.

is

a

- 131 -

A. Koranyi

Symmetric domains and symmetric spaces

Definition:

A bounded domain

D

is symmetric

if for every zeD there exists s C'G(D) such that (i) z -1 and (ii) z is an isolated fixed point of s . s s z z z If we equip D with its Bergman metric, it becomes a member of the following class of spaces: ~efinition:

symmetric if for every M

onto itself such that

of

s. P

A connected Riemannian space

of

L

n

(complex) di-

mensions we can talk about its underlying real manifold

point

is

s of PEM there exists an isometry p -1 s sand p is an isolated fixed point p P

Given a complex manifold

mensional).

M

Using a local coordinate system

PEL, every (real) tangent vector at

zl' ...• ' zn

p

(2n-dl

M

at a

can be written

in

the form

)'

J

v = LJ(a.~ J u Z. J

)

( j = 1, ... , n )

The mapping

a.--7'- ia. J J transformation J on M

+

.

induces a (real) linear

(for each p )

which is easily seen to 2 be independent of the coordinate system and is such that J = -I. P In this situation we say that J is a complex structure on M; p

L fold

P

is reconstructible from with complex structure

tiable map

f: M---7M'

M

and

J.

If

M'

is another mani-

J', it is immediate that a differen-

is holomorphic if and only if

dfoJ

=

J'o df

- 132 -

A. Koranyi

at every point of

(Cauchy - Riemann equations).

M

We say that

is a Hermitian space if

M

Riemannian space with a complex structure for every vector

J

such

M

is a

I

that

\ Jvl = v \

v. Clearly every domain

with Bergman metric is a

Hermitian space; every bounded symmetric domain is a Hermitian symmetric space, a notion defined as follows. Definition: mitian space phic isometry point of

A Hermitian symmetric space

M such that for sp

every -1

such that

sp = sp

P

Eo

and

is a Her-

M there exists a holomorp

is an isolated fixed

sp' In order to study bounded symmetric domains one looks at

Riemannian symmetric spaces first, then singles out the Hermitian ones, and finally the bounded domains.

Riemannian symmetric spaces mannian symmetric, let

r

transformation on Hence if small

M; p

Let

M

be Rie-

then since

is a geodesic with

t, at the moment).

r p

(ds) is an involutive linear pp is an isolated fixed point, (dSp)p=-I.

O) = p,

then

It is immediate that

sp r(t) =

r(

-t)

(for

Tt = s ((t/2)s r(O)

,

is a one-parameter (local) group of isometries, called transvections based at rallel

p, which

has

r

as one

translation of vectors along

of its orbits, (and induces pa-

y).

Now a simple process of

"continuatfon" shows that every geodesic can be continued indefinitely, so

M is complete.

- 133 -

A. Koranyi

Let

p, q E. M.

sic segment joining them; carries

p

to

q

This shows that on

let

r

By completenes s there is a geodebe its midpoint.

Then

g

=

s s

q r

and is on a one-parameter group of transvections. I (M) (the identity component of I(M)) is transitive

o

M. (These remarks show that Riemannian or Hermitian

symmetric spaces and bounded symmetric domains may be defined in new equivalent ways.

We formulate the new definition here for

domains: A bounded domain if it is homogeneous and for some

such that

s

-1

and

s

Let We fix a point (i. e.

K =

0 If:.

1g E

subgroup of

Z

is symmetric

point

if and only

to D there exists

Z0

is an isolated fixed point of

M,

and

and

denote by

J).

s t:: G(D)

s.)

M be Riemannian symmetric and let

GIg

G,

o

D

G

I (M). o K the isotropy group at 0 =

o =

0

M

can be identified with the coset space

K

is compact, contains no normal

G/K. g

~

s og So

is an involutive automorphism of

G;

it is easy to see that its fixed point set has the same identity component

as

K.

{!

of

It induces an involutive automorphism

G.

Denoting by ~

The fixed point set of G' is the

(-1) -eigenspace of

as a vector space direct sum.

(5'

(Q

k, ,

exp t Y}

that

with

G=KP =P K

Ye .

fl·

the Lie algebra of

we have

K

!if =n + ce

It is now not hard to check that the

one -parameter groups of transvections based at

t

of· the Lie algebra

Denoting

P =

0

1exp

are exactly the

Y lYE

~J

it follows

- 134 -

A. Koranyi

r

It is also easy to see that d

1/ ( Y ) = dt

(exp t Y ) • Oft = 0

commuting with the action of for all to

11'

k 6 K).

by

P , we

dratic form

Q

K

defined by

:~~Mo

is a vector space isomorphism

(i. e.

'If

ad(k) 115' = (dk) 0

0

f

~

Transporting the Riemannian structure from get an

M

o - invariant positive definite qua-

ad(K)

oni? Definition : An ortogonal involutive Lie algebra (oiLa)

is a triple

(f!, 6',

f!

(i)

such that

Q)

is a real Lie algebra

(ii) G' is an involutive automorphism of always denote its eigenspac~5 for then have

[If)

1f'J C 1e

JB

(iv)

Q

quadratic form on

and

-1

[f( 11{] C?

I

(iii)

1

contains nc, is an

by

Ie

#".

and

(We shall

ff;

we

[1f, ~?] ctt)~

I

non- zero ideal of

-;;!'

ad(;& ) -invariant positive definite

%" .

We have associated an oiLa to every Riemannian symmetric space. Conversely, given any oiLa reconstruct a

simply connected

K

G

corresponding to

define a Riemannian metric on

we can always

R.iemannian symmetric space from

it by taking the simply connected group the analytic subgroup

(:f/,6') Q)

·C/K = M.

but this causes no difficulty. Details are

with Lie algebra

18 '

and using

(It may be that

ff7, Q to

Gf

I (M), o

e. g. in [22J p. 242).

In general there are several symmetric spaces corresponding to an oiLa of

(since non-connected

groups may sometimes be used instead

K), but in the cases that interest us it will turn out that the

spaces are automatically simply connected.

- 135 -

A. Koranyi

Orthogonal involutive Lie algebras. is called Euclidean

[111 ff] = 0

if

space is Euclidean). It is called and

ad( Ie)

J

Q)

(the corresponding simply connected irreducible

acts irreducibly on

1fJ r5

The oiLa

if it is not Euclidean

--r.

By rather simple algebraic reasoning

(e. g. [22]

p. 235-237) the following decomposition theorem can be proved:

(if, 6")

Given any oiLa

17 = 1:$-- -~!J;

where each

Euc'lidean for

j= 0,

irreducible for

is semisimple,

1tj = [ft j' ~j]

Q) ,~

(~,G'lfti,

QI9j) For

j:> O.

and

is a direct

Q I~j

sum

is an oiLa,

j'7 0, ' j

is a scalar multiple

of the restriction of the Killing form to 4'.. Let defInes its

dual

transporting that one

~f "

e'

(f!,

(f!*.

and

1I'if is

l3J 0 (this is trivially

J = ad(Z) extends to a complex linear transformation on

Q' (Z)

= i.

We also introduce a basis for

c

0 implies

stable under Proof.

ad

by. J2 = -I, C{(Z) =

Similarly

If'

JE_c:i = -iE_ 0( •

defining, for all £y€

JY 0{ = - Xc{ 1.

le c

Z('t

and ~ - are abelian subalgebras of

.

ad(Z) has eigenvalues

corresponding eigenspaces

r.::t ,;lec.

~ i, 0 on

By a standard

-er c

with

arg~ment

Jacobi identity), if X, Yare eigenvectors for the eigenvalues

(the

PI

- 141 -

A, Koranyi

J, f

K

KA (x)

'f- 1A(x)

k E K }.

The other assertions are trivial, Note.

One can study the orbit of exp i {/[,cG u

a way similar to lemma that

~ (1:+)

6.

in

With the help of this one can show

in dense in

M* (

rJ 16

p. 286-287).

Some remarks on classification.

It is clear now that

in order to classify bounded symmetric domains (up to isomorphi.sm) it suffices to classify all pairs (Y,G") where

b

is an involutive automorphism, and

A;,

is not semisimple. The classification of all dition on

ft?

is known

out those for which

Ie

(e. g.

[8

JCh.

[23].

is compact simple,

the fixed point set of G' ,

({f,

XI{

HD(

~ r----'" -yo(

Proof. algebra as in Lemma

l.-

i ),

computation. Define

.6),

and

with respect to

1+

f-

Yrf/'

Ho(

span a 3-dimensional

the elemnt corresponding to

.2-4;

is

SL (2, ([' )

(C(E:

XC>(,

and acts as stated ,

CC(

by a simple

as the real space spanned by all

as the orthogonal complement of

in

1- I

iH I)(

in

B. Lemma 2.

it acts trivially on

J+.

ad(c)

interchanges

i ; - and

Ja

- 149 -

A

Proof.

ex (H) = 0

for all c{ <E

II

Corollary.

ad (c) to

y([

(or projection)

H~ C I~

since

1++ J ct

i

j-) ct

to

with respect to of the ordinary

J{)u~

J U'Ccr is /

Hence

HE

I

t-

DI, x~1 0 =

is a Cartan subalgebra

JiZ([ carrying

is an automorphism of

(r)+JCI(J(£,

system of

For the second note that

(O(Ei1) ,

LJ

off·

rlranyi

The first statement follows from Lemma

1 and strong orthogonality of implies

T7

It follows that the root

the same as the restriction system onto

-root

~-

B",

The norm

111

([

liT

has the structure of a

Ii

of any linear transfor-

will mean the norm with respect to this structure, (Note

J ((3 E rj)

is an orthonormal

system, and that

ad(G U )'

- 151 -

A. Koranyi

in particular

ad(c)

and

ad(K),

Proof.

D is invariant under

the norm on the right, since and

ad(k)

is unitary.

By Lemma

r 1(E)

Now

~ bo( HO(, since

ad (c)

and and

ad(r 1 (ad(k)E)) = ad(k) ad (j-1(E))ad(kr 1

ad(r 1(E))

E E A(O) ,

+

E E: D('\ or; .

0

bo( EO«( lb ol )-t )- M"'-
(u, v)

(ii)

12 ,

y €

is defined

be a regular cone

V l' and let

if it is non-empty,

(iii)

~

(u, u) E

I2.

(iv)

rP

(u, u) '"

a

(conjugation with respect to

for all implies

The generalized halfplane

W),

u c, V 2' u '"

a

corresponding to

Sl

and

is then defined by

(This notion is due to Pyatetskii-Shapiro; of type

I

there is no

or

II

rJ>,

he calls D

a

Siegel

or not

If

depending on whether

n 2 '" 0

and D

'tube domain'

is simply the

over

domain n2

D. )

=

O.

- 163 -

A. Koranyi

The distinguished boundary B =

i (z1'

and holomorphic

on

f

then sup

D ,

is a minimal subset of

D

is

z2) E V 1 X V 21 1m z1 -

It is not hard to show directly that if D

of

D

(z2' z2)

o} .

is bounded continuous on

If I =

I

f I ' and B B This will, however

~up

with this property.

D

P

be an immediate consequence of some later results.

n

admits a group

D

morphism, whose elements are pairs

of affine holomorphic auto-

g= (a, c) E W X V 2' acting by

~(c,c)

+a+2ip(z2,c)+i

g

+c It is easy to check that

of step

2

(if

n

n2 = 0

is simply transitive on

B,

it is even Abelian).

It is of interest to consider the group

and the group

~ D

is called

G (Q))

It and

G(Q,

g1

affine-homogeneous

is transitive on

~)

and nilpotent

f2. ;

¢ (u, v)

if

G(Q)

G(0.,P)

=

= P(g2 u , g2 v ),

pr 1G(n,~)

(the

l1~,v J .

projection onto

in this case the group generated by

is tr.ansitive on D.

All we do in this section is valid for any generalized halfplane.

Let us note that

one-paramet er group

G(n,

I(e 2t 1 ,

~)

etI)

always contains at least the

ItE

JR' S '(

In the following we assume that with their standard complex Euclidean structure. unimportant, Haar measures.

V1 = a;nl , V 2 = a;n2 This is actually

quite

all it is good for is to fix the normalization of certain

- 164 -

A. Koranyi

We define the measure

J WXV

g

~

on

B

B

(z2' z2)' z2) dX l dX 2 dY2

It ---;;. B

is a diffeomorphism

0

action of

n.

(3

Lifting

16

to

,equivariant

For

0

:=

f

< R

is a Banach space isomorphism particular,

H2(D)

LP(;\,({,) 2 -:; ,

r(~,C()d:t do( z

n

L 2( Q X R 2) ------".. H2(D)

(In

is a Hilbert space. )

The proof is too long and technical to be given here. We should note, however, that if

D

is the upper halfplane in

then this is a classical result of Paley and Wiener. case

(n 2

=

0)

it was proved by

Bochner

[2)

case in a somewhat different form and with only a

Q;,

In the tube The general sketchy

proof

is in [6] . The present version in due to (to be published) .

E. M. Stein and the al1ihor

- 166 -

A. Koranyi Theorem 2.

f E H2(D), then

If

Sketch of proof: on By

B

by setting

Y1

=

Let

P

(z2,z2)

j

f exists in L 2(B). t B be the function defined

f

lim

,tt~R

inOthe formula

of Theorem 1 .

using the Plancherel theorem twice, one sees that

J

n2Ir(~,Cl())2 (1_e-47I(tYf~dO(

dl

.Q.' X R From here the statement follows by the Lebesgue dominated convergen('e theorem. Remark

This theorem

to compute inner products of functions in compute the inner product in

? L ~(B)

gives a manageable way H 2(D);

one only has to

of their boundary functions

The following short discussion is of a quite general nature; it is probably rather familiar to the reader who has studied the question of the existence of the Bergman kernel and Bergman metric. Let

H

be a Hilbert space of complex-valued func-

tions defined on a set E. kernel

for

H

S : E x E ---

E --7' R

by

- 170 -

A. Koranyi

'""t . P (g) = P(0,g .. and for

f:

B~ 1

LP(B).

is a rather standard argument;

(is one variable this is called the

it is due to

E. M. Stein

[20]

F.

is the Poisson integral of

f

,then

J2

HP(D)

M. Riesz

!I F t II

p

;:.llf

II

if

isometrically

into

F

\~

f

LP(B)

F€ HP(D)

p L (B)

(immediate from the Remark after Theorem

This shows that the map

is

.

L (B)

t E

and

From this theorem it follows that,

Note .

for all

p =1

The case of

p.337 .

4).

imbeds the Banach space (if

1 ~ P ~ 00 ) .

In the case of the one-variable upper halfplane the set of Poisson integrals of all

(say, bounded)

functions

on

characterized as the set of (bounded) harmonic functions on It would be interesting to have an analogous If

D

B

can be D

result in the general case.

is symmetric, we shall show in sec.5

that every Poisson

integral is harmonic in the sense of symmetric spaces (but not every harmonic function D

is a Poisson integral).

nothing along these lines is known.

In the case of a general

- 173 -

A. Koranyi

The Cayley transform of a bounded symmetric domain.

5.

Let in

1j

+

be a bounded symmetric domain imbedded

D

c•D

We want to describe

in the standard way (sec. 2).

in the case of the unit disc this amounts to rotating the Riemann

~ around a horizontal axis,and thereby mapZ + i Z L>.. ping the unit disc onto the upper halfplane, r-r iz + 1 ,which sphere by the angle of

is equivalent to the usual Cayley transform). A priori it is not clear that

the action of

c

on

7;"

+

is defined as

S(- 0

i. e.

B'Z: (exp(ad(V)). Etl ' E A ) > 0 ,

E En.

for all

This means exactly that

E.1 E..Q I, finishing the proof. Lemma 2 .

a:

ad(c)2 = i ad(X).

For this we have to note that

irreducible representation spaces on which Hence

~

ad(c)2 = exp (ad (

which in turn implies that

+

7t .

e - 2'

(since

=

1

iX))

( using at

-

ad(c)

ad( 4-iX)

T

[u, x]

- [U,

=

Lemma

tp

-cc-

3

U C ~2- .

i on it

>

Geometry of Homogeneous Bounded Domains (C.I.M.E. Summer Schools, 45)

Homogeneous Bounded Domains and Siegel Domains

Theory of Complex Homogeneous Bounded Domains

Theory of complex homogeneous bounded domains

The Geometry of Complex Domains

The geometry of complex domains

Stochastic geometry CIME lecs Martina Franca

Stereodynamics (C.I.M.E. Summer Schools, 56)

Asymptotic expansions for pseudodifferential operators on bounded domains

Asymptotic expansions for pseudodifferential operators on bounded domains

Mirror Geometry of Lie Algebras, Lie Groups and Homogeneous Spaces

Maps of bounded rationality

Mirror Geometry of Lie Algebras, Lie Groups and Homogeneous Spaces

Mirror Geometry of Lie Algebras, Lie Groups, and Homogeneous Spaces

Noncommutative geometry (CIME lectures, 2000, LNM1831, Springer 2004)

The geometry of curvature homogeneous pseudo-Riemannian manifolds

Mirror Geometry of Lie Algebras, Lie Groups and Homogeneous Spaces

October journal No.45 Summer (1988)

Controllability and Observability (C.I.M.E. Summer Schools, 46)

Stochastic Differential Equations (C.I.M.E. Summer Schools, 77)

Dynamical Systems (C.I.M.E. Summer Schools, 78)

Wave Propagation (C.I.M.E. Summer Schools, 81)

Wave Propagation: (C.I.M.E. Summer Schools, Vol. 81)

Teoria Della Turbolenza (C.I.M.E. Summer Schools, 14)

Differential Topology (C.I.M.E. Summer Schools, 73)

Algebraic Surfaces (C.I.M.E. Summer Schools, 76)

Complex Analysis (C.I.M.E. Summer Schools, 62)

Complex Analysis (C.I.M.E. Summer Schools, 62)

Materials with Memory (C.I.M.E. Summer Schools, 74)

Potential Theory (C.I.M.E. Summer Schools, 49)

Spectral Analysis (C.I.M.E. Summer Schools, 64)

Geometry of Homogeneous Bounded Domains (C.I.M.E. Summer Schools, 45)

Homogeneous Bounded Domains and Siegel Domains

Theory of Complex Homogeneous Bounded Domains

Theory of complex homogeneous bounded domains

The Geometry of Complex Domains

The geometry of complex domains

Stochastic geometry CIME lecs Martina Franca

Stereodynamics (C.I.M.E. Summer Schools, 56)

Asymptotic expansions for pseudodifferential operators on bounded domains

Asymptotic expansions for pseudodifferential operators on bounded domains

Mirror Geometry of Lie Algebras, Lie Groups and Homogeneous Spaces

Maps of bounded rationality

Mirror Geometry of Lie Algebras, Lie Groups and Homogeneous Spaces

Mirror Geometry of Lie Algebras, Lie Groups, and Homogeneous Spaces

Noncommutative geometry (CIME lectures, 2000, LNM1831, Springer 2004)

The geometry of curvature homogeneous pseudo-Riemannian manifolds

Mirror Geometry of Lie Algebras, Lie Groups and Homogeneous Spaces

October journal No.45 Summer (1988)

Controllability and Observability (C.I.M.E. Summer Schools, 46)

Stochastic Differential Equations (C.I.M.E. Summer Schools, 77)

Dynamical Systems (C.I.M.E. Summer Schools, 78)

Wave Propagation (C.I.M.E. Summer Schools, 81)

Wave Propagation: (C.I.M.E. Summer Schools, Vol. 81)

Teoria Della Turbolenza (C.I.M.E. Summer Schools, 14)

Differential Topology (C.I.M.E. Summer Schools, 73)

Algebraic Surfaces (C.I.M.E. Summer Schools, 76)

Complex Analysis (C.I.M.E. Summer Schools, 62)

Complex Analysis (C.I.M.E. Summer Schools, 62)

Materials with Memory (C.I.M.E. Summer Schools, 74)

Potential Theory (C.I.M.E. Summer Schools, 49)

Spectral Analysis (C.I.M.E. Summer Schools, 64)

Recommend Documents