Lecture Notes in Mathematics Edited by A. Dold, B. Eckmann and E Takens
1422 V. Villani (Ed.)
ComplexGeometryand Analysis Proceedings of the InternationalSymposium in honour of EdoardoVesentini held in Pisa (Italy), May 23-27, 1988 II IIII
Springer-Verlag Berlin Heidelberg NewYork London ParisTokyo Hong Kong
Editor Vinicio Villani Dipartimento di Matematica, Universit& di Pisa Via Buonarroti, 2, 56100 Pisa, Italy
Mathematics Subject Classification (1980): 32-xx, 53-xx, 47-xx ISBN 3-540-52434-7 Springer-Verlag Berlin Heidelberg N e w Y o r k ISBN 0-387-52434-7 Springer-Vertag N e w Y o r k Berlin Heidelberg
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Foreword
This volume contains the texts of the main talks delivered at the International Symposium on Complex Geometry and Analysis held in Pisa, May 23-27, 1988. The Symposium was organized on the occasion of the sixtieth birthday of Edoardo Vesentini, by some of his former students, in appreciation of his many contributions to mathematics, of his teaching and advice. The aim of the lectures was to describe the present situation, the recent developments and research trends in several relevant topics in Complex Geometry and Analysis, that is in those fields in which the mathematical activity ofE. Vesentini is most fruitful and inspiring. The contributors are distinguished mathematicians who have actively collaborated with the mathematical school in Pisa over the past thirty years. The organizers would like to thank all the supporting institutions, and, in particular, the Cornitato per la Matematica (CNR) and the Gruppo Nazionale di Geometria Analitica ed Analisi Complessa (MPI).
The Organizing Committee V. Villani (chairman) T. Franzoni G. Gentili G. Gigante S. Levi E Ricci G. Tomassini
Contents
Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M . F . ATIYAH Hyperk~hler Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. CALABI A ~ n e Differential Geometry and Holomorphic Curves . . . . . . . . . . . . . . . . . . . . .
III
1
15
J . W . COGDELL, I.I. PIATETSKI-SHAPIRO The Meromorphic Continuation of Kloosterman-Selberg Zeta Functions . . . . 23 G. DETHLOFF, H. GRAUERT Deformation of Compact Riemann Surfaces Y of Genus p with Distinguished Points P 1 , . . . , P m 6 Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
S. KOBAYASHI On Moduli of Vector Bundles
45
.............................................
A. KORXNYI, H.M. REIMANN Quasiconformal Mappings on C R Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
R. NAGEL On the Stability of Positive Semigroups Generated by Operator Matrices .. 77 R. NARASIMHAN The Levi Problem on Algebraic Manifolds
.................................
85
H.H. SCHAEFER A Banach-Steinhaus Theorem for Weak* and Order Continuous Operators, 93 J . - P , VIGU~ Fixed Points of Holomorphic Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
101
List of participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
107
HYPERKAHLER
MANIFOLDS
Michael Atiyah Mather~atical Institute, 24-29 St. Giles, Oxford OXI 3LB, England, U.K.
§i.
Introduction
In r e c e n t variety very
years
of contexts,
interesting
of this survey
and D e f i n i t i o n s
hyperk~hler manifolds and
class
lecture
of m a n i f o l d s
is to justify
turned up in a wide
clear
with a rich
these claims
that they form a
theory.
by giving
The p u r p o s e
an overall
of the field.
I shall begin properties.
by reviewing
Then
construction
This
content.
In
the basic d e f i n i t i o n s
in §2 I will d e s c r i b e
of [7] w h i c h
painlessly.
shows
enables that
§3 I will c o n c e n t r a t e
are of special
interest
describe
the b e a u t i f u l
In §4 I e x p l a i n of h y p e r k ~ h l e r
manifolds.
magnetic monopoles §5 I d e s c r i b e
hyperk[hler
studied
moduli
in
§2
of R. P e n r o s e
and i l l u s t r a t e
it for
which
the c l a s s i c a l
d u e to P.B.
I will
Kronheimer
yet m o r e
the m o d u l i
quite
non-trivial
In particular,
are of special
theory
examples
manifolds
including
spaces give
In p a r t i c u l a r
the twistor
manifolds
reasons
equations.
family of e x a m p l e s
how Y a n g - M i l l s
quotient
has a d e f i n i t e l y
on 4 - d i m e n s i o n a l
for v a r i o u s
to physics via E i n s t e i n ' s
and e l e m e n t a r y
the h y p e r k ~ h l e r
us to c o n s t r u c t m a n y
the theory
relation
in
have
it is now becoming
[9].
examples
spaces
of
interest.
Finally
as it applies
to
the case of m o n o p o l e
spaces.
As the name rather a generalization briefly r e c a l l i n g Riemannian gonal
transformation
covariant
constant.
condition
for
group
I
X
of the t a n g e n t This c o n d i t i o n
so that
in
hyperk~hler
so it is best may
X
b u n d l e with implies
is a c t u a l l y
manifolds
are
to start by
be defined
w i t h an almost c o m p l e x
a K~hler m a n i f o l d
contained
suggests
that a K[hler m a n i f o l d
manifold
Equivalently
obviously
of K~hler m a n i f o l d s ,
as a
structure
I
12 = -i)
the usual
(orthowhich
is
integrability
a complex manifold.
is a R i e m a n n i a n m a n i f o l d
with holonomy
U(n)
The
c SO(2n)
importance
algebraic
manifolds
It is a l s o metric
of K ~ h l e r m a n i f o l d s
is c l o s e d
particular
or p r o j e c t i v e )
that the
2-form
and n o n - d e g e n e r a t e ,
Next
let us recall (over
i 2 = ]~ 2
=
R)
that
by the
k2 =
ij = -ji = k
A hyperk~hler with
K~hler to
metrics.
I
and
the
manifolds
are
in
the algebra
for
x,y,z
manifold
satisfying
constant.
~ R
group
X
Sp(k)
H
of q u a t e r n i o n s
j, k
with
is
the relations
lies
K
(i .i)
is n o w d e f i n e d (orthogonal
we may
of
identities
the tangent
structure.
symplectic
as a R i e m a n n i a n
transformations
algebra
say that
H-module in t h e
,
+ z 2)
the q u a t e r n i o n
Briefly constant
of
i,
-i
I, J,
covariant
manifold
the t a n g e n t
and covariant
spaces
Equivalently
to
X
have a
the h o l o n o m y
group
c SO(4k)
Clearly we
carry
associated
that
etc.
succinctly,
endowed
bundle)
always
~I
so t h a t K ~ h l e r
symbols
(xi + y j + zk) 2 = -(x 2 + y2
X
in the f a c t
symplectic.
generated
or m o r e
(affine
significant
lies m a i n l y
see t h a t
generally
by c h o o s i n g X
has
the role
the s t r u c t u r e
in p a r t i c u l a r of
I
can
I
and
a complex
be r e p l a c e d
ignoring
Kihler
J, K
structure.
More
by
I X = xI + yJ + zK
where
I =
This
(x,y,z)
shows
metrized
that
~ R3
X
by points
has l
with
12 = x 2 + y2 + z2
a whole of t h e
family
2-sphere,
=
of c o m p l e x and t h a t
1
.
structures, the metric
is
para-
K~hlerian
for all these complex
terminology
Note.
Although
cumbersome Because nions,
logical
and d e s c r i p t i v e
and a b e a u t i f u l
they
involve
symplectic
stage that
class
so m a n y
geometry,
they should
Unfortunately
of H a m i l t o n ' s theoretical
be c h r i s t e n e d
the h y p e r k ~ h l e r
The 3 o p e r a t o r s constant
structures. ~I
becomes
Since
Unfortunately
~J'
as we
geometry
provides
role
shall
play
geometry.
is u n i q u e l y
family
of complex
yield
3
3 symplectic
structure
defined
2-form d e f i n i n g
metrics
by
I
then
while
a "holomorphic
than K~hler
is t o t a l l y
false:
Clearly
quaternionic
hyperk~hler
quaternionic are s t r e n g t h -
hyperk~hler
scale factor)
are m a n y
K~hler m e t r i c s geometry
(and e v e n t u a l l y
clearer
by its
result
is
to algebra)
in the twistor
picture
§5.
space
Hk
with
standard m e t r i c
is a h y p e r k ~ h l e r
are not v e r y
interesting,
p o i n t for the c o n s t r u c t i o n
see in the next
seem to exist.
that
sense h y p e r k ~ h l e r
analysis
This b e c o m e s
act orthogonally)
or flat e x a m p l e s
In this
to complex
in
not
that the c o r r e s p o n d i n g there
play a
geometry"
an i r r e d u c i b l e
(up to a c o n s t a n t Note
explain
does
in r e v e r s e
algebraic
should
for this v i e w - p o i n t
fact:
structures.
geometry.
in c o m p l e x
for the n o n - e x i s t e n t
important
determined
role
manifolds
algebraic
geometry
argue
The a r g u m e n t s
tightly related
whic h we shall
hyperk~hler
algebraic one can
on a fixed complex m a n i f o l d .
starting
at one
(expecially
to the K~hler metric,
an important
that
a substitute
metric
shall
giving
in "quaternlonic
see,
by the following
i, j, k
used
w i t h the m e t r i c
~K
holomorphic
speculate
quaternionic
In fact,
more
fate.
(quater-
I proposed
too w i d e l y
combined
associated
K~hler m a n i f o l d s
for K~hler
interests
structure.
important
algebraic
main
is rather
a better
A pity!
~I'
a closed
one m i g h t
similarly
ened
form
deserves
physics)
u s a g e was
I, J, K
2-forms
(i,i)
symplectic"
geometry
the
"Hamiltonian manifolds".
If we fix on the c o m p l e x
is the
~j + i~ K
explains
the t e r m i n o l o g y
of m a n i f o l d s
by physicists) to be eradicated.
covariant
This
structures.
"hyperk~hler"
section.
manifold.
(where
These
but they provide
of n o n - l i n e a r
examples
linear
the as we
§2.
The ~ u o t i e n t
It will structure theory
construction
be clear
from
the d e f i n i t i o n s
is a v e r y r e s t r i c t e d
as having
certainly my
only a m i l d
initial
the d i s c o v e r y
in [7] of a v e r y
construction"
which generates
in a n a t u r a l
way:
analogue
version
Moreover
of c l a s s i c a l
quotient
replaces
action
a complex to form
space
C*,
=
structure
is not
function
Izl 2
symplectic
(2.2) known
invariant
theory".
construction
by c o n s i d e r i n g
standard S1
way
in a l g e b r a i c
S1
geometry
from
Cn
(a "bad"
point)
and
S1
to the unit
sphere
geometry S 2n-l,
is to so that
(2.2) a natural metric,
so transparant. lies
The
in symplectic
Cn
structure
inherits
on
to the c o m p l e x m u l t i p l i c a -
real d i f f e r e n t i a l
inherits
on
in
the
of the c i r c l e g r o u p
the origin
using
viewed
of
Cn
the H a m i l t o n i a n a natural
but the complex
link b e t w e e n geometry.
as a Hamiltonian,
given
symplectic
sl-action. structure,
and
the
with respect
by its s t a n d a r d
flow of the
the c o m p l e x
In fact
to the
hermitian metric, The q u o t i e n t
a procedure
well
in c l a s s i c i a l m e c h a n i c s .
This
simple
(connected) G
the h y p e r k ~ h l e r
(2.1)
of
Pn-i
view-points
generates
sense
= s2n-i/S 1
In this g u i s e
metric
is the
space
procedure
the action
Pn-i
manifolds
(Cn - 0)/C*
An e q u i v a l e n t restrict
The
then r e m o v e
to form t h e p r o j e c t i v e
Pn-i
C n.
is to c o m p l e x i f y
by
is the g e o m e t r i c
the q u o t i e n t
is p r o vi d e d
the
"quotient
construction which
"quaternionic
by reviewing
changed
of h y p e r k ~ h l e r
In this
(scalar multiplication)
vector
tive g r o u p
quotient
theory.
The p r o t o t y p e
a quotient
to d i s m i s s
That was
and b e a u t i f u l
this q u o t i e n t
of a K~hler
therefore
geometry.
standard
simple
vast numbers
invariant
tend
interest.
but m y v i e w was r a d i c a l l y
the n o n - e x i s t e n t
Let m e begin K~hler
and one m i g h t
specialized
reaction,
quaternionic
one,
in §i that a h y p e r k [ h l e r
preserves
example generalizes
Lie g r o u p
G
to the a c t i o n
on a K~hler m a n i f o l d
both the m e t r i c
and the c o m p l e x
X.
of any c o m p a c t We a s s u m e
structure,
hence
that also
the symplectic
structure.
Under m i l d
conditions
there
is then a
moment map
: X + g*
where
g*
(2.3)
is the dual
are H a m i l t o n i a n one-parameter
subgroups
equivariant. = 0)
=
inherits
~
assume -i
X
variety
~ E g*
symplectic
a projective
the c o o r d i n a t e
ring of
theory"
morphisms of
X
of
X.
: X +g*
this
X
X.
Using
quotient
define of
symplectic I
A1l
also inherits
X~
this
form X
which makes
a
[83.
variety
then
this
(with K[Puler class
X
is the p r o j e c t i v e the G - i n v a r i a n t
is part of
part
of
"geometric
by Mumford.
the h y p e r k i h l e r G
case,
be a c o m p a c t
the 3 symplectic assumptions)
so let
X
Lie g r o u p
be a
of auto-
structures
~I'
~J'
~K
3 moment maps
DI'
~J'
~K
moment map
@R 3
Let
~ c g* ~ R 3
value
of
~.
be fixed by
Then
G
and
the m a n i f o l d
(~)/G
has 3 induced metric,
we take
Then the m a n i f o l d
Clearly
the
into a single q u a t e r n i o n i c
is a r e g u l a r
= -i
of
by
to be G-
(frequently ~.
is e s s e n t i a l l y
and let
is G - e q u i v a r i a n t .
assume
with
algebraic
ring
(under m i l d
we can combine
whic h
G
for
structure
as d e v e l o p e d
manifold
we g e t
which
by
structure.
embedding)
We are n o w r e a d y for hyperk[hler
The c o m p o n e n t s
is assumed
value
the K~h__~ler q u o t i e n t
whose c o o r d i n a t e
invariant
~
be fixed
Together
is a p r o j e c t i v e
from
Also
an almost complex
K~hler m a n i f o l d ,
If
G.
G.
to the flows defined
(~)/G
a natural
then d e f i n e s
of
corresponding
it is a regular
a R i e m a n n i a n metric.
coming
of
N o w let
and
X
of the Lie algebra
functions
symplectic
structures
a hyperk~h~er
[73.
which,
structure.
together This
w i t h the induced
is the l % ~ e r k ~ h l e r
The complex
structure
tive description. symplectic
~j + i~ K
(c)-z (~j
G
and
+ i~K)
X
c g*
l)
2)
If e
X
is c o m p l e t e
these
I pointed
out
kahler manifolds. representation construct H k.
G
quotient
Since
there
manifolds. yields
§3.
are many
interesting
to
G
Then
quotient
quotient X
X
on by
Y
,
where
manifolds
are
models.
is a l s o
will
have
complete.
singularities
manifold.
spaces
= A u t ( H k)
situation
manifolds
choices
preserves
acted
supersymmetric
then
G + Sp(k) in the
construction
symplectic
we c a n
on
of g r o u p s
and r e p r e s e n t a t i o n
we
will
to v e r y m a n y
is a c i r c l e
or t o r u s
action
try to G
lead
the
are hyper-
is a n y
where
from
Hk
of
hyperkahler
the construction
examples.
a hyperk~hler is 4,
i.e.
the quaternionic as such,
interest
analogues
a hyperkihler
dimensional
examples
are
with
"gravitational
obviously
or
is t h e As
the
lowest
are
in a s e n s e
algebraic
are also
space-time
and,
of
curves special
since
same as a Kahlersolutions
such manifolds
of
the
have been
the q u a n t i z a t i o n
of g r a v i t y .
instantons".
questions
important
scarce
They
manifold.
equations
I have not discussed
these are
of
4-manifold
in c o n n e c t i o n
t o as
i.
surfaces
attention.
Einstein)
Einstein
by pysicists
So far
4k
These
of R i e m a n n
special
(or s e l f - d u a l
are referred
ness but
has dimension dimension
4 is t h e d i m e n s i o n
(positive definite) studied
manifold
quaternionic
deserve
because
= SU(2),
Einstein
They
~
Gc
4-dimensiona i examples
Since
Sp(1)
if
hyperkahler
Even when
dimension
and,
of
X
hyperkahler
§i t h e q u a t e r n i o n i c
we a r e
see t h a t t h e q u o t i e n t
indicates
t o an i n c o m p l e t e
in
of
of c ~ .
~ ~ g* ~ R 3
hyperkahler
value
Hence
of
[7]
of
map
Y
an a l t e r n a -
a holomorphic
action
moment
subaanifold
in r e l a t i o n
leads
defines
s a m e as the K i h l e r
of
the
a regular
be s e e n from
holomorphic
a holomorphic
the
title
can
~j + i~ K
The
are the 3 components
is n o t
As
gives
X
that
X.
to physicists
and removing
of
is a c o m p l e x
As the
of i n t e r e s t
If
on
is c l e a r l y
~I,~J,~K
Notes.
Recall
structure
this and
I
of c o m p a c t n e s s
aspects.
and e s s e n t i a l l y
Compact consist
or c o m p l e t e 4-
of f l a t
tori
and the
K3
surfaces
has been e s t a b l i s h e d
where
the e x i s t e n c e
by S.T.
Yau with
of a K ~ h l e r - E i n s t e i n
his proof
metric
of the C a l a b i
conjecture.
If we c o n s i d e r would
be c o m p l e t e
non-compact
manifolds
this c a n be i n t e r p r e t e d One c l a s s
(referred
Euclidean)
requires
wher e
F c Sp(1)
Since can occur Plat o n i c hedral, known
which
ALE
= SU(2)
double
just the d o u b l e
regular
asymptotically
to b e h a v e at
~
class
flat.
of s l i g h t l y d i f f e r e n t
spaces:
the m a n i f o l d
simplest
are a s y m p t o t i c a l l y
in a number
to as
the next
In fact ways.
locally
like
(R4-O)/F
is a f i n i t e group.
Sp(1) are
manifolds
solids
octahedral
in
covers
covers
R 3,
namely
and i c o s a h e d r a l
to be linked,
SO(3)
of the
the cyclic,
groups.
in a subtle way,
the g r o u p s
symmetry groups
to the
F
dihedral,
These groups s~ply-laced
which
of the tetra-
are w e l l Lie g r o u p s
An, D n, E 6, E 7, E 8 The c o n s t r u c t i o n of F has been theory.
worked
He c o n s t r u c t s
a judicious
choice
theory of
his m a n i f o l d s
uniformly,
F,
which
F
arose.
which arise Kronheimer
with an open
and
for all
F,
of
~
"regular"
for all choices
with
representation.
being
the r e g u l a r
F ÷ Sp(1) metrics
for the v a l u e
represent-
= SU(2)
from
have m o d u l i
of the m c ~ e n t map.
space can be n a t u r a l l y
points
beautiful
quotients
in terms of the r e p r e s e n t -
the h y p e r k i h l e r
that the m o d u l i
set of
symplectic
representation
Moreover,
frcra the c h o i c e proves
G
spaces
[93 in a v e r y
as h y p e r k i h l e r
the key i n g r e d i e n t s
ation and the 2 - d i m e n s i o n a l
of ALE
Kronheimer
of Lie g r o u p
Thes e are d e t e r m i n e d ation
and c l a s s i f i c a t i o n out by P.B.
identified
in the q u o t i e n t
(h ~ R 3 ) / W
w her e W
h
is the C a r t a n
algebra
of the c o r r e s p o n d i n g
Lie g r o u p a n d
is its Weyl group.
If we c o n s i d e r "quaternionic
these
algebraic
to complex
algebraic
H2
"quaternionic
of our
rank H 2
4-dimensional
curves"
curves.
(= dim h)
hyperk~hler
they are a n a l o g o u s H1
curves"
of complex so that
manifolds
as
in m a n y r e s p e c t s
curves
is r e p l a c e d
by
is a n a l o g o u s
to the g e n u s .
us to the c y c l i c richer
s i n c e we h a v e
exceptional matrices for~s
The Hanson,
An
Also
of
family
relation
resolutions
then we might higher
Let
X
and
double
A
be
the
P
over
forms
X
with values
similar space
operators
over
group
H,
(rather
: A +
in
A
p a r t of
self-dual
Yang-Mills
the hyperkahler
M = -1
g°
The
work
has an
and
like algebraic
be p o s s i b l e
spaces
I shall
let
I, J, A
on
A We
for
to c o n -
bundles
G
be a c o m p a c t for
Lie
a fixed
space modelled K
over
now explain.
o n l-
operators
induce
an ~ - d i m e n s i o n a l
affine
metric.
structures.
Moreover preserving
the gauge its a f f i n e ,
can therefore
hyperkihler
moment
consider map
R3
hyperk~hler
computation of
X)
equations
quotients.
(with a p p r o p r i a t e shows
the curvature.
quotient
(O)/O
are
it s h o u l d
which makes
the n o n - c o m p a c t n e s s
self-dual
2-
of E g u c h i -
on deformations
is a n a f f i n e
the u-dimensional
a little
constant
to w o r k
of a l l G - c o n n e c t i o n s A
acts naturally
(Lie G ) * ~
is
3
by period
Krcnheimer's
as m o d u l i
4-manifold,
space
then try to construct
In f a c t over
due
Also
that
a compatible
and q u a t e r n i o n i c formally)
known
to be t r u e as
Then
on
G = Aut(P)
metric
and
X.
with
and t h e
3 covariant
4-manifolds
examples
out
G-bundle on
restrict case
points.
conjecture
turns
(D n)
are determined the
would
spaces
be a h y p e r k ~ h l e r let
family
and H i t c h i n .
dimensional
This
analogy
the q u a t e r n i o n i c
.
that hyperk[hler
struct
direct but
t h a t of B r i e s k o r n [ 5 ]
moduli
curves
"curves"
n infinite
were previously
with
If we a c c e p t
)
the m o d u l i
H2
of r a t i o n a l
Yang-Mills
A
we i n t e g r a t e
Gibbons-Hawking
intimate
group
another
cases.
a basis
The most (type
in all c a s e s :
over
§4.
groups
Thus
that
~
~ = 0
which define
care
being
is e s s e n t i a l l y becomes
instantons
the on
taken the
(anti)X,
and
is j u s t t h e
instanton
moduli
space
so e x t e n s i v e l y
studied
in g e n e r a l
by Donaldson.
There simplest
are various
arise
instanton
moduli
of t h e m o d u l i spaces
for
cases
X = R4
spaces
Of c o u r s e
in
less
the analysis
works
have
here families
that
the
and
R4/F
Then we can
let
F'-invariant
instantons the
extreme
Let
same,
and C h e r n
reduces
us
[63,
gives
an a l g e b r a i c
description
leading
The is, case
has
one
the
analyt-
Neverthe-
valid
so t h a t we
arise
naturally
spaces.
principle
2 different
(dual)
speaking
be a s u b g r o u p dual
instantons
Nahm"s
of t h e
(or c h a r a c t e r
However,
of N a h m
form
group)
of
from
the Lie groups quantities
like
classes.
0 ,
then
and
In f a c t N a h m ' s
to the main of the
F' = R 4
result
instanton
moduli
space
finite-dimensional
principle,
in [1],
moduli has
as
[3] w h i c h
space.
In
2 hyperk~hler
described
in [6] a n d
~-dimensional.
F = R
we a r e
to t h e m o n o p o l e
hyperkihler
H x
the monopole ness
amounts
the instanton
descriptions,
for rather is
F =
ignores
interchanging
to algebra.
by D o n a l d s o n
When
part
moduli
which
Roughly
F c R4
the duality
shown
the other
has
F-invariant
case when
therefore
space
and vice-versa.
of t h e L i e g r o u p
this case
monopole
spaces.
affine
quotient.
construct
the rank
quotient
the
sl-invariant
and
duality
be t h e P o n t r j a g i n
are n o t
F'-invariance
the
gives
remain
manifolds,
a very mysterious
same moduli
r'
formal
of ~ - d i m e n s i o n a l
involved
In t h e
is v e r y
and t h e c o n c l u s i o n s
as f o l l o w s .
R a × Zb .
the
the m a g n e t i c
with ~-dimensional
as a h y p e r k ~ h l e r
goes
while
gives
of h y p e r k ~ h l e r
is a c t u a l l y
which means
principle
arise
quotients
presentations
of w h i c h
The first
in [1],
S1 x R3
this description that
There
interest,
[2].
ical difficulties
as h y p e r k ~ l e r
special S1 × R3
studied
s p a c e for
studied
of or
in t h e moduli
metric
reasons,
(H-0)/Z 2
with
moduli
spaces
spaces
on the
basic
an i m p o r t a n t
case
originally of
instanton
incomplete.
have c~plete
by N a h m
and
[2].
moduli For
the flat metric.
physical
studied
as
and
of
R4
the f i r s t
On the other
metrics
interpretation
spaces
example
hand
this complete-
explained
in [2].
10
These monopole
spaces
kahler manifolds stage
just
properties
§5.
are
therefore
I will
say t h a t
they
to t h e A L E
return
have
an i n t e r e s t i n g to t h e m
somewhat
in
class
§5.
of h y p e r -
Let me
different
at t h i s
asymptotic
spaces.
Tw!sto_r S ~ a c ~
Twistor physics into
with
spaces
were
the aim
an a l t e r n a t i v e
be b r o u g h t Penrose
success
framework
theory.
equation
basic
where
In f a c t
represents
idea
a family
by R.
complex
the
part
is v e r y
simple.
we can put all
these
structure
on t h e
Pl
its natural
complex
space
is a h o l o m o r p h i c
complex
structures
If complex extends
o
horizontal (i.e.
ion
(involving
it t u r n s
Z
map to
can the
involving
and
also
the
on
on
out
X x PI"
that
If we and give
we g e t a c o m p l e x
so t h a t t h e p r o j e c t i o n of t h e g e n e r a l X1
form
then
Thus
X
(x,~)
or r e a l
theory
space.
(1)
~
is t h e
(x,~(l))
structure
are holomorphic
of
a holomorphic
the twistor
$2 = P1
conjugation
manifold
by
X 1 = X × {I}
is c a l l e d
Xl
{x} × P1
a bit more
data,
on t h e f i b r e s only
is e n t i r e l y
Returning vide
into
curves
on
Z,
and a r e
The real
o-invariant ) .
structures
which
case
a hyperk~hler
fibre
say t h a t t h e
structure
sections
and g e o m e t r y
parametrized
together
In t e r m s
structures.
to a complex
By adding tic
we can
II
Z = X x P1
map.
is t h e a n t i p o d a l
conjugate ~
I1
structure
Z + P1
of c o m p l e x
space
fit naturally
4-dimensional
Since
structures
the complex
family
analysis
of the m o t i v a t i o n
put
total
theoretical
programme.
of c o m p l e x
on t h e
into
from Minkowski
manifolds
I c S 2 = P1 (C)
structure
Penrose
problems
Hyperk~hler
of t h e P e n r o S e
The has
introduced
of t r a n s l a t i n g
into play.
twistor
Einstein's
X
and
holomorphic
equivalent
to o u r g e n e r a l
a substitute
essentially Xl
for
philosophy
c a n n o w be
to develop
a theory
of
the
data
and the real
to the h y p e r k i h l e r
idea
quaternion
holomorphic
we end u p w i t h a t w i s t o r structure
metric
that hyperk~hler
algebraic
non-cc~mtative
the
Instead
quaternionic
of
o) X.
manifolds
varieties
sun~narized a s f o l l o w s .
symplecdescript-
pro-
twistor
of t r y i n g
analysis
we use
11
ordinary by
complex
~ ~ P1
The
'
analysis
suggests
For
example
given
each fibre
its
k-fold
symmetric
Z(k)
÷ PI"
For
needed
Consider
uses
studied
in
[23 w e c a n
twistor
This means
that
of
points
Z
i.e.
that
particle,
of
However
when
k
is v i e w e d
space.
Translated
k-monopole, which
The
however
further.
complex
structure
space R3
M1
of
and a
"phase" earlier,
is o b t a i n e d
indicated
of
Z(k)
÷ P1
from
above.
(which
correspond
curves
[4].
standard
construction
k-monopoles)
when
SU(2))
of the
meeting
far
in
R3
to
each fibre
provided close
separated
in
does
enables one
S1 x R 3
to
fibre ,
then
as a s u p e r -
take
of
k
the
field
says
simple
not resemble
us
magnetic
t h e k - m o n o p o l e l o s e s its
this
k
the
a r e far a p a r t .
is r e p r e s e n t e d of
of
I recall
localized
non-linear
picture
case,
ideas.
approximately these
together
like a union
If w e f i x
by k - s e c t i o n s
"soliton"
be v i e w e d
the twistor
looks
picture
of
to
a s an a p p r o x i m a t e l y
get
a k-section
twistor
its
as we o b s e r v e d Z(k)
is j u s t a c o m p l i c a t e d
into
approximately
general
stage
and
in t h e
sections
related
can
the particles
identity
with
by a version
holomorphic
such particles
particle
space
of k - m o n o p o l e s
and a k-soliton
position
of B e a u v i l l e
(for
in
to be h o l o -
coincident).
is i n t i m a t e l y
a l-monopole
is,
of
space
The desingular-
the m o d u l i
Mk
try
Xl(k)
o u t to w o r k
surfaces.
monopoles
product
and h e n c e
have
turns
I.
we c o u l d
twistor
Xl(k)
a "location"
space
symmetric
Z ÷ PI'
(possibly
space
has
Its t w i s t o r
Mk
a new
X = S1 x R 3
with
on
hyperk~hler
Z ÷ Pl
in t h e w o r k
M 1 = S1 × R3
This representation twistor
X
the horizontal of
as
the case
moduli
of
k-fold
points
k-sections k
schemes
identify
manifold.
desingularized
represent
are complex
such a monopole
space
space
procedure
of t h e m a g n e t i c
The k-monopole
a hyperk~hler the
Xl
Hilbert
In t e r m s
l-monopoles: angle.
the
par a m e t r i z e d
desingularization
the n e w f i b r e s This
,
dependence
of g e n e r a t i n g
so a s to o b t a i n
to w o r k
in p a r t i c u l a r
flat metric.
a twistor
manifolds.
so t h a t
new ways
by a suitable
product
this
symplectic
dim X = 4 ,
X1
C ÷ H
the holomorphic
picture
to r e p l a c e
ization
all e m b e d d i n g s
also
twistor
manifolds.
morphic
for
and r e m e m b e r
that
by a k - s e c t i o n sections.
separate
soliton
Z ÷ Pl'
a k-section
in a
In
sections.
idea
one
i.e.
if we f i x
does
indeed
cut
a
12
this
fibre
determine
in the
k-monopole This on
as
just
(so t h a t nearby
Since I like
to
century by t h e
The
however
dependence the
solitons
one
are
that
which
Let me
on the
conclude
with
monopole
space,
Because
there
a natural
covering,
M2
mass)
and
ables
relative
able
is
is t h e
another to
4-dimensional
studied
in
[23.
Asymptotically
(2)
The
fundamental
It a d m i t s not
(4)
Its is
2
Property the metric
(3)
integrals. The
the dynamics
theorem
of
[2].
of
M o2
~
is
the
by
order
intrigued
solitons.
first
space up
non-
M2
to
a double
centre
of
measures
vari-
a very
remark-
extensively
properties.
bundle
over
quaternion
isometries;
structures,
of
it is
basic
a circle
SO(3)
is
and
its
is
19th
been
the
which Mo 2
dichotomy
the
that,
in t h e
in t h e
with
about
case
strong
have
moduli
out
structure
but
2 and
group
this
rotates
its
R3 - 0 ,
of o r d e r
action
8,
does
them,
double
covering
surface
= 1
and
uniquely
parameters. of
2
- zy
at
of
group
of
like
complex
algebraic
would
(representing
manifold
looks
action the
fundamental the
x
elliptic
an
preserve
involved
The manifold
some
group
M1
4-manifold
centre.
it
it t u r n s
of
is
particle/wave
remarks
general
monopoles.
separated
but
2-monopole
centre
are
picture)
brief
these
the
of c o m p l e x far
in c o n n e c t i o n
the
hyperk~hler
(i)
(3)
a few
namely
product
Here
choice
was much
play
of
single
n a t u r e of light,
hyperk~hler the
k
the
and
think
in t h e
of
who
quaternions
trivial
on t h e
version
way
of
soliton
Hamilton,
controversies
coincident),
in t h i s
is w e a k
usual case.
role
can
"superposition"
interactive)
think
(possibly
We
depends
.
we r e c o v e r
(or
points
an e x a c t
description S1 x R 3
k
k-section.
the
and
hyperkihler
there
Except
geodesics slowly
is an for
on
moving
an M o2
property explicit overall have
monopoles
an
essentially
formula scale
for
there
determine it
are
interpretation and
this
is t h e
involving no
free
in t e r m s main
13
I hope manifolds which
this
has
brief
turn up n a t u r a l l y
quaternions
they
and
with
they can be viewed
that q u a t e r n i o n s
sketchy
survey of h y p e r k ~ h l e r
are an interesting
in a v a r i e t y
into g e o m e t r y
and tie up p a r t i c u l a r l y Perhaps
and rather
shown that
analysis ideas
c l a s s of m a n i f o l d s
of places.
They bring
in a deep
and r e m a r k a b l e
from t h e o r e t i c a l
as a j u s t i f i c a t i o n
have a f u n d a m e n t a l
for
r o l e to play
way
physics.
Hamilton's
belief
in m a t h e m a t i c s
and
physics.
Refer ence s 1.
M.F. Atiyah, G e o m e t r y of Y a n g - M i l l s Fields, Lezioni F e r m i a n e A c c a d e m i a N a z i o n a l e dei Lincei & Scuola N o r m a l e Superiore, P i s a (1979).
2.
M.F. A t i y a h and N.J. Hitchin, The g e o m e t r y magnetic monopoles, Princeton University
3.
M.F. Atiyah, N.J. Hitchin, V.G. D r i n f e l d and Y.I. Manin, C o n s t r u c t i o n of fnstantons, Phys. L e t t e r s 65A (1978) 185-7.
4.
A.
5.
E. Brieskorn, S i n g u l a r e l e m e n t s of s e m i - s i m p l e algebraic groups, A c t e s C o n g r e s Intern. Math. 1970, Vol. 2, 279-284.
6.
S.K. D o n a l d s o n , Commun. Math.
7.
N.J. Hitchin, A. Karlhede, U. L i n s t r o m and M. Ro~ek, H y p e r k~hler m e t r i c s and S u p e r s y m m e t r y , Commun. Math. Phys. 108 (1987) , 535-589.
8.
F.C. Kirwan, C o h o m o l o g y of q u o t i e n t s in symplectic and algebraic geometry, M a t h e m a t i c a l Notes 31, P r i n c e t o n Univ. Press (1984).
9.
P.B. Kronheimer, I n s t a n t o n s g r a v i t a t i o n n e l s et s i n g u l a r i t ~ s Klein, C.R. Acad. Sc. Paris 303 (1986), 53-55.
and d y n a m i c s of P r e s s (1988) .
Beauville, V a r i 6 t 4 s K ~ h l e r i e n n e s dont la p r e m i e r e c l a s s e de C h e r n est nulle, J. Diff. Geom. 18 (1983), 755-782.
I n s t a n t o n s and g e o m e t r i c invariant Phys. 93 (1984), 453-460.
theory,
de
Aftine Differential Geometry and Holomorphic Curves by Eugenio Calabi University of Pennsylvania
Consider a smooth, immersed, locally strongly convex surface in euclidean 3-space IR3 oriented so that the second fundamental form ]Ie is positive definite everywhere, and denote by K, and dA, its Gaussian curvature (K, > 0) and element of euclidean area respectively. The following objects are of interest in this note: (a) the positive density
dA
=
K~/4 dAe;
(0.1)
Ke-1/411e;
(0.2)
(b) the positive definite quadratic form
g
=
(c) the linear functional Y* on the tangent space of IR3 at each point of the surface, whose value at each vector ff is
Y*(~) = K~I/4(Nc,~,
(0.3)
where N, denotes the unit normal vector to the surface. The importance of dA, g and Y* is due to the fact that they are unchanged if one replaces the given euclidean structure on IR3 by any affinely equivalent one inducing the same orientation and volume form; they are in fact the most elementary among the basic unlmodular afflne invaHants of the surface. The quadratic form (0.2) is used as an a~nely invariant Riemannian metric on the surface, and is called the Berwald-Blaschke metr/c; the positive density (0.1) expresses the corresponding afl~nely invariant element of area, and the linear functional Y*, whose null-plane is the tangent plane at each point of the surface, is called the aff/ne co-normal vector to the surface. One of the more obvious questions in alpine differential geometry is the one concerning locally strongly convex surfaces that are extremals for the affinely invariant area
/ d A = / K~/4 dA,
(0.4)
under interior deformations. The Euler-Lagrange equation for this variational problem is equivalent to the system of equations
= 0,
(0.5)
16 where A is the Laplace-Beltrami operator associated to the Berwald-Blaschke metric, applied here independently to each component of the aft~ne co-normal. The techniques of complex analysis are useful in connection with the Riemannian geometry of surfaces, in the first place because they allow us to simplify many of the otherwise tedious calculations. Accordingly, we shall introduce local complex parameters ( for the surface, as well as their complex conjugates 7: they are characterized by the orientation of the surface and by the conformal class of the Berwald-Blaschke metric (or, equivalently, of the euclidean second fundamental form II,). Let the immersion functions of the surface be represented locally as a differentiable mapping X of a parameter domain fl C • into IR3 (without loss of generality we shall always assume ~ to be simply connected): x
=
,
e a).
The relation of the complex structure of the surface, represented by the holomorphy class of the parameter ~, and the affine invariants (0.1), (0.2), (0.3) is expressed as follows, where (.4, B, C) denotes the determinant functional of any ordered triple of vectors .4, B, C', and subscripts denote partial derivations with respect to the indicated parameters:
(x¢, x o x ~ ) = 0;
(xc,x¢,x~¢) =
0;
- i ( x c , x o x c ~ ) > o. The Berwald-Blaschke metric then is expressed by d8 z = 2 F
Id~l 2,
where F--
the affine area element is given by dA = i F d~ A d~, and the afflne co-normal by Y* = i F -~ X¢ x X~. The Euler-Lagrange equations (0.5) can be expressed equivalently by the condition that - ~1Y • is the imaginary part of an immersed holomorphic curve Z: fl --* C s, uniquely determined by X up to a real translation: Y*(~,~) = i(Z(~) - Z(~)).
(0.6 /
Conversely, Y* determines X uniquely up to a real translation, as stated more precisely below (see [21/:
17
Proposition 0.1: Given any immersed, holomorphic curve Z:f~ --, ~ 3 there exists a iocalIy strongly convex surface X : f ] --* iF{.s with Y* = i ( Z - "Z) as its co-normal indicatrix and a Berwald-Blaschke metr/c cortformally related to ~ if and only if Z satisfies the following inequality, in which Z' = dZ/ d(: (Z - Z, Z', Z') > 0.
(0.7)
Furthermore, X is uniquely determined by Z up to a translation, and can be expressed explicitely in terms of Z by
X -- -i(Z
+ f Z x dZ- f Z x dZ).
(0.8)
The left-hand member of (0.7) expresses the value of F in terms of Z and ~. The inequality (0.7), disregarding a possible change of sign due to a switch of orientation by interchanging the rbles of ( and 7, has the following geometric interpretation in terms of the holomorphic curve Z: for each point Z(~) of the holomorphic curve Z, the complex tangent line to it and the line obtained from it by complex conjugation are mutually skew, i.e., not coplanar. A locally strongly convex surface X satisfying the Euler-Lagrange equation (0.5), or equivalently (0.6) and (0.8), is called an af/~ne maximal surface. Certain important subclasses of a ~ n e maximal surfaces have corresponding equivalent descriptions in terms of their representation by holomorphic curves Z as follows:
Proposition 0.2: A strongly locally convex surface X is a trivial solution of the equation (0.5) characterizing it as an a//ine maximal surface~ i.e., an elliptic paraboloid, if and only if the corresponding holomorphic curve Z describes a complex al//ne line, necessarily skew to the complex conjugate line.
Proposition 0.3: An attune max/real surface X belongs to the special subclass of such surfaces called improper a~ne sphere~, characterized by the property that their co-normal inclicatrix traces a plane region, ff and only if the corresponding holomorphic curve Z lies in a complex a///ne 2-plane in Us that is parallel to, but distinct from, its complex conjugate. The representation of an affine maximal surface in terms of a class of holomorphic curves in C 3 may be interpreted as an affine counterpart of the WeierstrassSchwartz representation of minimal surfaces in ]R3 in terms of "minimal" holomorphic curves in ~3. The affine analogue of the Plateau problem (which has not been studied yet) consists of determining ma immersion of a closed disk in IlZ3, with boundary mapped onto a prescribed closed smooth curve, and tangent planes at boundary points in smoothly varying given directions (all compatible with at least one locally strongly convex immersion of the whole disk) in such a way as to maximize, whenever possible, the affinely invariant area (0.4) among all the allowable immersions.
18
For a solution of this problem it will be necessary to find a priori estimates for the location of interior point images, as well as of the direction of the tangent plane and of other invariants depending on derivatives of up to third order. One of the conceivable methods that would implicitely furnish estimates for third derivatives in terms of lower order ones consists of solving a "Bernstein problem" for affine maximal surfaces. In its simplest form, this problem amounts to formulating reasonably weak asymptotic conditions at the boundary of the parameter domain, under which any affine maximal surface satisfying them is a trivial solution, i.e., an elliptic paraboloid. The crucial idea in this version of the Bernstein problem exploits the fact that the Gaussian curvature K of the Bernwald-Blaschke metric for an afllne maximal surface is non-negative everywhere, and its vanishing identically characterizes the trivial solutions. In fact, for F ( ( , ~ ) = (Z - Z , Z ' , Z 7) according to (0.7), we have K -- - F -1 02 log_F _ F_aA-~, o oi where A --- A(¢,~) = ( Z - - Z , Z ' , Z " ) , and A = 0 easily implies that Z ~ x Z" -- 0 or, equivalently, that Z traces holomorphically a complex afflne line in C a. T h e o r e m 0.4: Let Z be an immersed holomorphic curve in ¢3, parametrized by ( E f~ C ~, such that the reM valued density
= (Z - -2, Z', Z-V) is strictly positive for each ~ C 12. If Z satisfies the following two conditions: (a) Z is complete with respect to the induced metric ds
=
(o.9)
(b) Z lies in a complex affine plane in C a, then Z traces a complex atone line in C a. Proof: If the complex curve is complete with respect to the metric (0.9), and since its curvature K is known to be non-negative everywhere, it follows from the theorem of Blanc and Fiala that it is of parabolic type; in particular it implies that the global uniformizing parameter domain ~ is the full complex line C. If Z(C) lies in a complex a ~ n e plane II C C a, it follows that II # H, for if II coincided with H, then every tangent line to the curve would be coplanar with its conjugate, and hence (Z - Z, Z w,Z') would be identically zero. We now compactify C 3 to the complex projective 3-space IP3(C) in the usual way, and denote by II and II the projective completion of the plane containing Z(C) and its conjugate
19 respectively, and consider the complex projective line A = II (3 H C IPa(~). Since A = A, A consists of a circle (real projective line) of real points, and two disjoint disks (real hyperbolic planes) H+ and H _ of imaginary points. Consider the holomorphic map ~o: C ~ A which assigns to each ( E C the point of intersection of the tangent line to Z ( ¢ ) at ( with A. Since ~0(() # ~o(() (the latter is the intersection of A with the conjugate to the tangent line), it follows that the image of C by ~ lies entirely either in //+ or H_. It follows from Liouville's theorem that ~0 is necessarily a constant map, from which one deduces immediately that Z ( ¢ ) traces a complex affine line, proving the assertion, q . e . d . A corollary of the theorem just proved is the "weak Bernstein theorem" previously announced in [1]. C o r o l l a r y 0.5: If X is an immersed, locally strongly convex, a//~ne maximal surface in ]R3 which is (a) complete with respect to the BernwMd-B1aschke metric, and (b) globally convex as an (open) surface in ]R3, then X traces an elliptic paraboloid. Proof: Since X is an open, globally convex surface in IRs, there exists at least one real afflne line L in IRs such that, for every point of the surface, the line parallel to L at that point intersects there the surface transversally; denoting by ~7 a fixed, non-zero vector representing the direction of L, or its opposite, we see that, without loss of generality, Y*(ff) > 0 everywhere on the surface, where Y* is the a:ffine co-normal. But the surface is of parabofic type, by the same argument as in Theorem 0.4, and Y*(ff) is harmonic and everywhere positive: hence Y*(9') is constant. Since i ( Z - Z).TYis constant, it follows that Z .6'is constant, so that Z lies in a complex affine plane, and the argument follows as in Theorem 0.4, q . e . d . The unsatisfactory feature of the statement of this corollary as a "Bernstein theorem" is that it is hard to conceive of the two completeness assumptions, the metric one in terms of the Berwald-Blaschke metric and the affine one in terms of global convexity of the s u r f a c e , as being both independently essential, especially since neither of the two assumptions seems to be exploited to its fullest extent. Howewer, we have not succeded so far in deducing either one of these two assumptions as a consequence of the other one. On the other hand we can prove another somewhat sharper version of the corollary, resulting by substituting another sort of a restrictive assumption, which still seems unnecessarily strong, as follows: T h e o r e m 0.6: Let Z be an entire, immersed holomorphic curve, i.e., Z: C -+ ~3 with Z ' ( ( ) 7~ 0 everywhere. Assume that, for any two points (1, (2 E ~ the complex tangent line to the curve at ( is skew with respect to the complex conjugate to the tangent line at (2, i.e.,
(z(6)- z(6),
# o.
20 Then Z( ~) lies in a complex attlne line. Proof: We first compose the immersion Z with the natural imbedding of C 3 in ]p3(~) by adjoining the constant 1 before the three components of Z, and treating the four resulting functions as homogeneous coordinates representing an immersion in ]p3(~), denoted by Zp. Then the six homogeneous Plficker coordinates of the tangent lines to the curve are represented, up to a scalar factor, by Zp A Z~ and can be described equivalently by the six components of the pair of vector functions (Z', Z x ZI). L e m m a 0.'g: The linear span in C 6 of the va/ues of the six Plficker coordinates Zp ^ Z~ of the tangent//nes of Z ( ¢ ) is at most three-dimensional over ¢. Proof of the Lemma: Denote by k the complex dimension of the linear span of Zp A Z~ (a priori, 1 < k < 6). Consider the inequality (0.10), at first for a fixed 6 , as ~1 varies over ¢. This can be rewritten as the (4 x 4)-determinant equation
zp(6) ^ z (6) ^ zp(6) ^ z g 6 ) # 0; for fixed ~2 this implies one homogeneous linear constraint on Zp A Z~: • --* ills; in other words, the image of ~ in IpS(c) represented by treating Zp A Z~ as homogeneous coordinates is disjoint from the hyperplane dual to the point Zp A Z~(G). We now repeat the argument taking all values of ~2, at least in a set that is dense in some open subdomain of ~. This shows that the image of • in ]pS(c) under ~1 ~-~ (Zp A Z~)(~I) avoids an infinite collection of hyperplanes, whose common intersection is a projective subspace of IpS(¢), exactly (5 - k)-dimensional. It follows from the Jensen-Nevanlinna theory, or even the most elementary generalization of Picard's theorem to higher dimensions, that (Zp A Z~)(¢) is contained in the join of one of its point with that (5 - k)-dimensional projective subspace. But that number again is a t / e a s t equal to k - 1. This implies that 2k < 7, and so it follows that k _< 3, as claimed, q.e.d. To complete the proof of Theorem 0.6, we observe that the image of ~ in IPS(¢) represented by ZpAZ~ or equivalently by Z'@(Z × Z') lies in the intersection of a complex projective 2-subplane with the nonsingular quadric described by the identity Z' • (Z x Z') = 0. Such an intersection is either: (a) a projective 2-plane in the Grassmannian of all projective lines in 1P3(¢) (the quadric in lPS(¢) is just that), consisting of all lines through a fixed point in 1P3(¢), and this is clearly absurd; (b) a projective 2-plane in the same Grassmannian consisting of all lines in a 2subplane of ]p3(¢); this would reduce Z ( ¢ ) to a plane curve in ¢3, where the conclusion follows by Theorem 0.4; (c) a reducible quadric curve if the 2-plane is tangent to but not contained in the 4-dimensional quadric; in this case, since the image of ~ is irreducible, it would be a line in IPS(¢), and this would again lead to an absurdity;
21 (d) finally, it could be a generic intersection. This would mean that the image of ¢ in ]p5(¢) is relatively open in an irreducible conic section: the family of lines in lPa(¢) represented by this curve in the Grassmannian quadric in ] p S ( , ) consists of the rulings on an irreducible quadric surface in ~ 3 ( C ) , either singular or nonsingular. Since neither of these two cases can occur as the family of tangents to an immersed holomorphic curve, the theorem is completely proved, q.e.d. A subsequent study on this topic should lead to a quantitative counterpart of the "Bernstein theorem" just established, or possibly of a sharper version of it. This means to seek a result of the following type. Consider a locally strongly convex, affine maximal surface in ]Rs that is not trivial, as measured quantitatively, for instance, by a positive value Ko of the curvature of the Berwald-Blaschke metric at some interior point p0; one would want to estimate an upper bound for K0 in terms of the distance, defined geodesicaUy or otherwise, of p0 from the nearest singularity or "ideal boundary point"; the latter measure is to be interpreted as a quantitative expression of the extent to which the conditions for the Bernstein theorem are not satisfied.
References. [1] E. Calabi: Hypersurfaces with maximal affinely invariant area. Amer. 3. Math. 104 (1982), 91-126. [2] E. Calabi: On aft/he max/raM surfaces. Technical report, Technische Universit£t, Berlin, 1987 (to appear).
The Meromorphic J.W.
Continuation of Kloosterman-Selberg
Zeta Functions
COGDELL1 AND I . I . PIATETSKI-SHAPIRO 2
Introduction. Let S(n) denote the classical Kloosterman sum
s(n) =
Z z,y(mod n) zy--l(mod n)
The Kloosterman-Selberg zeta function of level N is then
ZN(s) =
oo ~
s(n) n2 s .
n~--O
n=--O(rnod N)
From the trivial estimate [S(n)l < n on the K l o o s t e r m a n sums one sees t h a t ZN(S ) defines an analytic function for Re(s) > 1. In his celebrated paper of 1965 on the estimation of Fourier coefficients of m o d u l a r forms Selberg studied the analytic properties of Z g ( s ) and related functions [5]. He showed t h a t Z g ( s ) has a meromorphic continuation to all of C and that the location of the poles are related to the axithmetic of the congruence subgroups
c
d
e S L 2 ( Z ) lc-0(mod
)
.
In particular, he showed t h a t the poles of ZN(S) in Re(s) >_ ~1 are in a one-to-one correspondence with the discrete eigenfunctions of the Laplacian A on F0(N)\Y) having non-vanishing first Fourier coefficient. If ~o is such a discrete eigenfunction with A ~ = A~ then, writing A = ¥1 _p2 , ZN (s) will have a pole at s = ~1 + p . Selberg did not restrict himself to F0(N), but dealt with a larger class of Fuchsian groups. The above is indicative of the type of results he discussed. 1Department of Mathematics, Oklahoma State University, Stillwater, OK 74078 Partially supported by an NSF grant. 2Department of Mathematics, Yale University, New Haven, CT 06520 and School of Mathematics, The Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv, Israel Partially supported by an NSF grant.
24 In order to prove this result, Selberg constructed Poincar~ series depending on a complex p a r a m e t e r s. He found a beautiful proof of the meromorphic continuation of these Poincar~ series using a shift equation. He showed that the Fourier coefficients of these Poincar~ series had the form
C¢(8) = ~
)~n(8)ZN(8 2v n)
n
and from this formula he deduced the meromorphic continuation of ZN (s). This type of m e t h o d is difficult to generalize to higher dimensions. In this paper we would like to indicate generalizations of this type result in two directions. First we consider the Kloosterman-Selberg zeta function Z r ( s ) for discrete subgroups I~ acting on Lobaehevskii space of arbitrary dimension, i.e., discrete subgroups F of SO(n, 1). For these zeta functions we will give a meromorphic continuation and locate the poles. We also consider arbitrary Fuchsian groups IF' of the first kind in S L 2 (R), and for these we will derive a precise formula for Z r (s) from which the continuation follows. Note that in both cases, as in Selberg's original results, it is not merely the meromorphic continuation of Z r (s) which is of interest but the actual location of the poles of this continuation. The m e t h o d which we use allows us to give a direct integral representation of Z r ( s ) as the MeUin-Whittaker transform of some s m o o t h function P(g) on F \ G . The analytic continuation of this Mellin-Whittaker transform is more complicated. One of the tools which we use is a trick suggested by J. Bernstein which presents a generalization of Selberg's shift argument. Our basic approach is based on a "soft Kuznetsov formula" which we can formulate for discrete subgroups r of any semi-simple Lie group G over R such that the quotient F \ G is not c o m p a c t but still has finite volume. We expect that these techniques will eventually lead to the meromorphic continuation of the Kloosterman-Selberg zeta functions in this general setting. We would like to thank J. Bernstein for many helpful discussions. 1. T h e S o f t K u z n e t s o v
Formula.
Let G be any semi-simple Lie group over • and let F C G be a discrete subgroup such that I ' \ G is non-compact but has finite volume. Let P C G be a parabolic subgroup, P = MU with M its Levi component and U its unipotent radical. Let M ° be the intersection of the kernels of IX[ for all continuous characters X : M --* R x . We say that P is F-rational if (F fq U)\U is compact. In this case if we let r M o be the projection of r D p0 onto M °, then FM o \ M ° again has finite volume. A smooth function T on F \ G is called cuspidal if for every I~-rational parabolic P = M U we have
~(ug)du -- 0 (rnu)\v
25
for all g E G. Consider the space L2(F\G). This gives a unitary representation of G acting by right translation. L2(F\G) may then be decomposed into irreducible unitary representations of G. As a preliminary decomposition we have 2 L2(F\G) = L02(r\a) ¢ L,~s(F\G) • L¢on,(r\G).
L2(F\G) is the invariant subspace spanned by all cuspidal functions on F\G. It further decomposes into a discrete direct sum of irreducible representations, each isomorphism class occurring with finite multiplicity. L~es(r\G ) is the invariant subspace spanned by residues of Eisenstein series. It decompsoes into a discrete 2 sum of irreducible representations. Finally, Leont (F\G) is a direct sum of continuous direct sums of irreducible representations built from Eisenstein series induced off of equivalence classes of the F-rational parabolics. Set L~is(r\G ) = Lo2(r\a) @ Lr2s (F\G). Then we will write this decomposition as
L (r\c) = (,~cLL.(r\a) •
• Z
Z
/~(v,
r)d#(r)
{P} rCL~(F°M\M°)
where ~r(T,r) is an induced representation from P to G. Now fix N the unipotent subgroup of some F-rational parabolic. Let A = F MN, so that A \ N is compact. Let ¢ be a non-trivial character of N which is trivial on A. Let S(N\G, ¢) denote the space of smooth functions f on G satisfying
f(ng) : ¢(n)f(g) Then for
n e N.
f E S(N\G,¢) we may form the Poincar6 series ~e~\r
Under suitable regularity conditions on f (for example, compact support mod N) Pf (g) converges to a smooth function in L 2 (F\G). Then Pf (g) will have a spectral decomposition according to the decomposition of L 2 (F\G) above which we will now describe. For irreducible ~r occurring discretely in L2(F\G) and smooth ~ E 7r, let
W~(g) = / ~ ( n g ) ¢ - l ( n ) d n A\N be the associated Whittaker function. Then f will define a functional on ~r via
~-* / f(g)W~(g)dg. N\G
26
Then there will exist a smooth vector 2 ~ ( f ) E 7r representing this functional, i.e., such that
(F~(S),~)~\~ = f S(glW~(~}~9 N\G
where the inner product on the left is the natural Petersson inner product on L 2 ( F \ G ) . Similarly, one can introduce the projections Fr(~,~)(f) to the irreducible representations r ( r , r ) occurring in the continuous spectrum. The Poincar6 series then has a decomposition
Ps=
F~(S)+ Z
Z
~rCL~.¢(F\G)
~
f F~(~,~>(S)d.(r).
{P} rcL~(rMO \M ° )
Let us now introduce the Kloosterman sums relevant to this situation. With N and ¢ fixed as above, let ~t(F) = { g 6 G ] F N N g N • ¢ } . Then for g ~ a ( r ) and 7 e N g N n r we may write 7 = n1(7)an2(7) with ~ ( 7 ) e N . The ¢ - K l o o s t e r m a n sum associated to g ~ ~ ( F ) is then
Kl¢(g)
=
E ¢ ( n l (7))¢(n2 (7)). ~EA\FnNgN/A
Furthermore, for g e a ( F ) and .f e
S(N\G, ¢ )
let us define
./f(gn)¢ -l(n)dn
M(g,f) =
Ng\N
where Ng is the stabilizer of g in N under conjugation. If ~ is any smooth function in L2(F\G) then we may compute its C-Fourier coefficient, by which we m e a n C ¢ ( ~ ) = W~(1)
= ./ ~(n)¢-l(n)dn. A\N
If we do this for our Poincar6 series Pl(g) we may compute in two ways. First, if we unfold the Poincar6 series we arrive at C¢ ( P I ) =
E geN\ft(g)/N
gl¢(g)M(g,.f).
27
On the other hand, from the spectral expansion we have
rCL~di.¢(F\G)
{p}
r
Equating these two expressions gives the soft Kuznetsov formula, namely, (1.1)
~
gl¢(g)M(g,f) =
~eN\n(r)/N
2. T h e K l o o s t e r m a n - S e l b e r g
~_, C¢(F~(f)) ~CL~,.o(r\c)
Z e t a F u n c t i o n s for L o b a c h e v s k i i Space.
Now let G = SO(r + 1,1). The symmetric space of G is then r + 1 dimensional Lobachevskii space L~+I ~---{ ;
E Rr + l
I Xl2 -{- "'"
2
-Jl-Xr+ 1 < I } .
G will act on this space via linear fractional transformations. The b o u n d a r y of this space is B~, the unit sphere in R "+1 . If z E B~ is on the b o u n d a r y of Lobachevskii space, then the stabilizer of x in G will be a minimal parabohc subgroup P = M U with abelian u n i p o t e n t radical U ~- R r and Levi component M ~_ R × × S O ( r ) . Let F C G be a discrete subgroup of the type considered in Section 1. Then I ~ gives a discrete group of motions on Lobachevskii space. A point x E Br on the b o u n d a r y of Lobachevskii space is called a cusp of F if the associated parabolic subgroup P is F-rational. Fix now a cusp for F and write the associated parabolic as P = M N with N the u n i p o t e n t radical. By the B r u h a t decomposition for G we have G --- P U N w P with w the non-trivial element of the Weyl group of G. Hence we have canonical representatives for the double cosets N \ G / N via N \ G / N ~- M U wM. Set now
M ( r ) = {m e M [ F M g w m N ~ ¢}. Letting A _-- r n N as before we see t h a t we may take representatives for N\f~(r)/N of the form wm with m E M(r). Since M - R x x S O ( r ) , we m a y write m -- amhm with am E R x and h,~ E S O ( r ) . Fix now a non-trivial character ¢ of A \ N . Also fix a finite dimensional representation r of H = S O ( r ) and let ~ ( h ) be a m a t r i x coefficient of r. Then the relevant Kloosterman-Selberg zeta function for this situation is
Zr(r,s)= ~ rnEM(F)
Ktc(wm)~,.(h,-,,)la,,,I-('+~ ).
28
By comparing with the appropriate Eisenstein series to get the trivial b o u n d on Kloosterman sums, it follows t h a t the series for Z r ( r , s ) converges absolutely for
Re(s) > To analyze this function using the soft Kuznetsov formula, we need an appropriate Poincar6 series or equivalently an appropriate f E S(N\G,¢). To construct f we recall the B r u h a t decomposition G = P U gwP. Let # E C ~ ( R x ) and v E ,5(R~). Since N ~ R ~ we m a y consider r, as an element of S ( N ) . T h e n on the big B r u h a t cell set
f(n~ wn2m)
= ¢(n~ )t,(n2 )#(am ) ~
(hm)
and on the small cell set f(p) = 0 for p E P. Now, if we let y E R x C M, then R(y)f E S(N\G,¢) where R(y) denotes right translation by y. The associated Poincar$ series PR(~)f (g) is in L 2 ( F \ G ) and the soft Kuznetsov formula gives
(2.1) Kl¢(wm)M(wm,R(y)f) = meM(r)
~
C~(F~(R(y)f))
~cn~ise ( r \ a )
+ ~ Z f C¢(F~(~,,)(R(y)f))d#(r). By an apphcation of Sobolev's lemma we can show this absolutely converges pointwise in y. F u r t h e r m o r e , we may compute the Mellln trmasform term by term. If, for any s m o o t h function ~o E L2(r\G) we set
M
Then our soft Kuznetsov yields OO
(2.2)
Z
Kl¢(wm) f M(wm, R(y)J')y~-'--2d×y
mEM(F)
7rCL~i,¢( r \ c )
0
{P} aG/~
absolutely convergent for Re(s) > 2 + 1. interpreted as a c o m p u t a t i o n of IpI (r, s).
Note t h a t the left h a n d side m a y be
Suppose t h a t our character ¢ is given as follows. As N _ R ' , for n G N let n denote the associated vector in R". Let ~ E R ~ be such t h a t ¢ ( n ) =
e2~i~'n
For
29
v E 8([{"), let P denote the Fourier transform with respect to ¢ - 1 . T h e n a simple c o m p u t a t i o n gives oo
/M(wm,
R(y)f)yS-~dXy
o ?. am oo
where for # E C F ( i ~ x) we let #(s) = f # ( y ) y ' d X y .
By the discreteness of F,
o
there exists a constant C such t h a t for m E M ( F ) , lain ] > C. Since hm E S O ( r ) , a compact group, we see that { ~ h ~ I m e / ( r ) } is contained in a compact subset of R". Hence there exists v E ,-q(Rr) such t h a t ~, is identically one on this set. If we make this choice of v, then the left h a n d side of (2.2) becomes/5(s - 2 ) Z r (T, S). Hence we have PROPOSITION 1. For an appropriate choice o f v E ,9(R") and a n y / ~ E C ~ ( R × ) we have
for Re(s) > ~ + 1, where f is defined in terms of # and v as above.
As we noted above, the left h a n d side of Proposition I m a y be interpreted as
Zp,(r,,) = p(s - ~ ) Z r ( r , s ) for an appropriate choice of ]. Hence we have reduced the m e r o m o r p h i c continuation of Z r ( r , s) to the seemingly more general problem of meromorphically continuing I ~ ( r , s ) for W any s m o o t h function in L 2 ( F \ G ) . The analytic continuation of I , ( r , s) for s m o o t h ~ is based on the following two lemmas.
Ind~(I
LEMMA 1. Let ~o E 7r C Ip ® ~) be smooth. ( I I~ ~ o is a representation of P via m n ~-* lamlaa(hm)). Then I~(r, s) has a meromorphic continuation to all of C with possible poles at s =- i p - k for k = 0 , 1 , 2 , - . . . l f p ~ 0 these poles are simple. I~(T, S) Can have a pole at : k p - k only if T is a constituent of a ® R ~ where R is the standard representation of H -- SO(r) on R".
The point of this l e m m a is t h a t if T hes in an irreducible representation lr, then formulas of Wallach [6] and Casselman [1] give explicit asymptotics for W~(m) as a,~ --* 0. From these asymptotics, the continuation of I ~ ( r , s) is routine.
30
The second lemma is a shift equation due to J. Bernstein. We use such a shift equation for each generator of the center of the universal enveloping algebra U($) of $, the Lie algebra of G. For the Casimir operator this shift equation takes the following form. LEMMA 2. With appropriate normlizations, let D be the Casimir operator of G, DH the Casimir of H. For 7r an irreducible representation of G and 7" an irreducible representation of H let 7r(D) = A~ and T ( D H ) --~ )t r be the Casimir eigenvalues. Let ~ 6 7r be smooth. Then -47ri Iw(r,s) = s2 + rs + A~ + 2(r - 2)A. E
Iy,~(T ® R , s + 1)
i=1
where Y/are a basis for the Lie Mgebra of N - the opposite unipotent subgroup and I y ~ ( T ® R , s + 1) is defined with an appropriate matrix element OfT @ R. This lemma is proven by computing IDa(r, s) in two ways, once as A~I~(T, s) and the other by explicitly writing D = E X i X ~ for {Xi} an appropriate basis of $ and computing the action term by term. The other generators of the center of U($) axe treated similarly. Again, we use heavily that ~ actually lies in an irreducible representation. Now let us again look at ~ an arbitrary smooth function in L2(F\G). Then will have a spectral decomposition
~rCLa2i,¢(F\G)
{P} a 6 / f
and an application of Sobolev's lemma gives that for Re(s) sufficiently large
Ir
{p}
o-
For 7~ = P f , this is exactly Proposition 1. Let
rcL~i,o (F\G)
{P} Then to continue I ~ ( T , S ) i t is sufficient to continue I d ( T , S ) a n d I ~ ( r , s ) s e p e r a t e l y . For now, we will restrict our attention to I~a(7-,s). Using the above Lemmas, we have:
31
PROPOSITION 2. I d ( r , s ) has a meromorphic continuation to aJ1 of C. It has possible double poles at s = - k /'or k = 0 , 1 , . - . and at most single poles/'or s = 2 + p - k which can only occur i/. there exists 7r C Ldls¢(F\G ) such that 7r C Ind([ [P®~r) and ~" is a constituent o/. a ® R ~ . The continuation of I~(r, s) is achieved by a similar m e t h o d and gives possible poles at the poles of the Eisenstein series for F. If we now return to our original problem, combining the above with Proposition 1 gives the following result. THEOREM 1. The Kloosterman-Selberg zeta function Z r (% s) has a meromorphic continuation to nil of C. It has possible poles t'or Re(s) <_ ~ located at O) the points s = r k p - k if there exists a 7r C L21s¢(r\G) such that 7r C Ind([ I°®a) with r a constituent of a ® R k (ii) the points s = O , - 1 , - 2 , . . . (iii) the poles of the Eisenstein series for F. The complete details of the continuation as well as other results for Lobachevskii space are to be found in [2].
3. T h e K l o o s t e r m a n - S e l b e r g First Kind.
Zeta Function
for Fuchsian G r o u p s of the
Now let G = SL2 (R) and take F C G to be an arbitrary Fuchsian group of the first b kind. Let B = { ( 0 a-l)} be the B o r e l s u b g r o u p of G and N = {(10 1 ) } its u n i p o t e n t radical. For simplicity let us assume that F N N = A =
0
1
I
%
n E Z~. This can always be arranged by conjugation within SL2(R).
Also take
x ) ) : e 2 ~ r i . for our additive character. 0 1 Before considering the Kloosterman-Selberg zeta function, let us show how our soft Kuznetsov formula gives a version of the Kuznetsov trace formula. We will then use this formula to derive a precise formula for the KIoosterman-Selberg zeta function. In the soft Kuznetsov formula, we form our Poincar~ series P f out of a function f(g) defined using the Bruhat decomposition G = B U N w B where
¢
((1
/
w=
\( 01
--In)
T a k e / ~ E C ~ ( R x) and v E S ( R ) a n d set
U]
f(g) =
{ ¢(xl)v(x2)/z(y) g = (10 xl)o 1 0
gEB
1
0
0 )
y-1
32 Hence f(g) is essentially s u p p o r t e d in a neighborhood of w on the big B r u h a t cell. If we substitute this function into (1.1) then the left h a n d side is c o m p u t a b l e as in Section 2. If we now set M(r)={cER and let
×
lNw
0) NNP¢¢}
t C0
C_ 1
A(x) ----P(x=)tt(~) then the left h a n d side of (1.1) becomes E
Kl¢(C) A(1)
tct'
c6M(F) where we have set Kl¢(c) = Kl¢(w
(co) 0
c -1
)"
On the other hand, for ~r E L2(F\G) occurring discretely we m a y now explicitly compute the Fourier coefficients of the projections F~ (f) from the equality (F,~(/),~)r\G = (P/,~)r\v
~ 6 zr.
We c o m p u t e ( P f , T ) r \ a in terms of the Kirillov model of 7r on L2(R x) associated with the character ¢ [3]. Since our f is essentially s u p p o r t e d near w, the computation will involve the BesseI function of ~r, which we denote by J=(y), which gives the action of w in the Kirillov model of It. W h a t we finally find is that, again letting A(x) = ~ ( x 2 ) z ( ~ ) as above, C¢ (Fr (f)) = c(Tr, F) f A(x)J~ (x)d × x ~x where c0r, F ) is an arithmetic constant of proportionality between the two Ginvariant inner products on 7r, namely the Petersson inner p r o d u c t from L 2 ( F \ G ) and the Kirillov inner p r o d u c t from the model on L2(• x ). The constants c(Tr, F) are explicitly c o m p u t a b l e in terms of classical Fourier coefficients and the p a r a m e t e r of the representation. A similar c o m p u t a t i o n is valid for the continuous s p e c t r u m of L2(F\G). P u t t i n g these together, we arrive at the Kuznetrov trace formula for F. THEOREM 2. Let v 6 S(g~),# 6 C ~ ( R x) and set A(x) oo
oo
oo
CUSPS~O0
--00
+r
=
~(x2)#( ";). 1 Then
33
Using this formula, we can arrive at an exphcit formula for the KloostermanSelberg zeta function
Kl¢(c)
Zr(s)=
~
ceM(r)
lel='
and hence its m e r o m o r p h i c continuation. We return to (1.1) and instead of comput1 0 ing I / V P ' ( ( 0 1))we c o m p u t e I,V p , ( ( 0 0 form. For Re(s) > > 0, the formula in T h e o r e m 2 then becomes
Kl¢(c) B ( 1 ) = eEM(r)
c(~,r)J(B,~,s)
Z 'rcL~i,c (r\a) OO
+ Z
c(~(~),r)J(B,~0"), s) G
cusps
where now f l ( B , r , s) is the Bessel-Mellin transform GO
f f ( B , vr, s) = f
B '~Y)"J ,~(Y)Y ' ' :~'-2d×-y
--00
and B(x) = b(¢2) is an even Schwartz function. If we use Mellin inversion to express B in terms of its MeUin transform /3, then this formula can be written in a distributional form, namely
. zr(~) = ~ .
~
c(~, r)L(s)
~reL~i.c(P\G)
Now the MeUin transform of the Bessel function J , ( s ) is the v-factor %~(s) of the representation ~r, which is expressible in terms of the L - and s - functions of Jacquet and Langlands [3]. Hence the above gives a distributional equality (3.1) ~CL~I,¢ (F\G)
cusps
The right h a n d side of this equation can be shown to converge to an analytic function in a half-plane and in this half-plane the distributional equality becomes an equality of analytic functions. T h e V,~(s) occurring in the right side are expressible in terms
34
of the archimedean L-function L(Tr, s) of the representation (that is, classical Ffunctions) and can be seen to have a m e r o m o r p h i c continuation. This then gives the m e r o m o r p h i c continuation of Zr (s) and gives the precise location of its poles in terms of the spectral decomposition of L 2 ( F \ G ) . 0 To make this more precise, let ~r½+i, denote the unramified principal series representation of SL2 (R) with p a r a m e t e r 1 +iv, ~r° the c o m p l i m e n t a r y series of p a r a m e t e r r. If 7r E L L c ( F \ G ), then let al (~r) denote the first Fourier coefficient of the classical m o d u l a r form of lowest weight corresponding to ~r, normalized to have square norm one. Let Z~($~1 +iv) denote the first Fourier coefficient of the Eisenstein series formed at the cusp a off the representation lr°+i ~. T h e n formula (3.1) gives the following result. THEOREM 3. For Re(s) > ~1 we have Zr(s) =
E
ta1(~)12 sin(~:s)r(s 1 4(27r)2~_lch(Trr ) - ~ + ir)F(s
1 2
ir)
rcL~i.¢ (F\G) "/r --~~ °21.{_i~.
+
E
lal(~)l 2 4(27r)2,_1 sin(~r ) s i n ( ~ s ) r ( s
- 1 +
r)r(s - r)
~CL~i.¢ ( r \ G ) 7r___r°
1 1 r(s) 4~ ( 2 ~ ) ~ - 1 r(1 - s) _
__~°~
(_1)l(2~ ( ( 22s-1 r+ )1)
l=0
c
sin ( Trs )
+
eM~(
iz~(
F)
K/~(c) J2~+1 (~)) sin(Trs)r(s + ~)F(slcl
+ i~')12r(s - ~- + i r ) r ( s
2
1-
~)
ir)~-~
cusps cr
From this, the m e r o m o r p h i c continuation and the location of the poles can be read off rather easily. In his original paper on the sum formula Kuznetsov gave such an expression for Z r ( s ) for F = SL2(i[) and this result is a generalization of his formula. C o m p l e t e results and proofs for this section will appear in a forthcoming paper. REFERENCES 1. W. Casselman, Canonical eztensions of Harish-Chandra modules to representations of G, Preprint. 2. J.W. Cogdell, J.-S. Li, I.I. Piatetski-Shapiro and P. Sarnak, Poincard series for SO(n, 1), In preparation.
35 3. R. Godement, Notes on Jacquet-Langlands Theory, Lecture Notes, I.A.S.. 4. N.V. Kuznetsov, Petersson's conjecture for cusp forms of weight zero and Linnik's conjecture. Sums of Kloosterman sums~ Math. USSR Sbornik 39 (1981)~ 299-342. 5. A. Selberg, On the estimation of Fourier coefficients of modular forms, Proc. Symp. Pure Math VIII, A.M.S., Providence, R.I. (1965), 1-15. 6. N.R. Wallach, Asymptotic ezpansions of generalized matriz entries of representations of real reductive groups. Lie Group Representations L~ Lecture Notes in Mathematics 1024, 287-369.
Deformation of Compact Riemann Surfaces Y of Genus p with Distinguished Points P1,-.., EY
Dedicated to E. Vesentini to the occasion of his 60th birthday
Gerd Dethloff and Hans Grauert Mathematisches Institut, Universit£t Ghttingen Bunsenstr. 3-5 D-3400 Ghttingen
1
Introduction
In [G] an idea is given how in very general algebraic spaces X a jetmetric A can be constructed which has the following property: If C C X is a local complex curve then Air is a hermitean metric with negative Gaussian curvature K , where K is bounded away from zero independantly of C and X is complete in resp. of this metric. If such a metric exists then X is hyperbolic in the sense of Kobayashi, c.f. [Kob]. The construction in [G] uses families of osculating algebraic curves along C which may be punctured if X is not compact. In this paper we show more generally for families of punctured Riemann surfaces: The corresponding family of the universal coverings is real bianalytically and on the fibers holomorphicMly a cartesian product. Especially the family of the hyperbolic metrics on the punctured Riemann surfaces is a real analytic family. These results are obtained by proving that the so called Fricke coordinates of a (at least real analytic) family of punctured Riemann surfaces (see [Abi]) depend real analytically on the fiber parameter. This result probably was more or less known before by completely different methods, using quasi conformal mappings and the Teichmfiller theory (c.f. [Abi], he uses the result that the Fricke coordinates are real analytic coordinates for the Teichmfiller space without proof. From this result our results can be obtained quite easily by the theory of Teichm~ller spaces). Our proof uses the general theory of deformation of complex spaces in higher dimensions. So it is shown that this meanwhile established theory also applies to the special case of deformation of punctured P~iemann surfaces. We want to thank G. Schumacher for valuable hints for relevant literature.
38
2
Deformation
of
Surfaces
Infinitesimal
and
Punctured
Compact
Riemann
Deformation
Assume t h a t X, B are real analytic manifolds and that ,r : X --* B is a surjective, proper, smooth real analytic map. Further assume that there exists a special atlas .4 of charts of X of the following form: (w~, w2, z~,..., z ~ ) : U (C X ) -~ ~ + ~ s.th. for every x E U (z~,...z2,~) are real analytic coordinates around 7r(x) E B ; and s.th. for two such charts the w-components depend hotomorphically on each other for fixed z-components. T h e atlas `4 makes every fiber Xt := ~r-i(t), t E B a compact complex submanifold of X . We assume that every X~ is a Riemann surface of genus p. Moreover we assume that there are given real analytic cross sections s~(t), ..., s,~(t) in X over B s.th. for every t E B the points si(t),...,s,~(t) are disjoint. D e f i n i t i o n 2.1 X is called a real analytic deformation of Riemanu surfaces of genus p with m distinguished points. If additionally X, B are complex manifolds, .4 is a holomorphic atlas, and 7r : X -* B and si,..., sm are holomorphic, then X is called a holomorphic deformation of Riemann surfaces of genus p with m distinguished points. In all what follows we mean by real analytic resp. holomorphic deformation these special kinds of deformations. Next we define the infinitesimal deformations of such deformations in a point t E B . We first assume that the deformation is real analytic. Let g~ be the tangent bundle defined by the charts of .4, restricted to X , , and Y~ be the subbundle of those tangent vectors along the fibers. Let Ot resp. q~ be the sheaves of holomorphic cross sections in ~-, resp. g,. Further let (9F be the subsheaf generated by those sections of 0 t
which are -.ero in
...,
We take a covering 5/ = {[/1,..., Ul} of open subsets Ui,..., Uz in X~ s.th. for every 6 TB,t, there exist fields 77~ E H°(U~, ~t]v~) with l)
-
2) U~ is tangent along the si(t), ..., sra(~). Then the z?~, u --- 1,...,I define a O-cochain {U~} E C°(U, q t ) in the Cech complex defined by the sections of the sheaf k9, and the covering /d. It yields a cocycle d{~?~} E Zl(ld, 6);) which, by passing to the limit, yields a cohomology class p,(~) E H i ( X , , 6)~). This is uniquely determined and p, : TB,, -* H~(X,, 6);) is a real linear map. If the deformation is holomorphic then we define completely analogous using the holomorphic instead of the mixed real analytic and hotomorphic structure. We get a complex linear map p, : TB,t --* H l ( X t , 6)~) • D e f i n i t i o n 2.2 The map p~ : TB,t -+ H i ( X t , 6);) is called the infinitesimal deformation of the real analytic resp. holomorphie deformation.
39 D e f i n i t i o n 2.3 A real analytic reap. holomorphic deformation is called arable in t E B (with reap. to sl,...,sm ) if it is a cartesian product over the double point in t , i.e. if there are finitely many charts (w(O z) : U~ --* ~2,~+2, i = 1,...,t out of the atlas A with:
~) x~ c U~=1U~ a~k ~J = 0 for j ---- 1,..., m , zk being any component of z
# For any coordinate tran#ormation ~U) = ~U)(~(% ~) we have ~ ( t )
= 0
We have the following T h e o r e m 2.4 A real analytic reap. holomorphic deformation is stable in t E B with reap. to ~ , . . . , ~ iff p,(~) - o in T.,,. To prove that, it is first shown by using the long exact cohomology sequence belonging to the short exact sheaf sequence
t h a t p,(~) - 0 ifffor all ~ e TB,~ there exists afield U • q4(X,) with v.(77) _= ~. Now it can easily be shown t h a t the existence of such fields is equivalent with the existence of special charts in .4 like in Definition 2.3. For more details c.£. [KS,I]. E]
3
Existence of Semiuniversal Deformation
D e f i n i t i o n 3.1 A real analytic (reap. holomorphic) deformation 7r : X --* B is called aemiuniveraal in t E B if: 1) If ¢ : Z ---* G ia an other real analytic (reap. holomorphic) deformation with 0 e G and Zo = X t then there is a real analytic (reap. holomorphic) map a : U(O) --~ B with a(O) = t a.th. X o a = X ×~ U is isomorphic to Zlrr under an isomorphism which ia on Zo = X t the identity. 2) The total derivative da : TG,o --+ Ts,t is uniquely determined. T h e semiuniversal deformation is uniquely determined always. Remark: It is possible to prove that it is universal, i.e. t h a t a itself is uniquely determined. L e m m a 3.2 If Y is a compact Riemann surface of genus p with distinguished points P1, ..., P,~ E Y there exists a holomorphic aemiuniveraaI deformation ~r : X --* B C ¢ '~ with Y = X o , Pi = s~(O),i = 1 , . . . , m and n = d i m e H l ( X 0 , O ; ) . Proof:. We have H2(Y, 0~) = O. So if we forget about the distinguished points in Y for a moment there exists a seminuiversal deformation of compact Pdem~nn surfaces (in the ordinary sense) ~'~ : X ' --~ B I with X~ = Y and bijective infinitesimal deformation P0:Ts,,0 --* H I ( X ~ , ®0), cf. [KNS], [KS,II].
40 If we now additionally have holomorphic cross sections sl, ..., s,~ through PI,--., pro, we get r := max{0, m - max{3 - 2p, 0}} additional deformation p a r a m e t e r s , which are independent f r o m the others. So we have a holomorphic deformation (again of our kind, cf. Definition 2.1) 7r : X -~ B ' x 6g~ s.th. X0 = Y and p0 : TB,x~,0 --~ H i ( X 0 , O~) again is bijective. The last two statements can be proved by applying the theorem of K i e m a n n - R o c h . Now we still have to show that this family is complete in t = 0 since then, by the bijectivity of p0, it is semiuniversal in the zero p o i n t . This can be proved like in [KS,II], but we also can get this immediately by using a general t h e o r e m stated in [F1]. [] 3.3 If ~r : X --* B is semiuniversal in t E B for holomorphic deformations, then it there also is semiuniversal for real analytic deformations.
Lemma
Pro@ Assume t h a t • : Z --* G is a real analytic deformation with 0 E G and Z0 = Xt • We m a y assume t h a t G C ~ z with coordinates xl, ..., x l . We take a small domain G C ~l with complex coordinates :cl,..., xl s.th. G = G n JT~l and everything can be extended holomorphically onto G: Thus we get a holomorphic deformation {J : Z ~ G and disjoint cross sections s'l,..., s ~ . Since zr : X -~ B is semiuniversal for holomorphic families we obtain a holomorphic m a p & : U(0) --* B with &(0) = t s.th. ,~lu = X o & and d& is uniquely determined. By restricting a = &luna we have Zlcrnc = X o a . Since d& m a p s infinitesimal deform a t i o n onto the same infinitesimal deformation and this remains true under restriction of TO,o to Ta.0 and ~ to a the m a p do is uniquely determined. []
4
Real Analytic Triviality the Universal Coverings
of
the
Family
of
We now wish to construct a real analytic deformation • : Z --* G with Z0 = Y , where Y is again a given compact R i e m a n n surface with distinguished points P1,---, P-~, and with bijective P0 : Ta,o --~ HI(Zo, 0~) and on G real analytic Fricke coordinates (c.f. [Abi] or see below). We need some preparations: We put Y ' = Y \ { P 1 , . . . , P , ~ } and assume from now on p > 2 or p = l and m _ > l or p = 0 and m > 3. T h e n the universal covering of Y' is the u p p e r half plane H , and Y~ can be represented as the quotient of H by a Fuchsian group F which is finitely generated and which acts fixpointfree and properly discontinuous on H . Let II1 be the f u n d a m e n t a l group of y I . It is generated by loops e l , a~, ..., ap, a~, ill, ..-, fl,~ with the following properties: 1) T h e intersection n u m b e r of a~, a* is 1. 2) All the other loops only have the base point in common. T h e i r intersection n u m b e r is zero. If m = 0 then the loops e l , a~, ..., aT, a T generate II1. There is exactly one relation
4] K m ~ 0 H1 is generated by a l , ...,fl,~ and is free with the 2p + m - 1 generators al,---,tim-1 - We have an isomorphism X : H~ --+ F. Let × ( ~ ) _ a~z + b~
a ; z + b;
X ( / ~ ) - ¢~z + b'~
where the determinants always are one and all coefficients are real. Since the X(a~), X(c~*) are hyperbolic, the X(fl~) are parabolic ([Abi,p.42]), the relation has to be satisfied and H has a three parameter group we have 2(3p - 3 + rn) real parameters from which all the coefficients can be recovered ([Abi,p.64]). (Both references don't depend on the Teichmfiller theory or quasi conformal mappings.) Now we can construct the desired real analytic deformation: We take G C ~Tsp-3+'~ with 0 E G and define 2p-t-rn automorphisms a~(t)z + b~(t)
a;(t)z + b;(t)
~,(t)z + b'~(t)
c,(¢)z + d,(t)
c~(t)z + d;(t)
ff~(t)z + d,(t)
(1)
of H with coefficients depending real anatyticMly on t s.th. the coefficients satify our conditions (hence we only m a y define 2(3p - 3 + m) coefficient functions, the others are fixed by these conditions), the quotient of H by these automorphisms for t = 0 yields Y ' , and s.th. for no ~ E To,0, ~ ~ 0 the derivatives of the coefficient functions in ~-direction vanish simultaneously. Let Ft be the group generated by the automorphisms of (1) for ¢ E G and FG := F~, t E G. Then (possibly after having made G smaller) for all t E G the group F, is properly discontinuous, fixpointfree and finitely generated: Let Do be a fixed fundamental region of F0. If G is small enough one can (possibly after having changed the generators of Fa for a moment), by the action of 'one letter words' w(t) E Fa in a small neighborhood of Do, find 'fundamental regions' D~ which depend real analytically on t. Then we always have w(t)(]9~) VI/)~ = ~ (where /9~ denotes the open kernel of Dt ), for longer words g(t) E F a , of course, t h a t equation needn't hold. H is covered by the translates of D, given by the elements of Y~ for every t E G. We have to show that the open kernels of those translates are disjoint. Assume t h a t for some to two of them aren't disjoint. Then there also exists a g0(t0) E F,o with £}*0gl g0(t0)(D.', o) ¢ 0. Let ~ : [0,1] ~ G with ~(0) = to, ~(1) = 0 be any curve. If g0(0)(D0) M Do = 0 we obtain by multiplying rn > I 'letters' wj from the left: There exist open intervals I0,...,Iz C /R with LJ~=0/~ D [0,1], 0 E I0, 1 E I~, [0, 1] V11~~ 0 for all i = 0, ..., l and g~(t) := w,(t)...., wl(t), go(t) E Fa, i = 1, ..., l with g,(t)(D,) M ~)~ 7~ 0 for t E ~(I~ V1[0, 1]). We have g~(0) = id since F0 was properly discontinuous and fixpointfree. Since the relations are kept while deforming. Fo to F, we have g, = id on G. But then we have g,_~(*)(D,) n D, = wF~(t)(D,) n D, = 0 by construction of D, for all t E G, which is a contradiction. So we have g0(0)(/)0)M/J0 ~ ~, hence go(0) = id and then by the same argument as above we have go = id on G which proves our assertion. Therefore ( H x G)/Fa is a well defined family of Riemann surfaces with H/Fo = Y' on which H x G canonically yields real analytic and on the fibers holomorphic charts.
42 In order to get a real analytic deformation ~ : Z --+ G like in Definition 2.1 from (H x G)/I~G, we have to 'fill in the punctures'. If .5' resp. S' denote the underlying topological spaces of Y resp. o f Y ' = Y\{P1,...,P,,~} and s~(t):= (P~,~) E S x G , i = 1,...,m, that means that we can find real analytic and on the fibers holomorphic charts (with resp. to those charts on (H x G ) / F a , the underlying topological space of which is by construction S' x G since the real analytic and on the fibers complex structure of ( H x G ) / F a can be given through the automorphism X* : II1(5'1) --* r , ) around atl points s~(t) E S x G, ~ E G,i = 1,...,m, with the following properties: All those chaxts give an atlas .4 (cf. Definition 2.t) on S x G in such a way that ( H x G ) / F a = Z \ {sl(G), ..., s,~(G)}, where Z denotes S x G with the structure given by v4, the fibers of Z are compact Riemann surfaces of genus p, and the s~(~), t E G are real analytic cross sections in Z . Let Q -- si(t0) be arbitrary. Let V be a small neighborhood of Q in S x G . Then by the isomorphism X: :IIl(SI) -~ r , the deck transformation fl~(t) covers small simple loops in S' x {t} around the 'puncture' st(t). fit(t) has exactly one fixed point F(t) E 0A (where we identify /-/ and A through a fixed biholomorphic map) which depends real analytically on t E G. So there exists a family of rotations in the fibers depending real analytically on ~ which maps H x G real bianalytically and on the fibers holomorphicatly onto itself s.th. the border points F(~) are mapped to infinity. Now fit(t) is a translation z --* z + b(t), where b(t) depends real analytically on ~. So the automorphism (w, z) --* ( b ( t ) - l w , z) on H x G again is real bianalytic and on the fibers holomorphic. At last we map /-/x G to /k x G by (w, z) -> (ei~', z). Then the invers image of V(Q) in A x G is the intersection of an open neighborhood W((0, to)) C A x G with (A \ {0}) × G which, by construction and the properties of/3t(t), is mapped real bianalytically and on the fibers holomorphically to
v \
So we have constructed the desired charts and hence the real analytic deformation, the Fricke coordinates of which, i.e. the 2(31o - 3 + rn) coefficient functions in (1) which weren't given by the restructions, depend by construction real analytically on G. Since by the theorem of 1R.iemann-Roch we have dirr~ Ht(Zo, 6)~) --- 3p - 3 + m , we only have to show that the map P0 : TG,O--~ Hi(Z0, E)~) is injective. ~2(3p--3-{-m) 0 Let ~ = 2_,u=1 a~,b-(~ ~ E TG,o, ~ ¢ O, L -~ .It = (al, ..., a2(3p-a+m))$, s E .~} V1 G be the real line in ~-direction, and 0t be the double point in 0 E L. If we restrict the transformations (1) to 01 their deformation is by construction not zero. Hence the restriction of Z IL to 01 has nonvanishing infinitesimal deformation: The complex structure on the fibers is parametrized by the elements of I~G up to the automorphism group of H . Since in the family (1) of generators of I?a the influence of this automorphism group already has been thrown out by a reduction of the number of free parameters (by our conditions for the coefficient functions) this means that the complex structures of the fibers axe 'changing over 01'. Hence ZIL cannot be stable over 01, since otherwise there would exist real analytic and on the fibers biholomorphic coordinates which would make ZI/. a cartesian product over 01, but then the complex structures of the fibers couldn't 'change over 01 '. Hence the infinitesimal deformation of ZIL in the zero point isn't zero. Since ~ E Ta,o, ~ ~ 0 was arbitrary this means that the infinitesimal deformation of
43 : Z -+ G is i n j e c t i v e in t h e zero p o i n t . desired properties.
So o u r d e f o r m a t i o n
t~ : Z ---+ G
h a s all
W e n o w c a n p r o v e t h e following: 4 . 1 Let ~r' : X ' --+ B' be a real analytic deformation (like in Definition 2.1). Then its Fricke coordinates (more ezactly those of ~r' : X ' \ (sl,..., s k ) -+ B' ) depend real analytically on B ' . Especially the universal covering f(' of X ' \ {s'~,...,s') is real bianalytically and on the fibers holomorphically equivalent to H × B t, where H denotes the upper half plane, and the hyperbolic metric on the fibers of Z ' \ {s~, ..., s ' ) depends real analytically on the fiber parameter ~ C B I .
Theorem
Proof. L e t t E B t b e a r b i t r a r y . T h e n w i t h L e m m a 3.2 t h e r e e x i s t s a h o l o m o r p h i c s e m i u n i v e r s a l d e f o r m a t i o n 7r : X -+ B C (P~ w i t h X0 = X~ a n d n = dirn~ HI(Xo, ®~). F r o m L e m m a 3.3 we k n o w t h a t lr : X -+ B is also s e m i u n i v e r s a l for r e a l a n a l y t i c deformations. Together with our preceding construction this means: 1) T h e r e e x i s t s a r e a l a n a l y t i c m a p p i n g a ' : U(t)(C B t) --+ t3 s.th. X ' l u ~ X o a t . 2) T h e r e e x i s t s a r e a l a n a l y t i c d e f o r m a t i o n • : Z -+ G like c o n s t r u c t e d a b o v e w i t h Z0 -- X0 a n d w i t h a n in t h e zero p o i n t b i j e c t i v e i n f i n i t e s i m a l d e f o r m a t i o n . T h e r e f u r t h e r e x i s t s a a r e a l a n a l y t i c m a p a : V ( 0 ) -+ B w i t h Z[v -~ X o a . Since da m a p s i n f i n i t e s i m a l d e f o r m a t i o n to t h e s a m e i n f i n i t e s i m a l d e f o r m a t i o n a n d t h e i n f i n i t e s i m a l d e f o r m a t i o n of • : Z --+ G in 0 is b i j e c t i v e , da is i n j e c t i v e . Since d i m e B = dirn¢ HI(Xo, G~) = d i m e G t h e m a p da is b i j e c t i v e . B y t h e i n v e r s f u n c t i o n t h e o r e m a -1 : B --+ G exists l o c a l l y a r o u n d 0 E B . So we h a v e X ~ Z l o c a l l y a r o u n d 0 t h r o u g h t h e r e a l b i a n a l y t i c m a p a . E s p e c i a l l y t h e F r i c k e c o o r d i n a t e s of X d e p e n d r e a l a n a l y t i c a l l y on B since t h e F r i c k e c o o r d i n a t e s o f Z d e p e n d r e a l a n a l y t i c a l l y o n G , b o t h a r o u n d t h e zero p o i n t . Since b y 1) t h e f a m i l y ~rt : X t --+ B t c a n b e o b t a i n e d b y lifting f r o m ~r : X -+ B t h e F r i c k e c o o r d i n a t e s o f X t d e p e n d r e a l a n a l y t i c a l l y o n S t" T h e o t h e r b o t h a s s e r t i o n s a r e a n i m m e d i a t e c o n s e q u e n c e o f t h i s fact.
[]
R e m a r k : t n t h e cases of d e f o r m a t i o n s w i t h p = 0, m < 2 a n d i0 = 1, m = 0 (c.f. D e f i n i t i o n 2.1) we c a n n o t d e f o r m b y using a F u c h s i a n g r o u p . In t h e case p = 0, m _< 2 t h e d e f o r m a t i o n is t r i v i a l , a n d for p = 1, ra = 0 t h e d e f o r m a t i o n is p a r a m e t r i z e d b y the upper half plane H.
44
References - [Abi]: Abikoff, W.: The Real Analytic Theory of Teichmiiller Spaces. Berlin, Heidelberg, New York. Springer 1980 - [F1]: Flenner, H.: Uber Deformationen holomorpher Abbildungen. Habilitationsschrift. Osnabr/ick 1978 [G]: Grauert, H.: Jetmetriken und hyperbolische Geometrie. Math. Z. 200 (1989) [KNS]: Kodaira, K., Nierenberg, L., Spencer, D.C.: On the Existence of Deformations of Complex AnaJytic Structures. Ann. of Math. 68, 450-459 (1958) - [KS,I]: Kodaira, K., Spencer, D.C.: On Deformation of Complex Analytic Structure I, II. Ann. of Math 67, 328-468 (1958) - [KS,II]: Kodaira, K., Spencer, D.C.: A Theorem of Completeness for Complex Analytic Fiber Spaces. Acta Math. 100,281-294 (1958) - [Kob]: Kobayashi, S.: Intrinsic Distances, Measures and Geometric Function Theory. Bull. Am. Math. Soc. 82, 357-416 (1976)
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On Moduli of Vector Bundles Shoshichi K o b a y a s h i D e p a r t m e n t of Mathematics, University of California, Berkeley
1. I n t r o d u c t i o n .
We wish to discuss here moduli spaces of simple vector bundles on a compact K~hler manifold M from differential geometric viewpoints, placing emphasis on the case where M is symplectic Kg~hler. In order to explain the construction of such moduli spaces, we consider first an analogous construction of the moduli space of complex structures on a differentiable manifold M. Let ¢4(M) be the set of almost complex structures J on M. Each almost complex structure J gives rise to a decomposition d = d' + d" of exterior differentiation d. Then J is integrable if and only if d"o d" = O. Let C ( M ) b e the set of integrable almost complex structures on M. The group :D(M) of diffeomorphisms of M acts on .A(M) and C(M). Then the moduli space of complex structures on M is given by C(M)/~(M). Now, fix a C °o complex vector bundle E of rank r over a compact complex manifold M. A compatible almost complex structure in E is given by a differential operator D " sending each section of E to an E-valued (0,1)-form on M and satisfying conditions similar to those for covariant differentiation. The integrability condition is given by D JtoDtt = 0. We denote the set of integrable D" by ~"(E). As an analogue to the group of diffeomorphisms of M we consider this time the group GL(E) of automorphisms of E (inducing the identity transformation on the base manifold M). Then the group GL(E) acts on ~ " ( E ) , and the space of orbits, 7"I"(E)/GL(E), is the moduli space of holomorphic structures in E . However, in order to provide this moduli space with a good complex analytic structure, we have to restrict our consideration to simple holomorphic structures. A holomorphic vector bundle is said to be simple if it admits no (sheaf) endomorphisms other than scalar multiplications. The set of integrable D " that give rise to simple holomorphic structures is denoted 7~"(E). The moduli space A~I(E) = ~"(E)/GL(E) of simple holomorphic structures in E is a (possibly non-Hausdorff) complex analytic space (in generally, not reduced). By considering only stable holomorphic structures in E or Einstein-Hermitian structures in E we obtain a Hausdorff moduli space. Given an Hermitian structure h in a C °~ complex vector bundle E over a compact KEhler manifold M, the moduli space ~ ( E , h) of irreducible Einstein-Hermitian connections in (E, h) carries a natural K/thler metric while AA(E) may not. In spite of apparent advantages of Ad(E, h) over Ad(E), we consider here
46 mainly the latter since it is much easier to keep track of various holomorphic objects on A~I(E) than those on A74(E, h). It is also much easier to explain the algebraic concept of simple vector bundle t h a n that of Einstein-Hermitian vector bundle. This is somewhat analogous to the situation for a compact homogeneous K~hler manifold V which can be expressed either as a homogeneous space V = K / L of a compact Lie group K ofisometries or as a homogeneous space V = G / H of a complex Lie group G of holomorphic transformations; while metric properties of V can be seen more easily from K / L , it is often more convenient to use G / H in order to study holomorphic objects on V or algebraic properties of V. While K / L and G / H are diffeomorphic in a natural way, A~/(E, h) is open in A:4(E) and does not exactly agree with ¢Q(E). However, it is often sufficient to study A~I(E) in order to understand .~i(E, h). We generally use the notation of [9]. 2. H o l o m o r p h i c S t r u c t u r e s o f V e c t o r B u n d l e s . [5,6,9,10] We fix an n-dimensional compact K~hler manifold M and a differentiable complex vector bundle E of rank r over M. We wish to consider the set of holomorphic vector bundle structures in E compatible with the given complex structure of M and the complex vector bundle structure of E. Let AP,q(E) be the space of C ~ differential (p, q)-forms on M with values in E, and set At(E) = ~
AP'q(E) •
p+q=r
If E is a trivial line bundle, these spaces are denoted simply Ap,q and A r. Let 7)H(E) denote the set of C-linear maps
D" : A°(Z) --* A°'I(E) satisfying
D"(fs) = (d"f)s + f . D"8
for
s e A°(E), f • A °.
Every D ~ extends uniquely to a C-linear map
D" : AP'q(E) -'~ AP'q+I(E) satisfying
D"(¢a)=d"¢^a+(-1)r+SehD"a
for
a e A P ' q ( E ) , ¢ E A ~'8.
The total space of a C c¢ complex vector bundle E over a complex manifold M has no
natural almost complex structure. But each D" E 7)~'(E) gives rise, in a natural manner, to an almost complex structure on E that is compatible with the complex structure of M and the complex vector space structure of the fibers. Then the almost complex structure is integrable if and only if D ~t satisfies the integrability condition: D " o D II = 0. The holomorphic vector bundle defined by E and an integrable D ~ will be denoted E D". Then an integrable D ~ is nothing but d ~t of the holomorphic vector bundle E D''. Let ~"(E) C ~)"(E) be the subset consisting of integrable D".
47 We know that 7)"(E) is a complex affine space (of infinite dimension). In fact, if we
fix Dg E ~ ' ( E ) , then for every D" E ~ " ( E ) the difference a = D" - Dg is a map a : A°(E) --* A ° ' I ( E ) which is linear over A °, and a can be regarded as an element of A 0,1 (End(E)). Conversely, for any element a E A°'I(End(E)),D" = Dg + a is an element of :D'(E). Thus, once Dg is chosen and fixed as the origin, :D"(E) can be identified with the complex vector space
A°A(Znd(E)). However, TI'(E) is not aza affine subspace of I)"(E). In fact, the formula D" o D " = D ~ o D ~ + D ~ o a + a o D ~ + c ~ A o t shows that when Dg is integrable, the integrability condition for D " is given by
D~a + a A a = O, which is quadratic in a. Let GL(E) denote the group of C ~ bundle automorphisms of E (which induce the identity transformation on M). The space of Coo sections of the endomorphism bundle End(E) = E ~ E* will be denoted g l ( E ) and is considered as the Lie algebra of GL(E). The group GL(E) acts on 7)"(E) by
Dr' ~ D ul = f - 1 o D" o f = D" + f - l d n f , where D" e 7)"(E) and f e GL(E). Then GL(E) sends 7"/'(E) into itself. Two holomorphic structures ~ n ,1 , n~ -2 E 7-/'(E) of E are considered equivalent if they are in the same GL(E)-orbit. The space .hA(E) = 7~"(E)/GL(E) of GL(E)-orbits with the C°°-topology is the moduli space of holomorphic structures in E. Let [D"] denote the point of AA(E) represented by D" E 7"l'(E). In order to describe the tangent space to Ad(E) at [ D ' ] , let D~' = D " + a7 be a curve in 7-/"(E), where 0 7 e A°a(End(E)) and a g = 0. Then
D .a. t +
a~~A a~~ = 0 .
Differentiating the above equation with respect to t at t = 0, we obtain Drt o~u ~ O, tl
where a " = Ota t It=0. If D~' is obtained by a 1-parameter family of transformations f t E GL(E), i.e.,
DI[ =
ft 1 o Dtt o ft
with f0 = IE, then
D~I = D" + '~t,"
where
'~t" = fi - 1 D I!It-
48 If we set
O~tl= Otc~tHIt=O,
f = •tftlt=o,
then
a"= D"f. Hence, the tangent space T[D,,](A,~(E)) is given by H0,1(M ' End(ED,,)) = {a" e A°'l(End(E));D"a " = 0}
{D"f; f e A°(End(E))} provided that M(E) is nonsingular at [D"]. (Here, H°,I(M, End(ED")) is the (0,1)-th d~-cohomology of the bundle End(E D'') in the sense of Dolbeanlt). Every holomorphic vector bundle E I)" admits nonzero endomorphisms, namely clE, c E t2. If these are the only endomorphisms of E D', i.e., if H ° ( M , O(End(ED"))) = 12, then E D'' is said to be simple. It is convenient to introduce the subbundle End°(E D') of End(E D'') consisting of trace-free endomorphisms. Then E D'' is simple if artd only if
H°(M, O(End°(ED"))) = O. Let ~ " ( E ) be the subset of TI"(E) consisting of D " such that E D'' is simple, and let
,(4( E) = 7:I"(E)/GL( E). Then we have T h e o r e m (2.1). The moduli space ,~A(E) of simple holomorphic structures on E is a (possibly non-Hausdorff and non-reduced) complex analytic space. It is nonsingular at [D"] /f H2(M, (9(End°(ED"))) = 0, and its tangent space at [D"] is isomorphic to HI(M,O(End(ED"))). Even if [D tr] is a singular point, Hi(M, O(End(ED"))) is still isomorphic to the Zariski tangent space of A~(E) at [D"]. R e m a r k . Fix an Hermitian structure h in E. Let 79(E, h) denote the set of connections D = D ~ + D " in E preserving h,7"l(E,h) the subset of consisting of D such that D " o D" = O,TIE(E,h) the set of Einstein-Hermitian connections, and 7~e(E, h) the set of irreducible Einstein-Hermitian connections. Let U(E,h) denote the subgroup of GL(E) consisting of automorphisms of E preserving h. We set A4(E, h) = 7-le(E, h)/U(E, h) and 2~4(E, h) = ~e(E, h)/V(E,h). Then the moduli space AS~(E, h) of irreducible EinsteinHermitian connections is Hausdorff and open in A~(E) and carries a natural K~hler metric (on its nonsingular part). For details on Einstein- Hermitian connections, see [9]. If L is a trivial llne bundle over M, then the moduli space A,t(L) is by definition the Picard variety Pic°(M) of M. The group Pic°(M) acts on AA(E) essentially freely. Fixing a holomorphic structure D ~ in a C °° complex vector bundle E, we set
C = E D''. Then each element Z: E Pic°(M) gives rise to a holomorphic vector bundle £ • Z: with the same underlying C °~ complex vector bundle E. Clearly, C and C • Z: have the same associated projective bundle, i.e., P ( £ ) = P ( E . £:).
49 Conversely, if two holomorphic vector bundles C and C' have an isomorphic assocaited projective bundle P ( £ ) ~ P(Cr), then E r -~ ~ •/: for some holomorphic line bundle £:. If £ and C' have the same underlying C °O vector bundle E , then this line bundle £ is in the Picard variety P i c ° ( M ) . It may happen that C - £ is isomorphic to C. Following Mukai, we set
~°(E) = {~ E Pic°(M); E. £ ~- ~}. If L: • S°(E), then det(E) ~- det(C). £r. Hence, ~0(~) is contained in the r-torsion of Pic°(M). It follows that the moduli space 2k4(E) is fibered by P i c ° ( M ) / E ° ( C ) . On the other hand, by fixing a simple holomorphic structure D~ • 7-/~'(E) and setting ~o = E D'°', we can define a fibering A//(E) --* P i c ° ( M ) by
£ ~-~ det(C), det(Co) -1. We defined two complementary fiberings of A~(E). The decomposition
H I ( M , O ( E n d ( C ) ) ) ~_ H 1 ( M , O ( E n d ° ( £ ) ) ) + H I ( M , O ) defines a splitting of the tangent bundle of ATe(E) into two complementary subbundles. These subbundles correspond to the two fiberings above. R e m a r k . On A74(E, h), the decomposition above defines two parallel distributions of ¢Q(E,h). The following construction is consistent with the second fibering above.
Each nonzero holomorphic 1-form ~ on M induces, in a natural way, a nonzero closed holomorphic 1-form ~ on ](.4(E). For each tangent vector (~ E A°'l(End(E)) at D r' • T~"(E), we set
= f.
A tr( ) A
where ~ is the Kdhler form of M. Then ~ is a holomorphic 1-form on Z)"( E). Its restriction to TI"(E) projects down to a holomorphic I-form (a on A~I(E). In fact, we show (i) that if a comes from the infinitesimal action of gl(E) so that a = D " f for some f • A°(End(E)), then ~3(cz) = 0. Since tr(~) = t r ( D " f ) = d"tr(f), we have
~(~) = - JM d(~. t r ( f ) A ~n-1) = O. We show (ii) that ~ is invariant under the action of G L ( E ) , If f • G L ( E ) , then f sends a • A ° , l ( E n d ( Z ) ) ~_ TD,,(I)"(E)) to f - l c ~ f • A°,l(End(E)) ~ TD,,S(7)"(E)). Since t r ( f - l c ~ f ) = tr(a), it follows that
This construction generalizes to a holomorphic p-form.
Each holomorphic p-form qa on M induces, in a natural way, a closed holomorphic
p-form
on Xa(E).
50 For example, if q0 is a holomorphic 2-form on M, then we set
for a , ~ E A°'l(End(E)) ~- TD,,(1)"(E)). J.f~ is a p-form, then in order to define qb(al,--. ,up) for a l , . . . , a p E A°'l(End(E)) TD,(~)"(E)), we need to skew-symmetrize tr(al A . . . A up) in the definition. If we use t r ( a l ) A . . . A tr(ap) in place of tr(al A . . . A uP), we would get only a p-form arising from the fibering A t ( E ) -~ Pic°(M). 3. O n C u r v a t u r e o f M o d u l i S p a c e s o f B u n d l e s o v e r C u r v e s . [1,3,5,8,9] Let M be a compact Riemann surface of genus g, and let E be a C ~° complex vector bundle of rank r over M. It follows from Theorem 2.1 that the moduli space A~i(E) of simple holomorphic structures on E is a (possibly non-Hausdorff) nonsingular complex manifold of dimension r2(g - 1) + 1. For a fixed line bundle £, the moduli space A~4(E, £) of holomorphic structures with prescribed determinant bundle L: has dimension r2(g - 1) + 1 - g = (r 2 - 1)(g - 1). Since D"o D" = 0 automatically when dimM = 1, we have :D"(E) = 7-/t'(E). It follows that 7~"(E) and hence AbI(E) are connected. Fix an Hermitian structure h in E. Then the moduli space A~4(E, h) of irreducible Einstein-Hermitian connections in E (or equivalently, the moduli space of stable holomorphic structures in E) is Hausdorff and open in A~t(E). Its natural K~hler metric has nonnegative holomorphic sectional curvature and hence nonnegative scalar curvature. If the holomorphic sectional curvature is identically zero, then ~ e ( E , h) is totally geodesic (and hence flat) in H ( E , h) = :D(E, h), and it follows that either M is an elliptic curve or the rank of E is 1. Except in these trivial eases, the scalar curvature of M(E, h) is strictly positive somewhere. It follows that except in the trivial cases where A~i(E, h) is a complex torus, the phricanonical genera of ATi(E, h) axe all zero provided A~t(E, h) is compact. In the decomposition
H'(M, O(End(e))) ~_ Hi(M, O(End°(C))) + HI(M, O) of the tangent space at D" E He(E, h), the holomorphic sectional curvature vanishes in the direction of Hi(M, 0). The question remains whether it is positive in the direction of
Hl( M, O( End°( ED") )). Although we can compute the curvature of 2~I(E, h) explicitly using the Gauss equation for CR-submersions [8], the fact that the holomorphic sectional curvature is nonnegative can be understood intuitively from the general principle that the holomorphic curvature decreases with a holomorphic subbundle and increases with a quotient bundle. Thus, being a holomorphic quotient of a flat space ?-l"(E) = 7)"(E), the moduli space A,t(E) should have nonnegative holomorphic sectional curvature. However, if dimM _> 2, then H~(E) is a complex submanifold of:D"(E) and hence has nonpositive holomorphic sectional curvature. So passing to the quotient A4(E), we should get both negative and positive terms in the expression for its holomorphic sectional curvature. In other words, when
51 d i m M >>2, all we can say is that the holomorphic sectional curvature of ,~A(E) is no more negative than that o f / - / ' ( E ) . 4. H o l o m o r p h i c S y m p l e c t i c S t r u c t u r e s . [9,12] We consider now K~ihler manifolds with holomorphic symplectic structure or hyperKiihler manifolds. While Atiyah's article in this volume treats hyper-K~ihler manifolds more from the real or quaternionic viewpoint, we shall emphasize the complex analytic viewpoint. For the quaternionic apporach to Theorems 4.1 and 4.2, the reader is referred to Atiyah's article and references therein. A hyper-Kiihler manifold is a Kiihler manifold with a nongenerate holomorphic, parallel 2-form. In terms of holonomy, it is a K~hler manifold of even dimension, say 2m, whose holonomy group is contained in Sp(m). Such a Kiihler manifold is automatically Ricci-flat. The classical de Rham decomposition of a Riemannian manifold into irreducible factors takes a much more precise form in the special case of a Ricci-flat compact Kiihler manifold. In fact,[7]: If X is a compact Ricci-flat Kiihler manifold, then it has a finite, unramified covering X = T x Ml x . . . x
Mj x Nl x . . . x
Nk,
where T is a flat, complex torus, the Mi are compact, simply connected irreducible hyperKghIer manifolds, and the Ni are compact, simply connected irreducible Kfihler manifolds which have no holomorphic forms except for the form trivializing the canonical line bundle. In terms of holonomy, the Mi have holonomy Sp(m~),(where 2mi = dimMi), and the N~ have holonomy SU(ni), (where ni = dimN~), while T has trivial holonomy. Thirty years ago, Wakakuwa [16] wrote a paper on topology of hyper-K~ihler manifolds. Among other things, he proved that all odd dimensional Betti numbers are divisible by 4. However, for lack of interesting examples, Wakakuwa's paper has been long forgotten. Ten years ago, Bogomolov announced that there are no irreducible compact hyper-Kiihler manifolds in dimension greater than 4. However, the first examples of 4-dimensional compact irreducible hyper-K~.hler manifolds were discovered by Fujiki in 1981. He started with a K3 surface S and showed that a holomorphic (symplectic) 2-form of S induces a holomorphic symplectic form on the 4-dimensional complex manifold obtained by blowing up the 2-fold symmetric product S(2) along its diagonal,see [4]. His construction has been further generalized by Beauville to yield other examples,(see [2],[13]). However, there are still relatively few examples of compact hyper-K~itfler manifolds. It seems natural to consider more generally holomorphic 2-forms which may be degenerate along proper subvarieties. These generalized holomorphic symplectic manifolds occur as non-singular models of k-fold symmetric products X(k) of holomorphic symplectic manifolds X. They occur also as moduli spaces of simple or stable holomorphic vector bundles over a compact holomorphic symplectic manifolds as we see in this section. In order to discuss such moduli spaces, we consider infinite dimensional holomorphic symplectic manifolds. Let V be a complex Banach manifold. A holomorphic symplectic structure on V is given by a holomorphic 2-form wy which is closed and non-degenerate in the following sense: ~v(X,Y)=O
V}" E T x V
:a X =O.
52 This condition means that the linear map T , V -~ T*V defined by wv is injective (but not necessarily bijective). Let G be a complex Banach Lie group acting on V as a group of holomorphic transformations preserving the symplectic form w v . Let g be its Lie algebra and g* the dual (Banach) space of g. A holomorphic m o m e n t u m map for the action of G on V is a hoIomorphic map # : V --* g* such that
< A, d # ( X ) > = w ( A , , X )
A E g, X E T , V ,
for
where Ax E TxV is the vector defined by A through the action of G and d# : T x V ~ g* is the differential of # at x E V. We impose the following three conditions (a), (b), (c) on #. (a) Assume that # is G-equivariant in the sense that ,(g(x)) = (adg)*(,(x))
for
g e V, x e V,
where g ~-* (adg)* is the coadjoint representation of G. (b) Assume that 0 E g* is a weakly regular value of # in the sense that (b.1) #-1(0) is a complex submaaifold of V, (j : # - 1 ( 0 ) ~ V denoting the imbedding), (5.2) the inclusion dj : T x ( # - l ( 0 ) ) C Ker(d#x) is an equality for every x E #-1(0). (c) Assume that the action of G is free on # - 1 ( 0 ) and that at each point x E #-1(0) there is a holomorphic slice S~ C #-1(0) for the action, i.e., a complex submanifold Sx through x which is transversal to the orbit G(x) in the sense that
Tx(,-l(0)) = Tx(S~) •
T~(G(x)).
The quotient space W = # - I ( O ) / G is called the reduced phase space. If we take S , sufficiently small, then the projection lr : #-1(0) --+ W defines a homeomorphism of Sx onto an open subset ~r(Sx) of W and introduces a local coordinate system in W. This makes W into a (not necessarily Hausdorff) complex manifold. If the action of G on # - 1 ( 0 ) is proper, then W is Hausdorff. We state now the holomorphic version of the symplectic reduction theorem of MarsdenWeinstein. T h e o r e m (4.1). Let V be a complex Banach manifold with form wv. Let G be a complex Banach Lie group acting on V there is a holomorphic momentum map tt satisfying (a), (b) and holomorphic sympIectic form ww on the reduced phase space W r*~w = j*~v
on
a hoIomorphic symplectic leaving w v invariant. If (c), there exists a unique = # - I ( 0 ) / G such that
~-1(o).
We apply the reduction theorem to prove T h e o r e m (4.2). Let M be a compact complex manifold with a holomorphic symplectic fOrTl~ OdM and E a C °° complex vector bundle over M. Then ¢dM induces, in a natural way,
a holomorphic symplectic form on the nonsingular part of ](A(E). This theorem was first proved by Mukai [12] when M is an abelian surface or a K3 surface.
53 For the proof of Theorem 4.2, we apply the reduction theorem to
Y = L2(7)"(E)),
G = L~+I(GL(E))/C*,
g = L~+I(gl(E))/C,
where L~ and Lk+l, 2 (k > dimM), denote the Sobolev completion. We define a holomorphic symplectic form wy on V by
wV(a,/3)= / M t r ( a A ~ ) A w ~ l A ~ 4 - 1
a, Z6TD,,(V),
where ce and ~ axe considered as elements of L~(AO,I(End(E))) the complex dimension of M. We define a holomorphic m o m e n t u m map p : V -4 g* by
=--/Mtr(aoD"oD")Acv~Aff2~l-1
,
~- TD,(V) and 2m denotes
aEg, D"eY.
W'e note that #-1(0) = {D" e V; P " o D" = 0} =
L2k(H"(E)).
R e m a r k . We showed in Section 2 that every holomorphic p-form ~ on a compact K~ihler manifold M induces a holomorphic p-form q5 on the modnli space ATe(E). We notice that the construction of q5 in Section 2 is quite similar to that of 0ag above. In fact, the only difference is that ( 2 m - 1)-th power ,~2m-1 of the K~ihler form/~ is used in Section 2 in place o f w ~ -1 A ~ r -1 above. It is not hard to check that these two constructions give the same holomorphic 2-form (up to a constant factor) on Abf(E) when M is hyper-K~ihler. The construction in Section 2 shows that if WM is a generalized holomorphic symplectic form, we still get a generalized holomorphic symplectic form on A~t(E).
5. Examples. Let M be a complex torus of dimension n. The holomorphic vector bundles over an elliptic curve have been completely determined by Atiyah [1]. We consider therefore the two-dimensional case. In general, if M is a compact K~ihler surface with nowhere vanishing holomorphic 2form WM, (i.e., if M is a 2-dimensional complex torus or a K3 surface), and if E is a C oo complex vector bundle of rank r over M, then by Serre duality H2(M, O(End°(ED"))) is dual to H ° ( M , O(End°(ED"))) for any D " e H " ( E ) . Hence, at [D'q E A~i(E) not only H°(M,O(End°(ED"))) but also H2(M,O(End°(ED"))) vanishes. So in this case the moduli spa£e A74(E) of simple holomorphic structures in E is nonsingulax. From the Riemann-Roch formula we obtain its dimension:
dimH1(M, O(End(ED"))) = A ( E ) + r2h °'1 + 2 - 2r 2 where A(E) =
2rc2(E) - (r - 1)cl(E) 2,
provided that A74(E) is nonempty. By Theorem 4.2, A~(E) is holomorphic symplectic. In particular, the dimension above is even, i.e., (r - 1)cl(E) 2 must be even if A[4(E) is nonempty.
54 Now we assume that M is a complex 2-dimensional t o m s . Then the moduli space 2Q(E) has dimension + 2, again provided that it is nonempty. Assuming that E D" is semistable for some D" E 7"/n(E), we have (Bogomolov's inequality) > 0. If the equality holds here, the moduli space 2t;I(E) is a surface with a holomorphic symplectic form. We claim that this surface is a torus. In fact, as we saw in Section 2, the action of Pic°(M) alone gives a 2-dimensional family of holomorphic structures. So, the component of A;i(E) containing S = E D'' is isomorphic to Pic°(M)/E°(S). In general, if M is a complex torus of dimension n, every element x E M defines a translation v~ : M ~ M which sends y E M to x + y E M. Given a holomorphic vector bundle E over M, every translation rx, x E M induces a holomorphic vector bundle r~E. Thus, M acts on the moduli space A~(E) of simple holomorphic structures in E. We know from Section 2 that PicO(M) acts on A;i(E) essentially effectively (for any compact K~aler manifold M). If M is a torus, then M × Pic°(M) acts on A;I(E). Namely, every (x, Z:) E M × Pic°(M) maps (x,L:) : S ~ r ~ S . L:-1. Following Mukai [11], we set ¢ ° ( S ) = {(x, £) E M x Ric°(M); r~S ~- g . £}. Since il}°($) is the isotropy subgroup of M x Pic°(M) at S E A74(E), the quotient group ( M x Pic°(M))/il?°(S) embeds into A74(E) as the ( M x Pic°(M))-orbit through S. In particular, we have
n <_ 2n - dim~°(S) < d i m H l ( M , O ( E n d ( S ) ) ) . The bundle $ is said to be semihomogeneous if for every x E M there is a holomorphic line bundle £ such that Clearly, S is semihomogeneous if and only if dim~°(S) = n. From the inequa~ties above it follows that if d i m H l ( M , O(End(S))) = n, then S is semihomogeneous. An extensive study of semihomogeneous vector bundles has been made by Mukai. One of his main theorem states [11] T h e o r e m (5.1). For a simple holomorphic vector bundle S on an abelian variety M, of dimension n, the following conditions are mutually equivalent. (1) d i m H l ( M , O(End($))) = n; (2) dimHi(M, O(End(S))) = (•) for i = 1 , . . . , n; (3) S is semihomogeneous; (4) End(C) is homogeneous, i.e., r ' E n d ( S ) ~- End(C) for all x E M; (5) there exist an isogeney ~r : N ~ M and a line bundle l: over N such that S ~- ~r.£.
55 We r e t u r n to the two-dimensional case. The case of a vector bundle E over a 2dimensional torus with A(E) = 0 discussed above is the extreme case where dirn~.°(£) = 2=n. T h e other extreme is reached by a Picard bundle. Let C be a nonsingular curve of genus 2, and J its :Iacobian variety. Let C {m) denote the m-fold symmetric p r o d u c t of C. Fixing a point x0 E C, we define a m a p ~ : C(m) --* J by
~(xl,"',Xm)
-- Xl + ' " T
xm - mxo
for
xl,...,xm
E C.
For rn >_ 3, 7~ : C(rn) --~ J is a projective bundle over J, and by U m e m u r a [14] there is a stable vector bundle C of rank m - 1 over J with A(£) = 2 such t h a t C(m) = P(C). Hence, d i m H i ( M , O ( E n d ( ~ ) ) ) = 4. Let E denote the C a complex vector bundle underlying £. According to U m e m u r a [15], the moduli space ,VI(E) is isomorphic to J x P i c ° ( J ) . So this is the case where ~0(£) is trivial. We consider another example also due to U m e m u r a [15]. Let A be an abelian surface with a prinicipal polarization F-, i.e., an ample line bundle with d i m H °(A, L:) = 1. (By the Riemann- Roch formula, this latter condition is equivalent to cl(F-) 2 = 2). Let r > 2, and let F-o, F - l , " ' , F-r be line bundles algebraically equivalent to E with nonzero holomorphic sections ~Po,~ 1 , " ' , ~ar. Let ~(F-o, F-t,---, F-r) be the coherent sheaf over A defined by the exact sequence:
0 --* (9 -* F-o • £1 (~ --- @ £~ --' ~(£o, F-l,'", F-~) -* 0, where the second arrow is given by 1 ~-* ( ~ o , ~ 1 , " ",~r). T h e isomorphism class of £(F-0, F-l," • ", F-r) does not depend on the choice of ~o, ~1, • "', ~rFrom the W h i t n e y sum formula and Cl(F-i) 2 = 2 we obtain
)~(~(F-0, F-l,'" ", F'r)) ---- 2(?" -~- 1). Hence, we expect t h a t the dimension of the moduli space of C(F-0, F-l, • • ', F-r) to be 2 ( r + 2 ) . Using the translations rx0, vx~,..., vxr by elements x0, x l , . - . , xr E A, we set ~0,~,... ~, = C(~;oF-,~; F - , - . . , ~ ; F-). T h e n , for generic xo, x l , . . . , x r under the m a p
E A, £xo,x~,...,x~ is an £-stable vector bundle of rank r, and
(F-, XO, X i , ' " ", ~r) ~
~xO,xl,...,xr
the algebraic variety P i c ° ( A ) × A('+1) is birationally isomorphic to a c o m p o n e n t of the moduli space of L:-stable vector bundle containing a generic £x0,x~,...,x~. (Here, A (r+l) denotes the (r + 1)-fold symmetric p r o d u c t of A). R e m a r k . H E is a C ~ complex vector bundle of rank r over a 4-dimensional complex torus M , we can still express the dimension of its moduli space in terms of Chern classes c~ = ci(E) of E. Let D " E 7 ~ ' ( E ) and C = E D''. Since C is simple, we have d i m H ° ( i , O(End(C))) = 1. At a nonsingular point [D"], we have H 2 ( M , O(End°(C))) =
56 0 so that dimH2(M, O(End(C))) = 6. Hence, by the Riemann-Roch formula and Serre duality we obtain
dimHl(M, O(Znd(C))) = 4 + v(E), where
1f
v( E) = - 2 /M ch(End( E)), or, in terms of ci = ci(E), /~(E) = 2 ~ { - ( r - 1)614 + 4rc2c2 - 2(r + 6)c22 - 4(r - 3)CLC3 + 4rc4}. We do not know if v ( E ) is nonnegative for a semistable E over any compact 4dimensional KEhler manifold. Mukai [12] has given some interesting examples of stable vector bundles over a K3 surface. For example, let M be a nonsingular complete intersection of three hyperquadrics in P5 such that all hyperquadrics containing M are of rank 5. Let h E H2(M, Z) be the cohomology class given by a hyperplane section. Then the moduli space of h-stable vector bundles £ of rank 2 over M with Cl(C) = h and c2(g) = 4 is a K3 surface that can be described as a two-fold covering of P2 ramified over a curve of degree 6. In another example of Mukai, M is a nonsingnlar complete intersection of a hyperquadric and a hypercubic in P4. Assume that M contains no lines. Then the moduli space of h-stable vector bundles £ of rank 2 over M with Cl(C) = - h and c2(g) = 4 is a symplectic K~ilfler manifold of dimension 4, Hilb2M, discovered by Fujiki [4]. As we stated at the bigining of Section 4, every compact hyper-K~hler manifold X has a finite, unramified covering X which factors into a product of (even-dimensional) complex torus and a simply connected hyper-K~hler manifold. The above examples seem to indicate that, for the moduli space AA(E) or .AA(E, h), its torus factor comes only from the torus factor of M.
References [1]. M.F. Atiyah, Vector bundles over an elliptic curves, Proc. London Math. Soc. (3)
7 (195 ), 414-452. [2]. A. BeanviUe, Vari~t~s k~hl~riennes dont la premiere classe de Chern est nulle, J. of Diff. Geometry 18 (1983), 755-782. [3]. K. Cho, Positivity of the curvature of the Weil-Petersson metric on the moduli space of stable vector bundles, P h . D . thesis, Harvard Univ., 1985. [4]. A. Fujiki, On primitively symplectic compact K~hler V-manifolds of dimension four, in Classificationof Algebraic and Analytic Manifolds, pp. 71-250, Progr. Math. 39, Birkh~user, 1983 [5]. M. Itoh, Geometry of anti-self-dual connections and Kuranishi map, J. Math. Soc. J a p a n 40 (1988), 9-33.
57 [6]. H. J. Kim, Moduli of Hermite-Einstein vector bundles, Math. Z. 195 (1987), 143-150. [7]. S. Kobayashi, Recent results in complex differential geometry, Jber. d. Dt. Math.Verein. 83 (1981), 147-158. [8]. S. Kobayashi, Submersions of CR submanifolds, Tohoku Math. J. 39 (1987), 95-100. [9]. S. Kobayashi,Differential Geometry of Complex Vector Bundles, Iwanami Shoten/ Princeton U. Press, 1987. [10]. M. Lfibke and C. Okonek, Moduli spaces of simple bundles and HermitianEinstein connections, Math. Ann. 276 (1987), 663-674. [11]. S. Mukai, Semi-homogeneous vector bundles on an abelian variety, J. Math. Kyoto Univ. 18 (1978), 239-272. [12]. S. Mukai, Symplectic structure of the moduli space of sheaves on an abelian or K3 surface, Invent. Math. 77 (1984), 101-116. [13]. A. 3. Smith, Symplectic KEhler manifolds, Ph. D. thesis, Univ. of Calif., Berkeley, 1987. [14]. H. Umemura, On a property of symmetric products of a curve of genus 2, Proc. Intl. Symp. on Algebraic Geometry, Kyoto 1977, pp.709-721. [15]. H. Umemura, Moduli spaces of the stable vector bundles over abelian surfaces, Nagoya Math. J. 77 (1980), 47-60. [16]. H. Wakakuwa, On Riemannian manifolds with homogeneous holonomy group Sp(n), Tohoku Math. :I. 10 (1958), 274-303. This work has been partially supported by NSF Grant DMS-8801371.
Q u a s i c o n f o r m a l M a p p i n g s on C R Manifolds A d a m Kor inyi 1 Hans Martin Reimann 2 1
C R Manifolds and the Definition of Quasiconformal Mapping
A nondegenerate differential one-form 0 on a (2n + 1)-dimensional manifold M defines a contact structure on M. The contact structure is the subbundle H M of the tangent bundle T M consisting of the horizontal vectors:
H M = { X E T M : O(X) = 0} The form/9 is only determined up to a nonvanishing scalar factor and there is a nondegeneracy condition stating that O A (dO) n is a nonvanishing volume form. It is sufficient to have 0 defined locally. But it will be assumed that M is orientable which implies that 0 can be defined globally. If in addition to the contact structure there is given a smoothly varying automorphism J: H M ..~ H M with the property that j 2 = - I (i.e. a "complex structure"), then M has the structure of a CR manifold. The complexified subbundle H e M decomposes into the eigenspaces for J with eigenvalues + i and - i H e M = T 1'° + T °'x, T 1'° N T °'1 = {0} T 1,° is the space of holomorphic and T °'1 = T 1,° the space of antiholomorphic vectors. Locally, the space T 1,° can be defined by complex_one-forms 0 a E (Tl'°) *, c~ = 1 , . . . , n such that together with their complex conjugates/9 ~ = 0 a and with/9 they form a basis of the complexified cotangent space T~M. The CR manifold M is called integrable if
dO =_ dO° ' = 0
( m o d 0 , / 9 1 , . . . , 0 n)
for all a. We will assume throughout that this integrability condition is satisfied. For a fixed choice of the contact form 0 the forms 0 n can then be determined such that dO = ig~O '~ A 0# (Webster [15]) with ( g ~ ) a Hermitian matrix. In particular dO(Z, W ) = 0 for z, w E T 1'°, and d6 is J-invariant
dO(JX, J Y ) = dO(X,Y) for X , Y e H e M . The Levi form L is the Hermitian form on H e M which is uniquely determined by
L(Z, Z) = - i d O ( Z , Z ) for Z e T 1'° 1Lehman College and Graduate Center, City University of New York 2University of Bern, Switzerland
60 For real vectors X, Y the Levi form can expressed as
L(X, Y ) = dO(X, J r ) = dO(Y, J X ) . I f Z a , a = 1 , . . . , n is the basis o f T 1,° dual to 0a, t~ = 1 , . . . , n then L(Za, Z[3) = ga-~. The Levi form depends on the choice of the contact form 0. However if 0 and a0 are contact forms defining the same contact structure, then the associated Levi forms are the same up to the scalar factor a. The CR manifold M is strictly pseudoconvex if the Levi form is positive on H M . Replacing 0 by A0 on a strictly pseudoconvex CR manifold then requires that a be positive. Note also that the nondegeneracy condition 0 A (dO)n ~ 0, which usually is not part of the definition of a CR manifold, is automatically satisfied on strictly pseudoconvex CR manifolds. The smooth quasiconformal mappings f : M ~ M ~between strictly pseudoconvex integrable CI~ manifolds will be defined as C2-contact diffeomorphisms which are of bounded distortion. If 0 and 0~ are contact forms on M and M * with associated Levi forms L and L ~ then the diffeomorphism f: M ~ M' is a contact transformation if f*O~ = aO with some nonvanishing scalar function a. We will restrict all considerations to the case a > 0. DEFINITION A smooth K-quasiconformal mapping f: M --~ M ~ between strictly pseudo-
convex CR manifolds is a C2-contact transformation such that f*O' = aO with a > 0 and a K - 1 L ( X , X ) <_L ' ( f , X , f , X ) <_ A K L ( X , X ) for all X E H M . It is proved in [10] that for smooth mappings of the Heisenberg group to itself, this definition is equivalent to the original definition of quasiconformality due to Mostow. The smoothness requirement can in fact be relaxed considerably. We will come back to this question in a subsequent paper. The distortion condition can be formulated in an equivalent way which puts in evidence that quasiconformal mappings change the complex structure. Under the contact transformation f : M --* M' the complex structure J ' : H M ' --* H M ' pulls back to an automorphism j , _.. ( f , ) - I o J ' o f , on H M , which again satisfies (J*)~ = - I . If L* denotes the Levi form with respect to J, and the contact form O, then
L'(f,X, f.X) =
dO'(f,X, f . ( f ,
lJ' f . X ) ) = AdO(X, J ' X ) = aL*(X, X).
Therefore, f is K-quasiconformal if and only if
g-ln(x,x)
< L * ( X , X ) <_K L ( X , X )
for all X • H M. If f is quasiconformal and 0 ~ Z E_ T °'1 then f._(~) cannot be contained in (T') 1,° because this would lead to L ' ( f . Z , f . Z ) = -idO~(f.Z, f . Z ) = - a L ( Z , Z), contradicting the positive definiteness of L. The subspace
T 2'° = {Z • H e M : f . Z • (T') 1'°}
61
can therefore be represented as T,L° = {Z - p Z : Z E T 1'°} where/t: T 1,° --* T 1'° is a complex antilinear mapping. The J-invariance of dO~ implies the symmetry of #: )~dO(Z - fiZ, Z ' - I~Z ') = d O ' ( f . ( Z - p Z , Z' - p Z ' ) ) = 0 for all Z, Z ~ E T 1'°, and hence dO(Z, I.tZ') - dO( Z', p Z ) = O. The mapping # will be called the complex dilatation of the quasiconformal mapping. It is uniquely determined by the subspace T.L°. In particular, i f f j : M --~ M(J), j = 1, 2 are two quasiconformal mappings with the same complex dilatation p, then the mapping f2 o f~-1 is a quasiconformal mapping which preserves the complex structure, hence a conformal mapping. Its complex dilatation is 0. In the case that the CR manifolds are embedded as hypersurfaces in C", the mapping f2 o f l 1 locally extends to a biholomorphic mapping. (Webster [14]; see also [10] for the connection with the complex dilatation.) As will be shown, a contact transformation f: M --* M ~ is K-quasiconformal if and only if [[/~[[ = sup L(IJZ, I~Z) 1/2 < K - 1 L(Z,Z)=I -- K "~ 1" Observe that [[/~[[ only depends on the position of T.L° in the space H e M of horizontal tangent vectors. For the proof of the statement, it therefore suffices to consider the special case where M = M ~ -- N is the tteisenberg group. The proof will be given in the next section. 2
The Beltrami Equation
Strongly pseudoconvex C R manifolds naturally arise as boundaries of the strongly pseudoconvex domains in C n+l . If p is a defining function for such a domain D D = {z E C n + I : p(z) < 0} and dp(z) • 0 for z E OD, then the differential O = ( O p - Op)/2i defines the contact structure on M = OD. The complex structure of the underlying space C "+1 induces the complex structure of H M . T h e Heisenberg group N = OD n
D = {Z e C'~+1 : p ( Z ) = ~2Zn+l - ~--~ [zj[ 2 > 0} 0
serves as a model case. It is holomorphically equivalent via the Cayley transform to the unit sphere S with one point removed. The sphere S is distinguished by the property that the dimension of its (holomorphic) automorphism group is maximal (E. Cartan [2]). In this section we consider quasiconformal mappings from N onto strictly pseudoconvex hypersurfaces in C n+l and in particular from N onto N. We first give a different interpretation of the complex dilatation for such mappings f = ( f l , . . . , f n + l ) : N __+ C-+1.
62
In particular, we show that f satisfies a system of differential equations which we call, in analogy to the one dimensional case of a quasiconformal mapping f : C ~ C, a Beltrami equation. For this purpose, introduce the basis
o = 0°
:
oz
=
~ ~:=~ z,,dz,, - ~ ~ : : 1 z~dz,, + dtl2 ~= 1,...,n
dza
in the space of one-forms on N (the points of N are parameterized by (z,t) E C '~ x R, t = N z , + l ) . The dual basis is then given by the vector fields T = 2 ~ , Z~ = _2_oot~+ i z - ~ , Za, and { Z 1 , . . . ,Zr~) is an orthonormal basis of the space T 1,° of holomorphic vectors with respect to the Levi form. T h e image of Ze` under the tangent mapping f , decomposes into a hoIomorphic and an antiholomorphie part: f . Z ~ = V,~ + We,, Vc,, We, E T I'° with
vo = .+1 ~(zoP
)OOz , k
k=l
=
.+1
)0 o
k----1
Since f is assumed to be a contact diffeomorphism between N and the strictly pseudoconvex hypersurfaee M = f ( N ) C C n+a, it satisfies (eft the restriction on the type of the contact forms and the contact mappings introduced earlier) S*O~ = ~0 (~ > 0) where 0 ~ is a contact form on M. It then follows that the vectors V1,..., V~ span the whole space of h_._olomorphie tangent vectors. Otherwise there would exist Z E T 1'°, Z ¢ 0, with f . Z = W E T °'1 and this would contradict the strict pseudoconvexity:
0 < -)~iO(Z, Z) .= - i O ' ( W , W ) < O. There exists therefore a complex antilinear mapping # : T 1'° ~ T 1'° with W~ = ~ $ = a U~/~Vz, (a = 1 , . . . , n). Observe now that the vectors Za - ~ #a,~gO (or = 1 , . . . , n) are mapped into Tl,°:
= Ve` - ~ , ~ / ~ , ~ / ~ , ~ V ~ E T 1'° The m a p p i n g / t therefore coincides with the complex dilatation/~ introduced in the first section. n In view of the definition for Va and W~, the equation Wa = ~ 0 = 1 #~,~Vfl can be written as a general Beltrami equation
(~ = 1 , . . . , n + 1 ) .
63
We have thus shown that any contact diffeomorphism f : N ---* M from N onto a strictly pseudoconvex hypersurface M satisfies such a Beltrami equation with/~ symmetric. Later on we will show that if f if K-quasiconformal then in addition [1,112 =
L(uz,.z)
sup 0#ZeTX,0
L(Z,Z)
< - \K + I]
"
Conversely, assume that the diffeomorphism f : N --* M satisfies the Beltrami equation, with the matrix (#a,#) symmetric. Then f is a contact diffeomorphism. In fact if n+l k..---1
n+l
":o:
/.,
k=1
then f . Z ~ = V~ + W~ and f.Z-~ = "~ + Wa span a J-invariant subspace, since W~ = ~/~a,~V#. Hence there exists a holomorphic differential form r on C "+1, r # 0, such that r(Va) = 0, a = 1 , . . . , n . Furthermore, since f , is regular, a suitable linear combination /9' of 1/2(v + ~) and 1 / 2 ( r - r) can be chosen such that ( f . r ) ( 9 ' ) = 1. It then follows that f*O'lN = O. If in addition it is assumed that II/tll < 1, then M = f N is strictly pseudoconvex, i.e. the Levi form L ~ associated to/Y is positive definite:
Lt ( f , ( Z - I~Z), f ,( Z - p Z ) = -idO'(f.(_.Z - I ~ Z ) , f . ( Z -- I~Z)) = -idO(Z, Z) - idO(,Z, ttZ) = L ( Z , Z ) - L(I~Z, ltZ) > L(Z, Z)(1 -I1~11) >0. As will be shown next, the condition llull _< ( K - 1)(K + 1) -1 then implies that f is K-quasiconformal. THEOREM The contact diffeomorphism f: N --* M between strictly pseudoconvex integrable CR manifolds N and M is K-quasiconformal if and only if its complex dilatation I~ satisfies K-1
1111 ___+-----i" K
As already remarked in the first section, it suffices to consider a contact diffeomorphism of the Heisenberg group N onto itself. With respect to the basis { e l , . . . , e2n}, e~
=
X~=o--~+2y~ °
64 the tangent mapping f , restricted to H N is given by a multiple of a symplectic mapping d•(f.X, f , Y ) = )~dO(X, Y ) , (X, Y e H N ) . The mapping (1/A)f. restricted to H N can thus be represented by an element g in \ 0 --I Sp(2n, R) = {g • GI(R2"): g - l j g = j } , where J = ( I 0 ) (with I the identity %
f
matrix in R"). By complexification g extends to a complex linear mapping of (::2,~. The subspace T z'° C C 2" is spanned by the vectors 1/2(ca - ie,,+a) = Za (o~ = 1 , . . . , n). Upon setting g(ea - ie,~+a) = g(Za) = Va + W,~ with Va, Wa • T 1'°, the complex dilatation of g, /t = / t ( g ) , is defined by W~ = ~ p / ~ , ~ Vp. We will now make use of the Caftan decomposition G = K A + K of G = Sp(2n, R). The subgroup K = {k • G : k t = k - z } is identified with U(n). T z,° is an invariant subspace for k E K and ~: := klT~.o is unitary on T 1'° ~ C a. The subgroup A is of the form
X={a= /
etn :ta • R ; a =
e-tZ
1,...,n}
e--t~
and the elements in A + satisfy t l k t2 _> ... _ tn _> 0. From the definition of/~ it follows that ( tanhtz ) /~(a) = ".. tanh tn I.t(gk)
--
,(kg)
=
tC-ll.t(g)7
a • A;k • K;g • G
and consequently for g = klak2, #(g) = l~(kxak2) = k;llJ(a)-~2. (In particular,/J(g) is symmetric.) The norm of/~(g): T 1,° -* T 1'° (with g = klak2 (k~, k2 E K , a E A +) is then given by
I1 '11 =
sup I~(g)ZI = m a x t a n h t ~ = tanhtl. ZET,,o;IZI=I a
The condition of quasiconformality is expressed by the distortion inequality
K- I I < Igxl < KI I for
all x
E R2".
But since maxl~l=l [gxl e t' 1 + tanh tl 1 + II~ll minl~l= 1 Igxl - e -tl - 1 - t a n h t l - -1 - -I I ~ l l
65 the distortion inequality is equivalent to K+I II#l]-< K - l " This completes the proof of the theorem. It would now be interesting to know which functions tt arise as complex dilatations of quasiconformal mappings. Let us change the point of view slightly. Assume that f: N ---+C '*+i is a K-quasiconformal mapping of the Heisenberg group onto some strictly pseudoconvex hypersurface M = f N C C '*+i. Denote the complex dilatation by # and introduce a new complex structure J, on N by J. = ( f . ) - I o J ' o f . , where J ' is the complex structure of the image domain (or equivalently the complex structure on C "+i) and f. the tangent mapping restricted to H N . The complex structure Jr, is completely determined by/~. The Heisenberg group N with the contact form
og
and this new complex structure J, is a CR manifold, denoted by N~. A basis for the holomorphic differentials is given by 7 ~ = dza + ~ #~,,~dzz (a = 1 , . . . , n) and the dual basis for the vector fields is
If tt is the complex dilatation of a K-quasiconformal mapping, then tt is symmetric and satisfies K-1
II~I[-< g+l" But in addition it also has to satisfy the (Frobenius) integrability condition d8 - d7 ~ - 0 (mod 0 , 7 i , . . . ,7'*) for all a. This can easily be verified by a direct calculation. Assume now that it satisfies the above conditions. The question whether a quasiconformal mapping f: N ---* C '*+1 with this complex dilatation exists is then equivalent to the question whether N~ is embeddable into C '*+1. The statement that the CR manifold N~ is equivalent to N precisely means that there exists a quasiconformal mapping f : N --+ N with complex dilatation it. Both the embedding and the equivalence problem have been treated extensively in the literature. The embedding problem was first posed by Kohn [9] and subsequently solved to a large extent by Kuranishi [12]. According to Kuranishi's result, every smooth strictly pseudoconvex, integrable CR manifold of real dimension 2n + 1 >_ 9 arises locMly as the boundary of a strictly pseudoconvex domMn in C "+i. Akahori [1] extended this result to dimension 7. A greatly simplified proof has recently been given by Webster [16]. The problem remains open for dimension 5, whereas counterexamples in dimension 3 are due to Nirenberg [13]. These are constructed as perturbations of the CR structure of the Heisenberg group. More generally aacobowitz and TrOves [8] have shown that analytically small perturbations of strictly pseudoconvex realizable CR structures of dimension 3 result in nonembeddable CR manifolds.
66
The equivalence problem was solved by E. Cartan [2] for dimension 3. He constructed a fifth order relative CR invariant whose vanishing is the only (necessary and sufficient) condition for equivalence of a given 3-dimensional CR structure with the standard structure of the Heisenberg group. Cartan's method also applies to the higher dimensional case. In [3] Chern and Moser derived the corresponding relative CR invariants and thus solved the local equivalence problem. The observation that for dimensions 2n + 1 > 5 the higher order invariants can all be reduced to the invariants of second order is due to Webster [15]. Let us briefly summarize these results for 2n + 1 _> 5. On the integrable, strictly pseudoconvex CR manifold M fix a contact form 0 and a basis 0I,..., On of T 1,° such that dO = +igc-~Oot A 0~ with ( g ~ ) Hermitian and positive definite. The connection forms w~ot and torsion forms rot are the uniquely defined forms satisfying dO s = O~'Aw~-FOA7"ot ~= 1,...,n -- 0 (rood 0~) and the symmetry condition d g ~ - o.,~,'rg.t~ - wz'r got~ = 0
(the summation convention is being used). If the curvature forms are defined by fl~
= &v~ ~ - w~ ~ A w~ ~ - iO~ A rot + ir~ A 04
where the Levi form ga~ and its inverse g ' a are used to lower and raise indices, then f~-~ = R~-~,'iO ~' A 0"~ + A ~ A 0
for some one-forms AZ~ and curvature tensor components RZ~a~. The curvature satisfies the following symmetry conditions:
/~,~
-
/~
RZ~
-
R~,~.
(:=R~;)
The pseudoconformal curvature tensor S given by Chern and Moser [3] can then be expressed in terms of R (Webster [16, p. 35]), a
=
_/~ a _
Rot ~ p~
ot R a n+2(R~ Z + ~ gp;+~npH+~R~;) R
(n + 1)(n + 2) (
+
with R H "= ROtotpa' - R = Rp p. For dimensions 2n + 1 _> 5 the strictly pseudoconvex integrable CR manifold M is then locally equivalent to the Heisenberg group if and only if the pseudoconformal curvature tensor S vanishes. For dimension 3, Cartan's result gives a fifth order relative CR invariant. This means that the complex dilatation # has to satisfy a certain fifth order differential equation in order for N u to be locally equivalent to N. This differential equation can be calculated explicitly.
67
3
The Fourier Transform Method
For the lowest dimensional case, the embedding problem seems to be particularly difficult. :lacobowitz and TrOves [8] show that the complex dilatations/~ such that the CR manifold N~ is not embeddable are dense. More precisely, assume that f l and f2 are C 1-solutions of the Beltrami equation ~ f = I.tZf satisfying df ~A d f ~ • O. Then locally f = (f~, f2): N --* C 2 defines an embedding of the CR manifold N , into C 2. In this situation, there exists a real vector field Q and a C~-function g vanishing to infinite order at a_prescribed point p E N such that any two solutions f l and f2 of the perturbed equation Z f = I.tZf + g Q f necessarily satisfy df 1 A df 2 = 0 at p E N. In the example of Nirenberg [13] the perturbed Lewy operator does not admit any homogeneous solutions at all, except constants. The question therefore arises, for which tt the Beltrami equation Z f = I.tZf has nonconstant solutions. In this section we will exhibit a class of functions/~ with the property that the Beltrami equation "Zf = i.tZf on the lowest dimensional Heisenberg group N = N 1 admits nonconstant solutions. It will be assumed throughout that /~ is a measurable function with compact support satisfying
sup It~(P)l =
I1~11oo = ess
pEN
~ < 1.
As in the classical case, the existence proof is based on the integral representation for solutions of the homogeneous equation Z f = g. For g in the Schwartz class S(N), this equation only has a solution if the integrability condition
Ng( ) =
(=
f
(v-lu)g(v)dv) = 0
N
with
1 -S(z,t)
-
-1
~r2(iz12 + i t ) 2 -
~r2(t -
ilzl2)2
is satisfied (g has to be orthogonal to the kernel of Z, i.e. to the space of boundary values of antiholomorphic functions). The canonical solution (i.e. the one orthogonal to the kernel of Z) can then be represented in the form f = g * k with k(z,t)
1
z
ilzP)
~r2 (t + i l z P ) ( t -
(see Greiner-Kohn-Stein [~). Furthermore, differentiation in the direction of Z shows
that for y e S ( N ) , Z f = Z y • b with 2i ""
--2
z
Jo' ~z,~)= Z k ( z , t ) = ~r2 (t + ilzl2)(t - ilzl 2)
68
This kernel is homogeneous of degree - 4 and satisfies the cancellation property b(z, t) d~ = 0 II(z,t)ll=l The associated convolution operator B: ---* g • b therefore extends to a bounded operator on all LP-spazes, 1 < p < c~ (Kor£nyi-V~gi [11]). Similarly, the convolution operator K: g ---* g * k is homogeneous of degree ~ = - 3 . It extends to a bounded mapping K: LP(N) --+ Lq(N) if 1/q = l i p - 1/4 and 1 < p_.< q < c~ (Folland-Stein [5, p. 448]). If now f is a solution of the Beltrami equation Z f = p Z f then outside supp/~, f is the boundary value of a holomorphic function ( Z f = 0). Therefore, if nontrivial solutions f of the Beltrami equation have to be found, it is natural to impose an asymptotic condition on f by prescribing holomorphic behavior at ~ . This can be achieved by specifying a holomorphic function h (i.e. a smooth function satisfying Zh = 0) and requiring f - h to lie in the Sobolev space W [ ( N ) of locally integrable functions n
W [ ( N ) = {g e L~o¢(N) = Zg, Zg E LP}. The typical examples in connection with quasiconformal mappings are h(z, t) = z and
h(z, t) = t + ilzl 2. Set now g = Z f and insert the expression z/=
z(/-
h) + Z h = Bg + Z h
in the Beltrami equation so as to obtain
g = pBg + #Zh. This equation has the formal solution CO
g = E(pB)n(l~Zh). o
Since it is assumed that/~ E L ~ ( N ) and that supp/J is compact, the function #Zh is in /.2(N). The sum then converges in LP(N) if the operator norm of/~B (as an operator in LP(N)) is smaller than 1. A solution to the Beltrami equation can then be obtained if g satisfies the integrability condition ~g = 0. In this case, the solution is given by f = Kg + h and f - h = W~(N). In general, the integrability condition Sg = 0 will not hold. This accounts for the nonembeddable CR manifold N u. However, by using the Fourier transform, it is possible to single out a class of functions ~u for which the present method will provide nontrivial solutions for the Beltrami equation. Observe finally that it is not necessary to require that /~ have compact support. It suffices to have ltZh E 1.2. As in the Euclidean case, the convolution operators S, K and B can be represented as multipliers on the Fourier transforms. It is our aim to calculate these multipliers. Explicit
69
expressions for the matrix elements occurring in the Fourier transforms will then be used to control the operator ttB occurring in the solution of the Beltrami equation. This will be possible at least for certain it. The Fourier transform on the Heisenberg group is well known. Our formulas are based on Faraut [4]. The Bargmann representation (2)~, T~) is the unitary representation on the space 2)~ of holomorphic function in C with scalar product
(+, + ) :
j exp(-n,il~;l')<~(~;)7(e)< c
which is given by
exp(-A/2(it + Izl 2) - mzC)¢(C + z) exp(-AI2(it - Izl 2) + ~C)¢(~ + z)
Tx(z,t)¢(~) =
~ > O, ~ < O.
Note that T~(z, t) = T_x(z,-t). The induced representation of the Lie algebra is
T~(Z) = ~-g~, T~(Z) = ~(,
Ta(-Z) = - A ( , T~(-2) = ~-~,
Tx(T) = iAz T~(T) = _L~,
A>0 ~< 0
By direct calculation it can be verified that the representations are unitary. Observe that the present formulas slightly differ from the formulas in [4], since a different multiplication law for the Heisenberg group is used. For a function f in the Schwartz class S(H) on the Heisenberg group, the Fourier transform at A (0 ~ A E R) is the operator on 2ix given by TA(f)¢ = f
f(z, t)TA(z, t)¢ dx dy dt
H
this operator is of trace class, its norm (the "trace norm") is defined by
IIT~($)II 2 = tr T~(f)TA(f) The inversion formula and the Plancherel formula for the Fourier transform are ([4])
f(z,t) = 4 ~ i trT~(z't)Tx(f)IAldA R
and
'i
[[f[I 2 = 4~r2
lJTx(f)ll21AI dA
R
where [[f[I is the L2-norm of f. On the basis of these formulas the Fourier transform can be extended to functions in L2(N).
70
Similarly to the Euclidean situation, the Fourier transform of the convolution f * g(w) -- i f ( v ) g ( v - l w ) d v
f,g E S(N)
N t.i
is given by the product T ~ ( f • g) : Tx(f)Tx(g).
We next calculate the matrix elements of the operators T~ with respect to the orthonormal basis {en} - { ~ ( n } in 1~. For A > 0 they are given by t~,,,~
=
(T~(z,t)em,en)
= A ,V~ fn'm' exp(-I~ll<;l')×. c
× exp(-,x/2(it + Izl ~) - .x7"¢)(~; + z)'7-" d~ =
A
. exp(--iM/2) exp(-A[z[2/2) x
AA
x / exp(--A[([2) exp(--Az()(( + z)m"~ d(. C
Denote the coefficient of (n in the power series development of
exp(-~zC)(C + z) m by Cnm. It can be determined from the expression exp(-Az¢)(C + z)m = E j=0
j!
\ k] k=0
if the summation is only extended over the indices (j, k) such that j + k = n,
min(n'rn)(--~7,)n--k(~) k=O
So i f m > n t h e n Cnm =
j=0
(--~z) jj! m! - t l n - - J ) ! ( m - - n + J ) ! Z m - n+j. = z m - n L ~ - " ( A l z l ~ ) .
In this formula, the expression
L~(x)
(--1)'
=
k=0
F(n + a + l~Xk
k!N-Zl!r@+~+
a>-l,
x_>O
for the Laguerre polynomials is used. It can directly be derived from the definition
LZ= .e=x-" ~
(e-=x"+~).
71
For m k n, the matrix elements are now
~.,~(z,t)
=
exp(-iAt/2) exp(-Alzl2/2)z'~-"L~-"(Alzl2)x
f exp(-'~l~;l')¢'Tzd~'
X'~]/~"~)~V n! m! .
c =
~/Y-~",(v"~)'-"exp(-i,W2)exp(-;~lzl2/2)D~-"(~l:l~). v
T#t'
whereas for m < n,
t.~(z,t)
(T~(~,0e~,e.) = (e~,T;(z,t)e.) (~,T~(-z,-t)~.) = t~,.(-z,-t)
= =
~(-v~) x exp(-i)~t/2)
× exp(-Alzl2/2)L~-~(Alzl2).
Finally, for X < 0,
The differential operators Z, Z and T act as multiplication operators on the Fourier transform. In the case A > 0 the relations
T~ ( Z )em
=
~
~/-;S__" ~m
T~(7)e~
=
- a ~/~:-~+1 = - x / a ( m +
V -~-C~
=
~q-~e~- i
V~.,~
1)e~+~
give ztX,m ~t~,~
= =
( T x ( z , t ) T x ( Z ) e r n , e n ) = v/"A--mtXn,m_l x - X / X ( m + 1)tn,m+ 1
The corresponding formulas for )~ < 0 are
zt~,,~
=
x/IAl(m + 1)t~,,,~+z
Tt.,m
:
-T-.,m.
I
T h e action of Z and Z on the Fourier transform can now be described by a matrix multiplication. Set
0 M()~) = V ~
v~ o
: v~(v~,j+l)
72
and we use M* (A)to denote the transposed matrix. If the operator TA(z, t) is represented as a matrix (t~,m(z,t)) then the above equations show that
ZTx(z,t) -ZT~(z,t)
= =
TA(z,t). M*(A) - T a ( z , t ) . M(A)
ifA > 0,
and
ZT~(z,t) = T~(z,t) . M(A) if)~ < 0. ZT~(z,t) = T~(z,t) . M*(A) These are the formulas which were derived by Greiner [6]. Prom now on, the Fourier transform T)~(f) of a function f at A will also be represented as a matrix ](A) with coefficients fm~,,~.The inversion formula leads to the integral representations for Z f and Z f: OK)
Zf(z,t)
--
f trT;,(z,t)](A)M(A)lAldA
I
47r2
0 0
1
+'47~ f trT~,(z,t)](A)M*(A)lAld)' -Zf(z,t)
--
_ 1_2_ f t r r ; ( z , t ) / ( A ) M . ( A ) l A [ d A 4 x 2
0 0
f
+-24r~ ~
trT;(z,t)f(A)M(A)lAIdA
O0
A
The multipliers S, /~" and /) associated to the integral operators can be read off these A
formulas (¢f. Greiner [6], where this calculus was developed). Thus S is given by
A>O,
= o
o
:=C
o
A<0.
. . ,
The matrix 0 0 0 ~/~
M-I(A) = V~
0
is a right inverse for M, but only a partial left inverse. The relations
M M -1 M-1M M*(M-1) * (M-1)*M *
= = = =
I I- C I-C I
73
A
hold. If g E S(N) satisfies the integrability condition ~S = 0 then the function f(z,t)
--
-4-~ f tr T;(z, t)#(~)(M-1)'(:~)l~l da 0 0
+4-~'s~ f trT;(z,t)O(A)M-I(A)IAIdA --00
is the canonical solution (i.e. orthogonal to the kernel of Z) of the inhomogeneous equation Z f = g. The Fourier transform of the integral operator K is thus /¢(A) = { -(M-1)*A M-I(A)
X > 0, A < 0.
Finally, the Fourier transform of B is /}(A) = { - ( M - 1 ) * M M-1M *
A > O, X < O.
The coefficients are 6m-~,n~/~Z~
X < 0.
In particular, the operator norm of B: L2(N) ~ L2(N) is v~. If however B is restricted to the subspace n ~ ( Y ) = {f e L2(N): ](A) = 0 for a.e. A > 0} then the norm of B:L2_(N) ~ L~(N) is 1. Clearly, both L2_(N) and its orthogonal complement L2+(N) are invariant under B. Under the rotation group the space L2(N) decomposes into the mutually orthogonal subspaces V k = {f e L2(N): f(zei¢,t) = eikCf(z,t)}, ke Z (with z = Izle~¢). From the explicit form of the matrix elements of the representations it follows that the Fourier coefficients i x , , of the element f E U k satisfy fmAn ^x = 0 for a.a. A>0ifm-nCk,]X,n =0fora.a.A<0ifn-mCk, a n d B m a p s U k i n t o U k+2. The complete orthogonal sums
= Ou k<j
then obey the following multiplication law: f .p e Dj+m if f e Dj and p e Dm fqL~(N). Furthermore the spaces L~ (N) and L~ (N) are multiplication invariant in the sense that f . p e L+(2 N) if f e L~(N) and p E n2+(Y) fl L ~ ( N ) (and similarly for L 2_( g ) ) . THEOREM Assume that h is holomorphic (Zh = O) and p E L°°(N). If either of the conditions 13 # e L~.(N), pZh e L~(N) and IIPlI~ < 2) p e 9-2, #Zh e D-1 and tip[leo < 72,
74
3) it E D-2 N L~_(N), itZh E D-1 N L2_(N) and [[itlloo < 1, is satisfied then the Beltrami equation Z f = itZ f has a unique solution f such that f - h = W?( N). For the proof we have to show that the formal series OO
g =
(itB)°(itZh) 0 m
converges in L2(N) and satisfies the integrability condition Sg = O. In the first two cases, the condition [[it[[oo < 1 / ~ implies the convergence of the series in L2(N). The norm of B: L2(N) ~ L2(N) is V~ and the operator norm of # B is thus smaller than 1. In the third case, the operator B must be restricted to the invariant subspace L 2_ (N). The condition it E D - 2 N L 2_ (N) then implies that L 2_ (N) is also invariant under multiplication by it. When restricted to L ~_ (N) the norm of #B is smaller than 1. I f # E D_~ gl L 2_ ( N ) then the spaces Dk are invariant under the operator itB, since B maps Dk into Dk+2 and the multiplication operator maps Dk+2 into Dk. If in addition it e L~_(N) then 0 - 2 N L _2 ( N ) is invariant under itB. This shows that, in all three cases of the theorem, the function g = ~ o ( i t B ) " ( i t Z h ) lies in the same space as itZh. But this space is chosen in such a way that all its elements satisfy the integrability condition, and so the proof is finished. We finally remark that B is bounded on all LY-spaces (1 < p < ~ ) . Therefore the series ~ o ( i t B ) n converges for some p > 4 if Ilitl[~ is small enough. In this case the solutions f of the Beltrami equation will be locally Hhlder continuous (as a consequence of the Sobolev inequalities).
Acknowledgements: The authors are indebted to S. Webster for his patience in explaining some of the concepts used in the second section. A. Kor£nyi is partially supported by the National Science Foundation (DMS 8701530) and H. M. Reimann is partially supported by the Swiss National Foundation.
4
REFERENCES [1] T. AKAHORI. A new approach to the local embedding theorem of CR structures for n > 4. Memoirs of the Amer. Math. Soc. 366 (1987).
[2]
E. CARTAN. Sur la g~ometrie pseudo-conforme des hypersurfaces de l'espace de deux variables complexes. Annali di Matematica 11 (1932), 17-90. (Oeuvres II, 1231-1304).
[3] S. S. CHERN ~ d. MOSER. Real hypersurfaces in complex manifolds. Acta Math. 133 (1974), 219-271. [4] J. FARAUT. Analyse harmonique et fonctions sp~ciales. In Deux cours d'analyse harmonique: Ecole d'dtg d'analyse harmonique de Tunis, 1984. Birkh~user, 1987.
75 [5] G. B. FOLLAND ~5 E. M. STEIN. Estimates for the 0b complex and analysis on the Heisenberg group. Comm. Pure Appl. Math. 27 (1974), 429-522. [6] P. C. GREINEm On the Laguerre calculus of left-invaria~t convolution operators on the Iteisenberg group. Exposg XI, Sgminaire Goulaouic-Meyer-Schwartz (1980-81). [7] P. C. GREINER, J. J. KOHN, L: E. M. STEIN. Necessary and sufficient conditions for the solvability of the Lewy equation. Proc. Nat. Acad. of Sciences, USA 72 (1975), 3287-3289. [8] H. JACOBOWlTZ Lz F. TRI~VES. Nonrealizable CR structures. Inventiones Math. 66 (1982), 231-249. [9] J. J. KOttN. Boundaries of complex manifolds. In Proc. Conf. Complex Analysis 1964, Mincapolis. Springer-Verlag, 1965. [10] A. KOR~NYI &; H. M. REIMANN. Quasiconformal mappings on the Heisenberg group. Inventiones Math. 80 (1985), 309-338. [11] A. KORANYI &: S. VXGI. Singular integrals on homogeneous spaces and some problems of classical analysis. Annali Norm. Sup. Pisa 25 (1971), 575-648. [12] M. KURANISnI. Strongly pseudoeonvex CR structures on small balls I, II, III. Ann. Math. 115 (1982), 451-500; 116 (1982), 1-64; 116 (1982), 249-330, 1982. [13] L. NIRENBERG. Lectures on linear partial differential equations, CMBS 17. Providence, RI: Amer. Math. Sot., 1973. [14] S. WEBSTER. Pseudo-Itermitian structures on a real hypersurface. J. Diff. Geom. 13 (1978), 25-41. [15] [16]
. Analytic discs and the regularity of CR mappings of real submanifolds. Proe. Syrup. in Pure Math. 41 (1984), 199-208. _. On the proof of Kuranishi's embedding theorem. Preprint.
On the stability of positive semigroups generated by operator matrices by RAINER NAGEL Mathematisches Institut der Universit/~t Tiibingen, Auf der Morgenstelle 10, D-7400 Tiibingen
Dedicated to E. Vesentini on the occasion of his 60 th birthday.
A b s t r a c t . We consider unbounded operator matrices generating positive semigroups on products of Banach lattices. Generalizing the concept of an M-matrix (see [2]) we characterize the stability of the generated semigroup by simple criteria.
1. I n t r o d u c t i o n In many, but not all cases (see [12], A-III and A-IV) a strongly continuous semigroup (T(t))t>o of bounded linear operators on a Banach space is uniformly exponentially stable, i.e. limt-.oo e-"llT(t)l ] = 0 for some e > 0, if and only if the spectral bound
s(A) := sup{ReA : A e a(A)} of its generator A satisfies
s(A) < O. For example, this holds for positive semigroups on the Banach lattices C0(X), LI(#) and L2(#) ([12], B-IV, Thm. 1.4, C-IV, Thm. 1.1 and [8]). In addition, the Perron-Frobenius spectral theory for positive semigroups facilitates considerably the determination of s(A). As one of the fundamental spectral properties of positive semigroups we mention that the spectral bound s(A) always belongs to the spectrum a(A) ([12], C-III, Thm. 1.1), hence one obtains
s(A) = sup{A: A C a(d) ~IR}. For positive semigroups arising from delay equations [7] or population equations [9] these facts yielded simple and useful stability criteria. On the other hand, for positive matrix semigroups on the Banach lattice IRn one has the following well known characterization of stability (see [2] or [10]). T h e o r e m . Let A = (aij)n×n be a real matrix satisfying aij >_ 0 for i # j. Then the semigroup (e At)t>_o generated by A is positive and the following assertions are equivalent.
78 (a) (eAt)t>0 is uniformly exponentially stable.
(b) s(A) < 0. (c) detA # 0 and A -1 G 0. (d) All principal minors of - A are greater than zero. The purpose of this paper is to extend the above theorem to systems of evolution equations, i.e. to semigroups on products of infinite dimensional Banach spaces. At the same time our proof immediately suggests an algorithm for possible numerical applications. We use the basic results from the theory of one-parameter semigroups (see [1], [6] or [14]) and refer to [4], [12] or [15] for additional information on positive operators and positive semigroups. 2. S y s t e m s o f e v o l u t i o n e q u a t i o n s ( k n o w n results) Consider a system of linear evolution equations of the form
(SE)
vi(t) = ~ A q v j ( t ) ,
vi(0) = f / f o r i = 1 , . . . , n
j=l
for functions vi(.) on IR+ with values in (possibly different) Bmiach spaces Ei and for linear (possibly unbounded) operators Aij from Ej into El. Using the conventions from linear algebra this system can be written as
(SE)
i~(t) = `4u(t),
u(O) = uo
for a function u(.) with values in g := E1 x . . . x E , and an operator matrix ,4 := (Aij)nxn. In the following we always assume that the operators Aii with domains D(Aii ) are generators of strongly continuous semigroups (Ti(t))t>_o on Ei and that Aij are bounded for all i ¢ j. Then the operator matrix A with domain D(A) := D(An) x . . . x D(An,) generates a strongly continuous semigroup (T(t))t>o on g (use [6], Thm. I. 6. 4), (SE) is well posed in the sense of [6] and the solutions are given by the semigroup. For results on operator matrices with unbounded off-diagonal entries we refer to [11]. We are now interested in exponential stability of the solutions of (SE), i.e. of the semigroup (T(t))t>_o. As explained above it suffices in maay cases to determine the spectral bound s(`4) of the operator matrix `4. To that purpose we use the standard spectral theoretic notations (see [12], A-III) and the following results. 2.1 C h a r a c t e r i s t i c o p e r a t o r f u n c t i o n s (see [13]). Take g := E x F to be the product of two Banach spaces E and F. L e t ` 4 : = ( C
DB) b e a 2 x 2 o p e r a t o r m a t r i x o n g w i t h
domain D(`4) = D(A) x D(D) and bounded off-diagonal elements B, C. For ,k q~ a(A) mid R(,~, A) := (A - A) -I we define the characteristic operator function A(A) := D + CR()~, A)B mid have the following characterization of spectral values of .4 (not belonging to a(A)): e °(`4)
¢=.
e
79 If ~ 6 a ( A ) u a(A) then the resolvent R(~, A) := (~ - A ) -1 is given by the operator matrix
(,)
( R(A, A + BR(A, D)C) R(A,A) = \ R ( A , D + R(A,A)B)CR(A,A)
R(A, A + BR(A, D)C)BR(A, D) R(A,D+CR(A,A)B) /"
Next we are interested in systems (SE) having positive solutions for every positive initial value, or equivalently, in semigroups (T(t))~>_0 leaving invariant a positive cone in g. To that purpose we assume that the spaces Ei a r e Banach lattices (as, e.g., C(X), LP(#)) and consider the canonical product cone in g. Then one of the many classical characterizations of positive matrix semigroups on IR" (see [5] or [2], [10]) can be generalized as follows. 2.2 C h a r a c t e r i z a t i o n of positive m a t r i x s e m i g r o u p s (see [11]). For the operator matrix .4 = (Aij)nxn on the product g := E1 x ... x En of Banach lattices Ei the following assertions are equivalent. (a) The semigroup (T(t))t>_o generated by A is positive. (b) (i) Each Aii, i = 1,..., n, generates a positive semigroup on El. (ii) Each Aij, i ¢ j, is a positive operator from Ej into Ei.
3. Characterization of stability We start with the study of 2 x 2-operator matrices and only then extend the results to matrices of arbitrary size. LeE E and F be two Banach lattices and take an operator matrix A ---
D
A and D are generators while B and C are bounded. In addition, let A and D generate positive semigroups while B and C are positive. Then ,4 generates a semigroup (T(t))t>__o on g := E x F which is positive by 2.1. In order to characterize the stability of the solutions of the system (SE) corresponding to A we show %(A) < 0". The subsequent considerations are based on results from the PerronFrobenius theory of positive semigroups. In particular we need the following lemma (see [12], C-III, Thm.l.1). 3.1 L e m m a . Let A be the generator of a strongly continuous semigroup of positive operators on some Banach lattice. Then (i) s(A) e a(A)
and
(ii) 8(A) = inf{;~ e IR: R(A, A) >. 0}. In particular, for positive semigroups one obtains immediately the following surprisingly simple stability criterium which is an abstract version of the maximum principle. 3.2 Stability Criterion. For the generator A of a positive semigroup on a Banach lattice the following assertions are equivalent.
80
(a)
s(A) < O,
(b) 0 • o(A) and A -1 < 0. These results will now be applied to semigroups generated by the above operator matrix. A C
3.3 T h e o r e m . For the operator matrix ,A =
B)
generating a positive semigroup
on E x F the following assertions are equivalent. (a)
4,A) < 0,
(a')
0 • e(,A) and ,A-1 < O,
(b)
s(A) < 0 and s(D - C A - 1 B ) < O,
(~')
0 • o(A) fl o(D - C A - ~ B ) and A -~ _ 0, (D - C A - 1 B ) -~ < O,
(e)
s(D) < 0 and s ( A - B D - 1 C ) < O,
(c')
0 • o(D) n ~(A - B D - 1 C ) and D -1 < 0, (A - B D - I C ) -~ < O.
P r o o f . The equivalence of the primed and unprimed versions is an immediate consequence of the Stability Criterion 3.2. Since (c) is obtained from (b) by changing E x F to F x E (and vice versa) it suffices to show the equivalence of (a) and (b). (a) :=> (b). Since B und C are positive operators the semigroup generated by .4 dominates the semigroup generated by ,A0 := ( A0
0 ) (use the series expansion of the perturbed D
semigroup, e.g., [6], I. 6.5). The integral representation of the resolvent ([12], C-tII, Thm. 1.2) yields 0 _< R(.~, ,A0) _< R(),, ,4)for ), > s(Ao), hence s(Ao) <_ s(A). Therefore 8(.4) < 0 implies sup(s(A),s(D)) = s(*Ao) < 0 and A -~ < 0, D -1 _< 0. From (a') we infer that R(0, .4) = -,A-~ exists and is a positive operator on E x F. Moreover we have from [13], Thm. 2. 4 (compare with (,) in 2. 1) the matrix representation
(**)
_,A-1 = ( - A - l ( Id'-~"B(D-CA-1B)-ICA -1 A-1B(D-CA-1B) -1 ) (D - C A - I B ) - I C A -1
- ( D - C A - 1 B ) -1
"
This implies that each entry and in particular - ( D - C A - 1 B ) -1 is a positive operator. Thus we have shown (b) and (b'). (b) => (a). We use again the matrix representation (**) of the resolvent R(A, ,4) in A = 0. In fact, (b) means that - A -1 and - ( D - C A - I B ) -1 are positive. Therefore all entries in (**) exist and are positive which shows (a') and (a). • In order to obtain a criterion for matrices A of n Banach lattices we use the notation
=
(Aij)nxn on the product g := E1 x -.. x E,,
Alk / Ak:=(Aij)kxk,
B~:=
" Ak-1 k /
andCa:=(Akl,...,Akj,_l).
81 As before we assume that the diagonal elements Aii generate positive semigroups o n Ei while the off-diagonal elements Aij, i ¢ j, are positive and bounded from Ej into El. Clearly, for 1 < k < n we can write
f Ak-1
At := Ai1 and .Ak --- \
Ck
Bk )
Akk
"
Here ¢1k-1, resp., Akk generate positive semigroups on E1 x -.. x Ek-1, resp., Ek and Bk, resp., Ck are positive and bounded from Ek into E1 x . . - x Ek-1, resp., from E1 x - . - x Ek-1 into Ek. Therefore we can apply the above theorem to each of the 2 x 2-matrices ,4k, 1 < k <_ n and obtain the following corollary. 3.4 C o r o l l a r y . For the above operator matrix .4 = (Aij)nxn generating a positive semigroup on g := E1 x ..- x En the following assertions are equivalent.
(a) s(.4) < 0,
(a') 0 E e(A) and A -1 _< 0, (b)
s(All) < 0 and s(Akk -CkA-~I_IBk) <
0 for k = 2 , . . . ,n,
(b') 0 E ~(All) Cl p(Akk --CkA-kll~3k) and A-Z] < O, (Akk k=2,...,n.
-- C k A k ' l l B k ) - 1 ~ 0 for
In order to evaluate the above stability criterion we state the following observations. 3.5 R e m a r k . (i) The stability of the semigroup generated by A on the product space C = E1 x ... x E,~ can be determined by properties of certain operators on the (smaller) factor spaces Ek. (ii) Given the inverse of Akk --Ck.A-~I_IBk as in (b') there are explicit formulas for .A~-1 (see (**)). Hence criterion (b), resp. (b')is of algorithmic nature.
4. Applications Our abstract criterion will now be applied to certain more concrete situations. First we retrieve some classical results on scalar "M-matrices" (see [5] or [2], [10]). 4.1 S c a l a r M a t r i c e s . The real matrix ,4 = (aij)nxn generates a positive semigroup on IRn if and only if aij >__0 for i ¢ j . This semigroup is exponentially stable if and only if
(a)
s(.A) < O,
which by Cor. 3.4 is equivalent to (b)
alI<
0 and
akk
-- Ck~d[k.llBk < 0 for k = 2 , . . . ,n.
Since det,Ak = (akk -- Ck.A'~I_IBk) • detc4k-1 we obtain the wellknown stability condition using the principal minors of ¢1 : (c)
(--1) k+I- det.Ak < 0 for k = 1 , . . . , n .
82 4.2 R e a c t i o n - D i f f u s i o n S y s t e m s . Let f~ be a region in IRa and consider the Laplacian A on E := C0(f~) with appropriate domain D(A) such that A generates a positive semigroup on E. Choose operators Aij E £.(E), 1 <_ i,j <_ n, satisfying 0 < Aij for i # j and 0 ~_ e tA`` for all t _> 0 (compare [12], C-II, Thm. 1. 11). Many linear reaction-diffusion systems (e.g., [3]) can now be expressed by an operator matrix
.4 := diag(ai • A),~x~ + (Aij),~x,~ on the product space g := E ~ and with coefficients ai E I ~ . Clearly, .4 generates a positive semigroup on g which is uniformly exponentially stable if and only if s(.4) < 0 (use [12], B-IV, Thm. 1.4). By Cot. 3.4 this is equivalent to the property that certain bounded perturbations of ak • A, k = t , . . . , n, have negative spectral bound. In the simple case that A i d -- bij • Id, bij real, this becomes equivalent to s(diag(ai, c~),~x~ + (bij)~x~) < O,
(*)
where cr := s(A). Hence we can apply the criterion from 4.1. 4.3 S e m i g r o u p s o n L 2 x IR. In many situations (e.g. [16] or [13], Ex. 2.5) one studies semigroups on product spaces where one factor space is one or finite dimensional. In order to apply our stability criterion we assume here that E = L2(#) and that A with domain D(A) generates a positive semigroup on L2(#). For positive functions g, h E L2(#) and 6 E IR we consider .4:= (A
'
\ f fhd# + 5x /
with domain D(A) = D(A) x IR which clearly generates a positive semigroup in E x IR. Its stability is determined by the spectral bound s(A) (use [8]) and hence s(.4) < 0 is equivalent to
s(A) < 0
and
5-/hA-lgd#
REFERENCES
[1] A. BELLENI-MORANTE, "Applied Semigroups and Evolution Equations", Oxford University Press 1979. [2] A. BERMAN AND J. PLEMMONS, "Nonnegative Matrices in the Mathematical Sciences", Academic Press, New York 1979. [3] V. CAPASSO AND L. MADDALENA, Convergence to equilibrium states for a reactiondiffusion system modelling the spatial spread of a class of bacterial and viral diseases, J. Math. Biology 13 (1981), 173-184. [4] PH. CLEMENT, H. J. A. M. HEIJMANS et al., "One-parameter Semigroups", CWJ Monographs 5, North Holland, Amsterdam 1987.
83 [5] M. FIEDLER AND V. PTAK, On matrices with non-positive off-diagonal elements and positive principal minors, Czech. Math. J. 12 (1962), 382-400. [6] J. A. GOLDSTEIN, "Semigroups of Linear Operators and Applications", Oxford University Press, New York 1985. [7] A. GRABOSCH, Translation semigroups and their linearizations on spaces of integrable functions, Trans. Amer. Math. Soc. (to appear) [8] G. GREINER AND R. NAGEL, On the stability of stongly continuous semigroups of positive operators on L2(#), Annali Scuola Normale Sup. Pisa 10 (1983), 257-262. [9] G. GREINER AND R. NAGEL, Growth of cell populations via one-parameter semigroups of positive operators. In: J. A. Goldstein, S. Rosencrans, G. Sod (eds.): "Mathematics Applied to Science", Academic Press 1988, p. 79-105. [10] H. MINC, "Nonnegative Matrices", Wiley-Interscience 1988. [11] R. Nagel, Well-posedness and positivity for systems of linear evolution equations, Conferenze del Seminario di Matematica Bari 203 (1985), 1-29. [12] R. NAGEL (ed.), "One-Parameter Semigroups of Positive Operators", Lecture Notes Math. 1184, Springer-Verlag 1986. [13] R. NAGEL, Towards a "matrix theory" for unbounded operator matrices, Math. Z. (to appear)
[14] A. PAZY, "Semigroups of Linear Operators and Applications to Partial Differential Equations", Springer-Verlag 1983. [15] H. H. SCHAEFER, [16] E. VESENTINI, 306.
"Banach Lattices and Positive Operators", Springer-Verlag 1974. Semigroups of holomorphic isometries, Adv. Math. 65 (1987), 272-
The Levi Problem on Algebraic Manifolds
by Raghavan Narasimhan University of Chicago
For Edoardo Vesentini
In the early '60s, A. Andreotti and E. Vesentini began a systematic attempt to extend the vanishing theorems of Kodaira and Nakano to non-compact manifolds. This led, among other things, to their paper [1] on what is now often referred to as the O-method. Their approach is more closely tied to differential geometry than the parallel, more analytic, method of HSrmander [5]. T. Napier and K. Pinney of the University of Chicago have used the O-method to obtain new results concerning variants of two well-known problems in complex analysis. I shall attempt to state these results and to outline some very interesting open problems that arise from this work. In what follows, if L is a line bundle and p an integer ~_ 1, L p stands for the pth tensor power L ®v of g. 1. T h e L e v i P r o b l e m . It is well-known that (relatively compact) strongly pseudoconvex domains on any complex manifold are holomorph-convex. Howewer, weakly pseudoconvex domains, even with smooth (or analytic) boundary, need not carry any non-constant holomorphic functions (Grauert [3, 4]; see also [8]). The first result is an attempt to restore global convexity in this situation (at least on projective manifolds). DEFINITION 1. Let D CC M be a relatively compact open subset of a complex manifold M and let L be a holomorphic line bundle on M. We say that D is convex with respect to L if the following holds: Let a E OD, and U a neighbourhood of a in M on which there exists a holomorphic section s C F(U,L) without any zeros. Then, for any infinite sequence {x~}~>l of points ~v C D, converging to a, there exists a holomorphic section f E F ( D , L ) such that the function f / s on U n D is unbounded on the sequence {x~}~_>~o, u0 large. One then has the following T h e o r e m 1.1: (K. Pinney) Let D be a weatdy pseudoconvex domain with smooth boundary on the projective manlfold M , and let L be a positive (i.e., ample) line bundle on M . Then there exists an integer po > 0 such that D is convex with respect to L p for p _> P0. There are two main steps in the proof.
86 Step 1. Let D be a (relatively compact) weakly pseudoconvex domain in the K£hler manifold M. If D has smooth boundary, then there exists a complete K/ihler metric on D. The smoothness of OD can be replaced by the condition that OD be C 2 (both in Step I and Theorem 1.1). Theorem 1.1 is probably true without this assumption. Step 1 enables one to use the 0-method of Andreotti and Vesentini. Their basic remark may be formulated as follows: when the base manifold carries a complete K/ihler metric, the a priori estimates needed to solve the equation 0 u = f need only be checked for C ~° forms with compact support. This reduces the problem to finding lower bounds for (Au, u), A being the Laplace operator associated with a metric on a line bundle, and this latter expression can be estimated in terms of curvature forms (WeitzenbSck formula). For the second step, let k be an integer such that for any a E OD there exist s l , . . . ,s,~ C F ( M , L k) (where n = d i m c M ) which vanish at a and such that the hypersurfaces sj = 0 intersect transversally. Step 2. Let ~: O ~ -* L k be the morphism ~ ( a l , . . . , a , ~ ) = ~] a i s j. Then there exists an integer 10 > 0 such that for every integer I > l0 the image of ~ ® I : F ( D , L ~) --. F ( D , L ~+k) contains all sections of the form f i b , where f E F(M, L k+l) and vanishes at the common zeroes of s l , . . . , s,~ inside D. Theorem 1.1 is an easy consequence of this result. It seems very likely that one can use Theorem 1.1 to prove holomorphconvexity in certain cases. I shall state the following as a conjecture; special cases have already been proved.
CONJECTUI~¢,. Let M be a projective manifold with c~(M) > O. Let D be a pseudoconvex domain with smooth boundary on M. Suppose that there exists a complete K£hler metric g on D such that - a g < RJc(g) < - b g ,
where a and b are positive constants. Then D is Stein. This would enable one to show that several of the examples constructed by Grauert [4, 8] do not carry a complete K£hler-Einstein metric although they are pseudoconvex and do not contain compact analytic sets of positive dimension. One should also mention, in this connection, the theorem of Elencwajg on the holomorph-convexity of pseudoconvex domains on (not necessarily compact) manifolds M of positive holomorphic sectional curvature [2].
2. T h e S h a f a r e v i c h C o n j e c t u r e . This conjecture is the following: let M be a projective algebraic manifold. The universal covering M of M is then holomorph-convex. It should be pointed out at once that a similar statement for arbitrary coverings of M is false.
87 EXAMPLE. Consider the vectors el = (1,0), e2 = (0,1) and es = (a,fl) in C 2. Let A be the subgroup generated by el, e2 and e3. If 1, a and fl are linearly independent over Z, then there is no non-constant holomorphic function f on C 2 with f ( z + A) = f ( z ) for every A • A; so there are no non-constant holomorphic functions on M = C2/A. Clearly, such manifolds M occur as coverings of projective two-dimensional complex tori. DEFINITION 2. Let M be a compact complex manifold and ~': M ---} M a covering manifold. Let L be a holomorphic line bundle on M. We say that M is convex with respect to L if the following holds: Let U 1 , . . . , U N be open sets on M such that L[u i is trivial and M = (.JUj. Let Vj CC Ui be open sets on M such that [.JVj = M . Let sj • F ( U j , L ) be a holomorphic section without zeros on Ui (j = 1 , . . . , N ) . Then, given an infinite discrete sequence z , • M (v _> 1), there exists f ~ r ( M , ~r*L) and an index j such that the function z ~-, f(~:)/si(Tr(x)) on 7 r - l ( U j ) i s unbounded on the (infinite)
set
J
• Vj).
This is equivalent to requiring the following: for any hermitian metric h on L and any sequence {zv} as above, there exists f e F(M,,r*L) such that h * ( f ( z , , ) ) is unbounded, where h* is the metric on 7r*L induced by h. T. Napier has proved the following theorem; I had proved special cases of the theorem for canonically ample varieties (in which case one can obtain explicit bounds for P0 in terms of the dimension). T h e o r e m 2.1: (T. Napier) Let M be a projective manifold, let ~r: M --~ M be any covering manifold, and let L be a positive ]line bundle on M . Then there exists an integer Po such that M is convex with respect to L p for p >_ Po. The proof is based again on the 0-method which is used to construct sections of L p with given values on a suitable subsequence of {x~}. Napier's result is actually considerably stronger than Theorem 2.1; it deals with so-called "manifolds with bounded geometry" and the sections constructed have exponential growth. We turn next to the Shafarevich conjecture itself. The results obtained so far are all in the case when M is a smooth projective surface (i.e., d i m c M = 2). T h e o r e m 2.2: I f M is a smooth, projective, elliptic surface, then its universad covering is holomorph-convex. This theorem was also proved (independently) by Gurjar and Shastry of the Tata Institute of Fundamental Research (unpublished). The main steps in the proof are as follows. Let M ~.~ C be a holomorphic map onto a smooth curve C such that the generic fibre is a smooth elliptic c u r v e . L e t w: M --, M be the universal covering. The first step is to show that if M is not compact and F 1 , . . . , F k are the /c
singular fibres, then, if k > 0, 7r-1 ( M \ U F~) is Stein.
The proof of this is
1=1
analogous to the so-called Serre conjecture over Riemann surfaces.
88 The main step is to show that there exist neighbourhoods Uz of FI (where I ranges from 1 to k) such that 7r-l(Ut) is holomorph-convex. This is done by a case by case analysis using Kodaira's classification of singular fibres and their neighbourhoods. This result can also be proved using the techniques of Napier mentioned below in connection with Theorem 2.3. The final step is a Heftungslemma of the kind used in the Serre conjecture (about fibrations with Stein fibres over a Stein base) in the case when base and fibre have dimension 1. The other results known are due to T. Napier and can be summarized as follows: T h e o r e m 2.3: Let M be a smooth projective surface, and let 7r: M ~ M be the universal covering. Assume that the tlrst Betti number bl ( M) is non-zero, and that does not contain any connected, non-compact 1-complex dimensional subspace aH of whose irreducible components are compact. Then M is holomorph-convex. Napier's proof starts by considering the canonical map of X into its Albanese variety Alb(X): X ' ~ Alb(X). If bl(X) > 0, then a is non constant. Consider the singular fibres of a when d i m a ( X ) = 1, and the fibres of positive dimension (which get mapped onto a point in Alb(X)) when d i m a ( X ) = 2. Let C 1 , . . . , CN be the connected components of these curves. One has only to assume that if 7r:)~ ~ X is the universal covering of X, no connected component of 7r-l(Ci) is non-compact but has only compact irreducible components. It is not hard to show that ~r-l(X \ [J C~) is holomorph-eonvex (in fact Stein except in certain simple cases). The main diftlculty is in the proof of the following result: Let X be a smooth surface and let p: A" ~ X be any covering manifold. Let C be a connected curve on X. Assume that if p - l ( C ) has a non-compact connected component C, then C has at least one non-compact irreducible component. Then, there exists an open set U D C such that p - l ( U ) is holomorph-convex.* The proof of this result involves a subtle construction of plurisubharmonic functions. The essence is contained in the following result. Let p: )~ ~ X be a covering of the smooth projective surface X and let C be a connected curve such that p-1 (C) has no connected component all of whose irreducible components are compact. Let E be the union of those irreducible components D of C such that p - l ( D ) has only compact irreducible components. Let {a~}v=l,2 .... be any sequence of numbers tending sufficiently rapidly to oc (more precisely: a~+l - av is sufficiently large). Then, there exists a neighbourhood U of C in X and neighbourhoods V, W of E (V C W C U) and a continuous plurisubharmonic function ~oon p-a(U) such that: 1) (z e p-~(U) I~(z) < e} CC X for every c ~ rt; * Napier's proof of this result has been written up in full in his doctoral dissertation which is being submitted to the Mathematische AnnMen.
89 2) for every v _> 1 the function ~ is strongly plurisubharmonic on (a neighbourhood of) the set e I = 3) ~(p-l(W)) ~ {a~}~>l = q5 and ~0 is locally constant on p - l ( Y ) ; 4) ~0 is strongly plurisubharmonic on p-l(U \ W). It is easily seen that it suffices to prove this result after blowing up points on X finitely m a n y times. It can therefore be assumed that C has the form
C = A1
U
...
U
Ak
U
E1
U
""
U
El,
where the Aj are pairwise disjoint smooth curves such that no connected component of p-l(Aj) is compact, while the curves Ei (possibly singular) are pairwise disjoint and p - l ( E i ) has only compact connected components; moreover, each Aj meets at least one of the Ei and all such intersections occur transversally at smooth points of El. Let p-l(Ei) = U E~" be the decomposition into (compact) connected comv>l l
ponents. Let E ~ =
U El-
The most difficult point of the proof of the result
i=I
stated above consists in the following: 1
Let {tv}v>l be a sequence of real numbers, tv -* oo. Let Xo C C\ [3 Ei. There i=1
exist neighbourhoods V of C, W of E (depending on x0) and a neighbourhood n of x0 such that W C V, fl C V and there exists a continuous plurisubharmonic function ¢ on p - l ( V ) such that (a) ~b maps p - l ( W ) into {t~)~>_l; (b) {x e p - l ( W ) O p--l(~'~) I ¢ ( ~ ) < c} c c £ for every c 6 R; (c) ¢ ( E = for all > 1. The main ingredients in the proof are Siu's theorem about the existence of Stein neighbouxhoods for a Stein subvariety of any analytic space and the fact that on an open Pdemann surface one can approximate simultaneously holomorphic functions given on a locally finite, pairwise disjoint sequence of discs on the surface. Full details of these constructions will be found in [7]. The condition that M contains no connected non-compact curve C M1 of whose irreducible components are compact is clearly necessary: any holomorphic function on M is constant on C. Thus, Theorem 2.3 shows that for smooth projective surfaces M with bl(M) > 0 the Shafarevich conjecture is equivalent to the following: Let C be a connected curve on M, and C = U c j its decomposition into irreducible components. Assume that the image of the natural map 7r~(Cj) -* 7rl ( M ) is fin/re for each j; then the image of 7rl(C) in 7rl(M) is finite. It turns out (from Napier's proof of Theorem 2.3) that this last statement needs to be checked only for certain special curves C which either occur as the full fibre of a holomorphic map of M onto a smooth curve, or which can be blown
90 down to a point, so that, in either case, the self-intersection C 2 is non-positive. Thus, it would seem that the Shafarevich conjecture leads to a statement which is "dual" to a remarkable conjecture made by M.V. Nori [9]: NORI'S CONJECTURE. Let D = ~ n j C i be an effective divisor on the smooth projective surface M (the Cj are irreducible curves). Assume that D 2 > 0 and let Cj be the normalization of Cj. Then the normal subgroup of ~'1 (M) generated by the images of the ~'1 (Cj) has finite index in 7rl (M). In view of Theorem 2.3, one has to consider, for the Shafarevich conjecture, smooth projective surfaces M for which ~'I(M) is infinite but bl(M) = 0. One way to construct such surfaces is the following. Let D be an irreducible hermitian symmetric domain in C '~, and let F be a discrete torsion free group of analytic automorphisms of D such that X = D / F is compact. Then, if the rank of D is greater than 1, we have bl(X) = 0 (by a theorem of Matsushima [6]). Let Y be any projective manifold with b~(Y) = 0. Then, by imbedding X x Y in p N and taking successive generic hyperplane sections, one obtains, because of the Lefschetz theorem, a smooth projective surface M with ~rl(M) ~ 7rl(X) x ~'I(Y), which then has the required properties. This leads to the following QUESTION. Let M be a smooth projective surface with ~rl(M) infinite and bl(M) = O. Does M admit a non-constant holomorphic map into a compact quotient of a bounded domain by a discrete group of automorphisms? For surfaces M which admit such holomorphic maps, the universal covering M is again holomorph-convex if it does not contain connected non-compact curves with only compact irreducible components. One way to approach this question might be to ask when a projective surface parametrizes a non-trivial complex variation of Hodge structures. It is along these lines that Carlos Simpson has characterized compact K/£hler manifolds which are uniformized by hermitian symmetric domains. Moreover, work of Hitchin and Kevin Corlette shows that it would be sufficient to construct rigid finite dimensional representations of 7rl(M). Thus, one is led to the following question: Let M be a smooth projective surface, and let F -- 7rl(M). Assume that the commutator subgroup [P, F] has finite index in F. Does there exist a non-trivial representation p: F ---* G L ( V ) on a finite dimensional vector space V such that any representation pt: F --~ G L ( V ) sufficiently close to p is, in fact, equivalent to p? Very little is known at present about these questions, or about Nori's conjecture and its "dual".
91 References
[1] A. Andreotti, E. Vesentini: Carleman estimates for the Laplace-Beltrami equal tion on complex manifolds. Pubbl. Math. Inst. Hautes Etudes Scient. 25 (1965), 81-130. [2] G. Elencwajg: Pseudoconvexit~ locale dans les vari~t~s kiihle'riennes. Annales Inst. Fourier 25 (1975), 295-314. [3] H. Grauert: On Levi's problem and the imbedding of real analytic manifolds. Annals of Math. 68 (1958), 460-472. [4] It. Grauert: Bemerkenswerte pseudokonvexe Mannigfaltigkeiten. Math. Zeit. 81 (1963), 377-391. [5] L. H6rmander: L 2 estimates and existence theorems for the 0 operator. Acta Math. 113 (1965), 89-152. [6] Y. Matsushima: On the iqrst Betti number of compact quotients of higher d/mensional symmetric spaces. Annals of Math. T5 (1962), 312-330. [7] T. Napier: C o n v e x i t y p r o p e r t i e s of coverings o f s m o o t h p r o j e c t i v e varieties. Ph.D. Thesis, University of Chicago, 1989. [8] R. Narasimhan: The Levi problem in the theory of functions of several complex variables. In P r o c e e d i n g s of t h e i n t e r n a t i o n a l congress of m a t h e m a t ics, Stockholm, 1962, 385-388. [9] M.V. Nori: Zariski's conjecture and re/ated problems. Ann. Sci. ]~cole Norm. Sup. Paris 16 (1983), 305-344.
A Banach-Stelnhaus Theorem for Weak* and Order Continuous Operators
by H.H. Schaefer Mathematisches Institut der Eberhard-Karls-Universit£t Auf der Morgenstelle 10, D-7400 Tfibingen
Dedicated to Prof. E. Ve~entini on hi~ ~iztieth birthday
Introduction The classical theorem of Banach-Steinhaus asserts the following: if E is a linear topological Baire space, F a topological vector space, and if (T,~) is a sequence of continuous linear maps E ~ F such that (T,~z) converges in F for each z E E, then the limiting map T : E ~ F is continuous. The proof (see, for example, [2, §15; 13] or [6, m.4]) shows the result to be a fairly easy consequence of another theorem of Banach, to the effect that under the above assumptions on E and F every simply bounded subset H C L ( E , F ) is equicontinuous; this latter fact is better known as the principle of uniform boundedness. A closer analysis shows this principle to rest on the fact that in a Balre space, a closed, circled and absorbing subset has non-void interior; in the case of locally convex spaces E, F it suffices that every convex, circled, closed and absorbing set be a neighborhood of zero. A locally convex space E with this property is customarily called barreled (tonne1~); thus if E is a barreled and F any locally convex space, the principle of uniform boundedness is valid in £(E, F) by f/at. The merit, of course, of this concept lies in the fact that the class of barreled spaces includes but is substantially larger than the class of (locally convex) Baire spaces. Suffice it to say that barreledness is stable under the formation of inductive topologies and products of arbitrary families (see, for example, [6, IV.4.3 Cor. 3]). There are, however, situations in analysis where a theorem of Banach-Steinhaus type is desirable (and indeed valid) but where equicontinuity arguments necessarily fail. For example, if (A,~) is a sequence of matrix operators on £oo that converges pointwise for the weak* topology, is the limiting operator A again a matrix operator? The answer is positive (see Proposition 4.1 below), but equicontinuity of (A,~) with respect to the weak* topology will not hold except in very special cases. Generally, when dealing with weak or weak* topologies, equicontinuity of a set of continuous linear maps is an excessively restrictive requirement (see Section 2 below). Therefore, the Banach-Steinhaus theorem proved in this paper (see Theorem 3.3 below and [8, Thm. B]) rests on a rather different approach. It exploits, on the one hand, the close relationship between weak* and
94 order continuous linear maps on certain classes of Riesz spaces, notably L°°(/z) (see Section 1); on the other hand, it is based on the fact that in the order dual of a wide class of Riesz spaces (see Definition 3.1 below) that includes all Dedekind er-complete spaces, bands are weak* sequentially complete [7, 1[.5.10]. This, in turn, results from the fact that in these spaces a sufficient number of principal ideMs are Grothendieck spaces in the usual sense [5]. Thus the prime ingredients of Theorem 3.3 below are the Grothendieck property and a new dual characterization of order continuity [8]. (Perhaps it should be mentioned that a Banach-Steinhaus theorem, unrelated to order structures, is valid for sequences of weak* continuous operators on E ~ whenever E is a weakly sequentially complete Banach space; but this will be fairly obvious to the interested reader). Thus the present paper is an extension and elaboration of ideas first developed in [8]. Notation and terminology generally follows [6] and [7], except that we denote the order dual of a Riesz space E by E*. 1. W e a k * a n d O r d e r C o n t i n u o u s O p e r a t o r s For arbitrary Riesz spaces E and F , we denote by E~ and F* their respective order continuous duals and by L,~(E, F) the space of all (order bounded) order continuous linear maps E ~ F (cf. [1], [4] and [7]). From [8] we recall the following basic result, where E* is the (Riesz) space of all order bounded linear forms on E. T h e o r e m 1.1: H E is any Riesz space and J is an ideal orE* separating E, then the positive cone E+ = {a: C E [ • _> 0} is tr(E, J)-closed. An elementary proof of Theorem 1.1 can be found in [8], but its validity can also be seen as follows. The topology o(E, J) on E of uniform convergence on the order intervals of J (sometimes called the weak Riesz topology of E with respect to J ) is a locally convex Hausdorff topology for which the lattice operations are continuous, and which is consistent with the duality (E, J) because order intervals of J are a ( E * , E)-compact. Thus E+ is o(E, J)-closed and hence, being convex, tr(E, J)-closed. I As a first application, we note the following consequence of Theorem 1.1 which is related to the classical convergence theorem of U. Dini. P r o p o s i t i o n 1.2: Let A be a directed (<) subset orE which converges to a C E for tr(E, J), J being an ideal of E* that separates E. Then A converges for
~r(E, E*); in particular, if E is a Banach lattice with order continuous norm, then lim A = a for the norm topology. Proof: In fact, since E+ is ~(E, J)-closed by Theorem 1.1, we have a = sup A [7, ]I.5.8] whence it follows that a = l i m A for tr(E,E*). The second assertion now follows from [7, 11.5.9 and ]I.5.10]. [ For another consequence of Theorem 1.1 that will be needed below we recall that a linear operator T: E ~ F is called regular if T is the difference of two positive operators. The linear space L~(E, F) of all regular operators is a Dedekind
95 complete Pdesz space (under its natural ordering) whenever F is Dedekind complete [1], [7]; in this case, T is regular iff it is order bounded. As usual, T* denotes the algebraic adjoint of T.
Proposition 1.3: Let E and F be P,Jesz spaces, with F Dedekind complete and separated by F*=. For an operator T E Lr(E, F), the following assertions are equivalent: (a) T is order continuous; (b) T is o(E, E*) - o(F, F*) continuous; (c) T*(F~,) C S*. For the proof, we refer to [8, Prop. 2]. It follows from elementary considerations that whenever T: E ~ F is a regular linear operator, its adjoint T*: F* --~ E* (and, in particular, the restriction of T* to F~) is order continuous; in fact, this is so because T* is weak* continuous (cf. proof of Proposition 1.2 above). But when is the converse true, i.e., when do the weak* continuous operators F* ---* E* agree with the order continuous operators? It turns out that this occurs exactly when the range space F is perfect. Perfect Riesz spaces [1], [3] are the precise generalizations of KSthe's vollkommene Folgenr~ume, cf. [2]. We recall their definition [1]. D e f i n i t i o n 1.4: A Pdesz space E is said perfect if the evaluation m a p E --* (E*)* is a (surjective) Riesz isomorphism. Note t h a t this definition implies E to be Dedekind complete and separated by E~; if E is any Pdesz space, the Pdesz spaces E* and E* are perfect [1, 33 D]. A well known special class of perfect spaces is the class of weakly sequentially complete Banach lattices (KB-spaces; cf. [7, 11.10.6]); this shows that not all perfect spaces are dual Riesz spaces. We now obtain this characterization. T h e o r e m 1.5: Let F be a Dedekind complete Riesz space separated by F~. These assertions are pairwise equivalent: (a) F is perfect. (b) F is complete for the weak R/esz topology o(F, F~). (c) A linear form f on F; is weak* (i.e., a(F*, F)-) continuous whenever f is weak* continuous on order bounded subsets of F,~. (d) For any R/esz space E, a regular linear operator T: F~ ~ E* is weak* continuous iff T is order continuous.
Proof: (a)==~(b): The completion F of F with respect to o(F, F*) can be identified with the space of those linear forms f on F~ which are o~(F=*, F)-continuous on each order interval of F=* [6, IV.6.2]. But dearly, each such f is order continuous on F~,, hence we have f E F and thus F = F. (b)~(a): By a basic theorem of Riesz theory ([1, 32 C] or [7, ]I.4.12]) the evaluation m a p F --. (F~*)~ is injective and onto an order dense ideal; thus each
96 0 < f E (F,~)* is the pointwise (on F,~) limit of an increasing (directed <) family (f~) in F+. Since such a family is necessarily a Cauchy family for o(F, F~,), it follows that f E F; hence F is perfect. (b)v=:v(c) is just Grothendieek's completeness theorem [6, 1V.6.2]. ( a ) - - > ( d ) : It was observed that weak* continuity of T implies order continuity. Conversely, suppose T: F,~ --* E is order continuous. If now E, F denote the order continuous duals of E*, F,~, respectively, it follows that T*(E') C F'. But F = _P by assumption; whence it follows that T*(E) C F, i.e., that T is weak* continuous. (d):=~(a): Let f be an order continuous linear form on F*; for E* = IR, f is an order continuous operator F,~ ~ IR. By hypothesis, f E F whence (F~)~, = F , i.e., F is perfect. [ Let us note this classical example, where # is a (positive) measure defined on a ~-finite (or at least localizable) measure space, so that L~(~t) = Ll(~t) '. C o r o l l a r y 1.6: Let T denote a linear, regular (equivalently, order bounded) operator from L°~(#) into a dual Riesz space E* (such as Lq(~), 1 < q <_ +o¢). Then T is weak* continuous if and only if T is order continuous. The result follows from Theorem 1.5 by observing that F = L 1(#) is perfect. 2. E q u i c o n t i n u i t y for t h e W e a k R i e s z T o p o l o g y We are interested in the question if a theorem of Banach-Steinhaus type holds for order continuous operators on suitable classes of Riesz spaces; as shown by Theorem 1.5 and Corollary 1.6 of the preceding section, any possible result will apply as well to weak* continuous operators on a large class of dual Riesz spaces. As explained in the Introduction, the classical Banach-Steinhaus theorem for sequences of continuous linear operators between Banach spaces rests on the equicontinuity (or uniform boundedness) principle. However, this principle can rarely be applied to weak or weak* topologies; in fact, the only locally convex spaces which are barreled for the weak topology are the spaces IR A or C A and certain subspaces (A is any index set). Generally, if E and F are locally convex spaces, equicontinuity of a family H C L(E, F) for the weak topologies means that the family H* of adjoints maps every finite dimensional subspace of F winto a (common) finite dimensional subspace of E t in a uniformly bounded manner; dearly, this is too severe a restriction for interesting applications. In this section, we shall briefly investigate the equicontinuity concept for families of order continuous operators between Riesz spaces E and F; in view of Proposition 1.3, the appropriate topologies for this undertaking are the weak Pdesz topologies o(E, E*) and o(F, F*). Recall that L,~(E, F) stands for the space of all (order bounded and) order continuous linear maps from E into F. P r o p o s i t i o n 2.1: Let E, F denote Pdesz spaces with F Dedekind complete and separated by F~. A family H C L,~(E, F) is equicontinuous i'or the topologies
97
o(E,E*) and o(F,F*) if and o n l y i f f o r every ~ E F* the set {T*~ I T E H} is order bounded in E*. Proof: If ~ and ~ run through the respective positive cones of F* and E*, the polars Y = [-~o,50] ° C F and U = [ _ ¢ , ¢ ] o C E constitute 0-neighborhood bases for o(F,F*) and o(Z,E*), respectively. Moreover, by Proposition 1.3 the order continuity of each T E H is equivalent to the continuity of T for these topologies. Thus by well known facts of operator and duality theory (see [6, hi.4.1 and 1V.2.3]), equicontinuity of H is equivalent to requiring that for each ~ E (F*)+ there exists ¢ E (S*)+ such that H* ([-50, ~]) C [ - ¢ , ¢ ] , i.e., that H* ([-9~, ~]) be order b o u n d e d in E~. On the other hand, since F is Dedekind complete, L,~(E, F) is a Riesz space and from standard formulas for the modulus ITt of T (cf. [7, p. 229, (2)]) it follows (with the above notation) that T(U) C Y iff [T[(U) C V. Thus equicontinuity of H is equivalent to the order boundedness of {T*~0 [ T E H} for each ~ E F*. [ From the preceding proposition it is now easy to obtain a Banach-Steinhaus type result based on an equicontinuity principle. However, for all its generality, it is a meager result because the restriction placed on H is still too severe for m a n y applications. Thus in the subsequent section, we shall look for a different approach. C o r o l l a r y 2.2: Let E, F be R/esz spaces as in Proposition 2.1, and {T~ [ a E A} a directed family of order continuous operators E ~ F. If for each ~ E F* the set {T*~ I a E A} is order bounded in E* and if for all z E E lima T~z =: T z exists in F for ~r(F, F~), then T: E --~ F is order continuous.
Proof: By a general result on equicontinuous sets of linear mappings [6, 111.4.3], the closure of the set {T~ ] a E A} in L(E, F) (for the topology of simple convergence where F carries the weak topology ~r(F, F~*)), is equicontinuous. Thus T is continuous for the topologies o(E, E~) and o(F, F*) and hence, by Proposition 1.3, order continuous. [ 3. A Banach-Steinhaus
Theorem
A Banach space E is called a Grothendieck space if every weak* convergent sequence in E' converges weakly (i.e., for cr(E', E")). The first non-trivial (that is, non-reflexive) example of a such a space was given by Grothendieck (t953): every Dedekind complete space C(K) (K compact) has the said property. Other examples are t h e spaces C(K) where K is quasi-Stonian (equivalently, C(K) Dedekind a-complete) (Ando 1961) or an F space (Seever 1968); still other examples are the Baire classes Ba, a > 1, over a compact space (Dashiell 1981) and certain classes of Banach lattices discovered by R£biger [5]. (We refer to [5] for an excellent survey of the history of the subject and many interesting restflts concerning the relation of Grothendieck spaces and Banach lattices). Let E be a Pdesz space. If (and only if) E is relatively uniformly complete [4], each principal ideal Ez (z > 0) is a Banach space under the norm whose closed
98 unit ball is the order interval [-~, z], and isometrically isomorphic to some C ( K ) (Kakutani-Krein). D e f i n i t i o n 3.1: A RJesz space E is said to be of Grothendieck type (or an ROspace) if for all z in a cofinal subset of E+, the principal ideal E , is a Grothendieck space.
Thus, in particular, each Dedekind or-complete Pdesz space is an RG-space. T h e following l e m m a is now the key for the Banach-Steinhaus theorem we are going to prove. L e m m a 3.2: If E is an RG-space, every band in the order dual E* of E is ~( E*, E)-sequentially complete. The proof is identical to the proof (due to the author) of [7, 1[.10.5], where the Dedekind a-completeness of E was used only to infer that E is an RG-space. Therefore, the somewhat technical proof of [7, 1[.10.5] will not be repeated here. T h e o r e m 3.3: Let E be an RG-space, and let F be a Riesz space separated byF*. (i) I f ( T , ) is a sequence of positive, order continuous operators E --~ F such that for each z E E , (T,,z) converges in F for or(F, F,~), then T: z ~-~ lim,~ T,~z is order continuous. (it) /f, in addition, F is Dedekind complete and (T,~) is a sequence of order continuous operators converging pointwise (for a(F, F~ ) ) to an order bounded operator T: E ~ F, then T is order continuous. Proof: (i) Let 0 < ~o E F* be arbitrary; then (T*~) is a tr(E*,E)-convergent sequence contained in E~. By L e m m a 3.2, T*~ =: lim= T,~o is in E*; whence it follows that T*(F*) C E*. Since T is positive (note that, by Theorem 1.1, the positive cone F+ is tr(F,F*)-closed), Proposition 1.3 implies that T is order continuous. (ii) T h e preceding argument is again applicable, since T is order bounded, and hence Proposition 1.3 is valid for T. [
Simple examples (such as E = F = l 2) show that in assertion (ii) above, the hypothesis t h a t T be order bounded cannot be omitted. 4. E x a m p l e s .
In conclusion, we want to illustrate the usefulness of T h e o r e m 3.3 above by two examples. First, we consider the Riesz space £0% A linear operator A: £oo ~ £~o is called a matrix operator if (Ax)i = Y~°=l ctik~k for each i E IN, z = ( ~ ) E £~o and suitable sequences (ctik)ket~ (necessarily contained in £1). It is well known that an endomorphism A of loo is a matrix operator if and only if A is order continuous.
99
P r o p o s i t i o n 4.1: Let (A,~) be a sequence of matrix operators on £oo. If
exists for all pairs (*,z) E t °o x £1, then A is a matrix operator whose entries are the limits, as n --* 0% of the respective entries of A,~. Proof: Since for each • E goo the sequence (A,~x) is weak* and hence norm bounded, it follows that A is a norm bounded operator on £oo. But, as is well known, every norm bounded operator on too is order bounded; thus the assertion follows from Theorem 3.3.(ii). (Noting that, by Theorem 1.5, weak* continuity of an endomorphism of £oo is equivalent to order continuity, the result also follows using the weak sequential completeness of £1). ] Our second example is somewhat more delicate. Let F be a weakly sequentially complete Banach lattice; then F is a perfect Riesz space (Definition 1.4) whose order continuous dual F~ agrees with its Banach (and order) dual F'. Suppose (T,)tctt+ is a one-parameter semigroup of positive linear operators on F ' which is weak* operator continuous, i.e., such that for all pairs ( f , g ) E F × F ' the mapping t ~ (Ttg, f) is continuous on ]R+. Then it is somewhat tedious but not difficult to verify that (T,) satisfies estimates IIT,]I _< Me ~'* for suitable M , . , • IR, and that for g in a weak* dense linear subspace D(Z), Z g =: limt--.0 t - l ( T t g - g) exists for # ( F ' , F ) . Z is a weak* closed linear operator whose spectrum #(Z) lies in aleft half-plane {), • ¢ ] Re), _< s(Z)}; for ReA > s(Z), the resolvent ( ) , - Z ) -1 exists (as a bounded linear operator on F') and it is positive for A • IR, A > s(Z). We claim: P r o p o s i t i o n 4.2: In addition to the preceding assumptions suppose that for some A • IR, ~ > s(Z), the resolvent (A - Z) -1 is order continuous. Then each operator T, (t • IR+) is weak* continuous, and we have T, = S~ (t • IR+) for a (unique) Co-semigroup (St) on F. Proof: By Theorem 1.5, weak* and order continuity are equivalent properties for order bounded operators on f ' ( = F*). Moreover, by Theorem 3.3.(i) the cone of positive, order continuous operators on F' is complete under the operator norm; therefore, analytic continuation shows that (A - Z) -1 is order continuous for all A, ), > s(Z). Now the well known inversion formula Tt = l i
1 - -tZ n
holds for the weak* operator topology on F', by virtue of the continuity of t ~-~ Tt. Thus from Theorem 3.3.0) it follows that each Tt is order and hence o'(F', F)continuous. Thus Tt = S~ (t • 1R+) for a unique positive semigroup (St) on F which is obviously weakly continuous. But this is well known to imply the continuity of (St) for the strong operator topology; consequently, (St) is a C0-semigroup
onF.
I
100
The preceding result was obtained in [8] for the classical case F = Lx(p), F'
=
=
References. [1] Fremlin, D.H.: Topological Riesz Spaces and M e a s u r e T h e o r y . Cambridge University Press, Cambridge, 1974. [2] KSthe, G.: Topological Vector Spaces, I. Springer, New York, 1969. [3] Luxemburg, W.A.J. and Zaanen, A.C.: Notes on Banach ~ n c t i o n Spaces, V/-V//. Proc. Nederl. Acad. Wetensch. (A) 66 (1963), 655-681. [4] Luxemburg, W.A.J. and Zaanen, A.C.: R.iesz Spaces, I. North Holland, Amsterdam-London, 1971. [5] R~biger, F.: Beltriige zur Strukturtheorie der Grothendieck-R£ume. Sitz. Ber. Heidelberger Akad. Wiss. Nr. 4 (1985), Springer. [6] Schaefer, H.H.: Topological Vector Spaces. GTM 3, 5th printing, Springer, NewYork, 1986. [7] Schaefer, H.H.: B a n a c h Lattices and Positive O p e r a t o r s . Springer, New York, 1974. [8] Schaefer, H.H.: Dual Characterization of Order Continuity and Some Applications. Archiv d. Math. 49 (1988).
Fixed Points of Holomorphic Mappings
by Jean-Pierre Vigufi Universit~ de Poitiers
1. I n t r o d u c t i o n Let D be a bounded domain in (IJ'~ (or, more generally, in a complex Banach space E). Let f : D ~ D be a holomorphic mapping. The set Mix y =
e I) I
=
has been studied by many people. Let us recall first the following theorem proved by E. Vesentini [8 and 9]: T h e o r e m 1.1: Let B be the open unit ball of a complex Banach space E. Suppose that every point z belonging to the boundary OB of B is a complex extreme point of-B. Let f: B ~ B be a holomorphic mapping such that f(O) = O. Then Fix f = B N F, where is the eigenspace of the derivative if(O) of f at the origin for the eigenvalue 1. Moreover, if E, is re[lexive, there exists a projection p: E --~ F of norm l. So, B N F is the image of a linear retraction B --~ B n F.
The proof is based on the notion of complex geodesic. In fact, E. Vesentini [9] proved that, given x C D, there exists a unique complex geodesic through the origin and ~, and, more or less, this argument concludes the proof. But, in general, complex geodesics are not unique, and Vesentini's proof cannot be generalized. For example, the case of the bidisc A × A has been studied by M. Herv~ [6] and E. Vesentini [8], and they proved the following result: T h e o r e m 1.2: Let f : A x A --. A x A be a holomorphic mapping. The set Fix f is one of the following sets: 1. the e m p t y set ¢ ; 2. one point; 3. there exists a holomorphic mapping ~: A ~ A such that
102
or
F i x f = {(~1,~2) E A × A [ ~ I = ~(~2)}; 4. A x A . So, in this example, the set Fix f is not a linear subspace, but it is always a connected submanifold. Now, we are going to give the results of this talk, and, first, we will begin with the finite-dimensional case.
2. B o u n d e d d o m a i n s in ¢'~ We begin with the following result: T h e o r e m 2.1: ([13]) Let D be a bounded domain in C '~ and let f: D ~ D be a holomorphic mapping. Then Fix f is a complex submanifold of D. If a E Fix f, its tangent space T a ( F i x f ) is e q u a / t o
The proof of this result uses ideas of H. Cartan [3] and E. Bedford [1]. Let a E Fix f , and let us consider the sequence fP = f o - . . o f (p times) of iterates of f . We can find a sequence of integers pj ~ +o¢ such that qj = Pj+I - Pj and rj = pj+l - 2pj converge to +oc and that fPJ converges to a holomorphic map F (uniformly on compact subsets of D). Now, by taking subsequences of the sequences qj and rj, we can suppose that f~i ~ p,
f~J ~ G.
By shrinking D if necessary, we can suppose that p, F and G send D to D. Then, by composition, one proves easily the following relations:
poF=Fop=F,
FoG---GoF=p,
fop=pof.
We deduce that
p2 = p o p = p o F o G =
FoG=p.
So, p is a holomorphic retraction, and, by a result of H. C a f t a n [4], there exists a local coordinate chart u defined on a neighbourhood U of a, such that u(a) = 0 and t h a t u o p o u -1 is a linear projection. We have proved that p(D) is a submanifold of D containing Fix f , and it is easy to prove that f is a biholomorphic automorphism of p(D). It is clear t h a t p(D) is a hyperbolic manifold ([5]), and we can apply the following result of H. Cartan [2]:
103
2.2: Let X be a complex hyperbolic manifold of finite dimension n, and let a be a point of X . Let f E A u t ( X ) be a biholomorphic automorphism of X such that f ( a ) = a. Then there exists a local coordinate chart u defined in a neighbourhood U of a such that u(a) = 0 and that u o f o u -1 is a linear automorphism of ¢'~. Theorem
This theorem applied to lip(D) proves that F i x F is a submanifold of D. Of course, Fix f is not connected in general; for example, consider the annulus
A={¢c ¢11/2< I¢1 <2}, and the automorphism f of A defined by f ( ( ) = 1/(. In fact, as proved by P. Mazet and J.-P. Vigu6 [7], the components of Fix f do not always have the same dimensions. 3. B o u n d e d
convex
d o m a i n s in ¢=
Now, if we suppose that D is a bounded convex domain in ¢'~, I can prove that the set F i x f is connected ([11 and 12]). In fact, we have the more precise result: T h e o r e m 3.1: ([12]) Let D be a bounded convex doma/n in ¢'~. Let f : D ~ D be a holomorphic mapping and let us assume that Fix f is not empty. Then there exists a holomorphic retraction ¢: D -* Fix f . Idea of the proof. We consider T,~ defined by rt-1
n
p~O
~,~ is a holomorphic mapping from D to D, and, by Montel's theorem, we can find a subsequence ~v,~, converging to ~ (uniformly on compact subsets of D). ~ is holomorphic, and, as D is taut, ~ is a holomorphic mapping from D to D. Let a E Fix f . By elementary linear algebra considerations, one proves that ~o'(a) is a linear projection onto
e = {. c ¢" I s'(a).v =v}. Now, let us define
Using Cauchy's inequalities, one proves that Cr, converges uniformly on compact subsets of D to a holomorphic mapping ¢ such that ¢ ( D ) C Fix f , and ¢[Fix ! = id [Fix/. The theorem is proved.
104
4. B o u n d e d d o m a i n s in r e f l e x i v e B a n a c h s p a c e s The results of this section have been proved in collaboration with P. Mazet [7]. The first idea we use to generalize these results to the case of bounded domains in reflexive Banach spaces is to consider weak topology and weak hmits of sequences. However, it does not seem possible to generalize the proof of Theorem 2.1 for the following reason: if f,~ (respectively, g,,) weakly converges to f (respectively, g), in general, f,~ o g,~ does not converge to f o g. Fortunately, it is possible to generalize the proof I gave for bounded convex domains in ~'~, and we prove the following theorem: T h e o r e m 4.1: Let D be a bounded convex domain in a reflex/re Banach space E. Let a C D, and let f: D ~ D be a holomorphic mapping such that f ( a ) = a. Then the set Fix f is a complex direct submanifold of D, tangent in a at
F={vcElf'(a).v=v }, and there exists a holomorphic retraction ¢: D ~ Fix f . Idea of the proof: As in the finite-dimensional case, we define 1 ,~-1 n
p:0
Let us consider on the set H ( D , D) of holomorphic functions from D to D the topology of uniform weak convergence on finite-dimensional compact subsets of D. It is more or less standard that H ( D , D ) is compact, and so, we can find ~ adherent to the sequence ~,~. is a holomorphic mapping from D to D; it is clear t h a t o f = ~,
Fixf C Fix~,
and using the continuity of f ' ( a ) for the weak topology, we prove that f'(a)
o
:
So, ~t(a) is a projection onto F . Now, we consider the sequence of iterates ¢,~ = ~'~ of ~. If g is holomorphic in a neighbourhood of a, we note g =
ep(g) p=0
the development of g in series of homogeneous polynomials at a. We prove the following lemma.
105
L e m m a 4.2: For every n > 0 and p < n we have P p ( f o ~on) = Pp(~on) = Pp(tor'+l). We have already proved this lemma for n -- 1, and the proof is by induction on
r~.
Using Cauchy's inequalities for bounded mappings, this lemma implies that !b, converges to a limit @ uniformly on a ball of center a and of radius small enough. But, by [10], it implies that ~bn converges to a limit tb for the topology of local uniform convergence. So, ~b • H ( D , D), and we have f o ~b = tb, ~b~ = !b, Fix f = Fix ~b. As ~b is a holomorphic retraction, the theorem is a consequence of H. Caftan [4]. Now, if we do not suppose that D is convex, we can also define ~o,~ and ~0. The only difference is that ~0 does not send D to D. However, if a is a fixed point of f , we prove in [7] that there exists a neighbourhood U of a such that ~(U) C U, and, with some small changes, we can generalize the proof to this case. T h e o r e m 4.3: (P. Mazet and J.-P. Vigu~ [7]) Let D be a bounded domaJn in a reflexive Banach space E , and let f : D --~ D be a holomorphic mapping. Then the set Fix f is a complex direct submanffold o[ D. 5. A n e x a m p l e
To conclude this talk, I shall give an example [7] which proves that the conclusion of Theorems 4.1 and 4.3 is not true for every Banach space E. Let c0(IN) be the Banach space of sequences converging to 0 at infinity. Let B be the open unit ball of c0(IN). Let n • IN, and let f be a holomorphic mapping from the polydisk A '~ into itself. Let us define F: B ~ B in the following way: (Zp)p~N = F ( ( z p ) p ~ ) , where (Zo,...,Zn-1)
(z.,.
-~ (;g0,---,Zn-1),
= f(;go, Z2,~+~ = z,~+~,
Vk >_ O.
It is easy to check that Fix F = Z ( f ) x {0}, where z(s) =
{(z0,...,;g._l) •
I S(z0,... ,z._l) = 0}
is the zero set of f . It is clear that z ( S ) and Fix(F) are not, in general, submanifolds. There are also examples (P. Mazet and J.-e. Vigu~ [7]) in which E is a dual space.
106
References
[1] E. Bedford: On the automorphism group of a Stein manifold. Math. Ann. 266 (1983), 215-227. [2] H. Caftan: Les ironctions de deux variables complexes et le problbme de la representation analytique. J. Math. Pures Appl. 11 (1931), 1-114. [3] H. Cartan: Sur les fonctions de plusieurs variables complexes. L'itEration des transformations intErieures d'un domaine bornE. Math. Z. 35 (1932), 760-773. [4] H. Caftan: Surles retractions d'une variEtE. C.R. Acad. So. Paris 303 (1986), 715-716. [5] T. Franzoni, E. Vesentini: H o l o m o r p h l c m a p s a n d invarlant distances. Mathematical Studies 40, North-Holland, Amsterdam, 1980. [6] M. I-Ierv~: Quelques propridtEs des applications analytiques d'une boule & m dimensions dans elle-m~me. 3. Math. Pures Appl. 42 (1963), 117-147. [7] P. Mazet, J.-P. Vigu~: Points f~xes d'une application holomorphe d'un domaine borne dans lui-m~me. To appear. [8] E. Vesentini: Complex geodesics. Comp. Math. 44 (1981), 375-394. [9] E. Vesentini: Complex geodesics and holomorphic maps. Symp. Math. 26 (1982), 211-230. [10] J.-P. Vigu~: Le groupe des automorphismes analTtiques d'un domaine borne d'un espace de Banach complexe. Application aux domaines bornEs symdtriques. Ann. Sc. Ec. Norm. Sup. 9 (1976), 203-282. [11] J.-P. Vigu~: GEodEsiques complexes et points fixes d'applications holomorphes. Adv. Math. 52 (1984), 241-247. [12] J.-P. Vigu~: Points t~xes d'applications holomorphes clans un domaine borne convexe de C "~. Tr. Am. Math. Soc. 289 (1985), 345-353. [13] J.-P. Vigu~: Sur les points fixes d'applications holomorphes. C.R. Acad. Sc. Paris 303 (1986), 927-930.
List o f P a r t i c i p a n t s M a r c o A b a t e - - Scuola N o r m a l e Superiore, Pisa G i u s e p p e A c c a s c i n a - - Universit~ di Pisa F r a n c e s c a A c q u i s t a p a c e - - Universit~ di Pisa A n t o n i o A m b r o s e t t i - - Scuola N o r m a l e Superiore, Pisa V i n c e n z o A n c o n a - - Universit£ di Firenze Michael F. A t i y a h - - O x f o r d University B e r n a r d H. A u p e t i t - - Universit~ Laval Stefan A. B a u e r - - Universit£t G S t t i n g e n B r u n o Bigolin - - Universit£ C a t t o l i c a del S. Cuore, Brescia Fabrizio Brogtia - - Universit£ di Pisa C l a u d i o B u z z a n c a - - Universit£ di P a l e r m o E u g e n i o Calabi - - University of P e n n s y l v a n i a L e o n a r d o Cangeliti - - Universit£ di R o m a Mario C a r l o t t i - - Universit£ di Pisa Fabrizio C a t a n e s e - - Universit£ di Pisa F r a n c o C a z z a n i g a - - Universit£ di Milano M a u r o C h i a r e t t i - - Universit£ di R o m a A l b e r t o C o n t e - - C o m i t a t o per la M a t e m a t i c a del C N R , R o m a Paolo Cragnolini - - Universit£ di Pisa P a o l o de B a r t o l o m e i s - - Universit£ di Firenze M a f i a D e d b - - Universit£ di Cagliari G i l b e r t o Dini - - Universit£ di Firenze Luigia Diterlizzi - - Universit£ di Bari J a m e s Eeels - - University of W a r w i c k S a n d r o F a e d o - - Universit£ di Pisa F r a n c o Favilli - - Universit£ di Pisa Massimo F e r r a r o t t i - - Universit£ di Pisa E l i s a b e t t a F o r t u n a - - Universit£ di Pisa Tullio Franzoni - - Universit£ di Pisa M a r g h e r i t a Galbiati - - Universitk di Pisa L a u r a G e a t t i - - IEI, Consiglio Nazionale delle Ricerche, P i s a G r a z i a n o Gentili - - SISSA, Trieste Francesco Gherardelli - - Universit£ di Firenze Giuliana G i g a n t e - - Universit£ di P a r m a Michel Goze - - Universit~ de Haute-Alsace, Mulhouse Hans G r a u e r t - - Universit£t G g t t i n g e n R e n a t a Grimaldi - - Universit£ di P a l e r m o Francesco G u a r a l d o - - Universit£ di R o m a Shoshichi K o b a y a s h i - - University of California, Berkeley J e r z y J. K o n d e r a k - - I n t e r n a t i o n a l Centre for T h e o r e t i c a l Physics, Trieste A d a m K o r £ n y i - - L e h m a n College, City University of New York
t08
S a n d r o Levi - - Universit£ di Milano P a t r i z i a Macrl - - Universit£ di R o m a A b n e d a c e r M a k h l o u f - Universit~ de Haute-Alsace, Mulhouse M a r c o M a n e t t i - - Scuola N o r m a l e Superiore, Pisa Stefano M a r c h i a f a v a - - Universit£ di R o m a E r m a n n o M a r c h i o n n a - - Universit£ di Milano C e s a r i n a M a r c h i o n n a Tibiletti - - Universit~t di Milano E n z o Martinelli - - Universit£ di R o m a F r a n c e s c a Menozzi - - Universit~t di Pisa M a u r o Meschiari - - Universit£ di M o d e n a G i u s e p p e M o d i c a - - Universit£ di Firenze Flavio M o s c a - - Universit£ di Pisa V e n k a t e s h a M.K. M u r t h y - - Universit£ di Pisa Emilio Musso - - Universit£ de L ' A q u i l a M a u r o Nacinovich - - Universit£ di Pisa Rainer Nagel - - Universit£t Tfibingen A n t o n e l l a Nannicini - - Universit£ di Firenze R a g h a v a n N a r a s i m h a n - - University of C h i c a g o Michael A. O ' C o n n o r - - IBM, Y o r k t o w n Heights, NY Paolo Oliverio - - Universit£ della C a l a b r i a A n n a M a r i a P a s t o r e - - Universit/t di Bari I.I. P i a t e t s k i - S h a p i r o - - Tel Aviv University Paolo Piccinni - - Universit£ di Salerno Fabio P o d e s t £ - - Scuola N o r m a l e Superiore, Pisa Carlo P u c c i - - Universit£ di Firenze G i u s e p p e Puglisi - - Universit£ di Pisa Luigi A. R a d i c a t i di Brozolo - - Scuola N o r m a l e Superiore, P i s a Fulvio Ricci - - Politecnico di Torino H u g o Rossi - - University of U t a h Delfina R o u x - - Universit£ di Milano Simon M. S a t a m o n - - Oxford University Mario Satvetti - - Universit£ di Pisa Helmut H. Schaefer - - Universit/~t T f i b i n g e n Sergio S p a g n o l o - - Universit£ di Pisa David R. Speiner - - UCL, Louvain-la-Neuve, Belgique L£szl6 L. S t a c h 6 - - Bolyai I n s t i t u t e , Szeged Francesco Succi - - Universit£ di R o m a C o r r a d o Tanasi - - Universit£ di P a l e r m o C e s a r i n a Tibiletti - - Universit£ di Milano A l b e r t o Tognoli - - Universit£ di T r e n t o G i u s e p p e Tomassini - - Scuola N o r m a l e Superiore, Pisa F r a n c o Tricerri - - Universit£ di Firenze Sergio Venturini - - Scuola N o r m a l e Superiore, Pisa
109
E d o a r d o Vesentini - - Scuola Normale Superiore, Pisa J e a n - P i e r r e Vigu~ - - Universitfi de Paris ~'~ Vinicio Viltani - - Universit~ di Pisa Georges G. Weill - - Universit~ de Tours Gian Carlo Wick - - Universit~ di Torino Paolo Z a p p a - - Universit~ di C a m e r i n o
