Some Recent Results in C o m p l e x M a n i f o l d Theory Related to
V a n i s h i n g Theorems for the S e m i p o s i t i v e Case
Yum-Tong
Department
Harvard
Cambridge,
To put
this
survey
some rather general
complex
maps,
spaces,
holomorphic
holomorphic
example,
tries
holomorphic
that
two
biholomorphic
method
holomorphic
principle
then
objects
a harmonic
and,
for some
(2) The method
holomorphic
(3) Grauert's
strongly
holomorphic
of using
sections
bumping
pseudoconvex
the
tries
first
to
one
that a
n+l good
To p r o v e
produce
objects?
a
So
in
powers
is the use of the D i r i c h l e t
from
them.
the
surfaces
Examples
maps.
higher-dimensional
object
is i m p o s s i b l e
theorem
of
of p o s i t i v e
and
of
the
of Eel ls-Sampson
of h a r m o n i c
to c o n s t r u c t
[13].
and then getting
on open Riemann
and a holomorphic
vanishing
domains
To p r o v e
line bundle.
are the r e s u l t s
cases,
technique
to £ n
to produce
one
objects
functions
case,
of high
For
such h o l o m o r p h i c
functions
special
their
is b i h o l o m o r p h i c
An e x a m p l e
object
and
objects.
methods:
objects
one-dimensional
holomorphic
bundles
such
one tries
[40] on the e x i s t e n c e
gap between
vector
make
or in general
like
functions.
to Pn'
harmonic
holomorphic
objects
construct
one p r o d u c e
harmonic
let me first
manifolds
biholomorphic,
from them.
of h a r m o n i c
except
to
manifold
the f o l l o w i n g
and S a c n s - U h l e n b e c k
the
are
of c o n s t r u c t i n g
obtaining
unlike
has
holomorphic
How does
to c o n s t r u c t
construction
holomorphic
of a s u i t a b l e
manifolds
map.
perspective,
holomorphic
One
n suitable
far we h a v e m a i n l y
(i) The
with
is b i h o l o m o r p h i c
sections
complex
U.S.A.
To study c o m p l e x
that a c o m p l e x
to p r o d u c e
manifold
MA 02138,
functions,
to p r o v e
complex
works
sections.
University
in the proper
remarks.
one
Siu
of M a t h e m a t i c s
[i0]
However,
case
is v e r y
the
wide
to bridge.
Kodaira
to
line b u n d l e s
holomorphic
construct
[24].
functions
on
170
(4)
The
method
functions
on
Vesentini
are
of C a r t a n - S e r r e
a Stein
manifold
strongly
and
positive
with
if the c u r v a t u r e
quadratic
smooth
in
pseudoconvex
smooth
of
if
function
it
r as a H e r m i t i a n
line
vectors
~ of all
than
1 is
gives
to
coefficient
section
powers
in the
a
holomorphic
In
above
case
the
no
of
in
using
objects
case
require
is
of
producing
form
is used,
case
of
quadratic
special
be
is used.
between
[45,
of
a
46,
then
are
fiber~
on
its
k th
of L*
is a
sections
case
of
of
producing
domain.
complex
objects
Hessian
pseudoconvex
harmonic
51,
some
form
in
of
the
domain.
In
holomorphic
However,
certain
the
less
if L is
function
L because
it the c u r v a t u r e
47,
some
Hessian
only
holomorphic
the
for
lengths
the
with
a Hermitian
holomorphic
to c o n s t r u c t
gap
up to n o w
positive-definiteness
along
a strongly
form
is a w i d e
known
or
objects
r < 0
is
is a
strongly
complex
if a n d
to be
domain
be
manifold,
pseudoconvex
bundle
said
the
of
and
metric
by
If L
powers
a
to
A holomorphic
is
to
a holomorphic
So producing
line
the
that
in L*
expansion
harmonic
there
and m e t h o d s
domain
bundle
bundles
compact
said
complex
o f L.
methods
positive-definite
dimensional
gap
of
line
L* of L w h o s e
series
quadratic
function
method
is
such
of
space
existing
to the H e r m i t i a n
a compact
that
to
to h a v e
or a S t e i n
fibers
positive-definite.
of a s t r o n g l y
a positive
its
relatively
bundle
has
previously
its boundary
sections
line
functions
positive-definite
defining
power
positive
the
near
observed
of t h e k t h p o w e r
of
A
so
A and B
for the c o n s t r u c t i o n .
along
pseudoconvex
[14]
one
of p o s i t i v e
gradient
over
holomorphic
of £n)
manifold
of the dual
Grauert
rise
is
bundle
a strongly
positive.
form.
defined
form
set
metric
nonzero
Andreotti-
scratch
and h o w t h e y are r e l a t e d .
complex
is
r with
ho]omorphic
the
a
but
manifolds
form associated
positive-definite
boundary
such
the n o t i o n s
a Hermitian
holomorphic
[30],
from
objects,
are e s s e n t i a l
domains
construct
t h e u s e of T h e o r e m s
submanifold
on
explain
to
(Morrey
objects
like
holomorphic
B and
pseudoconvex
~
[20]).
methods
functions
me b r i e f l y
bundle
of
domains
holomorphic
other
(i.e. a complex
A
holomorphic
Let
line
produce
also
to c o n s t r u c t
Theorems
global
L 2 estimates
pseudoconvex
[25], H ~ r m a n d e r
methods
There
apply
using
[i], K o h n
These
speak.
of
strongly
objects
in
the h i g h e r -
and
holomorphic
52]
quadratic
to b r i d g e
form
the
coming
171
from
the
when
the q u a d r a t i c
curvature
only
positive
certain
wonder
are
This
used
semidefinite
w h y one
situations
sectional
only
bother
be
it
semidefinite
case
me
strictly
give
two
to
to
be
negativity.
here.
One
domain.
are used
strictly
are
There
that
Another
positive
though
is by far m u c h m o r e c o m p l i c a t e d
in
some
of the
is
(like
definite
positive
In
One may
case.
is
in p r o o f s
semidefinite
be
situation
objects
like the s e m i n e g a t i v i t y
l i m i t of s t r i c t l y
out
the
positive-definite.
benign
symmetric
objects
first
turn
discusses
holomorphic
to s t u d y the s e m i d e f i n i t e
a bounded
the
assumed
may
of
semidefinite,
for
talk
certain
Let
of h o l o m o r p h i c
method),
survey
in p r o d u c i n g
instead
reasons.
curvature
limits
result
should
of
are n a t u r a l l y
continuity
can
forms
c a s e s we m a y e v e n a l l o w
a number
when
tensor.
that
in the
objects
the
final
definite.
The
t h a n the d e f i n i t e
case.
In
this
vanishing
talk
theorems
we
will
for
the
M o r e s p e c i f i c a l l y we w i l l
(i) The
construction
curvature
form
not
of h o l o m o r p h i c
strictly
An
conjecture
characterizing
[49,
application
recent
case
sections
positive
is
a
or
proof
Moishezon
results
and
their
even
of
the
for
line
with
bengin
manifolds
in p a r t i c u l a r
K~hler m a n i f o l d s
the r e s u l t s
of J o s t - Y a u
Kohn's
Sube] liptic
applications
estimates
of
school
with
[22]
(iii)
negativity
line
seminegative
and Mok
[29]
on
of p o l y d i s c s .
[26,
6]
and
their
to v a n i s h i n g theorems for s e m i p o s i t i v e bundles.
I. Producing_ S e c t i o n s
We w a n t
for S e m i p o s i t i v e
to d i s c u s s
line b u n d l e
how one
whose
may e v e n be n e g a t i v e somewhere.
study
with
by s e m i p o s i t i v e
compact quotients
of
bundles
Grauert-Riemenschneider
the s t r o n g r i g i d i t y of i r r e d u c i b l e
a Hermitian
concerning
applications.
50].
(ii) The strong r i g i d i t y of c o m p a c t
curvature,
some
d i s c u s s the f o l l o w i n g three topics:
somewhere.
bundles
survey
semidefinite
is
to
prove
the
Bundles
can p r o d u c e
curvature
form
The o r i g i n a l
so-cal
led
holomorphic
is o n l y
sections
semipositive
for
or
m o t i v a t i o n for this kind
Grauert-Riemenschneider
172
conjecture[15,
manifolds
whose
p.277].
Kodaira[24]
by the e x i s t e n c e
curvature
of
form is p o s i t i v e
Riemenschneider attempts
characterized
a Hermitian
definite.
to g e n e r a l i z e
projective
holomorphic
algebraic
line
The c o n j e c t u r e
bundle
of G r a u e r t -
Kodaira's r e s u l t to the case of
M o i s h e z o n manifolds. A M o i s h z o n m a n i f o l d is a c o m p a c t c o m p l e x m a n i f o l d
with
the
property
function
field
that
equals
that such m a n i f o l d s
a projective
are
algebraic
a M o i s h e z o n space
the transcendence
its c o m p l e x
precisely
degree
dimension.
space
coherent
is s i m i l a r l y
is
Moishezon
analytic
sheaf
f o r m is p o s i t i v e
Hermitian
metric
those w h i c h
set
of
points
regular.
where
the
can be
Conjecture
of
[28]
transformed
into
is d e f i n e d
is
curvature
is p o s i t i v e
on
it
that a c o m p a c t
a
torsion-free
a Hermitian
metric
by g o i n g to the
form
the proof
free
and
whose
Here a
linear space
is d e f i n e d
locally
only
the
on the
space
of t h e c o n j e c t u r e
is
is h o w to
case.
Grauert-Riemenschneider.
admits
The c o n c e p t of
on an o p e n d e n s e s u b s e t .
the c u r v a t u r e
with
special
laanifold w h i c h
form
showed
asserts
exists
one with
definite
sheaf
The difficulty
p r o v e the f o l l o w i n g
there
rank
for a s h e a f
to the sheaf and
meromorphic
defined.
if
of
curvature
associated
its
m a n i f o l d by p r o p e r modification.
The c o n j e c t u r e of G r a u e r t - R i e m e n s c h n e i d e r
complex
of
Moishezon
a Hermitian
Let
M be
holomorphic
definite
a compact
line
complex
bundle
on an o p e n d e n s e
L whose
s u b s e t G of M.
Then M is Moishezon.
Since
the
conjecture
a number
of
obtained
[38,57,53,12,35]
the
other
blow-ups,
be
which
~
K~hler,
proof
Kodaira's
vanishing
of
identity
his
vanishing
differential
for
equations
Grauert-Riemenschneider.
of L is n o t p o s i t i v e
and
the
that a proof
and e m b e d d i n g
K~hler m a n i f o l d s .
then R i e m e n s c h n e i d e r
theorem
Moishezon
[39]
of
introduced,
by
have
of
theorems,
proving
stating
can be o b t a i n e d
been
the
by using
or L 2 e s t i m a t e s
If the m a n i f o l d M is a s s u m e d to
observed
embedding
solutions
was
spaces
difficulty
conjecture
in such a way
for c o m p l e t e
of
circumvent
Grauert-Riemenschneider
characterizations
of
of G r a u e r t - R i e m e n s c h n e i d e r
characterizations
that
theorems
second-order
Kodaira's
together
elliptic
original
with
the
partial
[2] a l r e a d y y i e l d s right away the c o n j e c t u r e of
If the set of points where
definite
is of c o m p l e x
the c u r v a t u r e
dimension
form
z e r o [38] or o n e
173
[44] or
if some
of
curvature
the
additional
Riemenschneider
can
used degenerate
assumptions
form
of
rather
K~hler
to
deal
general
with
the
holomorphic
are imposed
[47] ,
easily
metrics
the G r a u e r t - R i e m e n s c h n e i d e r
fail
L
the
be
proved.
to o b t a i n
conjecture.
fundamental
sections
for
of
Recently
However,
all
of
line b u n d l e
Grauert-
Peternell
some p a r t i a l
question
a
on the e i g e n v a l u e s
conjecture
[33]
results
about
the a b o v e r e s u l t s
how
not
to
produce
strictly
in
positive
definite.
Recently
nonstrictly
used
the
to
special
case
to g i v e
conjecture
the
familiar
and
where
M-G
is
making
line
bundle
Siegel
to obtain
[49,
complex
50]
the
of
technique
give
[49,50].
the
a
was
To make
condition
precise
M-G
Singer
manifold
[3]),
prove
that
sufficiently
suffices
large.
positive
Lk
many
to s h o w
Thus
the
is
number
~
proof
of
number
also
of
of
iater
theorem
function
complex
its
sections
Thimm's
meromorphic
its
and
of
the Schwarz
by e s t i m a t i n g
holomorphic
[41]
later
imitates
at a s u f f i c i e n t
the
in
version
method
by
[54]
field
of
dimension.
a
In
forms
with
coefficients
coupled
with
the
of
description
easier
to understand,
measure
for
of
zero
the
of
case
the
h ckn
of
index
for
enough
a
theorem
meromorphic
functions
d i m H 0 ( M , L k)
is
reduced
>
to
and for q ~ 1 one has
general
cortstant
dimension
M
ckn/2
for
proving
that
of
compact
of A t i y a h -
positive
to m a k e
of
impose
theorem
sections
dim Hq(M,
method
first
the
theorem
some
holomorphic
the
we
in M. By
large, w h e r e n is the c o m p l e x
admits
that
of using
to h ar m o n i c
dim Hq(M,L k)
problem
theory
its use was
is a c o n s e q u e n c e
~=o(-l)q
it was
It was
The
Serre
brief
(which
c w h e n k is s u f f i c i e n t l y
To
by
for
[19, 3].
Hirzebruch-Riemann-Roch
complex
and
the d e s c r i p t i o n
that
to
exceed
applied
M.
[50].
order
the
cannot
line b u n d l e
more
of
in
of a function
alternative
Hirzebruch-Riemann-Roch
We
used
degree
manifold
in a h o l o m o r p h i c
to h i g h
was
an
transcendence
compact
zero
number
applied
[49]. T h e r e
case and a stronger
vanishing
it v a n i s h
holomorphic
[43]
measure
in a n a l y t i c
sections
of G r a u e r t - R i e m e n s c h n e i d e r
of the general
technique
the
of
holomorphic
introduced
conjecture
a
that
was
the
the identical
Such
obtaining
Grauert-Riemenschneider
technique
lemma to p r o v e
points.
of
a proof
of
of
line b u n d l e s
a proof
the
order
method
positive
give
refined
a new
to
of M.
give
Moishezon,
it
k sufficiently
L k)
for
any
given
~ ~k n
for
a
174
k sufficiently
large.
of H q ( M , L k) b y
Lk-valued
T
one
obtains
space
a
K~hler)
show
and
for
manifold
that
any
vanish
closed
have
linear
and
of
zero
can
making
chooses
a local
is
consequence
harmonic
The
that
it
ball
any
The
why
one
forms
and
uses
is
there
from
the
no
via
M-G
the
is of
such
norm
a
of
, where
a
f is
function
this
e -k#
is a n
obstacle,
as de
by
of d e a l i n g
lemma
of the a b s o l u t e
corresponding
point,
constants
instead
Schwarz
one
vanishes
at that
above
cocycles
a nonzero
why
factor
as w e l l
is
positive
square
centered
by
as we please
IfI2e -k%
property
is
Since
reason
The
¢
that
log p l u r i s u b h a r m o n i c
and
L.
k -I/2
zero
is
plurisubharmonic
of L so t h a t
from
in
(which
prescribed
pointwise
of
would
form
forms
o f W as s m a l l
To o v e r c o m e
of r a d i u s
reason
functions
Schwarz
is
a
value
lemma
forms.
works
eigenvalue
lemma.
harmonic
method
nonempty.
a
the S c h w a r z
trivialization
of t h e
holomorphic
is
metric
of k. T h e
with
~
there
orders.
large.
the
~-
cohomology
points
harmonic
o f L k is o f t h e f o r m
and
below
of
than
that
the
of unity
its
map
order
H q ( M , L k) is d o m i n a t e d
vanishing
Hermitian
The on the
independent
for
the
bounded
directly
of
to
is
to
linear
fixed
of a h a r m o n i c
otherwise
the volume
lemma
that
in
lattice
a basis
sufficiently
section
to a p p l y i n g
at a point.
e -k#
k
is c h o s e n
dim
kn),
form
norm
of
smaller
the
a partition
the
that
from
via
of
necessarily
Schwarz
is so s m a l l
harmonic
of
the r e q u i r e d
function
corresponding
the
of W t i m e s
E
form
apart
technique
(not
the
of
to the
k -I/2
to an a p p r o p r i a t e
its n o r m
number
coming
choose
points
holormorphic
obstacle
the
volume
holomorphic
local
of
in M, w e c a n m a k e
after
of
a harmonic
It f o l l o w s
times
having
forms
distances
uses
elements
L 2 estimates
Hermitian
and
points
minimality
cocycles
therefore
lattice
from
that
the
class.
to the
the
of h a r m o n i c
with
compact
frora it b y u s i n g
map and
number
a
(7.14)]
lattice
than
constant
measure
of
coming
contradicting
comparable
case
p.429,
the
smaller
combination
space
constructed
form
fixed
the
and represent
using
of p o i n t s
otherwise
its cohomology
a
from
lattice
By
W of M-G. Then one uses the usual
the
at a l l
metric
forms.
identically,
a norm
class,
a
[16,
cocycle
vanishing
must
map
Take
neighborhood
Bochner-Kodaira
M a Hermitian
harmonic
linear
of c o c y c l e s .
in a s m a l l
Give
Let
of
outlined
in
R
the
the
be
above
general
the
set
curvature
can be refined
case
of
form
where
points
~
in t h e f o l l o w i n g
G
is
of
of
M
L
only
where
does
assumed
the
not
w a y so
to
be
smallest
exceed
some
175
positive
number
coordinate
polydisc
choose
C
>
a global
k .
For
every
D with
point
coordinates
trivialization
0
in
R
one
can
Zl,...,z n c e n t e r e d
of L o v e r
choose
at
O and
a
can
D s u c h t h a t for s o m e c o n s t a n t
0
n
I $(Pi ) -
for
PI'
$(P2) I <_ C
P2
in D.
( x Izi(Pi)
Moreover,
chosen
to
be
number
of
such
coordinate
only
on
depending
chooses
the
the
the
times
but
total
k kn
have
the
Theorem
1.
number
theorem
M
be
the
for
more
~
Then
of
than
~ and
the
z2,...,z n.
a constant
small,
for q > i one
d i m H 0 ( M , L k) is n o l e s s
k is s u f f i c i e n t l y
m
one
along
sufficiently
number
be
finite
constant
apart
directions
no
D can
a
intersect.
By c h o o s i n g
positive
some
( k k ) -I/2
is
therefore
of
R by
large.
than
Thus
we
[49, 50].
a
compact
and
complex
bundle
manifold
over
M whose
is s t r i c t l Z p o s i t i v e
and
L
curvature
be
form
at s o m e point.
a
is
Then
M
manifold.
result
vanishing
bundle
Cover
them
are
alon 9 the
c when
line
everzwher 9 semipositive
By the
and
R.
that
of
- zi(P212)
polyradius
of
points
R.
any given
holomorphic
is a M o i s h e z o n
of
~ Izi(Pi)
i:2
the
O
they
apart
+
so
m
that
lattice
~k n
positive
Let
Hermitian
than
so
volume
<
C and
points
polydiscs
more
of
for
following
both
all
k -I/2
the
that
ck n for some
no
are
d i m H q ( M , L k)
for
points
number
times
we conclude
has
n
lattice
z I direction
Now
same
- zi(P 2) I2
of G r a u e r t - R i e m e n s c h n e i d e r ,
of H q ( M , L K M) f o r q > I ,
where
one h a s as a c o r o l l a r y
K M is t h e
canonical
line
of M.
In c o n j u n c t i o n
I would
like
dimensional
Moishezon
positive
integral
homologous
p.443]
with
to m e n t i o n
to
zero.
manifold
linear
between
result
noncompact
of M o i s h e z o n
of P e t e r n e l l
is p r o j e c t i v e
with
of
algebraic
Hironaka's
analog
us
the complete
threefolds
of
Theorem
example
a 3-
if i n it n o
curves
[18
is
andl7,
manifold
of the difference
and Moishezon
the
that
Moishezon
picture
1 is
manifolds,
[34]
irreducible
non-projective-algebraic
gives
projective-algebraic
The
result
combination
Together
of a 3 - d i m e n s i o n a l
Peternell's
the c h a r a c t e r i z a t i o n
the r e c e n t
threefolds.
following
conjecture
176
which
is s t i l l
Conjecture.
manifold
and
open.
Let
such
is
fl be
that
strictly
function
approaching
P.
Theorem
1
on
to y i e l d
bundles
An example
Theorem
bundle
used
results
whose
about
2__
t F__o~ ~ y e r y
bundle
over
numbers
form
such
that
is the
positive
Assume
Cn
where
(i +
torsion
We
Theorem
we allow
less
is
the
a
in
set
1 c a n be f u r t h e r
holomorphic
sections
to be n e g a t i v e
n there
fl
of
refined
for
somewhere
exists
following
~r_~o~r_~ty.- L e t
n and
subset
of
form of
L be
M and
line
[50].
a constant
a,
b ~[
line
positive
a as a l o w e r
bound
Cn
M be a compact
a Hermitian
L admits
- b as a l o w e r
of
(b2/a)n(volume
is t h e f i r s t
the
2 is
theorem
natural.
of this
describe
class
bound
at ever Z point
of
kind
below
method
the constant
interesting.
metric.
There
of
M-G) <
Cl(L)n
of L. T h e n
dim H0(M,
L k) is
>
large.
of the m a n i f o l d
Hermitian
not
2. T h e
Chern
k sufficientl~
the m e t r i c
formulations
far
log+(b/a)) n
is a c o r r e s p o n d i n g
Theorem
exists
sequence
following.
the curvature
k n / 2 ( n !) for
When
there
point
i.
dimension
admits
a complex
there
some
where
<
of
at e v e r y
P. T h e n
along
of T h e o r e m
of
subset
that
Cl(L)
Cl(L)n
case
length
integer
the
complex
of G a n d
point
is a l l o w e d
M. L e t G b e a n o ~ _ n
at ever Z point
M-G.
the
the e x i s t e n c e
de__2mgnding o n l Z o n n w i t h
of
to
of L w i t h
of s u c h r e s u l t s
manifold
some
open
pseudoconvex
to i n f i n i t y
in t h e p r o o f
curvature
K~hler
at
going
corresponds
in t h e d u a l
The method
~
compact
is w e a k l y
pseudoconvex
holomorphic
vectors
a relatively
its b o u n d a r y
with
The
is H e r m i t i a n
instead
the c o n s t a n t
C n depending
inequality
should
be
of K~hler,
o n the
in
the
assumption
better
and
more
of
natural
of r e s u l t s .
the
described
refinement
above
C n to d e p e n d
can
needed
to
readily
yield
o n M, b u t
The r e a s o n w h y the a b o v e
get
a
Theorem
then Theorem
method
proof
of
2 if
2 would
can only
yield
be
a
177
C n depending
space
o n M is t h a t
of h a r m o n i c
equations,
obtained
solve
one
in
this
forms
has
the
of
the
coefficients
are
If. S t r o n g
way
Rigidity
K~hler
manifold
analog
manifolds
known
of
M with
K~ller
of
existence
because
the
The
~f
By
using
~f
reason
is
or
why
that
homogeneous
it
comes
under
the
we
can
assumed
the
only
pulling
That
is
negative
Ricci
tensor
the
of
in a s u i t a b l e
f to
~f
of
the
the
the B o c h n e r - K o d a i r a
of
N does
why
the
positive
not
of
of
of M,
enter
and
the
also
~f
or
is
M
because
under
image
operator.
of
of
is
The
of the
M has
the
f.
Yf
In o t h e r
to the d u a l
tensor
that
of
of M.
formula
~f
the
is
that
condition
tensor
picture.
~f
conclude
in
to
The
of E e l l s -
either
two
technique
curvature
compact
curvature
we
of
star
are
the strong
(0,1)-forms
bundle
is a p p l i e d
or
the
K~hler
the
Bochner-Kodaira
of the Hodge
if a n y
sense
result
of
vanishing
curvature
M.
equivalence.
b y the
and of degree
technique
version
reason
M from
of the s e c t i o n a l
the
of
as
to obtain
of the curvature
from
the
cohomology
Compact
Bochner-Kodaira
conclude
term
back
instead
[31].
(l,0)-tangent
of
to
lemma
regarded
of the bundle
because
two in
the complexified
bundle.
the
technique
the B o c h n e r - K o d a i r a
w o r d s w e are a p p l y i n g
product
the
biholomorphic
be
The way
map
the
and
stronq~ Z rigid
is
negative
form
We
K~dller M a n i f o l d s
M "which is a h o m o t o p y
f of
curvature
that
so
C n independent
can
m a p is g u a r a n t e e d
vanishes
of degree
from
Actually
under
it
47].
side
Dolbeault
rigidity
of the n o n p o s i t i v i t y
Yf
the
to
formula.
with
Compact
to be
M. T o
equations
Schwarz
the
rigidity
46,
the
constants
(0,1)-covariant
harmonic
the
Curved
a harmonic
to
of t h e t e n s o r
pullback
harmonic.
either
[45,
Y
the c o n s t a n t
tensor
rigid
solving
manifold
the
one
from
is s a i d
the
the
the
of
the
apply
strong
curvature
N homotopic
M. As a s e c t i o n
N and
Mostow's
of s u c h a h a r m o n i c
Sampson[10]
then
Strong
M is to c o n s i d e r
manifold
on
on
of
passing
and
from
Bochner-Kodaira
form
homotopic
it.
to be strongly
rigidity
and
avoid
manifold
to
the
besides
unity
the inhomogeneous
of S e m i n e g a t i v e l y
antibiholomorphic
of
heavily
from
and c a n m a k e
K~hler
cocycles,
coefficients
we
a correspondence
of the e s t i m a t e
harmonic
the
cohomology
A compact
complex
use
holomorphic
This
to the C e c h
other
of
of
very
form
with estimates
between
solutions
depend
we make
of
differences.
partition
a
harmonic
solve
the
space
use
process
locally
differences
to the
to
problem,
derivative
in c o n s t r u c t i n g
to be
reason
most
why
general
178
formulation
following
Theorem
of
this
theorem
3.
A
kind
compact
K~hler
stronq! ~ rigid
if t h e r e
the
~roBerties:
following
positive
semidefinite
(p,0)-forms
on
of
results
manifold
exists
p such
that
vectors
(i)
The
in the s e n s e
M !~ positive
orthogonal
As a c o r o l l a r y
because
the
domain
we have
smallest
bounded
rigidity
following
of
(l,0)-forms
[32]
Complex
M
is
a n d the b u n d l e
of
2 ~ ~a__~k~2
with
combined
dimension
quotient
table
the
at
giving
twc
the c o m p l e x
assumptions
of
3
Dimension
Smallest
p
III n
n(n+l)/2
n(n-l)/2
IV n
n
and
each
6
VI
27
ii
of the
Zhong
method
antiholomrorphicity
smallest
also
of K o d a i r a ' s
the
two e x c e p t i o n a l
domains
[58].
yields
the
ho ] omorphicity
map from a compact
]]9 i s > 2 p +I at some point
This method can be r e g a r d e d
tensor
p for
of any h a r m o n i c
rank o v e r
+i
+i
2
16
the c u r v a t u r e
for
(m-L) (n-1)+l
V
be c o n s i d e r e d
rigid,
dimension
Theorem
(n-2) (n-3)/2
version
two
bounded
is s t r o n g l g
mn
into M whose
of
of an i r r e d u c i b l e
least
n(n-l)/2
This
[32].
exceeding
of M in the d i r e c t i o n
II n
by
on
s p a c e of M d o e s n o t c o n t a i n
Ira,n
computed
n ~[
than n with
domain.
Type
were
the
dimension
p less
i__nnth___~es e n s e
tangent
dimension
satisfying
symmetric
The v a l u e s
is
vanishes.
any compact
the
complex
number
bundle
curvature
subspace
of
of N a k a n o
subspaces
of c o m p l e x
p
M
definite
the b i s e c t i o n a l
one from each
symmetric
strong
a positive
(ii) At a n [ p o i n t of M the c o m p l e x
two n o n t r i v i a l
on
[47].
vanishing
is not needed
as c o r r e s p o n d i n g
Though
for this
of the q u a s i l i n e a r
strict
method,
to the s t r i c t l y
or
manifold
[47].
as an a p p l i c a t i o n
theorem.
K~hler
negativity
this method
definite
of
should
case rather
179
than
the
semidefinite
case
of
the u s e of the c o m p l e x i f i e d
is
in c o d i m e n s i o n
The
which
only
are
results
by
of
[21]
partial
[47]
to
We would
compact
compact
of
Let
compact
the
of
theorem
K~hler
the
[22]
a
Q
we make
polydisc
about
the
of
map
and
a
This
quotient
the
of a
vanishing
theorem.
Jost-
case
obtained
and
solved
to
a
hyperbolic
the
simple
Riemann
K~hler
complex
from
sense
the
version
observations
but
of
a
rather
maps
from
surfaces.
manifold
M to a
dimension
the
about
case
holomorphic
immediate
some
[29] c o m p l e t e l y
more streamlined
of
of
as
by
case
discuss
Dn
covered
remaining
a compact
polydisc
are
domains
rigidity
compact
technique
existence
from
not
some general
into compact
conclusions
are
and M o k
Bochner-Kodaira
manifolds
quotient
of
remaining
Jost-Yau
f be a harmonic
following
case
through
required
symmetric
strong
two.
l i k e to s k e t c h a s l i g h t l y
quotient
surprising
of
and which
least
this
[29]. F i r s t
application
of b o u n d e d
property
at
because
vanishing
one.
of an irreducible
semidefinite
Recently
in
the
dimension
considered
results.
of the proof
the
the
the
in d i m e n s i o n
[31]
is t h e c a s e
theorem,
operator
quotients
enjoy
result
complex
first
this case.
to
vanishing
star
than
of c o m p a c t
Mostow's
of
corresponds
Yau
case
rather
expected
suggested
polydisc
one
the
Hodge
n.
The
Bochner-Kodaira
technique.
(i) f is p l u r i h a r m o n i c
local
complex
(ii)
~fi ^
curve
~ f--~
in t h e
is z e r o
it is e x p r e s s e d
component
discs.
From
structure
section
the
(0,i)
implies
with
for
conclusion
f of the
direction
that
such
the
is
1 <__i <__n, w h e r e
in t e r m s
(ii)
of a h o l o m o r p h i c
itself
for
above
(l,0)-tangent
is d e f i n e d
restriction
of f to any
in M is h a r m o n i c .
of f when
under
that
of
it
bundle
vector
local
identically
(0,i)
covariant
identically
a holomorphic
zero,
vector
i th component
coordinates
follows
that
the
along
pullback
T I'0 of Q can be endowed
bundle
to be holomorphic
is
f i is t h e
in the f o l l o w i n g
if i t s c o v a r i a n t
zero.
This
exterior
which
bundle
because
differentiation
same
the
way. A l o c a l
is the i n t e g r a b i l i t y
The
n
f * T ~ '0
with
derivative
c a n be d o n e
structure.
the
in
(ii)
composed
condition
argument
can
180
be
applied
T~'lof
to the
Q to g i v e
if e v e r y
element
component
disc
vector
Dullback
are
the d i r e c t
which are the p u l l b a c k s
Q defined
a case
l e t L i be
of c o n c l u s i o n
f,T~,0
× ~i and
is
bundle
the
of
D n to itself.
is
holomorphic
For
any
group
is
a
of ~
near
points
exists
family
one
of
Li O
points
~-
holomorphic
L i' be
of
In such
the
at
Assume
of
Li
that
line
of
where
every
~i
element
component
O
1-form
with
some
~fi
~
and
disc
afi
local
complex
spaces
of
codimension
points
where
to g i v e
If in a d d i t i o n
the
because
agree with
M
of
is
a
not
whose
be
by
we
of
df i is two
(ii) the
the fibers
of the
a
of
complex
Such
the
kernel
by
a
local
over
local
of
have
afi.
divided
zero h o l o m o r p h i c
of
exterior
theorem
vanish
defined
can
of L i,
By the
submanifolds
one
rank
on
annihilate
afz
a nowhere
bundle
1-form.
does
tangent
consideration,
foliation
section
f*T$'l × ~I,
s i of the dual
where
of
whose
function
under
bundle
discs.
O ~.
product
foliation
also
holomorphic
line bundles
n,
section
holomorphic
Frobenius,
of
individual
1 <i<
(local)
a
holomorphic
holomorphic
of the tangent
f*T I'0Q a n d
on M.
each
section
its
af z
two
component
section
holomorphic
equals
codimension
fixed
holomorphic
derivative
holomorphic
of
1-forms
of Q maps
for each
section
local
si ~fz
these
~f is a h o l o m o r p h i c
holomorphic
Then
~fi
(i)
3f is a h o l o m o r p h i c
fundamental
(iii)
of
line s u b b u n d l e s
subbundles
bundle
Moreover,
of Q m a p s e a c h i n d i v i d u a l
each
of the i n d i v i d u a l
line
structure.
f*T~'t
of
Because
the
group
then
(0,1)-tangent
bundle
sum of the n h o l o m o r p h i c
of the
by the d i r e c t i o n s
subbundles
of the
itelf,
f of the
vector
of the f u n d a m e n t a l
of D n to
bundles
f*T Q
0'I under
it a h o l o m o r p h i c
~
leaves
locally
a
of
local
section
at the
of the
defined
map
--w-
fi. T h e
same
(ii) the
with
~fl
consideration
c a n be a p p l i e d
holomorphic
foliation
defined
the h o l o m o r p h i c
foliation
of
can be d i v i d e d
holomorphic
rather
local
by
local
sections
straightforward
following
theorem
[48].
~fl
holomorphic
of L i ~
by
~
discussions
to
the
~fz. A l s o
kernel
at points
functions
and L [ ~
lead
us
~
because
of
where
of
~fl agrees
both
~fi and
to g i v e nowhere
respectively.
immediately
zero
These
to
the
181
Theorem
4. L e t
hyperbolic
from
M
M t_~o R w h i c h
a holomor~hi~
surface
to
be
Riemann
a compact
surface
is n o n z e r o
ma_~
g
that
on the
from
S and a harrnonic
K~hler
such
M
manifold
there
second
into
ma[ h from
a
and
exists
R be
a compact
a continuous
homology.
compac~
Then
there
h~perbolic
S ~o R such
map
f
exists
Riemann
t h a t hog is h o m o t o p i [
f.
The
Riemann
foliation
Eells-Sampson
harmonic
of
we
and
complex
or
surface
described
can
therefore
with
the
h
Let
V be
property
with
set
we
have
that
M-V,
fiber
of
under
the
2. O n M - Z
of
leaf
f passing
metric
subvarieties
every
leaf
of
obtained
of
the
the
the
the
or
(0,i)
Bishop's
one
by u s i n g
we
over
that
consisting
foliation.
model
of the extensions
can
The
of
the
the
of
to
the
limit
of
we conclude
extended
Since
to a
Z is of
of R e m m e r t - S t e i n
M
is
covered
by
of the e x t e n s i o n s
Rieraann
of the quotient
of t h e
be
leaves
<
the
in c o m m o n
Because
on
in M.
the t h e o r e m
conclude
is
with
branch
to limit,
one
of R.
c a n be e x t e n d e d
M.
a
~
~
above
[4]
foliation
of
bundle
a point
in
theorem
of codimension
[37]
has
therefore
and by passing
subvarieties
of df
either
to g i v e
product
described
of
f is
subvariety
function
tangent
rank
that
where
real-codimension-two
and
result
complex
points
tensor
foliation
holomorphic
holomorphic
of
of c o d i m e n s i o n
volume
the
of a l l
the
holomorphic
the
generality
Z be
foliation
of
using
> 2 in M,
of
of
By
holomorphic
(I,0)
that point
as the nonsingular
are the branches
local
M where
with
subvariety
family
the
of
leaf
subvarieties
holomorphic
leaves
by
of bounded
codimension
extending
a
agrees
M,
complex-analytic
complex
points
subvariety
of
Let
the
way.
loss
section
f of t h e
through
a complex-analytic
K~hler
by a n y
a holomorphic
whenever
the
without
local
from
following
2 in M c o n s i s t i n g
holomorphic
pullback
constructed
the
real-analytic.
codimension
zero
is
in
assume
~f c a n n o t be d i v i d e d
nowhere
that
S
above
surface
the
of the
S is
of M w h o s e
on
now
points
of t h e h o l o m o r p h i c
foliation.
The
existence
compact
rather
of
hyperbolic
can conclude
K~hler
surprising
a continuous
the existence
manifold
by going'to
Riemann
the
to a c o m p a c t
respective
aspect
map
of
from
surface
Theorem
a compact
nonzero
on
of a n o n t r i v i a l
hyperbolic
universal
the
is
that
surface.
we obtain
from
manifold
second
holomorphic
Riemann
covers
4
K~hler
the
to
homology
map
from
a
we
the
In p a r t i c u l a r
a nontrivial
182
bounded
holomorphic
manifold.
bounded
So
a
universal
of compact
manifold.
Instead
the
Riemann
general
of
could
existence
curved
would
of a b a l l
that
rare
K~hler
maps
difficult
to exist.
of c o m p l e x
As
Riemann
surface
branched
covers
are
of
quotients
(a) Is
1 < m < n are
integers.
of
the b a l l s
true
that
there
exists
no
it
true
that
every
holomorphic
is
It
Let
no
functions,
map
its
for negatively
expect
compact
such
quotients
known examples
so
hyperbolic
[27]
by
taking
is n o t k n o w n
whether
of m a p s b e t w e e n
compact
besides
of c o m p l e x
it
compact
exists,
not
Livn~
examples
Suppose
quotients
do
for
by
surfaces.
dimensions
a
into any compact
constructed
of d i f f e r e n t
K~hler
there
map
Even
we
fact,
maps
similar
the
the k i n d of c o n t i n u o u s
l e a s t t w o the o n l y
elliptic
there are
to
holomorphic
general
of
as
curvature.
now
continuous
of b a l l s
Problem.
compact
ones
certain
in o t h e r d i m e n s i o n s
at
of
such
holomorphic
map
until
to e s t a b l i s h .
in
holomorphic
the
continuous
a
a matter
dimension
are expected
bounded
bounded
such
K~hler
of the compact
Since
existence
manifolds
that admit nontrivial
a
used.
if
which
of negative
property
the
constructing
manifolds
nontrivial
the
of
of
functions,
of
is
and
be r a t h e r
compact
continuous
far
mean
rather
manifolds
of a nontrivial
existence
surface
cover
method
holomorphic
curvature
constructing
only
is
univeral
general
complex
K~hler
use any
hyperbolic
want
on
of b o u n d e d
the existence
we do not
the
known
functions
cover
way
on
no
number
to conclude
this
is
large
function
we
function
there
holomorphic
to a d m i t
Here
far
the o b v i o u s
ones.
M a n d N be r e s p e c t i v e l y
dimensions
surjective
m a n d n.
holomorphic
map
from N
to M?
(b) Is
totally
geodesic
Yau
conjectured
uniquenss
results
different
dimensions.
proper
holomorphic
boundary
are
automorphisms
maps
from
Unfortunately
holomorphic
known
that
for
For
maps
the
Problem
proper
n ~
obvious
2-bal
until
maps
boundary
(b)
of
1
3 Webster
ones.
now
between
regularity.
to
M in N m u s t
[56]
have
3-bal
are
balls
In o u r
no
1
a
general
of d i f f e r e n t
case
the
showed
to
that,
the
results
dimensions
proper
map,
the
of
only
C 3 u p to the
four proper
up
of
balls
that
(n+l)-ball
[ii]
C 3
a consequence
between
showed
to the
Faran
be
maps
there are only
the
there
should
holomorphic
f r o m the n - b a l l
of the two balls,
the
embedding
image?
up
to
holomorphic
boundary
about
proper
without
though
any
without
183
any
known
comes
boundary
from
property
maps
may
We
be used
now
irreducible
Theorem
introduce
manifold,
and
equivalence.
components
an
two
a product
the
aF i
vector
bundles
orbit
aF i as
the
of
f is
that
of
above
some
of
Q
the
point.
class
the
f
is
points
a
of
represented
projection
of
is
function.
that
for
at every
V
must
by V would
the
A
n
i = 1 and
1 < i<n
loss of
try
the
to g e t
length
of
loss
than
and
(iii)
we
of a nowhere
the
the
the
pole-set
to z e r o b y
q:D n
rank
otherwise
+
f.
write
zero smooth
V
of
of
g is a
the
at a maximum
real
- 2,
can
The pole-set
Laplacian
be m a p p e d
and
M).
Without
2n
cover
on
a F 1 is g r e a t e r
precisely
universal
We
1 ! i !
for every
if it is n o n e m p t y .
equivalence,
be
D.
holomorphic
each
point.
h of g we get a contradiction
homotopy
of Q
of
(ii)
Thus
we
group
disc
the
for
the
(nondiscrete)
ly o n M. W i t h o u t
length
By
For
covers,
(l,0)-forms
that
l o c a l ly g is t h e p r o d u c t
by
is a
equivalence,
considering
value
be d e c o m p o s e d
polydiscs.
of
as
fails
otherwise
Since
D n with
i ! n, F i i__{s
fundamental
to s h o w
cannot
regular
+
1 ~
1-dimensional
identical
M
of
sections
in
h.
K~hler
a homotopy
M
that cannot
(rather
a homotopy
that
a meromorphic
empty,
is
F:
for each
the
the
hypersurface
be
M and
of which
complex-analytic
the absolute
of
of
irreducible
is a c o m p a c t
t__oo Q w h i c h
one
each
a F-l-cannot a g r e e
a F 1 at
so that
and
it
rigidity
both M and Q by finite
t h a t the a s s e r t i o n
Since
of
that
additional
its proof.
Q is a n
M
quotients
in D. W e h a v e
we can assume
aF
--f
function
group
~ F 1 vanishes
length
length
means
irreducibility
we a s s u m e
of g e n e r a l i t y
M
cover
holomorphic
o f G i is d e n s e
a contradiction.
8F 1 a n d t h e
strong
sketch
2,
b ~ F. T h e n
GI,...,Gn,
on M described
aF i o r
generality
n h
from
of g e n e r a l i t y
automorphism
of
either
~F 1 = g
ma~
universal
by replacing
loss
and
consequence
the
and
[29]). S u p p o s e
D n with
quotient
of n groups
of
on
lower-dimensional
theorem,
regard
the
Mok
this
o_[r a n t i h o l o m o r p h i c .
can assume without
every
the
irreducible
of
of t h i s
subgroup
and
n-disc
property
Hopefully
regularity.
theorem
a harmonic
M be
holomorphic
Here
is
an
additional
of p o l y d i s c s
(F 1 ..... F n) b e i n d u c e d
as a p r o d u c t
proof
f is
Let
the
[22]
of
the
quotients.
of b o u n d a r y
quotients
5 (Jost-Yau
has
compact
instead
compact
colnp_act ~ u ~ t i e n t
either
regularity,
between
Let
of
log
point
f
the
p:M
on
V
of
of
the
homology
+
M be
D be the p r o j e c t i o n
184
onto
the
first
component.
discussed
above
the
components
of
F(p-I(v))
one
of
the
points.
It
of D w h i c h
density
Mok
K~hler
of
[29]
the
M
follows
an
rank
fiber
that
holomorphic
foliation
be
constant
along
closed
subset
proper
of q w h e n e v e r
q(F(p-I(v)))
under
the
group
is
the
it c o n t a i n s
a proper
G I. T h i s
closed
contradicts
that
for
any
irreducible
covers
harmonic
compact
each
induced
map
from
quotient
of
a compact
the
n-disc
of t h e n c o m p o n e n t s
by
it
is
either
of t h e
holomorphic
antiholomorphic.
Ill.
Vanishing
Theorems
So far
the
conditons
The
all
make
recent
Catlin,
based
form
is
vanishing
of
others
the
b~ Subelliptic
theorems
of the pointwise
theory
and
on
use
Obtained
[26, 5, 6]
local
property
semidefinite.
Kohn
of boundary
regularity
of a w e a k l y
pseudoconvex
Let
weakly
say
there
~
be
an
open
pseudoconvex
that
subset
developed
to g e t
form.
by
Kohn,
vanishing
when
the
theory
to
deal
of the
~
thoerems
curvature
with
equation
the
in
boundary.
of
£n
whose
point
estimate
a neighborhood
his
curvature
form
for solutions
at a b o u n d a r y
a subelliptic
exists
curvature
developed
with
of the curvature
it p o s s i b l e
the
the c a s e
bundles
multipliers
makes
of
for
Estimates
property
subel liptic
question
We
must
÷ D. T h e
2n s o m e w h e r e ,
universal
the
M
of G 1 in D.
showed
to
real
map between
FI:
of
on
an entire
orbit
also
manifold
hop
is i n v a r i a n t
of e v e r y
(n h 2) w i t h
or
fibers
of D n contains
its
subset
the
Because
function
U of
boundary
x 0. L e t
holds
x 0 and
for
is
1 ~q ~n
smooth
be an
(0,q)-forms
constants
~
>
and
integer.
at
x 0 if
0 and
C > 0
such that
Ir~fl 2 < c l l l ~ t l
2 + IIY%ll 2 ,
11~tl21
C
for
all
the
domain
smooth
Sobolev
duced
the
U is
said
(0,q)-form
of
~*,
E-norm.
In o r d e r
concept
to
be
where
of
a
~ on
U{~[
II
II m e a n s
to o b t a i n
a subelliptic
subelliptic
with
compact
the
L 2 norm
subelliptic
multiplier.
multiplier
if
support
belonging
and
II~ m e a n s
II
estimates
A smooth
there
Kohn
intro-
function
exist
to
the
f on
positive
¢
185
a n d C so t h a t
IIf~l12
for
all
9 . The
( l ! i,j ~n-l)
~ .
frame
The
starting
simplicity
we describe
(ii)
determinant
The
~
of
the
matrix
(iv)
of
g
integer
is
m,
dimension
a
frame
The
other
smooth
q,
then
results
c a s e is s i m i l a r .
(cij)l < i , j <__n_l
belongs
t o If,
the determinant
field
of
n-l-k
formulation
of
carried
showed
out
that
consists
(I,0)
columns
is
the
to
formed
I I.
in the
of the c o m p o n e n t s
vectors
are
any
with
that
no
tangential
n-l-k
of
(hl,...,h n)
if t h e b o u n d a r y
local
and
function
a
of
~
to
columns
of
if f b e l o n g s
to
of
estimate
~
is of t y p e
open
definition
<
for
t at
hold
of
a
of
complex
to the ideal
Iq o f
estimate
[9] c o n t r i b u t e d
result.)
Recently
of smooth
(0,1)-forms
type
for
positive
subelliptic
holds
finite
disc
to
Catlin
boundary
at x 0 if
a t x 0 in t h e s e n s e
subelliptic
type
x 0 if f o r e v e r y
1-dimensional
some
is r e a l - a n a l y t i c
the case
is of f i n i t e
statements
D'Angelo's
the
for
for
subvariety
1 belongs
in Kohn's
investigation
(Similar
from
Ifl
consequence
the a s s u m p t i o n s
~
£
complex-analytic
as
a subelliptic
(0,q)-forms.l
The boundary
Igl m
that
a t x 0. ( D i e d e r i c h - F o r n a e s s
the
[7,8].
in t h e s e n s e
to I I.
if t h e b o u n d a r y
D'Angelo
radical
the constant
the
h =
of
notational
zero-set
belongs
multipliers
holds
for
For
whose
function
g also
(0,q)-forms
of
matrix
of the
for
and only
is the f o l l o w i n g
gradient
cij
to the b o u n d a r y
multipliers.
to I I. T h e i th c o l u m n
contains
subelliptic
and
the
[26] s h o w e d
x 0 and
[5,6]
Let
~ n e a r x in t e r m s
tangential
theory
lq.
I I.
to its real
then
Kohn
near
of
~
of
ideal
(cij)l < i , j < n - l "
I 1 equals
I 1 and
to
nonzero
fl,...,fk b e l o n g
~fi i n t e r m s
boundary
vectors
of K o h n ' s
r with
way belongs
the
(i,0)
an
the case q = 1 and the general
belongs
(iii) W h e n e v e r
following
form
lq of s u b e l l i p t i c
function
of
of
point
the ideal
boundary
multipliers
f o r m of t h e b o u n d a r y
field
concerning
(i) A s m o o t h
+ 11-#'9112 + 11~112)
subelliptic
be the L e v i
an o r t h o n o r m a l
of
< ctlt~-~tl2
E
is
estimates
as
follows.
holomorphic
D to ~n with
map
h(0)
=
186
x 0 the
vanishing
minimum
of
the
the
of
of
£
general
is
fo~
near
x
Catlin's
number
~
roh
at
orders
let
t(x)
upper
result
that
of
¢
so far o n l y
exceed
0. A t
but
satisfies
x of
that
that
the
t(x)
in
t(x) ~ t ( x 0 ) n - i / 2 n - 2
estimate
of t(x)
be bigger
the
point
such
showed
at x 0
for x n e a r
than
show that subelliptic
times
every
subelliptic
the m a x i m u m
cannot
t
number
D'Angelo
E in the
reciprocal
showed
not
smallest
~ t(x).
The order
the
b u t he c a n
the
semicontinuous,
x0 .
to be
0 does
of hl,...,h n at
be
a t x is o f t y p e
not
is e x p e c t e d
of
vanishing
boundary
boundary
order
the
x 0.
expected
estimates
hold
for
2
an
¢
of
We
the
order
now
vanishing
study
of
means.
theorems.
Let
dual
V be
vectors
bundle
of L* of
the
concludes
length
Hq(£,p*V)
expansion
section
L k.
When
the
contains
is s u c h
of
all
covariant
directions
over
because
metric
the
its
is
under
o n W. If we g i v e
derivatives
vanish.
of
the
Hence
if
is
L is
local
Kohn's
boundary
of
If t h e r e
q-dimensional
to
some
represented
M a Hermitian
form
the
power
a
by
the
local
respect
the
one
holomorphic
subvariety.
p is a
curvature
we have
then
in t h e
o f L*
hold
of
the
(0,q)-forms
real-analytic,
with
metric
be
of all
® L k) v a n i s h e s
of L* of a l o c a l
zero-section
that
Hermitian
for
k th c o e f f i c i e n t
L
M
by representing
complex-analytic
property
the
+
points,
that Hq(M,V
(0,q)-forms
its p r o j e c t i o n
is c o n s t a n t
of W m u s t
of
for
q-dimensional
the
p:L*
of
It f o l l o w s
vanishing
a Hermitian
of L* c o n s i s t i n g
estimates
one
such
by
is s e m i p o s i t i v e .
~
every
theorems
derived
L be
i. If s u b e l l i p t i c
estimates
L
M. L e t
subset
in t h e
with
which
form
to get
the m e t h o d
be
of
curvature
coordinate
W
example
M whose
bundle
from
readily
and
fiber
local
can
an
a point
trivialization
function
by
near
a subvariety,
subvariety
them
manifold
at
be used
to v a n i s h i n g
is f i n i t e - d i m e n s i o n a l
Hermitian
can
complex
forms.
large,
subelliptic
no
approach
in the
defined
of
result
<
-t(x0)n
estimates
be the open
of
by harmonic
k sufficiently
function
~
power
c a n be f o r m u l a t e d
of
here
over
boundary
that
Some
vector
o f L. L e t
the
Grauert's
theorems
a compact
a holomorphic
for
series
follow
illustrate
M be
line
cohomology
for
We
bundle
hold
We
to
subelliptic
estiraates.
Let
holomorphic
the
of v a n i s h i n g
subelliptic
other
t(x 0) r a i s e d
how
theorems.
[14]. A n u m b e r
of
of
following
metric,
L
along
theorem.
local
by
a
then
the
187
Theorem
6. L e t
metric,
L
metric,
curvature
i~s p o s i t i v e
following
of
the
some m+2
vectors
sufficientl~
one
This
the
kind
for
0
x 2~
M
the
subspace
restriction
then
0
o__ff 0
for
some
evaluated
to r e m a r k
of
in
at
the first C h e r n c l a s s
of
The
assumptions
However,
case.
on c o m p a c t
[42,
certain
case
of
a
nondegeneracy
q
= i,
vector
Theorem
holomorphic
5
sections
and Grauert's
criterion
line b u n d l e
L must be
is no a p p l i c a t i o n
yet
estimates,
projective
theorems
p.549]
Viehweg
and
[55]
= 0 for
and K M denotes
are
line
M
than
only
to
of
every
q > i,
the c a n o n i c a l
weaker
apply
by
by
bundles
of such a kind
manifold
Cl(LIC) > 0 for
H q ( M , L k K M)
obtained
for
An e x a m p l e
algebraic
motivated
If L is a h o l o m o r p h i c
algebraic
results
the
smooth
Hermitian
of the c o h o m o l o g y
from s u b e l l i p t i c
of v a n i s h i n g
following.
involved
such
the
may turn out to be fruitful.
> 0 and
then
in
When
there
than ampleness.
Cl(L)n
M,
on
for h o l o m o r p h i c
above
derived
[23] , a n d
projective
that
that
of p r o d u c i n g
ampleness
is the
a compact
C
us
get v a n i s h i n g
form satisfy
that
kind
weaker
theorems
n such
tells
this a p p r o a c h
is a n o t h e r
pseudoconvex
assumption
examples
theorems
denotes
algebraic
be
Hq(M,V O L k) = 0 for k
to show that the
of known
conditions
M.
the
of
Then
0
Sup~gse
linear
o f E),
weakly
described
Lemma]
lack
like
curve
for
result
bundles
criterion
conditions.
conjugate
manifolds.
satisfying
analytic
that
formulated
[36] , K a w a m a t a
dimension
[6]
the m e t h o d
line
there
over
M. Let
point
comple ~
can be
Ramanujam
bundle
ever Z
pseudoconvex
in the future
vanishing
over
integer.
derivative
real-analytic
of
[14, p.347,
We w o u l d
of
at
a Hermitian
a real-analytic
bundle
such
covariant
theorems
by u s i n g
Though
Seshadri's
x
we can s t i l l
for the kind of v a n i s h i n g
manifolds
at
of the c u r v a t u r e
noncompact
semipositive
hopefully
suppose
result
drop
Similar
of a m p l e n e s s
ample.
m th
line b u n d l e
c a n be p r o v e d
for
the
Catlin's
if the d e r i v a t i v e s
and
M with
E is a q - d i m e n s i o n a l
from E and E is not zero.
can
L.
semipositive
bundles
and
with
large.
using
conditions.
over
q be a positive
(l,0)-vectors
m
manifold
yector
(where E is the c o m p l e x
integer
of
If
all
bundle
Let
semidefinite
t_oo E ×E is zero
boundary,
line
com[lex
V a holomorphic
of L.
is true.
positive
By
a compact
and
form
s~ace
metric
be
a holomorphic
Hermitian
the
M
local
the
line
complex
complex-
where
Cl(-)
line b u n d l e
curvature
projective
188
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