holomorphic sections of high powers of positive line bundles [24]. (3) Grauert's bumping technique to construct holomorphic functions on strongly pseudoconvex ...
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 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
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)
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
