Division Theorems and Twisted Complexes - SUNY, Stony Brook

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DIVISION THEOREMS AND TWISTED COMPLEXES
DROR VAROLIN
Abstract.We prove new Skoda-type division,or ideal membership,theorems.We work
in a geometric setting of line bundles over Kahler manifolds that are Stein away from an
analytic subvariety.(This includes complex projective manifolds.) Our approach is to
combine the twisted Bochner-Kodaira Identity,used in the Ohsawa-Takegoshi Theorem,
with Skoda's basic estimate for the division problem.Techniques developed by McNeal
and the author are then used to provide many examples of new division theorems.Among
other applications,we give a modication of a recent result of Siu regarding eective nite
generation of certain section rings.
Contents
1.Introduction 1
2.Corollaries of Theorem 1 4
3.Hilbert Space Theory 10
4.Classical L
2
identities and estimates 13
5.Twisted versions of L
2
identities and estimates 15
6.Proof of Theorem 1 17
References 19
1.Introduction
A classical problem both in commutative algebra and in several complex variables is the
ideal membership,or division problem.In the setting of commutative algebra,the decisive
result is Hilbert's Nullstellensatz.On the other hand,in several complex variables the
most basic division problem for bounded holomorphic functions is open:given a collection
of bounded holomorphic functions g
1
;:::;g
p
on the unit ball B  C
n
(n  2) such that
P
jg
i
j
2
= 1,are there bounded holomorphic functions h
1
;:::;h
p
such that
P
h
i
g
i
= 1?The
case n = 1 is the famous Corona Theorem of Carleson.In higher dimensions,the closest
result thus far is an L
2
version,also known as the celebrated Division Theorem of Skoda.
In this paper,we use the method of the twisted Bochner-Kodaira Identity together with
Skoda's Basic Estimate to establish a generalization of Skoda's Division Theorem.
Skoda's Theorem has many applications.Some recent applications in algebraic geometry
appear in the work of Ein and Lazarsfeld [1],where Skoda's Theorem plays a key role in
giving an eective version of the Nullstellensatz.Siu has used Skoda's Theorem to prove
results about eective global generation of multiplier ideals,which was a key tool in his
approach to establishing the deformation invariance of plurigenera [3].Siu has also used
Partially supported by NSF grant DMS-0400909.
1
Skoda's Theorem in his approach to the problem of nite generation of the canonical ring
[4].
At the same time,several authors have been establishing results that showthe fundamental
role of the Ohsawa-Takegoshi extension theorem and its variants in the areas of analytic
methods in algebraic geometry and of several complex variables.The applications of the
Ohsawa-Takegoshi Theorem are too numerous to mention in this introduction.
The results of Skoda and Ohsawa-Takegoshi are similar in nature and proof.Both re-
sults use the Bochner-Kodaira Identity.However,in the Ohsawa-Takegoshi technique,the
Bochner-Kodaira Identity is"twisted".
In Skoda's Theorem,a functional analysis argument is used that is similar to the well-
known Lax-MilgramLemma.We recall,perhaps with slight modication,Skoda's functional
analysis in Section 3.As usual,the functional analysis requires us to establish an a priori
estimate.This estimate is obtained from the Bochner-Kodaira Identity together with a non-
trivial and very sharp inequality due to Skoda,referred to in this paper as Skoda's Inequality.
The resulting a priori estimate is referred to here as Skoda's Basic Estimate.
The main idea of this paper is to introduce twisting into Skoda's Basic estimate.More
precisely,we twist the Bochner-Kodaira Identity before applying Skoda's Inequality.The
result is a series of divison theorems whose estimates are dierent from the original result of
Skoda.
Because many recent applications of L
2
theorems have been to complex and algebraic ge-
ometry,we will state our results in the more general language of singular metrics and sections
of holomorphic line bundles on so-called essentially Stein manifolds:a Kahler manifold X is
said to be essentially Stein if there exists a complex subvariety V  X such that XV is a
Stein manifold.For example,X could be a Stein manifold,in which case we can take V =;,
or X could be a smooth projective variety,in which case V could be the intersection of X
with a hyperplane in some projective space in which X is by hypothesis embedded.A third
interesting class of examples is a holomorphic family of algebraic manifolds bered over the
unit disk or over a more general Stein manifold.
We x on X two holomorphic line bundles E!X and F!X,with singular metrics
e
'
E
and e
'
F
respectively,and suppose given a collection of sections
g
1
;:::;g
p
2 H
0
(X;E)
where p  1 is some integer.
Taking cue from Skoda [5],we seek to determine which sections f 2 H
0
(X;F +K
X
) can
be divided by g = (g
1
;:::;g
p
),in the sense that there exist sections
h
1
;:::;h
p
2 H
0
(X;F E +K
X
)
satisfying the equality
f =
p
X
i=1
h
i
g
i
:
Moreover,if f satises some sort of L
2
estimate,what can we say about estimates for
h
1
;:::;h
p
?We shall refer to this question as the division problem.
To state our main result,it is useful to introduce the following denition.
2
Definition 1.1.For a triple (;F;q) where :[1;1)!R,F:[1;1)![1;1) are C
2
functions and q > 0 is an integer,dene the auxiliary function
(x) = x +F(x):
We call (;F;q) a Skoda Triple if
(1) (x)
0
(x) +1 +F
0
(x)  0 and (x)
00
(x) +F
00
(x)  0:
Given a Skoda Triple (;F;q),we dene
B(x) = 1 +
((x)
0
(x) +1 +F
0
(x))
q(x)
and A(x):=
(
(1+F
0
(x))
2
(F
00
(x)+(x)
00
(x))
1 +F
0
6 0
0 1 +F
0
 0
:
Remark.Note that
B
(B 1)
=
q +
0
+1 +F
0
(
0
+1 +F
0
)
=
1

+
q

0
+1 +F
0
Our main result is the following.
Theorem 1.Let X be an essentially Stein manifold of complex dimension n,L;E!X
holomorphic line bundles with singular Hermitian metrics e

and e

respectively,and
g
1
;:::;g
p
2 H
0
(X;E).Let
q = min(p 1;n) and  = 1 log(jgj
2
e

):
Let (;F;q) be a Skoda triple.Set  = (),A = A() and B = B(),with (x),A(x) and
B(x) as in Denition 1.1.Assume that
p
1@

@  Bq
p
1@

@:
If  non-constant,or if  cannot be dened on R and still satisfy (1) of Denition 1.1,
assume further that
jgj
2
e

:=
p
X
j=1
jg
j
j
2
e

< 1:
Then for every section f 2 H
0
(X;L+K
X
) such that
Z
X
B
(B 1)
jfj
2
e
()
e

(jgj
2
e

)
q+1
< +1
there exist sections h
1
;:::;h
p
2 H
0
(X;LE +K
X
) such that
p
X
j=1
h
j
g
j
= f
and
Z
X
jhj
2
e
()
e
( )
( +A)(jgj
2
e

)
q

Z
X
B
(B 1)
jfj
2
e
()
e

(jgj
2
e

)
q+1
:
Theorem 1 implies a large number of division theorems.However,as stated,Theorem
1 does not give a solution to the division problem unless we input a Skoda triple (always
with q = min(n;p  1)).By choosing dierent Skoda triples (;F;q) (with the same q =
min(n;p 1) from Theorem 1),one can obtain numerous division theorems as corollaries of
Theorem 1.In the next section we establish some of these corollaries,and hope the reader
3
is convinced that such corollaries are easy to come by,or equivalently,that Skoda triples are
easy to nd.
2.Corollaries of Theorem 1
Example.Fix  > 0.Let q = min(n;p 1),F(x) = 1 x and (x) = ( 1)q(x 1).
Then  = 1 so by denition A = 0,and B = .Thus we obtain the following geometric
reformulation of the famous theorem of Skoda.
Theorem 2.1.Let X be an essentially Stein manifold of complex dimension n,F;E!X
holomorphic line bundles with singular Hermitian metrics e

and e

respectively,and
g
1
;:::;g
p
2 H
0
(X;E).Assume that
p
1@

@  q
p
1@

@:
Then for any f 2 H
0
(X;K
X
+F) such that
Z
X
jfj
2
e

(jgj
2
e

)
q+1
< +1
there are p sections h
1
;:::;h
p
2 H
0
(X;K
X
+F E) such that
X
k
h
k
g
k
= f and
Z
X
jhj
2
e
( )
(jgj
2
e

)
q


 1
Z
X
jfj
2
e

(jgj
2
e

)
q+1
:
Remark.The proof of Theorem2.1 that we give essentially reduces to Skoda's original proof
since,as we shall see,when  is constant the twisting of the

@-complex becomes trivial.
Example.In his paper [4],Siu derived from Skoda's Theorem the following result in the
case of algebraic manifolds.We give a slightly dierent proof of Siu's result,in the more
general setting of essentially Stein manifolds.
Theorem 2.2.Let X be an essentially Stein manifold of complex dimension n,L!X
a holomorphic line bundle,and H!X a holomorphic line bundle with non-negatively
curved singular Hermitian metric e
'
.Let k  1 be an integer and x sections G
1
;:::;G
p
2
H
0
(X;L).Dene the multiplier ideals
J
k+1
= I(e
'
jGj
2(n+k+1)
) and J
k
= I(e
'
jGj
2(n+k)
):
Then
H
0
(X;((n +k +1)L +H +K
X
)
J
k+1
) =
p
M
j=1
G
j
H
0
(X;((n +k)L +H +K
X
)
J
k
):
Proof.By taking G
p
=:::G
n
= 0,we may assume that q = n.Take  = (n +k)=n,so that
q = n + k.We are going to use Theorem 2.1 with F = (n + k + 1)L + H,E = L and
g
i
= G
i
.Fix a metric e

for L having non-negative curvature current,(for example,one
could take  = log jG
1
j
2
) and let ='+(n +k +1).Then
p
1@

@ q
p
1@

@ =
p
1@

@'+
p
1@

@  0:
4
Suppose f 2 H
0
(X;O
X
((n + k + 1)L + H + K
X
)
J
k+1
).By Theorem 2.1 there exist
h
1
;:::;h
p
2 H
0
(X;O
X
((n +k)L +H +K
X
)) such that
P
j
h
j
G
j
= f.Moreover,
Z
X
jhj
2
e
'
jGj
2(n+k)
=
Z
X
jhj
2
e
( )
(jgj
2
e

)
n


 1
Z
X
jfj
2
e

(jgj
2
e

)
n+1
=
n +k
n +k 1
Z
X
jfj
2
e
'
jGj
2(n+k+1)
< +1:
Thus h
i
2 J
k
locally (and much more).The proof is complete.
The rest of the results we present here were motivated in part by our desire to say some-
thing about the case k = 0 in Theorem 2.2.Of course,Theorem 2.2 as stated is not true
in general if k = 0.But at the end of this section,we will state and prove a result similar
to Theorem 2.2 (namely Theorem 2.8) that does address the question of when sections of
(n +1)L +H +K
X
are expressible in terms of sections of nL +H +K
X
.
The next example again does not make use of twisting.
Example.For any"> 0 and any positive integer q the triple (;F;q) where (x) ="log x
and F(x) = 1  x is a Skoda Triple.Indeed,
0
(x) ="=x and 
00
(x) = "=x
2
< 0,while
1 +F
0
(x) = 0 and F
00
(x)  0  0,and thus (1) holds.Furthermore,
 = 1;A = 0;B = 1 +
"
q
and
B
(B 1)
= 1 +
q
"
;
and thus
Bq  q +";while
B
(B 1)

q +"
"
:
Applying Theorem 1 to the Skoda Triple ("log x;0;q) where q = min(n;p 1) gives us the
following result.
Theorem 2.3.Let the notation be as in Theorem 2.1.Fix"> 0 and assume that jgj
2
e

< 1
on X.Set q = min(n;p 1) and  = 1 log(jgj
2
e

).Suppose that
p
1@

@  (q +")
p
1@

@:
Then for every f 2 H
0
(X;F +K
X
) such that
Z
X
jfj
2
e


1+"
(jgj
2
e

)
q+1
< +1;
there exist sections h
1
;:::;h
p
2 (H
0
(X;F E +K
X
) such that
p
X
j=1
h
j
g
j
= f and
Z
X
jhj
2
e
( )

"
(jgj
2
e

)
q

q +"
"
Z
X
jfj
2
e


1+"
(jgj
2
e

)
q+1
:
By allowing twisting,we can improve the estimates of Theorem 2.3,at the cost of a little
more curvature from e

.
5
Example.For any"> 0 and any positive integer q the triple (;F;q) where (x) ="log x
and F(x)  0 is a Skoda Triple.Indeed,x
0
(x) ="and 
00
(x) = "=x
2
< 0,while
1 +F
0
(x) = 1 > 0 and F
00
(x)  0  0,and thus (1) holds.Moreover
A(x) =
(x)
"
;B(x) =
q(x) +1 +"
q(x)
and
(B 1)
B
=
(1 +")
q +
1+"

;
and thus since   1,
Bq  q +1 +";while
B
(B 1)

q +1 +"
1 +"
:
Applying Theorem 1 to the Skoda Triple ("log x;0;q) where q = min(n;p 1) gives us the
following result.
Theorem 2.4.Let the notation be as in Theorem 2.1.Fix"> 0 and assume that jgj
2
e

< 1
on X.Set q = min(n;p 1) and  = 1 log(jgj
2
e

).Suppose that
p
1@

@  (q +1 +")
p
1@

@:
Then for every f 2 H
0
(X;F +K
X
) such that
Z
X
jfj
2
e


"
(jgj
2
e

)
q+1
< +1;
there exist sections h
1
;:::;h
p
2 (H
0
(X;F E +K
X
) such that
p
X
j=1
h
j
g
j
= f and
Z
X
jhj
2
e
( )

"
(jgj
2
e

)
q


q +1 +"
"
Z
X
jfj
2
e


"
(jgj
2
e

)
q+1
:
Remark.Note that if,for example,X is a bounded pseudoconvex domain in C
n
and we
take E = O and   0,then Theorem 2.4 is a strict improvement over Theorem 2.3.Thus
twisting can sometimes get us stronger results.
Though less general,Theorems 2.1,2.3 and 2.4 are aesthetically more pleasing than The-
orem 1,because the integrands in the conclusions of the former are more natural.Another
method for obtaining such natural integrands is through the use of the notion of denomina-
tors introduced in [2] by McNeal and the author.
Definition 2.5.Let D denote the class of functions R:[1;1)![1;1) with the following
properties.
(D1) Each R 2 D is continuous and increasing.
(D2) For each R 2 D the improper integral
C(R):=
Z
1
1
dt
R(t)
is nite.
For  > 0,set
G

(x) =
1
1 +

1 +

C(R)
Z
x
1
dt
R(t)

;
6
and note that this function takes values in (0;1].Let
F

(x):=
Z
x
1
1 G

(y)
G

(y)
dy:
(D3) For each R 2 D there exists a constant  > 0 such that
K

(R):= sup
x1
x +F

(x)
R(x)
is nite.
A function R 2 D is called a denominator.
The following key lemma about denominators was proved in [2].
Lemma 2.6.If R 2 D and
R
1
1
dt
R(t)
= 1,then the function F = F

given in Denition 2.5
satises
x +F(x)  1;(2a)
1 +F
0
(x)  1;and(2b)
F
00
(x) < 0:(2c)
Moreover,F satises the ODE
(3) F
00
(x) +

(1 +)R(x)
(1 +F
0
(x))
2
= 0;x  1;
where  is a positive number guaranteed by Condition (D3) of Denition 2.5.
Using Lemma 2.6 we can prove the following theorem.
Theorem 2.7.Let X be an essentially Stein manifold of complex dimension n,L;E!X
holomorphic line bundles with singular Hermitian metrics e

and e

respectively,and
g
1
;:::;g
p
2 H
0
(X;E).Let
q = min(p 1;n) and  = 1 log(jgj
2
e

):
Let R 2 D with constant  > 0 and function F = F

determined by Denition 2.5,and set
B = 1 +
1+F
0
()
q(+F())
.Assume that
p
1@

@  qB
p
1@

@ and jgj
2
e

:=
p
X
j=1
jg
j
j
2
e

< 1:
Then for every section f 2 H
0
(X;L+K
X
) such that
Z
X
jfj
2
e

(jgj
2
e

)
q+1
< +1
there exist sections h
1
;:::;h
p
2 H
0
(X;LE +K
X
) such that
p
X
j=1
h
j
g
j
= f
and
Z
X
jhj
2
e
( )
(jgj
2
e

)
q
R()
 (1 +q)

(1 +)

C(R) +K

(R)
Z
X
jfj
2
e

(jgj
2
e

)
q+1
:
7
Remark.For the reader that does not like the appearance of the function B in the statement
of Theorem 2.7,we note that qB is always bounded above by q +2 +.Indeed,note that
qB = q +
1+F
0

.Now,in the notation of Denition 2.5,
 (1 +F
0
(x)) =
Z
x
1
dy
G

(y)

1
G

(x)
 (1 +)
since,from the denition of G

,
1
G

 (1 +) while
1
G

 0.Thus
(4) qB  q +1 +
1 +

 q +2 +:
The reason we did not hypothesize that
p
1@

@  (q +2 +)
p
1@

@ is that often can do
better than the bound (4).It is often easy to estimate
1+F
0

in specic examples.
Proof of Theorem 2.7.Let F be the function associated to the denominator R via Lemma
2.6.Observe that in view of Lemma 2.6,(0;F;q) is a Skoda triple.Let ,A and B be the
functions associated to (0;F;q) in Denition 1.1.We claim that
(5)
 +A
R

(1 +)

C(R) +K

(R):
Indeed,=R  K

(R) by property (D3),while A=R 
(1+)

C(R) by the denition of A and
the ODE of Lemma 2.6.(Note that if R 2 D,then C(R)R 2 D and
R
1
1
(C(R)R(t))
1
dt = 1.)
Finally,observe that
(6)
B
(B 1)
= 1 +
q
(1 +F
0
)
 1 +q:
Thus
Z
X
jhj
2
e
( )
(jgj
2
e

)
q
R()

Z
X
 +A
R()
jhj
2
e
()
e
( )
( +A)(jgj
2
e

)
q


(1 +)

C(R) +K

(R)
Z
X
B
(B 1)
jfj
2
e
()
e

(jgj
2
e

)
q+1
 (1 +q)

(1 +)

C(R) +K

(R)
Z
X
jfj
2
e

(jgj
2
e

)
q+1
:
In going from the second to the third line,we used Theorem 1 and The inequality (5),and
in going from the third line to the last we used (6).The proof is complete.
8
The following is a table of denominators and estimates for their corresponding constants.
(7)
R(x) =
(1+)

C(R) +K

(R) 
(i) e
s(x1)
(1+)
2
s
+1
(ii) x
2
(2+)
2
(1+)
2
4
(iii) x
1+s
1+
s
+

s

(1+)s
(1+s)

s+1
(iv) R
N
(x)
(1+)(1+s)
s
In entry (iv),
R
N
(x) = x

N2
Y
j=1
L
j
(x)
!
(L
N1
(x))
1+s
;
where
E
j
= exp
(j)
(1) and L
j
(x) = log
(j)
(E
j
x):
Remark.To dene denominators yielding Skoda triples (;F;q) with both  and F non-
trivial seems a little more complicated.The corresponding ODE that determines the asso-
ciated function  does have solutions,but since this (rst order) ODE is not autonomous,it
is harder to get explicit properties of the associated function .
We can now state and prove our Siu-type division theorem for the case k = 0,under an
additional assumption on the line bundle L.
Theorem 2.8.Let X be an almost Stein manifold of complex dimension n,L!X a
holomorphic line bundle,H!X a holomorphic line bundle with non-negatively curved
singular Hermitian metric e
'
.Fix sections G
1
;:::;G
p
2 H
0
(X;L) and a singular Hermitian
metric e

for L having non-negative curvature,and such that
jGj
2
e

< 1 on X:
Fix a denominator R 2 D such that the associated function B satises
p
1@

@'+(n +1 nB)
p
1@

@  0:
Dene the multiplier ideals
I
1
= I

e
'
jGj
2(n+1)

and I
0
= I

e
'
jGj
2n
R(1 log jGj
2
+)

:
9
Then
H
0
(X;((n +1)L +H +K
X
)
I
1
) =
p
M
j=1
G
j
H
0
(X;(nL +H +K
X
)
I
0
):
Proof.By taking G
p
=:::G
n
= 0,we may assume that q = n.We are going to use Theorem
2.7 with F = (n +1)L +H,E = L and g
i
= G
i
.Let ='+(n +1).Then
p
1@

@ qB
p
1@

@ =
p
1@

@'+(n +1 qB)
p
1@

@  0:
Suppose f 2 H
0
(X;O
X
((n +1)L +H +K
X
)
I
1
).Then
Z
X
jfj
2
e

(jgj
2
e

)
n+1
=
Z
X
jfj
2
e
'
jGj
2(n+1)
< +1:
By Theorem 2.7 there exist h
1
;:::;h
p
2 H
0
(X;O
X
(nL +H +K
X
)) such that
P
h
i
G
i
= f.
Moreover,
Z
X
jhj
2
e
'
jGj
2n
R(1 log jGj
2
+)
=
Z
X
jhj
2
e
( )
(jgj
2
e

)
2n
R()
.
Z
X
jfj
2
e

(jgj
2
e

)
n+1
< +1:
Thus h
i
2 I
0
locally.The proof is complete.
3.Hilbert Space Theory
Summation convention.We use the the complex version of Einstein's convention,where
one sums over (i) repeated indices,one upper and one lower,and (ii) an index and its complex
conjugate,provided they are both either upper or lower indices.In addition,we introduce
into our order of operations the rules
ja
i
b
i
j
2
= ja
1
b
1
+:::j
2
;while ja
i
j
2
jb
i
j
2
= ja
1
j
2
jb
1
j
2
+::::
The functional analysis.Let H
0
,H
1
,H
2
and F
1
be Hilbert spaces with inner products
(;)
0
,(;)
1
,(;)
2
and (;)

respectively.Suppose we have a bounded linear operator
T
2
:H
0
!H
2
and closed,densely dened operators T
1
:H
0
!H
1
and S
1
:H
1
!F
1
satisfying
S
1
T
1
= 0:
Let K = Kernel(T
1
).We consider the following problem.
Problem 3.1.Given  2 H
2
,is there an element  2 K such that T
2
 = ?If so,what
can we say about jj
0
?
Problem 3.1 was solved by Skoda in [5].The dierence between Skoda's solution and the
one we present here is that Skoda identied the subspace of all  for which the problem can
be solved,whereas we aim to solve the problems one  at a time.This is a dierence in
presentation only;the two approaches are equivalent.
Proposition 3.2.Let  2 H
2
.Suppose there exists a constant C > 0 such that for all
u 2 T
2
(K ) and all  2 Domain(T

1
),
(8) j(;u)
2
j
2
 C

jT

2
u +T

1
j
2
0
+jS
1
j
2


:
Then there exists  2 K such that T
2
 =  and jj
2
0
 C.
10
Proof.In (8) we may restrict our attention to  2 Domain(T

1
)\Kernel(S
1
).Note that (i)
since S
1
T
1
= 0,the image of T

1
agrees with the image of the restriction of T

1
to Kernel(S
1
),
and (ii) the image of T

1
is dense in K
?
.Thus the estimate (8) may be rewritten
(9) j(;u)j
2
 Cj[T

2
u]j
2
;
where we denote by [ ] the projection to the quotient space H
0
=K
?
and the norm on the
right hand side is the norm induced on H
0
=K
?
in the usual way.As is well known,with
this norm H
0
=K
?
is isomorphic to the closed subspace K.(The isomorphism sends any,
and thus every,member of u+K
?
to its orthogonal projection onto K.) We dene a linear
functional`:[Image(T

2
)]!C by
`([T

2
u]) = (;u)
2
:
Then by (9)`is continuous with norm 
p
C.By extending`constant in the directions
parallel to [Image(T

2
)]
?
in H
0
=K
?
,we may assume that`is dened on all of H
0
=K
?
with norm still bounded by
p
C.The Riesz Representation Theorem then tells us that`
is represented by inner product with respect to some element  of H
0
=K
?
which we can
identify with K at this point.Evidently we have jj
2
0
 C and
(T
2
;u)
2
= (;T

2
u +K
?
) =`([T

2
u]) = (;u)
2
:
The proof is complete.
Hilbert spaces of sections.Let Y be a Kahler manifold of complex dimension n and
H!Y a holomorphic line bundle equipped with a singular Hermitian metric e
'
.Given a
smooth section f of H +K
Y
!Y,we can dene its L
2
-norm
jjfjj
2
'
:=
Z
Y
jfj
2
e
'
:
This norm does not depend on the Kahler metric for Y.Indeed,we think of a section of
H + K
Y
as an H-valued (n;0)-form.Then the functions jfj
2
e
'
transforms like the local
representatives of a measure on Y,and may thus be integrated.
We dene
L
2
(Y;H +K
Y
;e
'
)
to be the Hilbert space completion of the space of smooth sections f of H +K
Y
!Y such
that jjfjj
2
'
< +1.
More generally,we have Hilbert spaces of (0;q)-forms with values in H+K
Y
.Given such
a (0;q)-form ,dened locally by
 = 

J
dz
J
;
where J = (j
1
;:::;j
q
) 2 f1;:::;ng
q
is a multiindex and dz
J
= dz
j
1
^:::^ dz
j
q
and the 

J
are
skew-symmetric in J,we set

J
= g
j
1

k
1
:::g
j
q

k
q


K
and jj
2
= 
J


J
;
where g
j

k
is the inverse matrix of the matrix g
j

k
of the Kahler metric g of Y.It follows that
the functions
jj
2
e
'
11
transform like the local representatives of a measure,and may thus be integrated.We then
dene
jjjj
2
'
=
Z
Y
jj
2
e
'
:
We now dene the Hilbert space
L
2
0;q
(Y;H +K
Y
;e
'
)
to be the Hilbert space closure of the space of smooth (0;q)-forms  with values in H +K
Y
such that jjjj
2
'
< +1.Of course,these norms depend on the Kahler metric g as soon as
q  1.
We are only going to be (explicitly) interested in the cases q = 0 and q = 1,although
q = 2 will enter in an auxiliary way.
Choices.In employing Proposition 3.2,we shall consider the following spaces.
H
0
:= (L
2
(
;K
X
+F E;e
'
1
))
p
H
1
:= (L
2
(0;1)
(
;K
X
+F E;e
'
1
))
p
H
2
:= L
2
(
;K
X
+F;e
'
2
)
F
1
:= (L
2
(0;2)
(
;K
X
+F E;e
'
1
))
p
Next we dene our operators T
1
and T
2
.Let
T:L
2
(
;K
X
+F E;e
'
1
)!L
2
0;1
(
;K
X
+F E;e
'
1
)
be the densely dened operator whose action on smooth forms with compact support is
Tu =

@u:
As usual,the domain of T consists of those u 2 L
2
(
;K
X
+ F  E;e
'
1
) such that

@u,
dened in the sense of currents,is represented by an of L
2
(0;1)
(
;K
X
+ F  E;e
'
1
)-form
with values in F +K
X
.We let
T
1
:H
0
!H
1
be dened by
T
1
(
1
;:::;
p
) = (T
1
;:::;T
p
):
We remind the reader that T has formal adjoint T

= T

'
1
given by the formula
T

 = e
'
1
@

(e
'
1


):
It follows that
T

1
(
1
;:::;
p
) =

e
'
1
@


e
'
1


1

;:::;e
'
1
@


e
'
1


p

:
We will also use the densely dened operators
S:L
2
0;1
(
;K
X
+F E;e
'
1
)!L
2
0;2
(
;K
X
+F E;e
'
1
)
dened by

@ on smooth forms,and the associated operator
S
1
(
1
;:::;
p
) = (S
1
;:::;S
p
):
However,we will not need the formal adjoint of S.
Next we let
T
2
:H
0
!H
2
12
be dened by
T
2
(h
1
;:::;h
p
) = h
i
g
i
:
We have
(T

2
u;h)
0
= (u;T
2
h)
2
=
Z


u
h
i
g
i
e
'
2
=
Z


e
('
2
'
1
)
g
i
u

h
j
e
'
1
;
And thus
T

2
u =

e
('
2
'
1
)
g
1
u;:::;e
('
2
'
1
)
g
p
u

:
Remark.Let us comment on the meaning of this a priori local formula.The sections g
j
,
1  j  p,are sections of E,and e
('
2
'
1
)
is a metric for F (F E) = E.Thus for each
j,jg
j
j
2
e
('
2
'
1
)
is a globally dened function,and so the expressions
e
('
2
'
1
)
g
j
= e
('
2
'
1
)
jg
j
j
2
=g
j
transform like sections of E.Since u takes values in K
X
+F,the expressions
e
('
2
'
1
)
g
j
u
transform like sections of K
X
+F E,which is what we expect.
4.Classical L
2
identities and estimates
In this section we collect some known L
2
identities.
The Bochner-Kodaira Identity.Let
be a domain in a complex manifold with smooth,
R-codimension-1 boundary @
.Fix a proper smooth function  on a neighborhood of

such that

= f < 0g and j@j  1 on @
:
Let H!
be a holomorphic line bundle with singular Hermitian metric e
'
.
The following identity is a basic fact known as the
Bochner-Kodaira Identity:
For any smooth (0;1)-form  with values in K
X
+H and lying in the domain of

@

,
Z


j e
'
@

(

e
'
)j
2
e
'
+
Z


j

@j
2
e
'
=
Z






(@

@

')e
'
+
Z


j

rj
2
e
'
+
Z
@





(@

@

)e
'
:
Remark.The formal case,in which the boundary termdisappears is due to Kodaira,and is
a complex version of earlier work of Bochner.With the boundary termincluded,the identity
above is due to C.B.Morrey.(For higher degree forms,it is due to Kohn.)
Skoda's Identity.For u 2 T
2
(Kernel T
1
) and  = (
1
;:::;
p
) 2 Domain(T

1
) we have
(T

2
u;T

1
)
0
= (T
1
(T

2
)u;)
1
=
Z


u
n


k
@

(g
k
e
('
2
'
1
)
)
o
e
'
1
:
13
It follows that if  is also in Domain(S
1
) then
jjT

1
 +T

2
ujj
2
0
+jjS
1
jj
2

= jjT

1
jj
2
0
+jjS
1
jj
2

+jjT

2
ujj
2
0
+2Re (T

2
u;T

1
)
0
=
p
X
k=1

jjT


k
jj
2
'
1
+jjS
k
jj
2
'
1

+
Z


e
2('
2
'
1
)
jgj
2
juj
2
e
'
1
+2Re
Z


u
n


k
@

(g
k
e
('
2
'
1
)
)
o
e
'
1
:
By applying the Bochner-Kodaira identity,we obtain the identity we have called
Skoda's Identity:
jjT

1
 +T

2
ujj
2
0
+jjS
1
jj
2

=
Z


e
('
2
'
1
)
jgj
2
juj
2
e
'
2
(10)
+2Re
Z


u
n


k
@

(g
k
e
('
2
'
1
)
)
o
e
'
1
+
Z


(

k


k
@

@

'
1
)e
'
1
+jj

rjj
2

+
Z
@

(

k


k
@

@

)e
'
1
:
Here jj

rjj
2

= jj

r
1
jj
2
'
1
+:::+jj

r
p
jj
2
'
1
.
Skoda's inequality.To obtain an estimate from Skoda's identity,one makes use of the
following inequality of Skoda.
Theorem 4.1 (Skoda's Inequality).[5] Let g = (g
1
;:::;g
p
) be holomorphic functions on a
domain U  C
n
,and let q = min(n;p 1).Then
q(

k


k
@

@

log jgj
2
)  jgj
2




k
@

(g
k
jgj
2
)


2
Remark.When passing to a global setting,it is helpful to keep in mind that g
k
jgj
2
trans-
forms like a section of the anti-holomorphic line bundle 
E,and thus @

(g
k
jgj
2
))dz

trans-
forms like a (
E)-valued (1;0)-form.In particular,both sides of Skoda's inequality consist
of globally dened functions.
Skoda's Basic Estimate.From Theorem 4.1 and Skoda's Identity (10),we immediately
obtain the following slight extension of a theorem of Skoda.
Theorem 4.2 (Skoda's Basic Estimate).Let X be an essentially Stein manifold,E;F!X
holomorphic line bundle with singular metrics e

and e

respectively,
 X a pseudo-
convex domain,B:
!(1;1) a function,and g
1
;:::;g
p
2 H
0
(X;E) holomorphic sections.
Set
q = min(n;p 1);'
1
= + +q log(jgj
2
e

) and'
2
='
1
+log jgj
2
:
14
For any p-tuple of F E-valued (0;1)-forms  = (
1
;:::;
p
) 2 Domain(T

1
)\Domain(S
1
)
and any u 2 T
2
(Kernel(T
1
)) we have the estimate
jjT

1
 +T

2
ujj
2
0
+jjS
1
jj
2

(11)

Z



B 1
B

juj
2
e
'
2
+
Z




k


k
(@

@

Bq@

@

) e
'
1
+
Z




k


k

@

@

 q(B 1)@

@

log(jgj
2
e

)

e
'
1
:
Proof.We are going to use Skoda's Identity (10).First note that,by the Cauchy-Schwartz
Inequality,for any open set U we have
2Re
Z
U
u
n


k
@

(g
k
e
('
2
'
1
)
)
o
e
'
1
= 2Re
Z
U
ujgj
1
n
jgj

k
@

(g
k
jgj
2
)
o
e
'
1
 
Z
U
1
B
juj
2
jgj
2
e
'
1

Z
U
Bjgj
2



k
@

(g
k
jgj
2
)


2
e
'
1
 
Z
U
1
B
juj
2
e
'
2

Z
U
Bq(

k


k
@

@

log jgj
2
)e
'
1
;(12)
where the second inequality follows from Skoda's Inequality (Theorem 4.1).Since the inte-
grands are globally dened,we may replace U by
.Substituting'
1
= + q log jgj
2
,
combining the inequality (12) with Skoda's Identity (10) and dropping the positive terms
jj

rjj
2

and
Z
@



k


k
@

@

e
'
1
nishes the proof.
5.Twisted versions of L
2
identities and estimates
The twisted Bochner-Kodaira Identity.Let'be the weight function in the Bochner-
Kodaira-Hormander identity.Suppose given a second weight function ,and set  = e
'
.
Then
p
1@

@'=
p
1@

@ 
p
1@

@

+
p
1@ ^

@

2
;
and from the Bochner-Kodaira identity we have
Z


j e

@

(

e

) 
1


@

j
2
e

+
Z


j

@j
2
e

=
Z







@

@

 @

@

 +
1
(@

)(@

)

e

+
Z


j

rj
2
e

+
Z
@





@

@

e

:
Expanding the rst term,we obtain the so-called
15
Twisted Bochner-Kodaira Identity:
jj
p
T


jj
2

+jj
p
Sjj
2

=
Z


(@

@

 @

@

) 



e

+2Re
Z




(@

)
T


e

+
Z


j

rj
2
e

+
Z
@





@

@

e

:
Twisted version of Skoda's Identity.We shall now twist the weights'
1
and'
2
by the
same factor .Let
 = e

1
'
1
= e

2
'
2
:
Then
@

@'
1
= @

@
1

@

@

+
@ ^

@

2
;
T

'
1
 = e
'
1
@

(e
'
1


) = 
e

1

@

(e

1


)
= e

1
@

(e

1


) 


@



= T


1
 
1


@

:
The operator T

2
remains unchanged.We thus calculate that
jjT

1;'
1
 +T

2
ujj
2
'
1
= jj
p
T

1;
1
 
p

1


@

 +
p
T

2
ujj
2

1
= jj
p
T

1;
1
 +
p
T

2
ujj
2

1
2Re
Z


(

k
@

)
(T

1;
1
 +T

2
u)
k
e

1
+
Z



1


k


k
(@

)(@

)e

1
;
that
Z


u
n


k
@

(g
k
e
('
2
'
1
)
)
o
e
'
1
=
Z


u
n


k
@

(g
k
e
(
2

1
)
)
o
e

1
and that
e
'
1
@

@'
1
= e

1
(@

@
1
@

@ 
1
@ ^

@):
Substitution of these three calculations into Skoda's Identity (10) yields the
Twisted version of Skoda's Identity:
jj
p
T

1;
1
 +
p
T

2
ujj
2

1
+jj
p
S
1
jj
2

= 2Re
Z


(

k
@

)
(T

1;
1
 +T

2
u)
k
e

1
(13)
+
Z


e
(
2

1
)
jgj
2
juj
2
e

2
+2Re
Z


u
n


k
@

(g
k
e
(
2

1
)
)
o
e

1
+
Z




k


k
(@

@


1
@

@

)e

1
+




p


r




2

+
Z
@

(

k


k
@

@

)e

1
:
Twisted version of Skoda's Basic Estimate.The following Lemma is trivial.
Lemma 5.1.T
2
(Kernel(T
1
)) = (T
2

p
 +A)(Kernel(T
1

p
 +A)):
Next we have the following result.
16
Theorem 5.2 (Twisted version of Skoda's Basic Estimate).Let X be an essentially Stein
manifold,E;F!X holomorphic line bundle with singular metrics e

and e

respectively,

 X a smoothly bounded pseudoconvex domain,;A:
!(0;1) and B:
!(1;1)
functions,and g
1
;:::;g
p
2 H
0
(X;E) holomorphic sections.Set
q = min(n;p 1);
1
= + +q log(jgj
2
e

) and 
2
= 
1
+log jgj
2
:
For any p-tuple of F E-valued (0;1)-forms  = (
1
;:::;
p
) 2 Domain(T

1
)\Domain(S
1
)
and any u 2 (T
2

p
 +A)(Kernel(T
1

p
 +A)) we have the estimate
jj
p
 +AT

1
 +
p
 +AT

2
ujj
2
0
+jj
p
S
1
jj
2

(14)

Z




B 1
B

juj
2
e

2
+
Z




k


k
(@

@

Bq@

@

) e

1
+
Z




k


k

@

@

 @

@

 
(@

)(@

)
A

e

1
:
+
Z




k


k

q(B 1)@

@

log(jgj
2
e

)

e

1
:
Proof.First,by Lemma 5.1 we may make use of (13).
By the pseudoconvexity of
,we may drop the last term on the right hand side of (13).
The second last term is clearly non-negative and may thus also be dropped.
Since 
2

1
= log jgj
2
,we see that
2Re
Z


u
n


k
@

(g
k
e
(
2

1
)
)
o
e

1
 
Z



B
juj
2
e

2

Z


Bq

k


k
(@

@

log jgj
2
)e

1
:
Moreover,by the Cauchy-Schwarz inequality we have
2Re
Z


(

k
@

)
(T

1
 +T

2
u)
k
e

1
 
Z




k


k
(@

)(@

)
A
e

1

Z


AjT

1
 +T

2
uj
2
e

1
:
Applying these inequalities to (13) easily yields (14).
6.Proof of Theorem 1
We now plan to use Proposition 3.2 to prove Theorem 1.
An a priori estimate.Assume that

 fjgj
2
e

< 1g:
With q = min(p 1;n),x a Skoda Triple (;F;q),and let
:= 1 log(jgj
2
e

):
Following Denition 1.1,let
:= () =  +F() and A:= A() =

(F
00
() +
00
())
(1 +F
0
())
2

1
:
17
Let
 = ():
Then if  is non-constant,we have

p
1@

@ 
p
1@

@ 
p
1
A
@ ^

@
= (
0
() +1 +F
0
())(
p
1@

@) 

F
00
() +
00
() +
(1+F
0
())
2
A

p
1@ ^

@
= (
0
() +1 +F
0
())(
p
1@

@):
The last equality follows from the denition of A.On the other hand,if  is constant,
we need not apply the twisted Skoda estimate;we can just appeal to the original Skoda
estimate,which would yield (by convention)

p
1@

@ 
p
1@

@ 
p
1
A
@ ^

@ = 
0
()(
p
1@

@) 
00
()
p
1@ ^

@
 (
0
() +1 +F
0
())(
p
1@

@)
Now,@

@ = @

@ log(jgj
2
e

).Thus from (14) we obtain the a priori estimate
jj
p
 +AT

1
 +
p
 +AT

2
ujj
2

1
+jj
p
S
1
jj
2

(15)

Z



B 1
B
juj
2
e

2
+
Z




k


k
(@

@

Bq@

@

)e

1
+
Z


(q(B 1) +(
0
() +1 +F
0
())) 

k


k
(@

@

log(jgj
2
e

))e

1
:
From the denition of B (See Denition 1.1) we have
q(B 1) = 
0
() +1 +F
0
():
Then
B =
q +
0
() +1 +F
0
())
q
;
and thus

(B 1)
B
=
(
0
() +1 +F
0
())
q +
0
() +1 +F
0
()
:
Thus we obtain from (15) the estimate
jj
p
 +AT

1
 +
p
 +AT

2
ujj
2

1
+jj
p
S
1
jj
2

(16)

Z


(B 1)
B
juj
2
e

2
+
Z




k


k
(@

@

Bq@

@

)e

1
:
Conclusion of the proof of Theorem 1.Now suppose that
p
1@

@  Bq
p
1@

@:
It follows from (16) that
j(f;u)

1
j
2

Z


B
(B 1)
jfj
2
e
()
e

jgj
2q+2

Z


(B 1)
B
juj
2
e

2
18
In view of Proposition 3.2,we nd sections H
1
;:::;H
p
such that

@(
p
 +AH
i
) = 0;(g
i
H
i
)
p
 +A = f
and
Z


jHj
2
e
()
e

jgj
2q

Z


B
(B 1)
jfj
2
e
()
e

jgj
2q+2
:
Letting h
i
=
p
 +AH
i
,we obtain
Z


jhj
2
e
()
e

( +A)jgj
2q

Z


B
(B 1)
jfj
2
e
()
e

jgj
2q+2
:
Since the estimates are uniform,we may let
!XV.Since V has measure zero,we may
replace X V by X.This completes the proof of Theorem 1.
Acknowledgment.I am indebted to Je McNeal,from whom I have learned a lot about
the method of twisted estimates and with whom I developed the theory of denominators.I
am grateful to Yum-Tong Siu for proposing that I study Skoda's Theorem and try to say
something non-trivial about the case  = 1,which I hope I have done at least somewhat.
References
[1] Ein,L.,Lazarsfeld,R.,A geometric eective Nullstellensatz.Invent.Math.137 (1999),no.2,427{448.
[2] McNeal,J.,Varolin,D.,Analytic Inversion of Adjunction.L
2
extension theorems with gain.Ann.Inst.
Fourier (Grenoble) 57 (2007),no.3,703{718.
[3] Siu,Y.-T.,Extension of Twisted Pluricanonical Sections with Plurisubharmonic Weight and Invariance
of Semipositively Twisted Plurigenera for Manifolds Not Necessarily of General Type.Complex geometry
(Gottingen,2000),223{277,Springer,Berlin,2002.
[4] Siu,Y.-T.,Multiplier Ideal Sheaves in Complex and Algebraic Geometry.Sci.China Ser.A 48 (2005),
suppl.,1{31.
[5] Skoda,H.,Application des techniques L
2
la theorie des ideaux d'une algebre de fonctions holomorphes
avec poids.Ann.Sci.

Ecole Norm.Sup.(4) 5 (1972),545{579.
Department of Mathematics
Stony Brook University
Stony Brook,NY 11794-3651
E-mail address:dror@math.sunysb.edu
19