A WEAK KAWAMATA-VIEHWEG VANISHING THEOREM ON
COMPACT KÄHLER MANIFOLDS
THOMAS ECKL
1. Introduction
Using a subtle version of the Bochner technique, J.-P. Demailly and Th. Peternell
were able to prove one instance of the Kawamata-Viehweg Vanishing Theorem for
(even singular) Kähler manifolds:
Theorem 1.1 ([DP02, Thm. 0.1]). Let X be a normal compact Kähler space of
dimension n and L a nef line bundle on X. Assume that L2 6= 0. Then
H q (X, KX + L) = 0
for q ≥ n − 1.
The aim of this paper is to use the same methods to prove a weak version of
Kawamata-Viehweg Vanishing on compact Kähler manifolds for all q > n − ν(L),
which also works for pseudo-effective line bundles. It is only a weak version, because
we have to tensorize KX + L with the upper regularized multiplier ideal sheaf
of a singular hermitian metric on L:
Definition 1.2. Let X be a compact Kähler manifold of dimension n and L a
pseudo-effective line bundle on X. Let hmin be a hermitian metric with minimal
singularities among all positive singular hermitian metrics on L. Then the upper
regularized multiplier ideal sheaf J+ (L) is defined as
[
J+ (L) :=
J (h1+
min ).
→0
This multiplier ideal J+ (L) is certainly not optimal: there are examples of nef
line bundles where it is not trivial, see [DPS94, Ex.1.7]. At least, it is conjectured (and true in dimension 1 and 2) that it equals the ordinary multiplier ideal
J (L) := J (hmin ).
So the main result of this paper will be
Theorem 1.3. Let X be a compact Kähler manifold of dimension n and L a
pseudo-effective line bundle on X of numerical dimension ν = ν(L). Then
H q (X, O(KX + L) ⊗ J+ (L)) = 0
for q ≥ n + 1 − ν(L).
It will be proven as a corollary of
1991 Mathematics Subject Classification. 32J25.
Key words and phrases. pseudo-effective line bundles, upper regularized multiplier ideal, weak
Kawamata-Viehweg Vanishing.
1
2
THOMAS ECKL
Theorem 1.4. Let X be a compact Kähler manifold of dimension n and L a
pseudo-effective line bundle on X of numerical dimension ν = ν(L) with a positive
hermitian metric hmin with minimal singularities on L. For every 0 > 0 there
exists an 0 < < 0 such that the homomorphism
0
1+
q
H q (X, O(KX + L) ⊗ J (h1+
min )) → H (X, O(KX + L) ⊗ J (hmin ))
0
1+
induced by the inclusion J (h1+
min ) ⊂ J (hmin ) vanishes for q ≥ n + 1 − ν(L).
This theorem implies theorem 1.3, since the ascending chain of ideal sheaves
J (h1+
min ), → 0, gets stable at some point by the Noetherian property, hence
for 0 small enough
0
1+
J (h1+
min ) = J (hmin ) = J+ (L).
But then the homomorphism between the cohomology groups becomes an isomorphism, and the involved vector spaces must be 0.
The Bochner technique consists of using the Bochner-Kodaira-Nakano inequality
to prove the vanishing of certain cohomolgical classes, see [Dem00, §4] for an application to classical vanishing theorems. The main obstacle to use this technique
in our case is that the Bochner inequality is only true for smooth metrics. This
was circumvented by Demailly in [Dem82] who observed that for a compact Kähler
manifold X and Z ⊂ X an analytic subset, X \Z has a complete Kähler metric. Using Hörmander’s confirmation of the Bochner inequality for complete metrics (and
elements of certain function spaces) in [Hör65] Demailly constructed sequences of
complete metrics converging to the original Kähler metric and got vanishing results
by going to the limit. This limit process is successful despite of the metrics changing
all the time, since there is a uniform bound for all the occuring norms, and because
one can compare the different metrics.
So in our case we start with a careful construction of singular hermitian metrics
h on L which are smooth outside an analytic subset Z (section 2). They are
composed of a (sufficiently small) part controlling the eigenvalues of the curvature
form (they are essential for the Bochner inequality) and another part controlling
the multiplier ideal sheaf (and hence the singularities) of the metric. The first is
produced by applying the Calabi-Yau theorem as in Boucksom’s thesis [Bou02], the
latter is constructed with the equisingular approximation theorem of [DPS01].
Then we use the Bochner technique for complete metrics on X \ Z converging
to the starting Kähler metric ω and go to the limit, using the uniform estimate
in section 5 and comparing the different metrics following the results in section 3
(which more or less repeat the inequalities in [Dem82, §3]).
Remark. Since the technical details of the strategy described above are quite intricate and treated in a very summary way in [DP02], the author decided to give
all the steps in full details, for his own safety and for the convenience of the not so
experienced reader – of course without claiming any originality. The expert may
skip section 3 alltogether and skim over the functional analytic details in section 4.
2. The construction of the metrics
As explained in the introduction, we construct metrics ĥ for arbitrarily small composed of two parts, and the first part is produced by using
Theorem 2.1 (Approximative Singular Calabi-Yau theorem). Let
R X be a compact
Kähler manifold of dimension n with Kähler form ω such that X ω n = 1, and let
A WEAK KAWAMATA-VIEHWEG VANISHING
3
α ∈ H 1,1 (X, R) be a big class containing a Kähler current ≥ δω. Then for every
> 0 there exists a closed positive current T ∈ α with analytic singularities such
that
T (x)n ≥ (1 − δ )v(α)ω(x)n
almost everywhere, δ → 0 if tends to 0, and the multiplier ideal J (T ) contains
J (Tmin ) for a current Tmin in α with minimal singularities in α[δω].
Here, v(α) denotes the volume of the class α, defined by Boucksom [Bou02, 3.1.6]
as
Z
n
v(α) := sup
Tac
,
T
X
where the T ’s run through all closed positive currents in α and Tac is the absolute
continuous part of the Lebesgue decomposition of T ([Bou02, 3.1.1]). Since α is
big, there is a closed positive current in α bigger than ω for some , and v(α) is
positive.
The proof of the theorem is contained in Boucksom’s construction of the current T
(with arbitrary singularities) solving the Monge-Ampère equation
Tac (x)n = v(α)ω(x)n
almost everywhere ([Bou02, Thm. 3.1.23]). His idea was of course to use the
ordinary form of the Calabi-Yau theorem (where α contains a Kähler form and T
will be a form satisfying the Monge-Ampère equation everywhere). To be able to
do this, he proved a singular version of Fujita’s theorem about the approximative
Zariski decomposition ([Bou02, Thm. 3.1.24]):
Theorem 2.2 (Singular Fujita decomposition). Let α ∈ H 1,1 (X, R) be a big class.
e →X
For all > 0 there exists a sequence of blow ups with smooth centers µ : X
and a decomposition
µ∗ α = β + {E},
where β is a Kähler class and E is an effective R− divisor such that |v(α)−v(β)| < and J (µ∗ [Ek ]) ⊃ J (Tmin ).
Proof. The additional statement compared to [Bou02, Thm. 3.1.24] is the inclusion
of the multiplier ideal sheaves. To get it, we must go through the proof of Boucksom:
It starts with a sequence of Kähler currents Tk ∈ α with analytic singularities such
that
Z
n
Tk,ac
→ v(α).
X
These Tk may be chosen in such a way that J (Tk ) ⊃ J (Tmin ): Apply Theorem
3.16 in [Eck03] to α − δω and get a sequence of currents Tk0 ∈ α − δω with analytic
singularities such that
Z
0
(Tk,ac
)n → ν(α − δω).
X
Note that by construction the potentials of Tk0 are not less than the potential of a
0
0
current Tmin
with minimal singularities in (α − δω)[0], hence J (Tk0 ) ⊃ J (Tmin
).
0
0
Now set Tk := Tk + δω and Tmin := Tmin + δω.
Next, we resolve the singularities of Tk and find a sequence of blow ups in smooth
e → X, such that the Siu decomposition ([Dem00, (2.18)])
centers, µk : X
µ∗ Tk = [Ek ] + Rk
4
THOMAS ECKL
consists of the integration current of an effective R-divisor Ek and a smooth positive
residue current Rk . Since Tk is big, µ∗ Tk is also big [Bou02, Prop.1.2.5], hence there
e
is a δ > 0 such that for a Kähler form ω
e on X
µ∗ Tk = Rk + [Ek ] ≥ δe
ω.
This inequality remains valid, when we subtract an integration current of divisors,
henceR Rk ≥ δω. R But Rk is smooth, hence is a Kähler form. Furthermore, we
n
have X Tk,ac
= Xe Rkn . Finally, as Tk is a current with analytic singularities, its
potential may be locally written as
X
φk = θk + c · log(
|fi |2 ),
where θk is a C ∞ function, and an easy calculation shows that
X
µ∗ φk = µ∗ θk + c · log(
|fi0 |2 ) + c · log |gk |2 ,
P 02
where gk is a local equation of Ek and
|fi | vanishes nowhere. Consequently,
J (µ∗ [Ek ]) = J (Tk ), and
µ∗ α = {Rk } + {Ek }
is the desired decomposition, if we choose k big enough.
To prove the approximative singular Calabi-Yau theorem, let us take a modification
e → X and a decomposition µ∗ α = β + {E} belonging to as above. Let ω
µ:X
e
∗
e
be a Kähler form on X, and set ω
eδ = µ ω + δe
ω . For all δ > 0, ω
eδ is a Kähler form
e hence by the usual Calabi-Yau theorem, we can find a Kähler form θ,δ ∈ β
on X,
such that
v(β)
θ,δ (x)n = R n ω
e (x)n
ω
eδ
e Now set T := µ∗ (θ,δ + [E ]) which is a closed positive current with
for all x ∈ X.
analytic singularities in µ(E ). Furthermore, choosing δ small enough and using
the properties of β and θ,δ , we see that
T (x)n ≥ (1 − δ )v(α)ω(x)n
almost everywhere, and δ → 0 if tends to 0. Since the multiplier ideals only
depend on [E ], the inclusion of multiplier ideals in the theorem remains true. The construction of the second part of ĥ uses
Theorem 2.3 (Equisingular Approximation). Let T = α + i∂∂φ be a closed (1, 1)current on a compact hermitian manifold (X, ω), where α is a smooth closed (1, 1)form and φ a quasi-plurisubharmonic function. Let γ be a smooth real (1, 1)- form
such that T ≥ γ. Then one can write φ = limν→+∞ φν where
(a) φν is smooth in the complement X \ Zν of an analytic subset Zν ⊂ X;
(b) (φν ) is a decreasing sequence, and Zν ⊂ Zν+1 for all ν;
(c) for every t > 0
Z
(e−2tφ − e−2tφν )dVω
X
is finite for ν large enough and converges to 0 as ν → +∞;
(d) J (tφv ) = J (tφ) for ν large enough (“equisingularity”);
(e) Tν = α + i∂∂φν satisfies Tν ≥ γ − ν ω, where limν→+∞ ν = 0.
A WEAK KAWAMATA-VIEHWEG VANISHING
5
Proof. See [DPS01] or [Dem00, (15.2.1)] and especially the remark after the proof.
Now, let X be a compact n-dimensional Kähler manifold with Kähler form ω and let
L be a pseudo-effective line bundle having a hermitian metric h∞ with a curvature
form Θh∞ (L) of arbitrary sign. Let T be a closed current with analytic singularities
in c1 (L)[−ω] such that
v(c1 (L) + ω) n
ω ,
2
as in the approximative singular Calabi-Yau theorem above. In particular
J (T + ω) ⊃ J (Tmin ) where Tmin is a current with minimal singularities in
(c1 (L) + ω − ω)[0] = c1 (L)[0].
There exists a hermitian metric h = h∞ e−2φ on L, such that
(T + ω)n ≥
Θh (L) = T
([Bon95],[Eck03, Lem.4.1]). Next, we observe that
v(c1 (L) + ω) ≥ n−l · (c1 (L)l .ω n−l )≥ 0
for all 0 ≤ l ≤ n ([Bou02, p.86]), hence there is a constant C > 0 such that
v(c1 (L) + ω) ≥ Cn−ν(L) .
Let hmin = h∞ e−2ψ be a metric with Θh (L) ≥ 0 and minimal singularities, and let
ψ ↓ ψ be an equisingular regularization of ψ, such that
e
h := h∞ e−2ψ
satisfies Θeh ≥ −ω in the sense of currents. The metrics considered in the follwing
are given by
b
h1+s
= h∞ exp(−2(δ(1 + s)φ + (1 − δ)(1 + s)ψ ))
where δ > 0 is a sufficiently small number which will be fixed later. Note that b
h1+s
is really a metric on L, since h∞ remains unchanged.
b
is smooth outside an analytic subset Z ⊂ X. Its multiplier ideal is controlled
h1+s
by subadditivity ([Dem00, (14.2)]):
(1−δ)(1+s)
J (b
h1+s
) ⊂ J (hδ(1+s) ) · J (e
h(1−δ)(1+s)
) ⊂ J (e
h(1−δ)(1+s)
) = J (hmin
)
because of the equisingularity. Locally, the Hölder inequality shows that
R
|f |2 e−2[δ(1+s)φ +(1−δ)(1+s)ψ ] dVω ≤
R
R
( |f |2 e−2(1+s)φ dVω )δ · |f |2 e−2(1+s)ψ dVω )1−δ ,
hence
1+s
b 1+s ),
J (h1+s
) ∩ J (h1+s
min ) = J (hmin ) ⊂ J (h
where the first equality comes from the properties of h .
Finally, we check how the eigenvalues of the curvature form are controlled: By
construction,
Θbh + 2ω
= δ(Θh (L) + ω) + (1 − δ)(Θeh (L) + ω) + ω
≥ δ(Θh (L) + ω) + ω.
6
THOMAS ECKL
At each point x ∈ X \ Z , we may choose coordinate systems (zj )1≤j≤n resp.
(wj )1≤j≤n which diagonalize simultaneously the hermitian forms ω(x) and T + ω
resp. Θbh + 2ω, in such a way that
X
X ()
dzj ∧ dz j , (T + ω)(x) = i
ω(x) = i
λj dzj ∧ dz j
1≤j≤n
1≤j≤n
resp.
ω(x) = i
X
dwj ∧ dwj , (Θbh + 2ω)(x) = i
1≤j≤n
()
λ1
X
b() dwj ∧ dwj .
λ
j
1≤j≤n
()
λn
b()
λ
1
b()
λ
n .
Let
≤ ... ≤
and
≤ ... ≤
Changing from zj to wj by a unitary
()
transformation, the λj ’s remain the same, and the inequality between the currents
from above implies
b() ≥ δλ() + ,
λ
j
j
by Weyl’s monotonicity principle [Bha01, p.291]. On the other hand, the MongeAmpère inequality satisfied by T tells us that
()
n−ν(L)
λ1 · · · λ()
n ≥C ·
almost everywhere on X.
3. Comparisons of the metrics
Let Z be the analytic subset such that b
h is smooth on X \ Z . By [Dem82,
Prop.1.6], for every > 0 there is a sequence of complete Kähler metrics (ω,t ) on
X \ Z converging from above against ω = ω.
n,q
Let Dc,
be the space of all (n, q)- forms with values in L and coefficients in
1+s
2
J (ĥ ) ⊗ C ∞ (X) and compact support in X \ Z . Let Ln,q
,t be the L - completion
n,q
of Dc, with respect to the norm
Z
2
k u k,t :=
|u|2Vn,q ω ⊗ĥ1+s dVω,t ,
,t
X\Z
Vn,q
including the case t = 0, where
ω,t ⊗ ĥ1+s
denotes the metric on (n, q)- forms
∞
1+s
with values in L and coefficients in J (ĥ ) ⊗ C (X) naturally induced by ω,t and
ωn
,t
.
ĥ1+s
. The volume form dVω,t equals n!
The operator ∂ defines a linear, closed, densely defined operator
n,q+1
∂ ,t : Ln,q
.
,t → L,t
An element u ∈ Ln,q
,t is in the domain D∂ ,t if ∂(u), defined in the sense of disn,q+1
tribution theory, belongs to L,t
. That ∂ ,t is closed follows from the fact that
differentiation is a continuous operation in distribution theory, and the domain is
n,q
dense since it contains Dc,
.
∗
Since ∂ ,t is densely defined, there is an adjoint operator ∂ ,t , and because ∂ ,t is
closed,
∗
(∂ ,t )∗ = ∂ ,t .
∗
(cf. [SN67, p.29]). Let D∂ ∗ denote the domain of the operator ∂ ,t in Ln,q
,t .
,t
Let Λ be the adjoint of the operator L which multiplicates with ω, that is
Lα = ω ∧ α,
hΛα|βi,t = hα| ω ∧ βi,t
A WEAK KAWAMATA-VIEHWEG VANISHING
7
for all forms α, β ∈ Ln,q
,t . (The scalar product above is taken in every point
z ∈ X \ Z .)
If θ is a real (1, 1)- form we define for all q = 1, . . . , n a sesquilinear form θq on the
fibers of Ωn,q
X\Z ⊗ L by setting in every point z ∈ X \ Z
θq (α, β) = hθΛα|βi,t
Ωn,q
X\Z ,z
⊗ Lz . If θ = Θ,t is the curvature form of the metric ĥ1+s
for all α, β ∈
on
L, a term of the form hθΛα|βi,t occurs in the Bochner-Kodaira inequality:
Z
∗
2
2
k ∂ ,t u k,t + k ∂ ,t u k,t ≥
hΘ,t Λu|ui,t dVω,t .
X
n,q
On Dc,
, this inequality is valid by the usual computations ([Dem00, (4.7)]).
Hörmander ([Hör65, Lem. 5.2.1]) showed that for the complete metric ω,t (t > 0)
n,q
the forms in Dc,
are dense in D∂ ,t ∩ D∂ ∗ w.r.t. the graph norm
,t
u
7 k u k2,t
→
∗
+ k ∂ ,t u k2,t + k ∂ ,t u k2,t .
Hence we have the Bochner inequality for all u ∈ D∂ ,t ∩ D∂ ∗ .
,t
To really apply the Bochner technique we still need some comparative inequalities
between the different metrics ω,t :
0
Lemma 3.1. k u k,t0 ≤k u k,t for all u ∈ Ln,q
,t and all 0 ≤ t ≤ t .
Proof. This is just Lemma 3.3 in [Dem82].
Consequently, we have a linear continuous operator fq :
k fq k≤ 1.
Ln,q
,t
→
Ln,q
,t0
with norm
Lemma 3.2. The diagram
Ln,q
,t
∂ ,t
Ln,q+1
,t
fq
Ln,q
,t0
#
∂ ,t0
fq+1
Ln,q+1
,t0
is commutative.
n,q
Proof. Let v,t be any element of Ln,q
,t such that ∂ ,t is defined. Since Dc, is dense
(n)
n,q
in D∂ ,t there is a sequence of smooth forms (v,t )n∈N in Dc,
such that
(n)
(n)
v,t → v,t , ∂v,t → ∂ ,t v,t
n,q
strongly in the (, t)- norm. Now, the derivation of smooth forms in Dc,
w.r.t. ∂ ,t
and ∂ ,t0 do not differ. So the two limits above exist and are the same for the (, t0 )norm because of lemma 3.1.
Again, let θ be any real (1, 1)- form. If α ∈ Ln,q
,t we define |α|θ in every point
z ∈ X \ Z as the smallest number ≥ 0 (perhaps infinite) such that
|hα|βi,t |2 ≤ |α|2θ hθΛα|βi,t
for all β ∈ Ln,q
,t .
Lemma 3.3. The (n, n)- form |α|2θ dVω,t decreases if t increases.
8
THOMAS ECKL
Proof. This is just Lemma 3.2 in [Dem82].
Lemma 3.4. For all β ∈ Ln,q
,t , we have
Z
Z
1
|β|2θ dVω,t ≤
|β|2ω,t dVω,t ,
X
X λ1 + . . . + λq
where λ1 , . . . , λq are the q smallest eigenvalues of θ with respect to ω,t .
Proof. There is an orthonormal base dz1 , . . . , dzn of Ωn,q
X\Z ,z such that we can write
n
ω,t =
iX
dzj ∧ dz j ,
2 j=1
n
iX
θ=
λj dzj ∧ dz j , λj ∈ R.
2 j=1
β ∈ Ln,q
,t may be written as
β=
X
βJ dz1 ∧ . . . ∧ dzn ∧ dz J ⊗ e,
|J|=q
where e is any section in L which makes the dz1 ∧ . . . ∧ dzn ∧ dz J ⊗ e orthonormal
in Ln,q
,t . We verify
X X
cj ∧ . . . ∧ dzn ∧ dz J ⊗ e
Λβ = 2
(−1)n−j βjJ dz1 ∧ . . . ∧ dz
|J|=q−1 1≤j≤n
cj meaning that we omit dzj ), and
(dz
X
θq (β, β) = hθΛβ|βi,t = 2n+q
X
λj βjJ βjJ = 2n+q
X X
(
λj )|βJ |2 .
|J|=q j∈J
|J|=q−1 1≤j≤n
Now,
|β|2θ
=
supu
|hβ|ui,t |2
hθΛu|ui,t
= |β|2,t · supu
where u =
P
≤ supu
P
P
|J|=q
|J|=q (
P
|β|2,t ·|u|2,t
hθΛu|ui,t
|uJ |2
λj )|uJ |2
j∈J
= |β|2,t · supu
|u|2,t
hθΛu|ui,t
=
1
≤ |β|2,t λ1 +...+λ
,
q
uJ dz1 ∧ . . . ∧ dzn ∧ dz J ⊗ e as above. The lemma follows.
|J|=q
4. The Bochner technique
Let
n,q
K,t
2
n,q
be the L - completion of Dc,
w.r.t. the metric
Z
k u k2K n,q :=
(|u|2Vn,q ω ⊗ĥ1+s + |∂u|2Vn,q ω
,t
X\Z
,t
1+s
,t ⊗ĥ
)dVω,t ,
n,q
in the space of all forms with L2loc coefficients, and let K,t
be the corresponding
sheaf of germs of locally L2 sections on X (the local L2 condition should hold on
X and not only on X \ Z ).
n,q
Lemma 4.1. For all > 0, the L2 Dolbeault complex (K,0
, ∂ ,0 ) is a fine resolution
1+s
of the sheaf KX ⊗ L ⊗ J (ĥ ).
A WEAK KAWAMATA-VIEHWEG VANISHING
9
Proof. X \ Z can be covered by open subsets U ⊂ X which are Stein, lie relatively
compact in another Stein open subset U 0 ⊂ X, and on which L is trivial. On these
U ’s we can show the ∂- Poincaré lemma.
First, as Stein sets, U ⊂⊂ U 0 may be embedded as analytic subsets into some
CN . Hence we can find a smooth plurisubharmonic function ψ on U 0 such that
i∂∂ψ ≥ 2λω for some constant λ > 0 on U (ω,0 = ω is smooth on U ). Furthermore,
|ψ| is bounded on U by some constant M > 0. Subtracting M we still have a
plurisubharmonic function, which we also call ψ, satisfying
−2M ≤ ψ ≤ 0 and i∂∂ψ ≥ 2λω.
This implies that the metrics b
h1+s
and b
h1+s
e−2ψ are comparable on U , and
Θbh1+s
e−2ψ ≥ λω, for λ sufficiently big.
Since b
h1+s
and b
h1+s
e−2ψ are comparable, we can interchange them
R in the metric for
n,q
n,q
K,0 (U ). So, given g ∈ K,0
(U ) with ∂ ,0 (g) = 0, we know that X |g|2,0 dVω < ∞,
R
hence also X |g|2,ψ dVω < ∞ with the new metric, and by lemma 3.4
Z
Z
1 2
2
|g|,ψ dVω < ∞.
|g|Θb 1+s −2ψ dVω ≤
h
e
qλ
X
X
Therefore, we can apply theorem 4.1 of [Dem82]: There exists a (n, q − 1)- form f
with L2loc coefficients in U such that
∂ ,0 (f ) = g
and
Z
|f |2,0 dVω ≤
X
Z
|f |2,ψ dVω ≤
X
Z
X
1 2
|g| dVω ≤ e2M
qλ ,ψ
Z
|g|2,0 dVω .
X
2
Finally, the L condition forces sections holomorphic on X \ Z to extend holomorphically across Z ([Dem82, Lem.6.9]). The L2 condition implies that the coefficients lie in J (ĥ1+s
). Consequently, the complex is a resolution, and it is fine
because of the existence of partition of unities.
0
Now, let us take a cohomology class {β} ∈ H q (X, O(KX + L) ⊗ J (h1+s
min )). If U is
a covering of X with Stein open subsets Uα , the class {β} may be represented by
a Čech cocycle
0
q
1+s
(βα0 ···αq )α0 ···αq ∈ C q (U, O(KX + L) ⊗ J (h1+s
)).
min )) ⊂ C (U, O(KX + L) ⊗ J (ĥ
Let (ψα ) be a C ∞ partition of unity subordinate to U. Taking the usual De RhamWeil isomorphisms between Čech and Dolbeault cohomology, we obtain a closed
n,q
(n, q)- form in K,0
of the form
X
β=
βα0 ···αq ∂ψα0 ∧ . . . ∧ ∂ψαq .
α0 ,...,αq
In particular, this form has coefficients in J (ĥ1+s
) ⊗ C ∞ (X).
We want
n,q
to show that β is a boundary in K,0 for some > 0, hence
{β} = 0 ∈ H q (X, O(KX + L) ⊗ J (ĥ1+s
)). This implies theorem 1.4 because of
(1−δ)(1+s)
1+s
the inclusion J (ĥ ) ⊂ J (hmin
).
10
THOMAS ECKL
n,q
The reasoning starts as follows: β is also an element of K,t
for any t ≥ 0, because
n,q
2
of lemma 3.1. Every L form u ∈ D∂ ∗ ⊂ K,t may be written as u = u1 + u2 with
,t
∗
∗
u1 ∈ ker ∂ ,t and u1 ∈ (ker ∂ ,t )⊥ = im ∂ ,t ⊂ ker ∂ ,t ,
since ∂ ,t is a closed operator, hence ker ∂ ,t is closed. Using β ∈ ker ∂ and the two
inequalities in lemma 3.3 and 3.4, we get (Θ,t denotes the curvature form of ĥ1+s
on X \ Z , plus 2ω,t )
R
| β, u ,t |2 = | β, u1 ,t |2 = | X\Z hβ, u1 i,t dVω,t |2 ≤
R
≤ ( X\Z |hβ, u1 i,t |dVω,t )2 ≤
p
R
≤ ( X\Z βΘ,t · hΘ,t Λu1 |u1 i,t dVω,t )2 ≤
≤
R
≤
R
≤
R
2
βΘ
dVω,t ·
,t
R
X\Z
2
βΘ
dVω,0 ·
,0
R
X\Z
X\Z
hΘ,t Λu1 |u1 i,t dVω,t ≤
X\Z
hΘ,t Λu1 |u1 i,t dVω,t ≤
1
|β|2,0 dVω,0
X\Z λ̂(,0) +···+λ̂(,0)
q
1
·
R
X\Z
hΘ,t Λu1 |u1 i,t dVω,t .
∗
u1 is an element of D∂ ,t ∩ D∂ ∗ , since u1 ∈ ker ∂ ,t , u2 ∈ ker ∂ ,t and u1 = u − u2 .
,t
Consequently, we can apply the Bochner inequality on u1 . As ∂u1 = 0 we get that
the second integral on the right hand side is bounded above by
∗
∗
k ∂ ,t u1 k2,t + 2q k u1 k2,t ≤ k ∂ ,t u k2,t + 2q k u k2,t ,
and finally
2
Z
|hβ, ui,t | ≤
1
(,t)
X λ̂1
+ ··· +
2
2q k u k,t comes in
∗
(,t)
λ̂q
|β|2,t dVω,t (k ∂ ,t u k2,t + 2q k u k2,t ),
where the term
because Θ,t differs from the curvature form of
1+s
ĥ by 2ω,t .
R
Using the uniform bound C = X (,0) 1 (,0) |β|2,0 dVω,0 we apply the Hahnλ̂1
+···+λ̂q
Banach theorem as in [Dem82]: β, u ,t defines a linear form on the range of
the densely defined operator
∗
n,q−1
T : Ln,q
⊕ Ln,q
,t → L,t
,t , u 7→ ∂ ,t u + 2qu
1
(with domain DT = D∂ ∗ ). Hence there exists an f,t = v,t ⊕ 2q
w,t ∈ Ln,q−1
⊕Ln,q
,t
,t
,t
such that
∗
β, u ,t = f,t , ∂ ,t u + 2qu ,t .
Consequently, β = T ∗ f,t = ∂ ,t v,t + w,t with
1
k v,t k2,t +
k w,t k2,t ≤ C .
2q
n,q
Furthermore, ∂ ,t w,t = ∂β = 0, and v,t , w,t are both contained in K,t
.
The estimates of section 3 tell us that the metrics of the v,t and w,t in the
L2 space Ln,q−1
resp. Ln,q
,t0
,t0 are uniformly bounded for all t0 ≥ t > 0. Since
k β k,t0 ≥ | k ∂ ,t v,t k,t0 − k w,t k,t0 |, the same is true for ∂ ,t v,t . Consequently,
A WEAK KAWAMATA-VIEHWEG VANISHING
11
the sequences of these elements converge weakly as t → 0, and we have three weak
limits in the respective spaces:
n,q
n,q
0
v,t * v ∈ Ln,q−1
,t0 , ∂ ,t v,t * v ∈ L,t0 , w,t * w ∈ L,t0 .
Note that these weak limits are the same for every choice of t0 > 0: The spaces Ln,q
,t0
n,q
all contain the dense subset Dc,
, hence weak convergence is transmitted through
the continuous maps between them.
Claim: ∂ ,t0 v = v0 .
Proof. On the one hand, we have
∂ ,t v,t , u ,t0 → v0 , u ,t0
for all u ∈ D∂ ∗ because of the weak convergence. On the other hand, the com,t0
mutativity of the diagram in lemma 3.2 and again the weak convergence show that
∗
∂ ,t v,t , u ,t0 = ∂ ,t0 v,t , u ,t0 = v,t , ∂ ,t0 u ,t0 →
∗
v , ∂ ,t0 u ,t0 = ∂ ,t0 v , u ,t0 .
Since D∂ ∗
,t0
is dense in Ln,q
,t0 the claim follows.
By standard properties of weak convergence,
k v k,t0 ≤ lim inf k v,t k,t0 ,
t→0
and similarly for k ∂ ,t0 v k,t0 and k w k,t0 . Consequently, these three norms are
uniformly bounded by C for t0 > 0.
Now, we restrict the integral defining the (, t0 )- norm to compact subsets
K ⊂ X \ Z . Of course, we get k v k,t0 ,K ≤k v k,t0 , hence the new (, t0 , K)norms of v are still uniformly bounded by C in t0 > 0. Furthermore, as ωt ↓ ω,
we see that k v k,t0 ,K →k v k,0,K , and monotone convergence tells us that k v k,0
exists and is ≤ C . The same is true for k ∂ ,0 v k,0 and k w k,0 .
As β = ∂ ,t v,t + w,t for all t, β = ∂ ,0 v + w remains true. Furthermore,
n,q−1
n,q
∂ ,0 w,t = ∂ ,t w,t = 0. Hence, v and w belong to K,0
resp. K,0
.
For the last step we note that the almost plurisubharmonic weights ψ defining h̃
form a decreasing sequence, and consequently,
k u k,0 ≤k u k0 ,0
∀u ∈ Kn,q
0 ,0 ,
if 0 < . This implies k w k0 ,0 ≤k w k,0 for some fixed 0 > 0 and < 0 . Since
k w k,0 ≤ C = 2q · C ,
we conclude using the estimate in section 5 that k w k0 ,0 → 0 for → 0. But
the norm of w measures the distance of β from the closure of the subspace of
boundaries in Kn,q
. So it only remains to show
0 ,0
of boundaries in the Dolbeault complex
Lemma 4.2. The subspace Bn,q
⊂ Kn,q
0 ,0
0 ,0
n,q
(K0 ,0 , ∂) is closed.
Proof. Let Zn,q
⊂ Kn,q
be the space of cocycles with respect to ∂, and let
0 ,0
0 ,0
n,q
n,q
Z0 ,0 ⊂ K0 ,0 be the corresponding sheaf. Let U be a covering of X with Stein
12
THOMAS ECKL
open subsets as in the proof of exactness of the Dolbeault complex in lemma 4.1.
By the usual DeRham-Weil isomorphism,
H q (Kn,•
)=
0 ,0
Zn,q
0 ,0
∂Kn,q−1
0 ,0
=
Z 0 (U, Zn,q
)
0 ,0
∂Z 0 (U, Kn,q−1
)
0 ,0
.
So we have to prove that ∂Z 0 (U, Kn,q−1
) is closed in Z 0 (U, Zn,q
) with respect to
0 ,0
0 ,0
2
the L norms on every set U in U.
Note first that
∂ : C 0 (U, Kn,q−1
) → C 0 (U, Zn,q
)
0 ,0
0 ,0
is continuous, by definition of the norms, and surjective, by exactness. Hence ∂ is
an open map, by Banach’s open mapping theorem. Its kernel is C 0 (U, Zn,q−1
).
0 ,0
n,q−1
n,q−1
0
0
Next, C 0 (U, Zn,q−1
)
and
Z
(U,
K
)
are
closed
in
C
(U,
K
),
since
∂
is a
0 ,0
0 ,0
0 ,0
closed operator, and equality is conserved when going to the limit. Consequently,
the sum of these two spaces is closed, too, and its complement is open. But then
∂(Z 0 (U, Kn,q−1
) + C 0 (U, Zn,q−1
)) = ∂(Z 0 (U, Kn,q−1
))
0 ,0
0 ,0
0 ,0
is closed in C 0 (U, Zn,q
) and also in Z 0 (U, Zn,q
).
0 ,0
0 ,0
5. The uniform estimate
The aim of this section is to prove
Lemma 5.1. Let 0 < s < s0 . For every smooth (n, q)- form β with values in L and
0
∞
coefficients in J (h1+s
min ) ⊗ C ,
Z
q
|β|2,0 dVω,0 ,
(,0)
(,0)
X λ̂1
+ · · · + λ̂q
tends to 0 for → 0.
(,0)
(,0)
Before starting with the proof, set λ̂j := λ̂j
and λj := λj
Now, by construction we know that λ̂j ≥ δλj + , and
.
λqq λq+1 . . . λn ≥ λ1 . . . λn ≥ Cn−ν ,
hence
1
1
≤
≤ C −1/q −(n−ν)/q) (λq+1 . . . λn )1/q .
λ1 + . . . + λq
λq
We infer
q
γ :=
λ̂1 + . . . + λ̂q
≤ min(1,
q
) ≤ min(1, C 0 δ −1 −(n−ν)/q) (λq+1 . . . λn )1/q ).
δλq
We notice that
Z
Z
λq+1 . . . λn dVω ≤
(Θh (L) + ω)n−q ∧ ω q ≤ (c1 (L) + {ω})n−q {ω}q ≤ C 00 ,
X
X
hence the functions (λq+1 . . . λn )1/q are uniformly bounded in L1 norm as tends
to 0. Since 1 − (n − ν)/q > 0 by hypothesis, we conclude that γ converges almost
everywhere to 0 as tends to 0. On the other hand,
|β|2ĥ = |β|2h∞ e−2(δ(1+s)φ +(1−δ)(1+s)ψ ) ≤ |β|2h∞ e−2(δ(1+s)φ ) e−2(1−δ)(1+s)ψ .
0
Our assumption that the coefficients of β lie in J (h1+s
min ) implies that there exists a
R
0
p0 > 1 such that X |β|2h∞ e−2p (1−δ)(1+s)ψ dVω < ∞, no matter how small δ is. Let
A WEAK KAWAMATA-VIEHWEG VANISHING
13
p ∈ (1, ∞) be the conjugate exponent such that p1 + p10 = 1. By Hölder’s inequality,
we have
Z
Z
Z
0
0
0
2
2
−2pδ(1+s)φ
1/p
γ |β|ĥ dVω ≤ ( |β|h∞ e
dVω ) ( γp |β|2h∞ e−2p (1+s)(1−δ)ψ dVω )1/p .
X
X
X
As γ ≤ 1, the Lebesgue dominated convergence theorem shows that
Z
0
0
γp |β|2h∞ e−2p (1+s)(1−δ)ψ dVω
X
converges to 0 as tends to 0.
For the first integral, we argue as follows: The φ may be constructed such that
the Lelong numbers ν((1 + s)φ , x) are bounded from above by the Lelong numbers
ν((1 + s)ψ, x) in every point x ∈ X (see again Theorem 3.16 in [Eck03]). On the
other hand, there is a constant C such that ν((1 + s)ψ, x) < C for all points x ∈ X,
by [Bou02, Lem.3.11]. Hence, ν( 1+s
C φ , x) < 1, and
Z
e−(2/C)(1+s)φ dVω < ∞,
X
by Skoda’s lemma [Dem00, (5.6)]. Adding sufficiently big constants to φ we can
even reach that
R the integrals above are uniformly bounded. By choosing δ ≤ 1/(pC),
the integral X |β|2h∞ e−2pδ(1+s)φ dVω remains bounded and we are done.
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Thomas Eckl, Institut für Mathematik, Universität Bayreuth, 95440 Bayreuth, Germany
E-mail address: [email protected]
URL: http://btm8x5.mat.uni-bayreuth.de/~eckl