Singular Demailly–Skoda-type theorem
Takahiro Inayama(Tokyo University of Science)

2022/08/05

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Positivity for smooth Hermitian verctor bundles

1

Positivity for smooth Hermitian verctor bundles

2

Demailly–Skoda theorem

3

Singular Hermitian metrics and main theorems

4

Applications

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Positivity for smooth Hermitian verctor bundles

Setting
X : complex manifold of dim n
E → X : holomorphic vector bundle of rank rh : (possibly singular) Hermitian metric on E
Θh : Chern curvature of (E, h)(z1 , . . . , zn ) : holomorphic local coordinate of X
(e1 , . . . , er ) : orthonormal frame of E at some ﬁxed point x0 ∈ X√−1Θh =

X

cj k̄λµ dzj ∧ dz̄k ⊗ e⋆λ ⊗ eµ .

1≤j,k≤n1≤λ,µ≤r3 / 32

Positivity for smooth Hermitian verctor bundles

Deﬁnition(E, h) → X : smooth Hermitian vetor bundleX√−1Θh =
cj k̄λµ dzj ∧ dz̄k ⊗ e⋆λ ⊗ eµ .
1≤j,k≤n1≤λ,µ≤reh
We naturally identify the curvature tensor with a Hermitian form Θ
on TX ⊗ E deﬁned byXe h (τ, τ ) =Θ
cj k̄λµ τjλ τ̄kµ1≤j,k≤n1≤λ,µ≤rP∂
for τ = j,λ τjλ ∂z⊗ eλ ∈ TX ⊗ E at x0 ∈ X, wherej(e1 , . . . , er ) is orthonormal.
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Positivity for smooth Hermitian verctor bundles

Deﬁnition (Griﬃths and Nakano positivity)
Keep the notation.
We say that:1 h is Nakano positive ifX
e h (τ, τ ) =Θ
cj k̄λµ τjλ τ̄kµ > 0
for allP
non-zero elements τ ∈ TX ⊗ E, where∂τ = j,λ τjλ ∂z⊗ eλ .
j2

h is Griﬃths positive if
e h (v ⊗ s, v ⊗ s) =ΘX

cj k̄λµ vj v̄k sλ s̄µ > 0

for all non-zeroelementsPv ⊗ s ∈ TX ⊗ E,
P decomposable∂
where v = j vj ∂zj ∈ TX and s = λ sλ eλ ∈ E.
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Positivity for smooth Hermitian verctor bundles

Note that Nakano positivity is stronger than Griﬃths positivity.
For example, the following vanishing theorem holds:
Let X be a projective manifold and (E, h) → X be a smoothNakano positive vector bundle. Then the following qth cohomology
group vanishes
H q (X, KX ⊗ E) = 0
for q > 0, which is known as the Nakano vanishing theorem.
However, for n > 1
H n−1 (Pn , KPn ⊗ TPn ) ̸= 0.
Here TPn admits a Griﬃths positive smooth Hermitian metric.
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Positivity for smooth Hermitian verctor bundles

We can also obtain the following L2 -estimate for Nakano positive
vector bundles.
TheoremLet (X, ω) be a complete Kähler manifold and (E, h) → X be ae E,h (τ, τ ) ≥ δ|τ |2 forNakano positive vector bundle such that Θω,hδ > 0 and τ ∈ TX ⊗ E. Then for any ∂-closed E-valued(n, q)-form u with ﬁnite L2 -norm, there exists an E-valued(n, q − 1)-form α satisfying ∂α = u andZZ
12|α|ω,h dVω ≤|u|2ω,h dVω .
δq X
XThis type of result is often called Hörmander’s or a Hörmander-type
L2 -estimate due to the original work of Hörmander.
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Positivity for smooth Hermitian verctor bundles

Nakano positivity,

cohomology vanishing7

×'Hörmander’s L2 -estimate
2

×

Griﬃths positivity

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Demailly–Skoda theorem

1

Positivity for smooth Hermitian verctor bundles

2

Demailly–Skoda theorem

3

Singular Hermitian metrics and main theorems

4

Applications

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Demailly–Skoda theorem

Theorem (Demailly–Skoda’79)
If (E, h) is a Griﬃths (semi-)positive smooth Hermitian vector
bundle, (E ⊗ det E, h ⊗ det h) is Nakano (semi-)positive.
Thanks to the generalizations by Berndtsson’09 and
Liu–Sun–Yang’13, it is also known that if (E, h) is Griﬃths(semi-)positive, (S m E ⊗ det E, S m h ⊗ det h) is Nakano(semi-)positive for all m ∈ N, where S m E is the mth symmetricpower of E.
In this talk, we only focus on the case that m = 1 for simplicity.

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Demailly–Skoda theorem

The Demailly–Skoda theorem has many important applications.
Theorem
Let X be a projective manifold and (E, h) → X be a Griﬃths
positive vector bundle. Then the qth cohomology group vanishes
H q (X, KX ⊗ E ⊗ det E) = 0
for q > 0.
This theorem is the so-called Griﬃths vanishing theorem. By using
the Demailly–Skoda theorem and the Nakano vanishing theorem, we
can easily see that this result holds since (E ⊗ det E, h ⊗ det h) is
Nakano positive.
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Demailly–Skoda theorem

We would like to consider the generalization of the Demailly–Skoda
theorem for singular Hermitian vector bundles.
In the next section, I will explain a notion of singular Hermitian
metrics.

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Singular Hermitian metrics and main theorems

1

Positivity for smooth Hermitian verctor bundles

2

Demailly–Skoda theorem

3

Singular Hermitian metrics and main theorems

4

Applications

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Singular Hermitian metrics and main theorems

Deﬁnition
Let E → X be a holomorphic vector bundle over a complex
manifold X. We say that h is a singular Hermitian metric on E if:
1 h is a measurable map from X to the space of non-negative
Hermitian forms on the ﬁbers, that is, |u|h is a measurable
function whenever U ⊂ X is open and u ∈ H 0 (U, E).
2 h is ﬁnite and positive deﬁnite almost everywhere.
Roughly speaking, a singular Hermitian metric is just a measurable
metric. When a metric has either positive or negative curvature, it
has stronger regularity properties than being measurable.

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Singular Hermitian metrics and main theorems

It is known that the Chern curvature current Θh cannot be
deﬁned with measure coeﬃcients in general (Rauﬁ’15).
For this reason, we cannot generally deﬁne the positivity or
negativity of a singular Hermitian metric on a vector bundle byusing the Chern curvature current.
We need to deﬁne positivity and negativity notions without
using the Chern curvature current.

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Singular Hermitian metrics and main theorems

There is a deﬁnition of Griﬃths positivity and negativity for singular
Hermitian metrics in the following form.
Deﬁnition
Let h be a singular Hermitian metric on E. We say that:2
2
1 h is Griﬃths semi-negative if |u|h (or log |u|h ) is psh, where0∀u ∈ H (U, E) and U ⊂ X is an open subeset.
⋆⋆
2 h is Griﬃths semi-positive if the dual metric h on E is
Griﬃths semi-negative.
I show some example.
There is also an approach to Nakano positivity for singular Hermitian
metrics.
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Singular Hermitian metrics and main theorems

Based on the work of Hosono–I.(’21) and
Deng–Ning–Wang–Zhou(’22), we can deﬁne Nakano positivity for
singular Hermitian metrics.
Deﬁnition (I.’22)
Let h be a Griﬃths semi-positive singular Hermitian metric on avector bundle E → X over a complex manifold X. We say that h
is Nakano semi-positive if ∀(Ω, ι) Stein coordinate such that E|ι(Ω)
is trivial, ∀ωΩ Kähler form, ∀ψ ∈ C ∞ (Ω) ∩ SPSH(Ω) and∀∂-closed u ∈ L2(n,q) (Ω, ι⋆ E; ω, (ι⋆ h)e−ψ ), there exists
α ∈ L2(n,q−1) (Ω, ι⋆ E; ωΩ , (ι⋆ h)e−ψ ) satisfying ∂α = u andZΩ|u|2ωΩ ,ι⋆ h e−ψ dVωΩ ≤Z√⟨[ −1∂∂ψ⊗IdE , ΛωΩ ]−1 f, f ⟩ωΩ ,ι⋆ h e−ψ dVωΩ .
Ωι : Ω → X is a holomorphic injective map.
[·, ·] is the graded Lie bracket.
ΛωΩ is the adjoint operator of LωΩ = ωΩ ∧ ·.
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Singular Hermitian metrics and main theorems

Apart from the detailed deﬁnitions, we have shown the following
theorem.
Theorem (I.’22)
Let h be a Griﬃths semi-positive singular Hermitian metric on E.
Then h ⊗ det h is Nakano semi-positive in the sense of singular
Hermitian metrics on E ⊗ det E.
This is a singular version of the Demailly–Skoda theorem. For
example, we can prove the following Hörmander-type L2 -estimate.

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Singular Hermitian metrics and main theorems

Theorem (I.’20)
Let (X, ω) be a Stein or projective manifold and E → X be aholomorphic vector bundle. We also let h be a strictly Griﬃths
δω -positive singular Hermitian metric on E for δ > 0. Then for any∂-closed E ⊗ det E-valued (n, q)-form u with ﬁnite L2 -norm,
there is an E ⊗ det E-valued (n, q − 1)-form α such that∂α = u andZZ
12|α|ω,h⊗det h dVω ≤|u|2ω,h⊗det h dVω .
δqr X
X
We say that h is strictly Griﬃths δω -positive for δ > 0 if for any
openKähler potential φ of ω on U , that is,√ set U in X and for any
δφ−1∂∂φ = ω on U , he is Griﬃths semi-positive. This is aesingular version
δ|v|2ω |s|2h for v ∈ TX
√ of ΘE,h (v ⊗√s, v ⊗ s) ≥ √and s ∈ E ( −1Θheδφ = −1Θh − δ −1∂∂φ ⊗ IdE ).
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Applications

1

Positivity for smooth Hermitian verctor bundles

2

Demailly–Skoda theorem

3

Singular Hermitian metrics and main theorems

4

Applications

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Applications

Before discussing the applications of the main theorems, we introduce
a notion of a higher rank analogue of a multiplier ideal sheaf, which
is a subsheaf of the locally free sheaf OX (E) associated with E.
Deﬁnition
Let (E, h) be a singular Hermitian vector bundle over a complex
manifold X. Then we deﬁne the higher rank analogue of themultiplier ideal sheaf E(h) by
E(h)x = {s ∈ OX (E) | |s|2h is locally integrable around x ∈ X}.
The multiplier ideal sheaf was originally introduced by Nadel in thecase that E is a line bundle.

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Applications

Deﬁnition (Nadel’90)
Let (L, h = e−φ ) be a singular Hermitian line bundle over a complex
manifold X. Then we deﬁne the multiplier ideal sheaf I(h) by
I(h)x = {fx ∈ OX,x | |f |2h = |f |2 e−φ is L1loc around x}.

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Applications

As a corollary of Hörmander’s L2 -estimate for(E ⊗ det E, h ⊗ det h), we obtain the following vanishing theorem.
Theorem (I.’22)
Let (X, ω) be a projective manifold and (E, h) be a strictly
Griﬃths δω -positive singular Hermitian metric for δ > 0. Then theqth cohomology group with coeﬃcients in the sheaf of germs ofholomorphic sections of KX ⊗ E(h ⊗ det h) vanishes for q > 0
H q (X, KX ⊗ E(h ⊗ det h)) = 0.
If h is smooth, E(h ⊗ det h) = OX (E ⊗ det E). Hence, we cansee that this theorem is a generalization of the Griﬃths vanishingtheorem: H q (X, KX ⊗ E ⊗ det E) = 0.
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Applications

Corollary
Keep the setting. If the Lelong number ν(− log det h, x) of− log det h satisﬁes ν(− log det h, x) < 1 for all x ∈ X, weobtainH q (X, KX ⊗ E ⊗ det E) = 0
for q > 0.
The Lelong number is an index that reﬂects a certain singularity of a
psh function. The deﬁnition is as follows:ν(− log det h, x) = lim inf
z→x− log det h(z)log |z − x|.

Note that − log det h is psh since (det E, det h) is semi-positive.
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Applications

Corollary
Keep the setting. If the Lelong number ν(− log det h, x) of− log det h satisﬁes ν(− log det h, x) < 1 for all x ∈ X, weobtainH q (X, KX ⊗ E ⊗ det E) = 0
for q > 0.
Note that if ν(− log det h, x) < 1, thanks to the result of
Skoda(’72), E(h ⊗ det h) = OX (E ⊗ det E).

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Applications

Whether the higher rank analogue of the multiplier ideal sheaf iscoherent or not is an important problem. In the case that E is a line
bundle, Nadel showed that a multiplier ideal sheaf associated with asemi-positive singular Hermitian metric is coherent. However, ingeneral (that is, the case that E is a vector bundle), the coherence of
it is only known in few cases.

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Applications

As an application of the main theorem, we can show the following
result.
Theorem (I.’22)
Let h be a Griﬃths semi-positive singular Hermitian metric on E.
Then we have that:
1 E(h ⊗ det h) is coherent.
2 E(h) is coherent if the unbounded locus L(det h) of det h isdiscrete.

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Applications(Strategy of the proof)
Nadel proved the coherence of the multiplier ideal sheaf I(h) basedon the technique of Hörmander’s L2 -estimate for (L, h).
Hence, by using the L2 -estimate for (E ⊗ det E, h ⊗ det h), we
can prove the coherence of E(h ⊗ det h).

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Applications

Here we recall the L2 -estimate for (E ⊗ det E, h ⊗ det h).
Theorem (I.’20)
Let (X, ω) be a Stein or projective manifold and E → X be aholomorphic vector bundle. We also let h be a strictly Griﬃths
δω -positive singular Hermitian metric on E for δ > 0. Then for any∂-closed E ⊗ det E-valued (n, q)-form u with ﬁnite L2 -norm,
there is an E ⊗ det E-valued (n, q − 1)-form α such that∂α = u andZZ12|u|2ω,h⊗det h dVω .
|α|ω,h⊗det h dVω ≤δqrXX

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Applications(Strategy of the proof)
Nadel proved the coherence of the multiplier ideal sheaf I(h) basedon the technique of Hörmander’s L2 -estimate.
Hence, by using the L2 -estimate for (E ⊗ det E, h ⊗ det h), we
can prove the coherence of E(h ⊗ det h).
The second case is technically complicated, but we still use
Hörmander’s L2 -estimate to prove our case.

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Applications

In summary, the Demailly–Skoda theorem has been important in thesmooth setting. We can see that this is also true for singular
Hermitian metrics. Indeed, we can show the following results (here welet (E, h) be a Griﬃths positive singular Hermitian vector bundle):2
1 Hörmander-type L -estimate for (E ⊗ det E, h ⊗ det h).
q
2 H (X, KX ⊗ E(h ⊗ det h)) = 0.
3 the coherence of E(h ⊗ det h).
4 the coherence of E(h) when h has isolated singularities.

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Applications

Thank you very much.

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