Positivity of singular Hermitian metrics on vector bundles - - PowerPoint PPT Presentation

Positivity of singular Hermitian metrics on vector bundles

稲山 貴大 (INAYAMA Takahiro) (the University of Tokyo) September 08, 2019

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1

Introduction

2

The first case

3

The second case

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Intro.

Setting

X: complex manifold (dim X = n) E → X: holomorphic vector bundle of rank r h: singular Hermitian metric (sHm) on E Θh: Chern curvature current of h

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Intro.

Setting

X: complex manifold (dim X = n) E → X: holomorphic vector bundle of rank r h: singular Hermitian metric (sHm) on E Θh: Chern curvature current of h (Def.) h is a measurable map from X to the space of non-negative Hermitian forms on the fibers, i.e. · |u|2

h : measurable (∀u ∈ H0(U, E), ∀U ⊂ X open),

· 0 < det h < +∞ a.e. on each fiber. Θh := ∂(h−1∂h) locally.

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Example

Example (Raufi ’15)

Let ∆ := {z ∈ C | |z| < 1}, E := ∆ × C2, and h = ( 1 + |z|2 z z |z|2 ) .

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Example

Example (Raufi ’15)

Let ∆ := {z ∈ C | |z| < 1}, E := ∆ × C2, and h = ( 1 + |z|2 z z |z|2 ) . Then, h is a sHm on E, and has a singularity at the origin. In fact, at z = 0, h = ( 1 ) . h is Griffiths semi-negative ( ⇐ ⇒ log |u|2

h is psh for ∀u ∈ O(E)).

Θh is not a current with measure coefficients.

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Motivation

· For this reason, we cannot generally define the positivity or negativity of a sHm on a vector bundle by using the curvature current. · For example, we do not know the definition of the Nakano positivity in the singular setting. · There are two ways to study positivity notions of a sHm.

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Motivation

· For this reason, we cannot generally define the positivity or negativity of a sHm on a vector bundle by using the curvature current. · For example, we do not know the definition of the Nakano positivity in the singular setting. · There are two ways to study positivity notions of a sHm.

1 Find some sufficient conditions that curvature currents can be defined

with measure coefficients.

2 Seek new positivity notions without using curvature currents. 5 / 14

1

Introduction

2

The first case

3

The second case

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The first one

Theorem (Raufi ’15)

Let h be a Griffiths semi-negative sHm. If det h > ϵ (ϵ > 0), then

1 θh = h−1∂h ∈ L2

loc(X), and

2 Θh has measure coefficients. 7 / 14

The first one

Theorem (Raufi ’15)

Let h be a Griffiths semi-negative sHm. If det h > ϵ (ϵ > 0), then

1 θh = h−1∂h ∈ L2

loc(X), and

2 Θh has measure coefficients.

· In other words, if a Griffiths semi-negative sHm h is non-degenerate, Θh has measure coefficients. · In general, a Griffiths semi-negative sHm possibly degenerates, that is, {det h = 0} ̸= ∅. · We find the condition that the curvature current can be defined with measure coefficients over the degeneracy set {det h = 0}.

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Main Theorem 1

Theorem (I. ’19)[I]

Let h be a Griffiths semi-negative sHm. Suppose that (i) ν(log det h, x) < 1 − ϵ for x ∈ X, and (ii) √−1∂∂ log det h ∈ L1+δ

loc

for 0 ≤ ϵ < 1, δ > 0. Then we have

1 θh ∈ L 2 2−ϵ

loc , and

2 if (ϵ + 1)(δ + 1) ≥ 1, Θh has measure coefficients. 8 / 14

Main Theorem 1

Theorem (I. ’19)[I]

Let h be a Griffiths semi-negative sHm. Suppose that (i) ν(log det h, x) < 1 − ϵ for x ∈ X, and (ii) √−1∂∂ log det h ∈ L1+δ

loc

for 0 ≤ ϵ < 1, δ > 0. Then we have

1 θh ∈ L 2 2−ϵ

loc , and

2 if (ϵ + 1)(δ + 1) ≥ 1, Θh has measure coefficients.

We can also prove a version of the above theorem in the case that h is Griffiths semi-positive. Namely, if the singularity of h is ”mild”, the curvature current has measure coefficients over the degeneracy set {det h = 0}.

[I] T. Inayama, Curvature Currents and Chern Forms of Singular Hermitian Metrics on Holomorphic Vector Bundles, J. Geom. Anal. (DOI: 10.1007/s12220-019-00164-9)

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Further applications

We introduce a notion of the sHm with minimal singularities. (Line bundle cases) · For every pseudo-effective line bundle L → X, there exist sHms hmin with minimal singularities such that √−1Θhmin ≥ 0 from the results of [Demailly-Peternell-Schneider ’01]. · Let φmin be the local weight of hmin = e−φmin. It is known that φmin satisfies like the conditions described in our theorem.

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Further applications

We introduce a notion of the sHm with minimal singularities. (Line bundle cases) · For every pseudo-effective line bundle L → X, there exist sHms hmin with minimal singularities such that √−1Θhmin ≥ 0 from the results of [Demailly-Peternell-Schneider ’01]. · Let φmin be the local weight of hmin = e−φmin. It is known that φmin satisfies like the conditions described in our theorem. For example, If X is a projective manifold and L → X is a nef and big line bundle, φmin has zero Lelong numbers everywhere [DPS ’01]. · ∂∂-Laplacian conditions of φmin are obtained by many people (cf. [Berman ’18], [Chu-Tosatti-Weinkove ’18]).

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(Vector bundle cases) · If the sHm with minimal singularities hmin on vector bundles is constructed, we can expect that hmin satisfies the above regularity properties and Θhmin has measure coefficients from our theorem. · However, we do not know anything about hmin (existence, construction, property, etc...). · There are various problems in this field. Construct hmin on a Griffiths semi-positive vector bundle. Is det hmin a sHm with minimal singularities on det E? Does Θhmin has measure coefficients? · · ·

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1

Introduction

2

The first case

3

The second case

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The second case

Theorem (Deng-Wang-Zhang-Zhou ’18)

Let X := Ω be a bounded domain in Cn. Assume that for any z ∈ Ω, any a ∈ Ez \ {0} with |a|h < +∞, and any m ∈ N, there is a fm ∈ H0(Ω, E ⊗m) such that fm(z) = a⊗m and satisfies the following condition: ∫

Ω

|fm|2

h⊗m ≤ Cm|a⊗m|2 h⊗m,

where Cm are constants independent of z ∈ Ω and satisfy the growth condition limm→∞ 1

m log Cm = 0.

Then if |ξ|h⋆ is u.s.c. for any ξ ∈ O(E ⋆), (E, h) is Griffiths semi-positive.

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The second case

Theorem (Deng-Wang-Zhang-Zhou ’18)

Let X := Ω be a bounded domain in Cn. Assume that for any z ∈ Ω, any a ∈ Ez \ {0} with |a|h < +∞, and any m ∈ N, there is a fm ∈ H0(Ω, E ⊗m) such that fm(z) = a⊗m and satisfies the following condition: ∫

Ω

|fm|2

h⊗m ≤ Cm|a⊗m|2 h⊗m,

where Cm are constants independent of z ∈ Ω and satisfy the growth condition limm→∞ 1

m log Cm = 0.

Then if |ξ|h⋆ is u.s.c. for any ξ ∈ O(E ⋆), (E, h) is Griffiths semi-positive. · The above theorem implies that an Ohsawa-Takegoshi type condition is a new positivity notion which is stronger than or equivalent to the Griffiths semi-positivity.

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Main Theorem 2

Theorem (Hosono-I. ’19)[HI]

This Ohsawa-Takegoshi type positivity is weaker than Nakano semi-positivity in smooth settings.

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Main Theorem 2

Theorem (Hosono-I. ’19)[HI]

This Ohsawa-Takegoshi type positivity is weaker than Nakano semi-positivity in smooth settings. To be precise, we define the Ohsawa-Takegoshi type positivity in global settings, and show the existence of (E, h) such that (E, h) is positively curved in the Ohsawa-Takegoshi sense, whereas (E, h) is not Nakano semi-positive. We have the following inclusion relations: { Nakano semi-positivity } ⊊ { Ohsawa-Takegoshi type positivity } ⊆ { Griffiths semi-positivity }

[HI] G. Hosono, T. Inayama, A converse of H¨

• rmander’s L2-estimate and new positivity notions

for vector bundles, arXiv:1901.02223.

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Thanks

Thank you for your kind attention!

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