担当区分：筆頭著者,　責任著者  

Abstract

In this article, we get properties for singular (dual) Nakano semi-positivity and obtain vanishing theorems involving $L^2$-subsheaves on weakly pseudoconvex manifolds by $L^2$-estimates and $L^2$-type Dolbeault isomorphisms.
As applications, Fujita's conjecture type theorem with singular Hermitian metrics is presented.

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担当区分：筆頭著者,　責任著者  

Abstract

The characterization of Nakano semi-positivity by $L^2$-estimate is already known by Deng-Ning-Wang-Zhou's recent work. Stimulated by this work, we study the positivity properties of the curvature operator for holomorphic Hermitian vector bundles.
Applying our results, we give a new characterization of Nakano semi-negativity.

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担当区分：筆頭著者,　責任著者   掲載種別：研究論文（学術雑誌）   出版者・発行元：Springer Science and Business Media LLC  

Abstract

In this article, we obtain the Bogomolov–Sommese type vanishing theorem involving multiplier ideal sheaves for big line bundles. We define a dual Nakano semi-positivity of singular Hermitian metrics with$$L^2$$-estimates and prove a vanishing theorem which is a generalization of the Bogomolov–Sommese type vanishing theorem to holomorphic vector bundles.

DOI： 10.1007/s00209-023-03252-3

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その他リンク： https://link.springer.com/article/10.1007/s00209-023-03252-3/fulltext.html

担当区分：筆頭著者,　責任著者   掲載種別：研究論文（学術雑誌）   出版者・発行元：Faculty of Mathematics, Kyushu University  

Abstract

We study the cohomology groups of vector bundles on neighborhoods of non-pluriharmonic loci in $q$-complete Kähler manifolds and in compact Kähler manifolds. Applying our results, we show variants of the Lefschetz hyperplane theorem.

DOI： 10.2206/kyushujm.75.323

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担当区分：筆頭著者,　責任著者  

Abstract

In this paper, we investigate various positivity for singular Hermitian metrics such as Griffiths, $\omega$-trace and RC, where $\omega$ is a Hermitian metric, and show that these quasi-positivity notions induce $0$-th cohomology vanishing, rational conected-ness, etc. Here, $\omega$-trace positivity of smooth Hermitian metrics $h$ on holomorphic vector bundles $E$ represents the positivity of $tr_\omega i\Theta_{E,h}$.

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Abstract

We investigate the positivity properties of the direct image $f_\ast(K_{X/Y}\otimes L)$ of the adjoint line bundle associated with a big and nef line bundle $L$, under a smooth fibration $f:X\to Y$ between projective varieties. We show that the vector bundle $f_\ast(K_{X/Y}\otimes L)$ is big.

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担当区分：筆頭著者,　責任著者  

Abstract

Let $f:X\to Y$ be a surjective projective map and $L$ be a holomorphic line bundle on $X$ equipped with a (singular) semi-positive Hermitian metric $h$. In this article, by studying the canonical metric on the direct image sheaf of the twisted relative canonical bundles $K_{X/Y}\otimes L\otimes \mathscr{I}(h)$, we obtain that this metric has dual Nakano semi-positivity when $h$ is smooth and there is no deformation by $f$ and that this metric has locally Nakano semi-positivity in the singular sense when $h$ is singular.

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担当区分：筆頭著者,　責任著者  

Abstract

In this article, we first establish an $L^2$-type Dolbeault isomorphism for the sheaf of logarithmic differential forms twisted by the multiplier ideal sheaf. By using this isomorphism and $L^2$-estimates equipped with a singular Hermitian metric, we obtain logarithmic vanishing theorems involving multiplier ideal sheaves on compact Kähler manifolds with simple normal crossing divisors.

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