Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

Abstract

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

DOI： 10.1007/s12220-025-02068-3

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Other Link： https://link.springer.com/article/10.1007/s12220-025-02068-3/fulltext.html

Authorship：Lead author,　Corresponding author   Publishing type：Research paper (scientific journal)   Publisher：Cambridge University Press (CUP)  

Abstract

Let $f:X\to Y$ be a surjective projective map, and let 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.

DOI： 10.1017/nmj.2024.20

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Authorship：Lead author,　Corresponding author  

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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Authorship：Lead author,　Corresponding author  

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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Authorship：Lead author,　Corresponding author   Publishing type：Research paper (scientific journal)   Publisher：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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Other Link： https://link.springer.com/article/10.1007/s00209-023-03252-3/fulltext.html

Authorship：Lead author,　Corresponding author  

Abstract.

Let $X$ be a weakly pseudoconvex manifold and $L\longrightarrow X$ be a holomorphic line bundle with a singular positive Hermitian metric $h$. In this article, we provide a points separation theorem and an embedding for the adjoint linear subsystem including the multiplier ideal sheaf $\mathscr{I}(h^m)$, with respect to an appropriate set excluding a singular locus of $h$. We also show that the adjoint bundle of $L$ is big. To handle analytical methods, we first establish an approximation of singular Hermitian metrics on relatively compact subsets from Demailly's approximation that preserves ideal sheaves and is compatible with blow-ups. Using the blow-ups obtained from this approximation, we provide the embedding and big-ness for the adjoint bundle $K_X\otimes L^{\otimes m}\otimes\mathscr{I}(h^m)$ on each sublevel set $X_c$, and a singular holomorphic Morse inequality including ideal sheaves. Furthermore, we establish the approximation theorem for holomorphic sections of the adjoint bundle including the multiplier ideal sheaf, i.e. $K_X\otimes L\otimes\mathscr{I}(h)$, as a key result in the process of globalizing. Using these results, we can achieve points separation on each $X_c\setminus Z_c$, where $Z_c$ is an analytic subset obtained as a singular locus of the approximation, and then globalize this to provide embeddings. Finally, as an application of this blow-ups, we present a global singular Nakano vanishing theorem.

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Authorship：Corresponding author  

Abstract

In this paper, we consider a proper K\"ahler fibration $f \colon X \to Y$
and a singular Hermitian line bundle $(L, h)$ on $X$ with semi-positive curvature.
We prove that the canonical $L^2$-metric on the direct image sheaf $f_{*}(\mathcal{O}_{X}(K_{X/Y}+L) \otimes \mathcal{I}(h))$ is singular Nakano semi-positive in the sense that the $\overline{\partial}$-equation can be solved with optimal $L^{2}$-estimate. Our proof does not rely on the theory of Griffiths positivity for the direct image sheaf.

DOI： 10.48550/arXiv.2407.11412

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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

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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