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  

In this article, we first establish an $L^2$-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, including results of Bogomolov–Sommese type.

DOI： 10.1007/s00209-025-03906-4

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その他リンク： https://link.springer.com/article/10.1007/s00209-025-03906-4

担当区分：責任著者  

Accepted for publication in Anal. PDE.
In this paper, we consider a proper Kähler fibration $f : X \to Y$ 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.

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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：Mathematical Institute, Tohoku University  

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.

DOI： 10.2748/tmj.20231128

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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：Cellule MathDoc/Centre Mersenne  

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.

DOI： 10.5802/afst.1815

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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

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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その他リンク： https://link.springer.com/article/10.1007/s12220-025-02068-3/fulltext.html

担当区分：筆頭著者,　最終著者,　責任著者  

In this paper, a Hard Lefschetz theorem for logarithmic sheaves twisted by pseudo-effective line bundles is established, extending a theorem of Demailly, Peternell, and Schneider.

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担当区分：筆頭著者,　最終著者,　責任著者  

In this paper, we generalize the Steenbrink vanishing theorem for ample line bundles on complex projective varieties by extending it to big line bundles on compact complex spaces with multiplier ideal sheaves.

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担当区分：筆頭著者,　最終著者,　責任著者  

In this paper, in order to develop a more general $L^2$-theory for the $\overline{\partial}$-operator on complex spaces, we introduce appropriate notions of singular Griffiths/Nakano positivity on complex spaces and establish various properties of these notions. By applying these results, we provide $L^2$-Dolbeault fine resolutions and cohomological isomorphisms, and $L^2$-existence theorems. As an application, we obtain Nakano-Nadel vanishing theorems on weakly pseudoconvex complex spaces.

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担当区分：筆頭著者,　最終著者,　責任著者  

In this paper, in order to develop a more general $\overline{\partial}$-theory for the -operator on complex spaces, we provide $L^2$-Dolbeault fine resolutions and isomorphisms, and $L^2$-estimates, for holomorphic line bundles on complex spaces equipped with singular Hermitian metrics. As applications, we obtain several generalizations of the Nadel vanishing theorem.

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担当区分：筆頭著者,　最終著者,　責任著者  

In this paper, by refining approximation theorems for holomorphic sections of adjoint line bundles, it is proved that the regular locus of a weakly pseudoconvex complex space admitting a positive line bundle can be embedded into a complex projective space. As an application of approximation theorems, it is shown that the Union problem can be solved for weakly pseudoconvex complex manifolds.

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開催年月日： 2026年5月 　 　

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