Injectivity theorems with multiplier ideal sheaves for higher direct images under K"ahler morphisms

Versions of injectivity and extension theorems

We give an analytic version of the injectivity theorem by using multiplier ideal sheaves, and prove some extension theorems for the adjoint bundle of dlt pairs. Moreover, by combining techniques of the minimal model program, we obtain some results for semi-ampleness related to the abundance conjecture in birational geometry and the Strominger-Yau-Zaslow conjecture for hyperKähler manifolds.

Global division of cohomology classes via injectivity

We note that the vanishing and injectivity theorems of Kollár and Esnault-Viehweg can be used to give a quick algebraic proof of a strengthening of the Ein-Lazarsfeld Skoda-type division theorem for global sections of adjoint line bundles vanishing along suitable multiplier ideal sheaves, and to extend it to higher cohomology classes as well. For global sections, this is also a slightly more general statement of the algebraic translation of an analytic result of Siu. Along the way we write down an injectivity statement for multiplier ideals, and its standard consequences.

Extension of cohomology classes and holomorphic sections defined on subvarieties

In this paper, we obtain two extension theorems for cohomology classes and holomorphic sections defined on analytic subvarieties, which are defined as the supports of the quotient sheaves of multiplier ideal sheaves of quasi-plurisubharmonic functions with arbitrary singularities. The first result gives a positive answer to a question posed by Cao-Demailly-Matsumura, and unifies a few well-known injectivity theorems. The second result generalizes and optimizes a general $L^2$ extension theorem obtained by Demailly.

A vanishing theorem of Koll'ar-Ohsawa type

For proper surjective holomorphic maps from K"ahler manifolds to analytic spaces, we give a decomposition theorem for the cohomology groups of the canonical bundle twisted by Nakano semi-positive vector bundles by means of the higher direct image sheaves, by using the theory of harmonic integrals developed by Takegoshi. As an application, we prove a vanishing theorem of Koll'ar-Ohsawa type by combining the L^2-method for the dbar-equation.

被乘子子模层扭曲的 Le Potier 型同构

A Le Potier-type isomorphism twisted with multiplier submodule sheaves

本文建立乘子子模层扭曲的 Le Potier 型同构，连接奇异 Hermite 向量丛与重言线丛，并据此推出 Kollár 型内射、Nadel 型消灭及奇异全纯 Morse 不等式。

伴随理想层的一个扩张定理

An extension theorem in terms of adjoint ideal sheaves

本文在紧Kähler流形上，借鉴Chan-Choi-Matsumura的单射定理研究方法，通过调和理论与残数计算，证明了以伴随理想层表述的定性扩张定理，可保证线丛值全纯顶形式的存在性。

snc空间上真Kähler态射下高阶直像的内射性定理

Injectivity theorems for higher direct images under proper Kähler morphisms on snc spaces

该研究针对snc空间上的真Kähler态射，将Fujino内射性定理推广到相对情形，利用调和积分等技术解决非紧性问题并得到应用。

A Nadel vanishing theorem for metrics with minimal singularities on big line bundles

The purpose of this paper is to establish a Nadel vanishing theorem for big line bundles with multiplier ideal sheaves of singular metrics admitting an analytic Zariski decomposition (such as, metrics with minimal singularities and Siu's metrics). For this purpose, we apply the theory of harmonic integrals and generalize Enoki's proof of Koll'ar's injectivity theorem. Moreover we investigate the asymptotic behavior of harmonic forms with respect to a family of regularized metrics.

通过带解析伴随理想层的扩张定理对藤田自由性猜想的解决

A solution to Fujita's freeness conjecture via an extension theorem with analytic adjoint ideal sheaves

本文利用伴随理想层对乘子理想层不可积轨迹的精细控制，并结合基于内射性定理技术的扩张定理，直接处理 lc 奇点及其极小 lc 中心，从而在 Angehrn-Siu 思路下解决了藤田有效自由性猜想。