General Vanishing and Regularity in Derived Categories, Adjoint Ideals and Extension Theorems

PI将学习代数几何中的各种主题，包括二次几何中的扩张定理、线性级数、派生范畴中的正则性条件以及交换簇中的极小上同调类。特别是，他建议进一步发展源自Siu、Takayama和Hacon-McKernan工作的线丛的部分的扩展结果。他还利用同调代数和交换代数将光滑射影簇派生范畴上的一般消失型滤子与Kashiwara构造的倒置T-结构联系起来。他将利用傅里叶-Mukai理论方法来研究交换簇极小类的子簇的分类。所提出的一些研究问题，如关于交换簇的极小类的研究和线丛截面的扩张问题，在各自的方向上都是最有特色的。他们还关注邻近领域的专家，并可能导致与PI专业领域以外的人进行互动。作为已证明的语句，它们将具有相当大的兴趣，例如通过最近将延拓定理应用于最小模型程序来记录。其他的，比如关于派生范畴的结果，形成了一种新的理论，尽管还没有完全理解，但它已经导致了对现代和古典问题的大量应用。

The PI will study various topics in Algebraic Geometry, including extension theorems in birational geometry, linear series, regularity conditions in derived categories, and minimal cohomology classes in abelian varieties. In particular, he proposes to further develop extension results for sections of line bundles originating in work of Siu, Takayama and Hacon-McKernan. He also proposes to relate a generic vanishing type filtration on the derived category of a smooth projective variety to the perverse T-structure constructed by Kashiwara, by means of homological and commutative algebra.He will approach the classification of subvarieties of minimal class in abelian varieties via Fourier-Mukai-theoretic methods.Some of the proposed research problems, like the study of minimal classes on abelian varieties and the extension problem for sections of line bundles, are among the most distinguished in their respective directions. They also concern experts in adjacent fields, and will likely lead to interactions with people outside of the PI's area of expertise. They would be of considerable interest as proved statements, as documented for example by recent applications of extension theorems to the minimal model program. Others, like the results on derived categories, form a newly emerging theory still to be fully understood, but which has already led to numerous applications to both modern and classical questions.