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

派生范畴、伴随理想和可拓定理中的一般消失性和正则性

• 批准号: 0758253
• 负责人: Mihnea Popa
• 金额: $ 14.27万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0758253&HistoricalAwards=false
• 关键词: General; Vanishing; Regularity; Derived; Categories

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.

PI 将研究代数几何中的各种主题，包括双有理几何中的可拓定理、线性级数、派生范畴中的正则条件以及阿贝尔簇中的最小上同调类。特别是，他建议进一步开发源于 Siu、Takayama 和 Hacon-McKernan 工作的线束部分的扩展结果。他还建议通过同调和交换代数，将光滑射影簇的派生范畴上的一般消失型过滤与 Kashiwara 构建的反常 T 结构联系起来。他将通过傅立叶-向井理论方法来处理阿贝尔簇中最小类的子类的分类。提出的一些研究问题，例如阿贝尔簇上最小类的研究 变体和线束截面的扩展问题，在各自的方向上都是最突出的。 它们还涉及邻近领域的专家，并可能导致与 PI 专业领域之外的人进行互动。作为已证明的陈述，它们将引起相当大的兴趣，例如最近将可拓定理应用于最小模型程序所记录的那样。其他的，比如派生类别的结果，形成了一种仍有待充分理解的新兴理论，但它已经在现代和古典问题上得到了广泛的应用。