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扩展定理，线性级数，导出类别中的正则性条件以及交换簇中的最小上同调类。特别是，他建议进一步发展延伸成果的部分线束起源于工作的萧，高山和Hacon-McKernan。他还建议通过同调和交换代数将光滑投射簇的导出范畴上的一般消失型过滤与Kashiwara构造的反常T-结构联系起来。他将通过Fourier-Mukai-theory方法探讨交换簇中最小类的子簇的分类。提出的一些研究问题，如阿贝尔簇上的极小类的研究和线丛截面的扩张问题，在它们各自的方向上都是最杰出的。 它们还涉及邻近领域的专家，并可能导致与PI专业领域以外的人进行互动。他们将是相当大的兴趣，证明声明，作为记录，例如最近的应用程序的扩展定理的最小模型程序。其他的结果，如关于派生范畴的结果，形成了一个新出现的理论，仍然有待于充分理解，但它已经导致了对现代和古典问题的大量应用。