Cohomological and singularity invariants via Hodge modules and derived equivalences

通过 Hodge 模和导出等价的上同调和奇点不变量

• 批准号: 1405516
• 负责人: Mihnea Popa
• 金额: $ 19.5万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-01 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1405516&HistoricalAwards=false
• 关键词: Cohomological; singularity; invariants; via; Hodge

This mathematical research project concerns algebraic geometry and related topics. Some of the problems under study in the project involve the extension of tools of algebraic geometry using algebraic structures called Hodge modules. Others involve checking invariance of various topological quantities under derived equivalences. All parts of the project will have a broad range of applications, further knowledge in the field, initiate interaction among people of different mathematical backgrounds, and produce problems suitable for research involvement of Ph.D. students. This project studies cohomological, numerical, and singularity invariants of projective manifolds by looking at their derived categories of coherent sheaves, and by applying techniques from generic vanishing and mixed Hodge module theory. The PI would like to develop a vanishing, injectivity and extension package to be added to Saito's Kodaira-type vanishing theorem for Hodge modules, and to use this in order to attack problems on the variation of families of varieties of varieties of general type or Kodaira dimension zero. He is also interested in studying generalizations of multiplier ideals coming from the Hodge filtration on localized D-modules, and to use this for attacking a conjecture on singularities of theta divisors. Another aim is to study singularities in the minimal model program by establishing a connection between Saito's theory and the filtered de Rham complex of singular varieties. In the direction of derived categories, the main topic under study is comparison of the cohomological invariants and the geometry of varieties with equivalent bounded derived categories of coherent sheaves, a topic of interest both in mirror symmetry and in birational geometry. Continuing work on the behavior of the Picard variety and of certain Hodge numbers under derived equivalence, the PI plans to address problems like the invariance of the canonical cohomology and the invariance of cohomological support loci coming from generic vanishing theory. He is also interested in studying similar questions in the singular setting, extending some previous work in the case of varieties with quotient singularities.

这个数学研究项目涉及代数几何和相关主题。 一些问题正在研究中的项目涉及扩展工具的代数几何使用代数结构称为霍奇模块。其他涉及检查不变性的各种拓扑量下导出的等价。该项目的所有部分将有广泛的应用，进一步了解该领域，启动不同数学背景的人之间的互动，并产生适合博士研究参与的问题。学生这个项目研究上同调，数值和奇异不变量的射影流形，通过寻找他们的派生类别的相干层，并通过应用技术从一般消失和混合霍奇模理论。PI希望开发一个消失，内射性和扩展包，以添加到齐藤的Kodaira型消失定理霍奇模块，并使用此，以攻击问题的变化的家庭品种的品种一般类型或科代拉零维。他还感兴趣的研究推广乘数理想来自霍奇过滤本地化的D-模块，并利用这一攻击猜想的奇异性θ因子。另一个目的是研究奇异性的最小模型程序之间建立联系齐藤的理论和过滤德拉姆复杂的奇异品种。在派生范畴的方向，正在研究的主要课题是比较的上同调不变量和几何的品种与等价的有界派生类别的相干层，一个感兴趣的话题都在镜像对称和双有理几何。继续研究Picard簇和某些Hodge数在导出等价下的行为，PI计划解决诸如典型上同调的不变性和来自一般消失理论的上同调支撑轨迹的不变性等问题。他也有兴趣在研究类似的问题在奇异的设置，延长一些以前的工作的情况下，品种与商奇点。