Cohomological and singularity invariants via Hodge modules and derived equivalences

这个数学研究项目涉及代数几何和相关主题。该项目研究的一些问题涉及使用称为Hodge模块的代数结构扩展代数几何工具。另一些则涉及在推导的等价下检查各种拓扑量的不变性。该项目的所有部分将具有广泛的应用范围，进一步了解该领域，启动不同数学背景的人之间的互动，并产生适合博士生研究参与的问题。本项目研究射影流形的上同调不变量、数值不变量和奇异不变量，方法是观察射影流形的相干束的派生范畴，并应用泛型消失和混合Hodge模理论的技术。PI想要开发一个消失、注入和扩展包，将其添加到Saito的Hodge模块的Kodaira型消失定理中，并利用它来解决一般类型或Kodaira零维的变种族的变化问题。他还对研究由局部d模的Hodge滤波得到的乘子理想的推广很感兴趣，并用它来攻击关于除数奇点的一个猜想。另一个目的是通过建立齐藤理论与奇异变体的过滤de Rham复形之间的联系来研究极小模型规划中的奇异性。在派生范畴的方向上，主要研究了相干束的上同调不变量的比较和具有等价有界派生范畴的变异几何，这是镜像对称和两族几何中都感兴趣的一个课题。继续研究派生等价下皮卡德变换和某些霍奇数的行为，PI计划解决诸如正则上同调的不变性和来自泛型消失理论的上同调支持座的不变性等问题。他也对奇异情况下的类似问题感兴趣，扩展了以前在商奇点变异情况下的一些工作。

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.