光滑拟射影复代数簇的 jump loci 与 L^2 类不变量

光滑拟射影复代数簇的拓扑性质是复几何与代数几何的核心课题之一。Jump loci 与 L^2 类不变量是研究拓扑空间同伦性质的有力工具。本项目拟研究光滑拟射影复代数簇的拓扑性质，特别是 jump loci 与 L^2 类不变量。具体来说，本项目将研究 L^2 类不变量与 jump loci 之间的关系，光滑拟射影复代数簇 L^2 类不变量与复系数、正特征域系数的 jump loci，超平面配置补空间的 L^2 类不变量与组合信息的关系。做为应用，我们拟对超平面配置 Milnor 纤维同调群 torsion 部分大小给出一个刻画，并以此解决 Denham-Suciu 提出的一个公开问题。

The topology of smooth complex quasi-projective variety is one of the central topics in complex geometry and algebraic geometry. Jump loci and L^2 type invariants are important and useful tools for studying homotopy type of topological spaces. This project is devoted to study the jump loci and L^2 type invarinats for smooth complex quasi-projective varieties. In particular, we will study the relation between L^2 type invarinats and jump loci, the jump loci with complex coefficients and positive characteristic field coefficients for smooth complex quasi-projective varieties and its application to L^2 type invarints, the relation between L^2 type invariants and combinatric data for hyperplane arrangements. As an application, we will study the homology torsion part of the Milnor fiber for central hyperplane arrangement and try to answer an open question proposed by Denham and Suciu.