Application of homotopy theoretic methods in algebraic geometry by means of generalized Deligne-Beilinson cohomology and cobordism

实光滑流形的Cheeger和Simons的微分上同调群将实微分形式的几何信息与整数上同调的全局拓扑信息相结合。Hopkins和Singer对这一构造提出了一种新的观点，可以将这一思想推广到任何由谱给出的上同理论，并得到广义微分上同理论。这种构造的例子是微分k理论和微分实矩阵。对于复数代数变种，delign - beilinson上同调将全纯微分形式的几何信息与整数上同调的全局拓扑信息相结合。与Michael Hopkins合作项目的基本思想是以类似的方式发展复代数变种的广义解析delig - beilinson上同调理论。特别地，我们想构造光滑复变的广义解析协共理论。本课题的最终目标是建立一个从整数Chow群到广义协整理论商的复变异的新的循环映射，类比于Totaro的思想，通过复协整的商将经典的循环映射分解为上同调。此外，我们希望得到一个广义的Abel-Jacobi映射，这将使我们对Walker定义的经典和形态Abel-Jacobi映射有新的认识。这与我最初研究有限域上变量的l进积分循环映射的项目密切相关。

Differential cohomology groups of Cheeger and Simons for real smooth manifolds combine the geometric information of real differential forms with the global topological information of integer cohomology. Hopkins and Singer developed a new perspective on this construction that allows to generalize this idea to any cohomology theory, given by a spectrum, and to obtain generalized differential cohomology theories. Examples for this construction are differential K-theory and differential real bordism.For complex algebraic varieties, Deligne-Beilinson cohomology combines the geometric information of holomorphic differential forms with the global topological information of integer cohomology. The basic idea for the joint project with Michael Hopkins is to develop in a similar way generalized analytic Deligne-Beilinson cohomology theories for complex algebraic varieties. In particular, we would like to construct a generalized analytic cobordism theory for smooth complex varieties.The final goal of the project consists in the construction of a new cycle map for complex varieties from integer Chow groups to a quotient of our generalized cobordism theory in analogy to the idea of Totaro to factor the classical cycle map to cohomology through a quotient of complex cobordism. Moreover, we hope to obtain a generalized Abel-Jacobi map that should allow us to get new insights in the classical and the morphic Abel-Jacobi map defined by Walker.This is closely related to my initial project to study the l-adic integral cycle map for varieties over finite fields by means of the cycle map in etale cobordism.