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

Cheeger和Simons关于真实的光滑流形的微分上同调群联合收割机了真实的微分形式的几何信息和整数上同调的整体拓扑信息。霍普金斯和辛格开发了一个新的角度对这一建设，使这一想法推广到任何上同调理论，给出了一个频谱，并获得广义微分上同调理论。对于复代数簇，Deligne-Beilinson上同调结合了全纯微分形式的几何信息和整数上同调的整体拓扑信息.基本思想的联合项目与迈克尔霍普金斯是发展以类似的方式广义分析Deligne-Beilinson上同调理论的复杂代数品种。特别是，我们想构建一个广义解析cobordism理论的光滑复杂的品种。该项目的最终目标包括在建设一个新的循环映射的复杂品种从整数周群的商我们的广义cobordism理论在类似的想法Totaro因素的经典循环映射的上同调通过商的复杂cobordism。此外，我们还希望得到一个广义Abel-Jacobi映射，这将使我们对步行者定义的经典Abel-Jacobi映射和形态Abel-Jacobi映射有新的认识，这与我最初的项目--利用有限域上簇的l-adic积分循环映射研究有限域上簇的l-adic积分循环映射密切相关.

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.