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

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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.