Gereon Quick

Hodge filtered complex bordism

We construct Hodge filtered cohomology groups for complex manifolds that combine the topological information of generalized cohomology theories with geometric data of Hodge filtered holomorphic forms. This theory provides a natural generalization of Deligne cohomology. For smooth complex algebraic varieties, we show that the theory satisfies a projective bundle formula and

Unstable splittings in Hodge filtered Brown–Peterson cohomology

\A^1

Profinite Homotopy Theory

-homotopy invariance. Moreover, we obtain transfer maps along projective morphisms.

Continuous group actions on profinite spaces

We construct Hodge filtered function spaces associated to infinite loop spaces. For Brown–Peterson cohomology, we show that the corresponding Hodge filtered spaces satisfy an analog of Wilson’s unstable splitting. As a consequence, we obtain an analog of Quillen’s theorem for Hodge filtered Brown–Peterson cohomology for complex manifolds.