Hodge filtered complex bordism

TL;DR

This paper introduces Hodge filtered cohomology groups for complex manifolds, integrating topological and geometric data, generalizing Deligne cohomology, and establishing key properties like projective bundle formula and homotopy invariance.

Contribution

It constructs a new cohomology theory combining topological and Hodge geometric data, extending Deligne cohomology with proven invariance and transfer properties.

Findings

Defines Hodge filtered cohomology groups for complex manifolds.

Proves the theory satisfies a projective bundle formula.

Establishes $ ext{A}^1$-homotopy invariance and transfer maps.

Abstract

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 $\A^1$-homotopy invariance. Moreover, we obtain transfer maps along projective morphisms.