Geometric Hodge filtered complex cobordism

TL;DR

This paper develops a geometric cycle model for Hodge filtered complex cobordism, providing an explicit isomorphism with the abstract model for complex manifolds, advancing the understanding of cobordism theories with Hodge filtration.

Contribution

It introduces a geometric cycle model for Hodge filtered complex cobordism and establishes an explicit isomorphism with existing abstract models for complex manifolds.

Findings

Constructed a geometric cycle model for Hodge filtered complex cobordism.

Established an explicit isomorphism with Hopkins-Quick's abstract model.

Applied a refined Pontryagin-Thom construction to achieve the results.

Abstract

We construct a geometric cycle model for a Hodge filtered extension of complex cobordism for every smooth manifold with a filtration on its de Rham complex with complex coefficients. Using a refinement of the Pontryagin-Thom construction, we construct an explicit isomorphism between our geometric model and the abstract model of Hodge filtered complex cobordism of Hopkins-Quick for every complex manifold with the Hodge filtration.