Bott-Chern homology, Bott-Chern differential cohomology and the Hodge conjecture

TL;DR

This paper introduces a new version of the Hodge conjecture within Bott-Chern cohomology, constructs refined Bott-Chern classes for holomorphic bundles, and provides a proof leveraging real holomorphic chains and differential cohomology techniques.

Contribution

It defines Bott-Chern differential cohomology, constructs refined classes for holomorphic bundles, and offers a proof of the conjecture using advanced geometric and cohomological methods.

Findings

Proposed a Bott-Chern cohomology version of the Hodge conjecture

Constructed refined Bott-Chern classes for holomorphic vector bundles

Provided a proof of the conjecture using real holomorphic chains

Abstract

We propose a version of the Hodge conjecture in Bott-Chern cohomology and using results from characterizing real holomorphic chains by real rectifiable currents to provide a proof for this question. We define a Bott-Chern differential cohomology and use atomic section theory of Harvey and Lawson to construct refined Bott-Chern classes for holomorphic vector bundles in this differential cohomology. These refined Bott-Chern classes transform naturally to standard Chern classes, Bott-Chern classes and Cheeger-Simons' refined Chern classes.