On algebraic Chern classes of flat vector bundles

TL;DR

This paper investigates the torsion properties of higher Chern classes of flat vector bundles under certain monodromy conditions, extending to deformations, Higgs bundles, and positive characteristic cases, with implications for Bloch's conjecture.

Contribution

It establishes new torsion results for algebraic Chern classes of flat bundles under specific assumptions, including deformations and related Higgs bundle cases.

Findings

Higher Chern classes are torsion under certain monodromy conditions.

Results extend to flat bundles deforming to such bundles on quasi-projective varieties.

Provides insights into Bloch's conjecture and positive characteristic analogues.

Abstract

We show that under some assumptions on the monodromy group some combinations of higher Chern classes of flat vector bundles are torsion in the Chow group. Similar results hold for flat vector bundles that deform to such flat vector bundles (also in case of quasi-projective varieties). The results are motivated by Bloch's conjecture on Chern classes of flat vector bundles on smooth complex projective varities but in some cases they give a more precise information. We also study Higgs version of Bloch's conjecture and analogous problems in the positive characteristic case.