Regulators of canonical extensions are torsion: the smooth divisor case

TL;DR

This paper generalizes Reznikov's theorem to show that Chern-Simons and Deligne Chern classes are torsion for flat bundles on smooth quasi-projective varieties with smooth divisors at infinity.

Contribution

It extends the torsion property of Chern classes to the setting of smooth quasi-projective varieties with smooth divisors at infinity, using canonical extensions.

Findings

Chern-Simons classes are torsion in this setting.

Deligne Chern classes lift to torsion classes.

Generalization of Reznikov's theorem to new geometric context.

Abstract

In this paper, we prove a generalization of Reznikov's theorem which says that the Chern-Simons classes and in particular the Deligne Chern classes (in degrees $>1$) are torsion, of a flat bundle on a smooth complex projective variety. We consider the case of a smooth quasi--projective variety with an irreducible smooth divisor at infinity. We define the Chern-Simons classes of Deligne's canonical extension of a flat vector bundle with unipotent monodromy at infinity, which lift the Deligne Chern classes and prove that these classes are torsion.