A report on "Regulators of canonical extension are torsion; the smooth divisor case"

TL;DR

This paper extends Reznikov's theorem to the setting of smooth quasi-projective varieties with divisors at infinity, proving that Chern-Simons classes of the canonical extension of flat bundles are torsion.

Contribution

It generalizes Reznikov's torsion result to the case of smooth divisors at infinity, defining and proving torsion properties of Chern-Simons classes for canonical extensions.

Findings

Chern-Simons classes of canonical extensions are torsion.

The result generalizes Reznikov's theorem to new geometric settings.

Provides a framework for understanding characteristic classes in quasi-projective varieties.

Abstract

In this note, we report on a work jointly done with C. Simpson on 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 vector 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 the Deligne's \textit{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. The details of the proof can be found in arxiv:0707.0372 [math.AG].