Application of Cheeger-Gromov theory to the $l^2$-cohomology of harmonic Higgs bundles over covering of finite volume complete manifolds

TL;DR

This paper applies Cheeger-Gromov theory to study the $l^2$-cohomology of harmonic Higgs bundles over infinite coverings of finite volume complete manifolds, revealing spectral sequence degeneration.

Contribution

It extends Cheeger-Gromov theory to harmonic Higgs bundles on coverings of K"ahler manifolds, demonstrating spectral sequence degeneration at $E_2$.

Findings

$l^2$-cohomology of harmonic Higgs bundles is well-behaved on certain coverings.

The $l^2$-Dolbeault to DeRham spectral sequence degenerates at $E_2$.

Application to Zariski open sets of K"ahler manifolds.

Abstract

We review and apply Cheeger-Gromov theory on $l^2$-cohomology of infinite coverings of complete manifolds with bounded curvature and finite volume. Applications focus on $l^2$-cohomology of (pullback of) harmonic Higgs bundles on some covering of Zariski open sets of K\"ahler manifolds. The $l^2-$Dolbeault to DeRham spectral sequence of these Higgs bundles is seen to degenerate at $E_2$.