Some mixed Hodge structure on l^2-cohomology of covering of K\"ahler manifolds

TL;DR

This paper develops methods to compute the l^2-cohomology of covering manifolds derived from K"ahler manifolds with divisors, establishing a natural mixed Hodge structure on these groups in certain categories.

Contribution

It introduces a novel approach to compute l^2-cohomology for covering manifolds and demonstrates the existence of a natural mixed Hodge structure on these groups.

Findings

l^2-cohomology groups admit a mixed Hodge structure

Graded pieces are given by Gysin maps

Methods applicable to coverings of K"ahler manifolds

Abstract

We give methods to compute l^2-cohomology groups of a covering manifolds obtained by removing pullback of a (normal crossing) divisor to a covering of a compact K\"ahler manifold. We prove that in suitable quotient categories, these groups admit natural mixed Hodge structure whose graded pieces are given by expected Gysin maps.