$L^2$-cohomology of a variation of Hodge structure for an infinite covering of an open curve ramified at infinity

TL;DR

This paper investigates the $L^2$ cohomology of polarized complex variations of Hodge structures on infinite Galois coverings of open Riemann surfaces, demonstrating a pure Hodge structure after tensorization with affiliated operators.

Contribution

It extends the understanding of $L^2$ cohomology for branched coverings of Riemann surfaces with Poincaré-like metrics, establishing a pure Hodge structure in this context.

Findings

$L^2$ cohomology admits a pure Hodge structure after tensorization.

Results apply to branched coverings with asymptotic Poincaré metrics.

Advances the theory of Hodge structures on infinite coverings.

Abstract

Let $X$ be a compact Riemann surface, $\Sigma$ a finite set of points and $M = X\setminus \Sigma$. We study the $L^2$ cohomology of a polarized complex variation of Hodge structure on a Galois covering of the Riemann surface of finite type $M$. In this article we treat the case when the covering comes from a branched covering of $X$, and where $M$ is endowed with a metric asymptotic to a Poincar\'e metric. We prove that after tensorisation with the algebra of affiliated operators, the $L^2$ cohomology admits a pure Hodge structure.