Mixed Hodge complexes and L^2-cohomology for local systems on ball quotients

TL;DR

This paper investigates the $L^2$-cohomology of local systems on non-compact arithmetic ball quotients, providing vanishing and non-vanishing results, and explores the associated mixed Hodge structures in higher dimensions.

Contribution

It introduces new results on the $L^2$-cohomology of local systems on ball quotients and extends the understanding of mixed Hodge structures in this context.

Findings

Established vanishing and non-vanishing theorems for $L^2$-cohomology.

Generalized results to higher-dimensional ball quotients.

Analyzed the mixed Hodge structure on sheaf cohomology.

Abstract

We study the $L^2$--cohomology of certain local systems on non-compact arithmetic ball quotients $X=\Gamma \backslash \B_n$, in particular vanishing and non--vanishing results. We also give generalizations to higher dimensional ball quotients and study the mixed Hodge structure on the sheaf cohomology of a local system with the $L^2$-cohomology contributing to the lowest weight part.