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

Abstract

We study the $L^2$-cohomology of certain local systems on non-compact arithmetic ball quotients $X=\Gamma \setminus {\mathbb{B}}_n$. In the case of a ball quotient surface $X$ we show that vanishing theorems for $L^2$-cohomology are intimately related to vanishing theorems of the type

for $m\ge n\ge 1$ on the toroidal compactification $(\overline{X},D)$. We also give generalizations to higher dimensional ball quotients and study the mixed Hodge structure on the sheaf cohomology of a local system in general with the $L^2$-cohomology contributing to the lowest weight part.