Summary: We study the $L^2$--cohomology of certain local systems on non-compact arithmetic ball quotients $X=\\Gamma \\B_n$. 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 $$ H^0(\overline{X}, S^n \Omega^1_{\overline{X}}(\log D) \otimes{\mathcal O}_{\overline{X}}(-D) \otimes (K_{\overline{X}}+D)^{-m/3})=0 $$ 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.

14G35, 14F17, 32M15, 32Q30

Shimura variety, uniformization, ball quotient, Higgs bundle, mixed Hodge theory, monodromy representation, abelian variety