Reflexive symmetric differentials and quotients of bounded symmetric domains

Abstract

For each classical irreducible bounded symmetric domain $\mathcal{D}$, Klingler has computed the minimum number $m_{\mathcal{D}}$ such that any smooth projective quotient $X=\mathcal{D}/\Gamma$, for $\Gamma\in\textrm{Aut}^0(\mathcal{D})$, satisfies $H^0(X,\mathrm{Sym}^i\Omega^1_X)=0$ for $0<i<m_{\mathcal{D}}$. In this article, we extend Klingler's result to the case when $X$ is normal and projective. This, together with a normal version of Arapura's result about the relationship between the vanishing of global symmetric differentials on $X$ and the rigidity of finite dimensional representations of $\pi_1(X)$, gives rigidity statements for representations of $\pi_1(X)$ and $\pi_1(X_{reg})$ in a low dimensional range, when $X$ is a normal projective quotient of a bounded symmetric domain.