Reflexive symmetric differentials and quotients of bounded symmetric domains
Aryaman Patel
TL;DR
The work extends Klingler's vanishing bounds for global symmetric differentials from smooth projective quotients to normal projective quotients of irreducible bounded symmetric domains, yielding rigidity statements for low-rank representations of the fundamental group. It uses automorphic vector bundles and Mok's curvature-based vanishing, together with Selberg's lemma to pass to smooth quasi-etale covers, and it provides a normal analogue of Arapura–Klingler–Zuo type results relating vanishing to representation rigidity. The bounds depend on the domain type via $m_{\mathcal{D}}$, with explicit values for the classical types, and the authors carefully treat connected vs. disconnected automorphism groups, including polydisk cases. Consequently, for a normal projective quotient $X$ of an irreducible bounded symmetric domain, $\pi_1(X)$ and $\pi_1(X_{reg})$ have rigid representations in dimensions up to $m_{\mathcal{D}}-1$, and the results extend to non-Archimedean settings and to ball quotient scenarios with prime $n+1$ in several cases, linking geometric vanishing to topological rigidity.
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 $X=\mathcal{D}/Γ$, for $Γ\in\textrm{Aut}^0(\mathcal{D})$, satisfies $H^0(X,\mathrm{Sym}^iΩ^1_X)=0$ 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 $π_1(X)$, gives rigidity statements for representations of $π_1(X)$ and $π_1(X_{reg})$ in a low dimensional range, when $X$ is a normal projective quotient of a bounded symmetric domain.