Residual finiteness for central extensions of lattices in $\operatorname{PU}(n,1)$ and negatively curved projective varieties

We study residual finiteness for cyclic central extensions of cocompact arithmetic lattices $\Gamma \lt \operatorname{PU}(n,1)$ of simple type. We prove that the preimage of $\Gamma$ in any connected cover of $\operatorname{PU}(n,1)$, in particular the universal cover, is residually finite. This follows from a more general theorem on residual finiteness of extensions whose characteristic class is contained in the span in $H^2(\Gamma, \mathbb{Z})$ of the Poincaré duals to totally geodesic divisors on the ball quotient $\Gamma \setminus \mathbb{B}^n$. For $n \geq 4$, if $\Gamma$ is a congruence lattice, we prove residual finiteness of the central extension associated with any element of $H^2(\Gamma, \mathbb{Z})$.

Our main application is to existence of cyclic covers of ball quotients branched over totally geodesic divisors. This gives examples of smooth projective varieties admitting a metric of negative sectional curvature that are not homotopy equivalent to a locally symmetric manifold. The existence of such examples is new for all dimensions $n \geq 4$.

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M. Stover was partially supported by Grant Number DMS-1906088 from the National Science Foundation.

Received 26 August 2021

Received revised 12 September 2021

Accepted 18 September 2021

Published 25 October 2022