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

Abstract

We study residual finiteness for cyclic central extensions of cocompact arithmetic lattices $\hat{I}“<\mathrm{PU}(n,1)$ simple type. We prove that the preimage of $\hat{I}“$ in any connected cover of $\mathrm{PU}(n,1)$, 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 $H^2(\hat{I}“,\mathbb{Z})$ of the Poincaré duals to totally geodesic divisors on the ball quotient $\hat{I}“\backslash {\mathbb{B}}^n$. For $n\ge 4$, if $\hat{I}“$ is a congruence lattice, we prove residual finiteness of the central extension associated with any element of $H^2(\hat{I}“,\mathbb{Z})$. 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 $n\ge 4$.