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

TL;DR

This paper proves residual finiteness for certain central extensions of lattices in complex hyperbolic space and applies these results to construct negatively curved projective varieties that are not homotopy equivalent to locally symmetric manifolds, especially in dimensions four and higher.

Contribution

It establishes residual finiteness for cyclic central extensions of cocompact arithmetic lattices in PU(n,1) and applies this to construct new negatively curved projective varieties in higher dimensions.

Findings

Residual finiteness of preimages in connected covers of PU(n,1)

Residual finiteness for extensions with characteristic classes in span of Poincaré duals

Existence of negatively curved projective varieties not homotopy equivalent to locally symmetric manifolds

Abstract

We study residual finiteness for cyclic central extensions of cocompact arithmetic lattices $\Gamma < \mathrm{PU}(n,1)$ simple type. We prove that the preimage of $\Gamma$ 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 $H^2(\Gamma, \mathbb{Z})$ of the Poincar\'e duals to totally geodesic divisors on the ball quotient $\Gamma \backslash \mathbb{B}^n$. For $n \ge 4$, 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 $n \ge 4$.