 
Summary: Asphericity of moduli spaces via curvature
Daniel Allcock*
26 March 2001
Abstract.
We show that under suitable conditions a branched cover satisfies the same upper curvature bounds
as its base space. First we do this when the base space is a metric space satisfying Alexandrov's
curvature condition CAT(Ÿ) and the branch locus is complete and convex. Then we treat branched
covers of a Riemannian manifold over suitable mutually orthogonal submanifolds. In neither setting
do we require that the branching be locally finite. We apply our results to hyperplane complements
in several Hermitian symmetric spaces of nonpositive sectional curvature in order to prove that
two moduli spaces arising in algebraic geometry are aspherical. These are the moduli spaces of the
smooth cubic surfaces in C P 3 and of the smooth complex Enriques surfaces.
x1. Introduction
It is wellknown that taking branched covers usually introduces negative curvature. One can see
this phenomenon in elementary examples using Riemann surfaces, and the idea also plays a role in
the construction [8] of exotic manifolds with negative sectional curvature. In this paper we work
in the setting of Alexandrov's comparison geometry; for background see [3]. In this setting we will
establish the persistence of upper curvature bounds in branched covers. A simple way to build a
cover b
Y of a space b
