Asphericity of moduli spaces via curvature

We show that under suitable conditions a branched cover satisses 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() 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 nite. 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. It is well-known 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 X branched over b X is to take any covering space Y of b X ? and deene b Y = Y. We call b Y a simple branched cover of b X over. Our rst result, theorem 2.1, states that if b X satisses Alexandrov's CAT() condition and is complete and satisses a convexity condition then the natural metric on b Y also satisses CAT(). The question which motivated this investigation is whether the moduli space of smooth cubic surfaces in C P 3 is aspherical (i.e., has contractible universal cover). It is, and our argument also establishes the analogous result for the moduli space of smooth complex Enriques surfaces. To prove these claims, we use the fact that each of these moduli spaces is known to be covered by a Hermitian symmetric space with nonpositive sectional curvature, minus an arrangement of complex hyperplanes. In each case the hyperplanes have the property that any two of them are orthogonal wherever they meet. In section 3 we show that such a hyperplane complement is aspherical. The result (theorem 3.1) is more general because the symmetric space structure is not needed. Theorem 5.3 of 4] morally …