Curvature of Innnite Branched Covers; Application to Moduli of Cubic Surfaces

We study branched coverings in several contexts, proving that under suitable circumstances the cover satisses the same upper curvature bounds as the base space. The rst context is of branched covers of an arbitrary CAT() space over an arbitrary complete convex subset. The second context is of a certain sort of branched cover of a Riemannian manifold over a family of mutually orthogonal submanifolds. We impose no conditions that the branching be locally nite. We apply our results to hyperplane complements in complex hyperbolic space and to the moduli space of smooth cubic surfaces in C P 3 .