  We study branched covering spaces in several contexts, proving that under
suitable circumstances the cover satisfies the same upper curvature bounds as
the base space. The first context is of a branched cover of an arbitrary metric
space that satisfies Alexandrov's curvature condition CAT(k), 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. In neither setting do we require that the branching be locally
finite. We apply our results to hyperplane complements in several complex
manifolds of nonpositive sectional curvature. This implies that two moduli
spaces arising in algebraic geometry are aspherical, namely that of the smooth
cubic surfaces in complex projective 3-space and that of the smooth complex
Enriques surfaces.
