Completions, branched covers, Artin groups, and singularity theory

Abstract

We study the curvature of metric spaces and branched covers of Riemannian manifolds, with applications in topology and algebraic geometry. Here curvature bounds are expressed in terms of the $CAT\left(\chi \right)$ inequality. We prove a general $CAT\left(\chi \right)$ extension theorem, giving sufficient conditions on and near the boundary of a locally $CAT\left(\chi \right)$ metric space for the completion to be $CAT\left(\chi \right)$. We use this to prove that a branched cover of a complete Riemannian manifold is locally $CAT\left(\chi \right)$ if and only if all tangent spaces are $CAT\left(0\right)$ and the base has sectional curvature bounded above by $\chi$. We also show that the branched cover is a geodesic space. Using our curvature bound and a local asphericity assumption we give a sufficient condition for the branched cover to be globally $CAT\left(\chi \right)$ and the complement of the branch locus to be contractible.

We conjecture that the universal branched cover of ${\mathbb{C}}^n$ over the mirrors of a finite Coxeter group is $CAT\left(0\right)$. This is closely related to a conjecture of Charney and Davis, and we combine their work with our machinery to show that our conjecture implies the Arnol$'$d–Pham–Thom conjecture on $K(\pi ,1)$ spaces for Artin groups. Also conditionally on our conjecture, we prove the asphericity of moduli spaces of amply lattice-polarized K3 surfaces and of the discriminant complements of all the unimodal hypersurface singularities in Arnol$'$d’s hierarchy.