## Duke Mathematical Journal

### Completions, branched covers, Artin groups, and singularity theory

Daniel Allcock

#### 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 $\operatorname {CAT}(\chi )$ inequality. We prove a general $\operatorname {CAT}(\chi )$ extension theorem, giving sufficient conditions on and near the boundary of a locally $\operatorname {CAT}(\chi )$ metric space for the completion to be $\operatorname {CAT}(\chi )$. We use this to prove that a branched cover of a complete Riemannian manifold is locally $\operatorname {CAT}(\chi )$ if and only if all tangent spaces are $\operatorname {CAT}(0)$ 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 $\operatorname {CAT}(\chi )$ 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 $\operatorname {CAT}(0)$. 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.

#### Article information

Source
Duke Math. J., Volume 162, Number 14 (2013), 2645-2689.

Dates
Revised: 1 March 2013
First available in Project Euclid: 6 November 2013

https://projecteuclid.org/euclid.dmj/1383760701

Digital Object Identifier
doi:10.1215/00127094-2380977

Mathematical Reviews number (MathSciNet)
MR3127810

Zentralblatt MATH identifier
1294.53036

#### Citation

Allcock, Daniel. Completions, branched covers, Artin groups, and singularity theory. Duke Math. J. 162 (2013), no. 14, 2645--2689. doi:10.1215/00127094-2380977. https://projecteuclid.org/euclid.dmj/1383760701

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