Duke Mathematical Journal

Completions, branched covers, Artin groups, and singularity theory

Daniel Allcock

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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 ( χ ) inequality. We prove a general CAT ( χ ) extension theorem, giving sufficient conditions on and near the boundary of a locally CAT ( χ ) metric space for the completion to be CAT ( χ ) . We use this to prove that a branched cover of a complete Riemannian manifold is locally CAT ( χ ) if and only if all tangent spaces are CAT ( 0 ) and the base has sectional curvature bounded above by χ . 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 ( χ ) and the complement of the branch locus to be contractible.

We conjecture that the universal branched cover of C n over the mirrors of a finite Coxeter group is 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 ( π , 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
Received: 2 August 2012
Revised: 1 March 2013
First available in Project Euclid: 6 November 2013

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1383760701

Digital Object Identifier
doi:10.1215/00127094-2380977

Mathematical Reviews number (MathSciNet)
MR3127810

Zentralblatt MATH identifier
1294.53036

Subjects
Primary: 51K10: Synthetic differential geometry
Secondary: 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 57N65: Algebraic topology of manifolds 20F36: Braid groups; Artin groups 14B07: Deformations of singularities [See also 14D15, 32S30]

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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