November 2013 Completions, branched covers, Artin groups, and singularity theory
Daniel Allcock
Duke Math. J. 162(14): 2645-2689 (November 2013). DOI: 10.1215/00127094-2380977


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.


Download Citation

Daniel Allcock. "Completions, branched covers, Artin groups, and singularity theory." Duke Math. J. 162 (14) 2645 - 2689, November 2013.


Received: 2 August 2012; Revised: 1 March 2013; Published: November 2013
First available in Project Euclid: 6 November 2013

zbMATH: 1294.53036
MathSciNet: MR3127810
Digital Object Identifier: 10.1215/00127094-2380977

Primary: 51K10
Secondary: 14B07 , 20F36 , 53C23 , 57N65

Rights: Copyright © 2013 Duke University Press


This article is only available to subscribers.
It is not available for individual sale.

Vol.162 • No. 14 • November 2013
Back to Top