Asphericity of moduli spaces via curvature.
type: Item
instance of: scholarly article
title: Asphericity of moduli spaces via curvature.
zbMATH Open document ID: 1067.53028
author: Daniel Allcock
published in: Journal of Differential Geometry
publication date: 2002-08-15T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.jdg/1090341260
review text: A simply way to build a cover \(\widehat Y/\) of a space \(\widehat X\) branched over \(\Delta\subseteq\widehat X\) is to take any covering space \(Y\) of \(\widehat X\setminus\Delta\) and define \(\widehat Y= Y\cup\Delta\). The cover \(\widehat Y\) is called a simple branched cover of \(\widehat X\) over \(\Delta\). The author proves that if \(\widehat X\) satisfies Alexandrov's curvature condition \(\text{CAT}(k)\) and \(\Delta\) is complete and satisfies a convexity condition, then the natural metric on \(\widehat Y\) also satisfies \(\text{CAT}(k)\). In other words, the branched cover satisfies the same upper curvature bounds as its base space. Then the author studies branched covers of a Riemannian manifold over suitable mutually orthogonal submanifolds. The results are applied to hyperplane complements in several Hermitian symmetric spaces of nonpositive sectional curvature. This permits to prove that the module spaces of the smooth cubic surfaces in \(\mathbb{C} P^3\) and of the smooth complex Enriques surfaces are aspherical (i.e., has contractible universal cover).
reviewed by: Alberto Cavicchioli
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 57N65
zbMATH DE Number: 1782657
zbMATH Keywords: branched cover
zbMATH Keywords: Alexandrov space
zbMATH Keywords: cubic surface
zbMATH Keywords: Enriques surface
Wikidata QID: Q115203119
MaRDI profile type: Publication
OpenAlex ID: W1482387846
DOI: 10.4310/JDG/1090341260
description: scientific article; zbMATH DE number 1782657