Nonpositive curvature of blow-ups
type: Item
instance of: scholarly article
title: Nonpositive curvature of blow-ups
zbMATH Open document ID: 0924.53033
published in: Selecta Mathematica. New Series
publication date: 1999-11-08T00:00:00Z
review text: Let \(M_{\mathbb{C}}\) be a complex manifold of dimension \(n\) and let \(D_{\mathbb{C}}\) be a union of smooth complex codimension-one submanifolds of \(M_{\mathbb{C}}\). A ``real version'' of \((M_{\mathbb{C}},D_{\mathbb{C}})\) is a pair \((M,D)\) such that \(M\) is the fixed point set of a smooth involution on \(M_{\mathbb{C}}\) which is locally isomorphic to complex conjugation and such that \(D=D_{\mathbb{C}}\cap M\). In many interesting cases, such a real version exists. In this paper, the authors are interested in the geometry and topology of \((M,D)\). More precisely, we have the following. The set \(D\) is locally isomorphic to an arrangement of hyperplanes, and it cuts \(M\) into regions which are combinatorially equivalent to convex polytopes. If this cellulation is simple, that is, if \(n\) edges meet at every vertex, then so is its dual cellulation. Moreover, the two cellulations have a common subdivision by cubes. The authors study this cubical complex, equipped with its natural piecewise Euclidean structure in which each cell is identified with a regular Euclidean cube whose edges are of length one. Gromov has shown that for a cubical complex, there is a simple combinatorial condition that is necessary and sufficient for the complex to be nonpositively curved in the sense of Alexandrov. The condition is that the link of each vertex is a ``flag complex''. A consequence of this nonpositive curvature (also due to Gromov) is that the universal cover of the space is contractible. One of the main points in this paper is to show that for many interesting examples, Gromov's condition holds.
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 52C35
Mathematics Subject Classification ID: 14M25
zbMATH DE Number: 1271759
zbMATH Keywords: aspherical
zbMATH Keywords: blow up
zbMATH Keywords: hyperplane arrangement
zbMATH Keywords: zonotope
author: Michael W. Davis
author: Tadeusz Januszkiewicz
author: Richard A. Scott
reviewed by: Athanase Papadopoulos
MaRDI profile type: Publication
full work available at URL: s000290050039
OpenAlex ID: W2156724902
DOI: 10.1007/S000290050039
description: scientific article; zbMATH DE number 1271759