Steinness of the universal covering of the complement of a 2-dimensional complex singularity
type: Item
instance of: scholarly article
title: Steinness of the universal covering of the complement of a 2-dimensional complex singularity
zbMATH Open document ID: 1039.32038
author: Mihnea Colţoiu
author: Mihai-Marius Tibar
published in: Mathematische Annalen
publication date: 2003-07-02T00:00:00Z
review text: Consider a 2-dimensional singularity \((X,x)\) in \(\mathbb{C}^n\). If we intersect \(X\) with a small ball around the singularity and remove the singular point itself, we get a space \(V\). The paper studies the universal cover of \(V\), namely it is shown that the universal cover is a Stein manifold if and only if the fundamental group \(G\) of \(V\) is infinite. The \(G\) finite case follows from results of Mumford and Prill. The infinite case is easier if the first homology group is infinite also. In this case the authors study the exceptional divisor at a resolution of the singularity. If its graph is a tree the result follows from a work of Napier; if there is a cycle the authors use it to construct an infinite covering with suitable properties. The other case when the homology is finite but the fundamental group is infinite is reduced to the first case using results of Luecke and Heil.
reviewed by: Richárd Rimányi
Mathematics Subject Classification ID: 32S45
Mathematics Subject Classification ID: 32Q28
Mathematics Subject Classification ID: 14E20
Mathematics Subject Classification ID: 32E10
zbMATH DE Number: 1945195
zbMATH Keywords: complement of singularity
zbMATH Keywords: universal covering
zbMATH Keywords: Stein manifold
MaRDI profile type: Publication
full work available at URL: s00208-003-0412-2
OpenAlex ID: W2242131142
DOI: 10.1007/S00208-003-0412-2
description: scientific article; zbMATH DE number 1945195