Steinness of the universal covering of the complement of a 2-dimensional complex singularity
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.
scientific article; zbMATH DE number 1945195