On the disk theorem

Abstract

We construct an example of a 2-dimensional Stein normal space X with one singular point x 0 such that X \{x 0 } is simply connected and it satisfies the disk condition. This answers a question raised by Fornaess and Narasimhan. We also prove that any increasing union of Stein open sets contained in a Stein space of dimension 2 satisfies the disk condition. Starting from the above example we exhibit, without using deformation theory, a new type of 2-dimensional holes which cannot be filled.

The disk property. A short survey

“…This is not true anymore on Stein complex spaces (hence if one allows singularities). For examples in this sense, see [5] and [11].…”