Stein domains in complex surfaces (Q1414834)

Let \(S\) be a closed connected real surface and \(\pi : S \rightarrow X\) a smooth embedding or immersion of \(S\) into a complex surface \(X\). The author finds conditions (related to the number of complex points of \(\pi ( S ) \) in \( X)\) under which \(\pi\) can be \(C^{0}\) approximated by an isotopic immersion \(\pi_{1} : S \rightarrow X\) whose image has a basis of open Stein neighbourhoods in \(X\) which are homotopy equivalent to \(\pi_{1}(S)\). Some very interesting examples and precise results for \(\mathbb{C} P^{2}\) are presented. Open problems related to the subject are also given.