2 Dimensional Counterexamples to Generalizations of the Levi Problem.
Let be a complex space of dimension , not necessarily reduced, whose cohomology groups are of finite dimension (as complex vector spaces). We show that is Stein (resp., -convex) if, and only if, is holomorphically spreadable (resp., is holomorphically spreadable at infinity). This, on the one hand, generalizes a known characterization of Stein spaces due to Siu, Laufer, and Simha and, on the other hand, it provides a new criterion for -convexity.