On the Levi problem with singularities

Is a complex space X which is the union of an increasing sequence X1 ⊂ X2 ⊂ X3 ⊂ · · · of open Stein subspaces itself a Stein space ? From the begining this question has held great interest in Stein theory. The special case when {Xj}j≥1 is a sequence of Stein domains in I C n had been proved long time ago by Behnke and Stein [2]. In 1956, Stein [13] answered positively the question under the additional hypothesis that X is reduced and every pair (Xν+1, Xν) is Runge. In the general case X is not necessarily holomorphically-convex. Fornaess [7], gave a 3-dimensional example of such situation. In 1977, Markoe [10] proved the following: Let X be a reduced complex space which the union of an increasing sequence X1 ⊂ X2 ⊂ · · · ⊂ Xn ⊂ · · · of Stein domains. Then X is Stein if and only if H(X,OX) = 0. M. Coltoiu has shown in [3] that if D1 ⊂ D2 ⊂ · · · ⊂ Dn ⊂ · · · is an increasing sequence of Stein domains in a normal Stein space X, then D = ⋃