Stein morphisms and Riemann domains over Stein spaces (Q5966462)

The authors results are closely related to a conjecture of \textit{J. P. Serre} [``Quelques problèmes globaux relatifs aux variétés de Stein'', Colloque sur les fonctions de plusieurs variables, Bruxelles, Liège, Paris, 57-68 (1953; Zbl 0053.053)] that a holomorphic bundle with a Stein base and a Stein fiber is Stein. A counterexample for Serre's problem was given in 1977 by \textit{H. Skoda} [Invent. Math. 43, 97-107 (1977; Zbl 0365.32018)]. \textit{J. P. Demailly} [Lect. Notes Math. 694, 15- 41 (1978; Zbl 0418.32011)] showed that the first cohomology group of the bundle space of the structure sheaf of this counterexample is not Hausdorff in the canonical topology. Based on this result the author's two main theorems are: Theorem 1. Let X be a complex space having a Stein morphism. Then X is Stein iff \(H^ 1(X,{\mathcal O}_ X)\) is Hausdorffsch. Theorem 2. Let X be a Riemann domain over a Stein space. Then X is Stein iff \(H^ q(X,{\mathcal O}_ X)=0\) for \(q=1,2,...,\dim X-1.\)