Locally Stein domains over holomorphically convex manifolds

Abstract

Let π : Y −→ X be a domain over a complex space X . Assume that π is locally Stein. Then we show that Y is Stein provided that X is Stein and either there is an open set W containing X sing with π − 1 ( W ) Stein or π is locally hyperconvex over any point in X sing . In the same vein we show that, if X is q -complete and X has isolated singularities, then Y results q -complete.