On the Projection of Stein Domains in Holomorphic Fiber Bundles

Abstract

If \(\pi :E\rightarrow X\) is a locally trivial holomorphic fiber bundle, with positive dimensional connected fiber and with the total space E being a Stein manifold, and \(D\subset X\) is an arbitrary connected open subset, then there exists an open, connected Stein subset \(\widetilde{D}\subset E\) such that \(\pi (\widetilde{D})=D\).