A Stein Domain with Smooth Boundary Which Has a Product Structure

Abstract

It is well known that the unit ball in is not biholomorphically equivalent to the bidisc. Its first proof is due to H. Cartan [1], although it is customally called Poincare's theorem. H. Rischel [3] extended this theorem by proving that there exists no surjective proper holomorphic map from a strictly pseudoconvex domain B to a product domain E. Recently, A. Huckleberry and E. Ormsby [2] generalized it to the case where B is a bounded domain in C with smooth boundary and E is the total space of a holomorphic fiber bundle. It will be natural to ask whether we can generalize it to the case where B is a relatively compact Stein domain with smooth boundary in a complex manifold and E is the total space of a fiber bundle. The purpose of the present note is to show an example of a Stein domain B in a compact complex manifold which has the following properties.