Covering spaces of families of compact Riemann surfaces
Let \(\pi:\tilde X\to X\) be a covering space, where \(X\) is a holomorphically convex complex space. In general \(\tilde X\) is not holomorphically convex, but \(\tilde X\) is Stein if \(X\) is Stein. The author looks at covering spaces of holomorphically convex complex surfaces whose Cartan-Remmert reduction mappings are submersions of rank one. The main result is the following. Let \(X\) be a connected complex manifold of dimension two, \(\pi:\tilde X\to X\) a connected covering space and \(Y\) an open Riemann surface. Suppose that there exists a surjective proper holomorphic map \(\psi:X\to Y\) with connected fibers. Let \(C\) be a fiber of \(\psi,\tilde C\) a connected component of \(\pi^{-1}(C)\) and \(\Gamma\) the image of \(H_ 1(\tilde C,\mathbb{Z})\) in \(H_ 1(C,\mathbb{Z})\). If the rank of \(\Gamma\) is smaller than the genus of \(C\), then \(\tilde X\) is Stein.
scientific article; zbMATH DE number 86201