Covering spaces of families of compact Riemann surfaces
type: Item
instance of: scholarly article
title: Covering spaces of families of compact Riemann surfaces
zbMATH Open document ID: 0765.32009
DOI: 10.1007/BF01934340
author: Terrence Napier
published in: Mathematische Annalen
publication date: 1993-01-12T00:00:00Z
full work available at URL: https://eudml.org/doc/165016
review text: 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.
Mathematics Subject Classification ID: 32E05
Mathematics Subject Classification ID: 32E10
Mathematics Subject Classification ID: 30F10
Mathematics Subject Classification ID: 32H35
zbMATH DE Number: 86201
zbMATH Keywords: covering spaces
zbMATH Keywords: holomorphically convex complex surfaces
zbMATH Keywords: Cartan-Remmert reduction
zbMATH Keywords: Riemann surface
zbMATH Keywords: proper holomorphic map
reviewed by: Alessandro Tancredi
MaRDI profile type: Publication
cites work: Levisches Problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten
cites work: Theta functions on Riemann surfaces
cites work: Q3931657
cites work: On Levi's problem and the imbedding of real-analytic manifolds
cites work: Convexity properties of coverings of smooth projective varieties
cites work: The Levi problem for complex spaces. II
cites work: Stetige streng pseudokonvexe Funktionen
cites work: Überlagerungen holomorph-vollständiger komplexer Räume
OpenAlex ID: W2088741463
description: scientific article; zbMATH DE number 86201