Holomorphic affine bundles on the complement of an analytic set
type: Item
instance of: scholarly article
title: Holomorphic affine bundles on the complement of an analytic set
zbMATH Open document ID: 0944.32010
DOI: 10.1007/PL00004669
author: Makoto Abe
published in: Mathematische Zeitschrift
publication date: 2000-09-18T00:00:00Z
review text: The author works at the subject going back to Serre on which conditions one can prove the Stein-property of \(E\) where \(E\to X\) is a holomorphic fiber bundle. There are examples of holomorphic affine \(\mathbb{C}\)-bundles \(E\) over \(\mathbb{C}\setminus\{(0,0)\}\) being Stein. The author states several such examples. He proves the following theorem: Let \(S\) be a normal Stein space and \(A\) an irreducible analytic subset of \(S\) of codimension 2. Let \(X:=S-A\) and let \(\pi:E\to X\) be a holomorphic affine \(\mathbb{C}\)-bundle on \(X\) which satisfies the condition that for every \(p\in A\) there exists a neighborhood \(V\) of \(p\) such that \(\pi:\pi^{-1}(V-A)\to V-A\) is quasi-trivial. If \(E\to X\) is not isomorphic to a holomorphic line bundle, then \(E\) is Stein. In this connection an affine \(\mathbb{C}\)-bundle \(E\to X\) is called quasi-trivial if a holomorphic vector bundle \(\widetilde E\) of rank 2 defined by the data of \(E\) is trivial.
Mathematics Subject Classification ID: 32E10
Mathematics Subject Classification ID: 32L05
zbMATH DE Number: 1336838
zbMATH Keywords: holomorphic affine bundles
zbMATH Keywords: Stein spaces
reviewed by: H.-J. Reiffen
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1336838