Holomorphic affine bundles on the complement of an analytic set
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.
scientific article; zbMATH DE number 1336838