A converse to the Oka principle over certain complex Banach manifolds
Conditions that can be viewed as a converse to the Grauert-Oka principle and that ensure that a complex manifold be Stein have been given by \textit{J. Kajiwara} [Mem. Fac. Sci., Kyushu Univ., Ser. A. 33, 83--93 (1979; Zbl 0404.32007)]. A slightly time later \textit{J. Leiterer} [Math. Nachr. 89, 181--183 (1979; Zbl 0445.32012)] characterized Steinness, in a similar spirit, when holomorphic properties are determined by topological conditions. For example, he showed that an open subset \(D \subset M\) of a Stein manifold is itself Stein if and only if \(H^1(D,{\mathcal O})=0\) and every holomorphic vector bundle that is topologically trivial over \(D\) is holomorphically trivial over \(D\). The purpose of the present paper is to give analogues of Leiterer's conditions for certain complex Banach manifolds. An essential ingredient is the notion of plurisubharmonic domination due to \textit{L. Lempert} [J. Am. Math. Soc. 17, No. 2, 361--372 (2004; Zbl 1042.32013)].
scientific article; zbMATH DE number 5597497