The Grauert-Oka principle in a Banach space
The main result here is the following. Theorem: Let \(M\) be a Stein Banach manifold modeled on a separable Banach space \(X\) with the bounded approximation property, let \(\Gamma \to M\) be a holomorphic Banach Lie group bundle over \(M,\) and let \({\mathcal O}^\Gamma \to M\)\ (resp. \({\mathcal C}^\Gamma\)) \(\to M\) be the sheaf of germs of holomorphic (resp. continuous) sections of \(\Gamma.\) Then the natural map \(H^1(M, {\mathcal O}^\Gamma) \to H^1(M, {\mathcal C}^\Gamma)\) of first cohomology sets is bijective. The proof of this and related results are non-trivial, relying on a classical paper of \textit{H. Cartan} [Sympos. Int. Topologia Algebraica 97--121 (1958; Zbl 0121.30503)] and several papers of L. Lempert. It is worth noting that this paper begins with a short, but helpful, introduction as well as a somewhat apt quotation of Schiller.
scientific article; zbMATH DE number 7315378
scientific article