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The Grauert-Oka principle in a Banach space. (English) Zbl 07315378
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 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.
MSC:
32K12 Holomorphic maps with infinite-dimensional arguments or values
46G20 Infinite-dimensional holomorphy
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References:
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