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Analytic cohomology groups of infinite dimensional complex manifolds. (English) Zbl 1361.32012

Given a cohesive sheaf \(\mathcal{S}\) over a complex Banach manifold \(M\), the author endows the cohomology groups \(H^{q}(M,\mathcal{S})\) of \(M\) and \(H^{q}(\mathfrak{U},\mathcal{S})\) of open covers \(\mathfrak{U}\) of \(M\) with a locally convex topology. The main result of the paper is the following:
Suppose that \(M\) is a locally Stein manifold, \(\mathcal{S}\rightarrow M\) a separated cohesive sheaf, \(\mathfrak{U}\) a cover of \(M\) by Stein open sets. Then the canonical map \[ H^{q}(\mathfrak{U},\mathcal{S})\rightarrow \check{H^{q}}(\mathfrak{U},\mathcal{S}) \] is an isomorphism of topological vector spaces, for \(q \in \mathbb N\).

MSC:

32C35 Analytic sheaves and cohomology groups
32Q28 Stein manifolds
46G20 Infinite-dimensional holomorphy
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References:

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