Representing analytic cohomology groups of complex manifolds

2013/12/30 by Lempert, Laszlo
#14F25 #32E10 #32L10 #46G20 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1312.7775

Abstract

Consider a holomorphic vector bundle L→ X and an open cover \frak U=\U_a\colon a∈ A\ of X, parametrized by a complex manifold A. We prove that the sheaf cohomology groups H^q(X,L) can be computed from the complex C^\bullet_hol (\frak U,L) of cochains (f_{a0… aq})_{a0,…, aq∈ A} that depend holomorphically on the a_j, provided S=\(a,x)∈ A× X\colon x∈ U_a\ is a Stein open subset of A× X. The result is proved in the setting of Banach manifolds, and is applied to study representations on cohomology groups induced by a holomorphic action of a complex reductive Lie group on L.

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• Analytic cohomology groups of infinite dimensional complex manifolds
• Smoothly Parameterised Cech Cohomology of Complex Manifolds
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