nLab Cartan theorem B

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Contents

Context

Complex geometry

Differential cohomology

Contents

Statement

Theorem

Cartan’s theorem B

On a Stein manifold Σ\Sigma and for AA an analytic coherent sheaf on Σ\Sigma then all the positive-degree abelian sheaf cohomology groups of Σ\Sigma with coefficients in AA vanish:

H 1(Σ,A)=0. H^{\bullet \geq 1}(\Sigma, A) = 0 \,.

(Serre 1954, Thm. B on p. 214, recalled for instance as (Forstnerič 11, theorem 2.4.1)

Theorem

Also all positive-degree Dolbeault cohomology groups vanish:

H ¯ ,1(Σ)=0. H^{\bullet, \bullet \geq 1}_{\bar \partial}(\Sigma) = 0 \,.

(recalled for instance in (Forstnerič 11, theorem 2.4.6, Gunning-Rossi).

References

Named after Henri Cartan.

Last revised on October 22, 2023 at 10:25:54. See the history of this page for a list of all contributions to it.