nLab Cartan theorem B




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 \,.

This is recalled for instance as (Forstnerič 11, theorem 2.4.1)


Also all positive-degree Dolbeault cohomology groups vanish:

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

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


Named after Henri Cartan.

  • Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, Prentice-Hall Inc., Englewood Cliffs, N.J., (1965)

  • Franc Forstnerič, Stein manifolds and holomorphic mappings – The homotopy principle in complex analysis, Springer 2011

Last revised on May 28, 2014 at 06:29:00. See the history of this page for a list of all contributions to it.