nLab Cartan theorem B

Statement

Theorem

Cartan’s theorem B

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

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

(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:

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

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

References

Named after Henri Cartan.

• Jean-Pierre Serre: Thm. B on p. 214 of: Cohomologie et fonctions de variables complexes, Séminaire Bourbaki 2 71 (1954) [numdam:SB_1951-1954__2__213_0]

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

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