geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
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:
(Serre 1954, Thm. B on p. 214, recalled for instance as (Forstnerič 11, theorem 2.4.1)
Also all positive-degree Dolbeault cohomology groups vanish:
(recalled for instance in (Forstnerič 11, theorem 2.4.6, Gunning-Rossi).
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
Last revised on October 22, 2023 at 10:25:54. See the history of this page for a list of all contributions to it.