Representing Analytic Cohomology Groups of Complex Manifolds
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 24 (2015) no. 1, p. 21-38

Consider a holomorphic vector bundle LX and an open cover 𝔘={U a :aA} 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 hol (𝔘,L) of cochains (f a 0 ...a q ) a 0 ,...,a q A that depend holomorphically on the a j , provided S={(a,x)A×X:xU 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.

On considère un fibré vectoriel holomorphe LX et un recouvrement ouvert 𝔘={U a :aA} de X, où A est une variété complexe non singulière. On démontre alors que les groupes de cohomologie H q (X,L) sont isomorphes aux groupes de cohomologie du complexe C hol (𝔘,L) des cochaînes (f a 0 ...a q ) a 0 ,...,a q A qui dépendent d’une façon holomorphe des a j , à condition que S={(a,x)A×X:xU a }A×X soit un ouvert de Stein. Ce résultat est démontré dans le cadre des variétés de Banach. On finit en donnant une application à l’étude des opérations holomorphes d’un groupe réductif complexe sur L.

@article{AFST_2015_6_24_1_21_0,
     author = {Lempert, L\'aszl\'o},
     title = {Representing Analytic Cohomology Groups of Complex Manifolds},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 24},
     number = {1},
     year = {2015},
     pages = {21-38},
     doi = {10.5802/afst.1440},
     zbl = {1318.32015},
     mrnumber = {3325949},
     language = {en},
     url = {http://www.numdam.org/item/AFST_2015_6_24_1_21_0}
}
Lempert, László. Representing Analytic Cohomology Groups of Complex Manifolds. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 24 (2015) no. 1, pp. 21-38. doi : 10.5802/afst.1440. http://www.numdam.org/item/AFST_2015_6_24_1_21_0/

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