Analytic cohomology of complete intersections in a Banach space

Patyi, Imre

Analytic cohomology of complete intersections in a Banach space
[Cohomologie d'une intersection complète dans un espace de Banach]

Patyi, Imre

Annales de l'Institut Fourier, Tome 54 (2004), p. 147-158 / Harvested from Numdam

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Résumé
Abstract

On démontre par exemple que dans un espace de Hilbert séparable au-dessus d’une intersection complète lisse $M$ tous les fibrés vectoriels holomorphes sont acycliques, et le faisceau idéal de $M$ est au-dessus des voisinages pseudoconvexes ouverts de $M$ assez petit.

Let $X$ be a Banach space with a countable unconditional basis (e.g., $X = ℓ_{2}$ ), $Ω \subset X$ an open set and $f_{1}, ..., f_{k}$ complex-valued holomorphic functions on $Ω$ , such that the Fréchet differentials $d f_{1} (x), ..., d f_{k} (x)$ are linearly independant over $ℂ$ at each $x \in Ω$ . We suppose that $M = {x \in Ω : f_{1} (x) = ... = f_{k} (x) = 0}$ is a complete intersection and we consider a holomorphic Banach vector bundle $E \to M$ . If $I$ (resp. $𝒪^{E}$ ) denote the ideal of germs of holomorphic functions on $Ω$ that vanish on $M$ (resp. the sheaf of germs of holomorphic sections of $E$ ), then the sheaf cohomology groups $H^{q} (Ω, I)$ , $H^{q} (M, 𝒪^{E})$ vanish for all $q \geq 1$ .

Publié le : 2004-01-01

Zbl 1080.32017 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.2013
Classification: 32L20, 32L10, 46G20
Mots clés: cohomologie analytique, intersection complète

@article{AIF_2004__54_1_147_0,
     author = {Patyi, Imre},
     title = {Analytic cohomology of complete intersections in a Banach space},
     journal = {Annales de l'Institut Fourier},
     volume = {54},
     year = {2004},
     pages = {147-158},
     doi = {10.5802/aif.2013},
     zbl = {1080.32017},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2004__54_1_147_0}
}

Patyi, Imre. Analytic cohomology of complete intersections in a Banach space. Annales de l'Institut Fourier, Tome 54 (2004) pp. 147-158. doi : 10.5802/aif.2013. http://gdmltest.u-ga.fr/item/AIF_2004__54_1_147_0/

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