Analytic cohomology of complete intersections in a Banach space
Annales de l'Institut Fourier, Volume 54 (2004) no. 1, p. 147-158
Let X be a Banach space with a countable unconditional basis (e.g., X=ℓ 2 ), Ω⊂X an open set and f 1 ,...,f k complex-valued holomorphic functions on Ω, such that the Fréchet differentials df 1 (x),...,df k (x) are linearly independant over ℂ at each x∈Ω. We suppose that M={x∈Ω:f 1 (x)=...=f k (x)=0} is a complete intersection and we consider a holomorphic Banach vector bundle E→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≥1.
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.
DOI : https://doi.org/10.5802/aif.2013
Classification:  32L20,  32L10,  46G20
Keywords: analytic cohomology, complete intersections 
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {54},
     number = {1},
     year = {2004},
     pages = {147-158},
     doi = {10.5802/aif.2013},
     zbl = {1080.32017},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2004__54_1_147_0}
}

Patyi, Imre. Analytic cohomology of complete intersections in a Banach space. Annales de l'Institut Fourier, Volume 54 (2004) no. 1, pp. 147-158. doi : 10.5802/aif.2013. http://www.numdam.org/item/AIF_2004__54_1_147_0/

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