Analytic cohomology of complete intersections in a Banach space

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., $X={\ell }_2$), $\Omega \subset X$ an open set and $f_1,...,f_k$ complex-valued holomorphic functions on $\Omega$, such that the Fréchet differentials $d{f}_1\left(x\right),...,d{f}_k\left(x\right)$ are linearly independant over $ℂ$ at each $x\in \Omega$. We suppose that $M=\{x\in \Omega :f_1\left(x\right)=...=f_k\left(x\right)=0\}$ 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 $\Omega$ that vanish on $M$ (resp. the sheaf of germs of holomorphic sections of $E$), then the sheaf cohomology groups $H^q(\Omega ,I)$, $H^q(M,{𝒪}^E)$ vanish for all $q\ge 1$.