Analytic cohomology of complete intersections in a Banach space

Abstract

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.