Abstract

Let $E$ be a holomorphic Banach bundle over a compact complex manifold, which can be defined by a cocycle of holomorphic transition functions with values of the form $\mathrm{id}+K$ where $K$ is compact. Assume that the characteristic fiber of $E$ has the compact approximation property. Let $n$ be the complex dimension of $X$ and $0\le q\le n$. Then: If $V\rightarrow X$ is a holomorphic vector bundle with $H^q =0$, then $\dim H^q<\infty $. In particular, if $\dim H^q = 0$, then $\dim H^q <\infty $