A finiteness theorem for holomorphic Banach bundles
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 1, p. 15-37
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 $V\to X$ is a holomorphic vector bundle (of finite rank) with ${H}^{q}\left(X,V\right)=0$, then $dim{H}^{q}\left(X,V\otimes E\right)<\infty$. In particular, if $dim{H}^{q}\left(X,𝒪\right)=0$, then $dim{H}^{q}\left(X,E\right)<\infty$.
Classification:  32F10,  32C37
@article{ASNSP_2007_5_6_1_15_0,
author = {Leiterer, J\"urgen},
title = {A finiteness theorem for holomorphic Banach bundles},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 6},
number = {1},
year = {2007},
pages = {15-37},
zbl = {1178.32016},
mrnumber = {2341512},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2007_5_6_1_15_0}
}

Leiterer, Jürgen. A finiteness theorem for holomorphic Banach bundles. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 1, pp. 15-37. http://www.numdam.org/item/ASNSP_2007_5_6_1_15_0/

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