A finiteness theorem for holomorphic Banach bundles

Leiterer, Jürgen

Leiterer, Jürgen

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 1, pp. 15-37.

Résumé

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 $id + K$ 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 $0 \leq q \leq n$ . Then: If $V \to X$ is a holomorphic vector bundle (of finite rank) with $H^{q} (X, V) = 0$ , then $dim H^{q} (X, V \otimes E) < \infty$ . In particular, if $dim H^{q} (X, 𝒪) = 0$ , then $dim H^{q} (X, E) < \infty$ .

MR Zbl

Classification : 32F10, 32C37

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     author = {Leiterer, J\"urgen},
     title = {A finiteness theorem for holomorphic {Banach} bundles},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {15--37},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 6},
     number = {1},
     year = {2007},
     mrnumber = {2341512},
     zbl = {1178.32016},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2007_5_6_1_15_0/}
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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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