A finiteness theorem for holomorphic Banach bundles

Jürgen Leiterer

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2007)

  • Volume: 6, Issue: 1, page 15-37
  • ISSN: 0391-173X

Abstract

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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 ≤ q ≤ n . Then: If V → X is a holomorphic vector bundle (of finite rank) with H q ( X , V ) = 0 , then dim H q ( X , V ⊗ E ) < ∞ . In particular, if dim H q ( X , 𝒪 ) = 0 , then dim H q ( X , E ) < ∞ .

How to cite

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Leiterer, Jürgen. "A finiteness theorem for holomorphic Banach bundles." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 6.1 (2007): 15-37. <http://eudml.org/doc/272252>.

@article{Leiterer2007,
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 (of finite rank) with $H^q (X,V)=0$, then $\dim H^q(X,V\otimes E)&lt;\infty $. In particular, if $\dim H^q (X, \{\mathcal \{O\}\}) = 0$, then $\dim H^q(X,E) &lt;\infty $.},
author = {Leiterer, Jürgen},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {holomorphic Banach bundles; finiteness of cohomology},
language = {eng},
number = {1},
pages = {15-37},
publisher = {Scuola Normale Superiore, Pisa},
title = {A finiteness theorem for holomorphic Banach bundles},
url = {http://eudml.org/doc/272252},
volume = {6},
year = {2007},
}

TY - JOUR
AU - Leiterer, Jürgen
TI - A finiteness theorem for holomorphic Banach bundles
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2007
PB - Scuola Normale Superiore, Pisa
VL - 6
IS - 1
SP - 15
EP - 37
AB - 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 (of finite rank) with $H^q (X,V)=0$, then $\dim H^q(X,V\otimes E)&lt;\infty $. In particular, if $\dim H^q (X, {\mathcal {O}}) = 0$, then $\dim H^q(X,E) &lt;\infty $.
LA - eng
KW - holomorphic Banach bundles; finiteness of cohomology
UR - http://eudml.org/doc/272252
ER -

References

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  2. [2] K. Clancey and I. Gohberg, “Factorization of Matrix Functions and Singular Integral Operators", OT 3, Birkhäuser, 1981. Zbl0474.47023MR657762
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  4. [4] I. Gohberg, The factorization problem for operator functions (Russian), Izv. Akad. Nauk SSSR Ser. Mat.28 (1964), 1055–1082. Zbl0202.42603MR174994
  5. [5] I. Gohberg and J. Leiterer, The local principle for the factorization problem of continuous operator functions (Russian), Rev. Roumaine Math. Pures. Appl.18 (1973), 1585–1600. Zbl0274.47011MR336408
  6. [6] G. Henkin and J. Leiterer, “Theory of Functions on Complex Manifolds", Akademie-Verlag Berlin, 1984, and Birkhäuser, 1984. Zbl0573.32001MR774049
  7. [7] G. Henkin and J. Leiterer, “Andreotti-Grauert Theory by Integral Formulas", Progress in Mathematics, Vol. 74, Birkhäuser, 1988. Zbl0654.32002MR986248
  8. [8] J. Leiterer, From local to global homotopy formulas for ∂ ¯ and ∂ ¯ b , In: “Geometric Complex Analysis", J. Noguchi et al. (eds.), World Scientific Publishing Co., 1996, 385–391. Zbl1025.32500MR1453619
  9. [9] J. Lindenstrauss and L. Tzafriri, “Classical Banach Spaces" I and II, Springer, 1996. Zbl0852.46015MR500056

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