Analytic cohomology of complete intersections in a Banach space

Imre Patyi[1]

  • [1] University of California at Riverside, Department of Mathematics, Riverside CA 92521-0135 (USA)

Annales de l’institut Fourier (2004)

  • Volume: 54, Issue: 1, page 147-158
  • ISSN: 0373-0956

Abstract

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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 d f 1 ( x ) , ... , d f 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 .

How to cite

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Patyi, Imre. "Analytic cohomology of complete intersections in a Banach space." Annales de l’institut Fourier 54.1 (2004): 147-158. <http://eudml.org/doc/116102>.

@article{Patyi2004,
abstract = {Let $X$ be a Banach space with a countable unconditional basis (e.g., $X=\ell _2$), $\Omega \subset X$ an open set and $f_1,\ldots ,f_k$ complex-valued holomorphic functions on $\Omega $, such that the Fréchet differentials $df_1(x),\ldots ,df_k(x)$ are linearly independant over $\mathbb \{C\}$ at each $x\in \Omega $. We suppose that $M=\lbrace x\in \Omega :f_1(x)=\ldots =f_k(x)=0\rbrace $ is a complete intersection and we consider a holomorphic Banach vector bundle $E\rightarrow M$. If $I$ (resp.$\{\mathcal \{O\}\}^E$) denote the ideal of germs of holomorphic functions on $\Omega $ that vanish on $M$ (resp. the sheaf of germs of holomorphic sections of $E$), then the sheaf cohomology groups $H^q(\Omega ,I)$, $H^q(M,\{\mathcal \{O\}\}^E)$ vanish for all $q\ge 1$.},
affiliation = {University of California at Riverside, Department of Mathematics, Riverside CA 92521-0135 (USA)},
author = {Patyi, Imre},
journal = {Annales de l’institut Fourier},
keywords = {analytic cohomology; complete intersections},
language = {eng},
number = {1},
pages = {147-158},
publisher = {Association des Annales de l'Institut Fourier},
title = {Analytic cohomology of complete intersections in a Banach space},
url = {http://eudml.org/doc/116102},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Patyi, Imre
TI - Analytic cohomology of complete intersections in a Banach space
JO - Annales de l’institut Fourier
PY - 2004
PB - Association des Annales de l'Institut Fourier
VL - 54
IS - 1
SP - 147
EP - 158
AB - Let $X$ be a Banach space with a countable unconditional basis (e.g., $X=\ell _2$), $\Omega \subset X$ an open set and $f_1,\ldots ,f_k$ complex-valued holomorphic functions on $\Omega $, such that the Fréchet differentials $df_1(x),\ldots ,df_k(x)$ are linearly independant over $\mathbb {C}$ at each $x\in \Omega $. We suppose that $M=\lbrace x\in \Omega :f_1(x)=\ldots =f_k(x)=0\rbrace $ is a complete intersection and we consider a holomorphic Banach vector bundle $E\rightarrow M$. If $I$ (resp.${\mathcal {O}}^E$) denote the ideal of germs of holomorphic functions on $\Omega $ that vanish on $M$ (resp. the sheaf of germs of holomorphic sections of $E$), then the sheaf cohomology groups $H^q(\Omega ,I)$, $H^q(M,{\mathcal {O}}^E)$ vanish for all $q\ge 1$.
LA - eng
KW - analytic cohomology; complete intersections
UR - http://eudml.org/doc/116102
ER -

References

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