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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

The construction of a $ \bar \partial $-simple covering


Author: Wilhelm Stoll
Journal: Proc. Amer. Math. Soc. 27 (1971), 101-106
MSC: Primary 32.40
DOI: https://doi.org/10.1090/S0002-9939-1971-0273059-4
MathSciNet review: 0273059
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Abstract: Let $ M$ be a complex manifold. It is shown by simple means that an arbitrary fine open covering $ \mathfrak{U} = {\{ {U_i}\} _{i \in I}}$ of $ M$ exists such that for every form $ \omega $ of class $ {C^\infty }$ and bidegree $ (p,q)$ with $ \bar \partial \omega = 0$ on $ {U_{{i_0}}}\bigcap \cdots \bigcap {{U_{{i_p}}}} $ there exists a form $ \psi $ of class $ {C^\infty }$ on $ {U_{{i_0}}}\bigcap \cdots \bigcap {{U_{{i_p}}}} $ such that $ \partial \psi = \omega $ provided $ q \geqq 1$.


References [Enhancements On Off] (What's this?)

  • [1] Henri Cartan and Jean-Pierre Serre, Un théorème de finitude concernant les variétés analytiques compactes, C. R. Acad. Sci. Paris 237 (1953), 128–130 (French). MR 0066010
  • [2] Pierre Dolbeault, Sur la cohomologie des variétés analytiques complexes, C. R. Acad. Sci. Paris 236 (1953), 175–177 (French). MR 0052771
  • [3] Raghavan Narasimhan, Analysis on real and complex manifolds, Advanced Studies in Pure Mathematics, Vol. 1, Masson & Cie, Éditeurs, Paris; North-Holland Publishing Co., Amsterdam, 1968. MR 0251745
  • [4] Séminaires de H. Cartan, 1953–1954. Chapters XVI–XIX and Séminaire Bourbaki, Mathematics Department, Massachusetts Institute of Technology, Cambridge, Mass., 1955 (French). MR 0079332

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0273059-4
Keywords: $ \partial $-acyclic, $ \partial $-simple, $ \partial E$-acyclic, $ \partial E$-simple
Article copyright: © Copyright 1971 American Mathematical Society