Exterior products of forms and the cohomology ring of a complex space

In recent publications, we have defined complexes of differential forms on analytic spaces which are resolutions of the constant sheaf. These complexes were used to prove the existence of a mixed Hodge structure on the cohomology of analytic spaces which possess kählerian hypercoverings, in particul...

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Bibliographic Details
Published in:Bulletin des sciences mathématiques 2006-09, Vol.130 (6), p.525-552
Main Authors: Ancona, Vincenzo, Gaveau, Bernard
Format: Article
Language:English
Subjects:
Cohomologie
Espaces analytiques
Exact sciences and technology
Formes différentielles
Global analysis, analysis on manifolds
Mathematical analysis
Mathematics
Sciences and techniques of general use
Several complex variables and analytic spaces
Structures de Hodge
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
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Summary:In recent publications, we have defined complexes of differential forms on analytic spaces which are resolutions of the constant sheaf. These complexes were used to prove the existence of a mixed Hodge structure on the cohomology of analytic spaces which possess kählerian hypercoverings, in particular, projective algebraic varieties. We define an exterior product on these forms, which induces the cup product on the cohomology of analytic spaces. The main difficulty is to prove that this exterior product is functorial with respect to morphisms of analytic spaces. This exterior product can be used to prove that the cup product is compatible with the mixed Hodge structure on the cohomology. Nous avons récemment défini des complexes de formes différentielles sur des espaces analytiques qui sont des résolutions du faisceau constant. Ces complexes ont été utilisés pour démontrer l'existence d'une structure de Hodge mixte sur la cohomologie des espaces analytiques possédant un hyperrecouvrement kählérien, en particulier les variétés algébriques projectives. Nous définissons maintenant un produit extérieur sur ces formes qui induit le cup produit en cohomologie. La difficulté principale est de démontrer que ce produit extérieur est fonctoriel par rapport aux morphismes d'espaces analytiques. Il peut alors être utilisé pour montrer que le cup produit est compatible avec la structure de Hodge mixte de la cohomologie.
ISSN:0007-4497
1952-4773
DOI:10.1016/j.bulsci.2005.10.004