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Hodge theory and holomorphic De Rham complexes for certain analytic spaces. I. (Théorie de Hodge et complexes de De Rham holomorphes pour certains espaces analytiques. I.) (French) Zbl 0928.32004
For a complex analytic space with a smooth singular locus and a smooth exceptional divisor, the following constructions are given: a sequence of de Rham complexes, Hodge filtrations with associated spectral sequences, holomorphic de Rham complexes with the usual properties in Hodge theory. Specifically, (i) for every natural number \(r\), a fine de Rham complex of differential forms is constructed, with a type definition for the forms; (ii) the associated spectral sequence converges to the graded de Rham cohomology; (iii) for every \(r\), a holomorphic de Rham complex is defined whose cohomology is the first term of the mentioned spectral sequence; (iv) for sufficiently great \(r\), the graded de Rham cohomology does not depend on \(r\).
Remark: There is the question of defining the analog of harmonic forms, i.e. is it possible to define canonical representatives in the cohomology classes of the considered analytic spaces?

32C15 Complex spaces
32C35 Analytic sheaves and cohomology groups
Full Text: DOI
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