Hodge theory and holomorphic De Rham complexes for certain analytic spaces. I. (Q1387726)

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?