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Hodge filtrations and filtrations by the order of pole for singular hypersurfaces. (Filtrations de Hodge et par l’ordre du pôle pour les hypersurfaces singulières.) (French) Zbl 0743.14028
Let \(D\) be a subvariety of a smooth complete complex variety \(X\) and set \(U=X\backslash D\). The Hodge type of \(D\) is the largest integer \(t\) for which the Hodge-Deligne filtration \(F\) on the cohomology of \(U\) with compact supports satisfies \(F^ tH^ i_ c(U)=H^ i_ c(U)\) for all \(i\). When \(D\) is a hypersurface in a projective space (or, more generally, in a Grassmannian), its Hodge type is computed using a basic relation between the Hodge-Deligne filtration and the filtration given by the order of the poles of meromorphic differential forms on \(U\) having poles along \(D\). The case of a general subvariety in a projective space is treated by H. Esnault [Math. Ann. 288, No. 3, 549-551 (1990)] and by H. Esnault, M. Nori and V. Srinivas [“Hodge type of projective varieties of low degree” (preprint 1991)]. For other related results see A. Dimca [Am. J. Math. 113, No. 4, 763-771 (1991); see the following review].
Reviewer: A.Dimca (Sydney)

MSC:
14J70 Hypersurfaces and algebraic geometry
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14F40 de Rham cohomology and algebraic geometry
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