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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
##### Keywords:
Hodge type; Hodge-Deligne filtration; hypersurface
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##### References:
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