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Hossein Movasati
Mixed Hodge structure of affine hypersurfaces
(Structures de Hodge mixtes d’hypersurfaces affines)
Annales de l'institut Fourier, 57 no. 3 (2007), p. 775-801, doi: 10.5802/aif.2276
Article PDF | Reviews MR 2336829 | Zbl 1123.14007
Class. Math.: 14C30, 32S35
Keywords: Mixed Hodge structures of affine varieties, Gauss-Manin connection

Résumé - Abstract

In this article we give an algorithm which produces a basis of the $n$-th de Rham cohomology of the affine smooth hypersurface $f^{-1}(t)$ compatible with the mixed Hodge structure, where $f$ is a polynomial in $n+1$ variables and satisfies a certain regularity condition at infinity (and hence has isolated singularities). As an application we show that the notion of a Hodge cycle in regular fibers of $f$ is given in terms of the vanishing of integrals of certain polynomial $n$-forms in $\mathbb{C}^{n+1}$ over topological $n$-cycles on the fibers of $f$. Since the $n$-th homology of a regular fiber is generated by vanishing cycles, this leads us to study Abelian integrals over them. Our result generalizes and uses the arguments of J. Steenbrink for quasi-homogeneous polynomials.

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