Mixed Hodge structure of affine hypersurfaces
[Structures de Hodge mixtes d’hypersurfaces affines]
Annales de l'Institut Fourier, Tome 57 (2007) no. 3, pp. 775-801.

Dans cet article nous donnons un algorithme qui produit une base du n-ième groupe de cohomology de De Rham de l’hypersurface affine lisse f -1 (t) compatible avec la structure de Hodge mixte, où f est un polynôme en n+1 variables et satisfait une condition de régularité à l’infini (en particulier, il a des singularités isolées). Comme application nous montrons que la notion de cycle de Hodge dans une fibre régulière de f est donnée par l’annulation des intégrales de certaines n-formes polynomiales dans n+1 sur des n-cycles topologiques dans les fibres de f. Puisque l’homologie de degré n d’une fibre régulière est engendrée par les cycles évanescents, cela conduit à étudier des intégrales abéliennes obtenues en intégrant sur ceux-ci. Notre résultat généralise et utilise les arguments de J. Steenbrink pour les polynômes quasi-homogènes.

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 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.

DOI : 10.5802/aif.2276
Classification : 14C30, 32S35
Keywords: Mixed Hodge structures of affine varieties, Gauss-Manin connection
Mot clés : problème d’appartenance, idéaux de polynômes, courant résidu, représentation intégrale
Movasati, Hossein 1

1 Instituto de Matemática Pura e Aplicada, IMPA Estrada Dona Castorina, 110 22460-320, Rio de Janeiro (Brazil)
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Movasati, Hossein. Mixed Hodge structure of affine hypersurfaces. Annales de l'Institut Fourier, Tome 57 (2007) no. 3, pp. 775-801. doi : 10.5802/aif.2276. http://www.numdam.org/articles/10.5802/aif.2276/

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