Mixed Hodge structure of affine hypersurfaces

Movasati, Hossein

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 $f^{-1}\left(t\right)$ 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 ${ℂ}^{n+1}$ 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 $f^{-1}\left(t\right)$ 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 ${ℂ}^{n+1}$ 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.

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

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