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Movasati, Hossein
Mixed Hodge structure of affine hypersurfaces. Annales de l'institut Fourier, 57 no. 3 (2007), p. 775-801
Texte intégral djvu | pdf | Analyses MR 2336829 | Zbl 1123.14007
Class. Math.: 14C30, 32S35
URL stable: http://www.numdam.org/item?id=AIF_2007__57_3_775_0
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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 $\mathbb{C}^{n+1}$ 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.
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