In this article we give an algorithm which produces a basis of the -th de Rham cohomology of the affine smooth hypersurface compatible with the mixed Hodge structure, where is a polynomial in 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 is given in terms of the vanishing of integrals of certain polynomial -forms in over topological -cycles on the fibers of . Since the -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.
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 compatible avec la structure de Hodge mixte, où est un polynôme en 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 est donnée par l’annulation des intégrales de certaines -formes polynomiales dans sur des -cycles topologiques dans les fibres de . Puisque l’homologie de degré 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.
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
@article{AIF_2007__57_3_775_0, author = {Movasati, Hossein}, title = {Mixed {Hodge} structure of affine hypersurfaces}, journal = {Annales de l'Institut Fourier}, pages = {775--801}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {57}, number = {3}, year = {2007}, doi = {10.5802/aif.2276}, mrnumber = {2336829}, zbl = {1123.14007}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2276/} }
TY - JOUR AU - Movasati, Hossein TI - Mixed Hodge structure of affine hypersurfaces JO - Annales de l'Institut Fourier PY - 2007 SP - 775 EP - 801 VL - 57 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2276/ DO - 10.5802/aif.2276 LA - en ID - AIF_2007__57_3_775_0 ER -
%0 Journal Article %A Movasati, Hossein %T Mixed Hodge structure of affine hypersurfaces %J Annales de l'Institut Fourier %D 2007 %P 775-801 %V 57 %N 3 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2276/ %R 10.5802/aif.2276 %G en %F AIF_2007__57_3_775_0
Movasati, Hossein. Mixed Hodge structure of affine hypersurfaces. Annales de l'Institut Fourier, Volume 57 (2007) no. 3, pp. 775-801. doi : 10.5802/aif.2276. https://aif.centre-mersenne.org/articles/10.5802/aif.2276/
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