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Mixed Hodge structure of affine hypersurfaces

Hossein Movasati
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Abstract

In this article we introduce the mixed Hodge structure of the Brieskorn module of a polynomial $f$ in $\C^{n+1}$, where $f$ satisfies a certain regularity condition at infinity (and hence has isolated singularities). We give an algorithm which produces a basis of a localization of the Brieskorn module which is compatible with its mixed Hodge structure. 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 1977 for quasi-homogeneous polynomials.

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Hossein Movasati. 2006-08-30. Mixed Hodge structure of affine hypersurfaces. https://arxiv.org/abs/math/0407064

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