Mixed Hodge structure of affine hypersurfaces

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In this article we introduce the mixed Hodge structure of the Brieskorn module of a polynomial f in 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 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.