Mixed Hodge structure of affine hypersurfaces
arXiv:math/0407064
Abstract
In this article we introduce the mixed Hodge structure of the Brieskorn module of a polynomial in $\C^{n+1}$, where 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 is given in terms of the vanishing of integrals of certain polynomial -forms in $\C^{n+1}$ 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 1977 for quasi-homogeneous polynomials.
22 pages, minor corrections, To appear in Ann. Ins. Fourier