2004/07/05 by Hossein Movasati, Movasati, Hossein
Mathematics · #14C30 #32S35 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.math/0407064
openalex publication_date 2004/07/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article we introduce the mixed Hodge structure of the Brieskorn module of a polynomial f in \Cn+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 \Cn+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.