J ul 2 00 4 Mixed Hodge structure of global Brieskorn modules

In this article we introduce the mixed Hodge structure of the Brieskorn module of a polynomial f in C, 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 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 in [St77] for quasi-homogeneous polynomials. 0 Introduction To study the monodromy and some numerical invariants of a singularity f : (Cn+1, 0) → (C, 0), E. Brieskorn in [Br70] introduced a OC,0 module H ′ and the notion of GaussManin connection on H ′. Later J. Steenbrink [St76], inspired by P. Deligne’s theory of mixed Hodge structures (see [De71] and two others with the same title) on algebraic varieties defined over complex numbers and W. Schmid’s limit Hodge structure (see [Sc73]) associated to a fibration, introduced the notion of the limit mixed Hodge structure for a germ of a singularity f . In this direction M. Saito’s mixed Hodge module theory (see [Sa86]) is another development. Steenbrink’s definition has to do more with the mixed Hodge structure of the singular fiber of f (in Deligne’s sense) and not the regular fibers. In the case of a polynomial f in Cn+1, on the one hand the n-th cohomology of a regular fiber carries Deligne’s mixed Hodge structure and on the other hand we have the Brieskorn module H ′ of f which contains the information of the n-th de Rham cohomology of regular fibers. In this article we define the mixed Hodge structure of H ′ and we study it for certain tame polynomials. At the beginning my purpose was to find explicit descriptions of arithmetic properties of Hodge cycles for hypersurfaces in projective spaces. Such descriptions for CM-Abelian varieties are well-known but in the case of hypersurfaces we have only descriptions for Fermat varieties (see [Sh79]). As an application we will see that it is possible to write down the property of being a Hodge cycle in terms of the vanishing of certain Abelian integrals over cycles generated by vanishing cycles (see the example at the end of this Introduction). Such Abelian integrals also appear in the context of holomorphic foliations/differential equations (see [Mo04, Mo04] and the references there). We state below the precise statements of our results in this article. Math. classification: 14C30, 32S35