Preprint A543/2007
Mixed Hodge structure of affine hypersurfaces

DDIC

**Keywords: **
Mixed Hodge Structure

In this article
we give an algorithm which produces a basis
of the $n$-th de Rham cohomology of the
affine smooth hypersurface $f^{-1}(t)$
compatible with the mixed Hodge structure,
where $f$ is a polynomial in $n+1$ variables and
satisfies a certain regularity
condition at infinity (and hence has isolated
singularities).
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 in \cite{st77} for quasi-homogeneous polynomials.