On the mixed hodge structure associated hypersurface singularities

Abstract

Let f:Cn+1→Cf:Cn+1→C be a germ of hypersurface with isolated singularity. One can associate to f a polarized variation of mixed Hodge structure HH over the punctured disc, where the Hodge filtration is the limit Hodge filtration of W. Schmid and J. Steenbrink. By the work of M. Saito and P. Deligne the VMHS associated to cohomologies of the fibers of ff can be extended over the degenerate point 00 of disc. The new fiber obtained in this way is isomorphic to the module of relative differentials of ff denoted ΩfΩf . A mixed Hodge structure can be defined on ΩfΩf in this way. The polarization on \mathcal{H} deforms to Grothendieck residue pairing modified by a varying sign on the Hodge graded pieces in this process. This also proves the existence of a Riemann-Hodge bilinear relation for Grothendieck pairing and allow to calculate the Hodge signature of Grothendieck pairing.