On the mixed Hodge structure associated hypersurface singularities

Let f: cn+1 → c be a germ of hypersurface with isolated singularity. one can associate to f a polarized variation of mixed hodge structure h over the punctured disc,where the hodge lltration is the limit hodge lltration of w. schmid and j. steenbrink. by the work of m. saito and p. deligne the vmhs associated to cohomologies of the lbers of f can be extended over the degenerate point 0 of disc. the new lber obtained in this way is isomorphic to the module of relative dikerentials of f denoted ωf . a mixed hodge structure can be delned on ωf in this way. the polarization on h deforms to grothendieck residue pairing modiled 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.