Infinitesimal Variation Functions for Families of Smooth Varieties

In this paper we introduce some variation functions associated to the rank of the Infinitesimal Variations of Hodge Structure for a family of smooth projective complex curves. We give some bounds and inequalities and, in particular, we prove that if X is a smooth plane curve, then, there exists a first order deformation xi is an element of H-1 (T-x) which deforms X as plane curve and such that xi. : H-0 (w(X)) -> H-1 (O-X) is an isomorphism. We also generalize the notions of variation functions to higher dimensional case and we analyze the link between IVHS and the weak and strong Lefschetz properties of the Jacobian ring of a smooth hypersurface.