Infinitesimal Variation Functions for Families of Smooth Varieties

Abstract

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 in H^1(T_X) which deforms X as plane curve and such that xi cdot :H^0(omega _X)rightarrow H^1({mathcal 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.