Infinitesimal variation functions for families of smooth varieties

In this paper we introduce some {\it 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 $X$, then there exists a first order deformation $\xi\in H^1(T_X)$ which deforms $X$ as plane curve, such that $\xi\cdot:H^0(\omega_X)\to H^1(\mathcal{O}_{X})$ is an isomorphism. We also generalize the notions of variation functions to the 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.