Infinitesimal Variation Functions for Families of Smooth Varieties

Abstract

AbstractIn 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 $$\xi \in H^1(T_X)$$ ξ ∈ H 1 ( T X ) which deforms X as plane curve and such that $$\xi \cdot :H^0(\omega _X)\rightarrow H^1({\mathcal O}_{X})$$ ξ · : H 0 ( ω 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.