Infinitesimal variation functions for families of smooth varieties

TL;DR

This paper introduces variation functions related to the Infinitesimal Variations of Hodge Structure for smooth varieties, providing bounds, inequalities, and links to Lefschetz properties, with specific results for plane curves.

Contribution

It defines and analyzes variation functions for families of smooth varieties, extending to higher dimensions and connecting to Lefschetz properties of Jacobian rings.

Findings

Existence of a deformation for plane curves inducing an isomorphism in cohomology

Bounds and inequalities for variation functions

Links between IVHS and Lefschetz properties of Jacobian rings

Abstract

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.