Numerical characterization of the hard Lefschetz classes of dimension two, II: supercritical collections of free divisor classes

TL;DR

This paper characterizes the kernel of Lefschetz-type operators induced by supercritical free divisor classes on smooth projective varieties, resolving an open problem and connecting to inequalities in convex geometry.

Contribution

It provides a characterization of the kernel space for supercritical collections of free divisor classes, solving an open problem and linking algebraic geometry with convex inequalities.

Findings

Characterization of the kernel space for supercritical free divisor classes

Algebro-geometric proof of extremals in the Alexandrov-Fenchel inequality

Characterization of extremals of the Khovanskii-Teissier inequality

Abstract

For $(n-2)$ free divisor classes on a smooth projective variety of dimension $n$, the product of these free divisor classes induces a Lefschetz type operator acting on the N\'{e}ron-Severi space or the cohomology group of $(1,1)$ classes. We give a characterization of this kernel space, when the collection of these free divisor classes is supercritical. This resolves Shenfeld-van Handel's open problem in this setting. As consequences, we provide an algebro-geometric proof of the characterization of the extremals of the Alexandrov-Fenchel inequality for a supercritical collection of rational convex polytopes; we also give a characterization of the extremals of the Khovanskii-Teissier inequality given by the intersection numbers of two arbitrary free divisor classes.