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

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 Néron-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.