Numerical characterization of the hard Lefschetz classes of dimension two, II: supercritical collections of free divisor classes
Jiajun Hu, Jian Xiao
TL;DR
The paper resolves Shenfeld–van Handel's open problem in the setting of supercritical collections of free divisor classes by proving that the kernel of the Lefschetz-type operator $\\mathbb{L}=L_1\\cdots L_{n-2}$ acting on $(1,1)$-classes is exactly the span of prime divisors annihilated by $\\mathbb{L}$. The authors develop a self-contained argument beginning with a base case on threefolds and then implement an induction via hyperplane sections to handle arbitrary dimension, leveraging positivity, Lorentzian proportionality, and null-locus analysis. As consequences, they provide an algebro-geometric proof of the extremals for the Alexandrov–Fenchel inequality for supercritical rational polytopes and a characterization of extremals for the Khovanskii–Teissier inequality in terms of linear combinations of divisor classes with vanishing mixed intersections. The results deepen the bridge between Hodge-theoretic Lefschetz phenomena and convex-geometric extremals, with potential toric-geometry interpretations and applications to positivity questions in algebraic geometry.
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 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.