Algebraic reverse Khovanskii--Teissier inequality via Okounkov bodies

Abstract

Let $X$ be a projective variety of dimension $n$ over an algebraically closed field of arbitrary characteristic and let $A,B,C$ be nef divisors on $X$. We show that for any integer $1\le k\le n-1$, $$(B^k\cdot A^{n-k})\cdot (A^k\cdot C^{n-k})\ge \frac{k!(n-k)!}{n!}(A^n)\cdot (B^k\cdot C^{n-k}).$$ The same inequality in the analytic setting was obtained by Lehmann and Xiao for compact Kähler manifolds using the Calabi--Yau theorem, while our approach is purely algebraic using (multipoint) Okounkov bodies. We also discuss applications of this inequality to Bézout-type inequalities and inequalities on degrees of dominant rational self-maps.

v1: 12 pages; v2: 13 pages, typos corrected, added a section discussing the equality case for surfaces; final version to appear in Math. Z

References in corpus (1)

• Sufficient Bigness Criterion for Differences of Two Nef Classes

Cited by in corpus (1)

• On the additivity of Newton-Okounkov bodies