Local numerical equivalences and Okounkov bodies in higher dimensions

TL;DR

This paper investigates how local properties of Okounkov bodies reflect the local numerical equivalence classes of divisors on higher-dimensional varieties, extending previous surface results.

Contribution

It demonstrates that the set of Okounkov bodies at a fixed point uniquely determines the local numerical equivalence class of divisors in higher dimensions.

Findings

Okounkov bodies encode local numerical equivalence information.

Extension of Roé's surface results to higher dimensions.

Different proof techniques from previous work.

Abstract

We continue to explore the numerical nature of the Okounkov bodies focusing on the local behaviors near given points. More precisely, we show that the set of Okounkov bodies of a pseudoeffective divisor with respect to admissible flags centered at a fixed point determines the local numerical equivalence class of divisors which is defined in terms of refined divisorial Zariski decompositions. Our results extend Ro\'{e}'s work on surfaces to higher dimensional varieties although our proof is essentially different in nature.