Okounkov bodies associated to abundant divisors and Iitaka fibrations

TL;DR

This paper investigates the properties of Okounkov bodies associated with abundant divisors, demonstrating they encode all numerical data and linking their volumes to geometric features like base loci and Iitaka fibrations.

Contribution

It establishes that valuative Okounkov bodies of abundant divisors fully capture their numerical properties and introduces criteria for when valuative and limiting bodies coincide.

Findings

Valuative Okounkov bodies encode all numerical properties of abundant divisors.

Asymptotic base loci can be recovered from Okounkov bodies.

Conditions are provided for the equality of valuative and limiting Okounkov bodies.

Abstract

The aim of this paper is to study the Okounkov bodies associated to abundant divisors. As a main result, we prove that the valuative Okounkov bodies of an abundant divisor encode all the numerical properties. We apply this result to recover the asymptotic base loci of an abundant divisor from the valuative Okounkov bodies. We also give a criterion of when the valuative and limiting Okounkov bodies of an abundant divisor coincide by comparing their Euclidean volumes. To obtain these results, we prove some variants of Fujita's approximations for Okounkov bodies using Iitaka fibrations.