Okounkov bodies associated to pseudoeffective divisors

TL;DR

This paper extends the concept of Okounkov bodies to pseudoeffective divisors on smooth projective varieties, introducing new convex bodies that capture their asymptotic properties, thus broadening the scope of previous work on big divisors.

Contribution

It introduces valuative and limiting Okounkov bodies for pseudoeffective divisors, extending the theory beyond big divisors and defining new subvarieties related to these divisors.

Findings

Defined valuative and limiting Okounkov bodies for pseudoeffective divisors

Extended the theory of Okounkov bodies to a broader class of divisors

Connected convex bodies to asymptotic properties of pseudoeffective divisors

Abstract

An Okounkov body is a convex subset in Euclidean space associated to a big divisor on a smooth projective variety with respect to an admissible flag. In this paper, we introduce two convex bodies associated to pseudoeffective divisors, called the valuative Okounkov bodies and the limiting Okounkov bodies, and show that these convex bodies reflect the asymptotic properties of pseudoeffective divisors as in the case with big divisors. Our results extend the works of Lazarsfeld-Musta\c{t}\u{a} and Kaveh-Khovanskii. For this purpose, we define and study special subvarieties, called the Nakayama subvarieties and the positive volume subvarieties, associated to pseudoeffective divisors.