Okounkov bodies for ample line bundles with applications to multiplicities for group representations

TL;DR

This paper constructs rational polyhedral Okounkov bodies for ample line bundles on complex varieties and applies these geometric insights to analyze asymptotic properties of group representations.

Contribution

It introduces a method to produce rational polyhedral Okounkov bodies for ample line bundles and applies this to the asymptotic study of group representation multiplicities.

Findings

Okounkov bodies are rational polytopes for certain varieties.

Global Okounkov bodies form rational polyhedral cones under specific conditions.

Application to asymptotic analysis of group representation multiplicities.

Abstract

Let $\mathscr{L} \rightarrow X$ be an ample line bundle over a complex normal projective variety $X$. We construct a flag $X_0 \subseteq X_1 \subseteq \cdots \subseteq X_n=X$ of subvarieties for which the associated Okounkov body for $\mathscr{L}$ is a rational polytope. In the case when $X$ is a homogeneous surface, and the pseudoeffective cone of $X$ is rational polyhedral, we also show that the global Okounkov body is a rational polyhedral cone if the flag of subvarieties is suitably chosen. Finally, we provide an application to the asymptotic study of group representations.