On global Okounkov bodies of spherical varieties

TL;DR

This paper introduces the global Okounkov moment cone for projective spherical varieties, proving it is rational polyhedral under mild conditions, which implies the same for the associated global Okounkov body.

Contribution

It generalizes the concepts of the global Okounkov body and moment body for spherical varieties, establishing their rational polyhedral nature.

Findings

Global Okounkov moment cone is rational polyhedral.

Global Okounkov body is rational polyhedral under certain conditions.

Provides a unified framework for understanding convex bodies of spherical varieties.

Abstract

We define and study the global Okounkov moment cone of a projective spherical variety X, generalizing both the global Okounkov body and the moment body of X defined by Kaveh and Khovanskii. Under mild assumptions on X we show that the global Okounkov moment cone of X is rational polyhedral. As a consequence, also the global Okounkov body of X, with respect to a particular valuation, is rational polyhedral.