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

Abstract

Let $\mathcal{L}\to X$ 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 $\mathcal{L}$ 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.

arXiv commentsupersedes the preprint arXiv:1007.1915