Momentum polytopes of projective spherical varieties and related K\"ahler geometry

TL;DR

This paper uses combinatorial methods to classify polarized projective spherical varieties and Fano spherical varieties, and characterizes when multiplicity free Hamiltonian manifolds are K"ahler based on their momentum polytopes.

Contribution

It provides a new classification framework for spherical varieties and a criterion for K"ahler structures in multiplicity free Hamiltonian manifolds.

Findings

Classification of polarized projective spherical varieties

Classification of Fano spherical varieties

Necessary and sufficient conditions for K"ahler multiplicity free manifolds

Abstract

We apply the combinatorial theory of spherical varieties to characterize the momentum polytopes of polarized projective spherical varieties. This enables us to derive a classification of these varieties, without specifying the open orbit, as well as a classification of all Fano spherical varieties. In the setting of multiplicity free compact and connected Hamiltonian manifolds, we obtain a necessary and sufficient condition involving momentum polytopes for such manifolds to be K\"ahler and classify the invariant compatible complex structures of a given K\"ahler multiplicity free compact and connected Hamiltonian manifold.