Momentum polytopes of projective spherical varieties and related Kähler geometry

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.