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 Kähler and classify the invariant compatible complex structures of a given Kähler multiplicity free compact and connected Hamiltonian manifold.

中文翻译：

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射影球形变体的动量多面体和相关的Kähler几何

我们应用球面变体的组合理论来表征极化射影球面变体的动量多面体。这使我们能够在不指定开放轨道的情况下得出这些变体的分类，以及所有Fano球形变体的分类。在无多重紧型和连通哈密顿流形的设置中，我们获得了一个包含动量多边形的充要条件，以使这些歧管成为Kähler，并对给定Kähler无多重紧型和连通哈密顿流形的不变相容复杂结构进行分类。