K{\"a}hler geometry of horosymmetric varieties, and application to Mabuchi's K-energy functional

We introduce a class of almost homogeneous varieties contained in the class of spherical varieties and containing horospherical varieties as well as complete symmetric varieties. We develop K{"a}hler geometry on these varieties, with applications to canonical metrics in mind, as a generalization of the Guillemin-Abreu-Donaldson geometry of toric varieties. Namely we associate convex functions with hermitian metrics on line bundles, and express the curvature form in terms of this function, as well as the corresponding Monge-Amp{`e}re volume form and scalar curvature. We then provide an expression for the Mabuchi functional and derive as an application a combinatorial sufficient condition of properness similar obtained by Li, Zhou and Zhu on group compactifications.