AbstractWe 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 Kähler 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–Ampère volume form and scalar curvature.
We provide an expression for the Mabuchi functional and derive as an
application a combinatorial sufficient condition of properness similar to
one obtained by Li, Zhou and Zhu on
group compactifications. This finally translates to a sufficient criterion of
existence of constant scalar curvature Kähler metrics thanks to the recent work of
Chen and Cheng. It yields infinitely many new examples of explicit Kähler classes
admitting cscK metrics.
Kähler geometry of horosymmetric varieties, and application to Mabuchi’s K-energy functional
