Kähler geometry of horosymmetric varieties, and application to Mabuchi’s K-energy functional

Abstract

<jats:title>Abstract</jats:title><jats:p>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 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.</jats:p>

Ricci flat Kähler metrics on rank two complex symmetric spaces

“…The structure of -variety on the boundary divisors (and more generally all orbits) is also known from [DP83]: there exist a parabolic subgroup such that is a -equivariant fibration → ∕ whose fiber is the wonderful compactification of the symmetric space ∕ ( ) where is a Levi subgroup of . They are examples of horosymmetric varieties [Del17b].…”

“…It is natural to impose a condition of invariance under the action of a maximal compact subgroup of , and we furthermore assume that the Kähler form is ̄ -exact (note that the invariance condition implies the second condition provided the symmetric space is not Hermitian by [AL92]). Then using the general setup of [Del17b], one derives easily that the Ricci flat equation translates as follows.…”

“…There is a natural action of on the symmetric space ∕ ( ), and we build from this data a rank one horosymmetric space ∕ 2 under the action of by parabolic induction: ∕ 2 is the quotient of × ∕ ( ) by the diagonal action of given by ⋅ ( , ) = ( −1 , ⋅ ). In order to match with the conventions of [Del17b], if we let denote the Levi subgroup of containing , then the involution of corresponding to ∕ 2 in the definition of [Del17b] is the involution 2 defined at the Lie algebra level by 2 = on , and 2 equal to the identity on the other factors ( ) and . The horosymmetric space thus constructed is actually exactly the open -orbit in the -stable prime divisor 2 of the wonderful compactification of ∕ ( ) corresponding to the root 2 , as one may deduce from [DP83], or with some different details, from [Del17b].…”

“…We find an ansatz to desingularize this singular cone using the other boundary divisor 1 and in particular the Stenzel metric on the fibers of the open orbit of this other boundary divisor, which gives the desingularization in the 'collapsed directions'. It is justified by analyzing the explicit examples produced by the first author and Gauduchon with the Kähler geometry techniques developed by the second author to study horosymmetric spaces [Del17b] (as both symmetric spaces and the open orbits of divisors in their wonderful compactifications are horosymmetric). There is no canonical choice of behavior on the respective divisors: we obtain examples where only one choice works, and examples where both choices work, thus providing two Ricci-flat Kähler metrics with different asymptotic behavior.…”

“…The article is organized as follows. In Section 2 we introduce the relevant combinatorial data associated to symmetric spaces, their wonderful compactifications, and derive from [Del17b] the translation of the Ricci flat equation as a real two-variables Monge-Ampère equation. In Section 3, we state a numerical criterion of existence of solutions to a one-variable ODE which arises as the equation ruling the existence of positive Kähler-Einstein metrics on rank one horosymmetric spaces or simple variants of this.…”