Greatest Ricci lower bounds of projective horospherical manifolds of Picard number one

TL;DR

This paper computes the greatest Ricci lower bounds for certain projective horospherical manifolds, revealing how close they can get to being Kähler--Einstein, especially in the case of odd symplectic Grassmannians.

Contribution

It introduces a method to calculate Ricci lower bounds for Picard number one horospherical manifolds using moment polytope barycenters, extending recent theoretical work.

Findings

Greatest Ricci lower bounds vary across manifolds.

Bounds can approach zero for large n in odd symplectic Grassmannians.

Automorphism groups of nonhomogeneous cases are non-reductive.

Abstract

A horospherical variety is a normal $G$-variety such that a connected reductive algebraic group $G$ acts with an open orbit isomorphic to a torus bundle over a rational homogeneous manifold. The projective horospherical manifolds of Picard number one are classified by Pasquier, and it turned out that the automorphism groups of all nonhomogeneous ones are non-reductive, which implies that they admit no K\"{a}hler--Einstein metrics. As a numerical measure of the extent to which a Fano manifold is close to be K\"{a}hler--Einstein, we compute the greatest Ricci lower bounds of projective horospherical manifolds of Picard number one using the barycenter of each moment polytope with respect to the Duistermaat--Heckman measure based on a recent work of Delcroix and Hultgren. In particular, the greatest Ricci lower bound of the odd symplectic Grassmannian $\text{SGr}(n,2n+1)$ can be arbitrarily close to zero as $n$ grows.