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

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 Kähler--Einstein metrics. As a numerical measure of the extent to which a Fano manifold is close to be Kähler--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.

22 pages, 1 figure, 3 tables

References in corpus (4)

• Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kahler metrics, I
• On the construction of Nadel multiplier ideal sheaves and the limiting behavior of the Ricci flow
• The Yau-Tian-Donaldson conjecture for cohomogeneity one manifolds
• Kähler-Einstein metrics on smooth Fano symmetric varieties with Picard number one