Mabuchi Solitons and Relative Ding Stability of Toric Fano Varieties

Abstract

As a generalization of Kahler-Einstein metrics for Fano manifolds with nonvanishing Futaki invariant, Mabuchi solitons are critical points of a Calabi-type energy functional. We study their existence on toric Fano varieties and the underlying algebraic stability notion: relative Ding stability. As a toy model for a YTD type correspondence, a new feature is the emergence of a non-uniformly stable case. We show a partial coercivity for the modified Ding functionals in this case, and obtain singular Mabuchi solitons via a variational approach. In the unstable case, we determine the maximal destabilizer which is a simple convex function over the moment polytope, and establish a Moment-Weight equality which connects the infimum of a Calabi-type energy and the Berman-Ding invariant.

44 pages, 1 figure. Final version. Many minor revisions. To appear on Int. Math. Res. Not. IMRN

References in corpus (7)

• A variational approach to the Yau-Tian-Donaldson conjecture
• Stability and coercivity for toric polarizations
• The inverse Monge-Ampere flow and applications to Kahler-Einstein metrics
• Mabuchi metrics and properness of the modified DING functional
• Relative Algebro-Geometric stabilities of Toric Manifolds
• Relative Ding and $K$-stability of toric Fano manifolds in low dimensions
• Remarks on modified Ding functional for toric Fano manifolds

Cited by in corpus (10)

• Coupled complex Monge-Ampère equations on Fano horosymmetric manifolds
• The inverse Monge-Ampere flow and applications to Kahler-Einstein metrics
• Relative Ding Stability and an Obstruction to the Existence of Mabuchi Solitons
• Mabuchi metrics and properness of the modified DING functional
• Geometric flow, Multiplier ideal sheaves and Optimal destabilizer for a Fano manifold
• Generalized Kähler Einstein metrics and uniform stability for toric Fano manifolds
• Relative Ding and $K$-stability of toric Fano manifolds in low dimensions
• Remarks on modified Ding functional for toric Fano manifolds
• Asymptotic Chow semistability implies Ding polystability for Gorenstein toric Fano varieties
• Examples on Loewy filtrations and K-stability of Fano varieties with non-reductive automorphism groups