Relative Ding Stability and an Obstruction to the Existence of Mabuchi Solitons

Abstract

Mabuchi solitons generalize Kähler-Einstein metrics on Fano manifolds, which constitute a Yau-Tian-Donaldson type correspondence with relative Ding stability. Comparing with Kähler-Ricci solitons, there is a distinct necessary condition for the existence. We show this condition can be implied by the uniformly relative Ding stability. For this we study the inner product of ${\mathbb{C}}^{\ast }$-actions on equivariant test-configurations and obtain an integration formula over the total space. To analyze the uniform stability, by adapting Okounkov body construction to the setting of torus action, we give a convex-geometry description for the reduced non-Archimedean J-functionals.

40 pages, 1 figure. Final version. To appear in J. Geom. Anal

References in corpus (9)

• On the constant scalar curvature Kähler metrics, general automorphism group
• Geodesic rays and stability in the cscK problem
• $C^{1,1}$ regularity for degenerate complex Monge-Ampère equations and geodesic rays
• The inverse Monge-Ampere flow and applications to Kahler-Einstein metrics
• Mabuchi metrics and properness of the modified DING functional
• Orthogonal projection of a test configuration to vector fields
• Relative Ding and $K$-stability of toric Fano manifolds in low dimensions
• On the Yau-Tian-Donaldson conjecture for generalized Kähler-Ricci soliton equations
• Mabuchi's soliton metric and relative D-stability

Cited by in corpus (2)

• A uniform version of the Yau-Tian-Donaldson correspondence for extremal Kähler metrics on polarized toric manifolds
• Relative uniform $K$-stability over models implies existence of extremal metrics