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

TL;DR

This paper explores the relationship between relative Ding stability and the existence of Mabuchi solitons on Fano manifolds, introducing new geometric tools to analyze stability conditions and obstructions.

Contribution

It establishes that uniformly relative Ding stability implies a necessary condition for Mabuchi solitons, using convex geometry and integration formulas for test-configurations.

Findings

Uniform relative Ding stability implies a necessary condition for Mabuchi solitons.

Derived an integration formula for $ ext{C}^*$-actions on test-configurations.

Provided a convex-geometry interpretation of non-Archimedean J-functionals.

Abstract

Mabuchi solitons generalize K\"{a}hler-Einstein metrics on Fano manifolds, which constitute a Yau-Tian-Donaldson type correspondence with relative Ding stability. Comparing with K\"{a}hler-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 $\mathbb{C}^{*}$-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.