# Mabuchi Solitons and Relative Ding Stability of Toric Fano Varieties

## TL;DR

This paper investigates the existence of Mabuchi solitons on toric Fano varieties, linking algebraic stability notions with variational methods to find singular solutions and analyze stability cases.

## Contribution

It introduces a new stability notion called relative Ding stability for toric Fano varieties and establishes a variational approach to find singular Mabuchi solitons.

## Key findings

- Partial coercivity of modified Ding functionals in stable cases
- Identification of maximal destabilizer in unstable cases
- Moment-Weight equality connecting energy infimum and Berman-Ding invariant

## 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.