Relative Ding and $K$-stability of toric Fano manifolds in low dimensions

TL;DR

This paper classifies the uniform relative Ding stability of toric Fano threefolds and fourfolds using Mabuchi constants derived from combinatorial data, and explores the relationship between relative K-stability and Ding stability.

Contribution

It provides a complete classification of relative Ding stability for low-dimensional toric Fano manifolds and clarifies the distinction between relative K-stability and Ding stability.

Findings

Classified all toric Fano threefolds and fourfolds by stability.

Calculated Mabuchi constants from moment polytope data.

Identified differences between K-stability and Ding stability in specific cases.

Abstract

The purpose of this paper is to clarify all of the uniformly relatively Ding stable toric Fano threefolds and fourfolds as well as unstable ones. The key player in our classification result is the Mabuchi constants, which can be calculated by combinatorial data of the associated moment polytopes due to the work of Yao [33]. In Tables 1-3, we give the list of uniform relative Ding stability of all toric Fano manifolds in dimension up to four with the values of the Mabuchi constants. As an application of our main theorem (Theorem 1.1), we clarify the difference between relative $K$-stability and relative Ding stability by considering some specific toric Fano manifolds (Corollaries 1.6 and 1.9). In the proof of Corollary 1.9, we used Bott tower structure of relatively Ding unstable toric Fano manifolds.