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

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.