The delta invariant and the various GIT-stability notions of toric Fano varieties

Abstract In this paper, we provide purely combinatorial proofs of the following two theorems: (1) If a Gorenstein toric Fano variety is asymptotically Chow semistable then it is Ding polystable with respect to toric test configurations. (2) For a smooth toric Fano variety X, the delta invariant δ(X) defined in [FO18] coincides with the greatest Ricci lower curvature R(X). Although it seems that these statements are known to the experts, our combinatorial proofs given here are more direct and elementary approach than the original sources [Ber16, BJ17]. We further verify relative Chow stability for Gorenstein toric del Pezzo surfaces using the combinatorial criterion developed in [YZ19] and specifying the symmetry of the associated polytopes as well. We also clarify the reductivity of automorphism group of toric Fano 3-folds by computing the set of Demazure roots for each. All the results are listed in Table 3 with the values of δ(X) and R(X).