Asymptotic Chow semistability implies Ding polystability for Gorenstein toric Fano varieties

TL;DR

This paper demonstrates that asymptotic Chow semistability implies Ding polystability for Gorenstein toric Fano varieties, extending previous results to varieties with Gorenstein singularities and exploring stability differences through examples.

Contribution

It extends the known stability implications to Gorenstein toric Fano varieties, introduces additivity of the Mabuchi constant for product varieties, and constructs examples illustrating stability distinctions.

Findings

Asymptotic Chow semistability implies Ding polystability for Gorenstein toric Fano varieties.

Additivity of the Mabuchi constant for product toric Fano varieties.

Constructed examples showing differences between relative K-stability and Ding stability.

Abstract

In this paper, we prove that a Gorenstein toric Fano variety $(X, -K_{X})$ is asymptotically Chow semistable then it is Ding polystable with respect to toric test configurations (Theorem 1.3). This extends the known result obtained by others (Theorem 1.2) to the case where $X$ admits Gorenstein singularity. We also show the additivity of the Mabuchi constant for the product toric Fano varieties in Proposition 1.5 based on the author's recent work (Ono, Sano and Yotsutani in arXiv:2305.05924). Applying this formula to certain toric Fano varieties, we construct infinitely many examples that clarify the difference between relative K-stability and relative Ding stability in a systematic way (Proposition 1.4). Finally, we verify relative Chow stability for Gorenstein toric del Pezzo surfaces using the combinatorial criterion developed in (Yotsutani and Zhou in Tohoku Math. J. 71 (2019), 495-524.) and specifying the symmetry of the associated polytopes as well.