Asymptotic Chow stability of symmetric reflexive toric varieties

TL;DR

This paper investigates the asymptotic Chow stability of symmetric reflexive toric varieties, providing criteria and examples that distinguish stable, semistable, and unstable cases, with implications for geometric stability theory.

Contribution

It introduces new criteria for asymptotic Chow polystability of symmetric reflexive toric varieties and provides explicit examples illustrating these stability conditions.

Findings

Some symmetric reflexive toric varieties are not asymptotic Chow semistable.

Weakly symmetric reflexive toric varieties with regular triangulation are asymptotic Chow polystable.

Examples of asymptotic Chow polystable varieties that are not special are provided.

Abstract

In this note, we study the asymptotic Chow stability of toric varieties. We provide examples of symmetric reflexive toric varieties that are not asymptotic Chow semistable. On the other hand, we also show that any weakly symmetric reflexive toric varieties which have regular triangulation (special) are asymptotic Chow polystable. After that, we provide another criteria that can show a symmetric reflexive toric variety is asymptotic Chow polystable. In particular, we give two examples that are asymptotic Chow polystable, but not special. We also provide some examples of special polytopes, mainly in 2 or 3 dimensions, and some in higher dimensions.