Chow stability of $\lambda$-stable toric varieties

TL;DR

This paper introduces a new stability notion called $$-stability for polarized toric varieties, linking it to Chow stability and K-stability, and provides criteria for asymptotic Chow polystability.

Contribution

It defines $$-stability as a generalization of uniform K-stability and establishes conditions under which $$-stability implies asymptotic Chow polystability for toric varieties.

Findings

$$-stability generalizes uniform K-stability.

Criteria for asymptotic Chow polystability are provided.

K-semistable toric varieties with vanishing Futaki-Ono invariant are asymptotically Chow polystable.

Abstract

For a given polarized toric variety, we define the notion of $\lambda$-stability which is a natural generalization of uniform K-stability. At the neighbourhoods of the vertices of the corresponding moment polytope $\Delta$, we consider appropriate triangulations and give a sufficient criteria for a $\lambda$-stable polarized toric variety $(X,L)$ to be asymptotically Chow polystable when the obstruction of asymptotic Chow semistability (the Futaki-Ono invariant) vanishes. As an application, we prove that any K-semistable polarized smooth toric variety $(X,L)$ with the vanishing Futaki-Ono invariant is asymptotically Chow polystable.