Asymptotic Chow stability of uniformly K-stable toric varieties

Abstract

For a polarized toric variety, we provide a sufficient criterion ensuring that a uniformly K-stable polarized toric variety $(X,L)$ is asymptotically Chow polystable, under the assumption that the obstruction to asymptotic Chow semistability (the Futaki-Ono invariant) vanishes. Our approach is based on a detailed study of triangulations of neighborhoods of the vertices of the associated moment polytope $\Delta$. As an application, we prove that every uniformly K-stable polarized smooth toric variety $(X,L)$ with vanishing Futaki-Ono invariant is asymptotically Chow polystable.

Second version: The title has been slightly changed. We have strengthened the main theorem and corrected certain logical gaps in the original proof. In particular, we have significantly improved the technical details regarding the triangulations of vertex cones, which are used to estimate the integrals over each simplex