Algebro-geometric semistability of polarized toric manifolds

TL;DR

This paper establishes a precise criterion for Chow semistability of polarized toric manifolds derived from Delzant polytopes, linking it to K-semistability in the context of toric degenerations without relying on Riemann-Roch or test configurations.

Contribution

It provides a necessary and sufficient condition for Chow semistability of polarized toric manifolds with maximal torus action, connecting it to K-semistability.

Findings

Chow semistability criterion for polarized toric manifolds

Asymptotic Chow semistability implies K-semistability for toric degenerations

No use of Riemann-Roch theorem or test configurations in proofs

Abstract

Let $\Delta\subset \mathbb{R}^n$ be an $n$-dimensional integral Delzant polytope. It is well-known that there exist the $n$-dimensional compact toric manifold $X_\Delta$ and the very ample $(\mathbb{C}^\times)^n$-equivariant line bundle $L_\Delta$ on $X_\Delta$ associated with $\Delta$. In the present paper, we give a necessary and sufficient condition for Chow semistability of $(X_\Delta,L_\Delta^i)$ for a maximal torus action. We then see that asymptotic (relative) Chow semistability implies (relative) K-semistability for toric degenerations, which is proved by Ross and Thomas, without any knowledge of Riemann-Roch theorem and test configurations.