Algebro-geometric semistability of polarized toric manifolds

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.