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 a very ample $(\mathbb{C}×)^n$-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 $( X_{\Delta}, {L^i}_{\Delta})$ 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.