A necessary condition for Chow semistability of polarized toric manifolds

Abstract

Let $\Delta \subset R^n$ be an $n$-dimensional Delzant polytope. It is well-known that there exist the $n$-dimensional compact toric manifold $X_\Delta$ and the very ample $(C^\times)^n$-equivariant line bundle $L_\Delta$ on $X_\Delta$ associated with $\Delta$. In the present paper, we show that if $(X_\Delta,L_\Delta^i)$ is Chow semistable then the sum of integer points in $i\Delta$ is the constant multiple of the barycenter of $\Delta$. Using this result we get a necessary condition for the polarized toric manifold $(X_\Delta,L_\Delta)$ being asymptotically Chow semistable. Moreover we can generalize the result in [4] to the case when $X_\Delta$ is not necessarily Fano.