A necessary condition for Chow semistability of polarized toric manifolds

Abstract

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