A necessary condition for Chow semistability of polarized toric manifolds

Abstract

Let \Delta\subset \mathbb{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 (\mathbb{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 of Futaki, Sano and the author to the case when X_\Delta is not necessarily Fano.