A necessary condition for Chow semistability of polarized toric manifolds

TL;DR

This paper establishes a necessary geometric condition for the Chow semistability of polarized toric manifolds, linking the sum of lattice points in scaled polytopes to barycenters, and extends previous results to non-Fano cases.

Contribution

It provides a new necessary condition for Chow semistability of polarized toric manifolds, generalizing prior results to include non-Fano cases.

Findings

Sum of lattice points in scaled polytope is proportional to barycenter if Chow semistable.

Necessary condition for asymptotic Chow semistability of polarized toric manifolds.

Extension of previous stability results to non-Fano 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.