Some characterizations of the complex projective space via Ehrhart polynomials

Abstract

Let [Formula: see text] be the Ehrhart polynomial associated to an integral multiple [Formula: see text] of the standard simplex [Formula: see text]. In this paper, we prove that if [Formula: see text] is an [Formula: see text]-dimensional polarized toric manifold with associated Delzant polytope [Formula: see text] and Ehrhart polynomial [Formula: see text] such that [Formula: see text], for some [Formula: see text], then [Formula: see text] (where [Formula: see text] is the hyperplane bundle on [Formula: see text]) in the following three cases: (1) arbitrary [Formula: see text] and [Formula: see text], (2) [Formula: see text] and [Formula: see text] and (3) [Formula: see text] under the assumption that the polarization [Formula: see text] is asymptotically Chow semistable.