Some characterizations of the complex projective space via Ehrhart polynomials.

Let P λ Σ n be the Ehrhart polynomial associated to an integral multiple λ of the standard simplex Σ n ⊂ ℝ n . In this paper, we prove that if (M , L) is an n -dimensional polarized toric manifold with associated Delzant polytope Δ and Ehrhart polynomial P Δ such that P Δ = P λ Σ n , for some λ ∈ ℤ + , then (M , L) ≅ (ℂ P n , O (λ)) (where O (1) is the hyperplane bundle on ℂ P n ) in the following three cases: (1) arbitrary n and λ = 1 , (2) n = 2 and λ = 3 and (3) λ = n + 1 under the assumption that the polarization L is asymptotically Chow semistable. [ABSTRACT FROM AUTHOR]