Some characterizations of the complex projective space via Ehrhart polynomials

Let $P_{λΣ_n}$ be the Ehrhart polynomial associated to an intergal multiple $λ$ of the standard symplex $Σ_n \subset \mathbb{R}^n$. 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 $P_Δ=P_{λΣ_n}$, for some $λ\in \mathbb{Z}^+$, then $(M, L)\cong (\mathbb{C} P^n, O(λ))$ (where $O(1)$ is the hyperplane bundle on $\mathbb{C} P^n$) in the following three cases: 1. arbitrary $n$ and $λ=1$, 2. $n=2$ and $λ=3$, 3. $λ=n+1$ under the assumption that the polarization $L$ is asymptotically Chow semistable.

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