Some characterizations of the complex projective space via Ehrhart polynomials

TL;DR

This paper characterizes complex projective spaces by their Ehrhart polynomials, showing that certain Ehrhart polynomial equalities imply the manifold is a complex projective space with a specific polarization.

Contribution

It proves that if a polarized toric manifold's Ehrhart polynomial matches that of a scaled standard simplex, then the manifold is a complex projective space with a corresponding line bundle, under specific conditions.

Findings

Identifies conditions under which Ehrhart polynomial equality characterizes complex projective space.

Establishes that the manifold is isomorphic to C P^n with a hyperplane bundle for certain Ehrhart polynomial cases.

Provides new characterizations of complex projective space via Ehrhart polynomial properties.

Abstract

Let $P_{\lambda\Sigma_n}$ be the Ehrhart polynomial associated to an intergal multiple $\lambda$ of the standard symplex $\Sigma_n \subset \mathbb{R}^n$. In this paper we prove that if $(M, L)$ is an $n$-dimensional polarized toric manifold with associated Delzant polytope $\Delta$ and Ehrhart polynomial $P_\Delta$ such that $P_{\Delta}=P_{\lambda\Sigma_n}$, for some $\lambda \in \mathbb{Z}^+$, then $(M, L)\cong (\mathbb{C} P^n, O(\lambda))$ (where $O(1)$ is the hyperplane bundle on $\mathbb{C} P^n$) in the following three cases: 1. arbitrary $n$ and $\lambda=1$, 2. $n=2$ and $\lambda =3$, 3. $\lambda =n+1$ under the assumption that the polarization $L$ is asymptotically Chow semistable.