On the equality case in Ehrhart's volume conjecture

TL;DR

This paper proves the uniqueness of projective space as the maximal degree toric Kähler-Einstein Fano manifold, confirming a convex-geometric aspect of Ehrhart's volume conjecture and discussing its generalizations.

Contribution

It establishes the uniqueness of projective space in the context of Ehrhart's conjecture for toric manifolds, extending previous results to a convex-geometric setting.

Findings

Projective space is the only toric manifold with maximal degree.

Confirmed the convex-geometric statement related to Ehrhart's conjecture.

Discussed a generalized Ehrhart's conjecture involving Ricci curvature.

Abstract

Ehrhart's conjecture proposes a sharp upper bound on the volume of a convex body whose barycenter is its only interior lattice point. Recently, Berman and Berndtsson proved this conjecture for a class of rational polytopes including reflexive polytopes. In particular, they showed that the complex projective space has the maximal anticanonical degree among all toric Kaehler-Einstein Fano manifolds. In this note, we prove that projective space is the only such toric manifold with maximal degree by proving its corresponding convex-geometric statement. We also discuss a generalized version of Ehrhart's conjecture involving an invariant corresponding to the so-called greatest lower bound on the Ricci curvature.