A volume stability theorem on toric manifolds with positive Ricci curvature

Abstract

In this short note, we will prove a volume stability theorem which says that if an $n$-dimensional toric manifold $M$ admits a $\mathbb {T}^n$ invariant Kähler metric $\omega$ with Ricci curvature no less than $1$ and its volume is close to the volume of $\mathbb {CP}^n$, $M$ is bi-holomorphic to $\mathbb {CP}^n$.