The sharp volume gap for K\"ahler manifolds with positive Ricci curvature

Abstract

We prove a sharp volume gap estimate: if an $n$-dimensional compact K\"ahler manifold $(X, \omega)$ satisfies $\mathrm{Ric}(\omega)\ge (n+1)\omega$ and $X\not\cong \mathbb{P}^n$, then $\mathrm{vol}(X, \omega)\le \frac{2n^n}{(n+1)^n}\mathrm{vol}(\mathbb{P}^n,\omega_{\mathrm{FS}})=\frac{2^{n+1} \, \pi^n \, n^n}{(n+1)^n}$. Moreover $\mathrm{vol}(X, \omega)= \frac{2^{n+1} \, \pi^n \, n^n}{(n+1)^n}$ occurs if and only if $(X, \omega)$ is biholomorphically isometric to the K\"ahler-Einstein metric on the quadric hypersurface $Q^n$ or on the product $\mathbb{P}^1\times \mathbb{P}^{n-1}$. We also obtain sharp volume gap estimates for K-semistable toric log Fano pairs.