The sharp volume gap for Kähler manifolds with positive Ricci curvature

Abstract

We prove a sharp volume gap estimate: if an $n$-dimensional compact Kähler manifold $(X,\omega )$ satisfies $\mathrm{Ric}(\omega )\ge (n+1)\omega$ and $X\ncong {\mathbb{P}}^n$, 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 Kähler-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.

29 pages, comments are welcome!