Author

Kewei Zhang

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Preprint Aug 2026

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

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.

Chi Li, Minghao Miao, Kewei Zhang · 0 citations

Preprint Aug 2026

Quantized Volume Comparison for Fano Manifolds, II

In this note, the second author's quantized volume comparison conjecture is solved: If $X$ is a $K$-semistable Fano manifold of dimension $n$, then for every integer $m\geq1$, \[ h^0(X,-mK_X)\leq h^0(\mathbb P^n,-mK_{\mathbb P^n}) =\binom{n+m(n+1)}{n}, \] and equality for one $m$ characterizes projective space. Somewhat surprisingly, the same statement actually holds whenever $T_X$ is slope semistable with respect to $-K_X$. The idea is to apply a jet-dimension counting trick to a filtration of subsheaves induced by $H^0(X,-mK_X)$. Then the slope semistability condition yields the desired dimension bound.

Kaixuan Lyu, Kewei Zhang · 0 citations