On the volume of K-semistable Fano manifolds

TL;DR

The paper establishes a sharp upper bound ${\rm vol}(X)=(-K_X)^n\le 2n^n$ for $n$-dimensional K-semistable Fano manifolds that are not ${\mathbb P}^n$, with equality only for ${\mathbb P}^1\times{\mathbb P}^{n-1}$ or a smooth quadric $Q\subset {\mathbb P}^{n+1}$. It introduces a novel link between K-semistability and minimal rational curves by analyzing two divisorial valuations arising from weighted blowups along minimal rational curves, and uses this to bound volumes via the delta-invariant and Seshadri constants. The argument combines a general submanifold-with-trivial-normal-bundle framework, careful treatment of singular minimal rational curves, and a detailed analysis of cases determined by the minimal anticanonical degree $l_X$, culminating in a complete proof (Theorem thm-main) and several related consequences, including toric and ODP-analytic implications. The results offer a conceptual explanation for the two extremal equality cases and provide a robust algebraic-geometry route to sharp volume bounds with potential extensions to singular and toric settings.

Abstract

We prove that the anti-canonical volume of an $n$-dimensional K-semistable Fano manifold that is not $\mathbb{P}^n$ is at most $2n^n$. Moreover, the volume is equal to $2n^n$ if and only if $X\cong \mathbb{P}^1\times \mathbb{P}^{n-1}$ or $X$ is a smooth quadric hypersurface $Q\subset \mathbb{P}^{n+1}$. Our proof is based on a new connection between K-semistability and minimal rational curves.

Figures (1)

• Figure 1: $n=7, 7\ge d\ge 2$

Theorems & Definitions (62)

• Theorem 1.1
• Theorem 1.2: See Theorem \ref{['thm:vtoric']}
• Corollary 1.3: See Corollary \ref{['cor:polytope']}
• Theorem 1.4
• Theorem 2.1
• Definition 2.2
• Conjecture 2.3
• Lemma 2.4
• proof
• Corollary 2.5