On the volume of K-semistable Fano manifolds

TL;DR

This paper establishes an upper bound on the anti-canonical volume of K-semistable Fano manifolds, characterizes the cases of equality, and introduces a new link between K-semistability and minimal rational curves.

Contribution

It provides a sharp volume bound for K-semistable Fano manifolds and characterizes the extremal cases, connecting stability with geometric structures.

Findings

Anti-canonical volume of non-projective K-semistable Fano manifolds is at most 2n^n.

Equality holds iff the manifold is a product of projective spaces or a smooth quadric hypersurface.

New connection established between K-semistability and minimal rational curves.

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.