On the volume of K-semistable Fano manifolds

Abstract: We prove that the volume of an $n$-dimensional K-semistable Fano manifold that is not $\mathbb{P}<sup>n$ is at most $2n^n$. Moreover, the equality holds only if $$X\cong \mathbb{P}^1\times \mathbb{P}<sup>{n-1}$$ or $X$ is a smooth quadric hypersurface $Q\subset \mathbb{P}<sup>{n+1}$. Our proof is based on a new connection between K-semistability and minimal rational curves. More generally, we show that the volume of a K-semistable Fano manifold with a minimal rational curve of degree $d$ is bounded above by the volume of $$\mathbb{P}^{d-1}\times \mathbb{P}<sup>{n-d+1}$$.