On Fano manifolds of Picard number one with big automorphism groups

Abstract

Let $X$ be an $n$-dimensional smooth Fano complex variety of Picard number one. Assume that the VMRT at a general point of $X$ is smooth irreducible and non-degenerate (which holds if $X$ is covered by lines with index $>(n+2)/2$). It is proven that $\dim \mathfrak{aut}(X)>n(n+1)/2$ if and only if $X$ is isomorphic to ${\mathbb{P}}^n,{\mathbb{Q}}^n$ or $\mathrm{Gr}(2,5)$. Furthermore, the equality $\dim \mathfrak{aut}(X)=n(n+1)/2$ holds only when $X$ is isomorphic to the 6-dimensional Lagrangian Grassmannian $\mathrm{Lag}(6)$ or a general hyperplane section of $\mathrm{Gr}(2,5)$.