On Fano manifolds of Picard number one with big automorphism groups

TL;DR

This paper classifies smooth Fano manifolds with Picard number one that have large automorphism groups, showing they are isomorphic to well-known homogeneous varieties like projective space, quadrics, or Grassmannians.

Contribution

It provides a complete characterization of Fano manifolds with maximal automorphism groups, identifying specific varieties based on automorphism group dimension.

Findings

Automorphism group dimension exceeds n(n+1)/2 only for projective space, quadrics, or Gr(2,5).

Equality in automorphism dimension occurs for Lag(6) and certain hyperplane sections.

Classifies Fano manifolds with big automorphism groups under geometric conditions.

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 ${\rm Gr}(2,5)$. Furthermore, the equality $\dim \mathfrak{aut}(X) = n(n+1)/2$ holds only when $X$ is isomorphic to the 6-dimensional Lagrangian Grassmannian ${\rm Lag}(6)$ or a general hyperplane section of ${\rm Gr}(2,5)$.