On the automorphisms of hyperplane sections of generalized Grassmannians

TL;DR

This paper investigates whether automorphisms of smooth hyperplane sections of rational homogeneous spaces can be lifted to the ambient Lie group, providing positive results with specific exceptions and detailed descriptions in certain cases.

Contribution

It establishes conditions under which automorphisms of hyperplane sections lift to the Lie group and fully describes automorphisms for adjoint varieties, extending previous work.

Findings

Automorphisms lift in most cases, with known exceptions.

Complete description of automorphisms for adjoint varieties.

Extension of prior results to the exceptional group G2.

Abstract

Given a smooth hyperplane section $H$ of a rational homogeneous space $G/P$ with Picard number one, we address the question whether it is always possible to lift an automorphism of $H$ to the Lie group $G$, or more precisely to Aut$(G/P)$. Using linear spaces and quadrics in $H$, we show that the answer is positive up to a few well understood exceptions related to Jordan algebras. When $G/P$ is an adjoint variety, we show how to describe Aut$(H)$ completely, extending results obtained by Prokhorov and Zaidenberg when $G$ is the exceptional group $G_2$.