Boundedness properties of automorphism groups of forms of flag varieties

TL;DR

This paper investigates the automorphism groups of forms of admissible flag varieties over fields of characteristic zero, establishing conditions under which these groups are either bounded or the varieties are ruled.

Contribution

It proves that forms of admissible flag varieties are either ruled or have bounded automorphism groups, extending understanding of automorphism group properties in algebraic geometry.

Findings

Automorphism groups are bounded or varieties are ruled.

Most flag varieties have automorphism groups isomorphic to projective general linear groups.

Boundedness depends on the structure of the variety and the field characteristics.

Abstract

We call a flag variety admissible if its automorphism group is the projective general linear group. (This holds in most cases.) Let $K$ be a field of characteristic $0$, containing all roots of unity. Let the $K$-variety $X$ be a form of an admissible flag variety. We prove that $X$ is either ruled, or the automorphism group of $X$ is bounded, meaning that, there exists a constant $C\in\mathbb{N}$ such that if $G$ is a finite subgroup of $Aut_K(X)$, then the cardinality of $G$ is smaller than $C$.