Boundedness properties of automorphism groups of forms of flag varieties

Attila Guld

Published 2018-06-14Version 1

Let $K$ be a field of characteristic $0$, containing all roots of unity. We call a flag variety admissible if the sequence of nonnegative integers $d_1<...<d_r$ corresponding to the dimensions of the linear subspaces forming an arbitrary flag of the variety is subject to the following condition. There exists an $i\in\{1,..,r\}$ such that $d_i+d_{r-i+1}\neq \dim V$, where $V$ is the underlying vector space of the flag variety. This condition excludes an additional $\mathbb{Z}/2\mathbb{Z}$-symmetry of the flag variety. 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 $C$.