When is the automorphism group of an affine variety linear?

TL;DR

This paper investigates when the automorphism group of an affine variety can be linear, showing that under certain conditions, it cannot be embedded into a general linear group, and characterizing when the automorphism group is algebraic.

Contribution

It provides new criteria for the non-linearity of automorphism groups of affine varieties and characterizes when these groups are algebraic, extending understanding of their structure.

Findings

Aut_{alg}(X) is not linear if it is sufficiently rich and dim X ≥ 2.

Aut(X) is algebraic only if its connected component is a torus or a direct limit of unipotent groups.

For uncountable fields, the birational transformation group cannot be isomorphic to automorphism groups of affine varieties.

Abstract

Let $Aut_{alg}(X)$ be the subgroup of the group of regular automorphisms $Aut(X)$ of an affine algebraic variety $X$ generated by all connected algebraic subgroups. We prove that if $dim X \ge 2$ and if $Aut_{alg}(X)$ is rich enough, $Aut_{alg}(X)$ is not linear, i.e., it cannot be embedded into $GL_n(K)$, where $K$ is an algebraically closed field of characteristic zero. Moreover, $Aut(X)$ is isomorphic to an algebraic group as an abstract group only if the connected component of $Aut(X)$ is either the algebraic torus or a direct limit of commutative unipotent groups. Finally, we prove that for an uncountable $K$ the group of birational transformations of $X$ cannot be isomorphic to the group of automorphisms of an affine variety if $X$ is endowed with a rational action of a positive-dimensional linear algebraic group.