Automorphism groups of affine varieties consisting of algebraic elements

TL;DR

This paper proves that for affine varieties, if the neutral component of the automorphism group consists of algebraic elements, then it is a nested union of algebraic subgroups, extending previous results.

Contribution

It establishes that the neutral component of the automorphism group is a direct limit of algebraic subgroups under certain conditions, improving earlier findings.

Findings

The neutral component is nested, i.e., a direct limit of algebraic subgroups.

Connected ind-groups containing a nested subgroup with certain power conditions are equal.

Provides a criterion for the structure of automorphism groups of affine varieties.

Abstract

Given an affine algebraic variety $X$, we prove that if the neutral component $\mathrm{Aut}^\circ(X)$ of the automorphism group consists of algebraic elements, then it is nested, i.e., is a direct limit of algebraic subgroups. This improves our earlier result. To prove it, we obtain the following fact. If a connected ind-group $G$ contains a closed connected nested ind-subgroup $H\subset G$, and for any $g\in G$ some positive power of $g$ belongs to $H$, then $G=H.$