Automorphism groups of affine varieties consisting of algebraic elements

TL;DR

This work studies the ind-group structure of automorphism groups of affine varieties and identifies conditions under which the connected component ${ m Aut}^ olinebreak[0]{ m c}(X)$ is nested and decomposes as ${f T} times{ m U}(X)$, where ${f T}$ is a maximal torus and ${ m U}(X)$ is the (abelian) unipotent part generated by ${f G}_a$-actions. The authors extend earlier results on ${ m Aut}_{alg}(X)$ to ${ m Aut}^ olinebreak[0]{ m c}(X)$ by proving a general ind-group theorem: if every element of a connected ind-group $G$ has a positive power in a closed connected nested ind-subgroup $H$, then $G=H$. They then apply this to show that if ${ m Aut}^ olinebreak[0]{ m c}(X)$ consists of algebraic elements, it equals ${f T} times{ m U}(X)$, with ${ m U}(X)$ abelian and ${f T}$ a maximal torus; in the rigid case (no ${f G}_a$-actions), ${ m Aut}^ olinebreak[0]{ m c}(X)$ is essentially torus-dominated. This clarifies the structure of large automorphism groups of affine varieties and provides a lever for understanding their abelian and unipotent components in terms of nesting and torus actions.

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 $G=H.$

Theorems & Definitions (28)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Proposition 2.3
• Definition 2.4: FK
• Remark 2.5
• Theorem 3.1
• Lemma 3.2
• proof
• Definition 3.3