On the geometry of the automorphism groups of affine varieties

TL;DR

This survey explores the structure of automorphism groups of affine varieties, focusing on ind-varieties and ind-groups, and introduces new results on their local closedness and subgroup properties.

Contribution

It provides a comprehensive overview of ind-varieties and ind-groups with detailed proofs, and presents new findings on automorphism groups' local closedness and subgroup examples.

Findings

Aut(X) is always locally closed in End(X)

Example of a closed subgroup with the same Lie algebra as a connected ind-group

Discussion of non-tame automorphisms of affine 3-space

Abstract

This article is a survey on ind-varieties and ind-groups introduced by Shafarevich in 1965, with a special emphasis on automorphism groups of affine varieties and actions of ind-groups on ind-varieties. We give precise definitions and complete proofs, including several known results. The survey contains many examples and also some questions which came up during our work on the subject. Among the new results we show that for an affine variety X the automorphism group Aut(X) is always locally closed in the ind-semigroup End(X) of all endomorphisms, and we give an example of a strict closed subgroup of a connected ind-group which has the same Lie algebra, based on the work of Shestakov-Umirbaev on the existence of non-tame automorphisms of affine 3-space.