Structure of connected nested automorphism groups

TL;DR

The paper develops a structural theory for Aut(X) as an ind-group, focusing on nested unipotent subgroups and their maximal dJ-like descriptions. It introduces dJ-like subalgebras and subgroups as natural generalizations of triangular de Jonquières structures, proves their closedness, and shows that connected nested subgroups are closed, with a Levi-type decomposition for connected cases. It extends recent work on maximal commutative unipotent subgroups by providing a constructive link to locally free G_a^k-subgroups and the R_X(V) framework, and it establishes a maximality criterion for dJ-like subgroups via extendability of locally free subgroups. The results unify topological and algebraic perspectives on Aut(X), clarifying how maximal nested unipotent subgroups sit inside de Jonquières-type contexts and how commutative unipotent subgroups arise from invariant function fields.

Abstract

A nested group is an increasing union of a sequence of algebraic groups. In this paper, we describe maximal nested unipotent subgroups of $\mathrm{Aut}(X)$, where $X$ is an affine variety. It turns out that they are similar to the group of triangular automorphisms of $\mathbb{A}^n$. We show that if an abstract subgroup of $\mathrm{Aut}(X)$ consists of unipotent elements, then it is closed if and only if it is nested. This implies that a connected nested subgroup of $\mathrm{Aut}(X)$ is closed, answering a question of Kraft and Zaidenberg (2022, arXiv:2203.11356). We also extend the recent description of maximal commutative unipotent subgroups by Regeta and van Santen (2024, arXiv:2112.04784), offering a direct construction method and relating them to our description.

Theorems & Definitions (130)

• Theorem 1.1: KZ
• Theorem 1.2: Theorem \ref{['th:nested=closed']}
• Theorem 1.4: Theorem \ref{['th:nested-closed']}
• Theorem 1.5: RvS
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Definition 3.1: Skutin
• Theorem 3.2: Skutin
• Corollary 3.3