ALGEBRAIC TRANSFORMATION GROUPs SEMINAR

Abstract: We study unipotent subgroups of the automorphism group of an affine variety X. In the case of an affine space Aⁿ, the unipotent de Jonquières subgroup of triangular automorphisms serves as an infinite-dimensional analogue of upper-triangular matrices U(n) in the matrix group GL(n). It is well known that any maximal unipotent subgroup of GL(n) is conjugate to U(n). However, this statement fails for triangular automorphisms in Aut(Aⁿ).

Our key result is a structural description of maximal unipotent subgroups of Aut(X), which generalizes the notion of the de Jonquières subgroup. This result allows us to give an affirmative answer to the question by  H.Kraft and M.Zaidenberg (arXiv:2203.11356): we show that connected nested subgroups of Aut(X) are closed with respect to the ind-topology. Here a group is called nested if it is a limit of an ascending sequence of algebraic subgroups. Specifically, a unipotent subgroup is closed in Aut(X) if and only if it is nested. The closure of a unipotent subgroup is a nested unipotent subgroup.

We illustrate these results with explicit examples in dimensions 2 and 3. The talk is based on arXiv:2312.08359.