Speakers

Title: Structure of unipotent automorphism groups

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.

Title: Flexibility of affine spherical varieties

Abstract: A smooth point x of an algebraic variety X is called flexible if the tangent space TₓX is generated by the tangent vectors to the orbits of Gₐ-actions passing through the point x. A variety X is called flexible if all its smooth points are flexible. Let SAut(X) denote the subgroup of Aut(X) generated by all Gₐ-subgroups. It was proven by Arzhantsev,  Flenner,  Kaliman,  Kutzschebauch, and Zaidenberg that an affine variety X is flexible if and only if the group SAut(X) acts transitively on the set of smooth points of X. Moreover, in this case, the group SAut(X) acts m-transitively on the set of smooth points for any m.

Flexible varieties must have a sufficiently large automorphism group. Therefore, algebraic varieties on which an algebraic group acts with an open orbit are natural candidates for flexibility. Among examples of flexible varieties are affine toric varieties without non‑constant invertible regular functions, affine cones over flag varieties, and smooth almost homogeneous spaces of semisimple groups.

Affine toric varieties and affine cones over flag varieties are examples of spherical varieties. Spherical varieties are normal varieties equipped with an action of a reductive group, such that a Borel subgroup has an open orbit. It was conjectured that affine spherical varieties are flexible if and only if they have no non-constant invertible regular functions. In the talk I will explain why this conjecture holds. Furthermore, I will show that for any affine spherical variety, the group Aut(X) acts transitively on the set of smooth points. The talk will be based on the arXiv:2512.07031.