63. Александр Перепечко Automorphism groups of affine varieties without non-algebraic elements

Given an affine algebraic variety X, we study when the neutral component Aut°(X) of the automorphism group consists of algebraic elements. It is conjectured that the following conditions on Aut°(X) are equivalent: - all unipotent elements (hence all Ga-actions on X) commute, - it consists of algebraic elements, - it is nested, i.e., a direct limit of algebraic subgroups, - it is a semidirect product of an algebraic torus and an abelian unipotent group. Earlier we proved the conjecture for the group generated by connected algebraic subgroups instead of Aut°(X). In this talk we present our further development: we proved that Aut°(X) consists of algebraic elements if and only if it is nested. To prove it, we obtained the following fact: if a connected ind-group G contains a closed connected ind-subgroup H⊂G with a geometrically smooth point, and for any g∈G some power of g belongs to H, then G=H. We will also discuss possible approaches to the conjecture and related questions. The talk is based on the joint work with Andriy Regeta.