Automorphism groups of affine varieties without non-algebraic elements

Abstract

Given an affine algebraic variety X, we prove that if the neutral component Aut • (X) of the automorphism group consists of algebraic elements, then it is nested, i.e., is a direct limit of algebraic subgroups. This improves our earlier result [5]. To prove it, we obtain 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.

Automorphism groups of rigid affine surfaces: the identity component

“…Some partial results on these conjectures can be found e.g. in [1, Theorem 2.1], [24,Proposition 5], [25,26], [27,Theorems 4.4 and 4.7], [28,Theorem 1.3 and Section 7], and [36,37]. Conjecture 1.0.1 holds, for instance, for toric affine varieties, see [4,Theorem 3].…”