On the geometry of the automorphism groups of affine varieties

Abstract

This article is a survey on ind-varieties and ind-groups introduced by Shafarevich in 1965, with a special emphasis on automorphism groups of affine varieties and actions of ind-groups on ind-varieties. We give precise definitions and complete proofs, including several known results. The survey contains many examples and also some questions which came up during our work on the subject. Among the new results we show that for an affine variety X the automorphism group Aut(X) is always locally closed in the ind-semigroup End(X) of all endomorphisms, and we give an example of a strict closed subgroup of a connected ind-group which has the same Lie algebra, based on the work of Shestakov-Umirbaev on the existence of non-tame automorphisms of affine 3-space.

References in corpus (1)

• Nonlinearity of some subgroups of the planar Cremona group

Cited by in corpus (12)

• Infinite transitivity, finite generation, and Demazure roots
• Algebraically generated groups and their lie algebras
• Some finiteness results on triangular automorphisms
• When is the automorphism group of an affine variety nested?
• On the characterization of Danielewski surfaces by their automorphism group
• Covariants, Invariant Subsets, and First Integrals
• Borel subgroups of the plane Cremona group
• Characterizing quasi-affine spherical varieties via the automorphism group
• Topologically integrable derivations and additive group actions on affine ind-schemes
• On Automorphisms of the Tame Polynomial Automorphism Group in Positive Characteristic
• On finite-dimensional homogeneous Lie algebras of derivations of polynomial rings
• Polynomial automorphisms, quantization and Jacobian conjecture related problems