Flexible varieties and automorphism groups

Abstract

Given an affine algebraic variety X of dimension at least 2, we let SAut (X) denote the special automorphism group of X i.e., the subgroup of the full automorphism group Aut (X) generated by all one-parameter unipotent subgroups. We show that if SAut (X) is transitive on the smooth locus of X then it is infinitely transitive on this locus. In turn, the transitivity is equivalent to the flexibility of X. The latter means that for every smooth point x of X the tangent space at x is spanned by the velocity vectors of one-parameter unipotent subgroups of Aut (X). We provide also different variations and applications.

Final version; to appear in Duke Math. J

References in corpus (7)

• Criteria for the density property of complex manifolds
• Flexibility of affine cones over del Pezzo surfaces of degree 4 and 5
• The homotopy principle in complex analysis: a survey
• Affine Toric SL(2)-embeddings
• Makar-Limanov invariant, Derksen invariant, flexible points
• Smooth Affine Surfaces with Non-Unique C*-Actions
• Algebraic density property of Danilov-Gizatullin surfaces

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