A Gromov–Winkelmann type theorem for flexible varieties

An affine variety X of dimension ≥ 2 is called flexible if its special automorphism group SAut( X ) acts transitively on the smooth locus Xreg [1]. Recall that SAut( X ) is the subgroup of the automorphism group Aut( X ) generated by all one-parameter unipotent subgroups [1]. Given a normal, flexible, affine variety X and a closed subvariety Y in X of codimension at least 2, we show that the pointwise stabilizer subgroup of Y in the group SAut( X ) acts infinitely transitively on the complement X Y , that is, m -transitively for any m ≥ 1 . More generally we show such a result for any quasi-affine variety X and codimension ≥ 2 subset Y of X . In the particular case of X = A^n , n ≥2 , this yields a Theorem of Gromov and Winkelmann [8], [18].