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 X-reg. Recall that SAut. (X) is the subgroup of the automorphism group Aut. (X) generated by all one-parameter unipotent subgroups [2]. 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 prove 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].