The Gromov-Winkelmann theorem for flexible varieties

An affine variety $X$ of dimension $\ge 2$ is called {\em flexible} if its special automorphism group SAut$(X)$ acts transitively on the smooth locus $X_{reg}$ \cite{AKZ}. Recall that the special automorphism group SAut$(X)$ is the subgroup of the automorphism group Aut$(X)$ generated by all one-parameter unipotent subgroups \cite{AKZ}. 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 $X\backslash Y$, that is, $m$-transitively for any $m\ge 1$. More generally we show such a result for any quasi-affine variety $X$ and codimension $\ge 2$ subset $Y$ of $X$. In the particular case of $X=\AA^n$, $n\ge 2$, this yields a Theorem of Gromov and Winkelmann \cite{Gr1}, \cite{Wi}.