A Gromov–Winkelmann type theorem for flexible varieties

Abstract

An affine variety $X$ of dimension ≥ 2 is called flexible if its special automorphism group SAut($X$) acts transitively on the smooth locus $X_{\mathrm{reg}}$ [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 $X\setminus Y$, that is, $m$-transitively for any $m\ge 1$. More generally we show such a result for any quasi-affine variety $X$ and codimension ≥ 2 subset $Y$ of $X$.

In the particular case of $X={\mathbb{A}}^n$, $n\ge 2$, this yields a Theorem of Gromov and Winkelmann [8], [18].