The Gromov-Winkelmann theorem for flexible varieties

TL;DR

This paper extends the Gromov-Winkelmann theorem by proving that for flexible affine varieties, the pointwise stabilizer subgroup acts infinitely transitively on the complement of a codimension at least 2 subvariety, generalizing previous results.

Contribution

It generalizes the Gromov-Winkelmann theorem to broader classes of flexible varieties and establishes infinite transitivity of stabilizer subgroups on complements of codimension at least 2.

Findings

Pointwise stabilizer subgroup acts infinitely transitively on the complement of Y

Generalization from affine space to quasi-affine varieties

Extension of Gromov-Winkelmann theorem to flexible varieties

Abstract

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}.