# A Gromov–Winkelmann type theorem for flexible varieties

### Hubert Flenner

Ruhr-Universität Bochum, Germany### Shulim Kaliman

University of Miami, Coral Gables, USA### Mikhail Zaidenberg

Université Grenoble I, Saint-Martin-D'hères, France

## 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 $m ≥ 1$. 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 ≥ 2, this yields a Theorem of Gromov and Winkelmann [8], [18].

## Cite this article

Hubert Flenner, Shulim Kaliman, Mikhail Zaidenberg, A Gromov–Winkelmann type theorem for flexible varieties. J. Eur. Math. Soc. 18 (2016), no. 11, pp. 2483–2510

DOI 10.4171/JEMS/646