Journal of the European Mathematical Society

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Volume 18, Issue 11, 2016, pp. 2483–2510
DOI: 10.4171/JEMS/646

Published online: 2016-10-12

A Gromov–Winkelmann type theorem for flexible varieties

Hubert Flenner[1], Shulim Kaliman[2] and Mikhail Zaidenberg[3]

(1) Ruhr-Universität Bochum, Germany
(2) University of Miami, Coral Gables, USA
(3) Université Grenoble I, Saint-Martin-D'hères, France

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 infi nitely 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].

Keywords: Affine varieties, group actions, one-parameter subgroups, transitivity

Flenner Hubert, Kaliman Shulim, Zaidenberg Mikhail: A Gromov–Winkelmann type theorem for flexible varieties. J. Eur. Math. Soc. 18 (2016), 2483-2510. doi: 10.4171/JEMS/646