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


An affine variety XX of dimension ≥ 2 is called flexible if its special automorphism group SAut(XX) acts transitively on the smooth locus XregX_{\mathrm {reg}} [1]. Recall that SAut(XX) is the subgroup of the automorphism group Aut(XX) generated by all one-parameter unipotent subgroups [1]. Given a normal, flexible, affine variety XX and a closed subvariety YY in XX of codimension at least 2, we show that the pointwise stabilizer subgroup of YY in the group SAut(XX) acts infi nitely transitively on the complement XYX \setminus Y, that is, mm-transitively for any m1m ≥ 1. More generally we show such a result for any quasi-affine variety XX and codimension ≥ 2 subset YY of XX.

In the particular case of X=AnX = \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