### 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_{reg}. Recall that SAut(X) is the subgroup of the automorphism group Aut(X) generated by all one-parameter unipotent subgroups [2]. 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 \ Y, that is, m-transitively for any m ≥ 1. More generally we prove such a result for any quasi-affine variety X and codimension ≥ 2 subset Y of X. In the particular case of X = double-struck A^{n}, n ≥ 2, this yields a theorem of Gromov and Winkelmann [8], [18].

Original language | English (US) |
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Pages (from-to) | 2483-2510 |

Number of pages | 28 |

Journal | Journal of the European Mathematical Society |

Volume | 18 |

Issue number | 11 |

DOIs | |

State | Published - 2016 |

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### Keywords

- Affine varieties
- Group actions
- One-parameter subgroups
- Transitivity

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Journal of the European Mathematical Society*,

*18*(11), 2483-2510. https://doi.org/10.4171/JEMS/646