### 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) |
---|---|

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 |

### Fingerprint

### 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

**A Gromov-Winkelmann type theorem for flexible varieties.** / Flenner, Hubert; Kaliman, Shulim; Zaidenberg, Mikhail.

Research output: Contribution to journal › Article

*Journal of the European Mathematical Society*, vol. 18, no. 11, pp. 2483-2510. https://doi.org/10.4171/JEMS/646

}

TY - JOUR

T1 - A Gromov-Winkelmann type theorem for flexible varieties

AU - Flenner, Hubert

AU - Kaliman, Shulim

AU - Zaidenberg, Mikhail

PY - 2016

Y1 - 2016

N2 - An affine variety X of dimension ≥ 2 is called flexible if its special automorphism group SAut(X) acts transitively on the smooth locus Xreg. 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 An, n ≥ 2, this yields a theorem of Gromov and Winkelmann [8], [18].

AB - An affine variety X of dimension ≥ 2 is called flexible if its special automorphism group SAut(X) acts transitively on the smooth locus Xreg. 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 An, n ≥ 2, this yields a theorem of Gromov and Winkelmann [8], [18].

KW - Affine varieties

KW - Group actions

KW - One-parameter subgroups

KW - Transitivity

UR - http://www.scopus.com/inward/record.url?scp=84991634660&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84991634660&partnerID=8YFLogxK

U2 - 10.4171/JEMS/646

DO - 10.4171/JEMS/646

M3 - Article

VL - 18

SP - 2483

EP - 2510

JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

SN - 1435-9855

IS - 11

ER -