A Gromov-Winkelmann type theorem for flexible varieties

Hubert Flenner, Shulim Kaliman, Mikhail Zaidenberg

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Abstract

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].

Original languageEnglish (US)
Pages (from-to)2483-2510
Number of pages28
JournalJournal of the European Mathematical Society
Volume18
Issue number11
DOIs
StatePublished - 2016

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Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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