Duke Mathematical Journal

Flexible varieties and automorphism groups

I. Arzhantsev, H. Flenner, S. Kaliman, F. Kutzschebauch, and M. Zaidenberg

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Given an irreducible affine algebraic variety $X$ of dimension $n\ge2$, we let $\operatorname {SAut}(X)$ denote the special automorphism group of $X$, that is, the subgroup of the full automorphism group $\operatorname{Aut}(X)$ generated by all one-parameter unipotent subgroups. We show that if $\operatorname {SAut}(X)$ is transitive on the smooth locus $X_{\mathrm{reg}}$, then it is infinitely transitive on $X_{\mathrm{reg}}$. In turn, the transitivity is equivalent to the flexibility of $X$. The latter means that for every smooth point $x\in X_{\mathrm{reg}}$ the tangent space $T_{x}X$ is spanned by the velocity vectors at $x$ of one-parameter unipotent subgroups of $\operatorname{Aut}(X)$. We also provide various modifications and applications.

Article information

Duke Math. J. Volume 162, Number 4 (2013), 767-823.

First available in Project Euclid: 15 March 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14R20: Group actions on affine varieties [See also 13A50, 14L30] 32M17: Automorphism groups of Cn and affine manifolds
Secondary: 14L30: Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17]


Arzhantsev, I.; Flenner, H.; Kaliman, S.; Kutzschebauch, F.; Zaidenberg, M. Flexible varieties and automorphism groups. Duke Math. J. 162 (2013), no. 4, 767--823. doi:10.1215/00127094-2080132. http://projecteuclid.org/euclid.dmj/1363355693.

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