Infinite transitivity and special automorphisms

It is known that if the special automorphism group $\mathrm{SAut}(X)$ of a quasiaffine variety $X$ of dimension at least $2$ acts transitively on $X$, then this action is infinitely transitive. In this paper we question whether this is the only possibility for the automorphism group $\mathbb{Aut}(X)$ to act infinitely transitively on $X$. We show that this is the case, provided $X$ admits a nontrivial $\mathbb{G}_a$ or $\mathbb{G}_m$-action. Moreover, $2$-transitivity of the automorphism group implies infinite transitivity.

Keywords

quasiaffine variety, automorphism, transitivity, torus action, rigidity

2010 Mathematics Subject Classification

Primary 14J50, 14M17. Secondary 13A50, 14L30, 14R20.

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The author’s research was supported by the grant RSF-DFG 16-41-01013.

Received 28 October 2016

Received revised 14 May 2017

Accepted 5 June 2017

Published 30 April 2018