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International conference "Birational and affine geometry"
April 26, 2012 10:00–10:50, Moscow, Steklov Mathematical Institute of RAS

Flexible varieties and Automorphism groups

Sh. Kaliman

University of Miami
 Video records: Flash Video 336.5 Mb Flash Video 2,047.1 Mb MP4 336.5 Mb

Abstract: Given an affine algebraic variety $X$ of dimension $n\geqslant 2$, we let $\mathrm{SAut}(X)$ denote the special automorphism group of $X$ i.e., the subgroup of the full automorphism group $\mathrm{Aut}(X)$ generated by all one-parameter unipotent subgroups. We show that if $\mathrm{SAut}(X)$ is transitive on the smooth locus $X_{reg}$ then it is infinitely transitive on $X_{reg}$. In turn, the transitivity is equivalent to the flexibility of $X$. The latter means that for every smooth point $x\in X_{reg}$ the tangent space $T_x X$ is spanned by the velocity vectors at $x$ of one-parameter unipotent subgroups of $\mathrm{Aut}(X)$. We provide also different variations and applications.