arXiv subjects

I. Arzhantsev

Publications and source records attributed to I. Arzhantsev.

5 recordsLinked to original sources

arXiv2023

Varieties covered by affine spaces, uniformly rational varieties and their cones

It was shown in [S. Kaliman, M. Zaidenberg, Gromov ellipticity of cones over projective manifolds, Math. Res. Lett. (to appear), arXiv:2303.02036 (2023)] that the affine cones over flag manifolds and rational smooth projective surfaces are elliptic in the sense of Gromov. The latter remains true after successive blowups of points on these varieties. In the present note we extend this to smooth projective spherical varieties (in particular, toric varieties) successively blown up along smooth subvarieties. The same also holds, more generally, for uniformly rational projective varieties, in particular, for projective varieties covered by affine spaces. It occurs also that stably uniformly rational complete varieties are elliptic.

math.AG↗

arXiv2020

Tits type alternative for groups acting on toric affine varieties

Given a toric affine algebraic variety $X$ and a collection of one-parameter unipotent subgroups $U_1,\ldots,U_s$ of $\mathop{\rm Aut}(X)$ which are normalized by the torus acting on $X$, we show that the group $G$ generated by $U_1,\ldots,U_s$ verifies the following alternative of Tits' type: either $G$ is a unipotent algebraic group, or it contains a non-abelian free subgroup. We deduce that if $G$ is $2$-transitive on a $G$-orbit in $X$, then $G$ contains a non-abelian free subgroup, and so, is of exponential growth.

math.AG↗

arXiv2012

Flexible varieties and automorphism groups

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

math.AG↗

arXiv2012

Infinite transitivity on affine varieties

In this note we survey recent results on automorphisms of affine algebraic varieties, infinitely transitive group actions and flexibility. We present related constructions and examples, and discuss geometric applications and open problems.

math.AG↗

arXiv2011

Acyclic curves and group actions on affine toric surfaces

We show that every irreducible, simply connected curve on a toric affine surface X over the field of complex numbers is an orbit closure of a multiplicative group action on X. It follows that up to the action of the automorphism group Aut(X) there are only finitely many non-equivalent embeddings of the affine line in X. A similar description is given for simply connected curves in the quotients of the affine plane by small finite linear groups. We provide also an analog of the Jung-van der Kulk theorem for affine toric surfaces, and apply this to study actions of algebraic groups on such surfaces.

math.AG↗