Infinite Transitivity on Affine Varieties

Abstract

Abstract. 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.

Flexible affine cones over del Pezzo surfaces of degree 4

“…Moreover, in [3], it is proved that the suspensions over flexible varieties are also flexible. One can find some other examples of flexible varieties in [1,2].…”

Infinite transitivity for Calogero-Moser spaces

“…Using 2-transitivity, find an automorphism mapping (X i , Y i ) to (X i , Y i ) and (X j , Y j ) to (X ′ j , Y j ). As above, we decompose it into automorphisms of the forms (1) and (2) and regard it as an element of a one-parameter family of automorphisms with t = 1 (not a subgroup!). We want all the matrices to remain diagonalizable, this forbids a finite number of values of t. We do not want to break edges that were constructed earlier, so for each old edge kl, as we did in Remark 2, we express the condition that (X k and X l have no common eigenvalue)&(Y k and Y l have no common eigenvalue) as a polynomial condition on t that holds for t = 0.…”

“…For a variety X, one can generate a group by all the one-parameter unipotent subgroups of Aut(X). This subgroup denoted by SAut(X) is treated in [9,3,1,2]. It is shown in [1] that infinite transitivity of SAut(X) on the smooth locus reg(X) for dim X 2 is equivalent to simple transitivity and is equivalent to flexibility property which means that the tangent space T x X in every smooth point x ∈ X is generated by tangent vectors to the orbits of one-parameter unipotent subgroups.…”

“…The key ingredient of the proof is that, whenever X-components of the given points have pairwise coprime minimal polynomials, the given points can be moved independently via automorphisms of the form (2).…”

Approximation and interpolation of regular maps from affine varieties to algebraic manifolds

“…This is Gromov's term. The property could also be called relative ellipticity 2. An algebraic manifold is a connected smooth algebraic variety over C, by definition quasi-compact in the Zariski topology.…”