Shulim Kaliman

Flexible varieties and automorphism groups

Given an irreducible affine algebraic variety X of dimension n≥2 , we let SAut(X) denote the special automorphism group of X , that is, 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 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∈X reg the tangent space T x X is spanned by the velocity vectors at x of one-parameter unipotent subgroups of Aut(X) . We also provide various modifications and applications.

Affine modifications and affine hypersurfaces with a very transitive automorphism group

We study the modificationA→A′ of an affine domainA which produces another affine domainA′=A[I/f] whereI is a nontrivial ideal ofA andf is a nonzero element ofI. First appeared in passing in the basic paper of O. Zariski [Zar], it was further considered by E. D. Davis [Da]. In [Ka1] its geometric counterpart was applied to construct contractible smooth affine varieties non-isomorphic to Euclidean spaces. Here we provide certain conditions (more general than those in [Ka1]) which guarantee preservation of the topology under a modification.As an application, we show that the group of biregular automorphisms of the affine hypersurfaceX⊂Ck+2, given by the equationuv=(p(x1,...,xk) wherep∈C[x1,...,xk],k≥2, actsm-transitively on the smooth part regX ofX for anym∈N. We present examples of such hypersurfaces diffeomorphic to Euclidean spaces.

Exotic analytic structures and Eisenman intrinsic measures

Using Eisenman intrinsic measures we prove a cancellation theorem. This theorem allows to find new examples of exotic analytic structures onCn under which we understand smooth complex affine algebraic varietiers which are diffeomorphic toR2n but not biholomorphic toCn. We also develop a new method of constructing these structures which enables us to produce exotic analytic structures onC3 with a given number of hypersurfaces isomorphic toC2 and a family of these structures with a given number of moduli.

Algebraic volume density property of affine algebraic manifolds

We introduce the notion of algebraic volume density property for affine algebraic manifolds and prove some important basic facts about it, in particular that it implies the volume density property. The main results of the paper are producing two big classes of examples of Stein manifolds with volume density property. One class consists of certain affine modifications of ℂn equipped with a canonical volume form, the other is the class of all Linear Algebraic Groups equipped with the left invariant volume form.

A Note on Locally Nilpotent Derivations and Variables of k(X,Y,Z)

We strengthen certain results concerning actions of

On the Danilov-Gizatullin Isomorphism Theorem

\left( \mathbb{C},\,+ \right)

Smooth affine surfaces with non-unique ℂ*-actions

on

Simple birational extensions of the polynomial algebra ℂ^{[3]}

{{\mathbb{C}}^{3}}

Infinite transitivity on affine varieties

and embeddings of

On the Density and the Volume Density Property

{{\mathbb{C}}^{2}}