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.Comment: Final version; to appear in Duke Math.

Knotted holomorphic discs in

We construct knotted proper holomorphic embeddings of the unit disc i

Linearization of holomorphic families of algebraic automor- phisms of the affine plane

Let $G$ be a reductive group. We prove that a family of polynomial actions of $G$ on $\mathbb{C}^2$, holomorphically parametrized by an open Riemann surface, is linearizable. As an application, we show that a particular class of reductive group actions on $\mathbb{C}^3$ is linearizable. The main step of our proof is to establish a certain restrictive Oka property for groups of equivariant algebraic automorphisms of $\mathbb{C}^2$

An interpolation theorem for proper holomorphic embeddings

Given a Stein manifold X of dimension n>1, a discrete sequence a_j in X, and a discrete sequence b_j in C^m where m > [3n/2], there exists a proper holomorphic embedding of X into C^m which sends a_j to b_j for every j=1,2,.... This is the interpolation version of the embedding theorem due to Eliashberg, Gromov and Schurmann. The dimension m cannot be lowered in general due to an example of Forster

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 \C^n equipped with a canonical volume form, the other is the class of all Linear Algebraic Groups equipped with the left invariant volume form.Comment: 35 page

Eigentliche Wirkungen von Liegruppen auf reell-analytischen Mannigfaltigkeiten

Available from TIB Hannover: RR 5293(5)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Sufficient conditions for holomorphic linearisation

Let G be a reductive complex Lie group acting holomorphically on X = ℂn. The (holomorphic) Linearisation Problem asks if there is a holomorphic change of coordinates on ℂn such that the G-action becomes linear. Equivalently, is there a G-equivariant biholomorphism Φ: X → V where V is a G-module? There is an intrinsic stratification of the categorical quotient QX, called the Luna stratification, where the strata are labeled by isomorphism classes of representations of reductive subgroups of G. Suppose that there is a Φ as above. Then Φ induces a biholomorphism φ: QX → QV which is stratified, i.e., the stratum of QX with a given label is sent isomorphically to the stratum of QV with the same label. The counterexamples to the Linearisation Problem construct an action of G such that QX is not stratified biholomorphic to any QV.Our main theorem shows that, for most X, a stratified biholomorphism of QX to some QV is sufficient for linearisation. In fact, we do not have to assume that X is biholomorphic to ℂn, only that X is a Stein manifold.Frank Kutzschebauch, Finnur Lárusson, Gerald W. Schwar