Holomorphic families of non-equivalent embeddings and of holomorphic group actions on affine space

We construct holomorphic families of proper holomorphic embeddings of \C^k into \C^n ($0<k<n-1$), so that for any two different parameters in the family no holomorphic automorphism of \C^n can map the image of the corresponding two embeddings onto each other. As an application to the study of the group of holomorphic automorphisms of \C^n we derive the existence of families of holomorphic \C^*-actions on \C^n ($n\ge 5$) so that different actions in the family are not conjugate. This result is surprising in view of the long standing Holomorphic Linearization Problem, which in particular asked whether there would be more than one conjugacy class of \C^* actions on \C^n (with prescribed linear part at a fixed point)

Exponential factorizations of holomorphic maps

We show that any element of the special linear group $SL_2(R)$ is a product of two exponentials if the ring $R$ is either the ring of holomorphic functions on an open Riemann surface or the disc algebra. This is sharp: one exponential factor is not enough since the exponential map corresponding to $SL_2(\mathbb{C})$ is not surjective. Our result extends to the linear group $GL_2(R)$.Comment: 9 page

On Kazhdan's Property (T) for the special linear group of holomorphic functions

We investigate when the group $SL_n(\mathcal{O}(X))$ of holomorphic maps from a Stein space $X$ to SL_n (\C) has Kazhdan's property (T) for $n\ge 3$. This provides a new class of examples of non-locally compact groups having Kazhdan's property (T). In particular we prove that the holomorphic loop group of SL_n (\C) has Kazhdan's property (T) for $n\ge 3$. Our result relies on the method of Shalom to prove Kazhdan's property (T) and the solution to Gromov's Vaserstein problem by the authors.Comment: 5 page

Criteria for the density property of complex manifolds

In this paper we suggest new effective criteria for the density property. This enables us to give a trivial proof of the original Anders\'en-Lempert result and to establish (almost free of charge) the algebraic density property for all linear algebraic groups whose connected components are different from tori or \C_+. As another application of this approach we tackle the question (asked among others by F. Forstneri\v{c}) about the density of algebraic vector fields on Euclidean space vanishing on a codimension 2 subvariety.Comment: to appear in Invent. Mat

On the present state of the Andersen-Lempert theory

In this survey of the Andersen-Lempert theory we present the state of the art in the study of the density property (which means that the Lie algebra generated by completely integrable holomorphic vector fields on a given Stein manifold is dense in the space of all holomorphic vector fields). There are also two new results in the paper one of which is the theorem stating that the product of Stein manifolds with the volume density property possesses such a property as well. The second one is a meaningful example of an algebraic surface without the algebraic density property. The proof of the last fact requires Brunella's technique.Comment: 40 page