Mid Sweden University

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 >= 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).

We investigate whether a Stein manifold M which allows proper holomorphic embedding into ℂ ⁿ can be embedded in such a way that the image contains a given discrete set of points and in addition follow arbitrarily close to prescribed tangent directions in a neighbourhood of the discrete set. The infinitesimal version was proven by Forstnerič to be always possible. We give a general positive answer if the dimension of M is smaller than n/2 and construct counterexamples for all other dimensional relations. The obstruction we use in these examples is a certain hyperbolicity condition.

In this paper we suggest new effective criteria for the density property. This enables us to give a trivial proof of the original Andersén-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. Forstnerič) about the density of algebraic vector fields on Euclidean space vanishing on a codimension 2 subvariety.

We study the number of equivalence classes of proper holomorphic embeddings of a Stein space X into C-n. In this paper we prove that if the automorphism. group of X is a Lie group and there exists a proper holomorphic embedding of X into C-n, 0 < dim X < n, then for any k >= 0 there exist uncountably many non-equivalent proper holomorphic embeddings Φ: X x C-k hooked right arrow C-n x C-k. For k = 0 all embeddings will be proved to satisfy the additional property of C-n\Φ)(X) being (n-dim X)-Eisenman hyperbolic. As a corollary we conclude that there are uncountably many non-equivalent proper holomorphic embeddings of C-k into C-n whenever 0 < k < n.

We study hypersurfaces of Cn+2 ¯x,u,v given by equations of form uv = p( ¯x) where the zero locus of a polynomial p is smooth reduced. The main result says that the Lie algebra generated by algebraic completely integrable vector fields on such a hypersurface coincides with the Lie algebra of all algebraic vector fields. Consequences of this result for some conjectures of affine algebraic geometry and for the Oka-Grauert-Gromov principle are discussed.

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.

We calculate the invariant subspaces in the linear representation of the group of algebraic automorphisms of ℂn on the vector space of algebraic vector fields on ℂn and more generally we do this in a setting with parameter. As an application to the field of Several Complex Variables we get a new proof of the Andersén–Lempert observation and a parametric version of the Andersén–Lempert theorem. Further applications to the question of embeddings of ℂk into ℂn are announced.