Andrea Iannuzzi

Quotients of the unit ball of ℂn for a free action of ℤ

LetBn be the unit ball of ℂn and ℤ ≅ Γ ⊂ AutBn be generated by a parabolic element of AutBn. We show that the quotientBn/Γ is biholomorphic to a holomorphically convex domain of ℂn, whose automorphism group is explicity described. It follows thatBn/ℤ is Stein for any free action of ℤ.

Maximal complexifications of certain homogeneous Riemannian manifolds

Let M = G/K be a homogeneous Riemannian manifold with dim C G C = dim R G, where G C denotes the universal complexification of G. Under certain extensibility assumptions on the geodesic flow of M, we give a characterization of the maximal domain of definition in TM for the adapted complex structure and show that it is unique. For instance, this can be done for generalized Heisenberg groups and naturally reductive homogeneous Riemannian spaces. As an application it is shown that the case of generalized Heisenberg groups yields examples of maximal domains of definition for the adapted complex structure which are neither holomorphically separable nor holomorphically convex.

Induced local actions on taut and Stein manifolds

Let G = (R, +) act by biholomorphisms on a taut manifold X. We show that X can be regarded as a G-invariant domain in a complex manifold X* on which the universal complexification (C, +) of G acts. If X is also Stein, an analogous result holds for actions of a larger class of real Lie groups containing, e.g., abelian and certain nilpotent ones. In this case the question of Steinness of X* is discussed.

Maximal complexifications of certain Riemannian homogeneous manifolds.

A characterization of maximal domains of existence of adapted complex structures for Riemannian homogeneous manifolds under certain extensibility assumptions on their geodesic flow is given. This is applied to generalized Heisenberg groups and naturally reductive Riemannian homogeneous spaces. As an application it is shown that the case of generalized Heisenberg groups yields examples of maximal domains of definition for the adapted complex structure which are are neither holomorphically separable, nor holomorphically convex.

On hyperbolicity of SU(2)-equivariant, punctured disc bundles over the complex affine quadric

Given a holomorphic line bundle over the complex affine quadric Q2, we investigate its Stein, SU(2)-equivariant disc bundles. Up to equivariant biholomorphism, these are all contained in a maximal one, say Ωmax. By removing the zero section from Ωmax one obtains the unique Stein, SU(2)-equivariant, punctured disc bundle over Q2 which contains entire curves. All other such punctured disc bundles are shown to be Kobayashi hyperbolic.