On Free Holomorphic C-actions on Cⁿ and Stein Homogeneous Manifolds.

Jörg Winkelmann

We study free holomorphic actions of (C,+) on Cⁿ and the induced quotient spaces. In particular, many non-trivial examples are constructed.

-- There is a free algebraic triangular C-action by quadratic transformations on C⁵ such that the quotient is diffeomorphic to C⁴ but not biholomorphic to C⁴ . In fact the quotient X is a non-Stein quasi--affine variety: X=Q\E where Q is a smooth four-dimensional quadric and E is a smooth subvariety of codimension two.

This variety X can furthermore be obtained as a quotient of an affine-linear (C² ,+)--action on C⁶ .

-- There is a free algebraic triangular C-action on C⁴ with all orbits closed such that the quotient is non-Hausdorff.

-- There is a free holomorphic C-action on C⁵ such that the quotient is a complex manifold with a one-dimensional compact complex submanifold.

-- There is a free holomorphic C-action on C⁵ with some non-closed orbits.

For a special class of free holomorphic C-actions, namely those given in the form (t,v) -> v+tF(v) it is proven that the quotient is at least Hausdorff and a Stein manifold.

One of the examples is applied to the question of characterization of Stein manifolds. It is used to demonstrate that a certain criterion of Matsushima is only necessary and not sufficient for a homogeneous complex manifold to be Stein.

Math. Ann. 286, 593--612 (1990).