Math. Z. 204 , 117--127 (1990)
Abstract.
In this article we want to investigate under which assumptions the complement
of an analytic subset of $ C^n$ is homogeneous. Here we distinguish between
two different notations of homogenity. We call a complex manifold $X$
\sl $\Aut$--homogeneous \rm if the group $\Aut(X)$
of all holomorphic automorphisms of $X$ acts transitively on $X$.
It is called \sl $G$--homogeneous \rm if there exists a finite-dimensional real
Lie group $G$ acting smoothly and
transitively on $X$ by biholomorphic transformations.
We are particularly interested in complex manifolds $X$ which are
$\Aut$--homogeneous but not $G$--homogeneous.
W.~Kaup was the first to find such an example. He showed that
the complement of a certain discrete subset of $ C^n$ (with $n\ge 2$)
has this property.
The main idea is the following:
Let $A$ be a sufficiently nice and small analytic subset of $ C^n$.
Since $Aut_{\cal O}( C^n)$ is \sl very \rm large the group
$S(A)=\{g\in Aut_{\cal O}( C^n): g(A)=A\}$ should act
transitively on $ C^n\setminus A$.
On the other hand a manifold can be $G$--homogeneous
only if it satisfies certain restrictive topological
assumptions. A simply-connected manifold which is homogeneous under a
Lie group action must be homeomorphic to a real vector bundle over a
quotient of compact Lie groups by a result of Mostow.
Therefore in general $ C^n\setminus A$ will \sl not \rm be $G$--homogeneous
even if it is $\Aut$--homogeneous.
Our main results are the following:
Let $A$ be an algebraic subset with $\codimc(A)\ge 2$.
Then $\Aut(X)$ acts transitively on $X= C^n\setminus A$.
There are discrete subsets $D$ in $ C^n$ such that $X= C^n\setminus D$
has no automorphism except the identity map (This is a result of Rosay
and Rudin).
Let $X=\{(z,w)\in C^2: zw\ne 1\}$. Then $X$ is a Stein manifold
on which $\Aut(X)$ acts transitively although it is not homomgeneous
with respect to a holomorphic action of a Lie group.
(This answers a question of Kaup).
Let $A$ be an analytic subset in $ C^n$ and assume that
$X= C^n\setminus A$ admits a non-constant holomorphic map to a
hyperbolic manifold (e.g. $ C\setminus\{0,1\}$).
Then $\Aut(X)$ does not act transitively on $X$.
Let $A$ be a union of affine hyperplanes in $ C^n$. Then
$X= C^n\setminus A$ is homogeneous under the $\Aut(X)$-action if and only if
$X\simeq( C^*)^k\times C^{n-k}$, i.e.~ if and only if $A$ is the union of
at most $n$ affine hyperplanes in general position.
Let $A$ be an algebraic hypersurface containing $n+1$ hyperplanes
in general position. Then $\Aut(X)$ is finite.
Let $X= C^*\times C^*\setminus\{(1,1),(2,1),(1,3)\}$. Then $X$
is a quasiaffine variety with no algebraic automorphism except the
identity map, but $\Aut(X)$ acts transitively on $X$.