manuscripta mathematica 68, 117--134 (1990).
Abstract.
In this article we consider the Kobayashi-pseudodistance on
homogeneous manifolds.
S.~Kobayashi has conjectured and Nakayima has proven
that a homogeneous
complex manifold is hyperbolic (\ie the
Kobayashi-pseudo-distance is a distance)
if and only if this manifold is biholomorphic to a bounded homogeneous
domain.
We want to study homogeneous complex manifolds $X$ for which the
Kobayashi-pseudodistance is not a distance.
First let us introduce some notations and conventions.
A homogeneous complex manifold in our sense is a complex manifold $X$ for
which there exists a real Lie group $G$ which acts smoothly and by
biholomorphic transformations transitively on $X$.
(Note that if
a complex Lie group acts transitively on $X$,
the Kobayashi-pseudodistance
is totally degenerate.)
For the sake of convenience we will always assume that $G$ is simply connected
and acts almost effectively on $X$.
For a Lie group $G$ we denote the Lie algebra by $\a g$, the commutator group
by $G'$ and the connected component of the neutral element by $G^0$.
We introduce two equivalence relations:
\item{(i)}
$x\sim y\iff d_X(x,y)=0$,
\item{(ii)}
$x\simh y\iff f(x)=f(y)$ for any holomorphic map $f:X\to Y$ to any
hyperbolic manifold $Y$.
We will call the quotients by these equivalence relations Kobayashi-- resp.
hyperbolic quotient.
For non--homogeneous complex manifolds quotients by these two equivalence
relations are in general not manifolds.
For example
let $X=\{(z,w)\in\s C^2\sth \abs{zw}<1,~\abs{w}<1\}$.
Then $(z_1,w_1)\sim(z_2,w_2)$ iff $(z_1,w_1)\simh(z_2,w_2)$
iff $(z_1,w_1)=(z_2,w_2)$
or
$z_1=z_2=0$.
It follows that for this manifold $X$ the quotients are not everywhere
locally compact.
Now we want to present our main results.
\Theorem 1
Let $X$ be a homogeneous complex manifold.
Then there exists a bounded homogeneous domain $\Omega$ and a holomorphic
map $f:X\to\Omega$ such that
any holomorphic map from $X$ to any hyperbolic manifold
fibers through $\Omega$.
Furthermore
$f$ is surjective and equivariant for all
automorphisms of $X$.
\endproclaim
\proclaim{Remark}
For this result we need only that the group of all automorphisms of $X$
acts transitively, \ie it is not necessary to assume that a \/{\rm Lie} group
acts transitively on $X$.
\endproclaim
\proclaim{Corollary}
The hyperbolic reduction coincides with the bounded holomorphic reduction,
\ie $x\simh y$ if and only if $f(x)=f(y)$ for every bounded holomorphic
function $f$ on $X$.
\endproclaim
\Theorem 2
Let $X$ be a homogeneous complex manifold.
Then there exists a {\/\rm real} homogeneous differentiable manifold $Y$
with a Riemannian metric $h$, positive real constants $C_1 > C_2 > 0 $
and a differentiable fiber bundle $\pi:X\to Y$ such that $\pi$ is
equivariant for all automorphisms of $X$ and moreover
$$
C_2\cdot d_h(\pi(x_1),\pi(x_2)) \le d_X(x_1,x_2) \le C_1 \cdot
d_h(\pi(x_1),\pi(x_2))
$$
for all $x_1$, $x_2$ in $X$, where $d_h$ denotes the distance on $Y$
induced by $h$.
\endproclaim
Of course one would like $Y$ to be a complex manifold such that $\pi$
is holomorphic.
This is the case in all examples known to us. However there exists a
non--homogeneous surface for which the Kobayashi--quotient is a
\sl real \rm one--dimensional manifold (see Example 1). This seems to indicate that in
general $Y$ might be even real odd--dimensional.
So far we know only for the following four special cases that $Y$ can be
endowed with a complex structure such that $\pi$ becomes a holomorphic map.
\footnote*{Both a previous preprint by the author and \cite{Gi} contain claims
that $Y$ is a complex manifold under less rigid assumptions.
However the proofs contain serious gaps.}
\Theorem 3
Let $X$ be a complex manifold homogeneous under the action of a Lie group $G$.
The Kobayashi--quotient $Y$ admits a complex structure such that the
projection $\pi:X\to Y$ is holomorphic if one of the following conditions
is fulfilled:
\item{1)}
If $\dimc(X)\le 3$.
\item{2)}
If $G$ (the Lie group acting transitively on $X$) is solvable,
$\pi_1(X)$ is finite or cyclic and the induced $G$\=action on $Y$
is almost effective.
\item{3)}
If $X$ is a flag domain.
\item{4)}
If the fibers of $\pi:X\to Y$ are compact complex--analytic subsets in $X$.
In the cases 1), 3) and 4) the Kobayashi--pseudodistance of $X$ coincides
with the pull--back via $\pi$ of the Kobayashi--pseudodistance of $Y$, \ie
$d_X(p,q)=d_Y(\pi(p),\pi(q))$ for all $p,q\in X$.
In particular $Y$ is hyperbolic in these cases.
\endproclaim
\Theorem 4
Let $X=G/H$ be a homogeneous complex manifold.
Assume that $G$ is solvable.
Then $X$ is hyperbolic if and only if every holomorphic map from $\s C$ to $X$
is constant.
\endproclaim
Now let us discuss the fibers.
\Theorem 5
Let $F$ denote a fiber of $f:X\to\Omega$ and $S$ a fiber of
$\pi:X\to Y$.
Then $F$ is a connected homogeneous complex manifold and never hyperbolic
unless it is a point.
$S$ is a connected real homogeneous manifold. If $G$ is solvable then the
complex--analytic Zariski--closure $\bar S$ of $S$ in $X$
is not hyperbolic unless it is a point.
\endproclaim
It should be noted that for non--homogeneous complex manifolds some
equivalence classes might be disconnected, \eg consider the manifold
$X=X_1\setminus A$ with $X_1=\{(z,w)\in\s C^2\sth \Im(w)>0\}$
and $A=\{(z,w)\sth z\in\s R\hbox{ \rm and }w=i\}$.
Furthermore for non--homogeneous complex manifolds these equivalence classes
may be hyperbolic analytic subsets. (Consider the example of Eisenman--Taylor
(\cite{K1}) where $X=\{(z,w)\sth \abs{zw}<1,\ \abs w <1\}
\setminus\{\abs z\le 1,\ w=0\}$).
\proclaim{Corollary}
Let $X$ be a homogeneous complex manifold, $Z$ a hyperbolic manifold and
$\phi:X\to Z$ a holomorphic map.
Assume that one fiber of $\phi$ is hyperbolic.
Then $X$ is a bounded homogeneous domain.
\endproclaim
Although by the above result the Kobayashi-pseudodistance on $F$ is at least
partially degenerate it may occur that it is not totally degenerate.
In fact $X=\{(x,w,z)\in\s C^3\sth \Im(w)\Im (x)-\Im(z)>0,\Im(x)>0\}$ is a
homogeneous complex manifold for which the fiber of the Kobayashi--reduction
is biholomorphic to $\Delta_1\times\s C$.