On non-compact complex nil-manifolds

Abstract

Given a complex manifold X, it is natural to consider the quotient space X~ ~, where p~q whenever f(p)=f(q) for every fe(9(X). In the best of all worlds one would hope that X / ~ is a Stein space, and that there are no non-constant holomorphic functions on the equivalence classes of ~ . One would also hope to explain the latter phenomenon by compactness or Levi curvature. A classical result of this type is the Remmert Reduction Theorem: If X is an irreducible, holomorphically convex analytic space, then X / ~ is a Stein analytic space and the natural holomorphic separation map F : X ~ X / ~ is proper [15]. In general it is not possible to prove any such reduction theorem. In fact there are naturally occuring manifolds X where there is no natural way to put a complex structure on X~ ~ [5]. Grauert's examples [5] all contain non-compact divisors on which every holomorphic function is constant. Off these divisors the holomorphic functions separate points and give local coordinates. If one assumes that X is homogeneous, then of course X can't be one of Grauert's examples. For the purposes of this paper, homogeneity means that a complex Lie group G acts holomorphically and transitively on X. In other words X is holomorphically equivalent to G/H, where G is a connected complex Lie group and H is a closed subgroup of G. One would hope to obtain some form of a reduction theorem in the homogeneous setting. It is true that the holomorphic separation map of a complex homogeneous manifold can always be realized as a holomorphic fibration over a holomorphically separable homogeneous manifold. However, we do not even know if a holomorphically separable homogeneous manifold always has an envelope of holomorphy. Clearly it need not be Stein, e.g. ~2\{(0,0)}. Also the fiber of the holomorphic separation map may have nonconstant hotomorphic functions (see [2]). In particular cases, some results have been obtained. Morimoto [13] showed that if G is a connected, complex Lie group, then it has a closed, connected, central,