Complex Orbits of Solvable Groups

The following structure theorems are proved: An orbit of a real solvable Lie group in projective space that is a complex submanifold is isomorphic to C x (C*)m x Í2 , where Í2 is an open orbit of a real solvable Lie group in a projective rational variety. Also, any homogeneous space of a complex Lie group that is isomorphic to C" can be realized as an orbit in some projective space. As a consequence, left-invariant complex structures on real solvable Lie groups are always induced from complex orbits in projective space. In [19] it is shown that an orbit of a complex solvable Lie group in complex projective space P" must be isomorphic as a manifold to C x (C*)m . The purpose of this note is to prove a similar theorem for an orbit of a real solvable Lie group which is a complex submanifold of P", not necessarily locally closed. We call such an orbit a complex solvable orbit. The simplest example is the upper half plane H in P . In the classification of 2-dimensional homogeneous surfaces [ 17] other examples turn up: the two-dimensional ball B , H x H, C2\R2, and P2\(B2 U A) where A is a projective line tangent to B2. Of course, any homogeneous bounded domain possesses a transitive solvable group of automorphisms. Moreover there exists a realization of the bounded domain as a convex unbounded domain such that the elements of the solvable group act as affine transformations and, therefore, can be extended to a projective space compactification [6]. Complex solvable orbits in projective space are in particular Kahler manifolds, although the Kahler structure need not be invariant under the group. If the Kahler structure is invariant then the manifold falls under a recently completed classification and must be isomorphic to a direct product of a homogeneous bounded domain, an abelian complex Lie group (C"/T where T is a discrete subgroup of C" ), and a compact homogeneous rational manifold (the quotient of a semisimple complex Lie group by a parabolic subgroup) [5]. The proof of this fundamental structure theorem is long and difficult and is the culmination of the efforts of many mathematicians over the course of three decades, see [7, 4] for surveys. The case of solvable groups had already been Received by the editors June 19, 1989 and, in revised form, January 8, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 32M10; Secondary 14L30. © 1990 American Mathematical Society 0002-9939/90 $1.00+ $.25 per page