Complex orbits of solvable groups

Abstract:

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 ${{\mathbf {C}}^k} \times {({{\mathbf {C}}^ * })^m} \times \Omega$, where $\Omega$ 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 ${{\mathbf {C}}^n}$ 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.