The Classification of Three-Dimensional Homogeneous Complex Manifolds

Abstract

A complex manifold X is called homogeneous if there exists a connected complex or real Lie group G acting transitively on X as a group of biholomorphic transformations. The goal is a general classification of homogeneous complex manifolds. Since the class of homogeneous complex manifolds is much too big for any serious attempt of complete classification, it is necessary to impose further conditions. For example E. Cartan classified in [Ca] symmetric homogeneous domains in ℂn. Here we will require that X is of small dimension. For dim ℂ(X) = 1 the classification follows from the uniformization Theorem. In 1962 J. Tits classified the compact homogeneous complex manifolds in dimension two and three [Ti1]. In 1979 J. Snow classified all homogeneous manifolds X = G/H with dim ℂ(X) ≤ 3, G being a solvable complex Lie group and H discrete [SJ1]. The classification of all complex-homogeneous (i.e. G is a complex Lie group) twodimensional manifolds was completed in 1981 by A. Huckleberry and E. Livorni [HL]. Next, in 1984 K. Oeljeklaus and W. Richthofer classified all those homogeneous two-dimensional complex manifolds X = G/H where G is only a real Lie group [OR]. The classification of three-dimensional complex-homogeneous manifolds was completed in 1985 [W1]. Finally in 1987 the general classification of the three-dimensional homogeneous complex manifolds was given by our Dissertation [W2]. The purpose of this note is to describe these manifolds and briefly outline the methods involved in the classification.