Two-orbit varieties with smaller orbit of codimension two

Abstract

Let G be a connected algebraic group and let X be an irreducible normal algebraic variety on which G acts regularly, where all our objects are defined over the base field 112. If G acts on X with an open orbit f~, then X is called an (algebraic) almost-homogeneous variety. Several approaches have been used to obtain a classification of such varieties under some additional assumptions. For example, a classification - even of complex analytic almost-homogeneous spaces - has been carried out in the case when the comple- ment ofis of dimension smaller or equal to one ((11) and (15)). The work of Luna and Vust ((16)) is fundamental for a good understanding especially in the case when G is a reductive connected group and a Borel subgroup of G acts on X with an open orbit in fL The open orbit f2 is then called a spherical homogeneous space and various results have been obtained for embeddings of such spaces (see e.g. (16), (5), (6), (19), (20)). Two-orbit varieties, i.e. complete almost-homogeneous varieties where the complement E of the open orbit is again an orbit of G, are in some sense the "elementary building blocks" of group actions. Important results for this approach have been obtained by Ahiezer ((1), (2), (3), see also (12)). In his papers, all complete two-orbit varieties X where the smaller orbit is a hypersurface are classified. The classification problem for codim~ E > 1 seems difficult. We hope that progress in the case where E is of codimension two which we present here will lead to some general results. In this paper we consider two-orbit varieties X with smaller orbit E of codimension two such that the acting group G is reductive. Of course, there are two-orbit spaces of this latter kind which arise from one of Ahiezer's examples by blowing down the smaller orbit. Simple examples of these are fibre bundles over homogeneous rational manifolds where the fibre is a Hirzebruch surface. These may be blown down to bundles where the fibre is a Hirzebruch cone. Carefully looking at Ahiezer's classification, it is in principle possible to determine all two-orbit varieties with smaller orbit of codimension two which may be obtained by blowing down the hypersurface orbit in one of Ahiezer's examples. This is why we concentrate on those varieties which don't arise in this way. The orbits of the maximal compact subgroup of G are then of codimension greater than one. We call this the Case 2. We obtain a complete picture of all two-orbit varieties arising in this case (see Theorem A and Theorem B). In short, each of these varieties is a fibre bundle G x Pl Z with fibre Z over the homogeneous rational manifold G/P1, and the construction of this bundle may be explicitly described. Conversely, the precise description of the fibre Z and