TOWARD A CLASSIFICATION OF ALMOST HOMOGENEOUS MANIFOLDS II — BAD MODELS

Abstract

In this paper, we continue to deal with singular extremal rays with a Gm x SL (2, C) action which we started in [9] (see also [6]). We shall classify all Gm x SL(2, C) actions in [10] and state our results in Sec. 5 of this paper (see also [6]). Our idea is that once we have a singular extremal ray we can obtain enough information on the whole manifold which eventually leads to a classification. In this paper we concentrate on the most difficult case (5b) in the Proposition 4. For this reason we leave the proofs of Theorem 1 and Lemma 3 to [10]. But we give the proof of Theorem 3 for intuition of the whole process. Section 4 is the core of this paper where we construct the House Models and solve the (5b) case of Proposition 4 (Theorem 6). Once again (other than the proof of the Theorem 1 in [9]) we use non-algebraic argument for Theorem 3, a method we used very often and successfully. We also find in Sec. 4 of this paper that minimal projective 3-fold house models can have an arbitrary large second Betti number. Their effective curve cones are fully understood. The last section contains a criterion for projectivity which is very useful in our situations, and I cannot find it in the literature (Theorem 8). Most of this work was finished in Fall 1991. The original work was written in one paper as in [6 Chap. 1], but we now separate it into [9, 10] and this paper to make it more readable.