Towards a Mori theory on compact Kaehler threefolds, III

Abstract

We prove abundance for a minimal Kaehler threefold which is not both simple and non-Kummer. Recall that a variety is simple if there is no compact subvariety of positive dimension through a sufficiently general point . Furthermore we prove that a smooth compact Kaehler threefold whose canonical bundle is not nef, carries a contraction unless (possibly) the manifold is simple non-Kummer. It is generally conjectured that simple threefolds must be Kummer.