Towards a Mori theory on compact Kähler threefolds III

Abstract

Based on the results of the first two parts to this paper, we prove that the canonical bundle of a minimal Kähler threefold (i.e. KX is nef) is good, i.e. its Kodaira dimension equals the numerical Kodaira dimension, (in particular some multiple of KX is generated by global sections); unless X is simple. "Simple" means that there is no compact subvariety through the very general point of X and X not Kummer. Moreover we show that a compact Kähler threefold with only terminal singularities whose canonical bundle is not nef, admits a contraction unless X is simple with Kodaira dimension -∞.