Towards a Mori theory on compact Kähler threefolds. III (Q2773553)

Based on the results and methods of the first two parts of this series of papers [see \textit{F. Campana} and \textit{Th. Peternell}, Math. Nachr. 187, 29-59 (1997; Zbl 0889.32027) and \textit{Th. Peternell}, Math. Ann. 311, No. 4, 729-764 (1998; Zbl 0919.32016)] the author continues his studies of compact Kähler threefolds \(X (\mathbb Q\)-factorial, at most terminal singularities) with regard to a solution of the abundance conjecture of Kawamata and to the existence of a contraction of \(X\) if \(X\) is smooth and the canonical bundle \(K_{X}\) is not nef. The abundance conjecture says that some power of the canonical bundle of a minimal model \(X\) is globally generated. This was known in the projective case and is proved here for all \(X\) except for those which are simple (no positive-dimensional proper subvariety through any very general point of \(X\)) and simultaneously non-Kummer. It is not known whether the latter threefolds exist.NEWLINENEWLINENEWLINEThe other main result of the paper says that a smooth \(X\) with \(K_{X}\) not nef admits a contraction (i.e. a regular map \(\varphi\) from \(X\) onto a normal variety \(Y\) with connected fibers such that \(-K_{X}\) is \(\varphi\)-ample and \(b_{2}(X)=b_{2}(Y)+1\)) if \(X\) is not simple or if \(\kappa(X)\geq 0\). Like in the projective case the contraction is constructed from a non-splitting family of rational curves on \(X\).