Towards a Mori theory on compact Kähler threefolds. II
This paper continues the studies of a former article by \textit{F. Campana} and \textit{Th. Peternell} [Math. Nachr. 187, 29-59 (1997; Zbl 0889.32027)], devoted to the development of a structure theory for compact Kähler threefolds analogous to the so-called Mori theory for projective algebraic varieties. The weak minimal model conjecture for compact Kähler manifolds states that each such manifold is either uniruled or birationally equivalent to a compact Kähler variety with at most terminal singularities and numerically effective canonical bundle, a so-called minimal modal. The abundance conjecture of Kawamata says that a minimal model is always good in the sense that some power of the canonical bundle is globally generated. The author shows that Kummer manifolds without non-constant meromorphic functions have good minimal models. The main result of the paper deals with the existence of extremal contractions for a non-algebraic compact Kähler threefold \(X\) which is not a minimal model: If \(X\) has a good model or can be approximated algebraically or has Kodaira dimension \(\kappa(X)=2\) then \(X\) contains a rational curve \(C\) with \(K_X\cdot C<0\) and allows an extremal contraction which is either of fibre type (a \(\mathbb{P}_1\)-bundle or a conic bundle over a non-algebraic Kähler variety) or is birational and contracts an irreducible rational divisor to a point.
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