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Mori fibre spaces for Kähler threefolds. (English) Zbl 1338.32019
Let $$X$$ be a normal $$\mathbb Q$$-factorial compact Kähler threefold with at most terminal singularities. In a former paper [Invent. Math. 203, No. 1, 217–264 (2016; Zbl 1337.32031)] the authors established the existence of a minimal model for $$X$$ if $$X$$ is not uniruled. In the present article they solve the remaining case by studying the MMP for a uniruled $$X$$ which is not projective. The last assumption implies that the base of the MRC fibration $$X\dashrightarrow Z$$ is two-dimensional. The main theorem states the existence of a bimeromorphic map $$X\dashrightarrow X'$$ consisting of contractions of extremal rays and flips such that $$X'$$ admits a fibration $$\varphi:X'\rightarrow S$$ onto a normal compact $$\mathbb Q$$-factorial Kähler surface with at most klt singularities. Moreover $$-K_{X'}$$ is $$\varphi$$-ample and $$\rho(X'/S)=1$$. The specific structure of the MRC fibration and the results and methods of the above quoted paper are essential for the strategy of the proof.

##### MSC:
 32J27 Compact Kähler manifolds: generalizations, classification 14E30 Minimal model program (Mori theory, extremal rays) 32J17 Compact complex $$3$$-folds 32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
##### Keywords:
MMP; rational curve; Zariski decomposition; Kähler manifold
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