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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)
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