On the 4-dimensional minimal model program for Kähler varieties

Abstract

In this article we establish the following results: Let (X,B) be a dlt pair, where X is a Q-factorial Kähler 4-fold – (i) if X is compact and KX+B∼QD≥0 for some effective Q-divisor, then (X,B) has a log minimal model, (ii) if (X/T,B) is a semi-stable klt pair, W⊂T a compact subset and KX+B is effective over W (resp. not effective over W), then we can run a (KX+B)-MMP over T (in a neighborhood of W) which ends with a minimal model over T (resp. a Mori fiber space over T). We also give a proof of the existence of flips for analytic varieties in all dimensions and the relative MMP for projective morphisms between analytic varieties.