On the 4-dimensional minimal model program for Kähler varieties

In this article we establish the following results: Let (X, B) be a dlt pair, where X is a Q-factorial K & auml;hler 4-fold - (i) if X is compact and K-X + B similar to(Q) D >= 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 subset of T a compact subset and K-X + B is effective over W (resp. not effective over W), then we can run a (K-X + 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. (c) 2024 Elsevier Inc. All rights reserved.