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 K X + B ∼ 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 ⊂ 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. ContentsOMPROKASH DAS, CHRISTOPHER HACON, AND MIHAI P ȂUN 4. Relative MMP for projective morphisms 38 Part 3. MMP in dimension 4 40 5. Cone and Contraction Theorems 40 5.1. Cone and contraction theorems in dimension 3 40 6. Termination of flips for effective pairs 45 7. MMP for κ(X, K X + B) ≥ 0 47 8. MMP for Semi-stable pairs 59 8.1. Relative cone theorem for 4-folds 60 References 71