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 $\mathbb{Q}$-factorial Kähler $4$-fold -- (i) if $X$ is compact and $K_X+B{\sim }_{\mathbb{Q}}D\ge 0$ for some effective $\mathbb{Q}$-divisor, then $(X,B)$ has a log minimal model, (ii) if $(X/T,B)$ is a semi-stable klt pair, $W\subset T$ 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.

Final version