On the $4$-dimensional minimal model program for K\"ahler varieties

TL;DR

This paper advances the minimal model program for 4-dimensional Kähler varieties, establishing existence of minimal models and flips, and extending MMP techniques to analytic varieties.

Contribution

It proves the existence of log minimal models for compact Q-factorial Kähler 4-folds and extends MMP results to analytic varieties, including flips and relative MMP.

Findings

Existence of log minimal models for certain Kähler 4-folds.

Extension of MMP to analytic varieties with flips.

Development of relative MMP for projective morphisms in analytic setting.

Abstract

In this article we establish the following results: Let $(X, B)$ be a dlt pair, where $X$ is a $\mathbb Q$-factorial K\"ahler $4$-fold -- (i) if $X$ is compact and $K_X+B\sim_{\mathbb Q} D\geq 0$ 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.