$P$-trivial MMP, Zariski decompositions and minimal models for generalised pairs

TL;DR

This paper develops a $P$-trivial minimal model program for generalized pairs, establishing conditions under which minimal models exist, including for dimension 3, extending previous results in the field.

Contribution

It introduces a $P$-trivial MMP framework for generalized pairs and proves the existence of minimal models under new nef and decomposition conditions, generalizing prior work.

Findings

Established $P$-trivial MMP steps for generalized pairs.

Proved minimal model existence under nef and Nakayama-Zariski decomposition conditions.

Extended results to three-dimensional generalized lc pairs.

Abstract

We develop a theory of $P$-trivial MMP whose each step is $P$-trivial for a given nef divisor $P$. As an application, we prove that, given a projective generalised klt pair $(X,B+M)$ with data $M'$ being just a nef $\mathbb{R}$-divisor, if $K_X+B+M$ birationally has a Nakayama-Zariski decomposition with nef positive part, and either if $M'$ or the positive part is log numerically effective, then it has a minimal model. Furthermore, we prove this for generalised lc pairs in dimension $3$. This is a generalisation of the main theorem of [Birkar-Hu14]. We also prove some related results.