Remarks on the existence of minimal models of log canonical generalized pairs

TL;DR

This paper proves the termination of the minimal model program for certain log canonical generalized pairs and establishes the existence of minimal models under various conditions, expanding the understanding of their structure.

Contribution

It demonstrates that the MMP with scaling terminates for NQC log canonical generalized pairs without the $Q$-factorial assumption, and proves existence of minimal models under additional conditions.

Findings

MMP with scaling terminates for non-$Q$-factorial pairs.

Existence of minimal models for pseudo-effective pairs under certain conditions.

Results extend minimal model theory to broader classes of generalized pairs.

Abstract

Given an NQC log canonical generalized pair $(X,B+M)$ whose underlying variety $X$ is not necessarily $\mathbb{Q}$-factorial, we show that one may run a $(K_X+B+M)$-MMP with scaling of an ample divisor which terminates, provided that $(X,B+M)$ has a minimal model in a weaker sense or that $K_X+B+M$ is not pseudo-effective. We also prove the existence of minimal models of pseudo-effective NQC log canonical generalized pairs under various additional assumptions, for instance, when the boundary contains an ample divisor.