On log minimal models and Zariski decompositions II

TL;DR

This paper explores the connection between log minimal models and Zariski decompositions, establishing conditions under which a log canonical pair admits a minimal model, and introduces polarized pairs for further birational geometry study.

Contribution

It demonstrates that certain Zariski decompositions imply the existence of log minimal models for projective log canonical pairs and introduces polarized pairs to extend the framework.

Findings

Log minimal models exist when $K_X+B$ has a Nakayama-Zariski decomposition with nef positive part.

Existence of log minimal models when $K_X+B$ is big and has Fujita or CKM Zariski decompositions.

Introduction of polarized pairs $(X,B+P)$ for studying birational geometry.

Abstract

We continue our study of the relation between log minimal models and various types of Zariski decompositions. Let $(X,B)$ be a projective log canonical pair. We will show that $(X,B)$ has a log minimal model if either $K_X+B$ birationally has a Nakayama-Zariski decomposition with nef positive part, or that $K_X+B$ is big and birationally it has a Fujita or CKM Zariski decomposition. Along the way we introduce polarized pairs $(X,B+P)$ where $(X,B)$ is a usual projective pair and $P$ is nef, and study the birational geometry of such pairs.