On log minimal models and Zariski decompositions II

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.