Polarized pairs, log minimal models, and Zariski decompositions

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 if $K_X+B$ is big and birationally has a Fujita–Zariski or Cutkosky–Kawamata–Moriwaki–Zariski decomposition. Along the way we introduce polarized pairs $(X,B+P)$, where $(X,B)$ is a usual projective pair and where $P$ is nef, and we study the birational geometry of such pairs.