Polarized pairs, log minimal models, and Zariski decompositions

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<jats:title>Abstract</jats:title><jats:p>We continue our study of the relation between log minimal models and various types of Zariski decompositions. Let (<jats:italic>X,B</jats:italic>) be a projective log canonical pair. We will show that (<jats:italic>X,B</jats:italic>) has a log minimal model if either <jats:italic>K<jats:sub>X</jats:sub></jats:italic> + <jats:italic>B</jats:italic> birationally has a Nakayama–Zariski decomposition with nef positive part, or if <jats:italic>K<jats:sub>X</jats:sub></jats:italic> +<jats:italic>B</jats:italic> is big and birationally has a Fujita–Zariski or Cutkosky–Kawamata–Moriwaki–Zariski decomposition. Along the way we introduce polarized pairs (<jats:italic>X,B</jats:italic> +<jats:italic>P</jats:italic>), where (<jats:italic>X,B</jats:italic>) is a usual projective pair and where <jats:italic>P</jats:italic> is nef, and we study the birational geometry of such pairs.</jats:p>

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