On existence of log minimal models and weak Zariski decompositions

Abstract

We first introduce a weak type of Zariski decomposition in higher dimensions: an $\R$-Cartier divisor has a weak Zariski decomposition if birationally and in a numerical sense it can be written as the sum of a nef and an effective $\R$-Cartier divisor. We then prove that there is a very basic relation between Zariski decompositions and log minimal models which has long been expected: we prove that assuming the log minimal model program in dimension $d-1$, a lc pair $(X/Z,B)$ of dimension $d$ has a log minimal model if and only if $K_X+B$ has a weak Zariski decomposition$/Z$.