On existence of log minimal models and weak Zariski decompositions

TL;DR

This paper introduces a new concept of weak Zariski decompositions in higher dimensions and establishes a fundamental link between these decompositions and the existence of log minimal models under the assumption of the log minimal model program in lower dimensions.

Contribution

It defines a weak Zariski decomposition in higher dimensions and proves its equivalence to the existence of log minimal models assuming the LMMP in one lower dimension.

Findings

Weak Zariski decomposition can be characterized in higher dimensions.

Existence of log minimal models is equivalent to having a weak Zariski decomposition.

The result depends on assuming the LMMP in dimension d-1.

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$/Z$.