Abstract
We first introduce a weak type of Zariski decomposition in higher dimensions: an \({\mathbb {R}}\) -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 \({\mathbb {R}}\) -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 (in our sense) if and only if K X + B has a weak Zariski decomposition/Z.
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Birkar, C. On existence of log minimal models and weak Zariski decompositions. Math. Ann. 354, 787–799 (2012). https://doi.org/10.1007/s00208-011-0756-y
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DOI: https://doi.org/10.1007/s00208-011-0756-y