On existence of log minimal models and weak Zariski decompositions

2009/07/29 by Caucher Birkar, Birkar, Caucher
Mathematics · #Advanced Topology and Set Theory #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #math.AG

paper · pdf · doi:10.48550/arxiv.0907.5182

arxiv created 2009/07/29 · openalex publication_date 2009/07/29 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Minimal models, flips and finite generation : a tribute to V.V. SHOKUROV and Y.-T. SIU

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