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On existence of log minimal models and weak Zariski decompositions. (English) Zbl 1255.14016
One of the main problems in birational geometry is the existence of minimal models and Mori fibre spaces. More precisely, given an LC pair \((X,B)\), it is conjectured that it admits a minimal model if the log canonical divisor \(K_X+B\) is pseudoeffective and a Mori fiber space if this is not the case.
It has been long believed that the crucial point of the proof is the nonvanishing conjecture, that roughly says that if \(K_X+B\) is pseudoeffective, then it is in fact effective.
By using an inductive argument on the dimension, in [Compos. Math. 145, No. 6, 1442–1446 (2009; Zbl 1186.14015)] and [J. Reine Angew Math. 658, 99–113 (2011; Zbl 1226.14021)], the author of this paper reduced, in fact, the existence of minimal models and Mori fibre spaces to (a weak form of) the nonvanishing conjecture. The aim of this paper is to work towards a further reduction.
In particular the notion of weak Zariski decomposition is introduced: a pseudoeffective divisor is said to admit a weak Zariski decomposition if, up to birational morphisms, it can be written as the sum of an effective and a nef divisor. In particular every effective divisor trivially admits a decomposition of this type.
The idea is to reduce the existence of minimal models to the existence of weak Zariski decompositions for pseudoeffective log canonical divisors, by using again an inductive approach.
Actually a stronger inductive hypothesis is needed, namely the full log minimal model program, that is existence and termination of arbitrary log flips (for DLT pairs).
The main theorem roughly says that if we assume the log minimal model program in dimension \(d-1\), then an LC pair \((X,B)\) of dimension \(d\) has a minimal model, provided that \(K_X+B\) admits a weak Zariski decomposition.
As a corollary a new proof of the existence of minimal models (and Mori fiber spaces) in dimension 4, first proved in [V. V. Shokurov, “Letters of a bi-rationalist: VII. Ordered termination”, Proc. Steklov Inst. Math. 264, 178–200 (2009), arXiv:math/0607822], is given.
Note that all results also hold in the relative setting.

MSC:
14E30 Minimal model program (Mori theory, extremal rays)
14C20 Divisors, linear systems, invertible sheaves
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