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On existence of log minimal models. II. (English) Zbl 1226.14021
Let \((X,B)\) be a projective log canonical pair (e.g. \(X\) is smooth and \(B=\sum b_iB_i\) is a sum of smooth codimension one subvarieties meeting transversely with \(0\leq b_i\leq 1\)). \((X,B)\) is pseudo-effective if there is a sequence of effective divisors \(M_i\geq 0\) such that \(K_X+B= \lim_{i\to \infty} M_i\). According to the Minimal Model Conjecture, if \((X,B)\) is pseudo-effective, then it has a minimal model \(\phi :X\dasharrow Z\) (in particular \(K_Z+\phi _* B\) is nef so that \((K_Z+\phi _* B)\cdot C\geq 0\) for any curve \(C\subset X\)) and if \((X,B)\) is not pseudo-effective, then it has a Mori fiber space (in particular there is a birational map \(\phi :X\dasharrow Z\) and a morphism \(Z\to W\) such that \(-(K_Z+\phi _* B)\) is ample over \(W\)). The Weak Nonvanishing Conjecture says that any pseudo-effective log canonical pair \((X,B)\) is effective so that \(K_X+B\equiv M\geq 0\).
In this paper, the author shows the important result that the Weak Nonvanishing Conjecture implies the Minimal Model Conjecture and that if \((X,B)\) is a \(\mathbb Q\)-factorial dlt pair, then the birational map \(\phi :X\dasharrow Z\) to the minimal model (or Mori fiber space) is given by a finite sequence of divisorial contractions and flips.
For part I, cf. [Compos. Math. 146, No. 4, 919–928 (2010; Zbl 1197.14011)].

MSC:
14E30 Minimal model program (Mori theory, extremal rays)
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[2] DOI: 10.1112/S0010437X09004564 · Zbl 1197.14011 · doi:10.1112/S0010437X09004564
[3] DOI: 10.2140/ant.2009.3.951 · Zbl 1194.14021 · doi:10.2140/ant.2009.3.951
[4] DOI: 10.1090/S0894-0347-09-00649-3 · Zbl 1210.14019 · doi:10.1090/S0894-0347-09-00649-3
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[6] DOI: 10.1070/IM1993v040n01ABEH001862 · Zbl 0785.14023 · doi:10.1070/IM1993v040n01ABEH001862
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