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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:
 1.4e+31 Minimal model program (Mori theory, extremal rays)
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References:
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