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On existence of log minimal models. (English) Zbl 1197.14011
Let $$(X,B)$$ be a log canonical pair such that $$K_X+B$$ is pseudo-effective (resp. is not pseudo-effective), then the log minimal model program conjecture predicts that there is a birational morphism $$\phi:X\dasharrow Y$$ to a $$\mathbb Q$$-factorial divisorially log terminal pair $$(Y,B_Y:=\phi _*B+\text{Ex}(\phi ^{-1}))$$ such that $$a(D,X,B)<a(D,Y,B_Y)$$ for any $$\phi$$-exceptional divisor $$D$$ and $$K_Y+B_Y$$ is nef (resp. there is a morphism $$Y\to T$$ with $$\dim Y> \dim T$$ and $$-(K_Y+B_Y)$$ is ample over $$T$$).
Several special cases of the above conjecture were proven in [J. Am. Math. Soc. 23, No. 2, 405–468 (2010; Zbl 1210.14019)].
In this paper the author proves that the log minimal model program conjecture in dimension $$d-1$$ implies the log minimal model conjecture for $$d$$-dimensional log canonical pairs $$(X,B)$$ such that $$K_X+B$$ is numerically equivalent to an effective $$\mathbb R$$-divisor. In particular log minimal models exist for these pairs in dimension $$5$$.

##### MSC:
 1.4e+31 Minimal model program (Mori theory, extremal rays)
##### Keywords:
minimal models; Mori fiber spaces; LMMP with scaling
Full Text:
##### References:
 [8] doi:10.2307/1990969 · Zbl 0649.14023 · doi:10.2307/1990969
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