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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:
14E30 Minimal model program (Mori theory, extremal rays)
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[8] doi:10.2307/1990969 · Zbl 0649.14023 · doi:10.2307/1990969
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