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Existence of log canonical flips and a special LMMP. (English) Zbl 1256.14012
The minimal model program is one of the main tools in the study of higher dimensional complex projective varieties. One of the features of this program is that one is naturally led to consider varieties (or even pairs) with mild singularities such as klt (Kawamata log terminal) or lc (log canonical) singularities. One may think of a klt (resp. lc) pair \((X,B)\) as a natural generalization of a pair consisting of a smooth manifold \(X\) and a \(\mathbb Q\)-divisor \(B=\sum b_iB_i\) where \(B_i\) are smooth codimension \(1\) subvarieties intersecting transversely and the \(b_i\) are non-negative rational numbers such that \(b_i<1\) (resp. \(b_i\leq 1\)). In recent years there has been remarkable progress in this area and in particular many features of the minimal model program for klt singularities are now understood such as the existence of flips and divisorial contractions and the existence of minimal models for varieties of general type [C. Birkar et al., J. Am. Math. Soc. 23, No. 2, 405–468 (2010; Zbl 1210.14019)]. The case of lc singularities is expected to be much harder. In the paper under review, the author proves an important special case of the lc minimal model program. He shows that if \(f:X\to Z\) is a surjective morphism, \((X,B+A)\) is a lc pair such that \(K_X+B+A\sim _{\mathbb Q,Z}0\) then the lc minimal model program holds for \((X,B)\) over \(Z\). (More precisely he shows that \((X,B)\) has a good minimal model or a Mori fiber space over \(Z\), and if \((X,B)\) is \(\mathbb Q\)-factorial and dlt, then any \(K_X+B\) minimal model program over \(Z\) with scaling of an ample divisor terminates.) In particular this result implies the existence of lc flips.

MSC:
14E30 Minimal model program (Mori theory, extremal rays)
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