The moduli \(b\)-divisor of an lc-trivial fibration. (English) Zbl 1094.14025

The paper under review studies lc-trivial fibrations. These are morphisms \(f\:(X,B)\to Y\) with relatively trivial log canonical class \(K_X+B\). In this context there is a discriminant divisor that measures the singularities of the log pair \((X,B)\) over codimension one points of \(Y\) and a moduli divisor \(M_Y\), first appeared in [Y. Kawamata, in: Birational algebraic geometry. Conf. Algebraic Geometry, in memory of Wei-Liang Chow (1911–1995), Baltimore 1996. Contemp. Math. 207, 79–88 (1997; Zbl 0901.14004)]. The paper proves a number of technical results related to semi-ampleness of \(M_Y\). As an application one gets the following interesting application. Let \((X,B)\) be a projective log variety with Kawamata log terminal singularities such that \(K_X+B\) is numerically trivial, then there exists a positive integer such that \(b(K_X+B)\sim0\), and the Albanese map \(X\to\text{ Alb}(X)\) is a surjective morphism with connected fibres.


14J10 Families, moduli, classification: algebraic theory
14E30 Minimal model program (Mori theory, extremal rays)
14N30 Adjunction problems


Zbl 0901.14004
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