## 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.

### MSC:

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

Zbl 0901.14004
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