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On the moduli b-divisors of lc-trivial fibrations. (Sur les b-diviseurs de modules des fibrations lc-triviales.) (English. French summary) Zbl 1314.14030
Let $$(X,B)$$ be a log canonical pair and $$f:X\to Y$$ be a projective surjective morphism between normal projective varieties with connected fibers. If $$K_X+B\sim _{\mathbb Q}f^*L$$ for some $$\mathbb Q$$-divisor $$L$$ on $$Y$$, then one expects that $$L\sim _{\mathbb Q}K_Y+B+M$$ where $$(Y,B)$$ is a log canonical pair, $$B$$ measures the singularities of $$(X,B)\to Y$$ and $$M$$ is a semiample divisor that measures the variation (moduli) of the fibers of $$(X,B)\to Y$$. (In practice, it is necessary to replace $$Y$$ and $$X$$ by appropriate resolutions and so one should view $$M$$ as a $$b$$-divisor.) By a result of F. Ambro [Compos. Math. 141, No. 2, 385–403 (2005; Zbl 1094.14025)], a slightly weaker result is known to be true for klt pairs, namely that (up to replacing $$X$$ and $$Y$$ by appropriate birational models) $$M$$ nef and abundant which means that there is a morphism $$h:Y\to Z$$ and a big and nef divisor $$H$$ on $$Z$$ such that $$M\sim_{\mathbb Q}h^*H$$. In this paper, the authors prove that Ambro’s result also holds for log canonical pairs.

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