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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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