The moduli b-divisor of an lc-trivial fibration

By Florin Ambro. Compositio Mathematica, 2003

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We study positivity properties of the moduli (b-)divisor associated to a relative log pair (X, B)/Y with relatively trivial log canonical class.

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Existence of log canonical flips and a special LMMP (2011, background): “Since KX + B ∼R 0/S, by Ambro [3] and Fujino-Gongyo [16, Theorem 3.”
Singularities on the base of a Fano type fibration (2012, methodology): “The difficult part of the theorem is to deal with the discriminant part BZ since by applying a result of Ambro [3] we can control the moduli part MZ (actually we have to understand the discriminant b-divisor BZ rather than just BZ).”
Extension theorems, non-vanishing and the existence of good minimal models (2010, unknown): “By [Ambro05, 0.2], we may write KW + τh∗∆ = r ∗(KV + BV ) where (V,BV ) is klt.”
Anti-pluricanonical systems on Fano varieties (2016, background): “The claim then follows from [13] (this relies on [2] in the klt case).”