Let $(X,B)$ be a Calabi-Yau pair, that is, $(X,B)$ is lc (or klt for simplicity) and $K_X+B \sim_\mathbb{Q} 0$. Given a fibration $f:X \to Y$, there is an induced generalised pair $(Y,B_Y+M_Y)$ with $B_Y$ the discriminant part and $M_Y$ the moduli part. Do we know any information about $M_Y$ other than it is b-nef and abundant? Is there an easy example that $M_Y \nsim_\mathbb{Q} 0$?
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$\begingroup$ Have you looked at Ambro's "Shokurov’s Boundary Property" (J.Diff Geom, 2004)? He gives an example on the first page which may be interesting to you. $\endgroup$– Eric CantonJun 26, 2019 at 19:34
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