Let $(X,B)$ be a projective log canonical pair (here I mean $B \geq 0$). Assume that the coefficients of $B$ are rational, and that $K\_X+B \equiv 0$. Is it true that $K\_X + B \sim\_\mathbb{Q} 0$? I know it is true if $(X,B)$ is klt, and "The moduli b-divisor of an lc-trivial fibration" by Ambro has a proof of it. I am curious whether the result extends to singularities that are lc and not klt, and, if so, where I could find a reference.



$K_X+B \equiv 0$ implies $K_X + B \sim_\mathbb{Q} 0$?
For lc pair or slc pair, it is true. This is Gongyo’s result. See [J. ALGEBRAIC GEOMETRY 22 (2013) 549–564].


BTW, the relative version is also true, which is not a trivial generalization of the absolute case. It is proved by Hacon and Xu [On Finiteness of B-representation and Semi-log Canonical Abundance]. 


