Main Thm A

THEOREM 1.2 Let X be a proper variety of dimension n, and let .X; / be a log canonical pair such that .KX C / is nef. If P ai Si is a decomposition of complexity c less than 1, then c 0 and there is a divisor D such that .X; D/ is a toric pair, where D and all but 2c components of D are elements of the set 1Si j 1 i ko. (paper) (Brown et al., 2018, p. 924)