complexity

Definition 1.1 Let X be a proper variety of dimension n, and let .X; / be a log pair. A decomposition of is an expression of the form X ai Si ; where Si 0 are Z-divisors and ai 0, 1 i k. The complexity of this decomposition is n C r d , where r is the rank of the vector space spanned by S1; S2; : : : ; Sk in the space of Weil divisors modulo algebraic equivalence and d is the sum of a1; a2; : : : ; ak. The complexity c D c.X; / of .X; / is the infimum of the complexity of any decomposition of . (paper) (Brown et al., 2018, p. 923)