A Base Point Free Theorem of Reid Type, II

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Let X be a complete algebraic variety over C. We consider a log variety (X,Δ) that is weakly Kawamata log terminal. We assume that Kₓ+Δ is a Q-Cartier Q-divisor and that every irreducible component of Δ is Q-Cartier. A nef and big Cartier divisor H on X is called nef and log big on (X,Δ) if H |B is nef and big for every center B of non-"Kawamata log terminal" singularities for (X,Δ). We prove that, if L is a nef Cartier divisor such that aL-(Kₓ+Δ) is nef and log big on (X,Δ) for some a ∈ N, then the complete linear system | mL | is base point free for m ≫ 0.