Numerical Semiampleness of Nef Adjoint Divisors

Abstract — v1

We prove the Generalised Abundance Conjecture: if $(X,B)$ is a projective klt $\mathbb Q$-pair over an algebraically closed field of characteristic zero, $K_X+B$ is pseudo-effective, $M$ is a nef $\mathbb Q$-Cartier divisor on $X$, and $K_X+B+M$ is nef, then $K_X+B+M$ is numerically equivalent to a semiample $\mathbb Q$-Cartier divisor on $X$.