The abundance conjecture for klt log pairs

Work over $\mathbb{C}$. Let $(X,\Delta)$ be a projective klt log pair of dimension $n\geq 2$, and let $\kappa(X,\Delta)$ denote the Kodaira dimension of $K_X+\Delta$. A divisor is semi-ample if some positive multiple is base-point-free. The abundance conjecture. If $\kappa(X,\Delta)\geq 0$ and $K_X+\Delta$ is nef, then $K_X+\Delta$ is semi-ample.

This is a central conjecture in minimal model theory. The paper proves abundance in cases of sufficiently large Kodaira dimension, while the statement in the indicated generality remains open.