Semiampleness conjecture on klt Calabi--Yau pairs

Let $(X,\Delta)$ be a projective klt pair of dimension $n$ over a field $k$ such that

$$K_X+\Delta\equiv 0.$$

Let $L$ be a nef $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor on $X$. A divisor is num-semiample if it is numerically equivalent to a semiample $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor.

Semiampleness conjecture on klt Calabi--Yau pairs. The divisor $L$ is num-semiample; that is, there exists a $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor $L'$ on $X$ such that

$$L\equiv L'$$

and $L'$ is semiample.

The conjecture generalizes abundance to nef divisors on klt Calabi--Yau pairs. It is known in characteristic zero for projective surfaces, certain classes of Calabi--Yau threefolds, and hyperkähler fourfolds, while this paper proves it for projective surfaces over fields of positive characteristic.