On the canonical bundle formula in positive characteristic

Abstract: Let $f: X \to Z$ be a fibration from a normal projective variety $X$ of dimension $n$ onto a normal curve $Z$ over a perfect field of characteristic $p>0$. Let $(X, B)$ be a dlt pair such that the induced pair on a general fibre is log canonical. Assuming the LMMP and the existence of log resolutions in dimension $\leq n$, we prove that, when $K_X+B$ is $f$-nef, the moduli part is nef up to a birational map $Y \dashrightarrow X$. As a corollary, we prove positivity of the moduli part in the $K$-trivial case, i.e. when $K_X+B \sim_{\mathbb{Q}} f^*L$ for some $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor $L$ on $Z$. In particular, consider a dlt pair $(X, B)$ of dimension $3$ over a perfect field of characteristic $p>5$ such that the induced pair on a general fibre is log canonical, then the canonical bundle formula holds unconditionally.