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 $p>2$. 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 $\le n$, we prove that, when $K_X+B$ is $f$-nef, the moduli part is nef up to a birational map $Y⇢X$. As a corollary, we prove positivity of the moduli part in the $K$-trivial case, i.e. when $K_X+B \sim_{\Q} f^*L$ for some $\Q$-Cartier $\Q$-divisor $L$ on $Z$. In particular, consider a dlt pair $(X,B)$ of dimension $3$ over an algebraically closed field of characteristic $p>5$ such that the induced pair on a general fibre is log canonical, then the canonical bundle formula holds unconditionally.

(v3) 54 pages, revised version, updated with suggestions from referee, to appear in Journal of the London Mathematical Society