On the canonical bundle formula in positive characteristic

Abstract Let be a fibration from a normal projective variety of dimension onto a normal curve over a perfect field of characteristic . Let be a dlt pair such that the induced pair on a general fibre is log canonical. Assuming the log minimal model program and the existence of log resolutions in dimension , we prove that, when is ‐nef, the moduli part is nef up to a birational map . As a corollary, we prove positivity of the moduli part in the ‐trivial case, that is, when for some ‐Cartier ‐divisor on . In particular, consider a dlt pair of dimension 3 over an algebraically closed field of characteristic such that the induced pair on a general fibre is log canonical, then the canonical bundle formula holds unconditionally.