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On the canonical bundle formula in positive characteristic
[Submitted on 31 May 2023 (v1), last revised 22 May 2026 (this v · 2026-05-25 · via math updates on arXiv.org

Mathematics > Algebraic Geometry

arXiv:2305.19841 (math)

[Submitted on 31 May 2023 (v1), last revised 22 May 2026 (this version, v3)]

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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 $\leq n$, 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 $K_X+B \sim_{\Q} f^*L$ 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.

Submission history

From: Marta Benozzo [view email]
[v1] Wed, 31 May 2023 13:30:45 UTC (413 KB)
[v2] Thu, 14 Nov 2024 14:07:59 UTC (70 KB)
[v3] Fri, 22 May 2026 14:32:17 UTC (67 KB)

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