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.
| Comments: | (v3) 54 pages, revised version, updated with suggestions from referee, to appear in Journal of the London Mathematical Society |
| Subjects: | Algebraic Geometry (math.AG) |
| MSC classes: | 14E30, 14G17, 14D06 |
| Cite as: | arXiv:2305.19841 [math.AG] |
| (or arXiv:2305.19841v3 [math.AG] for this version) | |
| https://doi.org/10.48550/arXiv.2305.19841 arXiv-issued DOI via DataCite |
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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