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On the canonical bundle formula and log abundance in positive characteristic


We show that a weak version of the canonical bundle formula holds for fibrations of relative dimension one. We provide various applications thereof, for instance, using the recent result of Xu and Zhang, we prove the log non-vanishing conjecture for three-dimensional klt pairs over any algebraically closed field k of characteristic \(p>5\), which in turn implies the termination of any sequence of three-dimensional flips in the pseudo-effective case. We also show the log abundance conjecture for threefolds over k when the nef dimension is not maximal, and the base point free theorem for threefolds over \(\mathbb {{\overline{F}}}_p\) when \(p>2\).

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  1. See Remark 3.9 for a clarification.


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I would like to express my special gratitude to Paolo Cascini for his substantial help, encouragement, and support. I am grateful to the referee for his very valuable comments and for reading the article thoroughly. Further, I would like to thank Hiromu Tanaka for numerous discussions, encouragement, and advice on proving log non-vanishing. I also thank Yoshinori Gongyo, Diletta Martinelli, Yusuke Nakamura, Johannes Nicaise, Zsolt Patakfalvi, Joe Waldron, Chenyang Xu, and Lei Zhang for comments and helpful suggestions. The paper has been motivated by the work [32] started at the Pragmatic research school in Catania 2013.

The author was supported by the Engineering and Physical Sciences Research Council [EP/L015234/1].

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Correspondence to Jakub Witaszek.

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Communicated by Vasudevan Srinivas.

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Witaszek, J. On the canonical bundle formula and log abundance in positive characteristic. Math. Ann. 381, 1309–1344 (2021).

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Mathematics Subject Classification

  • 14E30
  • 14E05