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Weak Akizuki–Nakano vanishing theorem for singular globally F-split 3-folds

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Abstract

In this paper, we prove that a weak form of the Akizuki–Nakano vanishing theorem holds on singular globally F-split 3-folds. Making use of this vanishing theorem, we study deformations of globally F-split Fano 3-folds and the Kodaira vanishing theorem for thickenings of locally complete intersection globally F-regular 3-folds.

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Notes

  1. The latter condition is satisfied for example if \(\mathcal {L}\) is very ample.

  2. This condition is satisfied for example if \(|-K_X|\) is base point free and \(p \geqslant 5\).

  3. Globally F-split varieties are often called simply F-split. However, we do not use this terminology in this paper, because it may be confused with locally F-split varieties.

  4. such a singularity is also called an \(A_1\)-singularity.

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Acknowledgements

The authors are grateful to Kenta Hashizume, Tatsuro Kawakami, Yujiro Kawamata, Yoichi Miyaoka, Shigefumi Mori, Noboru Nakayama, Taro Sano, Vasudevan Srinivas and Bernd Ulrich for valuable comments and discussions. They also thank the anonymous referee for helpful comments.

Funding

The first author was supported by RIKEN iTHEMS Program. The second author was partially supported by JSPS KAKENHI Grant Numbers JP15H03611, JP16H02141, and JP17H02831.

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Authors and Affiliations

  1. Faculty of Mathematics, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan

    Kenta Sato

  2. Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan

    Shunsuke Takagi

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  1. Kenta Sato
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  2. Shunsuke Takagi
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Correspondence to Kenta Sato.

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Dedicated to Professor Bernd Ulrich on the occasion of his sixty-fifth birthday.

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Sato, K., Takagi, S. Weak Akizuki–Nakano vanishing theorem for singular globally F-split 3-folds. manuscripta math. (2024). https://doi.org/10.1007/s00229-023-01529-9

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