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On Severi type inequalities

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Abstract

We study Severi type inequalities for big line bundles on irregular varieties via cohomological rank functions. We show that these Severi type inequalities are related to some naturally defined birational invariants of the general fibers of the Albanese morphisms. As an application, we show that the volume of an irregular threefold of general type is at least \(\frac{3}{8}\). We also show that the volume of a smooth projective variety X of general type and of maximal Albanese dimension is at least \(2(\dim X)!\). Moreover, if \({{\,\mathrm{vol}\,}}(X)=2(\dim X)!\), the canonical model of X is a double cover of a principally polarized abelian variety \((A, \Theta )\) branched over some divisor \(D\in |2\Theta |\).

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Acknowledgements

We thank Fabrizio Catanese, Yong Hu, Martí Lahoz, Giuseppe Pareschi and Tong Zhang for stimulating conversations. We thank Olivier Debarre for reading the draft carefully and suggestions. Parts of this work were written during the author’s visits to the Graduate School of Mathematical Sciences in the University of Tokyo and National Center for Theoretical Sciences in Taipei and we thank Jheng-Jie Chen, Jungkai Alfred Chen, Yoshinori Gongyo, and Yusuke Nakamura for the warm hospitality. Finally we are grateful to the referee for her/his careful reading and comments.

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  1. Shanghai Center for Mathematical Sciences, Xingjiangwan Campus, Fudan University, Shanghai, 200438, People’s Republic of China

    Zhi Jiang

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Correspondence to Zhi Jiang.

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Communicated by Vasudevan Srinivas.

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The author is partially supported by NSFC Grant Nos. 11871155 and 11731004.

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Jiang, Z. On Severi type inequalities. Math. Ann. 379, 133–158 (2021). https://doi.org/10.1007/s00208-020-02082-6

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  • DOI: https://doi.org/10.1007/s00208-020-02082-6

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