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Miyaoka-type inequalities for terminal threefolds with nef anti-canonical divisors

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Abstract

In this paper, we study Miyaoka-type inequalities on Chern classes of terminal projective 3-folds with nef anti-canonical divisors. Let X be a terminal projective 3-fold such that −KX is nef. We show that if c1(X) · c2(X) ≠ 0, then \(c_{1}(X)\cdot c_{2}(X)\geqslant {1\over 252}\); if further X is not rationally connected, then \(c_{1}(X)\cdot c_{2}(X)\geqslant {4\over 5}\) and this inequality is sharp. In order to prove this, we give a partial classification of such varieties along with many examples. We also study the nonvanishing of c1(X)dim X − 2 ·c2(X) for terminal weak Fano varieties and prove a Miyaoka-Kawamata-type inequality.

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Acknowledgements

This work was supported by National Natural Science Foundation of China for Innovative Research Groups (Grant No. 12121001) and National Key Research and Development Program of China (Grant No. 2020YFA0713200). The first author was supported by Grant-in-Aid for Early Career Scientists (Grant No. 22K13907). The second author is a member of Laboratory of Mathematics for Nonlinear Sciences, Fudan University. The authors thank Professors Vladimir Lazić (University of Saarland), Shin-ichi Matsumura (Tohoku University), Thomas Peternell (University of Bayreuth), and Shilin Yu (**amen University) for useful discussions and suggestions. The first author thanks Kento Fujita (Osaka University) for useful comments. The third author thanks **aobin Li (RingCentral) for his computer support. The authors thank the referees for their helpful comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Graduate School of Science, Osaka University, Osaka, 560-0043, Japan

    Masataka Iwai

  2. Shanghai Center for Mathematical Sciences & School of Mathematical Sciences, Fudan University, Shanghai, 200438, China

    Chen Jiang

  3. Department of mathematics, Sun Yat-sen University, Guangzhou, 510275, China

    Haidong Liu

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  1. Masataka Iwai
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  2. Chen Jiang
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  3. Haidong Liu
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Correspondence to Chen Jiang.

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Iwai, M., Jiang, C. & Liu, H. Miyaoka-type inequalities for terminal threefolds with nef anti-canonical divisors. Sci. China Math. (2024). https://doi.org/10.1007/s11425-023-2230-6

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