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Strictly nef divisors on K-trivial fourfolds

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In this paper, we prove the ampleness conjecture and Serrano’s conjecture for strictly nef divisors on K-trivial fourfolds, particularly, hyperkähler fourfolds. Specifically, for a smooth projective fourfold X of Calabi–Yau type with vanishing irregularity, we show that strictly nef divisors on X are ample.

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References

  1. Barth, W., Hulek, K., Peters, C.A.M., Van de Ven, A.: Compact complex surfaces, 2nd edn. In: Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics, vol. 4. Springer, Berlin (2004)

  2. Beauville, A.: Variétés Kähleriennes dont la première classe de Chern est nulle. J. Differ. Geom. 18(4), 755–782 (1983)

    Article  MATH  Google Scholar 

  3. Birkar, C., Cascini, P., Hacon, C.D., McKernan, J.: Existence of minimal models for varieties of log general type. J. Am. Math. Soc. 23(2), 405–468 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Birkenhake, Ch., Lange, H.: Complex abelian varieties. In: Grundlehren der Mathematischen Wissenschaften, vol. 302. Springer, Berlin (2004)

    Google Scholar 

  5. Boucksom, S., Demailly, J.-P., Pǎun, M., Peternell, Th.: The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension. J. Algebraic Geom. 22(2), 201–248 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Campana, F., Chen, J.A., Peternell, Th.: Strictly nef divisors. Math. Ann. 342(3), 565–585 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Campana, F., Oguiso, K., Peternell, Th.: Non-algebraic hyperkähler manifolds. J. Differ. Geom. 85(3), 397–424 (2010)

    Article  MATH  Google Scholar 

  8. Demailly, J.-P.: Regularization of closed positive currents and intersection theory. J. Algebraic Geom. 1(3), 361–409 (1992)

    MathSciNet  MATH  Google Scholar 

  9. Demailly, J.-P.: Analytic methods in algebraic geometry. In: Surveys of Modern Mathematics, vol. 1. International Press; Higher Education Press, Somerville; Beijing (2012)

    Google Scholar 

  10. Demailly, J.-P., Peternell, Th., Schneider, M.: Pseudo-effective line bundles on compact Kähler manifolds. Int. J. Math. 12(6), 689–741 (2001)

    Article  MATH  Google Scholar 

  11. Diverio, S., Fontanari, C., Martinelli, D.: Rational curves on fibered Calabi–Yau manifolds. Doc. Math. 24, 663–675 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fujiki, A., Pontecorvo, M.: Non-upper-semicontinuity of algebraic dimension for families of compact complex manifolds. Math. Ann. 348(3), 593–599 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fujino, O.: Finite generation of the log canonical ring in dimension four. Kyoto J. Math. 50(4), 671–684 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fujino, O.: Non-vanishing theorem for log canonical pairs. J. Algebraic Geom. 20(4), 771–783 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fujino, O.: On Kawamata’s theorem. In: Classification of Algebraic Varieties. EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, pp. 305–315 (2011)

  16. Fujino, O.: Foundations of the minimal model program. In: MSJ Memoirs, vol. 35. Mathematical Society of Japan, Tokyo (2017)

    Google Scholar 

  17. Fukuda, S.: On numerically effective log canonical divisors. Int. J. Math. Math. Sci. 30(9), 521–531 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Greb, D., Lehn, C., Rollenske, S.: Lagrangian fibrations on hyperkähler manifolds—on a question of Beauville. Ann. Sci. Éc. Norm. Supér. (4) 46(3), 375–403 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hwang, J.-M., Weiss, R.M.: Webs of Lagrangian tori in projective symplectic manifolds. Invent. Math. 192(1), 83–109 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Han, J., Liu, W.: On numerical nonvanishing for generalized log canonical pairs. Doc. Math. 25, 93–123 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hartshorne, R.: Ample subvarieties of algebraic varieties. In: Lect. Notes Math., vol. 156. Springer, Heidelberg (1970)

    Google Scholar 

  22. Huybrechts, D.: Compact hyper-Kähler manifolds: basic results. Invent. Math. 135(1), 63–113 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  23. Huybrechts, D.: The Kähler cone of a compact hyperkähler manifold. Math. Ann. 326(3), 499–513 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  24. Jiang, X.: On the pluricanonical maps of varieties of intermediate Kodaira dimension. Math. Ann. 356(3), 979–1004 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kawamata, Y.: Pluricanonical systems on minimal algebraic varieties. Invent. Math. 79(3), 567–588 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kollár, J.: Deformations of elliptic Calabi–Yau manifolds. In: Recent Advances in Algebraic Geometry. London Math. Soc., Lecture Note Ser., vol. 417, pp. 254–290. Cambridge Univ. Press, Cambridge (2015)

    Chapter  Google Scholar 

  27. Kollár, J., Mori, S.: Birational geometry of algebraic varieties. In: Cambridge Tracts in Mathematics, vol. 134. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  28. Lazarsfeld, R.: Positivity in algebraic geometry I, II. In: Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 48, 49. Springer, Berlin (2004)

    Google Scholar 

  29. Lazić, V., Oguiso, K., Peternell, Th.: Nef line bundles on Calabi–Yau threefolds I. Int. Math. Res. Not. IMRN (19), 6070–6119 (2020)

  30. Lazić, V., Oguiso, K., Peternell, Th.: The Morrison–Kawamata cone conjecture and abundance on Ricci flat manifolds. In: Uniformization, Riemann–Hilbert Correspondence, Calabi–Yau Manifolds and Picard–Fuchs Equations, Adv. Lect. Math. (ALM), vol. 42, pp. 157–185. Int. Press, Somerville (2018)

    Google Scholar 

  31. Lazić, V., Peternell, Th.: Rationally connected varieties-on a conjecture of Mumford. Sci. China Math. 60(6), 1019–1028 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lazić, V., Peternell, Th.: Abundance for varieties with many differential forms. Épijournal Géom. Algébrique 2, 35 (2018). (Art. 1)

    MathSciNet  MATH  Google Scholar 

  33. Lazić, V., Peternell, Th.: On generalised abundance, I. Publ. Res. Inst. Math. Sci. 56(2), 353–389 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  34. Liu, H., Svaldi, R.: Rational curves and strictly nef divisors on Calabi–Yau threefolds. Doc. Math. (to appear). arXiv:2010.12233

  35. Miyaoka, Y.: The Chern classes and kodaira dimension of a minimal variety. Adv. Stud. Pure Math. 10, 449–476 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  36. Oguiso, K.: On algebraic fiber space structures on a Calabi–Yau 3-fold. Int. J. Math. 4(3), 439–465 (1993). (with an appendix by Noboru Nakayama)

    Article  MathSciNet  MATH  Google Scholar 

  37. Serrano, F.: Strictly nef divisors and Fano threefolds. J. Reine Angew. Math. 464, 187–206 (1995)

    MathSciNet  MATH  Google Scholar 

  38. Verbitsky, M.: HyperKähler SYZ conjecture and semipositive line bundles. Geom. Funct. Anal. 19(5), 1481–1493 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  39. Wilson, P.M.H.: The Kähler cone on Calabi–Yau threefolds. Invent. Math. 107(3), 561–583 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  40. Wilson, P.M.H.: The existence of elliptic fibre space structures on Calabi–Yau threefolds. Math. Ann. 300(4), 693–703 (1994)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank Prof. Vladimir Lazić for carefully reading a draft of this paper and pointing out several mistakes. The discussions with him significantly improved this paper. The first author is very grateful to Prof. Chen Jiang for his useful conversations and reading several drafts of this paper. He would like to thank Professors Osamu Fujino, Zhengyu Hu, Roberto Svaldi, and Zhiyu Tian for useful discussions and suggestions and to thank Prof. Wenfei Liu for pointing out that [20, Question 3.5] is stronger than Conjecture 1.5. He also would like to thank Prof. Thomas Peternell for pointing out a disastrous gap in a draft of this paper. The first author is supported by the NSFC (Grant no. 12001018). The second author is supported by Grant-in-Aid for Scientific Research (B) \(\sharp \) 21H00976 and Fostering Joint International Research (A) \(\sharp \)19KK0342 from JSPS.

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

    Haidong Liu

  2. Mathematical Institute, Tohoku University, 6-3, Aramaki Aza-Aoba, Aoba-ku, Sendai, 980-8578, Japan

    Shin-ichi Matsumura

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  1. Haidong Liu
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Correspondence to Shin-ichi Matsumura.

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Liu, H., Matsumura, Si. Strictly nef divisors on K-trivial fourfolds. Math. Ann. 387, 985–1008 (2023). https://doi.org/10.1007/s00208-022-02461-1

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