Skip to main content
SpringerLink
Log in

Holomorphic sectional curvature, nefness and Miyaoka–Yau type inequality

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

On a compact Kähler manifold, we introduce a notion of almost nonpositivity for the holomorphic sectional curvature, which by definition is weaker than the existence of a Kähler metric with semi-negative holomorphic sectional curvature. We prove that a compact Kähler manifold of almost nonpositive holomorphic sectional curvature has a nef canonical line bundle, contains no rational curves and satisfies some Miyaoka-Yau type inequalities. In the course of the discussions, we attach a real value to any fixed Kähler class which, up to a constant factor depending only on the dimension of manifold, turns out to be an upper bound for the nef threshold.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Barth, W., Hulek, K., Peters, C., Van de Ven, A.: Compact complex surfaces, 2nd edn. Springer, Berlin (2004)

    Book  Google Scholar 

  2. Boucksom, S., Eyssidieux, P., Guedj, V., Zeriahi, A.: Monge-Ampère equations in big cohomology classes. Acta. Math. 205(2), 199–262 (2010)

    Article  MathSciNet  Google Scholar 

  3. Cao, H.-D.: Deformation of Kähler metrics to Kähler–Einstein metrics on compact Kähler manifolds. Invent. Math. 81(2), 359–372 (1985)

    Article  MathSciNet  Google Scholar 

  4. Chow, B., Lu, P., Ni, L.: Hamilton’s Ricci flow, Graduate Studies in Mathematics, 77. American Mathematical Society, Science Press, New York, Providence, RI (2006)

    Google Scholar 

  5. Demailly, J.-P., Păun, M.: Numerical characterization of the Kähler cone of a compact Kähler manifold. Ann. Math. 159(3), 1247–1274 (2004)

    Article  MathSciNet  Google Scholar 

  6. Diverio, S., Trapani, S.: Quasi-negative holomorphic sectional curvature and positivity of the canonical bundle. J. Differ. Geom. 111(2), 303–314 (2019)

    Article  MathSciNet  Google Scholar 

  7. Greb, D., Kebekus, S., Peternell, T., Taji, B.: The Miyaoka–Yau inequality and uniformisation of minimal models (2015). arXiv:1511.08822

  8. Guenancia, H., Taji, B.: Orbifold stability and Miyaoka–Yau inequality for minimal pairs (2016). arXiv:1611.05981

  9. Heier, G., Lu, S., Wong, B.: On the canonical line bundle and negative holomorphic sectional curvature. Math. Res. Lett. 17(6), 1101–1110 (2010)

    Article  MathSciNet  Google Scholar 

  10. Heier, G., Lu, S., Wong, B.: Kähler manifolds of semi-negative holomorphic sectional curvature. J. Differ. Geom. 104(3), 419–441 (2016)

    Article  Google Scholar 

  11. Heier, G., Lu, S., Wong, B., Zheng, F.: Reduction of manifolds with semi-negative holomorphic sectional curvature. Math. Ann. 372(3–4), 951–962 (2018)

    Article  MathSciNet  Google Scholar 

  12. Höring, A., Peternell, T.: Minimal model for Kähler threefolds. Invent. Math. 203(1), 217–264 (2016)

    Article  MathSciNet  Google Scholar 

  13. Kawamata, Y.: On the length of an extremal rational curve. Invent. Math. 105, 609–611 (1991)

    Article  MathSciNet  Google Scholar 

  14. Matsuki, K.: Introduction to the Mori program. Universitext, Springer, New York (2002)

    Book  Google Scholar 

  15. Miyaoka, Y.: On the Chern numbers of surfaces of general type. Invent. Math. 42, 225–237 (1977)

    Article  MathSciNet  Google Scholar 

  16. Nomura, R.: Kähler manifolds with negative holomorphic sectional curvature, Kähler–Ricci flow approach. Int. Math. Res. Not. IMRN 21, 6611–6616 (2018)

    Article  Google Scholar 

  17. Nomura, R.: Miyaoka–Yau inequality on compact Kähler manifolds with semi-positive canonical bundle (2018). arXiv:1802.05425

  18. Phong, D.H., Šešum, N., Sturm, J.: Multiplier ideal sheaves and the Kähler–Ricci flow. Commun. Anal. Geom. 15(3), 613–632 (2007)

    Article  Google Scholar 

  19. Royden, H.L.: Ahlfors–Schwarz lemma in several complex variables. Comment. Math. Helv. 55(4), 547–558 (1980)

    Article  MathSciNet  Google Scholar 

  20. Song, J., Tian, G.: Bounding scalar curvature for global solutions of the Kähler–Ricci flow. Am. J. Math. 138(3), 683–695 (2016)

    Article  Google Scholar 

  21. Song, J., Wang, X.: The greatest Ricci lower bound, conical Einstein metrics and the Chern number inequality. Geom. Topol. 20, 49–10 (2016)

    Article  MathSciNet  Google Scholar 

  22. Tang, K.: On real bisectional curvature and Kähler–Ricci flow. Proc. Am. Math. Soc. 147(2), 793–798 (2019)

    Article  Google Scholar 

  23. Tian, G., Zhang, Z.: On the Kähler–Ricci flow on projective manifolds of general type. Chin. Ann. Math. Ser. B 27(2), 179–192 (2006)

    Article  Google Scholar 

  24. Tosatti, V., Yang, X.: An extension of a theorem of Wu–Yau. J. Differ. Geom. 107(3), 573–579 (2017)

    Article  MathSciNet  Google Scholar 

  25. Tosatti, V., Zhang, Y.G.: Infinite-time singularities of the Kähler–Ricci flow. Geom. Topol. 19(5), 2925–2948 (2015)

    Article  MathSciNet  Google Scholar 

  26. Tsuji, H.: Existence and degeneration of Kähler–Einstein metrics on minimal algebraic varieties of general type. Math. Ann. 281, 123–133 (1988)

    Article  MathSciNet  Google Scholar 

  27. Tsuji, H.: Stability of tangent bundles of minimal algebraic varieties. Topology 27(4), 429–442 (1988)

    Article  MathSciNet  Google Scholar 

  28. Wong, B.: The uniformization of compact Kähler surfaces of negative curvature. J. Differ. Geom. 16(3), 407–420 (1981)

    Article  Google Scholar 

  29. Wong, P.-M., Wu, D., Yau, S.-T.: Picard number, holomorphic sectional curvature and ampleness. Proc. Am. Math. Soc. 140(2), 621–626 (2012)

    Article  MathSciNet  Google Scholar 

  30. Wu, D., Yau, S.-T.: Negative holomorphic curvature and positive canonical bundle. Invent. Math. 204(2), 595–604 (2016)

    Article  MathSciNet  Google Scholar 

  31. Wu, D., Yau, S.-T.: A remark on our paper “Negative holomorphic curvature and positive canonical bundle”. Commun. Anal. Geom. 24(4), 901–912 (2016)

    Article  Google Scholar 

  32. Yang, X., Zheng, F.: On real bisectional curvature for Hermitian manifolds. Trans. Am. Math. Soc. 371(4), 2703–2718 (2019)

    Article  MathSciNet  Google Scholar 

  33. Yau, S.-T.: Calabi’s conjecture and some new results in algebraic geometry. Proc. Nat. Acad. Sci. USA 74(5), 1798–1799 (1977)

    Article  MathSciNet  Google Scholar 

  34. Yau, S.-T.: A general Schwarz lemma for Kähler manifolds. Am. J. Math. 100(1), 197–203 (1978)

    Article  Google Scholar 

  35. Yau, S.-T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation, I. Commun. Pure Appl. Math. 31, 339–411 (1978)

    Article  Google Scholar 

  36. Zhang, Y.G.: Miyaoka-Yau inequality for minimal projective manifolds of general type. Proc. Am. Math. Soc. 137(8), 2749–2754 (2009)

    Article  MathSciNet  Google Scholar 

  37. Zheng, F.: Complex differential geometry. In: AMS/IP studies in advanced mathematics, vol. 18. American Mathematical Society, Providence (2000)

Download references

Acknowledgements

The author is grateful to Professors Huai-Dong Cao and Gang Tian for their interest in this work, and constant encouragement and support. The author also thanks Professors Gang Tian for conversations on Chern number inequality, Valentino Tosatti for discussions on papers [6, 24, 30], Chengjie Yu and Tao Zheng for comments and the referees for careful readings and a number of helpful comments and corrections.

Author information

Authors and Affiliations

  1. School of Mathematics and Hunan Province Key Lab of Intelligent Information Processing and Applied Mathematics, Hunan University, Changsha, 410082, China

    Yashan Zhang

Authors
  1. Yashan Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yashan Zhang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The author is partially supported by Fundamental Research Funds for the Central Universities (No. 531118010468) and National Natural Science Foundation of China (No. 12001179).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, Y. Holomorphic sectional curvature, nefness and Miyaoka–Yau type inequality. Math. Z. 298, 953–974 (2021). https://doi.org/10.1007/s00209-020-02636-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-020-02636-z

Keywords

Mathematics Subject Classification

Search

Navigation