Quasi-Negative Holomorphic Sectional Curvature and Ampleness of the Canonical Class

Part of the Springer INdAM Series book series (SINDAMS, volume 21)


This note is an extended version of a 50 min talk given at the INdAM Meeting “Complex and Symplectic Geometry”, held in Cortona from June 12th to June 18th, 2016. What follows was the abstract of our talk.

Let X be a compact Kähler manifold with a Kähler metric whose holomorphic sectional curvature is strictly negative. Very recent results by Wu–Yau and Tosatti–Yang confirmed an old conjecture by S.-T. Yau which claimed that under this curvature assumption X should be projective and canonically polarized. We will explain how one can relax the assumption on the holomorphic sectional curvature to the weakest possible, i.e. non positive and strictly negative in at least one point, in order to have the same conclusions. We shall also try to motivate this generalization by arguments coming from birational geometry, such as the abundance conjecture.

The results presented here were originally contained in the joint work with Diverio and Trapani (Quasi-negative holomorphic sectional curvature and positivity of the canonical bundle, 2016, ArXiv e-prints 1606.01381v3).


Canonical bundle Holomorphic sectional curvature Kobayashi’s conjecture Monge–Ampère equation Ricci curvature 



We would like to warmly thank the organizers Daniele Angella, Costantino Medori, Adriano Tomassini for the beautiful and stimulating environment of the INdAM Meeting “Complex and Symplectic Geometry”, in Cortona. A particular thought goes to Paolo De Bartolomeis, who sadly passed away on November 29th, 2016.

The author is partially supported by the ANR project “GRACK”, ANR-15-CE40-0003, and the ANR project “Foliage”, ANR-16-CE40-0008.


  1. 1.
    M. Berger, Sur les variétés d’Einstein compactes, in Comptes Rendus de la IIIe Réunion du Groupement des Mathématiciens d’Expression Latine (Namur, 1965) (Librairie Universitaire, Louvain, 1966), pp. 35–55Google Scholar
  2. 2.
    Campana, Frédéric; Höring, Andreas, Peternell, Thomas. Abundance for Kähler threefolds. Ann. Sci. Éc. Norm. Supér. (4) 49(4), 971–1025 (2016)Google Scholar
  3. 3.
    O. Debarre, Higher-Dimensional Algebraic Geometry. Universitext (Springer, New York, 2001)CrossRefMATHGoogle Scholar
  4. 4.
    J.-P. Demailly, Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials, in Algebraic Geometry—Santa Cruz 1995. Proceedings of Symposia in Pure Mathematics, vol. 62 (American Mathematical Society, Providence, RI, 1997), pp. 285–360Google Scholar
  5. 5.
    J.-P. Demailly, M. Păun, Numerical characterization of the Kähler cone of a compact Kähler manifold. Ann. Math. (2) 159(3), 1247–1274 (2004)Google Scholar
  6. 6.
    S. Diverio, Segre forms and Kobayashi–Lübke inequality. Math. Z. 283(3–4), 1033–1047 (2016)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    S. Diverio, S. Trapani, Quasi-negative holomorphic sectional curvature and positivity of the canonical bundle (2016). ArXiv e-prints 1606.01381v3Google Scholar
  8. 8.
    V. Guedj, A. Zeriahi, Intrinsic capacities on compact Kähler manifolds. J. Geom. Anal. 15(4), 607–639 (2005)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    G. Heier, S.S.Y. Lu, B. Wong, On the canonical line bundle and negative holomorphic sectional curvature. Math. Res. Lett. 17(6), 1101–1110 (2010)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    G. Heier, S.S.Y. Lu, B. Wong, Kähler manifolds of semi-negative holomorphic sectional curvature. J. Differ. Geom. (2014, to appear). ArXiv e-printsGoogle Scholar
  11. 11.
    A. Höring, T. Peternell, I. Radloff, Uniformisation in dimension four: towards a conjecture of Iitaka. Math. Z. 274(1–2), 483–497 (2013)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    R. Lazarsfeld, Positivity in Algebraic Geometry. I & II. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vols. 48 and 49 (Springer, Berlin, 2004)Google Scholar
  13. 13.
    R. Nomura, Kähler manifolds with negative holomorphic sectional curvature, Kähler-Ricci flow approach (2016). ArXiv e-printsGoogle Scholar
  14. 14.
    H.L. Royden, The Ahlfors-Schwarz lemma in several complex variables. Comment. Math. Helv. 55(4), 547–558 (1980)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    S. Takayama, On the uniruledness of stable base loci. J. Differ. Geom. 78(3), 521–541 (2008)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    V. Tosatti, X. Yang, An extension of a theorem of Wu-Yau. J. Differ. Geom. (2015, to appear). ArXiv e-printsGoogle Scholar
  17. 17.
    P.-M. Wong, D. Wu, S.-T. Yau, Picard number, holomorphic sectional curvature, and ampleness. Proc. Am. Math. Soc. 140(2), 621–626 (2012)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    D. Wu, S.-T. Yau, A remark on our paper “Negative Holomorphic curvature and positive canonical bundle” (2016). ArXiv e-printsGoogle Scholar
  19. 19.
    D. Wu, S.-T. Yau, Negative holomorphic curvature and positive canonical bundle. Invent. Math. 204(2), 595–604 (2016)MathSciNetCrossRefMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Dipartimento di Matematica “Guido Castelnuovo”SAPIENZA Università di RomaRomaItaly

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