Morse cohomology estimates for jet differential operators

  • Jean-Pierre DemaillyEmail author
  • Mohammad Reza Rahmati


We provide detailed holomorphic Morse estimates for the cohomology of sheaves of jet differentials and their dual sheaves. These estimates apply on arbitrary directed varieties, and a special attention has been given to the analysis of the singular situation. As a consequence, we obtain existence results for global jet differentials and global differential operators under positivity conditions for the canonical or anticanonical sheaf of the directed structure.


Directed variety Jet bundle Jet differential Jet metric Holomorphic Morse inequalities Canonical sheaf 

Mathematics Subject Classfication

32H30 32L10 14J17 14J40 53C55 


  1. 1.
    Bloch, A.: Sur les systèmes de fonctions uniformes satisfaisant à l’équation d’une variété algébrique dont l’irrégularité dépasse la dimension. J. de Math. 5, 19–66 (1926)zbMATHGoogle Scholar
  2. 2.
    Bloch, A.: Sur les systèmes de fonctions holomorphes à variétés linéaires lacunaires. Ann. Ecole Normale 43, 309–362 (1926)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bonavero, L.: Inégalités de Morse holomorphes singulières. C. R. Acad. Sci. Paris Sér. I Math. 317, 1163–1166 (1993)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Demailly, J.-P.: Champs magnétiques et inégalités de Morse pour la \(d^{\prime \prime }\)-cohomologie. Ann. Inst. Fourier (Grenoble) 35, 189–229 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Demailly, J.-P.: Singular hermitian metrics on positive line bundles. In: Hulek, K., Peternell, T., Schneider, M., Schreyer, F. (eds.) Lecture Notes in Math. Proceedings of the Bayreuth conference “Complex algebraic varieties”, April 2–6, 1990, \({\rm n}^{\circ }\) 1507, pp. 87–104. Springer (1992)Google Scholar
  6. 6.
    Demailly, J.-P.: Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials. In: Kollár, J., Lazarsfeld, R. (eds.) Algebraic Geometry–Santa Cruz 1995, Proceedings Symposia in Pure Math, vol 62, pp. 285–360. American Mathematical Society Providence, RI (1997)Google Scholar
  7. 7.
    Demailly, J.-P.: Holomorphic Morse inequalities and the Green–Griffiths–Lang conjecture. Pure Appl. Math. Q. 7, 1165–1208 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Demailly, J.-P.: Hyperbolic algebraic varieties and holomorphic differential equations. Expanded version of the lectures given at the annual meeting of VIASM. Acta Math. Vietnam 37, 441–512 (2012)MathSciNetGoogle Scholar
  9. 9.
    Demailly, J.-P.: Towards the Green–Griffiths–Lang conjecture. In: Baklouti, A., El Kacimi, A., Kallel, S., Mir N. (eds) Conference “Analysis and Geometry”, Tunis, March 2014, in honor of Mohammed Salah Baouendi, pp. 141–159. Springer (2015)Google Scholar
  10. 10.
    Demailly, J.-P.: Recent results on the Kobayashi and Green–Griffiths–Lang conjectures. Contribution to the 16th Takagi lectures in celebration of the 100th anniversary of K.Kodaira’s birth, November 2015, to appear in the Japanese Journal of Mathematics. arXiv: Math.AG/1801.04765
  11. 11.
    Green, M., Griffiths, P.: Two applications of algebraic geometry to entire holomorphic mappings. In: The Chern symposium, 1979, proc. internal. sympos. Berkeley, CA, 1979, pp. 41–74. Springer, New York, (1980)Google Scholar
  12. 12.
    Lang, S.: Hyperbolic and diophantine analysis. Bull. Am. Math. Soc. 14, 159–205 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Merker, J.: Low pole order frames on vertical jets of the universal hypersurface. Ann. Inst. Fourier (Grenoble) 59, 1077–1104 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Păun, M.: Vector fields on the total space of hypersurfaces in the projective space and hyperbolicity. Math. Ann. 340, 875–892 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Siu, Y.T.: Hyperbolicity in complex geometry. The legacy of Niels Henrik Abel, pp. 543–566. Springer, Berlin (2004)CrossRefGoogle Scholar

Copyright information

© Unione Matematica Italiana 2018

Authors and Affiliations

  1. 1.Université de Grenoble-Alpes, Institut Fourier (Mathématiques), UMR 5582 du C.N.R.S.GièresFrance
  2. 2.Institute of Algebraic GeometryGottfried Wilhelm Leibniz Universität HannoverHannoverGermany

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