Ohsawa-Takegoshi Extension Theorem for Compact Kähler Manifolds and Applications

  • Junyan CaoEmail author
Part of the Springer INdAM Series book series (SINDAMS, volume 21)


Our main goal in this article is to prove an extension theorem for sections of the canonical bundle of a weakly pseudoconvex Kähler manifold with values in a line bundle endowed with a possibly singular metric. We also give some applications of our result.


Kähler manifolds Ohsawa-Takegoshi extension Singular metric 



I would like to thank H. Tsuji who brought me attention to this problem during the Hayama conference 2013. I would also like to thank M. Păun for pointing out several interesting applications, and a serious mistake in the first version of the article. I would also like to thank J.-P. Demailly and X. Zhou for helpful discussions. Last but not least, I would like to thank the anonymous referee for excellent suggestions about this work.


  1. 1.
    B. Berndtsson, Integral formulas and the Ohsawa-Takegoshi extension theorem. Sci. China Ser. A 48(Suppl.), 61–73 (2005)Google Scholar
  2. 2.
    B. Berndtsson, L. Lempert, A proof of the Ohsawa–Takegoshi theorem with sharp estimates. J. Math. Soc. Jpn. 68(4), 1461–1472 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    B. Berndtsson, M. Păun, Bergman kernels and the pseudoeffectivity of relative canonical bundles. Duke Math. J. 145(2), 341–378 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    B. Berndtsson, M. Păun, Bergman kernels and subadjunction. arXiv: 1002.4145v1Google Scholar
  5. 5.
    J. Bertin, J.-P. Demailly, L. Illusie, C. Peters, Introduction to Hodge Theory. SMF/AMS Texts and Monographs, vol. 8 (2002)Google Scholar
  6. 6.
    Z. Błocki, Suita conjecture and the Ohsawa-Takegoshi extension theorem. Invent. Math. 193(1), 149–158 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    B.-Y. Chen, A simple proof of the Ohsawa-Takegoshi extension theorem. ArXiv e-prints 1105.2430Google Scholar
  8. 8.
    J.-P. Demailly, Multiplier ideal sheaves and analytic methods in algebraic geometry, in School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000). ICTP Lecture Notes, vol. 6 (ICTP, Trieste, 2001), pp. 1–148Google Scholar
  9. 9.
    J.-P. Demailly, Analytic Methods in Algebraic Geometry. Surveys of Modern Mathematics, vol. 1 (International Press, Boston, 2012)Google Scholar
  10. 10.
    J.-P. Demailly, Extension of holomorphic functions defined on non reduced analytic subvarieties. arXiv:1510.05230v1. Advanced Lectures in Mathematics Volume 35.1, the legacy of Bernhard Riemann after one hundred and fifty years, 2015Google Scholar
  11. 11.
    J.-P. Demailly, T. Peternell, A Kawamata-Viehweg vanishing theorem on compact Kähler manifolds. J. Differ. Geom. 63(2), 231–277 (2003)CrossRefzbMATHGoogle Scholar
  12. 12.
    H. Flenner, Ein Kriterium für die Offenheit der Versalität. Math. Z. 178(4), 449–473 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Q. Guan, X. Zhou, A solution of L 2 extension problem with optimal estimate and applications. Ann. Math. 181(3), 1139–1208 (2015). arXiv:1310.7169Google Scholar
  14. 14.
    Q. Guan, X. Zhou, A proof of Demailly’s strong openness conjecture. Ann. Math. 182(2), 605–616 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    P.H. Hiêp, The weighted log canonical threshold. C.R. Math. 352(4), 283–288 (2014)Google Scholar
  16. 16.
    Y. Kawamata, Pluricanonical systems on minimal algebraic varieties. Invent. Math. 79(3), 567–588 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    L. Manivel, Un théorème de prolongement L 2 de sections holomorphes d’un fibré hermitien. Math. Z. 212(1), 107–122 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    T. Ohsawa, On the extension of L 2 holomorphic functions. II. Publ. Res. Inst. Math. Sci. 24(2), 265–275 (1988)CrossRefzbMATHGoogle Scholar
  19. 19.
    T. Ohsawa, K. Takegoshi, On the extension of L 2 holomorphic functions. Math. Z. 195(2), 197–204 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Y.-T. Siu, Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semipositively twisted plurigenera for manifolds not necessarily of general type, in Complex Geometry: Collection of Papers Dedicated to Hans Grauert (Springer, Berlin, 2002), pp. 223–277zbMATHGoogle Scholar
  21. 21.
    H. Tsuji, Extension of log pluricanonical forms from subvarieties. arXiv 0709.2710Google Scholar
  22. 22.
    L. Yi, An Ohsawa-Takegoshi theorem on compact Kähler manifolds. Sci. China Math. 57(1), 9–30 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    L. Zhu, Q. Guan, X. Zhou, On the Ohsawa–Takegoshi L2 extension theorem and the Bochner–Kodaira identity with non-smooth twist factor. J. Math. Pures Appl. 97(6), 579–601 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer International Publishing AG, a part of Springer Nature 2017

Authors and Affiliations

  1. 1.Institut de Mathématiques de JussieuUniversité Paris 6, Case 247ParisFrance

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