Siu’s lemma, optimal \(L^2\) extension and applications to twisted pluricanonical sheaves

  • Xiangyu ZhouEmail author
  • Langfeng ZhuEmail author


We prove a general version of Siu’s lemma for plurisubharmonic functions with nontrivial multiplier ideal sheaves and use it to prove an optimal \(L^2\) extension theorem and an optimal \(L^{\frac{2}{m}}\) extension theorem for Kähler fibrations. These results are used to prove the positivity of twisted relative pluricanonical bundles and their direct images for Kähler fibrations and to answer a comparison question about singular metrics of twisted pluricanonical bundles.

Mathematics Subject Classification

Primary 32D15 32Q15 32U05 Secondary 32A55 32L10 32W05 



  1. 1.
    Angehrn, U., Siu, Y.-T.: Effective freeness and point separation for adjoint bundles. Invent. Math. 122(2), 291–308 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Berndtsson, B.: The extension theorem of Ohsawa–Takegoshi and the theorem of Donnelly–Fefferman. Ann. Inst. Fourier (Grenoble) 46(4), 1083–1094 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Berndtsson, B.: Curvature of vector bundles associated to holomorphic fibrations. Ann. Math. (2) 169(2), 531–560 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Berndtsson, B.: The openness conjecture and complex Brunn–Minkowski inequalities. In: Complex geometry and dynamics, Abel Symp. vol. 10. Springer, Cham, pp. 29–44 (2015)Google Scholar
  5. 5.
    Berndtsson, B., Lempert, L.: A proof of the Ohsawa–Takegoshi theorem with sharp estimates. J. Math. Soc. Jpn. 68(4), 1461–1472 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Berndtsson, B., Păun, M.: Bergman kernels and the pseudoeffectivity of relative canonical bundles. Duke Math. J. 145, 341–378 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Berndtsson, B., Păun, M.: Bergman kernels and subadjunction (2010). arXiv:1002.4145v1
  8. 8.
    Błocki, Z.: Suita conjecture and the Ohsawa–Takegoshi extension theorem. Invent. Math. 193(1), 149–158 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Cao, J.Y.: Ohsawa–Takegoshi extension theorem for compact Kähler manifolds and applications. In: Complex and symplectic geometry, Springer INdAM Series 21, Springer, Cham, pp. 19–38 (2017)Google Scholar
  10. 10.
    Demailly, J.-P.: Estimations \(L^2\) pour l’opérateur \(\bar{\partial }\) d’un fibré vectoriel holomorphe semi-positif au dessus d’une variété kählérienne complète. Ann. Sci. Ecole Norm. Sup. 15, 457–511 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Demailly, J.-P.: Regularization of closed positive currents of type \((1,1)\) by the flow of a Chern connection. Contributions to complex analysis and analytic geometry. Asp. Math. E 26, 105–126 (1994) (Friedr. Vieweg, Braunschweig)Google Scholar
  12. 12.
    Demailly, J.-P.: On the Ohsawa–Takegoshi-Manivel \(L^{2}\) extension theorem. Complex analysis and geometry (Paris, 1997). Progr. Math. 188, 47–82 (2000) (Birkhäuser, Basel)Google Scholar
  13. 13.
    Demailly, J.-P.: Multiplier ideal sheaves and analytic methods in algebraic geometry. School on vanishing theorems and effective results in algebraic geometry (Trieste, 2000), ICTP Lect. Notes, vol. 6, Abdus Salam Int. Cent. Theoret. Phys., Trieste, pp. 1–148 (2001)Google Scholar
  14. 14.
    Demailly, J.-P.: Analytic methods in algebraic geometry. Surveys of Modern Mathematics, vol. 1. International Press, Somerville, Higher Education Press, Beijing (2012)Google Scholar
  15. 15.
    Demailly, J.-P.: Extension of holomorphic functions defined on non reduced analytic subvarieties. The legacy of Bernhard Riemann after one hundred and fifty years, vol. I. Adv. Lect. Math. (ALM), vol. 35.1. Int. Press, Somerville, pp. 191–222 (2016)Google Scholar
  16. 16.
    Demailly, J.-P.: Complex analytic and differential geometry (2012).
  17. 17.
    Deng, F.S., Zhang, H.P., Zhou, X.Y.: Positivity of direct images of positively curved volume forms. Math. Z. 278(1–2), 347–362 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Deng, F.S., Zhang, H.P., Zhou, X.Y.: Positivity of character subbundles and minimum principle for noncompact group actions. Math. Z. 286(1–2), 431–442 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Fornæss, J.E.: Several complex variables. arXiv:1507.00562v1
  20. 20.
    Griffiths, P., Harris, J.: Principles of Algebraic Geometry, Pure and Applied Mathematics. Wiley, New York (1978)zbMATHGoogle Scholar
  21. 21.
    Guan, Q.A., Zhou, X.Y.: Optimal constant problem in the \(L^2\) extension theorem. C. R. Math. Acad. Sci. Paris 350(15–16), 753–756 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Guan, Q.A., Zhou, X.Y.: Optimal constant in an \(L^2\) extension problem and a proof of a conjecture of Ohsawa. Sci. China Math. 58(1), 35–59 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Guan, Q.A., Zhou, X.Y.: A solution of an \(L^2\) extension problem with an optimal estimate and applications. Ann. Math. (2) 181(3), 1139–1208 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Guan, Q.A., Zhou, X.Y.: A proof of Demailly’s strong openness conjecture. Ann. Math. (2) 182(2), 605–616 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Guan, Q.A., Zhou, X.Y.: Strong openness of multiplier ideal sheaves and optimal \(L^2\) extension. Sci. China Math. 60(6), 967–976 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Guan, Q.A., Zhou, X.Y., Zhu, L.F.: On the Ohsawa-Takegoshi \(L^2\) extension theorem and the twisted Bochner-Kodaira identity. C. R. Math. Acad. Sci. Paris 349(13–14), 797–800 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Gunning, R.C., Rossi, H.: Analytic functions of several complex variables. Prentice-Hall Inc., Englewood Cliffs (1965)zbMATHGoogle Scholar
  28. 28.
    Hacon, C.D., Popa, M., Schnell, C.: Algebraic fiber spaces over abelian varieties: around a recent theorem by Cao and Păun. Local and global methods in algebraic geometry. Contemp. Math. 712, 143–195 (2018) (Am. Math. Soc., Providence, RI)Google Scholar
  29. 29.
    Hörmander, L.: An introduction to complex analysis in several variables, 3rd edition. North-Holland Mathematical Library, vol. 7. North-Holland Publishing Co., Amsterdam (1990)zbMATHGoogle Scholar
  30. 30.
    Manivel, L.: Un théorème de prolongement \(L^2\) de sections holomorphes d’un fibré vectoriel. Math. Z. 212, 107–122 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    McNeal, J.D., Varolin, D.: Analytic inversion of adjunction: \(L^{2}\) extension theorems with gain. Ann. Inst. Fourier (Grenoble) 57(3), 703–718 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Nadel, A.M.: Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature. Ann. Math. (2) 132(3), 549–596 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Narasimhan, M.S., Simha, R.R.: Manifolds with ample canonical class. Invent. Math. 5, 120–128 (1968)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Ohsawa, T.: On the extension of \(L^{2}\) holomorphic functions III: negligible weights. Math. Z. 219(2), 215–225 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Ohsawa, T.: On the extension of \(L^{2}\) holomorphic functions V: effects of generalization. Nagoya Math. J. 161, 1–21 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Ohsawa, T.: \(L^2\) approaches in several complex variables, development of Oka–Cartan theory by \(L^2\) estimates for the \(\bar{\partial }\) operator, Springer Monographs in Mathematics. Springer, Tokyo (2015)CrossRefGoogle Scholar
  37. 37.
    Ohsawa, T., Takegoshi, K.: On the extension of \(L^{2}\) holomorphic functions. Math. Z. 195, 197–204 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Păun, M.: Singular Hermitian metrics and positivity of direct images of pluricanonical bundles. Algebraic geometry: Salt Lake City 2015. In: Proc. Sympos. Pure Math. 97.1, Am. Math. Soc., Providence, RI, pp. 519–553 (2018)Google Scholar
  39. 39.
    Păun, M., Takayama, S.: Positivity of twisted relative pluricanonical bundles and their direct images. J. Algebraic Geom. 27(2), 211–272 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Phong, D.H., Sturm, J.: Algebraic estimates, stability of local zeta functions, and uniform estimates for distribution functions. Ann. Math. (2) 152(1), 277–329 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Prill, D.: The divisor class groups of some rings of holomorphic functions. Math. Z. 121, 58–80 (1971)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Siu, Y.-T.: The Fujita Conjecture and the Extension Theorem of Ohsawa–Takegoshi, Geometric Complex Analysis (Hayama, 1995), pp. 577–592. World Sci. Publ., River Edge (1996)zbMATHGoogle Scholar
  43. 43.
    Siu, Y.-T.: Invariance of plurigenera. Invent. Math. 134(3), 661–673 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Siu, Y.-T.: Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semipositively twisted plurigenera for manifolds not necessarily of general type, Complex geometry, Collection of papers dedicated to Hans Grauert, pp. 223–277. Springer, Berlin (2002)zbMATHGoogle Scholar
  45. 45.
    Siu, Y.-T.: Invariance of plurigenera and torsion-freeness of direct image sheaves of pluricanonical bundles. Finite or infinite dimensional complex analysis and applications. Adv. Complex Anal. Appl. 2, 45–83 (2004) (Kluwer Acad. Publ., Dordrecht)Google Scholar
  46. 46.
    Siu, Y.-T.: Multiplier ideal sheaves in complex and algebraic geometry. Sci. China Ser. A 48, 1–31 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Yi, L.: An Ohsawa–Takegoshi theorem on compact Kähler manifolds. Sci. China Math. 57(1), 9–30 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Zhou, X.Y.: Extension theorems for special holomorphic functions. Geometry and nonlinear partial differential equations (Hangzhou, 2001). AMS/IP Stud. Adv. Math. 29, 235–237 (2002) (Amer. Math. Soc., Providence, RI)Google Scholar
  49. 49.
    Zhou, X.Y.: Some results related to group actions in several complex variables. In: Proceedings of the International Congress of Mathematicians, vol. II (Beijing, 2002). Higher Ed. Press, Beijing, 743–753 (2002)Google Scholar
  50. 50.
    Zhou, X.Y.: Invariant holomorphic extension in several complex variables. Sci. China Ser. A 49(11), 1593–1598 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Zhou, X.Y.: A survey on \(L^2\) extension problem. Complex geometry and dynamics. Abel Symp. 10, 291–309 (2015) (Springer, Cham)Google Scholar
  52. 52.
    Zhou, X.Y., Zhu, L.F.: Ohsawa–Takegoshi \(L^2\) extension theorem: revisited. Fifth International Congress of Chinese Mathematicians, Part 1, 2. AMS/IP Stud. Adv. Math. 51, 475–490 (2012) (Am. Math. Soc., Providence, RI)Google Scholar
  53. 53.
    Zhou, X.Y., Zhu, L.F.: A generalized Siu’s lemma. Math. Res. Lett. 24(6), 1897–1913 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Zhou, X.Y., Zhu, L.F.: An optimal \(L^2\) extension theorem on weakly pseudoconvex Kähler manifolds. J. Differ. Geom. 110(1), 135–186 (2018)zbMATHCrossRefGoogle Scholar
  55. 55.
    Zhu, L.F., Guan, Q.A., Zhou, X.Y.: On the Ohsawa–Takegoshi \(L^2\) extension theorem and the Bochner–Kodaira identity with non-smooth twist factor. J. Math. Pures Appl. (9) 97(6), 579–601 (2012)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Mathematics, AMSS, and Hua Loo-Keng Key Laboratory of MathematicsChinese Academy of SciencesBeijingChina
  2. 2.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingChina
  3. 3.School of Mathematics and StatisticsWuhan UniversityWuhanChina

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