Skip to main content
Log in

Optimal L2 Extension and Siu’s Lemma

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

In this paper, we discuss our most recent results on the optimal L2 extension problem and Siu’s lemma as applications of the strong openness property of multiplier ideal sheaves obtained by Guan and Zhou.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Berndtsson, B.: The extension theorem of Ohsawa–Takegoshi and the theorem of Donnelly–Fefferman. Ann. Inst. Fourier (Grenoble), 46(4), 1083–1094 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. Berndtsson, B.: The openness conjecture and complex Brunn–Minkowski inequalities. In: Complex Geometry and Dynamics, pp. 29–44, Abel Symp., 10, Springer, Cham, 2015

    Chapter  Google Scholar 

  3. Berndtsson, B., Păun, M.: Bergman kernels and the pseudoeffectivity of relative canonical bundles. Duke Math. J., 145, 341–378 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bierstone, E., Milman, P. D.: A simple constructive proof of canonical resolution of singularities. In: Effective Methods in Algebraic Geometry (Castiglioncello, 1990), pp. 11–30, Progress in Mathematics, 94, Birkhäuser, Boston, 1991

    Chapter  Google Scholar 

  5. Demailly, J. P.: Regularization of closed positive currents of type (1, 1) by the flow of a Chern connection. In: Contributions to Complex Analysis and Analytic Geometry, pp. 105–126, Aspects Math., Vol. E 26, Friedr. Vieweg, Braunschweig, 1994

    Google Scholar 

  6. Demailly, J. P.: On the Ohsawa–Takegoshi–Manivel L 2 extension theorem. In: Complex Analysis and Geometry (Paris, 1997), pp. 47–82, Progress in Mathematics, 188, Birkhäuser, Basel, 2000

    Chapter  Google Scholar 

  7. Demailly, J. P.: Multiplier ideal sheaves and analytic methods in algebraic geometry. In: School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000), pp. 1–148, ICTP Lect. Notes, 6, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001

    Google Scholar 

  8. Demailly, J. P.: Analytic Methods in Algebraic Geometry, Surveys of Modern Mathematics, 1, International Press, Somerville, MA; Higher Education Press, Beijing, 2012

    Google Scholar 

  9. Demailly, J. P.: Extension of holomorphic functions defined on non reduced analytic subvarieties. In: The Legacy of Bernhard Riemann after One Hundred and Fifty Years, Vol. I, pp. 191–222, Adv. Lect. Math. (ALM), 35.1, Int. Press, Somerville, MA, 2016

    MathSciNet  MATH  Google Scholar 

  10. Demailly, J. P., Kollár, J.: Semi-continuity of complex singularity exponents and Kähler–Einstein metrics on Fano orbifolds. Ann. Sci. École Norm. Sup., 34, 525–556 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Guan, Q. A., Zhou, X. Y.: A solution of an L 2 extension problem with an optimal estimate and applications. Ann. of Math. (2), 181(3), 1139–1208 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Guan, Q. A., Zhou, X. Y.: A proof of Demailly’s strong openness conjecture. Ann. of Math. (2), 182(2), 605–616 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero: I, II. Ann. of Math. (2), 79, 109–326 (1964)

    Article  MATH  Google Scholar 

  14. Hörmander, L.: An Introduction to Complex Analysis in Several Variables, third edition, North-Holland Mathematical Library, 7, North-Holland Publishing Co., Amsterdam, 1990

    MATH  Google Scholar 

  15. Manivel, L.: Un théorème de prolongement L 2 de sections holomorphes d’un fibré vectoriel. Math. Z., 212, 107–122 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  16. McNeal, J. D., Varolin, D.: Analytic inversion of adjunction: L 2 extension theorems with gain. Ann. Inst. Fourier (Grenoble), 57(3), 703–718 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Meng, X. K., Zhou, X. Y., Zhu, L. F.: A survey on optimal L 2 extension theorems and vanishing theorems. To appear in Proceeding of ICCM 2016

    Google Scholar 

  18. Nadel, A. M.: Multiplier ideal sheaves and Kähler–Einstein metrics of positive scalar curvature. Ann. of Math. (2), 132(3), 549–596 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ohsawa, T.: On the extension of L 2 holomorphic functions III: negligible weights. Math. Z., 219(2), 215–225 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ohsawa, T.: On the extension of L 2 holomorphic functions V: effects of generalization. Nagoya Math. J., 161, 1–21 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ohsawa, T., Takegoshi, K.: On the extension of L 2 holomorphic functions. Math. Z., 195, 197–204 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  22. Phong, D. H., Sturm, J.: Algebraic estimates, stability of local zeta functions, and uniform estimates for distribution functions. Ann. of Math. (2), 152(1), 277–329 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Siu, Y. T.: The Fujita conjecture and the extension theorem of Ohsawa–Takegoshi. In: Geometric Complex Analysis (Hayama, 1995), pp. 577–592, World Sci. Publ., River Edge, NJ, 1996

    Google Scholar 

  24. Siu, Y. T.: Multiplier ideal sheaves in complex and algebraic geometry. Sci. China Ser. A, 48(S1), 1–31 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Yi, L.: An Ohsawa–Takegoshi theorem on compact Kähler manifolds. Sci. China Math., 57(1), 9–30 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Zhou, X. Y.: A survey on L 2 extension problem. In: Complex Geometry and Dynamics, pp. 291–309, Abel Symp., 10, Springer, Cham, 2015

    Chapter  Google Scholar 

  27. Zhou, X. Y., Zhu, L. F.: Ohsawa–Takegoshi L 2 extension theorem: revisited. In: Fifth International Congress of Chinese Mathematicians, Part 1, 2, pp. 475–490, AMS/IP Stud. Adv. Math., 51, Amer. Math. Soc., Providence, RI, 2012

    Google Scholar 

  28. Zhou, X. Y., Zhu, L. F.: A generalized Siu’s lemma. Math. Res. Lett., 24(6), 1897–1913 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  29. Zhou, X. Y., Zhu, L. F.: An optimal L 2 extension theorem on weakly pseudoconvex Kähler manifolds. To appear in J. Differential Geom.

  30. Zhou, X. Y., Zhu, L. F.: Optimal L 2 extension of sections from subvarieties in weakly pseudoconvex manifolds. Preprint

  31. Zhou, X. Y., Zhu, L. F.: Siu’s lemma, optimal L 2 extension and applications. Preprint

  32. 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)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Lang Feng Zhu.

Additional information

In Memory of Professor Qikeng Lu (1927–2015)

The first author is supported by the National Natural Science Foundation of China (Grant Nos. 11688101 and 11431013); the second author is supported by the National Natural Science Foundation of China (Grant Nos. 11201347 and 11671306)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhou, X.Y., Zhu, L.F. Optimal L2 Extension and Siu’s Lemma. Acta. Math. Sin.-English Ser. 34, 1289–1296 (2018). https://doi.org/10.1007/s10114-018-7539-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-018-7539-2

Keywords

MR(2010) Subject Classification

Navigation