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L2 Extension and Effectiveness of Lp Strong Openness Property

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Abstract

In this note, we present an L2 extension approach to the effectiveness result of Lp strong openness property of multiplier ideal sheaves.

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References

  1. Bao, S. J., Guan, Q. A.: L2 extension and effectiveness of strong openness property. submitted (2021)

  2. Berndtsson, B.: Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains. Ann. Inst. Fourier (Grenoble), 56(6), 1633–1662 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Berndtsson, B.: The openness conjecture for plurisubharmonic functions. arXiv: 1305.5781, (2013)

  4. Berndtsson, B., Lempert, L.: A proof of the Ohsawa-Takegoshi theorem with sharp estimates. J. Math. Soc. Japan, 68(4), 1461–1472 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Blocki, Z.: Suita conjecture and the Ohsawa–Takegoshi extension theorem. Invent. Math., 193, 149–158 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Demailly, J. -P.: Transcendental proof of a generalized Kawamata–Viehweg vanishing theorem. In: Geometrical and Algebraical Aspects in Several Complex Variables (Cetraro, 1989), Vol. 8 of Sem. Conf., 81–94 (1991)

  7. Demailly, J. -P.: Complex analytic and differential geometry. Electronically accessible at https://www-fourier.ujf-grenoble.fr/∼demailly/manuscripts/agbook.pdf

  8. 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), volume 6 of ICTP Lect. Notes, 1–148 (2001)

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

    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. (4), 34(4), 525–556 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fornæss, J. E.: Several complex variables. arXiv: 1507.00562 (2015)

  12. Guan, Q. A.: A sharp effectiveness result of Demailly’s strong openness conjecture. Adv. Math., 348, 51–80 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  13. Guan, Q. A.: General concavity of minimal L2 integrals related to multiplier sheaves. arXiv:1811.03261v4 (2019)

  14. Guan, Q. A., Yuan, Z.: Effectiveness of strong openness property in Lp. arXiv: 2106.03552v3 (2021)

  15. 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 

  16. Guan, Q. A., Zhou, X. Y.: Strong openness conjecture and related problems for plurisubharmonic functions. arXiv: 1401.7158 (2014)

  17. Guan, Q. A., Zhou, X. Y.: Effectiveness of Demailly’s strong openness conjecture and related problems. Invent. Math., 202(2), 635–676 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Guan, Q. A., Zhou, X. Y.: A solution of an L2 extension problem with an optimal estimate and applications. Ann. of Math. (2), 181(3), 1139–1208 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Maitani, F., Yamaguchi, H.: Variation of Bergman metrics on Riemann surfaces. Math. Ann., 330(3), 477–489 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. 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 

  21. Ohsawa, T.: On the extension of L2 holomorphic functions. V. Effects of generalization. Nagoya Math. J., 161, 1–21 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ohsawa, T.: Analysis of Several Complex Variables. Translations of Mathematical Monographs, Vol. 211, American Mathematical Society, Providence, RI, 2002. Translated from the Japanese by Shu Gilbert Nakamura, Iwanami Series in Modern Mathematics

    MATH  Google Scholar 

  23. Ohsawa, T.: L2 Approaches in Several Complex Variables. Development of Oka–Cartan Theory by L2 Estimates for the \(\bar{\partial}\) operator. Springer Monographs in Mathematics, Springer, Tokyo (2015) ix+196 pp. ISBN: 978-4-431-55746-3; 978-4-431-55747-0

    Chapter  MATH  Google Scholar 

  24. Ohsawa, T.: L2 Approaches in Several Complex Variables. Towards the Oka–Cartan Theory with Precise Bounds. Second edition of [23]. Springer Monographs in Mathematics, Springer, Tokyo (2018) xi+258 pp. ISBN: 978-4-431-56851-3; 978-4-431-56852-0

    Chapter  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Tian, G.: On Kähler–Einstein metrics on certain Kähler manifolds with C1(M) > 0. Invent. Math., 89(2), 225–246 (1987)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We would like to thank Zheng Yuan and Zhitong Mi for checking this article. And we thank the referees for their time and comments.

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Authors and Affiliations

  1. School of Mathematical Sciences, Peking University, Beijing, 100871, P. R. China

    Shi Jie Bao & Qi An Guan

Authors
  1. Shi Jie Bao
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  2. Qi An Guan
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Corresponding author

Correspondence to Qi An Guan.

Additional information

The second author was supported by NSFC (Grant Nos. 11825101, 11522101 and 11431013)

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Bao, S.J., Guan, Q.A. L2 Extension and Effectiveness of Lp Strong Openness Property. Acta. Math. Sin.-English Ser. 39, 814–826 (2023). https://doi.org/10.1007/s10114-023-1368-7

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  • DOI: https://doi.org/10.1007/s10114-023-1368-7

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