Skip to main content

Log canonical thresholds and Monge-Ampère masses

Abstract

In this paper, we prove an inequality for log canonical thresholds and Monge-Ampère masses. The idea of proof is a combination of the Ohsawa-Takegoshi \(L^2\)-extension theorem and inequalities in Åhag et al. (Adv Math 222:2036–2058, 2009) and Demailly and Pham (Acta Math 212:1–9, 2014).

This is a preview of subscription content, access via your institution.

References

  1. Åhag, P., Cegrell, U., Czyz, R., Pham, H.H.: Monge-Ampère measures on pluripolar sets. J. Math. Pures Appl. 92, 613–627 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  2. Åhag, P., Cegrell, U., Kołodziej, S., Pham, H.H., Zeriahi, A.: Partial pluricomplex energy and integrability exponents of plurisubharmonic functions. Adv. Math. 222, 2036–2058 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  3. Berndtsson, B.: The openness conjecture and complex Brunn-Minkowski inequalities. Comp. Geom. Dyn. 10, 29–44 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  4. Blocki, Z.: The domain of definition of the complex Monge-Ampère operator. Am. J. Math. 128, 519–530 (2006)

    Article  MATH  Google Scholar 

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

    MathSciNet  Article  MATH  Google Scholar 

  6. Bedford, E., Taylor, B.A.: The Dirichlet problem for a complex Monge-Ampère equation. Invent. Math. 37, 1–44 (1976)

    MathSciNet  Article  MATH  Google Scholar 

  7. Bedford, E., Taylor, B.A.: A new capacity for plurisubharmonic functions. Acta Math. 149, 1–41 (1982)

    MathSciNet  Article  MATH  Google Scholar 

  8. Cegrell, U.: Pluricomplex energy. Acta Math. 180, 187–217 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  9. Cegrell, U.: The general definition of the complex Monge-Ampère operator. Ann. Inst. Fourier 54, 159–179 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  10. Corti, A.: Factoring birational maps of threefolds after Sarkisov. J. Alg. Geom. 4, 223–254 (1995)

    MathSciNet  MATH  Google Scholar 

  11. Corti, A.: Singularities of linear systems and \(3\)-fold birational geometry. Explic. Bir. Geom. 281, 259–312 (2000)

    MathSciNet  MATH  Google Scholar 

  12. Demailly, J.-P.: Nombres de Lelong généralisés, théorèmes d’intégralité et d’analyticité. Acta Math. 159, 153–169 (1987)

    MathSciNet  Article  MATH  Google Scholar 

  13. Demailly, J.-P.: Regularization of closed positive currents and Intersection Theory. J. Alg. Geom. 1, 361–409 (1992)

    MathSciNet  MATH  Google Scholar 

  14. Demailly, J.-P.: Monge-Ampère operators, Lelong numbers and intersection theory, Complex Analysis and Geometry, Univ. Series in Math., edited by V. Ancona and A. Silva, Plenum Press, New-York, (1993)

  15. Demailly, J.-P.: Estimates on Monge-Ampère operators derived from a local algebra inequality, in: Complex Analysis and Digital geometry, Proceedings of the Kiselmanfest 2006, Acta Universitatis Upsaliensis, (2009)

  16. Demailly, J.-P.: Extension of holomorphic functions defined on non reduced analytic subvarieties, arXiv:1510.05230 [math.CV] (2015)

  17. Demailly, J.-P., Pham, H.H.: A sharp lower bound for the log canonical threshold. Acta Math. 212, 1–9 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  18. Demailly, J.-P., Kollár, J.: Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds. Ann. Sci. Ecol. Norm. Sup. 34(4), 525–556 (2001)

    Article  MATH  Google Scholar 

  19. de Fernex, T., Ein, T., Mustaţǎ, M.: Bounds for log canonical thresholds with applications to birational rigidity. Math. Res. Lett. 10, 219–236 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  20. de Fernex, T., Ein, L., Mustaţǎ, M.: Multiplicities and log canonical thresholds. J. Alg. Geom. 13, 603–615 (2004)

    MathSciNet  Article  MATH  Google Scholar 

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

    MathSciNet  Article  MATH  Google Scholar 

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

    MathSciNet  Article  MATH  Google Scholar 

  23. Guan, Q., Zhou, X.: Multiplier ideal sheaves, jumping numbers, and the restriction formula, arXiv:1504.04209 [math.CV] (2015)

  24. Howald, J.: Multiplier ideals of monomial ideals. Trans. Am. Math. Soc. 353, 2665–2671 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  25. Kim, D.: Themes on Non-analytic Singularities of Plurisubharmonic Functions, Volume 144 of the series Springer Proceedings in Mathematics and Statistics, 197–206 (2015)

  26. Kiselman, C.O.: Un nombre de Lelong raffiné, Séminaire d’Analyse Complexe et Géométrie 1985-87, Fac. Sci. Tunis & Fac. Sci. Tech. Monastir, 61–70 (1987)

  27. Kiselman, C.O.: Attenuating the singularities of plurisubharmonic functions. Ann. Polon. Math. 60, 173–197 (1994)

    MathSciNet  Article  MATH  Google Scholar 

  28. Matsumura, S.: A Nadel vanishing theorem for metrics with minimal singularities on big line bundles. Adv. Math. 280, 188–207 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  29. Ohsawa, T., Takegoshi, K.: On the extension of \(L^2\) holomorphic functions. Math. Zeit. 195, 197–204 (1987)

    Article  MATH  Google Scholar 

  30. Pham, H.H.: The weighted log canonical threshold. C. R. Math. 352, 283–288 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  31. Rashkovskii, A.: Extremal cases for the log canonical threshold. C. R. Math. 353, 21–24 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  32. Skoda, H.: Sous-ensembles analytiques d’ordre fini ou infini dans \({\mathbb{C}}^n\). Bull. Soc. Math. Fr. 100, 353–408 (1972)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgements

The author is grateful to Professor Jean-Pierre Demailly, Dr. Nguyen Ngoc Cuong and the anonymous reviewers for valuable comments, which helped to improve the paper. This paper was partly written when the author was visiting the Vietnam Institute for Advanced Study in Mathematics (VIASM). He would like to thank the VIASM for support and providing a fruitful research environment and hospitality. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2014.01.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Hoang Hiep Pham.

Additional information

Communicated by Ngaiming Mok.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Pham, H.H. Log canonical thresholds and Monge-Ampère masses. Math. Ann. 370, 555–566 (2018). https://doi.org/10.1007/s00208-017-1518-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-017-1518-2

Mathematics Subject Classification

  • 14B05
  • 32S05
  • 32S10
  • 32U25