Mathematische Annalen

, Volume 370, Issue 1–2, pp 555–566 | Cite as

Log canonical thresholds and Monge-Ampère masses

  • Hoang Hiep PhamEmail author


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).

Mathematics Subject Classification

14B05 32S05 32S10 32U25 



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.


  1. 1.
    Åhag, P., Cegrell, U., Czyz, R., Pham, H.H.: Monge-Ampère measures on pluripolar sets. J. Math. Pures Appl. 92, 613–627 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Berndtsson, B.: The openness conjecture and complex Brunn-Minkowski inequalities. Comp. Geom. Dyn. 10, 29–44 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Blocki, Z.: The domain of definition of the complex Monge-Ampère operator. Am. J. Math. 128, 519–530 (2006)CrossRefzbMATHGoogle Scholar
  5. 5.
    Blocki, Z.: Suita conjecture and the Ohsawa-Takegoshi extension theorem. Invent. Math. 193, 149–158 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bedford, E., Taylor, B.A.: The Dirichlet problem for a complex Monge-Ampère equation. Invent. Math. 37, 1–44 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bedford, E., Taylor, B.A.: A new capacity for plurisubharmonic functions. Acta Math. 149, 1–41 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cegrell, U.: Pluricomplex energy. Acta Math. 180, 187–217 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cegrell, U.: The general definition of the complex Monge-Ampère operator. Ann. Inst. Fourier 54, 159–179 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Corti, A.: Factoring birational maps of threefolds after Sarkisov. J. Alg. Geom. 4, 223–254 (1995)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Corti, A.: Singularities of linear systems and \(3\)-fold birational geometry. Explic. Bir. Geom. 281, 259–312 (2000)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Demailly, J.-P.: Nombres de Lelong généralisés, théorèmes d’intégralité et d’analyticité. Acta Math. 159, 153–169 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Demailly, J.-P.: Regularization of closed positive currents and Intersection Theory. J. Alg. Geom. 1, 361–409 (1992)MathSciNetzbMATHGoogle Scholar
  14. 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)Google Scholar
  15. 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)Google Scholar
  16. 16.
    Demailly, J.-P.: Extension of holomorphic functions defined on non reduced analytic subvarieties, arXiv:1510.05230 [math.CV] (2015)
  17. 17.
    Demailly, J.-P., Pham, H.H.: A sharp lower bound for the log canonical threshold. Acta Math. 212, 1–9 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 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)CrossRefzbMATHGoogle Scholar
  19. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    de Fernex, T., Ein, L., Mustaţǎ, M.: Multiplicities and log canonical thresholds. J. Alg. Geom. 13, 603–615 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Guan, Q., Zhou, X.: A proof of Demailly’s strong openness conjecture. Ann. Math. 182, 605–616 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Guan, Q., Zhou, X.: Effectiveness of Demailly’s strong openness conjecture and related problems. Invent. Math. 202, 635–676 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Guan, Q., Zhou, X.: Multiplier ideal sheaves, jumping numbers, and the restriction formula, arXiv:1504.04209 [math.CV] (2015)
  24. 24.
    Howald, J.: Multiplier ideals of monomial ideals. Trans. Am. Math. Soc. 353, 2665–2671 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 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)Google Scholar
  26. 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)Google Scholar
  27. 27.
    Kiselman, C.O.: Attenuating the singularities of plurisubharmonic functions. Ann. Polon. Math. 60, 173–197 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Matsumura, S.: A Nadel vanishing theorem for metrics with minimal singularities on big line bundles. Adv. Math. 280, 188–207 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Ohsawa, T., Takegoshi, K.: On the extension of \(L^2\) holomorphic functions. Math. Zeit. 195, 197–204 (1987)CrossRefzbMATHGoogle Scholar
  30. 30.
    Pham, H.H.: The weighted log canonical threshold. C. R. Math. 352, 283–288 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Rashkovskii, A.: Extremal cases for the log canonical threshold. C. R. Math. 353, 21–24 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Skoda, H.: Sous-ensembles analytiques d’ordre fini ou infini dans \({\mathbb{C}}^n\). Bull. Soc. Math. Fr. 100, 353–408 (1972)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Institute of MathematicsVietnam Academy of Science and TechnologyHanoiVietnam

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