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Lelong Number and the Log Canonical Thresholds of Plurisubharmonic Functions on Analytic Subsets


The aim of this paper is to introduce the notion of Lelong number and the log canonical thresholds of plurisubharmonic functions on analytic subsets A in an open subset Ω of \(\mathbb {C}^{n}\). Next, we establish some results about the relationship between these quantities in the relation with the analyticity of A.

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

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  3. Bedford, E., Taylor, B.A.: A new capacity for plurisubharmonic functions. Acta Math. 149, 1–40 (1982)

    MathSciNet  Article  Google Scholar 

  4. Błocki, Z.: The complex Monge-Ampère Operator in Pluripotential Theory. Lectures Notes. (unpublish). Webside: (1998)

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

    MathSciNet  MATH  Google Scholar 

  6. Caffarelli, L., Kohn, J.J., Nirenberg, L., Spruck, J.: The Dirichlet problem for nonlinear second order elliptic equations. II. Complex Monge-Ampère, and uniformly elliptic, equations. Comm. Pure Appl. Math. 38, 209–252 (1985)

    MathSciNet  Article  Google Scholar 

  7. Cegrell, U.: Pluricomplex energy. Acta Mat. 180, 187–217 (1998)

    MathSciNet  Article  Google Scholar 

  8. Cegrell, U.: The general definition of the complex Monge-Ampère operator. Ann. Inst. Fourier. 54(1), 159–179 (2004)

    MathSciNet  Article  Google Scholar 

  9. Chirka, E.M.: Complex Analytic Sets. Kluwer Academic Publisher (1989)

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

    MathSciNet  Article  Google Scholar 

  11. Demailly, J. -P.: Monge-Ampère Operators, Lelong Numbers and Intersection Theory. Complex Analysis and Geometry. Univ. Series in Math. Edited by Ancona, V., Silva, A, Plenum Press, New-York (1993)

  12. Demailly, J.-P.: Complex Analytic and Differential Geometry. (2012)

  13. 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)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  15. Guedj, V., Zeriahi, A.: Degenerate Complex Monge-Ampère Equations. EMS Tracts in Mathematics, vol. 26. European Mathematical Society, Zürich (2017)

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

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  18. Fernex, T., Ein, L., Mustata, M.: Shokurov’s ACC conjecture for log canonical thresholds on smooth varieties. Duke Math. J. 152, 93–114 (2010)

    MathSciNet  Article  Google Scholar 

  19. Fornaess, J.E., Narasimhan, R.: The Levi problem on complex spaces with singularities. Math. Ann. 248, 47–72 (1980)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  21. Guan, Q., Li, Z.: A characterization of regular points by Ohsawa-Takegoshi extension theorem. J. Mat. Soc. Japan. 170, 403–408 (2018)

    MathSciNet  MATH  Google Scholar 

  22. Gunning, R.C., Rossi, H.: Analytic Functions of Several Complex Variables. Prentice-hall, Englewood Cliffs, NJ. (in Russian) (1965)

  23. Hacon, C.D., McKernan, J., Xu, C.: ACC for log canonical thresholds. Ann. of Math. 180, 523–571 (2014)

    MathSciNet  Article  Google Scholar 

  24. Hai, L.M., Hiep, P.H., Hung, V.V.: The log canonical threshold of holomorphic functions. Internat. J. Math. 23(11), 8pp (2012)

    MathSciNet  Article  Google Scholar 

  25. Hai, L.M., Hiep, P.H., Tung, T.: Estimates of level sets of holomorphic functions and applications to the weighted log canonical thresholds. The Journal of Geometric Analysis. (2020)

  26. Hiep, P.H.: The weighted log canonical threshold. C. R. Acad. Sci. Paris, Ser. I(352), 283–288 (2014)

    MathSciNet  Article  Google Scholar 

  27. Hiep, P.H.: Continuity properties of certain weighted log canonical thresholds. C. R. Acad. Sci. Paris, Ser. I(355), 34–39 (2017)

    MathSciNet  Article  Google Scholar 

  28. Hiep, P.H.: Log canonical thresholds and Monge-Ampère masses. Math. Ann. 370, 555–566 (2018)

    MathSciNet  Article  Google Scholar 

  29. Lelong, P.: Intégration sur un ensemble analytique complexe. Bull. Soc. Math. Fr. 85, 239–262 (1957)

    Article  Google Scholar 

  30. Hörmander, L.: Notions of Convexity. Birkhäuser Boston (1994)

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

    MathSciNet  Article  Google Scholar 

  32. Klimek, M.: Pluripotential Theory. New York (NY) the clarendon press (1991)

  33. Kołodziej, S.: The complex Monge-Ampère equation. Acta Math. 180, 69–117 (1998)

    MathSciNet  Article  Google Scholar 

  34. Kołodziej, S.: The Complex Monge-Ampère Equation and Pluripotential Theory. Memoirs of AMS. (2005)

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

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  37. Siu, Y.T.: Analyticity of sets associated to Lelong numbers and the extension of closed positive currents. Invent. Math. 27, 53–156 (1974)

    MathSciNet  Article  Google Scholar 

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The authors would like to thank the referees very much for carefully reading the paper and for valuable remarks and suggestions which led to the improvements of the exposition of the paper. The third author would like to thank IMU for supporting his Ph.D studies at Hanoi National University of Education through the IMU Breakout Graduate Fellowship.

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Correspondence to Le Mau Hai.

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Hai, L.M., Hiep, P.H. & Tung, T. Lelong Number and the Log Canonical Thresholds of Plurisubharmonic Functions on Analytic Subsets. Acta Math Vietnam 47, 223–241 (2022).

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  • Pure k-dimensional analytic subsets
  • Irreducible analytic subset
  • Ramified coverings
  • Plurisubharmonic functions on analytic subsets
  • Lelong number
  • The log canonical thresholds

Mathematics Subject Classification (2010)

  • 32C25
  • 32C30
  • 32U25
  • 32W20