Advertisement

Lelong numbers of bidegree (1, 1) currents on multiprojective spaces

  • Dan Coman
  • James HeffersEmail author
Article
  • 5 Downloads

Abstract

Let T be a positive closed current of bidegree (1, 1) on a multiprojective space \(X={\mathbb P}^{n_1}\times \cdots \times {{\mathbb {P}}}^{n_k}\). For certain values of \(\alpha \), which depend on the cohomology class of T, we show that the set of points of X where the Lelong numbers of T exceed \(\alpha \) have certain geometric properties. We also describe the currents T that have the largest possible Lelong number in a given cohomology class, and the set of points where this number is assumed.

Keywords

Positive closed currents Plurisubharmonic functions Lelong numbers 

Mathematics Subject Classification

Primary 32U25 Secondary 32U05 32U40 

Notes

References

  1. 1.
    Coman, D.: Entire pluricomplex Green functions and Lelong numbers of projective currents. Proc. Am. Math. Soc. 134, 1927–1935 (2006)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Coman, D., Guedj, V.: Quasiplurisubharmonic Green functions. J. Math. Pure Appl. 92(9), 456–475 (2009)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Coman, D., Truong, T.T.: Geometric properties of upper level sets of Lelong numbers on projective spaces. Math. Ann. 361, 981–994 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Demailly, J.P.: Regularization of closed positive currents and intersection theory. J. Algebraic Geom. 1, 361–409 (1992)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Demailly, J.P.: Monge-Ampère operators, Lelong numbers and intersection theory, in complex analysis and geometry, pp. 115–193. Plenum, New York (1993)zbMATHGoogle Scholar
  6. 6.
    Favre, C., Guedj, V.: Dynamique des applications rationnelles des espaces multiprojectifs. Indiana Univ. Math. J. 50, 881–934 (2001)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Fornæss, J.E., Sibony, N.: Complex dynamics in higher dimension. II, Modern methods in complex analysis (Princeton, NJ, 1992), 135–182, Ann. of Math. Stud., 137, Princeton University Press, Princeton, (1995)Google Scholar
  8. 8.
    Guedj, V., Zeriahi, A.: Intrinsic capacities on compact Kähler manifolds. J. Geom. Anal. 15, 607–639 (2005)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Heffers, J.J.: A property of upper level sets of Lelong numbers of currents on \({\mathbb{P}}^2\). Int. J. Math. 28(14), 1750110, 18 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Heffers, J.J.: On Lelong numbers of positive closed currents on \({{\mathbb{P}}}^n\). Complex Var. Elliptic Equ. 64, 352–360 (2019)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hörmander, L.: Notions of convexity. Birkhäuser, Basel (1994)zbMATHGoogle Scholar
  12. 12.
    Sibony, N.: Dynamique des applications rationnelles de \({{\mathbb{P}}}^k\), Dynamique et géométrie complexes (Lyon, 1997), 97–185, Panor. Synthèses, 8, Soc. Math. France, Paris, (1999)Google Scholar
  13. 13.
    Siu, Y.T.: Analyticity of sets associated to Lelong numbers and the extension of closed positive currents. Invent. Math. 27, 53–156 (1974)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsSyracuse UniversitySyracuseUSA
  2. 2.Department of MathematicsUniversity of MichiganAnn ArborUSA

Personalised recommendations