Advertisement

Mathematische Annalen

, Volume 361, Issue 3–4, pp 981–994 | Cite as

Geometric properties of upper level sets of Lelong numbers on projective spaces

  • Dan ComanEmail author
  • Tuyen Trung Truong
Article

Abstract

Let \(T\) be a positive closed current of unit mass on the complex projective space \(\mathbb P^n\). For certain values \(\alpha <1\), we prove geometric properties of the set of points in \(\mathbb P^n\) where the Lelong number of \(T\) exceeds \(\alpha \). We also consider the case of positive closed currents of bidimension (1,1) on multiprojective spaces.

Mathematics Subject Classification

Primary 32U25 Secondary 32U05 32U40 

References

  1. 1.
    Coman, D.: Entire pluricomplex Green functions and Lelong numbers of projective currents. Proc. Am. Math. Soc. 134, 1927–1935 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Coman, D., Guedj, V.: Quasiplurisubharmonic Green functions. J. Math. Pures Appl. (9) 92, 456–475 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Demailly, J.P.: Regularization of closed positive currents and intersection theory. J. Algebraic Geom. 1, 361–409 (1992)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Demailly, J.P.: Monge–Ampère operators, Lelong numbers and intersection theory. In: Complex Analysis and Geometry, pp. 115–193, Plenum, New York (1993)Google Scholar
  5. 5.
    Eisenbud, D., Harris, J.: On varieties of minimal degree (a centennial account). Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), pp. 3–13. In: Proceedings of Symposia in Pure Mathematics, vol. 46, part 1. American Mathematical Society, Providence (1987)Google Scholar
  6. 6.
    Fornæss, J.E., Sibony, N.: Oka’s inequality for currents and applications. Math. Ann. 301, 399–419 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Griffiths, Ph., Harris, J.: Principles of algebraic geometry. Pure and Applied Mathematics. Wiley-Interscience, New York (1978)Google Scholar
  8. 8.
    Guedj, V.: Dynamics of polynomial mappings of \({\mathbb{C}}^2\). Am. J. Math. 124, 75–106 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Guedj, V., Zeriahi, A.: Intrinsic capacities on compact Kähler manifolds. J. Geom. Anal. 15, 607–639 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Meo, M.: Inégalités d’auto-intersection pour les courants positifs fermés définis dans les variétés projectives. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26, 161–184 (1998)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Siu, Y.T.: Analyticity of sets associated to Lelong numbers and the extension of closed positive currents. Invent. Math. 27, 53–156 (1974)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Vigny, G.: Lelong–Skoda transform for compact Kähler manifolds and self-intersection inequalities. J. Geom. Anal. 19, 433–451 (2009)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of MathematicsSyracuse UniversitySyracuseUSA
  2. 2.School of MathematicsKorea Institute for Advanced StudySeoulRepublic of Korea

Personalised recommendations