Skip to main content
SpringerLink
Log in

Local regularity of super-potentials and equidistribution of positive closed currents on \({\mathbb {P}}^k\)

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

In this paper, we introduce and study natural notions of local continuity/boundedness of super-potentials on \({\mathbb {P}}^k\). Next, we prove an equidistribution theorem of positive closed (p, p)-currents of \({\mathbb {P}}^k\), whose super-potentials are continuous/bounded near an invariant analytic subset, for holomorphic endomorphisms of \({\mathbb {P}}^k\). We also consider the case of regular polynomial automorphisms of \(\mathbb C^k\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ahn, T.: Equidistribution in higher codimension for holomorphic endomorphisms of \(\mathbb{P}^k\). Trans. Am. Math. Soc. 368, 3359–3388 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  2. Briend, J.-Y., Duval, J.: Deux caractérisations de la measure d’équilibre d’un endomorphisme de \(\mathbb{P}^k(\mathbb{C})\). Publ. Math. Inst. Hautes Études Sci. 93, 145–159 (2001)

    Article  MATH  Google Scholar 

  3. Demailly, J.-P.: Complex Analytic and Differential Geometry. http://www.fourier.ujf-grenoble.fr/~demailly

  4. Dinh, T.-C.: Analytic multiplicative cocycles over holomorphic dynamical systems. Complex Var. Elliptic Equ. 54, 243–251 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dinh, T.-C., Sibony, N.: Green currents for holomorphic automorphisms of compact Kähler manifolds. J. Am. Math. Soc. 18(2), 291–312 (2005)

    Article  MATH  Google Scholar 

  6. Dinh, T.-C., Sibony, N.: Distribution des valeurs de transformations méromorphes et applications. Comment. Math. Helv. 81, 221–258 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dinh, T.-C., Sibony, N.: Equidistribution towards the Green current for holomorphic maps. Ann. Sci. Éc. Norm. Supér. 41, 307–336 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dinh, T.-C., Sibony, N.: Super-potentials of positive closed currents, intersection theory and dynamics. Acta Math. 203, 1–82 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dinh, T.-C., Sibony, N.: Equidistribution speed for endomorphisms of proejctive spaces. Math. Ann. 347, 613–626 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dinh, T.-C., Sibony, N.: Super-potentials for currents on compact Kähler manifolds and dynamics of automorphisms. J. Algebr. Geom. 19, 473–529 (2010)

    Article  MATH  Google Scholar 

  11. Favre, C.: Dynamique de applications rationelles. PhD thesis, Université de Paris-Sud, Orsay (2000)

  12. Favre, C.: Multiplicity of holomorphic functions. Math. Ann. 316, 355–378 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Favre, C., Jonsson, M.: Brolin’s theorem for curves in two complex dimensions. Ann. Inst. Fourier (Grenoble) 53, 1461–1501 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Favre, C., Jonsson, M.: Eigenvaluations. Ann. Sci. École Norm. Sup. 40, 309–349 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Freire, A., Lopes, A., Mañé, R.: An invariant measure for rational maps. Bol. Soc. Brasil. Mat. 14, 45–62 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fornæss, J.E., Sibony, N.: Complex dynamics in higher dimensions. In: Gauthier, P.M., Sabidussi, G. (eds.) Complex Potential Theory (Montreal, PQ, 1993), volume 439 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., pp. 131–186. Kluwer Academic Publishers (1994)

  17. Fornæss, J.E., Sibony, N.: Complex dynamics in higher dimension. II. In: Thomas, B., David, C., D’Angelo, J.P., Siu, Y.-T. (eds.) Modern Methods in Complex Analysis (Princeton, NJ, 1992), volume 137 of Annals of Mathematics Studies, pp. 135–182. Princeton University Press (1995)

  18. Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley Classics Library. Wiley, New York (1994) (reprint of the 1978 original edition)

  19. Gignac, W.: Measures and dynamics on Noetherian spaces. J. Geom. Anal. 24, 1770–1793 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Griffiths, P.: Complex differential and integral geometry and curvature integrals associated to singularities of complex analytic varieties. Duke Math. J. 45, 427–512 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  21. Guedj, V.: Equidistribution towards the Green current. Bull. Soc. Math. Fr. 131, 359–372 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Harvey, R., Polking, J.: Extending analytic objects. Commun. Pure Appl. Math. 28, 701–727 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ljubich, M.: Entropy properties of rational endomorphisms of the Riemann sphere. Ergod. Theory Dyn. Syst. 3, 351–385 (1983)

    Article  MathSciNet  Google Scholar 

  24. Malgrange, B.: Ideals of Differentiable Functions. Oxford University Press, London (1967)

    MATH  Google Scholar 

  25. Parra, R.: Currents and equidistribution in holomorphic dynamics. PhD thesis, Univ. of Michigan (2011)

  26. Russakovskii, A., Shiffman, B.: Value distribution for sequences of rational mappings and complex dynamics. Indiana Univ. Math. J. 46, 897–932 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  27. Sibony, N.: Dynamique des applications rationnelles de \(\mathbb{P}^k\). Panor. Synthèses 8, 97–185 (1999)

    MathSciNet  MATH  Google Scholar 

  28. Taflin, J.: Equidistribution speed towards the Green current for endomorphisms of \(\mathbb{P}^k\). Adv. Math. 227, 2059–2081 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author would like to thank John Erik Fornæss and Viet Anh Nguyen for helpful discussions and suggestions. The author expresses his thanks to Ngaiming Mok for suggestions. The author also would like to thank the referee for careful reading and for suggestions. Referee’s comments and suggestions were helpful in improving and clarifying the arguments.

Author information

Authors and Affiliations

  1. KIAS, 85 Hoegiro, Dongdaemun-gu, Seoul, 02455, Republic of Korea

    Taeyong Ahn

Authors
  1. Taeyong Ahn
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Taeyong Ahn.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ahn, T. Local regularity of super-potentials and equidistribution of positive closed currents on \({\mathbb {P}}^k\) . Math. Ann. 371, 1163–1190 (2018). https://doi.org/10.1007/s00208-017-1579-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-017-1579-2

Search

Navigation