Skip to main content
Log in

Upper level sets of Lelong numbers on \(\mathbb P^2\) and cubic curves

Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

Let T be a positive closed current of bidimension (1, 1) with unit mass on \(\mathbb P^2\) and \(V_{\alpha }(T)\) be the upper level sets of Lelong numbers \(\nu (T,x)\) of T. For any \(\alpha \ge \frac{1}{3}\), we show that \(|V_{\alpha }(T){\setminus } C|\le 2\) for some cubic curve C.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

References

  1. Coman, D.: Entire Pluricomplex green functions and Lelong numbers of projective currents. Proc. Am. Math. Soc. 134, 1927–1935 (2005)

    Article  MathSciNet  Google Scholar 

  2. Coman, D., Nivoche, S.: Plurisubharmonic functions with singularities and affine invariants for finite sets in \(\mathbb{C}^n\). Math. Ann. 322(2), 317–332 (2002)

    Article  MathSciNet  Google Scholar 

  3. Coman, D., Truong, T.T.: Geometric properties of upper level sets of Lelong numbers on projective spaces. Math. Ann. 361, 981–994 (2015)

    Article  MathSciNet  Google Scholar 

  4. Demailly, J.P., Monge-Ampére operators, Lelong numbers and intersection theory. In: Complex analysis and geometry, pp. 115–193. Plenum, New York (1993)

  5. Heffers, J.J.: A property of upper level sets of Lelong numbers of currents on \(\mathbb{P}^2\). Int. J. Math. 28(14), 1750110 (2017)

    Article  MathSciNet  Google Scholar 

  6. Heffers, J.J.: On Lelong numbers of positive closed currents on \( \mathbb{P}^n\). Complex Var. Elliptic Equ. 64, 352–360 (2019)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We would like to thank to the referee for his/her careful reading and suggestions. The second author is supported by TÜBİTAK 3501 Proj. No. 120F084 and TÜBİTAK 2518 Proj. No. 119N642.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ozcan Yazici.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kisisel, A.U.O., Yazici, O. Upper level sets of Lelong numbers on \(\mathbb P^2\) and cubic curves. Math. Z. 300, 2917–2930 (2022). https://doi.org/10.1007/s00209-021-02907-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-021-02907-3

Keywords

Mathematics Subject Classification

Navigation