Skip to main content
SpringerLink
Log in

Cohomology of non-pluriharmonic loci

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

Let D be a pseudoconvex domain in \(\mathbb {C}^{n}\) for \(n \ge 4\). Let \(\varphi \) be an exhaustive plurisubharmonic function on D. We show that the direct limit of the cohomology of open sets which contain the support of \(i \partial \overline{\partial }\varphi \) is equal to the cohomology of D in low degrees.

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. Bedford, E., Kalka, M.: Foliations and complex Monge–Ampère equations. Commun. Pure Appl. Math. 30, 543–571 (1977)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Berndtsson, B.: The extension theorem of Ohsawa–Takegoshi and the theorem of Donnelly–Fefferman. Ann. Inst. Fourier (Grenoble) 46, 1083–1094 (1996)

    Article  MathSciNet  Google Scholar 

  4. Berndtsson, B.: Charpentier, Ph: A Sobolev mapping property for the Bergman kernel. Math. Z. 235, 1–10 (2000)

    Article  MathSciNet  Google Scholar 

  5. Bott, R.: On a theorem of Lefschetz. Michigan Math. J. 6, 211–216 (1959)

    Article  MathSciNet  Google Scholar 

  6. Demailly, J.P.: Complex analytic and differential geometry, OpenContent Book. Version of Thursday June 21, Available at the authors web page (2012)

  7. Forstnerič, F.: Complements of Runge domains and holomorphic hulls. Michigan Math. J. 41(No2), 297–308 (1994)

    Article  MathSciNet  Google Scholar 

  8. Hörmander, L.: An introduction to complex analysis in several variables, Third edition, North-Holland Mathematical Library, 7. North-Holland Publishing Co., Amsterdam (1990)

    Google Scholar 

  9. Lefschetz, S.: L’analysis situs et la géométry algébriue. Collection de monographies sur la théorie des fonctions. Gauthier-Villars, Paris (1924)

    Google Scholar 

  10. Tiba, Y.: On a convex level set of a plurisubharmonic function and the support of the Monge–Ampère current. Ann. Polonici Math. 121, 251–262 (2018)

    Article  MathSciNet  Google Scholar 

  11. Tiba, Y.: Shilov boundaries of the pluricomplex Green function’s level sets. Int. Math. Res. Notices 24, 13717–13727 (2015)

    Article  MathSciNet  Google Scholar 

  12. Tiba, Y.: The extension of holomorphic functions on a non-pluriharmonic locus. arXiv preprint: arXiv:1706.01441v2

Download references

Acknowledgements

This work was supported by the Grant-in-Aid for Scientific Research (KAKENHI No. 17K14200).

Author information

Authors and Affiliations

  1. Department of Mathematics, Ochanomizu University, 2-1-1 Otsuka, Bunkyo-ku, Tokyo, Japan

    Yusaku Tiba

Authors
  1. Yusaku Tiba
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yusaku Tiba.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tiba, Y. Cohomology of non-pluriharmonic loci. Math. Z. 293, 1403–1414 (2019). https://doi.org/10.1007/s00209-019-02273-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-019-02273-1

Mathematics Subject Classification

Search

Navigation