The Journal of Geometric Analysis

, Volume 25, Issue 1, pp 329–335 | Cite as

A Complex Surface Admitting a Strongly Plurisubharmonic Function but No Holomorphic Functions

  • Franc Forstnerič


We find a domain \(X\subset \mathbb {C}\mathbb {P}^{2}\) with a strongly plurisubharmonic function such that every holomorphic function on X is constant.


Holomorphic function Strongly plurisubharmonic function Stein manifold Adjunction inequality 

Mathematics Subject Classification (2000)

32E10 32E40 32F05 32F25 57R17 



I wish to acknowledge support by the research program P1-0291 from ARRS, Republic of Slovenia. I thank Karl Oeljeklaus for communicating the question answered in this note, and Stefan Nemirovski for providing the example in Remark 1.5. Last but not least, I thank the organizers of the conference Geometric Methods in Several Complex Variables (Lille, France, 25–26 October 2012) for their kind invitation and hospitality.


  1. 1.
    Diederich, K., Fornæss, J.-E.: Pseudoconvex domains: an example with nontrivial Nebenhülle. Math. Ann. 225, 275–292 (1977) CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Forstnerič, F.: Complex tangents of real surfaces in complex surfaces. Duke Math. J. 67, 353–376 (1992) CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Forstnerič, F.: Stein Manifolds and Holomorphic Mappings (The Homotopy Principle in Complex Analysis). Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol. 56. Springer, Berlin (2011) CrossRefMATHGoogle Scholar
  4. 4.
    Forstnerič, F., Laurent-Thiébaut, C.: Stein compacts in Levi-flat hypersurfaces. Trans. Am. Math. Soc. 360, 307–329 (2008) CrossRefMATHGoogle Scholar
  5. 5.
    Gompf, R.E.: Handlebody construction of Stein surfaces. Ann. Math. (2) 148, 619–693 (1998) CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Gompf, R.E.: Stein surfaces as open subsets of \(\mathbb {C}^{2}\). J. Symplectic Geom. 3, 565–587 (2005) CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Grauert, H.: On Levi’s problem and the embedding of real-analytic manifolds. Ann. Math. (2) 68, 460–472 (1958) CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Lisca, P., Matić, G.: Tight contact structures and Seiberg–Witten invariants. Invent. Math. 129, 509–525 (1997) CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Nemirovski, S.: Complex analysis and differential topology on complex surfaces. Usp. Mat. Nauk, 54, 47–74 (1999). English transl.: Russian Math. Surveys, 54(4), 729–752 (1999) CrossRefGoogle Scholar
  10. 10.
    Range, M., Siu, Y.-T.: \(\mathcal {C}^{k}\) approximation by holomorphic functions and \(\overline{\partial}\)-closed forms on \(\mathcal {C}^{k}\) submanifolds of a complex manifold. Math. Ann. 210, 105–122 (1974) CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Siu, J.-T.: Every Stein subvariety admits a Stein neighborhood. Invent. Math. 38, 89–100 (1976) CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Stout, E.L.: Polynomial Convexity. Birkhäuser, Boston (2007) MATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2013

Authors and Affiliations

  1. 1.Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia
  2. 2.Institute of Mathematics, Physics and MechanicsLjubljanaSlovenia

Personalised recommendations