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

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Abstract

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.

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References

  1. Diederich, K., Fornæss, J.-E.: Pseudoconvex domains: an example with nontrivial Nebenhülle. Math. Ann. 225, 275–292 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  2. Forstnerič, F.: Complex tangents of real surfaces in complex surfaces. Duke Math. J. 67, 353–376 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Book  MATH  Google Scholar 

  4. Forstnerič, F., Laurent-Thiébaut, C.: Stein compacts in Levi-flat hypersurfaces. Trans. Am. Math. Soc. 360, 307–329 (2008)

    Article  MATH  Google Scholar 

  5. Gompf, R.E.: Handlebody construction of Stein surfaces. Ann. Math. (2) 148, 619–693 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  6. Gompf, R.E.: Stein surfaces as open subsets of \(\mathbb {C}^{2}\). J. Symplectic Geom. 3, 565–587 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. Grauert, H.: On Levi’s problem and the embedding of real-analytic manifolds. Ann. Math. (2) 68, 460–472 (1958)

    Article  MATH  MathSciNet  Google Scholar 

  8. Lisca, P., Matić, G.: Tight contact structures and Seiberg–Witten invariants. Invent. Math. 129, 509–525 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  11. Siu, J.-T.: Every Stein subvariety admits a Stein neighborhood. Invent. Math. 38, 89–100 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  12. Stout, E.L.: Polynomial Convexity. Birkhäuser, Boston (2007)

    MATH  Google Scholar 

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Acknowledgements

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.

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Authors and Affiliations

  1. Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia

    Franc Forstnerič

  2. Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000, Ljubljana, Slovenia

    Franc Forstnerič

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  1. Franc Forstnerič
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Correspondence to Franc Forstnerič.

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Communicated by Kang-Tae Kim.

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Forstnerič, F. A Complex Surface Admitting a Strongly Plurisubharmonic Function but No Holomorphic Functions. J Geom Anal 25, 329–335 (2015). https://doi.org/10.1007/s12220-013-9430-9

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  • DOI: https://doi.org/10.1007/s12220-013-9430-9

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