Skip to main content
SpringerLink
Log in

Pseudoconvex domains with smooth boundary in projective spaces

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

Given a pseudoconvex domain U with \(\mathscr {C}^1\)-boundary in \({\mathbb {P}}^n,\) \(n\geqslant 3,\) we show that if \(H^{2n-2}_{\mathrm{dR}}(U)\,\,\ne \,0,\) then there is a strictly psh function in a neighborhood of \(\partial U.\) We also solve the \({{{\overline{\partial }}}}\)-equation in \(X={\mathbb {P}}^n\setminus U,\) for data in \(\mathscr {C}^\infty _{(0,1)}(X).\) We discuss Levi-flat domains in surfaces. If Z is a real algebraic hypersurface in \({\mathbb {P}}^2,\) (resp a real-analytic hypersurface with a point of strict pseudoconvexity), then there is a strictly psh function in a neighborhood of Z.

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. Andreotti, A., Grauert, H.: Théorème de finitude pour la cohomologie des espaces complexes. (French). Bull. Soc. Math. France 90, 193–259 (1962)

    Article  MathSciNet  Google Scholar 

  2. Berndtsson, B., Sibony, N.: The \({\overline{\partial }}\)-equation on a positive current. Invent. math. 147(2), 371–428 (2002)

    Article  MathSciNet  Google Scholar 

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

  4. Dinh,T.-C., Nguyen, V.-A., Sibony, N.: Unique Ergodicity for foliations on compact Kähler surfaces. arXiv:1811.07450 math.CV (math.DS), 50 pages (2018)

  5. Dinh, T.-C., Sibony, N.: Pull-back currents by holomorphic maps. Manuscr. Math. 123(3), 357–371 (2007)

    Article  MathSciNet  Google Scholar 

  6. Dinh, T.-C., Sibony, N.: Rigidity of Julia sets for Hénon type maps. J. Mod. Dyn. 8(3–4), 499–548 (2014)

    MathSciNet  MATH  Google Scholar 

  7. Dinh, T.-C., Sibony, N.: Some open problems on holomorphic foliation theory. Acta Mathematica Vietnamica 45(1), 103–112 (2020)

    Article  MathSciNet  Google Scholar 

  8. Elencwajg, G.: Pseudo-convexité locale dans les variété kählériennes. Ann. Inst. Fourier (Grenoble)25, no. 2, xv, 295-314 (1975)

  9. Fornæss, J.E., Sibony, N.: Oka’s inequality for currents and applications. Math. Ann. 301(3), 399–419 (1995)

    Article  MathSciNet  Google Scholar 

  10. Fornaess, J.E., Sibony, N.: Harmonic currents of finite energy and laminations. Geom. Funct. Anal. 15(5), 962–1003 (2005)

    Article  MathSciNet  Google Scholar 

  11. Fornaess, J.E., Sibony, N.: Riemann surface laminations with singularities. J. Geom. Anal. 18(2), 400–442 (2008)

    Article  MathSciNet  Google Scholar 

  12. Greene, R.E., Wu, H.: On Kähler manifolds of positive bisectional curvature and a theorem of Hartogs. Abh. Math. Sem. Univ. Hamburg 47, 171–185 (1978)

    Article  MathSciNet  Google Scholar 

  13. Hörmander, L.: An introduction to complex analysis in several variables. Third edition. North-Holland Mathematical Library, 7. North-Holland Publishing Co., Amsterdam, (1990). xii+254 pp

  14. Kohn, J.J.: Global regularity for \({\overline{\partial }}\) on weakly pseudo-convex manifolds. Trans. Am. Math. Soc. 181, 273–292 (1973)

    MathSciNet  MATH  Google Scholar 

  15. Malgrange, B.: Ideals of differentiable functions. Tata Institute of Fundamental Research Studies in Mathematics, No. 3 Tata Institute of Fundamental Research, Bombay; Oxford University Press, London (1967) vii+106 pp

  16. Ohsawa, T., Sibony, N.: Bounded p.s.h. functions and pseudoconvexity in Kähler manifold. Nagoya Math. J.149, 1-8 (1998)

  17. Sibony, N.: Pfaff systems, currents and hulls. Math. Z. 285(3–4), 1107–1123 (2017)

    Article  MathSciNet  Google Scholar 

  18. Sibony, N.: Levi problem in complex manifolds. Math. Ann. 371(3–4), 1047–1067 (2018)

    Article  MathSciNet  Google Scholar 

  19. Siu, Y.-T.: Nonexistence of smooth Levi-flat hypersurfaces in complex projective spaces of dimension \(\ge 3.\)Ann. of Math. (2)151, no. 3, 1217-1243 (2000)

  20. Slodkowski, Z.: Local maximum property and q-plurisubharmonic functions in uniform algebras. J. Math. Anal. Appl. 115, no. 1, 105?130 (1986)

  21. Sullivan, D.: Cycles for the dynamical study of foliated manifolds in complex manifolds. Invent. Math. 36, 225–255 (1975)

    Article  Google Scholar 

  22. Takeuchi, A.: Domaines pseudoconvexes infinis et la métrique riemannienne dans un espace projectif. J. Math. Soc. Jpn. 16, 159–181 (1964)

    Article  Google Scholar 

Download references

Acknowledgements

It is a pleasure to thank Bo Berndtsson and Tien-Cuong Dinh for their insightful comments and the referee for his questions.

Author information

Authors and Affiliations

  1. Université Paris-Saclay, CNRS, Laboratoire de Mathématiques d’Orsay, 91405, Orsay, France

    Nessim Sibony

  2. Korea Institute for Advanced Study (KIAS), 85 Hoegiro, Dongdaemun-gu, Seoul, 02455, Republic of Korea

    Nessim Sibony

Authors
  1. Nessim Sibony
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nessim Sibony.

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

Sibony, N. Pseudoconvex domains with smooth boundary in projective spaces. Math. Z. 298, 625–637 (2021). https://doi.org/10.1007/s00209-020-02613-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-020-02613-6

Keywords

Mathematics Subject Classification

Search

Navigation