Skip to main content
SpringerLink
Log in

Integral formulas and the ohsawa-takegoshi extension theorem

  • Published:
Science in China Series A: Mathematics Aims and scope Submit manuscript

Abstract

We construct a semiexplicit integral representation of the canonical solution to the\(\bar \partial \)-equation with respect to a plurisubharmonic weight function in a pseudoconvex domain. The construction is based on a construction related to the Ohsawa-Takegoshi extension theorem combined with a method to construct weighted integral representations due to M. Andersson.

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. Henkin, G., Leiterer, J., Theory of Functions on Complex Manifolds, Mathematische Lehrbüher und Monographien, II. Abteilung: Mathematische Monographien (Mathematical Textbooks and Monographs, Part II: Mathematical Monographs), 60, Berlin: Akademie-Verlag, 1984, 1–226.

    Google Scholar 

  2. Range, R. M., Holomorphic Functions and Integral Representations in Several Complex Variables, Graduate Texts in Mathematics, Vol. 108, New York: Springer-Verlag, 1986.

    Google Scholar 

  3. Berndtsson, B., Weighted integral formulas, in Several Complex Variables (Stockholm, 1987/1988), Math. Notes, 38, Princeton: Princeton Univ. Press, 1993, 160–187.

    Google Scholar 

  4. Berndtsson, B., Andersson, M., Henkin-Ramirez formulas with weight factors, Ann. Inst. Fourier (Grenoble), 1982, 32(3): v-vi, 91–110.

    MATH  MathSciNet  Google Scholar 

  5. Cumenge, A., Sharp estimates for\(\bar \partial \) on convex domains of finite type, Ark. Mat., 2001, 39(1): 1–25.

    Article  MATH  MathSciNet  Google Scholar 

  6. Andersson, M., Integral representation with weights, I, Math. Ann., 2003, 326(1): 1–18.

    Article  MATH  MathSciNet  Google Scholar 

  7. Liu, K. -F., Holomorphic equivariant cohomology, Math. Ann., 1995, 303(1): 125–148.

    Article  MATH  MathSciNet  Google Scholar 

  8. Zhang, W., Lectures on Chern-Weil Theory and Witten Deformations, Nankai Tracts in Mathematics, 4, River Edge: World Scientific Publishing Co., Inc., 2001.

    Google Scholar 

  9. Duistermaat, J. J., Heckman, G., On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math., 1982, 69(2): 259–268.

    Article  MATH  MathSciNet  Google Scholar 

  10. Ohsawa, T., Takegoshi, K., On the extension ofL 2 holomorphic functions, Math. Z., 1987, 195(2): 197–204.

    Article  MATH  MathSciNet  Google Scholar 

  11. Andersson, M., Residue currents and ideals of holomorphic functions, Bull. Sci. Math., 2004, 128(6): 481–512.

    Article  MATH  MathSciNet  Google Scholar 

  12. Berndtsson, B., The extension theorem of Ohsawa-Takegoshi and the theorem of Donnelly-Fefferman, Ann. Inst. Fourier (Grenoble), 1996, 46(4): 1083–1094.

    MathSciNet  Google Scholar 

  13. Siu, Y. -T., Complex-analyticity of harmonic maps, vanishing and Lefschetz theorems, J. Differential Geom., 1982, 17(1): 55–138.

    MATH  MathSciNet  Google Scholar 

  14. Berndtsson, B., Uniform estimates with weights for the\(\bar \partial \)-equation, J. Geom. Anal., 1997, 7(2): 195–215.

    MATH  MathSciNet  Google Scholar 

  15. Manivel, L., Un théoréme de prolongementL 2 de sections holomorphes d’un fibré hermitien (in French) (AnL 2 extension theorem for the holomorphic sections of a Hermitian bundle), Math. Z., 1993, 212(1): 107–122.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Chalmers University of Technology and the University of Göteborg, S-412 96, Göteborg, Sweden

    Bo Berndtsson

Authors
  1. Bo Berndtsson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bo Berndtsson.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Berndtsson, B. Integral formulas and the ohsawa-takegoshi extension theorem. Sci. China Ser. A-Math. 48 (Suppl 1), 61–73 (2005). https://doi.org/10.1007/BF02884696

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02884696

Keywords

Search

Navigation