Skip to main content
SpringerLink
Log in

A Dolbeault-Grothendieck lemma on complex spaces via Koppelman formulas

  • Published:
Inventiones mathematicae Aims and scope

Abstract

Let X be a complex space of pure dimension. We introduce fine sheaves of (0,q)-currents, which coincides with the sheaves of smooth forms on the regular part of X, so that the associated Dolbeault complex yields a resolution of the structure sheaf . Our construction is based on intrinsic and quite explicit semi-global Koppelman formulas.

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. Ancona, V., Gaveau, B.: Differential forms and resolutions on certain analytic spaces. I. Irreducible exceptional divisor. Bull. Sci. Math. 116(3), 307–324 (1992)

    MathSciNet  MATH  Google Scholar 

  2. Ancona, V., Gaveau, B.: Differential forms and resolutions on certain analytic spaces. II. Flat resolutions. Can. J. Math. 44(4), 728–749 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ancona, V., Gaveau, B.: Differential forms and resolutions on certain analytic spaces. III. Spectral resolution. Ann. Math. Pures Appl. 166(4), 175–202 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ancona, V., Gaveau, B.: Bochner-Martinelli formulas on singular complex spaces. Int. J. Math. 21(2), 225–253 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Andersson, M.: Integral representation with weights II, division and interpolation formulas. Math. Z. 254, 315–332 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Andersson, M.: A residue criterion for strong holomorphicity. Ark. Mat. 48(1), 1–15 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Andersson, M.: Coleff-Herrera currents, duality, and Noetherian operators. Bull. Soc. Math. France (to appear). Available at arXiv:0902.3064 [math.CV]

  9. Andersson, M., Samuelsson, H.: Koppelman formulas and the \(\bar{\partial}\)-equation on an analytic space. arXiv:0801.0710

  10. Andersson, M., Samuelsson, H.: Koppelman formulas and the \(\bar{\partial}\)-equation on an analytic space. J. Funct. Anal. 261, 777–802 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Andersson, M., Wulcan, E.: Residue currents with prescribed annihilator ideals. Ann. Sci. Éc. Norm. Super. 40, 985–1007 (2007)

    MathSciNet  MATH  Google Scholar 

  12. Andersson, M., Wulcan, E.: Decomposition of residue currents. J. Reine Angew. Math. 638, 103–118 (2010)

    MathSciNet  MATH  Google Scholar 

  13. Andersson, M., Wulcan, E.: On the effective membership problem on singular varieties. arXiv:1107.0388

  14. Andersson, M., Samuelsson, H., Sznajdman, J.: On the Briançon-Skoda theorem on a singular variety. Ann. Inst. Fourier (Grenoble) 60(2), 417–432 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Barlet, D.: Le faisceau ω X sur un espace analytique X de dimension pure. In: Fonctions de Plusieurs Variables Complexes, III, Sém. François Norguet, 1975–1977. Lecture Notes in Math., vol. 670, pp. 187–204. Springer, Berlin (1978)

    Chapter  Google Scholar 

  16. Björk, J.-E., Samuelsson, H.: Regularizations of residue currents. J. Reine Angew. Math. 640, 101–115 (2010)

    MathSciNet  Google Scholar 

  17. Demailly, J.-P.: Complex analytic and algebraic geometry. Monograph, Grenoble. Available at http://www-fourier.ujf-grenoble.fr/~demailly/books.html

  18. Diederich, K., Fornæss, J.E., Vassiliadou, S.: Local L 2 results for \(\bar{\partial}\) on a singular surface. Math. Scand. 92, 269–294 (2003)

    MathSciNet  MATH  Google Scholar 

  19. Eisenbud, D.: Commutative Algebra. With a View Toward Algebraic Geometry. Graduate Texts in Mathematics, vol. 150. Springer, New York (1995)

    MATH  Google Scholar 

  20. Fornæss, J.E., Gavosto, E.A.: The Cauchy-Riemann equation on singular spaces. Duke Math. J. 93, 453–477 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Fornæss, J.E., Øvrelid, N., Vassiliadou, S.: Semiglobal results for \(\bar{\partial}\) on a complex space with arbitrary singularities. Proc. Am. Math. Soc. 133(8), 2377–2386 (2005)

    Article  MATH  Google Scholar 

  22. Hatziafratis, T.E.: Integral representation formulas on analytic varieties. Pac. J. Math. 123(1), 71–91 (1986)

    MathSciNet  MATH  Google Scholar 

  23. Henkin, G., Passare, M.: Abelian differentials on singular varieties and variations on a theorem of Lie-Griffiths. Invent. Math. 135(2), 297–328 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  24. Henkin, G., Polyakov, P.: The Grothendieck-Dolbeault lemma for complete intersections. C. R. Acad. Sci., Ser I, Math. 308(13), 405–409 (1989)

    MathSciNet  MATH  Google Scholar 

  25. Henkin, G., Polyakov, P.: Residual d-bar-cohomology and the complex Radon transform on subvarieties of CPn. arXiv:1012.4438

  26. Lärkäng, R., Samuelsson, H.: Various approaches to products of residue currents. Preprint, Göteborg (2010). Available at arXiv:1005.2056 [math.CV]

  27. Lejeune-Jalabert, M.: Remarque sur la classe fondamentale d’un cycle. C. R. Acad. Sci., Ser I, Math. 292, 801–804 (1981)

    MathSciNet  MATH  Google Scholar 

  28. Lundqvist, J.: A local Grothendieck duality theorem for Cohen-Macaulay ideals. Math. Scand. (to appear). Available at arXiv:1201.4749 [math.CV]

  29. Malgrange, B.: Sur les fonctions différentiables et les ensembles analytiques. Bull. Soc. Math. Fr. 91, 113–127 (1963)

    MathSciNet  MATH  Google Scholar 

  30. Nagase, M.: Remarks on the L 2-Dolbeault cohomology groups of singular algebraic surfaces and curves. Publ. Res. Inst. Math. Sci. 26(5), 867–883 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  31. Ohsawa, T.: Hodge spectral sequences on compact Kähler spaces. Publ. Res. Inst. Math. Sci. 23(2), 265–274 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  32. Øvrelid, N., Vassiliadou, S.: Some L 2 results for \(\bar{\partial}\) on projective varieties with general singularities. Am. J. Math. 131, 129–151 (2009)

    Article  Google Scholar 

  33. Pardon, W., Stern, M.: L 2-\(\bar{\partial}\)-cohomology of complex projective varieties. J. Am. Math. Soc. 4(3), 603–621 (1991)

    MathSciNet  MATH  Google Scholar 

  34. Pardon, W., Stern, M.: Pure Hodge structure on the L 2-cohomology of varieties with isolated singularities. J. Reine Angew. Math. 533, 55–80 (2001)

    MathSciNet  MATH  Google Scholar 

  35. Ruppenthal, J.: Zur Regularität der Cauchy-Riemannschen Differentialgleichungen auf komlexen Räumen. Diplomarbeit, Bonn (2003), and dissertation published in Bonner Math. Schr. 380 (2006)

  36. Ruppenthal, J.: L 2-theory for the \(\bar{\partial}\)-operator on compact complex spaces. ESI-Preprint 2202, available at arXiv:1004.0396 [math.CV]

  37. Ruppenthal, J., Zeron, E.S.: An explicit \(\bar{\partial}\)-integration formula for weighted homogeneous varieties II, forms of higher degree. Mich. Math. J. 59, 283–295 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  38. Samuelsson, H.: Analytic continuation of residue currents. Ark. Mat. 47(1), 127–141 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  39. Scheja, G.: Riemannsche Hebbarkeitssätze für Cohomologieklassen. Math. Ann. 144, 345–360 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  40. Spallek, K.: Über Singularitäten analytischer Mengen. Math. Ann. 172, 249–268 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  41. Stout, E.L.: An integral formula for holomorphic functions on strictly pseudoconvex hypersurfaces. Duke Math. J. 42, 347–356 (1975)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Division of Mathematics, Chalmers University of Technology and the University of Gothenburg, 412 96, Göteborg, Sweden

    Mats Andersson & Håkan Samuelsson

Authors
  1. Mats Andersson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Håkan Samuelsson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mats Andersson.

Additional information

The first author was partially supported by a grant from the Swedish Research Council. The second author wishes to thank the Department of Mathematics, University of Oslo, where part of his work was done.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Andersson, M., Samuelsson, H. A Dolbeault-Grothendieck lemma on complex spaces via Koppelman formulas. Invent. math. 190, 261–297 (2012). https://doi.org/10.1007/s00222-012-0380-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-012-0380-9

Mathematics Subject Classification (2000)

Search

Navigation