Advertisement

Approximation and Extension of Whitney CR Forms

  • Mauro Nacinovich
Part of the The University Series in Mathematics book series (USMA)

Abstract

In this Chapter I want to outline some results on the approximation and extension of functions and forms that are defined in the sense of Whitney on closed subsets of a complex manifold and that satisfy the Cauchy-Riemann equations. Attention to such objects was first drawn by the study of the tangential Cauchy-Riemann complex, as in Andreotti and Hill [3] by the use of a Mayer-Vietoris sequence relating the cohomology of this complex to that of the ambient space. However, the Whitney cohomology appears as a natural object to be considered in more general situations and provides a unique framework to understand different situations, encompassing the classical Oka extension theorem and the question of approximating functions defined on a totally real subset by holomorphic functions in the ambient space. The results are obtained for zero sets of ideals of holomorphic functions that satisfy the conditions of Lojasiewicz, and I think it would be interesting to pursue the question of the general characterization of the object that I introduce here.

Keywords

Exact Sequence Holomorphic Function Closed Subset Complex Manifold Ambient Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    E. Amar, Cohomologie complexe et applications, J. London Math. Soc. 29,127–140 (1984).zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    A. Andreotti and H. Grauert, Théorèmes de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France 90, 193–259 (1962).zbMATHMathSciNetGoogle Scholar
  3. 3.
    A. Andreotti and C. D. Hill, E. E. Levi convexity and the Hans Lewy problem. I, II, Ann. Scuola Norm. Sup. Pisa 26, 325–363, 747–806, (1927).MathSciNetGoogle Scholar
  4. 4.
    D. Catlin, Boundary behaviour of holomorphic functions on pseudoconvex domains, J. Diff. Geom. 15, 605–625 (1980).zbMATHMathSciNetGoogle Scholar
  5. 5.
    J. Chaumat and A. M. Chollet, Ensembles pics pour A 8 (D), Ann. Inst. Fourier (Grenoble) 29, 171–200 (1979).zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    M. Hakim and N. Sibony, Ensembles pics dans des domains strictement pseudoconvexes, Duke Math. J. 45, 601–617 (1978).zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    M. Hakim and N. Sibony, Spectre de A(Ω0304) pour des domains bornés faiblement pseudoconvexes reguliers, J. Fund. Anal. 37, 127–135 (1980).zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    G. M. Henkin and J. Leiterer, Andreotti-Grauert Theory by Integral Formulas, Birkhäuser, Berlin (1988).zbMATHGoogle Scholar
  9. 9.
    J. J. Kohn, Global regularity for don weakly pseudoconvex manifolds, Trans. Am. Math. Soc. 181, 273–292 (1973).zbMATHMathSciNetGoogle Scholar
  10. 10.
    B. Malgrange, Ideals of Differentiable Functions, Oxford University Press (1966).zbMATHGoogle Scholar
  11. 11.
    M. Nacinovich, Poincaré lemma for tangential Cauchy-Riemann complexes, Math. Ann. 268, 449–471 (1984).zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    M. Nacinovich, On boundary Hilbert differential complexes, Ann. Polon. Math. 46, 213–235 (1985).zbMATHMathSciNetGoogle Scholar
  13. 13.
    M. Nacinovich, On strict Levi q-convexity and q-concavity on domains with piecewise smooth boundaries, Math. Ann. 281, 459–482 (1988).zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    M. Nacinovich, On a theorem of Airapetyan and Henkin, preprint, Dip. Mat. Pisa No. 431 (1989).Google Scholar
  15. 15.
    M. Nacinovich and G. Valli, Tangential Cauchy-Riemann complexes on distributions, Ann. Mat. Pura Appl. (4)146, 123–160 (1987).zbMATHMathSciNetGoogle Scholar
  16. 16.
    L. Schwartz, Théorie des Distributions, Hermann, Paris (1966).zbMATHGoogle Scholar
  17. 17.
    J. C. Tougeron, Ideaux de Fonctions Différentiables, Springer, Berlin (1972).zbMATHGoogle Scholar
  18. 18.
    H. Whitney, Analytic extension of differentiable functions defined in closed sets, Trans. Am. Math. Soc. 36, 63–89 (1934).CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • Mauro Nacinovich
    • 1
  1. 1.Dipartimento di MatematicaUniversità di PisaPisaItaly

Personalised recommendations