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Distributions and regular holonomic systems

  • Jan-Erik Björk
Chapter
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Part of the Mathematics and Its Applications book series (MAIA, volume 247)

Summary

The first section treats analytic D-module theory on real analytic manifolds and some basic results concerned with extendible distributions is presented in section 2 as a preparation to section 3. There we prove that every regular holonomic D X -module on a complex manifold is locally a cyclic module generated by a distribution on the underlying real manifold. The main result is Theorem 7.3.5 which gives an exact functor from RH(D X ) into the category of regular holonomic modules on the conjugate complex manifold defined by

We refer to К X as Kashiwara’s conjugation functor. Reversing the roles between X and there exists the conjugation functor from . We prove that the compsed functor is the identity on RH(D X ).

Distributions whose cyclic D X -modules are regular holonomic will be called regular holonomic distributions. Various examples of regular holonomic distributions are given in subsequent sections. In particular we mention the principal value distribu?tions defined by
where Ψ is any test-form on X R and fO(X). Meromorphic continuations of distributions are also discussed.

In the final sections we use the conjugation functor to exhibit an inverse functor to the de Rham functor in the Riemann-Hilbert correspondence. The inverse functor is obtained from a temperate Hom-functor composed with the 6-complex. This leads to properties of regular holonomic modules which go beyond those in Chapter V. The main results occur at the end of section 9. The last section contains a discussion about D-module theory related to Hodge theory.

Keywords

Complex Manifold Left Ideal Direct Image Natural Morphism Real Manifold 
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.

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Notes

  1. Herrera, M. and Liebermann, D., Residues and principal values on a complex space, Invent. Math. 194 (1971).Google Scholar
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    Barlet, D., Fonctions de type trace, Ann. l’Inst. Fourier 33 (1983), 43–76.MathSciNetzbMATHCrossRefGoogle Scholar
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    Kashiwara, M., The Riemann-Hilbert problem for holonomic systems, Publ. RIMS Kyoto 20 (1984), 319–365.MathSciNetzbMATHCrossRefGoogle Scholar
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    Kashiwara, M., Distributions and regular holonomic D-modules on complex manifolds, Adv. Stud. Pure Math. 8 (1986), 199–206.MathSciNetGoogle Scholar
  5. Barlet, D. and Kashiwara, M., Le réseau L2 d’un système holonôme regulier, Invent. Math. 86 (1986), 33–62.MathSciNetGoogle Scholar
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    Kashiwara, M. and Kawai T., Second microlocalization and asymptotic expansions, Lecture Notes in Physics 126, Springer, 1979, pp. 21–77.Google Scholar
  7. Barlet, D. and Maire, H. M., Asymptotic expansion of complex integrals via Mellin transforms, J. of Functional Anal. 83 No. 2 (1989), 233–257.MathSciNetzbMATHCrossRefGoogle Scholar
  8. Atiyah, M. F., Resolutions of singularities and divisions of distributions, Comm. Pure and Appl. Math. 23 (1970), 145–150.MathSciNetzbMATHGoogle Scholar
  9. Kashiwara M., Kawai, T. and Kimura, T., Foundations of algebraic analysis, Princeton Univ. Press, 1986.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1993

Authors and Affiliations

  • Jan-Erik Björk
    • 1
  1. 1.Department of MathematicsStockholm UniversityStockholmSweden

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