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Microdifferential operators

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

Summary

This chapter is devoted to a study of the ring ε X of micro-differential operators on the cotangent bundle T*(X) of a complex manifold. The construction of ε X is presented in the first section. The sheaf of rings ε X is coherent and the stalks are regular Auslander rings with global homological dimension equal to d X . Let π: T*(X) →X be the projection. Then π −1 D X is a subring of ε X . If M ∈ coh(D X ) there exists the microlocalisation
$$\varepsilon (M) = {\varepsilon _X} \otimes {\pi ^{ - 1}}M$$
.

A basic result is the equality SS(M) = Supp(ε(M)) for every coherent D X -module. This result and various facts about the sheaf ε X and its coherent modules are explained in the first two sections where the presentation is expositary and details of proofs often are omitted. For more detailed studies of the sheaf ε X we refer to [Schapira 2], [Kashiwara-Kawai-Kimura] and [Björk 1]. Coherent ε X -modules with regular singularities along analytic sets are studied in section 3 and 4. In section 5 we construct automorphisms on coherent ε X -modules with regular singularities along a non-singular and conic hypersurface in T*(X). They are called micro-local monodromy operators.

Holonomic ε X -modules are studied in section 6. The support of every M ∈ hol(ε X ) is a conic Lagrangian. The case when this Lagrangian is in a generic position is of particular importance. The main result in section 6 is that there is an equivalence of categories between germs of holonomic ε X -modules whose supports are in generic positions with a subcategory of germs of holonomic D X -modules.

Section 7 is devoted to regular holonomic ε X -modules and their interplay with regular holonomic D X -modules. The main result for analytic D-module theory is that a holonomic D X -module is regular holonomic in the sense of Chapter V if and only if its micro-localisation is regular holonomic. This result is deep because the characteristic variety of a regular holonomic D X -module is complicated with many singularities in general. The fact that the microlocalisation of every Deligne module is microregular is for example non-trivial.

The micro-local regularity of regular holonomic D X -modules is used in Section 8 to exhibit b-functions where the second member is controlled for generators of regular holonomic D X -modules. This is an important result when one wants to analyze solutions to regular holonomic systems.

Section 9 treats the sheaf ε X R of micro-local operators. Applications of micro-local analysis to D X -modules occur in Section 10 where we prove a local index formula for holonomic D X -modules. The final section contains results about analytic wavefront sets of regular holonomic distributions.

Keywords

Principal Symbol Chapter VIII Regular Singularity Good Filtration Monodromy Operator 
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

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    Kashiwara, M., Algebraic study of systems of partial differential equations, Univ. Tokyo, 1970.Google Scholar
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    Andronikof, E., The Kashiwara conjugation functor and wave front sets of regular holonomic distributions on a complex manifold, Inventiones Math. To appear.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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