Microdifferential operators

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


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.


Principal Symbol Chapter VIII Regular Singularity Good Filtration Monodromy Operator 
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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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