Analytic *D*-Modules and Applications
pp 333-400 |
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# Microdifferential operators

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## Summary

*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*

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## Preview

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## Notes

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