Microdifferential operators

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 ^R _X 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.