Applications to O-modules and D-modules

Summary

By its definition, a complex manifold X is endowed with the sheaf of rings O _X of holomorphic functions. The structure of O _X , and the theory of O _X -modules, are now well-understood, and it is not our aim to explain this theory from the start. We shall content ourselves with a few basic facts concerning the algebraic structure of O _X , its flabby dimension and the operations on O _X . References are made to Banica-Stanasila [1], Cartan [2], Hörmander [1], Serre [1].

Next, we introduce the sheaf of rings D _X of finite order holomorphic differential operators on X. Here again the theory of D _X -modules is now well-understood, and we shall be rather brief on this subject, having in mind to make understood the main notions, in particular that of characteristic variety, and to explain the operations on D _X -modules. We shall also recall the classical Cauchy-Kowalewski theorem and its extension to D _X -modules, and we shall derive the formula:

$$SS\left( {R\mathcal{H}\mathcal{O}{{\mathcal{M}}_{{\mathcal{D}X}}}\left( {\mathcal{M},{{\mathcal{O}}_{X}}} \right)} \right) \subset char\left( \mathcal{M} \right),$$

(11.0.1)

, where char(M) is the characteristic variety of the D _X-module M. (In fact, this inclusion is an equality, cf. §4.) As an application of (11.0.1) one immediately obtains, with the help of the results of VIII §5 the constructibility of the complex $RHo{m_{{D_X}}}\left( {M,{O_X}} \right)$ when M is holonomic, and one also easily proves that this complex is perverse.

For a more detailed exposition of the theory of Q _X -modules, we refer to Björk [1], Kashiwara [5] and Schapira [2].

Then we study “microlocally” the sheaf O _X . After having introduced the ring E ^ℝ _X of microlocal operators we sketch the proof of an important theorem which asserts that one can locally “quantize” holomorphic contact transformations over O _X . We end this chapter by introducing the sheaf L _M of Sato micro-functions on a real analytic manifold M. Using (11.0.1), and the results of Chapters V and VI, it is an easy exercise to recover many classical results of the theory of linear partial differential equations, in particular those concerning elliptic equations or the analytic wave front set, micro-hyperbolic systems and propagation of singularities.

As it should be clear, the aim of this chapter is not to give a complete or systematic treatment of the theory of analytic (micro-)differential equations, but rather to introduce the reader to it, and in particular to make him better understand the basic paper of Sato-Kawai-Kashiwara [1], under the light of the theory of micro-support of sheaves.

In this chapter all sheaves, unless otherwise specified, are sheaves of ℂ-vector spaces.