Contact transformations and pure sheaves

Summary

In this chapter, we perform contact transformations for sheaves. We begin by extending the notion of kernel introduced in III §6 to a microlocal situation and we develop the microlocal calculus of kernels.

Let X and Y be two manifolds, Ω, and Ω _γ two open subsets of T*X and T*Y respectively. We show that under suitable hypotheses, the functors Φ _K and Ψ _K , are well-defined from D ^b(Y; Ω _γ ) to D ^b(X; Ω _X ) and from D ^b(X; Ω _X ) to D ^b(Y; Ω _Y ) respectively, and give equivalences of categories. Moreover, these equivalences are compatible with the functor μhom. Next, if χ: Ω _X ≃ Ω _Y is a contact transformation, we show that it is always possible after shrinking Ω _X and Ω _Y to construct an equivalence D ^b(X; Ω _X ) ≃ D ^b(Y; Ω _Y ), using these kernels. Now let M and N be two hypersurfaces of X and Y respectively, and assume the contact transformation χ interchanges T ^* _M X ⋂ Ω _X and T ^* _N Y ⋂ Ω _Y . If the graph of χ is associated to the conormal bundle to a hypersurface S of X × Y, and if one chooses the sheaf A _S as kernel K, then one proves that Φ _K (A _N ) ≃ A _M [d] in D ^b(X; p), (p є Ω _X ), where d is a shift that we calculate using the inertia index.

This calculation leads to the notion of pure sheaves along a smooth Lagrangian manifold Λ, a sheaf-theoretical analogue of the notion of Fourier distributions of Hörmander [2], [4], or of that of simple holonomic systems of Sato-Kawai-Kashiwara [1]. In case Λ = T ^* _M X, for a closed submanifold M of X, a pure sheaf F along A at p is nothing but the image in D ^b(X; p) of L _M [d], where L is an A-module (hence L _M is the sheaf on X supported by M and constant on M with stalk L) and d is a shift. (In such a case one says F is pure with shift d + 1/2 codim M.) When the rank of the projection π: Λ → X is not constant any more, the shift of F may “jump” and its calculation requires the full machinery of the inertia index. We end this chapter by calculating the shift of the composite of two kernels, and the shift of a pure sheaf, after taking its direct or inverse image.

The contents of this chapter are not necessary for the understanding of the rest of the book, with the exception of Chapter X §3 and Chapter XI §4.

We keep convention 4.0. Moreover, unless otherwise specified, all submanifolds of cotangent bundles are supposed to be locally conic.