Poincaré-Verdier duality and Fourier-Sato transformation

Summary

In this chapter we introduce two fundamental tools for the study of sheaves on manifolds. First, following Verdier [1], we construct a right adjoint f ^! to the functor Rf _!. If f: Y → X is a continuous map of locally compact spaces satisfying suitable conditions, and if F (resp. G) belongs to D ⁺(A _X ) (resp. D ⁺(A _Y )), one gets the formula:

$$Hom(R{f_!}G,F) = Hom(G,{f^!}F)$$

(3.0.1)

.

This generalizes the classical Poincaré duality. In fact if X is reduced to a single point, if F = A and if Y is a topological manifold, one calculates f ^! A (called “the dualizing complex” on Y), and shows it is isomorphic to the orientation sheaf on Y, shifted by the dimension. With the functor f ^! in hands, it is then possible to derive many nice new formulas of sheaf theory.

Secondly, we make a detailed study of the Fourier-Sato transformation, an operation which interchanges conic sheaves (in the derived category) on a vector bundle and conic sheaves on the dual vector bundle.

In the course of this chapter we study sheaves on topological manifolds: orientation sheaf, flabby dimension, cohomologically constructible sheaves, and we also introduce the γ-topology, as a preparation to Chapter V.

For another approach to the contents of §1–§4 one may also consult Borel et al [1], Gelfand-Manin [1], Iversen [1] and [SHS].

Much of the material of this chapter may be considered as classical, though it had not received a systematic treatment beforehand.