# How to understand the Fourier-Sato transform and microlocalization functors?

Given a smooth real vector bundle $$\pi: E \to M$$ I can look at the (bounded from below) derived category of sheaves on $$E$$. Since $$E$$ admits a very natural action of $$\mathbb{R}^{\geq 0}$$ by scaling, and I can ask that a sheaf be constant after pulling back to the orbits of this action (i.e. rays in the fibers of $$E$$). Sheaves that satisfy this property are called conic sheaves.

There is an operation we can define which takes conic sheaves on $$E$$ to conic sheaves on the dual bundle $$E^*$$ called the Fourier-Sato transform, defined as follows: Define $$P' = \{(x,y) \in E \times_M E^*: \langle x,y \rangle \leq 0\}$$. Given a conic sheaf $$G$$ we define the Fourier-Sato transform to be $$\hat{G} = R\pi^*_!(\pi^{-1}G)_{P'}$$, where $$\pi^*$$ denotes the projection map of the dual bundle.

Now given a sheaf $$F$$ on an arbitrary manifold $$M$$, we can define the specialization of $$F$$ with respect to a given submanifold $$N$$, which is a conic sheaf on the normal bundle to $$N$$, denoted by $$\nu_N(F)$$. The specialization functor is defined by first forming the normal deformation of $$N$$ in $$M-$$a sort of a "real blow-up" of the submanifold $$N-$$ creating a family over $$\mathbb{R}$$ equipped with a map to $$M$$ and whose fiber at $$0$$ is the normal bundle $$T_NM$$; and then doing some push-pull operations akin to the nearby cycles functor.

Putting the specialization functor $$\nu_N$$ and the Fourier-Sato transform together gives us the microlocalization functor: $$\mu_N(F) = \nu_N(F)$$^.

At this point, I have an okay understanding of the specialization functor as something that takes a sheaf $$F$$ and asks how it behaves infinitesimally close to $$N$$ in the normal directions to $$N$$, like we're kind of taking dough and stretching it thin in only normal directions, then asking what our sheaf looks like in this stretched picture. I also have an okay understanding of the Fourier-Sato transform as being defined the way it is to imitate the classical Fourier transform by switching skyscraper and constant sheaves and perhaps have restriction to $$P'$$ play the role of $$e^{-ix \cdot \xi}$$.

My question is: how can I see that the microlocalization functor somehow organizes or is related to the microlocal information of my sheaf like its name suggests? I'd appreciate at least some pseudo-geometric intuition in the style of what I've written above Moreover, is my intuition as written above good or misleading?

These definitions are all contained in chapters 3 and 4 of Sheaves on Manifolds by Kashiwara, Schapira, but going through their proofs often leaves me, at best, unsure of the ideas behind the proof, and, at worst, hopelessly confused even if I follow individual steps in the proofs. Any help?

I'm including the algebraic geometry tag on this because I'm aware there are related constructions in the algebraic world.

• What does "microlocal" mean to you? – Will Sawin Oct 10 '19 at 2:32
• I suppose "microlocal" means "relating to the directions in the cotangent bundle where interesting changes in my sheaf happen" to me. I guess I could also phrase it as "relating to the microsupport of my sheaf." – Mathmank Oct 10 '19 at 2:50
• All I can help you with is that the dual to the normal bundle is contained in the cotangent bundle and that the stalk of the Fourier-Sato transform at a point should basically measure changes by taking the cohomology with compact supports of a halfspace. – Will Sawin Oct 10 '19 at 3:10