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.

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    $\begingroup$ What does "microlocal" mean to you? $\endgroup$ – Will Sawin Oct 10 '19 at 2:32
  • $\begingroup$ 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." $\endgroup$ – Mathmank Oct 10 '19 at 2:50
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    $\begingroup$ 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. $\endgroup$ – Will Sawin Oct 10 '19 at 3:10
  • $\begingroup$ I find the cases of characteristic sheaves on open/closed balls very instructive. Try to plug in such a sheaf into the machine and see what you get at the end. After that you can try more general open/closed (perhaps some mildly singular) submanifolds. The general constructible sheaf always has filtrations whose quotients look (at least locally) like these so once you understand these examples you should be in good shape. $\endgroup$ – Saal Hardali Feb 18 '20 at 23:09
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    $\begingroup$ Well, given that the microlocal content of that book does not really get started until chapter 5, you may have to read a bit further. One answer to your question is: later on in there you learn that $\mu_N(F) = \mu hom(A_N, F)$. The $\mu hom$ is a sheaf on the cotangent bundle, whose stalk at a given point is in a sense explained in chapter 6 the `microlocal hom' between those sheaves at that point. $\endgroup$ – Vivek Shende Mar 30 '20 at 7:56

I know comes from far in the future, and you've probably figured it out by now, but I wanted to add to Vivek's comment.

Along a complex hypersurface $\{f=0\}$ that is non-singular at point $p$, there are identifications $\left [ \mu_{\{f=0\}}(F) \right ]_{(p;d_p f)} = (\varphi_f F)_p = R\Gamma_{\{Re(f) \geq 0\}}(F)_p.$ (see Prop 8.6.3 and Ex VIII.13 of The Book (using Kashiwara-Schapira's shift convention for $\varphi_f$). Each of these objects can be interpreted as a sort of "microlocal test functor" for $F$ at the covector $(p,d_p f)$, in that they microlocally detect cohomological changes of $F$.

More geometrically, if $F$ is $\mathbb{C}$-constructible along a complex analytic submanifold $S \subseteq X$, then the stalk $\mu_S(F)_{(p;\xi)}$ at a non-degenerate covector $(p;\xi) \in T_S^*X$ computes the normal Morse data of $F$ along $S$ (in the sense of Goresky-MacPherson's stratified Morse theory), and the isomorphism-type of this stalk is independent of the chosen non-degenerate covector (Jörg Schürmann has a nice discussion of this perspective in section 5.3.3. of "Topology of singular spaces and constructible sheaves")

Also, requiring these microlocal stalks to always be concentrated in cohomological degree $-\dim S$ in $D^b(k)$ along any stratum $S$ is equivalent to the perversity conditions (for the middle perversity), via Theorem 10.3.12 of "sheaves on manifolds".

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