nLab microlocalization

Idea

Microlocalization is a tool invented by Mikio Sato to study linear partial differential equations (as a part of his algebraic analysis program) not only locally in space but also locally in momentum variable. It is a purely algebraic theory that was also continued in parallel by analysts, like Hormander, giving the domain of microlocal analysis.

Construction

The original construction is based on the use of the specialization functor and Fourier-Sato transformation. In this section, we will discuss the construction in a general setting, i.e., over an arbitrary field of characteristic $0$. In the real situation, one usually refines the construction by using conic sheaves (i.e., sheaves invariant with respect to the natural $\mathbb{R}_+$-action) to get information about the oriented direction of propagation of singularities of sheaves of solutions of analytic partial differential systems. The construction we describe does not treat this refined information.

Let $Z\hookrightarrow X$ be a closed subspace of a given analytic manifold, defined by a sheaf of ideals $\mathcal{I}$, with normal bundle denoted $T_Z X$ and conormal bundle denoted $T^*_Z X$. One defines the deformation to the normal bundle as the (analytic space associated to the) relative scheme over $X$ given by

$$\widetilde{T_Z X}:=Spec_X(\oplus_{i\in \mathbb{Z}} z^{-i} \mathcal{I}^i)^{an}$$

with $\mathcal{I}^i=\mathcal{O}_X$ for $i\leq 0$. There is a projection $p:\widetilde{T_Z X}\to X$ and a projection $\tau:\widetilde{T_Z X}\to \mathbb{A}^1$. The fiber at $0$ of $\tau$ is denoted $s:T_Z X\to \widetilde{T_Z X}$, and its fiber at $t\neq 0$ is $X$. The fiber of $p$ on the open subset $(X\backslash Z)$ is $(X\backslash Z)\times \mathbb{A}^1-\{0\}$.

The specialization of a sheaf $F\in D^b(k_X)$ is the sheaf $\nu_Z(F)\in D^b(k_{T_Z X})$ defined as

$$\nu_Z(F):=s^*p^*F.$$

The Fourier-Sato transform is the functor

$$\Phi:D^b(k_{T_Z X})\to D^b(k_{T^*_Z X})$$

defined by

$$\Phi(G):=\mathbb{R}p_{2!}p_1^*G$$

where $p_1:T_Z X\times_Z T^*_Z X\to T_Z X$ and $p_2:T_Z X\times_Z T^*_Z X\to T^*_Z X$ are the two natural projections.

The $Z$-microlocalization functor is the functor

$$\mu_Z:=\Phi\circ \nu_Z:D^b(k_X)\to D^b(k_{T^*_Z X}).$$

The microlocalization functor on a variety $M$ is defined as the $Z$-microlocalization associated to the closed immersion $Z=M\subset M\times M=X$. Since $T^*_{\Delta_M} (M\times M)\cong T^*M$, this gives a functor

$$\mu:D^b(k_M)\to D^b(k_{T^*M}).$$

Denoting $q_1,q_2:M\times M\to M$ the natural projection, we defined the microlocal homomorphisms $\mu hom(F,G)$ between two complexes of sheaves $F$ on $G$ on $X$ by

$$\mu hom(F,G):=\mu_{\Delta_M}\mathbb{R} Hom(q_2^{-1} F,q_1^{!}G).$$

If $\pi:T^*_{\Delta_M}(M\times M)\to M$ is the natural projection, we have

$$\pi_*\mu hom(F,G)\cong \mathbb{R}Hom(F,G).$$

Related subjects

index theory

microlocal formulation of index theory

global analytic index theory

derived microlocalization

algebraic microlocalization

References

Sato’s theory of microlocalization was first described in the setting of D-modules:

• M. Kashiwara, Kawai, Kimura: foundations of algebraic analysis.

It was then extended to a purely sheaf theoretical theory in

• Masaki Kashiwara, Pierre Schapira, Sheaves on manifolds, Grundlehren 292, Springer (1990) [doi:10.1007/978-3-662-02661-8]

This theory of microlocalization of (ind)-sheaves (and also sub-analytic sheaves) was developped in the following works:

• Masaki Kashiwara, Pierre Schapira, Ind-sheaves, distributions and microlocalization, describes the program.
• Masaki Kashiwara, Pierre Schapira, Florian Ivorra, Ingo Waschkies Microlocalization of ind-sheaves, gives the main results

  and proofs.

• Masaki Kashiwara, Pierre Schapira Ind-sheaves, SMF, gives a complete account of the theory.
• Luca Prelli?, Microlocalization of sub-analytic sheaves, gives the theory in the sub-analytic setting.

A good overview of the theory can by found at:

• Pierre Schapira Derived categories for the analyst (2010)