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

T ZX˜:=Spec X( iz i i) an\widetilde{T_Z X}:=Spec_X(\oplus_{i\in \mathcal{I}} z^{-i} \mathcal{I}^i)^{an}

with i=𝒪 X\mathcal{I}^i=\mathcal{O}_X for i0i\leq 0. There is a projection p:T ZX˜Xp:\widetilde{T_Z X}\to X and a projection τ:T ZX˜𝔸 1\tau:\widetilde{T_Z X}\to \mathbb{A}^1. The fiber at 00 of τ\tau is denoted s:T ZXT ZX˜s:T_Z X\to \widetilde{T_Z X}, and its fiber at t0t\neq 0 is XX. The fiber of pp on the open subset (X\Z)(X\backslash Z) is (X\Z)×𝔸 1{0}(X\backslash Z)\times \mathbb{A}^1-\{0\}.

The specialization of a sheaf FD b(k X)F\in D^b(k_X) is the sheaf ν Z(F)D b(k T ZX)\nu_Z(F)\in D^b(k_{T_Z X}) defined as

ν Z(F):=s *p *F.\nu_Z(F):=s^*p^*F.

The Fourier-Sato transform is the functor

Φ:D b(k T ZX)D b(k T Z *X)\Phi:D^b(k_{T_Z X})\to D^b(k_{T^*_Z X})

defined by

Φ(G):=p 2!p 1 *G\Phi(G):=\mathbb{R}p_{2!}p_1^*G

where p 1:T ZX× ZT Z *XT ZXp_1:T_Z X\times_Z T^*_Z X\to T_Z X and p 2:T ZX× ZT Z *XT Z *Xp_2:T_Z X\times_Z T^*_Z X\to T^*_Z X are the two natural projections.

The ZZ-microlocalization functor is the functor

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

The microlocalization functor on a variety MM is defined as the ZZ-microlocalization associated to the closed immersion Z=MM×M=XZ=M\subset M\times M=X. Since T M *(M×M)T *MT^*_M (M\times M)\cong T^*M, this gives a functor

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

Denoting q 1,q 2:M×MMq_1,q_2:M\times M\to M the natural projection, we defined the microlocal homomorphisms μhom(F,G)\mu hom(F,G) between two complexes of sheaves FF on GG on XX by

μhom(F,G):=μ Δ MHom(q 2 1F,q 1 !G).\mu hom(F,G):=\mu_{\Delta_M}\mathbb{R} Hom(q_2^{-1} F,q_1^{!}G).

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

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

Index theory

Microlocal formulation of index theory

Global analytic index theory

Derived 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 by M. Kashiwara and P. Schapira to a purely sheaf theoretical theory in

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

A good overview of the theory can by found at:

Revised on November 10, 2014 10:40:36 by Frédéric Paugam (134.157.13.171)