REGULARIZATION FOR HOLONOMIC D-MODULES

REGULARIZATION FOR HOLONOMIC D-MODULES

ANDREA D’AGNOLO AND MASAKI KASHIWARA

Abstract. On a complex manifold, the embedding of the category of regular holonomic D-modules into that of holonomic D-modules has a left quasi-inverse functor M 7→ M reg , called regularization. Re- call that M reg is reconstructed from the de Rham complex of M by the regular Riemann-Hilbert correspondence. Similarly, on a topo- logical space, the embedding of sheaves into enhanced ind-sheaves has a left quasi-inverse functor, called here sheafification. Regularization and sheafification are intertwined by the irregular Riemann-Hilbert correspondence. Here, we study some of the properties of the sheafi- fication functor. In particular, we provide a stalk formula for the sheafification of enhanced specialization and microlocalization.

Contents

1. Introduction 1

2. Notations and complements 3

3. Sheafification 9

4. Stalk formula 16

5. Specialization and microlocalization 19

Appendix A. Complements on enhanced ind-sheaves 25 Appendix B. Complements on weak constructibility 26

References 30

1. Introduction

Let X be a complex manifold. The regular Riemann-Hilbert correspon- dence (see [6]) states that the de Rham functor induces an equivalence

2010 Mathematics Subject Classification. Primary 32C38, 14F05.

Key words and phrases. irregular Riemann-Hilbert correspondence, enhanced per- verse sheaves, holonomic D-modules.

The research of A.D’A. was partially supported by GNAMPA/INdAM. He ac- knowledges the kind hospitality at RIMS of Kyoto University during the preparation of this paper.

The research of M.K. was supported by Grant-in-Aid for Scientific Research (B) 15H03608, Japan Society for the Promotion of Science.

1

between the triangulated category of regular holonomic D-modules and that of C-constructible sheaves. More precisely, one has a diagram (1.1) D ^b _hol (D _X )

DR

))

D ^b _rh (D _X )

OO

ι

OO

DR

∼ // D ^b

C-c (C X )

Φ

oo

where ι is the embedding (i.e. fully faithful functor) of regular holonomic D-modules into holonomic D-modules, the triangle quasi-commutes, DR is the de Rham functor, and Φ is an (explicit) quasi-inverse to DR.

The regularization functor reg : D ^b _hol (D _X ) − → D ^b _rh (D _X ) is defined by M _reg := Φ(DR(M)). It is a left quasi-inverse to ι, of transcendental nature. Recall that (ι, reg) is not a pair of adjoint functors ¹ . Recall also that reg is conservative ² .

Let k be a field and M be a good topological space. Consider the natural embeddings D ^b (k _M ) ^{// ι //} D ^b (I k _M ) ^{// e //} E ^b _st (I k _M ) of sheaves into ind-sheaves into stable enhanced ind-sheaves. One has pairs of adjoint functors (α, ι) and (e, Ish), and we set sh := α Ish:

sh : E ^b _st (I k _M ) −→ D ^{Ish b} (I k _M ) − → D ^{α b} (k _M ).

We call Ish and sh the ind-sheafification and sheafification functor, re- spectively. The functor sh is a left quasi-inverse of e ι.

For k = C and M = X, the irregular Riemann-Hilbert correspondence (see [1]) intertwines ³ the pair (ι, reg) with the pair (e ι, sh). In particular, the pair (e ι, sh) is not a pair of adjoint functors in general.

With the aim of better understanding the rather elusive regulariza- tion functor, in this paper we study some of the properties of the ind- sheafification and sheafification functors.

More precisely, the contents of the paper are as follows.

In §2, besides recalling notations, we establish some complementary results on ind-sheaves on bordered spaces that we need in the following.

Further complements are provided in Appendix A.

Some functorial properties of ind-sheafification and sheafification are obtained in §3. In §4, we obtain a stalk formula for the sheafification of a pull-back by an embedding. Then, these results are used in section §5 to obtain a stalk formula for the sheafification of enhanced specialization and microlocalization. In particular, the formula for the specialization puts in a more geometric perspective what we called multiplicity test functor in [2, §6.3].

1 By saying that (ι, reg) is a pair of adjoint functors, we mean that ι is the left adjoint of reg.

2 In fact, if M reg ' 0 then DR(M) ' DR(M reg ) ' 0, and hence M ' 0.

3 Using formula (3.1) below, this follows from [1, Corollary 9.6.7].

Finally, we provide in Appendix B a formula for the sections of a weakly constructible sheaf on a locally closed subanalytic subset, which could be of independent interest.

2. Notations and complements

We recall here some notions and results, mainly to fix notations, refer- ring to the literature for details. In particular, we refer to [8] for sheaves, to [11] (see also [5, 3]) for enhanced sheaves, to [9] for ind-sheaves, and to [1] (see also [10, 7, 3]) for bordered spaces and enhanced ind-sheaves.

We also add some complements.

In this paper, k denotes a base field.

A good space is a topological space which is Hausdorff, locally compact, countable at infinity, and with finite soft dimension.

By subanalytic space we mean a subanalytic space which is also a good space.

2.1. Bordered spaces. The category of bordered spaces has for objects the pairs M = (M, C) with M an open subset of a good space C. Set

◦

M := M and M := C. A morphism f : M −

→ N is a morphism f :

M −

→ N

of good spaces such that the projection Γ _f − → M is proper. Here, Γ

_f denotes the closure in

∨

M ×

∨

N of the graph Γ f of

◦

f . Note that M 7→

∨

M is not a functor. The functor M 7→ M is right adjoint

to the embedding M 7→ (M, M ) of good spaces into bordered spaces. We will write for short M = (M, M ).

Note that the inclusion k M :

◦

M − → M factors into

(2.1) k _M :

◦

M ⁱ

^// M ^j

^//

∨

M.

By definition, a subset Z of M is a subset of

◦

M. We say that Z ⊂ M is open (resp. closed, locally closed) if it is so in

◦

M. For a locally closed subset Z of M, we set Z ∞ = (Z, Z) where Z is the closure of Z in

∨

M.

Note that U ∞ ' (U, M) for U ⊂ M open.

We say that Z is a relatively compact subset of M if it is contained in a compact subset of

∨

M. Note that this notion does not depend on the choice of

∨

M. This means that if N is a bordered space with N ' M and

◦

N = M, then Z is relatively compact in M if and only if it is so in N.

An open covering {U _i } _i∈I of a bordered space M is an open covering of

◦

M which satisfies the condition: for any relatively compact subset Z of M there exists a finite subset I ⁰ of I such that Z ⊂ ^[

i∈I

U _i .

We say that a morphism f : M − → N is

(i) an open embedding if f is a homeomorphism from

M onto an open

subset of

◦

N,

(ii) borderly submersive if there exists an open covering {U _i } _i∈I of M such that for any i ∈ I there exist a subanalytic space S _i and an open embedding g _i : (U _i ) ∞ − → S _i × N with a commutative diagram of bordered spaces

(U _i ) ∞ //

g



M

 f

S i × N _p

i

// N, where p _i is the projection,

(iii) semiproper if Γ _f − → N is proper,

(iv) proper if it is semiproper and

◦

f : M −

→ N is proper,

(v) self-cartesian if the diagram M

f

//

i



◦

N

i



M ^{f //} N

is cartesian.

Recall that, by [1, Lemma 3.3.16], a morphism f : M − → N is proper if and only if it is semiproper and self-cartesian.

2.2. Ind-sheaves on good spaces. Let M be a good space.

We denote by D ^b (k M ) the bounded derived category of sheaves of k- vector spaces on M . For S ⊂ M locally closed, we denote by k S the extension by zero to M of the constant sheaf on S with stalk k.

For f : M − → N a morphism of good spaces, denote by ⊗, f ⁻¹ , Rf _! and RHom , Rf ∗ , f ^! the six operations. Denote by  the exterior tensor and by D _M the Verdier dual.

We denote by D ^b (I k _M ) the bounded derived category of ind-sheaves of k-vector spaces on M , and by ⊗, f ⁻¹ , Rf _!! and R Ihom , Rf ∗ , f ^! the six operations. Denote by  the exterior tensor and by D M the Verdier dual.

There is a natural embedding ι M : D ^b (k M ) − → D ^b (I k M ). It has a left adjoint α M , which in turn has a left adjoint β M . The commutativity of these functors with the operations is as follows

(2.2)

 f ⁻¹ Rf ∗ f ^! Rf _!!

ι ◦ ◦ ◦ ◦ ×

α ◦ ◦ ◦ × ◦

β ◦ ◦ × × ×

where “◦” means that the functors commute, and “×” that they don’t.

2.3. Ind-sheaves on bordered spaces. Let M be a bordered space.

Setting D ^b (k _M ) := D ^b (k

M )/D ^b (k

M\

◦

M ), one has D ^b (k _M ) ' D ^b (k

M ).

The bounded derived category of ind-sheaves of k-vector spaces on M is defined by D ^b (I k _M ) := D ^b (I k

M )/D ^b (I k

M\

◦

M ). There is a natural embedding

ι _M : D ^b (k

M ) ' D ^b (k _M ) − → D ^b (I k _M ) induced by ι

M . It has a left adjoint

α M : D ^b (I k M ) − → D ^b (k

M ),

which in turn has a left adjoint β _M . One sets RHom := α _M R Ihom , a functor with values in D ^b (k

M ).

For F ∈ D ^b (k

M ), we often simply write F instead of ι M F in order to make notations less heavy.

For operations, we use the same notations as in the case of good spaces.

Recall (see [1, Proposition 3.3.19] ⁴ ) that Rf _!! ' Rf ∗ if f is proper, (2.3)

f ^! ' f ^! k

N ⊗ f ⁻¹ if f : M − → N is borderly submersive.

(2.4)

The last statement implies

f ^! commutes with α if f is borderly submersive.

(2.5)

With notations (2.1), (2.4) implies that

(2.6) i ⁻¹ _M ' i _M ^! , j _M ⁻¹ ' j _M ^! . The quotient functor D ^b (I k

M ) − → D ^b (I k M ) is isomorphic to j _M ⁻¹ ' j _M ^! and has a left adjoint Rj _{M !!} and a right adjoint Rj _M∗ , both fully faithful.

The functors ι M , α M and β M are exact. Moreover, ι M and β M are fully faithful. This was shown in [9] in the case of good spaces. The general case reduces to the former by the

Lemma 2.1. One has (i) ι _M := j _M ⁻¹ ι

M Rk _M∗ ' Ri _M∗ ι

M , (ii) α _M ' k ⁻¹ _M α

M Rj _{M !!} ' α

M i ⁻¹ _M , (iii) β _M ' Ri _{M !!} β

M . Proof. One has

j _M ⁻¹ ι

M Rk _M∗ '

(∗) j _M ⁻¹ Rk _M∗ ι

M ' j _M ⁻¹ Rj _M∗ Ri _M∗ ι

M ' Ri _M∗ ι

M , where (∗) follows from (2.2).

This proves (i). Then (ii) and (iii) follow by adjunction. 

4 The statement of this proposition is erroneous. The first isomorphism in loc. cit.

may not hold under the condition that f is topologically submersive. However, it

holds if f is borderly submersive. The second isomorphism, i.e. (2.4), holds under the

condition that f is topologically submersive.

For bordered spaces, the commutativity of the functor α with the op- erations is as follows.

Lemma 2.2. Let f : M − → N be a morphism of bordered spaces.

(i) There are a natural isomorphism and a natural morphism of func- tors

◦

f ⁻¹ α _N ' α _M f ⁻¹ , α _M f ^! − → f

^! α _N ,

and the above morphism is an isomorphism if f is borderly sub- mersive.

(ii) There are natural morphisms of functors R

◦

f _! α _M − → α _N Rf _!! , α _N Rf ∗ − → R f

_∗ α _M , which are isomorphisms if f is self-cartesian.

(iii) For K ∈ D ^b (I k _M ) and L ∈ D ^b (I k _N ) one has α _M×N (K  L) ' (α M K)  (α N L).

Proof of Lemma 2.2. (i-a) By Lemma 2.1 (ii) and (2.2), one has f

⁻¹ α _N '

◦

f ⁻¹ α

N i ⁻¹ _N ' α

M

◦

f ⁻¹ i ⁻¹ _N ' α

M i ⁻¹ _M f ⁻¹ ' α _M f ⁻¹ ’

(i-b) By Lemma 2.1 (ii), the morphism is given by the composition α

M i ⁻¹ _M f ^! −→ ∼

(∗) α

M

◦

f ^! i ⁻¹ _N −−−→ ^(∗∗) f

^! α

N i ⁻¹ _N .

Here, (∗) follows from (2.6), and (∗∗) follows by adjunction from

◦

f ^! − →

◦

f ^! ι

N α

N ' ι

M

◦

f ^! α

N , with the isomorphism due to (2.2).

If f is borderly submersive, (∗∗) is an isomorphism by (2.5).

(ii-a) By Lemma 2.1 (ii), the morphism is given by R

◦

f _! α

M i ⁻¹ _M ' α

N R

◦

f _!! i ⁻¹ _M −−−→ α ^(∗)

N i ⁻¹ _N Rf _!! . Here (∗) follows by adjunction from Ri _{N !!} R

◦

f _!! i _M ^! ' Rf _!! Ri _{M !!} i _M ^! − → Rf _!! , recalling (2.6).

If f is self-cartesian, this is an isomorphism by cartesianity.

(ii-b) By Lemma 2.1 (ii) and (2.2), the morphism is given by the compo- sition

α

N i ⁻¹ _N Rf ∗

−→ α (∗)

N R

◦

f _∗ i ⁻¹ _M ' R f

_∗ α

M i ⁻¹ _M . Here (∗) follows from Lemma A.3.

Recall (2.6). If f is self-cartesian, then (∗) is an isomorphism by carte- sianity.

(iii) follows from α _M ' α

M i ⁻¹ _M and (2.2). 

2.4. Enhanced ind-sheaves. Denote by t ∈ R the coordinate on the affine line, consider the two-point compactification R := R ∪ {−∞, +∞}, and set R ^∞ := (R, R). For M a bordered space, consider the projection

π _M : M × R ^∞ − → M.

Denote by E ^b (I k _M ) := D ^b (I k _M×R

)/π ⁻¹ _M D ^b (I k _M ) the bounded derived category of enhanced ind-sheaves of k-vector spaces on M. Denote by Q : D ^b (I k _M×R

) − → E ^b (I k _M ) the quotient functor, and by L ^E and R ^E its left and right adjoint, respectively. They are both fully faithful.

For f : M − → N a morphism of bordered spaces, set f _R := f × id _R

: M × R ^∞ − → N × R ^∞ .

Denote by ⊗, Ef ^{+ −1} , Ef _!! and RIhom ⁺ , Ef ∗ , Ef ^! the six operations for enhanced ind-sheaves. Recall that ⊗ is the additive convolution in the ⁺ t variable, and that the external operations are induced via Q by the corresponding operations for ind-sheaves, with respect to the morphism f _R . Denote by  the exterior tensor and by D ^{+ E} the Verdier dual.

We have

L ^E Q(F ) ' (k _{t>0} ⊕ k _{t60} ) ⊗ F ⁺ and (2.7)

R ^E Q(F ) ' RIhom ⁺ (k _{t>0} ⊕ k _{t60} , F ).

(2.8)

The functors R Ihom ^E and RHom ^E , taking values in D ^b (I k M ) and D ^b (k

M ), respectively, are defined by

R Ihom ^E (K ₁ , K ₂ ) := Rπ _M∗ R Ihom (F ₁ , R ^E K ₂ ) (2.9)

' Rπ _M∗ R Ihom (L ^E K ₁ , F ₂ ), RHom ^E (K ₁ , K ₂ ) := α _M R Ihom ^E (K ₁ , K ₂ ), (2.10)

for K i ∈ E ^b (I k _M ) and F _i ∈ D ^b (I k _M×R

) such that K _i = Q F _i (i = 1, 2).

There is a natural decomposition E ^b (I k _M ) ' E ^b ₊ (I k _M ) ⊕ E ^b ₋ (I k _M ), given by K 7→ (Q k _{t>0} ⊗K)⊕ (Q k ⁺ _{t60} ⊗K). Denote by L ^{+ E} _± and R ^E _± the left and right adjoint, respectively, of the quotient functor Q : D ^b (I k _M×R

) − → E ^b _± (I k M ).

There are embeddings

 ^± _M : D ^b (I k _M )  E ^b ± (I k _M ), F 7→ Q(k _{±t>0} ⊗ π _M ⁻¹ F ),

and one sets  _M (F ) :=  ⁺ _M (F ) ⊕  ⁻ _M (F ) ∈ E ^b (I k _M ). Note that  _M (F ) ' Q(k {t=0} ⊗ π _M ⁻¹ F ).

2.5. Stable objects. Let M be a bordered space. Set k {t0} := “lim −→ ”

a→+∞

k _{t>a} ∈ D ^b (I k _M×R

),

k ^E _M := Q k _{t0} ∈ E ^b ₊ (I k _M ).

An object K ∈ E ^b ₊ (I k _M ) is called stable if k ^E _M ⊗ K ∼ ⁺ −→ K. We denote by E ^b _st (I k _M ) the full subcategory of E ^b ₊ (I k _M ) of stable objects. The embed- ding E ^b _st (I k _M )  E ^b + (I k _M ) has a left adjoint k ^E _M ⊗ ∗, as well as a right ⁺ adjoint RIhom ⁺ (k ^E _M , ∗).

There is an embedding

e _M : D ^b (I k _M )  E ^b st (I k _M ), F 7→ k ^E _M ⊗  ⁺ _M (F ) ' Q(k {t0} ⊗ π _M ⁻¹ F ).

Notation 2.3. Let S ⊂ T be locally closed subsets of M.

(i) For continuous maps ϕ ± : T − → R such that −∞ 6 ϕ ⁻ 6 ϕ + <

+∞, set

E ^ϕ _S|M

^Bϕ

:= Q k {x∈S, −ϕ

(x)6t<−ϕ

(x)} ∈ E ^b ₊ (I k M ), E ^ϕ _S|M

^Bϕ

:= k ^E _M ⊗ E ^{+ ϕ} _S|M

^Bϕ

∈ E ^b _st (I k _M ),

where we write for short

{x ∈ S, − ϕ + (x) 6 t < −ϕ ⁻ (x)}

:= {(x, t) ∈

◦

M × R ; x ∈ S, −ϕ + (x) 6 t < −ϕ ⁻ (x)}, with < the total order on R. If S = T , we also write for short {−ϕ ₊ (x) 6 t < −ϕ − (x)} := {x ∈ T, −ϕ ₊ (x) 6 t < −ϕ − (x)}.

(ii) For a continuous map ϕ : T − → R, consider the object of E ^b + (I k _M ) E ^ϕ _S|M := Q k {x∈S, t+ϕ(x)>0} ∈ E ^b ₊ (I k _M ),

E ^ϕ S|M := k ^E _M ⊗ E ^{+ ϕ} _S|M ∈ E ^b _st (I k M ).

where we write for short

{x ∈ S, t + ϕ(x) > 0} = {(x, t) ∈ M × R ; x ∈ S, t + ϕ(x) > 0}.

If S = T , we also write for short

{t + ϕ(x) > 0} := {x ∈ T, t + ϕ(x) > 0}.

Note that one has E ^ϕ _S|M ' E ^ϕB−∞ _S|M , and that there is a short exact sequence

0 − → E ^ϕ _S|M

^Bϕ

− → E ^ϕ _S|M

− → E ^ϕ _S|M

− → 0 in the heart of E ^b (I k _M ) for the natural t-structure.

2.6. Constructible objects. A subanalytic bordered space is a bor- dered space M such that

◦

M is an open subanalytic subset of the suban- alytic space

∨

M. A morphism f : M − → N of subanalytic bordered spaces is a morphism of bordered spaces such that Γ f is subanalytic in

∨

M ×

∨

N.

By definition, a subset Z of M is subanalytic if it is subanalytic in M.

Let M be a subanalytic bordered space.

Denote by D ^b _w-R-c (k M ) the full subcategory of D ^b (k

M ) whose objects F are such that Rk _M∗ F (or equivalently, Rk _{M !} F ) is weakly R-constructible, for k _M : M −

→ M the embedding. We similarly define the category D

^b

R-c (k _M ) of R-constructible sheaves.

Denote by E ^b _w-R-c (I k _M ) the strictly full subcategory of E ^b (I k _M ) whose objects K are such that for any relatively compact open subanalytic subset U of M, one has

π _M ⁻¹ k _U ⊗ K ' k ^E _M ⊗ Q F ⁺

for some F ∈ D ^b _w-R-c (k _M×R

). In particular, K belongs to E ^b _st (I k M ).

We similarly define the category E ^b _R-c (I k M ) of R-constructible enhanced ind-sheaves.

3. Sheafification

In this section, we discuss what we call here ind-sheafification and sheafification functor, and prove some of their functorial properties. Con- cerning constructibility, we use a fundamental result from [10, §6].

3.1. Associated ind-sheaf. Let M be a bordered space. Let i ₀ : M − → M × R ^∞ be the embedding x 7→ (x, 0).

Definition 3.1. Let K ∈ E ^b (I k M ) and take F ∈ D ^b (I k _M×R

) such that K ' Q F . We set

Ish _M (K) :=R Ihom ^E (Q k _{t=0} , K)

' Rπ _M∗ R Ihom (k _{t>0} ⊕ k _{t60} , F ) ' Rπ _M∗ R Ihom (k _{t=0} , R ^E K) ' Rπ _{M !!} R Ihom (k _{t=0} , R ^E K) ' i ₀ ^! R ^E K ∈ D ^b (I k M )

(see [1, Lemma 4.5.16]), and call it the associated ind-sheaf (in the de- rived sense) to K on M. We will write for short Ish = Ish M , if there is no fear of confusion.

Note that one has

Ish(K) ' R Ihom ^E (Q k _{t>0} , K) for K ∈ E ^b ₊ (I k _M ), Ish(K) ' R Ihom ^E (k ^E _M , K) for K ∈ E ^b _st (I k _M ).

Lemma 3.2. The following are pairs of adjoint functors (i) (, Ish) : D ^b (I k _M ) ^{ //} E ^b (I k _M )

Ish

oo ,

(ii) ( ⁺ , Ish) : D ^b (I k _M ) ^{// }

+

// E ^b ₊ (I k _M )

Ish

oo ,

(iii) (e, Ish) : D ^b (I k M ) ^{// e //} E ^b _st (I k M )

Ish

oo .

Proof. (i) For F ∈ D ^b (I k M ) and K ∈ E ^b (I k M ) one has

Hom _E

(I k

) ((F ), K) ' Hom _D

(I k

) (π ⁻¹ F ⊗ k {t=0} , R ^E K) ' Hom _D

(I k

) (F, Rπ ∗ R Ihom (k {t=0} , R ^E K)) ' Hom _D

(I k

) (F, Ish(K)).

(ii) and (iii) follow from (i), noticing that there are pairs of adjoint func- tors (∗ ⊗ Q k ⁺ _{t>0} , ι) and (∗ ⊗ k ^{+ E} _M , ι):

E ^b (I k _M )

∗ ⊗Q k

// E ^b ₊ (I k _M )

oo ι

oo

∗ ⊗k

// E ^b _st (I k _M ).

oo ι

oo

Here we denote by ι the natural embeddings.  Lemma 3.3. Let f : M − → N be a morphism of bordered spaces.

(i) There are a natural morphism and a natural isomorphism of func- tors

f ⁻¹ Ish _N − → Ish _M Ef ⁻¹ , f ^! Ish _N ' Ish _M Ef ^! ,

and the above morphism is an isomorphism if f is borderly sub- mersive.

(ii) There are a natural morphism and a natural isomorphism of func- tors

Rf _!! Ish _M − → Ish _N Ef _!! , Rf ∗ Ish _M ' Ish _N Ef ∗ ,

and the above morphism is an isomorphism if f is proper.

(iii) For K ∈ E ^b (I k _M ) and L ∈ E ^b (I k _N ), there is a natural morphism Ish(K)  Ish(L) − → Ish(K  L). ⁺

Proof. Recall that one sets f _R := f × id _R

: M × R ^∞ − → N × R ^∞ . (i) Let L ∈ E ^b (I k _N ) and set G := R ^E L ∈ D ^b (I k _N×R

).

(i-a) One has

f ⁻¹ Ish _N (L) ' f ⁻¹ Rπ _{N !!} R Ihom (k {t=0} , G) ' Rπ _{M !!} f ⁻¹

R R Ihom (k {t=0} , G)

−

→

(∗) Rπ M∗ R Ihom (k {t=0} , f _R ⁻¹ G)

−

→

(∗∗) Rπ M∗ R Ihom (k {t=0} , R ^E Ef ⁻¹ L) ' Ish _M (Ef ⁻¹ L).

Here, (∗) follows from [1, Proposition 3.3.13], and (∗∗) from Lemma A.4.

If f is borderly submersive, then (∗) is an isomorphism by [1, Propo- sition 3.3.19] and (∗∗) is an isomorphism by Lemma A.4.

(i-b) Recall that f _R ^! G ' R ^E (Ef ^! L). One has

f ^! Ish _N (L) = f ^! Rπ _N∗ R Ihom (k _{t=0} , G) ' Rπ _M∗ f _R ^! R Ihom (k {t=0} , G) ' Rπ _M∗ R Ihom (k {t=0} , f ^!

R G) ' Rπ _M∗ R Ihom (k {t=0} , R ^E (Ef ^! L)) ' Ish _M (Ef ^! L).

(ii) Let K ∈ E ^b (I k M ) and set F := R ^E K ∈ D ^b (I k _M×R

).

(ii-a) One has

Ish _N (Ef _!! K) = Rπ _{N !!} R Ihom (k {t=0} , R ^E Ef _!! K)

← − Rπ _{N !!} R Ihom (k _{t=0} , Rf _{R !!} F ) '

(∗)

Rπ _{N !!} Rf _{R !!} R Ihom (k _{t=0} , F )

←− Rf ∼ _!! Rπ _{M !!} R Ihom (k _{t=0} , F )

= Rf !! (Ish M (K)).

Here (∗) follows from [9, Lemma 5.2.8].

(ii-b) Since R ^E (Ef ∗ K) ' Rf _R∗ F , one has

Ish _N (Ef ∗ K) ' Rπ _N∗ R Ihom (k _{t=0} , Rf _R∗ F ) ' Rπ _M∗ Rf _R∗ R Ihom (k _{t=0} , F ) ' Rf ∗ Rπ M∗ R Ihom (k {t=0} , F ).

If f is proper, f _! ' f ∗ .

(iii) Set F := R ^E K ∈ D ^b (I k _M×R

) and G := R ^E L ∈ D ^b (I k _N×R

). Recall that F  G := Rm ^{+ !!} (F  G), where

m : M × R ∞ × N × R ∞ − → M × N × R ∞ (x, t ₁ , y, t ₂ ) 7→ (x, y, t ₁ + t ₂ ).

Then, one has Ish(K)  Ish(L)

' Rπ _M∗ R Ihom (k {t

=0} , F )  Rπ N∗ R Ihom (k {t

=0} , G)

−

→ R(π _M × π _N ) ∗ R Ihom (k {t

=0} , F )  R Ihom (k ^{t

=0} , G)

−

→ Rπ _M×N∗ Rm ∗ R Ihom (k {t

=0}  k ^{t

=0} , F  G)

−

→ Rπ _M×N∗ R Ihom Rm _!! (k {t

=0}  k ^{t

=0} ), Rm _!! (F  G)
' Rπ _M×N∗ R Ihom (k {t=0} , F  G), ⁺

One concludes using the natural morphism F  G − ⁺ → R ^E (K  L). ⁺ 

3.2. Associated sheaf. Let M be a bordered space.

Definition 3.4. Let K ∈ E ^b (I k _M ).

(i) We set

sh _M (K) := RHom ^E (Q k _{t=0} , K)

= α _M Ish _M (K) ∈ D ^b (k

M ),

and call it the associated sheaf (in the derived sense) to K on

◦

M.

We will write for short sh = sh M , if there is no fear of confusion.

(ii) We say that K is of sheaf type (in the derived sense) if it is in the essential image of

e _M ι _M : D ^b (k

M )  E ^b (I k _M ), One has

sh M (K) ' RHom ^E (Q k _{t>0} , K), for K ∈ E ^b ₊ (I k M ), sh _M (K) ' RHom ^E (k ^E _M , K), for K ∈ E ^b _st (I k _M ).

(3.1)

Lemma 3.5. One has sh _M ' sh

M Ei ⁻¹ _M .

Proof. Recall that i ⁻¹ _M ' i _M ^! . Using Lemma 2.1 (ii), one has α _M Ish _M ' α

M i _M ^! R Ihom ^E (Q k {t=0} , K)

= α

M i _M ^! Rπ _M∗ R Ihom (k {t=0} , R ^E K) ' α

M Rπ

M∗ i _M×R ^!

R Ihom (k {t=0} , R ^E K) ' α

M Rπ

M∗ R Ihom (k {t=0} , i _M×R ^!

R ^E K) ' α

M Rπ

M∗ R Ihom (k {t=0} , R ^E Ei _M ^! K) ' α

M R Ihom ^E (Q k _{t=0} , Ei _M ^! K).

 Let M be a bordered space, and consider the natural morphisms of good spaces

◦

M × R − → ^k M × R

− → ^π M.

We write t for points of R := R ∪ {−∞, +∞}.

An important tool in this framework is given by

Proposition 3.6 ([10, Corollary 6.6.6]). Let M be a bordered space.

Then, for F ∈ D ^b (k

M×R ) one has

sh _M (k ^E _M ⊗ Q F ) ' Rπ ⁺ ∗ (k {−∞<t6+∞} ⊗ Rk ∗ F ).

Consider the natural morphism j : M × R ^∞ − → M × R, and the em-

beddings i _±∞ : M − → M × R, x 7→ (x, ±∞). Using the above proposition

and [1, Proposition 4.3.10, Lemma 4.3.13], we get

Corollary 3.7. Let M be a bordered space. Then, for F ∈ D ^b (k

M×R ) one has

sh M (k ^E _M ⊗ Q F ) ' i ^{+ −1} _+∞ Rj ∗ L ^E ₊ Q F ' i ⁻¹ _−∞ Rj ∗ R ^E ₊ Q F [−1]

' Rπ _∗ L ^E ₊ Q F ' Rπ _! R ^E ₊ Q F.

Consider the functors

(3.2) D ^b (k

M ) ^//

e

ι

// E ^b (I k _M ).

sh

oo

As explained in the Introduction, (e _M ι _M , sh _M ) is not an adjoint pair of functors in general.

Proposition 3.8. Consider the functors (3.2).

(i) sh _M is a left quasi-inverse to e _M ι _M .

(ii) The property of being of sheaf type is local ⁵ on M, and K ∈ E ^b (I k _M ) is of sheaf type if and only if K ' e _M ι _M sh _M (K)
.

Proof. (i) By Proposition 3.6, for L ∈ D ^b (k

M ), one has sh _M e _M ι _M (L) ' sh _M k ^E _M ⊗ Q(k ⁺ {t=0} ⊗ π ⁻¹ ι _M L)

' Rπ ∗ k {−∞<t6+∞} ⊗ k _{t=0} ⊗ π ⁻¹ L
' Rπ _! k _{t=0} ⊗ π ⁻¹ L

' Rπ _! k _{t=0}
⊗ L ' L.

(ii) follows from (i). 

By Lemmas 2.2 and 3.3, one gets

Lemma 3.9. Let f : M − → N be a morphism of bordered spaces.

(i) There are natural morphisms of functors

◦

f ⁻¹ sh N − → sh M Ef ⁻¹ , sh M Ef ^! − → f

^! sh N ., which are isomorphisms if f is borderly submersive.

(ii) There are natural morphisms of functors R

◦

f _! sh _M − → sh _N Ef _!! , sh _N Ef ∗ − → R f

_∗ sh _M .

The first morphism is an isomorphism if f is proper. The sec- ond morphism is an isomorphism if f is self-cartesian, and in particular if f is proper.

5 Saying that a property P(M) is local on M means the following. For any open

covering {U _i } _i∈I of M, P(M) is true if and only if P (U _i ) _∞
is true for any i ∈ I.

(iii) For K ∈ E ^b (I k M ) and L ∈ E ^b (I k N ), there is a natural morphism sh(K)  sh(L) − → sh(K  L). ⁺

Example 3.10. Let M = R x , U = {x > 0}. By Corollary 3.7 one has R ^E E ^1/x _{U |M} ' k {x>0, xt<−1} [1], R ^E E ^−1/x _{U |M} ' k {x>0, xt<1} [1], sh(E ^1/x _{U |M} ) ' k {x>0} , sh(E ^−1/x _{U |M} ) ' k _{x>0} .

Note that, denoting by i : {0} − → M the embedding, one has

i ^! (sh(E ^1/x _{U |M} )) 6' sh(Ei ^! (E ^1/x _{U |M} )), i ⁻¹ (sh(E ^−1/x _{U |M} )) 6' sh(Ei ⁻¹ (E ^−1/x _{U |M} )).

In fact, on one hand one has i ^! (sh(E ^1/x _{U |M} )) ' k[−1] and Ei ^! (E ^1/x _{U |M} ) ' 0, and on the other hand one has i ⁻¹ (sh(E ^−1/x _{U |M} )) ' k and Ei ⁻¹ (E ^−1/x _{U |M} ) ' 0.

Note also that sh is not conservative, since sh(E ^2/xB1/x U |X ) ' 0.

Example 3.11. Let X ⊂ C ^z be an open neighborhood of the origin, and set

•

X = X \ {0}. The real oriented blow-up p : X ₀ ^rb − → X with cen- ter the origin is defined by X ₀ ^rb := {(r, w) ∈ R >0 × C ; |w| = 1, rw ∈ X}, p(r, w) = rw. Denote by S ₀ X = {r = 0} the exceptional divisor.

Let f ∈ O _X (∗0) be a meromorphic function with pole order d > 0 at the origin. With the identification

•

X ' {r > 0} ⊂ X ₀ ^rb , the set I :=

S ₀ X \ {z ∈

•

X ; Re f (z) > 0} is the disjoint union of d open non-empty intervals. Here {·} is the closure in X ₀ ^rb . Then, recalling Notation 2.3,

sh(E ^{Re f} _X|X

) ' sh(Ep ∗ E ^{Re f ◦p} _X|X

) ' Rp ∗ sh(E ^{Re f ◦p} _X|X

) ' Rp _! k _{It X}

.

Recall that, for k = C, the Riemann-Hilbert correspondence of [ 1] as- sociates the meromorphic connection d − df with E ^{Re f} _X|X

by the functor DR ^E _X .

3.3. (Weak-) constructibility. An important consequence of Proposi- tion 3.6 is

Proposition 3.12 ([10, Theorem 6.6.4]). Let M be a subanalytic bordered space. The functor sh _M induces functors

sh _M : E ^b _w-R-c (I k _M ) − → D ^b _w-R-c (k _M ), sh _M : E ^b

R-c (I k _M ) − → D ^b

R-c (k _M ).

Proposition 3.13. Let M be a subanalytic bordered space. For K ∈ E ^b _R-c (I k M ) there is a natural isomorphism

sh _M (D ^E _M K) ∼ −→ D

M (sh _M K).

Proof. Recall that sh _M ' sh

M Ei ⁻¹ _M and Ei ⁻¹ _M ' Ei _M ^! . Since Ei ⁻¹ _M D ^E _M ' D ^E

M Ei ⁻¹ _M , we may assume that M = M = M is a subanalytic space.

(i) Let us construct a natural morphism

sh(D ^E K) − → D(shK).

By adjunction, it is enough to construct a natural morphism sh(D ^E K) ⊗ sh(K) − → ω _M .

Note that we have a morphism

D ^E K ⊗ K − ⁺ → ω _M ^E .

Let δ : M − → M × M be the diagonal embedding, so that D ^E K ⊗ K ' ⁺ Eδ ⁻¹ (D ^E K  K). There are natural morphisms ⁺

sh(D ^E K) ⊗ sh(K) ' δ ⁻¹ sh(D ^E K)  sh(K)

−

→

(∗) δ ⁻¹ sh(D ^E K  K) ⁺

−

→

(∗∗) sh Eδ ⁻¹ (D ^E K  K) ⁺

−

→ sh(ω _M ^E ) ' ω _M ,

where (∗) is due to Lemma 3.9 (iii), and (∗∗) is due to Lemma 3.9 (i).

(ii) By (i), the problem is local on M . Hence, we may assume that K ' k ^E _M ⊗ Q F for F ∈ D ^{+ b}

R-c (k _{M ×R}

). Considering the morphisms k : M × R ^∞ −→ M × (R ∪ {±∞}, R) ⁱ

−→ M × R. ^j

Since

k {−∞<t6+∞} ⊗ Rk ∗ F ' Rj ⁺ _! Ri ⁺ _∗ F ' Rj _∗ ⁻ Ri ⁻ _! F, Proposition 3.6 gives

sh _M (K) ' Rπ ∗ Rj _! ⁺ Ri ⁺ _∗ F ' Rπ _∗ Rj _∗ ⁻ Ri ⁻ _! F.

By [1, Proposition 4.8.3] one has D ^E _M (k ^E _M ⊗ Q F ) ' k ^{+ E} _M ⊗ Q a ^{+ −1} D _{M ×R}

F , where a : M × R ^∞ − → M × R ∞ is given by a(x, t) = (x, −t). Then, one has

sh _M (D ^E _M K) ' sh _M (k ^E _M ⊗ Q a ^{+ −1} D _{M ×R}

F )

' Rπ ∗ Rj _! ⁺ Ri ⁺ _∗ a ⁻¹ D _{M ×R}

F

' Rπ ∗ Rj _! ⁻ Ri ⁻ _∗ D _{M ×R}

F

' D _M (Rπ ∗ Rj _∗ ⁻ Ri ⁻ _! F )

' D _M (sh _M (K)).

 Lemma 3.14. Let M and N be bordered spaces. Let F ∈ D ^b _R-c (k _M ) and L ∈ E ^b (I k _N ). Then

sh((F )  L) ' F  sh(L). ⁺ Proof. For G := R ^E L ∈ D ^b (I k _N×R

), one has

sh((F )  L) ' α ^{+ M×N} Rπ _M×N∗ R Ihom (k {t=0} , F  G) '

(a) α _M×N Rπ _M×N∗ F  R Ihom (k ^{t=0} , G)
'

(b) α _M×N F  Rπ N∗ R Ihom (k {t=0} , G)
' F  α ^N Rπ N∗ R Ihom (k {t=0} , G),

where (a) follows from [1, Corollary 2.3.5] and (b) follows from Proposi-

tion A.2 in Appendix. 

4. Stalk formula

As we saw in the previous section, sheafification does not commute with the pull-back by a closed embedding, in general. We provide here a stalk formula for the sheafification of such a pull-back, using results from Appendix B.

4.1. Restriction and stalk formula. Let M be a subanalytic bordered space. Recall Notation 2.3.

Let N ⊂ M be a closed subanalytic subset, denote by i : N ∞ → M the − embedding. To illustrate the difference between sh Ei ⁻¹ and i ⁻¹ sh note that on one hand, by [2, Lemma 2.4.1], for K ∈ E ^b ₊ (I k M ) and y 0 ∈ N one has ⁶

i ⁻¹ sh(K)

y

' sh(K) _y

' lim

U 3y −→

RHom ^E (E ⁰ _{U |M} , K), where U runs over the open neighborhoods of y 0 in

◦

M. On the other hand,

Proposition 4.1. Let ϕ : M − → R ^∞ be a morphism of subanalytic bor- dered spaces, set N := ϕ

⁻¹ (0) ⊂ M, and denote by i : N ∞ − → M the embedding. For y ₀ ∈ N and K ∈ E ^b _w-R-c (I k _M ) one has

sh(Ei ⁻¹ K) _y

' lim −→

U 3y

δ,ε − → 0+

RHom ^E (E ^0B−δ|

ϕ(x)|

U |M , K),

6 Recall from [2 , §2.1] that, for any c, d ∈ Z, small filtrant inductive limits exist in

D ^[c,d] (k), the full subcategory of D ^b (k) whose objects V satisfy H ^j (V ) = 0 for j < c

or j > d. That is, uniformly bounded small filtrant inductive limits exist in D ^b (k).

where U runs over the open neighborhoods of y ₀ in M. Here, we set

−δ| ϕ(x)|

^−ε = −∞ for ϕ(x) = 0.

More generally, for T ⊂ N a compact subset one has (4.1) RΓ (T ; sh(Ei ⁻¹ K)) ' lim −→

U ⊃T δ,ε − → 0+

RHom ^E (E ^0B−δ|

ϕ(x)|

U |M , K),

where U runs over the open neighborhoods of T in

◦

M.

Proof. Let us prove the isomorphism (4.1).

Since T ⊂ N ⊂

◦

M is compact, we may assume that M =

◦

M =: M is a subanalytic space.

(i) On the right hand side of (4.1), we may assume that U runs over the open subanalytic neighborhoods of T in M . Up to shrinking M around T , we can assume that there exists F ∈ D ^b _w-R-c (k _{M ×R}

) such that K ' k ^E _M ⊗ Q F . ⁺

For c ∈ R, and U an open relatively compact subanalytic subset of M containing T , set

U _c,δ,ε := {(x, t) ∈ U × R ; t + c < δ|ϕ(x)| ^−ε }.

Note that L ^E E cBc−δ|ϕ(x)|

U |M ' k _U

⊗ k _{t>−c} . Then, one has RHom ^E (E ^{0B−δ|ϕ(x)|} _{U |M}

, K) ' lim −→

c − → +∞

RHom ^E (Q k _{t>−c} ⊗ E ^{+ 0B−δ|ϕ(x)|} _{U |M}

, Q F ) ' lim −→

c − → +∞

Hom (L ^E E cBc−δ|ϕ(x)|

U |M , F ) ' lim −→

c − → +∞

Hom (k _U

⊗ k _{t>−c} , F ) '

(∗) lim −→

c − → +∞

Hom (k _U

, k {t>−c} ⊗ F ) ' lim −→

c − → +∞

RΓ U _c,δ,ε ; k _{t>−c} ⊗ F ) ' lim

c − → −→ +∞

RΓ U _c,δ,ε ∩ {t > −c}; k {t>−c} ⊗ F ).

Here (∗) follows from the same argument used in the proof of the second isomorphism in [10, (6.6.2)].

(ii) Let us deal with the left hand side of (4.1). Consider the natural maps

N × R ^{∞ i}

^//

π



M × R ^∞

π



N ^{i //} M

and set, for S = M, N ,

k _S×{t>∗} := “lim −→ ”

c − → +∞

k S×{t>−c} ∈ D ^b (I k _S×R

).

Noticing that Ei ⁻¹ K ' k ^E _N ⊗ Q i ^{+ −1}

R F , by [10, Proposition 6.6.5] one has sh(Ei ⁻¹ K) ' α _N Rπ _{N ∗} k N ×{t>∗} ⊗ i ⁻¹

R F
' α _N Rπ _{N ∗} i ⁻¹

R k M ×{t>∗} ⊗ F
.

Hence

RΓ (T ; sh(Ei ⁻¹ K)) ' lim −→

V

RΓ (V ; sh(Ei ⁻¹ K)
' lim −→

c,V

RΓ (V ; Rπ _{N ∗} i ⁻¹

R (k M ×{t>−c} ⊗ F )
' lim −→

c,V

RΓ V × R; i ⁻¹ _R (k M ×{t>−c} ⊗ F )
' lim −→

c,V

RΓ V × {t > −c}; i ⁻¹ _R (k M ×{t>−c} ⊗ F )
' lim −→

c,V,W

RΓ W ; k M ×{t>−c} ⊗ F

where c − → +∞, V runs over the system of open relatively compact subanalytic neighborhoods of T in N , and W = W _c,V runs over the system of open subanalytic subsets of M × {t ∈ R; +∞ > t > −c}, such that W ⊃ V × {t ∈ R; t > −c}. Here, the last isomorphism follows from Corollary B.3.

(iii) For c ∈ R consider the following inductive systems: I c is the set of tuples (U, δ, ε) as in (i); J _c is the set of tuples (V, W ) as in (ii). We are left to show the cofinality of the functor φ : I _c − → J _c , (U, δ, ε) 7→

(U ∩ N, U _c,δ,ε ∩ {t > c}).

Given (V, W ) ∈ J c , we look for (U, δ, ε) ∈ I c such that U ∩ N ⊂ V and U c,δ,ε ∩ {t > −c} ⊂ W . Let U be a subanalytic relatively compact open neighborhood of T in M such that U ∩ N ⊂ V . With notations as in Lemma B.1 , set X = M × {t ∈ R | t > −c}, W = W , T = U × {t ∈ R | t > −c}, f(x, t) = ϕ(x) and

g(x, t) = (t + c + 1) ⁻¹ .

Note that g(x, +∞) = 0. Since (B.1) is satisfied, Lemma B.1 (ii) provides C > 0 and n ∈ Z >0 such that

{(x, t) ∈ U × R ; t > −c, Cg(x, t) ⁿ > |ϕ(x)|} ⊂ W.

Then

{(x, t) ∈ U × R ; t > −c, C(t + c + 1) ⁻ⁿ > |ϕ(x)|} ⊂ W.

One concludes by noticing that the set on the left hand side contains

U _c,δ,ε ∩ {t > −c} for δ = C ^1/n and ε = 1/n. 

5. Specialization and microlocalization

Using results from the previous section, we establish here a stalk for- mula for the natural enhancement of Sato’s specialization and microlo- calization functors, as introduced in [4].

5.1. Real oriented blow-up transforms. Let M be a real analytic manifold and N ⊂ M a closed submanifold. Denote by S _N M the sphere normal bundle. Consider the real oriented blow-up M _N ^rb of M with center N , which enters the commutative diagram with cartesian square

(5.1) S _N M ^{ i //}

σ 

M _N ^rb

p 

(M \ N ) Jj ∞ j

ww

j ? _

oo

N ^

i

//



M.

Note that (M \ N ) ∞ := (M \ N, M ) ' (M \ N, M _N ^rb ).

Recall the blow-up transform of [4, §4.4]

Eν ^rb : E ^b (I k M ) − → E ^b (I k S

M ), K 7→ Ei ⁻¹ Ej ∗ Ej _N ⁻¹ K.

A sectorial neighborhood of θ ∈ S _N M is an open subset U ⊂ M \ N such that S _N M ∪ j(U ) is a neighborhood of θ in M _N ^rb . We write U 3 θ to

indicate that U is a sectorial neighborhood of θ. We say that U ⊂ M \ N is a sectorial neighborhood of Z ⊂ S N M , and we write U ⊃ Z, if U is a

sectorial neighborhood of each θ ∈ Z.

Lemma 5.1. Let ϕ : M − → R be a subanalytic continuous map such that N = ϕ ⁻¹ (0). Let K ∈ E ^b _w-R-c (I k _M ). For θ ₀ ∈ S _N M , one has

sh Eν _N ^rb (K)

θ

' lim −→

δ,ε,U

RHom ^E (E ^{0B−δ|ϕ(x)|} _{U |M}

, K),

where δ, ε − → 0+ and U 3 θ

₀ . More generally, if Z ⊂ S _N M is a closed subset one has

RΓ Z; sh(Eν _N ^rb (K))
' lim −→

δ,ε,U

RHom ^E (E ^{0B−δ|ϕ(x)|} _{U |M}

, K) where δ, ε − → 0+ and U ⊃ Z.

Proof. Let us prove the last statement.

Note that in M _N ^rb one has S _N M = (ϕ ◦ p) ⁻¹ (0). Hence, by Proposi- tion 4.1,

RΓ Z; sh(Eν _N ^rb (K))
' lim −→

δ,ε, e U

RHom ^E (E ^{0B−δ|ϕ(p(˜ x))|}

U |M e

, Ej ∗ Ej _N ⁻¹ K),

where e U ⊂ M _N ^rb runs over the neighborhoods of i(Z). Then RΓ Z; sh(Eν _N ^rb (K))
' lim −→

δ,ε, e U

RHom ^E (Ej _{N !!} Ej ⁻¹ E ^{0B−δ|ϕ(p(˜ x))|}

U |M e

, K) ' lim −→

δ,ε, e U

RHom ^E (Ej _{N !!} E ^{0B−δ|ϕ(x)|}

j

( e U )|(M \N )

, K) ' lim −→

δ,ε, e U

RHom ^E (E ^{0B−δ|ϕ(x)|}

j

(j

( e U ))|M , K).

One concludes by noticing that U ⊃ Z if and only if U = j

_N (j ⁻¹ ( e U )) for

some neighborhood e U of i(Z) in M _N ^rb . 

5.2. Sheafification on vector bundles. Recall from [4, §2.2] that any morphism p : M − → S, from a good space to a bordered space, admits a bordered compactification p ∞ : M ∞ − → S such that (M ∞ ) ^◦ = M and p ∞

is semiproper. Moreover, such a bordered compactification is unique up to isomorphism.

Let τ : V − → N be a vector bundle. Denote by V ∞ its bordered com- pactification, and by o : N − → V the zero section.

The natural action of R >0 on V extends to an action of the bordered group ⁷ (R ^× >0 ) ∞ :=(R ^>0 , R) on V ^∞ . Denote by E ^b

(R

)

(I k V

) the category of conic enhanced ind-sheaves on V ∞ (see [4, §4.1]).

Lemma 5.2. For K ∈ E ^b

(R

)

(I k V

), one has

o ⁻¹ sh(K) ' sh(Eo ⁻¹ K), o ^! sh(K) ' sh(Eo ^! K).

Proof. We shall prove only the first isomorphism since the proof of the second is similar.

With the identification N ' o(N ) ⊂ V , set

•

V = V \ N . Consider the commutative diagram, associated with the real oriented blow-up of V with center N ,

S N V



  // (V _N ^rb ) ∞

 p

˜

γ // S N V

σ

'' N



  ^o // V ∞

τ

33

(

•

V ) ∞ j ? _

oo

γ

OO

τ

//

˜

hh 

N.

Here (

•

V ) ∞ denotes the bordered compactification of

•

V − → V ∞ . Consider the distinguished triangle

Ej _!! Ej ⁻¹ K − → K − → Eo ∗ Eo ⁻¹ K −→ . ⁺¹

7 a group object in the category of bordered spaces

One has

o ⁻¹ sh(Eo ∗ Eo ⁻¹ K) '

(∗) o ⁻¹ o ∗ sh(Eo ⁻¹ K) ' sh(Eo ⁻¹ K).

where (∗) holds since o is proper. Hence, we can assume K ' Ej !! Ej ⁻¹ K

and, since Eo ⁻¹ K ' 0, we have to show o ⁻¹ sh(K) ' 0.

Recall that Ej ⁻¹ K ' Eγ ⁻¹ K ^sph for K ^sph := Eγ ∗ Ej ⁻¹ K. Then one has K ' Ej _!! Eγ ⁻¹ K ^sph

' Ep _∗ E˜  _!! E˜  ⁻¹ E˜ γ ⁻¹ K ^sph ' Ep ∗ k _V

N

\S

V ⊗ E˜ γ ⁻¹ K ^sph
.

Thus, recalling that o ⁻¹ sh(K) ' Rτ _∗ sh(K) since sh(K) is conic, o ⁻¹ sh(K) ' Rτ ∗ sh Ep ∗ (k _V

N

\S

V ⊗ E˜ γ ⁻¹ K ^sph )
'

(∗) Rτ ∗ Rp ∗ sh k _V

N

\S

V ⊗ E˜ γ ⁻¹ K ^sph
' Rσ _∗ R˜ γ _∗ sh k _V

N

\S

V ⊗ E˜ γ ⁻¹ K ^sph
, where (∗) holds since p is proper. It is then enough to show

R˜ γ ∗ sh(k _V

N

\S

V ⊗ E˜ γ ⁻¹ K ^sph ) ' 0.

Since ˜ γ is borderly submersive and ˜ γ ^! k _S

_V ' k _V

N

\S

V , one has by (2.4) k _V

N

\S

V ⊗ E˜ γ ⁻¹ K ^sph ' E˜ γ ^! K ^sph . Hence one obtain

R˜ γ ∗ sh(k _V

N

\S

V ⊗ E˜ γ ⁻¹ K ^sph ) ' R˜ γ ∗ sh(E ˜ γ ^! K ^sph ) '

(∗) R˜ γ ∗ γ ˜ ^! sh(K ^sph ) 'R˜ γ ∗ RHom k _V

N

, ˜ γ ^! sh(K ^sph )
'RHom R˜ γ _! k _V

N

, sh(K ^sph )
.

where (∗) follows from Lemma 3.9 (i). Then the desired result follows from R˜ γ ! k _V

N

' 0. 

5.3. Specialization and microlocalization. Let us recall from [4] the natural enhancement of Sato’s specialization and microlocalization func- tors.

Let M be a real analytic manifold and N ⊂ M a closed submanifold.

Consider the normal and conormal bundles

T N M ^{τ //} N ^{oo $} T _N ^∗ M,

and denote by (T N M ) ∞ and (T _N ^∗ M ) ∞ the bordered compactification of τ and $, respectively.

Denote by (p, s) : M _N ^nd − → M × R the normal deformation of M along N (see [8, §4.1]). Setting Ω := s ⁻¹ (R >0 ), one has morphisms

(5.2) (T _N M ) ∞   ⁱ // (M _N ^nd ) ∞ oo ^j ? _ Ω ∞ ^p

// M ,

where (M _N ^nd ) ∞ is the bordered compactification of p, and p _Ω = p| _Ω . The enhanced Sato’s specialization functor is defined by

Eν _N : E ^b (I k _M ) − → E ^b

(R

)

(I k _(T

_{M )}

), K 7→ Ei ⁻¹ Ej _∗ Ep ⁻¹ _Ω K.

Sato’s Fourier transform have natural enhancements (see e.g. [4, §5.2]) (·) ^∧ : E ^b ₊ (I k _(T

_{M )}

) − → E ^b ₊ (I k _(T

N

M )

),

L (·) : E ^b ₊ (I k _(T

_{M )}

) − → E ^b ₊ (I k _(T

N

M )

),

and we denote by (·) ^∨ and ^L (·) their respective quasi-inverses. Recall that (·) ^∧ and (·) ^∨ take values in conic objects, and that ^L (·) and ^L (·) send conic objects to conic objects.

Finally, Sato’s microlocalization functor have a natural enhancement Eµ _N : E ^b ₊ (I k _M ) − → E ^b ₊ (I k _(T

N

M )

) ∩ E ^b

(R

)

(I k _(T

N

M )

),

defined by Eµ _N (K) := ^L Eν _N (K). Recall that Eµ _N (K) ' Eν _N (K) ^∧ by [4, Proposition 5.3].

Consider the natural morphisms

S N M (

•

T N M ) ∞

oo γ ^u // (T N M ) ∞ ^{oo o} N ,

where

•

T N M is the complement of the zero-section, and o is the embed- ding of the zero-section. Here (

•

T _N M ) ∞ denotes the bordered compacti- fication of

•

T N M − → (T N M ) ∞ . Recall that one has Eγ ⁻¹ ◦ Eν _N ^rb ' Eu ⁻¹ ◦ Eν N .

Recall from [8, §4.1] that the normal cone C _N (S) ⊂ T _N M to S ⊂ M along N is defined by C N (S) := T _N M ∩ p ⁻¹ _Ω (S), where (·) denotes the closure in M _N ^nd .

Proposition 5.3. Let ϕ : M − → R be a continuous subanalytic function such that N = ϕ ⁻¹ (0). For v ₀ ∈ T _N M , ξ ₀ ∈ T _N ^∗ M , and K ∈ E ^b _w-R-c (I k _M ), one has

sh Eν _N (K)

v

' lim −→

δ,ε,U

RHom ^E (E ^{0B−δ|ϕ(x)|} _{U |M}

, K), (i)

sh Eµ _N (K)

ξ

' lim

δ,ε,W,Z −→

RHom ^E (E ^{0B−δ|ϕ(x)|} _{W ∩Z|M}

, K),

(ii)

where δ, ε − → 0+, U runs over the open subsets of M such that v 0 ∈ / C _N (M \ U ), W runs over the open neighborhoods of $(ξ ₀ ) in M , and Z runs over the closed subsets of M such that

C N (Z) $(ξ

) ⊂ {v ∈ (T N M ) $(ξ

) ; hv, ξ 0 i > 0} ∪ {0}.

Proof. (i-a) Assume that v ₀ ∈

•

T _N M , and set θ ₀ = γ(v ₀ ). Then, one has sh Eν _N (K)

v

'

(∗) sh Eu ⁻¹ Eν _N (K)

v

' sh Eγ ⁻¹ Eν _N ^rb (K)

v

'

(∗∗) sh Eν _N ^rb (K)

θ

,

where (∗) and (∗∗) follow from Lemma 3.9 (i). Then, the statement follows from Lemma 5.1 (i), by noticing that U 3 θ

₀ if and only if v ₀ ∈ / C _N (M \ U ).

(i-b) Assume that v 0 = o(y ₀ ) for y ₀ ∈ N , where o : N − → T _N M is the embedding of the zero section. Then, Lemma 5.2 gives

sh Eν _N (K)

o(y

) ' o ⁻¹ sh(Eν _N (K))

y

' sh(Eo ⁻¹ Eν _N (K))

y

'

(∗) sh(Ei ⁻¹ _N K)

y

,

where (∗) follows from [4, Lemma 4.8 (i)], with i N : N − → M denoting the embedding. Then the statement follows from Proposition 4.1.

(ii) For F ∈ D ^b

R

(k T

M ) one has

RHom ^E ((F ), Eµ _N (K)) = RHom ^E ((F ), ^L Eν _N (K)) ' RHom ^E ( ^L (F ), Eν _N (K)) ' RHom ^E ((F ^∨ ), Eν _N (K)).

Hence

sh Eµ _N (K)

ξ

' lim −→

V 3ξ

RHom ^E ((k _V ), Eµ _N (K)) ' lim −→

V 3ξ

RHom ^E ((k ^∨ _V ), Eν _N (K)) '

(∗) lim −→

V 3ξ

RHom ^E ((k V

), Eν N (K)),

where V runs over the conic open neighborhoods of ξ 0 in T _N ^∗ M , and

V ^◦ := {v ∈ T _N M ; hv, ξi > 0, ∀ξ ∈ V } denotes the polar cone. Here

(∗) follows by noticing that ξ ₀ has a fundamental system of open conic

neighborhoods V ⊂ T _N ^∗ M such that τ | _V has convex fibers.