## 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}

^{//} M ^{j}

^{M}

^{//}

∨

## 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

^{0}

## 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

i## 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

M◦

## N

### i

N## 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}

b### (I k

M### ) ((F ), K) ' Hom _{D}

b### (I k

_{M×R∞}

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

b### (I k

M### ) (F, Rπ ∗ R Ihom (k {t=0} , R ^{E} K)) ' Hom _{D}

b### (I k

M### ) (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

^{+}

_{{t>0}}

### // E ^{b} _{+} (I k _{M} )

### oo ι

### oo

### ∗ ⊗k

^{+}

^{E}

_{M}

### // 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

_{1}

### =0} , F ) Rπ N∗ R Ihom (k {t

_{2}

### =0} , G)

## −

## → R(π _{M} × π _{N} ) ∗ R Ihom (k {t

_{1}

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

2### =0} , G)

## −

## → Rπ _{M×N∗} Rm ∗ R Ihom (k {t

_{1}

### =0} k ^{{t}

2### =0} , F G)

## −

## → Rπ _{M×N∗} R Ihom Rm _{!!} (k {t

_{1}

### =0} k ^{{t}

2### =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

M### ι

M### // E ^{b} (I k _{M} ).

### sh

M### 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}

^{•}rb 0

## ) ' Rp ∗ sh(E ^{Re f ◦p} _{X|X}

^{•}rb 0

## ) ' 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

0## ' sh(K) _{y}

_{0}

## ' lim

### U 3y −→

0## 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}

_{0}

## ' lim −→

### U 3y

0### δ,ε − → 0+

## RHom ^{E} (E ^{0B−δ|}

### ϕ(x)|

◦^{−ε}

### 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)|

◦^{−ε}

### 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)|

^{−ε}

### U |M ' k _{U}

_{c,δ,ε}

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

^{−ε}

## , K) ' lim −→

### c − → +∞

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

^{−ε}

## , Q F ) ' lim −→

### c − → +∞

## Hom (L ^{E} E cBc−δ|ϕ(x)|

^{−ε}

### U |M , F ) ' lim −→

### c − → +∞

## Hom (k _{U}

_{c,δ,ε}

## ⊗ k _{{t>−c}} , F ) '

### (∗) lim −→

### c − → +∞

## Hom (k _{U}

_{c,δ,ε}

## , 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}

^{R}

^{//}

### π

N## M × R ^{∞}

### π

M## 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

N### ww

### j ? _

### oo

## N ^{ }

### i

N### //

## 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

_{N}

### 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

0## ' lim −→

### δ,ε,U

## RHom ^{E} (E ^{0B−δ|ϕ(x)|} _{U |M}

^{−ε}

## , 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}

^{−ε}

## , 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))|}

^{−ε}

### U |M e

_{N}

^{rb}

## , 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))|}

^{−ε}

### U |M e

_{N}

^{rb}

## , K) ' lim −→

### δ,ε, e U

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

^{−ε}

### j

^{−1}

### ( e U )|(M \N )

∞## , K) ' lim −→

### δ,ε, e U

## RHom ^{E} (E ^{0B−δ|ϕ(x)|}

^{−ε}

### j

N### (j

^{−1}

### ( 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

^{×}

_{>0}

### )

∞## (I k V

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

## Lemma 5.2. For K ∈ E ^{b}

### (R

^{×}

_{>0}

### )

∞## (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}

^{rb}

N

### \S

N### 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}

^{rb}

N

### \S

_{N}

### V ⊗ E˜ γ ^{−1} K ^{sph} ) '

### (∗) Rτ ∗ Rp ∗ sh k _{V}

rb
N

### \S

_{N}

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

rb
N

### \S

_{N}

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

## R˜ γ ∗ sh(k _{V}

rb
N

### \S

_{N}

### V ⊗ E˜ γ ^{−1} K ^{sph} ) ' 0.

## Since ˜ γ is borderly submersive and ˜ γ ^{!} k _{S}

_{N}

_{V} ' k _{V}

rb
N

### \S

_{N}

### V , one has by (2.4) k _{V}

rb
N

### \S

_{N}

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

## R˜ γ ∗ sh(k _{V}

rb
N

### \S

_{N}

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

### (∗) R˜ γ ∗ γ ˜ ^{!} sh(K ^{sph} ) 'R˜ γ ∗ RHom k _{V}

rb
N

## , ˜ γ ^{!} sh(K ^{sph} ) 'RHom R˜ γ _{!} k _{V}

^{rb}

N

## , sh(K ^{sph} ).

## where (∗) follows from Lemma 3.9 (i). Then the desired result follows from R˜ γ ! k _{V}

^{rb}

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}

^{Ω}

### // 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

^{×}

_{>0}

### )

∞## (I k _{(T}

_{N}

_{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}

_{N}

_{M )}

_{∞}

## ) − → E ^{b} _{+} (I k _{(T}

^{∗}

N

### M )

∞## ),

### L (·) : E ^{b} _{+} (I k _{(T}

_{N}

_{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

^{×}

_{>0}

### )

∞## (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

0## ' lim −→

### δ,ε,U

## RHom ^{E} (E ^{0B−δ|ϕ(x)|} _{U |M}

^{−ε}

## , K), (i)

## sh Eµ _{N} (K)

### ξ

0## ' lim

### δ,ε,W,Z −→

## RHom ^{E} (E ^{0B−δ|ϕ(x)|} _{W ∩Z|M}

^{−ε}

## , 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

0### ) ⊂ {v ∈ (T N M ) $(ξ

0### ) ; 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

0## '

### (∗) sh Eu ^{−1} Eν _{N} (K)

### v

0## ' sh Eγ ^{−1} Eν _{N} ^{rb} (K)

### v

0## '

### (∗∗) sh Eν _{N} ^{rb} (K)

### θ

0## ,

## 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

0### ) ' o ^{−1} sh(Eν _{N} (K))

### y

0## ' sh(Eo ^{−1} Eν _{N} (K))

### y

0## '

### (∗) sh(Ei ^{−1} _{N} K)

### y

0## ,

## 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

^{×}>0

## (k T

_{N}

^{∗}

### 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

0## ' lim −→

### V 3ξ

0## RHom ^{E} ((k _{V} ), Eµ _{N} (K)) ' lim −→

### V 3ξ

0## RHom ^{E} ((k ^{∨} _{V} ), Eν _{N} (K)) '

### (∗) lim −→

### V 3ξ

0## RHom ^{E} ((k V

^{◦}