RIEMANN-HILBERT CORRESPONDENCE FOR HOLONOMIC D-MODULES

HOLONOMIC D-MODULES

ANDREA D’AGNOLO AND MASAKI KASHIWARA

Abstract. The classical Riemann-Hilbert correspondence establishes an equivalence between the triangulated category of regular holo- nomic D-modules and that of constructible sheaves.

In this paper, we prove a Riemann-Hilbert correspondence for holonomic D-modules which are not necessarily regular. The con- struction of our target category is based on the theory of ind-sheaves by Kashiwara-Schapira and influenced by Tamarkin’s work. Among the main ingredients of our proof is the description of the structure of flat meromorphic connections due to Mochizuki and Kedlaya.

Contents

1. Introduction 1

2. Notations and complements 7

3. Bordered spaces 17

4. Enhanced ind-sheaves 29

5. Review of tempered functions 71

6. Exponential D-modules 76

7. Normal form of holonomic D-modules 82

8. Enhanced tempered functions 88

9. Riemann-Hilbert correspondence 93

References 113

1. Introduction

1.1. On a complex manifold, the classical Riemann-Hilbert correspon- dence establishes an equivalence between the triangulated category of

Date: August 12, 2015.

2010 Mathematics Subject Classification. 32C38, 35A27, 32S60.

Key words and phrases. irregular Riemann-Hilbert problem, irregular holonomic D-modules, ind-sheaves, Stokes phenomenon.

The first author was partially supported by grant CPDA122824/12, Padova Univer- sity, and by the project “Differential methods in Arithmetic, Geometry and Algebra”, Fondazione Cariparo.

The second author was partially supported by Grant-in-Aid for Scientific Research (B) 22340005, Japan Society for the Promotion of Science.

1

regular holonomic D-modules and that of constructible sheaves (see [11]).

Here D denotes the sheaf of differential operators.

In particular, flat meromorphic connections with regular singularities correspond to local systems on the complementary of the singular locus (see [5]).

1.2. The problem of extending the Riemann-Hilbert correspondence to cover the case of holonomic D-modules with irregular singularities has been open for 30 years. Some results in this direction have appeared in the literature.

In the one-dimensional case, classical results of Levelt-Turittin and Hukuhara-Turittin describe the formal structure and the asymptotic ex- pansion on sectors of flat meromorphic connections which are not nec- essarily regular. Using these descriptions, Deligne and Malgrange estab- lished a Riemann-Hilbert correspondence on a complex curve for holo- nomic D-modules with a fixed set of singular points (see [6]). See also the work of Babbitt-Varadarajan [1].

Recently, Mochizuki [22, 23] and Kedlaya [19, 20] extended the results of Levelt-Turittin and Hukuhara-Turittin to higher dimensions. Namely, they proved that any flat meromorphic connection becomes “good” after blowing-ups. Sabbah [27] obtained an analogue of the construction by Deligne and Malgrange on a complex manifold for “good” flat meromor- phic connections with a fixed singular locus.

1.3. In this paper, we prove a Riemann-Hilbert correspondence for holo- nomic D-modules on a complex manifold. The construction of our target category is based on the theory of ind-sheaves by Kashiwara-Schapira [15]

and influenced by the work of Tamarkin [28]. The description of the structure of flat meromorphic connections by Mochizuki and Kedlaya is one of the key ingredients of our proof.

Let us explain our results in greater detail.

1.4. Let X be a complex manifold. As we have already mentioned, the Riemann-Hilbert correspondence of [11] establishes an equivalence between the triangulated category D

(D

) of regular holonomic D

- modules and the triangulated category D

(C

) of C-constructible sheaves on X. More precisely, there are functors

(1.1) D

(D

)

DRX

// D

(C

)

ΨX

oo

quasi-inverse to each other. Here, DR

(L) := Ω

⊗

X

L is the holo-

morphic de Rham complex with Ω

the sheaf of holomorphic differential

forms of highest degree, and Ψ

(L) := T hom (D

L, O

)[d

] is the com-

plex of holomorphic functions tempered along the dual D

L of L.

In particular, a regular holonomic D

-module L can be reconstructed from DR

(L).

Let M be an irregular holonomic D

-module, and consider the reg- ular holonomic D

-module M

:= Ψ

(DR

(M)). Since DR

(M) ' DR

(M

), it follows that M cannot be reconstructed from DR

(M).

1.5. The theory of ind-sheaves, that is, of ind-objects in the category of sheaves with compact support, was initiated and developed by Kashiwara- Schapira [15]. In such a framework, one can consider the complex O

of tempered holomorphic functions, which is an object of the derived cate- gory of ind-sheaves D

(I C

). This is related to the functor Ψ

in (1.1), since one has RHom (F, O

) ' T hom (F, O

) for any R-constructible sheaf F .

Set Ω

= Ω

⊗

X

O

. For a holonomic D

-module M, the tempered de Rham complex DR

(M) := Ω

⊗

X

M has been introduced and studied in [16] and studied further in [24, 25]. This complex retains some information on the irregularity of M. For example, let ϕ ∈ O

(∗Y ) be a meromorphic function with poles at a hypersurface Y , and denote by E

the exponential D

-module generated by e

(see Definition 6.1.1).

Then one has (1.2)

DR

(E

) ' R Ihom (C

, “lim −→ ”

a→+∞

C

)[dim X], where Ihom denotes the inner-hom functor of ind-sheaves and C

denotes the extension by zero to X of the constant sheaf on X \ Y . Since DR

(E

) ' DR

(E

), one cannot reconstruct M from DR

(M).

1.6. Denote by τ ∈ C ⊂ P the affine variable in the complex projec- tive line P. In this paper, we will show that M can be reconstructed from the tempered de Rham complex DR

(M 

E

C|P

), an object of D

(I C

). In the case where X is a complex curve, we outlined a proof of this fact in [3]. The proof in the general case follows from the arguments in the present paper. However, in this paper we take as target category a modification of D

(I C

). As we now explain, this is related to a construction by Tamarkin [28] (see also Guillermou-Schapira [7] for an exposition and some complementary results).

1.7. On a real analytic manifold M , the microlocal theory of sheaves

by Kashiwara-Schapira [13] associates to an object of D

(C

) its micro-

support, a closed conic involutive subset of the cotangent bundle T

M .

In his study of symplectic topology, Tamarkin [28] uses the techniques

of [13] in order to treat involutive subsets of T

M which are not neces-

sarily conic. To this end, he adds a real variable t ∈ R and, denoting by

(t; t

) ∈ T

R the associated symplectic coordinates, considers the quo- tient category D

(C

)/C

by the category C

consisting of objects microsupported on {t

≤ 0}.

An important observation in [28] is that there are equivalences

⊥

C

' D

(C

)/C

' C

(1.3)

between the quotient category and the left and right orthogonal cate- gories. Moreover, such categories can be described without using the notion of microsupport. For example, C

is the full subcategory of D

(C

) of objects whose convolution with C

vanishes.

1.8. Back to our complex manifold X, recall that we aim to recon- struct a holonomic D

-module M from the tempered de Rham complex DR

(M 

E

C|P

). As we explain in §1.13 below, a special important case is when M = E

for ϕ ∈ O

(∗Y ). Then, (1.2) implies that the tempered de Rham complex is described in terms of the ind-sheaf (1.4) “lim −→ ”

a→+∞

C

.

Here t = Re τ is the real part of the affine coordinate τ of the com- plex projective line P. We are thus led to replace the target category D

(I C

) with what we call the category of enhanced ind-sheaves and denote by E

(I C

). This is a quotient category of D

(I C

), where P is the real projective line.

Let us describe the category E

(I C

) in greater detail.

1.9. As a preliminary step, we introduce the notion of bordered space.

A bordered space is a pair (M, ˇ M ) of a topological space ˇ M and an open subset M ⊂ ˇ M , and we associate the triangulated category D

(I C

):=

D

(I C

)/D

(I C

) to it. There is a natural fully faithful embedding D

(C

) ⊂ D

(I C

).

The main example for us is the bordered space R

:= (R, P). This notion appears naturally when we deal with ind-sheaves such as (1.4).

For example, for ϕ = 0 such an ind-sheaf becomes trivial when restricted to D

(I C

), but is a non trivial object of D

(I C

).

1.10. We define the category E

(I C

) of enhanced ind-sheaves by E

(I C

) = D

(I C

)/{K ; K ' π

L for some L ∈ D

(I C

)}.

Here π : X × R

− → X is the projection. This is related with Tamarkin’s construction as follows. We set

E

(I C

) := D

(I C

)/I C

,

where I C

is the full subcategory of objects whose convolution with C

vanishes. As in (1.3), we have

⊥

I C

' E

(I C

) ' I C

.

Replacing C

with C

one obtains the category E

(I C

). It turns out that

E

(I C

) ' E

(I C

) ⊕ E

(I C

).

This is the target category of our Riemann-Hilbert correspondence. It is a triangulated tensor category whose tensor product is given by the convolution ⊗ in the t variable.

1.11. Set C

:= “lim −→ ”

a→+∞

C

. We say that an object K of E

(I C

) is stable if K ' C

⊗ K.

There is a natural fully faithful embedding of the category of ind- sheaves into the category of stable enhanced ind-sheaves

e : D

(I C

) −→ E

(I C

), F 7→ C

⊗ π

F.

Denote by D

R-c

(C

) the full subcategory of D

(C

) whose objects have R-constructible cohomology groups. We say that an object K of E

(I C

) is R-constructible if, for any relatively compact subanalytic open subset U ⊂ X, there exists F ∈ D

(C

) such that

π

C

⊗ K ' C

⊗ F.

Note that such a K is a stable object, and that R-constructibility is a local property on X. We denote by E

R-c

(I C

) the full subcategory of E

(I C

) consisting of R-constructible objects.

1.12. We can now state our Riemann-Hilbert correspondence.

The objects of E

(I C

) which play a role analogous to the objects O

and Ω

of D

(I C

) are

O

:= i

RHom

P

(E

, O

)[2], Ω

:= Ω

⊗

X

O

.

where i : X × R

− → X × P is the embedding. It turns out that O

and Ω

are stable objects endowed with a natural D

-module structure.

Denote by D

(D

) the full subcategory of D

(D

) consisting of ob- jects with holonomic cohomologies. We define the enhanced de Rham functor

DR

: D

(D

) −→ E

(I C

), M 7→ Ω

⊗

X

M

and the reconstruction functor

Ψ

: E

(I C

) −→ D

(D

), K 7→ Hom

(D

K, O

)[d

],

where Hom

is the hom-functor between enhanced ind-sheaves, with val- ues in sheaves on X, and D

is a natural duality functor of E

R-c

(I C

).

Our main result can be stated as follows.

Theorem. (i) The functor DR

is fully faithful and takes values in E

(I C

).

(ii) there is an isomorphism

M ∼ −→ Ψ

DR

(M)

functorial in M ∈ D

(D

). In particular, one can reconstruct M from DR

(M).

We prove the compatibility of DR

with duality. We also prove com- patibility with the classical Riemann-Hilbert correspondence (1.1). More precisely, there is a quasi-commutative diagram:

D

(D

)

^//



D

C-c

(C

)

^//



D

(D

)



D

(D

)

E

X

// E

(I C

)

E

X

// D

(D

).

1.13. A key ingredient in our proofs is the following (see Lemma 7.3.7).

Let P

(M) be a statement concerning a complex manifold X and a holonomic D

-module M. For example,

P

(M) = “M ∼ −→ Ψ

(DR

(M))”.

In order to prove P

(M), the results of Mochizuki [22, 23] and Ked- laya [19, 20] allow one, heuristically speaking, to reduce to the case when M = E

for ϕ ∈ O

(∗Y ).

1.14. Recall that irregular holonomic modules are subjected to the Stokes phenomenon. In §9.8 we describe with an example how the Stokes data are encoded topologically in our construction.

1.15. The contents of this paper are as follows.

Section 2 fixes notations regarding sheaves, ind-sheaves and D-modules.

References are made to [13, 15, 12]. We also state some complementary results which are of use in later sections.

In Section 3, we introduce the notion of bordered space and of ind- sheaves on it, and develop the formalism of operations in this context.

We also discuss a natural t-structure in the triangulated category of ind- sheaves on a bordered space.

In Section 4, we introduce the category E

(I C

) of enhanced ind- sheaves, mentioned in §1.10, and develop the formalism of operations in this framework. We also introduce the notion of R-constructible objects in E

(I C

).

Section 5 recalls from [14, 15] the construction and main properties of

the ind-sheaves of tempered distributions Db

on a real analytic manifold

M , and of tempered holomorphic functions O

on a complex manifold X.

As explained above, this is a fundamental ingredient of our construction.

In Section 6, we prove the isomorphism (1.2). The fundamental ex- ample where X = C 3 z and ϕ(z) = 1/z has been already treated in [16].

Mochizuki and Kedlaya’s results on the structure of flat meromorphic connections are recalled in Section 7. There, we give a precise formulation of the heuristic argument mentioned in §1.13.

Section 8 introduces and studies the enhancement O

of O

mentioned in §1.12, along with the enhancement Db

of Db

.

Our main results, mentioned in §1.12, are stated and proved in Sec- tion 9.

Acknowledgments. We thank Pierre Schapira, who taught us that the ind-sheaf O

of tempered holomorphic functions is an appropriate lan- guage for the study of irregular holonomic D-modules.

We also thank Takuro Mochizuki for his explanations on the structure of irregular holonomic D-modules.

The first author acknowledges the kind hospitality at RIMS, Kyoto University, during the preparation of this paper.

Finally, we wish to thank the anonymous referee for his/her careful reading of our manuscript and his/her suggestions to simplify the proof of Proposition 4.9.6.

2. Notations and complements

We fix here some notations regarding sheaves, ind-sheaves and D- modules, and state some complementary results that we will need in later sections. Our notations follow those in [13, 15, 12], to which we refer for further detail.

Let us say that a topological space is good if it is Hausdorff, locally compact, countable at infinity and has finite flabby dimension.

In this paper, we take a field k as base ring. However, after minor modifications, one can take any regular ring as base ring.

For a category C, we denote by C

the opposite category of C. For a ring A, we denote by A

the opposite ring of A.

2.1. Sheaves. Let M be a good topological space. Denote by Mod(k

) the abelian category of sheaves of k-vector spaces on M , and by D

(k

) its bounded derived category.

For a locally closed subset S ⊂ M , denote by k

the extension by zero to M of the constant sheaf on S.

For f : M − → N a morphism of good topological spaces, denote by ⊗, RHom , f

, Rf

, Rf

, f

the six Grothendieck operations for sheaves.

Denote by  the exterior product.

We define the duality functor D

of D

(k

) by

D

F = RHom (F, ω

) for F ∈ D

(k

),

where ω

denotes the dualizing complex. If M is a C

-manifold of di- mension d

, one has ω

' or

[d

], where or

denotes the orientation sheaf.

2.2. Ind-sheaves. The theory of ind-sheaves has been introduced and developed in [15].

Let C be a category and denote by C

the category of contravariant functors from C to the category of sets. Consider the Yoneda embedding h : C − → C

, X 7→ Hom

(∗, X). The category C

admits small inductive limits. Since h does not commute with inductive limits, one denotes by

“lim −→ ” instead of lim −→ the inductive limits taken in C

. An ind-object in C is an object of C

isomorphic to “lim −→ ”

i∈I

X(i) for some functor X : I − → C with a small filtrant category I. Denote by Ind(C) the full subcategory of C

consisting of ind-objects in C

.

Let M be a good topological space. The category of ind-sheaves on M is the category I (k

) := Ind(Mod

(k

)) of ind-objects in the category Mod

(k

) of sheaves with compact support. Denote by D

(I k

) the bounded derived category of I (k

).

There is a natural exact embedding ι

: Mod(k

) − → I (k

) given by F 7→ “lim −→ ”(k

⊗ F ), for U running over the relatively compact open subsets of M . The functor ι

has an exact left adjoint α

: I (k

) − → Mod(k

) given by α

(“lim −→ ” F

) = lim −→ F

. The functor α

has an exact fully faithful left adjoint β

: Mod(k

) − → I (k

). For example, if Z ⊂ M is a closed subset, one has

β

k

' “lim −→ ”

U

k

,

where U ranges over the family of open subsets of M containing Z.

For f : M − → N a morphism of good topological spaces, denote by

⊗, R Ihom , f

, Rf

, Rf

, f

the six Grothendieck operations for ind- sheaves. Denote by  the exterior product.

Since ind-sheaves form a stack, they have a sheaf-valued hom-functor Hom . One has RHom ' α

R Ihom .

We will need the following proposition to calculate R Ihom .

For a ≤ b in Z, denote by C

(Mod(k

)) the category of complexes of sheaves F

such that F

= 0 unless a ≤ k ≤ b.

Proposition 2.2.1. Let f : M − → N be a morphism of good topological

spaces. Let G ∈ D

(I k

) and let {F

}

be an inductive system in

C

(Mod(k

)) for some a ≤ b in Z. Assume that the pro-object

“lim ←− ”

n

Rf

R Ihom (F

, G) ∈ Pro(D

(I k

)) is represented by an object of D

(I k

). Then

Rf

R Ihom (“lim −→ ”

n

F

, G) ' “lim ←− ”

n

Rf

R Ihom (F

, G).

Proof. Set S

= L

k≤n

F

and denote by ˜ F

the mapping cone of the morphism S

− → S

. Note that the morphism ˜ F

− → F

induced by the projection S

− → F

is a quasi-isomorphism. Consider the morphism S

− → S

⊕ F

= S

obtained by id

and S

− → F

− → F

. This induces a morphism ˜ F

− → ˜ F

which has a cosection for any k and n.

Hence, replacing F

with ˜ F

, we may assume from the beginning that the morphism F

− → F

has a cosection for any k and n.

We may also assume that G

is a complex of quasi-injective sheaves, i.e. that the functor Hom (∗, G

) is exact in Mod(k

) for any n ∈ Z.

In order to prove that the morphism Rf

R Ihom (“lim −→ ”

n

F

, G

) −−→ “lim

←− ”

n

Rf

R Ihom (F

, G

)

is an isomorphism, it is enough to show that RHom (H, u) is a quasi- isomorphism for any H ∈ Mod(k

).

Set E

= Hom (F

⊗ f

H, G

). Then lim ←−

n

E

' RHom (H, Rf

R Ihom (“lim −→ ”

n

F

, G

)),

“lim ←− ”

n

H

(E

) ' H

RHom (H, “lim ←− ”

n

Rf

R Ihom (F

, G

)).

Hence we have to show that (2.2.1) H

(lim ←−

n

E

) − → “lim ←− ”

n

H

(E

) ' lim ←−

n

H

(E

)

is an isomorphism for any k. Since E

− → E

is an epimorphism and {H

(E

)}

satisfies the Mittag-Leffler condition, we conclude that (2.2.1)

is an isomorphism. 

Let us recall the results of [17, §15.4]. These provide useful tools to re- duce proofs of many results in the framework of ind-sheaves to analogous results in sheaf theory.

Recall that Mod

(k

) denotes the category of sheaves with compact support. Then D

(Mod

(k

)) is equivalent to the full triangulated sub- category of D

(k

) consisting of objects with compact support.

Proposition 2.2.2 (cf. [17, §15.4]). There exists a canonical functor J

: D

(I k

) −→ Ind D

(Mod

(k

))

which satisfies the following properties:

(i) For F ∈ D

(Mod

(k

)), and K ∈ D

(I k

), we have

Hom

(F, K) ∼ −→ Hom

J

(F ), J

(K)
.

(ii) The functor J

is conservative, i.e. a morphism u in D

(I k

) is an isomorphism as soon as J

(u) is an isomorphism.

(iii) J

(F ) ' F for any F ∈ D

(Mod

(k

)).

(iv) J

(“lim −→ ” F

) ' lim −→ J

(F

) for any filtrant inductive system {F

} in Mod(k

). Here, lim

−→ denotes the inductive limit in the category Ind(D

(Mod

(k

))).

(v) J

commutes with ⊗ and J

R Ihom (F, G) ' RHom (F, J

(G)) for F ∈ D

(k

) and G ∈ D

(I k

). Here, RHom (F, ∗) denotes the endofunctor of Ind(D

(Mod

(k

))) induced by the endofunctor RHom (F, ∗) of D

(Mod

(k

)).

(vi) H

J

(F ) ' H

F for any n ∈ Z and F ∈ D

(I k

). Here, H

on the right hand side is the cohomology functor D

(I k

) − → I (k

), and H

on the left hand side is the functor Ind(D

(Mod

(k

))) − →

Ind(Mod

(k

)) = I (k

) induced by the cohomology functor D

(Mod

(k

)) − → Mod

(k

).

(vii) Let f : M − → N be a continuous map. Then (a) J

◦ Rf

' Rf

◦ J

.

(b) J

◦f

' f

◦J

and J

◦f

' f

◦J

. Here, for u = f

, f

, we denote by the same letter the composition

Ind(D

(Mod

(k

))) −−→ Ind(D

(k

)) −→ Ind(D

(Mod

(k

))).

The last arrow is given by “lim −→ ”

i

F

7→ “lim −→ ”

i,U

(F

)

, where U ranges over the relatively compact open subsets of M .

Note that J

◦ Rf

' Rf

◦ J

does not hold in general.

As an example of application of Proposition 2.2.2, one has the following result.

Corollary 2.2.3. Let G ∈ D

(k

), K ∈ D

(I k

) and {F

} a filtrant inductive system in Mod(k

). If supp G is compact, then

Hom

(G, K ⊗ “lim −→ ”

i

F

) ' lim −→

i

Hom

(G, K ⊗ F

).

Proof. One has J

(K ⊗ “lim −→ ”

i

F

) ' lim −→

i

J

(K ⊗ F

) by Proposition 2.2.2 (iv) and (v). Then the assertion follows from Proposition 2.2.2 (i). 

Here is another application of Proposition 2.2.2.

Proposition 2.2.4. Let f : M − → N be a continuous map of good topo- logical spaces and K ∈ D

(I k

). Let U be an open subset of M and {V

}

an increasing sequence of open subsets of N . Assume that

U ∩ f

(V

) ⊂ f

(V

) for any n ∈ Z

. Then, setting L = “lim −→ ”

n

k

, there is an isomorphism k

⊗ f

K ⊗ f

L ∼ −→ k

⊗ f

(K ⊗ L).

Proof. Since the question is local on M , we may assume that U is rela- tively compact. By the assumption, U ∩ supp(f

(K ⊗k

)) ⊂ f

(V

).

Thus we have

k

⊗ f

(K ⊗ k

) ∼ ←− k

⊗ f

(K ⊗ k

) ⊗ f

k

,

−→ k

⊗ f

K ⊗ f

k

.

By applying J

and taking the inductive limit with respect to n in Ind(D

(Mod

(k

))), we obtain a morphism

lim −→

n

J

(k

⊗ f

(K ⊗ k

)) −→ lim −→

n

J

(k

⊗ f

K ⊗ f

k

).

By Proposition 2.2.2 (iv), (v) and (viib), this gives a morphism J

(k

⊗ f

(K ⊗ L)) −→ J

(k

⊗ f

K ⊗ f

L).

We can easily see that this is an inverse to the natural morphism J

(k

⊗ f

K ⊗ f

L) −→ J

(k

⊗ f

(K ⊗ L)).

Hence, the statement follows from Proposition 2.2.2 (ii).  We will use the following lemma only in Remark 4.7.13.

Lemma 2.2.5. Let M be a good topological space and {F

}

an inductive system in C

(Mod(k

)) for some a ≤ b in Z. Then

R Ihom (“lim −→ ”

n

F

, ω

) ∼ ←− R Ihom (lim −→

n

F

, ω

).

Here, lim −→

n

F

is the inductive limit of {F

}

in C

(Mod(k

)).

Proof. By dévissage, we may assume that the morphism F

− → F

has a cosection for each k and n, as in the proof of Proposition 2.2.1, and that all the sheaves F

are soft sheaves.

Then, for any G ∈ Mod

(k

),

RHom

(G, R Ihom (“lim −→ ”

n

F

, ω

)) ' RHom

(G ⊗ “lim −→ ”

n

F

, ω

) ' RHom

(RΓ

(M ; G ⊗ “lim −→ ”

n

F

), k).

Since G ⊗ F

are soft sheaves (see [13, Lemma 3.1.2]), RΓ

(M ; G ⊗ “lim −→ ”

n

F

) ' “lim −→ ”

n

Γ

(M ; G ⊗ F

).

Hence

RHom

(RΓ

(M ; G ⊗ “lim −→ ”

n

F

), k) ' Rπ “lim ←− ”

n

Γ

(M ; G ⊗ F

)

, where π : Pro(Mod(k)) − → Mod(k) is the functor of taking the projective limit (see [17, Corollary 13.3.16]). Since Γ

(M ; G ⊗ F

)

satisfies the Mittag-Leffler condition, one has

R

π “lim ←− ”

n

Γ

(M ; G ⊗ F

)

' 0 for any i 6= 0.

Hence

Rπ “lim ←− ”

n

Γ

(M ; G ⊗ F

)

' lim ←−

n

Γ

(M ; G ⊗ F

)

' (lim −→

n

Γ

(M ; G ⊗ F

))

' Γ

(M ; G ⊗ lim −→

n

F

)

' RHom (G, R Ihom (lim −→

n

F

, ω

)).

This implies that RHom (G, R Ihom (lim −→

n

F

, ω

)) ∼ −→ RHom (G, R Ihom (“lim −→ ”

n

F

, ω

)) for any G ∈ Mod

(k

), and hence we obtain the desired result.  2.3. R-constructible sheaves. The notion of subanalytic subset and of R-constructible sheaf, usually defined on real analytic manifolds, nat- urally extend to subanalytic spaces (cf. [13, Exercise IX.2]).

Definition 2.3.1. A subanalytic space (M, S

) is an R-ringed space which is locally isomorphic to (Z, S

), where Z is a closed subanalytic subset of a real analytic manifold, and S

is the sheaf of R-algebras of real valued subanalytic continuous functions. In this paper, we assume that subanalytic spaces are good topological spaces.

One naturally defines the category of subanalytic spaces. The mor- phisms are morphisms of R-ringed spaces.

Let M be a subanalytic space. One says that an object of D

(k

) is R-constructible if all of its cohomologies are R-constructible. Denote by D

(k

) the full subcategory of R-constructible objects of D

(k

). The category D

(k

) is triangulated and is closed under ⊗, RHom and the duality functor D

.

The following two propositions are classical results (see e.g. [13, Propo-

sitions 3.4.3, 3.4.4, 8.4.9]).

Proposition 2.3.2. Let M be a subanalytic space and F ∈ D

(k

).

Then the natural morphism

F − → D

D

F is an isomorphism.

Proposition 2.3.3. Let M be a subanalytic space and N a good topolog- ical space. Let p

: M × N − → M and p

: M × N − → N be the projections.

Then for any F ∈ D

(k

) and G ∈ D

(k

) the natural morphism p

D

F ⊗ p

G −→ RHom (p

F, p

G)

is an isomorphism.

Hence, by applying Corollary 2.2.3, we obtain the following proposi- tion.

Proposition 2.3.4. Let M , N , p

and p

be as in Proposition 2.3.3. Let F ∈ D

R-c

(k

) and G ∈ D

(I k

). Then there are isomorphisms p

k

⊗ p

G ∼ −→ p

G,

p

D

F ⊗ p

G ∼ −→ R Ihom (p

F, p

G).

Corollary 2.3.5. Let M be a subanalytic space and N a good topological space. Let F

, F

∈ D

R-c

(k

) and G

, G

∈ D

(I k

). Then the canonical morphism

RHom (F

, F

)  R Ihom (G

, G

) −→ R Ihom (F

 G

, F

 G

) is an isomorphism.

Proof. Let p

: M × N − → M and p

: M × N − → N be the projections.

We have

p

F

⊗ p

G

' R Ihom (p

D

F

, p

G

).

Hence

R Ihom (p

F

⊗ p

G

, p

F

⊗ p

G

)

' R Ihom (p

F

⊗ p

G

⊗ p

D

F

, p

G

) ' R Ihom (p

(F

⊗ D

F

), R Ihom (p

G

, p

G

)) '

(∗)

R Ihom (p

D

RHom (F

, F

), p

R Ihom (G

, G

)) ' p

RHom (F

, F

) ⊗ p

R Ihom (G

, G

).

Here, in (∗) we have used

F

⊗ D

F

' D

RHom (F

, F

),

which follows from

D

(F

⊗ D

F

) = RHom (F

⊗ D

F

, ω

)

' RHom (F

, RHom (D

F

, ω

)) ' RHom (F

, F

).

 2.4. Subanalytic ind-sheaves. Let M be a subanalytic space. An ind- sheaf on M is called subanalytic if it is isomorphic to a small filtrant ind- limit of R-constructible sheaves. Let us denote by I

(k

) the category of subanalytic ind-sheaves. Note that it is stable by kernels, cokernels and extensions in I (k

). An object of D

(I k

) is called subanalytic if all of its cohomologies are subanalytic. Denote by D

(I k

) the full subcategory of subanalytic objects in D

(I k

). It is a triangulated category.

Let Op

be the category of relatively compact subanalytic open sub- sets of M , whose morphisms are inclusions.

Definition 2.4.1 (cf. [14, 15]). A subanalytic sheaf F is a functor Op

− → Mod(k) which satisfies

(i) F (∅) = 0,

(ii) For U, V ∈ Op

, the sequence

0 −→ F (U ∪ V ) −−→ F (U ) ⊕ F (V )

−−→ F (U ∩ V )

is exact. Here r

is given by the restriction maps and r

is given by the restriction F (U ) − → F (U ∩V ) and the opposite of the restriction F (V ) − → F (U ∩ V ).

Denote by Mod(k

) the category of subanalytic sheaves.

The following result is proved in [15].

Proposition 2.4.2. The category I

(k

) of subanalytic ind-sheaves and the category Mod(k

) of subanalytic sheaves are equivalent by the functor associating with F ∈ I

(k

) the subanalytic sheaf

Op

3 U 7−→ Hom

M)

(k

, F ).

In particular, we have

Proposition 2.4.3. Let K ∈ D

(I k

). Then K ' 0 if and only if Hom

(k

[n], K) ' 0

for any n ∈ Z and any relatively compact subanalytic open subset U ⊂ M . We will need the following result.

∗

In [15 ], subanalytic ind-sheaves are called ind-R-constructible sheaves, and

I

(k

) and D

(I k

) are denoted by I

(k

) and D

(I k

), respectively.

Lemma 2.4.4. Let M be a subanalytic space and K ∈ D

(I k

).

Then K ' 0 if and only if Hom

(k

[n], K) ' 0 for any n ∈ Z and any relatively compact subanalytic open subset U ⊂ M × [0, 1] such that each fiber of U − → M is either empty or connected.

This follows from Proposition 2.4.3 and the following lemma.

Lemma 2.4.5. Any relatively compact subanalytic open subset of M × [0, 1] is a finite union of subanalytic open sets U such that each fiber of U − → M is either empty or connected.

For a similar statement, see [14, Lemma 3.6].

2.5. D-modules. Let X be a complex manifold. We denote by d

its (complex) dimension. Denote by O

and D

the sheaves of algebras of holomorphic functions and of differential operators, respectively. Denote by Ω

the invertible sheaf of differential forms of top degree.

Denote by Mod(D

) the category of left D

-modules, and by D

(D

) its bounded derived category. For f : X − → Y a morphism of complex manifolds, denote by ⊗

, Df

, Df

the operations for D-modules. De- note by 

the exterior product.

Let us denote by

r : D

(D

) ∼ −→ D

(D

) (2.5.1)

the equivalence of categories given by the functor M

= Ω

⊗

X

M.

Consider the dual of M ∈ D

(D

) given by D

M = RHom

X

(M, D

⊗

X

Ω

)[d

], where the shift is chosen so that D

O

' O

.

Denote by D

(D

), D

(D

) and D

(D

) the full subcategories of D

(D

) whose objects have coherent, quasi-good and good cohomolo- gies, respectively. Here, a D

-module M is called quasi-good if, for any relatively compact open subset U ⊂ X, M|

is the sum of a filtrant family of coherent (O

|

)-submodules. A D

-module M is called good if it is quasi-good and coherent.

Recall that to a coherent D

-module M one associates its character- istic variety char(M), a closed conic involutive subset of the cotangent bundle T

X. If char(M) is Lagrangian, M is called holonomic. For the notion of regular holonomic D

-module, refer e.g. to [12, §5.2].

Denote by D

(D

) and D

(D

) the full subcategories of D

(D

) whose objects have holonomic and regular holonomic cohomologies, re- spectively.

Note that D

(D

), D

(D

), D

(D

), D

(D

) and D

(D

)

are triangulated categories.

If Y ⊂ X is a closed hypersurface, denote by O

(∗Y ) the sheaf of meromorphic functions with poles at Y . It is a regular holonomic D

- module. For M ∈ D

(D

), set

M(∗Y ) = M ⊗

O

(∗Y ).

If Y is a closed submanifold of X, denoting by i : Y − → X the inclusion morphism, one sets

B

= Di

O

. (2.5.2)

Then B

is concentrated in degree zero, and is a regular holonomic D

- module.

For M ∈ D

(D

), denote by sing. supp(M) ⊂ X its singular sup- port, that is the set of points where char(M) := S

i∈Z

char(H

M) is not contained in the zero-section of T

X.

Proposition 2.5.1 ([12, Theorem 4.33]). Let f : X − → Y be a mor- phism of complex manifolds. Let M ∈ D

(D

) and N ∈ D

(D

). If supp(M) is proper over Y , then Df

M ∈ D

(D

) and there is an isomorphism

Rf

RHom

X

(M, Df

N )[d

] ' RHom

Y

(Df

M, N )[d

].

In particular,

Hom

(M, Df

N [d

]) ' Hom

(Df

M, N [d

]).

Proposition 2.5.2 ([12, Theorem 4.40]). If f : X − → Y is a smooth mor- phism of complex manifolds, then for M ∈ D

(D

) and N ∈ D

(D

) we have

Rf

RHom

X

(Df

N , M)[d

] ' RHom

Y

(N , Df

M)[d

].

In particular,

Hom

(Df

N , M[d

]) ' Hom

(N , Df

M[d

]).

A transversal Cartesian diagram is a commutative diagram

(2.5.3)

X

0

//

g⁰



Y



X

^//



Y

with X

' X ×

Y

and such that the map of tangent spaces T

X ⊕ T

Y

− → T

Y

is surjective for any x ∈ X

.

Proposition 2.5.3. Consider the transversal Cartesian diagram (2.5.3).

Then, for any M ∈ D

(D

) such that supp(M) is proper over Y ,

Dg

Df

M ' Df

Dg

M.

3. Bordered spaces

Let ˇ M be a good topological space, and M ⊂ ˇ M an open subset. For usual sheaves, the restriction functor F 7→ F |

induces an equivalence

D

(k

)/D

(k

) ∼ −→ D

(k

).

This is no longer true for ind-sheaves, as seen by the following example.

Example. Let ˇ M = R and M = ]0, 1[. Consider the ind-sheaf on ˇ M β

k

= “lim

−→ ”

U 30

k

,

where U ranges over the family of open neighborhoods of 0 ∈ ˇ M . Then β

k

|

' 0, but β

k

∈ D /

(I k

).

Therefore, in the framework of ind-sheaves one should consider the quotient category D

(I k

)/D

(I k

) attached to the pair (M, ˇ M ). We will call such a pair a bordered space.

In this section, we define the category of bordered spaces, develop the formalism of external operations, and define the natural t-structure on the derived category of ind-sheaves on a bordered space.

3.1. Quotient categories. Let D be a triangulated category and N ⊂ D a full triangulated subcategory. The quotient category D/N is defined as the localization D

of D with respect to the multiplicative system Σ of morphisms u fitting into a distinguished triangle

X −−→ Y −→ Z

−−−→

with Z ∈ N .

The right orthogonal N

and the left orthogonal

N are the full sub- categories of D

N

= {X ∈ D ; Hom

(Y, X) ' 0 for any Y ∈ N },

⊥

N = {X ∈ D ; Hom

(X, Y ) ' 0 for any Y ∈ N }.

The following result is elementary (cf. [17, Exercise 10.15]).

Proposition 3.1.1. Assume that

if X, Y ∈ D, Z ∈ N and Z ' X ⊕ Y , then one has X ∈ N . Then the following conditions are equivalent:

(i) the composition N

− → D − → D/N is an equivalence of categories, (ii) the embedding N − → D has a right adjoint,

(iii) the quotient functor D − → D/N has a right adjoint,

(iv) for any X ∈ D there is a distinguished triangle X

− → X − → X

−→

with X

∈ N and X

∈ N

.

Similar results hold for the left orthogonal.

3.2. Bordered spaces. Let M ⊂ ˇ M and N ⊂ ˇ N be open embeddings of good topological spaces. For a continuous map f : M − → N , denote by Γ

its graph in M × N , and by Γ

the closure of Γ

in ˇ M × ˇ N .

Definition 3.2.1. The category of bordered spaces is the category whose objects are pairs (M, ˇ M ) with M ⊂ ˇ M an open embedding of good topo- logical spaces. Morphisms f : (M, ˇ M ) − → (N, ˇ N ) are continuous maps f : M − → N such that

(3.2.1) Γ

− → ˇ M is proper.

The composition of (L, ˇ L) − → (M, ˇ

M ) − → (N, ˇ

N ) is given by f ◦ g : L − → N (see Lemma 3.2.3 below), and the identity id

is given by id

.

Remark 3.2.2. The properness assumption (3.2.1) is used in Lemma 3.3.10 below to prove the functoriality of external operations. It is satisfied in particular if either M = ˇ M or ˇ N is compact.

Lemma 3.2.3. Let f : (M, ˇ M ) − → (N, ˇ N ) and g : (L, ˇ L) − → (M, ˇ M ) be morphisms of bordered spaces. Then the composition f ◦ g is a morphism of bordered spaces.

Proof. Note that Γ

×

Γ

− → Γ

×

M − ˇ → ˇ L is proper. Hence Γ

×

Γ

− → L × ˇ ˇ N is proper. In particular, Im(Γ

×

Γ

− → ˇ L × ˇ N ) is a closed subset of ˇ L × ˇ N . Since it contains Γ

, it also contains Γ

. Since Γ

×

(Γ

×

Γ

) − → ˇ L is proper, Γ

− → ˇ L is proper. 

Note that the category of bordered spaces has (i) a final object ({pt}, {pt}),

(ii) fiber products.

In fact, the fiber product of f : (M, ˇ M ) − → (L, ˇ L) and g : (N, ˇ N ) − → (L, ˇ L) is represented by (M ×

N, Γ

×

Γ

).

Regarding a space M as the bordered space (M, M ), one gets a fully faithful embedding of the category of good topological spaces into that of bordered spaces.

Remark 3.2.4. For any bordered space (M, ˇ M ), using the identifications M = (M, M ) and ˇ M = ( ˇ M , ˇ M ), there are natural morphisms

M −→ (M, ˇ M ) −→ ˇ M .

Note however that id

does not necessarily induce a morphism (M, ˇ M ) − → M of bordered spaces.

If a continuous map f : M − → N extends to a continuous map ˇ f : ˇ M − → N , then f induces a morphism of bordered spaces (M, ˇ ˇ M ) − → (N, ˇ N ).

However the converse is not true. If f : (M, ˇ M ) − → (N, ˇ N ) is a morphism

of bordered spaces, the map f : M − → N does not extend to a continuous

map ˇ f : ˇ M − → ˇ N , in general. However, the next lemma shows how one

can always reduce to this case.

Lemma 3.2.5. Any morphism of bordered spaces f : (M, ˇ M ) − → (N, ˇ N ) decomposes as

(M, ˇ M ) ∼ ←−

q1

(Γ

, Γ

) − − →

q2

(N, ˇ N ),

where the first arrow is an isomorphism and the maps q

: Γ

− → M and q

: Γ

− → N extend to maps ˇ q

: Γ

− → ˇ M and ˇ q

: Γ

− → ˇ N .

Definition 3.2.6. The derived category of ind-sheaves on a bordered space (M, ˇ M ) is the quotient category

D

(I k

) := D

(I k

)/D

(I k

),

where D

(I k

) is identified with its essential image in D

(I k

) by the fully faithful functor Ri

' Ri

, for i : ˇ M \M − → ˇ M the closed embedding.

Remark 3.2.7. In the framework of subanalytic sheaves, an analogue of D

(I k

) is the derived category of sheaves on some site considered in Definitions 6.1.1 (iv) and 7.1.1 of [15].

Since the functor Ri

' Ri

has both a right and a left adjoint, it follows from Proposition 3.1.1 that there are equivalences

⊥

D

(I k

) ' D

(I k

) ' D

(I k

)

. Let us describe these equivalences more explicitly.

Lemma 3.2.8. For F ∈ D

(I k

), one has k

⊗ R Ihom (k

, F ) ∼ ←− k

⊗ F,

R Ihom (k

, k

⊗ F ) ∼ −→ R Ihom (k

, F ).

Proposition 3.2.9. Let (M, ˇ M ) be a bordered space.

(i) One has

D

(I k

) = {F ∈ D

(I k

) ; F ∼ −→ k

⊗ F }

= {F ∈ D

(I k

) ; k

⊗ F ' 0}

= {F ∈ D

(I k

) ; R Ihom (k

, F ) ∼ −→ F }

= {F ∈ D

(I k

) ; R Ihom (k

, F ) ' 0}.

(ii) One has

⊥

D

(I k

) = {F ∈ D

(I k

) ; k

⊗ F ∼ −→ F }

= {F ∈ D

(I k

) ; k

⊗ F ' 0}, and there is an equivalence

D

(I k

) ∼ −→

D

(I k

), F 7→ k

⊗ F,

with quasi-inverse induced by the quotient functor.

(iii) One has

D

(I k

)

= {F ∈ D

(I k

) ; F ∼ −→ R Ihom (k

, F )}

= {F ∈ D

(I k

) ; R Ihom (k

, F ) ' 0}, and there is an equivalence

D

(I k

) ∼ −→ D

(I k

)

, F 7→ R Ihom (k

, F ), with quasi-inverse induced by the quotient functor.

Corollary 3.2.10. For F, G ∈ D

(I k

) one has Hom

(F, G) ' Hom

(k

⊗ F, G)

' Hom

(F, R Ihom (k

, G)) ' Hom

(k

⊗ F, k

⊗ G)

' Hom

(R Ihom (k

, F ), R Ihom (k

, G)).

There is a quasi-commutative diagram of natural functors D

(k

) ^

^//



D

(I k

)



D

(k

) ^

ι_{(M, ˇM )}

// D

(I k

),

where the left vertical arrow is the functor of restriction to M , the right vertical arrow is the quotient functor, and the bottom arrow is the com- position

D

(k

) ' D

(k

)/D

(k

) −→ D

(I k

).

Notation 3.2.11. We sometimes write D

(k

) for D

(k

), when considered as a full subcategory of D

(I k

) by ι

.

3.3. Operations. Let us discuss internal and external operations in the category of bordered spaces.

Definition 3.3.1. The functors ⊗ and R Ihom in D

(I k

) induce well defined functors

⊗ : D

(I k

) × D

(I k

) − → D

(I k

), R Ihom : D

(I k

)

× D

(I k

) − → D

(I k

).

Lemma 3.3.2. For F

, F

∈ D

(I k

) one has

Hom

(k

, R Ihom (F

, F

)) ' Hom

(F

, F

).

Lemma 3.3.3. For F

, F

, F

∈ D

(I k

) one has

R Ihom (F

⊗ F

, F

) ' R Ihom (F

, R Ihom (F

, F

)),

Hom

(F

⊗ F

, F

) ' Hom

(F

, R Ihom (F

, F

)).