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Constructibility of tempered solutions of holonomic D-modules

In this paper we prove the constructibility on the subanalytic sites of the sheaves of tempered holomorphic solutions of holonomic D-modules on complex analytic manifolds. Such a result solves a conjecture of M. Kashiwara and P. Schapira (2003).

Introduction a fundamental result in the algebraic study of systems of partial differential equations. In its general form, it states that if M, N are holonomic D-modules on a complex manifold X, the cohomology groups of the complex RHom D X (M, N ) are locally constant sheaves with finite dimensional stalks when restricted to the strata of some complex analytic stratification. In particular, this implies (for N ≃ O X ) that the complex of holomorphic solutions of a holonomic D-module on a complex manifold is constructible. A part from its intrinsic importance, this results is at the base of the statement of the Riemann-Hilbert correspondence, one of the most important achievements in D-module theory. Such a correspondence states that the derived functor of holomorphic solutions establishes an equivalence between the bounded derived categories of complexes of D-modules with regular holonomic cohomology and the bounded derived category of complexes of sheaves with constructible cohomology. In his proof ( [13,14]), M. Kashiwara gave an explicit inverse to the functor of holomorphic solutions, the functor T Hom. Kashiwara defined it on the category of R-constructible sheaves. The objects of such a category satisfy the above conditions of locally constancy and finiteness on some subanalytic stratification. For various reasons, the category of R-constructible sheaves is more handy than the category of C-constructible sheaves. On the other hand the D-modules obtained by applying the functor T Hom to R-constructible sheaves which are not C-constructible are still not well understood.
Later ( [17,18]), M. Kashiwara and P. Schapira deepened the study of the functor T Hom (and its dual, the Whitney tensor) realizing it as the complex of sheaves on the subanalytic site X sa relative to the analytic manifold X of tempered holomorphic functions, denoted O t Xsa . Furthermore, in [19], they suggested the use of tempered holomorphic solutions of holonomic D-modules for approaching the general case of the Riemann-Hilbert correspondence. Since then, this latter idea produced significant contributions to the study of irregular D-modules. Let us briefly recall the classical results in this subject.
The local, 1-dimensional version of the irregular Riemann-Hilbert correspondence has been solved through the work of many mathematicians as P. Deligne, M. Hukuhara, A. Levelt, B. Malgrange, J.-P. Ramis, Y. Sibuya, H. Turrittin (see [7], [23] and [33]). Such a classical result consists in two steps: the analysis of the formal decompositions of meromorphic connections and their asymptotic lifts. The higher dimensional case is more complicated. The formal decomposition was conjectured by C. Sabbah in [32] and later proved by T. Mochizuki ([24,25]) in the algebraic case and by K. Kedlaya ([20], [21]) in the analytic one. Recently ( [6,5]), A. D'Agnolo and M. Kashiwara defined the functor of enhanced tempered solutions using tempered holomorphic functions. They proved that it is fully faithful on the category of holonomic D X -modules and they proved a reconstruction theorem for such category. Such a result is a generalization of the classical ones as it is of global nature. Anyway, for the moment being, the statement of a Riemann-Hilbert correspondence for holonomic D-modules is not clear. Indeed, despite of the important results characterizing the enhanced tempered solutions of holonomic D-modules given in [5], a topological description (as for perverse sheaves or C-constructible sheaves for the regular case) of the image category of the enhanced tempered solution functor has not been achieved.
In [19], the authors give a definition of R-constructibility for complexes of sheaves on the subanalytic sites and they conjectured the R-constructibility for tempered solutions of holonomic D-modules. This should be a first attempt to describe the image category of a functor of tempered solutions. The conjecture was proved for the case of complex curves ( [26]) and in a weaker version for complex manifold ( [27]). In the present article we prove the conjecture in its full generality.
Let us conclude by recalling that Kashiwara-Schapira's conjecture is strictly related to Kashiwara's functor T Hom and the D-modules obtained by applying it to R-constructible sheaves. The conjecture states that the the complex of solutions of these latter modules (which are not even coherent) with values in any holonomic D-module is a complex of sheaves on X with Rconstructible cohomology.
The present article is organized as follows. In the first section we review classical results on the subanalytic geometry, the sheaves on subanalytic sites, the D-modules, the tempered De Rham complexes and the elementary asymptotic decompositions of flat meromorphic connections. In the second section we prove our main result in two steps, first we prove Kashiwara-Schapira's conjecture for meromorphic connections, then for D-modules supported on their singular locus.
Acknowledgments: I wish to express my gratitude to M. Kashiwara, T. Mochizuki, C. Sabbah and P. Schapira for many essential discussions. Most of the results exposed in this paper were obtained during my stay at RIMS, Kyoto University, I acknowledge the kind hospitality I found there.
Funding: the research leading to these results has received funding from the 1 Notations and review

Subanalytic sites
For the theory of subanalytic sets we refer to [1,11,10]. For the convenience of the reader we recall here some definitions and the Rectilinearization Theorem in a case which we will need later.
Let M be a real analytic manifold.
there exist a neighborhood W of p, a finite set J, real analytic manifolds N r,j and proper real analytic maps f r,j : N r,j → W (r = 1, 2, j ∈ J) such that A subset is called subanalytic if it is subanalytic at any point of M. For any x 0 ∈ M there exist a real analytic map π : R n → X and a compact set K ⊂ R n such that induces an open embedding of π −1 (U) into X, (iv) ϕ • π(y 1 , . . . , y n ) = a · y r 1 1 · . . . · y rn n , the r j 's being non-negative integers and a ∈ R \ {0}.
For the theory of sheaves on topological spaces and for the results on derived categories we will use, we refer to [16]. For the theory of sheaves on the subanalytic site that we are going to recall now, we refer to [18] and [29].
Let X be a real analytic manifold countable at infinity. The subanalytic site X sa associated to X is defined as follows. An open subset U of X is an open set for X sa if it is relatively compact and subanalytic. The family of open sets of X sa is denoted Op c (X sa ). For U ∈ Op c (X sa ), a subset S of the family of open subsets of U is said an open covering of U in X sa if S ⊂ Op c (X sa ) and, for any compact K of X, there exists a finite subset For Y = X or X sa , k a commutative ring, one denotes by k Y the constant sheaf. For a sheaf of rings R Y , one denotes by Mod(R Y ) the category of sheaves of R Y -modules on Y and by D b (R Y ) the bounded derived category of Mod(R Y ).
With the aim of defining the category Mod(k Xsa ), the adjective "relatively compact" can be omitted in the definition of X sa . Indeed, in [18,Remark 6.3.6], it is proved that Mod(k Xsa ) is equivalent to the category of sheaves on the site whose open sets are the open subanalytic subsets of X and whose coverings are the same as X sa .
One denotes by ̺ : X −→ X sa , the natural morphism of sites given by the inclusion of Op c (X sa ) into the category of open subsets of X. We refer to [18] for the definitions of the functors ̺ * : Mod(k X ) −→ Mod(k Xsa ) and ̺ −1 : Mod(k Xsa ) −→ Mod(k X ) and for Proposition 1.1.3 below.
The functor ̺ ! is described as follows. If U ∈ Op c (X sa ) and F ∈ Mod(k X ), then ̺ ! (F ) is the sheaf on X sa associated to the presheaf U → F U .
Let us conclude this subsection by recalling the definition of R-sa-constructibility for sheaves on X sa . It is due to M. Kashiwara and P. Schapira ( [19]).
Denote by D b R−c (C X ) the full triangulated subcategory of D b (C X ) consisting of complexes whose cohomology modules are R-constructible sheaves. In what follows, for F ∈ D b (C Xsa ) and G ∈ D b R−c (C X ), we set for short Let us remark that the condition (1.1), defining the property of R-saconstructibility, implies that such property is local with respect to the topology of X sa and that it can be tested locally on X (i.e. with respect to G). Remark 1.1.5. The definition of R-sa-constructibility on X sa given above is quite abstract. A more geometrical description of such property, recalling the classical one for sheaves on analytic manifolds, would be of great interest.
With this aim in mind, it is worth recalling that, by means of classical topos theory, one can prove the existence of a topological space X sa such that Mod(C Xsa ) is equivalent to Mod(C Xsa ). The semi-algebraic case is deeply studied in [4]. Anyway, the different descriptions of X sa do not allow a straightforward generalization of the classical constructibility property.

D-modules
The results on D-modules we are going to use in this paper are well exposed in the literature, see for example [15] and [3]. Nonetheless, for the convenience of the reader, we prefer to recall them here.
Let X be a complex analytic manifold. One denotes by O X the sheaf of rings of holomorphic functions on X and by D X the sheaf of rings of linear partial differential operators with coefficients in O X .
Given two left D X -modules M 1 , M 2 , one denotes by M 1 D ⊗M 2 the internal tensor product and by · D ⊗ · its extension to the derived category of D Xmodules. Let us start by recalling the following Now, let T * X denote the cotangent bundle on X. We denote by Mod coh (D X ) the full subcategory of Mod(D X ) whose objects are coherent over D X . For M ∈ Mod coh (D X ) we denote by charM the characteristic variety of M. Recall that charM ⊂ T * X and that M is said holonomic if charM is Lagrangian. One denotes by Mod h (D X ) the full subcategory of Mod(D X ) consisting of holonomic modules.
One denotes by D b coh (D X ) (resp. D b h (D X )) the full subcategory of D b (D X ) consisting of bounded complexes whose cohomology modules are coherent (resp. holonomic) D X -modules. For M ∈ D b coh (D X ), set charM := ∪ j∈Z charH j (M).
Let π X : T * X → X be the canonical projection, T * X X the zero section of Now, we are going to recall the definition of a regular holonomic D-module. There are several equivalent definitions (see [15,Definition 5.2 and Proposition 5.5]), we chose the following one because, in our opinion, it is the most direct and easy to state.
where O X,x is the D X,x -module of formal power series at x. We denote by all of whose cohomology modules are regular. Now, let Z be a closed analytic subset of X. Let I Z be the coherent ideal consisting of the holomorphic functions vanishing on Z, for M ∈ Mod(D X ), one sets If S ⊂ X can be written as S = Z 1 \ Z 2 , for Z 1 and Z 2 closed analytic sets, then it can be proved that the following object is well defined . Recall that f is said smooth if the corresponding maps of tangent spaces T y Y → T f (y) X are surjective for any y ∈ Y . If f is a smooth map, then Df * is an exact functor.
We conclude describing the behavior of RΓ [X\Z] with respect to Df * . Proposition 1.2.4 below can be directly obtained using, for example, Proposition 2.5.27 and Theorem 2.3.17 of [3].
We start this subsection by recalling some results of [18] on tempered holomorphic functions and tempered De Rham complexes of holonomic Dmodules. Then we prove some general results on the R-sa-constructibility of the tempered De Rham complex with respect to the inverse image functors.
Given a complex analytic manifold X of dimension d X , one denotes by X R the real analytic manifold underlying X. Furthermore, one denotes by X the complex conjugate manifold, in particular O X is the sheaf of antiholomorphic functions on X. Denote by Db X R the sheaf of distributions on X R and, for a closed subset Z of X, by Γ Z (Db X R ) the subsheaf of sections supported by Z. One denotes by Db t Xsa the presheaf of tempered distributions on X sa defined by The sheaf Db t Xsa is strictly related to the Kashiwara's T Hom(·, Db X ) functor introduced in [13] and deeply studied [14]. For the definition see [14,Definition 3.13]. Let us simply recall [18, Proposition 7.2.6 (i)] Let Ω j X be the sheaf of differential forms of degree j. For sake of simplicity, let us write Ω X instead of Ω d X X . Set We are now going to recall a conjecture of M. Kashiwara and P. Schapira on the R subanalytic constructibility of tempered holomorphic solutions of holonomic D-modules and the result we obtained on curves.
In [26], we proved that Conjecture 1.3.3 is true on analytic curves.
Later we will use a version of the conjecture using the functor T Hom instead of the complex of sheaves O t . Let us show how to pass from one to the other with the following easy Proof. We have the following sequence of isomorphisms where we used [29, Proposition 1.1.16] in the third isomorphism.
As it will be useful later, let us give an explicit form to . Recall that one denotes by X R the real analytic manifold underlying X, by C ω X R the sheaf of real analytic functions on X R , by X the complex conjugate manifold of X, by O X (resp. D X , Ω X ) the sheaf of holomorphic functions (resp. linear differential operators with coefficients in O X , maximal degree differential forms) on X. Furthermore, we denote by Ω • X (resp. Ω •,• X R ) the complex (resp. double complex) of sheaves of differential forms on X (resp. we have that RHom C X (C U , DR t X M) is isomorphic to the total complex relative to the double complex where the ∇ above, in a local coordinate system z : V → C n , is defined by for f ∈ Db t Xsa , ω ∈ Ω k,h X R and m ∈ DM. Now, we prove some general results on the behaviour of the R-saconstructibility of a tempered De Rham complex with respect to the tensor product and the inverse image functor. We obtained similar results in [27], we adapted the proofs to the present case.
R−c (C X ), the following sequence of isomorphisms proves (1.8) In the previous series of isomorphisms we used Proposition 1.2.1 in the first isomorphism and Theorem 1.3.1(i) in the second isomorphism.
Let us recall two definitions. is an isomorphism on X \ Z.
(ii) Suppose that X ≃ C n and Z is defined by the equation x 1 · . . . · x k = 0. By a ramification map fixing Z we mean a map for some l ∈ Z >0 . Proposition 1.3.8. (i) Given Z = X a closed analytic hypersurface of X, f : Y → X a modification with respect to Z, (ii) Suppose we are in the situation of Definition 1.3.7 (ii). Let f : Y → X be a ramification fixing Z, There exists a local system L on X \ Z such that the following isomorphism holds Proof. (i) Let f : Y → X be a modification. The following sequence of isomorphisms proves the statement We have used Lemma 1.3.6 with R := RΓ [X\Z] O X in the second isomorphism, the fact that f | Y \f −1 (Z) is an isomorphism in the third isomorphism and Theorem 1.3.1(ii) in the last isomorphism.
(ii) There exists a local system L on U \ Z such that In Subsection 2.1 we will need also some results on D M -modules for M a real analytic manifold, i.e. modules over the sheaf of rings of linear differential operators of finite order with real analytic coefficients on M. We refer to [3, Section VII.1]. Let us conclude with some notation about D M -modules and their De Rham complexes. We denote with Ω ω the sheaf on M of differential forms of maximal degree with real analytic coefficients. For M ∈ Mod(D M ), we set .

Asymptotic decomposition of meromorphic connections
In this subsection we are going to recall some fundamental results on the asymptotic decompositions of flat meromorphic connections. The first results on this subject were obtained by H. Majima ([22]) and C. Sabbah ([31]). In particular, Sabbah proved that any good formal decomposition of a flat meromorphic connection on a complex surface admits an asymptotic lift on small multisectors. Let us recall that Sabbah conjectured that any flat meromorphic connection admits a good formal decomposition after a finite number of complex pointwise blow-up and ramification maps ( [32]). Such a conjecture was proved in the algebraic case by T. Mochizuki ([24,25]) and in the analytic one by K. Kedlaya ([20,21]). As in this paper we are not concerned with the formal decomposition of flat meromorphic connections and since the goodness property is not needed within the scope of our results, we are not going to give details on them. In this way, we will avoid to go into unessential technicalities for the rest of the paper. For this subsection we refer to [32,25,8]. Let X be an analytic manifold of dimension n and Z a divisor of X. Let us start by recalling some results about integrable connections on X with meromorphic poles on Z. As in the rest of the paper we will just need the case where Z is a normal crossing hypersurface, from now on, we will suppose such hypothesis.
Let Ω j X be the sheaf of j-forms on X. Let M be a finitely generated O X [ * Z]-module endowed with a C X -linear morphism ∇ : M → Ω 1 X ⊗ O X M satisfying the Leibniz rule, that is to say, for any In general, in the literature, a meromorphic connection is just a coherent O[ * Z]-module. As stated in Proposition 1.2.1 of [32], locally, the condition of being locally free is not restrictive. Since in this paper we deal with the local study of meromorphic connections and holonomic D-modules, we adopted Definition 1.4.1. For sake of shortness, in the rest of the paper, we will drop the adjective "flat". If there is no risk of confusion, given a meromorphic connection (M, ∇), we will simply denote it by M.
Let M 1 , M 2 be two locally free O X [ * Z]-modules, a morphism ϕ : We denote by M(X, Z) the category of meromorphic connections with poles along Z.
Let us recall some facts on meromorphic connections that will be useful in this paper, we refer to [3] or Sections I.1.2 and I.1.3 of [32].
It is well known that the category M(X, Z) is equivalent to the image in Let us now introduce the elementary asymptotic decompositions. Let us denote by X the real oriented blow-up of the irreducible components of Z and by π : X → X the composition of all these. Let us suppose that Z is locally defined by x 1 · . . . · x k = 0. Then, locally In what follows, S 1 will be identified with the unit circle in C, so S 1 = {e iϑ ∈ C; ϑ ∈ R}.
A multisector is an open subset S ⊂ X such that there exist a j , b j ∈ R, a j < b j , r j ∈ R >0 (j = 1, . . . , k), W an open neighborhood of 0 ∈ C n−k such that , we say that a multisector S is small with respect to (r, τ ) if S can be written as in (1.11) and, for any j = 1, . . . , k, b j − a j < τ , r j < r and for any x ∈ W , |x| < r.
Let x 1 , . . . , x n denote the antiholomorphic coordinates on X. Then, for j = k + 1, . . . , n (resp. j = 1, . . . , k) ∂ x j (resp. x j ∂ x j ) acts on C ∞ X (see [31] 2.12 or [32] 1.1.4). The sheaf of algebras on X of holomorphic functions with asymptotic development on Z, denoted A X , is defined as The sections of A X are holomorphic functions on X which admit an asymptotic development as explained in Proposition B.2.1 of [32]. Moreover, A X is a π −1 O X -module. (r, τ ) ∈ R 2 >0 such that for any multisector S ⊂ X small with respect to (r, τ ), there exists Y S ∈ Gl(rk M, A X (S)) such that the following diagram commutes Remark that the isomorphism in (1.12) depends on S and, in general, it does not give a global isomorphism.
The following Theorem is obtained by combining fundamental results of C. Sabbah ([31]), T. Mochizuki ([24, 25]) and K. Kedlaya ([20,21]). For any x 0 ∈ Z there exist a neighborhood W of x 0 and a modification with respect to W ∩ Z above x 0 , σ : Y → X such that σ −1 (Z) is a normal crossing divisor and there exists a ramification map η : X ′ → Y fixing σ −1 (Z) such that D(η • σ) * M| W admits an elementary A-decomposition.

R-sa-constructibility of the tempered De
Rham complex of holonomic D-modules In this section we are going to prove the following We will deal these two cases separately in the next subsections.

The case of RΓ [X\S(M)] M
In the present Subsection, first we prove the conjecture for elementary models, then for connections admitting an elementary A-decomposition and in the end for meromorphic connections.
Recall that, for Z a normal crossings hypersurface and ϕ ∈ O X [ * Z]/O X , we denote by L ϕ the meromorphic connection of rank 1 whose matrix in a basis is dϕ. As a D X -module, L ϕ is isomorphic to RΓ [X\Z] (D X exp(ϕ)). Remark that, with a harmless abuse, we use the same notation for the real analytic case. rh (D X ), we have that DR t X (L ϕ ⊗ R) is R-sa-constructible. Proof. First, let us remark that, by Lemma 1.3.6 it is sufficient to prove the statement with R ≃ O X .
Given U ∈ Op c (C n sa ), by Theorem 1.1.2 we have that for any x 0 ∈ R 2n there exist real analytic map g : R 2n → R 2n and a compact set K ⊂ R 2n such that 2n , a ∈ R × , k 1 , . . . , k 2n ∈ N. Since U is compact, it is sufficient to prove that Set g k := g| K . We have the following sequence of isomorphisms Set M := R 2n , V := g −1 (U) ∩ K and Z ′ := {(y 1 , . . . , y 2n ); y 1 · . . . · y 2n = 0}. Clearly one has Let us now consider, for We are going to prove that it does not depend on y 0 , this will conclude the proof. First, remark that

Now,
where in the third isomorphism, we used the fact that, for W ′ ⊂⊂ W , the restriction morphism Consider the map f : M → R, f (y 1 , . . . , y 2n ) = a −1 y k 1 1 · . . . · y k 2n 2n and set B := B(y 0 , ǫ). We have the following series of isomorphisms Using the well known solvability of the homogeneous and non homogeneous differential equations related to the operator t 2 d dt − 1 acting on tempered distributions on the real line, we have that for ǫ small enough the following isomorphisms hold Let us now consider the case of meromorphic connections admitting an elementary A-decomposition.
Proof. For sake of shortness, let us set Z := S(M). We want to prove that for any First, let us remark that it is sufficient to prove (2.2) for G = C U for U ∈ Op c (X sa ).
We have the following sequence of isomorphisms Where we used Lemma 1.3.6 with R := O X [ * Z] in the second isomorphism. Hence, it is sufficient to prove the (2. Let us briefly recall the meaning of the hypothesis (iii) of the statement we are proving. The problem being local, we can suppose that X ≃ C n and Z ≃ {(x 1 , . . . , x n ) ∈ C n ; x 1 · . . . · x k = 0}. Furthermore, as M ≃ RΓ [X\Z] M and as V ⊂ X \ Z we can suppose that both M and DM are meromorphic connections, in particular they are locally free O[ * Z]-modules of finite rank.
Let Ω • X be the complex of differential forms on X. Recall that we defined X as the real oriented blow-up of the irreducible components of Z and by π : X → X the composition of all these. Then, locally X ≃ (S 1 × R ≥0 ) k × C n−k . Now, as M has an elementary Adecomposition, there exist an elementary model (M el , ∇ el ) and (r, τ ) ∈ R 2 >0 such that for any multisector S ⊂ X small with respect to (r, τ ) there exists Y S ∈ Gl(rk M, A X (S)) giving an isomorphism of complexes Clearly, for any (r, τ ) ∈ R 2 >0 , there exists a finite family {S k } k∈K of multisectors small with respect to (r, τ ) such that, for any L ⊂ K, ∩ l∈L S l is a multisector small with respect to (r, τ ) and ∪ k∈K S k is an open neighborhood of π −1 (Z) ⊂ X.
Coming back to the proof of (2.2), as said above it is sufficient to prove it for G = C V , V ∈ Op c (X sa ), V ⊂ X \ Z. Given such a V , consider V k := V ∩ π(S k ) for {S k } k∈K a family of multisectors as above, then there exists W ∈ Op c (X sa ) such that W ∩Z = ∅ and V = ∪ k∈K V k ∪W . Now, since for any L ⊂ K, ∩ l∈L V l is contained in a multisector small with respect to (r, τ ), it follows that it is sufficient to prove (2.2) for G = C V , V ∈ Op c (X sa ), V ⊂ X \ Z and π −1 (V ) contained in a multisector S small with respect to (r, τ ).
Recall that we denote by X R the real analytic manifold underlying X, by C ω X R the sheaf of real analytic functions on X R . Furthermore, we denote by Ω •,• X R the double complex of sheaves of differential forms on X R with coefficients in C ω X R . In Subsection 1.3, in (1.7), we proved that RHom C X (C V , DR t X M) is isomorphic to the total complex relative to (1.7). Now, as T Hom(C π(S)\Z , Db) is a π * A X | S -module, it makes sens to consider Y S as an isomorphism which satisfies the natural commuting conditions with respect to ∇, ∇ el and ∂.
It follows that the total complex of (1.7) is isomorphic to the total complex relative to the double complex obtained from (1.7) replacing M with M el and ∇ with ∇ el .
The total complex of this last double complex is isomorphic to First, let us suppose that S(M) is a hypersurface.
For r > 1, we have the following distinguished triangle, By the inductive hypothesis, the second and the third terms of (2.3) have R-sa-constructible tempered De Rham complexes. It follows that DR t X (RΓ [X\S(M)] M) is R-sa-constructible.

The case of RΓ [S(M)] M
In this subsection we are going to prove the following Proof. Since the problem is local, we can add (or neglect when necessary) the additional hypothesis that the singular locus of M is connected. For sake of shortness, set Z := S(M). Let us prove our result with an induction on d := dim Z. Clearly, for any complex manifold Y and any N ∈ D b h (D Y ) such that dim S(N ) = 0, DR t Y RΓ [S(N )] N is R-sa-constructible. Now, suppose d := dim Z > 0 and that for any complex manifold Y and any N ∈ D b h (D Y ) such that dim S(N ) < d, DR t Y RΓ [S(N )] N is Rsa-constructible. By Proposition 2.1.3, it follows that DR t Y (N ) is R-saconstructible.
We want to prove that, for any G ∈ D b R−c (C X ) As the condition is local on X and as D b R−c (C X ) is generated by objects of the form C U , for U ∈ Op(X sa ), it is sufficient to prove (2.4) with G ≃ C U , U ∈ Op c (X sa ).
First, we have the following sequence of isomorphisms in D b (C X ) We used Theorem 1. It follows that it is sufficient to prove (2.4) with G ≃ C U ∩Z for all U ∈ Op c (X sa ). Now, we are going to use some results of analytic geometry concerning analytic sets, their regular and singular parts and their strict transform (see [9] or [2]); for the convenience of the reader we will briefly recall them here. Given an analytic set Z ⊂ X, a point z ∈ Z is said regular if there exists a neighborhood W of z such that Z ∩ W is a closed submanifold of W . It is well known that the set of regular points of an analytic set Z ⊂ X, denoted Z r , is a dense open subset of Z. Furthermore Z s := Z \ Z r is a closed analytic set called the singular part of Z whose dimension is smaller that the dimension of Z. Moreover, there exists a proper morphism of analytic manifolds π : X ′ → X and a closed submanifold Z ′ of X ′ , called the strict transform of Z, such that (2.5) π| Z ′ \π −1 (Zs) : Z ′ \ π −1 (Z s ) −→ Z \ Z s is an isomorphism. Now, it is sufficient to prove (2.4) with G ≃ C U ∩Zr and G ≃ C U ∩Zs for U ∈ Op c (X sa ). Let us treat these two cases separately.
The case of C U ∩Zr . Let π : X ′ → X be the map described above in (2.5) and let Z ′ be the strict transform of Z, set π Z ′ := π| Z ′ . Clearly, V := π −1 (U ∩ Z r ) is a relatively compact subanalytic subset of Z ′ such that C U ∩Zr ≃ Rπ Z ′ ! C V .
We have the following sequence of isomorphisms We used Theorem 1.3.1(ii) in the third isomorphism.
Since dim S(Dπ Z ′ * RΓ [Z] M) < dim Z ′ = d, by the inductive hypothesis we have that DR t Z ′ (Dπ Z ′ * RΓ [Z] M) is R-sa-constructible. The case of C U ∩Zs . We have the following sequence of isomorphisms where we used Lemma 1.3.6 with R := RΓ [Zs] O X in the second isomorphism and Theorem 1.2.3(iii) in the third isomorphism.
Since S(RΓ [Zs] M) ⊆ Z s and its dimension is strictly smaller than d, the conclusion follows by the inductive hypothesis.