On a topological counterpart of regularization for holonomic D-modules

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 $\mathcal{M}\mapsto\mathcal{M}_{\mathrm{reg}}$, called regularization. Recall that $\mathcal{M}_{\mathrm{reg}}$ is reconstructed from the de Rham complex of $\mathcal{M}$ by the regular Riemann-Hilbert correspondence. Similarly, on a topological 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 their properties. In particular, we provide a germ formula for the sheafification of enhanced specialization and microlocalization.


Introduction
Let X be a complex manifold.The regular Riemann-Hilbert correspondence (see [7]) states that the de Rham functor induces an equivalence between the triangulated category of regular holonomic D-modules and that of C-constructible sheaves.More precisely, one has a diagram 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)et 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: We call Ish and sh the ind-sheafification and sheafification functor, respectively.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 regularization 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 Section 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 Section 3. In Section 4, we obtain a stalk formula for the sheafification of a pull-back by an embedding.(At the level of D-modules, the interest of such a formula is due to the lack of commutation between the de Rham functor and the restriction functor.)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].
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.

Notations and complements
We recall here some notions and results, mainly to fix notations, referring to the literature for details.In particular, we refer to [9] for sheaves, to [13] (see also [5, 3]) for enhanced sheaves, to [10] for ind-sheaves, and to [1] (see also [12, 8, 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.
M is right adjoint to the embedding M → (M, M ) of good spaces into bordered spaces.We will write for short M = (M, M ).
Note that the inclusion k M : By definition, a subset Z of M is a subset of 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 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 of I such that Z ⊂ i∈I U i .
We say that a morphism f : M onto an open subset of Recall that, by [1, Lem.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 RH om, Rf * , f ! the six operations.Denote by the exterior tensor product and by D M the Verdier duality functor.
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 product and by D M the Verdier duality functor.
There is a natural embedding 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) where "•" means that the functors commute, and "×" that they don't.
The bounded derived category of ind-sheaves of k-vector spaces on M is defined by There is a natural embedding M ), we often simply write F instead of ι M F in order to make notations less heavy.
For bordered spaces, the commutativity of the functor α with the operations 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 functors and the above morphism is an isomorphism if f is borderly submersive.
(ii) There are natural morphisms of functors Proof (i)(a) By Lemma 2.1(ii) and (2.2), one has Here ( * ) follows by adjunction from this is an isomorphism by cartesianity.
(ii)(b) By Lemma 2.1(ii) and (2.2), the morphism is given by the composition  For M a bordered space, consider the projection Denote by 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 Denote by We have

F ). and
The functors R Ihom E and RH om E , taking values in D b (I k M ) and D b (k • M ), respectively, are defined by There is a natural decomposition given by Denote by L E ± and R E ± the left and right adjoint, respectively, of the quotient functor and one sets M (F ): 2.5.Stable objects.-Let M be a bordered space.Set There is an embedding where we write for short with < the total order on R. If S = T , we also write for short (ii) For a continuous map ϕ : where we write for short If S = T , we also write for short , and that there is a short exact sequence for the natural t-structure.M is an open subanalytic subset of the subanalytic space

Sheafification
In this section, we discuss what we call here ind-sheafification and sheafification functor, and prove some of their functorial properties.Concerning constructibility, we use a fundamental result from [12,  §6].[1, Lem.4.5.16]), and call it the associated ind-sheaf (in the derived sense) to K on M. We will write for short Ish = Ish M , if there is no fear of confusion.

Note that one has
-The following are pairs of adjoint functors (ii) and (iii) follow from (i), noticing that there are pairs of adjoint functors Here we denote by ι the natural embeddings.(i) There are a natural morphism and a natural isomorphism of functors and the above morphism is an isomorphism if f is borderly submersive.
(ii) There are a natural morphism and a natural isomorphism of functors and the above morphism is an isomorphism if f is proper.
, there is a natural morphism Here, ( * ) follows from [1, Prop.(i Here ( * ) follows from [10, Lem.5.2.8]. (ii Then, one has One concludes using the natural morphism J.É.P. -M., 2021, tome 8 3.2.Associated sheaf.-Let M be a bordered space. 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 One has Let M be a bordered space, and consider the natural morphisms of good spaces We write t for points of R := R ∪ {−∞, +∞}. An important tool in this framework is given by Proposition 3.6 ([12, Cor.6.6.6])Let M be a bordered space.Then, for Consider the natural morphism j : Consider the functors 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 (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 which are isomorphisms if f is borderly submersive.
(ii) There are natural morphisms of functors The first morphism is an isomorphism if f is proper.The second 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. sh(E . Note that, denoting by i : {0} → M the embedding, one has In fact, on one hand one has i !(sh(E U |M ) 0, and on the other hand one has i −1 (sh(E Note also that sh is not conservative, since sh(E Here {•} is the closure in X rb 0 .Then, (6) recalling Notation 2.3, Recall that, for k = C, the Riemann-Hilbert correspondence of [1]   6) The analogue result for ind-sheaves was obtained in [11, Prop.7.3] and [6, Prop.3.14], at the level of cohomology groups.
By adjunction, it is enough to construct a natural morphism Note that we have a morphism Let δ : M → M ×M be the diagonal embedding, so that * ) 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 where (a) follows from [1, Cor.2.3.5] and (b) follows from Proposition A.2 in Appendix A.

Stalk formula
As we saw in Example 3.10, 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, Lem.2.4.1], for K ∈ E b + (I k M ) and y 0 ∈ N one has (7) where U runs over the open neighborhoods of y 0 in • M. On the other hand, where U runs over the open neighborhoods of y 0 in • M. Here, we set −δ|  7) 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).More generally, for T ⊂ N a compact subset one has (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 and U an open relatively compact subanalytic subset of M containing T , set Here ( * ) follows from the same argument used in the proof of the second isomorphism in [12, (6.6.2)].
(ii) Let us deal with the left hand side of (4.1).Consider the natural maps (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 : Given (V, W ) ∈ J c , we look for (U, δ, ε) ∈ I c such that 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 g(x, t) = (t + c + 1) −1 .and Note that g(x, +∞) = 0. Since (B.1) is satisfied, Lemma B.1(ii) provides C > 0 and n ∈ Z >0 such that 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.

Specialization and microlocalization
Using results from the previous section, we establish here a stalk formula for the natural enhancement of Sato's specialization and microlocalization functors, as introduced in [4].
where δ, ε → 0+ and U where U ⊂ M rb N runs over the neighborhoods of i(Z).Then RΓ Z; sh(Eν rb One concludes by noticing that U Proof.-We shall prove only the first isomorphism since the proof of the second is similar. With the identification N o(N ) ⊂ V , set Here ( where ( * ) holds since o is proper.Hence, we can assume and, since Eo −1 K 0, we have to show ) a group object in the category of bordered spaces (ii) Let W ⊂ M be an open subanalytic subset, and assume that Then there exist ε > 0 and n ∈ Z >0 such that Hence, there exist ε > 0 and n ∈ Z >0 such that Z ⊂ {(t, u) ∈ R 2 ; ε|u| n |t|}.
This gives the statement. (ii

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 ∨
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 product and by D E the Verdier duality functor.

Lemma 3 .
3. -Let f : M → N be a morphism of bordered spaces.

J
.É.P. -M., 2021, tome 8 (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 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 rb 0 → X with center the origin is defined by X rb 0 :={(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 rb 0 , the set I := S 0 X {z ∈ • X ; Re f (z) 0} is the disjoint union of d open non-empty intervals.

(
J.É.P. -M., 2021, tome 8 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.
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.

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 rb N of M with center N , which enters the commutative diagram with cartesian square