Relative regular Riemann–Hilbert correspondence II

We develop the theory of relative regular holonomic $\mathcal {D}$-modules with a smooth complex manifold $S$ of arbitrary dimension as parameter space, together with their main functorial properties. In particular, we establish in this general setting the relative Riemann–Hilbert correspondence proved in a previous work in the one-dimensional case.


Introduction
In this article, we are concerned with holomorphic families of coherent D-modules on a complex manifold X of dimension d X , parametrized by a complex manifold S, that is, coherent modules over the ring D X×S/S of linear relative differential operators with respect to the projection p X : X × S → S (simply denoted by p when no confusion is possible). More specifically, we consider families for which the characteristic variety in the relative cotangent space (T * X) × S is contained in the product by S of a fixed closed conic Lagrangian analytic subset Λ ⊂ T * X. Following [FMFS21, MFS13, MFS19], we call these systems relative holonomic D X×S/S -modules.
Here are some examples.
(i) Deligne considered in [Del70] the case of vector bundles E on X × S with a flat relative connection ∇, and established an equivalence with the category of locally constant sheaves of coherent p −1 X O S -modules on X × S. In this case, the relative characteristic variety is the product of the zero section T * X X by S. (ii) For any holonomic D X -module M on X with characteristic variety Λ, the pullback q * M = O X×S ⊗ q −1 O X q −1 M by the projection q : X × S → X is naturally endowed with a D X×S/Smodule structure, and the relative characteristic variety of M is equal to Λ × S. For (E, ∇) as in example (i), the characteristic variety of q * M ⊗ O X×S E (equipped with its natural D X×S/S -module structure) is contained in Λ × S. (iii) Some integral transformations from objects on X to objects on S have kernels which are such flat bundles (E, ∇). One of them is the Fourier-Mukai transformation FM introduced by Laumon [Lau96] and Rothstein [Rot96], which attaches to any bounded complex of D-modules with coherent cohomology on an abelian variety A a bounded complex of O-modules with coherent cohomology on the moduli space A of line bundles with integrable connection on A (cf. [Sch15] and the references therein). It is obtained as the integral transform with kernel P on A × A associated to the Poincaré bundle on the product A × Pic 0 (A). By construction, P is equipped with a flat relative connection, i.e., is a D A×A /A -module. Then q * M ⊗ O A×A P is an instance of example (ii), and FM(M) is the pushforward D p * (q * M ⊗ O A×A P). It is an object of D b coh (O A ). (iv) Given a holonomic D X -module M and holomorphic functions f 1 , . . . , f p on X defining a divisor Y = { i f i = 0}, and setting S = C p with coordinates s 1 , . . . , s p , the S-analytic counterpart of [Mai23,Proposition 13] asserts that the D X×S/S -submodule generated by q * M · ( i f s i i ) in the twisted coherent D X×S/S ( * (Y × S))-module q * M( * (Y × S)) · ( i f s i i ) is relative holonomic.
(v) In his construction of moduli spaces for regular holonomic D-modules, Nitsure [Nit99] fixes a divisor with normal crossings in X and deforms pre-D-modules (extending the notion of vector bundle with flat logarithmic connection) relative to this divisor and its canonical stratification. The corresponding holomorphic family of regular holonomic D-modules has its characteristic variety adapted to this stratification, hence of the form Λ × S. (vi) Mixed twistor D-modules (cf. [Moc15]) are compound objects defined on the product of X by the complex line C, whose module components are holomorphic families of holonomic D-modules parametrized by S = C * degenerating at 0 ∈ C to a coherent module on the cotangent space T * X. On S, the characteristic variety of each holonomic D-module is, by definition, contained in a fixed Lagrangian variety Λ. Of particular interest are the regular mixed twistor D-modules, which have furnished the first example of families we are dealing with (cf. [MFS13]).
Our definition of relative holonomicity imposes the following: the only possible changes in the characteristic variety of the restricted D X -module to a fixed parameter, when the parameter varies, is a change of multiplicities on each irreducible component of Λ. This condition is reasonable, as shown by the previous examples.
In addition, not any relative holonomic D X×S/S -module K can serve as the kernel of an integral transformation as in example (iii), because it cannot be ensured that, for a holonomic In order to replace Bernstein-Sato theory, the main tool is the stability of regular holonomicity by D-module pullback [FMFS21, Theorem 2], that we generalize to the case where dim S can be arbitrary. (ii) If instead of considering D X -modules one considers flat meromorphic bundles on X (i.e., meromorphic connections) with fixed pole divisor, there are many works in the algebraic or formal setting, with base fields or rings that can be different from C. As the literature is vast, let us only mention [ABC20] and the more recent preprint [HdST21]. In the latter preprint, the authors extend Deligne's equivalence result [Del70,Theorem II.5.9] to an equivalence with a formal parameter. The corresponding equivalence in the present article would be that for D X×S/S -modules of D-type, as defined in § 4.
(iii) Wu [Wu21] has established a Riemann-Hilbert correspondence similar to that of Theorem 2 in the case of Alexander complexes that occur in example (iv).
Organization of the paper. In § 2, we review and complete various results on coherent D X×S/S -modules obtained in our previous works [MFS13,MFS19,FMF18,FMFS21]. We emphasize the behavior of holonomicity with respect to pullback and proper pushforward both with respect to X and to the parameter space S. Note that the parameter space S is always assumed to be a complex manifold, whereas one should be able to generalize various statements to any complex analytic space. For example, the sheaf D X×T /T is well-defined if T is a possibly singular and non-reduced complex analytic space with sheaf of functions O T . In this work, we restrict the setting to those complex spaces T that are embeddable in a smooth complex manifold S, with ideal I T ⊂ O S , and we regard D X×T /T -modules as D X×S/S -modules annihilated by I T .
In § 3, we complement various results of [FMFS21] on the regularity property. In particular, we give details on [FMFS21, Remark 1.11]. Furthermore, the relation with the usual notion of regularity of holonomic D X×S -modules is made precise in § 3.c. Stability under base pullback and proper base pushforward is established in § 3.b (as usual, under a goodness assumption for proper pushforward). The case of pushforward with respect to a proper morphism f : X → Y (with a goodness assumption) has already been treated in [MFS19], and stability by pullback, which is the content of Theorem 1 needs first a detailed analysis of holonomic D X×S/S -modules of D-type.
This analysis is performed in § 4. The reasoning made in [MFS19] when dim S = 1, relying on the property that the torsion-free quotient of a coherent O S -module is locally free, has to be adapted by using base changes with respect to S, so that the base functoriality properties considered in § § 2 and 3 are most useful. Theorem 1 is proved in § 4.e in a way similar to that done in [FMFS21]. In § 4.f, we give a characterization of relative regular holonomicity in terms of formal solutions.
Section 6 gives details on the main steps of the proof of Theorem 2. The strategy is similar to that in [MFS19], although we need various new technical details contained in § 5. On the one hand, the construction of the Riemann-Hilbert functor RH S X is now performed by using the partial subanalytic site X sa × S introduced in [MFP21]. On the other hand, the comparison between two definitions of Deligne's extension, one using the standard notion of moderate growth and the other obtained via the complex of tempered holomorphic functions on the partial subanalytic site X sa × S, is also done in § 5, relying once more on results of [MFP21]. We emphasize that the proof given here is simpler, when dim S = 1, than that given in [MFS19].

A review on relative coherent and holonomic D-modules
For complex analytic manifolds X and S, we denote the projections by p X : X × S −→ S and q S : X × S −→ X, 1416 Relative regular Riemann-Hilbert correspondence II and we use the notation p, q when there is no confusion possible. The sheaf of relative differential operators D X×S/S is naturally defined as (2.1)

2.a Coherence, goodness, and holonomicity
We adapt the definitions of [Kas03,§ 4.7], which we refer to for properties and proofs.
Definition 2.2. We say that an O X×S -module F is: any relatively compact open set U of X × S, F |U is the direct limit of an increasing sequence of O U -coherent submodules (equivalently, the direct limit of an inductive system of O U -coherent modules); We say that a D X×S/S -module M is: (ii) Let L be an O-quasi-coherent module supported on Y × S and localized along Y × S (i.e., L L( * Y )). Then L = 0.
(iii) Let M φ −→ N be a morphism of O-quasi-coherent modules which are localized along Y × S. If the restriction of φ at X * × S is an isomorphism (respectively, zero), then φ is an isomorphism (respectively, zero).
Proof. (i) By definition, any point of X × S has an open neighborhood U , which we can suppose to be a relatively compact open set, on which L |U = i L i is the direct limit of an increasing sequence of O-coherent submodules L i . As L i ( * Y ), being equal to (ii) The question is local. As L is an O-quasi-coherent module supported on Y × S, we can suppose (up to shrinking the neighborhood) that it is a direct limit L = i L i of an increasing sequence of O-coherent submodules L i which are supported on Y × S. Therefore, L i ( * Y ) = 0 for every i and, thus, L( * Y ) = i L i ( * Y ) = 0.
(iii) Let us denote by L and L the kernel and the cokernel of φ, respectively. They are O-quasi-coherent modules. If the restriction of φ to X * × S is an isomorphism, they are supported on Y × S. By applying the localization functor along Y , we get the following exact sequence of O-quasi-coherent modules (by the first point) By hypothesis, M( * Y ) M and N( * Y ) N, thus L( * Y ) L, L ( * Y ) L and by the previous point L and L are zero. If φ |X * ×S is zero, then L → M is an isomorphism on X * × S, hence it is an isomorphism and so φ is zero. 1417 Let D X×S/S ( * Y ) be the coherent sheaf of rings O X×S ( * Y ) ⊗ O X×S D X×S/S . Then any coherent D X×S/S -or D X×S/S ( * Y )-module is O-quasi-coherent. On the other hand, any O-quasicoherent D X×S/S -submodule N of a coherent D X×S/S -module M is D X×S/S -coherent. The next lemma follows by an easy adaptation of [Kas03,Proposition 4.23].
Characteristic varieties. To any object M of Mod coh (D X×S/S ) is associated, by means of local coherent O X×S -filtrations, its characteristic variety Char(M), which is contained in T * X × S. A coherent D X×S/S -module is holonomic if its characteristic variety is contained in Λ × S, with Λ closed complex analytic Lagrangian C * -homogeneous in T * X. The derived categories D b coh (D X×S/S ) and D b hol (D X×S/S ) and the characteristic variety Char(M) for such objects M are defined correspondingly.
The structure of the characteristic variety of a holonomic D X×S/S -module M is described in [FMF18, Lemma 2.10]: for each irreducible component Λ i of Λ (i ∈ I) there exists a locally finite family (T ij ) j∈J i of closed analytic subsets of S such that (2.6) The projection to X of Λ i is an irreducible closed analytic subset of X that we denote by Z i . These subsets form a locally finite family of closed analytic subsets of X. We have and we set that we call the X-support (which is a closed analytic subset of X) and the S-support of M (which may be not closed analytic if I is infinite), respectively. Anyway, we set dim Supp S (M) = max i,j dim T ij dim S. 1418 Any D X×S/S -coherent submodule or quotient module of M is holonomic and its characteristic variety is the union of some irreducible components of Char(M).
Lemma 2.8. The category Mod hol (D X×S/S ) of holonomic D X×S/S -modules is closed under taking extensions in the category Mod(D X×S/S ), and under taking sub-quotients in the category Mod coh (D X×S/S ).
We say that a local section m of a D X×S/S -module M is an S-torsion section if it is annihilated by some holomorphic function on S. The S-torsion submodule M t of M is the submodule consisting of S-torsion local sections. Note that if M is a holonomic D X×S/S -module, then the D X×S/S -submodule M t is holonomic because it is an O-quasi-coherent submodule of M. We say that M is S-torsion-free if M t = 0. We denote by M tf : M/M t the torsion-free quotient.
We recall that the duality functor D for D X×S/S -modules was considered in [MFS13,Definition 3.4] and that D b hol (D X×S/S ) is stable under duality which is an involution.

2.b
Behavior with respect to pullback, pushforward, and external product Notation 2.9.
(i) For a holomorphic map f : X → X , we also denote by f the morphism of complex manifolds f × Id : X × S → X × S and by D f * and D f * the pullback and pushforward functors in the derived category of relative D-modules. (ii) For a morphism π : S → S between analytic spaces, we denote by π * and Rπ * the natural extension to the category of relative D-modules of the similar functors defined on the categories of O-modules if it is (O-)good in some neighborhood of each fiber of f (respectively, π).
We recall results concerning the behavior with respect to a morphism of complex manifolds.
Proposition 2.11. Let M be a coherent D X×S/S -module.
(i) If f : X → X is a holomorphic map of complex manifolds, then, for each ∈ Z, Base pullback. Let π : S → S be a morphism of complex analytic manifolds. We also denote by π the induced map Id ×π : X × S → X × S. We have Relative regular Riemann-Hilbert correspondence II by the diagonal embedding δ : coh (D X×X×S/S ) and we have C := Char(M L D N) ⊂ Char(M) × S Char(N). This is seen by considering local resolutions of M (respectively, N) by free D X×S/S -modules, showing both inclusions C ⊂ T * X × Char(N) and C ⊂ Char(M) × T * X. Therefore, if M, N are holonomic, so is M L D N.

Regular holonomic D X×S/S -modules
In the special case of Lemma 2.14 where π is the inclusion i so : we recall a consequence of [Kas03,(A.10)].   We note that statement (iii) follows from statements (i) and (ii) together with Lemma 2.8, so that we focus on the latter properties. The proof in both cases is the same because it is based on the coherence of the rings involved. We provide it in the localized case.  Proof. We argue by induction on the amplitude of the complex M. We may assume that M ∈ D 0 coh (D X×S/S ( * Y )) and we consider the following distinguished triangle:

3.a Characterization of relative regular holonomicity
where τ 1 is the truncation functor with respect to the natural t-structure on D b coh (D X×S/S ( * Y )). Let us assume that M satisfies (Reg 1), hence by definition and induction, both H 0 M and τ 1 M satisfy (Reg 1). As remarked, H 0 M satisfies (Reg 2) too and by induction on the amplitude of M, τ 1 M satisfies (Reg 2).
Proof of Proposition 3.2. For d 0, we denote by (Reg 1) d , respectively, (Reg 2) d , the corresponding condition for dim S d. If dim S = 0, (i) 0 holds true, and (ii) 0 is proved, e.g., in [Bjö93,Theorem 5.3.4]. We thus assume from now on that d 1 and we proceed by induction on d := dim S, denoting by (i) d and (ii) d the statements of the proposition restricted to dim S d. We prove the following implications for d 1: Let us start with implication (a). Assuming that both (i) d−1 and (ii) d−1 hold, we have to prove that (Reg 2) d ⇒ (Reg 1) d . Owing to the induction hypothesis (i) d−1 we simply write (Reg) d−1 for either (Reg 1) d−1 or (Reg 2) d−1 .
We note that M ∈ D b coh (D X×S/S ( * Y )) satisfies (Reg 2) d if and only if for each smooth codimension-one germ (H, It is then enough to prove that, for such an M, Li * H H j M satisfies (Reg) d−1 for any j and H. We shall argue by induction on the amplitude of M. We may assume that M ∈ D 0 coh (D X×S/S ( * Y )) and we consider the distinguished triangle (3.4). We deduce an isomorphism and an exact sequence Let us now prove implication (b). The extension property in (ii) d is clear. Let us consider stability by sub-quotients in Mod coh (D X×S/S ( * Y )). Let M ∈ Mod coh (D X×S/S ( * Y )) satisfy (Reg) d . Given any short exact sequence As Let us denote by M the pullback of M 2 := tors H (M 2 ) in M and by M 2,tf the quotient M 2 /M 2 . The following commutative diagram is Cartesian and its columns and rows are short exact sequences. (3.8) As M 2,tf is i * H -acyclic, the exact sequence (3.7) for the middle column splits as We now prove that Li * H M 2 satisfies (Reg) d−1 , which will conclude the proof. As M 2 is D X×S/S ( * Y )-coherent, there exists locally an integer k 1 such that s k M 2 = 0. We prove by induction on k that any D X×S/S ( * Y )-coherent torsion quotient of a D X×S/S ( * Y )-coherent module satisfying (Reg) d , satisfies (Reg) d too.
If k = 1, we have H −1 Li * H M 2 H 0 Li * H M 2 , and, by (ii) d−1 the latter satisfies (Reg) d−1 , being a quotient of H 0 Li * H M . If k > 1, we argue with the following Cartesian commutative diagram, analogous to (3.8).  Proof. We note that (Reg 2) implies that D b rhol (D X×S/S ) is a full triangulated subcategory of D b hol (D X×S/S ). As the latter is a full triangulated subcategory of D b coh (D X×S/S ), the first assertion follows. Stability by duality follows from the same property in the absolute case (cf., e.g., [Bjö93,Theorem 5.4.15 (4)]), together with the isomorphism Li * so DM DLi * so M, which follows from Proposition 3.1.

3.b Stability of regular holonomicity under base pullback and base pushforward
For a proper morphism f : X → Y , it has been shown in [MFS19,Corollary 2.4 On the other hand, stability of regular holonomicity (and, hence, its coherence) by pullback D f * has been shown in [FMFS21] only if dim S = 1 as a consequence of the Riemann-Hilbert correspondence proved there. This will be obtained in general by the proof of Theorem 1 in § 4.e. In this section, we consider on the other hand the behavior with respect to base pullback and pushforward. Proof. We already know that Lπ * M belongs to D b hol (D X×S /S ) by Lemma 2.14. Regularity follows from the isomorphism of functors Li * s o Lπ * Li * π(s o ) for any s o ∈ S . Let t o be a fat point of S, that is, a complex subspace of S supported on a reduced point |t o | ∈ S. In other words, the ideal I to ⊂ O S , which satisfies I to ⊂ m |to| (which is the ideal associated to |t o | such that its fiber in |t o | is the maximal ideal of O S,|to| ), contains some power m k |to| . By abuse of notation we still denote by I to (respectively, m |to| ) the ideal I to,|to| (respectively, m |to|,|to| ) and also the sheaf we regard in a natural way as a D X -module because O S,|to| /I to is a finite-dimensional vector space. We define thereby the pullback functor . We note that, endowed with its natural structure of D X -module, i * to D X×S/S is coherent. As a consequence, if M is D X×S/S -coherent, then Li * to M has D X -coherent cohomology. In other words, , and the conclusion follows.
Setting 3.12. We consider a local setting where S is a polydisc Δ d written as Δ d−1 × Δ = S × Δ, and we denote by q : S → S the projection (s , t) → s , and we keep the same notation after taking the product with X. Recall (cf., e.g., [Kas03,Proposition A.14]) that the sheaves of rings Let h(s , t) = t k + k−1 i=0 h i (s )t i be a Weierstrass polynomial, with h i holomorphic on S and let T = h −1 (0). The equivalence between the categories of: extends in a natural way to O X×S , to D X×S/S and to D X×S/S ( * Y ) for a given hypersurface Y of X. For example, in one direction, if M is D X×S/S -coherent and supported on X × T , then M is q −1 D X×S /S -coherent and each local section is annihilated by some power of h. By taking local D X×S/S -generators of M, we conclude that there exists locally an integer 1 such that h M = 0. Conversely, if M satisfies the latter property, we can regard it as an h-nilpotent coherent By definition, the characteristic variety of a coherent q −1 D X×S /S -module is (locally) the support in (T * X) × S of the graded coherent q −1 gr F D X×S /S -module with respect to any local coherent q −1 F • D X×S /S -filtration. Such a module is said to be holonomic if this support is contained in Λ × S for some Lagrangian variety Λ ⊂ T * X.
Remark 3.13 (Holonomic and regular holonomic q −1 D X×S /S -modules). Given a coherent D X×S/S -module M supported on T as above, one checks that a coherent F • D X×S/S -filtration is also a coherent q −1 F • D X×S /S -filtration by the above correspondence. As a consequence, such On the other hand, we say that a holonomic Corollary 3.14. A coherent D X×S/S -module M supported on X × T is regular holonomic if and only if, when regarded as a q −1 D X×S /S -module, it is regular holonomic.
Proof. By Remark 3.13, we only need to check regularity, and we can suppose from the start that M is D X×S/S -holonomic supported on X × T . We assume first that M is q −1 D X×S /Sregular. We wish to prove that Li * so M is D X -regular holonomic for any s o ∈ S. It is enough to prove this for s o ∈ T . Let us choose coordinates (s 1 , . . . , is a finite union of fat points in S (defined by the ideal generated by h k (0, t)), one of which is supported at s o , we conclude that Li * so M is a regular holonomic D X -module by applying Corollary 3.11. The converse is proved similarly. Proof. A standard consequence of Proposition 2.15 is that Rπ * M belongs to D b hol (D X×S /S ). Furthermore, we note that the question is local with respect to X and to S .
Step 1: reduction to the case where S is a point. As π is projective, we can regard π (locally with respect to S ) as the composition of the inclusion S → P m × S and the projection P m × S → S for a suitable m. Moreover, we note that the result is easy if π is a closed embedding. We can thus assume that π is a projection S and consider the following Cartesian square.
Then we have where the isomorphism ( * ) follows by extension of scalars because M is a complex of O X×Smodules. By Proposition 3.10, Li * M is a complex with regular holonomic cohomologies, so that if we know the theorem for π , we deduce it for π. From now on, we assume that S is a point. In such a case, S is a projective space P m . As the question is local with respect to X and since S is compact, we can assume that the sets I and J i occurring in (2.6) are finite, so the S-support T of M is a closed analytic subset of S. We argue by induction on the dimension of the S-support of M.
Step 2: the case dim Supp S M = 0. If the S-support of M has dimension zero, it consists of a finite number of points, and it is enough to consider the case where the support consists of one point s o ∈ S. By a standard argument we may assume that M is concentrated in degree zero and locally we can assume that there exists k 1 such that, denoting by m so the maximal In this case, the theorem follows from Corollary 3.11.
Step 3: the case dim Supp S M 1. We recall that, in this step, π is the constant map on S = P m .
(i) Proof of the regularity of π * M. Let d 1. We now assume that the statement holds true for any complex M whose S-support has dimension < d and we aim at proving the same property for any complex M with S-support T of dimension d. We may then reduce again to the case where M is a single module.
We first prove that π * M is regular holonomic (instead of all modules R k π * M).

Relative regular Riemann-Hilbert correspondence II
One checks that the D X×S/S -submodule M of M consisting of local sections m such that the S-support of D X×S/S · m has dimension < d is holonomic (denoting by T <d the union of irreducible components of T of dimension < d, M is locally defined as the sheaf of local sections annihilated by some power of the ideal I T <d of T <d in O S ). By Proposition 3.2(iii), M is regular holonomic and, by the induction hypothesis, π * M is regular holonomic. It is thus enough to prove regularity of π * (M/M ), and we can likewise assume that (3.16) M has no non-zero coherent submodule with S-support of dimension < d.
Let us denote by Λ a Lagrangian subvariety of T * X such that the characteristic variety of M is contained in Λ × S. As T is compact, it has a finite number of irreducible components and we index by I d those which are of dimension d. We choose a point s i on the smooth part of each T i (i ∈ I d ). For the sake of simplicity, we denote by s o the finite set There is a natural morphism of D X -modules The proposition will be proved if we prove that this morphism is injective for k large enough, because the right-hand side is regular holonomic by Corollary 3.11. Its kernel N k is a coherent, hence holonomic, D X -submodule of π * M and the sequence (N k ) is decreasing with characteristic variety contained in Λ. It is thus stationary. Let N ⊂ π * M denote this constant value. We conclude that the map is zero for any k. As N = N k for k large enough, we aim at proving that and similarly by M (xo,so) that of M. It thus suffices to prove that, for all x o ∈ X, the germ M (xo,so) = k m k so M (xo,so) is zero. We note that M (xo,so) is of finite type over D (xo,so) because the latter ring is Noetherian, hence M (xo,so) is holonomic with characteristic variety contained in Λ xo × (S, s o ), where Λ xo is the germ of Λ along T * xo X. Furthermore, it satisfies M (xo,so) = m so M (xo,so) . The Nakayama-type argument given in the proof of [MFS19, Proposition 1.9(1)] shows that M (xo,so) = 0 for each x o . This concludes the proof of step 3(i).
(ii) Regularity of R k π * M. We use the result of step 3(i) in order to prove holonomicity and regularity of R k π * M for all k. 1427 For any ∈ Z, we consider the line bundle O S ( ) that we can realize as We apply the technical Lemma 3.18 below to M. We note that M( ) is D X×S/S regular holonomic: indeed, this is a local property on S, and locally . The latter isomorphism implies the regularity of R k+1 π * M for k 1, and, together with Corollary 3.9, it implies that of π * M and R 1 π * M.
A similar result is well-known to hold for a coherent O X×S -module, as a consequence of Grauert-Remmert's Theorems A and B (cf. [GR58] and [BS76,Theorem IV.2

.1]).
Proof. For any N 1, by iteration of Lemma 2.4(iii) we obtain a distinguished triangle on nb(x o ) × S: Let us choose N = 2m + 1 and let be an integer given by Grauert-Remmert's theorems for L i (i = −N, . . . , 0) in nb(x o ). We consider the same triangle obtained after tensoring with p −1 O S ( ). As π has cohomological dimension 2m, we find that R k π * N ( )[N ] = 0 for any k 0. On the other hand, on nb(x o ) (cf. (2.1)), and because the latter is in non-positive degrees, we conclude that Rπ * N( ) |nb(xo) π * N( ) |nb(xo) .

3.c Integrable regular holonomic D X×S/S -modules
The following proposition, answering a question of Lei Wu, shows that, under a suitable condition on the characteristic variety, a coherent D X×S/S -submodule of a regular holonomic D X×S -module is relatively regular holonomic. We call such D X×S/S -modules integrable because their relative connection can be lifted as an integrable connection, i.e., a D X×S -module structure.The condition on the characteristic variety is due to the restrictive definition of a holonomic D X×S/S -module (cf. after Lemma 2.4).
Proposition 3.19. Let X and S be complex manifolds and let N be a regular holonomic D X×Smodule. Assume that the characteristic variety of N is contained in Λ × T * S for some conic Let us first recall two properties (see (3.20) and (3.21)) of regular holonomic D X×S -modules that will be useful for the proof and do not depend on the assumption on Char N made in the proposition.
Let N be any regular holonomic D X×S -module and let M be a coherent D X×S/S -submodule of N. Any irreducible component of Char N projects to X × S as an irreducible closed analytic subset Z and, denoting by Z • the smooth part of Z, this irreducible component is the closure of the conormal bundle T * and is a conic analytic subset of T * (X × S/S).
(3.20) According to [Sab87,Theorem 3.2] we denote by s a local coordinate on S vanishing at s o . We postpone after the proof of Proposition 3.19 that of the following classical result.
(3.21) Under the above assumptions, the kernel and cokernel of s : M → M are regular holonomic D X -modules.
Proof of Proposition 3.19, step 1. We now add the assumption on Char N. We first show that M is relative holonomic with characteristic variety contained in Λ × S. We note that T * Z (X × S) is contained in Λ × T * S if and only if Z decomposes as the product Y × T for some irreducible closed analytic subsets Y ⊂ X and T ⊂ S and T * Y X is an irreducible component of Λ (this seen by considering first the smooth part Z • ). It is then easily seen that T * p|Z (X × S/S) = (T * Y X) × T, hence is contained in Λ × S. The conclusion follows from (3.20).
Proof of Proposition 3.19, step 2. It remains to show the relative regularity of M. We argue by induction on d = dim S. If d = 1, relative regularity is provided by (3.21), because we already have relative holonomicity by step 1. We thus assume that the statement of the proposition holds whenever dim S d − 1 and we assume dim S = d 2. The question being local, we fix a local coordinate s in S, defining a smooth hypersurface H := {s = 0}. Locally, we can assume that S = H × C. According to Proposition 3.2, we are reduced to proving the following. For the sake of clarity we set We denote by M 1 the D X ×S /S -submodule of the regular holonomic D X ×S -module N generated by M. It is D X ×S /S -coherent, so by (3.21), the kernel N 1 and the cokernel N 1 of s : M 1 → M 1 are D X -regular holonomic.
Claim. The characteristic varieties Char N 1 and Char N 1 are contained in Λ × T * H.
Proof. We apply (3.20) to M 1 ⊂ N and to the map X × S → S . As any irreducible component of Char N takes the form T The fiber of the composition T * q|T (H × S /S ) → T → S above s = 0 (s is the coordinate on S ) is contained in T * H and is Lagrangian: indeed, it has dimension dim H since T * s|T (H × S /S ) has dimension dim H + 1, and it is easily seen to be isotropic. We denote it by Λ T . The characteristic varieties of N 1 and N 1 are contained in the fiber above s = 0 of Char M 1 , hence in the union, over Y, T occurring in Char N, of the Lagrangian subsets T * Y X × Λ T , as claimed, because Λ is nothing but Y T * Y X. As a consequence, N 1 is a regular holonomic D X×H -module satisfying the assumption of the proposition, and ker(s : M → M) is a coherent D X×H/H -submodule of it. We can thus apply the induction hypothesis to conclude that (3.22) holds for ker s.
On the other hand, for each k 0, by applying (3.21) to s k M 1 , we obtain, according to the claim, that s k M 1 /s k+1 M 1 is a regular holonomic D X×H -module with characteristic variety contained in Λ × T * H. Then is also a regular holonomic D X×H/H -module. It remains to note that coherence implies that, locally on X, there exists k such that M ∩ s k+1 M 1 ⊂ sM, so that M/sM is a coherent quotient of a regular holonomic D X×H/H -module. Again by Proposition 3.2(iii) it is also a regular holonomic D X×H/H -module, concluding the proof of (3.22).
Proof of (3.21). We consider the Kashiwara-Malgrange V -filtration V • N of N relative to the function s (cf., e.g., [MS89,MM04]). This is an increasing filtration indexed by Z which satisfies, owing to the regularity of N, the following properties: • V k N is D X×S/S -coherent for any k ∈ Z (see [MS89, Theorem (4.12.1)]); • for each k ∈ Z, s(V k N) ⊂ V k−1 N and for k −1, the multiplication by s on V k N is injective with image V k−1 N (cf., e.g., [MS89, Proposition (4.5.2)]); • each gr V k N is a regular holonomic D X -module (cf., e.g., [MM04,). From the second point we deduce that, for k −1, Li * so V k N = i * so V k N = gr V k N, and by the third point the latter is D X -regular holonomic. For k 0, V k N is a successive extension of V −1 N by regular holonomic D X -modules gr V N. Therefore, for each k, the kernel and cokernel of s : V k N → V k N are D X -regular holonomic: this is proved by induction on k since it is clear for k < 0, and the inductive step follows by considering the snake lemma applied to the following commutative diagram of exact sequences.

Relative regular Riemann-Hilbert correspondence II
We deduce that sV k N/s V k is regular holonomic for any 1. As any D X×S/S -coherent submodule of N, locally on X × S, M is contained in V k N for some k 0 and, by the Artin-Rees lemma, sM contains M ∩ s V k N for some 1. Since ker(s : M → M) is contained in ker(s : V k N → V k N), the former is regular holonomic because the latter is so. On the other hand, by considering the inclusion and the quotient maps we conclude that (M ∩ sV k N)/sM is also D X -regular holonomic. Furthermore, by considering the exact sequence we conclude that M/sM is D X -regular holonomic.

4.a S-locally constant sheaves and their associated relative connections
Let X be a connected complex manifold and let L be an S-locally constant sheaf of p −1 O Smodules on X × S (cf. [MFS19, Appendix]). For any x o ∈ X, L is uniquely determined from a monodromy representation π 1 (X, To any S-locally constant sheaf L one can associate an exact sequence of sheaves of where L t denotes the subsheaf of p −1 O S -torsion and L tf the maximal p −1 O S -torsion-free quotient. Then L t and L tf are S-locally constant and the previous exact sequence yields (for any choice of G) to the exact sequence of O S -modules with G t and G tf defined similarly. The following is straightforward.
Lemma 4.2. Let π : S → S be a holomorphic map between complex manifolds.
We denote by d X×S/S : O X×S → Ω 1 X×S/S the relative differential associated to p. Let us recall the Riemann-Hilbert correspondence for coherent S-local systems proved in [Del70,Theorem 2.23 p. 14], in the particular case of a projection X × S → S, where X and S are complex manifolds. The induces an equivalence between the category of coherent S-locally constant sheaves of p −1 O S -modules and the category of coherent O X×S -modules E equipped with an integrable relative connection A quasi-inverse is given by (E, ∇) → E ∇ = ker ∇. The monodromy representation attached to the coherent S-locally constant sheaf E ∇ is also called the monodromy representation of ∇ on E. Let us emphasize a direct consequence.
is an isomorphism compatible with the integrable connections d X×S/S ⊗ Id and ∇.
(i) Let L be a coherent S-locally constant sheaf. If π is proper, then π * L is S -coherent and there exists a natural morphism E π * L → π * E L . (ii) Let L be a coherent S -locally constant sheaf. Then E π * L π * (E L ).
Proof. (i) As π is proper, π * L is a coherent S -locally constant sheaf on X × S (Lemma 4.2(i) and Grauert's theorem). The natural morphism O X×S → π * O X×S induces a composed morphism which is clearly compatible with the relative differential d ⊗ Id. We note that if π is surjective with connected fibers, the first morphism is an isomorphism because O X×S → π * O X×S is then an isomorphism. This is the case if for example S is a complex manifold and π is a proper modification of S .
(ii) The second point is straightforward.

4.b The Deligne extension of an S-locally constant sheaf
We recall Theorem 2.6 and extend Corollary 2.8 of [MFS19] to the case where dim S > 1.
Notation 4.5. Let Y be a hypersurface in X (assumed to be connected) and let us denote the inclusion by j : X * := X Y → X. Let L be a coherent S-locally constant sheaf on X * × S.
We simply set E = E L when the context is clear, that we consider as a left D X * ×S/S -module. We sometimes call L a coherent S-local system.
We assume from now on that Y = D is a divisor with normal crossings in X. Let : X → X denote the real blowing up of X along the components of D. We denote by j : X * → X the inclusion, so that j = • j. Let x o ∈ D, x o ∈ −1 (x o ) and let s o ∈ S. Choose local coordinates (x 1 , . . . , x n ) at x o such that D = {x 1 · · · x = 0} and consider the associated polar coordinates (ρ, θ, x ) := (ρ 1 , θ 1 , . . . , ρ , θ , x +1 , . . . , x n ) so that x o has coordinates ρ o = 0, θ o , x o = 0. We also denote by the induced map X × S → X × S and by p : X × S → S the projection.
Definition 4.7. The subsheaf A mod X×S of j * O X * ×S of holomorphic functions having moderate growth along −1 (D) is defined by the following two requirements: Relative regular Riemann-Hilbert correspondence II • for any x o ∈ D, x o ∈ −1 (x o ) and s o ∈ S, a germ h ∈ ( j * O X * ×S ) ( xo,so) is said to have moderate growth, i.e., to belong to A mod X×S, ( xo,so) , if there exist open sets

Let us already note for later use that these properties imply, for any
where the latter isomorphism follows from flatness of O X×S ( * D) over p −1 O S . Furthermore, property (ii) implies that, for any coherent O S -module G, the natural morphism [MFS19, Proposition A.12]). We thus have an identification (4.10) Indeed, for a polysector V ε ⊂ U ε and V ⊂ U (s o ), the natural morphism is an isomorphism, because the adjunction morphism p W * p −1 W G → G is an isomorphism for both W = V * ε and W = V ε (both being connected, cf., e.g., [MFS19, Proposition A.1(2)]). The assertion is obtained by passing from the pre-sheaf isomorphism to a sheaf isomorphism. 1433 Definition 4.11 (The Deligne extension). Let x o ∈ X, x o ∈ −1 (x o ) and s o ∈ S.
(i) A germ v of ( j * E L ) ( xo,so) is said to have moderate growth if for some open set U ε × U (s o ) as above, and for some (equivalently, any) identification L |U * ε ×U (so) , the corresponding germ in ( j * E L ) ( xo,so) has moderate growth. In particular, this holds for any local section of The subsheaf of j * E L consisting of local sections whose germs have moderate growth is denoted by E L . It satisfies j * E L = E L . It is called the Deligne extension of E L .
Remark 4.12. By definition, with the previous notation, v has moderate growth if and only if, on any such polysector U ε × U (s o ), for any isomorphism L| U * ε ×U (so) , for any family of local generators (g j ) of G on U (s o ), v can be written as j v j ⊗ g j with v j being holomorphic functions on the corresponding polysector in X * × S with moderate growth with respect to D.
As A mod X×S is stable by derivations with respect to X, E L is a D X×S/S ( * D)-submodule of j * E L .
Proof. We only prove parts (i) and (ii), and part (iii) will be a consequence of Theorem 4.15.
(i) A morphism of S-locally constant sheaves ϕ : L → L defines a morphism ϕ : E L → E L compatible with ∇, hence j * E L → j * E L compatible with ∇, and we only need to check that it sends E L to E L . This is straightforward from the definition.
(ii) From the commutative diagram (4.14) together with the natural morphism of Proposition 4.2(i), we obtain the morphism j * (E π * L , ∇) → π * j * (E L , ∇) and similarly j * (E π * L , ∇) → π * j * (E L , ∇). Let us consider an open subset U ε ⊂ X as in (4.8). Then U * ε is contractible and for any s o ∈ S , L |U * . In order to simplify the notation, we set S = U (s o ) and S = π −1 (U (s o )). We also set p = p Uε , so that p U * ε = p • j. According to (4.10), we are led to proving that the natural morphism Following the proof of Proposition 4.4(i), we are led to proving that the natural morphism j Relative regular Riemann-Hilbert correspondence II sends A mod Uε×S to π * A mod

Uε×S
. This follows from the definition of moderate growth, owing to the properness of π.

4.c Regular holonomicity of the Deligne extension of an S-locally constant sheaf
We continue to refer to Notation 4.5 and assume that Y = D has normal crossings.
Theorem 4.15. Assume that L is a coherent S-locally constant sheaf on X * × S. Then: (ii) the functor L → E L from the category of coherent S-locally constant sheaves on X * × S to that of D X×S/S -modules is fully faithful; (iii) as a D X×S/S -module, ( E L , ∇) is regular holonomic.
We make use of the following flattening result for a coherent O S -module (here, we only need a local version, but the corresponding global version also holds true).
Proposition 4.16. Near each s o ∈ S, there exists a projective modification π : S → S with S smooth such that the torsion-free quotient of π * G is O S -locally free.
Proof. We first apply the flattening theorem [Ros68,Theorem 3.5] to the coherent sheaf G. There exists thus a projective modification π : S → S such that π * G, when quotiented by its O S -torsion, is O S -locally free. We then apply resolution of singularities π : S → S of S in the neighborhood of the projective subset π −1 (s o ) (cf. [Hir64, § 7, Main Theorem I ]) and denote by π the morphism π • π , which answers the question.

Proof of Theorem 4.15(i).
We recall the proof of [MFS19, Theorem 2.6] for the O X×S ( * D)coherence. The problem is local on X × S. We thus assume that X × S is a small neighborhood of (x o , s o ) as above. In such a neighborhood, giving the local system is equivalent to giving . . , , and zero otherwise. Then the monodromy representation of ∇ on E G|X * ×S is given by T 1 , . . . , T , hence an isomorphism L ( E G|X * ×S ) ∇ , from which one deduces, according to Corollary 4.3, an isomorphism (4.17) It follows that E L E G|X * ×S and we are thus reduced to proving that E G|X * ×S = E G (we have trivialized the locally constant sheaf but the connection is not trivial anymore).
Remark 4.18. Let p −1 X * G be the constant S-local system on X * × S. Thus, locally on X × S, Let us prove the inclusion E G|X * ×S ⊂ E G . Let v ∈ (j * E G ) (xo,so) be locally defined on U × U (s o ). On any polysector U * ε × U (s o ), we can choose logarithms of x i (i = 1, . . . , ) and the L. Fiorot, T. Monteiro Fernandes and C. Sabbah

be a family of local generators of G on U (s o ). The entries of the matrices of
with respect to this family have moderate growth. If, on such a polysector, w writes j w j ⊗ g j , where w j are sections of A mod We conclude that, for any such polysector, the moderate growth condition on v is equivalent to The reverse inclusion also follows from the moderate growth of the entries Proof of Theorem 4.15(ii). We already know, by the Riemann-Hilbert correspondence of [Del70,Theorem 2.23], that the functor L → E L is fully faithful, and it is clearly exact, so it remains to prove that E L → E L is so.
). Our method is to reduce to the case where G is locally free of rank one and then prove Theorem 4.15(iii) for it. We argue by induction on the lexicographically ordered pair (dim S, rk G), where rk G is the rank of G at a general point of S. The case dim S = 0 and rk G arbitrary is well-known. We thus assume that d = dim S 1 and r = rk G 0, and that Theorem 4.15(iii) holds for ( E G , ∇) for Step 1: the case (d, 0). If rk G = 0, G is an O S -torsion module. As the question is local, we can assume that the support of G is contained in a hypersurface T ⊂ S locally presented as in the local Setting 3.12. Then G is q −1 O S -coherent and is endowed with an endomorphism t (i.e., multiplication by t), so that the natural ∇) when regarded as a q −1 D X×S /S -module. By the induction hypothesis, it is regular holonomic. By Corollary 3.14, we conclude that ( E G , ∇) is regular holonomic, so that Theorem 4.15(iii) holds for ( E G , ∇).
Step 2: reduction after proper surjective generically finite base change. Let π : S → S be a proper surjective generically finite morphism of complex manifolds and let p : X × S → S denote the projection. Let us assume that Theorem 4.15(iii) holds for (O X×S ( * D) ⊗ p −1 O S p −1 π * G, π * ∇). The purpose of step 2 is to show that, under such an assumption, Theorem 4.15(iii) holds for ( E G , ∇).
From now on, we assume dim S = d and rk G 1. Let G be the O S -torsion subsheaf of G. We note that the endomorphisms A i of G preserve G , so the exact sequence 0 → G → G → G → 0 with G being torsion-free gives rise to an exact sequence As Theorem 4.15(iii) holds for ( E G , ∇) by step 1, it is enough to prove Theorem 4.15(iii) for ( E G , ∇). In other words, we may assume that G is torsion-free.
By assumption, there exists a closed analytic subset T of codimension 1 in S such that, setting T = π −1 (T ), the morphism π : S T → S T is finiteétale. We have dim S = dim S and dim T < dim S.
Remark that we have, for any holomorphic map π : S → S, and denoting for clarity by π the map Id ×π : X × S → X × S, the following list of canonical isomorphisms: (4.20) We conclude that, for each j, we have L j π * ∇ x k ∂x k = d ⊗ Id + Id ⊗L j π * A k . Furthermore, under the assumption on π for this step, according to the projection formula for R π * applied to ( * ), we have By induction, if j = 0, Theorem 4.15(iii) holds for (O X×S ( * D) ⊗ p −1 O S p −1 L j π * G, L j π * ∇), according to the argument given in step 1, because L j π * G is supported on T , hence is a torsion module. Thus, Theorem 4.15(iii) holds for Lπ * E G (i.e., for each of its cohomology modules L j π * E G ) because for j = 0 it is the initial assumption. As E G is O-quasi-coherent (cf. Remark 2.5(i)), one deduces from Lemma 2.13 that each L j π * E G is π-good. Then, by Theorem 3.15, Theorem 4.15(iii) holds for Rπ * (Lπ * E G ), hence for Both modules are O X×S ( * D)-coherent, hence so is the kernel of this morphism. On S T , we claim that this morphism is injective: indeed, since π : S T → S T is finiteétale, the trace morphism tr π : π * O S T → O S T defined by tr π (ϕ)(s) = (1/ deg π) s ∈π −1 (s) ϕ(s ) satisfies tr π • ι = Id, if ι denotes the natural morphism O S T → π * O S T ; hence E G|S T is locally a direct summand of the right-hand side. As G, hence E G , is assumed to be L. Fiorot, T. Monteiro Fernandes and C. Sabbah O S -torsion-free, the kernel is O S -torsion-free, O X×S ( * D)-coherent and S-supported on T . It is thus zero and this morphism is injective.
Furthermore, E G is D X×S/S -coherent: indeed, as Theorem 4.15(iii) holds for H 0 (Rπ * (Lπ * E G )) and E G is O-good, it follows that E G is D X×S/S -coherent, and then regular holonomic by Proposition 3.2(iii), so that Theorem 4.15(iii) holds for E G .
Step 3: reduction to the case where G is O S -locally free. We choose π as in Proposition 4.16 and by step 2 we can assume from the start that the torsion-free quotient G of G is O S -locally free. By step 1, it is enough to prove Theorem 4.15(iii) for ( E G , ∇), i.e., we can assume (and we do assume from now on) that G is O S -locally free.
Step 4: the case where G is O S -locally free. We still work locally on S and we assume that G is O S -free of rank r 1. By the induction hypothesis, Theorem 4.15(iii) holds for any ( E G , ∇) with rk G < r. Locally let us fix an O S -basis of G and let A 1 (s) denote the matrix of A 1 in this basis. Let Σ ⊂ S × C be the zero locus of P := det(α 1 Id −A 1 ) and let σ : Σ → S denote the projection. As P is a Weierstrass polynomial with respect to the variable α 1 (considered as a coordinate on C) with coefficients in O S , σ is a finite morphism of degree deg σ and Σ is defined by the corresponding reduced Weierstrass polynomial. We note the following two properties related to σ and similarly to Id X ×σ. Let π : S → Σ be a resolution of singularities of Σ, so that the natural composed map (σ • π) : S → S is projective and generically finite. Let us set G = ker(α 1 Id −A 1 • σ • π) ⊂ (σ • π) * G, where we regard α 1 as a function S → C. Noting that σ • π is generically a local isomorphism, it follows by construction that rk G 1. We consider the exact sequence 0 → G → (σ • π) * G → G → 0 which satisfies 0 < rk G and rk G < rk G, and which is preserved by the endomorphisms (σ • π) * A i , so that it induces an exact sequence of D X×S/S -modules: If rk G < rk G, we can apply induction to ( E G , ∇), ( E G , ∇) and conclude that Theorem 4.15(iii) holds for (σ • π) * ( E G , ∇), hence for ( E G , ∇) according to step 2.
If rk G = rk G, then rk G = 0, so step 1 applies to ( E G , ∇), and we are reduced to proving Theorem 4.15(iii) for ( E G , ∇), i.e., we can assume that A 1 = α 1 Id. Iterating the argument, we are reduced to the case where A i = α i Id for i = 1, . . . , , where α 1 , . . . , α are holomorphic functions on S, and G is O S -locally free. By considering a local basis of G, it suffices to consider the case where rk G = 1.
Step 5: the case where G is O S -locally free of rank one. We now consider ( E G , ∇) = (O X×S ( * D), d X×S/S + i=1 α i (s)dx i /x i ). The argument for obtaining Theorem 4.15(iii) is then similar to that used in the proof of [MFS19, Corollary 2.8]. One can assume that, in the Relative regular Riemann-Hilbert correspondence II neighborhood of s o and for any i, α i (s) ∈ Z ⇒ α i (s) = 0. Then there exists a surjective morphism sending 1 to 1/x 1 · · · x , which is easily seen to be an isomorphism by the condition on (α i ) i=1,..., .
Proof of Proposition 4.13(iii). The statement is local, so, as a consequence of (4.17) in the proof of Theorem 4.15(i) and keeping the same notation, we are reduced to proving π * ( E G , ∇) ( E π * G , ∇), that is, which follows by taking the 0-cohomology in (4.19).

4.d D X×S/S -modules of D-type
Recall Notation 4.5. In this section, we exhibit a family of regular holonomic D X×S/S -modules, that we call of D-type and we prove in Proposition 4.26 a special case of the analogue in the relative setting of [KK81,Theorem 2.3.2] asserting that the restriction functor to the complement of the divisor is an equivalence of categories. The general case will be obtained in Theorem 6.17.
Let Y be a closed hypersurface of X.
Definition 4.23. We say that a coherent D X×S/S ( * Y )-module L is of D-type if: , equivalently, L |X * ×S is D X * ×S/S -holonomic with characteristic variety contained in the zero section; (b) for each s ∈ S, the cohomology of Li * s L is D X -regular holonomic (in particular, D X -coherent); in other words, L satisfies condition (Reg 2) and, thus, condition (Reg 1) (cf. Proposition 3.2).
We say that L is strict if L is p −1 O S -locally free.
We denote by Mod Y (D X×S/S ( * Y )) the full subcategory of Mod(D X×S/S ( * Y )) whose objects are coherent D X×S/S ( * Y )-modules of D-type.
Proof. The good behavior of Definition 4.23(a) by base pullback is clear. Let us check that of Definition 4.23(b). Arguing as in Lemma 2.14, we see that Lπ * L is an object of D b coh (D X×S /S ( * D)). For any s ∈ S we have Li * s Lπ * L Li * π(s ) L, so the complex Li * s Lπ * L has D X -regular holonomic cohomology. By Proposition 3.2(i), each Li * s L j π * L also has D X -regular holonomic cohomology.  Proof. We note that the question is local on X × S, so we may replace E L with E G|X * ×S as in (4.17). Namely, we have We note the following.
1439 L. Fiorot, T. Monteiro Fernandes and C. Sabbah • If L is strict, the second assertion follows from the first one, because E L is D X×S/S -regular holonomic (Theorem 4.15(iii)). • For any coherent D X×S/S ( * D)-module of D-type, the morphism ψ is injective. Indeed, according to the first point of Definition 4.23, the restriction of ψ to X * × S is an isomorphism. The assertion follows from the O-quasi-coherence of L (cf. Remark 2.5(ii)). We thus identify L with a D X×S/S ( * D)-submodule of j * E L . • Due to Definition 4.23(b), Proposition 4.25(ii) amounts to holonomicity of L (in particular, D X×S/S -coherence).
Proof of Proposition 4.25(i). We assume that G is O S -locally free of finite rank. We can mimic the end of the proof of [MFS19, Proposition 2.11] to directly show that ψ is an isomorphism L ∼ −→ E L because, although in [MFS19, Proposition 2.11] we assumed the D X×S/Scoherence of L, that proof works under the weaker assumption of its D X×S/S ( * Y )-coherence. This shows Proposition 4.25(i). In particular, this implies the D X×S/S -regular holonomicity of L.
Proof of Proposition 4.25(ii). We now prove the holonomicity of L by assuming only that G is O S -coherent. As in the proof of Theorem 4.15(iii), we argue by induction on the dimension of S. The case where S is a point is well-known [KK81,Theorem 2.3.2]. We assume that the result holds if dim S d − 1 (d 1) and that dim S = d.
Step 1: the case where G is a torsion O S -module. As we work locally, we can assume that the support of G is contained in a hypersurface T of S, having equation h = 0, endowed with a finite morphism q to S of dimension d − 1, and that h k G = 0 for some k 1. We claim that any local section m of L is annihilated by h k . Indeed, for any such m, h k m is zero on X * × S and we can apply the result of Remark 2.5(ii).
Therefore, Supp S L is contained in T . By the induction hypothesis and the equivalence recalled in Setting 3.12, we deduce that L is q −1 D X×S /S -holonomic, hence, arguing as in Remark 3.13, we conclude that L is D X×S/S -holonomic, as desired.
Step 2: the case where G tf is O S -locally free. We consider the exact sequence (4.1) and we assume that G tf is O S -locally free. We also consider the similar exact sequence 0 → L t → L → L tf → 0. Let us prove that L tf is of D-type. First, L t is easily seen to be D X×S/S ( * D)-coherent, hence so is L tf . Next, L tf satisfies the first point of Definition 4.23, with local system p −1 G tf .
For the second point, we note that the image of i * As a consequence, it is D X -regular holonomic because i * s L is so, and thus i * s L tf is regular holonomic. On the other hand, for j < 0, L j i * s L tf is O-quasi-coherent and supported on D by our assumption on G tf , so L j i * s L tf = 0 by Lemma 2.3(ii). In conclusion, L tf is of D-type, and we also deduce that L t is of D-type. By step 1 and Proposition 4.25(i), L is holonomic if G tf is O S -locally free.
Step 3: the general case. Let π : S → S be a projective modification as in Proposition 4.16 such that the S -torsion-free quotient of π * G tf is O S -locally free. The S -torsion-free quotient of π * G, being equal to it, is then O S -locally free.
By Lemma 4.24, Lπ * L has cohomology in Mod D (D X×S /S ( * D)). We can therefore apply steps 1 and 2 to deduce that π * L is D X×S /S -holonomic, and it is regular by Definition 4.23(b). It is moreover π-good (cf. Lemma 2.13). Hence, π * π * L is D X×S/S -regular holonomic (Theorem 3.15).
As L is D X×S/S ( * D)-coherent, the image L of the adjunction morphism L → π * π * L is D X×S/S -coherent, hence regular holonomic (Proposition 3.2(iii)) and its kernel is D X×S/S ( * D)coherent. It follows that the latter, which clearly satisfies the first point of Definition 4.23, also satisfies the second point because L and L do so (cf. Proposition 3.2). As π is biholomorphic above S T for some closed analytic subset T of codimension 2, this kernel satisfies the assumption of step 1. It is thus D X×S/S -holonomic, hence L is D X×S/S -holonomic, as desired.
Note that the assignment L → L = H 0 DR(L |X * ×S ) is a functor which takes values in the category of coherent S-locally constant sheaves, because the characteristic variety of L |X * ×S is contained in the zero section.
As the natural morphism L → j * E L sends isomorphically L to E L (Proposition 4.25(i)), the functor L → E L from the category of S-locally constant sheaves on X * × S which are p −1 O S -locally free of finite rank to that of regular holonomic D X×S/S -modules of D-type is essentially surjective. That it is fully faithful has been proved in Theorem 4.15(ii).

4.e Proof of Theorem 1
Although the next proposition is not general enough to prove Theorem 1, it will be one of the main tools for its proof.
Let f : X → Y be a morphism of real or complex analytic manifolds, we denote by D Y ←X/S and by D X→Y /S the relative transfer bi-modules.  Zo O S -module L. One can find a complex manifold X together with a divisor with normal crossings Y ⊂ X and a projective morphism f : X → X which induces a biholomorphism X Y ∼ −→ Z o (cf., e.g., [AHV18,Prologue,Theorem 4]). We set δ = dim Z − dim X = dim X − dim X 0. For each , we consider the D X ×S/S -module M := H D f * M. Although it is not yet known to be coherent, it is f -O-good in the sense of Definition 2.2 (Proposition 2.11(i)). By considering the filtration by the order of the pole along Y , one checks that M ( * Y ) is also f -O-good (cf. Remark 2.5(i)). If = δ, the sheaf-theoretic restriction of M to (X Y ) × S is zero, therefore M ( * Y ) = 0 owing to O-quasi-coherence (cf. Lemma 2.3(ii)). As O X ×S ( * Y ) is flat over O X ×S , we conclude that Let Y i (i = 1, . . . , p) be hypersurfaces of X defined as the zero set of holomorphic functions h i : X → C, set Y = i Y i and let N be a D X×S/S ( * Y )-module. We regard N as an O X×S ( * Y )-module with flat relative connection ∇, and for a tuple α = (α 1 , . . . , α p ) of holomorphic functions α i : S → C, we denote by Nh α the O X×S -module N, endowed with the flat relative connection ∇ + i α i Id ⊗dh i /h i . The functor N → Nh α is an auto-equivalence of the category Mod(D X×S/S ( * Y )), as well as of Mod coh (D X×S/S ( * Y )). We have a functorial isomorphism is also regular holonomic. Furthermore, the D X×S/S -submodule of q * (M( * Y ))f s generated by the image of q * M ⊗ 1 · f s is also regular holonomic, according to Proposition 3.2(iii), as it is clearly coherent (being locally of finite type in a coherent D X×S/S -module). This property is the S-analytic variant of [Mai23,Proposition 13], with the regularity assumption however.
As M is regular holonomic, it is good, and so is q * (M( * Y ))f s . Therefore, if X is compact, the D-module pushforward D p * [q * (M( * Y ))f s ] is an object of D b coh (O S ). This is the generalized Mellin transform of M with respect to (f 1 , . . . , f p ).

4.f Another characterization of regular holonomicity
For a closed analytic subset Y of X, we denote by We consider the exact sequence of sheaves supported on Y × S: [MFS13,Theorem 3.7]. On the one hand, by mimicking the proof when S is reduced to a point (cf., e.g., [Meb04,), one finds a natural isomorphism hence the S-C-constructibility of the latter complex. On the other hand, we have natural isomorphisms showing S-C-constructibility of the latter complex, and therefore that of RHom D X×S/S (M, Q Y ×S ). Proof of Theorem 4.38. We first remark that the theorem holds if S is reduced to a point, according to [KK81,(6.4.6) and (6.4.7)].
If Relative regular Riemann-Hilbert correspondence II

Construction of the relative Riemann-Hilbert functor RH S X
In this section, we extend the definition of the functor RH S X introduced in [MFS19] when dim S = 1 to the case dim S 2. We check that it satisfies properties similar to those explained in [MFS19, FMFS21].

5.a Reminder on the subanalytic site and complements
We first recall the main results in [MFP21]. We consider the site in the real analytic manifold S R given by the usual topology, that is, where the family Op(S) consisting of all open sets. On the other hand, we have the subanalytic site X sa underlying the real analytic manifold X R for which the family of open subsets Op(X sa ) consists of subanalytic open subsets in X R . Lastly, we let X sa × S be the subanalytic site underlying the real analytic manifold X R × S R , for which the family of open sets Op(X sa × S) consists of those which are finite unions of products In particular, when U is relatively compact I is finite but J needs not to be so even if V is relatively compact.
We have the following commutative diagram, where the arrows are natural morphisms of sites induced by the inclusion of families of open subsets.  [MFP21,§3]).
The following proposition generalizes [MFS19, Proposition 3.3]. Its proof is completely similar.  Remark 5.5. Recall the diagram (5.1). From the fact that a * is fully faithful and that a −1 a * = Id, we have ρ S * = a −1 ρ * (thus, Rρ S * = a −1 Rρ * ) as explained in [MFP21] before Proposition 3.1, we deduce that, for any open subset U × V ∈ Op c (X sa × S), the constant sheaf C U ×V on the site L. Fiorot, T. Monteiro Fernandes and C. Sabbah because the isomorphisms hold true with ρ instead of ρ S (cf. [MFP14,Lemma 3.6(2)]).
Remark 5.6. For any S, the site X sa × S is a ringed site both relatively to the sheaf ρ S * (p −1 X O S ) and to the sheaf ρ S! O X×S (cf. [KS06, p. 449]), and ρ S is a morphism of ringed sites in both cases. Thus, according to [KS06, Lemma 18.3.1, Theorem 18.6.9], ρ −1 [MFP21,Proposition 3.16]). Let π : S → S be a morphism of complex manifolds.
(5.6 * ) We have Lemma 5.7. Let π : S → S be a morphism. Then we have an isomorphism of functors on Relative regular Riemann-Hilbert correspondence II

5.b The construction of RH S X and behavior under pushforward
For this section, we refer to the notation introduced in [FMFS21, §2.2]. However, instead of making use of the morphism ρ of (5.1) as in [FMFS21, §2.2], we replace it with ρ S . Most proofs do not require any change.
We define the triangulated functor TH S X : This can be seen as follows: noting that (cf. (5.1)) ρ * = a * • ρ S * , a −1 a * = Id and a −1 O t,S, The proof is stepwise similar to that with dim S = 1 in [FMFS21, Lemma 2.5] using Proposition A.6. We omit it here and the detailed proof can be found in the arXiv version of this paper. Proposition 7.1 of [Kas84] (see also [KS96, Theorem 5.7 (5.12)]) has a relative version already used in [MFS19], the proof of which is given in the following.
L. Fiorot, T. Monteiro Fernandes and C. Sabbah Theorem 5.11. Let f : X → Y be a morphism of complex analytic manifolds, let F ∈ D b R-c (p −1 X O S ), and assume that f is proper on Supp F . Then there is a canonical isomorphism in D b (D X×S/S ) which is compatible in a natural way with the composition of morphisms: In view of Theorem 5.10, where we replace F by Rf * F , and by adjunction, we derive a natural morphism Lemma 5.14. The morphism μ F is an isomorphism.
The proof of this lemma, which is a relative version of [KS96, Theorem 4.4], is given in the appendix and some more details are also given in the arXiv version of this article. Applying ] to both terms of (5.13) the right term becomes RH S Y (Rf * F ). For the left-hand side of (5.13) we obtain Here part (a) follows from [Kas03,(A10)] and part (b) follows from the relative version of [Kas84,Lemma 7.2] which asserts that Therefore, by applying RHom D Y ×S/S (O Y ×S , •)[d Y ] to the left-hand side of (5.13) we obtain an isomorphism with D f ! RH S X (F ) which concludes the construction of the morphism (5.11 * ). Lemma 5.14 shows that it is an isomorphism.

5.c Riemann-Hilbert correspondence for Deligne's extensions
We recall that, for F ∈ D b (p −1 O S ) one defines D F := RHom p −1 O S (F, p −1 O S ) and DF := Let L be a coherent S-locally constant sheaf on X * × S. We consider the setting of Notation 4.5 and assume that Y = D has normal crossings in X.
Relative regular Riemann-Hilbert correspondence II Proposition 5.15. Let L be a coherent S-locally constant sheaf on X * × S and let E L be the associated Deligne extension. Then: • the complex of D X×S/S -modules RH S X (j ! D L) [−d X ] is isomorphic to E L and thus it is regular holonomic; Proof. We adapt the idea of the proof of [MFS19, Lemma 4.2]. Let us prove the second statement assuming the first one holds true. First, the following lemma is similar to [MFS19, Lemma 3.19].
We make use of the following result of [MFP21], the proof of which we recall with details as it is used in an essential way in the following.
Lemma 5.18 [MFP21,Proposition 3.32]. The complex E S L is concentrated in degree zero. Proof. Assume first that L is the constant local system p −1 X * G with G being O S -coherent. As the question is local on X × S, we can assume that G admits a finite resolution O • S → G by free O S -modules of finite rank. Then we have RHom(ρ S * j ! C X * ×S , (•)) ∼ −→ Rj * RHom(ρ S * C X * ×S , j −1 (•)) Rj * j −1 (•).
As a consequence, we obtain an isomorphism . It follows then from [MFP21,Proposition 3.24(1)] applied to C X * ×S = C X * C S , that the latter complex is isomorphic to T Hom(C X * ×S , O X×S ), so that, according to [Kas84,Lemma 7.5 (5.23) By the first part of the proof, we deduce that E S L is concentrated in degree zero. This concludes the proof of Proposition 5.18. Proof. First, applying the commutation of ρ −1 S with j −1 together with the analogue of [MFS19, Corollary 3.24] entails that E S L and E L coincide with E L when restricted to X * × S. On the one hand, by construction, E L is naturally a D X×S/S -submodule of j * E L . Let us check, on the other hand, that E S L is also naturally a D X×S/S -submodule of j * E L . From the natural morphism of functors Id → Rρ S * ρ −1 S on Mod(C Xsa×S ) we derive a natural morphism ρ −1 S Rj * → Rj * ρ −1 S : denoting for a moment by j S * the morphism in the subanalytic site, we have an isomorphism of functors Rj S * • Rρ S * ∼ −→ Rρ S * • Rj * (cf. [Pre08, Proposition 2.2.1(i)], as the unique morphism such that (6.3) M,M (β M ) = Idp Sol(M) . One classically deduces from the full faithfulness of p Sol in the absolute case (a consequence of the Riemann-Hilbert correspondence of [Kas84] and [Meb84]) that, for each s ∈ S, β Li * s M is an isomorphism. Therefore, by a Nakayama-type argument [MFS19, Proposition 1.9 and Corollary 1.10], β M is an isomorphism.
In order to check that β M is functorial with respect to M, that is, it defines a morphism of functors Id D b rhol (D X×S/S ) → RH S X • p Sol X , we consider as in [MFS19, p. 668], for a morphism ϕ : M → N, the following commutative diagram.
Hom D X×S/S (M, RH S X ( p Sol X (M))) Such a morphism (6.4) is given by [FMFS21,(14)]. In view of Proposition 5.9 we can argue as in [FMFS21], where it is also shown that, for each s o ∈ S, Li * so (6.4) can be identified with the morphism constructed in the absolute case by Kashiwara [Kas84,Corollary 8.6], hence it is an isomorphism. To conclude that (6.4) is an isomorphism, we apply a Nakayama-type argument [MFS13,Proposition 2.2].
To construct the morphism (6.4) we need to check a finiteness property. The proof of (6.2) is thus concluded with the proof of the following assertion.  . We argue by induction on the pair (dim S, dim Supp X M) ordered lexicographically. For that purpose we introduce the following notation.
• For d 0, we denote by (6.5) d , respectively, (6.6) d , the corresponding statement concerning any S satisfying dim S d. • For k 0, we introduce the assertion (6.6) d,k by requiring that the property holds for dim S d − 1 or dim S = d and dim Supp X M k.
We are thus reduced to proving the following.
We can reduce to the case M, N are regular holonomic D X×S/S -modules. Let Z denote the X-support of M. As the assertion is local, we can assume that there exists a hypersurface Y of X such that Y contains the singular locus of Z and dim Z ∩ Y < k. Recall that localization along Y × S preserves regular holonomicity (cf. Corollary 4.31).
Lemma 6.11. It is enough to prove the assertion (6.10) for M, N such that M = M( * Y ) and N = N( * Y ).
Proof. The assertion for M follows from the property that RΓ [Y ] M belongs to D b rhol (D X×S/S ) (Corollary 4.32) and has X-support of dimension < k.
For the assertion concerning N, we recall the argument given at the end of the proof of [FMFS21, Theorem 3]. It is enough to prove that (6.10) holds if N = RΓ [Y ] N. We have, by [MFS13,(3) As N has D X×S/S -coherent cohomology and is supported on Y × S, we have RHom D X×S/S (DN, (DM)( * Y )) = 0.