# Relative Regular Riemann-Hilbert correspondence

Luisa Fiorot  Dipartimento di Matematica “Tullio Levi-Civita” Università degli
Studi di Padova
Via Trieste, 63
35121 Padova Italy<EMAIL_ADDRESS>,  Teresa Monteiro Fernandes  Centro
de Matemática e Aplicações Fundamentais-CIO and Departamento de Matemática da
Faculdade de Ciências da Universidade de Lisboa, Bloco C6, Piso 2, Campo
Grande, 1749-016, Lisboa Portugal<EMAIL_ADDRESS>and  Claude Sabbah
CMLS, CNRS, École polytechnique, Institut Polytechnique de Paris, 91128
Palaiseau cedex, France<EMAIL_ADDRESS>
###### Abstract.

On the product of a complex manifold $X$ by a complex curve $S$ considered as
a parameter space, we show a Riemann-Hilbert correspondence between regular
holonomic relative $\mathscr{D}$-modules (resp. complexes) on the one hand and
relative perverse complexes (resp. $S$-$\mathbb{C}$-constructible complexes)
on the other hand.

###### Key words and phrases:

relative $\mathscr{D}$-module, De Rham functor, regular holonomic
$\mathscr{D}$-module

###### 2010 Mathematics Subject Classification:

14F10, 32C38, 35A27, 58J15

The research of L. Fiorot was supported by project PRIN 2017 Categories,
Algebras: Ring-Theoretical and Homological Approaches (CARTHA). The research
of T. Monteiro Fernandes was supported by supported by Fundação para a Ciência
e a Tecnologia, UID/MAT/04561/2019.

###### Contents

  1. 1 Review on the relative holonomic $\mathscr{D}_{X\times T/T}$-modules and constructible complexes
    1. 1.a Holonomic $\mathscr{D}_{X\times T/T}$-modules
    2. 1.b Relative constructible and perverse complexes
    3. 1.c The relative solution functor
    4. 1.d Regular holonomic complexes of $\mathscr{D}_{X\times S/S}$-modules
  2. 2 The relative Riemann-Hilbert functor $\operatorname{RH}^{S}$
    1. 2.a Relative subanalytic sites and relative subanalytic sheaves
    2. 2.b The functors $\operatorname{TH}^{S}$ and $\operatorname{RH}^{S}$
    3. 2.c Some functorial properties
    4. 2.d Behaviour of $\operatorname{RH}^{S}$ under finite ramification over $S$
  3. 3 Relative Riemann-Hilbert correspondence
    1. 3.a Proof of Theorems 1 and 3 in the torsion case
    2. 3.b Proof of Theorem 1 in the strict case
    3. 3.c End of the proof of Theorem 2
    4. 3.d End of the proof of Theorem 3
    5. 3.e Proof of Corollary 4

## Introduction

Let $X$ and $T$ be complex manifolds. Relative $\mathscr{D}$-modules are
modules over the sheaf $\mathscr{D}_{X\times T/T}$ of differential operators
relative to the projection $p_{X}:X\times
T\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}T$.
_Holonomic_ $\mathscr{D}_{X\times T/T}$-modules encode holomorphic families of
holonomic $\mathscr{D}_{X}$-modules parametrized by $T$ whose characteristic
variety is contained in a fixed Lagrangian subset $\Lambda\subset T^{*}X$.
This is a strong condition avoiding confluence phenomena. It describes
nevertheless the generic behaviour of a deformation of a holonomic
$\mathscr{D}_{X}$-module. From this point of view, it is natural to emphasize
those $\mathscr{D}_{X\times T/T}$-modules which are
$p^{-1}_{X}\mathscr{O}_{T}$-flat, that we call _strict_.

There exists a solution functor from the category of _holonomic_
$\mathscr{D}_{X\times T/T}$-modules to that of perverse
$\mathbb{C}$-constructible complexes of $p^{-1}_{X}\mathscr{O}_{T}$-modules,
objects defined in [15]. Strictness (i.e., flatness) on the
$\mathscr{D}_{X\times T/T}$-module side corresponds to the property that the
Verdier dual of a perverse complex (as a complex of
$p^{-1}_{X}\mathscr{O}_{T}$-modules) is also perverse.

In their previous work [17], Monteiro Fernandes and Sabbah have introduced the
notion of relative regular holonomic $\mathscr{D}_{X\times T/T}$-modules. For
example, if $T=\mathbb{C}^{*}$, a $\mathscr{D}_{X\times T/T}$-module
underlying a regular mixed twistor $\mathscr{D}$-module (cf.[13]) is regular
holonomic in this sense.

######  Convention and notation.

In this article, we mainly consider the case where the parameter space $T$ has
dimension one, and we emphasize this assumption by denoting it by $S$.
Therefore, throughout this article, $S$ will denote a complex curve. On the
other hand, we will set $d_{X}=\dim X$.

Our main result is a Riemann-Hilbert correspondence for regular holonomic
$\mathscr{D}_{X\times S/S}$-modules, in the following form. We let
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{rhol}}(\mathscr{D}_{X\times
S/S})$ denote the full subcategory of
$\operatorname{\mathsf{D}}^{\mathrm{b}}(\mathscr{D}_{X\times S/S})$ consisting
of complexes having regular holonomic cohomology modules (cf.Section 1.d for
details) and
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})$
the category of $\mathbb{C}$-constructible complexes of
$p^{-1}_{X}\mathscr{O}_{S}$-modules (cf.Section 1.b and [15]).

######  Theorem 1.

The functors

$\displaystyle\operatorname{{}^{\mathrm{p}}Sol}_{X}:\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{rhol}}(\mathscr{D}_{X\times
S/S})$
$\displaystyle\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})$
$\displaystyle\operatorname{RH}^{S}_{X}:\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})$
$\displaystyle\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{rhol}}(\mathscr{D}_{X\times
S/S})$

are quasi-inverse equivalences of categories.

The functor $\operatorname{{}^{\mathrm{p}}Sol}_{X}$ is the solution functor
shifted by the dimension of $X$ (cf.[15, §3.3] and Section 1.c), and
$\operatorname{RH}^{S}_{X}$ is the relative Riemann-Hilbert functor (cf.[17,
§3.4] and Section 2). One direction of the correspondence, namely
$\operatorname{Id}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{{}^{\mathrm{p}}Sol}_{X}\circ\operatorname{RH}^{S}_{X}$,
was proved in [17, Th. 3], and a particular case of this correspondence was
obtained as Theorem 5 in loc.cit., namely, if $\mathscr{M}$ underlies a
regular mixed twistor $\mathscr{D}$-module, then $\mathscr{M}$ can be
recovered from $\operatorname{{}^{\mathrm{p}}Sol}_{X}(\mathscr{M})$ up to
isomorphism by the formula
$\mathscr{M}\simeq\operatorname{RH}^{S}_{X}(\operatorname{{}^{\mathrm{p}}Sol}_{X}(\mathscr{M}))$.

The methods used in the present paper rely on the previous works [15], [17] as
well as [19], [14], [3]. The main tools in the proof given in [17] are the
good functorial properties satisfied by holonomic $\mathscr{D}$-modules
underlying mixed twistor $\mathscr{D}$-modules, which include stability under
pullback, localization along an hypersurface and direct image by projective
morphisms.

The main problem to extend these results to more general situations is the bad
behaviour of $\mathscr{D}_{X\times S/S}$-holonomicity by pullback in general
([16, Ex. 2.4]). We realized however that we can avoid these general arguments
for _regular_ holonomic $\mathscr{D}_{X\times S/S}$-modules. We replace them
by proving that regular holonomiciy behaves well with respect to pullback:

######  Theorem 2.

Let
$\mathscr{M}\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{rhol}}(\mathscr{D}_{X\times
S/S})$ and let
$f:Y\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X$ be a
morphism of complex manifolds, then
${}_{\scriptscriptstyle\mathrm{D}}f^{*}\mathscr{M}\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{rhol}}(\mathscr{D}_{Y\times
S/S})$.

Let us indicate the main points in the proof of Theorem 1 in Section 3. The
first tool is [17, Th. 3] which asserts that there exists a natural
transformation

$\operatorname{Id}_{\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})}\mathchoice{\xrightarrow{\textstyle\kern
4.0pt\alpha\kern
3.0pt}}{\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}}{}{}\operatorname{{}^{\mathrm{p}}Sol}_{X}\circ\operatorname{RH}^{S}_{X}$

providing a functorial isomorphism

$F\xrightarrow[\sim]{~{}\textstyle\alpha_{F}~{}}\operatorname{{}^{\mathrm{p}}Sol}_{X}(\operatorname{RH}^{S}_{X}(F))$

for any
$F\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})$.
We are then reduced to proving that there exists a natural transformation

$\operatorname{Id}_{\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{rhol}}(\mathscr{D}_{X\times
S/S})}\mathchoice{\xrightarrow{\textstyle\kern 4.0pt\beta\kern
3.0pt}}{\stackrel{{\scriptstyle\beta}}{{\longrightarrow}}}{}{}\operatorname{RH}^{S}_{X}\circ\operatorname{{}^{\mathrm{p}}Sol}_{X}$

such that, for any
$\mathscr{M}\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{rhol}}(\mathscr{D}_{X\times
S/S})$, denoting by

$\beta_{\mathscr{M}}:\mathscr{M}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{RH}^{S}_{X}(\operatorname{{}^{\mathrm{p}}Sol}_{X}(\mathscr{M})))$

the unique morphism such that
$\operatorname{{}^{\mathrm{p}}Sol}_{X}(\beta_{\mathscr{M}})\circ\alpha_{\operatorname{{}^{\mathrm{p}}Sol}_{X}(\mathscr{M})}=\operatorname{Id}_{\operatorname{{}^{\mathrm{p}}Sol}_{X}(\mathscr{M})}$,
we have

$\mathscr{M}\xrightarrow[\sim]{~{}\textstyle\beta_{\mathscr{M}}~{}}\operatorname{RH}^{S}_{X}(\operatorname{{}^{\mathrm{p}}Sol}_{X}(\mathscr{M})).$

The proof of the existence of such a $\beta$ is reduced to that of the
following result, which is equivalent to Theorem 1 by an argument already used
in [17, §4.3]:

######  Theorem 3.

For any
$\mathscr{M}\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{rhol}}(\mathscr{D}_{X\times
S/S})$ and for any
$F\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathbb{C}\textup{-c}}(p^{-1}_{X}\mathscr{O}_{S})$
the complex
$\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{X\times
S/S}}(\mathscr{M},\operatorname{RH}^{S}_{X}(F))$ belongs to
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})$.

The proof of Theorem 3 follows the ideas of [11, Lem. 4.1.4] (see also
Kashiwara’s proof [5, §8.3] in the absolute case).

Although we directly prove Theorem 1 in the torsion-free (strict) case and in
the torsion case separately, the general case needs Theorem 3 in order to
apply induction on the dimension of the support. The proof of Theorem 3 uses
the torsion-free case of Theorem 1 for the induction step and proceeds by
considering the case of $\mathscr{D}_{X\times S/S}$-modules of D-type (normal
crossing case).

We know (cf.[15, §3]) that the functor $\operatorname{{}^{\mathrm{p}}Sol}_{X}$
transforms duality on
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{hol}}(\mathscr{D}_{X\times
S/S})$ to Poincaré-Verdier duality on
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})$.
A consequence of the Riemann-Hilbert correspondence of Theorem 1 and the full
faithfulness of $\operatorname{RH}^{S}_{X}$ is the good behaviour of the
functor $\operatorname{RH}^{S}_{X}$ with respect to Poincaré-Verdier duality
on the one hand, and duality for $\mathscr{D}_{X\times S/S}$-modules on the
other hand.

######  Corollary 4.

For any
$F\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})$,
there exists an isomorphism in
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{rhol}}(\mathscr{D}_{X\times
S/S})$

$\boldsymbol{D}(\operatorname{RH}^{S}_{X}(F))\simeq\operatorname{RH}^{S}_{X}(\boldsymbol{D}F)$

which is functorial in $F$.

#### Acknowledgements

We are grateful to Luca Prelli for useful advising in Section 2.d. We thank
the anonymous referee for valuable comments which helped us to improve the
presentation of the article.

##  1\. Review on the relative holonomic $\mathscr{D}_{X\times T/T}$-modules
and constructible complexes

In this section, we review the main definitions and properties of the objects
entering the relative Riemann-Hilbert correspondence. We refer to [15, 17, 16]
for details. We also give supplementary properties that will happen to be
useful in the proof of the main results of this article.

###  1.a. Holonomic $\mathscr{D}_{X\times T/T}$-modules

We denote by $\mathscr{D}_{X\times T/T}$ the subsheaf of $\mathscr{D}_{X\times
T}$ of relative differential operators with respect to the projection

$p_{X}:X\times
T\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}T,$

that we simply denote by $p$ when there is no ambiguity. This is a Noetherian
sheaf of rings. A coherent $\mathscr{D}_{X\times T/T}$-module $\mathscr{M}$ is
said to be _holonomic_ if
$\operatorname{Char}(\mathscr{M})\subseteq\Lambda\times T$ for some closed
conic Lagrangian complex analytic subset $\Lambda$ of $T^{\ast}X$ (see [3,
Lem. 2.10] for a more precise description of the characteristic variety).

######  Example 1.1.

Let
$f_{1},\dots,f_{p}:X\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbb{C}$
be holomorphic functions. We let here $T=\mathbb{C}^{p}$ with coordinates
$s_{1},\dots,s_{p}$ and we consider the partially algebraic version
$\mathscr{D}_{X}[s_{1},\dots,s_{p}]$ of $\mathscr{D}_{X\times T/T}$. Let $M$
be a holonomic $\mathscr{D}_{X}$-module and let $m$ be a local section of $M$.
Extending [4] (which is the case $p=1$), one considers the
$\mathscr{D}_{X}[s_{1},\dots,s_{p}]$-submodule
$\mathscr{M}=\mathscr{D}_{X}[s_{1},\dots,s_{p}]\cdot m\cdot
f_{1}^{s_{1}}\cdots f_{p}^{s_{p}}$ of
$M[(\prod_{i}f_{i})^{-1}][s_{1},\dots,s_{p}]$. It is proved in [12, Prop. 13]
that $\mathscr{M}$ is relatively holonomic.

######  Example 1.2.

Set $S=\mathbb{C}^{*}$ with coordinate $z$. Any mixed twistor
$\mathscr{D}$-module, in the sense of [13], which consists of data
parametrized by $\mathbb{P}^{1}$, gives rise, when restricting the parameter
to $S=\mathbb{C}^{*}$, to a holonomic $\mathscr{D}_{X\times S/S}$-module (this
is by definition, cf.[18, Chap. 1]).

We denote by
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{hol}}(\mathscr{D}_{X\times
T/T})$ the full subcategory of
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{coh}}(\mathscr{D}_{X\times
T/T})$ whose complexes have holonomic cohomologies.

Given $t_{o}\in T$, let $i_{t_{o}}$ denote the inclusion
$X\times\\{t_{o}\\}\mathrel{\lhook\joinrel\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}X\times
T$. Following [15], we denote by

$Li^{*}_{t_{o}}:\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{hol}}(\mathscr{D}_{X\times
T/T})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{hol}}(\mathscr{D}_{X})$

the derived functor

(1)
$p^{-1}(\mathscr{O}_{T}/\mathfrak{m}_{t_{o}})\overset{L}{\otimes}_{p^{-1}_{X}\mathscr{O}_{T}}({\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}),$

where $\mathfrak{m}_{t_{o}}$ is the maximal ideal of functions vanishing at
$t_{o}$. Thanks to the variant of Nakayama’s lemma [17, Prop. 1.9 & Cor.
1.10], the family of functors $Li^{*}_{t_{o}}$ on
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{hol}}(\mathscr{D}_{X\times
T/T})$, for $t_{o}\in T$, is a conservative family, i.e., if
$\phi:\mathscr{M}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{N}$
is a morphism in
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{hol}}(\mathscr{D}_{X\times
T/T})$ such that $Li^{*}_{t_{o}}\phi$ is an isomorphism in
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{hol}}({\mathscr{D}}_{X})$ for
each $t_{o}\in T$ then $\phi$ is an isomorphism (or, equivalently, using the
mapping cone: if
$\mathscr{M}\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{hol}}(\mathscr{D}_{X\times
T/T})$ is such that $Li^{*}_{t_{o}}\mathscr{M}=0$ for each $t_{o}\in T$ then
$\mathscr{M}=0$).

Recall (cf.[15]) that a coherent $\mathscr{D}_{X\times T/T}$-module is said to
be _strict_ if it is $p^{-1}_{X}\mathscr{O}_{T}$-flat. If $\mathscr{M}$ is
strict, $Li^{*}_{t_{o}}\mathscr{M}$ consists of a single coherent
$\mathscr{D}_{X}$-module $i^{*}_{t_{o}}\mathscr{M}$ (in degree zero). For
example (recall that $\dim S=1$) a coherent $\mathscr{D}_{X\times S/S}$-module
$\mathscr{M}$ is strict if and only if it has no
$p^{-1}_{X}\mathscr{O}_{S}$-torsion. If $\mathscr{M}$ is possibly not strict,
we shall denote by $t(\mathscr{M})$ its (coherent) submodule consisting of
germs of sections which are torsion elements for the
$p^{-1}_{X}\mathscr{O}_{S}$-action, and
$f(\mathscr{M}):=\mathscr{M}/t(\mathscr{M})$ is called its strict (or torsion-
free) quotient. Therefore, the $\mathscr{D}_{X\times S/S}$-module
$\mathscr{M}$ is strict if and only if $\mathscr{M}\simeq f(\mathscr{M})$.

Given
$\mathscr{M}\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{hol}}(\mathscr{D}_{X\times
T/T})$, the functor

$\boldsymbol{D}(\mathscr{M}):=\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{X\times
T/T}}(\mathscr{M},\mathscr{D}_{X\times T/T}\otimes_{\mathscr{O}_{X\times
T}}\Omega^{\otimes^{-1}}_{X\times T/T})[d_{X}]$

provides a duality in
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{hol}}(\mathscr{D}_{X\times
T/T})$ but, contrary to the absolute case (i.e., $\dim T=0$), this functor is
not $t$-exact. For example, when the parameter space has dimension one, the
lack of exactness of the dual functor on
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{hol}}(\mathscr{D}_{X\times
S/S})$ is due to the fact that the dual of a torsion holonomic
$\mathscr{D}_{X\times S/S}$-module $\mathscr{M}$ is not concentrated in degree
zero: if $\mathscr{M}\simeq\nobreak t(\mathscr{M})$ we have
$\boldsymbol{D}(\mathscr{M})\simeq\mathscr{H}^{1}(\boldsymbol{D}(\mathscr{M}))[-1]$.
On the other hand, if $\mathscr{N}$ is a strict holonomic
$\mathscr{D}_{X\times T/T}$-module, then
$\boldsymbol{D}(\mathscr{N})\simeq\mathscr{H}^{0}(\boldsymbol{D}(\mathscr{N}))$
and $\mathscr{H}^{0}(\boldsymbol{D}(\mathscr{N}))$ is strict (see [17, Prop.
2]), that is, $\mathscr{N}$ is also dual holonomic: recall that a complex
$\mathscr{N}$ in
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{hol}}(\mathscr{D}_{X\times
T/T})$ is called _dual holonomic_ if it is in the heart of the $t$-structure
$\Pi$ (see [3, §2]) which, by definition, is the $t$-structure dual to the
canonical $t$-structure.

The following Lemma will be used in the next Theorem 1.4 which plays an
important role in Section 3.b.

######  Lemma 1.3.

Let $\mathscr{M}$ be a coherent (resp.holonomic) $\mathscr{D}_{X\times
T/T}$-module and let $Z$ be a closed analytic subset of $X$. Then
$\Gamma_{Z\times T}\mathscr{M}$ is $\mathscr{D}_{X\times T/T}$-coherent
(resp.holonomic).

###### Proof.

The question is local. Let $F_{\scriptscriptstyle\bullet}\mathscr{M}$ be a
good filtration of $\mathscr{M}$. According e.g. to [20, Prop. 1.9],
$\Gamma_{Z\times T}F_{k}\mathscr{M}$ is $\mathscr{O}_{X\times T}$-coherent for
each $k$, and thus $\mathscr{D}_{X\times T/T}\cdot\Gamma_{Z\times
T}F_{k}\mathscr{M}$ is a $\mathscr{D}_{X\times T/T}$-coherent submodule of
$\mathscr{M}$. Since $\Gamma_{Z\times
T}\mathscr{M}=\mathop{\textstyle\bigcup}\displaylimits_{k}\mathscr{D}_{X\times
T/T}\cdot\Gamma_{Z\times T}F_{k}\mathscr{M}$, we conclude that
$\Gamma_{Z\times T}\mathscr{M}$ is $\mathscr{D}_{X\times T/T}$-coherent since
$\mathscr{D}_{X\times T/T}$ is Noetherian. The statement for holonomic modules
obviously follows, since the characteristic variety of $\Gamma_{Z\times
T}\mathscr{M}$ is contained in that of $\mathscr{M}$. ∎

######  Theorem 1.4.

Let $\mathscr{M},\mathscr{N}$ be strict holonomic $\mathscr{D}_{X\times
S/S}$-modules. Then the $p^{-1}_{X}\mathscr{O}_{S}$-module
$\mathscr{H}\\!\mathit{om}_{\mathscr{D}_{X\times
S/S}}(\mathscr{M},\mathscr{N})_{(x,s)}$ is finitely generated for any point
$(x,s)\in X\times S$.

###### Proof.

We adapt the proof of [6, Th. 4.45]. Let $(X_{\alpha})_{\alpha\in A}$ be a
$\mu$-stratification of $X$ compatible with $\mathscr{M}$ and $\mathscr{N}$,
let $U$ be the union of strata of maximal dimension on which $\mathscr{M}$ and
$\mathscr{N}$ are not identically zero and let $Z$ denotes its closure in $X$.
Then $Z$ is a closed analytic subset of $X$. We argue by induction on the
dimension of $Z$. Set $Z^{\prime}=Z\smallsetminus U$. This is a closed
analytic subset of $Z$ of dimension $<\dim Z$. Let $i:U\hookrightarrow
X^{\prime}:=X\smallsetminus Z^{\prime}$ denote the inclusion. By Kashiwara’s
equivalence theorem in the relative setting (cf.[17, Th. 1.5]), we have

$\mathscr{H}\\!\mathit{om}_{\mathscr{D}_{X^{\prime}\times
S/S}}(\mathscr{M}_{|X^{\prime}\times S},\mathscr{N}_{|X^{\prime}\times
S})\simeq i_{*}\mathscr{H}\\!\mathit{om}_{\mathscr{D}_{U\times
S/S}}({}_{\scriptscriptstyle\mathrm{D}}i^{*}\mathscr{M}_{|X^{\prime}\times
S},{}_{\scriptscriptstyle\mathrm{D}}i^{*}\mathscr{N}_{|X^{\prime}\times S}).$

Moreover, for any open subset ${\widetilde{U}}$ in $X$ such that
${\widetilde{U}}\cap Z=U$, the restriction
$\mathscr{M}_{|{\widetilde{U}}\times S}$ is holonomic and its characteristic
variety is contained in the union of the sets
$T^{*}_{X_{\alpha}}{\widetilde{U}}\times S$ for any stratum $X_{\alpha}$ in
$U$, hence
${}_{\scriptscriptstyle\mathrm{D}}i^{*}\mathscr{M}_{|X^{\prime}\times
S},{}_{\scriptscriptstyle\mathrm{D}}i^{*}\mathscr{N}_{|X^{\prime}\times S}$
are $\mathscr{O}_{U\times S}$-coherent, and thus of the form
$\mathscr{O}_{U\times S}\otimes_{p_{U}^{-1}\mathscr{O}_{S}}\mathscr{F}$ and
$\mathscr{O}_{U\times S}\otimes_{p_{U}^{-1}\mathscr{O}_{S}}\mathscr{G}$ for
some $p_{U}^{-1}\mathscr{O}_{S}$-coherent $S$-local systems
$\mathscr{F},\mathscr{G}$ on $U\times S$, according to [1, Th. I.2.23(iii)].
Kashiwara’s equivalence implies that

$\mathscr{H}\\!\mathit{om}_{\mathscr{D}_{U\times
S/S}}({}_{\scriptscriptstyle\mathrm{D}}i^{*}\mathscr{M}_{|X^{\prime}\times
S},{}_{\scriptscriptstyle\mathrm{D}}i^{*}\mathscr{N}_{|X^{\prime}\times
S})\simeq\mathscr{H}\\!\mathit{om}_{p_{U}^{-1}\mathscr{O}_{S}}(\mathscr{F},\mathscr{G}),$

which is thus also a $p_{U}^{-1}\mathscr{O}_{S}$-coherent local system.
Therefore,

$\mathscr{H}\\!\mathit{om}_{\mathscr{D}_{X^{\prime}\times
S/S}}(\mathscr{M}_{|X^{\prime}\times S},\mathscr{N}_{|X^{\prime}\times
S})\simeq
i_{*}\mathscr{H}\\!\mathit{om}_{p_{U}^{-1}\mathscr{O}_{S}}(\mathscr{F},\mathscr{G})$

is $S$-$\mathbb{C}$-constructible. Let
$j\\!:\\!X^{\prime}\\!\mathrel{\lhook\joinrel\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}\\!X$
denote the open inclusion. According to [15, Cor. 2.8 & Prop. 2.20], the sheaf
$j_{*}\mathscr{H}\\!\mathit{om}_{\mathscr{D}_{X^{\prime}\times
S/S}}(\mathscr{M}_{|X^{\prime}\times S},\mathscr{N}_{|X^{\prime}\times S})$ is
$S$-$\mathbb{C}$-constructible. By considering the exact sequence

$0\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{H}\\!\mathit{om}_{\mathscr{D}_{X\times
S/S}}(\mathscr{M},\Gamma_{Z^{\prime}\times
S}\mathscr{N})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{H}\\!\mathit{om}_{\mathscr{D}_{X\times
S/S}}(\mathscr{M},\mathscr{N})\\\
\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}j_{*}j^{-1}\mathscr{H}\\!\mathit{om}_{\mathscr{D}_{X\times
S/S}}(\mathscr{M},\mathscr{N}),$

we are reduced to proving the result in the case where $\mathscr{N}$ is
replaced with $\mathscr{N}^{\prime}\\!:=\\!\Gamma_{Z^{\prime}\times
S}\mathscr{N}$, which is holonomic by Lemma 1.3. Thus $\mathscr{N}^{\prime}$
is strict if $\mathscr{N}$ is so. Since $\mathscr{M}$ is strict holonomic,
$\boldsymbol{D}\mathscr{M}$ is also strict holonomic (cf.[17, Cor. 1.12]),
hence so is $\Gamma_{Z^{\prime}\times S}\boldsymbol{D}\mathscr{M}$, as well as
$\mathscr{M}^{\prime}:=\boldsymbol{D}\Gamma_{Z^{\prime}\times
S}\boldsymbol{D}\mathscr{M}$. On the other hand, since $\mathscr{N}^{\prime}$
is strict holonomic, $\boldsymbol{D}\mathscr{N}^{\prime}$ satisfies the same
properties, and the natural morphism

$\mathscr{H}\\!\mathit{om}_{\mathscr{D}_{X\times
S/S}}(\boldsymbol{D}\mathscr{N}^{\prime},\Gamma_{Z^{\prime}\times
S}\boldsymbol{D}\mathscr{M})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{H}\\!\mathit{om}_{\mathscr{D}_{X\times
S/S}}(\boldsymbol{D}\mathscr{N}^{\prime},\boldsymbol{D}\mathscr{M})$

is an isomorphism. Since $\mathscr{N}^{\prime}$ and $\Gamma_{Z^{\prime}\times
S}\boldsymbol{D}\mathscr{M}$ are $\mathscr{D}_{X\times S/S}$-coherent, they
satisfy the biduality isomorphism and by [15, (3)] we conclude

$\mathscr{H}\\!\mathit{om}_{\mathscr{D}_{X\times
S/S}}(\mathscr{M}^{\prime},\mathscr{N}^{\prime})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathscr{H}\\!\mathit{om}_{\mathscr{D}_{X\times
S/S}}(\mathscr{M},\mathscr{N}^{\prime}).$

Now the induction hypothesis applies to the pair
$(\mathscr{M}^{\prime},\mathscr{N}^{\prime})$ with corresponding set
$Z^{\prime}$. ∎

We recall the following result in [3, Lem. 2.10]:

######  Proposition 1.5.

For any holonomic $\mathscr{D}_{X\times T/T}$-module $\mathscr{M}$ we have

$\mathrm{Char}(\mathscr{M})=\mathop{\textstyle\bigcup}\displaylimits_{i\in
I}\Lambda_{i}\times T_{i}$

for some closed $\mathbb{C}^{*}$-conic irreducible Lagrangian subsets
$\Lambda_{i}$ of $T^{*}X$ and some closed analytic subsets $T_{i}$ of $T$,
and, locally on $X$, the set $I$ is finite. Moreover
$p_{X}(\mathrm{Supp}(\mathscr{M}))=\mathop{\textstyle\bigcup}\displaylimits_{i\in
I}T_{i}$, hence it is an analytic subset of $T$, and

$\dim{\mathrm{Char}}(\mathscr{M})=\dim X+t,\quad\hbox{where }t=\dim
p_{X}(\mathrm{Supp}(\mathscr{M}))=\sup_{i\in I}\dim T_{i}.$

###  1.b. Relative constructible and perverse complexes

####  1.b.1. Relative local systems

Following [15] we say that a sheaf $p^{-1}_{X}\mathscr{O}_{T}$-module $F$ is
$T$-locally constant coherent if, for each point $(x_{0},t_{o})\in X\times T$
there exists a neighborhood $U=V_{x_{0}}\times T_{t_{o}}$ and a coherent sheaf
$G^{(x_{0},t_{o})}$ of $\mathscr{O}_{T_{t_{o}}}$-modules such that
$F_{|U}\cong p_{V_{x_{0}}}^{-1}(G^{(x_{0},t_{o})})$. We refer to [17, App.]
for basic properties.

By definition,
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{lc}\;\mathrm{coh}}(p^{-1}_{X}\mathscr{O}_{T})$
is the full subcategory of
$\operatorname{\mathsf{D}}^{\mathrm{b}}(p^{-1}_{X}\mathscr{O}_{T})$ whose
complexes have $T$-locally constant coherent cohomologies (notice that, for
such an $F$, $F_{|\\{x_{0}\\}\times
T}\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{coh}}(\mathscr{O}_{T})$).
We refer to [15, §2] for more properties.

####  1.b.2. Relative $\mathbb{R}$-constructibility

In the following, we will have to consider derived categories
$\operatorname{\mathsf{D}}^{\star}$ with $\star=\mathrm{b}$ or $\star={}-$. We
denote by
$\operatorname{\mathsf{D}}^{\star}_{\mathbb{R}\textup{-c}}(p^{-1}_{X}\mathscr{O}_{T})$
the full subcategory of
$\operatorname{\mathsf{D}}^{\star}(p^{-1}_{X}\mathscr{O}_{T})$ whose objects
$F$ admit a $\mu$-stratification $(X_{\alpha})$ of $X$ such that
$i_{\alpha}^{-1}(F)\in\operatorname{\mathsf{D}}^{\star}_{\mathrm{lc}\;\mathrm{coh}}(p_{X_{\alpha}}^{-1}\mathscr{O}_{T})$
for any $\alpha$. We refer to [15, §2] for details.

Objects of
$\operatorname{\mathsf{D}}^{-}_{\mathbb{R}\textup{-c}}(p^{-1}_{X}\mathscr{O}_{T})$
can be given a simple representative. Let us denote by $\mathscr{S}$ the full
additive subcategory of
$\mathsf{Mod}_{{\mathbb{R}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{T})$ whose
objects are sheaves which can be expressed as locally finite direct sums of
terms of the form
$\mathbb{C}_{\Omega}\boxtimes\mathscr{O}_{V}:=p^{-1}_{X}\mathscr{O}_{T}\otimes\mathbb{C}_{\Omega\times
V}$ for some relatively compact open subanalytic subsets $\Omega$ in $X$ and
$V$ in $T$. A morphism
$\varphi:\mathbb{C}_{\Omega}\boxtimes\mathscr{O}_{V}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbb{C}_{\Omega^{\prime}}\boxtimes\mathscr{O}_{V^{\prime}}$
is easily described: setting $\Omega^{\prime\prime}=\Omega\cap\Omega^{\prime}$
and $V^{\prime\prime}=V\cap V^{\prime}$, $\varphi$ is the extension by zero of
its restriction $\varphi_{|\Omega^{\prime\prime}\times
V^{\prime\prime}}:p^{-1}_{X}\mathscr{O}_{T}{}_{|\Omega^{\prime\prime}\times
V^{\prime\prime}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}p^{-1}_{X}\mathscr{O}_{T}{}_{|\Omega^{\prime\prime}\times
V^{\prime\prime}}$, which is the multiplication by a section of
$p^{-1}_{X}\mathscr{O}_{T}$ on each connected component of
$\Omega^{\prime\prime}\times V^{\prime\prime}$.

Following the terminology of [8, App. A], we say that an object $F$ of
$\mathsf{Mod}(p^{-1}_{X}\mathscr{O}_{T})$ is $\mathscr{S}$-coherent if there
exist $L\in\mathscr{S}$ and an epimorphism
$L\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}F$, and
if, for any morphism
$L^{\prime}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}F$
with $L^{\prime}$ in $\mathscr{S}$, there exist $L^{\prime\prime}$ in
$\mathscr{S}$ and a morphism
$L^{\prime\prime}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}L^{\prime}$
such that
$L^{\prime\prime}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}L^{\prime}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}F$
is exact. It follows from [17, Prop. 3.5] that the category of
$\mathscr{S}$-coherent objects of $\mathsf{Mod}(p^{-1}_{X}\mathscr{O}_{T})$ is
equal to $\mathsf{Mod}_{{\mathbb{R}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{T})$
and the category
$\operatorname{\mathsf{D}}^{-}_{\mathscr{S}\text{-}\mathrm{coh}}(\mathsf{Mod}(p^{-1}_{X}\mathscr{O}_{T}))$
is nothing but
$\operatorname{\mathsf{D}}^{-}_{{\mathbb{R}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{T})$.
On the other hand, one defines the category
$\operatorname{\mathsf{D}}^{-}_{\mathrm{coh}}(\mathscr{S})$ as in [8, p. 63],
with the identification $\mathbf{A}=\mathsf{Mod}(p^{-1}_{X}\mathscr{O}_{T})$
and $\mathbf{P}=\mathscr{S}$ (the functor $L$ of loc.cit. is here the
inclusion, $H$ is the restriction of
$\operatorname{Hom}_{\mathsf{Mod}(p^{-1}_{X}\mathscr{O}_{T})}$ to
$\mathscr{S}\times\mathsf{Mod}(p^{-1}_{X}\mathscr{O}_{T})$ and $\alpha$ is the
identity). In the present situation, we have
$\operatorname{\mathsf{D}}^{-}_{\mathrm{coh}}(\mathscr{S})=\operatorname{\mathsf{D}}^{-}(\mathscr{S})$.

######  Proposition 1.6 (cf.[8, Th. A.5]).

The natural functor

$L:\operatorname{\mathsf{D}}^{-}(\mathscr{S})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{\mathsf{D}}^{-}_{{\mathbb{R}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{T})$

is an equivalence of categories.

###### Proof.

We only need to check that the conditions for applying [8, Th. A.5] are
fulfilled, that is, that the pair
$(\mathsf{Mod}(p^{-1}_{X}\mathscr{O}_{T}),\mathscr{S})$ satisfies the
properties (A.1)–(A.4) for $(\mathbf{A},\mathbf{P})$ in loc.cit., and only
(A.3) is not obvious. It is proved in the lemma below. ∎

######  Lemma 1.7.

Let $F$ and $G$ be objects of $\mathsf{Mod}(p^{-1}_{X}\mathscr{O}_{T})$, let
$\psi:F\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}G$
be an epimorphism, let $\mathscr{T}\in\mathscr{S}$ and let
$g:\mathscr{T}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}G$
be given. Then there exist an object $\mathscr{T}^{\prime}\in\mathscr{S}$, an
epimorphism
$\psi^{\prime}:\mathscr{T}^{\prime}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{T}$
and a morphism
$g^{\prime}:\mathscr{T}^{\prime}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}F$
such that $\psi g^{\prime}=g\psi^{\prime}$.

###### Proof.

By $p^{-1}_{X}\mathscr{O}_{T}$-linearity we may reduce to the case
$\mathscr{T}={p^{-1}_{X}\mathscr{O}_{T}}\otimes\mathbb{C}_{\Omega\times V}$,
for some relatively compact open subanalytic subsets $\Omega,V$ respectively
in $X$ and $T$.

Let $e$ be the section $1\in\Gamma(\Omega\times V,\mathscr{T})$. By the
assumption on $\psi$, we can cover $\Omega$ (resp.$V$) by a locally finite
family of relatively compact open subanalytic sets $\Omega_{i}\subset\nobreak
X$ (resp.$V_{i}\subset T$), $i=1,\dots,m$, and find sections
$f_{i}\in\Gamma(\Omega_{i}\times V_{i},F)$ such that $\psi|_{\Omega_{i}\times
V_{i}}(f_{i})=g(e)|_{\Omega_{i}\times V_{i}}$. The morphism
$\mathbb{C}_{|\Omega_{i}\times
V_{i}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}F_{|\Omega_{i}\times
V_{i}}$ extends in a unique way as a morphism $\mathbb{C}_{\Omega_{i}\times
V_{i}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}F$
and, by $p^{-1}_{X}\mathscr{O}_{T}$-linearity, as a morphism
$p^{-1}_{X}\mathscr{O}_{T}\otimes\mathbb{C}_{\Omega_{i}\times
V_{i}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}F$.

Setting
$\mathscr{T}^{\prime}:=\mathop{\textstyle\bigoplus}\displaylimits_{i=1}^{m}p^{-1}_{X}\mathscr{O}_{T}\otimes\mathbb{C}_{\Omega_{i}\times
V_{i}}$, we obtain in this way a $p^{-1}_{X}\mathscr{O}_{T}$-linear morphism
$g^{\prime}:\mathscr{T}^{\prime}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}F$.
On the other hand, by the covering property, the natural morphism
$\mathbb{C}_{\Omega_{i}\times
V_{i}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbb{C}_{\Omega\times
V}$ which extends $\operatorname{Id}:\mathbb{C}_{|\Omega_{i}\times
V_{i}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbb{C}_{\Omega\times
V|\Omega_{i}\times V_{i}}$ induces an epimorphism
$\mathop{\textstyle\bigoplus}\displaylimits_{i=1}^{m}\mathbb{C}_{\Omega_{i}\times
V_{i}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbb{C}_{\Omega\times
V}$ and, by $p^{-1}_{X}\mathscr{O}_{T}$-linearity, an epimorphism
$\psi^{\prime}:\mathscr{T}^{\prime}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{T}$,
which clearly satisfies $\psi g^{\prime}=g\psi^{\prime}$. ∎

We note the following, to be used in the course of the proof of Lemma 2.5:

######  Remark 1.8.

Let $\Phi,\Psi$ be two triangulated functors from
$\operatorname{\mathsf{D}}^{-}_{\mathbb{R}\textup{-c}}(p^{-1}_{X}\mathscr{O}_{T})$
to a triangulated category $\mathsf{C}$. Any morphism of functors
$\eta_{\mathscr{S}}:\Phi_{\mathscr{S}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Psi_{\mathscr{S}}$
(with $\Phi_{\mathscr{S}}=\Phi\circ L$, $\Psi_{\mathscr{S}}=\Psi\circ L$) can
be extended to a morphism of functors
$\eta:\Phi\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Psi$.

####  1.b.3. Relative $\mathbb{C}$-constructibility

By definition (cf.[15, Def. 2.19]), the full subcategory
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathbb{C}\textup{-c}}(p^{-1}_{X}\mathscr{O}_{T})$
consists of objects of
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{R}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{T})$
whose microsupport is $\mathbb{C}^{*}$-conic. We call these objects _$T$
-$\mathbb{C}$-constructible complexes_.

For any $t_{o}\in T$, there is a functor

$Li^{*}_{t_{o}}:\operatorname{\mathsf{D}}^{\mathrm{b}}(p^{-1}_{X}\mathscr{O}_{T})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{\mathsf{D}}^{\mathrm{b}}(\mathbb{C}_{X})$

also defined by (1). It sends
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathbb{R}\textup{-c}}(p^{-1}_{X}\mathscr{O}_{T})$
to
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathbb{R}\textup{-c}}(\mathbb{C}_{X})$
and
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathbb{C}\textup{-c}}(p^{-1}_{X}\mathscr{O}_{T})$
to
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathbb{C}\textup{-c}}(\mathbb{C}_{X})$.

Recall (cf.[15, Prop. 2.2]) that a variant of Nakayama’s lemma holds for
complexes $F$ in
$\operatorname{\mathsf{D}}^{\mathrm{b}}(p^{-1}_{X}\mathscr{O}_{T})$ whose
cohomology objects $\mathscr{H}^{j}F$ have fibers $\mathscr{H}^{j}F_{(x,s)}$
of finite type over $\mathscr{O}_{T,s}$ for any $(x,s)\in X\times T$. As a
consequence, the family of functors $(Li^{*}_{t_{o}})_{{t_{o}}\in T}$ on
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathbb{R}\textup{-c}}(p^{-1}_{X}\mathscr{O}_{T})$
(resp.$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathbb{C}\textup{-c}}(p^{-1}_{X}\mathscr{O}_{T})$)
is a conservative family (in particular, if
$F\\!\in\\!\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathbb{C}\textup{-c}}(p^{-1}_{X}\mathscr{O}_{T})$
satisfies $Li^{*}_{t_{o}}F=0$ for each ${t_{o}}\in T$, then $F=0$).

####  1.b.4. Perversity

The category
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathbb{C}\textup{-c}}(p^{-1}_{X}\mathscr{O}_{T})$
is endowed with a perverse $t$-structure defined in [15, §2.7] as the relative
analogue to the middle perverse $t$-structure in the absolute case where $\dim
T=0$:

  * •

$\operatorname{{}^{\mathrm{p}}\mathsf{D}}^{\leqslant
0}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{T})$ is the full
subcategory of objects $F$ of
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{T})$
such that there exists an adapted $\mu$-stratification $(X_{\alpha})$ of $X$
for which
$i_{x}^{-1}F\in{\operatorname{\mathsf{D}}}_{\mathrm{coh}}^{\leqslant-
d_{X_{\alpha}}}(\mathscr{O}_{T})$ for any $x\in X_{\alpha}$ and any $\alpha$.

  * •

$\operatorname{{}^{\mathrm{p}}\mathsf{D}}^{\geqslant
0}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{T})$ is the full
subcategory of objects $F$ of
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{T})$
such that there exists an adapted $\mu$-stratification $(X_{\alpha})$ of $X$
for which $i_{x}^{!}F\in{\operatorname{\mathsf{D}}}_{\mathrm{coh}}^{\geqslant
d_{X_{\alpha}}}(\mathscr{O}_{T})$ for any $x\in X_{\alpha}$ and any $\alpha$.

The heart of this $t$-structure is the abelian category of _relative perverse
sheaves_ denoted by $\mathrm{perv}(p^{-1}_{X}\mathscr{O}_{T})$. We often omit
the word “relative”.

In analogy with the $\mathscr{D}_{X\times S/S}$-module counterpart ($S$ is a
curve), following [3, Prop. 3.12], we say that a perverse sheaf is _torsion_
if it belongs to the subcategory
$\mathrm{perv}(p^{-1}_{X}\mathscr{O}_{S})_{\mathrm{t}}$ of
$\mathrm{perv}(p^{-1}_{X}\mathscr{O}_{S})$ whose objects $F$ satisfy
$\operatorname{codim}p(\operatorname{Supp}F)\geqslant 1$ (cf.[3, Cor. 3.1] for
this condition), while a perverse sheaf is called _strictly perverse_ if it
belongs to the full subcategory
$\mathrm{perv}(p^{-1}_{X}\mathscr{O}_{S})_{\mathrm{tf}}$ of
$\mathrm{perv}(p^{-1}_{X}\mathscr{O}_{S})$ whose objects $F$ satisfy
$Li^{\ast}_{s}F\in\mathrm{perv}(\mathbb{C}_{X})$ for all $s\in S$. The
category $\mathrm{perv}(p^{-1}_{X}\mathscr{O}_{S})_{\mathrm{t}}$ is a full
thick abelian subcategory of the category
$\mathrm{perv}(p^{-1}_{X}\mathscr{O}_{S})$.

We denote by
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})_{\mathrm{t}}$
the thick subcategory of
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})$
whose objects have support in $X\times T$, where $T$ is a subset of $S$ with
$\dim T=0$ or, equivalently, whose perverse cohomologies belong to
$\mathrm{perv}(p^{-1}_{X}\mathscr{O}_{S})_{\mathrm{t}}$.

Given an object $F$ of
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})$,
the functor $\boldsymbol{D}$ defined by

$\boldsymbol{D}(F)=\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{p^{-1}_{X}\mathscr{O}_{S}}(F,p^{-1}_{X}\mathscr{O}_{S})[2d_{X}]$

provides a duality in
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})$,
which is however not $t$-exact with respect to the perverse $t$-structure. For
example, if $F$ is a torsion perverse sheaf, then
$\boldsymbol{D}(F)\simeq{}^{p}\mathscr{H}^{1}(\boldsymbol{D}(F))[-1]$ (it is a
perverse sheaf shifted in degree $1$), while if $F$ is a strictly perverse
sheaf, $\boldsymbol{D}(F)\simeq{}^{p}\mathscr{H}^{0}(\boldsymbol{D}(F))$ is
perverse too.

Let us recall that an object $F$ of
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})$
is called _dual perverse_ if it is in the heart of the $t$-structure $\pi$,
which by definition is the $t$-structure dual to the perverse $t$-structure
introduced in [15, §2.7]. By [17, Lem. 1.4], a complex
$F\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})$
is perverse and dual perverse if and only if it is strictly perverse.

###  1.c. The relative solution functor

The solution functor for a coherent $\mathscr{D}_{X\times T/T}$-module or an
object of
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{coh}}(\mathscr{D}_{X\times
T/T})$ is defined by

$\displaystyle\operatorname{{}^{\mathrm{p}}Sol}_{X}:\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{coh}}(\mathscr{D}_{X\times
T/T})$
$\displaystyle\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{\mathsf{D}}^{\mathrm{b}}(p^{-1}_{X}\mathscr{O}_{T})$
$\displaystyle\mathscr{M}$
$\displaystyle\mathchoice{\longmapsto}{\mapsto}{\mapsto}{\mapsto}\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{X\times
T/T}}(\mathscr{M},\mathscr{O}_{X\times T})[d_{X}]$

By [15, Th. 3.7], when restricted to
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{hol}}(\mathscr{D}_{X\times
T/T})$ the solution functor $\operatorname{{}^{\mathrm{p}}Sol}_{X}$ takes
values in
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{T})$
and by [3, Cor. 4.3], is $t$-exact with respect to the $t$-structure $\Pi$ in
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{hol}}(\mathscr{D}_{X\times
T/T})$ and the perverse one $p$ in
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{T})$.

###  1.d. Regular holonomic complexes of $\mathscr{D}_{X\times S/S}$-modules

In this section, we review the notion of relative regularity as introduced in
[17, §2.1] and recall the fundamental example of relative
$\mathscr{D}_{X\times S/S}$-modules of D-type.

######  Definition 1.9 (Regularity, [17, Def. 2.1]).

A holonomic $\mathscr{D}_{X\times S/S}$-module $\mathscr{M}$ is said to be
_regular_ if, for any $s_{o}\in S$, the object $Li^{*}_{s_{o}}\mathscr{M}$ of
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{hol}}(\mathscr{D}_{X})$ has
regular holonomic cohomologies.

######  Example 1.10.

  1. (a)

In Example 1.1 let us assume that $M$ is regular. Then the
$\mathscr{D}_{X\times S/S}$-module generated by $\mathscr{M}$ is regular.

  2. (b)

In Example 1.2, assume that the mixed twistor $\mathscr{D}$-module is regular
in the sense of [18, Def. 4.1.2]. Then the underlying holonomic
$\mathscr{D}_{X\times S/S}$-module $\mathscr{M}$ is regular.

According to [17, §2.1], we say that an object
$\mathscr{M}\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{hol}}(\mathscr{D}_{X\times
S/S})$ is _regular_ if each of its cohomology modules is regular.

######  Remark 1.11.

An object $\mathscr{M}$ of
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{hol}}(\mathscr{D}_{X\times
S/S})$ is regular if and only if, for each $s_{o}\in S$, the object
$Li^{*}_{s_{o}}\mathscr{M}$ of
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{hol}}(\mathscr{D}_{X})$ has
regular holonomic cohomology. Indeed, we argue by induction on the amplitude
of the complex $\mathscr{M}$. Without loss of generality, we may assume that
$\mathscr{M}\in\operatorname{\mathsf{D}}^{\geqslant
0}_{\mathrm{hol}}(\mathscr{D}_{X\times S/S})$ and we consider the following
distinguished triangle

$\mathscr{H}^{0}\mathscr{M}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{M}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\tau^{\geqslant
1}\mathscr{M}\mathchoice{\xrightarrow{\textstyle\kern 4.0pt+1\kern
3.0pt}}{\stackrel{{\scriptstyle+1}}{{\longrightarrow}}}{}{}$

(where $\tau^{\geqslant 1}$ is the truncation functor with respect to the
natural $t$-structure on
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{hol}}(\mathscr{D}_{X\times
S/S})$). We deduce
$\mathscr{H}^{-1}Li^{*}_{s_{o}}\mathscr{H}^{0}(\mathscr{M})\simeq\mathscr{H}^{-1}Li^{*}_{s_{o}}(\mathscr{M})$
and an exact sequence

$0\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{H}^{0}Li^{*}_{s_{o}}\mathscr{H}^{0}\mathscr{M}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{H}^{0}Li^{*}_{s_{o}}\mathscr{M}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{H}^{0}Li^{*}_{s_{o}}\tau^{\geqslant
1}\mathscr{M}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}0.$

(Note that $\mathscr{H}^{k}Li^{*}_{s_{o}}\mathscr{H}^{0}(\mathscr{M})=0$ for
$k\neq 0,-1$.) The assertion follows from the induction hypothesis and the
property that the category of regular holonomic $\mathscr{D}_{X}$-modules is
closed under sub-quotients in the category
$\mathsf{Mod}_{\mathrm{coh}}(\mathscr{D}_{X})$.

We recall the following result in [17]:

######  Proposition 1.12 (cf.[17, Cor. 2.4]).

Let $f:Y\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X$
be a morphism of complex manifolds and let us assume that $f$ is proper. Then,
for each
$\mathscr{M}\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{rhol}}(\mathscr{D}_{Y\times
S/S})$, the pushforward ${}_{\scriptscriptstyle\mathrm{D}}f_{*}\mathscr{M}$ is
an object of
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{rhol}}(\mathscr{D}_{X\times
S/S})$.

####  1.d.1. Proof of Theorem 2 for a smooth morphism

Let $f:Y\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X$
be a smooth morphism and let $\mathscr{M}$ be an object of
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{rhol}}(\mathscr{D}_{X\times
S/S})$. Due to the locality of the regular holonomic property, we may assume
that $f$ is a projection $Y=Z\times
X\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X$. In
that case,
$\mathscr{D}_{(Y\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X)\times
S}$ is $f^{-1}\mathscr{D}_{X\times S/S}$-flat, so
$\mathscr{H}^{j}{}_{\scriptscriptstyle\mathrm{D}}f^{*}\mathscr{M}\simeq{}_{\scriptscriptstyle\mathrm{D}}f^{*}\mathscr{H}^{j}\mathscr{M}$
for every $j$, and we can assume that
$\mathscr{M}=\mathscr{H}^{0}\mathscr{M}$. As in the absolute case one checks
that
$\operatorname{Char}({}_{\scriptscriptstyle\mathrm{D}}f^{*}\mathscr{M})=T^{*}_{Z}Z\times\operatorname{Char}(\mathscr{M})$,
so
${}_{\scriptscriptstyle\mathrm{D}}f^{*}\mathscr{M}\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{hol}}(\mathscr{D}_{Y\times
S/S})$. Moreover, the commutativity of

$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{i_{s}}$$\textstyle{f}$$\textstyle{Y\times
S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{f}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{i_{s}}$$\textstyle{X\times
S}$

implies that
$Li^{*}_{s}{}_{\scriptscriptstyle\mathrm{D}}f^{*}(\mathscr{M})\simeq{}_{\scriptscriptstyle\mathrm{D}}f^{*}Li_{s}^{*}\mathscr{M}$,
and the latter is known to be regular holonomic on $Y$. Therefore, Theorem 2
is proved for $f$ smooth.∎

####  1.d.2. $\mathscr{D}_{X\times S/S}$-modules of D-type

They are the fundamental examples of regular holonomic $\mathscr{D}_{X\times
S/S}$-modules, so we recall their definition. Let $D$ be a normal crossing
divisor in $X$ and let $j:X^{*}:=X\smallsetminus
D\mathrel{\lhook\joinrel\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}X$
denote the inclusion (we will also denote by $j$ the morphism
$j\times{\operatorname{Id}}_{S}$). Let $F$ be a coherent $S$-locally constant
sheaf on $X^{*}\times S$ and let $(V,\nabla)=(\mathscr{O}_{X^{*}\times
S}\otimes_{p^{-1}_{X}\mathscr{O}_{S}}\nobreak F,\mathrm{d}_{X/S})$ be the
associated coherent $\mathscr{O}_{X^{*}\times S}$-module with flat relative
connection. There exists a coherent $\mathscr{O}_{S}$-module $G$ such that, if
$U$ is any contractible open set of $X^{*}$, then $F_{|U\times S}\simeq
p_{U}^{-1}G$.

Let
$\varpi:\widetilde{X}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X$
denote the real oriented blowing up of $X$ along the components of $D$. Denote
by
$\widetilde{\jmath}:X^{*}\mathrel{\lhook\joinrel\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}\widetilde{X}$
the inclusion, so that $j=\varpi\circ\widetilde{\jmath}$. Let $x^{o}\in D$,
$\widetilde{x}^{o}\in\varpi^{-1}(x^{o})$ and let $s^{o}\in S$. Choose local
coordinates $(x_{1},\dots,x_{n})$ at $x^{o}$ such that $D=\\{x_{1}\cdots
x_{\ell}=0\\}$ and consider the associated polar coordinates
$(\boldsymbol{\rho},\boldsymbol{\theta},\boldsymbol{x}^{\prime}):=(\rho_{1},\theta_{1},\dots,\rho_{\ell},\theta_{\ell},x_{\ell+1},\dots,x_{n})$
so that $\widetilde{x}^{o}$ has coordinates $\boldsymbol{\rho}^{o}=\nobreak
0$, $\boldsymbol{\theta}^{o}$, $\boldsymbol{x}^{\prime o}=0$.

A local section $\widetilde{v}$ of
$(\widetilde{\jmath}_{*}V)_{(\widetilde{x}^{o},s^{o})}$ is said to have
_moderate growth_ if for some system of generator of $G_{s^{o}}$, and some
neighbourhood

$U_{\varepsilon}:=\\{\|\boldsymbol{\rho}\|<\varepsilon,\|\boldsymbol{x}^{\prime}\|<\varepsilon,\|\boldsymbol{\theta}-\boldsymbol{\theta}^{o}\|<\varepsilon\\}$

($\varepsilon$ small enough) on which it is defined, its coefficients on the
chosen generators of $G_{s^{o}}$ (these are sections of
$\mathscr{O}(U^{*}_{\varepsilon}\times U(s^{o}))$ for a small enough
neighbourhood $U(s^{o})$ of $s^{o}$ in $S$, and
$U^{*}_{\varepsilon}:=U_{\varepsilon}\smallsetminus\\{\rho_{1}\cdots\rho_{\ell}=0\\}$)
are bounded by $C\boldsymbol{\rho}^{-N}$, for some $C,N>0$. A local section
$v$ of $(j_{*}V)_{(x^{o},s^{o})}$ is said to have _moderate growth_ if for
each $\widetilde{x}^{o}$ in $\varpi^{-1}(x^{o})$, the corresponding germ in
$(\widetilde{\jmath}_{*}V)_{(\widetilde{x}^{o},s^{o})}$ has moderate growth.

On the other hand (cf.[17, Def. 2.10]), a coherent $\mathscr{D}_{X\times
S/S}$-module $\mathscr{L}$ is said to be of D-type with singularities along a
normal crossing divisor $D\subset X$ if it satisfies the following conditions:

  1. (a)

$\operatorname{Char}(\mathscr{L})\subset(\pi^{-1}(D)\times
S)\cup(T^{*}_{X}X\times S)$,

  2. (b)

$\mathscr{L}$ is regular holonomic and strict,

  3. (c)

$\mathscr{L}\simeq\mathscr{L}(\ast(D\times S))$.

The following result is proved in [17, Th. 2.6, Cor. 2.8 & Prop. 2.11]:

######  Theorem 1.13.

  1. (a)

The subsheaf $\widetilde{V}$ of $j_{*}V$ consisting of local sections having
moderate growth is stable by $\nabla$ and it is $\mathscr{O}_{X\times
S}(*D)$-coherent. Moreover, $\widetilde{V}$ is a regular holonomic
$\mathscr{D}_{X\times S/S}$-module with characteristic variety contained in
$\Lambda\times S$, where $\Lambda$ is the union of the conormal spaces of the
natural stratification of $(X,D)$.

  2. (b)

A coherent $\mathscr{D}_{X\times S/S}$-module $\mathscr{L}$ is of D-type on
$(X,D)$ if and only if it is isomorphic to some $\widetilde{V}$ as in (a).

##  2\. The relative Riemann-Hilbert functor $\operatorname{RH}^{S}$

In this section we recall the definition of the relative Riemann-Hilbert
functor
$\operatorname{RH}^{S}({\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}})$
introduced in [17] and state some supplementary results needed in the sequel.

###  2.a. Relative subanalytic sites and relative subanalytic sheaves

For details on this subject we refer to [14] and [2]. We also refer to [9] as
a foundational paper and to [10] for a detailed exposition on the general
theory of sheaves on sites.

Let $X$ and $T$ be real or complex analytic manifolds. One denotes by
$\operatorname{Op}(X\times T)$ the family of open subsets of $X\times T,$ by
$\operatorname{Op}((X\times T)_{\mathrm{sa}})\subset\operatorname{Op}(X\times
T)$ the family of open subanalytic sets;
$\mathscr{T}:=\operatorname{Op}^{\mathrm{c}}((X\times T)_{\mathrm{sa}})$
denotes the family of relatively compact open subanalytic subsets of $X\times
T$ and $\mathscr{T}^{\prime}\subset\mathscr{T}$ denotes the family of finite
unions of relatively compact open subanalytic sets of the form $U\times V$.

The product $X\times T$ is both a $\mathscr{T}$\- as well as a
$\mathscr{T}^{\prime}$-space. The associated sites $({X\times
T})_{\mathscr{T}}$ and $({X\times T})_{\mathscr{T}^{\prime}}$ are,
respectively, the subanalytic site $(X\times T)_{\mathrm{sa}}$, for which the
coverings of an element $\Omega\in\operatorname{Op}((X\times
T)_{\mathrm{sa}})$ are the locally finite coverings with elements in
$\mathscr{T}$, and the site denoted by $X_{\mathrm{sa}}\times
T_{\mathrm{sa}}$, for which the coverings of $\Omega\in\mathscr{T}^{\prime}$
are the coverings with elements in $\mathscr{T}^{\prime}$ which admit a finite
subcovering.

We shall denote by $\rho_{T}$ the natural functor of sites $\rho_{T}:X\times
T\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}(X\times
T)_{\mathrm{sa}}$ associated to the inclusion
$\operatorname{Op}_{\mathrm{sa}}(X\times T)\subset\operatorname{Op}(X\times
T)$. Accordingly, we shall consider the associated functors
$\rho_{T*},\rho_{T}^{-1},\rho_{T!}$.

We shall also denote by $\rho^{\prime}_{T}:X\times
T\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X_{\mathrm{sa}}\times
T_{\mathrm{sa}}$ the functor of sites associated to the inclusion
$\mathscr{T}^{\prime}\subset\operatorname{Op}(X\times T)$. Following [10] we
have functors $\rho^{\prime}_{T*}$ and $\rho^{\prime}_{T!}$ from
$\mathsf{Mod}(\mathbb{C}_{X\times T})$ to
$\mathsf{Mod}(\mathbb{C}_{X_{\mathrm{sa}}\times T_{\mathrm{sa}}})$. We simply
denote by $\rho$, resp.$\rho^{\prime}$, the previous morphism when there is no
ambiguity.

Subanalytic sheaves are defined on the subanalytic site of a real analytic
manifold, and relative subanalytic sheaves are defined on the subanalytic site
$X_{\mathrm{sa}}\times T_{\mathrm{sa}}$. We refer to [14] for the detailed
construction of the relative subanalytic sheaves
$\operatorname{\mathfrak{Db}}^{t,T}_{X\times T}$ (where $X$ and $T$ are real
analytic) and $\mathscr{O}^{t,T}_{X\times T}$ in the complex framework
(denoted $\operatorname{\mathfrak{Db}}^{t,T,\sharp}_{X\times T}$ and
$\mathscr{O}^{t,T,\sharp}_{X\times T}$ in [14]).

They are both $\rho^{\prime}_{!}\mathscr{D}_{X\times T}$-modules (either in
the real or the complex case) as well as a
$\rho^{\prime}_{*}p^{-1}_{X}\mathscr{O}_{T}$-modules when $T$ is complex.

If $\operatorname{\mathfrak{Db}}^{t}_{X\times T}$ denotes the subanalytic
sheaf of tempered distributions introduced by Kashiwara-Schapira in [9], we
have, for $U\in\operatorname{Op}(X_{\mathrm{sa}})$ and
$V\in\operatorname{Op}(T_{\mathrm{sa}})$

$\displaystyle\Gamma(U\times V;\operatorname{\mathfrak{Db}}^{t,T}_{X\times
T})$ $\displaystyle=\varprojlim_{W\Subset V}\Gamma(U\times
W;\operatorname{\mathfrak{Db}}^{t}_{X\times T})$
$\displaystyle\simeq\Gamma(X\times V;\rho^{\prime-1}\Gamma_{U\times
T}\operatorname{\mathfrak{Db}}^{t}_{X\times T})$
$\displaystyle\simeq\Gamma(X\times
V;\operatorname{\mathit{T}\mathscr{H}\\!\mathit{om}}(\mathbb{C}_{U\times
T},\operatorname{\mathfrak{Db}}_{X\times T})).$

Moreover, when $X$ is also complex, considering the complex conjugate
structure $\overline{X}$ on $X$ (resp.$\overline{T}$ on $T$) and the
underlying real analytic structure $X_{\mathbb{R}}$ (resp.$T_{\mathbb{R}}$),
we have

$\mathscr{O}^{t,T}_{X\times
T}=\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\rho^{\prime}_{!}\mathscr{D}_{\overline{X}\times\overline{T}}}(\rho^{\prime}_{!}\mathscr{O}_{\overline{X}\times\overline{T}},\mathscr{D}^{t,T}_{X\times
T}),$

where we omit the reference to the real structures.

###  2.b. The functors $\operatorname{TH}^{S}$ and $\operatorname{RH}^{S}$

####  2.b.1. The functor $\operatorname{TH}^{S}$

When $X$ is a real analytic manifold and $S$ is a complex curve, we define the
triangulated functor

$\operatorname{TH}^{S}_{X}:\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathbb{R}\textup{-c}}(p^{-1}_{X}\mathscr{O}_{S})^{\mathrm{op}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{\mathsf{D}}^{+}(\mathscr{D}_{X\times
S_{\mathbb{R}}/S})$

given by

$F\mathchoice{\longmapsto}{\mapsto}{\mapsto}{\mapsto}\operatorname{TH}^{S}_{X}(F):=\rho^{\prime-1}\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\rho^{\prime}_{*}p^{-1}_{X}\mathscr{O}_{S}}(\rho^{\prime}_{*}F,\operatorname{\mathfrak{Db}}^{t,S}_{X\times
S})$

where $\mathscr{D}_{X\times S_{\mathbb{R}}/S}$ denotes the sheaf of linear
differential operators with real analytic coefficients on $X\times
S_{\mathbb{R}}$ which commute with $p^{-1}_{X}\mathscr{O}_{S}$.

Recall that, as a consequence of [14, Prop. 4.7] we have

$\displaystyle\operatorname{TH}^{S}_{X}(p^{-1}_{X}\mathscr{O}_{S}\otimes\mathbb{C}_{H\times
V})$
$\displaystyle\simeq\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}(\mathbb{C}_{X\times
V},\operatorname{\mathit{T}\mathscr{H}\\!\mathit{om}}(\mathbb{C}_{H\times
S},\operatorname{\mathfrak{Db}}_{X\times S}))$ $\displaystyle\simeq
R\Gamma_{X\times
V}\operatorname{\mathit{T}\mathscr{H}\\!\mathit{om}}(\mathbb{C}_{H\times
S},\operatorname{\mathfrak{Db}}_{X\times
S})\quad(\text{\cite[cite]{[\@@bibref{}{KS1}{}{}, (2.6.9)]}})$

for any relatively compact locally closed, resp.open, subanalytic subsets $H$
of $X$, resp.$V$ of $S$. If $H=Z$ is closed, we have
$\operatorname{\mathit{T}\mathscr{H}\\!\mathit{om}}(\mathbb{C}_{Z\times
S},\operatorname{\mathfrak{Db}}_{X\times S})=\Gamma_{Z\times
S}\operatorname{\mathfrak{Db}}_{X\times S}$ by definition. We conclude:

(2)
$\operatorname{TH}^{S}_{X}(p^{-1}_{X}\mathscr{O}_{S}\otimes\mathbb{C}_{Z\times
V})\simeq\Gamma_{Z\times V}\operatorname{\mathfrak{Db}}_{X\times S}.$

On the other hand, if $H=\Omega$ is open, since
$\operatorname{\mathit{T}\mathscr{H}\\!\mathit{om}}(\mathbb{C}_{\Omega\times
S},\operatorname{\mathfrak{Db}}_{X\times S})$ is a c-soft sheaf, we obtain

(3)
$\operatorname{TH}^{S}_{X}(p^{-1}_{X}\mathscr{O}_{S}\otimes\mathbb{C}_{\Omega\times
V})\simeq\Gamma_{X\times
V}\operatorname{\mathit{T}\mathscr{H}\\!\mathit{om}}(\mathbb{C}_{\Omega\times
S},\operatorname{\mathfrak{Db}}_{X\times S}).$

####  2.b.2. The functor $\operatorname{RH}^{S}$

If $X$ is a complex manifold and $S$ is a complex curve,
$\operatorname{RH}^{S}_{X}:\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathbb{R}\textup{-c}}(p^{-1}_{X}\mathscr{O}_{S})^{\mathrm{op}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{\mathsf{D}}^{\mathrm{b}}(\mathscr{D}_{X\times
S/S})$ is given by the assignment

(4)
$\begin{split}F\mathchoice{\longmapsto}{\mapsto}{\mapsto}{\mapsto}\operatorname{RH}^{S}_{X}(F)&:=\rho^{\prime-1}\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\rho^{\prime}_{*}p^{-1}_{X}\mathscr{O}_{S}}(\rho^{\prime}_{*}F,\mathscr{O}^{t,S}_{X\times
S})[d_{X}]\\\
&\phantom{:}\simeq\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{\overline{X}\times\overline{S}}}(\mathscr{O}_{\overline{X}\times\overline{S}},\operatorname{TH}^{S}_{X}(F))[d_{X}],\end{split}$

the last isomorphism being called here “realification procedure” for short
(cf.[17, (3.16)]).

We collect below some results in [17] which will be useful in the sequel. The
first gives the behaviour of $Li^{*}_{s}$ with respect to
$\operatorname{RH}^{S}$.

######  Proposition 2.1 (cf.[17, Prop. 3.29]).

For each $s\in S$ there is an isomorphism of functors on
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{R}\textup{-c}}}(p_{X}^{-1}\mathscr{O}_{S})$

$Li^{*}_{s}\operatorname{RH}^{S}_{X}[-d_{X}]({\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}})\simeq\operatorname{\mathit{T}\mathscr{H}\\!\mathit{om}}(Li^{*}_{s}({\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}),\mathscr{O}_{X}),$

where $X$ is identified to $X\times\\{s\\}$ and $X_{\mathrm{sa}}$ is
identified to $X_{\mathrm{sa}}\times\\{s\\}$.

######  Theorem 2.2 (cf.[17, Th. 3]).

Let
$F\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p_{X}^{-1}\mathscr{O}_{S})$.
Then
$\operatorname{RH}^{S}_{X}(F)\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{rhol}}(\mathscr{D}_{X\times
S/S})$ and we have an isomorphism
$F\simeq\operatorname{{}^{\mathrm{p}}Sol}_{X}(\operatorname{RH}^{S}_{X}(F))$
which is functorial in
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p_{X}^{-1}\mathscr{O}_{S})$.

Let us fix a normal crossing divisor $D$.

######  Theorem 2.3 (cf.[17, Prop. 2.11 & Lem. 4.2]).

The category of holonomic $\mathscr{D}_{X\times S/S}$-modules $\mathscr{L}$ of
D-type with singularities along $D$ is equivalent to the category of locally
free $p_{X^{*}}^{-1}\mathscr{O}_{S}$-modules with $X^{*}:=X\smallsetminus D$
under the correspondence

$\mathscr{L}\mathchoice{\longmapsto}{\mapsto}{\mapsto}{\mapsto}\mathscr{H}^{0}\operatorname{DR}_{X}(\mathscr{L})_{|X^{*}\times
S},\qquad
F\mathchoice{\longmapsto}{\mapsto}{\mapsto}{\mapsto}\operatorname{RH}^{S}_{X}(j_{!}\boldsymbol{D}(F[d_{X}]))=\mathscr{L}.$

###  2.c. Some functorial properties

Let $f:Y\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X$
be a morphism of real or complex analytic manifolds. We denote similarly the
morphism $f\times\operatorname{Id}:Y\times
S\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X\times
S$. We consider in this section the corresponding pullback functor.

####  2.c.1. Pullback with respect to $X$ for $\operatorname{TH}^{S}$

######  Proposition 2.4.

For any morphism
$f:Y\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X$ of
real analytic manifolds there exists a morphism of functors from
$\operatorname{\mathsf{D}}^{-}_{\mathbb{R}\textup{-c}}(p^{-1}_{X}\mathscr{O}_{S})^{\mathrm{op}}$
to $\operatorname{\mathsf{D}}^{+}(\mathscr{D}_{X\times S_{\mathbb{R}}/S})$:

(5)
${}_{\scriptscriptstyle\mathrm{D}}f_{!}\operatorname{TH}^{S}_{Y}(f^{-1}{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{TH}^{S}_{X}({\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}).$

###### Proof.

We first define the desired morphism functorially on the category
$\mathscr{S}$ introduced in Section 1.b.2. We deduce from (3) that the objects
of $\mathscr{S}$ are acyclic for
$\operatorname{TH}^{S}_{X}({\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}})$
and for
$\operatorname{TH}^{S}_{Y}(f^{-1}{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}})$.

######  Lemma 2.5.

For any morphism
$f:Y\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X$ of
real analytic manifolds there exist functors

$\displaystyle{}_{\scriptscriptstyle\mathrm{D}}f_{!}\operatorname{TH}^{S}_{Y}(f^{-1}{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}})_{|\mathscr{S}^{\mathrm{op}}}:\mathscr{S}^{\mathrm{op}}$
$\displaystyle\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathsf{C}^{+}(\mathscr{D}_{X\times
S_{\mathbb{R}}/S})$
$\displaystyle\operatorname{TH}^{S}_{X}({\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}})_{|\mathscr{S}^{\mathrm{op}}}:\mathscr{S}^{\mathrm{op}}$
$\displaystyle\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathsf{C}^{+}(\mathscr{D}_{X\times
S_{\mathbb{R}}/S}).$

and there exists a canonical morphism of functors

${}_{\scriptscriptstyle\mathrm{D}}f_{!}\operatorname{TH}^{S}_{Y}(f^{-1}{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}})_{|\mathscr{S}^{\mathrm{op}}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{TH}^{S}_{X}({\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}})_{|\mathscr{S}^{\mathrm{op}}}$

inducing the morphism (5).

###### Proof.

Let $\Omega$, resp.$V$, be a relatively compact open subanalytic set in $X$,
resp.$S$ (here, we consider the real analytic structures in $X$ and $S$, also
usually identified by $-_{\mathbb{R}}$). We set $Z=X\smallsetminus\Omega$ and
we start by considering the sheaf
$G=p^{-1}_{X}\mathscr{O}_{S}\otimes\mathbb{C}_{Z\times V}$. According to (2),
we have $\operatorname{TH}^{S}_{X}(G)=\Gamma_{Z\times
V}\operatorname{\mathfrak{Db}}_{X\times S}$, regarded as a
$\mathscr{D}_{X_{\mathbb{R}}\times S_{\mathbb{R}}/S}$-module.

On the other hand, the integration of distributions induces a morphism

$\int_{f}:f_{!}(\operatorname{\mathfrak{Db}}_{Y\times
S}\otimes_{\mathscr{A}_{Y\times S}}\omega_{Y\times
S/S})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{\mathfrak{Db}}_{X\times
S}\otimes_{\mathscr{A}_{X\times S}}\omega_{X\times S/S}.$

One then mimics [8, Prop. 4.3] by replacing the transfer module
$\mathscr{D}_{Y\times
S\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X\times
S}$ for the morphism $Y\times
S\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X\times S$
in the absolute sense by the relative one
$\mathscr{D}_{(Y\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X)\times
S/S}:=\mathscr{A}_{Y\times S}\otimes_{f^{-1}\mathscr{A}_{X\times
S}}f^{-1}\mathscr{D}_{X\times S_{\mathbb{R}}/S}$. Let us set
$\operatorname{\mathfrak{Db}}_{Y\times
S}^{\scriptscriptstyle\vee}=\operatorname{\mathfrak{Db}}_{Y\times
S}\otimes_{\mathscr{A}_{Y\times S}}\omega_{Y\times S/S}$ and let
$\mathrm{Sp}_{{\scriptscriptstyle\bullet}}(\mathscr{D}_{(Y\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X)\times
S/S})$ denote the Spencer resolution of
$\mathscr{D}_{(Y\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X)\times
S/S}$ (we recall that, for $k\in\mathbb{N}$,
$\mathrm{Sp}_{k}(\mathscr{D}_{(Y\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X)\times
S/S})$ are locally free over $\mathscr{D}_{Y\times S_{\mathbb{R}}/S}$). The
terms of the complex in $\mathsf{C}^{+}(\mathscr{D}_{Y\times
S_{\mathbb{R}}/S})$

$\displaystyle\mathscr{C}_{f^{-1}Z\times V}$
$\displaystyle:=(\Gamma_{f^{-1}Z\times V}\operatorname{\mathfrak{Db}}_{Y\times
S}^{\scriptscriptstyle\vee})\otimes_{\mathscr{D}_{Y\times
S_{\mathbb{R}}/S}}\mathrm{Sp}_{{\scriptscriptstyle\bullet}}(\mathscr{D}_{(Y\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X)\times
S/S})$ $\displaystyle\hphantom{:}=\Gamma_{f^{-1}Z\times
V}\bigl{(}\operatorname{\mathfrak{Db}}_{Y\times
S}^{\scriptscriptstyle\vee}\otimes_{\mathscr{D}_{Y\times
S_{\mathbb{R}}/S}}\mathrm{Sp}_{{\scriptscriptstyle\bullet}}(\mathscr{D}_{(Y\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X)\times
S/S})\bigr{)}$

are thus c-soft sheaves. Hence, the object
${}_{\scriptscriptstyle\mathrm{D}}f_{!}\operatorname{TH}^{S}(p^{-1}_{Y}\mathscr{O}_{S}\otimes\mathbb{C}_{f^{-1}Z\times
V})$ is represented by the complex

$f_{!}\mathscr{C}_{f^{-1}Z\times V}\simeq\Gamma_{Z\times
V}f_{!}\bigl{(}\operatorname{\mathfrak{Db}}_{Y\times
S}^{\scriptscriptstyle\vee}\otimes_{\mathscr{D}_{Y\times
S_{\mathbb{R}}/S}}\mathrm{Sp}_{{\scriptscriptstyle\bullet}}(\mathscr{D}_{(Y\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X)\times
S/S})\bigr{)}$

in $\mathsf{C}^{+}(\mathscr{D}_{X\times S_{\mathbb{R}}/S})$. As in loc.cit.,
we get a morphism in $\mathsf{C}^{+}(\mathscr{D}_{X\times S_{\mathbb{R}}/S})$:

$\varphi_{Z\times V}:f_{!}\mathscr{C}_{f^{-1}Z\times
V}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Gamma_{Z\times
V}\operatorname{\mathfrak{Db}}_{X\times S}=\operatorname{TH}^{S}_{X}(G).$

We now consider the object
$L=p^{-1}_{X}\mathscr{O}_{S}\otimes\mathbb{C}_{\Omega\times V}$ of
$\mathscr{S}$. From the short exact sequence

$0\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbb{C}_{\Omega\times
V}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbb{C}_{X\times
V}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbb{C}_{Z\times
V}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}0,$

we _define_ $f_{!}\mathscr{C}_{f^{-1}(\Omega\times V)}$ as the cokernel of the
natural morphism $f_{!}\mathscr{C}_{f^{-1}Z\times
V}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}f_{!}\mathscr{C}_{f^{-1}X\times
V}$ in $\mathsf{C}^{+}(\mathscr{D}_{X\times S_{\mathbb{R}}/S})$.

Therefore, on the one hand, the complex $f_{!}\mathscr{C}_{f^{-1}(\Omega\times
V)}$ is a representative of
${}_{\scriptscriptstyle\mathrm{D}}f_{!}\operatorname{TH}^{S}_{Y}(f^{-1}L)$ and
$\operatorname{TH}^{S}_{X}(L)$ is the cokernel in
$\mathsf{Mod}(\mathscr{D}_{X\times S_{\mathbb{R}}/S})$ of

$\operatorname{TH}^{S}_{X}(p^{-1}_{X}\mathscr{O}_{S}\otimes\mathbb{C}_{Z\times
V})\mathrel{\lhook\joinrel\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}\operatorname{TH}^{S}_{X}(p^{-1}_{X}\mathscr{O}_{S}\otimes\mathbb{C}_{X\times
V}).$

On the other hand, by completing the commutative diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{f_{!}\mathscr{C}_{f^{-1}Z\times
V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\varphi_{Z\times
V}}$$\textstyle{f_{!}\mathscr{C}_{f^{-1}X\times
V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\varphi_{X\times
V}}$$\textstyle{f_{!}\mathscr{C}_{f^{-1}(\Omega\times
V)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{TH}^{S}_{X}(p^{-1}_{X}\mathscr{O}_{S}\otimes\mathbb{C}_{Z\times
V})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{TH}^{S}_{X}(p^{-1}_{X}\mathscr{O}_{S}\otimes\mathbb{C}_{X\times
V})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{TH}^{S}_{X}(L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

we define a morphism $\varphi_{\Omega\times
V}:f_{!}\mathscr{C}_{f^{-1}(\Omega\times
V)}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{TH}^{S}_{X}(L)$
in $\mathsf{C}^{+}(\mathscr{D}_{X\times S_{\mathbb{R}}/S})$. Functoriality
with respect to $\mathscr{S}$ follows from the description of the morphisms in
$\mathscr{S}$ (cf.Section 1.b.2). ∎

We can now end the proof of Proposition 2.4. According to Remark 1.8, it is
enough to extend the morphism obtained in Lemma 2.5 as a morphism of functors
from $\operatorname{\mathsf{D}}^{-}(\mathscr{S})$ to
$\operatorname{\mathsf{D}}^{+}(\mathscr{D}_{X\times S_{\mathbb{R}}/S})$.
Functoriality above leads to the definition of a morphism of functors from
$\mathsf{C}^{-}(\mathscr{S})$ to the category of double complexes indexed by
$\mathbb{N}^{2}$ of $\mathsf{Mod}(\mathscr{D}_{X\times S_{\mathbb{R}}/S})$,
and we obtain the desired morphism of functors by passing to the associated
simple complexes. ∎

####  2.c.2. Behaviour of $\operatorname{RH}^{S}$ by localization and pullback
with respect to $X$

######  Proposition 2.6.

Let $Y$ be a complex hypersurface of $X$. Then, for any
$F\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathbb{R}\textup{-c}}(p^{-1}_{X}\mathscr{O}_{S})$
there is a natural isomorphism

(6) $\operatorname{RH}^{S}_{X}(F)(\ast(Y\times
S))\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{RH}^{S}_{X}(F\otimes\mathbb{C}_{(X\smallsetminus
Y)\times S}).$

In particular, if
$F\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathbb{C}\textup{-c}}(p^{-1}_{X}\mathscr{O}_{S})$,

  1. (a)

$\operatorname{RH}^{S}_{X}(F)(\ast(Y\times S))$ belongs to
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{rhol}}(\mathscr{D}_{X\times
S/S})$.

  2. (b)

There is a natural isomorphism
$\operatorname{RH}^{S}_{X}(F\otimes\mathbb{C}_{Y\times
S})\\!\simeq\\!R\Gamma_{[Y\times S]}(\operatorname{RH}^{S}_{X}(F))$ and so
$R\Gamma_{[Y\times S]}(\operatorname{RH}^{S}_{X}(F))$ also belongs to
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{rhol}}(\mathscr{D}_{X\times
S/S})$.

  3. (c)

If the natural morphism
$\operatorname{RH}^{S}_{X}(F)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{RH}^{S}_{X}(F)(\ast(Y\times
S))$ is an isomorphism, then so is the natural morphism
$F\otimes\mathbb{C}_{(X\smallsetminus Y)\times
S}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}F$.

###### Proof.

Let $f=0$ be a local defining equation of $Y$. Since this is a local problem
we may start by assuming that
$F=p^{-1}_{X}\mathscr{O}_{S}\otimes\mathbb{C}_{\Omega\times S}$ for a
relatively compact open subanalytic subset $\Omega$ of $X$. Noting that $f$ is
invertible on
$\operatorname{\mathit{T}\mathscr{H}\\!\mathit{om}}(\mathbb{C}_{(\Omega\smallsetminus
Y)\times S},\operatorname{\mathfrak{Db}}_{X\times S})$, according to [5, Prop.
3.23], the natural restriction morphism

$\operatorname{\mathit{T}\mathscr{H}\\!\mathit{om}}(\mathbb{C}_{\Omega\times
S},\operatorname{\mathfrak{Db}}_{X\times S})(\ast(Y\times
S))\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{\mathit{T}\mathscr{H}\\!\mathit{om}}(\mathbb{C}_{(\Omega\smallsetminus
Y)\times S},\operatorname{\mathfrak{Db}}_{X\times S})$

is an isomorphism. Thus, applying Proposition 1.6, the natural
$\mathscr{D}_{\overline{X}\times\overline{S}}$-linear morphism
$\operatorname{TH}^{S}_{X}(F)(*(Y\times
S))\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{TH}^{S}_{X}(F\otimes\mathbb{C}_{(X\smallsetminus
Y)\times S})$ is an isomorphism for any
$F\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathbb{R}\textup{-c}}(p^{-1}_{X}\mathscr{O}_{S})$.
The existence of the morphism (6) and the fact that it is an isomorphism then
follow by (4) and functoriality.

The remaining statements (a) and (b) follow straightforwardly (see also [17,
Ex. 3.20]), while (c) is obtained by applying
$\operatorname{{}^{\mathrm{p}}Sol}_{X}$ to the isomorphism (6), and by using
Theorem 2.2.∎

######  Corollary 2.7.

For any
$F\\!\in\\!\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathbb{C}\textup{-c}}(p^{-1}_{X}\mathscr{O}_{S})$
and for any closed submanifold $Y$ of $X$, $R\Gamma_{[Y\times
S]}(\operatorname{RH}^{S}_{X}(F))$ is a complex with regular holonomic
$\mathscr{D}_{X\times S/S}$-cohomologies.

###### Proof.

The statement being local, we may assume that $Y$ is an intersection of smooth
hypersurfaces of $X$ and then conclude by Proposition 2.6(b) that
$R\Gamma_{[Y\times
S]}(\operatorname{RH}^{S}(F))\simeq\operatorname{RH}^{S}(F\otimes\mathbb{C}_{Y\times
S})$ which concludes the proof. ∎

######  Proposition 2.8.

Let $f:Y\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X$
be a smooth morphism of complex manifolds. Then there exists a natural
isomorphism in
$\operatorname{\mathsf{D}}^{\mathrm{b}}(f^{-1}\mathscr{D}_{X\times S/S})$,
functorial in
$F\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{R}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})$:

$\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{Y\times
S/S}}(\mathscr{D}_{(Y\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X)\times
S/S},\operatorname{RH}^{S}_{Y}(f^{-1}F))\simeq
f^{-1}\operatorname{RH}^{S}_{X}(F)[d_{Y}-d_{X}].$

The proof is performed by mimicking the proof of [8, Th. 5.8 (5.14)] using
Proposition 2.4.

The following result is the relative version of [8, Prop. 5.9 (5.20)] from
which we adapt the proof.

######  Proposition 2.9.

For any
$F\\!\in\\!\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathbb{R}\textup{-c}}(p^{-1}_{X}\mathscr{O}_{S})$
and for any morphism
$f\\!:\\!Y\\!\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\\!X$
of complex manifolds, there exists a natural morphism in
$\operatorname{\mathsf{D}}^{\mathrm{b}}(\mathscr{D}_{Y\times S/S})$:

(7)
${}_{\scriptscriptstyle\mathrm{D}}f^{*}\operatorname{RH}^{S}_{X}(F)[d_{Y}-d_{X}]\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{RH}^{S}_{Y}(f^{-1}F).$

Moreover, when
$F\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathbb{C}\textup{-c}}(p^{-1}_{X}\mathscr{O}_{S})$,
this morphism is an isomorphism.

###### Proof.

We start by decomposing $f$ as the graph embedding
$Y\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}Y\times
X$ followed by the projection $Y\times
X\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X$,
reducing to the case of (i) a closed immersion and (ii) a projection morphism.

Let us treat (i). We shall prove that

${}_{\scriptscriptstyle\mathrm{D}}f^{*}\operatorname{RH}^{S}_{X}(F)[d_{Y}-d_{X}]\simeq\operatorname{RH}^{S}_{Y}(f^{-1}F)$

by a natural isomorphism in
$\operatorname{\mathsf{D}}^{\mathrm{b}}(\mathscr{D}_{Y\times S/S})$
functorially in $F$. We start by noticing that

(8)
${}_{\scriptscriptstyle\mathrm{D}}f^{*}\operatorname{RH}^{S}_{X}(F\otimes\mathbb{C}_{(X\smallsetminus
Y)\times S})=0.$

To check this local statement we may assume, by induction on
$\operatorname{codim}Y$, that $Y$ is smooth of codimension one, and (8)
follows from Proposition 2.6. Hence we conclude that

$\displaystyle{}_{\scriptscriptstyle\mathrm{D}}f^{*}\operatorname{RH}^{S}_{X}(F)[d_{Y}$
$\displaystyle-
d_{X}]\simeq{}_{\scriptscriptstyle\mathrm{D}}f^{*}\operatorname{RH}^{S}_{X}(F\otimes\mathbb{C}_{Y\times
S})[d_{Y}-d_{X}]$
$\displaystyle\simeq{}_{\scriptscriptstyle\mathrm{D}}f^{*}{}_{\scriptscriptstyle\mathrm{D}}f_{*}\operatorname{RH}^{S}_{Y}(f^{-1}F)[d_{Y}-d_{X}]\quad\text{by
Proposition \ref{RHSloc}\eqref{RHSloc2}}$
$\displaystyle\simeq\operatorname{RH}^{S}_{Y}(f^{-1}F)\ \text{ since
${}_{\scriptscriptstyle\mathrm{D}}f^{*}{}_{\scriptscriptstyle\mathrm{D}}f_{*}[d_{Y}-d_{X}]\simeq\operatorname{Id}_{\operatorname{\mathsf{D}}^{\mathrm{b}}(\mathscr{D}_{Y\times
S/S})}$}.$

Let us now treat (ii). Let
$f:Y\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X$ a
projection of $Y=X\times Z$ on $X$. Recall that in that case we have a natural
transformation of functors on
$\operatorname{\mathsf{D}}^{\mathrm{b}}(\mathscr{D}_{Y\times S/S})$

$\mathscr{D}_{(Y\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X)\times
S/S}\otimes_{f^{-1}\mathscr{D}_{X\times
S/S}}\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{Y\times
S/S}}(\mathscr{D}_{(Y\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X)\times
S/S},\cdot)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{Id}.$

Then by Proposition 2.8 we obtain the canonical morphism (7).

Let us now assume that
$F\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathbb{C}\textup{-c}}(p^{-1}_{X}\mathscr{O}_{S})$.
Since the result is true when $f$ is a closed embedding, it remains to
consider the case where $f$ is a smooth morphism. Then $f$ is a non-
characteristic morphism for any
$\mathscr{M}\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{coh}}(\mathscr{D}_{X})$
in the sense of [7, Def. 11.2.11], hence, according to Theorem 11.3.5 of
loc.cit., we have a functorial isomorphism
$f^{-1}\operatorname{Sol}(\mathscr{M})\simeq\operatorname{Sol}({}_{\scriptscriptstyle\mathrm{D}}f^{*}\mathscr{M})$.
If
$\mathscr{M}\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{rhol}}(\mathscr{D}_{X})$
and $F=\operatorname{Sol}(\mathscr{M})$, according to the Riemann-Hilbert
correspondence in the absolute case (here denoted by $\operatorname{RH}$), we
have $\mathscr{M}\simeq\operatorname{RH}(F)$ and we conclude an isomorphism
${}_{\scriptscriptstyle\mathrm{D}}f^{*}\mathscr{M}\simeq\operatorname{RH}(f^{-1}F)$.
As a consequence,

$\displaystyle
Li_{s}^{*}{}_{\scriptscriptstyle\mathrm{D}}f^{*}\operatorname{RH}^{S}_{X}(F)$
$\displaystyle\simeq{}_{\scriptscriptstyle\mathrm{D}}f^{*}Li_{s}\operatorname{RH}^{S}_{X}(F)\simeq{}_{\scriptscriptstyle\mathrm{D}}f^{*}\operatorname{\mathit{T}\mathscr{H}\\!\mathit{om}}(Li_{s}^{*}F,\mathscr{O}_{X})[d_{X}]$
$\displaystyle\overset{(*)}{\simeq}\operatorname{\mathit{T}\mathscr{H}\\!\mathit{om}}(f^{-1}Li_{s}^{*}F,\mathscr{O}_{Y})[d_{X}]\simeq
Li_{s}^{*}\operatorname{RH}^{S}_{Y}(f^{-1}F)[d_{X}-d_{Y}],$

where $(*)$ holds by the absolute case recalled above and the compatibility of
our constructions with the similar ones in the absolute case. By applying the
variant of Nakayama’s Lemma (see 1.a) to the morphism (7) in
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{rhol}}(\mathscr{D}_{Y\times
S/S})$, we obtain that (7) is an isomorphism for any smooth morphism, and thus
for any morphism. ∎

Recall that, for
$F\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})$,
$\operatorname{RH}^{S}_{X}(F)$ is a strict regular holonomic module if and
only if $F$ is strictly perverse ([17, Prop. 2]).

######  Lemma 2.10.

Let
$F,G\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})$
and let us assume that $G$ is perverse and $F$ is strictly perverse. Then the
$p^{-1}_{X}\mathscr{O}_{S}$-module

$\mathscr{H}\\!\mathit{om}_{\mathrm{perv}(p^{-1}_{X}\mathscr{O}_{S})}(F,G):=\mathscr{H}^{0}R\mathscr{H}\\!\mathit{om}_{p^{-1}_{X}\mathscr{O}_{S}}(F,G)$

is strict.

###### Proof.

Let us consider a morphism
$\Phi:F\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}G$
of perverse sheaves such that, for some $s_{o}\in S$ and some local coordinate
$s$ vanishing at $s_{o}$, $s^{n}\Phi=\nobreak 0$. Then $\Phi$ is the zero
morphism away from $s_{o}$, so
$\operatorname{RH}^{S}_{X}(\Phi):\operatorname{RH}^{S}_{X}(G)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{RH}^{S}_{X}(F)$
is also zero away from $s_{o}$. This means that the image of
$\operatorname{RH}^{S}_{X}(\Phi)$ is a torsion submodule of the strict module
$\operatorname{RH}^{S}_{X}(F)$, so it is zero, hence $\Phi$ is zero since
$\operatorname{{}^{\mathrm{p}}Sol}_{X}\operatorname{RH}^{S}_{X}\simeq\operatorname{Id}$.
∎

###  2.d. Behaviour of $\operatorname{RH}^{S}$ under finite ramification over
$S$

Let $s_{0}\in S$, let $N$ be a natural number and let
$\delta:(S^{\prime},s^{\prime}_{0})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}(S,s_{0})$
be the ramification of center $s_{0}$ of degree $N$ (that is, there exist a
local chart on $S$ centered in $s_{0}$ and a local chart centered in
$s^{\prime}_{0}$ such that $\delta(s^{\prime})=s^{\prime N}$). For simplicity
we shall keep the notation $\delta$ also to denote the morphism
$\operatorname{Id}_{X}\times\delta:X\times
S^{\prime}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X\times
S$.

For a $\mathscr{D}_{X\times S/S}$-module $\mathscr{M}$, resp.an object
$F\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{R}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})$,
the pullback is defined by

${}_{\scriptscriptstyle\mathrm{D}}\delta^{*}\mathscr{M}:=\mathscr{O}_{X\times
S^{\prime}}\otimes_{\delta^{-1}\mathscr{O}_{X\times
S}}\delta^{-1}\mathscr{M}\quad\text{resp.}\kern
3.00003pt\delta^{*}F:=p^{-1}\mathscr{O}_{S^{\prime}}\otimes_{p^{-1}\delta^{-1}\mathscr{O}_{S}}\delta^{-1}F.$

We remark that if ${}_{\scriptscriptstyle\mathrm{D}}\delta_{*}$ denotes the
direct image in the sense of $\mathscr{D}_{X\times S/S}$-modules, then
${}_{\scriptscriptstyle\mathrm{D}}\delta_{*}=\delta_{*}$, so we simply denote
${}_{\scriptscriptstyle\mathrm{D}}\delta^{*},{}_{\scriptscriptstyle\mathrm{D}}\delta_{*}$
by $\delta^{*},\delta_{*}$. We also remark that $\mathscr{O}_{X\times
S^{\prime}}$ is flat over $\delta^{-1}\mathscr{O}_{X\times S}$ hence we have

$\delta_{*}\delta^{*}\mathscr{M}\simeq\delta_{*}\mathscr{O}_{X\times
S^{\prime}}\otimes_{\mathscr{O}_{X\times S}}\mathscr{M}\quad\text{resp.}\kern
3.00003pt\delta_{*}\delta^{*}F:=\delta_{*}\mathscr{O}_{S^{\prime}}\otimes_{\mathscr{O}_{S}}F,$

so that $\mathscr{M}$, resp.$F$, is a direct summand in
$\delta_{*}\delta^{*}\mathscr{M}$, resp.in $\delta_{*}\delta^{*}F$.

The first pullback induces a well-defined exact functor from
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{coh}}(\mathscr{D}_{X\times
S/S})$ to
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{coh}}(\mathscr{D}_{X\times
S^{\prime}/S^{\prime}})$, as already used in [17] in a particular situation
(proof of Corollary 2.8, where $\delta$ is denoted by $\rho$), and the second
one a well-defined functor
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{R}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{R}\textup{-c}}}(p^{-1}\mathscr{O}_{S^{\prime}})$.
The following results benefited from useful discussions with Luca Prelli.

######  Lemma 2.11.

There is a well-defined morphism in
$\operatorname{\mathsf{D}}^{\mathrm{b}}(\rho^{\prime}_{S*}\delta^{-1}\mathscr{D}_{X\times
S/S})$

(9) $\delta^{-1}\mathscr{O}_{X\times
S}^{t,S}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{O}_{X\times
S^{\prime}}^{t,S^{\prime}}.$

###### Proof.

It is sufficient to prove the existence of such a morphism when replacing
$\mathscr{O}^{t,S}_{X\times S}$ with $C^{\infty,t,S}_{X\times S}$. In that
case it is obtained by the composition of functions with $\delta$ — this does
not interfere with the growth conditions — yielding a
$\delta^{-1}\mathscr{D}_{X\times S/S}$-linear morphism since the operators in
$\mathscr{D}_{X\times S/S}$ do not involve derivations along $S$. ∎

######  Proposition 2.12.

With the notation as above, for any
$F\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{R}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})$,
there exists a morphism, functorial in $F$, in
$\operatorname{\mathsf{D}}^{\mathrm{b}}(\mathscr{D}_{X\times
S^{\prime}/S^{\prime}})$

$\Psi_{F}:\delta^{*}\operatorname{RH}^{S}_{X}(F)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{RH}^{S^{\prime}}_{X}(\delta^{*}F),$

which is an isomorphism if $F$ is an object of
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})$.

###### Proof.

Following the definition of the relative Riemann-Hilbert functor, we have to
construct a natural morphism:

(10) $\mathscr{O}_{X\times S^{\prime}}\otimes_{\delta^{-1}\mathscr{O}_{X\times
S}}\delta^{-1}{{\rho^{\prime}}_{S}^{-1}}\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\rho^{\prime}_{S*}p^{-1}_{X}\mathscr{O}_{S}}(\rho^{\prime}_{{S}*}F,\mathscr{O}^{t,S}_{X\times
S})\\\
\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\rho^{\prime-1}_{S^{\prime}}\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\rho^{\prime}_{S^{\prime}*}p^{-1}\mathscr{O}_{S^{\prime}}}(\rho^{\prime}_{S^{\prime}*}\delta^{*}F,\mathscr{O}^{t,S^{\prime}}_{X\times
S^{\prime}}).$

We have a natural isomorphism of functors on sites
$\delta^{-1}\rho^{\prime-1}_{S}\simeq\rho^{\prime-1}_{S^{\prime}}\delta^{-1}$
which yields a natural isomorphism in
$\operatorname{\mathsf{D}}^{\mathrm{b}}(\delta^{-1}\mathscr{D}_{X\times S/S})$

(11)
$\delta^{-1}{\rho^{\prime-1}_{S}}\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\rho^{\prime}_{S*}p^{-1}_{X}\mathscr{O}_{S}}(\rho^{\prime}_{S*}F,\mathscr{O}^{t,S}_{X\times
S})\\\
\simeq{\rho^{\prime-1}_{S^{\prime}}}\delta^{-1}\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\rho^{\prime}_{S*}p^{-1}_{X}\mathscr{O}_{S}}(\rho^{\prime}_{S*}F,\mathscr{O}^{t,S}_{X\times
S}).$

Recall that, for a morphism
$g:Z^{\prime}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}Z$
of manifolds and $\mathscr{A}$ a sheaf of rings on $Z$, we have a natural
morphism of bifunctors on
$\operatorname{\mathsf{D}}^{\mathrm{b}}(\mathscr{A})$ (cf.[7, (2.6.27]):

(12)
$f^{-1}\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{A}}({\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{f^{-1}\mathscr{A}}(f^{-1}({\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}),f^{-1}({\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}})).$

Since we are working with sheaves on Grothendieck topologies (see [2] and
[9]), we have the analogous of (12), that is, in the present situation, we
have a natural morphism in
$\operatorname{\mathsf{D}}^{\mathrm{b}}(\rho^{\prime}_{S^{\prime}!}\delta^{-1}\mathscr{D}_{X\times
S/S})$

$\delta^{-1}\\!\\!\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\rho^{\prime}_{S*}p^{-1}_{X}\mathscr{O}_{S}}(\rho^{\prime}_{S*}F,\mathscr{O}^{t,S}_{X\times
S})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\delta^{-1}\rho^{\prime}_{S*}p^{-1}_{X}\mathscr{O}_{S}}(\delta^{-1}\rho^{\prime}_{S*}F,\delta^{-1}\mathscr{O}^{t,S}_{X\times
S}),$

hence a natural morphism in
$\operatorname{\mathsf{D}}^{\mathrm{b}}(\delta^{-1}\mathscr{D}_{X\times S/S})$

(13)
${\rho^{\prime-1}_{S^{\prime}}}\delta^{-1}\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\rho^{\prime}_{S*}p^{-1}_{X}\mathscr{O}_{S}}(\rho^{\prime}_{S*}F,\mathscr{O}^{t,S}_{X\times
S})\\\
\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\rho^{\prime-1}_{S^{\prime}}}\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\delta^{-1}\rho^{\prime}_{S*}p^{-1}_{X}\mathscr{O}_{S}}(\delta^{-1}\rho^{\prime}_{S*}F,\delta^{-1}\mathscr{O}^{t,S}_{X\times
S}).$

According to the morphism (9) in
$\operatorname{\mathsf{D}}^{\mathrm{b}}(\delta^{-1}\mathscr{D}_{X\times
S/S})$, combining with (13) and the commutation
$\delta^{-1}\rho^{\prime}_{S*}\simeq\rho^{\prime}_{S^{\prime}*}\delta^{-1}$ we
derive a functorial chain of morphisms in
$\operatorname{\mathsf{D}}^{\mathrm{b}}(\delta^{-1}\mathscr{D}_{X\times S/S})$

$\rho^{\prime-1}_{S^{\prime}}\delta^{-1}\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\rho^{\prime}_{S*}p^{-1}_{X}\mathscr{O}_{S}}(\rho^{\prime}_{S*}F,\mathscr{O}^{t,S}_{X\times
S})\\\
\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\rho^{\prime-1}_{S^{\prime}}\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\delta^{-1}\rho^{\prime}_{S*}p^{-1}_{X}\mathscr{O}_{S}}(\delta^{-1}\rho^{\prime}_{S*}F,\mathscr{O}^{t,S^{\prime}}_{X\times
S^{\prime}})\\\ \hskip
65.44142pt\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\rho^{\prime-1}_{S^{\prime}}\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\rho^{\prime}_{S*}\delta^{-1}p^{-1}_{X}\mathscr{O}_{S}}(\rho^{\prime}_{S*}\delta^{-1}F,\mathscr{O}^{t,S^{\prime}}_{X\times
S^{\prime}})\\\
\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\rho^{\prime-1}_{S^{\prime}}\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\rho^{\prime}_{S^{\prime}*}p^{-1}\delta^{*}\mathscr{O}_{S}}(\rho^{\prime}_{S^{\prime}*}\delta^{*}F,\mathscr{O}^{t,S^{\prime}}_{X\times
S^{\prime}}),$

where the last term results by applying
$\mathscr{O}_{S^{\prime}}\otimes_{\delta^{-1}\mathscr{O}_{S}}(\cdot)$. We also
remark that the last term (which we will name $\mathscr{L}$ for simplicity) is
the right term of the desired morphism (10) and is already an object of
$\operatorname{\mathsf{D}}^{\mathrm{b}}(\mathscr{D}_{X\times S^{\prime}})$.
Hence, by applying to (11) the functor $\mathscr{O}_{X\times
S^{\prime}}\otimes_{\delta^{-1}\mathscr{O}_{X\times S}}(\cdot)$ we derive a
chain of natural morphisms in
$\operatorname{\mathsf{D}}^{\mathrm{b}}(\mathscr{D}_{X\times
S^{\prime}/S^{\prime}})$.

$\mathscr{O}_{X\times S^{\prime}}\otimes_{\delta^{-1}\mathscr{O}_{X\times
S}}\delta^{-1}\rho^{\prime-1}_{S^{\prime}}\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\rho^{\prime}_{S*}p^{-1}_{X}\mathscr{O}_{S}}(\rho^{\prime}_{S*}F,\mathscr{O}^{t,S}_{X\times
S})\\\
\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{O}_{X\times
S^{\prime}}\otimes_{\delta^{-1}\mathscr{O}_{X\times
S}}\mathscr{L}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{L}$

whose composition gives the desired morphism $\Psi_{F}$.

Assume now that
$F\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})$
and let us prove that $\Psi_{F}$ is an isomorphism.

Since in that case $\operatorname{RH}^{S}_{X}(F)$ is an object of
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{rhol}}(\mathscr{D}_{X\times
S/S})$, it is clear that $\delta^{*}\operatorname{RH}^{S}_{X}(F)$ is an object
of
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{rhol}}(\mathscr{D}_{X\times
S/S}^{\prime})$. The same holds true with
$\operatorname{RH}^{S^{\prime}}_{X}(\delta^{*}F)$ since $\delta^{*}F$ is an
object of
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S^{\prime}})$.

It is then sufficient to apply $Li^{*}_{s^{\prime}}$ to both sides of the
morphism $\Psi_{F}$ and apply Proposition 2.1, noting that
$Li^{*}_{s^{\prime}}\delta^{*}=Li^{*}_{s}$, where
$s\\!=\\!\delta(s^{\prime})$. This way, both members become, by reduction to
the absolute case, isomorphic to
$\operatorname{\mathit{T}\mathscr{H}\\!\mathit{om}}(Li^{*}_{s}F,\mathscr{O}_{X})[d_{X}]$.
∎

We shall now come back to the situation described at the beginning of this
section, and we keep the same notations.

######  Proposition 2.13.

Let $\mathscr{M},\mathscr{N}$ be $\mathscr{D}_{X\times S/S}$-modules and
assume that $\mathscr{M}$ is coherent. If the complex
$\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{X\times
S^{\prime}/S^{\prime}}}(\delta^{*}\mathscr{M},\delta^{*}\mathscr{N})$ belongs
to
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S^{\prime}})$,
then the complex
$\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{X\times
S/S}}(\mathscr{M},\mathscr{N})$ also belongs to
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})$.

###### Proof.

Since $\mathscr{N}$ is a direct summand of $\delta_{*}\delta^{*}\mathscr{N}$
it suffices to deduce from the assumption that
$\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{X\times
S/S}}(\mathscr{M},\delta_{*}\delta^{*}\mathscr{N})$ is an object of
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p_{X}^{-1}\mathscr{O}_{S})$.
Thanks to the adjunction morphism we have in
$\operatorname{\mathsf{D}}^{\mathrm{b}}(p^{-1}_{X}\mathscr{O}_{S})$

$\displaystyle\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{X\times
S/S}}(\mathscr{M},\delta_{*}\delta^{*}\mathscr{N})$
$\displaystyle\simeq\delta_{*}\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\delta^{-1}\mathscr{D}_{X\times
S/S}}(\delta^{-1}\mathscr{M},\delta^{*}\mathscr{N})$
$\displaystyle\simeq\delta_{*}\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{X\times
S^{\prime}/S^{\prime}}}(\delta^{*}\mathscr{M},\delta^{*}\mathscr{N}).$

The result is then a consequence of the assumption and the properness of
$\delta$. ∎

The following result will be used in Section 3. Recall (cf.[17, (3.17)]) that,
for
$F\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathbb{R}\textup{-c}}(p^{-1}_{X}\mathscr{O}_{S})$,
there exists a natural morphism

$\operatorname{RH}^{S}_{X}(F)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{p^{-1}_{X}\mathscr{O}_{S}}(F,\mathscr{O}_{X\times
S})[d_{X}].$

######  Lemma 2.14.

Let $\mathscr{L}$ be a $\mathscr{D}_{X\times S/S}$-module of D-type and let
$F$ be an object of
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathbb{R}\textup{-c}}(p^{-1}_{X}\mathscr{O}_{S})$
be such that $F\simeq F\otimes\mathbb{C}_{(X\smallsetminus D)\times S}$. Then
the morphism

$\beta_{\mathscr{L},F}:\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{X\times
S/S}}(\mathscr{L},\operatorname{RH}^{S}_{X}(F))\\\
\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{X\times
S/S}}(\mathscr{L},\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{p^{-1}_{X}\mathscr{O}_{S}}(F,\mathscr{O}_{X\times
S})[d_{X}])$

is an isomorphism.

###### Proof.

The case where $\mathscr{L}=\mathscr{O}_{X\times S}$ was proved in [17, Lem.
3.19]. We aim at reducing to this case. The statement has a local nature so we
choose local coordinates $x_{1},\dots,x_{n}$ in $X$ such that $D=\\{x\in X\mid
x_{1}\cdots x_{\ell}=0\\}$, and we assume that $S$ is a disc of small enough
radius with coordinate $s$. Let
$\delta:S^{\prime}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}S$
be a finite morphism ramified at $s=0$ only. According to Propositions 2.12
and 2.13, the assertion of the lemma holds for $(\mathscr{L},F)$ if it holds
for $(\delta^{*}\mathscr{L},\delta^{*}F)$. By Theorem 1.13(b) and the same
argument as in the proof of [17, Cor. 2.8], there exists $\delta$ such that
$\delta^{*}\mathscr{L}$ is isomorphic to

$\mathscr{D}_{X\times
S^{\prime}/S^{\prime}}\Big{/}\Bigl{(}\textstyle\sum_{i=1}^{\ell}\mathscr{D}_{X\times
S^{\prime}/S^{\prime}}(x_{i}\partial_{x_{i}}-\alpha_{i})+\sum_{i=\ell+1}^{n}\mathscr{D}_{X\times
S^{\prime}/S^{\prime}}\partial_{x_{i}}\Bigr{)}$

for some holomorphic functions $\alpha_{i}$, $i=1,\dots,\ell$, on
$S^{\prime}$. We will therefore assume that $\mathscr{L}$ is already of this
form. We can replace $F$ with a resolution as given by Proposition 1.6 and,
since the result is local, we may reduce to the case
$F=p^{-1}_{X}\mathscr{O}_{S}\otimes\mathbb{C}_{(\Omega\smallsetminus D)\times
S}$ (since $F\simeq F\otimes\mathbb{C}_{(X\smallsetminus D)\times S}$) for
some relatively compact open subanalytic subset $\Omega$ of $X$. We have

$\operatorname{RH}^{S}(F)\simeq\operatorname{\mathit{T}\mathscr{H}\\!\mathit{om}}(\mathbb{C}_{(\Omega\smallsetminus
D)\times S},\mathscr{O}_{X\times S})[d_{X}]$

and

$\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{p^{-1}_{X}\mathscr{O}_{S}}(F,\mathscr{O}_{X\times
S})[d_{X}]\simeq\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}(\mathbb{C}_{(\Omega\smallsetminus
D)\times S},\mathscr{O}_{X\times S})[d_{X}].$

Let us consider the automorphism $\Phi$ induced on
$\operatorname{\mathit{T}\mathscr{H}\\!\mathit{om}}(\mathbb{C}_{(\Omega\smallsetminus
D)\times S},\operatorname{\mathfrak{Db}}_{X\times S})$ and on
$\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}(\mathbb{C}_{(\Omega\smallsetminus
D)\times S},\operatorname{\mathfrak{Db}}_{X\times
S})\simeq\Gamma_{(\Omega\smallsetminus D)\times
S}(\operatorname{\mathfrak{Db}}_{X\times S})$ by multiplication by the real
analytic function
$|x_{1}|^{2\alpha_{1}(s)}|x_{2}|^{2\alpha_{2}(s)}\cdots|x_{\ell}|^{2\alpha_{\ell}(s)}$.
Then, in $\operatorname{\mathsf{D}}^{\mathrm{b}}(p^{-1}_{X}\mathscr{O}_{S})$,
$\Phi$ induces isomorphisms

$\Phi_{1}:\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{X\times
S/S}}(\mathscr{L},\operatorname{\mathit{T}\mathscr{H}\\!\mathit{om}}(\mathbb{C}_{(\Omega\smallsetminus
D)\times S},\operatorname{\mathfrak{Db}}_{X\times S}))\\\
\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{X\times
S/S}}(\mathscr{O}_{X\times
S},\operatorname{\mathit{T}\mathscr{H}\\!\mathit{om}}(\mathbb{C}_{(\Omega\smallsetminus
D)\times S},\operatorname{\mathfrak{Db}}_{X\times S}))$

and

$\Phi_{2}:\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{X\times
S/S}}(\mathscr{L},\Gamma_{(\Omega\smallsetminus D)\times
S}(\operatorname{\mathfrak{Db}}_{X\times S}))\\\
\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{X\times
S/S}}(\mathscr{O}_{X\times S},\Gamma_{(\Omega\smallsetminus D)\times
S}(\operatorname{\mathfrak{Db}}_{X\times S})).$

We derive a morphism in
$\operatorname{\mathsf{D}}^{\mathrm{b}}(p^{-1}_{X}\mathscr{O}_{S})$

$\Phi_{2}\mu\Phi_{1}^{-1}:\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{X\times
S/S}}(\mathscr{O}_{X\times
S},\operatorname{\mathit{T}\mathscr{H}\\!\mathit{om}}(\mathbb{C}_{(\Omega\smallsetminus
D)\times S},\operatorname{\mathfrak{Db}}_{X\times S}))\\\
\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{X\times
S/S}}(\mathscr{O}_{X\times S},\Gamma_{(\Omega\smallsetminus D)\times
S}(\operatorname{\mathfrak{Db}}_{X\times S}))$

which coincides with the natural one. By the realification procedure
(cf.Section 2.b.2), we are thus reduced to the case
$\mathscr{L}=\mathscr{O}_{X\times S}$, as wanted. ∎

##  3\. Relative Riemann-Hilbert correspondence

Proving Theorem 1 is equivalent to proving Theorem 3. Indeed, in one
direction, let us recall the method introduced in [17, §4.3] to deduce Theorem
1 from Theorem 3. According to [17, (3.17)], there exists a natural morphism
of bifunctors from
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{rhol}}(\mathscr{D}_{X\times
S/S})^{\mathrm{op}}\times\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{R}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})$
to $\operatorname{\mathsf{D}}^{\mathrm{b}}(p^{-1}_{X}\mathscr{O}_{S})$:

(14) $\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{X\times
S/S}}(\mathscr{M},\operatorname{RH}^{S}_{X}(F))\\\
\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{X\times
S/S}}(\mathscr{M},\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{p^{-1}_{X}\mathscr{O}_{S}}(F,\mathscr{O}_{X\times
S}[d_{X}]))\\\
\hfil\simeq\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{p^{-1}_{X}\mathscr{O}_{S}}(F,\operatorname{{}^{\mathrm{p}}Sol}_{X}(\mathscr{M})),$

where the last isomorphism is an application of [7, (2.6.7)]. Notice that the
right-hand side of (14) is an object of
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})$
provided that
$F\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})$.
In that case, by Theorem 3, the left-hand side is also an object of
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})$.
In particular,
$\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{X\times
S/S}}(\mathscr{M},\operatorname{RH}^{S}_{X}(F))_{(x,s)}$ has
$\mathscr{O}_{S,s}$-finitely generated cohomologies for any $(x,s)\in X\times
S$. By the variant of Nakayama’s lemma recalled in Section 1.b.3, and since
$Li_{s_{o}}^{*}\eqref{E:20}$ is an isomorphism for any $s_{o}\in S$ (this is
the absolute case, where the result is known (cf. [5, Cor. 8.6]), we conclude
that (14) is an isomorphism. Replacing $F$ with
$\operatorname{{}^{\mathrm{p}}Sol}_{X}\mathscr{M}$, we deduce an isomorphism
of functors

$\operatorname{Id}_{\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{rhol}}(\mathscr{D}_{X\times
S/S})}\mathchoice{\xrightarrow{\textstyle\kern 4.0pt\beta\kern
3.0pt}}{\stackrel{{\scriptstyle\beta}}{{\longrightarrow}}}{}{}\operatorname{RH}^{S}_{X}\circ\operatorname{{}^{\mathrm{p}}Sol}_{X},$

concluding Theorem 1 as explained in the introduction.

Conversely, Theorem 1 implies Theorem 3 since the former implies full
faithfulness of $\operatorname{{}^{\mathrm{p}}Sol}$, so we have a natural
isomorphism

$\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{X\times
S/S}}(\mathscr{M},\operatorname{RH}^{S}_{X}(F))\simeq\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{p^{-1}_{X}\mathscr{O}_{S}}(F,\operatorname{{}^{\mathrm{p}}Sol}_{X}\mathscr{M})$

and the right-hand side belongs to
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})$.

###  3.a. Proof of Theorems 1 and 3 in the torsion case

Recall that, according to Proposition 1.5 a holonomic $\mathscr{D}_{X\times
S/S}$-module $\mathscr{M}$ is torsion if and only if
$\operatorname{Supp}(M)\subseteq X\times T$ with $\dim T=0$. In that case we
have the following result.

######  Proposition 3.1.

Let $\mathscr{M}\in\mathrm{Mod}_{\mathrm{rhol}}(\mathscr{D}_{X\times S/S})$ be
a torsion $\mathscr{D}_{X\times S/S}$-module. Then
$\widetilde{\mathscr{M}}:=\mathscr{D}_{X\times S}\otimes_{\mathscr{D}_{X\times
S/S}}\mathscr{M}$ is a regular holonomic $\mathscr{D}_{X\times S}$-module.

###### Proof.

The statement being local, we may assume that
$\operatorname{Char}(\mathscr{M})=\Lambda\times\\{s_{o}\\}$, where $\Lambda$
is a Lagrangian $\mathbb{C}^{*}$-conic closed analytic subset in $T^{*}X$,
and, taking a local coordinates $s$ on $S$ vanishing at $s_{o}$, there exists
$n\in\mathbb{N}$ such that $s^{n}\mathscr{M}=0$. Since we are dealing with
triangulated categories, by an easy argument by induction on $n$ we may assume
that $n=1$. In that case, we have
$\mathscr{M}\simeq\mathscr{M}_{0}\boxtimes\mathscr{O}_{S}/\mathscr{O}_{S}s$,
where, by the assumption of relative regularity, $\mathscr{M}_{0}$ is a
regular holonomic $\mathscr{D}_{X}$-module satisfying
$\operatorname{Char}(\mathscr{M}_{0})=\Lambda$. By construction
$\widetilde{\mathscr{M}}\simeq\mathscr{M}_{0}\boxtimes\mathscr{D}_{S}/\mathscr{D}_{S}s$
and $\operatorname{Char}(\widetilde{\mathscr{M}})=\Lambda\times
T^{*}_{T}S=:\widetilde{\Lambda}$.

Therefore $\widetilde{\mathscr{M}}$ is a regular holonomic
$\mathscr{D}_{X\times S}$-module since the category of regular holonomic
$\mathscr{D}_{X\times S}$-modules is closed under external tensor product. ∎

We denote by
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{rhol}}(\mathscr{D}_{X\times
S/S})_{\mathrm{t}}$ the thick subcategory of
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{rhol}}(\mathscr{D}_{X\times
S/S})$ whose objects have support in $X\times T$ with $\dim T=0$.

######  Proposition 3.2.

The solution functor $\operatorname{{}^{\mathrm{p}}Sol}$ restricted to
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{rhol}}(\mathscr{D}_{X\times
S/S})_{\mathrm{t}}$ is an equivalence of categories

$\operatorname{{}^{\mathrm{p}}Sol}_{X}:\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{rhol}}(\mathscr{D}_{X\times
S/S})_{\mathrm{t}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})_{\mathrm{t}}$

with quasi-inverse the restriction of the functor $\operatorname{RH}^{S}_{X}$
to
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})_{\mathrm{t}}$.

###### Proof.

It is sufficient to prove that the restriction of $\operatorname{RH}^{S}_{X}$
to
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})_{t}$
is fully faithful. Indeed $\operatorname{{}^{\mathrm{p}}Sol}$ is essentially
surjective since, according to Theorem 2.2, for any
$F\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})$
we have
$F\simeq\operatorname{{}^{\mathrm{p}}Sol}_{X}\operatorname{RH}^{S}_{X}(F)$,
and in the case of a torsion object $F$ in
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})_{\mathrm{t}}$
we have
$\operatorname{RH}^{S}_{X}(F)\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{rhol}}(\mathscr{D}_{X\times
S/S})_{\mathrm{t}}$.

For the full faithfulness it is enough to prove that the morphism:

$\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{X\times
S/S}}(\mathscr{M},\operatorname{RH}^{S}_{X}(G))\\\
\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{X\times
S/S}}(\mathscr{M},\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{p^{-1}_{X}\mathscr{O}_{S}}(G,\mathscr{O}_{X\times
S})[d_{X}])$

is an isomorphism for any
$\mathscr{M}\\!\in\\!\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{rhol}}(\mathscr{D}_{X\times
S/S})_{\mathrm{t}}$ and for any
$G\\!\in\\!\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathbb{R}\textup{-c}}(p^{-1}_{X}\mathscr{O}_{S})$.

The cohomologies of $\mathscr{M}$ are regular holonomic $\mathscr{D}_{X\times
S/S}$-modules and, according to Proposition 3.1, $\mathscr{D}_{X\times
S}\otimes_{\mathscr{D}_{X\times S/S}}\mathscr{M}$ is a complex whose
cohomologies are regular holonomic.

Thanks to Proposition 1.6, we may assume that
$G\\!=\\!p^{-1}_{X}\mathscr{O}_{S}\otimes\mathbb{C}_{\Omega\times S}$ for some
open subanalytic subset $\Omega$ of $X$, hence

$\operatorname{RH}^{S}_{X}(G)=\operatorname{\mathit{T}\mathscr{H}\\!\mathit{om}}(\mathbb{C}_{\Omega\times
S},\mathscr{O}_{X\times S})[d_{X}],$

which is a complex with $\mathscr{D}_{X\times S}$-modules as cohomologies and
we get a chain of isomorphisms

$\displaystyle\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{X\times
S/S}}$
$\displaystyle(\mathscr{M},\operatorname{RH}^{S}_{X}(G))\simeq\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{X\times
S}}(\mathscr{D}_{X\times S}\otimes_{\mathscr{D}_{X\times
S/S}}\mathscr{M},\operatorname{RH}^{S}_{X}(G))$
$\displaystyle\overset{(\ast)}{\simeq}\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{X\times
S}}(\mathscr{D}_{X\times S}\otimes_{\mathscr{D}_{X\times
S/S}}\mathscr{M},\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}(\mathbb{C}_{\Omega\times
S},\mathscr{O}_{X\times S})[d_{X}])$
$\displaystyle\simeq\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{X\times
S/S}}(\mathscr{M},\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}(\mathbb{C}_{\Omega\times
S},\mathscr{O}_{X\times S})[d_{X}])$
$\displaystyle\simeq\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{X\times
S/S}}(\mathscr{M},\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{p^{-1}_{X}\mathscr{O}_{S}}(G,\mathscr{O}_{X\times
S})[d_{X}]),$

where isomorphism $(\ast)$ follows by [5, Cor. 8.6]. ∎

###  3.b. Proof of Theorem 1 in the strict case

######  Proposition 3.3.

Let $\mathscr{M},\mathscr{N}$ be regular holonomic $\mathscr{D}_{X\times
S/S}$-modules, $\mathscr{N}$ being strict. Then the natural morphism

$\beta_{\mathscr{M},\mathscr{N}}:\mathscr{H}\\!\mathit{om}_{\mathscr{D}_{X\times
S/S}}(\mathscr{M},\mathscr{N})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{H}\\!\mathit{om}_{\mathrm{perv}(p^{-1}_{X}\mathscr{O}_{S})}(\operatorname{{}^{\mathrm{p}}Sol}_{X}\mathscr{N},\operatorname{{}^{\mathrm{p}}Sol}_{X}\mathscr{M})$

is an isomorphism. In particular, if $\mathscr{M}$ is strict, taking
$\mathscr{N}:=\operatorname{RH}^{S}_{X}(\operatorname{{}^{\mathrm{p}}Sol}_{X}\mathscr{M})$,
there exists an isomorphism
$\beta_{\mathscr{M}}:\mathscr{M}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{RH}^{S}_{X}(\operatorname{{}^{\mathrm{p}}Sol}_{X}\mathscr{M})$
which is compatible with Kashiwara’s morphism in the absolute case.

###### Proof.

Since $\mathscr{N}$ is holonomic and strict,
$\operatorname{{}^{\mathrm{p}}Sol}_{X}\mathscr{N}$ is strictly perverse, after
[17, Prop. 2], and by Lemma 2.10 the target of
$\beta_{\mathscr{M},\mathscr{N}}$, which is $S$-$\mathbb{C}$-constructible, is
strict. By an argument similar to that of Lemma 2.10, the source of
$\beta_{\mathscr{M},\mathscr{N}}$ is also strict, and its fibers at any
$(x,s)\in X\times S$ are $\mathscr{O}_{S,s}$ modules of finite type according
to Theorem 1.4. Hence, in particular, for each $s\in S$ we have

$Li^{*}_{s}\mathscr{H}\\!\mathit{om}_{\mathscr{D}_{X\times
S/S}}(\mathscr{M},\mathscr{N})\simeq
i^{*}_{s}\mathscr{H}\\!\mathit{om}_{\mathscr{D}_{X\times
S/S}}(\mathscr{M},\mathscr{N})$

and

$Li^{*}_{s}(\mathscr{H}\\!\mathit{om}_{\mathrm{perv}(p^{-1}_{X}\mathscr{O}_{S})}(\operatorname{{}^{\mathrm{p}}Sol}_{X}\mathscr{N},\operatorname{{}^{\mathrm{p}}Sol}_{X}\mathscr{M}))\\\
\simeq
i^{*}_{s}\mathscr{H}\\!\mathit{om}_{\mathrm{perv}(p^{-1}_{X}\mathscr{O}_{S})}(\operatorname{{}^{\mathrm{p}}Sol}_{X}\mathscr{N},\operatorname{{}^{\mathrm{p}}Sol}_{X}\mathscr{M}).$

Since $i^{*}_{s}(\beta_{\mathscr{M},\mathscr{N}})$ is an isomorphism according
to the regularity assumption and the absolute case, the first statement
follows by the variants of Nakayama’s lemma (see Sections 1.a and 1.b.3).

For the second statement, we use the isomorphism $\alpha_{F}$ recalled in the
introduction for $F=\operatorname{{}^{\mathrm{p}}Sol}_{X}(\mathscr{M})$, so
that, for
$\mathscr{N}=\operatorname{RH}^{S}_{X}(\operatorname{{}^{\mathrm{p}}Sol}_{X}\mathscr{M})$,
we find
$\operatorname{{}^{\mathrm{p}}Sol}_{X}\mathscr{N}\simeq\operatorname{{}^{\mathrm{p}}Sol}_{X}\mathscr{M}$.
Note that $F$ is strictly perverse, according to [17, Prop. 2], and therefore
$\mathscr{N}$ is also strict (cf.[17, Cor. 4]). We have thus obtained an
isomorphism

$\operatorname{Hom}_{\mathscr{D}_{X\times
S/S}}(\mathscr{M},\operatorname{RH}^{S}_{X}(\operatorname{{}^{\mathrm{p}}Sol}_{X}\mathscr{M}))\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{Hom}_{\mathrm{perv}(p^{-1}_{X}\mathscr{O}_{S})}(\operatorname{{}^{\mathrm{p}}Sol}_{X}\mathscr{M},\operatorname{{}^{\mathrm{p}}Sol}_{X}\mathscr{M}),$

and we define $\beta_{\mathscr{M}}$ as being the unique morphism corresponding
to
$\operatorname{Id}\in\operatorname{Hom}_{\mathrm{perv}(p^{-1}_{X}\mathscr{O}_{S})}(\operatorname{{}^{\mathrm{p}}Sol}_{X}\mathscr{M},\operatorname{{}^{\mathrm{p}}Sol}_{X}\mathscr{M})$.
It is an isomorphism by checking the reduction to each $s_{o}\in S$. ∎

######  Corollary 3.4.

For any strict regular holonomic $\mathscr{D}_{X\times S/S}$-module
$\mathscr{M}$, the complexes $R\Gamma_{[Y\times S]}(\mathscr{M})$ and
$\mathscr{M}(\ast(Y\times S))$ belong to
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{rhol}}(\mathscr{D}_{X\times
S/S})$.

###### Proof.

Since $\mathscr{M}$ is strict we have a natural isomorphism
$\mathscr{M}\simeq\operatorname{RH}^{S}_{X}(\operatorname{{}^{\mathrm{p}}Sol}_{X}\mathscr{M})$
by Proposition 3.3. Thus $\mathscr{M}(\ast(Y\times
S))\simeq\operatorname{RH}^{S}_{X}(\operatorname{{}^{\mathrm{p}}Sol}_{X}(\mathscr{M})\otimes\mathbb{C}_{(X\smallsetminus
Y)\times S})$ and $\\!R\Gamma_{[Y\times
S]}(\mathscr{M})\\!\simeq\operatorname{RH}^{S}_{X}(\operatorname{{}^{\mathrm{p}}Sol}_{X}(\mathscr{M})\otimes\mathbb{C}_{Y\times
S})$ by Proposition 2.6 which concludes the proof. ∎

A special case of Corollary 4 can now be obtained (the general case is proved
in Section 3.e).

######  Corollary 3.5.

For each
$F\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})$
which is strictly perverse, there exists a functorial (when restricting to
this quasi-abelian category) isomorphism which is compatible with the
canonical morphism in the absolute case:

$\boldsymbol{D}(\operatorname{RH}^{S}_{X}(F))\simeq\operatorname{RH}^{S}_{X}(\boldsymbol{D}F),$

where in the left-hand side $\boldsymbol{D}$ stands for the duality functor in
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{hol}}(\mathscr{D}_{X\times
S/S})$ while in the right-hand side stands for the duality functor on
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})$.

###### Proof.

We consider the isomorphism $\beta_{\mathscr{M},\mathscr{N}}$ in the case of
$\mathscr{M}:=\boldsymbol{D}\operatorname{RH}^{S}_{X}(F)$ and
$\mathscr{N}:=\operatorname{RH}^{S}_{X}(\boldsymbol{D}F)$. Since
$\operatorname{{}^{\mathrm{p}}Sol}_{X}(\mathscr{M})\simeq\operatorname{{}^{\mathrm{p}}Sol}_{X}(\mathscr{N})\simeq\boldsymbol{D}F$,
the desired morphism will be the unique one corresponding to the identity of
$\boldsymbol{D}F$. The fact that it is an isomorphism is again a consequence
of the reduction to the absolute case for each $s_{o}$ in $S$. ∎

###  3.c. End of the proof of Theorem 2

We can argue by induction on the length of $\mathscr{M}$ and then reduce to
the cases of a projection and of a closed embedding. The first case was proved
in Section 1.d.1. Let us now consider a closed embedding
$i:Y\mathrel{\lhook\joinrel\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}X$.
The strict case is a consequence of Corollary 3.4. It remains to treat the
torsion case. If $\mathscr{M}=t(\mathscr{M})$ then (Proposition 3.2) there
exists
$F\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})$
such that $\mathscr{M}=\operatorname{RH}^{S}_{X}(F)$, hence (Proposition 2.9)
${}_{\scriptscriptstyle\mathrm{D}}i^{*}\mathscr{M}\simeq\operatorname{RH}^{S}_{Y}(i^{-1}F)$
and the latter object belongs to
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{rhol}}(\mathscr{D}_{Y\times
S/S})$.∎

###  3.d. End of the proof of Theorem 3

We refer to [11, Lem. 4.1.4] which contains the guidelines for the proof of
Theorem 3. In what follows, for a complex manifold $X$ and
$\mathscr{M}\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{rhol}}(\mathscr{D}_{X\times
S/S})$ we consider the statement

$P_{X}(\mathscr{M}):\quad\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{X\times
S/S}}(\mathscr{M},\operatorname{RH}^{S}_{X}(F))\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})\quad\forall\,F\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S}),$

in other words, $\mathscr{M}$ satisfies Theorem 3.

######  Lemma 3.6.

The statement $P$ satisfies the following properties.

  1. (a)

For any manifold $X$ and any open covering $(U_{i})_{i\in I}$ of $X$,

$P_{X}(\mathscr{M})\iff P_{U_{i}}(\mathscr{M}_{|U_{i}})\ \forall\,i\in I.$

  2. (b)

$P_{X}(\mathscr{M})\implies P_{X}(\mathscr{M}[n])\ \forall\,n\in\mathbb{Z}$.

  3. (c)

For any distinguished triangle
$\mathscr{M}^{\prime}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{M}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{M}^{\prime\prime}\mathchoice{\xrightarrow{\textstyle\kern
4.0pt+1\kern 3.0pt}}{\stackrel{{\scriptstyle+1}}{{\longrightarrow}}}{}{}$ in
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{hol}}(\mathscr{D}_{X\times
S/S})$,

$P_{X}(\mathscr{M}^{\prime})\wedge P_{X}(\mathscr{M}^{\prime\prime})\implies
P_{X}(\mathscr{M}).$

  4. (d)

For any regular relative holonomic $\mathscr{D}_{X\times S/S}$-modules
$\mathscr{M}$ and $\mathscr{M}^{\prime}$,

$P_{X}(\mathscr{M}\oplus\mathscr{M}^{\prime})\implies P_{X}(\mathscr{M}).$

  5. (e)

For any projective morphism
$f:X\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}Y$ and
any good regular holonomic $\mathscr{D}_{X\times S/S}$-module $\mathscr{M}$,

$P_{X}(\mathscr{M})\implies
P_{Y}({}_{\scriptscriptstyle\mathrm{D}}f_{*}\mathscr{M}).$

  6. (f)

If $\mathscr{M}=\mathscr{H}^{0}(\mathscr{M})$ is torsion, then
$P_{X}(\mathscr{M})$ is true.

###### Proof.

It is clear that $P_{X}({\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}})$
satisfies Properties 3.6(a), (b), (c), (d). Then Property (e) follows by
adjunction, Proposition 2.9 and by the stability of
$S$-$\mathbb{C}$-constructibility under proper direct image. Last, Property
(f) has been seen in Section 3.a. ∎

###### End of the proof of Theorem 3 (and hence that of Theorem 1).

We wish to prove that $P_{X}(\mathscr{M})$ is true for any $X$ and
$\mathscr{M}\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{rhol}}(\mathscr{D}_{X\times
S/S})$. As recalled in the introduction of Section 3, it is enough to show
that, for any
$F\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})$,
the cohomology modules of
$\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{X\times
S/S}}(\mathscr{M},\operatorname{RH}^{S}_{X}(F))_{(x,s)}$ are finitely
generated $\mathscr{O}_{S,s}$-modules for any $(x,s)\in X\times S$.

Let us set

$\operatorname{Char}(\mathscr{M}):=\mathop{\textstyle\bigcup}\displaylimits_{j}\operatorname{Char}(\mathscr{H}^{j}\mathscr{M})=\mathop{\textstyle\bigcup}\displaylimits_{i\in
I}\Lambda_{i}\times\nobreak T_{i}$

(see Proposition 1.5),
$\operatorname{Supp}(\mathscr{M})=\mathop{\textstyle\bigcup}\displaylimits_{i\in
I}Y_{i}\times T_{i}$ and
$Z_{\mathscr{M}}:=\mathop{\textstyle\bigcup}\displaylimits_{i\in I}Y_{i}$. We
proceed by induction on the dimension of $Z_{\mathscr{M}}$. If $\dim
Z_{\mathscr{M}}<0$ we have $\operatorname{Supp}(\mathscr{M})=\varnothing$,
which implies $\mathscr{M}=0$ and hence, by 3.6(f), $P_{X}(0)$ holds true.

Let us suppose $P_{X}(\mathscr{N})$ true for any
$\mathscr{N}\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{rhol}}(\mathscr{D}_{X\times
S/S})$ such that $\dim Z_{\mathscr{N}}<k$ (with $k\geqslant 0$) and let us
prove the truth of $P_{X}(\mathscr{M})$ for
$\mathscr{M}\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{rhol}}(\mathscr{D}_{X\times
S/S})$ with $\dim Z_{\mathscr{M}}=k$.

By 3.6(b) and (c), we are reduced to proving $P_{X}(\mathscr{M})$ in the case
where $\mathscr{M}$ is a regular holonomic $\mathscr{D}_{X\times S/S}$-module
with $\dim Z_{\mathscr{M}}=k$.

Following the notation of Section 1.a, let $t(\mathscr{M})$ (respectively
$f(\mathscr{M})$) be the torsion part (respectively the strict quotient) of
$\mathscr{M}$. According to 3.6(c) (applied to the distinguished triangle
$t(\mathscr{M})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{M}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}f(\mathscr{M})\mathchoice{\xrightarrow{\textstyle\kern
4.0pt+1\kern 3.0pt}}{\stackrel{{\scriptstyle+1}}{{\longrightarrow}}}{}{}$) and
to 3.6(f), we are reduced to proving $P_{X}(f(\mathscr{M}))$. Notice that
$\dim Z_{f(\mathscr{M})}\leqslant k$ since $Z_{f(\mathscr{M})}\subseteq
Z_{\mathscr{M}}$. If $\dim Z_{f(\mathscr{M})}<k$, $P_{X}(f(\mathscr{M}))$
holds true by induction. Hence we are reduced to proving $P_{X}(\mathscr{M})$
in the case where $\mathscr{M}$ is a strict regular holonomic
$\mathscr{D}_{X\times S/S}$-module such that $\dim Z_{\mathscr{M}}=k$.

In this case, locally (recall that $P_{X}(\mathscr{M})$ is a local statement
by 3.6(a)), there exists a hypersurface $Y$ in $X$ such that $Y\cap
Z_{\mathscr{M}}$ is of dimension $<k$ and $Y\smallsetminus Y\cap
Z_{\mathscr{M}}$ is smooth. We can now apply 3.6(c) to the triangle
$R\Gamma_{[Y\times
S]}(\mathscr{M})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{M}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{M}(\ast(Y\times
S))\mathchoice{\xrightarrow{\textstyle\kern 4.0pt+1\kern
3.0pt}}{\stackrel{{\scriptstyle+1}}{{\longrightarrow}}}{}{}$ (which is a
distinguished triangle in
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{rhol}}(\mathscr{D}_{X\times
S/S})$ by Corollary 3.4). By the induction hypothesis,
$P_{X}(R\Gamma_{[Y\times S]}(\mathscr{M}))$ holds true. We are thus reduced to
proving $P_{X}(\mathscr{M})$ for $\mathscr{M}\simeq\mathscr{M}(\ast(Y\times
S))$.

On the other hand, it is enough to check the property $P_{X}(\mathscr{M})$ for
those
$F\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})$
such that $F=F\otimes\mathbb{C}_{(X\smallsetminus Y)\times S}$. Indeed, let us
check that it holds for those $F$ such that $F=F\otimes\mathbb{C}_{Y\times
S}$. For any
$F\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p^{-1}_{X}\mathscr{O}_{S})$,
we have

$\displaystyle\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{X\times
S/S}}(\mathscr{M},$ $\displaystyle R\Gamma_{[Y\times
S]}(\operatorname{RH}^{S}_{X}(F)))$
$\displaystyle\simeq\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{X\times
S/S}}(\mathscr{M},R\Gamma_{[(Y\cap Z_{\mathscr{M}})\times
S]}(\operatorname{RH}^{S}_{X}(F)))$
$\displaystyle\simeq\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{X\times
S/S}}(\boldsymbol{D}R\Gamma_{[(Y\cap Z_{\mathscr{M}})\times
S]}(\operatorname{RH}^{S}_{X}(F)),\boldsymbol{D}\mathscr{M}),$

where the first isomorphism holds true because
$\operatorname{Supp}(\mathscr{M})\subset Z_{\mathscr{M}}\times S$ and the
second one follows by [15, (3)]. According to Proposition 2.6(b)

$\mathscr{N}:=\boldsymbol{D}R\Gamma_{[(Y\cap Z_{\mathscr{M}})\times
S]}(\operatorname{RH}^{S}_{X}(F))$

belongs to
$\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{rhol}}(\mathscr{D}_{X\times
S/S})$. Thus, by the induction hypothesis, since $\mathscr{N}$ is supported on
$(Y\cap Z_{\mathscr{M}})\times S$ while
$\boldsymbol{D}(\mathscr{M})\simeq\operatorname{RH}^{S}_{X}(\boldsymbol{D}\operatorname{{}^{\mathrm{p}}Sol}\mathscr{M})$
(by Proposition 3.3 and Corollary 3.5),
$\mathscr{H}^{j}\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{X\times
S/S}}(\mathscr{N},\boldsymbol{D}(\mathscr{M}))_{(x,s)}$, and thus
$\mathscr{H}^{j}\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{X\times
S/S}}(\mathscr{M},R\Gamma_{[Y\times
S]}(\operatorname{RH}^{S}_{X}(F)))_{(x,s)}$, is of finite type over
$\mathscr{O}_{S,s}$ for all $j\in\mathbb{Z}$ and $(x,s)\in X\times S$. We now
notice that, if $F=F\otimes\mathbb{C}_{Y\times S}$, then
$\operatorname{RH}^{S}_{X}(F)=R\Gamma_{[Y\times
S]}(\operatorname{RH}^{S}_{X}(F))$, so the same property holds for
$\operatorname{\mathit{R}\mathscr{H}\\!\mathit{om}}_{\mathscr{D}_{X\times
S/S}}(\mathscr{M},\operatorname{RH}^{S}_{X}(F))$, which is enough, as noticed
at the beginning of the proof.

We thus assume that $\mathscr{M}=\mathscr{M}(*(Y\times S))$ is strict, and
$F=F\otimes\mathbb{C}_{(X\smallsetminus Y)\times S}$. There exists a
commutative diagram

$\begin{array}[]{c}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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