# Riemann-Hilbert correspondence for Alexander complexes

Lei Wu  Lei Wu, KU Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium
<EMAIL_ADDRESS>
###### Abstract.

We establish a Riemann-Hilbert correspondence for Alexander complexes (also
known as Sabbah specialization complexes) by using relative holonomic
$\mathscr{D}$-modules in an equivariant way, which particularly gives a
“global” approach to the correspondence for Deligne’s nearby cycles. Using the
correspondence and zero loci of Bernstein-Sato ideals, we obtain a formula for
the relative support of the Alexander complexes.

###### 2010 Mathematics Subject Classification:

14F10; 13N10; 32C38; 32S60; 32S55; 14F43; 14L30.

###### Contents

  1. 1 Introduction
  2. 2 Sheafification of sheaves of modules over commutative rings
  3. 3 Relative maximal and minimal extensions along $F$
  4. 4 Alexander complex of Sabbah
  5. 5 Comparison

##  1\. Introduction

Let $f$ be a holomorphic function on a complex manifold $X$ with $D$ the
divisor of $f$ and let $U_{x}$ be a small open neighborhood of $x\in D$. Then
we consider the fiber product diagram

${\widetilde{U}_{x}=(U_{x}\setminus
D)\times_{\mathbb{C}^{*}}\widetilde{\mathbb{C}}^{*}}$${\widetilde{\mathbb{C}}^{*}}$${U_{x}\setminus
D}$${\mathbb{C}^{*}}$$\scriptstyle{exp}$$\scriptstyle{f}$

where $exp\colon\widetilde{\mathbb{C}}^{*}\to\mathbb{C}^{*}$ is the universal
cover of the punctured complex plane $\mathbb{C}^{*}$. The deck transformation
induces a $\mathbb{C}[\pi_{1}(\mathbb{C}^{*})]$-module structure on the
compactly supported cohomology group
$H^{i}_{c}(\widetilde{U}_{x},\mathbb{C})$, which is called the $i$-th (local)
_Alexander module_ of $f$. Taking $U_{x}$ sufficiently small, the Alexander
modules contain the information of the cohomology groups of the Milnor fibers
around $x$ together with their monodromy action. As $x$ varies along $D$, all
the local Alexander modules give a constructible complex of sheaves of
$\mathbb{C}[\pi_{1}(\mathbb{C}^{*})]$-modules, which recovers the Deligne
nearby cycle of the constant sheaf along $f$ (see [Bry86]). Sabbah [Sab90]
made generalizations for a finite union of holomorphic functions and obtained
what he called the _Alexander complexes_ (see §4 for construction).

In this article, we establish a “global” Riemann-Hilbert correspondence for
the Alexander complexes in a functorial way (Theorem 1.1). A “local”
correspondence for complete specializations of Alexander complexes was
constructed by Sabbah [Sab90, Theorem 5.1.2]; see also [BG12] for the
algebraic “local” case along a single regular function following the approach
of Beilinson-Bernstein [Bei87]. Riemann-Hilbert correspondence between nearby
cycles of regular holonomic $\mathscr{D}$-modules and Deligne nearby cycles of
complex perverse sheaves was constructed by Kashiwara [Kas83] and Malgrange
[Mal83] by using $V$-filtrations along a single holomorphic function. However,
there is no well-defined notion of “normalized” multi indexed $V$-filtrations
along a finite union of holomorphic functions in general (based on the work of
Sabbah [Sab87a, Sab87b]; see also [Wu20, Remark 4.22]). Thus, one would not be
able to generalize the method of Kashiwara and Malgrange to construct the
correspondence for Alexander complexes in general.

Roughly speaking, our approach to the “global” correspondence can be seen as a
hybrid of the method of Sabbah [Sab87a, Sab87b] and that of Beilinson-
Bernstein [Bei87]. To be more precise, the construction of the correspondence
relies heavily on the work of Maisonobe [Mai16a] in the study of Bernstei-Sato
ideals by using relative holonomic $\mathscr{D}$-modules over algebraic affine
spaces and its development in [WZ21, BVWZ21a, BVWZ21b], and on the theory of
analytic relative holonomic $\mathscr{D}$-modules and relative constructible
complexes developed in a series of fundamental papers [MFS13, MFS19, FF18,
FFS21]. The new input we need to establish the correspondence is a GAGA-type
principle between analytic and algebraic relative $\mathscr{D}$-modules and
between algebraic and analytic relative constructible sheaves, which is called
in this paper the sheafification of relative modules over commutative rings
$R$ (see §2). The other input is the construction of relative holomorphic
$\mathscr{D}$-modules that are $G$-equivariant with respect to the universal
covering of $(\mathbb{C}^{*})^{r}$ (see §3.4).

Let us briefly explain the terminology of sheafification. When the base space
$X$ is a point, the sheafification of $R$-modules is just the functor
transforming $R$-modules into quasi-coherent $\mathscr{O}$-modules on
$\textup{Spec }R$. For general base spaces, it is the relative analogue for
complexes of sheaves of $R$-modules. One then make the base change onto
$\textup{Spec}^{\textup{an}}R$ to obtain the analytic one for $R$ commutative
finite generated $\mathbb{C}$-algebras. See Remark 3.4 for the reason why the
analytic sheafification is necessary in proving Theorem 1.1 and Theorem 1.2.

###  1.1. Riemann-Hilbert correspondence for Alexander complex

Let $f_{i}$ be holomorphic functions on a complex manifold $X$ for
$i=1,\dots,r$ with $D_{i}$ the divisor of $f_{i}$. We write
$F=(f_{1},\dots,f_{r})$, and $D=\sum_{i}D_{i}$. Suppose that $\mathcal{M}$ is
a (left) holonomic $\mathscr{D}_{X}$-module. We write by

$\mathcal{M}(*D)=\mathcal{M}\otimes_{\mathscr{O}}\mathscr{O}_{X}(*D)$

the algebraic localization of $\mathcal{M}$ along $D$, where
$\mathscr{O}_{X}(*D)=\bigcup_{k\in\mathbb{Z}}\mathscr{O}_{X}(kD)$ is the sheaf
of meromorphic functions with poles along $D$.

To establish the Riemann-Hilbert correspondence for Alexander complexes of
Sabbah, we first construct the relative maximal (resp. minimal) extensions of
$\mathcal{M}$ along $F$, denoted by $\mathcal{M}(*D^{(r)}_{F})$ (resp.
$\mathcal{M}(!D^{(r)}_{F})$), which are both relative holonomic
$\mathscr{D}_{X\times\mathbb{C}^{r}/\mathbb{C}^{r}}$-modules where
$\mathscr{D}_{X\times\mathbb{C}^{r}/\mathbb{C}^{r}}$ is the sheaf of relative
holomorphic differential operators with respect to the projection $pr\colon
X\times\mathbb{C}^{r}\to\mathbb{C}^{r}$ (see Definition 2.4 for relative
holonomicity). More precisely, $\mathcal{M}(*D^{(r)}_{F})$ is the analytic
sheafification of the $\mathscr{D}_{X}[\mathbf{s}]$-module

$\mathcal{M}(*D)\otimes_{\mathbb{C}}\mathbb{C}[\mathbf{s}]\cdot
F^{\mathbf{s}},\quad\mathbf{s}=(s_{1},\dots s_{r})\textup{ the algebraic
coordinates of }\mathbb{C}^{r}$

and $\mathcal{M}(!D^{(r)}_{F})$ is the
$\mathscr{D}_{X\times\mathbb{C}^{r}/\mathbb{C}^{r}}$-dual of
$(\mathbb{D}\mathcal{M})(*D^{(r)}_{F})$, where

$F^{\mathbf{s}}=\prod_{i}f_{i}^{s_{i}}$

is a formal symbol and $\mathbb{D}(\mathcal{M})$ is the $\mathscr{D}_{X}$-dual
of $\mathcal{M}$. See §3.3 for details. It is worth mentioning that when
$\mathcal{M}=\mathscr{O}_{X}$, $\mathscr{O}_{X}(*D_{F}^{(r)})$ gives a non-
trivial example of relative Deligne meromorphic extensions of relative local
systems (see §4.2 and also [FS19, §2.2] and [NS96, §8]).

The key property for the maximal and minimal extensions is that there exists a
natural inclusion (Lemma 3.7)

(1.1) $\mathcal{M}(!D^{(r)}_{F})\hookrightarrow\mathcal{M}(*D^{(r)}_{F}),$

which is a “global”, sheafified and higher dimensional generalization of a
classical result of Beilinson and Bernstein (see [Gin86, Proposition 3.8.3]
and also [BG12]). We then define a relative holonomic $\mathscr{D}$-module on
$X\times\mathbb{C}^{r}$,

$\Psi_{F}(\mathcal{M})=\frac{\mathcal{M}(*D^{(r)}_{F})}{\mathcal{M}(!D^{(r)}_{F})}.$

We denote by

$Exp\colon\mathbb{C}^{r}\rightarrow(\mathbb{C}^{*})^{r},(\alpha_{1},\dots,\alpha_{r})\mapsto(exp(-2\pi\sqrt{-1}\alpha_{1}),\dots,exp(-2\pi\sqrt{-1}\alpha_{r}))$

the universal covering of $(\mathbb{C}^{*})^{r}$ and by

$\pi=(\textup{id},Exp)\colon X\times\mathbb{C}^{r}\rightarrow
X\times(\mathbb{C}^{*})^{r}$

the induced map. We write $G=\pi_{1}((\mathbb{C}^{*})^{r})$, the fundamental
group of $(\mathbb{C}^{*})^{r}$. The universal covering makes
$X\times\mathbb{C}^{r}$ a $G$-space with the quotient $G\backslash
X\times\mathbb{C}^{r}=X\times(\mathbb{C}^{*})^{r}$. The operation

$t_{i}(F^{\mathbf{s}})=f_{i}F^{\mathbf{s}}$

with $t_{i}$ representing the (counterclockwise) loops around the puncture of
each factor $\mathbb{C}^{*}$ of $(\mathbb{C}^{*})^{r}$, makes
$\mathcal{M}(*D^{(r)}_{F})$, $\mathcal{M}(!D^{(r)}_{F})$ and
$\Psi_{F}(\mathcal{M})$ all $G$-equivariant sheaves (see Theorem 3.8, Theorem
3.9 and Remark 3.10(1)).

######  Theorem 1.1.

With the notation as above, if $\mathcal{M}$ is a regular holonomic
$\mathscr{D}_{X}$-module, then we have a natural quasi-isomorphism

$\pi_{*}^{G}\big{(}\textup{DR}_{X\times\mathbb{C}^{r}/\mathbb{C}^{r}}(\Psi_{F}(\mathcal{M}))\big{)}\simeq\widetilde{\psi}_{F}(\textup{DR}_{X}(\mathcal{M})),$

where $\pi_{*}^{G}$ is the equivariant direct image functor,
$\textup{DR}_{X\times\mathbb{C}^{r}/\mathbb{C}^{r}}$ is the relative de Rham
functor on $X\times\mathbb{C}^{r}$ over $\mathbb{C}^{r}$ and
$\widetilde{\psi}_{F}(\textup{DR}_{X}(\mathcal{M}))$ is the analytic
sheafification of the Alexander complex
$\psi_{F}(\textup{DR}_{X}(\mathcal{M}))$.

The $\mathbb{C}[G]$-module structure of
$\pi_{*}^{G}\big{(}\textup{DR}_{X\times\mathbb{C}^{r}/\mathbb{C}^{r}}(\Psi_{F}(\mathcal{M}))\big{)}$
is induced by

$Exp_{*}^{G}(\mathscr{O}_{\mathbb{C}^{r}})\simeq\mathscr{O}_{(\mathbb{C}^{*})^{r}}$

and the natural inclusion

$\mathbb{C}[G]\simeq\mathbb{C}[t_{1}^{\pm 1},\dots,t_{r}^{\pm
1}]\hookrightarrow\mathscr{O}_{(\mathbb{C}^{*})^{r}},$

where in the inclusion $\mathbb{C}[G]$ is treated as a constant sheaf on
$(\mathbb{C}^{*})^{r}$. Theorem 1.1 is related to the local comparision for
Alexander complexes of Sabbah [Sab90, Theorem 5.1.2]. To be more precise, one
can take the (complete) localization of the quasi-isomorphism in Theorem 1.1
at a point ${\boldsymbol{\lambda}}\in(\mathbb{C}^{*})^{r}$ to obtain the local
comparison for $\psi_{F}(\textup{DR}_{X}(\mathcal{M}))$. Notice that
localizing
$\pi_{*}^{G}\big{(}\textup{DR}_{X\times\mathbb{C}^{r}/\mathbb{C}^{r}}(\Psi_{F}(\mathcal{M}))\big{)}$
at ${\boldsymbol{\lambda}}$ is equivalent to localizing
$\textup{DR}_{X\times\mathbb{C}^{r}/\mathbb{C}^{r}}(\Psi_{F}(\mathcal{M}))$ at
${\boldsymbol{\alpha}}$ for every ${\boldsymbol{\alpha}}\in
Exp^{-1}({\boldsymbol{\lambda}})$, since $G$ acts freely on $\mathbb{C}^{r}$
with $G\textbackslash\mathbb{C}^{r}=(\mathbb{C}^{*})^{r}$. In particular, when
$r=1$, localizing the quasi-isomorphism in Theorem 1.1 at
$\lambda\in\mathbb{C}^{*}$ gives the comparison between the $\alpha$-nearby
cycle of $\mathcal{M}$ and the $\lambda$-nearby cycle of
$\textup{DR}(\mathcal{M})$ along $f$ for every $\alpha\in Exp^{-1}(\lambda)$
(see [Kas83, Mal83, BG12] and also [Wu21]). Notice that the $G$-action in
Theorem 1.1 explains why the correspondence between $\alpha$-nearby cycles of
regular holonomic $\mathscr{D}$-modules and $\lambda$-nearby cycles of
perverse sheaves along a single holomorphic function is $\mathbb{Z}$-to-1.

Since the relative de Rham functor is $G$-equivariant (see §3.5), we have a
natural isomorphism

$\textup{DR}_{X\times(\mathbb{C}^{*})^{r}/(\mathbb{C}^{*})^{r}}\big{(}\pi_{*}^{G}(\Psi_{F}(\mathcal{M}))\big{)}\simeq\pi_{*}^{G}\big{(}\textup{DR}_{X\times\mathbb{C}^{r}/\mathbb{C}^{r}}(\Psi_{F}(\mathcal{M}))\big{)}.$

Then for regular holonomic $\mathscr{D}_{X}$-modules $\mathcal{M}$,
$\textup{DR}_{X\times(\mathbb{C}^{*})^{r}/(\mathbb{C}^{*})^{r}}(\bullet)$
gives a relative Riemann-Hilbert correspondence on
$X\times(\mathbb{C}^{*})^{r}$ over $(\mathbb{C}^{*})^{r}$

${\pi_{*}^{G}(\Psi_{F}(\mathcal{M}))}$${\widetilde{\psi}_{F}(\textup{DR}_{X}(\mathcal{M}))}$

with “$\longleftarrow$” induced by the Riemann-Hilbert correspondence for
regular holonomic $\mathscr{D}$-modules on $X$. See also [FFS21] for a
relative Riemann-Hilbert correspondence over curves and §5.4 for a related
open question.

Theorem 1.1 can be further refined to a comparision for higher-codimensional
Sabbah specialization complexes (see §4.1) as follows by using sheaves of
(algebraic) local cohomology.

######  Theorem 1.2.

In the situation of Theorem 1.1, for every subset
$I\subseteq\\{1,2,\dots,r\\}$ we have quasi-isomorphisms

$\displaystyle\pi_{*}^{G}\big{(}\textup{DR}_{X\times\mathbb{C}^{r}/\mathbb{C}^{r}}\mathbb{D}R\Gamma_{[D_{I}\times\mathbb{C}^{r}]}\big{(}(\mathbb{D}\mathcal{M})(!D^{(r)}_{F})\big{)}\big{)}$
$\displaystyle\simeq\textup{DR}_{X\times(\mathbb{C}^{*})^{r}/(\mathbb{C}^{*})^{r}}\mathbb{D}R\Gamma_{[D_{I}\times(\mathbb{C}^{*})^{r}]}\big{(}\pi_{*}^{G}\big{(}(\mathbb{D}\mathcal{M})(!D^{(r)}_{F})\big{)}\big{)}$
$\displaystyle\simeq\widetilde{\psi}_{D_{I}}(\textup{DR}_{X}(\mathcal{M})),$

where $\psi_{D_{I}}(\textup{DR}_{X}(\mathcal{M}))$ is the Sabbah
specialization complexes along $D_{I}=\bigcap_{i\in I}D_{i}$.

###  1.2. Relative Support

We now discuss the relative supports of $\Psi_{F}(\mathcal{M})$ and the
Alexander complex of $\textup{DR}(\mathcal{M})$ (see Definition 2.2 and
Proposition-Definition 2.7). Applying Theorem 1.1, we obtain:

######  Corollary 1.3.

Let $\mathcal{M}$ be a regular holonomic $\mathscr{D}_{X}$-module on a complex
manifold $X$ and let $F=(f_{1},\dots,f_{r})$ be an $r$-tuple of holomorphic
functions on $X$. Then

$\textup{supp}_{(\mathbb{C}^{*})^{r}}\psi_{F}(\textup{DR}(\mathcal{M}))=Exp\big{(}\textup{supp}_{\mathbb{C}^{r}}\Psi_{F}(\mathcal{M})\big{)}$

and

$\textup{supp}_{\mathbb{C}^{r}}\Psi_{F}(\mathcal{M})=Exp^{-1}\big{(}\textup{supp}_{(\mathbb{C}^{*})^{r}}\psi_{F}(\textup{DR}(\mathcal{M}))\big{)}.$

The action of $G$ on $\mathbb{C}^{r}$ induces an action of $G$ on its
algebraic coordinate ring $\mathbb{C}[\mathbf{s}]$, i.e.

$t_{i}\cdot s_{j}=\begin{cases}s_{j}+1,&\textup{ if }i=j\\\ s_{j},&\textup{ if
}i\not=j.\end{cases}$

Using Bernstein-Sato ideals (see §2.4 and §3.6), we obtain a geometric
description of the relative support of $\Psi_{F}(\mathcal{M})$:

######  Theorem 1.4.

For each pair $(\mathcal{M},F)$ as in Corollary 1.3, locally on a relative
compact open subset 111The condition “locally on a relative compact open
subset” (here and elsewhere it appears in this article) is to ensure
finiteness. Thus, if we are in the algebraic category, then this condition can
usually be eliminated. of $X$ there exist finite sets

$S(F,\mathcal{M})\subseteq\mathbb{N}_{\geq 0}^{r}\textup{ and
}\kappa(L)\subseteq\mathbb{C}\textup{ for each }L\in S(F,\mathcal{M})$

such that

$\textup{supp}_{\mathbb{C}^{r}}\Psi_{F}(\mathcal{M})=\bigcup_{g\in
G}\bigcup_{L\in
S(F,\mathcal{M})}\bigcup_{\alpha\in\kappa(L)}(g\cdot(L\cdot\mathbf{s}+\alpha)=0)\subseteq\mathbb{C}^{n},$

where $(g\cdot(L\cdot\mathbf{s}+\alpha)=0)$ denotes the divisor inside
$\mathbb{C}^{r}$ defined by $g\cdot(L\cdot\mathbf{s}+\alpha)=0$.

In the theorem above, $S(F,\mathcal{M})$ is the union of all (primitive)
slopes of the codimension-one components of the zero locus of a Bernstein-Sato
ideal of $\mathcal{M}$ along $F$, and for every $L\in S(F,\mathcal{M})$ the
finite set $\kappa(L)$ is the set of all $\alpha\in\mathbb{C}$ such that
$L\cdot\mathbf{s}+\alpha=0$ defines an irreducible codimension-one component
of the zero locus of the Bernstein-Sato ideal. See §3.6 for details.

Corollary 1.3 and Theorem 1.4 together give:

######  Corollary 1.5.

Locally on a relative compact open subset of $X$,

$\textup{supp}_{(\mathbb{C}^{*})^{r}}\psi_{F}(\textup{DR}(\mathcal{M}))=\bigcup_{L\in
S(F,\mathcal{M})}\bigcup_{\alpha\in\kappa(L)}({\bf
t}^{L}=exp(2\pi\sqrt{-1}\alpha))\subseteq(\mathbb{C}^{*})^{r},$

where ${\bf t}^{L}=\prod_{i}t_{i}^{l_{i}}$ with $L=(l_{1},\dots,l_{r})$ and
$({\bf t}^{L}=exp(2\pi\sqrt{-1}\alpha))$ is the divisor defined by ${\bf
t}^{L}=exp(2\pi\sqrt{-1}\alpha)$ in $(\mathbb{C}^{*})^{r}$. In particular,
$\textup{supp}_{(\mathbb{C}^{*})^{r}}\psi_{F}(\textup{DR}(\mathcal{M}))$is a
finite union of translated codimensional-one subtori inside
$(\mathbb{C}^{*})^{r}$.

The study of
$\textup{supp}_{(\mathbb{C}^{*})^{r}}\psi_{F}(\textup{DR}(\mathcal{M}))$ was
initiated by Sabbah [Sab90], where the support is shown to be included in a
union of translated subtori. In the case $\mathcal{M}=\mathscr{O}_{X}$, it is
proved in [BLSW17, Theorem 1.3] that
$\textup{supp}_{(\mathbb{C}^{*})^{r}}\psi_{F}(\mathbb{C}_{X})$ is a finite
union of torsion translated codimensional-one subtori from a purely
topological approach. Corollary 1.5 gives a precise description of
$\textup{supp}_{(\mathbb{C}^{*})^{r}}\psi_{F}(\textup{DR}(\mathcal{M}))$ in
general. If additionally $\mathcal{M}$ is quasi-unipotent along $F$ (for
instance $\mathcal{M}=\mathscr{O}_{X}$) , then $\kappa(L)\subseteq\mathbb{Q}$
for every $L$ and hence
$\textup{supp}_{(\mathbb{C}^{*})^{r}}\psi_{F}(\textup{DR}(\mathcal{M}))$ is a
finite union of torsion translated subtori inside $(\mathbb{C}^{*})^{r}$. In
consequence, Theorem 1.1, Corollary 1.3, Theorem 1.4 and Corollary 1.5
together give a generalization of [BVWZ21a, Theorem 1.5.1] as well as [BG12,
Theorem 2 and Theorem 3].

As mentioned above,
$\textup{supp}_{(\mathbb{C}^{*})^{r}}\psi_{F}(\mathbb{C}_{X})$ has a purely
topological interpretation by using (local) cohomology jumping loci [BLSW17].
But the cohomology jumping loci (of quasi-projective varieties or analytic
germ complements) might contain torsion-translated subtori of higher
codimension [BW15, BW17]. Then one can naturally ask how to give a
$\mathscr{D}$-module interpretation of such lower dimensional loci. Along this
line, we propose a lineariality conjecture (Conjecture 3.20) by using the
codimension filtration of Gabber-Kashiwara under the relative setting (see
§2.8). The lineariality conjecture is interesting because it would imply a
conjecture of Budur [Bud15] that the zero locus of the Bernstein-Sato ideal of
$F$ is a finite union of translated linear subspaces of $\mathbb{C}^{r}$
defined over $\mathbb{Q}$ (see Proposition 3.22).

###  1.3. Relative characteristic cycle and monodromy zeta function

Our next results are about understanding the irreducible divisor

$(g\cdot(L\cdot\mathbf{s}+\alpha)=0)\subseteq\mathbb{C}^{r}$

inside $\textup{supp}_{\mathbb{C}^{r}}\Psi_{F}(\mathcal{M})$ and their
relations with relative characteristic cycles and the monodromy zeta
functions.

We now assume $\mathcal{M}$ a regular holonomic $\mathscr{D}_{X}$-module and
$F=(f_{1},\dots,f_{r})$ an $r$-tuple of holomorphic functions on a complex
manifold $X$.

######  Theorem 1.6.

With assumptions above , we have the relative characteristic cycles

$\textup{CC}^{\textup{rel}}(\mathcal{M}(*D_{F}^{(r)}))=\textup{CC}^{\textup{rel}}(\mathcal{M}(!D_{F}^{(r)}))=\textup{CC}(\mathcal{M}(*D))\times\mathbb{C}^{r}\subseteq
T^{*}X\times\mathbb{C}^{r},$

where $\textup{CC}^{\textup{rel}}$ and CC denote the relative characteristic
cycle and the absolute one respectively and similarly for Ch. Moreover,
locally on a relative compact open subset $W$ of $X$, the relative
characteristic cycle of $\Psi_{F}(\mathcal{M})$ is an infinite sum

$\textup{CC}^{\textup{rel}}(\Psi_{F}(\mathcal{M}))=\sum_{g\in G}\sum_{L\in
S(F,\mathcal{M})}\sum_{\alpha\in\tilde{\kappa}(L)}\Lambda_{L,\alpha}\times(g\cdot(L\cdot\mathbf{s}+\alpha)=0)$

where each $\Lambda_{L,\alpha}$ is a Lagrangian cycle supported on
$\textup{Ch}(\mathcal{M}(*D))$ $($over $D\cap W)$, and $\tilde{\kappa}(L)$ is
$\kappa(L)$ modulo $L$-equivalence $($Definition 3.16 $)$.

The relative characteristic cycle of $\Psi_{F}(\mathcal{M})$ is a symmetric
infinite sum because of the G-action on $\Psi_{F}(\mathcal{M})$ (if a
$G$-equivariant sheaf is supported on a point, then it is supported on the
$G$-orbit of the point). If $\widetilde{W}$ is a relative compact open subset
of $X\times\mathbb{C}^{r}$, then

$\big{(}\sum_{g\in G}\sum_{L\in
S(F,\mathcal{M})}\sum_{\alpha\in\tilde{\kappa}(L)}\Lambda_{L,\alpha}\times(g\cdot(L\cdot
s-\alpha)=0)\big{)}\big{|}_{\widetilde{W}}$

becomes a finite sum since $X\times(g\cdot(L\cdot s+\alpha)=0)$ intersect
$\widetilde{W}$ for only finite many $g\in G$. See Proposition 3.19 for
formulas of $\Lambda_{L,\alpha}$ in terms of the characteristic cycles of very
general choices of one-dimensional nearby cycles associated to $\mathcal{M}$.

The relative characteristic cycle formula of $\mathcal{M}(*D^{(r)}_{F})$ in
Theorem 1.6, as well as the irregular case in Theorem 3.3, has an application
to the study of constructibility of the logarithmic de Rham complexes of
lattices of holonomic modules [Wu20, Theorem 1.1].

We now fix an arbitrary point $x\in X$. By Theorem 1.4 we can focus on
$S(F,\mathcal{M})$ and $\tilde{\kappa}(L)$ (or $\kappa(L)$) for $L\in
S(F,\mathcal{M})$ locally around $x$. By Theorem 1.6, both $S(F,\mathcal{M})$
and $\tilde{\kappa}(L)$ (or $\kappa(L)$) are constructible as $x$ varies on
$X$ and hence so are
$\textup{supp}_{(\mathbb{C}^{*})^{r}}\psi_{F}(\textup{DR}(\mathcal{M}))$ and
$\textup{supp}_{\mathbb{C}^{r}}\Psi_{F}(\mathcal{M})$. More precisely, we fix
a finite Whitney stratification

$X=\bigsqcup_{\beta}X_{\beta}\textup{ such that
}\textup{Ch}(\mathcal{M}(*D))\subseteq\bigsqcup_{\beta}T_{X_{\beta}}^{*}X,$

where $T_{X_{\beta}}^{*}X$ is the conormal bundle of the smooth strata
$X_{\beta}$. Then $S(F,\mathcal{M})$ and $\tilde{\kappa}(L)$ for $L\in
S(F,\mathcal{M})$ are locally constant with respect to the above
stratification. Moreover, for $L\in S(F,\mathcal{M})$ and
$\alpha\in\tilde{\kappa}(L)$ locally around $x$, we have

$\Lambda_{L,\alpha}=\sum_{\beta}m_{\beta}(L,\alpha)\overline{T^{*}_{X_{\beta}}X},\quad
m_{\beta}(L,\alpha)\in\mathbb{Z}_{\geq 0}.$

Next, we give a formula for the multiplicity $m_{\beta}(L,\alpha)$ by using
the monodromy zeta function. By construction,
$\psi_{F}(\textup{DR}(\mathcal{M}))$ is a $\mathbb{C}$-constructible complex
of sheaves of $\mathbb{C}[G]$-modules (see [KS13, §8.5] for the definition of
$\mathbb{C}$-constructibility). Hence, for each generic point $q$ of
Supp${}_{(\mathbb{C}^{*})^{r}}(\psi_{F}(\textup{DR}(\mathcal{M})))$

$\psi_{F,q}(\textup{DR}(\mathcal{M}))=\psi_{F}(\textup{DR}(\mathcal{M}))\otimes_{\mathbb{C}[G]}\mathbb{C}[G]_{q}$

is a $\mathbb{C}$-constructible complex of sheaves of
$\mathbb{C}[G]_{q}$-modules, where $\mathbb{C}[G]_{q}$ is the localization of
$\mathbb{C}[G]$ at the prime ideal $q$. We define

$\chi_{x}(\psi_{F}(\textup{DR}(\mathcal{M})),q)=\sum_{i}(-1)^{i}\textup{lg}(\mathcal{H}^{i}\psi_{F,q}(\textup{DR}(\mathcal{M}))|_{x})$

where lg denotes the length function, and

$\zeta_{x}(\psi_{F}(\textup{DR}(\mathcal{M})))=\sum_{q}\chi_{x}(\psi_{F}(\textup{DR}(\mathcal{M})),q)\bar{q}$

where $q$ goes over the set of all the generic points of
Supp${}_{(\mathbb{C}^{*})^{r}}(\psi_{F}(\textup{DR}(\mathcal{M})))$ (since
Supp${}_{(\mathbb{C}^{*})^{r}}(\psi_{F}(\textup{DR}(\mathcal{M})))$ is an
algebraic closed subset of $(\mathbb{C}^{*})^{r}$ by Corollary 2.21). By
$\mathbb{C}$-constructibility, $\chi_{x}(\psi_{F}(\textup{DR}(\mathcal{M}))$
is a $\mathbb{Z}$-valued constructible function on $X$ and
$\zeta_{x}(\psi_{F}(\textup{DR}(\mathcal{M})))$ is a constructible function on
$X$ valued in the abelian group of algebraic subvarieties in
$(\mathbb{C}^{*})^{r}$. The constructible function
$\zeta_{x}(\psi_{F}(\textup{DR}(\mathcal{M})))$ is the so-called _monodromy
zeta function_ of $\psi_{F}(\textup{DR}(\mathcal{M}))$ (cf. [Sab90, §2.5]).

We denote by $q(L,\alpha)\in\textup{Spec }\mathbb{C}[G]$ the generic point of
$({\bf t}^{L}-exp(2\pi\sqrt{-1}\alpha)=0)$ for $L$ and $\alpha$. With the help
of Corollary 1.5, we obtain a precise formula:

(1.2) $\zeta_{x}(\psi_{F}(\textup{DR}(\mathcal{M})))=\prod_{L\in
S(F,\mathcal{M})}\prod_{\alpha\in\tilde{\kappa}(L)}({\bf
t}^{L}-exp(2\pi\sqrt{-1}\alpha))^{\chi_{x}(\psi_{F}(\textup{DR}(\mathcal{M})),q(L,\alpha))},$

where we make the constructible function valued in rational functions.

Using a local index formula of Kashiwara-Dubson-Ginsburg [Gin86, Theorem 8.2],
we get a formula for $\chi_{x}(\psi_{F,q}(\textup{DR}(\mathcal{M})))$ and
$m_{\beta_{1}}(L,\alpha)$:

(1.3)
$m_{\beta_{1}}(L,\alpha)=\sum_{X_{\beta_{1}}\subseteq\bar{X}_{\beta}}(-1)^{\dim
X_{\beta}}c(X_{\beta_{1}},X_{\beta})\chi_{x_{\beta}}(\psi_{F}(\textup{DR}(\mathcal{M})),q(L,\alpha))),$

where

$c(X_{\beta_{1}},X_{\beta})=\begin{cases}\chi^{top}(B_{x_{\beta_{1}}}\cap
X_{\beta}\cap H),&\textup{ if }X_{\beta_{1}}\not=X_{\beta}\\\ 1,&\textup{ if
}X_{\beta_{1}}=X_{\beta}.\end{cases}$

with $B_{x_{\beta_{1}}}$ a small polydisc open neighborhood inside $X$ of some
$x_{\beta_{1}}\in X_{\beta_{1}}$, $H$ a linear subspace of $X$ sufficiently
close to $x_{\beta_{1}}$ of codimension $\dim X_{\beta_{1}}+1$ and
$\chi^{top}$ denoting the topological Euler characteristic.

###  1.4. Outline of the paper

In Section 2, we discuss sheafification of sheaves of modules on topological
spaces over commutative rings and discuss the properties of algebraic and
analytic relative supports. Section 3 is about the construction of maximal and
minimal extensions under the relative setting and their $G$-equivariance. In
Section 4, we recall the construction of the Alexander complexes of Sabbah. In
Section 5, we discuss comparisons in the sense of relative Riemann-Hilbert
correspondence and prove Theorem 1.1 and Theorem 1.2.

### Acknowledgement

The author would like to thank Nero Budur, Linquan Ma, Mircea Mustaţǎ, Ruijie
Yang and Peng Zhou for useful discussions and comments and for answering
questions.

The author was supported by the grant Methusalem METH/15/026 from KU Leuven.

##  2\. Sheafification of sheaves of modules over commutative rings

In this section, we discuss sheafifying sheaves of modules over commutative
rings in a standard way, motivated by the study of relative holonomic
$\mathscr{D}$-modules over algebraic affine spaces in [Mai16a] and the theory
of analytic relative holonomic $\mathscr{D}$-modules developed in [MFS13,
MFS19, FF18, FFS21]. Compared to the references above, we mainly and more
focus on studying algebraic and analytic relative supports and their
differences.

###  2.1. Pre-sheafification

Let $k$ be a field, and let $X$ be topological spaces with $\mathscr{A}$ a
sheaf of $k$-algebra on $X$ for some base field $k$. For a commutative
$k$-algebra $R$, we obtain a new sheaf of rings

$\mathscr{A}_{R}=\mathscr{A}\otimes_{k}R.$

We write by $\textup{Mod}(\mathscr{A}_{R})$ the abelian category of
$\mathscr{A}_{R}$-modules on $X$.222Throughout this paper, all the modules
over non-commutative rings are assumed to be the left ones, unless indicated
otherwise. For $\mathscr{M}\in\textup{Mod}(\mathscr{A}_{R})$, we define a
presheaf on $X\times\textup{Spec }R$ (use the product topology) by

$\widetilde{\mathscr{M}}^{\textup{pre}}(U\times\textup{Spec
}R_{f})=\mathscr{M}(U)\otimes_{R}R_{f}$

where $R_{f}$ is the localization of $R$ with respect to $f\in R$. Since
$\textup{Spec }R_{f}$ give a basis of the Zariski topology of $\textup{Spec
}R$, the presheaf above gives a sheaf on $X\times\textup{Spec }R$, denoted by
$\widetilde{\mathscr{M}}^{\textup{pre}}$. We call such procedure the _relative
pre-sheafification_ of $\mathscr{M}$. Consequently,
$\widetilde{\mathscr{M}}^{\textup{pre}}$ is an
$\widetilde{\mathscr{A}}_{R}^{\textup{pre}}$-module on $X\times\textup{Spec
}R$. One can see that when $X$ is a point, the relative pre-sheafification is
the $\sim$-functor in algebraic geometry transforming $R$-modules into quasi-
coherent sheaves on $\textup{Spec }R$.

######  Lemma 2.1.

The stalk of $\widetilde{\mathscr{M}}^{\textup{pre}}$ at $(x,m)\in
X\times\textup{Spec }R$ satisfies

$\widetilde{\mathscr{M}}^{\textup{pre}}_{(x,m)}=\mathscr{M}_{x}\otimes_{R}R_{m}=(\mathscr{M}\otimes_{R}R_{m})_{x}$

where $\mathscr{M}_{x}$ is the stalk of $\mathscr{M}$ at $x$ and similarly for
$(\mathscr{M}\otimes_{R}R_{m})_{x}$.

###### Proof.

Since localizations are colimits, we apply [Sta18, Part 1, Lemma 4.14.10]. ∎

Let $I\subseteq R$ be an ideal. The _relative pullback_ of $\mathscr{M}$ to
$\textup{Spec }R/I$ is defined as

$\tilde{i}^{*}_{\textup{pre}}(\widetilde{\mathscr{M}}^{\textup{pre}})=\widetilde{\mathscr{M}\otimes_{R}R/I}^{\textup{pre}}$

where $\tilde{i}\colon X\times\textup{Spec }R/I\hookrightarrow
X\times\textup{Spec }R$ is the closed embedding induced by $R\to R/I$. More
generally, if $h\colon\textup{Spec }S\to\textup{Spec }R$ is a morphism of
affine schemes, then one can define the _relative pullback functor_
$\tilde{h}^{*}_{\textup{pre}}$ similarly. If $m\subseteq R$ is a maximal
ideal, the pullback of $\mathscr{M}$ at the closed point of $m$ gives a
$\mathscr{A}$-module on $X$. Hence, $\widetilde{\mathscr{M}}$ is also called a
relative $\mathscr{A}$-module over $\textup{Spec }R$.

Let $\mathfrak{p}\subseteq R$ be a prime ideal. Then the _specialization_ of
$\mathscr{M}$ at $\mathfrak{p}$ is

$\mathscr{M}_{\mathfrak{p}}=\mathscr{M}\otimes_{R}R_{\mathfrak{p}}$

where $R_{\mathfrak{p}}$ is the localization of $R$ at $\mathfrak{p}$, which
is a $\mathscr{A}_{R_{\mathfrak{p}}}$-module on $X$. The _complete
specialization_ of $\mathscr{M}$ at $p$ is

$\hat{\mathscr{M}}_{\mathfrak{p}}=\mathscr{M}\otimes_{R}\hat{R}_{\mathfrak{p}}$

where $\hat{R}_{\mathfrak{p}}$ is the completion of $R_{\mathfrak{p}}$ with
respect to $\mathfrak{p}$. Then $\hat{\mathscr{M}}_{\mathfrak{p}}$ is an
$\hat{\mathscr{A}}_{R_{\mathfrak{p}}}$-module.

######  Definition 2.2.

For $\mathscr{M}\in\textup{Mod}(\mathscr{A}_{R})$, the relative support of
$\mathscr{M}$ is

$\textup{supp}_{\textup{Spec }R}\mathscr{M}=\\{\textup{maximal ideals
}m\in\textup{Spec }R\mid\mathscr{M}_{m}\not=0\\}.$

More generally, if $\mathscr{M}^{\bullet}\in D^{b}(\mathscr{A}_{R})$, the
bounded derived category of $\textup{Mod}(\mathscr{A}_{R})$, then we define

$\textup{supp}_{\textup{Spec
}R}\mathscr{M}^{\bullet}=\bigcup_{i}\textup{supp}_{\textup{Spec
}R}\mathcal{H}^{i}(\mathscr{M}^{\bullet}).$

###  2.2. Analytic sheafification

We now assume $X$ a complex manifold, $\mathscr{A}$ a sheaf of
$\mathbb{C}$-algebra and $R$ a commutative $\mathbb{C}$-algebra of finite
type. We write $\textup{Spec}^{\textup{an}}R$ the analytic scheme of
$\textup{Spec }R$.

For $\mathscr{M}\in\textup{Mod}(\mathscr{A}_{R})$, the analytic sheafification
of $\mathscr{M}$ is defined as

$\widetilde{\mathscr{M}}=\iota^{-1}(\widetilde{\mathscr{M}}^{\textup{pre}})\otimes_{\widetilde{pr}^{-1}_{X}\mathscr{O}_{\textup{Spec
}R}}pr_{X}^{-1}\mathscr{O}_{\textup{Spec}^{\textup{an}}R},$

where $\iota:X\times\textup{Spec}^{\textup{an}}R\to X\times\textup{Spec }R$ is
the natural inclusion map induced by
$\iota_{R}\colon\textup{Spec}^{\textup{an}}R\to\textup{Spec }R$ from GAGA
principle, $pr_{X}\colon
X\times\textup{Spec}^{\textup{an}}R\to\textup{Spec}^{\textup{an}}R$ is the
projection and $\widetilde{pr}_{X}$ is the composition of $pr_{X}$ and
$\iota_{R}$. The output $\widetilde{\mathscr{M}}$ gives a
$\widetilde{\mathscr{A}}_{R}$-module on $X\times\textup{Spec}^{\textup{an}}R$.

If $X$ is an algebraic scheme over $k$ and $R$ a commutative $k$-algebra, then
we can define the algebraic sheafification of
$\mathscr{M}\in\textup{Mod}(\mathscr{A}_{R})$ by

$\widetilde{\mathscr{M}}^{\textup{alg}}=\iota^{-1}_{\textup{alg}}(\widetilde{\mathscr{M}}^{\textup{pre}})$

where $\iota^{-1}_{\textup{alg}}:X_{R}\to X\times\textup{Spec }R$ is the
natural inclusion map with $X_{R}$ the scheme of $X$ over $R$.

###  2.3. Analytic sheafification for $\mathscr{D}_{X,R}$-module and relative
holonomicity

Now we assume $\mathscr{A}=\mathscr{D}_{X}$, the sheaf of holomorphic
differential operators on a complex manifold $X$, and $R$ a commutative finite
generated $\mathbb{C}$-algebra. In this case, we also want the analytic
sheafification to be consistent with the analytic structure sheaf of
$X\times\textup{Spec}^{\textup{an}}R$ (not only with that of the second
factor). For instance, the sheafification of an $\mathscr{O}_{X,R}$-module
should be its GAGA analytification. Thus, for
$\mathscr{M}\in\textup{Mod}(\mathscr{D}_{X,R})$ (or
$\mathscr{M}\in\textup{Mod}(\mathscr{O}_{X,R}$)) we define

$\widetilde{\mathscr{M}}=\iota^{-1}(\widetilde{\mathscr{M}}^{\textup{pre}})\otimes_{\iota^{-1}(\widetilde{\mathscr{O}}_{X,R}^{\textup{pre}})}\mathscr{O}_{X\times\textup{Spec}^{\textup{an}}R}.$

Therefore, the output $\widetilde{\mathscr{M}}$ in this case gives a
$\widetilde{\mathscr{D}}_{X,R}=\mathscr{D}_{X\times\textup{Spec}^{\textup{an}}R/\textup{Spec}^{\textup{an}}R}$-module,
i.e. a relative analytic $\mathscr{D}$-module. Let us refer to [Sch12, Chapter
III. 1.3] and [MFS13, §3] for the general theory of relative analytic
$\mathscr{D}$-modules; see also [Wu20, §2]). The relative pullback for
$\widetilde{\mathscr{M}}$ in this case is exactly the base change for relative
$\mathscr{D}$-modules [Wu20, §2.2].

######  Remark 2.3.

When $X$ is an smooth complex algebraic variety and $\mathscr{M}$ is an
algebraic $\mathscr{D}_{X,R}$-module, we define the algebraic sheafification
by

$\widetilde{\mathscr{M}}^{\textup{alg}}=\iota^{-1}_{\textup{alg}}(\widetilde{\mathscr{M}}^{\textup{pre}})\otimes_{\iota^{-1}(\widetilde{\mathscr{O}}_{X,R}^{\textup{pre}})}\mathscr{O}_{X_{R}},$

which gives an algebraic relative $\mathscr{D}$-module over $\textup{Spec }R$.
In this case, the functor $\sim_{\textup{alg}}$ has a quasi-inverse $p_{*}$:

$p_{*}(\widetilde{\mathscr{M}}^{\textup{alg}})\simeq\mathscr{M},$

where $p\colon X_{R}\to X$ is the natural projection.

######  Definition 2.4.

For $\mathscr{M}\in\textup{Mod}(\mathscr{D}_{X,R})$, we say $\mathscr{M}$
relative coherent over $\textup{Spec}^{\textup{an}}R$ if
$\widetilde{\mathscr{M}}$ is coherent over $\widetilde{\mathscr{D}}_{X,R}$. We
say $\mathscr{M}$ $($or
$\widetilde{\mathscr{M}}\in\textup{Mod}(\widetilde{\mathscr{D}}_{X,R}))$
relative holonomic if it is relative coherent and locally on a relative
compact open subset $W$ of $X$

$\textup{Ch}^{{\textup{rel}}}(\widetilde{\mathscr{M}}|_{W\times\textup{Spec}^{\textup{an}}R})\subseteq\Lambda\times\textup{Spec}^{\textup{an}}R,$

where $\Lambda$ is a $($possibly reducible$)$ conic Lagrangian subvariety
inside the cotangent bundle $T^{*}X$ $($over W$)$ and
$\textup{Ch}^{{\textup{rel}}}(\widetilde{\mathscr{M}})$ is the relative
characteristic variety.

Notice that coherence over $\mathscr{D}_{X,R}$ on $X$ implies relative
coherence, but not vice versa. The relative holonomicity above is the analytic
sheafification of the algebraic relative holonomicity [BVWZ21a, Definition
3.2.3], which in turn follows the analytic relative holonomicity defined in
[MFS13, §3.4]. By definition, the category of relative holonomic modules is
abelian.

The following lemma is well-known; see for instance [Mai16a, Proposition 8],
[FF18, Lemma 2.10]333Lemma 2.10 in [FF18] seems to be not completely correct
since locally on $X$ the index set $I$ can be infinite by Theorem 1.6. and
also [BVWZ21a, Proposition 3.2.5].

######  Lemma 2.5.

If $\widetilde{\mathscr{M}}\in\textup{Mod}(\widetilde{\mathscr{D}}_{X,D})$ is
relative holonomic, then for every pair $(W,V)$ with $W$ a relative compact
open subset of $X$ and $V$ a relative compact open subset of
$\textup{Spec}^{\textup{an}}R$, there exist a finite number of closed analytic
subvarieties $S_{w}\subseteq V$ such that

$\textup{Ch}^{\textup{rel}}(\widetilde{\mathscr{M}}|_{W\times
V})=\bigcup_{w}\Lambda_{w}\times S_{w},$

where $\Lambda_{w}$ are irreducible conic Lagrangian subvarieties in
$T^{*}X|_{W}$. If moreover $\mathscr{M}\in\textup{Mod}(\mathscr{D}_{X,R})$ is
coherent over $\mathscr{D}_{X,R}$ and relative holonomic over
$\textup{Spec}^{\textup{an}}R$, then

$\textup{Ch}^{\textup{rel}}(\widetilde{\mathscr{M}}|_{W\times\textup{Spec}^{\textup{an}}R})=\bigcup_{w}\Lambda_{w}\times
S_{w}$

with a finite number of closed analytic subvarieties
$S_{w}\subseteq\textup{Spec}^{\textup{an}}R$.

###### Proof.

For completeness, we present here a proof essentially due to Maisonobe
[Mai16a]. We write

$\textup{Ch}^{\textup{rel}}(\widetilde{\mathscr{M}}|_{W\times\textup{Spec}^{\textup{an}}R})\subseteq\Lambda\times\textup{Spec}^{\textup{an}}R$

with $\Lambda=\sum_{w}\Lambda_{w}$ a finite union of irreducible conic
Lagrangian subvarieties in $T^{*}X|_{W}$. Notice that the inclusion above does
not imply that
$\textup{Ch}^{\textup{rel}}(\widetilde{\mathscr{M}}|_{W\times\textup{Spec}^{\textup{an}}R})$
has finite many irreducible components. For every $p\in V$, by the relative
Bernstein inequality (see [Mai16a, Proposition 5] and also [Wu20, Theorem
2.2]), we have

$\textup{Ch}^{\textup{rel}}(\widetilde{\mathscr{M}})_{p}\coloneqq\textup{Ch}^{\textup{rel}}(\widetilde{\mathscr{M}})\cap
pr^{-1}(p)\subset\sum_{w}\Lambda_{w},$

where $pr\colon
T^{*}X\times\textup{Spec}^{\textup{an}}R\to\textup{Spec}^{\textup{an}}R$ is
the second projection. Then we define

$S_{w}=\\{p\in
V\mid\Lambda_{w}\subseteq\textup{Ch}^{\textup{rel}}(\widetilde{\mathscr{M}})_{p}\\},$

from which we obtain

$\textup{Ch}^{\textup{rel}}(\widetilde{\mathscr{M}}|_{W\times
V})=\bigcup_{w}\Lambda_{w}\times S_{w}.$

The set $S_{w}$ is analytic and closed since

$\\{\lambda_{w}\\}\times
S_{w}=\textup{Ch}^{\textup{rel}}(\widetilde{\mathscr{M}}|_{W\times V})\cap
pr_{1}^{-1}(\lambda_{w})$

with $\lambda_{w}$ a general point on $\Lambda_{w}$, where $pr_{1}\colon
T^{*}X\times\textup{Spec}^{\textup{an}}R\to T^{*}X$ is the first projection.

If additionally $\mathscr{M}$ is coherent over $\mathscr{D}_{X,R}$, we can
define

$S_{w}=\\{p\in\textup{Spec}^{\textup{an}}R\mid\Lambda_{w}\subseteq\textup{Ch}^{\textup{rel}}(\widetilde{\mathscr{M}})_{p}\\}.$

Since
$\textup{Ch}^{\textup{rel}}(\widetilde{\mathscr{M}}|_{W\times\textup{Spec}^{\textup{an}}R})$
is a closed analytic subvariety of $T^{*}W\times\textup{Spec}^{\textup{an}}R$,
we conclude that $S_{w}\subseteq\textup{Spec}^{\textup{an}}R$ is analytic and
closed. ∎

For a maximal ideal $m\in\textup{Spec }R$, we write
$p_{m}=\iota^{-1}_{R}(m)\in\textup{Spec}^{\textup{an}}R$. For $(x,p)\in
X\times\textup{Spec}^{\textup{an}}R$, $\widetilde{\mathscr{M}}_{(x,p)}$
denotes the (analytic) localization of $\widetilde{\mathscr{M}}$ at $(x,p)$.

######  Lemma 2.6.

If $\mathscr{M}$ is relative coherent over $\textup{Spec}^{\textup{an}}R$,
then

$\textup{supp}_{\textup{Spec }R}\mathscr{M}=\\{m\in\textup{Spec
}R\mid\widetilde{\mathscr{M}}_{(x,p_{m})}\not=0\textup{ for some $x\in
X$}\\}.$

###### Proof.

We first assume $R=\mathbb{C}[{\bf y}]$ for ${\bf y}=(y_{1},\dots,y_{l})$ for
some integral $l>0$. Since completion of local rings is faithfully flat, we
have

$\hat{\mathscr{M}}_{m}\not=0\Leftrightarrow\mathscr{M}_{m}\not=0.$

By definition, we have

$\hat{\mathscr{M}}_{m}\not=0\Leftrightarrow(\hat{\mathscr{M}}_{m})_{x}\textup{
for some }x\in X,$

where $(\hat{\mathscr{M}}_{m})_{x}$ is the localization of
$\hat{\mathscr{M}}_{m}$ at $x$ because $\hat{\mathscr{M}}_{m}$ is a sheaf on
$X$. We write by $\mathbb{C}\\{{\bf x}\\}$ the stalk of $\mathscr{O}_{X}$ at
$x$, and by $\mathbb{C}[[{\bf x}]]$ the ring of formal power series. Since
$\mathbb{C}[[{\bf x}]]$ is faithfully flat over $\mathbb{C}\\{{\bf x}\\}$, we
know

$(\hat{\mathscr{M}}_{m})_{x}\not=0\Leftrightarrow(\hat{\mathscr{M}}_{m})_{x}\otimes_{\mathbb{C}\\{{\bf
x}\\}}\mathbb{C}[[{\bf x}]]\not=0.$

We also have

$(\hat{\mathscr{M}}_{m})_{x}\otimes_{\mathbb{C}\\{{\bf x}\\}}\mathbb{C}[[{\bf
x}]]\simeq\mathscr{M}_{x}\otimes_{\mathbb{C}\\{{\bf
x}\\}\otimes_{\mathbb{C}}R}\mathbb{C}[[{\bf x}]][[{\bf y}]].$

Thanks to the fact that the completion of local rings is faithfully flat
again, we have

$\widetilde{\mathscr{M}}_{(x,p_{m})}\not=0\Leftrightarrow\widetilde{\mathscr{M}}_{(x,p_{m})}\otimes_{\mathbb{C}\\{{\bf
x,y}\\}}\mathbb{C}[[{\bf x,y}]]\not=0.$

By Lemma 2.1, we have

$\mathscr{M}_{x}\otimes_{\mathbb{C}\\{{\bf
x}\\}\otimes_{\mathbb{C}}R}\mathbb{C}[[{\bf x}]][[{\bf
y}]]\simeq\widetilde{\mathscr{M}}_{(x,p_{m})}\otimes_{\mathbb{C}\\{{\bf
x,y}\\}}\mathbb{C}[[{\bf x,y}]].$

Therefore, we conclude

$\mathscr{M}_{m}\not=0\Leftrightarrow\widetilde{\mathscr{M}}_{(x,p_{m})}\not=0\textup{
for some }x\in X.$

In general, considering a surjection

$\mathbb{C}[{\bf y}]\twoheadrightarrow R,$

we are done by similar arguments. ∎

######  Proposition-Definition 2.7.

Let $\mathscr{M}\in\textup{Mod}(\mathscr{D}_{X,R})$ be relative coherent over
$\textup{Spec}^{\textup{an}}R$. Then

$\textup{supp}_{\textup{Spec}^{\textup{an}}R}\mathscr{M}\coloneqq\iota_{R}^{-1}(\textup{supp}_{\textup{Spec
}R}\mathscr{M})=\textup{supp}_{\textup{Spec}^{\textup{an}}R}\widetilde{\mathscr{M}}\coloneqq
pr(\textup{Ch}^{\textup{rel}}(\widetilde{\mathscr{M}})),$

where $pr\colon
T^{*}X\times\textup{Spec}^{\textup{an}}R\to\textup{Spec}^{\textup{an}}R$ is
the projection.

###### Proof.

We focus ourselves on the characteristic variety

$\textup{Ch}^{\textup{rel}}(\widetilde{\mathscr{M}})\subseteq
T^{*}X\times\textup{Spec}^{\textup{an}}R$

locally over $W\times V$ by picking a good filtration
$F_{\bullet}\widetilde{\mathscr{M}}$. Then
$\textup{Ch}^{\textup{rel}}(\widetilde{\mathscr{M}})$ is the support of
$\widetilde{\textup{gr}}^{F}_{\bullet}\widetilde{\mathscr{M}}$ inside
$T^{*}X\times\textup{Spec}^{\textup{an}}V$, which means
$\textup{Ch}^{\textup{rel}}(\widetilde{\mathscr{M}})$ is a closed analytic
subvariety over $W\times V$. We observe by coherence

$\widetilde{\mathscr{M}}_{(x,p)}\not=0\Leftrightarrow(\widetilde{\textup{gr}}^{F}_{\bullet}\widetilde{\mathscr{M}})_{(x,\xi,p)}\not=0\textup{
for some $\xi\in T^{*}_{x}X$}.$

Thus, we have

$\widetilde{\mathscr{M}}_{(x,p)}\not=0\Leftrightarrow(x,\xi,p)\in\textup{Ch}^{\textup{rel}}(\widetilde{\mathscr{M}})\textup{
for some $\xi\in T^{*}_{x}X$}.$

By Lemma 2.6, we hence conclude

$\iota_{R}^{-1}(\textup{supp}_{\textup{Spec
}R}\mathscr{M})=pr(\textup{Ch}^{\textup{rel}}(\widetilde{\mathscr{M}}))$

locally over $W\times V$. ∎

###  2.4. Bernstein-Sato ideal

If $\mathscr{M}\in\textup{Mod}(\mathscr{D}_{X,R})$ is coherent over
$\mathscr{D}_{X,R}$, then we define

$B(\mathscr{M})={\textup{Ann}}_{R}(\mathscr{M})\subseteq R,$

calling it the Bernstein-Sato ideal of $\mathscr{M}$. We write by
$Z(B(\mathscr{M}))\subseteq\textup{Spec }R$ the zero locus of
$B(\mathscr{M})$. A priori, since $\mathscr{M}$ is not finite generated over
$R$ in general, we only have

$\textup{supp}_{\textup{Spec }R}\mathscr{M}\subseteq Z(B(\mathscr{M})),$

but we do not know if $\textup{supp}_{\textup{Spec }R}\mathscr{M}$ is an
algebraic subvariety.

######  Proposition 2.8.

If $\mathscr{M}\in\textup{Mod}(\mathscr{D}_{X,R})$ is coherent over
$\mathscr{D}_{X,R}$ and relative holonomic over
$\textup{Spec}^{\textup{an}}R$, then locally on a relative compact open subset
$W\subseteq X$ we have

$\textup{supp}_{\textup{Spec }R}\mathscr{M}|_{W}=Z(B(\mathscr{M}|_{W})).$

###### Proof.

For simplicity, we assume $X=W$. By [Mai16a, Proposition 9] (see also
[BVWZ21a, Lemma 3.4.1]), we have

$Z(B(\mathscr{M}))=pr(\textup{Ch}^{\textup{rel}}(\widetilde{\mathscr{M}})).$

Then the required equality follows from Proposition-Definition 2.7. ∎

By the proposition above and Proposition 2.5, we have:

######  Corollary 2.9.

If $\mathscr{M}\in\textup{Mod}(\mathscr{D}_{X,R})$ is coherent over
$\mathscr{D}_{X,R}$ and relative holonomic over
$\textup{Spec}^{\textup{an}}R$, then over a relative compact open subset
$W\subseteq X$

$\textup{Ch}^{\textup{rel}}(\widetilde{\mathscr{M}}|_{W\times\textup{Spec}^{\textup{an}}R})=\bigcup_{w}\Lambda_{w}\times
S_{w}$

with a finite number of closed algebraic subvarieties
$S_{w}\subseteq\textup{Spec }R$.

Applying Proposition-Definition 2.7, Lemma 2.5 and Proposition 2.8, we have:

######  Corollary 2.10.

Let $\mathscr{M}\in\textup{Mod}(\mathscr{D}_{X,R})$ be relative holonomic over
$\textup{Spec}^{\textup{an}}R$. Then

  1. (1)

$\textup{supp}_{\textup{Spec}^{\textup{an}}R}\mathscr{M}$ is analytically
closed locally over both $X$ and $\textup{Spec}^{\textup{an}}R$, i.e. over
each open subset $W\times V$ satisfying $W\subseteq X$ a relative compact open
subset and $V\subseteq\textup{Spec}^{\textup{an}}R$ a relative compact open
subset,

$\textup{supp}_{\textup{Spec}^{\textup{an}}R}(\mathscr{M}|_{W})\cap
V\subseteq\textup{Spec}^{\textup{an}}R$

is a closed analytic subvariety.

  2. (2)

if $\mathscr{M}$ is also coherent over $\mathscr{D}_{X,R}$, then
$\textup{supp}_{\textup{Spec }R}\mathscr{M}$ is algebraically closed locally
over $X$, i.e. over a relative compact open subset $W\subseteq X$

$\textup{supp}_{\textup{Spec }R}(\mathscr{M}|_{W})\subseteq\textup{Spec }R$

is a closed algebraic subvariety.

###  2.5. Sheafification for derived category

By construction, the analytic sheafification is a faithful and exact functor
from $\textup{Mod}(\mathscr{A}_{R})$ to
$\textup{Mod}(\widetilde{\mathscr{A}}_{R})$ for $\mathscr{A}$ in general
(similarly for the algebraic sheafification). We define
$\widetilde{\textup{Mod}}(\mathscr{A}_{R})$ as the subabelian category of
$\textup{Mod}(\widetilde{\mathscr{A}}_{R})$ as the image of
$\textup{Mod}(\widetilde{\mathscr{A}}_{R})$ under the analytic sheafification.
We write by $\widetilde{D^{b}}(\mathscr{A}_{R})$ the derived category of
$\widetilde{\textup{Mod}}(\mathscr{A}_{R})$ and by
$D^{b}(\widetilde{\mathscr{A}}_{R})$ the derived category of
$\textup{Mod}(\widetilde{\mathscr{A}}_{R})$. We can define the _analytic
relative pullback_ functor $\widetilde{h}^{*}$ for an affine morphism
$h:\textup{Spec }S\to\textup{Spec }R$ in an obvious way. We denote by
$\mathbf{L}\widetilde{h}^{*}$ its derived functor. By slightly abuse of
notations, we use $\widetilde{h}^{*}$ to denote the functor for both
$\widetilde{\textup{Mod}}(\mathscr{A}_{R})$ and
$\textup{Mod}(\widetilde{\mathscr{A}}_{R})$ and similarly for
$\mathbf{L}\widetilde{h}^{*}$ in the derived case.

For $\widetilde{\mathscr{D}}_{X,R}$-module, if $\mu\colon
T\to\textup{Spec}^{\textup{an}}R$ is a morphism between analytic schemes, we
use $\widetilde{u}^{*}$ (resp. $\mathbf{L}\widetilde{u}^{*}$) to denote the
(resp. derived) pullback functor of the natural morphism
$\widetilde{\mu}\colon X\times T\to X\times\textup{Spec}^{\textup{an}}R$
induced by base-change (see also [Wu20, §2.2]).

###  2.6. Sheaves of algebraic local cohomology

Let $X$ be a complex manifold and let $R$ be a commutative
$\mathbb{C}$-algebra of finite type. For $Z\subseteq X$ an analytic
subvariety, we use $R\Gamma_{[Z]}$ to denote the right derived functor of
sheaves of algebraic local cohomology along $Z$ (cf. [Bj93, Chapter II.5]). We
denote by $\widetilde{Z}=Z\times\textup{Spec}^{\textup{an}}R$. For
$\mathscr{M}\in\textup{Mod}(\mathscr{D}_{X,R})$, by construction we have a
natural isomorphism

$\widetilde{R\Gamma}_{[Z]}(\mathscr{M})\simeq
R\Gamma_{[\widetilde{Z}]}(\widetilde{\mathscr{M}}).$

The following lemma is standard and we leave its proof for interested readers.

######  Lemma 2.11.

Let $h\colon\textup{Spec }S\to\textup{Spec }R$ be an affine morphism. Then for
$\widetilde{\mathscr{M}}\in\textup{Mod}(\widetilde{\mathscr{D}}_{X,R})$ we
have a natural isomorphism

$\mathbf{L}\widetilde{h}^{*}R\Gamma_{[\widetilde{Z}]}(\widetilde{\mathscr{M}})\simeq
R\Gamma_{[Z\times\textup{Spec}^{\textup{an}}S]}(\mathbf{L}\widetilde{h}^{*}\widetilde{\mathscr{M}}).$

###  2.7. Duality for relative coherent $\mathscr{D}$-module

We keep notation as in §2.3. We assume $\dim_{\mathbb{C}}X=n$. For
$\mathscr{M}\in\textup{Mod}(\mathscr{D}_{X,R})$, if $\mathscr{M}$ is coherent
over $\mathscr{D}_{X,R}$, then we define

$\mathbb{D}_{R}(\mathscr{M})=\mathscr{R}hom_{\mathscr{D}_{X,R}}(\mathscr{M},\mathscr{D}_{X,R})\otimes_{\mathscr{O}}\omega^{-1}_{X}[n]\in
D^{b}(\mathscr{D}_{X,R}),$

where $\omega_{X}$ is the sheaf of holomorphic $n$-forms on $X$. If
$\widetilde{\mathscr{M}}$ is relative coherent over
$\textup{Spec}^{\textup{an}}R$, then we define

$\mathbb{D}_{\textup{Spec}^{\textup{an}}R}(\widetilde{\mathscr{M}})=\mathscr{R}hom_{\widetilde{\mathscr{D}}_{X,R}}(\widetilde{\mathscr{M}},\widetilde{\mathscr{D}}_{X,R})\otimes_{\widetilde{\mathscr{O}}}\widetilde{\omega^{-1}_{X}\otimes_{\mathbb{C}}R}[n]\in
D^{b}(\widetilde{\mathscr{D}}_{X,R}).$

Notice that by construction,

$\widetilde{\omega^{-1}_{X}\otimes_{\mathbb{C}}R}\simeq
pr_{X}^{*}\omega^{-1}_{X},$

where $pr_{X}\colon X\times\textup{Spec}^{\textup{an}}R\to X$ is the
projection. When $R=\mathbb{C}$, $\mathbb{D}_{\mathbb{C}}(\bullet)$ is just
the usual duality functor for coherent $\mathscr{D}_{X}$-modules. When $R$
(resp. $\textup{Spec}^{\textup{an}}R$) is obvious from the context, we use
$\mathbb{D}$ to denote $\mathbb{D}_{R}$ (resp.
$\mathbb{D}_{\textup{Spec}^{\textup{an}}R}$) for short.

By construction, we immediately obtain the following lemma, with its proof
skipped.

######  Lemma 2.12.

Let $h:\textup{Spec }S\to\textup{Spec }R$ be a finite type morphism of affine
schemes. For $\mathscr{M}\in\textup{Mod}(\mathscr{D}_{X,R})$, we have:

  1. (1)

if $\mathscr{M}$ is coherent over $\mathscr{D}_{X,R}$, then

$\mathbb{D}_{S}(\mathscr{M}\otimes^{\mathbf{L}}_{R}S)\simeq\mathbb{D}_{R}(\mathscr{M})\otimes^{\mathbf{L}}_{R}S,$

  2. (2)

if $\mathscr{M}$ is relative coherent over $\textup{Spec}^{\textup{an}}R$,
then

$\mathbb{D}_{\textup{Spec}^{\textup{an}}S}(\mathbf{L}\widetilde{h}^{*}\widetilde{\mathscr{M}})\simeq\mathbf{L}\widetilde{h}^{*}\mathbb{D}_{\textup{Spec}^{\textup{an}}R}(\widetilde{\mathscr{M}}).$

The following definition is the analytification of [BVWZ21a, Definition 3.3.1
and 4.3.4].

######  Definition 2.13.

For $\widetilde{\mathscr{M}}\in\textup{Mod}(\widetilde{\mathscr{D}}_{X,R})$
relative coherent over $\textup{Spec}^{\textup{an}}R$, we say
$\widetilde{\mathscr{M}}$ is $j$-Cohen-Macaulay for some $j\in\mathbb{Z}_{\geq
0}$ if $\widetilde{\mathscr{M}}\not=0$ and

$\mathscr{E}xt^{k}_{\widetilde{\mathscr{D}}_{X,R}}(\widetilde{\mathscr{M}},\widetilde{\mathscr{D}}_{X,R})=0\textup{
for every $k\not=j$}.$

If $\widetilde{\mathscr{M}}$ is $n$-Cohen-Macaulay, then for simplicity we
also use $\mathbb{D}(\widetilde{\mathscr{M}})$ to denote

$\mathscr{E}xt^{n}_{\widetilde{\mathscr{D}}_{X,R}}(\widetilde{\mathscr{M}},\widetilde{\mathscr{D}}_{X,R})\otimes_{\mathscr{O}}\widetilde{\omega^{-1}_{X}\otimes_{\mathbb{C}}R}.$

###  2.8. The relative codimension filtration of Gabber-Kashiwara

In this subsection, we let $R$ be a regular commutative finitely generated
$\mathbb{C}$-algebra integral domain. The following lemma is well-known (see
for instance [FF18, Lemma 2.9] and [BVWZ21a, §3.6]).

######  Lemma 2.14.

Let $\widetilde{\mathscr{M}}$ be relative coherent over
$\textup{Spec}^{\textup{an}}R$. If $\widetilde{\mathscr{M}}_{(x,p)}\not=0$ for
some point $(x,p)\in X\times\textup{Spec}^{\textup{an}}R$, then

$j(\widetilde{\mathscr{M}}_{(x,p)})+\dim_{\mathbb{C}}(\textup{Ch}^{{\textup{rel}}}(\widetilde{\mathscr{M}})\cap\pi_{R}^{-1}(\widetilde{W}))=2n+\dim
R,$

where $\pi_{R}\colon T^{*}X\times\textup{Spec}^{\textup{an}}R\to
X\times\textup{Spec}^{\textup{an}}R$ is the natural projection,
$\widetilde{W}$ is a small open neighborhood of $(x,p)$ and $j(\bullet)$
denotes the graded number for $(\mathscr{D}_{X,R})_{(x,p)}$-modules $($see for
instance [BVWZ21a, Definition 4.3.1]$)$.

Following the recipe in [Gin86, Gabber-Kashiwara theorem] and [Kas03, §2.4],
we now construct the Gabber-Kashiwara codimension filtration under the
relative setting. For $\mathscr{M}$ a coherent $\mathscr{D}_{X,R}$-module and
$k\in\mathbb{Z}_{\geq 0}$, we define a submodule

$T_{k}(\mathscr{M})=\\{m\in\mathscr{M}\mid\textup{codim}_{\mathbb{C}}\textup{Ch}^{\textup{rel}}(\widetilde{\mathscr{D}}_{X,R}\cdot
m)\geq k\\}\subseteq\mathscr{M},$

where $\widetilde{\mathscr{D}}_{X,R}\cdot m\subseteq\widetilde{\mathscr{M}}$
is the submodule generated by $m$. Then we have a finite decreasing
filtration, called the _relative codimension filtration_ ,

$\cdots\subseteq T_{k}(\mathscr{M})\subseteq
T_{k-1}(\mathscr{M})\cdots\subseteq T_{1}(\mathscr{M})\subseteq
T_{0}(\mathscr{M})=\mathscr{M}.$

In fact, by Theorem 2.16 and the relative Bernstein inequality [Wu20, Theorem
2.2], the relative codimension filtration always stops at $k=n+\dim R$.

If $\mathscr{M}$ is only relative coherent over $\textup{Spec}^{\textup{an}}R$
(or more generally $\widetilde{\mathscr{M}}$ is a coherent
$\widetilde{\mathscr{D}}_{X,R}$-module), we set

$T_{k}(\widetilde{\mathscr{M}})=\\{m\in\widetilde{\mathscr{M}}\mid\textup{codim}_{\mathbb{C}}\textup{Ch}^{\textup{rel}}(\widetilde{\mathscr{D}}_{X,R}\cdot
m)\geq k\\}\subseteq\widetilde{\mathscr{M}}.$

One can easily check that the relative codimension filtration is compatible
with analytic sheafification, that is, if $\mathscr{M}$ is coherent over
$\mathscr{D}_{X,R}$, then

$\widetilde{T_{k}(\mathscr{M})}=T_{k}(\widetilde{\mathscr{M}}).$

######  Definition 2.15.

For a non-zero coherent $\mathscr{D}_{X,R}$-moudle $\mathscr{M}$, we say
$\mathscr{M}$ pure of codimension $l$ if

  1. (1)

$T_{l}(\mathscr{M})=\mathscr{M}$ and

  2. (2)

$T_{k}(\mathscr{M})=0$ if $k<l$.

For a coherent $\widetilde{\mathscr{D}}_{X,R}$-module
$\widetilde{\mathscr{M}}$, we define its purity similarly.

By definition, a coherent submodule of a pure module is always pure of the
same codimension.

We have the following relative Gabber-Kashiwara theorem:

######  Theorem 2.16.

Let $\widetilde{\mathscr{M}}$ $($or $\mathscr{M})$ be a coherent
$\widetilde{\mathscr{D}}_{X,R}$-module $($resp. $\mathscr{D}_{X,R}$-module$)$.
Then we have:

  1. (1)

$T_{k}(\widetilde{\mathscr{M}})$ $($or $T_{k}(\mathscr{M}))$ is coherent over
$\widetilde{\mathscr{D}}_{X,R}$ $($resp. $\mathscr{D}_{X,R})$.

  2. (2)

$\textup{Ch}^{{\textup{rel}}}\frac{T_{k}(\widetilde{\mathscr{M}})}{T_{k+1}(\widetilde{\mathscr{M}})}$
is purely $k$-codimensional $($if not empty$)$, i.e. every irreducible
component is of codimension $k$.

  3. (3)

if
$\frac{T_{k}(\widetilde{\mathscr{M}})}{T_{k+1}(\widetilde{\mathscr{M}})}\not=0$
$($or $\frac{T_{k}(\mathscr{M})}{T_{k+1}(\mathscr{M})}\not=0)$, then it is
pure of codimension $k$.

###### Proof.

We follow the strategy in [Kas03, 2.4]. We use Lemma 2.14 to replace the role
of [Kas03, Theorem 2.19]. Then the proof of Theorem 2.18 and Theorem 2.24 in
_loc. cit._ give us all the required statements. If $\mathscr{M}$ is coherent
over $\mathscr{D}_{X,R}$, then we run the arguments in _loc. cit._ on $X$
replacing $\mathscr{D}_{X}$ by $\mathscr{D}_{X,R}$. If
$\widetilde{\mathscr{M}}$ is coherent over $\widetilde{\mathscr{D}}_{X,R}$, we
run the same arguments but replacing $X$ by
$X\times\textup{Spec}^{\textup{an}}R$ and $\mathscr{D}_{X}$ by
$\widetilde{\mathscr{D}}_{X,R}$. ∎

By Lemma 2.14 and Theorem 2.16, we immediately obtain:

######  Corollary 2.17.

If $\widetilde{\mathscr{M}}$ is pure of codimension $l$, then for $(x,p)\in
X\times\textup{Spec}^{\textup{an}}R$ if
$\widetilde{\mathscr{M}}_{(x,p)}\not=0$ then $\widetilde{\mathscr{M}}_{(x,p)}$
is $l$-pure over $(\widetilde{\mathscr{D}}_{X,R})_{(x,p)}$ in the sense of
[BVWZ21a, Definition 4.3.4].

If $\mathscr{M}$ is relative coherent over $\textup{Spec}^{\textup{an}}R$, we
set

$S_{k}(\mathscr{M})=\textup{supp}_{\textup{Spec}^{\textup{an}}R}\frac{T_{k}(\widetilde{\mathscr{M}})}{T_{k+1}(\widetilde{\mathscr{M}})}.$

Then by Corollary 2.10 and Theorem 2.16 (2), we have:

######  Corollary 2.18.

We have:

  1. (1)

If $\mathscr{M}$ is relative holonomic over $\textup{Spec}^{\textup{an}}R$,
then over each open subset $W\times V$ satisfying $W\subseteq X$ a relative
compact open subset and $V\subseteq\textup{Spec}^{\textup{an}}R$ a relative
compact open subset, $S_{k}(\mathscr{M})\subseteq V$ is a closed analytic
subvariety and if $S_{k}(\mathscr{M})\not=\emptyset$ then it is purely
$(k-n)$-codimensional.

  2. (2)

If $\mathscr{M}$ is coherent over $\mathscr{D}_{X,R}$ and relative holonomic
over $\textup{Spec}^{\textup{an}}R$, then over each relative compact open
subset $W\subseteq X$, $S_{k}(\mathscr{M})\subseteq\textup{Spec }R$ is a
closed algebraic subvariety and if $S_{k}(\mathscr{M})\not=\emptyset$ then it
is purely $(k-n)$-codimensional.

The following proposition links Cohen-Macaulay modules to pure modules.

######  Proposition 2.19.

If $\widetilde{\mathscr{M}}$ (or $\mathscr{M}$) is relative holonomic over
$\textup{Spec}^{\textup{an}}R$ and $(n+j)$-Cohen-Macaulay for some $j$, then
every non-zero coherent submodule of $\widetilde{\mathscr{M}}$ is pure of
codimension $j$. In particular, locally on every open subset $W\times V$
satisfying $W$ a relative compact open subset of $X$ and $V$ a relative
compact open subset of $\textup{Spec}^{\textup{an}}R$,

$\textup{supp}_{\textup{Spec}^{\textup{an}}R}\widetilde{\mathscr{M}}|_{W\times
V}\subseteq V$

is purely $j$-codimensional. If moreover $\mathscr{M}$ is coherent over
$\mathscr{D}_{X,R}$, then locally over every relative compact open subset
$W\subseteq X$, $\textup{supp}_{\textup{Spec
}R}\mathscr{M}\subseteq\textup{Spec }R$ is purely $j$-codimensional.

###### Proof.

We only need to prove $T_{k}(\widetilde{\mathscr{M}})=0$ for every $k>n+j$. We
assume on the contrary, $T_{k}(\widetilde{\mathscr{M}})_{(x,p)}\not=0$ for
some $k>n+j$ and some $(x,p)\in X\times\textup{Spec}^{\textup{an}}R$. For a
$(\widetilde{\mathscr{D}}_{X,R})_{(x,p)}$-module $M$, for simplicity we write

$E^{l}(M)=\textup{Ext}^{l}_{(\widetilde{\mathscr{D}}_{X,R})_{(x,p)}}(M,(\widetilde{\mathscr{D}}_{X,R})_{(x,p)}).$

By Lemma 2.14, the graded number

$j(T_{k}(\widetilde{\mathscr{M}})_{(x,p)})\geq k>n+j.$

Then $E^{n+j}(T_{k}(\widetilde{\mathscr{M}})_{(x,p)})=0$ and hence
$E^{n+j}(E^{n+j}(T_{k}(\widetilde{\mathscr{M}})_{(x,p)}))=0$. Taking double-
dual of the inclusion

$T_{k}(\widetilde{\mathscr{M}})_{(x,p)}\hookrightarrow\widetilde{\mathscr{M}}_{(x,p)}$

by Cohen-Macaulayness and naturality we obtain a commutative diagram

${T_{k}(\widetilde{\mathscr{M}})_{(x,p)}}$${\widetilde{\mathscr{M}}_{(x,p)}}$${E^{n+j}(E^{n+j}(T_{k}(\widetilde{\mathscr{M}})_{(x,p)}))}$${E^{n+j}(E^{n+j}(\widetilde{\mathscr{M}}_{(x,p)})).}$$\scriptstyle{\simeq}$

We then end up with a contradiction since
$E^{n+j}(E^{n+j}(T_{k}(\widetilde{\mathscr{M}})_{(x,p)}))=0.$

By purity,
$S_{n+j}(\widetilde{\mathscr{M}})=\textup{supp}_{\textup{Spec}^{\textup{an}}r}(\widetilde{\mathscr{M}})$.
The other required statements follow from Corollary 2.18. ∎

###  2.9. A relative Nakayama lemma

In this subsection, our goal is to prove a relative Nakayama lemma for
$\mathbb{C}$-constructible sheaves of $R$-modules (cf. [KS13, Definition
8.5.6]).

We assume $X$ a complex manifold, $\mathscr{A}=\mathbb{C}_{X}$ the constant
sheaf on $X$, and $R$ a noetherian commutative $\mathbb{C}$-algebra. We write
the derived category of $\mathbb{C}$-constructible (or constructible for
short) sheaves of $\mathbb{C}$-vector spaces by $D^{b}_{c}(\mathbb{C}_{X})$,
by $D^{b}_{c}(R_{X})$ the derived category of $\mathbb{C}$-constructible
sheaves of $R$-modules and by $\widetilde{D}^{b}_{c}(R_{X})$ the derived
category of the analytic sheafification of $\mathbb{C}$-constructible sheaves
of $R$-modules. We say a (bounded) complex is 0 in the derived category if it
is quasi-isomorphic to the 0 complex in the derived category.

One can check that when $R$ is a commutative $\mathbb{C}$-algebra of finite
type, if

$\mathcal{F}^{\bullet}\in D^{b}_{c}(R_{X}),$

then $\widetilde{\mathcal{F}}^{\bullet}$ is a relative constructible complex
over $\mathbb{C}^{l}$ supported on $X\times\textup{Spec}^{\textup{an}}R$ in
the sense of [MFS13, Definition 2.19.(2)], where we fix a closed embedding

$\textup{Spec }R\hookrightarrow\mathbb{C}^{l}$

for some $l\in\mathbb{Z}_{>0}$.

The following lemma follows directly from constructibility and Nakayama Lemma.
We leave its proof for interested readers.

######  Lemma 2.20.

Let $\mathcal{F}^{\bullet}$ be a $\mathbb{C}$-constructible complex of
$R$-modules on $X$. Then the following are equivalent:

  1. (1)

$\mathcal{F}^{\bullet}=0$ in $D^{b}_{c}(R_{X})$

  2. (2)

$\mathcal{F}^{\bullet}\otimes^{\mathbf{L}}_{R}R/m=0$ in
$D^{b}_{c}(\mathbb{C}_{X})$ for every maximal ideal $m\in\textup{Spec }R$

  3. (3)

$\widetilde{\mathcal{F}}^{\bullet}=0$ in $\widetilde{D^{b}_{c}}(R_{X})$.

The lemma above immediately implies:

######  Corollary 2.21.

If $\mathcal{F}^{\bullet}\in D^{b}_{c}(R_{X})$, then
$\textup{supp}_{\textup{Spec }R}\mathcal{F}^{\bullet}$ is algebraically closed
locally over $X$, i.e. locally on a relative compact open subset $W$ of $X$,

$\textup{supp}_{\textup{Spec
}R}\mathcal{F}^{\bullet}|_{W}\subseteq\textup{Spec }R$

is a closed algebraic subvariety.

##  3\. Relative maximal and minimal extensions along $F$

In this section, we construct the maximal and minimal extensions for
$\mathscr{D}$-modules under the relative setting analogous to the
${}^{p}j_{!}$ and ${}^{p}j_{*}$ extensions of affine (or Stein) open
inclusions $j$ for perverse sheaves [BBDG18].

###  3.1. Notations

We introduce notation for this whole section. Let $X$ be a complex manifold
with $\dim_{\mathbb{C}}X=n$. For some fixed $r\in\mathbb{Z}_{>0}$, we consider
the morphism

$\pi=(\textup{id},Exp)\colon X\times\mathbb{C}^{r}\rightarrow
X\times(\mathbb{C}^{*})^{r}$

where

$Exp\colon\mathbb{C}^{r}\rightarrow(\mathbb{C}^{*})^{r},(\alpha_{1},\dots,\alpha_{r})\mapsto(exp(-2\pi\sqrt{-1}\alpha_{1}),\dots,exp(-2\pi\sqrt{-1}\alpha_{r}))$

is the universal cover of $(\mathbb{C}^{*})^{r}$. We write
$G=\pi_{1}((\mathbb{C}^{*})^{r})$, the fundamental group of
$(\mathbb{C}^{*})^{r}$. The deck transformation of $Exp$ makes
$X\times\mathbb{C}^{r}$ a free $G$-space (we always use the discrete topology
of $G$). We also write

$R=\textup{Sym}^{\bullet}(\mathbb{C}^{r})^{\vee}\simeq\mathbb{C}[\mathbf{s}]$

the algebraic coordinate ring of $\mathbb{C}^{r}$, where
$\textup{Sym}^{\bullet}(\mathbb{C}^{r})^{\vee}$ is the symmetric algebra of
the dual vector space $(\mathbb{C}^{r})^{\vee}$ and the isomorphism is induced
from using the standard basis of $\mathbb{C}^{r}$. Then $G$ acts on $R$
induced by the $G$-action on $(\mathbb{C}^{r})^{\vee}$. In algebraic
coordinates, the $G$-action on $R$ is given by

(3.1) $t_{i}\cdot s_{j}=\begin{cases}s_{j}+1,&\textup{ if }i=j\\\
s_{j},&\textup{ if }i\not=j.\end{cases}$

where $t_{i}$ represents the (counterclockwise) loops around the puncture of
each factor $\mathbb{C}^{*}$ of $(\mathbb{C}^{*})^{r}$ and such $t_{i}$ induce
an isomorphism $G\simeq\mathbb{Z}^{r}$.

Let $F=(f_{1},\dots,f_{r})$ be an $r$-tuple of holomorphic functions on $X$
with $D_{i}$ the divisor of $f_{i}$ and let $\mathcal{M}$ be a holonomic
$\mathscr{D}_{X}$-module. We write by $D=\sum_{i}D_{i}$ and by

$\mathcal{M}(*D)=\mathcal{M}\otimes_{\mathscr{O}}\mathscr{O}_{X}(*D)$

the algebraic localization of $\mathcal{M}$ along $D$, where
$\mathscr{O}_{X}(*D)=\bigcup_{k\in\mathbb{Z}}\mathscr{O}_{X}(kD)$ is the sheaf
of meromorphic functions with poles along $D$. We write by

$\mathscr{M}=\mathcal{M}(*D)\otimes_{\mathbb{C}}\mathbb{C}[\mathbf{s}]\cdot
F^{\mathbf{s}}.$

We have natural actions

(3.2) $\theta\cdot(m\cdot F^{\mathbf{s}})=(\theta\cdot m)\cdot
F^{\mathbf{s}}+m\cdot\sum_{i=1}^{r}\frac{s_{i}\theta(f_{i})}{f_{i}}\cdot
F^{\mathbf{s}}$

for every section $m$ of $\mathcal{M}(*D)$ and for holomorphic vector fields
$\theta$ on $Y$, which makes $\mathscr{M}$ a $\mathscr{D}_{X,R}$-module. Since
$\mathcal{M}(*D)$ is holonomic, by [Mal04, Theorem 3.1] one can assume

$\mathcal{M}(*D)=\mathscr{D}_{X}\cdot\mathcal{M}_{0}$

for $\mathcal{M}_{0}$ a coherent $\mathscr{O}_{X}$-submodule of
$\mathcal{M}(*D)$. We then fix such an $\mathcal{M}_{0}$ throughout the
remaining of this paper and define the $\mathscr{D}_{X,R}$-submodules

$\mathscr{N}_{k}=\mathscr{D}_{X,R}(\mathcal{M}_{0}\cdot F^{\mathbf{s}-{\bf
k}})\subseteq\mathscr{M}$

generated by $\mathcal{M}_{0}\cdot F^{\mathbf{s}-{\bf k}}$ with ${\bf
k}=(k,k,\dots,k)$ for $k\in\mathbb{Z}$. By construction,
$\mathscr{D}_{X,R}(\mathcal{M}_{0}\cdot F^{\mathbf{s}})$ is coherent over
$\mathscr{D}_{X,R}$ but $\mathscr{M}$ might not be so, and

$\mathscr{N}_{k_{1}}\subseteq\mathscr{N}_{k_{2}}\textup{ for $k_{1}\leq k_{2}$
and }\lim_{k\to\infty}\mathscr{N}_{k}=\mathscr{M}.$

###  3.2. $G$-action and translation on $\mathscr{M}$

The $G$-action on $R$ induces a $G$-action on $\mathscr{M}$. In algebraic
coordinates, the $G$-action on $\mathscr{M}$ is given by

$t_{i}\cdot P(\mathbf{s})(m\cdot F^{\mathbf{s}})=P(\mathbf{s}+e_{i})(m\cdot
F^{\mathbf{s}+e_{i}})$

for $P(\mathbf{s})\in\mathscr{D}_{X,R}$, where $e_{i}\in\mathbb{Z}^{r}$ is the
$i$-th unit vector. The $G$-action on $\mathscr{M}$ induces an operation
$g\cdot\mathscr{N}_{k}\subset\mathscr{M}$, i.e.

$t_{i}\cdot\mathscr{N}_{k}=\mathscr{D}_{X}[\mathbf{s}]\cdot F^{\mathbf{s}-{\bf
k}+e_{i}}.$

Since $\mathbb{C}^{r}$ is homogeneous, for every
${\boldsymbol{\alpha}}\in\mathbb{C}^{r}$ the translation of
${\boldsymbol{\alpha}}$ induces an isomorphism

$\tau_{\boldsymbol{\alpha}}\colon X\times\mathbb{C}^{r}\to
X\times\mathbb{C}^{r}\quad(x,\mathbf{s})\mapsto(x,\mathbf{s}+{\boldsymbol{\alpha}}).$

Substituting $\mathbf{s}$ by $\mathbf{s}+{\boldsymbol{\alpha}}$ gives an
$\mathscr{D}_{X}$-linear isomorphism (but not $\mathbb{C}[\mathbf{s}]$-linear)

(3.3)
$\mathscr{N}_{k}\simeq\mathscr{D}_{X}[\mathbf{s}+{\boldsymbol{\alpha}}](\mathcal{M}_{0}\cdot
F^{\mathbf{s}+{\boldsymbol{\alpha}}-{\bf k}})$

which further induces an $pr_{X}^{-1}\mathscr{D}_{X}$-linear isomorphism

(3.4)
$\tau^{-1}_{{\boldsymbol{\alpha}}}\widetilde{\mathscr{N}}_{k}\simeq\widetilde{\mathscr{N}}_{k},$

where $pr_{X}\colon X\times\mathbb{C}^{r}\to X$ is the projection. The
standard basis on $\mathbb{C}^{r}$ induces an isomorphism

$\mathbb{Z}^{r}\simeq G,\quad{\bf a}=(a_{1},\dots,a_{r})\mapsto g_{\bf
a}=\sum_{i}a_{i}t_{i}.$

Then for ${\bf a}\in\mathbb{Z}^{r}$, by construction we have

(3.5) $\tau^{-1}_{\bf a}\widetilde{\mathscr{N}}_{k}=\widetilde{g_{\bf
a}\cdot\mathscr{N}_{k}}$

for every $k$.

###  3.3. Relative maximal and minimal extension

We now are ready to construct the relative maximal and minimal extensions.

We first recall a result of Sabbah [Sab87b] about the existence of generalized
$b$-functions.

######  Theorem 3.1 (Sabbah).

Let $\mathcal{M}$ be a holonomic $\mathscr{D}_{X}$-module (with a fixed
$\mathcal{M}_{0}$). Then, locally on a relative compact open subset
$W\subseteq X$, there exists a polynomial
$b(\mathbf{s})\in\mathbb{C}[\mathbf{s}]$ such that the following two
conditions are satisfied:

  1. (1)

$b(\mathbf{s})=\prod_{L\in\mathbb{Z}^{r}_{\geq
0}}\prod_{\alpha\in\mathbb{C}}(L\cdot\mathbf{s}+\alpha)^{n_{L,\alpha}}$ over
some finite sets of $L$ and $\alpha$,

  2. (2)

$b(\mathbf{s})\cdot\frac{\mathscr{N}_{0}}{\mathscr{N}_{-1}}=0$.

######  Proposition 3.2.

Let $\mathcal{M}$ be a holonomic $\mathscr{D}_{X}$-module. For every relative
compact open subset $W\subseteq X$ and for every relative compact open subset
$V\subseteq\mathbb{C}^{r}$, there exists an integer $l=l_{W,V}>0$ such that

  1. (1)

${\widetilde{\mathscr{M}}}|_{W\times
V}={\widetilde{\mathscr{N}}_{k}}|_{W\times V}$ for all $k\geq l$.

  2. (2)

${\widetilde{\mathscr{N}}_{-k}}|_{W\times
V}={\widetilde{\mathscr{N}}_{-l}}|_{W\times V}$ for all $k\geq l$

###### Proof.

By the construction of the analytic sheafification, we have

$b(\mathbf{s})\cdot\widetilde{\mathscr{N}_{0}/\mathscr{N}_{-1}}=0$

where $b(\mathbf{s})$ is the generalized $b$-function in Theorem 3.1. By
translation, for every $k\in\mathbb{Z}$ with ${\bf k}=(k,k,\dots,k)$, we have

(3.6) $b(g_{\bf
k}\cdot\mathbf{s})\cdot\widetilde{\mathscr{N}_{-k}/\mathscr{N}_{-k+1}}=0,$

Choose $l>0$ large enough (since $V$ is relative compact) such that $b(g_{\bf
k}\cdot\mathbf{s})|_{V}$ are invertible for all $|k|>l$. Then by (3.6), we
have

$\widetilde{\mathscr{N}}_{k}=\widetilde{\mathscr{N}}_{k-1}$

for all $|k|>l$. Part (2) then follows. Since
$\displaystyle{\lim_{k\to\infty}}\mathscr{N}_{k}=\mathscr{M}$, part (1) also
follows. ∎

The following theorem is a natural generalization of [Mai16a, Résultat] and
[BVWZ21b, Theorem 4.3.4].

######  Theorem 3.3.

Let $\mathcal{M}$ be a holonomic $\mathscr{D}_{X}$-module. Then $\mathscr{M}$
and $\mathscr{N}_{k}$ are relative holonomic over the complex manifold
$\mathbb{C}^{r}$ for all $k\in\mathbb{Z}$. Moreover,

$\textup{CC}^{\textup{rel}}(\widetilde{\mathscr{M}})=\textup{CC}^{\textup{rel}}(\widetilde{\mathscr{N}}_{k})=\textup{CC}(\mathcal{M}(*D))\times\mathbb{C}^{r}.$

###### Proof.

By Proposition 3.2 (1), $\mathscr{M}$ is relative coherent over
$\mathbb{C}^{n}$. Now we construct the relative characteristic cycles for
$\widetilde{\mathscr{N}}_{k}$ over a relative compact open subset $W\subseteq
X$. For simplicity we assume $X=W$. Then we define

$F_{p}(\mathscr{N}_{k})=(F_{p}\mathscr{D}_{X}\otimes_{\mathbb{C}}\mathbb{C}[\mathbf{s}])\cdot(\mathcal{M}_{0}\cdot
F^{\mathbf{s}-{\bf k}})$

which gives a relative good filtration of $\mathscr{N}_{k}$ over
$\mathbb{C}[\mathbf{s}]$ (cf. [BVWZ21a, §3.1]). Then
$\widetilde{F_{p}(\mathscr{N}_{k})}$ gives a relative good filtration for
$\widetilde{\mathscr{N}}_{k}$ over $\mathbb{C}^{r}$ (cf. [Wu20, §2.1]). Using
the isomorphism (3.4),
$\tau^{-1}_{\boldsymbol{\alpha}}\widetilde{F_{p}(\mathscr{N}_{k})}$ gives a
relative good filtration of
$\tau^{-1}_{\boldsymbol{\alpha}}\widetilde{\mathscr{N}}_{k}$ such that

$\textup{gr}^{F}_{\bullet}(\tau^{-1}_{\boldsymbol{\alpha}}\widetilde{\mathscr{N}}_{k})\simeq\tau^{-1}_{\boldsymbol{\alpha}}(\textup{gr}^{F}_{\bullet}\mathscr{N}_{k}),$

which implies $\textup{CC}^{{\textup{rel}}}(\widetilde{\mathscr{N}}_{k})$ is
invariant under translation by ${\boldsymbol{\alpha}}$. Since
${\boldsymbol{\alpha}}$ is arbitrary, we thus have

$\textup{CC}^{{\textup{rel}}}(\widetilde{\mathscr{N}}_{k})=\textup{CC}^{{\textup{rel}}}(\widetilde{\mathscr{N}}_{k})|_{X\times\\{{\bf
a}\\}}\times\mathbb{C}^{r}.$

We then pick ${\bf a}=(a,a,\dots,a)\in\mathbb{Z}^{r}$ with $a\ll 0$. On a
small neighborhood $V_{\bf a}$ of ${\bf a}$, by Proposition 3.2(1), we know

$\widetilde{\mathscr{N}}_{k}|_{W\times V_{\bf
a}}=\widetilde{\mathscr{M}}|_{W\times V_{\bf a}}.$

Since $\mathscr{M}$ is free over $\mathbb{C}[\mathbf{s}]$, one can repeatedly
apply [Wu20, Proposition 2.7] (by choosing general hyperplanes
$H_{1},H_{2},\dots,H_{r}$ passing ${\bf a}$) and conclude

$\textup{CC}^{\textup{rel}}(\widetilde{\mathscr{N}}_{k})|_{X\times\\{{\bf
a}\\}}=\textup{CC}^{\textup{rel}}(\widetilde{\mathscr{M}})|_{X\times\\{{\bf
a}\\}}=\textup{CC}(\widetilde{\mathscr{M}}|_{X\times\\{{\bf a}\\}}).$

But since ${\bf a}\in\mathbb{C}^{n}$ is a $\mathbb{Z}$-point,

$\widetilde{\mathscr{M}}|_{X\times\\{{\bf
a}\\}}\simeq\mathscr{M}\otimes_{\mathbb{C}[\mathbf{s}]}\mathbb{C}_{{\bf
a}}=\mathcal{M}(*D)$

where $\mathbb{C}_{\bf a}$ is the residue field of ${\bf a}\in\mathbb{C}^{n}$.
We therefore obtain

$\textup{CC}^{\textup{rel}}(\widetilde{\mathscr{N}}_{k})=\textup{CC}(\mathcal{M}(*D))\times\mathbb{C}^{r}$

for every $k$. Thanks to Proposition 3.2 again, we have

$\textup{CC}^{\textup{rel}}(\widetilde{\mathscr{M}})|_{X\times V_{\bf
a}}=\textup{CC}(\mathcal{M}(*D))\times V_{\bf a},$

since fixing $k$ and making $a\ll 0$ is equivalent to fixing $V_{\bf a}$ and
making $k\gg 0$. Thus, globally

$\textup{CC}^{\textup{rel}}(\widetilde{\mathscr{M}})=\textup{CC}(\mathcal{M}(*D))\times\mathbb{C}^{r}.$

∎

######  Remark 3.4.

From the proof Proposition 3.2, one can see that it is necessary to use the
analytic sheafification of $\mathscr{M}$. To be more precise, even if
$\mathcal{M}$ is an algebraic holonomic $\mathscr{D}_{X}$-module on a smooth
complex algebraic variety $X$, $\widetilde{\mathscr{M}}^{\textup{alg}}$ (cf.
Remark 2.3) is not relative coherent over $\textup{Spec }R$, since every
Zariski open subset of $\textup{Spec }R$ intersects the divisors $(b(g_{\bf
k}\cdot\mathbf{s})=0)$ for all but a finite number of $k\in\mathbb{Z}$, where
$b(\mathbf{s})$ is the generalized $b$-function as in Theorem 3.1.

We now write

$\widetilde{\mathscr{M}}=\mathcal{M}(*D^{(r)}_{F}),$

calling it the _relative maximal extension_ of $\mathcal{M}$ along $F$. Next
we construct the minimal one.

######  Proposition 3.5.

If $\mathcal{M}$ is holonomic, then
$\widetilde{\mathscr{M}}=\mathcal{M}(*D^{(r)}_{F})$ is $n$-Cohen-Macaulay.

###### Proof.

We fix a point $(x,p)\in X\times\mathbb{C}^{r}$. First, we know
$(\widetilde{\mathscr{D}}_{X,R})_{(x,p)}$ is Auslander regular (by for
instance [BVWZ21a, Theorem 4.3.2]). Then by Auslander regularity, the graded
number

$j(\mathscr{E}xt^{k}_{(\widetilde{\mathscr{D}}_{X,R})_{(x,p)}}(\widetilde{\mathscr{N}}_{0},(\widetilde{\mathscr{D}}_{X,R})_{(x,p)}))\geq
k$

for each $k$. Also, for all $k$ as right coherent
$\widetilde{\mathscr{D}}_{X,R}$-modules

$\mathscr{E}xt^{k}_{\widetilde{\mathscr{D}}_{X,R}}(\widetilde{\mathscr{N}}_{0},\widetilde{\mathscr{D}}_{X,R})$

are relative holonomic, as so is $\widetilde{\mathscr{N}}$ by Theorem 3.3. By
Lemma 2.14, we then know

$\mathscr{E}xt^{k}_{\widetilde{\mathscr{D}}_{X,R}}(\widetilde{\mathscr{N}}_{0},\widetilde{\mathscr{D}}_{X,R})=0\textup{
for }k<n$

and

$\dim_{\mathbb{C}}(\textup{Ch}^{\textup{rel}}(\mathscr{E}xt^{k}_{\widetilde{\mathscr{D}}_{X,R}}(\widetilde{\mathscr{N}}_{0},\widetilde{\mathscr{D}}_{X,R})))\leq
n+r-k,\textup{ for }n<k\leq n+r.$

By Lemma 2.5, we conclude that $\widetilde{\mathscr{N}}_{0}$ is $n$-Cohen-
Macaulay away from $W\times V$ for $V\subsetneq\mathbb{C}^{r}$ some closed
algebraic subvariety, where $W$ is a small neighborhood of $x\in X$. We now
translate $\mathscr{N}_{0}$ by $\tau_{-{\bf a}}$ for some ${\bf
a}\in\mathbb{Z}^{r}_{\geq 0}$ with each $a_{i}\gg 0$ so that
$(x,p)\not\in\tau_{{\bf a}}(W\times V)$. Thus, ${\tau^{-1}_{-{\bf
a}}\widetilde{\mathscr{N}}_{0}}$ is Cohen-Macaulay around $(x,p)$. But from
the proof of Proposition 3.2, such ${\tau^{-1}_{-{\bf
a}}\widetilde{\mathscr{N}}_{0}}$ is just $\widetilde{\mathscr{M}}$ around
$(x,p)$. ∎

######  Remark 3.6.

  1. (1)

After applying Theorem 3.3, Proposition 3.5 also follows from [MFS19,
Proposition 2]

  2. (2)

In general $\widetilde{\mathscr{N}}_{k}$ are not $n$-Cohen-Macaulay. But since
$\widetilde{\mathscr{N}}_{k}\subseteq\widetilde{\mathscr{M}}$ is a submodule
of a $n$-Cohen-Macaulay module, $\widetilde{\mathscr{N}}_{k}$ is always pure
of codimension $n$ for every $k$ (by Proposition 2.19).

We write by $\widetilde{\mathscr{M}}_{\mathbb{D}\mathcal{M}}$ the analytic
sheafification of
$(\mathbb{D}\mathcal{M})(*D)\otimes_{\mathbb{C}}\mathbb{C}[\mathbf{s}]\cdot
F^{-\mathbf{s}}$. We apply Proposition 3.5 for $\mathbb{D}\mathcal{M}$, and
thus see that $\widetilde{\mathscr{M}}_{\mathbb{D}\mathcal{M}}$ is $n$-Cohen-
Macaulay and relative holonomic over $\mathbb{C}^{r}$. We then define

$\mathcal{M}(!D_{F}^{(r)})=\mathbb{D}(\widetilde{\mathscr{M}}_{\mathbb{D}\mathcal{M}})$

which is a relative holonomic $\mathscr{D}$-module over $\mathbb{C}^{r}$,
thanks to Cohen-Macaulayness. In particular, if $r=0$, then we have

$\mathcal{M}(!D_{F}^{(0)})=\mathcal{M}(!D)=\mathbb{D}((\mathbb{D}\mathcal{M})(*D)).$

Notice that in the definition of $\mathcal{M}(!D_{F}^{(r)})$ we use the symbol
$F^{-\mathbf{s}}$ instead of $F^{\mathbf{s}}$ because $\mathbb{D}$ naturally
maps $F^{-\mathbf{s}}$ back to $F^{\mathbf{s}}$ (see [BVWZ21b, Lemma 5.3.1]).

Since $\mathcal{M}(!D_{F}^{(r)})|_{(X\setminus
D)\times\mathbb{C}^{r}}=\mathcal{M}(*D_{F}^{(r)})|_{(X\setminus
D)\times\mathbb{C}^{r}}$, the morphism
$\mathscr{O}_{X}\hookrightarrow\mathscr{O}_{X}(*D)$ induces a natural morphism

$\mathcal{M}(!D_{F}^{(r)})\longrightarrow\mathcal{M}(*D_{F}^{(r)}).$

######  Lemma 3.7.

For $\mathcal{M}$ a holonomic $\mathscr{D}_{X}$-module, if $r\geq 1$, then the
natural morphism

$\mathcal{M}(!D_{F}^{(r)})\hookrightarrow\mathcal{M}(*D_{F}^{(r)})$

is injective.

###### Proof.

We fix a small neighborhood $W\times V$ of a point $(x,p)\in
X\times\mathbb{C}^{r}$. By construction, $\mathcal{M}(*D_{F}^{(r)})$ has no
non-zero coherent submodule supported on $D\times\mathbb{C}^{r}$. By duality
and Lemma 2.14, $\mathcal{M}(!D_{F}^{(r)})$ has no non-zero coherent quotient
module supported on $D\times\mathbb{C}^{r}$. Then the image of
$\mathcal{M}(!D_{F}^{(r)})\to\mathcal{M}(*D_{F}^{(r)})$, denoted by
$\mathcal{M}(!*D_{F}^{(r)})$, is the minimal extension. Since the cokernel of
$\mathcal{M}(!D_{F}^{(r)})\to\mathcal{M}(*D_{F}^{(r)})$ is supported on
$D\times\mathbb{C}^{r}$, by analytic nullstellensatz,

$(\widetilde{\mathcal{M}_{0}\cdot F^{\mathbf{s}+{\bf a}}})|_{W\times
V}\subseteq\mathcal{M}(!*D_{F}^{(r)})|_{W\times V}$

for some ${\bf a}\in\mathbb{Z}^{r}_{\geq 0}$ with each $a_{i}\gg 0$, and thus

$(\tau^{-1}_{{\bf a}}\widetilde{\mathscr{N}}_{0})|_{W\times
V}\subseteq\mathcal{M}(!*D_{F}^{(r)})|_{W\times V}$

By minimality,

(3.7) $\mathcal{M}(!*D_{F}^{(r)})|_{W\times V}=(\tau^{*}_{{\bf
a}}\widetilde{\mathscr{N}}_{0})|_{W\times V}$

(since otherwise $\frac{\mathcal{M}(!*D_{F}^{(r)})|_{W\times
V}}{(\tau^{-1}_{{\bf a}}\widetilde{\mathscr{N}}_{0})|_{W\times V}}$ is a non-
zero coherent quotient module of $\mathcal{M}(!D_{F}^{(r)})$ supported on
$D\times\mathbb{C}^{r}$). Similar to the proof of Proposition 3.5,
$(\tau^{-1}_{{\bf a}}\widetilde{\mathscr{N}}_{0})|_{W\times V}$ is $n$-Cohen-
Macaulay and hence so is $\mathcal{M}(!*D_{F}^{(r)})|_{W\times V}$.

We use $\mathcal{K}$ to denote the kernel on $W\times V$. If
$\mathcal{K}\not=0$, then we have a short exact sequence

$0\to\mathcal{K}\to\mathcal{M}(!D_{F}^{(r)})|_{W\times
V}\to\mathcal{M}(!*D_{F}^{(r)})|_{W\times V}\to 0.$

Since $\mathcal{K}$ is relative holonomic, by Proposition 2.19 and Proposition
2.5,

$\textup{Ch}^{\textup{rel}}(\mathcal{K})=\Lambda_{\mathcal{K}}\times\mathbb{C}^{r}$

for some Lagrangian subvariety $\Lambda_{\mathcal{K}}\subseteq T^{*}X|_{W}$.
Taking the dual, we have

$0\to(\mathbb{D}\mathcal{M})(!*D_{F}^{(r)})|_{W\times
V}\to(\mathbb{D}\mathcal{M})(*D_{F}^{(r)})|_{W\times
V}\to\mathbb{D}(\mathcal{K})\to 0.$

By (3.7) (replacing $\mathcal{M}$ by $\mathbb{D}\mathcal{M}$) and Theorem 3.3,

$\textup{CC}((\mathbb{D}\mathcal{M})(!*D_{F}^{(r)})|_{W\times
V})=\textup{CC}((\mathbb{D}\mathcal{M})(*D_{F}^{(r)})|_{W\times V}).$

Since
$\textup{Ch}^{\textup{rel}}(\mathbb{D}(\mathcal{K}))=\Lambda_{\mathcal{K}}\times\mathbb{C}^{r}$,
we get a contradiction by counting multiplicities. ∎

An immediate consequence of the inclusion in the lemma above is that
$\mathcal{M}(!D_{F}^{(r)})$ is the _minimal extension_ (analogous to the
Deligne-Goresky-MacPherson extensions for perverse sheaves [HTT08, §8.2.1]).
Therefore, when $\mathcal{M}$ is holonomic, $\mathcal{M}(!D_{F}^{(r)})$
satisfies the property that it has no non-trivial submodule (or quotient
module) supported on $D\times\mathbb{C}^{r}$ in the category of relative
coherent $\widetilde{\mathscr{D}}_{X,R}$-modules. By this property, in the
situation of Proposition 3.2, we have

(3.8) $\mathcal{M}(!D_{F}^{(r)})|_{W\times
V}=\widetilde{\mathscr{N}}_{-k}|_{W\times V}$

for all $k\geq l_{W\times V}$.

###  3.4. Gluing data and $G$-equivariance

Now we discuss how to “glue” the local data in Proposition 3.2 to construct
the maximal and minimal extensions alternatively.

We keep assuming $\mathcal{M}$ a holonomic $\mathscr{D}_{X}$-module and $W$ a
relative compact open subset of $X$. We pick a locally finite covering

$\mathbb{C}^{r}=\bigcup_{\beta}V_{\beta}$

such that each $V_{\beta}$ is relatively compact. For instance, we can pick a
finite covering for the unit polydisc in $\mathbb{C}^{r}$ and then translate
the covering by $\mathbb{Z}^{r}$ to get such a covering of $\mathbb{C}^{r}$.
We can tautologically “glue” $\mathcal{M}(!D^{(r)}_{F})$ with the help of
Proposition 3.2 (2) as follows:

  1. (i)

For every $V_{\beta_{0}}$, we have

$\mathcal{M}(!D^{(r)}_{F})|_{W\times
V_{\beta_{0}}}=\widetilde{\mathscr{N}}_{-l_{W,V_{\beta_{0}}}}|_{W\times
V_{\beta_{0}}}$

  2. (ii)

We set

$k_{0}=\max\\{l_{W,V_{\beta_{i}}}\mid V_{\beta_{0}}\cap
V_{\beta_{i}}\not=\emptyset\\}.$

  3. (iii)

If $V_{\beta_{0}}\cap V_{\beta_{i}}\not=\emptyset$, we can “glue”
$\widetilde{\mathscr{N}}_{-l_{W,V_{\beta_{0}}}}|_{W\times V_{\beta_{0}}}$ and
$\widetilde{\mathscr{N}}_{-l_{W,V_{\beta_{i}}}}|_{W\times V_{\beta_{i}}}$
since

$\widetilde{\mathscr{N}}_{-l_{W,V_{\beta_{0}}}}|_{W\times(V_{\beta_{0}}\cap
V_{\beta_{i}})}=\widetilde{\mathscr{N}}_{-k_{0}}|_{W\times(V_{\beta_{0}}\cap
V_{\beta_{i}})}=\widetilde{\mathscr{N}}_{-l_{W,V_{\beta_{i}}}}|_{W\times(V_{\beta_{0}}\cap
V_{\beta_{i}})}.$

Notice that the cocycle condition of “gluing” is guaranteed by Condition (ii).
Using Proposition 3.2 (1), we can “glue” $\mathcal{M}(*D^{(r)}_{F})$
similarly.

######  Theorem 3.8.

Let $\mathcal{M}$ be a holonomic $\mathscr{D}_{X}$-module. Then both
$\mathcal{M}(*D^{(r)}_{F})$ and $\mathcal{M}(!D^{(r)}_{F})$ are
$G$-equivariant sheaves of abelian groups on $X\times\mathbb{C}^{r}$ and the
inclusion

$\mathcal{M}(!D^{(r)}_{F})\hookrightarrow\mathcal{M}(*D^{(r)}_{F})$

is a $G$-equivariant morphism.

###### Proof.

By (3.4) and (3.5), we have for every $g\in G$

$\widetilde{g\cdot\mathscr{N}_{k}}\simeq\widetilde{\mathscr{N}}_{k}$

In particular,

(3.9) $\widetilde{g\cdot\mathscr{N}_{k}}|_{W\times
V_{\beta_{0}}}\simeq\widetilde{\mathscr{N}}_{k}|_{W\times V_{\beta_{0}}}.$

The isomorphism above is obviously compatible with the “gluing” data for both
$\mathcal{M}(*D^{(r)}_{F})$ and $\mathcal{M}(!D^{(r)}_{F})$. Since $G$ acts on
$X\times\mathbb{C}^{r}$, for every $g\in G$ we have an isomorphism

$g:X\times\mathbb{C}^{r}\to X\times\mathbb{C}^{r}.$

The “gluing” data and (3.9) together give isomorphisms

(3.10)
$g^{-1}(\mathcal{M}(*D^{(r)}_{F}))\simeq\mathcal{M}(*D^{(r)}_{F})\textup{ and
}g^{-1}(\mathcal{M}(!D^{(r)}_{F}))\simeq\mathcal{M}(!D^{(r)}_{F}).$

Since we use the discrete topology of $G$, the isomorphisms above make
$\mathcal{M}(*D^{(r)}_{F})$ and $\mathcal{M}(!D^{(r)}_{F})$ $G$-equivariant
sheaves of abelian groups (one can easily check the cocycle condition). Here
we use the definition of $G$-equivariant sheaves in [BL94, Part I.0.2].

The inclusion
$\mathcal{M}(!D^{(r)}_{F})\hookrightarrow\mathcal{M}(*D^{(r)}_{F})$ is
$G$-equivariant because of the following commutative diagram

${\mathscr{N}_{l}}$${g\cdot\mathscr{N}_{l}}$${\mathscr{N}_{k}}$${g\cdot\mathscr{N}_{k})}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

for all $l<k$. ∎

For $\mathcal{M}$ a holonomic $\mathscr{D}_{X}$-module, we now define

$\Psi_{F}(\mathcal{M})=\frac{\mathcal{M}(*D^{(r)}_{F})}{\mathcal{M}(!D^{(r)}_{F})}.$

By Lemma 3.7, another interpretation of $\Psi_{F}(\mathcal{M})$ is

$\Psi_{F}(\mathcal{M})[-1]\stackrel{{\scriptstyle
q.i.}}{{\simeq}}R\Gamma_{[D\times\mathbb{C}^{r}]}(\mathcal{M}(!D^{(r)}_{F})).$

######  Theorem 3.9.

Let $\mathcal{M}$ be a holonomic $\mathscr{D}_{X}$-module. Then
$\Psi_{F}(\mathcal{M})$ is $G$-equivariant and relative holonomic over
$\mathbb{C}^{r}$. If moreover, $\Psi_{F}(\mathcal{M})\not=0$, then it is
$(n+1)$-Cohen-Macaulay.

###### Proof.

As a quotient of $\mathcal{M}(*D_{F}^{(r)})$, $\Psi_{F}(\mathcal{M})$ is
relative holonomic. Since the category of $G$-equivariant sheaves is abelian
and the inclusion

$\mathcal{M}(!D^{(r)}_{F})\hookrightarrow\mathcal{M}(*D^{(r)}_{F})$

is $G$-equivariant by Theorem 3.8, $\Psi_{F}(\mathcal{M})$ is $G$-equivariant.

To prove Cohen-Macaulayness, we consider the short exact sequence

$0\to\mathcal{M}(!D^{(r)}_{F})\rightarrow\mathcal{M}(*D^{(r)}_{F})\rightarrow\Psi_{F}(\mathcal{M})\to
0.$

By Theorem 3.3 and (3.8), we know

$\textup{CC}^{\textup{rel}}(\mathcal{M}(!D^{(r)}_{F}))=\textup{CC}^{\textup{rel}}(\mathcal{M}(*D^{(r)}_{F})).$

If $\Psi_{F}(\mathcal{M})\not=0$, by counting multiplicity we have

$\dim_{\mathbb{C}}\textup{Ch}^{\textup{rel}}(\Psi_{F}(\mathcal{M}))\leq
n+r-1.$

By Lemma 2.14,

$\mathscr{E}xt^{k}_{\widetilde{\mathscr{D}}_{X,R}}(\Psi_{F}(\mathcal{M}))=0\textup{
for $k\leq n$}.$

Taking the dual of the short exact sequence, since both
$\mathcal{M}(!D^{(r)}_{F})$ and $\mathcal{M}(*D^{(r)}_{F})$ are $n$-Cohen-
Macaulay, $\Psi_{F}(\mathcal{M})$ is $(n+1)$-Cohen-Macaulay. ∎

######  Remark 3.10.

(1) By Theorem 3.8 and Theorem 3.9, $\mathcal{M}(*D_{F}^{(r)})$,
$\mathcal{M}(!D_{F}^{(r)})$ and $\Psi_{F}(\mathcal{M})$ are $G$-equivariant
sheaves of abelian groups. Since the isomorphisms in (3.10) are not
$\mathscr{O}_{X\times\mathbb{C}^{r}}$-linear (as (3.3) is not
$\mathbb{C}[\mathbf{s}]$-linear), $\mathcal{M}(*D_{F}^{(r)})$,
$\mathcal{M}(!D_{F}^{(r)})$ and $\Psi_{F}(\mathcal{M})$ are not
$G$-equivariant $\widetilde{\mathscr{D}}_{X,R}$-modules. However, since $\pi$
is covering, the sheaf morphism

$\pi^{-1}(\mathscr{D}_{X\times(\mathbb{C}^{*})^{r}/(\mathbb{C}^{*})^{r}})\rightarrow\widetilde{\mathscr{D}}_{X,R}$

is an isomorphism locally and hence also a global isomorphism. In particular,
$\widetilde{\mathscr{D}}_{X,R}$ is G-equivariant (see [BL94, Part I.0.3]) and
its $G$-equivariance is induced from the deck transformation of the universal
covering of $(\mathbb{C}^{*})^{r}$. Equivalently, the $G$-equivariance of
$\widetilde{\mathscr{D}}_{X,R}$ is induced from the $G$-action on
$\mathbb{C}[\mathbf{s}]$ given by (3.1). Moreover, the $G$-equivariance of
$\mathcal{M}(*D_{F}^{(r)})$, $\mathcal{M}(!D_{F}^{(r)})$ and
$\Psi_{F}(\mathcal{M})$ is also induced from the $G$-action on
$\mathbb{C}[\mathbf{s}]$. Then, for $g\in G$,
$Q\in\pi_{*}(\widetilde{\mathscr{D}}_{X,R})$ and
$m\in\pi_{*}(\mathcal{M}(*D_{F}^{(r)}))$ (similarly for
$\mathcal{M}(!D_{F}^{(r)})$ and $\Psi_{F}(\mathcal{M})$), we have

$g\cdot(Q\cdot m)=(g\cdot Q)\cdot(g\cdot m).$

Therefore, the $G$-invariant parts $\pi^{G}_{*}(\mathcal{M}(*D_{F}^{(r)}))$
$\pi_{*}^{G}(\mathcal{M}(!D_{F}^{(r)}))$ and
$\pi_{*}^{G}(\Psi_{F}(\mathcal{M}))$ are
$\pi^{G}_{*}(\widetilde{\mathscr{D}}_{X,R})\simeq\mathscr{D}_{X\times(\mathbb{C}^{*})^{r}/(\mathbb{C}^{*})^{r}}$-modules.
Since $\pi$ is covering, they are automatically relative holonomic
$\mathscr{D}_{X\times(\mathbb{C}^{*})^{r}/(\mathbb{C}^{*})^{r}}$-modules.

(2) Since

(3.11)
$\mathbb{D}(\mathcal{M}(!D^{(r)}_{F}))\simeq(\mathbb{D}\mathcal{M})(*D^{(r)}_{F})\textup{
and
}\mathbb{D}(\mathcal{M}(*D^{(r)}_{F}))\simeq(\mathbb{D}\mathcal{M})(!D^{(r)}_{F}),$

$\mathbb{D}(\mathcal{M}(!D^{(r)}_{F}))$ (resp.
$\mathbb{D}(\mathcal{M}(*D^{(r)}_{F}))$) is also $G$-equivariant. Since
$\pi^{-1}\circ\pi^{G}_{*}\simeq\textup{id}$ (see [BL94, Part I.0.3]), we have

(3.12)
$\pi_{*}^{G}\mathbb{D}\big{(}\mathcal{M}(!D^{(r)}_{F})\big{)}\simeq\mathbb{D}\pi_{*}^{G}\big{(}\mathcal{M}(!D^{(r)}_{F})\big{)}$

and

(3.13)
$\pi_{*}^{G}\mathbb{D}\big{(}\mathcal{M}(*D^{(r)}_{F})\big{)}\simeq\mathbb{D}\pi_{*}^{G}\big{(}\mathcal{M}(*D^{(r)}_{F})\big{)}.$

Theorem 3.9 and (3.11) together imply:

######  Corollary 3.11.

For $\mathcal{M}$ a holonomic $\mathscr{D}_{X}$-module, we have a quasi-
isomorphism

$\mathbb{D}(\Psi_{F}(\mathcal{M}))\stackrel{{\scriptstyle
q.i.}}{{\simeq}}\Psi_{F}(\mathbb{D}\mathcal{M})[-1].$

Now, we discuss relative intermediate extensions and use them to study local
cohomology sheaves for $\mathcal{M}(!D_{F}^{(r)})$. For every subset
$I\subseteq\\{1,2,\dots,r\\}$, we denote

$D_{I}=\bigcap_{i\in I}D_{i}\textup{ and }D^{I}=\bigcup_{i\in I}D_{i}$

and

$\widetilde{D}_{I}=D_{I}\times\mathbb{C}^{r}\textup{ and
}\widetilde{D}^{I}=D^{I}\times\mathbb{C}^{r}.$

Then we define

$\mathcal{M}(!D_{F}^{(r)})(*\widetilde{D}^{I})\coloneqq\lim_{k\to+\infty}\mathcal{M}(!D_{F}^{(r)})\otimes_{\mathscr{O}}\mathscr{O}_{X\times\mathbb{C}^{r}}(k\widetilde{D}^{I}).$

If $J\subseteq I$, then similar to Theorem 3.8 we have a $G$-equivariant
inclusion

(3.14)
$\mathcal{M}(!D_{F}^{(r)})(*\widetilde{D}^{J})\hookrightarrow\mathcal{M}(!D_{F}^{(r)})(*\widetilde{D}^{I}).$

In fact, the inclusion in Theorem 3.8 factor through (3.14) when $J$ is empty.
Similar to Proposition 3.5 (see also [BVWZ21b, §5.3] for the algebraic case),

$\mathcal{M}(!D_{F}^{(r)})(*\widetilde{D}^{I})$

is $n$-Cohen-Macaulay. We thus define

$\mathcal{M}(*D_{F}^{(r)})(!\widetilde{D}^{I})\coloneqq\mathbb{D}\big{(}(\mathbb{D}\mathcal{M})(!D_{F}^{(r)})(*\widetilde{D}^{I})\big{)}.$

If $J\subseteq I$, then similar to (3.14) we have a $G$-equivariant inclusion

(3.15)
$\mathcal{M}(*D_{F}^{(r)})(!\widetilde{D}^{I})\hookrightarrow\mathcal{M}(*D_{F}^{(r)})(!\widetilde{D}^{J}).$

Furthermore, similar to (3.12) in Remark 3.10 (2), we have

(3.16)
$\pi_{*}^{G}\mathbb{D}\big{(}\mathcal{M}(!D_{F}^{(r)})(*\widetilde{D}^{J})\big{)}\simeq\mathbb{D}\pi_{*}^{G}\big{(}\mathcal{M}(!D_{F}^{(r)})(*\widetilde{D}^{J})\big{)}.$

By (3.14) and (3.15), for every subset $I$,
$\mathcal{M}(!D_{F}^{(r)})(*\widetilde{D}^{I})$ and
$\mathcal{M}(*D_{F}^{(r)})(!\widetilde{D}^{I})$ are _relative intermediate
extensions_ lying in between $\mathcal{M}(!D_{F}^{(r)})$ and
$\mathcal{M}(*D_{F}^{(r)})$.

######  Theorem 3.12.

Let $\mathcal{M}$ be a holonomic $\mathscr{D}_{X}$-module. Then for every
$I\subseteq\\{1,2,\dots,r\\}$, the complexes
$R\Gamma_{[\widetilde{D}_{I}]}\mathcal{M}(!D_{F}^{(r)})$ and
$\mathbb{D}R\Gamma_{[\widetilde{D}_{I}]}\big{(}(\mathbb{D}\mathcal{M})(!D_{F}^{(r)})\big{)}$
are $G$-equivariant.
Moreover,

$\pi_{*}^{G}R\Gamma_{[\widetilde{D}_{I}]}\big{(}\mathcal{M}(!D_{F}^{(r)})\big{)}\simeq
R\Gamma_{[D_{I}\times(\mathbb{C}^{*})^{r}]}\pi_{*}^{G}\big{(}\mathcal{M}(!D_{F}^{(r)})\big{)}$

and

$\pi_{*}^{G}\mathbb{D}R\Gamma_{[\widetilde{D}_{I}]}\big{(}(\mathbb{D}\mathcal{M})(!D_{F}^{(r)})\big{)}\simeq\mathbb{D}R\Gamma_{[D_{I}\times(\mathbb{C}^{*})^{r}]}\pi_{*}^{G}\big{(}(\mathbb{D}\mathcal{M})(!D_{F}^{(r)})\big{)}$

are complexes of relative holonomic
$\mathscr{D}_{X\times(\mathbb{C}^{*})^{r}/(\mathbb{C}^{*})^{r}}$-modules.

###### Proof.

By the definition of $R\Gamma_{[\widetilde{D}_{I}]}$,
$R\Gamma_{[\widetilde{D}_{I}]}\mathcal{M}(!D_{F}^{(r)})$ is represented by the
Koszul-type complex

$\displaystyle 0\to\mathcal{M}(!D_{F}^{(r)})\rightarrow\cdots$
$\displaystyle\rightarrow\bigoplus_{J\subseteq
I,|J|=i}\mathcal{M}(!D_{F}^{(r)})(*\widetilde{D}^{J})\rightarrow$
$\displaystyle\rightarrow\bigoplus_{J\subseteq
I,|J|=i+1}\mathcal{M}(!D_{F}^{(r)})(*\widetilde{D}^{J})\rightarrow\cdots\mathcal{M}(!D_{F}^{(r)})(*\widetilde{D}^{I})\to
0.$

with cohomogy degrees $0,1,2,\dots|I|$, and taking duality of the above
complex for $\mathbb{D}\mathcal{M}$,
$\mathbb{D}R\Gamma_{[\widetilde{D}_{I}]}\big{(}(\mathbb{D}\mathcal{M})(!D_{F}^{(r)})\big{)}$
is the complex

$\displaystyle
0\to\mathcal{M}(*D_{F}^{(r)})(!\widetilde{D}^{I})\rightarrow\cdots$
$\displaystyle\rightarrow\bigoplus_{J\subseteq
I,|J|=i}\mathcal{M}(*D_{F}^{(r)})(!\widetilde{D}^{J})\rightarrow$
$\displaystyle\rightarrow\bigoplus_{J\subseteq
I,|J|=i-1}\mathcal{M}(*D_{F}^{(r)})(!\widetilde{D}^{J})\rightarrow\cdots\mathcal{M}(*D_{F}^{(r)})\to
0.$

with cohomogy degrees $-|I|,-|I|+1,\dots,-1,0$. By using (3.14) and (3.15),
the two complexes as above are $G$-equivariant. The second required claim
follows from Remark 3.10 and (3.16). ∎

###  3.5. $G$-equivariance of relative de Rham complex

Let $\mathscr{E}$ be $G$-equivariant sheaf and a
$\mathscr{D}_{X\times\mathbb{C}^{r}/\mathbb{C}^{r}}$-module (see Remark 3.10).
For a differential operator $P\in\mathscr{D}_{X}$, since the $G$-action on
$\mathscr{E}$ is induced from the $G$-action on $\mathbb{C}^{r}$, we have

$g^{*}(P\cdot\mathscr{E})\simeq P\cdot(g^{*}\mathscr{E}),$

which means that the $P$-action on $\mathscr{E}$ is $G$-equivariant.

Now we fix a local coordinates $(x_{1},\dots,x_{n})$ of $X$ and denote by
$\partial_{x_{i}}$ the vector field of $x_{i}$. Then locally the relative de
Rham complex of $\mathscr{E}$ satisfying

(3.17)
$\textup{DR}_{X\times\mathbb{C}^{r}/\mathbb{C}^{r}}(\mathscr{E})\simeq\textup{Kos}(\mathscr{E};\partial_{x_{1}},\dots,\partial_{x_{r}}),$

where $\textup{Kos}(\mathscr{E};\partial_{x_{1}},\dots,\partial_{x_{r}})$ is
the Koszul complex of the $\partial_{x_{i}}$-action on $\mathscr{E}$. We thus
obtain:

######  Lemma 3.13.

Let $\mathscr{E}$ be a sheaf (or complexes) of
$\mathscr{D}_{X\times\mathbb{C}^{r}/\mathbb{C}^{r}})$-modules and
$G$-equivariant. Then the de Rham complex
$\textup{DR}_{X\times\mathbb{C}^{r}/\mathbb{C}^{r}}(\mathscr{E})$ is
$G$-equivariant and hence we have a natural isomorphism

$\pi_{*}^{G}\big{(}\textup{DR}_{X\times\mathbb{C}^{r}/\mathbb{C}^{r}}(\mathscr{E})\big{)}\simeq\textup{DR}_{X\times(\mathbb{C}^{*})^{r}/(\mathbb{C}^{*})^{r}}(\pi_{*}^{G}\mathscr{E}).$

###  3.6. Bernstein-Sato ideal along $F$

For every ${\bf a}=(a_{1},\dots,a_{r})\in\mathbb{Z}^{r}_{\geq 0}$, the
Bernstein-Sato ideal of $\mathscr{N}_{0}$ and ${\bf a}$ along $F$ is the ideal

$B_{F}(\mathscr{N}_{0},{\bf a})=B(\frac{\mathscr{N}_{0}}{g_{\bf
a}\cdot\mathscr{N}_{0}})\subseteq\mathbb{C}[\mathbf{s}].$

For simplicity, we write
$B_{F}(\mathscr{N}_{0})=B_{F}(\mathscr{N}_{0},{\mathbf{1}}_{r})$ with
${\mathbf{1}}_{r}=(1,1,\dots,1)\in\mathbb{Z}^{r}$. When
$\mathcal{M}_{0}=\mathscr{O}_{X}$, we further denote $B_{F}(\mathscr{N}_{0})$
by $B_{F}$ for short. Notice that $B_{F}$ is the major Bernstein-Sato ideal
studied in the literature; see for instance [Bud15, BVWZ21a, Mai16b].

Bernstein-Sato ideals along $F$ are natural generalization of the Bernstein-
Sato polynomial for a single holomorphic function (see [Kas77] and [Bud15,
§4.1]). By Theorem 3.1 (replacing $\mathscr{N}_{-1}$ by $g_{\bf
a}\cdot\mathscr{N}_{0}$), we have that, if $\mathcal{M}$ is holonomic, then
locally on a relative compact open subset $W\subseteq X$

$B_{F}(\mathscr{N}_{0},{\bf a})\not=0$ for all ${\bf a}\in\mathbb{Z}^{r}_{\geq
0}$ and ${\bf a}\not=\mathbf{0}$.

The following theorem is essentially due to Maisonobe [Mai16a]; here we
present it in a more generalized form.444In _loc. cit._ the author only
considered the cyclic modules $\mathscr{D}_{X}[\mathbf{s}]m\cdot
F^{\mathbf{s}}$ for sections $m$ of $\mathcal{M}$ (instead of
$\mathscr{N}_{0}$). See also [BVWZ21a] for $\mathcal{M}_{0}=\mathscr{O}_{X}$
under the algebraic setting.

######  Theorem 3.14.

Let $\mathcal{M}$ be a regular holonomic $\mathscr{D}_{X}$-module. Locally on
a relative compact open subset $W\subseteq X$, if
$\mathscr{N}_{0}/\mathscr{N}_{-1}$ is not zero on $W$ $(\Leftrightarrow W\cap
D\not=\emptyset)$, then $Z(B_{F}(\mathscr{N}_{0}))\subseteq\mathbb{C}^{r}$ is
of codimension-one $($but not purely in general $)$.

###### Proof.

If $W\cap D=\emptyset$, then obviously $\mathscr{N}_{0}=\mathscr{N}_{-1}$. We
now assume $W\cap D\not=0$ and then prove
$\mathscr{N}_{0}/\mathscr{N}_{-1}\not=0$ on $W$.

To this purpose, we need to treat $\mathscr{N}_{k}$ as a logarithmic
$\mathscr{D}$-module. Using the graph embedding (see [Wu20, §5]), the
inclusion

$\iota_{F,*}(\mathscr{N}_{0})\hookrightarrow\iota_{F,*}(\mathscr{M})\simeq\iota_{F,+}(\mathcal{M}(*D))$

makes $\iota_{F,*}(\mathscr{N}_{0})$ a lattice of the regular holonomic
$\mathscr{D}_{Y}$-module $\iota_{F,+}(\mathcal{M}(*D))$, where

$\iota_{F}\colon X\hookrightarrow Y=X\times\mathbb{C}^{r}_{\mathbf{u}},\quad
x\mapsto(x,f_{1}(x),\dots,f_{r}(x))$

is the graph embedding with ${\mathbf{u}}=(u_{1},\dots,u_{r})$ the coordinates
and $\iota_{F,+}$ denotes the $\mathscr{D}$-module pushforward functor of
$\iota_{F}$.

Since $\iota_{F}$ is a closed embedding, we have

$\frac{\mathscr{N}_{0}}{\mathscr{N}_{-1}}\not=0\Leftrightarrow\frac{\iota_{F,*}(\mathscr{N}_{0})}{\iota_{F,*}(\mathscr{N}_{-1})}\not=0.$

By the construction of $\iota_{F,+}$, we have

$\prod_{i=1}^{r}u_{i}\cdot\iota_{F,*}(\mathscr{N}_{0})=\iota_{F,*}(\mathscr{N}_{-1}).$

Therefore, we have a short exact sequence

$0\to\iota_{F,*}(\mathscr{N}_{0})\xrightarrow{\prod_{i=1}^{r}u_{i}}\iota_{F,*}(\mathscr{N}_{0})\longrightarrow\frac{\iota_{F,*}(\mathscr{N}_{0})}{\iota_{F,*}(\mathscr{N}_{-1})}\to
0.$

For simplicity, we write

$\mathscr{G}=\iota_{F,*}(\mathscr{N}_{0}),\quad\mathscr{H}=\frac{\iota_{F,*}(\mathscr{N}_{0})}{\iota_{F,*}(\mathscr{N}_{-1})}\quad\textup{
and }\quad v=\prod_{i=1}^{r}u_{i}.$

Since $\mathscr{G}$ is a lattice along the normal crossing divisor
$D_{Y}=(\prod_{i=1}^{r}u_{i}=0)$, by [Wu20, Theorem 1.6] the logarithmic
characteristic variety satisfies

$\textup{Ch}^{\log}(\mathscr{G}))=\overline{\widetilde{\iota}_{F}(\textup{Ch}(\mathcal{M}|_{X\setminus
D}))},$

where $\widetilde{\iota}_{F}$ is the induced embedding

$\widetilde{\iota}_{F}\colon T^{*}(X\setminus D)\hookrightarrow
T^{*}(Y\setminus D_{Y})\simeq T^{*}(X\setminus D)\times
T^{*}\mathbb{C}^{r}_{\mathbf{u}},\quad(x,\xi)\mapsto(x,\xi,F(x),dF(x))$

and the closure is taken inside the logarithmic cotangent bundle
$T^{*}(Y,D_{Y})$. In particular, $\textup{Ch}^{\log}(\mathscr{G})$ has no
component over $(v=0)$.

We now take a good filtration $F_{\bullet}\mathscr{G}$ over
$F_{\bullet}\mathscr{D}_{Y,D_{Y}}$ (cf. [Wu20, §3]). We thus have a filtered
morphism

$F_{\bullet}\mathscr{G}\xrightarrow{v}F_{\bullet}\mathscr{G}$

with the associated graded morphism

$\varphi\colon\textup{gr}^{F}_{\bullet}\mathscr{G}\xrightarrow{v}\textup{gr}^{F}_{\bullet}\mathscr{G}.$

Since the filtration is bounded from below, the filtered morphism gives a
convergent spectral sequence. By convergence (see for instance [Lau83, Lemma
3.5.13 (iii)]), we have

$[\textup{gr}^{F}_{\bullet}\mathscr{H}]=[\textup{Coker}\varphi]-[\ker\varphi]$

in the Grothendieck group $K_{0}$ generated by
$\textup{gr}^{F}_{\bullet}\mathscr{D}_{Y,D_{Y}}$-modules with support of
dimension $\leq n+r-1$, where $F_{\bullet}\mathscr{H}$ is the induced
filtration on the quotient. But one can check in $K_{0}$

$[\textup{Coker}\varphi]-[\ker\varphi]=[(\textup{gr}^{F}_{\bullet}\mathscr{G}/\mathcal{T})|_{(v=0)}]$

where $\mathcal{T}\subseteq\textup{gr}^{F}_{\bullet}\mathscr{G}$ is the
$v$-torsion subsheaf. Since $\textup{gr}^{F}_{\bullet}\mathscr{G}/\mathcal{T}$
has no $v$-torsion, we have

$\textup{supp}_{\textup{gr}_{\bullet}^{F}\mathscr{D}_{Y,D_{Y}}}((\textup{gr}^{F}_{\bullet}\mathscr{G}/\mathcal{T})|_{(v=0)})=\textup{supp}_{\textup{gr}_{\bullet}^{F}\mathscr{D}_{Y,D_{Y}}}(\textup{gr}^{F}_{\bullet}\mathscr{G}/\mathcal{T})|_{(v=0)}.$

Since

$\textup{Ch}^{\log}(\mathscr{G})=\textup{supp}_{\textup{gr}_{\bullet}^{F}\mathscr{D}_{Y,D_{Y}}}\textup{gr}^{F}_{\bullet}\mathscr{G}$

and $\textup{Ch}^{\log}(\mathscr{G})$ does not have component over $(v=0)$, we
have

$\textup{supp}_{\textup{gr}_{\bullet}^{F}\mathscr{D}_{Y,D_{Y}}}\textup{gr}^{F}_{\bullet}\mathscr{G}=\textup{supp}_{\textup{gr}_{\bullet}^{F}\mathscr{D}_{Y,D_{Y}}}(\textup{gr}^{F}_{\bullet}\mathscr{G}/\mathcal{T}).$

By [Gin86, Proposition 1.5.1], we conclude

$\textup{Ch}^{\log}_{n+r-1}(\mathscr{H})=\textup{Ch}^{\log}(\mathscr{G})|_{v=0}$

where $\textup{Ch}^{\log}_{n+r-1}(\mathscr{H})$ is the purely
$(n+r-1)$-dimensional part of the characteristic variety. But by construction,

$\textup{Ch}^{\log}(\mathscr{H})\subseteq\textup{Ch}^{\log}(\mathscr{G})|_{v=0}$

and hence

$\textup{Ch}^{\log}(\mathscr{H})=\textup{Ch}^{\log}(\mathscr{G})|_{v=0}.$

In fact, the above argument proves more generally

$\textup{CC}^{\log}(\mathscr{H})=\textup{CC}^{\log}(\mathscr{G})|_{v=0}.$

In particular, $\mathscr{H}\not=0$ and hence so is
$\mathscr{N}_{0}/\mathscr{N}_{-1}$. Then
$\mathscr{N}_{k}/\mathscr{N}_{-k}\not=0$ for all $k>0$. Since for $k\gg 0$
$\widetilde{\mathscr{N}}_{k}/\widetilde{\mathscr{N}}_{-k}$ is just
$\Psi_{F}(\mathcal{M})$ analytically locally over $\mathbb{C}^{r}$, we now
conclude that $\Psi_{F}(\mathcal{M})$ is $(n+1)$-Cohen-Macaulay by Theorem
3.9. By Lemma 2.14, this also implies

$\dim_{\mathbb{C}}(\textup{Ch}^{\textup{rel}}(\widetilde{\mathscr{N}}_{k}/\widetilde{\mathscr{N}}_{-k}))=n+r-1.$

Thus, by [Wu20, Lemma 2.3]

$\dim_{\mathbb{C}}(\textup{Ch}^{\textup{rel}}(\widetilde{\mathscr{N}}_{l}/\widetilde{\mathscr{N}}_{l-1}))=n+r-1$

for some $-k<l\leq k$. But by translation,

$\mathscr{N}_{l}/\mathscr{N}_{l-1}\simeq\mathscr{N}_{0}/\mathscr{N}_{-1}$

and hence

$\dim_{\mathbb{C}}(\textup{Ch}^{\textup{rel}}(\widetilde{\mathscr{N}}_{0}/\widetilde{\mathscr{N}}_{-1}))=n+r-1.$

The proof is then accomplished by Lemma 2.5 and Proposition 2.8. ∎

The theorem above and Theorem 3.1 together implies:

######  Corollary 3.15.

Let $\mathcal{M}$ be a regular holonomic $\mathscr{D}_{X}$-module. Locally on
a relative compact open subset $W\subseteq X$ intersecting $D$,
$Z_{r-1}(B_{F}(\mathscr{N}_{0}))$ is a finite union of translated hyperplanes
of the form $(L\cdot\mathbf{s}+\alpha=0)$ with $L\in\mathbb{Z}^{r}_{\geq 0}$
and $\alpha\in\mathbb{C}$, where $Z_{r-1}(B_{F}(\mathscr{N}_{0}))$ denotes the
pure $(r-1)$-dimensional part of $Z(B_{F}(\mathscr{N}_{0}))$.

The corollary above enables us to make the following definition:

######  Definition 3.16.

For $\mathcal{M}$ a regular holonomic $\mathscr{D}_{X}$-module, locally on a
relative compact open subset $W\subseteq X$, we define

$S(F,\mathcal{M})=\\{\textup{primitive
}L\in\mathbb{Z}^{r}\mid(L\cdot\mathbf{s}+\alpha=0)\subseteq
Z_{r-1}(B_{F}(\mathscr{N}_{0})),\textup{ for some }\alpha\in\mathbb{C}\\}$

and that for $L\in S(F,\mathcal{M})$

$\kappa(L)=\\{\alpha\in\mathbb{C}\mid(L\cdot\mathbf{s}+\alpha=0)\subseteq
Z_{r-1}(B_{F}(\mathscr{N}_{0}))\\}.$

We set

$\widetilde{\kappa}(L)=\kappa(L)/\sim_{L}$

where $\sim_{L}$ denotes the equivalence

$\alpha_{1}\sim_{L}\alpha_{2}\Longleftrightarrow\alpha_{2}=L\cdot{\bf
a}+\alpha_{1}\textup{ for some }{\bf a}\in\mathbb{Z}^{r}.$

Since $\mathscr{N}_{0}$ is depending on $\mathcal{M}_{0}$,
$B_{F}(\mathscr{N}_{0})$ is depending on $\mathcal{M}_{0}$. However, one can
easily check that

$S(F,\mathcal{M})$ is independent of the choices of $\mathcal{M}_{0}$.

See [Mai16a, Résultat 6] for a logarithmic interpretation of
$S(F,\mathcal{M})$ and also [BVWZ21b, Proposition 4.4.4].

###### Proof of Theorem 1.4.

For $p\in Z_{r-1}(B_{F}(\mathscr{N}_{0}))$ and for every $g\in G$, by
Proposition 2.8 we know

$(\frac{g\cdot\mathscr{N}_{0}}{g\cdot\mathscr{N}_{-1}})_{m_{g\cdot p}}\not=0,$

where $m_{g\cdot p}$ is the maximal ideal of $p$. We take $k\gg 0$ such that
$g\cdot\mathscr{N}_{0}/g\cdot\mathscr{N}_{-1}$ is a subquotient of
$\mathscr{N}_{k}/\mathscr{N}_{-k}$. Then

$(\frac{\mathscr{N}_{k}}{\mathscr{N}_{-k}})_{m_{g\cdot p}}\not=0.$

By Lemma 2.6, we then have for some $x\in X$

$(\frac{\widetilde{\mathscr{N}}_{k}}{\widetilde{\mathscr{N}}_{-k}})_{(x,g\cdot
p)}\not=0.$

But by Proposition 3.2 (1) and (3.8),
$\widetilde{\mathscr{N}}_{k}/\widetilde{\mathscr{N}}_{-k}$ is
$\Psi_{F}(\mathcal{M})$ locally around $(x,g\cdot p)$. We thus have

$\bigcup_{g\in G}\bigcup_{L\in
S(F,\mathcal{M})}\bigcup_{\alpha\in\kappa(L)}(g\cdot(L\cdot\mathbf{s}+\alpha)=0)\subseteq\textup{supp}_{\mathbb{C}^{r}}\Psi_{F}(\mathcal{M}).$

Conversely, we assume $\Phi_{F}(\mathcal{M})_{(x,p)}\not=0$ for some $x\in X$
and for $p\in\mathbb{C}^{r}$ away from

$\bigcup_{g\in G}\bigcup_{L\in
S(F,\mathcal{M})}\bigcup_{\alpha\in\kappa(L)}(g\cdot(L\cdot\mathbf{s}+\alpha)=0).$

Then for some $k\gg 0$

$\Phi_{F}(\mathcal{M})|_{W\times
V}=\frac{\widetilde{\mathscr{N}}_{k}}{\widetilde{\mathscr{N}}_{-k}}|_{W\times
V}\not=0$

with $W$ a small neighborhood of $x$ and $V$ a small neighborhood of $p$.
Since $\Phi_{F}(\mathcal{M})$ is $(n+1)$-Cohen-Macaulay, by Lemma 2.14 and
Proposition 2.19, we know

$\textup{supp}_{\mathbb{C}^{r}}\Phi_{F}(\mathcal{M})|_{W\times V}$

is purely of codimension one. But by Theorem 3.1 and Proposition 2.8, we know

$\textup{supp}_{\mathbb{C}^{r}}\frac{\mathscr{N}_{k}}{\mathscr{N}_{-k}}$

is contained in a union of translated linear hyperplanes in $\mathbb{C}^{r}$
of forms $(L\cdot\mathbf{s}+\alpha=0)$ with $L\in\mathbb{Z}^{r}_{\geq 0}$.
Therefore, we can assume $(L\cdot\mathbf{s}+\alpha=0)\cap V\ni p$ is a
component of $\textup{supp}_{\mathbb{C}^{r}}\Phi_{F}(\mathcal{M})|_{W\times
V}$. Then $(L\cdot\mathbf{s}+\alpha=0)$ must be a component of
$\textup{supp}_{\mathbb{C}^{r}}\mathscr{N}_{l}/\mathscr{N}_{l-1}$ for some
$-k<l\leq k$. But this means

$p\in\bigcup_{g\in G}\bigcup_{L\in
S(F,\mathcal{M})}\bigcup_{\alpha\in\kappa(L)}(g\cdot(L\cdot\mathbf{s}+\alpha)=0),$

which is a contradiction. ∎

###  3.7. Evaluating $\Psi_{F}(\mathcal{M})$

In this subsection, we discuss “evaluating” $\mathcal{M}(*D_{F}^{(r)})$,
$\mathcal{M}(!D_{F}^{(r)})$ and $\Psi_{F}(\mathcal{M})$ at points in
$\mathbb{C}^{r}$ and applying them to prove Theorem 1.6. We first introduce
some notations.

For ${\boldsymbol{\alpha}}=(\alpha_{1},\dots,\alpha_{r})\in\mathbb{C}^{r}$ and
for $i\in\\{1,\dots,r\\}$, we write

$\hat{\boldsymbol{\alpha}}_{i}=(\alpha_{1},\dots,\alpha_{i-1},0,\alpha_{i+1},\dots,\alpha_{r}).$

We also write by

$i_{{\boldsymbol{\alpha}}}\colon\\{{\boldsymbol{\alpha}}\\}\hookrightarrow\mathbb{C}^{r}.$

the closed embedding and set

$\delta_{\hat{\boldsymbol{\alpha}}_{i}}\colon\mathbb{C}\hookrightarrow\mathbb{C}^{r},\quad
x\mapsto(\alpha_{1},\dots,\alpha_{i-1},x,\alpha_{i+1},\dots,\alpha_{r}).$

For ${\boldsymbol{\alpha}}=(\alpha_{1},\dots,\alpha_{r})\in\mathbb{C}^{r}$,
the multivalued holomorphic function $F^{\boldsymbol{\alpha}}$ determines a
rank one local system $L_{\boldsymbol{\lambda}}$ on $X\setminus D$, that is,
the local system has local monodromy along each $D_{i}$ given by
multiplication by $\lambda_{i}$ with

$\lambda_{i}=exp(-2\pi\sqrt{-1}\alpha_{i}).$

By Riemann-Hilbert correspondence, $Rj_{*}L_{\boldsymbol{\alpha}}$ determines
a regular holonomic $\mathscr{D}_{X}$-module, denoted by
$\mathcal{V}_{\boldsymbol{\alpha}}$, where $j\colon X\setminus
D\hookrightarrow X$ the open embedding. We write

$\mathcal{M}_{\boldsymbol{\alpha}}=\mathcal{M}\otimes_{\mathscr{O}}\mathcal{V}_{\boldsymbol{\alpha}}.$

######  Proposition 3.17.

Let $\mathcal{M}$ be a holonomic $\mathscr{D}_{X}$-module. With notations as
above, for each ${\boldsymbol{\alpha}}\in\mathbb{C}^{r}$ and for each $i$, we
have

$\mathbf{L}\widetilde{i}_{{\boldsymbol{\alpha}}}^{*}(\mathcal{M}(*D_{F}^{(r)}))\stackrel{{\scriptstyle
q.i.}}{{\simeq}}\mathcal{M}_{\boldsymbol{\alpha}}(*D)\quad\textup{and}\quad\widetilde{\mathbf{L}}i_{{\boldsymbol{\alpha}}}^{*}(\mathcal{M}(!D_{F}^{(r)}))\stackrel{{\scriptstyle
q.i.}}{{\simeq}}\mathcal{M}_{\boldsymbol{\alpha}}(!D)$

and for each $0<i\leq r$

$\mathbf{L}\widetilde{\delta}_{\hat{\boldsymbol{\alpha}}_{i}}^{*}(\mathcal{M}(*D_{F}^{(r)}))\stackrel{{\scriptstyle
q.i.}}{{\simeq}}\mathcal{M}_{\hat{\boldsymbol{\alpha}}_{i}}(*D_{f_{i}}^{(1)})\quad\textup{
and
}\quad\mathbf{L}\widetilde{\delta}_{\hat{\boldsymbol{\alpha}}_{i}}^{*}(\mathcal{M}(!D_{F}^{(r)}))\stackrel{{\scriptstyle
q.i.}}{{\simeq}}\mathcal{M}_{\hat{\boldsymbol{\alpha}}_{i}}(!D_{f_{i}}^{(1)}).$

###### Proof.

By construction, $\mathscr{M}$ is flat over $R\simeq\mathbb{C}[\mathbf{s}]$.
Then

$\mathbf{L}\widetilde{i}_{{\boldsymbol{\alpha}}}^{*}(\mathcal{M}(*D_{F}^{(r)}))\stackrel{{\scriptstyle
q.i.}}{{\simeq}}\widetilde{i}_{{\boldsymbol{\alpha}}}^{*}(\mathcal{M}(*D_{F}^{(r)}))=\mathscr{M}\otimes_{R}\mathbb{C}_{\boldsymbol{\alpha}}=\mathcal{M}_{\boldsymbol{\alpha}}(*D),$

where $\mathbb{C}_{\boldsymbol{\alpha}}$ is the residue field of
${\boldsymbol{\alpha}}\in\mathbb{C}^{r}$. By using Lemma 2.12 and Cohen-
Macaulayness,

$\mathbf{L}\widetilde{i}_{{\boldsymbol{\alpha}}}^{*}(\mathcal{M}(!D_{F}^{(r)}))\stackrel{{\scriptstyle
q.i.}}{{\simeq}}\widetilde{i}_{{\boldsymbol{\alpha}}}^{*}(\mathcal{M}(!D_{F}^{(r)}))=\mathcal{M}_{\boldsymbol{\alpha}}(!D)$

follows similarly.

The other required statements for $i_{\hat{\boldsymbol{\alpha}}_{i}}$ can be
checked in a similar way. We leave details for interested readers. ∎

The proposition above has an immediate corollary:

######  Corollary 3.18.

Let $\mathcal{M}$ be a holonomic $\mathscr{D}_{X}$-module. Then for each
${\boldsymbol{\alpha}}\in\mathbb{C}^{r}$ and for each $i$, we have

$\mathbf{L}\widetilde{\delta}_{\hat{\boldsymbol{\alpha}}_{i}}^{*}\Psi_{F}(\mathcal{M})\stackrel{{\scriptstyle
q.i.}}{{\simeq}}\Psi_{f_{i}}(\mathcal{M}_{\hat{\boldsymbol{\alpha}}_{i}}).$

###### Proof of Theorem 1.6.

The relative characteristic cycle formula for $\mathcal{M}(*D_{F}^{(r)})$ has
been obtained in Theorem 3.3 even if $\mathcal{M}$ is only holonomic. By
(3.8), we also get the relative characteristic cycle of
$\mathcal{M}(!D_{F}^{(r)})$.

We now prove the relative characteristic cycle formula for
$\Psi_{F}(\mathcal{M})$. For $L\in S(F,\mathcal{M})$ and $\alpha\in\kappa(L)$,
we fix a relative compact open subset $V\subseteq\mathbb{C}^{r}$ intersecting
$(L\cdot\mathbf{s}+\alpha=0)$. By Theorem 1.4 and Proposition 2.5, we can
assume that

$\Lambda_{L,\alpha}\times(L\cdot\mathbf{s}+\alpha=0)|_{V}$

is the part of $\textup{CC}^{\textup{rel}}(\Psi_{F}(\mathcal{M}))|_{W\times
V}$ over $(L\cdot\mathbf{s}+\alpha=0)$, where $\Lambda_{L,\alpha}$ is a conic
Lagrangian cycle supported on $\textup{Ch}(\mathcal{M}(*D))$ over $D\cap W$.
Using the “gluing” date of $\mathcal{M}(*D_{F}^{(r)})$ and
$\mathcal{M}(!D_{F}^{(r)})$ in §3.4, we can move
$\Lambda_{L,\alpha}\times(L\cdot\mathbf{s}+\alpha=0)|_{V}$ along
$(L\cdot\mathbf{s}+\alpha=0)$ as we translate $V$ along
$(L\cdot\mathbf{s}+\alpha=0)$ such that
$\Lambda_{L,\alpha}\times(L\cdot\mathbf{s}+\alpha=0)|_{V}$ is always the part
of $\textup{CC}^{\textup{rel}}(\Psi_{F}(\mathcal{M}))|_{W\times V}$ over
$(L\cdot\mathbf{s}+\alpha=0)$. Therefore,

$\Lambda_{L,\alpha}\times(L\cdot\mathbf{s}+\alpha=0)$

is the part of
$\textup{CC}^{\textup{rel}}(\Psi_{F}(\mathcal{M}))|_{W\times\mathbb{C}^{r}}$
over $(L\cdot\mathbf{s}+\alpha=0)$.

For every $g\in G$, using the isomorphism

$G\simeq\mathbb{Z}^{r}$

we can assume $g=g_{\bf a}$ for some ${\bf a}\in\mathbb{Z}^{r}$. Since for
every $k>0$

$\tau^{*}_{\bf
a}(\widetilde{\mathscr{N}_{k}/\mathscr{N}_{-k}})\simeq\widetilde{\mathscr{N}_{k}/\mathscr{N}_{-k}},$

the part of
$\textup{CC}^{\textup{rel}}(\Psi_{F}(\mathcal{M}))|_{W\times\mathbb{C}^{r}}$
over $g\cdot(L\cdot\mathbf{s}+\alpha=0)$ is

$\Lambda_{L,\alpha}\times(L\cdot\mathbf{s}+\alpha=0).$

Therefore,

$\textup{CC}^{\textup{rel}}(\Psi_{F}(\mathcal{M}))=\sum_{g\in G}\sum_{L\in
S(F,\mathcal{M})}\sum_{\alpha\in\tilde{\kappa}(L)}\Lambda_{L,\alpha}\times(g\cdot(L\cdot
s+\alpha)=0).$

To avoid counting components repeatedly, the summation above is taken over
$\widetilde{\kappa}(L)$ (instead of $\kappa(L)$). ∎

We now give a formula for $\Lambda_{L,\alpha}$.

######  Proposition 3.19.

In the situation of Theorem 1.6, for each $L\in S(F,\mathcal{M})$ and for each
$\alpha\in\widetilde{\kappa}(L)$, then after picking a very general point

${\boldsymbol{\alpha}}=(\alpha_{1},\dots,\alpha_{r})\in(L\cdot\mathbf{s}+\alpha=0)$

for each $i\in\\{1,\dots,r\\}$ we have

$\Lambda_{L,\alpha}=\textup{CC}(\Psi_{f_{i}}(\mathcal{M}_{\hat{\boldsymbol{\alpha}}_{i}})|_{W\times
V_{\alpha_{i}}})$

where $V_{\alpha_{i}}$ is a sufficiently small neighborhood of
$\alpha_{i}\in\mathbb{C}$.

###### Proof.

Let us first explain the notation

$\textup{CC}(\Psi_{f_{i}}(\mathcal{M}_{\hat{\boldsymbol{\alpha}}_{i}})|_{W\times
V_{\alpha_{i}}}).$

In the case $r=1$, $\Psi_{f_{i}}(\mathcal{M}_{\hat{\boldsymbol{\alpha}}_{i}})$
is a relative holonomic $\mathscr{D}$-module over the complex line
$\mathbb{C}$. In this case,
$S(f_{i},\mathcal{M}_{\hat{\boldsymbol{\alpha}}_{i}})=\\{1\\}$. By Theorem
1.4, we then know $\Psi_{f_{i}}(\mathcal{M}_{\hat{\boldsymbol{\alpha}}_{i}}$)
is supported over an infinite discrete subset of $\mathbb{C}$, where the
discrete subset containing $\alpha_{i}$ (by Corollary 3.18). With
$V_{\alpha_{i}}$ small enough,
$\Psi_{f_{i}}(\mathcal{M}_{\hat{\boldsymbol{\alpha}}_{i}})|_{W\times
V_{\alpha_{i}}}$ is just supported over $\alpha_{i}$. More explicitly, by
construction
$\Psi_{f_{i}}(\mathcal{M}_{\hat{\boldsymbol{\alpha}}_{i}})|_{W\times
V_{\alpha_{i}}}$ is just

$\frac{(\mathscr{N}^{\hat{\boldsymbol{\alpha}}_{i}}_{k})_{m_{\alpha_{i}}}}{(\mathscr{N}^{\hat{\boldsymbol{\alpha}}_{i}}_{-k})_{m_{\alpha_{i}}}}$

for $k\gg 0$ (since both of them are killed by $(s+\alpha_{i})^{l}$ for some
$l>0$, sheafification does nothing), where
$\mathscr{N}^{\hat{\boldsymbol{\alpha}}_{i}}_{k}$ is the
$\mathscr{D}_{X,\mathbb{C}[s_{i}]}$-module $\mathscr{N}_{k}$ for
$\mathcal{M}_{\hat{\boldsymbol{\alpha}}_{i}}$ as in §3.1, and $m_{\alpha_{i}}$
is the maximal ideal of $\alpha_{i}\in\mathbb{C}$. But the latter module is
particularly a $\mathscr{D}_{X}$-module. Thus, by Lemma 2.5

(3.18)
$\textup{CC}^{{\textup{rel}}}(\Psi_{f_{i}}(\mathcal{M}_{\hat{\boldsymbol{\alpha}}_{i}})|_{W\times
V_{\alpha_{i}}})=\textup{CC}(\Psi_{f_{i}}(\mathcal{M}_{\hat{\boldsymbol{\alpha}}_{i}})|_{W\times
V_{\alpha_{i}}})\times\\{\alpha_{i}\\}.$

As a consequence,
$\Psi_{f_{i}}(\mathcal{M}_{\hat{\boldsymbol{\alpha}}_{i}})|_{W\times
V_{\alpha_{i}}}$ is a holonomic $\mathscr{D}_{X}$-module (in fact, it is also
regular if so is $\mathcal{M}$). Moreover, if we use the terminology in
[Kas83, Mal83, Wu21],
$\Psi_{f_{i}}(\mathcal{M}_{\hat{\boldsymbol{\alpha}}_{i}})|_{W\times
V_{\alpha_{i}}}$ is the $\alpha_{i}$-nearby cycle of
$\mathcal{M}_{\hat{\boldsymbol{\alpha}}_{i}}$.

We now prove the required identity. By the proof of Theorem 1.6, to obtain
$\Lambda_{L,\alpha}$ it suffices to consider a small neighborhood $V$
intersecting $(L\cdot\mathbf{s}+\alpha=0)$. By the definition of
characteristic cycles, we only need to consider a small neighborhood of a
general point ${\boldsymbol{\alpha}}\in V\cap(L\cdot\mathbf{s}+\alpha=0)$,
which is equivalent to globally considering a small neighborhood
$V_{{\boldsymbol{\alpha}}}$ of a very general point
${\boldsymbol{\alpha}}\in(L\cdot\mathbf{s}+\alpha=0)$. By Corollary 3.18,

$\widetilde{\delta}_{\hat{\boldsymbol{\alpha}}_{i}}^{*}(\Psi_{F}(\mathcal{M}))=\Psi_{f_{i}}(\mathcal{M}_{\hat{\boldsymbol{\alpha}}_{i}}).$

By Cohen-Macaulayness and Proposition 2.19, we can inductively apply [Wu20,
Proposition 2.7] (by picking $r-1$ general hyperplanes passing
${\boldsymbol{\alpha}}$) and conclude

(3.19)
$\widetilde{\delta}_{\hat{\boldsymbol{\alpha}}_{i}}^{*}(\textup{CC}^{\textup{rel}}(\Psi_{F}(\mathcal{M})))=\textup{CC}^{\textup{rel}}(\widetilde{\delta}_{\hat{\boldsymbol{\alpha}}_{i}}^{*}(\Psi_{F}(\mathcal{M})))=\textup{CC}^{\textup{rel}}(\Psi_{f_{i}}(\mathcal{M}_{\hat{\boldsymbol{\alpha}}_{i}})).$

By the very general choice of ${\boldsymbol{\alpha}}$, we have

$\textup{CC}^{\textup{rel}}(\Psi_{F}(\mathcal{M}))|_{W\times
V_{{\boldsymbol{\alpha}}}}=\Lambda_{L,\alpha}\times(L\cdot\mathbf{s}+\alpha=0),$

Taking
$V_{\alpha_{i}}=\delta_{\hat{\boldsymbol{\alpha}}_{i}}^{-1}(V_{{\boldsymbol{\alpha}}})$,
we finish the proof by (3.18) and (3.19). ∎

###  3.8. A lineariality conjecture

Let $\mathcal{M}$ be a regular holonomic $\mathscr{D}_{X}$-module. By
Proposition 3.5, the maximal extension $\mathcal{M}(*D_{F}^{(r)})$ is
$n$-Cohen-Macaulay. On the contrary, for some $k$ (and hence for all $k$ by
applying $\tau$-translation) $\widetilde{\mathscr{N}}_{k}$ are not $n$-Cohen-
Macaulay in general. Thus, we might have non-zero

$\mathscr{E}xt^{l}_{\widetilde{\mathscr{D}}_{X,R}}(\widetilde{\mathscr{N}}_{k},\widetilde{\mathscr{D}}_{X,R})\textup{
for }n\leq l\leq n+r$

and hence non-zero right $\mathscr{D}_{X,R}$-modules

$\mathscr{E}xt^{l}_{\mathscr{D}_{X,R}}(\mathscr{N}_{k},\mathscr{D}_{X,R})\textup{
for }n\leq l\leq n+r.$

Motivated by Corollary 3.15 and the relative condimension filtration in §2.8,
we conjecture:

######  Conjecture 3.20.

Let $\mathcal{M}$ be a regular holonomic $\mathscr{D}_{X}$-module. If locally
on a relative compact open subset $W\subseteq X$

$\mathscr{E}xt^{l}_{\mathscr{D}_{X,R}}(\mathscr{N}_{k},\mathscr{D}_{X,R})\not=0\textup{
for some }n\leq l\leq n+r,$

then we have:

  1. (i)

If
$S_{j}(\mathscr{E}xt^{l}_{\mathscr{D}_{X,R}}(\mathscr{N}_{k},\mathscr{D}_{X,R}))\not=\emptyset$,
then it is a finite union of translated $(j-n)$-codimensional linear subspaces
of $\mathbb{C}^{r}$.

  2. (ii)

If $\mathcal{M}$ underlies a $\mathbb{Q}$-mixed Hodge module $($for example
$\mathcal{M}=\mathscr{O}_{X})$, then all
$S_{j}(\mathscr{E}xt^{l}_{\mathscr{D}_{X,R}}(\mathscr{N}_{k},\mathscr{D}_{X,R}))$
are defined over $\mathbb{Q}$.

By Auslander regularity, Lemma 2.14 and Theorem 3.3, we have

$\textup{Ch}^{\textup{rel}}(\mathscr{E}xt^{n}_{\widetilde{\mathscr{D}}_{X,R}}(\widetilde{\mathscr{N}}_{k},\widetilde{\mathscr{D}}_{X,R}))=\textup{Ch}^{\textup{rel}}(\widetilde{\mathscr{N}}_{k})=\textup{Ch}(\mathcal{M}(*D))\times\mathbb{C}^{r}$

and that
$\mathscr{E}xt^{n}_{\mathscr{D}_{X,R}}(\mathscr{N}_{k},\mathscr{D}_{X,R})$ is
pure of codimension $n$. Thus, the conjecture above holds for $l=n$.

When $\mathcal{M}_{0}=\mathscr{O}_{X}$, Budur made a conjecture about the
structure of generators of the Bernstein-Sato ideal $B_{F}$ [Bud15, Conjecture
1.1]. The following is a weak version of the conjecture.

######  Conjecture 3.21 (Budur).

Each irreducible component of $Z(B_{F})$ is a translated linear subspace of
$\mathbb{C}^{r}$ defined over $\mathbb{Q}$.

By Sabbah [Sab87b] (or one can apply Corollary 3.15) and [Gyo93], codimension-
one irreducible components of $Z(B_{F})$ are translated linear subspaces of
$\mathbb{C}^{r}$ defined over $\mathbb{Q}$.

######  Proposition 3.22.

Conjecture 3.20 $\Longrightarrow$ Conjecture 3.21.

###### Proof.

We assume $E$ an irrducible component of $Z(B_{F})$ of codimension $2\leq
l\leq r$. We fix a point $(x,p)$ such that
$(\widetilde{\mathscr{N}}_{0}/\widetilde{\mathscr{N}}_{-1})_{(x,p)}\not=0$ (by
Proposition 2.8 and
$\widetilde{\mathscr{N}}_{0}/\widetilde{\mathscr{N}}_{-1}=\widetilde{\mathscr{N}_{0}/\mathscr{N}_{-1}}$)
and $p$ is a general point in $E$. By Corollary 2.9 and the dimension of $E$,
we can choose a small neighborhood $W$ of $x\in X$ and a Zariski open
neighborhood $V\subseteq\mathbb{C}^{r}$ of $p$ such that the codimension
filtration satisfies

$T_{j}(\widetilde{\mathscr{N}}_{0}/\widetilde{\mathscr{N}}_{-1})=\widetilde{\mathscr{N}}_{0}/\widetilde{\mathscr{N}}_{-1}$

for $j<n+l$ over $W\times V$. By Lemma 2.14 and Corollary 2.17, the Auslander
regularity tells

$\textup{codim}_{\mathbb{C}}\textup{Ch}^{\textup{rel}}(\mathscr{E}xt^{n+j}_{\widetilde{\mathscr{D}}_{X,R}}(\widetilde{\mathscr{N}}_{0}/\widetilde{\mathscr{N}}_{-1},\widetilde{\mathscr{D}}_{X,R})|_{W\times
V})>l\textup{ for $j>l$}.$

By shrinking $V$ in the Zariski topology (pick a more general $p\in E\cap V$
if necessary), we can assume that on $W\times V$

$T_{n+l}(\widetilde{\mathscr{N}}_{0}/\widetilde{\mathscr{N}}_{-1})=\widetilde{\mathscr{N}}_{0}/\widetilde{\mathscr{N}}_{-1}$

and $\widetilde{\mathscr{N}}_{0}/\widetilde{\mathscr{N}}_{-1}$ is
$(n+l)$-Cohen-Macaulay.

Now, we consider the short exact sequence (on $W$)

$0\to\mathscr{N}_{-1}\rightarrow\mathscr{N}_{0}\to\frac{\mathscr{N}_{0}}{\mathscr{N}_{-1}}\to
0.$

Taking analytic sheafification and dual, we have a long exact sequence

$\dots\to\mathscr{E}xt^{n+l-1}_{\widetilde{\mathscr{D}}_{X,R}}(\widetilde{\mathscr{N}}_{-1},\widetilde{\mathscr{D}}_{X,R})\to\mathscr{E}xt^{n+l}_{\widetilde{\mathscr{D}}_{X,R}}(\frac{\widetilde{\mathscr{N}}_{0}}{\widetilde{\mathscr{N}}_{-1}},\widetilde{\mathscr{D}}_{X,R})\rightarrow\mathscr{E}xt^{n+l}_{\widetilde{\mathscr{D}}_{X,R}}(\widetilde{\mathscr{N}}_{0},\widetilde{\mathscr{D}}_{X,R})\rightarrow\cdots.$

Since $\widetilde{\mathscr{N}}_{0}$ and $\widetilde{\mathscr{N}}_{-1}$ are
$n$-pure (since they are submodules of a $n$-Cohen-Macaulay module
$\widetilde{\mathscr{M}}$ by Proposition 3.5 and Proposition 2.19), by [Bj93,
A.IV.2.6], Corollary 2.17 and Lemma 2.14 we have

$T_{n+l+1}(\mathscr{E}xt^{n+l}_{\widetilde{\mathscr{D}}_{X,R}}(\widetilde{\mathscr{N}}_{0},\widetilde{\mathscr{D}}_{X,R}))=\mathscr{E}xt^{n+l}_{\widetilde{\mathscr{D}}_{X,R}}(\widetilde{\mathscr{N}}_{0},\widetilde{\mathscr{D}}_{X,R})$

and

$T_{n+l}(\mathscr{E}xt^{n+l-1}_{\widetilde{\mathscr{D}}_{X,R}}(\widetilde{\mathscr{N}}_{0},\widetilde{\mathscr{D}}_{X,R}))=\mathscr{E}xt^{n+l-1}_{\widetilde{\mathscr{D}}_{X,R}}(\widetilde{\mathscr{N}}_{0},\widetilde{\mathscr{D}}_{X,R}).$

Since $\widetilde{\mathscr{N}}_{0}/\widetilde{\mathscr{N}}_{-1}$ is
$(n+l)$-Cohen-Macaulay on $W\times V$, we know

$\textup{Ch}^{\textup{rel}}(\mathscr{E}xt^{n+l-1}_{\widetilde{\mathscr{D}}_{X,R}}(\frac{\widetilde{\mathscr{N}}_{0}}{\widetilde{\mathscr{N}}_{-1}},\widetilde{\mathscr{D}}_{X,R}))=\textup{Ch}^{\textup{rel}}(\frac{\widetilde{\mathscr{N}}_{0}}{\widetilde{\mathscr{N}}_{-1}})$

on $W\times V$. Therefore,

$\textup{Ch}^{\textup{rel}}(\mathscr{E}xt^{n+l-1}_{\widetilde{\mathscr{D}}_{X,R}}(\frac{\widetilde{\mathscr{N}}_{0}}{\widetilde{\mathscr{N}}_{-1}},\widetilde{\mathscr{D}}_{X,R}))\subseteq\textup{Ch}^{\textup{rel}}(\frac{T_{n+l}(\mathscr{E}xt^{n+l-1}_{\widetilde{\mathscr{D}}_{X,R}}(\widetilde{\mathscr{N}}_{0},\widetilde{\mathscr{D}}_{X,R}))}{T_{n+l+1}(\mathscr{E}xt^{n+l-1}_{\widetilde{\mathscr{D}}_{X,R}}(\widetilde{\mathscr{N}}_{0},\widetilde{\mathscr{D}}_{X,R}))}).$

By Proposition 2.8 and Proposition-Definition 2.7, we have

$E\subseteq
S_{n+l}(\mathscr{E}xt^{n+l-1}_{\widetilde{\mathscr{D}}_{X,R}}(\widetilde{\mathscr{N}}_{0},\widetilde{\mathscr{D}}_{X,R})),$

which proves the required implication. ∎

##  4\. Alexander complex of Sabbah

In this section, we recall the construction of the Alexander complexes in
[Sab90].

###  4.1. Alexander complex of Sabbah

We keep using notations in §3.1. The universal Alexander sheaf is defined by

$\psi^{\textup{univ}}=Exp_{!}(\mathbb{C}_{\mathbb{C}^{r}})$

where $Exp_{!}$ is the proper direct image functor of $Exp$ (cf. [KS13, 2.5])
and $\mathbb{C}_{\mathbb{C}^{r}}$ is the $\mathbb{C}$-constant sheaf on
$\mathbb{C}^{r}$. Since $Exp$ gives the universal covering of
$(\mathbb{C}^{*})^{r}$, $\psi^{\textup{univ}}$ is a locally free sheaf of
$\mathbb{C}[G]$-modules. We use the isomorphism

$\mathbb{C}[G]\simeq\mathbb{C}[t_{1}^{\pm},\dots,t_{r}^{\pm}]$

with $t_{i}$ representing the (counterclockwise) loops around the puncture of
each factor of $(\mathbb{C}^{*})^{r}$. Then the monodromy of
$\psi^{\textup{univ}}$ around each loop is induced by the
$t_{i}$-multiplication on $\mathbb{C}[t_{1}^{\pm},\dots,t_{r}^{\pm}]$. For
$\mathcal{F}^{\bullet}\in D^{b}_{c}(\mathbb{C}_{X})$, its _Alexander complex_
along $F$ (also known as _Sabbah specialization complex_) is defined as

$\psi_{F}(\mathcal{F}^{\bullet})=i_{D,*}i_{D}^{-1}Rj_{*}(\mathcal{F}^{\bullet}|_{X\setminus
D}\otimes_{\mathbb{C}}F^{-1}\psi^{\textup{univ}}),$

where $j\colon X\setminus D\hookrightarrow X$ is the open embedding and
$i_{D}\colon D\hookrightarrow X$ is the closed embedding. More generally, for
every subset $I\subseteq\\{1,2,\dots,r\\}$, the _Sabbah specialization
complex_ along $D_{I}=\bigcap_{i\in I}D_{i}$ is define as

$\psi_{D_{I}}(\mathcal{F}^{\bullet})=i_{D_{I},*}i_{D_{I}}^{-1}Rj_{*}(\mathcal{F}^{\bullet}|_{X\setminus
D}\otimes_{\mathbb{C}}F^{-1}\psi^{\textup{univ}}),$

where $i_{D_{I}}\colon D_{I}\hookrightarrow X$ is the closed embedding. Since
$\psi^{\textup{univ}}$ is locally free over $\mathbb{C}[G]$,
$\psi_{D_{I}}(\mathcal{F}^{\bullet})\in D^{b}_{c}(\mathbb{C}[G])$.

###  4.2. The universal relative local system

We define a locally free $pr^{-1}\mathscr{O}_{(\mathbb{C}^{*})^{r}}$-module on
the complex manifold $(\mathbb{C}^{*})^{r}\times(\mathbb{C}^{*})^{r}$ with
$pr\colon(\mathbb{C}^{*})^{r}\times(\mathbb{C}^{*})^{r}\to(\mathbb{C}^{*})^{r}$
the second projection, denoted by $\mathcal{L}^{\textup{univ}}$, by requiring
the following two conditions:

  1. (i)

$\mathcal{L}^{\textup{univ}}|_{W\times(\mathbb{C}^{*})^{r}}\simeq
pr^{-1}\mathscr{O}_{(\mathbb{C}^{*})^{r}}|_{W\times(\mathbb{C}^{*})^{r}}$ for
all simply connected open subsets $W\subset(\mathbb{C}^{*})^{r}$

  2. (ii)

for every
${\boldsymbol{\lambda}}=(\lambda_{1},\dots,\lambda_{r})\in(\mathbb{C}^{*})^{r}$,
$\widetilde{i}_{{\boldsymbol{\lambda}}}^{*}\mathcal{L}^{\textup{univ}}$ is the
local system on the first factor of
$(\mathbb{C}^{*})^{r}\times(\mathbb{C}^{*})^{r}$ satisfying that the monodromy
along each $t_{i}$ is given by multiplication by $\lambda_{i}$, where
$i_{{\boldsymbol{\lambda}}}\colon\\{{\boldsymbol{\lambda}}\\}\hookrightarrow(\mathbb{C}^{*})^{r}$
is the closed embedding into the second factor of
$(\mathbb{C}^{*})^{r}\times(\mathbb{C}^{*})^{r}$.

It is easy to check that the two conditions above uniquely determine
$\mathcal{L}^{\textup{univ}}$.

######  Lemma 4.1.

$\mathcal{L}^{\textup{univ}}\simeq\widetilde{\psi}^{\textup{univ}}.$

###### Proof.

Obviously, $\widetilde{\psi}^{\textup{univ}}$ satisfies the two conditions of
$\mathcal{L}^{\textup{univ}}$. ∎

##  5\. Comparison

###  5.1. The universal relative flat bundle

On $\mathbb{C}^{r}_{\mathbf{x}}\times\mathbb{C}^{r}$ with
$\mathbf{x}=(x_{1},\dots,x_{r})$ the coordinates, we write

$D_{\mathbf{x}}=(\prod_{i=1}^{r}x_{i}=0)\quad\textup{and}\quad
U=\mathbb{C}^{r}\setminus D.$

Then using the recipe from §3, we have the maximal extension
$\mathscr{O}_{\mathbb{C}^{r}_{\mathbf{x}}}(*D_{\mathbf{x}}^{(r)})$. We set

$\mathscr{O}^{\textup{univ}}=\mathscr{O}_{\mathbb{C}^{r}_{\mathbf{x}}}(*D_{\mathbf{x}}^{(r)})|_{U\times\mathbb{C}^{r}},$

calling it the universal relative flat bundle on $U\times\mathbb{C}^{r}$
because

(5.1)
$\widetilde{F}^{*}(\mathscr{O}_{\mathbb{C}^{r}_{\mathbf{x}}}(*D_{\mathbf{x}}^{(r)}))=\mathscr{O}_{X}(*D_{F}^{(r)})$

in the situation of §3.1, with $\widetilde{F}=(F,\textup{id})\colon
X\times\mathbb{C}^{r}\to\mathbb{C}^{r}\times\mathbb{C}^{r}$ the associated
morphism, or equivalently

$F^{*}(\mathscr{O}_{\mathbb{C}^{r}_{\mathbf{x}}}(*D_{\mathbf{x}})\otimes_{\mathbb{C}}\mathbb{C}[\mathbf{s}]\cdot{\mathbf{x}}^{\mathbf{s}})=\mathscr{O}_{X}(*D_{F})\otimes_{\mathbb{C}}\mathbb{C}[\mathbf{s}]\cdot
F^{\mathbf{s}}.$

######  Remark 5.1.

Since $\mathbf{x}$ is the complex coordinates of
$\mathbb{C}^{r}_{\mathbf{x}}$, it is a direct computation to get the relative
characteristic cycle:

$\textup{CC}^{{\textup{rel}}}\big{(}\mathscr{O}_{\mathbb{C}^{r}_{\mathbf{x}}}(*D_{\mathbf{x}}^{(r)})\big{)}=(\sum_{\beta}T^{*}_{\bar{X}_{\beta}}\mathbb{C}^{r}_{\mathbf{x}})\times\mathbb{C}^{r},$

where $\bigsqcup_{\beta}X_{\beta}=\mathbb{C}^{r}_{\mathbf{x}}$ is the
stratification naturally given by the coordinates $\mathbf{x}$. In particular,
$\mathscr{O}_{\mathbb{C}^{r}_{\mathbf{x}}}(*D_{\mathbf{x}}^{(r)})$ is relative
holonomic. In fact, it is regular relative holonomic (see [MFS13, §2.1] for
the definition of relative regularity). Therefore, if $\mathcal{M}$ is a
regular holonomic $\mathscr{D}_{X}$-module, then the relative holonomicity of
$\mathcal{M}(*D_{F}^{(r)})$ in Theorem 3.3 can also be deduced from [FFS21,
Theorem 2].

To avoid confusion, we set the morphism

$\pi_{\mathbf{x}}=(\textup{id},Exp)\colon U\times\mathbb{C}^{r}\rightarrow
U\times(\mathbb{C}^{*})^{r}.$

######  Lemma 5.2.

We have

$\textup{$\pi_{{\mathbf{x}},*}^{G}(\textup{DR}_{U\times\mathbb{C}^{r}/\mathbb{C}^{r}}(\mathscr{O}^{\textup{univ}}))\stackrel{{\scriptstyle
q.i.}}{{\simeq}}\mathcal{L}^{\textup{univ}}[n]$ and
$\mathscr{O}_{U\times\mathbb{C}^{r}}\otimes_{pr^{-1}\mathscr{O}_{\mathbb{C}^{r}}}\pi^{-1}_{\mathbf{x}}(\mathcal{L}^{\textup{univ}})\simeq\mathscr{O}^{\textup{univ}}$}.$

###### Proof.

Obviously,
$\pi_{{\mathbf{x}},*}^{G}(\textup{DR}_{U\times\mathbb{C}^{r}/\mathbb{C}^{r}}(\mathscr{O}^{\textup{univ}}))$
satisfies the two conditions of $\mathcal{L}^{\textup{univ}}$(up to the
$n$-shifts).

The quasi-isomorphism particularly gives us a morphism

$\mathscr{O}_{U\times\mathbb{C}^{r}}\otimes_{pr^{-1}\mathscr{O}_{\mathbb{C}^{r}}}\pi^{-1}_{\mathbf{x}}(\mathcal{L}^{\textup{univ}})\rightarrow\mathscr{O}^{\textup{univ}}.$

We pullback the morphism above by $\widetilde{i}_{{\boldsymbol{\alpha}}}$ for
every ${\boldsymbol{\alpha}}\in\mathbb{C}^{r}$ and get an isomorphism. Thus by
[FS19, Proposition 1.9.(1)], the morphism is isomorphic. ∎

###  5.2. Proof of Theorem 1.2

We first prove the following theorem.

######  Theorem 5.3.

Let $\mathcal{M}$ be a regular holonomic $\mathscr{D}_{X}$-module. Then
naturally

$\pi_{*}^{G}(\textup{DR}_{X\times\mathbb{C}^{r}/\mathbb{C}^{r}}(\mathcal{M}(*D_{F}^{(r)})))\stackrel{{\scriptstyle
q.i.}}{{\simeq}}\widetilde{Rj_{*}}(\textup{DR}(\mathcal{M})|_{X\setminus
D}\otimes_{\mathbb{C}}F^{-1}\psi^{\textup{univ}}).$

###### Proof.

By using the second isomorphism in Lemma 5.2 and (5.1), we get

(5.2)
$\textup{DR}_{X\times\mathbb{C}^{r}/\mathbb{C}^{r}}(\mathcal{M}(*D_{F}^{(r)}))|_{(X\setminus
D)\times\mathbb{C}^{r}}\simeq p_{1}^{-1}(\textup{DR}(\mathcal{M}|_{X\setminus