# Characteristic cycles associated to holonomic $\mathscr{D}$-modules

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

We study relative and logarithmic characteristic cycles associated to
holonomic $\mathscr{D}$-modules. As applications, we obtain: (1) an
alternative proof of Ginsburg’s log characteristic cycle formula for lattices
of regular holonomic $\mathscr{D}$-modules following ideas of Sabbah and
Briancon-Maisonobe-Merle, and (2) the constructibility of the log de Rham
complexes for lattices of holonomic $\mathscr{D}$-modules, which is a natural
generalization of Kashiwara’s constructibility theorem.

###### 2010 Mathematics Subject Classification:

14F10, 32S60, 14C17, 32S30, 14A21

##  1\. Introduction

The characteristic variety of a coherent $\mathscr{D}$-module with a good
filtration is the support of the associated graded module on the cotangent
bundle (see [Kas03]). Characteristic cycles can be obtained with
multiplicities taken into account. They can also be considered relative to
smooth morphisms (or holomorphic submersions under the analytic setting) and
from a logarithmic perspective. See §2.1 and §3 for definitions.

In this paper, we study characteristic cycles of relative
$\mathscr{D}$-modules associated to (regular) holonomic $\mathscr{D}$-modules.
We apply the relative characteristic cycles to studying the logarithmic
characteristic cycles for lattices of regular holonomic $\mathscr{D}$-modules
and the constructibility of logarithmic de Rham complexes.

###  1.1. Constructibility of log de Rham complexes for lattices

For holonomic $\mathscr{D}$-modules on complex manifolds, Kashiwara’s
constructibility theorem [Kas75] says that the de Rham complexes of the
holonomic modules are constructible (they are indeed perverse, see also
[HTT08, Theorem 4.6.6]). Our first two main results are constructibility and
perversity of log de Rham complexes of lattices.

Suppose that $(X,D)$ is a pair consisting of a complex manifold together with
a reduced normal crossing divisor $D$, called an _analytic smooth log pair_.
Let $\mathscr{D}_{X,D}$ be the sheaf of rings of holomorphic logarithmic
differential operators, that is, the sub-sheaf of $\mathscr{D}_{X}$ consisting
of differential operators preserving the defining ideal of $D$. Let
$\mathcal{M}$ be a coherent $\mathscr{D}_{X}$-module. We now consider a
$\mathscr{D}_{X,D}$-_lattice_ $\bar{\mathcal{M}}$ of $\mathcal{M}$, a special
$\mathscr{D}_{X,D}$-module associated to $\mathcal{M}$ (see §3.3 for
definition). Typical examples of lattices include Deligne lattices (cf. [WZ21,
§4.4]) and lattices given by the graph embedding construction of Malgrange
(see §5 for details).

######  Theorem 1.1.

Suppose that $(X,D)$ is an analytic smooth log pair and that $\mathcal{M}$ is
a holonomic $\mathscr{D}_{X}$-module. If $\bar{\mathcal{M}}$ is a
$\mathscr{D}_{X,D}$-lattice of $\mathcal{M}$, then the log de Rham complex
$\textup{DR}_{X,D}(\bar{\mathcal{M}})$ is constructible.

The above theorem naturally generalizes Kashiwara’s constructibility theorem
and answers the question at the beginning of [WZ21]. The constructible complex
$\textup{DR}_{X,D}(\bar{\mathcal{M}})$ is not perverse in general and the
stratification of the constructible complex
$\textup{DR}_{X,D}(\bar{\mathcal{M}})$ is determined by the stratification of
$\textup{Ch}(\mathcal{M}(*D))$ (see Remark 5.2).

######  Theorem 1.2.

In the situation of Theorem 1.1, $\textup{DR}_{X,D}(\bar{\mathcal{M}}(kD))$
are perverse locally on a relative compact open subset of $X$ $($or globally
when $X$ is algebraic$)$ for all $|k|\gg 0$. Moreover, if $\mathcal{M}$ is
regular holonomic, then locally on a relative compact open subset of $X$ $($or
globally when $X$ is algebraic$)$ we have natural quasi-isomorphisms

  1. (1)

$\textup{DR}_{X,D}(\bar{\mathcal{M}}(kD))\stackrel{{\scriptstyle
q.i.}}{{\simeq}}Rj_{*}\textup{DR}(\mathcal{M}|_{U})$,

  2. (2)

$\textup{DR}_{X,D}(\bar{\mathcal{M}}(-kD))\stackrel{{\scriptstyle
q.i.}}{{\simeq}}j_{!}\textup{DR}(\mathcal{M}|_{U})$

for all integral $k\gg 0$, where $j\colon U=X\setminus D\hookrightarrow X$ is
the open embedding.

Taking the lattice $\bar{\mathcal{M}}=\mathscr{O}_{X}$ in Theorem 1.2 (1), we
recover the Grothendieck comparison [Gro66],

$[\mathscr{O}_{X}\rightarrow\Omega^{1}_{X}(\log
D)\rightarrow\cdots\rightarrow\Omega^{n}_{X}(\log D)]\stackrel{{\scriptstyle
q.i.}}{{\simeq}}Rj_{*}\mathbb{C}_{U}[n],$

where $n$ is the dimension of $X$. See also [WZ21, Theorem 1.2]. Theorem 1.2
for lattices given by the graph embedding construction has been used in
studying the cohomology support loci of rank one complex local systems
[BVWZ21a, BVWZ21b].

The Kashiwara’s constructibility theorem has been extended to Riemann-Hilbert
correspondence, the regular case by Kashiwara and Mebkhout independently (see
for instance [HTT08, §7]) and the irregular case by D’Agnolo and Kashiwara
[DK16]. Under the logarithmic setting, Kato and Nakayama [KN99] and Ogus
[Ogu03] studied Riemann-Hilbert correspondence for log connections on smooth
log schemes, and Koppensteiner and Taplo [KT19] further studied the theory of
logarithmic $\mathscr{D}$-modules on smooth log schemes. Koppensteiner [Kop20]
(based on the work of Ogus) augmented the log de Rham complexes to graded
complexes on Kato-Nakayama spaces and proved a finiteness result for
logarithmic holonomic $\mathscr{D}$-modules. Since smooth log pairs are smooth
log schemes, lattices in this paper are special examples of log
$\mathscr{D}$-modules in [KT19]. Therefore, one can naturally ask whether
Theorem 1.1 together with Theorem 1.2 can be enhanced to a Riemann-Hilbert
correspondence on smooth log pairs in the logarithmic category. Notice that
our proof of Theorem 1.1 is logarithmic in nature, since it depends on the
natural stratification of the normal crossing divisor $D$, which gives
evidence of the existence of the log Riemann-Hilbert correspondence.

A similar logarithmic Riemann-Hilbert program for log holonomic modules has
also been proposed in [KT19]. However, lattices are not logarithmic holonomic
[KT19, Definition 3.22] in general and hence Theorem 1.1 is different from the
finiteness result in [Kop20]. It would be interesting to further understand
relations between lattices and logarithmic holonomic modules.

Our proof of Theorem 1.1 and Theorem 1.2 depends on “transforming” log
$\mathscr{D}$-modules to relative $\mathscr{D}$-modules and on the study of
relative characteristic cycles. Typical examples of non-trivial relative
$\mathscr{D}$-modules arise from the construction of the generalized
Kashiwara-Malgrange filtrations. Then we discuss relative characteristic
cycles associated to Kashiwara-Malgrange filtrations.

###  1.2. $V$-filtrations along slopes of smooth complete intersections and
their relative characteristic cycles

Suppose that $X$ is a smooth algebraic variety over $\mathbb{C}$ (or a complex
manifold) and that $Y\subseteq X$ is a smooth complete intersection of
codimension $r$, that is,

$Y=\bigcap_{j=1}^{r}H_{j}$

where $\sum_{j}H_{j}$ is a normal crossing divisor. Let $\mathscr{D}_{X}$ be
the sheaf of rings of algebraic (or holomorphic) differential operators.
Define a $\mathbb{Z}^{r}$-filtration on $\mathscr{D}_{X}$ along $Y$ by

(1)
$V_{\mathbf{s}}\mathscr{D}_{X}=\bigcap_{j=1}^{r}V_{s_{j}}^{H_{j}}\mathscr{D}_{X}$

for every $\mathbf{s}=(s_{1},s_{2}\dots,s_{r})\in\mathbb{Z}^{r}$, where
$V_{\bullet}^{H_{j}}\mathscr{D}_{X}$ is the Kashiwara-Malgrange filtration on
$\mathscr{D}_{X}$ along $H_{j}$ (see Definition 4.1). Following ideas of
Sabbah [Sab87a], for a nondegenerate slope $L$ in the dual cone
$(\mathbb{Z}_{\geq 0}^{r})^{\vee}$, we define the (generalized) Kashiwara-
Malgrange filtration on $\mathscr{D}_{X}$ along $L$ by

${}^{L}V_{L(\mathbf{s})}\mathscr{D}_{X}=\sum_{L(\mathbf{s}^{\prime})\leq
L(\mathbf{s})}V_{\mathbf{s}^{\prime}}\mathscr{D}_{X}.$

This gives a $\mathbb{Z}$-filtration ${}^{L}V_{\bullet}\mathscr{D}_{X}$ via
the isomorphism $\mathbb{Z}\simeq\mathbb{Z}^{r}/L^{\perp}$. For a coherent
$\mathscr{D}_{X}$-module $\mathcal{M}$, one can then define the Kashiwara-
Malgrange filtration (or $V$-filtration for short) on $\mathcal{M}$ along $L$
(see Definition 4.14). For degenerate slopes, one can reduce to the
nondegenerate ones by ignoring the unrelated $H_{j}$.

The $V$-filtration of a coherent $\mathscr{D}_{X}$-module $\mathcal{M}$ along
$L$ (if exists) contains the “deformation” information of $\mathcal{M}$. More
precisely, the Rees ring ${}^{L}R_{V}\mathscr{D}_{X}$ associated to
${}^{L}V_{\bullet}\mathscr{D}_{X}$ gives the sheaf of differential operators
relative to the (algebraic) normal deformation of $Y$ in $X$ along $L$,

$\varphi^{L}:\widetilde{X}^{L}\to\mathbb{C},$

where $\widetilde{X}^{L}$ is the ambient space of the deformation. Hence, the
Rees module ${}^{L}R_{V}\mathcal{M}$ associated to
${}^{L}V_{\bullet}\mathcal{M}$ is a $\mathscr{D}$-module relative to
$\varphi^{L}$. See §4.7 for details.

######  Theorem 1.3 (Sabbah).

Suppose that $\mathcal{M}$ is a holonomic $\mathscr{D}_{X}$-module and that
$Y$ is a smooth complete intersection of codimension $r$. Then $\mathcal{M}$
is specializable along every slope vector $L$ (i.e. the $V$-filtration on
$\mathcal{M}$ along $L$ uniquely exists). Moreover, if $\mathcal{M}$ is
regular holonomic and the slope vector $L$ is nondegenerate, then
$\textup{gr}^{{}^{L}V}_{\bullet}\mathcal{M}$ gives a regular holonomic
$\mathscr{D}$-module on $T_{Y}X$ and we have the following formulas for
characteristic cycles,

$\textup{CC}_{\widetilde{X}^{L}/\mathbb{C}}({}^{L}R_{V}\mathcal{M})=\overline{q^{*}_{L}(\textup{CC}(\mathcal{M}))}\subseteq
T^{*}(\widetilde{X}^{L}/\mathbb{C})$

and

$\textup{CC}(\widetilde{\textup{gr}}^{{}^{L}V}_{\bullet}\mathcal{M})=\overline{q^{*}_{L}(\textup{CC}(M))}|_{T^{*}T_{Y}X}\subseteq
T^{*}T_{Y}X,$

where $q_{L}:T^{*}(\widetilde{X}^{L}/\mathbb{C})\setminus T^{*}T_{Y}X\simeq
T^{*}X\times\mathbb{C}^{\star}\to T^{*}X$ is the natural projection.

One can also consider the Rees ring $R_{V}\mathscr{D}_{X}$ associated to the
$\mathbb{Z}^{r}$-filtration $V_{\bullet}\mathscr{D}_{X}$. Similar to
${}^{L}R_{V}\mathscr{D}_{X}$, $R_{V}\mathscr{D}_{X}$ can be seen as the sheaf
of differential operators relative to the (refined) normal deformation of $Y$
in $X$ (see §4.6),

$\varphi:\widetilde{X}\to\mathbb{C}^{r},$

with the $\mathbb{Z}^{r}$-grading on $R_{V}\mathscr{D}_{X}$ induced from the
natural toric structure on the base $\mathbb{C}^{r}$. Then $\varphi^{L}$, as
well as ${}^{L}R_{V}\mathscr{D}_{X}$, is obtained from $\varphi$ and
$R_{V}\mathscr{D}_{X}$ respectively through the base-change,

$\iota_{L}\colon\mathbb{C}\hookrightarrow\mathbb{C}^{r}$

induced by the one parameter subgroup of $L$. For a $\mathscr{D}_{X}$-module
$\mathcal{M}$ with a good filtration $U_{\bullet}\mathcal{M}$ over
$V_{\bullet}\mathscr{D}_{X}$, the associated Rees module $R_{U}\mathcal{M}$ is
then a coherent relative $\mathscr{D}$-module with respect to $\varphi$.

######  Theorem 1.4 (Sabbah).

Suppose that $\mathcal{M}$ is a regular holonomic $\mathscr{D}_{X}$-module and
that Y is a smooth complete intersection of codimension $r$. Let
$U_{\bullet}\mathcal{M}$ be a good $\mathbb{Z}^{r}$-filtration over
$V_{\bullet}\mathscr{D}_{X}$. Then we have

$\textup{CC}_{\widetilde{X}/\mathbb{C}^{r}}(R_{U}\mathcal{M})=\overline{q^{*}(\textup{CC}(\mathcal{M}))}\subseteq
T^{*}(\widetilde{X}/\mathbb{C}^{r})$

where
$q:T^{*}(\widetilde{X}^{L}/\mathbb{C})\setminus(\prod^{r}_{j=1}u_{j}=0)\simeq
T^{*}X\times(\mathbb{C}^{\star})^{r}\to T^{*}X$ is the natural projection and
$(u_{1},u_{2},\dots,u_{r})$ are coordinates of $\mathbb{C}^{r}$.

Theorem 1.3 and 1.4 are essentially due to Sabbah (see [Sab87b, §2]). Their
proof relies on the study of characteristic cycles for relative
$\mathscr{D}$-modules (see §4). When $L=(\underbrace{1,1,\dots,1}_{r})$ (use
the standard dual basis of $(\mathbb{Z}^{r})^{\vee}$), the characteristic
cycle formula for $\textup{gr}^{{}^{L}V}_{\bullet}\mathcal{M}$ in Theorem 1.3
is due to Ginsburg ([Gin86, Theorem 5.8]). But the precise characteristic
cycle formulas in Theorem 1.3 and 1.4 seem to be missing in the literature. It
is worth mentioning that Ginsburg use the characteristic cycle formula in
[Gin86, Theorem 5.8] to study index theorems. It would be interesting to know
whether the characteristic cycle formulas in Theorem 1.3 are related to index
theorems or even Fukaya categories (cf. [NZ09]).

###  1.3. Relative Riemann-Hilbert correspondence

We give an explicit relative Riemann-Hilbert correspondence for
${}^{L}R_{V}\mathcal{M}$. We assume $\mathcal{M}$ a regular holonomic
$\mathscr{D}_{X}$-module. The relative $\mathscr{D}$-module
${}^{L}R_{V}\mathcal{M}$ has fibers

$\mathbf{L}i_{\alpha}^{*}({}^{L}R_{V}\mathcal{M})\stackrel{{\scriptstyle
q.i.}}{{\simeq}}\mathcal{M}\textup{ for $0\not=\alpha\in\mathbb{C}$ and,
}\mathbf{L}i_{0}^{*}({}^{L}R_{V}\mathcal{M})\stackrel{{\scriptstyle
q.i.}}{{\simeq}}\widetilde{\textup{gr}}^{{}^{L}V}_{\bullet}\mathcal{M},$

where $\stackrel{{\scriptstyle q.i.}}{{\simeq}}$ denotes quasi-isomorphism and
$i_{\alpha}:\widetilde{X}^{L}_{\alpha}=(\varphi^{L})^{-1}(\alpha)\hookrightarrow\widetilde{X}^{L}$
is the closed embedding. Thus, ${}^{L}R_{V}\mathscr{D}_{X}$ deforms
$\mathcal{M}$ into $\textup{gr}^{{}^{L}V}_{\bullet}\mathcal{M}$. By Theorem
1.3, ${}^{L}R_{V}\mathcal{M}$ provides an example of relative regular
holonomic $\mathscr{D}$-modules (see Definition 2.4 and §2.2). Moreover, using
Lemma 2.9 the relative _de Rham complex_ ,

$\omega_{\widetilde{X}^{L}/\mathbb{C}}\otimes^{\mathbf{L}}_{\mathscr{D}}{}^{L}R_{V}\mathcal{M},$

has fibers $\textup{DR}(\mathcal{M})$ for $\alpha\not=0$ and the central fiber
$\textup{DR}_{T_{Y}X}(\widetilde{\textup{gr}}^{{}^{L}V}_{\bullet}\mathcal{M})$,
where $\omega_{\widetilde{X}^{L}/\mathbb{C}}$ is the relative canonical sheaf
of $\varphi_{L}$ (see §4.3 and Remark 4.13 for explicit formulas). But the
central fiber satisfies (applying Theorem 4.5 and Lemma 4.16)

$\textup{DR}_{T_{Y}X}(\widetilde{\textup{gr}}^{{}^{L}V}_{\bullet}\mathcal{M})\simeq\psi_{T_{Y}X}(Rj^{L}_{*}p^{-1}(\textup{DR}(M)))\eqqcolon\textup{Sp}^{L}_{T_{Y}X}(\textup{DR}(\mathcal{M}))$

where $\textup{Sp}^{L}_{T_{Y}X}$ is the _Verdier specialization_ along $L$ by
definition, $p:\widetilde{X}^{L}\setminus T_{Y}X\to X$ is the natural
projection and $j^{L}:\widetilde{X}^{L}\setminus
T_{Y}X\hookrightarrow\widetilde{X}^{L}$ is the open embedding. Thus, the
relative de Rham complex deforms $\textup{DR}(\mathcal{M})$ into its Verdier
specialization along $L$. In particular, the relative de Rham complex gives a
relative constructible complex (cf. [MFS16]).

In general, a relative regular Riemann-Hilbert correspondence for relative
regular holonomic $\mathscr{D}$-modules over curves is established in [FFS20].
However, the relative holonomicity in _loc. cit._ is more restricted.
Motivated by the above example, one might ask whether the relative Riemann-
Hilbert correspondence over curves can be extended to the case by using the
relative holonomicity in Definition 2.4.

In contrast to ${}^{L}R_{V}\mathcal{M}$, $R_{U}\mathcal{M}$ is not necessarily
relatively holonomic because of the main obstruction that $R_{U}\mathcal{M}$
is only torsion-free but not necessarily flat over $\mathbb{C}^{r}$.
Consequently, one cannot “normalize” $U_{\bullet}\mathcal{M}$ into a
$\mathbb{Z}^{r}$-indexed $V$-filtration in general. See Remark 4.22 for more
discussions.

######  Proposition 1.5.

In the situation of Theorem 1.4, if $R_{U}\mathcal{M}$ is flat over
$\mathbb{C}^{r}$, then $R_{U}\mathcal{M}$ is relative holonomic.

###  1.4. Logarithmic characteristic cycles of lattices

If $D=\sum_{i=1}^{r}H_{j}$ is a normal crossing divisor, then

$\mathscr{D}_{X,D}=V_{\vec{0}}\mathscr{D}_{X}$

where the latter is defined in Eq.(1). This means that if
$U_{\bullet}(\mathcal{M}(*D))$ is a good filtration over the
$\mathbb{Z}^{r}$-filtration $V_{\bullet}\mathscr{D}_{X}$, then each filtrant
$U_{\mathbf{s}}\mathcal{M}(*D)$ is a lattice of $\mathcal{M}$. This is an easy
relation between log $\mathscr{D}$-modules and relative $\mathscr{D}$-modules.
See §3.4 for a complicated relation between them. Our next main result is an
alternative proof of Ginsburg’s log characteristic cycle formula based on the
complicated relation.

######  Theorem 1.6 (Ginsburg).

Suppose that $(X,D)$ is an analytic smooth log pair and that $\mathcal{M}$ is
a regular holonomic $\mathscr{D}_{X}$-module. If $\bar{\mathcal{M}}$ is a
$\mathscr{D}_{X,D}$-lattice of $\mathcal{M}$, then

$\textup{CC}_{\log}(\bar{\mathcal{M}})=\overline{\textup{CC}}^{\log}(\mathcal{M}|_{U}),$

where $\overline{\textup{CC}}^{\log}(\mathcal{M}|_{U})$ denotes the closure of
$\textup{CC}(\mathcal{M}_{U})\subseteq T^{*}U$ inside the logarithmic
cotangent bundle $T^{*}(X,D)$.

Ginsburg’s proof of the above theorem under the algebraic setting in [Gin89,
Appendix A] uses microlocalization of $\mathscr{D}$-modules and resolution of
singularities. Our proof under the analytic setting relies on converting log
$\mathscr{D}$-modules to relative $\mathscr{D}$-modules, with some ideas due
to Sabbah and Briancon-Maisonobe-Merle.

Theorem 1.6 has a long history. A characteristic variety formula of the
holonomic system for $\prod_{l=1}^{N}(f_{l}+\sqrt{-1}{O})^{\lambda_{l}}$ first
appears in [KK79]. For the lattice $\mathscr{D}_{X}[\mathbf{s}]({\bf
f}^{\mathbf{s}}\cdot m)$ given by the graph embedding construction (see §5),
the formula for characteristic varieties is proved by Briancon-Maisonobe-Merle
[BMM02, Théorèm 2.1] and the characteristic cycle formula in this case is
obtained in [BVWZ21b]. Ginsburg [Gin89] made it in its most general form as in
Theorem 1.6 under the algebraic setting. See [Kas77, Gin86, Gin89, BVWZ21b,
Mai16a, Mai16b] for applications of Theorem 1.6. See also [CM99, CMNM02] for
related applications.

In a follow-up paper, Theorem 1.6 in its general form is used to obtain the
conclusion that the zero loci of Bernstein-Sato ideals for regular holonomic
$\mathscr{D}$-modules in general are always of codimension-one [Wu21, Theorem
3.11]. This conclusion in turn plays an important role in the establishment of
the Riemann-Hilbert correspondence for Alexander complexes (see [Wu21, §3]).

The following example shows that regularity in both Theorem 1.6 and Theorem
1.3 is needed.

######  Example 1.

We consider $\mathcal{M}=\mathbb{C}[t,1/t]\cdot e^{1/t}$, the algebraic
irregular holonomic $\mathscr{D}_{X}$-module generated by the function
$e^{1/t}$ for $X=\mathbb{C}$, where $t$ is the coordinate of the complex plane
$\mathbb{C}$. Let $H$ be the divisor $\\{0\\}\subseteq X$. Since

$t\partial_{t}\cdot e^{1/t}=-e^{1/t}/t,$

$\mathcal{M}$ is coherent over $\mathscr{D}_{X,H}$. Hence, $\mathcal{M}$ is a
$\mathscr{D}_{X,H}$-lattice of itself and its $V$-filtration along $H$ is the
trivial filtration with $\textup{gr}^{V}_{\bullet}\mathcal{M}=0.$ Moreover,
since

$(1+t^{2}\partial_{t})\cdot e^{1/t}=0,$

one can see that $\textup{Ch}_{\log}(\mathcal{M})$ has a component over $H$.

###  1.5. Key ideas in proving main results

For Theorem 1.1, we first use direct images of log $\mathscr{D}_{X}$-modules
(see §3.2) under the graph embedding of the defining functions of the normal
crossing divisor locally to reduce to the case for lattices
$\mathscr{D}_{X}[\mathbf{s}]({\bf f}^{\mathbf{s}}\cdot\mathcal{M}_{0})$. The
lattices $\mathscr{D}_{X}[\mathbf{s}]({\bf
f}^{\mathbf{s}}\cdot\mathcal{M}_{0})$ can be treated as relative
$\mathscr{D}$-modules over $\mathbb{C}[\mathbf{s}]$ with independent
parameters

$\mathbf{s}=(s_{1},s_{2},\dots,s_{r}).$

We then use the fact that $\mathscr{D}_{X}[\mathbf{s}]({\bf
f}^{\mathbf{s}}\cdot\mathcal{M}_{0})$ is indeed relative holonomic, see
Theorem 5.1, which generalizes a relative holonomicity result of Maisonobe
[Mai16a]. We then identify the log de Rham complex of
$\mathscr{D}_{X}[\mathbf{s}]({\bf f}^{\mathbf{s}}\cdot\mathcal{M}_{0})$ as the
fiber of the relative de Rham complexes over the origin
${\mathbf{0}}\in\mathbb{C}^{r}=\textup{Spec }\mathbb{C}[\mathbf{s}]$. Finally,
Theorem 1.1 follows from the relative holonomicity result and Kashiwara’s
constructibility theorem for complexes of $\mathscr{D}$-modules with holonomic
cohomologies. To prove Theorem 1.2, we reduce the required perversity to the
flatness of the twisted lattices

$\mathscr{D}_{X}[\mathbf{s}]({\bf f}^{\mathbf{s}}\cdot\mathcal{M}_{0}(kD))$

over a small (analytic) neighborhood of ${\mathbf{0}}\in\mathbb{C}^{r}$ for
$|k|\gg 0$ by applying Sabbah’s generalized $b$-functions.

The key point of the proof of Theorem 1.6 is to interpret log
$\mathscr{D}$-modules as certain relative $\mathscr{D}$-modules. More
precisely, we use what we call the log rescaled families (locally) to convert
lattices to relative $\mathscr{D}$-modules over the log factor. See §3.4 for
constructions. Finally we apply a relative characteristic variety formula of
Sabbah and Briancon-Maisonobe-Merle (see Lemma 3.4).

###  1.6. Relations to singularities in algebraic geometry

We first clarify the Bernsten-Sato polynomials (or $b$-functions) in this
paper and in the literature.

The $b$-functions in Definition 4.14 are the natural generalization of the
$b$-functions for the usual $V$-filtrations (see §4.1 and also [Bj93,
§III.7]). For a holomorphic function $f$ and for the lattices
$\mathscr{D}_{X}[s](f^{s}\cdot\mathcal{M}_{0})$ (that is, $r=1$), the
associated $b$-function is the monic polynomial $b(s)\in\mathbb{C}[s]$ of the
least degree such that

$b(s)\cdot\frac{\mathscr{D}_{X}[s](f^{s}\cdot\mathcal{M}_{0})}{\mathscr{D}_{X}[s](f^{s+1}\cdot\mathcal{M}_{0})}=0,$

where $\mathcal{M}_{0}$ is an $\mathscr{O}_{X}$-coherent submodule of a
holonomic $\mathscr{D}_{X}$-module. See [Bj93, III.2 and VI.1] for this case.
If $\mathcal{M}_{0}=\mathscr{O}_{X}$, the above definition gives particularly
the $b$-function for $f$. For the lattice $\mathscr{D}_{X}[\mathbf{s}]({\bf
f}^{\mathbf{s}}\cdot\mathcal{M}_{0})$ in general, since $s_{i}$ is identified
with $-t_{i}\partial_{t_{i}}$, Theorem 4.23 gives a polynomial
$b(\mathbf{s})\in\mathbb{C}[\mathbf{s}]$ such that

$b(\mathbf{s})\cdot\frac{\mathscr{D}_{X}[\mathbf{s}]({\bf
f}^{\mathbf{s}}\cdot\mathcal{M}_{0})}{\mathscr{D}_{X}[\mathbf{s}]({\bf
f}^{\mathbf{s}+\vec{1}}\cdot\mathcal{M}_{0})}=0,$

with $\vec{1}=(1,1,\dots,1)$. Such $b(\mathbf{s})$ are what we mean by
Sabbah’s generalized $b$-functions. This can be further generalized to the
definition of the Bernstein-Sato ideal $B_{\bf f}$ of ${\bf f}$ (see for
instance [Bud15]).

Now we discuss the relations in between $V$-filtrations (and/or $b$-functions)
and singularities in algebraic geometry. The $b$-function of $f$ provides a
useful tool to study singularities of the divisor of $f$, see for instance
[Kas77, ELS+04, BS05, Kol97, Ste88]. For multiple functions, Budur, Mustaţǎ
and Saito [BMS06] defined the Bernstein-Sato polynomials (or $b$-functions)
for arbitrary schemes in $X=\mathbb{C}^{n}$ by considering the $V$-filtration
along smooth subvarieties (see Definition 4.2).111The $V$-filtration in _loc.
cit._ is the $\mathbb{Q}$-indexed one. One can refine the $\mathbb{Z}$-index
to the $\mathbb{Q}$-index by a standard procedure using $b$-functions. More
precisely, they considered the $V$-filtration and $b$-functions along the
slope $L=(1,1,\dots,1)$ for the holonomic module $\iota_{{\bf
f}+}\mathscr{O}_{X}$, where ${\bf f}=(f_{1},\dots,f_{r})$ generate the ideal
of an affine scheme. They can reinterpret the log-canonical threshold as well
as other jumping coefficients of the multiplier ideals of the scheme [Laz] as
roots of the $b$-function of the $V$-filtration ([BMS06, Theorem 2]). By
Theorem 4.23, one can see that if $L=(1,\dots,1)$ is not a slope of an adapted
cone of certain good $\mathbb{Z}^{r}$-filtration on $\iota_{{\bf
f}+}\mathscr{O}_{X}$, then the $b$-function of the scheme is irrelevant to the
generalized $b$-function of Sabbah. In fact, this can be further refined in
terms of Bernstein-Sato ideals with the help of a result of Maisonobe. More
precisely, by [Mai16a, Résultat 4], if $L=(1,\dots,1)$ is not a slope of the
codimension one components of the zero locus of $B_{\bf f}$, then the
$b$-function of the scheme defined by ${\bf f}$ is irrelevant to the
Bernstein-Sato ideal of ${\bf f}$.

By Theorem 1.3, one can now consider the $V$-filtration and the $b$-function
of $\iota_{{\bf f}+}\mathscr{O}_{X}$ along an arbitrary slope $L$. It would be
interesting to know whether there exist algebro-geometric interpretations of
the jumping indices of the $V$-filtration and the roots of the $b$-function of
$\iota_{{\bf f}+}\mathscr{O}_{X}$ along $L$.

Mustaţǎ and Popa [MP19a, MP19b, MP20] defined Hodge ideals (see also [Sai16])
by considering the Hodge filtration of the $\mathscr{D}$-module
$\mathscr{O}_{X}(*D)$ using mixed Hodge modules. It is then natural to ask
whether there exist connections between $V$-filtrations of $\iota_{{\bf
f}+}\mathscr{O}_{X}(*D)$ along $L$ and Hodge ideals of $D$, where
$\prod_{i}f_{i}=0$ defines the divisor $D$.

###  1.7. Outline

In §2, we discuss the general theory of relative $\mathscr{D}$-modules. Then
we discuss log $\mathscr{D}$-modules and the proof of Theorem 1.6 in §3. §4 is
about the generalized $V$-filtrations and their relations with relative
$\mathscr{D}$-modules. Most of the results in §4 are essentially due to
Sabbah. We give a down-to-earth exposition in §4, for the reason that the
beautiful theory of multi-indexed filtrations of Sabbah seems to be not widely
known. Also, the construction of $V$-filtrations along arbitrary slopes seems
to be missing in the literature, to the best of our knowledge. For instance,
as mentioned above only the $V$-filtration along the slope $L=(1,\dots,1)$ was
studied in [BMS06]. Finally, we recall the graph construction of Malgrange and
prove the constructibility of log de Rham complexes in §5.

###  1.8. Convention

Throughout this paper, we discuss sheaves of modules on either algebraic or
analytic spaces (or both). If the underlying space is algebraic (resp.
analytic), then the sheaves of modules on it are all assumed to be algebraic
(resp. analytic). But, when we discuss constructible complexes (or perverse
sheaves) on a complex algebraic variety, we always use the Euclidean topology.
If $f:X\to Y$ is a morphism of (algebraic or analytic) schemes, we use
$f^{-1}$ and $f_{*}$ to denote the sheaf-theoretical inverse and direct image
functors respectively.

### Acknowledgement

The author thanks Peng Zhou and Nero Budur for useful discussions, Claude
Sabbah for answering questions and Yajnaseni Dutta and Ruijie Yang for useful
comments.

##  2\. Relative $\mathscr{D}$-modules

###  2.1. Relative characteristic cycles

We recall the theory of $\mathscr{D}$-modules under the relative setting. We
mainly follow [Sch12, Chapter III. 1.3]. Suppose that
$\varphi\colon\mathcal{X}\to\mathcal{S}$ is a smooth morphism (i.e. $d\varphi$
is surjective everywhere) of complex smooth algebraic varieties (or complex
manifolds). We write by $\mathscr{T}_{\mathcal{X}/\mathcal{S}}$ the sheaf of
vector fields tangent to the leaves of $\phi$. We then have an inclusion

$\mathscr{T}_{\mathcal{X}/\mathcal{S}}\hookrightarrow\mathscr{T}_{\mathcal{X}}.$

Then the sheaf of rings of relative differential operators associated to
$\varphi$ is defined to be the subalgebra

$\mathscr{D}_{\varphi}=\mathscr{D}_{\mathcal{X}/\mathcal{S}}\subseteq\mathscr{D}_{X}$

generated by $\mathscr{T}_{X/Y}$ and $\mathscr{O}_{X}$. Similar to the
absolute case, $\mathscr{D}_{\mathcal{X}/\mathcal{S}}$ is a coherent and
noetherian sheaf of rings. Modules over
$\mathscr{D}_{\mathcal{X}/\mathcal{S}}$ are called relative
$\mathscr{D}$-modules over $\mathcal{S}$. We also write by
$\Omega^{1}_{\mathcal{X}/\mathcal{S}}$ the relative cotangent sheaf which is
defined to be the $\mathscr{O}$-dual of
$\mathscr{T}_{\mathcal{X}/\mathcal{S}}$. Since
$\mathscr{D}_{\mathcal{X}/\mathcal{S}}$ is not commutative, we have both right
and left $\mathscr{D}_{\mathcal{X},\mathcal{S}}$-modules and the side-change
operator is given by tensoring
$\omega_{\mathcal{X}/\mathcal{S}}\coloneqq\wedge^{m}\Omega^{1}_{\mathcal{X}/\mathcal{S}}$
with its quasi-inverse by tensoring $\omega^{-1}_{\mathcal{X}/\mathcal{S}}$,
where $m=\dim\mathcal{X}-\dim\mathcal{S}$.

Since $\varphi$ is smooth, we have a short exact sequence of cotangent bundles

$0\to\mathcal{X}\times_{\mathcal{S}}T^{*}\mathcal{S}\to T^{*}X\to
T^{*}(\mathcal{X}/S)\to 0.$

The filtration $F_{\bullet}\mathscr{D}_{X}$ by the orders of differential
operators induces on $\mathscr{D}_{\mathcal{X}/\mathcal{S}}$ the order
filtration $F_{\bullet}\mathscr{D}_{\mathcal{X}/S}$. Then the associated
graded sheaf of rings $\textup{gr}_{\bullet}^{F}\mathscr{D}_{\mathcal{X}/S}$
gives the algebraic structure sheaf of $T^{*}(\mathcal{X}/S)$ by the
$\sim$-functor. In the analytic case, $\mathscr{O}_{T^{*}(\mathcal{X}/S)}$ is
a faithfully flat ring extension of
$\widetilde{\textup{gr}}_{\bullet}^{F}\mathscr{D}_{\mathcal{X}/S}$ by GAGA.

For a coherent $\mathscr{D}_{\mathcal{X}/\mathcal{S}}$-module $\mathscr{M}$, a
filtration $F_{\bullet}\mathscr{M}$ over
$F_{\bullet}\mathscr{D}_{\mathcal{X}/\mathcal{S}}$ is called _good_ if
$\textup{gr}^{F}_{\bullet}\mathscr{M}$ is coherent over
$\textup{gr}^{F}_{\bullet}\mathscr{D}_{\mathcal{X}/\mathcal{S}}$. Conversely,
if there exists a filtration $F_{\bullet}\mathscr{M}$ satisfying that
$\textup{gr}^{F}_{\bullet}\mathscr{M}$ is coherent over
$\textup{gr}^{F}_{\bullet}\mathscr{D}_{\mathcal{X}/\mathcal{S}}$, then
$\mathscr{M}$ is coherent over $\mathscr{D}_{\mathcal{X}/\mathcal{S}}$. Good
filtrations for coherent $\mathscr{D}_{\mathcal{X}/\mathcal{S}}$-modules exist
locally in the analytic category and globally in the algebraic category. We
define the relative characteristic variety by

$\textup{Ch}_{{\textup{rel}}}(\mathscr{M})=\textup{Ch}_{\mathcal{X}/\mathcal{S}}(\mathscr{M})\coloneqq\textup{supp}(\widetilde{\textup{gr}}^{F}_{\bullet}\mathscr{M})\subseteq
T^{*}(\mathcal{X}/\mathcal{S}),$

where we use the $\sim$-functor of the affine morphism

$\pi:T^{*}(\mathcal{X}/\mathcal{S})\to X.$

By construction, $\textup{Ch}_{\textup{rel}}(\mathscr{M})$ is a conic
subvariety of $T^{*}(\mathcal{X}/\mathcal{S})$, where “conic” means that it is
invariant under the natural $\mathbb{C}^{\star}$-action induced by the grading
on $\textup{gr}^{F}_{\bullet}\mathscr{D}_{\mathcal{X}/\mathcal{S}}$. Each
irreducible components of $\textup{Ch}_{{\textup{rel}}}(\mathscr{M})$ has a
multiplicity. Similar to the absolute case,
$\textup{Ch}_{{\textup{rel}}}(\mathscr{M})$ and the multiplicities are
independent of good filtrations. Then the relative characteristic cycle,
denoted by $\textup{CC}_{{\textup{rel}}}(\mathscr{M})$ is the associated
locally finite cycles of $\textup{Ch}_{{\textup{rel}}}(\mathscr{M})$ with
multiplicities. If $\varphi$ is an algebraic smooth morphism between smooth
varieties over $\mathbb{C}$, then $\textup{CC}_{{\textup{rel}}}(\mathscr{M})$
is an algebraic cycle inside the algebraic relative cotangent bundle
$T^{*}(\mathcal{X}/\mathcal{S})$.

Similar to the absolute case, for a relative differential operator
$P\in\mathscr{D}_{\mathcal{X}/\mathcal{S}}$ of order $k$, we can define its
principal symbol, which gives a section of homogeneous degree $k$ in
$\textup{gr}^{F}_{\bullet}\mathscr{D}_{\mathcal{X}/\mathcal{S}}$. By [Bj93,
3.24 Definition], we obtain the relative Poisson bracket on
$\textup{gr}^{F}_{\bullet}\mathscr{D}_{\mathcal{X}/\mathcal{S}}$ and hence the
relative Poisson bracket on $\mathscr{O}_{T^{*}(\mathcal{X}/\mathcal{S})}$ (by
faithful flatness). A subvariety of $T^{*}(\mathcal{X}/\mathcal{S})$ is called
(relative) _involutive_ if its radical ideal sheaf is closed under the Poisson
bracket. Then by Gabber’s involutive theorem (see [Bj93, Appendix III, 3.25
Theorem]), we obtain:

######  Theorem 2.1 (Gabber’s Involutivity).

Suppose that $\mathscr{M}$ is a coherent
$\mathscr{D}_{\mathcal{X}/\mathcal{S}}$-module $($left or right$)$. Then
$\textup{Ch}_{\textup{rel}}(\mathscr{M})$ is $($relative$)$ involutive.

Notice that the fibers of a relative involutive subvariety
$\mathcal{Z}\subseteq T^{*}(\mathcal{X}/\mathcal{S})$ are not necessarily
involutive. One reason is that the intersections

$\mathcal{Z}_{s}=\mathcal{Z}\cap T^{*}(\mathcal{X}/\mathcal{S})_{s}\subseteq
T^{*}\mathcal{X}_{s}$

are not always proper intersections for $s\in\mathcal{S}$. If additionally
$\mathcal{Z}$ is smooth over $\mathcal{S}$, then one can easily check that
$\mathcal{Z}_{s}\subseteq T^{*}\mathcal{X}_{s}$ is either empty or involutive.
However, we have the following relative Bernstein inequality:

######  Theorem 2.2 (Relative Bernstein Inequality of Maisonobe).

Suppose that $\mathscr{M}$ is a coherent
$\mathscr{D}_{\mathcal{X}/\mathcal{S}}$-module $($left or right$)$. If
$\textup{Ch}_{\textup{rel}}(\mathscr{M})_{s}$ is not empty for
$s\in\mathcal{S}$, then all the irreducible components of
$\textup{Ch}_{\textup{rel}}(\mathscr{M})_{s}$ are of dimension
$\geq\dim\mathcal{X}-\dim\mathcal{S}$.

###### Proof.

The proof is essentially the same as that of [Mai16a, Proposition 5], where
the author only discussed relative $\mathscr{D}$-modules over
$\mathcal{S}=\mathbb{C}^{r}$. For completeness, we sketch the proof in
general. We take a smooth point of $\textup{Ch}_{\textup{rel}}(\mathscr{M})$
and focus on an open neighborhood $W$ around it. By generic smoothness (or
Morse-Sard Theorem for critical values in the analytic case),
$\textup{Ch}_{\textup{rel}}(\mathscr{M})\cap W$ is smooth over an open
neighborhood $U$ of $\mathcal{S}$ (shrink $W$ if necessary). Then the relative
involutivity in Theorem 2.1 and the relative smoothness imply that
$(\textup{Ch}_{\textup{rel}}(\mathscr{M})\cap W)\cap T^{*}\mathcal{X}_{s}$ is
involutive in $T^{*}\mathcal{X}_{s}$ for $s\in\mathcal{S}$. In particular, the
dimension of

$(\textup{Ch}_{\textup{rel}}(\mathscr{M})\cap W)\cap T^{*}\mathcal{X}_{s}$

is $\geq\dim\mathcal{X}-\dim\mathcal{S}$. Therefore, the required statement
follows from the upper semicontinuity of the dimension of fibers of
$\textup{Ch}_{\textup{rel}}(\mathscr{M})$. ∎

The following lemma is the relative analogue of [Kas03, Proposition 2.10]. We
leave its proof for interested readers. See also [BVWZ21a, Lemma 3.2.2].

######  Lemma 2.3.

If

$0\to\mathscr{M}^{\prime}\longrightarrow\mathscr{M}\longrightarrow\mathscr{M}^{\prime\prime}\to
0$

is a short exact sequence of coherent
$\mathscr{D}_{\mathcal{X}/\mathcal{S}}$-modules, then

$\textup{Ch}_{\textup{rel}}(\mathscr{M})=\textup{Ch}_{\textup{rel}}(\mathscr{M}^{\prime})\cup\textup{Ch}_{\textup{rel}}(\mathscr{M}^{\prime\prime}).$

Following [Sab87b], we define the relative holonomicity as follows.

######  Definition 2.4 (Relative holonomicity).

A coherent $\mathscr{D}_{\mathcal{X}/\mathcal{S}}$-module $\mathscr{M}$ is
called relative holonomic over $\mathcal{S}$ (or $\mathscr{O}_{S}$) if its
characteristic variety $\textup{Ch}_{\textup{rel}}(\mathscr{M})$ is relative
Lagrangian, that is, the fiber $\textup{Ch}_{\textup{rel}}(\mathscr{M})_{s}$
is either empty or a (possibly reducible) Lagrangian subvariety in
$T^{*}\mathcal{X}_{s}$ for every $s\in\mathcal{S}$.222This relative
holonomicity is slightly more general than the ones in [Mai16a, MFS16,
BVWZ21a, BVWZ21b], where the latter requires additionally that the relative
Lagrangian subvarieties are independent of $s\in\mathcal{S}=\mathbb{C}^{r}$.

Following from Lemma 2.3 and Theorem 2.2, we immediately have:

######  Corollary 2.5.

Relative holonomicity is preserved by subquotients and extensions. In
particular, the category of relative holonomic modules is abelian.

###  2.2. Base change for relative $\mathscr{D}$-modules

We now discuss the base change for relative $\mathscr{D}_{X}$-modules. Suppose
we have the following commutative diagram,

(2)
${\mathcal{X}_{\mathcal{T}}}$${\mathcal{X}}$${\mathcal{T}}$${\mathcal{S}}$$\scriptstyle{\mu}$$\scriptstyle{\nu}$

so that
$\mathcal{X}_{\mathcal{T}}=\mathcal{X}\times_{\mathcal{S}}\mathcal{T}$.
Suppose $\mathscr{M}$ is a (left)
$\mathscr{D}_{\mathcal{X}/\mathcal{S}}$-module. We consider the
$\mathscr{O}$-pullback through $\mu$:

$\mu^{*}(\mathscr{M})=\mathscr{O}_{\mathcal{X}_{\mathcal{T}}}\otimes_{\mu^{-1}\mathscr{O}_{\mathcal{X}}}\mu^{-1}\mathscr{M}.$

Since
$\mu^{*}{\mathscr{D}_{\mathcal{X}/\mathcal{S}}}=\mathscr{D}_{\mathcal{X}_{\mathcal{T}}/\mathcal{T}}$,
$\mu^{*}(\mathscr{M})$ is naturally a relative $\mathscr{D}$-module over
$\mathcal{T}$. We then have the derived pullback functor $\mathbf{L}\mu^{*}$
for relative $\mathscr{D}$-modules. When the relative $\mathscr{D}$-module
structure is forgotten, it is exactly the derived $\mathscr{O}$-module
pullback functor. One can easily see that the derived functor
$\mathbf{L}\mu^{*}$ for relative $\mathscr{D}$-module preserves coherence,
thanks to the identification
$\mu^{*}{\mathscr{D}_{\mathcal{X}/\mathcal{S}}}=\mathscr{D}_{\mathcal{X}_{\mathcal{T}}/\mathcal{T}}$
again.

We write by $i_{s}\colon\mathcal{X}_{s}\hookrightarrow\mathcal{X}$ the closed
embedding of the fiber over $s\in\mathcal{S}$. A relative holonomic
$\mathscr{D}_{\mathcal{X}/\mathcal{S}}$-module $\mathcal{M}$ is _regular_ if
$\mathbf{L}i^{*}_{s}(\mathcal{M})$ is a complex of
$\mathscr{D}_{\mathcal{X}_{s}}$-modules with regular holonomic cohomology
sheaves for every $s\in\mathcal{S}$. The author is told by C. Sabbah that it
is not known whether the category of regular relative holonomic
$\mathscr{D}$-module is closed by taking subquotients.

For a closed subvariety $\mathcal{Z}\subseteq\mathcal{S}$, we denote by
$i_{\mathcal{Z}}\colon\mathcal{X}_{\mathcal{Z}}\hookrightarrow\mathcal{X}$ the
closed embedding with
$\mathcal{X}_{\mathcal{Z}}=\mathcal{X}\times_{\mathcal{S}}\mathcal{Z}$.

######  Lemma 2.6.

If $\mathscr{M}$ is relative holonomic over $\mathcal{S}$ and $\mathcal{Z}$ is
a smooth subvariety, then $\mathbf{L}i_{\mathcal{Z}}^{*}(\mathcal{M})$ is a
complex of relative holonomic cohomology sheaves over $\mathcal{Z}$. In
particular, $\mathbf{L}^{k}i_{s}^{*}\mathcal{M}$ is a holonomic
$\mathscr{D}_{\mathcal{X}_{s}}$-module for each $k$.

###### Proof.

It is obvious that $\mathbf{L}^{k}i_{\mathcal{Z}}^{*}(\mathscr{M})$ is
coherent over $\mathscr{D}_{\mathcal{X}_{\mathcal{Z}}/\mathcal{Z}}$ and that

$\textup{Ch}_{\textup{rel}}(\mathbf{L}^{k}i_{\mathcal{Z}}^{*}(\mathscr{M}))\subseteq\textup{Ch}_{\textup{rel}}(\mathscr{M})|_{\mathcal{Z}}\coloneqq\textup{Ch}_{\textup{rel}}(\mathscr{M})\cap\mathcal{X}_{\mathcal{Z}}.$

But $\textup{Ch}_{\textup{rel}}(\mathscr{M})|_{\mathcal{Z}}$ is relative
Lagrangian over $\mathcal{Z}$. Therefore,
$\mathbf{L}^{k}i_{\mathcal{Z}}^{*}(\mathscr{M})$ is relative holonomic by
Theorem 2.2. ∎

######  Proposition 2.7.

Suppose that $\phi:\mathcal{X}\to\mathcal{S}$ is smooth with
$\mathcal{H}\subset\mathcal{S}$ a smooth divisor, and that $\mathscr{M}$ is a
coherent $\mathscr{D}_{\mathcal{X}/\mathcal{S}}$-module. If $\mathscr{M}$ has
no torsion subsheaf supported on $\mathcal{X}_{\mathcal{H}}$ and if the cycle
$\textup{CC}_{\textup{rel}}(\mathscr{M})$ does not have components over
$\mathcal{X}_{\mathcal{H}}$, then

$\textup{CC}_{\textup{rel}}(i^{*}_{\mathcal{H}}\mathscr{M})=\textup{CC}_{\textup{rel}}(\mathscr{M})|_{\mathcal{X}_{\mathcal{H}}},$

where $i_{\mathcal{H}}:\mathcal{X}_{\mathcal{H}}\hookrightarrow\mathcal{X}$ is
the closed embedding with the fiber product
$\mathcal{X}_{\mathcal{H}}=\mathcal{X}\times_{\mathcal{S}}\mathcal{H}$.

###### Proof.

Since characteristic cycles are local, it is enough to assume $\mathcal{H}$
defined by a regular (or holomorphic) function $h$. The torsion-free
assumption implies that we have a short exact sequence

$0\to\mathscr{M}\xrightarrow{\cdot h}\mathscr{M}\rightarrow
i^{*}_{\mathcal{H}}(\mathscr{M})\to 0.$

Now we pick a good filtration $F_{\bullet}\mathscr{M}$ over
$\mathscr{D}_{\mathcal{X}/\mathcal{S}}$ and the filtration induces a filtered
complex

$F_{\bullet}\mathscr{M}\xrightarrow{\cdot h}F_{\bullet}\mathscr{M}.$

We then obtain a convergent spectral sequence with the $E^{0}$-page given by
the following length 2 complex

$\eta\colon\textup{gr}^{F}_{\bullet}\mathscr{M}\xrightarrow{\cdot
h}\textup{gr}^{F}_{\bullet}\mathscr{M}.$

Then, $i^{*}_{\mathcal{H}}(\mathscr{M})$ has an induced filtration
$F_{\bullet}(i^{*}_{\mathcal{H}}(\mathscr{M}))$ (good over
$\mathscr{D}_{\mathcal{X}_{\mathcal{H}}/\mathcal{H}}$). By convergence of the
spectral sequence (see for instance [Lau83, Lemme 3.5.13] and also [Sab87b,
3.7. Lemme]), we have

$[\textup{gr}^{F}_{\bullet}(i^{*}_{\mathcal{H}}(\mathscr{M}))]=[\textup{Coker}(\eta)]-[\textup{Ker}(\eta)]$

in the Grothendieck group $K_{0}$. Since the characteristic cycle is well
defined for $[\textup{gr}^{F}_{\bullet}(i^{*}_{\mathcal{H}}(\mathscr{M}))]$,
the required statement for characteristic cycles follows from [Bj93, Appendix
IV. 3.13 Proposition]. ∎

Suppose that $\mathscr{N}$ is a (left)
$\mathscr{D}_{\mathcal{X}_{\mathcal{T}}/\mathcal{T}}$-module (or more
generally a complex). We consider the derived pushforward functor
$\mathbf{R}\mu_{*}(\mathscr{N})$ with $\mu$ as in Diag.(2). Since
$\mu^{*}{\mathscr{D}_{\mathcal{X}/\mathcal{S}}}=\mathscr{D}_{\mathcal{X}_{\mathcal{T}}/\mathcal{T}}$,
we have

$\mathbf{R}\mu_{*}(\mathscr{N})\simeq\mathbf{R}\mu_{*}(\mu^{*}(\mathscr{D}_{\mathcal{X}/\mathcal{S}})\otimes_{\mathscr{D}_{\mathcal{X}_{\mathcal{T}}/\mathcal{T}}}\mathscr{N}),$

and hence the complex $\mathbf{R}\mu_{*}(\mathscr{N})$ is a complex of
$\mathscr{D}_{\mathcal{X}/\mathcal{S}}$-modules by adjunction.

######  Proposition 2.8.

Suppose that $\nu$ is a proper morphism, $\mathscr{N}$ is coherent over
$\mathscr{D}_{\mathcal{X}_{\mathcal{T}}/\mathcal{T}}$ $($or more generally a
complex of $\mathscr{D}_{\mathcal{X}_{\mathcal{T}}/\mathcal{T}}$-modules with
coherent cohomology sheaves$)$ and $\mathcal{N}$ $($or each of its cohomology
sheaves$)$ admits a good filtration over
$F_{\bullet}\mathscr{D}_{\mathcal{X}/\mathcal{S}}$. Then
$\mathbf{R}^{i}\mu_{*}(\mathscr{N})$ is coherent over
$\mathscr{D}_{\mathcal{X}/\mathcal{S}}$ for each $i\in\mathbb{Z}$.

###### Proof.

The idea of the proof of the required coherence result is similar to that of
the case for absolute $\mathscr{D}$-modules (see for instance [Bj93, Theorem
2.8.1]). We sketch here its proof for completeness.

By a standard procedure (see for instance the proof of [Bj93, Theorem 1.5.8]),
the required statement can be reduced to the case for

$\mathscr{N}=\mathscr{D}_{\mathcal{X}_{\mathcal{T}}/\mathcal{T}}\otimes_{\mathscr{O}}\mathscr{L}$

where $\mathscr{L}$ is a coherent $\mathscr{O}$-module. But this case follows
immediately from the Grauert’s direct image theorem for $\mathscr{O}$-modules
and the projection formula (since
$\mu^{*}{\mathscr{D}_{\mathcal{X}/\mathcal{S}}}=\mathscr{D}_{\mathcal{X}_{\mathcal{T}}/\mathcal{T}}$).
∎

It is worth mentioning that $\mathbf{R}\mu_{*}$ does not preserve relative
holonomicity under proper base changes in general (compared to Proposition
2.10). See §4.10 for related examples.

###  2.3. Relative de Rham complexes

We keep notations as in the previous subsection. Let $\mathscr{M}$ be a (left)
$\mathscr{D}_{\mathcal{X}/\mathcal{S}}$-module. The _relative de Rham complex_
of $\mathscr{M}$ is defined as

$\textup{DR}_{\mathcal{X}/\mathcal{S}}(\mathscr{M})\coloneqq\omega_{\mathcal{X}/\mathcal{S}}\otimes^{\mathbf{L}}_{\mathscr{D}_{\mathcal{X}/\mathcal{S}}}\mathscr{M}.$

######  Lemma 2.9.

We have a natural isomorphism

$\mathbf{L}i^{*}_{s}\circ\textup{DR}_{\mathcal{X}/\mathcal{S}}\simeq\textup{DR}_{\mathcal{X}_{s}}\circ\mathbf{L}i^{*}_{s}$

for $s\in\mathcal{S}$.

###### Proof.

The functor $\mathbf{L}i^{*}_{s}$ can be rewritten as
$\otimes^{\mathbf{L}}_{\mathscr{O}_{\mathcal{S}}}\mathbb{C}_{s}$, where
$\mathbb{C}_{s}$ is the residue field of $s\in\mathcal{S}$. Since sections of
$\mathscr{O}_{\mathcal{S}}$ contains in the center of
$\mathscr{D}_{\mathcal{X}/\mathcal{S}}$, the required statement follows. ∎

###  2.4. Relative direct images

We discuss direct image functors for $\mathscr{D}$-modules under the relative
setting. We fix a morphism $f$ over $\mathcal{S}$

${\mathcal{Y}}$${\mathcal{X}}$${\mathcal{S}}$$\scriptstyle{f}$$\scriptstyle{\varphi_{1}}$$\scriptstyle{\varphi}$

with $\varphi_{1}$ and $\varphi$ smooth. We recall the definition of the
relative direct image:

$f_{+}(\mathscr{N})\coloneqq\mathbf{R}f_{*}(\mathscr{M}\otimes_{\mathscr{O}}\omega_{f/\mathcal{S}}\otimes_{\mathscr{D}}f^{*}\mathscr{D}_{\mathcal{X}/\mathcal{S}})$

for a left $\mathscr{D}_{\mathcal{Y}/\mathcal{S}}$-module $\mathscr{N}$ (or
more generally a complex of left
$\mathscr{D}_{\mathcal{Y}/\mathcal{S}}$-modules), where
$\omega_{f/\mathcal{S}}=\omega_{\mathcal{Y}/\mathcal{S}}\otimes
f^{*}(\omega^{-1}_{\mathcal{X}/\mathcal{S}})$.

The morphism $f$ over $\mathcal{S}$ induces a relative Lagrangian
correspondence

${T^{*}(\mathcal{Y}/\mathcal{S})}$${\mathcal{Y}\times_{\mathcal{X}}T^{*}(\mathcal{X}/\mathcal{S})}$${T^{*}(\mathcal{X}/\mathcal{S})}$${\mathcal{S}}$$\scriptstyle{\phi_{1}}$$\scriptstyle{\varrho_{f}}$$\scriptstyle{\varpi_{f}}$$\scriptstyle{\phi}$

See for instance [HTT08, §2.4] for the absolute Lagrangian correspondence.

The following proposition is a generalization of [MFS16, Theorem 1.17(a)].

######  Proposition 2.10.

With above notations, let $\mathscr{N}$ be a coherent
$\mathscr{D}_{\mathcal{Y}/\mathcal{S}}$-module with a good filtration over
$F_{\bullet}\mathscr{D}_{\mathcal{Y}/\mathcal{S}}$.

  1. (1)

If $f$ is proper over $\mathcal{S}$, then $f_{+}(\mathscr{N})$ is a complex of
$\mathscr{D}_{\mathcal{X}/\mathcal{S}}$-modules with coherent cohomology
sheaves.

  2. (2)

If moreover $\mathscr{N}$ is relative holonomic, then $f_{+}(\mathscr{N})$ is
a complex of $\mathscr{D}_{\mathcal{X}/\mathcal{S}}$-modules with relative
holonomic cohomology sheaves.

###### Proof.

Since $(\mathscr{N},\mathbb{C}_{\mathcal{Y}})$ gives us a good relative
elliptic pair (see [SS94, Definition 2.14]), the first statement follows from
Theorem 4.2 in _loc. cit._ If moreover $\mathscr{N}$ is relative holonomic,
then by Corollary 4.3 in _loc. cit._ we have

$\textup{Ch}_{\textup{rel}}(\mathcal{H}^{i}f_{+}(\mathscr{N}))\subseteq\varpi_{f}(\varrho_{f}^{-1}(\textup{Ch}_{\textup{rel}}(\mathscr{N}))$

for each $i$. Thus, for $s\in\mathcal{S}$

$\textup{Ch}_{\textup{rel}}(\mathcal{H}^{i}f_{+}(\mathscr{N}))\cap\phi^{-1}(s)\subseteq\varpi_{f}(\varrho_{f}^{-1}(\textup{Ch}_{\textup{rel}}(\mathscr{N}))\cap\phi^{-1}(s)\subseteq\varpi_{f}(\varrho_{f}^{-1}(\textup{Ch}_{\textup{rel}}(\mathscr{N})\cap\phi_{1}^{-1}(s))).$

By definition $\textup{Ch}_{\textup{rel}}(\mathscr{N})\cap\phi_{1}^{-1}(s)$ is
Lagrangian and hence isotropic. By [Kas77, (4.9)],
$\varpi_{f}(\varrho_{f}^{-1}(\textup{Ch}_{\textup{rel}}(\mathscr{N})\cap\phi_{1}^{-1}(s))$
and hence
$\textup{Ch}_{\textup{rel}}(\mathcal{H}^{i}f_{+}(\mathscr{N}))\cap\phi^{-1}(s)$
are both isotropic. Thus,
$\textup{Ch}_{\textup{rel}}(\mathcal{H}^{i}f_{+}(\mathscr{N}))\cap\phi^{-1}(s)$
is Lagrangian by Theorem 2.2 and $\mathcal{H}^{i}f_{+}(\mathscr{N})$ is
relative holonomic. ∎

The following lemma is immediate by construction, and we skip its proof.

######  Lemma 2.11.

We have a natural isomorphism

$\mathbf{L}i^{*}_{s}\circ f_{+}\simeq(f_{s})_{+}\circ\mathbf{L}i^{*}_{s},$

where $f_{s}:\mathcal{Y}_{s}\to\mathcal{X}_{s}$ is the induced morphism over
$s\in\mathcal{S}$.

######  Corollary 2.12.

If $f$ is (relative) proper over $\mathcal{S}$, then $f_{+}$ preserves
relative regular holonomicity.

###### Proof.

If $f$ is proper over $\mathcal{S}$, then $f_{s}$ is proper for
$s\in\mathcal{S}$. Since $(f_{s})_{+}$ preserves regular holonomicity, the
required statement follows from Lemma 2.11. ∎

##  3\. Logarithmic $\mathscr{D}$-modules

In this section, we recall $\mathscr{D}$-modules under the logarithmic
setting. Let $X$ be a complex manifold of dimension $n$ and let $D$ be a
normal crossing divisor. We call such $(X,D)$ a (analytic) smooth log pair. We
write by $\mathscr{D}_{X,D}$ the subalgebra of $\mathscr{D}_{X}$ consisting of
differential operators preserving the ideal sheaf of $D$. In local coordinates
$(x_{1},x_{2},\dots,x_{n})$ on an open neighborhood $U$ with
$D|_{U}=(\prod_{i=1}^{r}x_{i}=0)$ for some $r\leq n$, $\mathscr{D}_{X,D}$ is
the subalgebra generated by $\mathscr{O}_{U}$ and

$x_{1}\partial_{x_{1}},\dots,x_{k}\partial_{x_{r}},\partial_{x_{r+1}},\dots,\partial_{x_{n}}.$

Since $D$ is normal crossing, $\mathscr{D}_{X,D}$ is a coherent and noetherian
sheaf of rings. Modules over $\mathscr{D}_{X,D}$ are called logarithmic (log)
$\mathscr{D}$-modules.

The order filtration $F_{\bullet}\mathscr{D}_{X}$ induces the order filtration
$F_{\bullet}\mathscr{D}_{X,D}$ such that the analytification of
$\textup{gr}^{F}_{\bullet}\mathscr{D}_{X,D}$ gives us the structure sheaf of
the log cotangent bundle $T^{*}(X,D)$. For a coherent
$\mathscr{D}_{X,D}$-module $\mathcal{M}$, one can define the log
characteristic variety

$\textup{Ch}_{\log}(\mathcal{M})\subseteq T^{*}(X,D),$

and the log characteristic cycle $\textup{CC}_{\log}(\mathcal{M})$ similar to
the relative case. When $D=\emptyset$, we use $\textup{Ch}(\mathcal{M})$
(resp. $\textup{CC}(\mathcal{M})$) to denote the characteristic variety (resp.
cycle) of the $\mathscr{D}_{X}$-module $\mathcal{M}$.

###  3.1. Log de Rham complexes

Suppose that $\mathcal{M}$ is a left $\mathscr{D}_{X,D}$-module on a smooth
log pair $(X,D)$. Similar to the absolute case, $\omega_{X}(D)$ is a right
$\mathscr{D}_{X,D}$-module, where $\omega_{X}$ is the sheaf of the top forms
on $X$. The _log de Rham complex_ of $\mathcal{M}$ is defined as

$\textup{DR}_{X,D}(\mathcal{M})\coloneqq\omega_{X}(D)\otimes^{\mathbf{L}}_{\mathscr{D}_{X,D}}\mathcal{M}.$

By [WZ21, Lemma 2.3], we have

$\textup{DR}_{X,D}(\mathcal{M})\simeq[\mathcal{M}\rightarrow\Omega^{1}(\log
D)\otimes_{\mathscr{O}}\mathcal{M}\rightarrow\cdots\rightarrow\Omega^{n}(\log
D)\otimes_{\mathscr{O}}\mathcal{M}]$

where the complex on the right-hand side starts from the degree $-n$-term and
$\Omega^{i}(\log D)$ denotes the sheaf of the degree $i$ log forms. In local
coordinates

$(x_{1},x_{2},\dots,x_{n})\textup{ with }D|_{U}=(\prod_{i=1}^{r}x_{i}=0)$

on an open neighborhood $U$ for some $r\leq n$, we further have

(3)
$\textup{DR}_{X,D}(\mathcal{M})\simeq\textup{Kos}(\mathcal{M};x_{1}\partial_{x_{1}},\dots,x_{r}\partial_{x_{r}},\partial_{x_{r+1}},\dots,\partial_{x_{n}}),$

where Kos denotes the Koszul complex of the actions
$x_{1}\partial_{x_{1}},\dots,x_{r}\partial_{x_{r}},\partial_{x_{r+1}},\dots,\partial_{x_{n}}$
on $\mathcal{M}$.

###  3.2. Direct image functor

Suppose that $f\colon(X,D)\rightarrow(Y,E)$ is a morphism of smooth log pairs,
that is, $f\colon X\rightarrow Y$ is a morphism of complex manifolds such that
$f^{-1}E\subseteq D$. Then we define the derived direct image functor for left
log $\mathscr{D}$-modules $\mathcal{M}$ by

$f^{\log}_{+}(\mathcal{M})=Rf_{*}(\mathcal{M}\otimes_{\mathscr{O}}\omega_{f}\otimes^{\mathbf{L}}_{\mathscr{D}_{X,D}}f^{*}\mathscr{D}_{Y,E}),$

where $\omega^{\log}_{f}$ is the canonical sheaf of $f$,

$\omega^{\log}_{f}=\omega_{X}(D)\otimes_{\mathscr{O}}f^{*}(\omega^{-1}_{Y}(E)).$

The above definition is compatible with the direct images in the relative case
in §2.4 if one take $D$ and $E$ empty.

######  Proposition 3.1.

Let $f\colon(X,D)\rightarrow(Y,E)$ be a morphism of smooth log pairs and let
$\mathcal{M}$ be a $($left$)$ $\mathscr{D}_{X,D}$-modules. Then

$Rf_{*}\textup{DR}_{X,D}(\mathcal{M})\simeq\textup{DR}_{Y,E}(f^{\log}_{+}\mathcal{M}).$

###### Proof.

The proof of this proposition is exactly the same as the non-log case (cf.
[HTT08, Theorem 4.2.5]). We leave the detail for interested readers. ∎

###  3.3. Lattices

We recall the definition of lattices in the analytic setting. Suppose that
$(X,D)$ is an analytic smooth log pair and $\mathcal{M}$ is a coherent
$\mathscr{D}_{X}$-module. We write the algebraic localization of $\mathcal{M}$
along $D$ by

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

where

$\mathscr{O}_{X}(*D)=\lim_{k\to\infty}\mathscr{O}_{X}(kD).$

Notice that in general $\mathcal{M}(*D)$ is not even coherent over
$\mathscr{D}_{X}$. However, if $\mathcal{M}$ is holonomic, then so is
$\mathcal{M}(*D)$, since holonomicity is preserved under tenser products over
$\mathscr{O}$. A coherent $\mathscr{D}_{X,D}$-submodule

$\bar{\mathcal{M}}\subseteq\mathcal{M}(*D)$

is called a $\mathscr{D}_{X,D}$-_lattice_ of $\mathcal{M}(*D)$ (or
$\mathcal{M}$) if $\bar{\mathcal{M}}|_{X\setminus D}=\mathcal{M}|_{X\setminus
D}$. By definition, lattices of $\mathcal{M}$ do not have torsion subsheaves
supported on $D$ (although, $\mathcal{M}$ might have). The prototype examples
of lattices are Deligne lattices of local systems (see [WZ21, §4.4]).

###  3.4. From log to relative

In this subsection, we discuss the connection between log
$\mathscr{D}$-modules and relative $\mathscr{D}$-modules. We focus locally on
$W$ a polydisc,

$W=\mathbb{D}^{k}_{x}\times\mathbb{D}_{t}^{r}\textup{ with coordinates
}(x_{1},\dots,x_{k},t_{1},\dots,t_{r})$

and a divisor $D$ given by $t_{1}\cdot t_{2}\cdots t_{r}=0$. In particular,
$(W,D)$ is a smooth log pair. We consider the log rescaled families

$\widetilde{W}_{j}=W\times\mathbb{D}_{y}^{j}$

with $y(j)=(y_{1},\dots,y_{j})$ the coordinates on $\mathbb{D}_{y}^{j}$ for
$0\leq j\leq r$ ($\widetilde{W}_{0}=W$), and the maps

$p_{j}\colon\widetilde{W}_{j}\rightarrow W,(x,t,y(j))\mapsto(x,e^{y(j)}t),$

where we abbreviate $(x_{1},\dots,x_{r})$ as $x$,
$(e^{y_{1}}t_{1},\dots,e^{y_{j}}t_{j},t_{j+1},\dots,t_{r})$ as $e^{y(j)}x$ and
etc. Then we have the commutative diagram for each $0\leq j<k$

(4)
${\widetilde{W}_{j}}$${\widetilde{W}_{j+1}}$${W}$$\scriptstyle{i_{j}}$$\scriptstyle{p_{j}}$$\scriptstyle{p_{j+1}}$

where the inclusion is

$i_{j}:\widetilde{W}_{j}\hookrightarrow\widetilde{W}_{j+1},(x,t,y(j))\mapsto(x,t,y(j),0).$

Let $U=\mathbb{D}^{k}_{x}\times(\mathbb{D}^{\star})^{r}_{t}\subset W$ with
$\mathbb{D}^{\star}$ the punctured disk. Then we define
$\widetilde{U}_{j}=p^{-1}(U)$, $\widetilde{D}_{j}=p^{-1}_{j}(D)$. Given a
$\mathscr{D}_{W,D}$-lattice $\bar{\mathcal{M}}$ of a $\mathscr{D}_{X}$-module
$\mathcal{M}$, we consider the pull-backs $\mathscr{M}_{j}\coloneqq
p^{*}_{j}\bar{\mathcal{M}}$ and write $p_{k}=p$,
$\mathscr{M}=\mathscr{M}_{k}$, $\widetilde{W}=\widetilde{W}_{k}$,
$\widetilde{U}_{k}=\widetilde{U}$ and $\widetilde{D}=\widetilde{D}_{k}$.

######  Lemma 3.2.

With notations above, let $\mathcal{M}$ be a coherent
$\mathscr{D}_{X}$-module. Then

  1. (1)

$p_{j}$ is smooth $($submersive$)$ for each $j$,

  2. (2)

$\mathscr{M}_{j}$ is a coherent
$\mathscr{D}_{\widetilde{W}_{j},\widetilde{D}_{j}}$-module for each $j$,

  3. (3)

$\mathscr{M}$ is a coherent
$\mathscr{D}_{\widetilde{W}/\mathbb{D}^{r}_{t}}$-module for the natural
projection

$\pi_{t}:\widetilde{W}\to\mathbb{D}^{r}_{t}.$

###### Proof.

Part (1) is obvious. For Part (2), we pick a good filtration
$F_{\bullet}\bar{\mathcal{M}}$ over $F_{\bullet}\mathscr{D}_{X,D}$. Since
$p_{j}$ is smooth, $p_{j}^{*}(F_{\bullet}\bar{\mathcal{M}})$ is a filtration
of $\mathscr{M}_{j}$ over $F_{\bullet}\mathscr{D}_{\widetilde{W}_{j},D_{j}}$.
By construction, we have

$t_{i}\partial_{t_{i}}\cdot(1\otimes p_{j}^{-1}(m))=1\otimes
p_{j}^{-1}(t_{i}\partial_{t_{i}}\cdot m)$

for sections $m$ of $\bar{\mathcal{M}}$. Therefore,
$p^{*}_{j}(\textup{gr}^{F}_{\bullet}\mathcal{M})$ is coherent over
$\textup{gr}^{F}_{\bullet}\mathscr{D}_{\widetilde{W}_{j},\widetilde{D}_{j}}$
and hence $p_{j}^{*}(F_{\bullet}\bar{\mathcal{M}})$ is a good filtration of
$\mathscr{M}_{j}$ over $F_{\bullet}\mathscr{D}_{\widetilde{W}_{j},D_{j}}$. In
particular, $\mathscr{M}_{j}$ is coherent over
$\mathscr{D}_{\widetilde{W}_{j},\widetilde{D}_{j}}$. For Part (3), one
observes that $t_{i}\partial_{t_{i}}-\partial_{y_{i}}$ annihilates $1\otimes
p_{j}^{-1}(m)$. Thus, $p^{*}(\textup{gr}^{F}_{\bullet}\mathcal{M})$ is
coherent over
$\textup{gr}^{F}_{\bullet}\mathscr{D}_{\widetilde{W}/\mathbb{D}^{r}_{t}}$. In
consequence, $\mathscr{M}$ is coherent over
$\mathscr{D}_{\widetilde{W}/\mathbb{D}^{r}_{t}}$. ∎

######  Theorem 3.3.

Let $\mathcal{M}$ be a regular holonomic $\mathscr{D}_{W}$-module and let
$\bar{\mathcal{M}}$ be a $\mathscr{D}_{W,D}$-lattice of $\mathcal{M}$. Then we
have:

  1. (1)

If $r=1$, then $\mathscr{M}$ is a relative regular holonomic
$\mathscr{D}_{\widetilde{W}/\mathbb{D}^{r}_{x}}$-module.

  2. (2)

If $\mathscr{M}$ is flat over $\mathbb{D}^{r}_{x}$ for some $r>1$, then
$\mathscr{M}$ is relative holonomic over $\mathbb{D}^{r}_{x}$.

If $r\geq 2$, then $\mathscr{M}$ is not necessarily relative holonomic. See
Example 2 in §5.

###  3.5. Proof of Theorem 1.6

Our first goal is to prove that, the characteristic variety
$\textup{Ch}_{\log}(\bar{\mathcal{M}})$ is the closure of
$\textup{Ch}(\mathcal{M}|_{U})$ in the log cotangent bundle $T_{\log}^{*}W$,
which is equivalent to prove that $\textup{Ch}_{\log}(\bar{\mathcal{M}})$ has
no irreducible components over $D$. Then the statement of characteristic
cycles follows immediately, since multiplicities are generically defined.

Our main tool is a technical result of Sabbah [Sab87b] (see also [BMM02]).
Consider a smooth submersion $\varphi:\mathcal{X}\to\mathcal{S}$. Let
$\mathcal{N}$ be a regular hononomic $\mathscr{D}_{\mathcal{X}}$-module,
$\mathcal{N}_{{\textup{rel}}}$ be a coherent
$\mathscr{D}_{\mathcal{X}/\mathcal{S}}$-submodule of $\mathcal{N}$, that
generates $\mathcal{N}$ as $D_{\mathcal{X}}$-module. Suppose
$\textup{Ch}(\mathcal{N})=\bigcup_{Y\in L}T_{Y}^{*}\mathcal{X}$, with a set
$L$ (locally finite) consisting of irreducible subvarieties of $\mathcal{X}$.
Let $L_{1}\subset L$ be a subset consisting of $Y$ such that $\varphi(Y)$
contains a non-empty open subset of $\mathcal{S}$. Denote by $Y^{\textup{sm}}$
the smooth locus of $Y$. By generic smoothness, we can assume
$Y^{\textup{sm}}$ is smooth over $\mathcal{S}$ (shrink $Y^{\textup{sm}}$ if
necessary). Then we obtain the relative conormal bundle of
$Y^{\textup{sm}}\hookrightarrow\mathcal{X}$ over certain open subset of
$\mathcal{S}$. We then write by
$T^{*}_{\varphi|_{Y}}(\mathcal{X}/\mathcal{S})$ the closure of the
(generically defined) relative conormal bundle, calling it the _relative
conormal space_ of $\varphi|_{Y}$. Notice that relative conormal spaces are
not necessarily relative Lagrangian.

######  Lemma 3.4.

[Sab87b, 3.2.Théorèm], [BMM02, Lemme 2.2] With notations as above, suppose
there is a non-constant holomorphic function $F:\mathcal{X}\to\mathbb{C}$,
such that $L\backslash L^{\prime}$ are contained in $F^{-1}(0)$, and
$\mathcal{N}$ are without $F$-torsion. Then, we have

$\textup{Ch}_{{\textup{rel}}}(\mathcal{N}_{{\textup{rel}}})=\bigcup_{Y\in
L_{1}}T^{*}_{\varphi|_{Y}}(\mathcal{X}/\mathcal{S}),$

where $T^{*}_{\varphi|_{Y}}(\mathcal{X}/\mathcal{S})$ is the relative conormal
space of $\varphi|_{Y}$.

Since characteristic cycles are local, it is enough to assume $X=W$, a
polydisc as in §3.4. To apply Lemma 3.4, we let
$\mathcal{N}=p^{*}(\mathcal{M}(*D))$,
$\mathcal{N}_{{\textup{rel}}}=\mathscr{M}$, $\mathcal{X}=\widetilde{W}$,
$\mathcal{S}=\mathbb{D}^{r}_{t}$, $\varphi=\pi_{t}$, and $F=t_{1}\cdots
t_{r}$. Suppose we have decomposition

$\textup{Ch}(\mathcal{M}|_{U})=\bigcup_{Y\in L_{1}}T^{*}_{Y}W$

for a set of closed strata $L_{1}$ in $U$. Then we may define a set of closed
strata $\widetilde{L}_{1}$ in $\widetilde{W}$, by sending $Y\in L_{1}$ to
$\widetilde{Y}=\overline{p^{-1}Y}\subset\widetilde{W}$. Since $p$ is
submersive, we have

$\textup{Ch}(p^{*}\widetilde{\mathcal{M}}(*D))|_{\widetilde{U}}=\bigcup_{\widetilde{Y}\in\widetilde{L}_{1}}T^{*}_{\widetilde{Y}}\widetilde{W}|_{\widetilde{U}}.$

######  Corollary 3.5.

$\textup{Ch}_{{\textup{rel}}}(\mathscr{M})=\bigcup_{\widetilde{Y}\in\widetilde{L}_{1}}T^{*}_{\pi_{t}|_{\widetilde{Y}}}(\widetilde{W}/\mathbb{D}^{k}_{x}).$

###### Proof.

By Lemma 3.4, we only need to look at strata of
$\textup{Ch}(p^{*}\mathcal{M}(*D))$ that project under $\pi_{x}$ to an open
set in $\mathbb{D}_{x}^{k}$, hence it suffices to consider the strata that
intersect with $\pi_{x}^{-1}((\mathbb{D}^{*}_{x})^{k})=\widetilde{U}$ (by the
construction of $p$). These are labeled exactly by $\widetilde{L}_{1}$. ∎

###### Proof of Theorem 3.3.

By Corollary 3.5, $\textup{Ch}_{\textup{rel}}(\mathscr{M})$ does not have
components over $t_{1}\cdots t_{r}=0$. For every point
$\alpha\in\mathbb{D}^{r}_{t}$, we pick $r$ general hyperplanes passing
$\alpha$. By flatness assumption, the relative holonomicity in Part (2)
follows from inductively applying Proposition 2.7. Since
$\mathscr{M}\subseteq\mathcal{N}=p^{*}(\mathcal{M}(*D))$, $\mathscr{M}$ is
torsion-free over $\mathbb{D}^{r}_{t}$. When $r=1$, torsion-freeness implies
flatness and hence the relative holonomicity of Part (1) follows. For
regularity in Part (1), one observes that

$i_{\alpha}^{*}(\mathscr{M})\simeq p_{\alpha}^{*}(\mathcal{M}|_{U})$

are regular holonomic $\textup{for }0\not=\alpha\in\mathbb{D}_{t}$, where the
morphism $p_{\alpha}$ is

$p_{\alpha}\colon\mathbb{D}_{x}^{k}\times\\{\alpha\\}\times\mathbb{D}_{y}\to\mathbb{D}_{x}^{k}\times\mathbb{D}_{t}\quad(x,\alpha,y)\mapsto(x,\alpha
e^{y}).$

When $\alpha=0$,

$i_{0}^{*}(\mathscr{M})\simeq p_{0}^{*}(\mathcal{M}|_{(t=0)}).$

But

$[\mathcal{M}|_{(t=0)}]=[\Phi_{t=0}(\mathcal{M})]$

in the Grothendieck group $K_{0}$ (by [Gin86, Proposition 1.1.2]). Since
regularity is well-defined for objects in $K_{0}$, $\mathcal{M}|_{(t=0)}$ is
regular holonomic by Theorem 4.5(1). The regularity in Part (2) is similar by
induction. ∎

We use $T^{*}_{\log}$ and $T^{*}_{\textup{rel}}$ to denote the log/relative
cotangent bundle.

######  Proposition 3.6.

We have:

  1. (1)

$\textup{Ch}_{\log}(\mathscr{M})=\iota(\textup{Ch}_{{\textup{rel}}}(\mathscr{M}))$
where

$T^{*}_{\log}\widetilde{W}=T^{*}\mathbb{D}_{x}^{k}\times
T^{*}_{\log}\mathbb{D}^{r}_{t}\times T^{*}\mathbb{D}^{r}_{y},\quad
T^{*}_{{\textup{rel}}}\widetilde{W}=T^{*}\mathbb{D}_{x}^{k}\times\mathbb{D}^{r}_{t}\times
T^{*}\mathbb{D}^{r}_{y}$ $\iota:T^{*}_{{\textup{rel}}}\widetilde{W}\to
T^{*}_{\log}\widetilde{W},\quad(x,t,y,\xi_{x},\xi_{y})\mapsto(x,t,y,\xi_{x},\widetilde{\xi}_{t}=\xi_{y},\xi_{y}),$

and $\widetilde{\xi}_{t}$ is the coefficient in front of $dt/t$.

  2. (2)

$\tilde{\iota}(\textup{Ch}_{\log}(\bar{\mathcal{M}}))=\textup{Ch}_{\log}(\mathscr{M})|_{\\{y=0\\}}$,
where $\tilde{\iota}\colon T^{*}_{\log}W\hookrightarrow
T^{*}_{\log}\widetilde{W}$ is given by

$(x,t,\xi_{x},\widetilde{\xi}_{t})\mapsto(x,t,\xi_{x},\widetilde{\xi}_{t},\xi_{y}=\widetilde{\xi}_{t}).$

###### Proof.

For Part(1), since for any section $s$ in $p^{-1}(\bar{\mathcal{M}})$,
$t_{i}\partial_{t_{i}}-\partial_{y_{i}}$ annihilate $s$, hence on the level of
the associated graded modules, we have $\widetilde{\xi}_{t_{i}}-\xi_{y_{i}}$
annihilate $\textup{gr}^{F}_{\bullet}(\mathscr{M})$.

Now we prove Part(2). By Lemma 3.2, we have

$\textup{gr}^{F}_{\bullet}(i_{r}^{*}(\mathscr{M}))\simeq
i_{r}^{*}(\textup{gr}^{F}_{\bullet}(\mathscr{M}))\simeq
p_{r-1}^{*}(\textup{gr}^{F}_{\bullet}\bar{\mathcal{M}}).$

Meanwhile, since $\textup{gr}^{F}_{\bullet}(\mathscr{M})$ and $\mathscr{M}$
both have no $y_{r}$-torsion, we have

$\widetilde{\textup{supp}}_{i^{*}_{r}(\textup{gr}^{F}_{\bullet}\mathscr{D}_{\widetilde{W},\widetilde{D}})}(i_{r}^{*}(\textup{gr}^{F}_{\bullet}(\mathscr{M})))=\textup{Ch}_{\log}(\mathscr{M})|_{\\{y_{r}=0\\}}\subseteq
T^{*}_{\log}\widetilde{W}|_{y_{r}=0}.$

Since $\widetilde{\xi}_{x_{r}}-\xi_{y_{r}}$ annihilates
$\textup{gr}^{F}_{\bullet}(\mathscr{M})$, we further have

$\widetilde{\textup{supp}}_{i^{*}_{r}(\textup{gr}^{F}_{\bullet}\mathscr{D}_{\widetilde{W},\widetilde{D}})}(i_{r}^{*}(\textup{gr}^{F}_{\bullet}(\mathscr{M})))=\widetilde{\iota}(\textup{Ch}_{\log}(\mathscr{M}_{r-1})).$

We then do induction backwards until we obtain Part (2). ∎

Finally, we describe the closure in log cotangent bundle.

######  Lemma 3.7.

Let $Y\subset U$ be a closed stratum, and let
$\widetilde{Y}=\overline{p^{-1}(Y)}\subset\widetilde{W}$. Then
$\tilde{\iota}(\overline{T_{Y}^{*}U}^{\log})=\iota(T^{*}_{\pi_{t}\mid\widetilde{Y}}(\widetilde{W}/\mathbb{D}_{t}^{r}))|_{\\{y=0\\}}$.

###### Proof.

Let us unpackage the definition of $\widetilde{Y}$. For each local function
$f(x,t)\in\mathscr{O}_{U}$ that vanishes on $Y$, we consider $f(x,e^{y}t)$ on
$\widetilde{W}$. The relative conormals with a general $t$ fixed,
$T^{*}_{\pi_{t}\mid\widetilde{Y}}(\widetilde{W}/\mathbb{D}_{t}^{r})$, are
generated by the relative differentials

$d_{/\pi_{t}}(p^{-1}f)=(\partial_{x}f)dx+t(\partial_{t}f)e^{y}dy.$

After we apply $\iota$, we get

$\iota(d_{/\pi_{t}}(p^{-1}f))=(\partial_{x}f)dx+t(\partial_{t}f)e^{y}\frac{dt}{t}+t(\partial_{t}f)e^{y}dy.$

Then we restrict to $\\{y=0\\}$, meaning setting $y_{i}=0$, and forget
$\xi_{y_{i}}$.

$\iota(d_{/\pi_{t}}(p^{-1}f))|_{\\{y=0\\}}=(\partial_{x}f)dx+t(\partial_{t}f)\frac{dt}{t}.$

Indeed, this gives us back the $df$ in the basis section of $T_{\log}^{*}W$,
i.e. $dx$ and $dt/t$.

The above argument works generically. Taking closure, the proof is done by the
construction of relative conormal spaces. ∎

Now the proof of Theorem 1.6 is accomplished by combining Proposition 3.6 and
Lemma 3.7.

##  4\. Relative $\mathscr{D}$-modules and $V$-filtrations

In this section, we discuss the Kashiwara-Malgrange filtrations for
$\mathscr{D}$-modules in the general sense of Sabbah by using relative
$\mathscr{D}$-modules. For simplicity, we focus on the algebraic category
unless stated otherwise, that is, all the underlying spaces and sheaves on
them are algebraic in this section. See Remark 4.12 for the analytic case.

###  4.1. Kashiwara-Malgrange filtrations

######  Definition 4.1.

Suppose that $X$ is a smooth complex variety and $Y$ is a smooth subvariety of
$X$ with its ideal sheaf denoted by $\mathscr{I}_{Y}$. Then the Kashiwara-
Malgrange filtration on $\mathscr{D}_{X}$ is a $\mathbb{Z}$-indexed increasing
filtration defined by

$V^{Y}_{k}\mathscr{D}_{X}\coloneqq\\{P\in\mathscr{D}_{X}|\quad
P\mathscr{I}_{Y}^{j}\subseteq\mathscr{I}_{Y}^{j-k}\textup{ for every
}j\in\mathbb{Z}\\}$

where $\mathscr{I}_{Y}^{j}=\mathscr{O}_{X}$ if $j\leq 0$.333In the literature,
some authors define the Kashiwara-Malgrange filtration on $\mathscr{D}_{X}$ as
the decreasing filtration, that is, $V_{Y}^{k}\mathscr{D}_{X}\coloneqq
V^{Y}_{-k}\mathscr{D}_{X}$. In particular, if $Y=H$ is a smooth hypersurface
$V^{Y}_{0}\mathscr{D}_{X}=\mathscr{D}_{X,H}$, the sheaf of logarithmic
differential operators along $H$.

We then define the associated Rees ring by

$R^{Y}_{V}\mathscr{D}_{X}\coloneqq\bigoplus_{k\in\mathbb{Z}}V^{Y}_{k}\mathscr{D}_{X}\cdot
u^{k}\subseteq\mathscr{D}_{X}[u,1/u],$

where the independent variable $u$ is used to help remember the grading.

######  Definition 4.2.

Suppose that $\mathcal{M}$ is a (left) $\mathscr{D}_{X}$-module. A
$\mathbb{Z}$-indexed increasing filtration $\Omega_{\bullet}\mathcal{M}$ is
_compatible_ with $V_{\bullet}\mathscr{D}_{X}$ if

$V^{Y}_{k}\mathscr{D}_{X}\cdot\Omega_{j}\mathcal{M}\subseteq\Omega_{k+j}\mathcal{M}\textup{
for all }k,j\in\mathbb{Z}.$

A compatible filtration $\Omega_{\bullet}\mathcal{M}$ is a _good_ filtration
over $V_{\bullet}\mathscr{D}_{X}$ if the associated Rees module

$R_{\Omega}\mathcal{M}\coloneqq\bigoplus_{k\in\mathbb{Z}}\Omega_{k}\mathcal{M}\cdot
u^{k}\subseteq\mathcal{M}[u,1/u]$

is coherent over $R^{Y}_{V}\mathscr{D}_{X}$. A good filtration
$V_{\bullet}\mathcal{M}$ is called the Kashiwara-Malgrange filtration on
$\mathcal{M}$ if there exists a monic polynomial $b(s)\in\mathbb{C}[s]$ with
its roots having real parts in $[0,1)$ so that

$b(\sum_{i}t_{i}\partial_{t_{i}}+k)\textup{ annihilates
}\textup{gr}^{V^{Y}}_{k}\mathcal{M}\coloneqq
V_{k}\mathcal{M}/V_{k-1}\mathcal{M}\textup{ for each }k\in\mathbb{Z},$

where $(t_{1}=t_{2}=\cdots=t_{r}=0)$ locally defines $Y$ and
$\partial_{t_{i}}$ are the local vector field along the smooth divisor
$(t_{i}=0)$. The monic polynomial $b(s)$ of the least degree is called the
Bernstein-Sato polynomial or $b$-function of $\mathcal{M}$ along $Y$.

One can check that the Kashiwara-Malgrange filtration on $\mathcal{M}$ is
unique if exists. It is obvious that the existence of the Kashiwara-Malgrange
filtration on $\mathcal{M}$ guarantees that $\mathcal{M}$ is coherent over
$\mathscr{D}_{X}$. A coherent $\mathscr{D}_{X}$-module $\mathcal{M}$ is called
_specializable_ along $Y$ if the Kashiwara-Malgrange filtration exists along
$Y$. Furthermore, it is called $R$-_specializable_ if the $b$-function $b(s)$
has roots in $R$, where $R$ is a subring of $\mathbb{C}$.

######  Theorem 4.3 (Kashiwara).

If $\mathcal{M}$ is holonomic over $\mathscr{D}_{X}$, then it is specializable
along every submanifold $Y\subseteq X$.

We will give a proof of the above fundamental theorem under more general
settings (cf. Theorem 4.15).

We now recall the definition of nearby cycles and vanishing cycles along
smooth hypersurfaces.

######  Definition 4.4.

Suppose that $H$ is a smooth hypersurface and it is defined by $(t=0)$
locally. We assume that $\mathcal{M}$ is specializable along $H$ with
$V_{\bullet}\mathcal{M}$ its Kashiwara-Malgrange filtration. Then the nearby
cycle of $\mathcal{M}$ along $H$ is defined by

$\Psi_{H}(\mathcal{M})\coloneqq\textup{gr}^{V}_{0}\mathcal{M}$

and the vanishing cycle is

$\Phi_{H}(\mathcal{M})\coloneqq\textup{gr}^{V}_{1}\mathcal{M}.$

Since the morphism

$t:\textup{gr}^{V}_{k}\mathcal{M}\longrightarrow\textup{gr}^{V}_{k-1}\mathcal{M}$

is an isomorphism of $\mathscr{D}_{H}$-modules for all $k\not=0$, we then have

$\Phi_{H}(\mathcal{M})\simeq\textup{gr}^{V}_{k}\mathcal{M}$

for $k>0$ and

$\Psi_{H}(\mathcal{M})\simeq\textup{gr}^{V}_{k}\mathcal{M}$

for $k\leq-1$. For a perverse sheaf $K$ (with complex coefficients) on $X$, we
use $\psi_{t}(K)$ and $\phi_{t}(K)$ to denote the nearby cycle and vanishing
cycle of $K$ along $H=(t=0)$ respectively. Let us refer to [KS13, §8.6] for
their definitions.

The following theorem of Kashiwara is the Riemann-Hilbert correspondence of
nearby and vanishing cycles.

######  Theorem 4.5.

[Kas83, Theorem 2] Suppose that $\mathcal{M}$ is regular holonomic and
$V_{\bullet}\mathcal{M}$ is the Kashiwara-Malgrange filtration along a smooth
hypersurface $H=(t=0)$. Then

  1. (1)

$\textup{gr}^{V}_{k}\mathcal{M}$ is a regular holonomic
$\mathscr{D}_{H}$-module for every $k\in\mathbb{Z}$,

  2. (2)

$\textup{DR}_{H}(\Psi_{H}(\mathcal{M}))\simeq\psi_{t}(\textup{DR}_{X}\mathcal{M})$
and
$\textup{DR}_{H}(\Phi_{H}(\mathcal{M}))\simeq\phi_{t}(\textup{DR}_{X}\mathcal{M}).$

###  4.2. Algebraic normal deformation

We now recall the normal deformation algebraically; see [KS13, §4.1] for the
topological construction.

Suppose that $Y\subseteq X$ is a smooth subvariety with the ideal sheaf
$\mathscr{I}_{Y}$. We algebraically define a space by

$\widetilde{X}_{Y}\coloneqq\textup{Spec
}[\bigoplus_{k\in\mathbb{Z}}\mathscr{I}^{-k}_{Y}\otimes u^{k}],$

where $u$ is an independent variable giving a $\mathbb{C}^{\star}$-action on
$\widetilde{X}_{Y}$. Then the natural inclusion
$\mathbb{C}[u]\hookrightarrow\bigoplus_{k\in\mathbb{Z}}\mathscr{I}^{-k}_{Y}\otimes
u^{k}$ gives rise to a smooth family

$\varphi_{Y}\colon\widetilde{X}_{Y}\to\mathbb{C}$

so that

  1. (1)

$\varphi_{Y}^{-1}(u)\simeq X$ if $0\not=u\in\mathbb{C}$;

  2. (2)

$\varphi_{Y}^{-1}(0)=\textup{Spec }[\bigoplus_{k\in\mathbb{Z}^{\geq
0}}\mathscr{I}^{k}_{Y}/\mathscr{I}_{Y}^{k+1}]\eqqcolon T_{Y}X$, the algebraic
normal bundle of $Y\subseteq X$.

By the construction of $T_{Y}X$ and $V^{Y}_{\bullet}\mathscr{D}_{X}$, one
easily observes

(5)
$\textup{gr}_{\bullet}^{V^{Y}}\mathscr{D}_{X}\simeq\pi_{*}\mathscr{D}_{T_{Y}X},$

where $\pi:T_{Y}X\to X$ is the natural affine morphism (see [Bj93, §II.10] for
the analytical case).

Under the identification (5), $\sum t_{i}\partial_{t_{i}}$ gives a global
section of $\mathscr{D}_{T_{Y}X}$ corresponding to the radial vector field of
the bundle $T_{Y}X$ (with respect to the natural $\mathbb{C}^{\star}$-action
on $T_{Y}X$), denoted by
$\sum\overline{t_{i}\partial}_{t_{i}}\in\textup{gr}^{V^{Y}}_{\bullet}\mathscr{D}_{X}$.
In particular, $\sum\overline{t_{i}\partial}_{t_{i}}$ is independent of
choices of $t_{1},\dots,t_{r}$.

Now we assume that $\mathcal{M}$ is a coherent $\mathscr{D}_{X}$-module with a
good filtration $\Omega_{\bullet}\mathcal{M}$ over
$V^{Y}_{\bullet}\mathscr{D}_{X}$. Since $\pi$ is affine, we get a coherent
$\mathscr{D}_{T_{Y}X}$-module
$\widetilde{\textup{gr}}^{\Omega}_{\bullet}\mathcal{M}$ after applying the
$\sim$-functor. We then say that $\textup{gr}^{\Omega}_{\bullet}\mathcal{M}$
is holonomic (resp. regular holonomic) over
$\textup{gr}^{V^{Y}}_{\bullet}\mathscr{D}_{X}$ if
$\widetilde{\textup{gr}}^{\Omega}_{\bullet}\mathcal{M}$ is so over
$\mathscr{D}_{T_{Y}X}$.

The deformation $\varphi_{Y}$ induces a deformation from $T^{*}X$ to
$T^{*}T_{Y}X$ as follows. We consider the relative cotangent bundle of
$\varphi_{Y}$:

$T^{*}\varphi_{Y}:T^{*}(\widetilde{X}_{Y}/\mathbb{C})\to\mathbb{C}.$

The fiber of $T^{*}\varphi_{Y}$ over $0$ is $T^{*}T_{Y}X$ and
$T^{*}(\widetilde{X}_{Y}/\mathbb{C})\setminus T^{*}T_{Y}X\simeq
T^{*}X\times\mathbb{C}^{\star}$.

The following lemma is essentially due to Sabbah (see [Sab87a, Lemme 2.0.1]).
We rephrase it algebraically.

######  Lemma 4.6 (Sabbah).

For a smooth subvariety $Y\subseteq X$, we have a natural isomorphism

$R^{Y}_{V}\mathscr{D}_{X}\simeq\mathscr{D}_{\widetilde{X}_{Y}/\mathbb{C}}.$

###### Proof.

We pick a (étale) local coordinates $(x_{1},\dots,x_{n-r},t_{1},\dots,t_{r})$
so that $(t_{1}=t_{2}=\cdots=t_{r}=0)$ defines $Y$, where $n=\dim X$ and $r$
is the codimension of $Y\subseteq X$ (cf. [HTT08, §A.5]). Since
$d(t^{k}_{i}\cdot u)=kt_{i}^{k-1}dt\cdot u$ (taking differentials over
$\mathbb{C}[u]$), we have a local decomposition of the relative cotangent
sheaf

$\Omega^{1}_{\widetilde{X}_{Y}/\mathbb{C}}=\bigoplus_{k\in\mathbb{Z}}\mathscr{I}_{Y}^{-k}(\bigoplus_{i=1}^{r}dt_{i}\otimes
u^{k-1}\bigoplus_{j=1}^{n-r}dx_{j}\otimes u^{k})$

and hence

(6)
$\mathscr{T}_{\widetilde{X}_{Y}/\mathbb{C}}=\bigoplus_{k\in\mathbb{Z}}\mathscr{I}_{Y}^{-k}(\bigoplus_{i=1}^{r}\partial_{t_{i}}\otimes
u^{k+1}\bigoplus_{j=1}^{n-r}\partial_{x_{j}}\otimes u^{k}).$

Since $\mathscr{T}_{\widetilde{X}_{Y}/\mathbb{C}}$ and
$\mathscr{O}_{\widetilde{X}_{Y}}$ generate
$\mathscr{D}_{\widetilde{X}_{Y}/\mathbb{C}}$, the required isomorphism then
follows. ∎

By Lemma 4.6, we immediately have:

######  Proposition 4.7.

$T^{*}(\widetilde{X}_{Y}/\mathbb{C})=\textup{Spec
}[\textup{gr}^{F}_{\bullet}R^{Y}_{V}\mathscr{D}_{X}],$

where $F_{\bullet}(R^{Y}_{V}\mathscr{D}_{X})$ is the order filtration for
relative differential operators induced from the order filtration on
$\mathscr{D}_{X}$.

###  4.3. Side-change for Rees modules

We discuss side-changes for $R^{Y}_{V}\mathscr{D}_{X}$-modules. The proof of
Lemma 4.6 implies that the relative canonical sheaf of $\varphi_{Y}$ is

$\omega_{\widetilde{X}_{Y}/\mathbb{C}}=\bigoplus_{k\in\mathbb{Z}}\omega_{X}\otimes_{\mathscr{O}_{X}}\mathscr{I}_{Y}^{-k}\otimes
u^{k-r},$

where we use $\omega_{-}$ to denote the canonical sheaf of the smooth variety
$-$. We then immediately have the side-change operators

$\omega_{\widetilde{X}_{Y}/\mathbb{C}}\otimes_{\mathscr{O}}(\bullet)\colon\textup{Mod}^{l}(R^{Y}_{V}\mathscr{D}_{X})\longrightarrow\textup{Mod}^{r}(R^{Y}_{V}\mathscr{D}_{X})$

and

$\omega^{-1}_{\widetilde{X}_{Y}/\mathbb{C}}\otimes_{\mathscr{O}}(\bullet)\colon\textup{Mod}^{r}(R^{Y}_{V}\mathscr{D}_{X})\longrightarrow\textup{Mod}^{l}(R^{Y}_{V}\mathscr{D}_{X}),$

where $\textup{Mod}^{l}(R^{Y}_{V}\mathscr{D}_{X})$ (resp.
$\textup{Mod}^{r}(R^{Y}_{V}\mathscr{D}_{X})$) is the abelian category of left
(resp. right) $R_{V}\mathscr{D}_{X}$-modules. Similar to the absolute case,
the side-change operators give an equivalence between
$\textup{Mod}^{l}(R^{Y}_{V}\mathscr{D}_{X})$ and
$\textup{Mod}^{r}(R^{Y}_{V}\mathscr{D}_{X})$.

Since
$\omega_{\widetilde{X}_{Y}/\mathbb{C}}\otimes_{\mathbb{C}[u]}\mathbb{C}_{0}\simeq\omega_{T_{Y}X}$
and
$R^{Y}_{V}\mathscr{D}_{X}\otimes_{\mathbb{C}[u]}\mathbb{C}_{0}\simeq\textup{gr}_{\bullet}^{V^{Y}}\mathscr{D}_{X}$
($\mathbb{C}_{0}$ is the residue field of $0\in\textup{Spec }\mathbb{C}[u]$),
we immediately have the following commutative diagram

(7)
${\textup{Mod}^{l}(R^{Y}_{V}\mathscr{D}_{X})}$${\textup{Mod}^{r}(R^{Y}_{V}\mathscr{D}_{X})}$${\textup{Mod}^{l}(\textup{gr}^{V^{Y}}_{\bullet}\mathscr{D}_{X})}$${\textup{Mod}^{r}(\textup{gr}^{V^{Y}}_{\bullet}\mathscr{D}_{X})}$$\scriptstyle{\omega_{\tilde{X}_{Y}/\mathbb{C}}\otimes\bullet}$$\scriptstyle{\bullet\otimes_{\mathbb{C}[u]}\mathbb{C}_{0}}$$\scriptstyle{\omega_{\tilde{X}_{Y}/\mathbb{C}}^{-1}\otimes\bullet}$$\scriptstyle{\bullet\otimes_{\mathbb{C}[u]}\mathbb{C}_{0}}$$\scriptstyle{\omega_{T_{Y}X}\otimes\bullet}$$\scriptstyle{\omega_{T_{Y}X}^{-1}\otimes\bullet}$

where $\textup{Mod}^{l}(\textup{gr}^{V^{Y}}_{\bullet}\mathscr{D}_{X})$ (resp.
$\textup{Mod}^{r}(\textup{gr}^{V^{Y}}_{\bullet}\mathscr{D}_{X})$) are the
abelian category of graded left (resp. right)
$\textup{gr}^{V}_{\bullet}\mathscr{D}_{X}$-modules.

Furthermore, since
$R_{V}\mathscr{D}_{X}\otimes_{\mathbb{C}[u]}\mathbb{C}_{1}\simeq\mathscr{D}_{X}$
where $\mathbb{C}_{1}$ is the residue field of $1\in\textup{Spec
}\mathbb{C}[u]$, we also have the following commutative diagram

(8)
${\textup{Mod}^{l}(R_{V}\mathscr{D}_{X})}$${\textup{Mod}^{r}(R_{V}\mathscr{D}_{X})}$${\textup{Mod}^{l}(\mathscr{D}_{X})}$${\textup{Mod}^{r}(\mathscr{D}_{X}).}$$\scriptstyle{\omega_{\tilde{X}_{Y}}\otimes\bullet}$$\scriptstyle{\bullet\otimes_{\mathbb{C}[u]}\mathbb{C}_{1}}$$\scriptstyle{\omega_{\tilde{X}_{Y}}^{-1}\otimes\bullet}$$\scriptstyle{\bullet\otimes_{\mathbb{C}[u]}\mathbb{C}_{1}}$$\scriptstyle{\omega_{X}\otimes\bullet}$$\scriptstyle{\omega_{X}^{-1}\otimes\bullet}$

###  4.4. Characteristic varieties of nearby cycles.

In this subsection, we calculate the characteristic cycles of nearby cycles.

Suppose that $\mathcal{M}$ is specializable along a smooth hypersurface
$H\subseteq X$. Then $\textup{gr}_{k}^{V}\mathcal{M}$ is both a coherent
$\mathscr{D}_{X,H}$-module and a coherent $\mathscr{D}_{H}$-module for every
$k\in\mathbb{Z}$.

######  Lemma 4.8.

If $\mathcal{M}$ is holonomic, then the Kashiwara-Malgrange filtrations
satisfy

$V_{k}\mathcal{M}=V_{k}(\mathcal{M}(*H))$

for every $k<0$, where $\mathcal{M}(*H)$ is the algebraic localization of
$\mathcal{M}$ along $H$. In particular, $V_{k}\mathcal{M}$ is a
$\mathscr{D}_{X,H}$-lattice of $\mathcal{M}(*H)$ for every $k<0$.

###### Proof.

We consider the exact sequence

$0\to\mathcal{T}\rightarrow\mathcal{M}\rightarrow\mathcal{M}(*H)\rightarrow\mathcal{Q}\to
0$

where $\mathcal{T}$ is the torsion subsheaf of $\mathcal{M}$ supported on
$\mathcal{H}$, namely $\mathcal{T}=\mathcal{H}^{0}_{H}(\mathcal{M})$, and
$\mathcal{Q}$ is the quotient module, or
$\mathcal{Q}=\mathcal{H}^{1}_{H}(\mathcal{M})$. Since $\mathcal{T}$ and
$\mathcal{Q}$ are supported on $H$, by Kashiwara’s equivalence (cf. [HTT08,
Theorem 1.6.1]) $V_{k}\mathcal{T}$ and $V_{k}\mathcal{Q}$ are zero for all
$k<0$. The exact sequence induces another exact sequence

$0\to V_{k}\mathcal{T}\rightarrow V_{k}\mathcal{M}\rightarrow
V_{k}\mathcal{M}(*H)\rightarrow V_{k}\mathcal{Q}\to 0$

for each $k$. We thus have obtained the required statement.

∎

The following theorem is equivalent to [Gin86, Theorem 5.5], where in _loc.
cit._ the nearby cycle is alternatively constructed following the algebraic
approach of Beilinson and Bernstein. We give it a proof by applying Theorem
1.6.

######  Theorem 4.9.

Suppose that $\mathcal{M}$ is a regular holonomic $\mathscr{D}_{X}$-module and
that $H\subseteq X$ is a smooth hypersurface. Then

$\overline{\textup{CC}(\mathcal{M}|_{U})}|_{H}\subseteq T^{*}(X,H)$

is a Lagrangian cycle in $T^{*}H\subseteq T^{*}(X,H)|_{H}$ with $U=X\setminus
H$. Furthermore, the nearby cycle $\Psi_{H}(\mathcal{M})$ has the
characteristic cycle

$\textup{CC}(\Psi_{H}(\mathcal{M}))=\overline{\textup{CC}(\mathcal{M}|_{U})}|_{H}\subseteq
T^{*}H.$

###### Proof.

Since characteristic cycles are local, it is enough to assume $H=(t=0)$ for
some local regular (or holomorphic) function $t$. Since
$\Psi_{H}(\mathcal{M})=\textup{gr}^{V}_{k}\mathcal{M}$ for $k<0$, we can focus
on the short exact sequence of
$V_{0}\mathscr{D}_{X}=\mathscr{D}_{X,H}$-modules

$0\to V_{-1}\mathcal{M}\xrightarrow{\cdot
t}V_{-1}\mathcal{M}\rightarrow\textup{gr}^{V}_{-1}\mathcal{M}\to 0.$

By Lemma 4.8, $V_{-1}\mathcal{M}$ is a $\mathscr{D}_{X,H}$-lattice of
$\mathcal{M}$. Then, by Theorem 1.6, we have

$\textup{CC}_{\log}(V_{-1}\mathcal{M})=\overline{\textup{CC}(\mathcal{M}|_{U})}.$

Similar to the proof of Proposition 2.7, considering the above short exact
sequence, we conclude that

$\textup{CC}_{\log}(\textup{gr}^{V}_{-1}\mathcal{M})=\overline{\textup{CC}(\mathcal{M}|_{U})}|_{H}\subseteq
T^{*}(X,H)$

and

$\dim(\overline{\textup{CC}(\mathcal{M}|_{U})}|_{H})=\dim X-1.$

Now we pick a good filtration $F_{\bullet}(\textup{gr}^{V}_{-1}\mathcal{M})$
over $F_{\bullet}\mathscr{D}_{H}$. Furthermore, we have a closed embedding

$T^{*}H\hookrightarrow T^{*}(X,H)$

defined by $\widetilde{\xi}_{t}=0$, where $\widetilde{\xi}_{t}$ is the symbol
of $t\partial_{t}$ in $\textup{gr}^{V}_{\bullet}\mathscr{D}_{X,H}$. Thus,
$F_{\bullet}(\textup{gr}^{V}_{-1}\mathcal{M})$ is also good over
$F_{\bullet}\mathscr{D}_{X,H}$. Since characteristic cycles are independent of
good filtrations, we therefore have

$\textup{CC}(\Psi_{H}(\mathcal{M}))=\textup{CC}_{\log}(\textup{gr}^{V}_{-1}\mathcal{M})=\overline{\textup{CC}(\mathcal{M}|_{U})}|_{H}\subseteq
T^{*}H.$

∎

###  4.5. Generalized Kashiwara-Malgrange filtrations

We discuss refinements of the Kashiwara-Malgrange filtration by using Sabbah’s
multi-filtrations.

Suppose that $X$ is a smooth complex variety and $Y\subseteq X$ a smooth
subvariety of codimension $r$ such that

$Y=\bigcap_{j=1}^{r}H_{j},$

where $H_{1},H_{2},\dots,H_{r}$ are smooth hypersurfaces intersecting
transversally (that is, the divisor $D=\sum_{j}H_{j}$ has simple normal
crossings). We then call $Y$ a _smooth complete intersection_ of
$H_{1},\dots,H_{r}$.

We use ${V}^{H_{j}}_{\bullet}\mathscr{D}_{X}$ to denote the Kashiwara-
Malgrange filtration of $\mathscr{D}_{X}$ along $H_{j}$ for $j=1,\dots,r$. For
$\mathbf{s}=(s_{1},s_{2},\dots,s_{r})\in\mathbb{Z}^{r}$, we set

$V_{\mathbf{s}}\mathscr{D}_{X}=\bigcap_{j=1}^{r}{}{V}^{H_{j}}_{s_{j}}\mathscr{D}_{X}.$

As the index $\mathbf{s}$ varies in $\mathbb{Z}^{r}$, we get an increasing
$\mathbb{Z}^{r}$-filtration of $\mathscr{D}_{X}$ with respect to the natural
partial order on $\mathbb{Z}^{r}$, denote by $V_{\bullet}\mathscr{D}_{X}$. One
can easily check

$V_{\mathbf{0}}\mathscr{D}_{X}=\mathscr{D}_{X,D},$

where the latter is the sheaf of rings of log differential operators. We write
the associated Rees ring by

$R_{V}\mathscr{D}_{X}:=\bigoplus_{\mathbf{s}\in\mathbb{Z}^{r}}V_{\mathbf{s}}\mathscr{D}_{X}\cdot\prod_{j=1}^{r}u_{j}^{s_{i}},$

where the product $\prod_{j=1}^{r}u_{j}^{s_{i}}$ is used to help us remember
the multi-grading of $R_{V}\mathscr{D}_{X}$.

For a coherent $\mathscr{D}_{X}$-module $\mathcal{M}$, similar to Definition
4.2, we say that a $\mathbb{Z}^{r}$-filtration $U_{\bullet}\mathcal{M}$ is
compatible with $V_{\bullet}\mathscr{D}_{X}$ if

$V_{\mathbf{s}}\mathscr{D}_{X}\cdot U_{\bf k}\mathcal{M}\subseteq U_{{\bf
k}+\mathbf{s}}\mathcal{M}$

for all ${\bf k},\mathbf{s}\in\mathbb{Z}^{r}$. Such a filtration
$U_{\bullet}\mathcal{M}$ is called good over $V_{\bullet}\mathscr{D}_{X}$ if
its associated Rees module $R_{U}\mathcal{M}$ is coherent over
$R_{V}\mathscr{D}_{X}$.

###  4.6. Refinement of normal deformation

We keep the notations as in the previous subsection. Suppose that $Y\subseteq
X$ is a smooth complete intersection of $H_{1},\dots,H_{r}$. We denote by
$\mathscr{I}_{H_{j}}$ the ideal sheaf of $H_{j}$ for $j=1,2,\dots,r$. Define

$\widetilde{X}\coloneqq\textup{Spec
}(\bigoplus_{k_{1},\cdots,k_{r}\in\mathbb{Z}}\bigotimes_{j}{\mathscr{I}_{H_{j}}^{-k_{j}}\otimes
u_{j}^{k_{j}}}).$

Then the natural inclusion

$\mathbb{C}[u_{1},\dots,u_{r}]\hookrightarrow\bigoplus_{k_{1},\cdots,k_{r}\in\mathbb{Z}}\bigotimes_{j}{\mathscr{I}_{H_{j}}^{-k_{j}}\otimes
u_{j}^{k_{j}}}$

gives rise to a smooth family

$\varphi\colon\widetilde{X}\to\mathbb{C}^{r}$

so that

  1. (1)

$\varphi^{-1}(u_{1},\dots,u_{r})\simeq X$ if
$(u_{1},\dots,u_{r})\in(\mathbb{C}^{\star})^{r}$;

  2. (2)

$\varphi^{-1}(\mathbf{0})=T_{Y}X$, the algebraic normal bundle of $Y\subseteq
X$.

The $\mathbb{Z}^{r}$-grading of
$\bigoplus_{k_{1},\cdots,k_{r}\in\mathbb{Z}}\prod_{j}{\mathscr{I}_{H_{j}}^{-k_{j}}\otimes
u_{j}^{k_{j}}}$ induces $(\mathbb{C}^{\star})^{k}$-actions on both
$\widetilde{X}$ and $T_{Y}X$. Since $Y$ is a complete intersection, the
obvious thing is that

$T_{Y}X=T_{H_{1}}X\times_{X}\cdots\times_{X}T_{H_{r}}X$

and hence $T_{Y}X\to Y$ is a split rank $r$ vector bundle. Moreover, the
induced $(\mathbb{C}^{\star})^{r}$-action on $T_{Y}X$ is given by rescaling
the fibers.

Similar to Lemma 4.6, we obtain:

######  Lemma 4.10.

We have a natural isomorphism

$R_{V}\mathscr{D}_{X}\simeq\mathscr{D}_{\widetilde{X}/\mathbb{C}^{r}}.$

From the above lemma, we immediately conclude that $R_{V}\mathscr{D}_{X}$ is a
coherent and noetherian sheaf of rings.

Similar to Proposition 4.7, we have:

######  Proposition 4.11.

$T^{*}(\widetilde{X}/\mathbb{C}^{r})=\textup{Spec
}[\textup{gr}^{F}_{\bullet}R_{V}\mathscr{D}_{X}],$

where $F_{\bullet}(R^{Y}_{V}\mathscr{D}_{X})$ is the order filtration for
relative differential operators induced from the order filtration on
$\mathscr{D}_{X}$.

######  Remark 4.12.

In the case that $X$ is a complex manifold and $Y$ is an analytic smooth
complete intersection, one can construct the complex manifold $\widetilde{X}$
similar to the topological construction in [KS13, §4.1] or by using open
blowups as in [Sab87a, §2.1]. Then
$\mathscr{D}_{\widetilde{X}/\mathbb{C}^{r}}$ is a faithfully flat ring
extension of $R_{V}\mathscr{D}_{X}$ by GAGA, or more precisely

$\mathscr{D}_{\widetilde{X}/\mathbb{C}^{r}}=\mathscr{O}_{\widetilde{X}}\otimes_{R_{V}\mathscr{O}_{X}}R_{V}\mathscr{D}_{X},$

where

$R_{V}\mathscr{O}_{X}=\bigoplus_{k_{1},\cdots,k_{r}\in\mathbb{Z}}\bigotimes_{j}{\mathscr{I}_{H_{j}}^{-k_{j}}\otimes
u_{j}^{k_{j}}}.$

As a consequence, all the results in this section can be extended to the
analytic case.

###  4.7. Specializability along arbitrary slopes

Let $L=(l_{1},\dots,l_{r})$ be a nonzero primitive covector in
$(\mathbb{Z}^{r}_{\geq 0})^{\vee}$. We also use $L$ to denote the ray
generated by the primitive vector. We call such $L$ a slope for the smooth
complete intersection $Y\subseteq X$ and that $L$ is non-degenerate if each
$l_{j}$ is not zero and degenerate otherwise. We set

$Y_{L}\coloneqq\bigcap_{l_{j}\not=0}H_{j}.$

By definition, if $L$ is non-degenerate, then $Y_{L}=Y$.

Given a nondegenerate slope $L$, we have a toric embedding

$\iota\colon\mathbb{C}\hookrightarrow\mathbb{C}^{k},\textup{ by
}u\mapsto(u^{l_{1}},\cdots,u^{l_{k}}).$

We can pull-back $\widetilde{X}$, to get a smooth family $\varphi_{L}$ in the
following Cartesian diagram:

${\widetilde{X}^{L}}$${\widetilde{X}}$${\mathbb{C}}$${\mathbb{C}^{r}}$$\scriptstyle{\iota_{L}}$$\scriptstyle{\varphi^{L}}$$\scriptstyle{\varphi}$

This can be constructed directly, as

$\widetilde{X}^{L}=\textup{Spec
}(\bigoplus_{k_{1},\cdots,k_{r}\in\mathbb{Z}}\bigotimes_{j}{\mathscr{I}_{H_{j}}^{-k_{j}}\otimes(u^{l_{j}})^{k_{j}}})$

and the fiber over $u=0$ is

$(\varphi^{L})^{-1}(0)=\widetilde{X}_{L}|_{u=0}=\textup{Spec
}[(\bigoplus_{k_{1},\cdots,k_{r}\in\mathbb{Z}}\bigotimes_{j}{\mathscr{I}_{H_{j}}^{-i_{j}}\otimes(u^{a_{j}})^{i_{j}}})\otimes_{\mathbb{C}[u]}\mathbb{C}[u]/(u)]\simeq
T_{Y}X.$

In other words, $\widetilde{X}^{L}$ gives a normal deformation along the slope
direction $L$. The isomorphism $\widetilde{X}^{L}|_{u=0}\simeq T_{Y}X$ induces
a $\mathbb{C}^{\star}$-action on $T_{Y}X$ :

(9)
$\lambda\cdot(y_{1},\dots,y_{n-r},\xi_{1},\dots,\xi_{r})=(y_{1},\dots,y_{n-r},\lambda^{l_{1}}\cdot\xi_{1},\dots,\lambda^{l_{r}}\cdot\xi_{r})$

for $\lambda\in\mathbb{C}^{\star}=\mathbb{C}\setminus\\{0\\}$ and
$(y_{1},\dots,y_{n-r},\xi_{1},\dots,\xi_{r})\in T_{Y}X$.

The construction of $\widetilde{X}^{L}$ induces the following Cartesian
diagram of relative cotangent bundles:

(10)
${T^{*}(\widetilde{X}^{L}/\mathbb{C})}$${T^{*}(\widetilde{X}/\mathbb{C}^{r})}$${\mathbb{C}}$${\mathbb{C}^{r}.}$$\scriptstyle{T^{*}\varphi^{L}}$$\scriptstyle{T^{*}\varphi}$$\scriptstyle{\iota}$

By Lemma 4.10, we see that $R_{V}\mathscr{D}_{X}$ is flat over
$\mathbb{C}[u_{1},u_{2},\dots,u_{r}]$. We then set:

${}^{L}R_{V}\mathscr{D}_{X}:=\iota_{L}^{*}(R_{V}\mathscr{D}_{X})=\mathscr{D}_{\widetilde{X}^{L}/\mathbb{C}}.$

In particular, ${}^{L}R_{V}\mathscr{D}_{X}$ is coherent and noetherian. If $L$
is degenerate, then one can replace $Y$ by $Y_{L}$ to reduce to the non-
degenerate case.

######  Remark 4.13.

Similar to $\omega_{\widetilde{X}_{Y}/\mathbb{C}}$ in §4.3, one can get the
explicit formula for the relative canonical sheaves:

$\omega_{\widetilde{X}/\mathbb{C}^{r}}=\bigoplus_{k_{1},\cdots,k_{r}\in\mathbb{Z}}\omega_{X}\otimes_{\mathscr{O}}(\bigotimes_{j}{\mathscr{I}_{H_{j}}^{-k_{j}}\otimes
u_{j}^{k_{j}-1}})\textup{ and
}\omega_{\widetilde{X}^{L}_{Y}/\mathbb{C}}=\iota_{L}^{*}(\omega_{\widetilde{X}/\mathbb{C}^{r}}).$

By construction, we have the explicit formula for
${}^{L}R_{V}\mathscr{D}_{X}$:

${}^{L}R_{V}\mathscr{D}_{X}=\bigoplus_{k\in\mathbb{Z}}{}^{L}V_{k}\mathscr{D}_{X}\cdot
u^{k}$

where by definition

${}^{L}V_{k}\mathscr{D}_{X}\coloneqq\sum_{\mathbf{s}\in\mathbb{Z}^{r},L\cdot\mathbf{s}=k}V_{\mathbf{s}}\mathscr{D}_{X}.$

The graded ring ${}^{L}R_{V}\mathscr{D}_{X}$ then induces an increasing
$\mathbb{Z}$-filtration ${}^{L}V_{\bullet}\mathscr{D}_{X}$ on
$\mathscr{D}_{X}$. We might call ${}^{L}V_{\bullet}\mathscr{D}_{X}$ the
Kashiwara-Malgrange filtration of $\mathscr{D}_{X}$ along the slope $L$. Since
$\varphi^{L}$ is smooth and $\widetilde{X}^{L}|_{u=0}\simeq T_{Y_{L}}X$, we
have that

(11)
$\textup{gr}^{{}^{L}V}_{\bullet}\mathscr{D}_{X}\simeq\pi_{*}\mathscr{D}_{T_{Y_{L}}\mathscr{D}_{X}}$

where $\pi:T_{Y_{L}}X\to Y_{L}\hookrightarrow X$ is the composition (which is
an affine morphism). The $\mathbb{Z}$-grading of
$\textup{gr}^{{}^{L}V}_{\bullet}\mathscr{D}_{X}$ corresponds to the
$\mathbb{C}^{\star}$-action in (9).

The $\mathbb{C}^{\star}$-action in (9) induces a radial vector field on
$T_{Y_{L}}X$, denoted by $v_{L}$. We assume that locally $H_{j}$ are defined
by $t_{j}=0$ where $t_{j}$ are among some local coordinate system
$(x_{1},\dots,x_{n-r},t_{1},\dots,t_{r})$. Then locally

$v_{L}=L\cdot(t_{1}\partial_{t_{1}},t_{2}\partial_{t_{2}},\dots,t_{r}\partial_{t_{r}}).$

######  Definition 4.14.

Suppose that $Y\subseteq X$ is a smooth complete intersection of
$H_{1},\dots,H_{r}$, $\mathcal{M}$ is a $($left$)$ $\mathscr{D}_{X}$-module
and $L$ is a slope. A $\mathbb{Z}$-indexed increasing filtration
$\Omega_{\bullet}\mathcal{M}$ is _compatible_ with
${}^{L}V_{\bullet}\mathscr{D}_{X}$ if

${}^{L}V_{k}\mathscr{D}_{X}\cdot\Omega_{j}\mathcal{M}\subseteq\Omega_{k+j}\mathcal{M}\textup{
for all }k,j\in\mathbb{Z}.$

A compatible filtration $\Omega_{\bullet}\mathcal{M}$ is a _good_ filtration
over ${}^{L}V_{\bullet}\mathscr{D}_{X}$ if the associated Rees module

${}^{L}R_{\Omega}\mathcal{M}\coloneqq\bigoplus_{k\in\mathbb{Z}}\Omega_{k}\mathcal{M}\cdot
u^{k}\subseteq\mathcal{M}[u,1/u]$

is coherent over ${}^{L}R_{V}\mathscr{D}_{X}$. A good filtration
${}^{L}V_{\bullet}\mathcal{M}$ is called the Kashiwara-Malgrange filtration on
$\mathcal{M}$ along the slope $L$ if there exists a monic polynomial
$b(s)\in\mathbb{C}[s]$ with its roots having real parts in $[0,1)$ so that

$b(v_{L}+k)\textup{ annihilates
}\textup{gr}^{{}^{L}V}_{k}\mathcal{M}\coloneqq{}^{L}V_{k}\mathcal{M}/{}^{L}V_{k-1}\mathcal{M}\textup{
for each }k\in\mathbb{Z}.$

The monic polynomial $b(s)$ of the least degree is called the Bernstein-Sato
polynomial or $b$-function of $\mathcal{M}$ along $L$.

Kashiwara-Malgrange filtrations along $L$ are unique if exist. For an
arbitrary slope $L$, a coherent $\mathscr{D}_{X}$-module $\mathcal{M}$ is
called $L$-_specializable_ if the Kashiwara-Malgrange filtration of
$\mathcal{M}$ along $L$ exists. When $L=(1,1,\dots,1)$, the Definition 4.14
coincides with Definition 4.2. When $L=\mathbf{e}_{j}$, the $j$-th unit vector
in $\mathbb{Z}^{r}_{\geq 0}$, it coincides with Definition 4.2 for the case
when $Y=H_{j}$. In particular, specializability along $L$ is compatible with
the specializability defined in §4.1. We can then similarly define
$R$-specializability along $L$ for $R$ a subring of $\mathbb{C}$.

Suppose that $U_{\bullet}\mathcal{M}$ is a good filtration of $\mathcal{M}$
over $V_{\bullet}\mathscr{D}_{X}$. Then for a slope $L$ we can obtain a
compatible filtration ${}^{L}U_{\bullet}\mathcal{M}$ over
${}^{L}V_{\bullet}\mathscr{D}_{X}$ defined by

(12)
$^{L}U_{k}\mathcal{M}\coloneqq\sum_{\mathbf{s}\in\mathbb{Z}^{r},L\cdot\mathbf{s}=k}U_{\mathbf{s}}\mathcal{M}.$

We denote the associated Rees module by

${}^{L}R_{U}\mathcal{M}\coloneqq\bigoplus_{k\in\mathbb{Z}}^{L}U_{k}\mathcal{M}\cdot
u^{k}.$

Since
${}^{L}R_{U}\mathcal{M}\simeq\iota_{L}^{*}(R_{U}\mathcal{M})/\mathcal{T}_{u}$
and the pullback functor for relative $\mathscr{D}$-modules preserves
coherence, we have that ${}^{L}U_{\bullet}\mathcal{M}$ is good over
${}^{L}V_{\bullet}\mathscr{D}_{X}$, where $\mathcal{T}_{u}$ is the $u$-torsion
subsheaf.

The following result is a natural generalization of Theorem 4.3, which is
first observed by Sabbah [Sab87a, §3.1]. We provide an alternative proof with
the idea essentially due to Björk.

######  Theorem 4.15.

Suppose that $\mathcal{M}$ is a holonomic $\mathscr{D}_{X}$-module and $L$ is
a slope. Then $\mathcal{M}$ is splecializable along $L$. Moreover, if
$\Omega_{\bullet}\mathcal{M}$ is a good filtration over
${}^{L}V_{\bullet}\mathscr{D}_{X}$, then
$\textup{gr}^{\Omega}_{\bullet}\mathcal{M}$ is holonomic over
$\textup{gr}^{{}^{L}V}_{\bullet}\mathscr{D}_{X}$.

###### Proof.

We take a good filtration $\Omega_{\bullet}\mathcal{M}$ over
${}^{L}V_{\bullet}\mathscr{D}_{X}$ locally. Such a filtration exists at least
locally by coherence. Then we apply [Bj93, Appendix IV. Theorem 4.10] and
conclude that
$j_{\textup{gr}^{{}^{L}V}_{\bullet}\mathscr{D}_{X}}(\textup{gr}^{\Omega}_{\bullet}\mathcal{M})=j(\mathcal{M}),$
where the first grade number is the graded number of
$\textup{gr}^{\Omega}_{\bullet}\mathcal{M}$ over
$\textup{gr}^{{}^{L}V}_{\bullet}\mathscr{D}_{X}$. Since $\mathcal{M}$ is
holonomic, by [Bj93, Appendix IV. Proposition 3.5(2)] $j(\mathcal{M})=n$, the
dimension of $X$. Hence
$j_{\textup{gr}^{{}^{L}V}_{\bullet}\mathscr{D}_{X}}(\textup{gr}^{\Omega}_{\bullet}\mathcal{M})=n$
and hence $\textup{gr}^{\Omega}_{\bullet}\mathcal{M}$ is holonomic over
$\textup{gr}^{{}^{L}V}_{\bullet}\mathscr{D}_{X}\simeq\pi_{*}\mathscr{D}_{T_{Y_{L}}X}$
(since $\dim T_{Y_{L}}X=\dim X=n$).

Now we consider the operator

$\theta_{L}\coloneqq\bigoplus_{k\in\mathbb{Z}}v_{L}+k$

on $\textup{gr}^{\Omega}_{\bullet}\mathcal{M}$. By construction, one can
easily check

$\theta_{L}\in\textup{End}_{\textup{gr}^{{}^{L}V}_{\bullet}\mathscr{D}_{X}}(\textup{gr}^{\Omega}_{\bullet}\mathcal{M}).$

Since $\textup{gr}^{\Omega}_{\bullet}\mathcal{M}$ is holonomic over
$\textup{gr}^{{}^{L}V}_{\bullet}\mathscr{D}_{X}$, we conclude that
$\theta_{L}$ admits a minimal polynomial $b(s)\in\mathbb{C}[\mathbf{s}]$. The
real parts of roots of $b(s)$ might not be contained in $[0,1)$. Namely, the
good filtration $\Omega_{\bullet}\mathcal{M}$ is not the Kashiwara-Malgrange
filtration along $L$. We then apply the procedure in the proof of [Kas83,
Theorem 1(1)] to adjust the roots of $b(s)$ and the filtration
$\Omega_{\bullet}\mathcal{M}$. The output of the procedure gives us the
Kashiwara-Malgrange filtration ${}^{L}V_{\bullet}\mathcal{M}$. By uniqueness,
the local construction glues to the global ${}^{L}V_{\bullet}\mathcal{M}$.
Therefore, $\mathcal{M}$ is specializable along $L$. ∎

###  4.8. Micro nearby cycles along arbitrary slopes

We keep notations as in the previous subsection and continue to assume
$Y\subseteq X$ a smooth complete intersection and $\mathcal{M}$ a holonomic
$\mathscr{D}_{X}$-module.

By Theorem 4.15, the Kashiwara-Malgrange filtration
${}^{L}V_{\bullet}\mathcal{M}$ of $\mathcal{M}$ along every slope $L$. For a
nondegenerate slope $L$, the module $p^{*}_{L}(\mathcal{M})$ on
$\widetilde{X}^{L}\setminus T_{Y}X$ gives rise to a holonomic
$\mathscr{D}_{\widetilde{X}^{L}}$-module, denoted by
$\widetilde{\mathcal{M}}_{L}$ (that is,
$\widetilde{\mathcal{M}}_{L}=j^{L}_{*}(\mathcal{M}_{L})$), where
$p_{L}\colon\widetilde{X}^{L}\setminus T_{Y}X\simeq
X\times\mathbb{C}^{\star}\to X$ is the natural projection and
$j^{L}\colon\widetilde{X}^{L}\setminus T_{Y}X\hookrightarrow\widetilde{X}^{L}$
is the open embedding.

######  Lemma 4.16.

Suppose that $\mathcal{M}$ is specializable along a slope $L$. Then

$\widetilde{\textup{gr}}^{{}^{L}V}_{\bullet}\mathcal{M}\simeq\Psi_{u=0}(\widetilde{\mathcal{M}}_{L}).$

###### Proof.

The $\mathbb{C}^{\star}$-action on $\widetilde{X}^{L}$ is induced by the
grading of $u$ by construction. Hence, the $\mathbb{C}^{\star}$-action induces
the action of the Euler vector field along the smooth divisor
$T_{Y}X=(u=0)\subseteq\widetilde{X}^{L}$. But the $\mathbb{C}^{\star}$-actions
on $\widetilde{X}^{L}$ and hence $T_{Y_{L}}X$ are both induced by the operator

$\bigoplus_{k\in\mathbb{Z}}(L\cdot(t_{1}\partial_{t_{1}},t_{2}\partial_{t_{2}},\dots,t_{r}\partial_{t_{r}})+k).$

The required statement then follows by definition. See also [BMS06, §1.3] for
the case $L=(1,1,\dots,1)$. ∎

By the above lemma and Theorem 4.5, we obtain:

######  Corollary 4.17.

If $\mathcal{M}$ is regular holonomic, then so is
$\textup{gr}^{{}^{L}V}_{\bullet}\mathcal{M}$.

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

${}^{L}R\Psi_{T_{Y}X}(\mathcal{M})\coloneqq\widetilde{\textup{gr}}^{{}^{L}V}_{\bullet}\mathcal{M}.$

Following terminology in [Sch12] (and motivated by Lemma 4.16), we call
${}^{L}R\Psi_{T_{Y}X}(\mathcal{M})$ the _micro nearby cycle_ of $\mathcal{M}$
along $L$.

######  Theorem 4.18.

Assume that $\mathcal{M}$ is a regular holonomic $\mathscr{D}_{X}$-module and
$L$ is a nondegenerate slope. Then

$\textup{CC}_{\tilde{X}_{H}/\mathbb{C}}({}^{L}R_{V}\mathcal{M})=\overline{q^{*}_{L}(\textup{CC}(\mathcal{M}))}\textup{
and
}\textup{CC}({}^{L}R\Psi_{T_{Y}X}(\mathcal{M}))=\overline{q^{*}_{L}(\textup{CC}(\mathcal{M}))}|_{T^{*}T_{H}X}.$

where $q_{L}:T^{*}(\widetilde{X}^{L}/\mathbb{C})\setminus T^{*}T_{Y}X\simeq
T^{*}X\times\mathbb{C}^{\star}\to T^{*}X$ is the natural projection.

###### Proof.

We apply Lemma 3.4 and let
$\mathcal{N}_{\textup{rel}}={}^{L}R_{V}\mathcal{M}$,
$\mathcal{N}=\widetilde{\mathcal{M}}_{L}$ and $F=u$. We thus have

$\textup{CC}_{\tilde{X}_{H}/\mathbb{C}}({}^{L}R_{V}\mathcal{M})=\overline{q^{*}_{L}(\textup{CC}(\mathcal{M}))}.$

To obtain

$\textup{CC}({}^{L}R\Psi_{T_{Y}X}(\mathcal{M}))=\overline{q^{*}_{L}(\textup{CC}(\mathcal{M}))}|_{T^{*}T_{H}X},$

we apply Proposition 2.7. ∎

Theorem 1.3 follows from combining Theorem 4.15, Corollary 4.17 and Theorem
4.18.

###  4.9. Sabbah’s toric base-change of $R_{U}\mathcal{M}$

We now recall Sabbah’s toric base-changes of Rees modules.

Let us first introduce some notations. We set

$M=\mathbb{Z}^{r},M^{+}=(\mathbb{Z}_{\geq
0})^{r},M_{\mathbb{Q}}=M\times\mathbb{Q}\textup{ and
}M^{+}_{\mathbb{Q}}=(\mathbb{Q}_{\geq 0})^{r}.$

Let $N$ be the dual lattice of $M$ and define $N^{+}$, $N_{\mathbb{Q}}$ and
$N^{+}_{\mathbb{Q}}$ similarly. We then fix a simplicial fan $\Sigma$ in
$N_{\mathbb{Q}}$. We assume that $\Sigma$ is given by a subdivision of
$N^{+}_{\mathbb{Q}}$. Then $\Sigma$ gives a toric variety
$\mathcal{S}_{\Sigma}$ (by making further subdivision, one can assume
$\mathcal{S}_{\Sigma}$ is smooth) and a projective birational morphism

$\nu_{\Sigma}\colon\mathcal{S}_{\Sigma}\rightarrow\mathbb{C}^{r}\simeq\textup{Spec
}\mathbb{C}[M^{+}].$

For a cone $\Gamma\in\Sigma$, we set

$\check{\Gamma}\coloneqq\\{m\in M\mid\langle m,n\rangle\geq 0,\forall
n\in\Gamma\\}.$

Then

$\mathcal{S}_{\Gamma}\coloneqq\textup{Spec
}\mathbb{C}[\check{\Gamma}]\hookrightarrow\mathcal{S}_{\Sigma}$

gives an open affine patch of $\mathcal{S}_{\Sigma}$. We denote a partial
ordering induced by $\Gamma$ on $M$ by

$\mathbf{s}\leq_{\Gamma}\mathbf{s}^{\prime}\Leftrightarrow\mathbf{s}^{\prime}-\mathbf{s}\in\check{\Gamma}$

and we say

$\mathbf{s}<_{\Gamma}\mathbf{s}^{\prime}\Leftrightarrow\mathbf{s}\leq_{\Gamma}\mathbf{s}^{\prime}\textup{
but not }\mathbf{s}^{\prime}\leq_{\Gamma}\mathbf{s}.$

We use $\mathcal{L}(\Gamma)$ to denote the finite set of all the primitive
generators of $\Gamma$ and write

$\mathcal{L}(\Sigma)\coloneqq\bigcup_{\Gamma\in\Sigma}\mathcal{L}(\Gamma).$

We suppose that $Y\subseteq X$ is a smooth complete intersection of
$H_{1},\dots,H_{r}$ and $\mathcal{M}$ a $\mathscr{D}_{X}$-module with a
$\mathbb{Z}^{r}$-filtration $U_{\bullet}\mathcal{M}$ that is good over
$V_{\bullet}\mathscr{D}_{X}$. It is now natural to make
$U_{\bullet}\mathcal{M}$ and $V_{\bullet}\mathscr{D}_{X}$ indexed by
$M\simeq\mathbb{Z}^{r}$. We consider the following fiber-product diagram

${\widetilde{X}_{\Sigma}\coloneqq\widetilde{X}\times_{\mathbb{C}^{r}}\mathcal{S}_{\Sigma}}$${\widetilde{X}}$${\mathcal{S}_{\Sigma}}$${\mathbb{C}^{r}.}$$\scriptstyle{\mu_{\Sigma}}$$\scriptstyle{\varphi_{\Sigma}}$$\scriptstyle{\varphi}$$\scriptstyle{\nu_{\Sigma}}$

Using the flattening theorem relative over toric varieties, Sabbah and Castro
proved the following fundamental theorem:

######  Theorem 4.19.

[Sab87a, A.1.1] Suppose that $U_{\bullet}\mathcal{M}$ is a good multi-
filtration over $V_{\bullet}\mathscr{D}_{X}$ along the smooth complete
intersection $Y\subseteq X$. Then there exists a simplicial fan $\Sigma$
subdividing $\mathbb{N}^{+}_{\mathbb{Q}}$ such that
$\widetilde{R_{U}\mathcal{M}}\coloneqq\mu^{*}_{\Sigma}R_{U}\mathcal{M}/\mathcal{T}$
is flat over $\mathcal{S}_{\Sigma}$, where $\mathcal{T}$ is the torsion
subsheaf of $\mu^{*}_{\Sigma}R_{U}\mathcal{M}$ supported over the exceptional
locus of $\nu_{\Sigma}$.

Such $\Sigma$ in the above theorem is called a fan _adapted_ to
$U_{\bullet}\mathcal{M}$. By construction, we have a natural inclusion of
$\mathbb{Z}^{r}$-graded modules

$R_{U}\mathcal{M}\hookrightarrow\mathcal{M}[u_{1},1/u_{1},\dots,u_{r},1/u_{r}]$

or equivalently

$R_{U}\mathcal{M}|_{\widetilde{X}_{N}}=\mathcal{M}[u_{1},1/u_{1},\dots,u_{r},1/u_{r}].$

If we have a cone $\Gamma\in\Sigma$, then we know

$\mu^{*}_{\Sigma}R_{U}\mathcal{M}|_{\widetilde{X}_{\Gamma}}=\mu^{*}_{\Gamma}R_{U}\mathcal{M}=\mathbb{C}[\check{\Gamma}]\otimes_{\mathbb{C}[M^{+}]}R_{U}\mathcal{M}.$

Since $\mu_{\Sigma}$ and $\mu_{\Gamma}$ are identical over $S_{N}$, using the
pullback functor we have an induced natural morphism

(13)
$\mathbb{C}[\check{\Gamma}]\otimes_{\mathbb{C}[M^{+}]}R_{U}\mathcal{M}\rightarrow\mathcal{M}[u_{1},1/u_{1},\dots,u_{r},1/u_{r}].$

The above morphism is neither necessarily surjective nor injective. What is
the its image? To answer this question, Sabbah introduced a refined filtration
for each cone $\Gamma\in\Sigma$ by

${}^{\Gamma}U_{\mathbf{s}}\mathcal{M}=\sum_{\mathbf{s}^{\prime}\leq_{\Gamma}\mathbf{s}}U_{\mathbf{s}^{\prime}}\mathcal{M},\quad\forall\mathbf{s}\in
M.$

If $\Gamma$ is a unimodular cone of dimesion $<k$, then one easily sees that
${}^{\Gamma}U_{\mathbf{s}}\mathcal{M}$ only depends on the image of
$\mathbf{s}$ in $M/\Gamma^{\perp}$. Hence, in the special case that $L$ is a
ray in $\Sigma$, we have

${}^{L}U_{\mathbf{s}}\mathcal{M}={}^{L}U_{L(\mathbf{s})}\mathcal{M},$

that is, ${}^{L}U_{\bullet}\mathcal{M}$ is indexed by $\mathbb{Z}\simeq
M/\Gamma^{\perp}$. Therefore, ${}^{L}U_{\bullet}\mathcal{M}$ coincides with
the $\mathbb{Z}$-indexed filtration defined by (12).

We write the associated Rees module by

${}^{\Gamma}R_{U}\mathcal{M}\coloneqq\bigoplus_{s\in
M}{}^{\Gamma}U_{\mathbf{s}}\mathcal{M}\cdot\prod_{j=1}^{r}u_{j}^{s_{j}}.$

One observes that ${}^{\Gamma}R_{U}\mathcal{M}$ is the image of the natural
morphism (13) and that its kernel is the torsion subsheaf
$\mathcal{T}|_{\widetilde{X}_{\Gamma}}$. Therefore, we have proved that

$\widetilde{R_{U}\mathcal{M}}|_{\widetilde{X}_{\Gamma}}\simeq{}^{\Gamma}R_{U}\mathcal{M}.$

Furthermore, Sabbah proved:

######  Lemma 4.20.

[Sab87a, 2.2.2.Lemme] If ${}^{\Gamma}R_{U}\mathcal{M}$ is flat over
$\mathbb{C}[\check{\Gamma}]$, then

${}^{\Gamma}U_{\mathbf{s}}\mathcal{M}=\bigcap_{L\in\mathcal{L}(\Gamma)}{}^{L}U_{L(\mathbf{s})}\mathcal{M}.$

It is obvious that
$\mathbb{C}[\check{\Gamma}]\otimes_{\mathbb{C}[M^{+}]}R_{U}\mathcal{M}$ is
coherent over ${}^{\Gamma}R_{V}\mathscr{D}_{X}$ and hence
${}^{\Gamma}R_{U}\mathcal{M}$ is coherent over
${}^{\Gamma}R_{V}\mathscr{D}_{X}$. However, it is not always the case that
${}^{\Gamma}R_{U}\mathcal{M}$ is coherent over $R_{V}\mathscr{D}_{X}$ and
hence ${}^{\Gamma}U_{\bullet}\mathcal{M}$ is not necessarily a good filtration
over $V_{\bullet}\mathscr{D}_{X}$. To fix this, Sabbah defined the
_saturation_ of $U_{\bullet}\mathcal{M}$ by

$\bar{U}_{\mathbf{s}}\mathcal{M}\coloneqq\bigcap_{\textup{primitive vectors
}L\in N^{+}}{}^{L}U_{L(\mathbf{s})}\mathcal{M}.$

######  Theorem 4.21 (Sabbah).

Suppose that $U_{\bullet}\mathcal{M}$ is a good multi-filtration over
$V_{\bullet}\mathscr{D}_{X}$ and $\Sigma$ is a simplicial fan adapted to
$U_{\bullet}\mathcal{M}$. Then $\bar{U}_{\bullet}\mathcal{M}$ is good over
$V_{\bullet}\mathscr{D}_{X}$ and

$\bar{U}_{\mathbf{s}}\mathcal{M}=\bigcap_{L\in\mathcal{L}(\Sigma)}{}^{L}U_{L(\mathbf{s})}\mathcal{M}.$

###### Proof.

We take a cone $\Gamma\in\Sigma$ and consider the natural surjection

$\mu^{*}_{\Gamma}R_{U}\mathcal{M}\rightarrow\widetilde{R_{U}\mathcal{M}}|_{\widetilde{X}_{\Gamma}}\simeq{}^{\Gamma}R_{U}\mathcal{M}.$

By Theorem 4.19, $\widetilde{R_{U}\mathcal{M}}$ is flat over
$\mathcal{S}_{\Sigma}$. By Lemma 4.20, we hence know

$\widetilde{R_{U}\mathcal{M}}|_{\widetilde{X}_{\Gamma}}=\bigoplus_{s\in
M}(\bigcap_{L\in\mathcal{L}(\Gamma)}{}^{L}U_{L(\mathbf{s})}\mathcal{M})\cdot\prod_{j=1}^{r}u_{j}^{s_{j}}.$

Since $\\{\widetilde{X}_{\Gamma}\\}_{\Gamma\in\Sigma}$ gives a covering of
$\widetilde{X}_{\Sigma}$, we hence obtain that

$\mu_{\Sigma*}(\widetilde{R_{U}\mathcal{M}})=\bigoplus_{s\in
M}(\bigcap_{L\in\mathcal{L}(\Sigma)}{}^{L}U_{L(\mathbf{s})}\mathcal{M})\cdot\prod_{j=1}^{r}u_{j}^{s_{j}}.$

Since $\nu_{\Sigma}$ is projective, by Proposition 2.8 we conclude that
$\mu_{\Sigma*}(\widetilde{R_{U}\mathcal{M}})$ is coherent over
$R_{V}\mathscr{D}_{X}$.

By construction, we know that

${}^{\Gamma}U_{\mathbf{s}}\mathcal{M}\subseteq\bigcap_{\textup{primitive
vectors }L\in\Gamma\cap N^{+}}{}^{L}U_{L(\mathbf{s})}\mathcal{M}$

where on the right hand side the intersection is over all primitive vectors in
$\Gamma\cap N^{+}$ (not just generators of $\Gamma$) and hence

${}^{\Gamma}U_{\mathbf{s}}\mathcal{M}=\bigcap_{L\in\mathcal{L}(\Gamma)}{}^{L}U_{L(\mathbf{s})}\mathcal{M}=\bigcap_{\textup{primitive
vectors }L\in\Gamma\cap N^{+}}{}^{L}U_{L(\mathbf{s})}\mathcal{M}$

thanks to Lemma 4.20 again. Therefore,

$\bar{U}_{\mathbf{s}}\mathcal{M}=\bigcap_{\Gamma\in\Sigma}{}^{\Gamma}U_{\mathbf{s}}\mathcal{M}=\bigcap_{L\in\mathcal{L}(\Sigma)}{}^{L}U_{L(\mathbf{s})}\mathcal{M}.$

and

$\mu_{\Sigma*}(\widetilde{R_{U}\mathcal{M}})=R_{\bar{U}}\mathcal{M}.$

Since we have proved that $\mu_{\Sigma*}(\widetilde{R_{U}\mathcal{M}})$ is
coherent over $R_{V}\mathscr{D}_{X}$, $\bar{U}_{\bullet}\mathcal{M}$ is good
over $V_{\bullet}\mathscr{D}_{X}$. ∎

######  Remark 4.22.

Let $\mathcal{M}$ be a holonomic $\mathscr{D}_{X}$-module. By Theorem 4.15, we
have ${}^{L}V_{\bullet}\mathcal{M}$ the Kashiwara-Malgrange filtration of
$\mathcal{M}$ for each slope $L$. We then fix an adapted fan $\Sigma$ to a
good multi-filtration $U_{\bullet}\mathcal{M}$. Now one can naively define

${}^{\Sigma}V_{\mathbf{s}}\mathcal{M}=\bigcap_{L\in\mathcal{L}(\Sigma)}{}^{L}V_{L(\mathbf{s})}\mathcal{M}\quad\forall\mathbf{s}\in
M,$

which gives a $\mathbb{Z}^{r}$-filtration ${}^{\Sigma}V_{\bullet}\mathcal{M}$
over $V_{\bullet}\mathscr{D}_{X}$. However, it is not necessarily true in
general that ${}^{\Sigma}V_{\bullet}\mathcal{M}$ is good over
$V_{\bullet}\mathscr{D}_{X}$ even if $\mathcal{M}$ is regular holonomic; see
[Sab87a, §3.3] for further discussions. This means that one cannot define
multi-indexed Kashiwara-Malgrange filtrations in general. On the contrary,
Bernstein-Sato polynomials can be generalized successfully to the multi-
indexed case (see Theorem 4.23).

Using Theorem 4.21, Sabbah proved the following beautiful result about the
existence of multi-variable $b$-function. We sketch its proof for
completeness.

######  Theorem 4.23 (Existence of Sabbah’s generalized $b$-functions).

Suppose that $\mathcal{M}$ is a holonomic $\mathscr{D}_{X}$-module with a
$\mathbb{Z}^{r}$-filtration $U_{\bullet}\mathcal{M}$ good over
$V_{\bullet}\mathscr{D}_{X}$ along a smooth complete intersection $Y\subseteq
X$ of $H_{1},\dots,H_{r}$. Then there exists a simplicial fan $\Sigma$
subdividing $N^{+}$ such that for every nonzero vector ${\bf a}\in M^{+}$
there exist polynomials $b^{\bf a}_{L}(s)\in\mathbb{C}[s]$ $($depending on
${\bf a}$$)$ for all slopes $L\in\mathcal{L}(\Sigma)$ so that locally

$\prod_{L\in\mathcal{L}(\Sigma)}b^{\bf
a}_{L}(L\cdot(t_{1}\partial_{t_{1}},\dots,t_{r}\partial_{t_{r}}))\textup{
annihilates }\frac{U_{\vec{0}}\mathcal{M}}{U_{-{\bf a}}\mathcal{M}},$

where $t_{j}$ are local defining functions of $H_{j}$.

###### Proof.

We take a simplicial fan $\Sigma$ adapted to $U_{\bullet}\mathcal{M}$, whose
existence is guaranteed by Theorem 4.19. By Theorem 4.21, the saturation of
$U_{\bullet}\mathcal{M}$ is

$\bar{U}_{\mathbf{s}}\mathcal{M}=\bigcap_{L\in\mathcal{L}(\Sigma)}{}^{L}U_{L(\mathbf{s})}\mathcal{M}\quad\forall\mathbf{s}\in
M.$

Since $\bar{U}_{\bullet}\mathcal{M}$ is good over
$V_{\bullet}\mathscr{D}_{X}$, there exists a vector ${\bf k}\in M^{+}$
depending on ${\bf a}$ such that

$\bar{U}_{-k}\mathcal{M}\subseteq U_{-{\bf a}}\mathcal{M}.$

On the other hand, similar to the proof of Theorem 4.15, we conclude that
$\textup{gr}^{{}^{L}U}_{\bullet}\mathcal{M}$ is holonomic over
$\textup{gr}^{{}^{L}V}_{\bullet}\mathscr{D}_{X}$ for each slope $L$.
Therefore, there exists $b_{L}(s)\in\mathbb{C}[s]$ such that
$b_{L}(\theta_{L})$ kills $\textup{gr}^{{}^{L}U}_{\bullet}\mathcal{M}$. In
particular, there exists $b^{\bf a}_{L}(s)\in\mathbb{C}[s]$ so that

$b^{\bf
a}_{L}(L\cdot(t_{1}\partial_{t_{1}},\dots,t_{r}\partial_{t_{r}}))\cdot{}^{L}U_{0}\mathcal{M}\subseteq{}^{L}U_{L(-{\bf
k})}\mathcal{M}$

for each $L\in\mathcal{L}(\Sigma)$. Since
$U_{\vec{0}}\mathcal{M}\subseteq\bar{U}_{\vec{0}}\mathcal{M}$ and

$\bar{U}_{-k}\mathcal{M}=\bigcap_{L\in\mathcal{L}(\Sigma)}{}^{L}U_{L(-{\bf
k})}\mathcal{M}\subseteq U_{-{\bf a}}\mathcal{M},$

the required statement follows. ∎

###  4.10. Relative characteristic cycles for $R_{U}\mathcal{M}$

We now prove Theorem 1.4. Assume that $\mathcal{M}$ be a regular holonomic
$\mathscr{D}_{X}$-module with a good filtration $U_{\bullet}\mathcal{M}$ over
$V_{\bullet}\mathscr{D}_{X}$. We write

$p\colon\varphi^{-1}((\mathbb{C}^{*})^{r})\simeq
X\times(\mathbb{C}^{*})^{r}\longrightarrow X$

the natural projection, and
$j\colon\varphi^{-1}((\mathbb{C}^{*})^{r})\hookrightarrow\widetilde{X}$ the
open embedding. We then set $\mathcal{N}=j_{*}(p^{*}\mathcal{M})$,
$\mathcal{N}_{\textup{rel}}=R_{U}\mathcal{M}$ and $F=\prod_{j}u_{j}$. Thus,
Theorem 1.4 follows from Lemma 3.4.

If additionally $R_{U}\mathcal{M}$ is flat over $\mathbb{C}^{r}$, then we pick
an arbitrary point $\alpha\in\mathbb{C}^{r}$ and general hyperplanes
$\mathcal{H}_{1},\dots,\mathcal{H}_{r}$ such that $\alpha$ is a smooth
complete intersection of these hyperplanes. Applying Lemma 3.4 and Proposition
2.7 inductively, we conclude that
$\textup{CC}_{\textup{rel}}(R_{U}\mathcal{M})$ is relative Lagrangian and
hence that $R_{U}\mathcal{M}$ is relative holonomic. We have thus proved
Proposition 1.5. Now we pick a simplicial fan adapted to
$U_{\bullet}\mathcal{M}$ as in Theorem 4.19. Then
$\widetilde{R_{U}\mathcal{M}}$ is flat over $S_{\Sigma}$. By a similar
argument, we can more generally prove:

######  Proposition 4.24.

In the situation of Theorem 4.19, if $\mathcal{M}$ is a regular holonomic
$\mathscr{D}_{X}$-module, then $\widetilde{R_{U}\mathcal{M}}$ is relative
holonomic over $S_{\Sigma}$.

Since the saturation $\bar{U}_{\bullet}\mathcal{M}$ is good over
$V_{\bullet}\mathscr{D}_{X}$ (Theorem 4.21), Theorem 1.4 specifically implies

$\textup{CC}_{\textup{rel}}(R_{\bar{U}}\mathcal{M})=\overline{q^{-1}(\textup{CC}(\mathcal{M}))}.$

By construction, $\overline{q^{-1}(\textup{CC}(\mathcal{M}))}$ is a relative
conormal space but not necessarily a relative Lagrangian in general (unless
$r=1$). Since

$R_{\bar{U}}\mathcal{M}={\mu_{\Sigma}}_{*}(\widetilde{R_{U}\mathcal{M}}),$

from Proposition 4.24, we see that the direct image functors for relative
$\mathscr{D}$-modules under proper base changes do not necessarily preserve
relative holonomicity (cf. §2.2).

##  5\. Graph embedding construction of Malgrange

Let ${\bf f}=(f_{1},f_{2},\dots,f_{r})$ be a $r$-tuple of regular (or
holomorphic) functions on a smooth complex variety (or a complex manifold)
$Y$. We write

$j_{\bf f}\colon U_{Y}\coloneqq
Y\setminus(\prod_{i=1}^{r}f_{i}=0)\hookrightarrow Y$

the open embedding. We consider the graph embedding

$\iota_{{\bf f}}\colon Y\hookrightarrow X\coloneqq Y\times\mathbb{C}^{r}\quad
x\mapsto(x,f_{1}(x),\dots,f_{r}(x)).$

Let $\mathcal{M}_{Y}$ be a holonomic $\mathscr{D}_{Y}$-module. We set
$\widetilde{\mathcal{M}}=\mathcal{M}_{Y}(*D_{Y})$ with the divisor
$D_{Y}=(\prod_{i=1}^{r}f_{i}=0)$, which is a holonomic
$\mathscr{D}_{Y}$-module. We assume

$\widetilde{\mathcal{M}}=\mathscr{D}_{Y}\cdot\mathcal{M}_{0}$

for some $\mathscr{O}_{Y}$-coherent submodule
$\mathcal{M}_{0}\subseteq\widetilde{\mathcal{M}}$. Following the idea of
Malgrange [Mal83], we have a coherent
$\mathscr{D}_{Y}[\mathbf{s}]=\mathscr{D}_{Y}\otimes_{\mathbb{C}}\mathbb{C}[\mathbf{s}]$-submodule

$\mathscr{D}_{Y}[\mathbf{s}]({\bf
f}^{\mathbf{s}}\cdot\mathcal{M}_{0})\subseteq\widetilde{\mathcal{M}}[\mathbf{s}]\cdot{\bf
f}^{\mathbf{s}}$

where $\mathbf{s}=(s_{1},s_{2},\dots,s_{r})$,

${\bf f}^{\mathbf{s}}=\prod_{i=1}^{r}f_{i}^{s_{i}},$

and the $\mathscr{D}_{Y}[\mathbf{s}]$-module structure is induced by

$\theta\cdot({\bf f}^{\mathbf{s}}\cdot m_{0})={\bf
f}^{\mathbf{s}}\cdot\theta(m_{0})+\sum_{i=1}^{r}s_{i}\frac{\theta(f_{i})}{f_{i}}{\bf
f}^{\mathbf{s}}\cdot m_{0}$

for vector fields $\theta$ on $Y$, where $m_{0}$ is a section of
$\mathcal{M}_{0}$. Since $\mathscr{D}_{Y}[\mathbf{s}]({\bf
f}^{\mathbf{s}}\cdot\mathcal{M}_{0})$ is both a
$\mathbb{C}[\mathbf{s}]$-module and a $\mathscr{D}_{Y}$-module, it is a
coherent relative $\mathscr{D}$-module over $\mathbb{C}[\mathbf{s}]$. However,
$\widetilde{\mathcal{M}}[\mathbf{s}]\cdot{\bf f}^{\mathbf{s}}$ is not coherent
over $\mathscr{D}_{Y}[\mathbf{s}]$.

We denote by $(t_{1},\dots,t_{r})$ the coordinates of $\mathbb{C}^{r}$. The
key point is that after identifying $s_{i}$ with $-t_{i}\partial_{t_{i}}$, we
have a $\mathscr{D}_{X}$-module isomorphism

$\iota_{{\bf f}+}(\widetilde{\mathcal{M}})\simeq\iota_{{\bf
f}*}({\widetilde{\mathcal{M}}}[\mathbf{s}]{\bf f}^{\mathbf{s}})$

with the $t_{i}$-action on $\iota_{{\bf
f}*}({\widetilde{\mathcal{M}}}[\mathbf{s}]{\bf f}^{\mathbf{s}})$ given by

$t_{i}\cdot(b(\mathbf{s}){\bf f}^{\mathbf{s}}\cdot
m_{0})=b(s_{1},\dots,s_{i-1},s_{i}+1,s_{i+1},\dots,s_{r})f_{i}{\bf
f}^{\mathbf{s}}\cdot m_{0}.$

Consequently, $\iota_{{\bf f}*}(\mathscr{D}_{Y}[\mathbf{s}]({\bf
f}^{\mathbf{s}}\cdot\mathcal{M}_{0}))$ is a $\mathscr{D}_{X,D}$-lattice of
$\iota_{{\bf f}+}(\widetilde{\mathcal{M}})$, where $D$ is the divisor defined
by $(t_{1}\cdots t_{r}=0)$. Since $\iota_{{\bf
f}*}(\mathscr{D}_{Y}[\mathbf{s}]({\bf f}^{\mathbf{s}}\cdot\mathcal{M}_{0}))$
is supported on the graph of $Y$, abusing notations, we also say
$\mathscr{D}_{Y}[\mathbf{s}]({\bf f}^{\mathbf{s}}\cdot\mathcal{M}_{0})$ is a
$\mathscr{D}_{X,D}$-lattice of $\iota_{{\bf f}+}(\widetilde{\mathcal{M}})$.
Then $\mathcal{M}_{0}$ generates a holonomic $\mathscr{D}_{X}$-module

$\mathcal{M}=\mathscr{D}_{X}\cdot\iota_{{\bf
f}*}\mathcal{M}_{0}\subseteq\iota_{{\bf f}+}\widetilde{\mathcal{M}}$

and $\mathscr{D}_{Y}[\mathbf{s}]({\bf f}^{\mathbf{s}}\cdot\mathcal{M}_{0})$
generates a $R_{V}\mathscr{D}_{X}$-module

$R_{V}\mathscr{D}_{X}\cdot\iota_{{\bf f}*}(\mathscr{D}_{Y}[\mathbf{s}]({\bf
f}^{\mathbf{s}}\cdot\mathcal{M}_{0})),$

where the latter induces a $\mathbb{Z}^{r}$-filtration on $\mathcal{M}$ with

$U_{\vec{0}}\mathcal{M}=\iota_{{\bf f}*}(\mathscr{D}_{Y}[\mathbf{s}]({\bf
f}^{\mathbf{s}}\cdot\mathcal{M}_{0}))\textup{ and
}U_{-\vec{1}}\mathcal{M}=\iota_{{\bf f}*}(\mathscr{D}_{Y}[\mathbf{s}]({\bf
f}^{\mathbf{s}+\vec{1}}\cdot\mathcal{M}_{0})).$

We then apply Theorem 4.23 and obtain the Sabbah’s generalized $b$-function
$b(\mathbf{s})\in\mathbb{C}[\mathbf{s}]$ for $\mathscr{D}_{Y}[\mathbf{s}]({\bf
f}^{\mathbf{s}}\cdot\mathcal{M}_{0})$ such that

$b(\mathbf{s})\cdot\dfrac{\mathscr{D}_{Y}[\mathbf{s}]({\bf
f}^{\mathbf{s}}\cdot\mathcal{M}_{0})}{\mathscr{D}_{Y}[\mathbf{s}]({\bf
f}^{\mathbf{s}+\vec{1}}\cdot\mathcal{M}_{0})}=0$

with $b(\mathbf{s})$ given by a product of polynomials of degree 1. Sabbah’s
generalized $b$-functions associated to graph embeddings can be further
generalized to notions of Bernstein-Sato ideals (see for instance [Bud15]).

In the graph embedding case, we can construct the log rescaled family globally
(cf. §3.4):

$p:\widetilde{X}\coloneqq Y\times\mathbb{C}^{r}_{t}\times\mathbb{C}_{y}^{r}\to
Y\times\mathbb{C}^{r}_{t},\quad(x,t,y)\mapsto(x,e^{y}t).$

We now give a counterexample of Theorem 3.3 without flatness when $r=2$:

######  Example 2.

We take $Y=\mathbb{C}^{2}$ with coordinates $(x_{1},x_{2})$ and ${\bf
f}=(x_{1}x_{2},x_{2})$. We consider the $\mathscr{D}_{X,D}$-lattice
$\bar{\mathcal{M}}=\mathscr{D}_{Y}[\mathbf{s}]{\bf f}^{\mathbf{s}}$. Its
$\mathscr{D}_{X,D}$-annihilator is

${\textup{Ann}}_{\mathscr{D}_{X,D}}(\bar{\mathcal{M}})=(x_{1}\partial_{x_{1}}+t_{1}\partial_{t_{1}},x_{2}\partial_{x_{2}}+t_{2}\partial_{t_{2}},t_{1}-x_{1}x_{2},t_{2}-x_{2}).$

Since $\bar{\mathcal{M}}$ and $\mathscr{M}=p^{*}(\iota_{{\bf
f}*}\bar{\mathcal{M}})$ are both acyclic,

$\textup{Ch}_{{\textup{rel}}}(\mathscr{M})=(x_{1}\xi_{x_{1}}=-\xi_{y_{1}},x_{2}\xi_{x_{2}}=-\xi_{y_{2}},e^{y_{1}}t_{1}=x_{1}x_{2},e^{y_{2}}t_{2}=x_{2}).$

Thus, the fiber of $\textup{Ch}_{{\textup{rel}}}(\mathscr{M})$ over
$(t_{1}=0,t_{2}=0)$ satisfies

$\textup{Ch}_{{\textup{rel}}}(\mathscr{M})\cap(t_{1}=0,t_{2}=0)=(x_{1}\xi_{x_{1}}=-\xi_{y_{1}},\xi_{y_{2}}=0,x_{2}=0)$

and hence the dimension of the fiber is $5>4$. Therefore,
$\textup{Ch}_{\textup{rel}}(\mathscr{M})$ is not relative Lagrangian over
$\mathbb{C}^{2}_{t}$.

The following theorem is a generalization of [Mai16a, Résultat 1]. See [Wu21,
Theorem 3.3] for the proof of a more generalized result and also [BVWZ21b,
Theorem 4.3.4]when $\mathcal{M}_{0}=\mathscr{O}_{X}$.

######  Theorem 5.1.

If $\widetilde{\mathcal{M}}$ is a holonomic $\mathscr{D}_{Y}$-module, then
every lattice $\mathscr{D}_{Y}[\mathbf{s}]({\bf
f}^{\mathbf{s}}\cdot\mathcal{M}_{0})$ is relative holonomic over
$\mathbb{C}[\mathbf{s}]$ and

$\textup{Ch}^{\textup{rel}}(\mathscr{D}_{Y}[\mathbf{s}]({\bf
f}^{\mathbf{s}}\cdot\mathcal{M}_{0}))=\textup{Ch}(\widetilde{\mathcal{M}})\times\mathbb{C}^{r}.$

###### Proof of Theorem 1.1.

Since constructibility is local, it is enough to assume that

$D=(\prod_{i=1}^{r}x_{i}=0)$

with local coordinates $(x_{1},x_{2},\dots,x_{n})$ and
$\widetilde{\mathcal{M}}=\mathcal{M}(*D)$ satisfies

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

for some coherent $\mathscr{O}_{X}$-submodule $\mathcal{M}_{0}$. Then we take
the graph embedding of smooth log pairs

$\iota_{\bf f}\colon(X,D)\hookrightarrow(Z,D_{Z})$

where ${\bf f}=(x_{1},x_{2},\dots,x_{r})$ and $Z=X\times\mathbb{C}^{r}$,
$D_{Z}=(t_{1}\cdots t_{r}=0)$. Similar to the non-log case (cf. [HTT08,
Example 1.5.23]), we have

$\iota^{\log}_{{\bf f}+}(\bar{\mathcal{M}})\simeq\iota_{{\bf
f}*}(\bar{\mathcal{M}}\otimes_{\mathscr{O}}\omega^{\log}_{\iota_{\bf
f}}\otimes_{\mathscr{D}_{X,D}}f^{*}\mathscr{D}_{Z,D_{Z}})$

and

$\iota^{\log}_{{\bf f}+}(\bar{\mathcal{M}})\hookrightarrow\iota_{{\bf
f}+}(\widetilde{\mathcal{M}}).$

Thus, $\iota^{\log}_{{\bf f}+}(\bar{\mathcal{M}})$ is a
$\mathscr{D}_{Z,D_{Z}}$-lattice of the holonomic $\mathscr{D}_{Z}$-module
$\iota_{{\bf f},+}(\widetilde{\mathcal{M}})$. But the graph embedding gives us
different lattices of $\iota_{{\bf f},+}(\widetilde{\mathcal{M}})$,

$\mathscr{D}_{X}[\mathbf{s}]({\bf f}\cdot\mathcal{M}_{0}(kD))$

for all $k\in\mathbb{Z}$. The lattices $\mathscr{D}_{X}[\mathbf{s}]({\bf
f}\cdot\mathcal{M}_{0}(kD))$ are relative holonomic over
$\mathbb{C}[\mathbf{s}]$ by Theorem 5.1. Meanwhile, we can compare lattices:

$\iota^{\log}_{{\bf f}+}(\bar{\mathcal{M}})\subseteq\iota_{{\bf
f}*}(\mathscr{D}_{X}[\mathbf{s}]({\bf
f}\cdot\mathcal{M}_{0}))\otimes_{\mathscr{O}_{Z}}\mathscr{O}_{Z}(kD_{Z})=\iota_{{\bf
f}*}(\mathscr{D}_{X}[\mathbf{s}]({\bf f}\cdot\mathcal{M}_{0}(kD)))$

for $k\gg 0$. The above inclusion and Corollary 2.5 together imply that
$\iota^{-1}_{\bf f}(\iota^{\log}_{{\bf f}+}(\bar{\mathcal{M}}))$ is relative
holonomic over $\mathbb{C}[\mathbf{s}]$. By Lemma 2.6,

$\iota^{-1}_{\bf f}(\iota^{\log}_{{\bf
f}+}(\bar{\mathcal{M}}))\otimes_{\mathbb{C}[\mathbf{s}]}^{\mathbf{L}}\mathbb{C}_{\mathbf{0}}\simeq\mathbf{L}i_{\mathbf{0}}^{*}(\iota^{-1}_{\bf
f}(\iota^{\log}_{{\bf f}+}(\bar{\mathcal{M}})))$

is a complex of $\mathscr{D}_{X}$-modules with holonomic cohomology sheaves,
where $i_{\mathbf{0}}\colon\\{\mathbf{0}\\}\hookrightarrow\textup{Spec
}\mathbb{C}[\mathbf{s}]$ is the closed embedding and $\mathbb{C}_{\mathbf{0}}$
is the residue field. By the construction of the log de Rham complex and
Proposition 3.1, we have

(14) $\iota_{{\bf
f}*}(\textup{DR}_{X,D}(\bar{\mathcal{M}}))\simeq\textup{DR}_{Z,D_{Z}}(\iota^{\log}_{{\bf
f}+}(\bar{\mathcal{M}}))\simeq\iota_{{\bf f}*}(\textup{DR}_{X}(\iota^{-1}_{\bf
f}(\iota^{\log}_{{\bf
f}+}(\bar{\mathcal{M}}))\otimes_{\mathbb{C}[\mathbf{s}]}^{\mathbf{L}}\mathbb{C}_{\mathbf{0}}))$

where the last quasi-isomorphism follows by identifying $s_{i}$ with
$-t_{i}\partial_{t_{i}}$. Since

$\iota^{-1}_{\bf f}(\iota^{\log}_{{\bf
f}+}(\bar{\mathcal{M}}))\otimes_{\mathbb{C}[\mathbf{s}]}^{\mathbf{L}}\mathbb{C}_{\mathbf{0}}$

is a complex of $\mathscr{D}_{X}$-modules with holonomic cohomology sheaves,
$\textup{DR}_{X,D}(\bar{\mathcal{M}})$ is constructible by Kashiwara’s
constructibility theorem (cf. [HTT08, Theorem 4.6.3]). ∎

######  Remark 5.2.

(1) From the proof of Theorem 1.1, $\textup{DR}_{X,D}(\bar{\mathcal{M}})$ is
not necessarily perverse in general, unless $\iota^{-1}_{{\bf f}}(\iota_{{\bf
f}+}^{\log}(\bar{\mathcal{M}}))$ is flat over a neighborhood of
${\mathbf{0}}\in\mathbb{C}^{r}$.
(2) The stratification for the constructible complex
$\textup{DR}_{X,D}(\bar{\mathcal{M}})$ is determined by the stratification of
$\textup{Ch}(\mathcal{M}(*D))$ by (14) and Theorem 5.1.

###### Proof of Theorem 1.2.

Part (1) is the analytification of [WZ21, Theorem 1.1] with the same proof. We
now prove Part (2). We keep the notations as in the proof of Theorem 1.1. By
picking some $k\gg 0,$ we have an inclusion of lattices

$\iota^{\log}_{{\bf f}+}(\bar{\mathcal{M}})\subseteq\iota_{{\bf
f}*}(\mathscr{D}_{X}[\mathbf{s}]({\bf
f}^{\mathbf{s}}\cdot\mathcal{M}_{0}(kD))).$

We then consider the short exact sequence of $\mathscr{D}_{Z,D_{Z}}$-modules

$0\to\iota^{\log}_{{\bf f}+}(\bar{\mathcal{M}})\rightarrow\iota_{{\bf
f}*}(\mathscr{D}_{X}[\mathbf{s}]({\bf
f}\cdot\mathcal{M}_{0}(kD)))\rightarrow\mathcal{Q}\to 0,$

where $\mathcal{Q}$ is defined to be the quotient module. Applying Theorem
4.23 to $\mathcal{Q}$, there exists $b(\mathbf{s})\in\mathbb{C}[\mathbf{s}]$
as a product of linear polynomials in $\mathbf{s}$ such that

$b(\mathbf{s})\cdot\mathcal{Q}=0.$

Using substitution, we have

$b(\mathbf{s}+\mathbf{l})\cdot\mathcal{Q}(-lD_{Z})=0$

for $\mathbf{l}=(l,l,\dots,l)\in\mathbb{Z}^{r}$ and for each $l$. Chose $l\gg
0$, so that $b(\mathbf{s}+\mathbf{l})$ does not vanishing at
${\mathbf{0}}\in\mathbb{C}^{r}$. Thus,
$\mathcal{Q}(-lD_{Z})\otimes\mathbb{C}_{\mathbf{0}}=0.$ Considering the above
short exact sequence, since

$\textup{DR}_{Z,D_{Z}}(\mathcal{Q}(-lD_{Z}))\simeq\iota_{{\bf
f}*}(\textup{DR}_{X}(\iota^{-1}_{{\bf
f}}(\mathcal{Q}(-lD_{Z}))\otimes\mathbb{C}_{\mathbf{0}}))$

we hence obtain a quasi-isomorphism

(15) $\textup{DR}_{Z,D_{Z}}(\iota^{\log}_{{\bf
f}+}(\bar{\mathcal{M}})(-lD_{Z}))\xrightarrow{q.i.}\textup{DR}_{Z,D_{Z}}(\iota_{{\bf
f}*}(\mathscr{D}_{X}[\mathbf{s}]({\bf f}\cdot\mathcal{M}_{0}((k-l)D))))$

for some $l\gg k$. By construction, we have

$\textup{DR}_{Z,D_{Z}}(\iota_{{\bf f}*}(\mathscr{D}_{X}[\mathbf{s}]({\bf
f}\cdot\mathcal{M}_{0}((k-l)D))))\simeq\iota_{{\bf
f}*}(\textup{DR}_{X}(\mathscr{D}_{X}[\mathbf{s}]({\bf
f}\cdot\mathcal{M}_{0}((k-l)D))\otimes^{\mathbf{L}}_{\mathbb{C}[\mathbf{s}]}\mathbb{C}_{\mathbf{0}})).$

We then apply [WZ21, Corollary 5.4], and by the quasi-isomorphism (15) obtain

$\textup{DR}_{Z,D_{Z}}(\iota^{\log}_{{\bf
f}+}(\bar{\mathcal{M}})(-lD_{Z}))\simeq\iota_{{\bf
f}*}(j_{!}(\textup{DR}(\mathcal{M}|_{U}))).$

By the projection formula,

$\iota^{\log}_{{\bf f}+}(\bar{\mathcal{M}})(-lD_{Z})\simeq\iota^{\log}_{{\bf
f}+}(\bar{\mathcal{M}}(-lD)).$

Since $\iota_{{\bf f}}$ is a closed embedding, the required quasi-isomorphism
then follows from Proposition 3.1.

We now prove the perversity statement without the regularity assumption. Using
the argument in proving [WZ21, Theorem 5.2 and 5.3] as well as the discussion
in [BVWZ21a, §3.6] in dealing with the local analytic case , one obtains that
the $\mathbb{C}[\mathbf{s}]$-modules $\mathscr{D}_{X}[\mathbf{s}]({\bf
f}^{\mathbf{s}}\cdot\mathcal{M}_{0}(kD))$ are flat over a Zariski neighborhood
of ${\mathbf{0}}\in\textup{Spec }\mathbb{C}[\mathbf{s}]$ for all $|k|\gg 0$.
Using Sabbah’s $b$-functions, we then conclude that $\iota^{-1}_{\bf
f}(\iota^{\log}_{{\bf f}+}(\bar{\mathcal{M}}(kD)))$ are flat over a Zariski
neighborhood of ${\mathbf{0}}\in\textup{Spec }\mathbb{C}[\mathbf{s}]$ for all
$|k|\gg 0$. Consequently, the required perversity follows. ∎

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