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

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).

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).

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

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.

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,

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

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

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 $L$,

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.

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),

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,

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 1.3 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]).

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

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,

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)

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.

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

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.

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.

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

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

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).

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

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

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 $D$.

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.

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.

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.

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

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

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

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

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

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:

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

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:

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].

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

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

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

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 $\mu$:

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 $\mathcal{X}_{\mathcal{Z}}=\mathcal{X}\times_{\mathcal{S}}\mathcal{Z}$.

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

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

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.

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

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

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

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

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

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

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