$t$-Structures for Relative $\mathcal{D}$-Modules and $t$-Exactness of the de Rham Functor

TL;DR

This paper investigates the behavior of $t$-structures in relative holonomic $ $-modules, revealing how duality affects $t$-exactness and establishing conditions under which key functors are $t$-exact.

Contribution

It explicitly describes dual $t$-structures in the relative setting and proves $t$-exactness of the solution, Riemann-Hilbert, and de Rham functors in certain cases.

Findings

Dual $t$-structures are explicitly characterized.

Solution and Riemann-Hilbert functors are $t$-exact in the 1-dimensional parameter case.

De Rham functor is $t$-exact for canonical and middle-perverse $t$-structures.

Abstract

This paper is a contribution to the study of relative holonomic $\mathcal{D}$-modules. Contrary to the absolute case, the standard $t$-structure on holonomic $\mathcal{D}$-modules is not preserved by duality and hence the solution functor is no longer $t$-exact with respect to the canonical, resp. middle-perverse, $t$-structures. We provide an explicit description of these dual $t$-structures. When the parameter space is 1-dimensional, we use this description to prove that the solution functor as well as the relative Riemann-Hilbert functor are $t$-exact with respect to the dual $t$-structure and to the middle-perverse one while the de Rham functor is $t$-exact for the canonical, resp. middle-perverse, $t$-structures and their duals.

Equations

$$\begin{array}{cc}{D}_{(\mathcal{T},\mathcal{F})}^{⩽0}=& \{C\in \mathcal{C}|H_D^1(C)\in \mathcal{T},H_D^i(C)=0\forall i>1\}\text{hfill}\\ D_{(\mathcal{T},\mathcal{F})}^{⩾0}=& \{C\in \mathcal{C}|H_D^0(C)\in \mathcal{F},H_D^i(C)=0\forall i<0\}\end{array}$$

$${\mathcal{H}}_{D_{(\mathcal{T},\mathcal{F})}}=\{C\in \mathcal{C}|H_D^1(C)\in \mathcal{T},H_D^0(C)\in \mathcal{F},H_D^i(C)=0\forall i\notin \{0,1\}\}.$$

$${\mathcal{H}}_{{\tilde{D}}_{(\mathcal{T},\mathcal{F})}}=\{C\in \mathcal{C}|H_D^0(C)\in \mathcal{T},H_D^{-1}(C)\in \mathcal{F},H_D^i(C)=0\forall i\notin \{0,-1\}\}.$$

$$D^{⩽0}\subseteq D_{(\mathcal{T},\mathcal{F})}^{⩽0}\subseteq D^{⩽1}\text{or equivalently}{D}^{⩾1}\subseteq D_{(\mathcal{T},\mathcal{F})}^{⩾0}\subseteq D^{⩾0}$$

$$D^{⩽-1}\subseteq {\tilde{D}}_{(\mathcal{T},\mathcal{F})}^{⩽0}\subseteq D^{⩽0}\text{or equivalently}{D}^{⩾0}\subseteq {\tilde{D}}_{(\mathcal{T},\mathcal{F})}^{⩾0}\subseteq D^{⩾-1}$$

$$(\mathcal{T},\mathcal{F}):=({\overline{D}}^{⩽-1}\cap {\mathcal{H}}_D,{\overline{D}}^{⩾0}\cap {\mathcal{H}}_D)(resp.(\mathcal{T},\mathcal{F}):=({\overline{D}}^{⩽0}\cap {\mathcal{H}}_D,{\overline{D}}^{⩾1}\cap {\mathcal{H}}_D)$$

$$D^{⩽0}\subseteq {\overline{D}}^{⩽0}\subseteq D^{⩽\ell }$$

$$\mathbf{D}(\mathcal{M}):={\mathrm{RHom}}_{{\mathcal{D}}_{X\times S/S}}(\mathcal{M},{\mathcal{D}}_{X\times S/S}{\otimes }_{{\mathcal{O}}_{X\times S}}{\Omega }_{X\times S/S}^{{\otimes }^{-1}})[n]$$

$$\begin{array}{cc}{}^{\Pi }{D}_{\mathrm{hol}}^{⩽0}({\mathcal{D}}_{X\times S/S})=& \{\mathcal{M}\in D_{\mathrm{hol}}^{\mathrm{b}}({\mathcal{D}}_{X\times S/S})|\mathbf{D}\mathcal{M}\in {}^P{D}_{\mathrm{hol}}^{⩾0}({\mathcal{D}}_{X\times S/S})\}\text{hfill}\\ {}^{\Pi }{D}_{\mathrm{hol}}^{⩾0}({\mathcal{D}}_{X\times S/S})=& \{\mathcal{M}\in D_{\mathrm{hol}}^{\mathrm{b}}({\mathcal{D}}_{X\times S/S})|\mathbf{D}\mathcal{M}\in {}^P{D}_{\mathrm{hol}}^{⩽0}({\mathcal{D}}_{X\times S/S})\}.\end{array}$$

$$\begin{array}{cc}{}^{\pi }{D}_{\mathrm{coh}}^{⩽0}({\mathcal{O}}_S)=& \{\mathcal{M}\in D_{\mathrm{coh}}^{\mathrm{b}}({\mathcal{O}}_S)|\mathrm{codim}\mathrm{Supp}({\mathcal{H}}^k(\mathcal{M}))⩾k\}\text{hfill}\\ {}^{\pi }{D}_{\mathrm{coh}}^{⩾0}({\mathcal{O}}_S)=& \{\mathcal{M}\in D_{\mathrm{coh}}^{\mathrm{b}}({\mathcal{O}}_S)|{\mathcal{H}}_{[Z]}^k({\mathcal{M}}_{|U})=0\; for\; any\; analytic\; closed\; subsetZ\\ & \text{hfill}of\; any\; open\; subsetU\subseteq Sandk<{\mathrm{codim}}_{U}Z\}.\end{array}$$

$$\mathcal{M}\in {}^{\Pi }{D}_{\mathrm{hol}}^{⩾0}({\mathcal{D}}_{X\times S/S})⟺\mathbf{D}\mathcal{M}\in {}^P{D}_{\mathrm{hol}}^{⩽0}({\mathcal{D}}_{X\times S/S})\text{hfill}$$

$$\stackrel{by\; (2)}{⟺}\mathcal{M}\in D_{\mathrm{hol}}^{\mathrm{b}}({\mathcal{D}}_{X\times S/S})and\; \forall s\in S,L{i}_s^{\ast }\mathbf{D}\mathcal{M}\stackrel{\ast }{\cong }\mathbf{D}L{i}_s^{\ast }\mathcal{M}\in D_{\mathrm{hol}}^{⩽0}({\mathcal{D}}_X)\text{hfill}$$

$$⟺\mathcal{M}\in D_{\mathrm{hol}}^{\mathrm{b}}({\mathcal{D}}_{X\times S/S})and\; \forall s\in S,L{i}_s^{\ast }\mathcal{M}\in D_{\mathrm{hol}}^{⩾0}({\mathcal{D}}_X)\text{hfill}$$

$${}^{\Pi }{D}_{\mathrm{hol}}^{⩽-\ell }({\mathcal{D}}_{X\times S/S})\subseteq {}^P{D}_{\mathrm{hol}}^{⩽0}({\mathcal{D}}_{X\times S/S})\subseteq {}^{\Pi }{D}_{\mathrm{hol}}^{⩽0}({\mathcal{D}}_{X\times S/S})$$

$$\mathbf{D}\mathcal{M}\in {}^P{D}_{\mathrm{hol}}^{[0,\ell ]}({\mathcal{D}}_{X\times S/S}).$$

$$\mathcal{M}\in {}^{\Pi }{D}_{\mathrm{hol}}^{⩽-\ell }({\mathcal{D}}_{X\times S/S})⟺\mathbf{D}\mathcal{M}\in {}^P{D}_{\mathrm{hol}}^{⩾\ell }({\mathcal{D}}_{X\times S/S})$$

$$\Rightarrow \forall s\in S,L{i}_s^{\ast }\mathbf{D}\mathcal{M}\cong \mathbf{D}L{i}_s^{\ast }\mathcal{M}\in D_{\mathrm{hol}}^{⩾0}({\mathcal{D}}_X)\iff \forall s\in S,L{i}_s^{\ast }\mathcal{M}\in D_{\mathrm{hol}}^{⩽0}({\mathcal{D}}_X)$$

$$⟺\mathcal{M}\in {}^P{D}_{\mathrm{hol}}^{⩽0}({\mathcal{D}}_{X\times S/S}).$$

$${\mathcal{F}}^{\bullet }:=\cdots 0⟶0⟶\mathrm{Coker}(d_{{\mathcal{L}}^{\ast \bullet }}^{-1})⟶{\mathcal{L}}^{\ast 1}\stackrel{d_{{\mathcal{L}}^{\ast \bullet }}^1}{⟶}{\mathcal{L}}^{\ast 2}⟶\cdots$$

$${\mathcal{L}}^{\ast ⩾1}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${}^{\Pi }{D}_{\mathrm{hol}}^{⩽-1}({\mathcal{D}}_{X\times S/S})\subset {}^P{D}_{\mathrm{hol}}^{⩽0}({\mathcal{D}}_{X\times S/S})\subset {}^{\Pi }{D}_{\mathrm{hol}}^{⩽0}({\mathcal{D}}_{X\times S/S})\subset {}^P{D}_{\mathrm{hol}}^{⩽1}({\mathcal{D}}_{X\times S/S})$$

$$({}^{\Pi }{D}_{\mathrm{hol}}^{⩽-1}({\mathcal{D}}_{X\times S/S})\cap {\mathrm{Mod}}_{\mathrm{hol}}({\mathcal{D}}_{X\times S/S}),{}^{\Pi }{D}_{\mathrm{hol}}^{⩾0}({\mathcal{D}}_{X\times S/S})\cap {\mathrm{Mod}}_{\mathrm{hol}}({\mathcal{D}}_{X\times S/S})).$$

$${\mathcal{H}}_{\Pi }=\{\mathcal{M}\in {}^P{D}_{\mathrm{hol}}^{[0,1]}({\mathcal{D}}_{X\times S/S})|{}^P{\mathcal{H}}^0(\mathcal{M})strict\; and{}^P{\mathcal{H}}^1(\mathcal{M})torsion\}$$

$$\mathrm{codim}\mathrm{Char}(\mathcal{E}{xt}_{{\mathcal{D}}_{X\times S/S}}^k(\mathcal{N},{\mathcal{D}}_{X\times S/S}))⩾k,$$

$$\mathrm{codim}\mathrm{Char}({\mathcal{H}}_{{\mathcal{D}}_{X\times S/S}}^k\mathbf{D}\mathcal{N})⩾k+d_X.$$

$$\mathrm{codim}\mathrm{Char}(\mathcal{E}{xt}_{{\mathcal{D}}_{X\times S}}^k({\mathcal{D}}_{X\times S}{\otimes }_{{\mathcal{D}}_{X\times S/S}}\mathcal{N},{\mathcal{D}}_{X\times S}))$$

$$=\mathrm{codim}\mathrm{Char}(\mathcal{E}{xt}_{{\mathcal{D}}_{X\times S/S}}^k(\mathcal{N},{\mathcal{D}}_{X\times S/S}){\otimes }_{{\mathcal{D}}_{X\times S/S}}{\mathcal{D}}_{X\times S})⩾k$$

$$\mathrm{Char}(\mathcal{E}{xt}_{{\mathcal{D}}_{X\times S/S}}^k(\mathcal{N},{\mathcal{D}}_{X\times S/S}){\otimes }_{{\mathcal{D}}_{X\times S/S}}{\mathcal{D}}_{X\times S})={\pi }^{-1}\mathrm{Char}(\mathcal{E}{xt}_{{\mathcal{D}}_{X\times S/S}}^k(\mathcal{N},{\mathcal{D}}_{X\times S/S}))$$

$$\mathrm{codim}\mathrm{Char}(\mathcal{E}{xt}_{{\mathcal{D}}_{X\times S/S}}^k(\mathcal{N},{\mathcal{D}}_{X\times S/S}))⩾k$$

$$\mathrm{Char}(\mathcal{M})=\bigcup _{i\in I}{\Lambda }_i\times T_i$$

$$\dim \mathrm{Char}(\mathcal{M})=\dim X+twheret=\dim p_X(\mathrm{Supp}(\mathcal{M}))=\underset{i\in I}{\sup }\dim T_i.$$

$$\mathrm{Supp}(\mathcal{M})=\mathrm{Char}\mathcal{M}\cap (T_X^{\ast }X\times S)$$

Full text

The research of L.Fiorot was supported by project BIRD163492 "Categorical homological methods in the study of algebraic structures" and project DOR1749402. The research of T.Monteiro Fernandes was supported by Fundação para a Ciência e a Tecnologia, UID/MAT/04561/2013.

1. 1 Torsion pairs, quasi-abelian categories and $t$-structures
2. 2 $t$-structures on $\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{hol}}(\mathscr{D}_{X\times S/S})$
3. 3 $t$-structures on $\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p_{X}^{-1}\mathscr{O}_{S})$

Introduction.

Let $X$ and $S$ be complex manifolds and let $p_{X}$ denote the projection of $X\times S\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}S$. We shall denote by $d_{X}$ and $d_{S}$ their respective complex dimensions and will often write $p$ instead of $p_{X}$ whenever there is no ambiguity.

An extensive study of holonomic and regular holonomic $\mathscr{D}_{X\times S/S}$-modules as well as of their derived categories was performed in [MFCS1] and [MFCS2]. Such modules are called for convenience respectively relative holonomic and regular relative holonomic modules. Relative holonomic modules are coherent modules whose characteristic variety, in the product $(T^{*}X)\times S$, is contained in $\Lambda\times S$ for some Lagrangian conic closed analytic subset $\Lambda$ of $T^{*}X$. Regular relative holonomic modules are holonomic modules whose restriction to the fibers of $p_{X}$ have regular holonomic $\mathscr{D}_{X}$-modules as cohomologies.

Another notion introduced in [MFCS1] was that of $\mathbb{C}$-constructibility over $p_{X}^{-1}\mathscr{O}_{S}$, leading to the (bounded) derived category of sheaves of $p^{-1}_{X}\mathscr{O}_{S}$-modules with $\mathbb{C}$-constructible cohomology, the $S$-$\mathbb{C}$-constructible complexes (this category is denoted by $\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathbb{C}\textup{-c}}(p_{X}^{-1}\mathscr{O}_{S})$), where a natural notion of perversity was also introduced. In loc.cit. it was proved that the essential image of the de Rham functor $\operatorname{DR}$ as well as of the solution functor $\operatorname{Sol}$, when restricted to the bounded derived category of $\mathscr{D}_{X\times S/S}$-modules with holonomic cohomology, is $\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathbb{C}\textup{-c}}(p_{X}^{-1}\mathscr{O}_{S})$. Recall that, denoting by $\operatorname{{}^{\mathrm{p}}Sol}(\mathscr{M})$ (resp.$\operatorname{{}^{\mathrm{p}}DR}(\mathscr{M}))$ the complex $\operatorname{Sol}(\mathscr{M})[d_{X}]$ (resp.$\operatorname{DR}(\mathscr{M})[d_{X}])$, these two functors satisfy a natural isomorphism of commutation with duality: $\boldsymbol{D}\operatorname{{}^{\mathrm{p}}Sol}(\cdot)\simeq\operatorname{{}^{\mathrm{p}}DR}(\cdot)$.

Under the assumption $d_{S}=1$, a right quasi-inverse functor to $\operatorname{{}^{\mathrm{p}}Sol}$, the functor $\operatorname{RH^{S}}$, was introduced in [MFCS2], so naturally $\operatorname{RH^{S}}$ is a functor from $\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathbb{C}\textup{-c}}(p_{X}^{-1}\mathscr{O}_{S})$ to the bounded derived category $\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{rhol}}(\mathscr{D}_{X\times S/S})$ of $\mathscr{D}_{X\times S/S}$-modules with regular holonomic modules. $\operatorname{RH^{S}}$ is the relative version of Kashiwara’s Riemann Hilbert functor $\mathrm{RH}$ (cf.[Ka3]) as explained in Section LABEL:subsec:relsubanalytic where we briefly recall its construction. Recall that the importance of this apparently restrictive assumption on $S$ is two-sided: for $d_{S}=1$, $\mathscr{O}_{S}$-flatness and absence of $\mathscr{O}_{S}$ torsion are equivalent, so we can split proofs in the torsion case and in the torsion free case; on the other hand, although we will not enter into details here, the construction of $\operatorname{RH^{S}}$ requires, locally on $S$, the existence of bases of the coverings of the subanalytic site $S_{sa}$ formed by $\mathscr{O}_{S}$-acyclic open subanaytic sets which is possible in the case $d_{S}=1$.

The main goal of this paper is to prove the $t$-exactness of $\operatorname{{}^{\mathrm{p}}Sol}$, $\operatorname{{}^{\mathrm{p}}DR}$ and $\operatorname{RH^{S}}$ with respect to the $t$-structures involved (for any $S$ in the first two cases and for $d_{S}=1$ in the case of $\operatorname{RH^{S}}$). Recall that when one replaces $\mathscr{O}_{S}$ by the constant sheaf $\mathbb{C}_{X}[[\hbar]]$ of formal power series in one parameter $\hbar$, so no longer in the relative case, these questions were studied and solved by A. D’Agnolo, S. Guillermou and P. Schapira in [D'AgnSGuillPSch].

Here, to be more precise, in the holonomic side we have the standard $t$-structure $P$ as well as its dual $\Pi$, which, contrary to the absolute case proved by Kashiwara in [Ka3], do not coincide if $d_{X}\geqslant 1,\,d_{S}\geqslant 1$ which is not surprising due to the possible absence of $\mathscr{O}_{S}$-flatness. Similarly, on the $\mathbb{C}$-constructible side, we have the perverse $t$-structure $p$ introduced in [MFCS1] and its dual $\pi$, which do not coincide if $d_{X},\,d_{S}\geqslant 1$ as well. Kashiwara’s paper [Ka4] provides a wide setting for this kind of problems covering the case $d_{X}=0$ (the $\mathscr{O}_{S}$-coherent case) as well as the standard $t$-structure on the $\mathbb{C}$-constructible case and the correspondent $t$-structure on $\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{rhol}}(\mathscr{D}_{X})$ via $\mathrm{RH}$. We took there our inspiration, adapting the ideas of several proofs.

In Theorems 2.11 and LABEL:PrOp:PerDual we completely describe $\Pi$ and $\pi$ for any $d_{X}$ and $d_{S}$. In particular, when $d_{S}=1$, we prove in Proposition 2.6 that $\Pi$ is obtained by left tilting $P$ with respect to a natural torsion pair (respectively $P$ is obtained by right tilting $\Pi$ with respect to a natural torsion pair) and we conclude in Corollary 2.7 that the category of strict relative holonomic modules is quasi-abelian ([S]). Similar results are deduced for $\pi$ and $p$ in Proposition LABEL:PrOp:cctor leading to the conclusion that perverse $S$-$\mathbb{C}$-constructible complexes with a perverse dual are the objects of a quasi-abelian category. Recall that the procedure of tilting a $t$-structure $({\operatorname{\mathsf{D}}}^{\leqslant 0},{\operatorname{\mathsf{D}}}^{\geqslant 0})$ on a triangulated category $\mathscr{C}$ with respect to a given torsion pair $(\mathscr{T},\mathscr{F})$ on its heart has been introduced by Happel, Reiten and Smalø in their work [HRS]. Following the notation of Bridgeland ([Brid] and [Brid2]) Polishchuk proved in [Po] that performing the left tilting procedure one gets all the $t$-structures $({\overline{\operatorname{\mathsf{D}}}}^{\leqslant 0},{\overline{\operatorname{\mathsf{D}}}}^{\geqslant 0})$ satisfying the condition ${\operatorname{\mathsf{D}}}^{\leqslant 0}\subseteq{\overline{\operatorname{\mathsf{D}}}}^{\leqslant 0}\subseteq{{\operatorname{\mathsf{D}}}}^{\leqslant 1}$. The relations between torsion pairs, tilted $t$-structures and quasi-abelian categories have been clarified in [BonVdBergh] and [F].

With these informations in hand we have the tools to prove, in Theorem LABEL:T:perv_1 that $\operatorname{{}^{\mathrm{p}}DR}$ is exact with respect to $P$ and $p$ (so, by duality, with respect to $\Pi$ and $\pi$) which gives a precision to the behaviour of $\operatorname{{}^{\mathrm{p}}DR}$ already studied in [MFCS2]. However, since it is not known if $\operatorname{RH^{S}}$ provides an equivalence of categories for general $d_{X}$, we do not dispose of a morphism of functors $\boldsymbol{D}\operatorname{RH^{S}}(\cdot)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{RH^{S}}(\boldsymbol{D}(\cdot))$ allowing us to argue by duality as in the $\mathbb{C}$-constructible framework. Nevertheless, by a direct proof, in Theorem LABEL:T:perv_2 we prove that $\operatorname{RH^{S}}$ is exact with respect to $p$ and $\Pi$ as well as to the dual structures $\pi$ and $P$.

We are deeply grateful to the referee for the pertinent corrections which helped us to improve our work.

1. Torsion pairs, quasi-abelian categories and $t$-structures

Let $\mathscr{C}$ be an additive category. In what follows any full subcategory $\mathscr{C}^{\prime}$ of $\mathscr{C}$ will be strictly full (i.e., closed under isomorphisms) and additive and we will use the notation $\mathscr{C}^{\prime}\subseteq\mathscr{C}$ to indicate such a subcategory. Any functor between additive categories will be an additive functor. In these terms given $\mathscr{C}_{i}\subseteq\mathscr{C}$ for $i\in\{1,2\}$ following [BBD, Definition 1.3.1] we will denote by $\mathscr{C}_{1}\cap\mathscr{C}_{2}$ the strictly full subcategory of $\mathscr{C}$ whose objects belong to both $\mathscr{C}_{1}$ and $\mathscr{C}_{2}$.

A torsion pair in an abelian category $\mathscr{A}$ is a pair $(\mathscr{T},\mathscr{F})$ of full subcategories of $\mathscr{A}$ satisfying the following conditions: $\operatorname{Hom}_{A}(T,F)=0$ for every $T\in\mathscr{T}$ and every $F\in\mathscr{F}$ ; for any object $A\in\mathscr{A}$ there exists a short exact sequence: $0\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}t(A)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}A\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}f(A)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}0$ in $\mathscr{A}$ such that $t(A)\in\mathscr{T}$ and $f(A)\in\mathscr{F}$. The class $\mathscr{T}$ is called the torsion class and it is closed under extensions, direct sums and quotients, while $\mathscr{F}$ is the torsion-free class and it is closed under extensions, subobjects and direct products. In particular, $\mathscr{T}$ is a full subcategory of $\mathscr{A}$ such that the inclusion functor $i_{\mathscr{T}}:\mathscr{T}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{A}$ admits a right adjoint $t:\mathscr{A}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{T}$ such that $ti_{\mathscr{T}}={\mathrm{id}}_{\mathscr{T}}$, and dually, the inclusion functor $i_{\mathscr{F}}:\mathscr{F}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{A}$ admits a left adjoint $f:\mathscr{A}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{F}$ such that $fi_{\mathscr{F}}={\mathrm{id}}_{\mathscr{F}}$.

In general, the categories $\mathscr{T}$ and $\mathscr{F}$ are not abelian categories but, as observed in [BonVdBergh, 5.4], they are quasi-abelian categories. Let us recall that an additive category $\mathscr{E}$ is called quasi-abelian if it admits kernels and cokernels, and the class of short exact sequences $0\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}E_{1}\stackrel{{\scriptstyle\alpha}}{{\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}}E_{2}\stackrel{{\scriptstyle\beta}}{{\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}}E_{3}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}0$ with $E_{1}\cong\mathrm{Ker}\beta$ and $E_{3}\cong\mathrm{Coker}\,\alpha$ is stable by pushouts and pullbacks. Both $\mathscr{T}$ and $\mathscr{F}$ admit kernels and cokernels such that: $\mathrm{Ker}_{\mathscr{T}}=t\circ\mathrm{Ker}_{\mathscr{A}}$, $\mathrm{Coker}_{\mathscr{T}}=\mathrm{Coker}_{\mathscr{A}}$, $\mathrm{Ker}_{\mathscr{F}}=\mathrm{Ker}_{\mathscr{A}}$ and $\mathrm{Coker}_{\mathscr{F}}=f\circ\mathrm{Coker}_{\mathscr{A}}$. Exact sequences in $\mathscr{T}$ (respectively in $\mathscr{F}$) coincide with short exact sequences in $\mathscr{A}$ whose terms belong to $\mathscr{T}$ (respectively $\mathscr{F}$) and hence they are stable by pullbacks and push-outs thus proving that $\mathscr{T}$ and $\mathscr{F}$ are quasi-abelian categories. For more details on quasi-abelian categories we refer to Schneiders work [S] and to [F]. We refer to [KS3] for generalities on triangulated categories.

Definition 1.1**.**

([HRS, Ch. I, Proposition 2.1], [Brid, Proposition 2.5]). Let ${\mathscr{H}}_{\operatorname{\mathsf{D}}}$ be the heart of a $t$-structure ${\operatorname{\mathsf{D}}}=({\operatorname{\mathsf{D}}}^{\leqslant 0},{\operatorname{\mathsf{D}}}^{\geqslant 0})$ on a triangulated category ${\mathscr{C}}$ and let $({\mathscr{T}},{\mathscr{F}})$ be a torsion pair on ${\mathscr{H}}_{\operatorname{\mathsf{D}}}$. Then the pair ${\operatorname{\mathsf{D}}}_{({\mathscr{T}},{\mathscr{F}})}:=(\operatorname{\mathsf{D}}^{\leqslant 0}_{({\mathscr{T}},{\mathscr{F}})},{\operatorname{\mathsf{D}}}^{\geqslant 0}_{({\mathscr{T}},{\mathscr{F}})})$ of full subcategories of $\mathscr{C}$

[TABLE]

is a $t$-structure on $\mathscr{C}$ whose heart is

[TABLE]

Following [Brid] we say that ${\operatorname{\mathsf{D}}}_{({\mathscr{T}},{\mathscr{F}})}$ is obtained by left tilting $\operatorname{\mathsf{D}}$ with respect to the torsion pair ${({\mathscr{T}},{\mathscr{F}})}$ while the $t$-structure ${\widetilde{\operatorname{\mathsf{D}}}}_{({\mathscr{T}},{\mathscr{F}})}:={\operatorname{\mathsf{D}}}_{({\mathscr{T}},{\mathscr{F}})}[1]$ is called the $t$-structure obtained by right tilting $\operatorname{\mathsf{D}}$ with respect to the torsion pair ${({\mathscr{T}},{\mathscr{F}})}$ and in this case the right tilted heart is:

[TABLE]

Remark 1.2**.**

([HRS]). Following the previous notations, whenever one performs a left tilting of $\operatorname{\mathsf{D}}$ with respect to a given torsion pair ${({\mathscr{T}},{\mathscr{F}})}$ on ${\mathscr{H}}_{\operatorname{\mathsf{D}}}$ one obtains the new heart ${\mathscr{H}}_{{\operatorname{\mathsf{D}}}_{({\mathscr{T}},{\mathscr{F}})}}$ and the starting torsion pair is “tilted” in the torsion pair $({\mathscr{F}},{\mathscr{T}}[-1])$ which is a torsion pair in ${\mathscr{H}}_{{\operatorname{\mathsf{D}}}_{({\mathscr{T}},{\mathscr{F}})}}$: the class ${\mathscr{F}}$ placed in degree zero is the torsion class for this torsion pair while the old torsion class ${\mathscr{T}}$ shifted by $[-1]$ becomes the new torsion-free class and, for any $M\in{\mathscr{H}}_{{\operatorname{\mathsf{D}}}_{({\mathscr{T}},{\mathscr{F}})}}$, the sequence $0\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}H^{0}(M)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}M\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}H^{1}(M)[-1]\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}0$ is the short exact sequence associated to the torsion pair $({\mathscr{F}},{\mathscr{T}}[-1])$.

Performing a right tilting of ${\operatorname{\mathsf{D}}}_{({\mathscr{T}},{\mathscr{F}})}$ with respect to the torsion pair $({\mathscr{F}},{\mathscr{T}}[-1])$ on ${\mathscr{H}}_{{\operatorname{\mathsf{D}}}_{({\mathscr{T}},{\mathscr{F}})}}$ one re-obtains the starting $t$-structure ${\operatorname{\mathsf{D}}}$ endowed with its torsion pair $({\mathscr{T}},{\mathscr{F}})$. In such a way the right tilting by $({\mathscr{F}},{\mathscr{T}}[-1])$ in ${\mathscr{H}}_{{\operatorname{\mathsf{D}}}_{({\mathscr{T}},{\mathscr{F}})}}$ is the inverse of the left tilting of $\operatorname{\mathsf{D}}$ with respect to ${({\mathscr{T}},{\mathscr{F}})}$ on ${\mathscr{H}}_{\operatorname{\mathsf{D}}}$.

Any $t$-structure ${\operatorname{\mathsf{D}}}_{({\mathscr{T}},{\mathscr{F}})}$ obtained by left tilting ${\operatorname{\mathsf{D}}}$ with respect to a torsion pair ${({\mathscr{T}},{\mathscr{F}})}$ in the heart ${\mathscr{H}}_{\operatorname{\mathsf{D}}}$ of a $t$-structure $\operatorname{\mathsf{D}}$ in $\mathscr{C}$ satisfies

[TABLE]

and hence the heart ${\mathscr{H}}_{{\operatorname{\mathsf{D}}}_{({\mathscr{T}},{\mathscr{F}})}}$ of the $t$-structure ${\operatorname{\mathsf{D}}}_{({\mathscr{T}},{\mathscr{F}})}$ satisfies ${\mathscr{H}}_{{\operatorname{\mathsf{D}}}_{({\mathscr{T}},{\mathscr{F}})}}\subseteq{\operatorname{\mathsf{D}}}^{[0,1]}:={\operatorname{\mathsf{D}}}^{\leqslant 1}\cap{\operatorname{\mathsf{D}}}^{\geqslant 0}$. Dually any $t$-structure ${\widetilde{\operatorname{\mathsf{D}}}}_{({\mathscr{T}},{\mathscr{F}})}:={\operatorname{\mathsf{D}}}_{({\mathscr{T}},{\mathscr{F}})}[1]$ obtained by right tilting ${\operatorname{\mathsf{D}}}$ with respect to a torsion pair ${({\mathscr{T}},{\mathscr{F}})}$ in the heart ${\mathscr{H}}_{\operatorname{\mathsf{D}}}$ of a $t$-structure $\operatorname{\mathsf{D}}$ in $\mathscr{C}$ satisfies

[TABLE]

and hence ${\mathscr{H}}_{{\widetilde{\operatorname{\mathsf{D}}}}_{({\mathscr{T}},{\mathscr{F}})}}\subseteq{\operatorname{\mathsf{D}}}^{[-1,0]}:={\operatorname{\mathsf{D}}}^{\leqslant 0}\cap{\operatorname{\mathsf{D}}}^{\geqslant-1}.$

Polishchuk in [Po, Lemma 1.2.2] proved the following:

Lemma 1.3**.**

In any pair of $t$-structures ${\operatorname{\mathsf{D}}},{\overline{\operatorname{\mathsf{D}}}}$ on a triangulated category $\mathscr{C}$ verifying the condition ${\operatorname{\mathsf{D}}}^{\leqslant 0}\subseteq{\overline{\operatorname{\mathsf{D}}}}^{\leqslant 0}\subseteq{{\operatorname{\mathsf{D}}}}^{\leqslant 1}$ (resp. ${\operatorname{\mathsf{D}}}^{\leqslant-1}\subseteq{\overline{\operatorname{\mathsf{D}}}}^{\leqslant 0}\subseteq{{\operatorname{\mathsf{D}}}}^{\leqslant 0}$), the $t$-structure ${\overline{\operatorname{\mathsf{D}}}}$ is obtained by left tilting (resp. right tilting) ${\operatorname{\mathsf{D}}}$ with respect to the torsion pair

[TABLE]

and in particular, for any $A\in{\mathscr{H}}_{\operatorname{\mathsf{D}}}$, the approximating triangle for the $t$-structure $\overline{{\operatorname{\mathsf{D}}}}$ is the short exact sequence for this torsion pair.

Remark 1.4**.**

In the work [FMT] and [V] the authors propose a generalization of the previous result. In [FMT, Theorem 2.14 and 4.3] the authors proved that, under some technical hypotheses, given any pair of $t$-structures ${\operatorname{\mathsf{D}}}$, $\overline{\operatorname{\mathsf{D}}}$ satisfying the condition:

[TABLE]

one can recover the $t$-structure ${\overline{\operatorname{\mathsf{D}}}}$ by an iterated procedure of left tilting of length $\ell$ starting with $\operatorname{\mathsf{D}}$. Equivalently the $t$-structure $\operatorname{\mathsf{D}}$ can be obtained by an iterated procedure of right tilting of length $\ell$ starting with ${\overline{\operatorname{\mathsf{D}}}}$.

In particular by [FMT, Lemma 2.10 (ii)] these hypotheses are fulfilled whenever, following the definition of Keller and Vossieck [KV] (cf. also [FMT, Definition 6.8]), the $t$-structure ${\overline{\operatorname{\mathsf{D}}}}$ is left ${\operatorname{\mathsf{D}}}$-compatible i.e. the class ${\overline{\operatorname{\mathsf{D}}}}^{\leqslant 0}$ is stable under the left truncations $\tau_{\operatorname{\mathsf{D}}}^{\leqslant k}$ of $\operatorname{\mathsf{D}}$ for any $k\in\mathbb{Z}$.

2. $t$-structures on $\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{hol}}(\mathscr{D}_{X\times S/S})$

Following the notation of the introduction, we denote by $\mathscr{D}_{X\times S/S}$ the subsheaf of $\mathscr{D}_{X\times S}$ of relative differential operators with respect to $p_{X}$ and by $\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{coh}}(\mathscr{D}_{X\times S/S})$ the bounded derived category of left $\mathscr{D}_{X\times S/S}$-modules with coherent cohomologies. As in the absolute case (in which $S$ is a point) the category $\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{coh}}(\mathscr{D}_{X\times S/S})$ is endowed with a duality functor: given $\mathscr{M}\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{coh}}(\mathscr{D}_{X\times S/S})$ we set

[TABLE]

(with $n=d_{X}$) where $\Omega_{X\times S/S}$ denotes the sheaf of relative differential forms of maximal degree, hence $\mathscr{M}\stackrel{{\scriptstyle\cong}}{{\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}}\boldsymbol{D}\boldsymbol{D}\mathscr{M}$ (since, as explained in [MFCS1, Proposition 3.2], any coherent $\mathscr{D}_{X\times S/S}$-module locally admits a free resolution of length at most $2n+\ell$ with $\ell=d_{S}$, see also [JEB]). In [MFCS1, 3.4] the authors proved that the dual of a holonomic $\mathscr{D}_{X\times S/S}$-module is an object in $\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{hol}}(\mathscr{D}_{X\times S/S})$ (the bounded derived category of left $\mathscr{D}_{X\times S/S}$-modules with holonomic cohomologies; [MFCS1, Corollary 3.6]). Hence the previous duality restricts into a duality in $\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{hol}}(\mathscr{D}_{X\times S/S})$, but despite the absolute case it is no longer true that the dual of a holonomic $\mathscr{D}_{X\times S/S}$-module is a holonomic $\mathscr{D}_{X\times S/S}$-module.

Due to the previous considerations, we can endow the triangulated category $\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{hol}}(\mathscr{D}_{X\times S/S})$ with two $t$-structures $P$ and $\Pi$: we denote by $P$ the natural $t$-structure and by $\Pi$ its dual $t$-structure with respect to the functor $\boldsymbol{D}$. Thus, by definition, complexes in ${}^{P}\operatorname{\mathsf{D}}_{\mathrm{hol}}^{\leqslant 0}(\mathscr{D}_{X\times S/S})$ (respectively ${}^{P}\operatorname{\mathsf{D}}_{\mathrm{hol}}^{\geqslant 0}(\mathscr{D}_{X\times S/S})$) are isomorphic in $\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{hol}}(\mathscr{D}_{X\times S/S})$ to complexes of $\mathscr{D}_{X\times S/S}$-modules which have zero entries in positive (respectively negative) degrees and holonomic cohomologies. The dual $t$-structure $\Pi$ is by definition:

[TABLE]

Remark 2.1**.**

We have the following statements:

1. (1)

If $S=\{pt\}$ then $\Pi=P$ (cf.[Ka2, 4.11]). 2. (2)

If $X=\{pt\}$ then $P$ is nothing more than the natural $t$-structure in $\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{coh}}(\mathscr{O}_{S})$ and $\Pi$ is its dual $t$-structure with respect to the functor $\boldsymbol{D}(\cdot):=\operatorname{\mathit{R}\mathscr{H}\mathit{om}}_{\mathscr{O}_{S}}(\cdot,\mathscr{O}_{S})$ described by Kashiwara in [Ka4, §4, Proposition 4.3] which we shall denote by $\pi$:

[TABLE]

Recall that, following [MFCS1], for $s\in S$ on denotes by $Li^{*}_{s}$ the derived functor $p_{X}^{-1}(\mathscr{O}_{S}/m)\overset{L}{\otimes}_{p_{X}^{-1}\mathscr{O}_{S}}(\cdot)$ where $m$ is the maximal ideal of functions vanishing at $s$.

Lemma 2.2**.**

Let consider the functors $Li^{*}_{s}:\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{hol}}(\mathscr{D}_{X\times S/S})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{hol}}({\mathscr{D}}_{X})$ with $s$ varying in $S$. The following holds true:

1. (1)

the complex $\mathscr{M}\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{hol}}(\mathscr{D}_{X\times S/S})$ is isomorphic to [math] if and only if $Li^{*}_{s}\mathscr{M}=0$ for any $s$ in $S$; 2. (2)

$Li^{*}_{s}\mathscr{M}\in\operatorname{\mathsf{D}}_{\mathrm{hol}}^{\leqslant k}({\mathscr{D}}_{X})$* for each $s\in S$ if and only if $\mathscr{M}\in{}^{P}\operatorname{\mathsf{D}}_{\mathrm{hol}}^{\leqslant k}(\mathscr{D}_{X\times S/S})$;* 3. (3)

if $Li^{*}_{s}\mathscr{M}\in\operatorname{\mathsf{D}}_{\mathrm{hol}}^{\geqslant k}({\mathscr{D}}_{X})$ for each $s\in S$ then $\mathscr{M}\in{}^{P}\operatorname{\mathsf{D}}_{\mathrm{hol}}^{\geqslant k}(\mathscr{D}_{X\times S/S})$; 4. (4)

$Li^{*}_{s}\mathscr{M}\in\operatorname{\mathsf{D}}_{\mathrm{hol}}^{\geqslant k}({\mathscr{D}}_{X})$* for each $s\in S$ if and only if $\mathscr{M}\in{}^{\Pi}\operatorname{\mathsf{D}}_{\mathrm{hol}}^{\geqslant k}(\mathscr{D}_{X\times S/S})$.*

Proof.

These statements are a slight generalization of [MFCS2, Corollary 1.11], with exactly the same idea of proof. In particular $(4)$ can be deduced by duality from $(2)$ since we can characterize the objets in ${}^{\Pi}\operatorname{\mathsf{D}}_{\mathrm{hol}}^{\geqslant 0}(\mathscr{D}_{X\times S/S})$ as follows:

[TABLE]

where the last equivalence holds true since in the absolute case the functor $\boldsymbol{D}$ on $\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{hol}}(\mathscr{D}_{X})$ is exact with respect to the natural $t$-structure. As a morphism, $\ast:Li^{*}_{s}\boldsymbol{D}\mathscr{M}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\boldsymbol{D}Li^{*}_{s}\mathscr{M}$ is an application of [Ka2, (A.10)] and it is an isomorphism because $Li^{*}_{s}$ is the derived tensor product of a coherent module ($p^{-1}\mathscr{O}_{S}/m$) over a coherent sheaf ($p^{-1}\mathscr{O}_{S}$). ∎

Lemma 2.3**.**

We have the double inclusion

[TABLE]

hence, given $\mathscr{M}$ a holonomic $\mathscr{D}_{X\times S/S}$-module, its dual satisfies

[TABLE]

Proof.

In general, if $\mathscr{M}\in{}^{P}\operatorname{\mathsf{D}}_{\mathrm{hol}}^{\leqslant 0}(\mathscr{D}_{X\times S/S})$, by the right exactness of $i^{*}_{s}$ we deduce that for any $s\in S$ the complex $Li^{*}_{s}\mathscr{M}$ belongs to $\operatorname{\mathsf{D}}_{\mathrm{hol}}^{\leqslant 0}(\mathscr{D}_{X})$ and hence $Li^{*}_{s}\boldsymbol{D}\mathscr{M}\cong\boldsymbol{D}Li^{*}_{s}\mathscr{M}\in\operatorname{\mathsf{D}}_{\mathrm{hol}}^{\geqslant 0}(\mathscr{D}_{X})$ thus, according to $(3)$ of Lemma 2.2, $\boldsymbol{D}\mathscr{M}\in{}^{P}\operatorname{\mathsf{D}}_{\mathrm{hol}}^{\geqslant 0}(\mathscr{D}_{X\times S/S})$ and so $\mathscr{M}\in{}^{\Pi}\operatorname{\mathsf{D}}_{\mathrm{hol}}^{\leqslant 0}(\mathscr{D}_{X\times S/S}).$

According to the definitions, Lemma 2.2 and by the $t$-exactness of the functor $\boldsymbol{D}$ in the absolute case, we have the following chain:

[TABLE]

∎

Following [Sa1], $p_{X}^{-1}\mathscr{O}_{S}$-flat holonomic $\mathscr{D}_{X\times S/S}$-modules are called strict. The following result will be useful in the sequel:

Lemma 2.4**.**

Let $\mathscr{N}\in{}^{P}\operatorname{\mathsf{D}}^{\leqslant 0}_{\mathrm{hol}}(\mathscr{D}_{X\times S/S})$. Then $\boldsymbol{D}\mathscr{N}$ is quasi-isomorphic to a bounded complex $\mathscr{F}^{\scriptscriptstyle\bullet}$ of coherent $\mathscr{D}_{X\times S/S}$-modules whose terms in negative degrees are zero while the terms in positive degrees are strict coherent $\mathscr{D}_{X\times S/S}$-modules. In particular $\mathscr{H}^{0}\boldsymbol{D}\mathscr{N}$ is torsion free.

Proof.

Since any coherent $\mathscr{D}_{X\times S/S}$-module locally admits a resolution of finite length by free $\mathscr{D}_{X\times S/S}$-modules of finite rank, any complex $\mathscr{N}\in{}^{P}\operatorname{\mathsf{D}}_{\mathrm{hol}}^{\leqslant 0}(\mathscr{D}_{X\times S/S})$ locally admits a resolution $\mathscr{L}^{\scriptscriptstyle\bullet}$ by free $\mathscr{D}_{X\times S/S}$-modules of finite rank such that $\mathscr{L}^{i}=0$ for any $i>0$ and for $i\ll 0$. Thus $\boldsymbol{D}\mathscr{N}$ can be represented locally by the complex $\mathscr{L}^{\ast{\scriptscriptstyle\bullet}}:=\mathscr{H}\mathit{om}_{\mathscr{D}_{X\times S/S}}(\mathscr{L}^{\scriptscriptstyle\bullet},\mathscr{D}_{X\times S/S}\otimes_{\mathscr{O}_{X\times S}}\Omega^{\otimes^{-1}}_{X\times S/S})[n]$ whose terms are free $\mathscr{D}_{X\times S/S}$-modules of finite rank and whose cohomology in negative degrees is zero. By the assumption

$\boldsymbol{D}\mathscr{N}\simeq{}^{P}\tau^{\geqslant 0}(\boldsymbol{D}\mathscr{N})\simeq\mathscr{F}^{\scriptscriptstyle\bullet}\in{}^{\Pi}\operatorname{\mathsf{D}}_{\mathrm{hol}}^{\geqslant 0}(\mathscr{D}_{X\times S/S})$ (since $\mathscr{N}\in{}^{P}\operatorname{\mathsf{D}}^{\leqslant 0}_{\mathrm{hol}}(\mathscr{D}_{X\times S/S})$) with

[TABLE]

where $\mathrm{Coker}(d^{-1}_{\mathscr{L}^{\ast{\scriptscriptstyle\bullet}}})$ is placed in degree [math]. It remains to prove that $\mathrm{Coker}(d^{-1}_{\mathscr{L}^{\ast{\scriptscriptstyle\bullet}}})$ is a strict coherent $\mathscr{D}_{X\times S/S}$-module.

Let us consider the distinguished triangle induced by the following short exact sequence of complexes of coherent $\mathscr{D}_{X\times S/S}$-modules:

[TABLE]

The triangle $Li^{*}_{s}\mathscr{L}^{\ast\geqslant 1}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}Li^{*}_{s}\mathscr{F}^{\scriptscriptstyle\bullet}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}Li^{*}_{s}(\mathrm{Coker}(d^{-1}_{\mathscr{L}^{\ast\scriptscriptstyle\bullet}}))\stackrel{{\scriptstyle+}}{{\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}}$ is distinguished, since each $\mathscr{L}^{i\ast}$ is strict, $Li^{*}_{s}\mathscr{L}^{\ast\geqslant 1}\in\operatorname{\mathsf{D}}_{\mathrm{coh}}^{\geqslant 1}(\mathscr{D}_{X})$ while $Li^{*}_{s}\mathscr{F}^{\scriptscriptstyle\bullet}\in\operatorname{\mathsf{D}}_{\mathrm{coh}}^{\geqslant 0}(\mathscr{D}_{X})$ in view of Lemma 2.2 $(4)$. Hence, for any $s\in S$, $Li^{*}_{s}(\mathrm{Coker}(d^{-1}_{\mathscr{L}^{\ast\scriptscriptstyle\bullet}}))\in\operatorname{\mathsf{D}}_{\mathrm{hol}}^{\geqslant 0}(\mathscr{D}_{X})$, so $\mathscr{H}^{j}Li^{*}_{s}(\mathrm{Coker}(d^{-1}_{\mathscr{L}^{\ast\scriptscriptstyle\bullet}}))=0,\forall\,j\neq 0$ since $Li^{*}_{s}(\mathrm{Coker}(d^{-1}_{\mathscr{L}^{\ast\scriptscriptstyle\bullet}}))\in\operatorname{\mathsf{D}}_{\mathrm{hol}}^{\leqslant 0}(\mathscr{D}_{X})$. According to [MFCS2, Lemma 1.13] we conclude that $\mathrm{Coker}(d^{-1}_{\mathscr{L}^{\ast\scriptscriptstyle\bullet}})$ is strict and so $\mathscr{H}^{0}\boldsymbol{D}\mathscr{N}$ is torsion free. ∎

Remark 2.5**.**

In accordance with Lemma 2.4, if $\mathscr{M}$ is a torsion module, $\mathscr{H}^{0}\boldsymbol{D}(\mathscr{M})$, being torsion free and a torsion module, is zero.

When $d_{S}=1$, it is well known that $p_{X}^{-1}\mathscr{O}_{S}$-flatness is equivalent to absence of $p_{X}^{-1}\mathscr{O}_{S}$-torsion, hence a holonomic $\mathscr{D}_{X\times S/S}$-module $\mathscr{M}$ is strict if and only if for any $f\in\mathscr{O}_{S}$ the morphism $\mathscr{M}\stackrel{{\scriptstyle f}}{{\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}}\mathscr{M}$ (multiplication by $f$) is a monomorphism.

In this case, for a given coherent $\mathscr{D}_{X\times S/S}$-module $\mathscr{M}$, we denote by $\mathscr{M}_{t}$ the coherent sub-module of sections locally annihilated by some $f\in\mathscr{O}_{S}$ and we denote by $\mathscr{M}_{tf}$ the quotient $\mathscr{M}/\mathscr{M}_{t}$. We denote by $\mathrm{Mod}_{\mathrm{hol}}(\mathscr{D}_{X\times S})_{t}$ the full subcategory of holonomic $\mathscr{D}_{X\times S/S}$-modules satisfying $\mathscr{M}_{t}\simeq\mathscr{M}$ and by $\mathrm{Mod}_{\mathrm{hol}}(\mathscr{D}_{X\times S/S})_{tf}$ the full subcategory of holonomic $\mathscr{D}_{X\times S/S}$-modules satisfying $\mathscr{M}\simeq\mathscr{M}_{tf}$. The properties of torsion pair in $\mathrm{Mod}_{\mathrm{hol}}(\mathscr{D}_{X\times S/S})$ are clearly satisfied by $(\mathrm{Mod}_{\mathrm{hol}}({\mathscr{D}_{X\times S/S}})_{t},\mathrm{Mod}_{\mathrm{hol}}({\mathscr{D}_{X\times S/S}})_{tf})$.

Moreover this torsion pair is hereditary i.e. the class of torsion modules (which coincides with the class of holonomic $\mathscr{D}_{X\times S/S}$-modules $\mathscr{M}$ satisfying $\dim p_{X}(\operatorname{Supp}(\mathscr{M}))=0$ plus the zero module) is closed under sub-objects and so it forms an abelian category.

Proposition 2.6**.**

If $d_{S}=1$, $\Pi$ is the $t$-structure obtained by left tilting $P$ with respect to the torsion pair $(\mathrm{Mod}_{\mathrm{hol}}({\mathscr{D}_{X\times S/S}})_{t},\mathrm{Mod}_{\mathrm{hol}}({\mathscr{D}_{X\times S/S}})_{tf})$ in $\mathrm{Mod}_{\mathrm{hol}}(\mathscr{D}_{X\times S/S})$ while $P$ is the $t$-structure obtained by right tilting $\Pi$ with respect to the torsion pair $(\mathrm{Mod}_{\mathrm{hol}}({\mathscr{D}_{X\times S/S}})_{tf},\mathrm{Mod}_{\mathrm{hol}}({\mathscr{D}_{X\times S/S}})_{t}[-1])$ in $\mathscr{H}_{\Pi}$.

Proof.

By Lemma 2.3 we have

[TABLE]

(the last inclusion on the right is obtained by shifting by $[-1]$ the first one) and hence, by Polishchuk’s result (Lemma 1.3), the $t$-structure $\Pi$ is obtained by left tilting $P$ with respect to the torsion pair

[TABLE]

Also by Lemma 2.3, if $\mathscr{M}$ is holonomic, then $\boldsymbol{D}\mathscr{M}\in{}^{P}\operatorname{\mathsf{D}}^{[0,1]}_{\mathrm{hol}}(\mathscr{D}_{X\times S/S})$, that is, $\boldsymbol{D}\mathscr{M}$ is concentrated in degrees [math] and $1$. The result will then be a consequence of the following statements:

• •

$(i)$ $\mathscr{M}$ is a strict holonomic module if and only if $\boldsymbol{D}(\mathscr{M})$ is concentrated in degree zero and strict.

• •

$(ii)$ If $\mathscr{M}$ belongs to $\mathrm{Mod}_{\mathrm{hol}}(\mathscr{D}_{X\times S/S})_{t}$ then $\boldsymbol{D}(\mathscr{M})$ is concentrated in degree $1$ and ${}^{P}\mathscr{H}^{1}(\boldsymbol{D}\mathscr{M})$ belongs to $\mathrm{Mod}_{\mathrm{hol}}(\mathscr{D}_{X\times S/S})_{t}$.

Item $(i)$ is contained in Proposition 2 of [MFCS2]. Therefore it remains to check item $(ii)$. Let $\mathscr{M}\in\mathrm{Mod}_{\mathrm{hol}}(\mathscr{D}_{X\times S/S})_{t}$. First we remark that, by the functoriality of the action of $p_{X}^{-1}\mathscr{O}_{S}$, all cohomology groups ${}^{P}\mathscr{H}^{j}(\boldsymbol{D}\mathscr{M})$ belong to $\mathrm{Mod}_{\mathrm{hol}}(\mathscr{D}_{X\times S/S})_{t}$. In accordance with Remark 2.5, ${}^{P}\mathscr{H}^{0}(\boldsymbol{D}\mathscr{M})=0$. This ends the proof of $(ii)$ and proves that $\mathscr{M}$ belongs to $\mathrm{Mod}_{\mathrm{hol}}(\mathscr{D}_{X\times S/S})_{t}$ if and only if $\boldsymbol{D}(\mathscr{M})$ is concentrated in degree $1$ and ${}^{P}\mathscr{H}^{1}(\boldsymbol{D}\mathscr{M})$ belongs to $\mathrm{Mod}_{\mathrm{hol}}(\mathscr{D}_{X\times S/S})_{t}$. This proves the first statement.

As a consequence, the heart of $\Pi$ can be described as

[TABLE]

and thus the $t$-structure $P$ is obtained by right tilting $\Pi$ with respect to $(\mathrm{Mod}_{\mathrm{hol}}({\mathscr{D}_{X\times S/S}})_{tf},\mathrm{Mod}_{\mathrm{hol}}({\mathscr{D}_{X\times S/S}})_{t}[-1])$ in $\mathscr{H}_{\Pi}$ (cf. [HRS] and Remark 1.2). ∎

Corollary 2.7**.**

If $d_{S}=1$ then the full subcategory of strict holonomic $\mathscr{D}_{X\times S/S}$-modules (thus holonomic $\mathscr{D}_{X\times S/S}$-modules with a strict holonomic dual) is quasi-abelian.

Therefore the problem of expliciting $\Pi$ only matters for $d_{S}\geqslant 2$ and $d_{X}\geqslant 1$. The following Lemmas permit to describe the $t$-structure $\Pi$ in terms of support conditions as done by Kashiwara in the case of $X=\{pt\}$ (cf.[Ka4]).

Lemma 2.8**.**

Let us consider $F:\mathscr{C}\longrightarrow\overline{\mathscr{C}}$ a triangulated functor between two triangulated categories $\mathscr{C}$ and $\overline{\mathscr{C}}$. Let $P:=({}^{P}\operatorname{\mathsf{D}}^{\leqslant 0},{}^{P}\operatorname{\mathsf{D}}^{\geqslant 0})$ be a bounded $t$-structure on $\mathscr{C}$ and ${}^{\overline{P}}\operatorname{\mathsf{D}}^{\leqslant 0}$ (resp. ${}^{\overline{P}}\operatorname{\mathsf{D}}^{\geqslant 0})$) a class on $\overline{\mathscr{C}}$ closed under extensions and shift by $[1]$ (resp. closed under extensions and shift by $[-1]$). The following statements hold true:

1. (1)

the functor $F({}^{P}\operatorname{\mathsf{D}}^{\leqslant 0})\subseteq{}^{\overline{P}}\operatorname{\mathsf{D}}^{\leqslant 0}$ if and only if $F(\mathscr{H}_{P})\subseteq{}^{\overline{P}}\operatorname{\mathsf{D}}^{\leqslant 0}$; 2. (2)

the functor $F({}^{P}\operatorname{\mathsf{D}}^{\geqslant 0})\subseteq{}^{\overline{P}}\operatorname{\mathsf{D}}^{\geqslant 0}$ if and only if $F(\mathscr{H}_{P})\subseteq{}^{\overline{P}}\operatorname{\mathsf{D}}^{\geqslant 0}$; 3. (3)

the previous conditions are simultaneously satisfied if and only if $F(\mathscr{H}_{P})\subseteq\mathscr{H}_{\overline{P}}$.

Proof.

Let us recall that by definition a $t$-structure $P:=({}^{P}\operatorname{\mathsf{D}}^{\leqslant 0},{}^{P}\operatorname{\mathsf{D}}^{\geqslant 0})$ on $\mathscr{C}$ is bounded if for any $X\in\mathscr{C}$ there exist $m\leqslant n\in\mathbb{Z}$ such that $X\in{}^{P}\operatorname{\mathsf{D}}^{\leqslant n}\cap{}^{P}\operatorname{\mathsf{D}}^{\geqslant m}$ and as remarked by Bridgeland in [Brid, Lemma 2.3] these $t$-structures are completely determined by their hearts (via its Postnikov tower).

The left to right implication is clear since $\mathscr{H}_{P}\subseteq{}^{P}\operatorname{\mathsf{D}}^{\leqslant 0}$ so let us suppose that $F(\mathscr{H}_{P})\subseteq{}^{\overline{P}}\operatorname{\mathsf{D}}^{\leqslant 0}$ and let us prove that $F({}^{P}\operatorname{\mathsf{D}}^{\leqslant 0})\subseteq{}^{\overline{P}}\operatorname{\mathsf{D}}^{\leqslant 0}$. Recall that for any $X\in{}^{{P}}\operatorname{\mathsf{D}}^{\leqslant 0}$ there exists a suitable $k\in\mathbb{N}$ such that $X\in{}^{{P}}\operatorname{\mathsf{D}}^{\leqslant 0}\cap{}^{P}\operatorname{\mathsf{D}}^{\geqslant-k}$. Let us proceed by induction on $k\in\mathbb{N}$. For $k=0$ we get $X\in\mathscr{H}_{P}$ and thus $F(X)\in{}^{\overline{P}}\operatorname{\mathsf{D}}^{\leqslant 0}$ by hypothesis. Let us suppose by inductive hypothesis that the first statement holds true for $k$ and let $X\in{}^{{P}}\operatorname{\mathsf{D}}^{\leqslant 0}\cap{}^{P}\operatorname{\mathsf{D}}^{\geqslant-k-1}$. By applying the functor $F$ to the distinguished triangle ${}^{P}H^{-k-1}(X)[k+1]\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{}^{P}\tau^{\geqslant-k}(X)\stackrel{{\scriptstyle+1}}{{\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}}$ we obtain $F({}^{P}H^{-k-1}(X))[k+1]\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}F(X)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}F({}^{P}\tau^{\geqslant-k}(X))\stackrel{{\scriptstyle+1}}{{\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}}$. By hypothesis $F({}^{P}H^{-k-1}(X))[k+1]\in{}^{\overline{P}}\operatorname{\mathsf{D}}^{\leqslant 0}[k+1]\subseteq{}^{\overline{P}}\operatorname{\mathsf{D}}^{\leqslant 0}$ (thanks to the fact that ${}^{\overline{P}}\operatorname{\mathsf{D}}^{\leqslant 0}$ is closed under $[1]$) and by inductive hypothesis $F({}^{P}\tau^{\geqslant-k}(X))\in{}^{\overline{P}}\operatorname{\mathsf{D}}^{\leqslant 0}$. Thus $F(X)\in{}^{\overline{P}}\operatorname{\mathsf{D}}^{\leqslant 0}$ since ${}^{\overline{P}}\operatorname{\mathsf{D}}^{\leqslant 0}$ is closed under extensions. The second statement follows similarly and the third is the consequence of the first and second ones. ∎

Lemma 2.9**.**

Let $\mathscr{N}$ be a coherent $\mathscr{D}_{X\times S/S}$-module. Then, for each $k$,

[TABLE]

in particular

[TABLE]

Proof.

According to the faithfull flatness of $\mathscr{D}_{X\times S}$ over $\mathscr{D}_{X\times S/S}$ and to [Ka2, Theorem 2.19 (2)], we have, for each $k$,

[TABLE]

[TABLE]

Since

[TABLE]

where $\pi:T^{*}X\times T^{*}S\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}T^{*}X\times S$ is the projection, we conclude that

[TABLE]

as desired. ∎

Lemma 2.10**.**

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

[TABLE]

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

[TABLE]

Proof.

Let $\mathscr{M}$ be a holonomic $\mathscr{D}_{X\times S/S}$-module, and let $\Lambda\subseteq T^{*}X$ be a Lagrangian analytic $\mathbb{C}^{*}$-conic (or conic, for short) closed subset such that ${\mathrm{Char}}(\mathscr{M})\subseteq\Lambda\times S$.

Let $\Lambda=\bigcup\limits_{i\in I}\Lambda_{i}$, with $\Lambda_{i}$ closed conic irreducible Lagrangian in $T^{*}X$, be the (locally finite) decomposition of $\Lambda$ in irreducible components.

Let us consider the family of the components $\Lambda_{j}$ such that $\operatorname{Char}{\mathscr{M}}\cap(\Lambda_{j}\times S)\neq\varnothing$. For simplicity, let us denote this family by $\{\Lambda_{1},\cdots,\Lambda_{K}\}$. By the assumption of irreducibility, for each irreducible component $W$ of $\operatorname{Char}\mathscr{M}$ there must exist a $\Lambda_{j}$ such that $W\subset\Lambda_{j}\times S$.

Let $W$ be any irreducible component of $\operatorname{Char}\mathscr{M}$ which is contained in $\Lambda_{1}\times S$. Then $W$ is conic involutive in the Poisson manifold $T^{*}X\times S$ and, for each $s\in S$, $W\cap p_{X}^{-1}(s)$ is contained in $\Lambda_{1}\times\{s\}$. According to [KMF, Cor.1.1.14], $W\cap p_{X}^{-1}(s)$ is still involutive in $T^{*}X\times\{s\}$. Since it is contained in $\Lambda_{1}\times\{s\}$, it must be a conic Lagrangian closed analytic set. Since $\Lambda_{1}$ is conic Lagrangian closed analytic irreducible, we must have either $W\cap p_{X}^{-1}(s)=\varnothing$ or $W\cap p_{X}^{-1}(s)=\Lambda_{1}\times\{s\}$. In particular $W=\Lambda_{1}\times\widetilde{T}_{1}$, for some closed subset $\widetilde{T}_{1}$ of $S$. To see that $\widetilde{T}_{1}$ is analytic (hence irreducible analytic) it suffices to fix a point $p\in\Lambda_{1}$, then $\{p\}\times\widetilde{T}_{1}=q_{X}^{-1}(p)\cap W$ where $q$ denotes the projection $T^{*}X\times S\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}T^{*}X$. Hence $\{p\}\times\widetilde{T}_{1}$ is analytic and so is $\widetilde{T}_{1}$.

By the preceding argument, the union of the family of irreducible components of $\operatorname{Char}\mathscr{M}$ contained in $\Lambda_{1}\times S$ is equal to $\cup_{l\in L_{1}}(\Lambda_{1}\times\widetilde{T}_{1,l})=\Lambda_{1}\times T_{1}$ for some finite family $(\widetilde{T}_{i,l})_{l\in L}$ of closed irreducible subsets in $S$. We can now apply this argument to each $\Lambda_{i}$ and the first part of the result follows.

Since

[TABLE]

we deduce that $p_{X}(\mathrm{Supp}(\mathscr{M}))=\bigcup\limits_{i}T_{i}$ and hence $t:=\dim p_{X}(\mathrm{Supp}(\mathscr{M}))=\sup\limits_{i\in I}\dim T_{i}$ hence $\dim\mathrm{Char}(\mathscr{M})=\dim X+t$ which ends the proof.

∎

We have now the tools to obtain the description of $\Pi$ for arbitrary $d_{S}$:

Theorem 2.11**.**

*The $t$-structure $\Pi$ on $\operatorname{\mathsf{D}}_{\mathrm{hol}}^{\mathrm{b}}(\mathscr{D}_{X\times S/S})$ can be described in the following way: *

[TABLE]

Proof.

Note that the statement is true in the absolute case since we get ${}^{\Pi}\operatorname{\mathsf{D}}_{\mathrm{hol}}^{\leqslant 0}(\mathscr{D}_{X})={}^{P}\operatorname{\mathsf{D}}_{\mathrm{hol}}^{\leqslant 0}(\mathscr{D}_{X})$ (an holonomic ${\mathscr{D}}_{X}$-module whose characteristic variety has codimension grater than $d_{X}$ is necessarily zero).

Step 1

Let us prove the equality $(\ast\ast)$. We start by proving the inclusion of the left hand side into the right one.

Let $W$ be a closed analytic subset of an open subset $U\subseteq S$ such that $\operatorname{codim}_{U}W\geqslant k$. Let us prove that $R\Gamma_{[X\times W]}(\boldsymbol{D}\mathscr{N}_{|X\times U})\in{}^{P}\operatorname{\mathsf{D}}^{\geqslant k}(\mathscr{D}_{X\times U/U})$ for any complex $\mathscr{N}\in{}^{P}\operatorname{\mathsf{D}}_{\mathrm{hol}}^{\leqslant 0}(\mathscr{D}_{X\times S/S})$. This will be a consequence of Lemma 2.4. Indeed, keeping the notation of the proof of this Lemma, we have

[TABLE]

since $R\Gamma_{[W]}(p_{X}^{-1}\mathscr{O}_{U})\in\operatorname{\mathsf{D}}^{\geqslant k}(p_{X}^{-1}\mathscr{O}_{U})$ and the terms of $\mathscr{F}^{\scriptscriptstyle\bullet}_{|X\times U}$ are strict coherent $\mathscr{D}_{X\times U/U}$-modules.

Let us now prove the inclusion of the right hand side in the left one, that is, let us prove that given $\mathscr{M}\in\operatorname{\mathsf{D}}_{\mathrm{hol}}^{\mathrm{b}}(\mathscr{D}_{X\times S/S})$ such that ${}^{P}\mathscr{H}^{k}_{[X\times W]}(\mathscr{M}_{|X\times U}))=0$ for any closed analytic subset $W$ of an open subset $U\subseteq S$ and $k<\operatorname{codim}_{S}W$ we get $\mathscr{M}\in{}^{\Pi}\operatorname{\mathsf{D}}_{\mathrm{hol}}^{\geqslant 0}(\mathscr{D}_{X\times S/S})$.

We note that the statement is local. In view of Lemma 2.2 $(4)$ it suffices to check that, for each $s\in S$, $Li^{*}_{s}\mathscr{M}\in\operatorname{\mathsf{D}}_{\mathrm{hol}}^{\geqslant 0}(\mathscr{D}_{X})$. We shall argue by induction on $d_{S}$. First suppose $d_{S}=1$. Let $s_{0}\in S$ and $s$ be a local coordinate on $S$ vanishing in $s_{0}$. By the same arguments of Lemma 2.8 we may assume that $\mathscr{M}$ is concentrated in degree [math]. Hence $\mathscr{M}$ is strict, since, if $s^{P}\mathscr{M}=0$ for some natural $P$, $\Gamma_{[X\times\{s_{0}\}]}(\mathscr{M})\neq 0$ contradicting the assumption on $\mathscr{M}$; so, according to Proposition 2.6 $\mathscr{M}\in{}^{\Pi}\operatorname{\mathsf{D}}_{\mathrm{hol}}^{\geqslant 0}(\mathscr{D}_{X\times S/S})$.

Let us now treat the general case. It will be a consequence of the following Lemma which is a variation of a formula proved in [MFCS3], page 153 (see also [Ka0]):

Lemma 2.12**.**

Let $X$ be an open subset in $\mathbb{C}^{n}$, let $S$ be an open set of $\mathbb{C}^{d}$ containing [math], with coordinates $(s_{1},\cdots,s_{d})$. Let us denote by $S_{j}$ the submanifold of $S$ of equations $s_{1}=0,\cdots,s_{j}=0$, for $j=1,\cdots,d$ and, for any $f$ holomorphic on $S_{j}$, denote by $L_{f}^{*}:=p_{X}^{-1}(\mathscr{O}_{S_{j}}/\mathscr{O}_{S_{j}}f)\otimes_{p_{X}^{-1}(\mathscr{O}_{S_{j}})}^{L}\,(\cdot)$ the corresponding derived functor. Then we have an isomorphism of functors on $\operatorname{\mathsf{D}}^{\mathrm{b}}(\mathscr{O}_{X\times S_{j}})$

[TABLE]

We consider the local situation where $s_{0}=0\in\mathbb{C}^{d}$. Following the notations of the preceding Lemma, let $W=S_{1}$. Let $S^{*}:=S\smallsetminus S_{1}$ and we denote by $R\Gamma_{[X\times S^{*}]}(\cdot)$ the functor of localization relatively to the hypersurface $X\times S_{1}$. As usual we may assume that $\mathscr{M}$ is concentrated in degree [math]. Since $S_{1}$ has equation $s_{1}=0$ we deduce, as in the case $d_{S}=1$, that $\mathscr{M}$ has no $s_{1}$ torsion, since, if $s_{1}^{P}\mathscr{M}=0$ for some natural $P$, $\Gamma_{[X\times S_{1}]}(\mathscr{M})\neq 0$ contradicting the assumption that $R\Gamma_{[X\times S_{1}]}(\mathscr{M})\in{}^{P}\operatorname{\mathsf{D}}^{\geqslant 1}(\mathscr{D}_{X\times S/S})$.

Let consider the distinguished triangle

[TABLE]

we have $Li^{*}_{s_{1}}R\Gamma_{[X\times S_{1}]}\mathscr{M}\in{}^{P}\operatorname{\mathsf{D}}^{\geqslant 0}(\mathscr{D}_{X\times S_{1}/S_{1}})$ and so $Li^{*}_{s_{1}}\mathscr{M}\in{}^{P}\operatorname{\mathsf{D}}^{\geqslant 0}(\mathscr{D}_{X\times S_{1}/S_{1}})$ since in this case $s_{1}$ is invertible on $R\Gamma_{[X\times S^{*}]}\mathscr{M}$ which is an object in ${}^{P}\operatorname{\mathsf{D}}^{\geqslant 0}(\mathscr{D}_{X\times S/S})$.

Moreover, given a closed analytic subset $W_{1}$ of $S_{1}$, we have, according to $(A)$

[TABLE]

Hence $Li^{*}_{s_{1}}\mathscr{M}$ belongs to ${}^{\Pi}\operatorname{\mathsf{D}}_{\mathrm{hol}}^{\geqslant 0}(\mathscr{D}_{X\times S_{1}/S_{1}})$ and we can proceed recursively to conclude the statement.

Step 2

By Lemma 2.10 we know that for any $\mathscr{M}\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{hol}}(\mathscr{D}_{X\times S/S})$ we have

[TABLE]

hence

[TABLE]

so we are reduced to prove that

[TABLE]

First we prove the inclusion:

[TABLE]

Let us argue by induction on $m$ such that $\mathscr{M}\in{}^{P}\operatorname{\mathsf{D}}^{\leqslant m}_{\mathrm{hol}}(\mathscr{D}_{X\times S/S})$ and that $\operatorname{codim}{\mathrm{Char}}({}^{P}\mathscr{H}^{k}(\mathscr{M}))\geqslant k+d_{X}$. For $m=0$ we have by Lemma 2.3 that ${}^{P}\operatorname{\mathsf{D}}_{\mathrm{hol}}^{\leqslant 0}(\mathscr{D}_{X\times S/S})\subset{}^{\Pi}\operatorname{\mathsf{D}}_{\mathrm{hol}}^{\leqslant 0}(\mathscr{D}_{X\times S/S})$. Let us suppose that any complex in ${}^{P}\operatorname{\mathsf{D}}^{\leqslant m}_{\mathrm{hol}}(\mathscr{D}_{X\times S/S})$ satisfying $\operatorname{codim}{\mathrm{Char}}({}^{P}\mathscr{H}^{k}(\mathscr{M}))\geqslant k+d_{X}$ belongs to ${}^{\Pi}\operatorname{\mathsf{D}}_{\mathrm{hol}}^{\leqslant 0}(\mathscr{D}_{X\times S/S})$ and let $\mathscr{M}\in{}^{P}\operatorname{\mathsf{D}}^{\leqslant m+1}_{\mathrm{hol}}(\mathscr{D}_{X\times S/S})$ satisfying $\operatorname{codim}{\mathrm{Char}}({}^{P}\mathscr{H}^{k}(\mathscr{M}))\geqslant k+d_{X}$. By inductive hypothesis we have that ${}^{P}\tau^{\leqslant m}\mathscr{M}\in{}^{\Pi}\operatorname{\mathsf{D}}_{\mathrm{hol}}^{\leqslant 0}(\mathscr{D}_{X\times S/S})$ and the distinguished triangle

[TABLE]

proves that $\mathscr{M}\in{}^{\Pi}\operatorname{\mathsf{D}}_{\mathrm{hol}}^{\leqslant 0}(\mathscr{D}_{X\times S/S})$ if and only if ${}^{P}\mathscr{H}^{m+1}(\mathscr{M})\in{}^{\Pi}\operatorname{\mathsf{D}}_{\mathrm{hol}}^{\leqslant-m-1}(\mathscr{D}_{X\times S/S})$. This last condition is satisfied in view of the assumption on $\mathscr{M}$ according to [Ka2, Theorem 2.19 (1)] together with the faithfull flatness of ${\mathscr{D}}_{X\times S}$ over $\mathscr{D}_{X\times S/S}$, which shows that $\boldsymbol{D}({}^{P}\mathscr{H}^{m+1}(\mathscr{M}))\in{}^{P}\operatorname{\mathsf{D}}_{\mathrm{hol}}^{\geqslant m+1}(\mathscr{D}_{X\times S/S})$.

Let us now prove the inclusion

[TABLE]

Recalling that ${}^{\Pi}\operatorname{\mathsf{D}}_{\mathrm{hol}}^{\leqslant 0}:=\boldsymbol{D}({}^{P}\operatorname{\mathsf{D}}_{\mathrm{hol}}^{\geqslant 0}(\mathscr{D}_{X\times S/S}))$ we can apply Lemma 2.8 with $F=\boldsymbol{D}$ and so we need only to prove that given $\mathscr{N}$ a holonomic $\mathscr{D}_{X\times S/S}$-module, $\boldsymbol{D}(\mathscr{N})$ satisfies

[TABLE]

and this holds true by Lemma 2.9. ∎

Remark 2.13**.**

We conclude by the previous Theorem 2.11 that the $t$-structure $\Pi$ is left $P$-compatible (cf. Remark 1.4) and so, according to Lemma 2.3 and to [FMT, Theorem 4.3], it can be recovered from $P$ via an iterated right tilting procedure of length $\ell$.

3. $t$-structures on $\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p_{X}^{-1}\mathscr{O}_{S})$

In $\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p_{X}^{-1}\mathscr{O}_{S})$ the natural dualizing complex is $p_{X}^{!}\mathscr{O}_{S}=p_{X}^{-1}\mathscr{O}_{S}[2d_{X}]$ and one defines the duality functor (cf. [MFCS1] for details) by setting

[TABLE]

Hence the canonical morphism $F\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\boldsymbol{D}\boldsymbol{D}(F)$ is an isormorphism for any $F\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p_{X}^{-1}\mathscr{O}_{S})$.

We are now concerned by the corresponding of Lemma 2.10 in the framework of $\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p_{X}^{-1}\mathscr{O}_{S})$. In the case of $d_{S}=1$ it can be deduced thanks to the functor $\operatorname{RH^{S}}$ which will be recalled later (cf. LABEL:subsec:relsubanalytic). Let us be more precise: for a complex $F$ of sheaves on $X\times S$, let $SS\,F$ denote its microsupport (cf. [KS1] for a detailed introduction to this notion). Given $F\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathbb{C}-c}(p^{-1}\mathscr{O}_{S})$, as proved in [MFCS2, Th.  3], we have a functorial isomorphism $F\simeq\operatorname{{}^{\mathrm{p}}Sol}\operatorname{RH^{S}}(F)$ where $\operatorname{RH^{S}}(F)$ is a complex with (regular) holonomic cohomology, hence $\operatorname{Char}(\operatorname{RH^{S}}(F))=SS\,F$. Therefore we conclude:

Corollary 3.1**.**

Let us assume that $d_{S}=1$ Let $F\in\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathbb{C}-c}(p^{-1}\mathscr{O}_{S})$. Let $\Lambda$ be a Lagrangian closed analytic $\mathbb{C}^{*}$-conic subset of $T^{*}X$ such that $SS\,F$ is contained in $\Lambda\times T^{*}S$. Then each closed irreducible component of $SS\,F\cap T^{*}S\times S$ is of the form $\Lambda_{j}\times T$ where $T$ is an analytic closed irreducible subset of $S$ and $\Lambda_{j}$ is a closed irreducible component of $\Lambda$. In particular, $p_{X}(\mathrm{Supp}\,F)$ is an analytic subset of $S$.

Definition 3.2**.**

[MFCS1, 2.7] The perverse $t$-structure $p$ on the triangulated category $\operatorname{\mathsf{D}}^{\mathrm{b}}_{{\mathbb{C}\textup{-c}}}(p_{X}^{-1}\mathscr{O}_{S})$ is given by

[TABLE]

or equivalently

[TABLE]

(See [KS1, Definition 8.3.19] for the definition of adapted $\mu$-stratification.)

Hence its dual $\pi$ with respect to the functor $\boldsymbol{D}$ is

[TABLE]