A description of monodromic mixed Hodge modules (Q2673073)

Let \(X\) be a smooth algebraic variety of finite type over \(\mathbb{C}\) and \(\mathbb{C}_{t}\) the affine line with the coordinate \(t\). We consider a graded polarizable mixed Hodge module \(\mathcal{M}=(M,F_{\bullet}M,K,W_{\bullet}K)\) on \(X\times \mathbb{C}_{t}\), where \(M\) is the underlying \(D\)-module, \(F_{\bullet}M\) is the Hodge filtration, \(K\) is the underlying \(\mathbb{Q}\)-perverse sheaf and \(W_{\bullet}K\) is the weight filtration. Here, in this paper, ``\(D\)-module'' means ``left \(D\)-module''. Since the underlying \(D\)-module of a mixed Hodge module is always regular holonomic, so is \(M\). Brylinski defined the Fourier-Laplace transformation \({}^F{M}\) of \(M\), whose underlying constructible sheaf is the Fourier-Sato transformation of \(K\). \({}^F{M}\) is a \(D\)-module on \(X\times \mathbb{C}_{\tau}\), where \(\mathbb{C}_{\tau}\) is the dual vector space of \(\mathbb{C}\). While \({}^F{M}\) is holonomic, it may not be regular in general. If \({}^F{M}\) is not regular, it can not be the underlying \(D\)-module of a mixed Hodge module. Brylinski showed that if \(M\) is monodromic, \({}^F{M}\) is regular. Therefore, it is natural to ask whether there is a natural mixed Hodge module structure on the Fourier-Laplace transformation \({}^F{M}\) of a monodromic \(D\)-module \(M\). We will identify quasi-coherent \(D_{X\times \mathbb{C}}\)-modules (resp. \(O_{X\times \mathbb{C}}\)-modules) with quasi-coherent \(D_{X}[t]\langle\partial_{t}\rangle(:=D_{X}\otimes_{\mathbb{C}}\mathbb{C}[t]\langle\partial_{t}\rangle)\)-modules (resp. \(O_{X}[t](:=O_{X}\otimes_{\mathbb{C}} \mathbb{C}[t])\)-modules) by using the pushforward and the pullback by the projection \(p: X\times \mathbb{C}\to X\), where \(\mathbb{C}[t]\langle\partial_{t}\rangle\) is the ring of differential operators on \(\mathbb{C}_{t}\). Assume that \(M\) is monodromic. For \(\beta\in \mathbb{Q}\), we define a \(D_{X}\)-submodule \(M^{\beta}\) of \(M\) (we regard \(M\) as a \(D_{X}[t]\langle\partial_{t}\rangle\)-module) as \[ M^{\beta}:=\bigcup_{l\geq 0}\mathrm{Ker}((t\partial_{t}-\beta)^{l}: M\to M). \] Since \(M\) is monodromic, we have a decomposition \[ M=\bigoplus_{\beta\in \mathbb{Q}}M^{\beta},\tag{1} \] where we endow the right hand side with a \(D_{X}[t]\langle\partial_{t}\rangle\)-structure by using the morphisms \(t: M^{\beta}\to M^{\beta+1}\) and \(\partial_{t}: M^{\beta}\to M^{\beta-1}\). The first result is the following: \textbf{Theorem 1.} For \(p\in \mathbb{Z}\), \(F_{p}M\subset M\) is decomposed as \[ F_{p}M=\bigoplus_{\beta\in \mathbb{Q}}F_{p}M^{\beta}, \] where we set \(F_{p}M^{\beta}:=F_{p}M\cap M^{\beta}\) and the \(O_{X}[t]\)-module structure of the right hand side is defined by the morphism \(t: F_{p}M^{\beta}\to F_{p}M^{\beta+1}\). In other words, the Hodge filtration \(F_{\bullet}M\) is decomposed with respect to the decomposition (1). Let us explain what this theorem means in terms of the nearby and vanishing cycles. We have \begin{itemize} \item \(t: M^{\beta}\to M^{\beta+1}\) is an isomorphism if \(\beta\neq -1\). \item \(\partial_{t} : M^{\beta}\to M^{\beta-1}\) is an isomorphism if \(\beta\neq 0\). \end{itemize} Therefore, we can reconstruct the \(D\)-module \(M\) from the data: \begin{itemize} \item[(i)] The \(D\)-modules \(M^{\alpha}\) on \(X\) for \(\alpha\in [-1,0]\cap \mathbb{Q}\). \item[(ii)] The nilpotent endomorphisms \(t\partial_{t}-\alpha: M^{\alpha}\to M^{\alpha}\) for \(\alpha\in (-1,0)\cap \mathbb{Q}\). \item[(iii)] The morphisms \(\partial_{t}: M^{0}\to M^{-1}\) and \(t: M^{-1}\to M^{0}\) such that \(t\partial_{t}\) and \(\partial_{t}t\) are nilpotent. \end{itemize} On the other hand, by the definitions of the nearby cycle (resp. the \(e(\alpha)(:=\exp(-2\pi\sqrt{-1}\alpha))\)-nearby cycle, the unipotent vanishing cycle) \(\psi_{t}M\) (resp. \(\psi_{t,e(\alpha)}M\), \(\phi_{t,1}M\)) of the \(D\)-module \(M\) along \(t=0\) (note that they are \(D\)-module on \(X\)), we have \[ \begin{aligned} &\psi_{t}M=\bigoplus_{\alpha\in (-1,0]\cap \mathbb{Q}}M^{\alpha},\\ &\psi_{t,e(\alpha)}M=M^{\alpha},\text{ and}\\ &\phi_{t,1}M=M^{-1}. \end{aligned} \tag{2} \] Moreover, through the identifications, \(\frac{-1}{2\pi\sqrt{-1}}\) times the logarithm of the unipotent part of the monodromy automorphism of the nearby and vanishing cycles are corresponding to \(t\partial_{t}-\alpha\) on \(M^{\alpha}\) for \(\alpha\in [-1,0]\), the can morphism \(\mathrm{can}: \psi_{t,1}M\to \phi_{t,1}M\) (resp. the var morphism \(\mathrm{var}: \phi_{t,1}M\to \psi_{t,1}M\)) corresponds to \(-\partial_{t}: M^{0}\to M^{-1}\) (resp. \(t: M^{-1}\to M^{0}\)). Therefore, we can say that the monodromic \(D\)-module is determined by the nearby and vanishing cycles of it along \(t=0\) with some morphisms between them. Moreover, under the identifications (2) the Hodge filtrations of \(\psi_{t}M\), \(\psi_{t,e(\alpha)}M\) and \(\phi_{t,1}M\) are \[ \begin{aligned} &F_{\bullet}\psi_{t}M=\bigoplus_{\alpha\in (-1,0]}F_{\bullet}M^{\alpha},\\ &F_{\bullet}\psi_{t,e(\alpha)}M=F_{\bullet}M^{\alpha},\text{ and}\\ &F_{\bullet}\phi_{t,1}M=F_{\bullet+1}M^{-1}, \end{aligned} \] where \(F_{\bullet}M^{\alpha}\) is \(F_{\bullet}M\cap M^{\alpha}\). By the strict specializability, we have \[ \begin{aligned} &t: F_{p}M^{\beta}\to F_{p}M^{\beta+1}\text{ is an isomorphism if }\beta> -1. \\ &\partial_{t} : F_{p}M^{\beta}\to F_{p+1}M^{\beta-1}\text{ is an isomorphism if }\beta< 0. \end{aligned} \] Therefore, Theorem 1 means that the Hodge filtration of the monodromic mixed Hodge module is determined from the ones of the nearby and vanishing cycles of it (up to shift). The natural question arises whether it is possible to reconstruct a monodromic mixed Hodge module on \(X\times \mathbb{C}\) from its nearby and vanishing cycles. The second main result gives an affirmative answer to this question. Consider a tuple \(((\mathcal{M}_{(-1,0]},T_{s},N),\mathcal{M}_{-1},c,v)\), where \(\mathcal{M}_{(-1,0]}\) and \(\mathcal{M}_{-1}\) are mixed Hodge modules on \(X\) and \(T_{s}: \mathcal{M}_{(-1,0]}\overset{\sim}{\to} \mathcal{M}_{(-1,0]}\), \(N : \mathcal{M}_{(-1,0]}\to \mathcal{M}_{(-1,0]}(-1)\), \(c: \mathcal{M}_{0}(:=\mathrm{Ker}(T_{s}-1)\subset \mathcal{M}_{(-1,0]})\to \mathcal{M}_{-1}\) and \(v: \mathcal{M}_{-1}\to \mathcal{M}_{0}(-1)\) are morphisms between mixed Hodge modules with the following properties: \begin{itemize} \item[(\(\star\)-i)] \(T_{s}\) commutes with \(N\). \item[(\(\star\)-ii)] The underlying \(D\)-module \(M_{(-1,0]}\) of \(\mathcal{M}_{(-1,0]}\) is decomposed as \[ M_{(-1,0]}=\bigoplus_{\alpha\in (-1,0]\cap \mathbb{Q}}M_{\alpha}, \] where \(M_{\alpha}:=\mathrm{Ker}(T_{s}-e(\alpha))\subset M_{(-1,0]}\). \item[(\(\star\)-iii)] \(vc: \mathcal{M}_{0}\to \mathcal{M}_{0}(-1)\) is \(-N\). \end{itemize} Since the weight filtration of \(\mathcal{M}_{(-1,0]}\) is a finite filtration, \(N\) is a nilpotent operator. Moreover, by (\(\star\)-ii), \(F_{p}M_{(-1,0]}\) (\(p\in \mathbb{Z}\)) is decomposed as \[ F_{p}M_{(-1,0]}=\bigoplus_{\alpha\in (-1,0]\cap \mathbb{Q}}F_{p}M_{\alpha}, \] where \(F_{p}M_{(-1,0],\alpha}=F_{p}M_{(-1,0]}\cap M_{\alpha}\). Let \(\mathscr{G}(X)\) be the category of such tuples. For a monodromic mixed Hodge module \(\mathcal{M}\) on \(X\times \mathbb{C}\), we obtain an object in \(\mathscr{G}(X)\) by setting \(\mathcal{M}_{(-1,0]}=\psi_{t}\mathcal{M}\) and \(\mathcal{M}_{-1}=\phi_{t,1}\mathcal{M}\). In this way, we can define a functor from the category \(\mathrm{MHM}^{p}_{\mathrm{mon}}(X\times \mathbb{C})\) of monodromic graded polarizable mixed Hodge modules on \(X\times \mathbb{C}\) to the category \(\mathscr{G}(X)\). Our second main result is the following. \textbf{Theorem 2.} There is an equivalence of categories \[ \mathrm{MHM}^{p}_{\mathrm{mon}}(X\times \mathbb{C})\simeq \mathscr{G}(X). \] We can naturally construct a monodromic mixed Hodge module on \(X\times \mathbb{C}\) from an object in \(\mathscr{G}(X)\). Let \(\mathcal{M}\) be a monodromic mixed Hodge module on \(X\times \mathbb{C}\). The Fourier-Laplace transformation \({}^F{M}\) of the underlying \(D\)-module \(M\) is again monodromic. Therefore, as explained above, its \(D\)-module structure is determined by \(({}^F{M})^{\alpha}(:=\bigcup_{l\geq 0}\mathrm{Ker}(t\partial_{t}-\alpha)^l\subset {}^F{M})\) for \(\alpha\in [-1,0]\cap \mathbb{Q}\) with some morphisms. By the definition of the Fourier-Laplace transformation, we have \[ ({}^F{M})^{\beta}\simeq M^{-\beta-1} \] as \(D\)-modules on \(X\) and we thus obtain \[ {}^F{M}(=\bigoplus_{\beta\in \mathbb{Q}}({}^F{M})^{\beta})=\bigoplus_{\beta\in \mathbb{Q}}M^{-\beta-1}. \] By setting \(\mathcal{M}_{(-1,0]}:=\phi_{t,1}\mathcal{M}\oplus \psi_{t,\neq 1}\mathcal{M}\) and \(\mathcal{M}_{-1}:=\psi_{t,1}\mathcal{M}(-1)\), we can naturally define an object in \(\mathscr{G}(X)\). By Theorem 2, we can construct a monodromic mixed Hodge module \({}^F{\mathcal{M}}\) on \(X\times \mathbb{C}_{\tau}\) whose underlying \(D\)-module is \({}^F{M}\). Through the identification \({}^F{M}^{\alpha}= M^{-\alpha-1}\), we have \[ F_{p}{}^F{M}^{\alpha}:= \begin{cases} F_{p+1}M^{-1} &(\alpha=0)\\ F_{p}M^{-\alpha-1} &(\alpha\in [-1,0)), \end{cases} \] for \(\alpha\in [-1,0]\) and \(p\in \mathbb{Z}\). The Hodge filtration \(F_{\bullet}{}^F{M}\) is decomposed by Theorem 1 and determined only by \(F_{\bullet}{}^F{M}^{\alpha}\) for \(\alpha\in [-1,0]\).