I am reading Ayoub's paper, The Motivic Nearby Cycles and the Conservation Conjecture,


In section 2.3, he talks a little about the limit of a variation of Hodge structures as a classical picture of nearby cycle functor. A question comes to me, is the limit mixed Hodge structure (assume they exist) motivic? I try to put up together the following argument,

To be precise, let $C$ be an open subset of $\text{Spec}\, \mathbb{Q}[t]$ that contains 0 and $X$ is a variety over $\mathbb{Q}$. $\pi$ is a fibration \begin{equation} \pi: X \rightarrow C \end{equation} whose only singular fiber is $\pi^{-1}(0)$. $C$ is smooth and its local ring at 0, $\mathcal{O}_{C,0}$, is a discrete valuation ring. The strict henselisation of $\mathcal{O}_{C,0}$ will be denoted by $\mathcal{O}_{C,0}^{sh}$. The space $B:=\text{Spec}\,\mathcal{O}_{C,0}^{sh}$ consists of two points, the closed point will be denoted by $s:=\text{Spec}(\mathbb{\overline{Q}})$ and the open point will be denoted by $\eta$. They form a henselian trait, \begin{equation} \eta \rightarrow B \leftarrow s \end{equation} There is a morphism from $B \rightarrow C$, which is the composition \begin{equation} \text{Spec}\,\mathcal{O}_{C,0}^{sh} \rightarrow \text{Spec}\,\mathcal{O}_{C,0} \rightarrow C \end{equation}

By pulling back $\pi$ over $B \rightarrow C$, we find, \begin{equation} f:X_B \rightarrow B \end{equation} with fibers $X_{\eta}$ and $X_s$ over the two points of $B$. From Ayoub's paper, there exists a nearby cycle functor from the motivic sheaves on $X_{\eta}$ to the motivic sheaves on $X_s$, \begin{equation} \textbf{R}\Psi_f:\text{DM}(X_{\eta},\mathbb{Q}) \rightarrow \text{DM}(X_s,\mathbb{Q}) \end{equation} that maps constuctible elements to constructible elements (? I am not sure about this, could someone explains it a little bit). The identity element $\mathbb{Q}(0)$ of $\text{DM}(X_{\eta},\mathbb{Q})$ is mapped to a constructible element $\textbf{R}\Psi_f(\mathbb{Q}(0))$ (? not sure about this) in $\text{DM}(X_s,\mathbb{Q})$. The structure morphism $f_s$ induces a morphism from $X_s$ to $\text{Spec}\,\mathbb{Q}$ which is the composition, \begin{equation} f_{s,\mathbb{Q}}: X_s \rightarrow \text{Spec}\,\overline{\mathbb{Q}} \rightarrow \text{Spec}\,\mathbb{Q} \end{equation} Therefore from Grothendieck's six operations, we find a constructible element $\mathcal{M}$ in the triangulated category $\text{DM}_{\text{gm}}(\mathbb{Q},\mathbb{Q})$, \begin{equation} \mathcal{M}=(f_{s,\mathbb{Q}})_*\, \textbf{R} \Psi_f(\mathbb{Q}(0)) \end{equation}

The Hodge realisation functor will be denoted by $R_{\sigma}$ for the natural embedding $\sigma : \mathbb{Q} \rightarrow \mathbb{C}$, \begin{equation} R_{\sigma}:\text{DM}_{gm}(\mathbb{Q},\mathbb{Q}) \rightarrow D^b(\text{MHS}_{\mathbb{Q}}) \end{equation}

The Hodge realisation of $\mathcal{M}$ is a bounded complex in the derived category of $\mathbb{Q}$-MHS and its $l$-th cohomology $H^l(R_{\sigma}(\mathcal{M}))$ is the limit MHS at 0.

I am not very sure about my argument, anyone could make is rigorous? Any stupid mistakes in it?


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