2.1 Mixed Hodge structure

Deligne in [Reference Deligne3] and [Reference Deligne4] proved the existence of mixed Hodge structures on the cohomology of a complex algebraic variety.

Theorem 2.2. ([Reference Deligne3], [Reference Deligne4])

Let $X$ be a complex algebraic variety. Then there exists a mixed Hodge structure on $H^{j}(X,\mathbb{C})$ . Moreover, the weight filtration satisfies

$$\begin{eqnarray}0=W_{-1}\subseteq W_{0}\subseteq \cdots \subseteq W_{2j}=H^{j}(X,\mathbb{Q})\end{eqnarray}$$

and the Hodge filtration satisfies

$$\begin{eqnarray}H^{j}(X,\mathbb{C})=F^{0}\supseteq F^{1}\supseteq \cdots \supseteq F^{j}\supseteq F^{j+1}=0.\end{eqnarray}$$

One can define a mixed Hodge structure on the compactly supported cohomology $H_{c}^{\ast }(X,\mathbb{Q})$ . The following theorem is basic property of mixed Hodge structures (for a proof, e.g., see [Reference Peters and Steenbrink13]).

Theorem 2.3. (Poincaré duality)

For a smooth connected $X$ , we have that Poincaré duality

$$\begin{eqnarray}H^{k}(X,\mathbb{Q})\times H_{c}^{2d-k}(X,\mathbb{Q})\longrightarrow H_{c}^{2d}(X)\cong \mathbb{Q}(-d)\end{eqnarray}$$

is compatible with mixed Hodge structures, where $\mathbb{Q}(-d)$ is the pure mixed Hodge structure on $\mathbb{Q}$ with weight $2d$ and Hodge filtration $F^{d}=\mathbb{Q}$ and $F^{d+1}=0$ .

3.1 $\unicode[STIX]{x1D706}$ -connection

We fix integers $g\geqslant 0,k>0$ and $n>0$ . We also fix a $k$ -tuple of partition of $n$ , denoted by $\boldsymbol{\unicode[STIX]{x1D707}}=(\unicode[STIX]{x1D707}^{1},\ldots ,\unicode[STIX]{x1D707}^{k})$ , that is, $\unicode[STIX]{x1D707}^{i}=(\unicode[STIX]{x1D707}_{1}^{i},\ldots ,\unicode[STIX]{x1D707}_{r_{i}}^{i})$ such that $\unicode[STIX]{x1D707}_{1}^{i}\geqslant \unicode[STIX]{x1D707}_{2}^{i}\geqslant \cdots \,$ are nonnegative integers and $\sum _{j}\unicode[STIX]{x1D707}_{j}^{i}=n$ . Let $\unicode[STIX]{x1D6F4}$ be a smooth complex projective curve of genus $g$ . We fix $k$ -distinct points $p_{1},\ldots ,p_{k}$ in $\unicode[STIX]{x1D6F4}$ and we define a divisor by $D:=p_{1}+\cdots +p_{k}$ . We put $\unicode[STIX]{x1D6F4}_{0}=\unicode[STIX]{x1D6F4}\setminus D$ . For integer $d$ , we put

$$\begin{eqnarray}\unicode[STIX]{x1D6EF}_{n}^{\boldsymbol{\unicode[STIX]{x1D707}},d}:=\left\{\left(\unicode[STIX]{x1D706},(\unicode[STIX]{x1D709}_{j}^{i})_{1\leqslant j\leqslant r_{i}}^{1\leqslant i\leqslant k}\right)\in \mathbb{C}\times \mathbb{C}^{r}\,\left|\,\unicode[STIX]{x1D706}d+\mathop{\sum }_{i,j}\unicode[STIX]{x1D707}_{j}^{i}\unicode[STIX]{x1D709}_{j}^{i}=0\right.\right\}\end{eqnarray}$$

where $r:=\sum r_{i}$ . We take $(\unicode[STIX]{x1D706},\boldsymbol{\unicode[STIX]{x1D709}})\in \unicode[STIX]{x1D6EF}_{n}^{\boldsymbol{\unicode[STIX]{x1D707}},d}$ where $\boldsymbol{\unicode[STIX]{x1D709}}=(\unicode[STIX]{x1D709}_{j}^{i})_{1\leqslant j\leqslant r_{i}}^{1\leqslant i\leqslant k}$ .

For $\unicode[STIX]{x1D706}=1$ , this is a regular singular $\boldsymbol{\unicode[STIX]{x1D709}}$ -parabolic connection of spectral type $\boldsymbol{\unicode[STIX]{x1D707}}$ (Definition 1.2). For $\unicode[STIX]{x1D706}=0$ and $\boldsymbol{\unicode[STIX]{x1D709}}=0$ , this is a parabolic Higgs bundle (Definition 1.1).

Remark 3.2. For $\unicode[STIX]{x1D706}\neq 0$ , we have

$$\begin{eqnarray}\deg E=\deg (\det (E))=-\mathop{\sum }_{i=1}^{k}\text{tr}\left(\text{Res}_{p_{i}}((\unicode[STIX]{x1D706}^{-1}\unicode[STIX]{x1D6FB}))\right)=-\mathop{\sum }_{i=1}^{k}\mathop{\sum }_{j=1}^{r_{i}}\frac{\unicode[STIX]{x1D707}_{j}^{i}\unicode[STIX]{x1D709}_{j}^{i}}{\unicode[STIX]{x1D706}}=d.\end{eqnarray}$$

We take rational numbers

$$\begin{eqnarray}0<\unicode[STIX]{x1D6FC}_{1}^{(i)}<\unicode[STIX]{x1D6FC}_{2}^{(i)}<\cdots <\unicode[STIX]{x1D6FC}_{r_{i}}^{(i)}<1\end{eqnarray}$$

for $i=1,\ldots ,k$ satisfying $\unicode[STIX]{x1D6FC}_{j}^{(i)}\neq \unicode[STIX]{x1D6FC}_{j^{\prime }}^{(i^{\prime })}$ for $(i,j)\neq (i^{\prime },j^{\prime })$ . We choose $\boldsymbol{\unicode[STIX]{x1D6FC}}=(\unicode[STIX]{x1D6FC}_{j}^{(i)})$ sufficiently generic.

Definition 3.3. A parabolic $\unicode[STIX]{x1D706}$ -connection $(\unicode[STIX]{x1D706},E,\unicode[STIX]{x1D6FB},\{l_{\ast }^{(i)}\}_{1\leqslant i\leqslant k})$ is $\boldsymbol{\unicode[STIX]{x1D6FC}}$ -stable (resp. $\boldsymbol{\unicode[STIX]{x1D6FC}}$ -semistable) if for any proper nonzero subbundle $F\subset E$ satisfying $\unicode[STIX]{x1D6FB}(F)\subset F\otimes \unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F4}}^{1}(D)$ , the inequality

$$\begin{eqnarray}\text{par}\,\unicode[STIX]{x1D707}(F)<\text{par}\,\unicode[STIX]{x1D707}(E)\quad (\text{resp. }\leqslant )\end{eqnarray}$$

holds. Here, the induced parabolic structure on a subbundle $F\subset E$ is the filtrations $F|_{p_{i}}=F|_{p_{i}}\cap l_{1}^{(i)}\supset F|_{p_{i}}\cap l_{2}^{(i)}\supset \cdots \supset F|_{p_{i}}\cap l_{r_{i}}^{(i)}\supset F|_{p_{i}}\cap l_{r_{i}+1}^{(i)}=0$ for each $p_{i}$ .

3.2 Construction of the moduli space

The argument in this subsection is almost same as in [Reference Inaba8]. The difference from [Reference Inaba8] is that we fix the $k$ -distinct points $\{p_{1},\ldots ,p_{k}\}$ , the flag $\{l_{\ast }^{(i)}\}$ is not necessarily full flag, and we construct the moduli space of $\boldsymbol{\unicode[STIX]{x1D6FC}}$ -semistable parabolic $\unicode[STIX]{x1D706}$ -connections instead of $\boldsymbol{\unicode[STIX]{x1D6FC}}$ -semistable parabolic connections.

We recall the definition of a parabolic $\unicode[STIX]{x1D6EC}_{D}^{1}$ -triple defined in [Reference Inaba, Iwasaki and Saito10]. Let $D$ be an effective divisor on a nonsingular curve $\unicode[STIX]{x1D6F4}$ . We define $\unicode[STIX]{x1D6EC}_{D}^{1}$ as ${\mathcal{O}}_{\unicode[STIX]{x1D6F4}}\oplus \unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F4}}^{1}(D)^{\vee }$ with the bimodule structure given by

$$\begin{eqnarray}\displaystyle & \displaystyle f(a,v)=(fa,fv)\quad (f,a\in {\mathcal{O}}_{\unicode[STIX]{x1D6F4}},~v\in \unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F4}}^{1}(D)^{\vee }), & \displaystyle \nonumber\\ \displaystyle & \displaystyle (a,v)f=(fa+v(f),fv)\quad (f,a\in {\mathcal{O}}_{\unicode[STIX]{x1D6F4}},~v\in \unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F4}}^{1}(D)^{\vee }). & \displaystyle \nonumber\end{eqnarray}$$

Note that to give a left ${\mathcal{O}}_{\unicode[STIX]{x1D6F4}}$ -homomorphism $\unicode[STIX]{x1D6F7}:\unicode[STIX]{x1D6EC}_{D}^{1}\otimes E_{1}\rightarrow E_{2}$ is equivalent to give an ${\mathcal{O}}_{\unicode[STIX]{x1D6F4}}$ -homomorphism $\unicode[STIX]{x1D719}:E_{1}\rightarrow E_{2}$ and a morphism $\unicode[STIX]{x1D6FB}:E_{1}\rightarrow E_{2}\otimes \unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F4}}^{1}(D)$ such that $\unicode[STIX]{x1D6FB}(fa)=\unicode[STIX]{x1D719}(a)\otimes df+f\unicode[STIX]{x1D6FB}(a)$ for $f\in {\mathcal{O}}_{\unicode[STIX]{x1D6F4}}$ and $a\in E_{1}$ . We also denote the parabolic $\unicode[STIX]{x1D6EC}_{D}^{1}$ -triple $(E_{1},E_{2},\unicode[STIX]{x1D6F7},F_{\ast }(E_{1}))$ by $(E_{1},E_{2},\unicode[STIX]{x1D719},\unicode[STIX]{x1D6FB},F_{\ast }(E_{1}))$ .

We take positive integers $\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2},\unicode[STIX]{x1D6FE}$ and rational numbers $0<\unicode[STIX]{x1D6FC}_{1}^{\prime }<\cdots <\unicode[STIX]{x1D6FC}_{l}^{\prime }<1$ . We assume $\unicode[STIX]{x1D6FE}\gg 0$ .

Definition 3.6. A parabolic $\unicode[STIX]{x1D6EC}_{D}^{1}$ -triple $(E_{1},E_{2},\unicode[STIX]{x1D719},\unicode[STIX]{x1D6FB},F_{\ast }(E_{1}))$ is $(\boldsymbol{\unicode[STIX]{x1D6FC}}^{\prime },\boldsymbol{\unicode[STIX]{x1D6FD}},\unicode[STIX]{x1D6FE})$ -stable (resp.  $(\boldsymbol{\unicode[STIX]{x1D6FC}}^{\prime },\boldsymbol{\unicode[STIX]{x1D6FD}},\unicode[STIX]{x1D6FE})$ -semistable) if for any subbundle $(F_{1},F_{2})\subset (E_{1},E_{2})$ satisfying $(0,0)\neq (F_{1},F_{2})\neq (E_{1},E_{2})$ and $\unicode[STIX]{x1D6F7}(\unicode[STIX]{x1D6EC}_{D}^{1}\otimes F_{1})\subset F_{2}$ , the inequality

$$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D6FD}_{1}\deg F_{1}(-D)+\unicode[STIX]{x1D6FD}_{2}(\deg F_{2}-\unicode[STIX]{x1D6FE}\,\text{rank}\,F_{2})+\unicode[STIX]{x1D6FD}_{1}\mathop{\sum }_{j=1}^{l}\unicode[STIX]{x1D6FC}_{j}^{\prime }\text{length}(F_{j}(E_{1})\cap F_{1}/(F_{j+1}(E_{1})\cap F_{1}))}{\unicode[STIX]{x1D6FD}_{1}\,\text{rank}\,F_{1}+\unicode[STIX]{x1D6FD}_{2}\,\text{rank}\,F_{2}} & & \displaystyle \nonumber\\ \displaystyle \underset{(\text{resp.}~\leqslant )}{{<}}\frac{\unicode[STIX]{x1D6FD}_{1}\deg E_{1}(-D)+\unicode[STIX]{x1D6FD}_{2}(\deg E_{2}-\unicode[STIX]{x1D6FE}\,\text{rank}\,E_{2})+\unicode[STIX]{x1D6FD}_{1}\mathop{\sum }_{j=1}^{l}\unicode[STIX]{x1D6FC}_{j}^{\prime }\text{length}((F_{j}(E_{1}))/(F_{j+1}(E_{1})))}{\unicode[STIX]{x1D6FD}_{1}\,\text{rank}\,E_{1}+\unicode[STIX]{x1D6FD}_{2}\,\text{rank}\,E_{2}} & & \displaystyle \nonumber\end{eqnarray}$$

holds.

Definition 3.8. We put ${\mathcal{C}}=\unicode[STIX]{x1D6F4}\times \unicode[STIX]{x1D6EF}_{n}^{\boldsymbol{\unicode[STIX]{x1D707}},d}$ , $S=\unicode[STIX]{x1D6EF}_{n}^{\boldsymbol{\unicode[STIX]{x1D707}},d}$ , $\tilde{p}_{i}=p_{i}\times \unicode[STIX]{x1D6EF}_{n}^{\boldsymbol{\unicode[STIX]{x1D707}},d}$ (for $i=1,\ldots ,k$ ) and ${\mathcal{D}}=\tilde{p}_{1}+\cdots +\tilde{p}_{k}$ . We define a functor ${\mathcal{M}}_{\text{Hod}}^{\boldsymbol{\unicode[STIX]{x1D6FC}},\boldsymbol{\unicode[STIX]{x1D707}},d}({\mathcal{C}}/S,{\mathcal{D}})$ of the category of locally Noetherian schemes over $S$ to the category of sets by

$$\begin{eqnarray}{\mathcal{M}}_{\text{Hod}}^{\boldsymbol{\unicode[STIX]{x1D6FC}},\boldsymbol{\unicode[STIX]{x1D707}},d}({\mathcal{C}}/S,{\mathcal{D}})(T):=\left\{(\unicode[STIX]{x1D706},E,\unicode[STIX]{x1D6FB},\{l_{\ast }^{(i)}\}_{1\leqslant i\leqslant k})\right\}/\sim ,\end{eqnarray}$$

for a locally Noetherian scheme $T$ over $S$ where:

1. (1) $E$ is a vector bundle on ${\mathcal{C}}_{T}$ of rank $n$ ;

2. (2) $\unicode[STIX]{x1D706}:E\rightarrow E$ is the ${\mathcal{O}}_{{\mathcal{C}}_{T}}$ -homomorphism defined by $a(x)\mapsto \unicode[STIX]{x1D706}(x)\cdot a(x)$ where $\unicode[STIX]{x1D706}(x)$ is the image of $x\in {\mathcal{C}}_{T}$ by the composition ${\mathcal{C}}_{T}\rightarrow S\rightarrow \mathbb{C}$ of the natural morphism and the projection;

3. (3) $\unicode[STIX]{x1D6FB}:E\rightarrow E\otimes \unicode[STIX]{x1D6FA}_{{\mathcal{C}}_{T}}^{1}(({\mathcal{D}}_{T}))$ is a relative $\unicode[STIX]{x1D706}_{T/S}$ -connection;

4. (4) $E|_{(\tilde{p}_{i})_{T}}=l_{1}^{(i)}\supset l_{2}^{(i)}\supset \cdots \supset l_{r_{i}}^{(i)}\supset l_{r_{i}+1}^{(i)}=0$ is a filtration by subbundles such that $(\text{Res}_{(\tilde{p}_{i})_{T}}(\unicode[STIX]{x1D6FB})-(\unicode[STIX]{x1D709}_{j}^{i})_{T})\subset l_{j+1}^{(i)}$ for $j=1,\ldots ,r_{i}$ , $i=1,\ldots ,k$ ;

5. (5) for any geometric point $t\in T$ , $\dim (l_{j}^{i}/l_{j+1}^{i})\otimes k(t)=\unicode[STIX]{x1D707}_{j}^{i}$ for any $i,j$ and $(\unicode[STIX]{x1D706},E,\unicode[STIX]{x1D6FB},\{l_{\ast }^{(i)}\})\otimes k(t)$ is $\boldsymbol{\unicode[STIX]{x1D6FC}}$ -stable.

The following proposition is shown by the same way as in the proof [Reference Inaba8, Theorem 2.1].

Proposition 3.9. There exists a relative coarse moduli scheme

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70B}:{\mathcal{M}}_{\text{Hod}}^{\boldsymbol{\unicode[STIX]{x1D707}}} & \longrightarrow & \displaystyle \unicode[STIX]{x1D6EF}_{n}^{\boldsymbol{\unicode[STIX]{x1D707}},d}\nonumber\\ \displaystyle (\unicode[STIX]{x1D706},E,\unicode[STIX]{x1D6FB},\{l_{\ast }^{(i)}\}_{1\leqslant i\leqslant r_{i}}) & \longmapsto & \displaystyle (\unicode[STIX]{x1D706},\boldsymbol{\unicode[STIX]{x1D709}})\nonumber\end{eqnarray}$$

of $\boldsymbol{\unicode[STIX]{x1D6FC}}$ -stable parabolic $\unicode[STIX]{x1D706}$ -connections of rank $n$ , of degree $d$ , and of type $\boldsymbol{\unicode[STIX]{x1D707}}$ . For simplicity, we drop $\boldsymbol{\unicode[STIX]{x1D6FC}}$ and $d$ from the notation of the moduli space.

If $n$ and $d$ are coprime, then ${\mathcal{M}}_{\text{Hod}}^{\boldsymbol{\unicode[STIX]{x1D707}}}$ is a relative fine moduli scheme, that is, there is a universal family over ${\mathcal{M}}_{\text{Hod}}^{\boldsymbol{\unicode[STIX]{x1D707}}}$ .

Proof. Fix a weight $\boldsymbol{\unicode[STIX]{x1D6FC}}$ which determines the stability of parabolic $\unicode[STIX]{x1D706}$ -connections. We take positive integers $\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2},\unicode[STIX]{x1D6FE}$ and rational numbers $0<\tilde{\unicode[STIX]{x1D6FC}}_{1}^{(i)}<\cdots <\tilde{\unicode[STIX]{x1D6FC}}_{r_{i}}^{(i)}<1$ satisfying $(\unicode[STIX]{x1D6FD}_{1}+\unicode[STIX]{x1D6FD}_{2})\unicode[STIX]{x1D6FC}_{j}^{(i)}=\unicode[STIX]{x1D6FD}_{1}\tilde{\unicode[STIX]{x1D6FC}}_{j}^{(i)}$ for any $i,j$ . We assume $\unicode[STIX]{x1D6FE}\gg 0$ . We take an increasing sequence $0<\unicode[STIX]{x1D6FC}_{1}^{\prime }<\cdots <\unicode[STIX]{x1D6FC}_{l}^{\prime }<1$ such that

$$\begin{eqnarray}\{\unicode[STIX]{x1D6FC}_{i}^{\prime }\mid 1\leqslant i\leqslant l\}=\{\tilde{\unicode[STIX]{x1D6FC}}_{j}^{(i)}\mid 1\leqslant i\leqslant k,1\leqslant j\leqslant r_{i}\}\end{eqnarray}$$

where we put $l=\sum _{i=1}^{k}r_{i}$ . We take any member $(\unicode[STIX]{x1D706},E,\unicode[STIX]{x1D6FB},\{l_{\ast }^{(i)}\}_{1\leqslant i\leqslant k})\in {\mathcal{M}}_{\text{Hod}}^{\boldsymbol{\unicode[STIX]{x1D6FC}},\boldsymbol{\unicode[STIX]{x1D707}},d}({\mathcal{C}}/S,{\mathcal{D}})(T)$ . For each $1\leqslant p\leqslant l$ , there exist $i,j$ satisfying $\tilde{\unicode[STIX]{x1D6FC}}_{j}^{(i)}=\unicode[STIX]{x1D6FC}_{p}^{\prime }$ . We put $F_{1}(E):=E$ and define inductively

$$\begin{eqnarray}F_{p}(E):=\text{Ker}(F_{p-1}(E)\longrightarrow E|_{(\tilde{p}_{i})_{T}}/l_{p})\end{eqnarray}$$

for $p=1,\ldots ,l$ . Here, we put $l_{p}:=l_{j}^{(i)}$ for the unique $(i,j)$ for which $\tilde{\unicode[STIX]{x1D6FC}}_{j}^{(i)}=\unicode[STIX]{x1D6FC}_{p}^{\prime }$ . We also put $d_{p}:=\text{length}((E/F_{p+1}(E))\otimes k(t))$ for $p=1,\ldots ,l$ and $t\in T$ . Then $(\unicode[STIX]{x1D706},E,\unicode[STIX]{x1D6FB},\{l_{\ast }^{(i)}\})\mapsto (E,E,\unicode[STIX]{x1D706}\text{id},\unicode[STIX]{x1D6FB},F_{\ast }(E))$ determines the morphism

$$\begin{eqnarray}\unicode[STIX]{x1D704}:{\mathcal{M}}_{\text{Hod}}^{\boldsymbol{\unicode[STIX]{x1D6FC}},\boldsymbol{\unicode[STIX]{x1D707}},d}({\mathcal{C}}/S,{\mathcal{D}})\longrightarrow \overline{{\mathcal{M}}_{n,d,\{d_{i}\}}^{\boldsymbol{\unicode[STIX]{x1D6FC}}^{\prime },\boldsymbol{\unicode[STIX]{x1D6FD}},\unicode[STIX]{x1D6FE}}}({\mathcal{C}}/S,{\mathcal{D}})\end{eqnarray}$$

where $\overline{{\mathcal{M}}_{n,d,\{d_{i}\}}^{\boldsymbol{\unicode[STIX]{x1D6FC}}^{\prime },\boldsymbol{\unicode[STIX]{x1D6FD}},\unicode[STIX]{x1D6FE}}}({\mathcal{C}}/S,{\mathcal{D}})$ is the moduli functor of $(\boldsymbol{\unicode[STIX]{x1D6FC}},\boldsymbol{\unicode[STIX]{x1D6FD}},\unicode[STIX]{x1D6FE})$ -stable $\unicode[STIX]{x1D6EC}_{D}^{1}$ -triples whose coarse moduli scheme exists by Theorem 3.7. Then we have that a certain subscheme ${\mathcal{M}}_{\text{Hod}}^{\boldsymbol{\unicode[STIX]{x1D707}}}$ of $\overline{{\mathcal{M}}_{n,d,\{d_{i}\}}^{\boldsymbol{\unicode[STIX]{x1D6FC}}^{\prime },\boldsymbol{\unicode[STIX]{x1D6FD}},\unicode[STIX]{x1D6FE}}}({\mathcal{C}}/S,{\mathcal{D}})$ is just the coarse moduli scheme of ${\mathcal{M}}_{\text{Hod}}^{\boldsymbol{\unicode[STIX]{x1D6FC}},\boldsymbol{\unicode[STIX]{x1D707}},d}({\mathcal{C}}/S,{\mathcal{D}})$ in the same way as in [Reference Inaba, Iwasaki and Saito10, Theorem 2.1] and [Reference Inaba8, Theorem 2.1].

If $n$ and $d$ are coprime, then there is a universal family on ${\mathcal{M}}_{\text{Hod}}^{\boldsymbol{\unicode[STIX]{x1D707}}}\times \unicode[STIX]{x1D6F4}$ (see [Reference Huybrechts and Lehn7, Theorem 4.6.5] and the proof of [Reference Inaba8, Theorem 2.1]).◻

We denote the fibers of ${\mathcal{M}}_{\text{Hod}}^{\boldsymbol{\unicode[STIX]{x1D707}}}$ over $\unicode[STIX]{x1D706}=0$ and $\unicode[STIX]{x1D706}=1$ by ${\mathcal{M}}_{\text{Dol}}^{\boldsymbol{\unicode[STIX]{x1D707}}}$ and ${\mathcal{M}}_{\text{DR}}^{\boldsymbol{\unicode[STIX]{x1D707}}}$ , respectively. Let ${\mathcal{M}}_{\text{Hod}}^{\boldsymbol{\unicode[STIX]{x1D707}}}(\unicode[STIX]{x1D706},\boldsymbol{\unicode[STIX]{x1D709}})$ be the fiber of $(\unicode[STIX]{x1D706},\boldsymbol{\unicode[STIX]{x1D709}})$ . Let ${\mathcal{M}}_{\text{DR}}^{\boldsymbol{\unicode[STIX]{x1D707}}}(\boldsymbol{\unicode[STIX]{x1D709}})$ and ${\mathcal{M}}_{\text{Dol}}^{\boldsymbol{\unicode[STIX]{x1D707}}}(\mathbf{0})$ be the fibers of $(1,\boldsymbol{\unicode[STIX]{x1D709}})$ and $(0,\mathbf{0})$ , respectively. The fiber ${\mathcal{M}}_{\text{DR}}^{\boldsymbol{\unicode[STIX]{x1D707}}}(\boldsymbol{\unicode[STIX]{x1D709}})$ is the moduli space of $\boldsymbol{\unicode[STIX]{x1D6FC}}$ -stable regular singular $\boldsymbol{\unicode[STIX]{x1D709}}$ -parabolic connections of spectral type $\boldsymbol{\unicode[STIX]{x1D707}}$ (constructed in [Reference Inaba and Saito9]), and the fiber ${\mathcal{M}}_{\text{Dol}}^{\boldsymbol{\unicode[STIX]{x1D707}}}(\mathbf{0})$ is the moduli space of $\boldsymbol{\unicode[STIX]{x1D6FC}}$ -stable parabolic Higgs bundles of rank $n$ and of degree $d$ (constructed as a hyperkähler quotient using gauge theory in [Reference Konno11] or as a closed subvariety of the moduli space of parabolic Higgs sheaves constructed in [Reference Yokogawa19]).

The following proposition is shown by the same way as in the proofs [Reference Arinkin1, Lemma 4], [Reference Inaba8, Theorem 2.1], [Reference Inaba and Saito9] and [Reference Simpson17, Lemma 6.1].

Proposition 3.10. The morphism

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70B}:{\mathcal{M}}_{\text{Hod}}^{\boldsymbol{\unicode[STIX]{x1D707}}} & \longrightarrow & \displaystyle \unicode[STIX]{x1D6EF}_{n}^{\boldsymbol{\unicode[STIX]{x1D707}},d}\nonumber\\ \displaystyle (\unicode[STIX]{x1D706},E,\unicode[STIX]{x1D6FB},\{l_{\ast }^{(i)}\}_{1\leqslant i\leqslant r_{i}}) & \longmapsto & \displaystyle (\unicode[STIX]{x1D706},\boldsymbol{\unicode[STIX]{x1D709}})\nonumber\end{eqnarray}$$

is smooth. Moreover, ${\mathcal{M}}_{\text{Hod}}^{\boldsymbol{\unicode[STIX]{x1D707}}}$ is nonsingular.

Proof. At first, we prove that $\unicode[STIX]{x1D70B}:{\mathcal{M}}_{\text{Hod}}^{\boldsymbol{\unicode[STIX]{x1D707}}}\rightarrow \unicode[STIX]{x1D6EF}_{n}^{\boldsymbol{\unicode[STIX]{x1D707}},d}$ is smooth. Let ${\mathcal{M}}_{\text{Hod}}^{1}$ be the moduli space of tuples $(\unicode[STIX]{x1D706},L,\unicode[STIX]{x1D6FB}_{L})$ where $L$ is a line bundle of degree $d$ on $\unicode[STIX]{x1D6F4}$ and $\unicode[STIX]{x1D6FB}_{L}:L\rightarrow L\otimes \unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F4}}^{1}(D)$ is a $\unicode[STIX]{x1D706}$ -connection. We put

$$\begin{eqnarray}\unicode[STIX]{x1D6EF}^{k,d}:=\left\{(\unicode[STIX]{x1D706},(\unicode[STIX]{x1D709}^{i}))\in \mathbb{C}\times \mathbb{C}^{k}\,\left|\,\unicode[STIX]{x1D706}d+\mathop{\sum }_{i=1}^{k}\unicode[STIX]{x1D709}^{i}=0\right.\right\}.\end{eqnarray}$$

Let $\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D706}=1}^{k,d}$ be the subset of $\unicode[STIX]{x1D6EF}^{k,d}$ where $\unicode[STIX]{x1D706}=1$ and let ${\mathcal{M}}_{\text{DR}}^{1}$ be the inverse image of the subset $\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D706}=1}^{k,d}$ . Since ${\mathcal{M}}_{\text{DR}}^{1}\rightarrow \unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D706}=1}^{k,d}$ is smooth (see [Reference Inaba8] and [Reference Inaba and Saito9]), ${\mathcal{M}}_{\text{Hod}}^{1}\rightarrow \unicode[STIX]{x1D6EF}^{k,d}$ is smooth (see [Reference Simpson17, Lemma 6.1]). We consider the morphism

$$\begin{eqnarray}\displaystyle & & \displaystyle \text{det}:{\mathcal{M}}_{\text{Hod}}^{\boldsymbol{\unicode[STIX]{x1D707}}}\longrightarrow {\mathcal{M}}_{\text{Hod}}^{1}\times _{\unicode[STIX]{x1D6EF}^{k,d}}\unicode[STIX]{x1D6EF}_{n}^{\boldsymbol{\unicode[STIX]{x1D707}},d}\nonumber\\ \displaystyle & & \displaystyle (\unicode[STIX]{x1D706},E,\unicode[STIX]{x1D6FB},\{l_{j}^{(i)}\})\longmapsto ((\unicode[STIX]{x1D706},\text{det}(E),\text{det}(\unicode[STIX]{x1D6FB})),\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D706},E,\unicode[STIX]{x1D6FB},\{l_{j}^{(i)}\})).\nonumber\end{eqnarray}$$

It is sufficient to show that the morphism $\text{det}$ is smooth. Let $A$ be an Artinian local ring over ${\mathcal{M}}_{\text{Hod}}^{1}\times _{\unicode[STIX]{x1D6EF}^{k,d}}\unicode[STIX]{x1D6EF}_{n}^{\boldsymbol{\unicode[STIX]{x1D707}},d}$ with the maximal ideal $m$ and $I$ be an ideal of $A$ such that $mI=0$ . Let $(\unicode[STIX]{x1D706},L,\unicode[STIX]{x1D6FB})\in {\mathcal{M}}_{\text{Hod}}^{1}(A)$ and $(\unicode[STIX]{x1D706},\boldsymbol{\unicode[STIX]{x1D709}})\in \unicode[STIX]{x1D6EF}_{n}^{\boldsymbol{\unicode[STIX]{x1D707}},d}(A)$ be the elements corresponding to the morphism

$$\begin{eqnarray}\text{Spec}\,A\longrightarrow {\mathcal{M}}_{\text{Hod}}^{1}\times _{\unicode[STIX]{x1D6EF}^{k,d}}\unicode[STIX]{x1D6EF}_{n}^{\boldsymbol{\unicode[STIX]{x1D707}},d}.\end{eqnarray}$$

We take any member $(\unicode[STIX]{x1D706},E,\unicode[STIX]{x1D6FB},\{l_{j}^{(i)}\})\in {\mathcal{M}}_{\text{Hod}}^{\boldsymbol{\unicode[STIX]{x1D707}}}(A/I)$ such that

$$\begin{eqnarray}\det (\unicode[STIX]{x1D706},E,\unicode[STIX]{x1D6FB},\{l_{j}^{(i)}\})\cong ((\unicode[STIX]{x1D706},L,\unicode[STIX]{x1D6FB}),(\unicode[STIX]{x1D706},\boldsymbol{\unicode[STIX]{x1D709}}))\otimes A/I.\end{eqnarray}$$

It is sufficient to show that $(\unicode[STIX]{x1D706},E,\unicode[STIX]{x1D6FB},\{l_{j}^{(i)}\})$ may be lifted to a flat family $(\tilde{\unicode[STIX]{x1D706}},\tilde{E},\tilde{\unicode[STIX]{x1D6FB}},\{\tilde{l}_{j}^{(i)}\})$ over $A$ such that $\det (\tilde{\unicode[STIX]{x1D706}},\tilde{E},\tilde{\unicode[STIX]{x1D6FB}},\{\tilde{l}_{j}^{(i)}\})\cong ((\unicode[STIX]{x1D706},L,\unicode[STIX]{x1D6FB}),(\unicode[STIX]{x1D706},\boldsymbol{\unicode[STIX]{x1D709}}))$ . The obstructions lie in the hypercohomology $\mathbb{H}^{2}(\unicode[STIX]{x1D6F4},{\mathcal{F}}_{0}^{\bullet }\otimes I)$ . Here, ${\mathcal{F}}_{0}^{\bullet }$ is the complex of sheaves defined by ${\mathcal{F}}_{0}^{i}=0$ for $i\neq 0,1$ ,

$$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{F}}_{0}^{0}:=\left\{s\in {\mathcal{E}}nd(E\otimes A/m)\,\left|\,\begin{array}{@{}l@{}}\text{Tr}(s)=0\text{ and }\\ s_{|_{p_{i}\otimes A/m}}(l_{j}^{i})_{A/m}\subset (l_{j}^{i})_{A/m}~\text{for any}~i,j\end{array}\right.\right\}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle {\mathcal{F}}_{0}^{1}:=\left\{s\in {\mathcal{E}}nd(E\otimes A/m)\otimes \unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F4}}^{1}(D)\,\left|\,\begin{array}{@{}l@{}}\text{Tr}(s)=0\text{ and }\\ \text{Res}_{p_{i}\otimes A/m}(s)(l_{j}^{i})_{A/m}\\ \quad \subset (l_{j+1}^{i})_{A/m}~\text{for any}~i,j\end{array}\right.\right\}, & \displaystyle \nonumber\end{eqnarray}$$

and $d:{\mathcal{F}}_{0}^{0}\rightarrow {\mathcal{F}}_{0}^{1}$ maps $s$ to $\unicode[STIX]{x1D6FB}\circ s-s\circ \unicode[STIX]{x1D6FB}$ . From the spectral sequence $H^{q}({\mathcal{F}}_{0}^{p})\Rightarrow \mathbb{H}^{p+q}({\mathcal{F}}_{0}^{\bullet })$ , there is an isomorphism

$$\begin{eqnarray}\mathbb{H}^{2}({\mathcal{F}}_{0}^{\bullet })\cong \text{Coker}\left(H^{1}({\mathcal{F}}_{0}^{0})\xrightarrow[]{\{}H^{1}({\mathcal{F}}_{0}^{1})\right).\end{eqnarray}$$

Since $({\mathcal{F}}_{0}^{0})^{\vee }\otimes \unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F4}}^{1}\cong {\mathcal{F}}_{0}^{1}$ and $({\mathcal{F}}_{0}^{1})^{\vee }\otimes \unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F4}}^{1}\cong {\mathcal{F}}_{0}^{0}$ , we have

$$\begin{eqnarray}\displaystyle \mathbb{H}^{2}({\mathcal{F}}_{0}^{\bullet }) & \cong & \displaystyle \text{Coker}\left(H^{1}({\mathcal{F}}_{0}^{0})\xrightarrow[]{\{}H^{1}({\mathcal{F}}_{0}^{1})\right)\nonumber\\ \displaystyle & \cong & \displaystyle \text{Ker}\left(H^{1}({\mathcal{F}}_{0}^{1})^{\vee }\xrightarrow[]{\{}H^{1}({\mathcal{F}}_{0}^{0})^{\vee }\right)^{\vee }\nonumber\\ \displaystyle & \cong & \displaystyle \text{Ker}\left(H^{0}(({\mathcal{F}}_{0}^{1})^{\vee }\otimes \unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F4}}^{1})\xrightarrow[]{\{}H^{0}(({\mathcal{F}}_{0}^{0})^{\vee }\otimes \unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F4}}^{1})\right)^{\vee }\nonumber\\ \displaystyle & \cong & \displaystyle \text{Ker}\left(H^{0}({\mathcal{F}}_{0}^{0})\xrightarrow[]{\{}H^{0}({\mathcal{F}}_{0}^{1})\right)^{\vee }.\nonumber\end{eqnarray}$$

We take any element $s\in \text{Ker}(H^{0}({\mathcal{F}}_{0}^{0})\xrightarrow[]{\{}H^{0}({\mathcal{F}}_{0}^{1}))$ , which may be regarded as an element of $\text{End}((\unicode[STIX]{x1D706},E,\unicode[STIX]{x1D6FB},\{l_{j}^{(i)}\}))$ . Since $(\unicode[STIX]{x1D706},E,\unicode[STIX]{x1D6FB},\{l_{j}^{(i)}\})$ is $\boldsymbol{\unicode[STIX]{x1D6FC}}$ -stable, the endomorphism $s$ is a scalar multiplication. By $\text{Tr}(s)=0$ , we have $s=0$ . Hence, $\mathbb{H}^{2}({\mathcal{F}}_{0}^{\bullet })=0$ .

Secondly, we prove that ${\mathcal{M}}_{\text{Hod}}^{\boldsymbol{\unicode[STIX]{x1D707}}}$ is nonsingular (see [Reference Inaba, Iwasaki and Saito10, Remark 6.1]). It is enough to show $\unicode[STIX]{x1D706}:{\mathcal{M}}_{\text{Hod}}^{\boldsymbol{\unicode[STIX]{x1D707}}}\rightarrow \mathbb{C}$ given by $(\unicode[STIX]{x1D706},E,\unicode[STIX]{x1D6FB},\{l_{\ast }^{(i)}\})\mapsto \unicode[STIX]{x1D706}$ is smooth. In this case, the obstructions of the extensions lie in the hypercohomology $\mathbb{H}^{2}(\unicode[STIX]{x1D6F4},{\mathcal{F}}_{0}^{\bullet ,+}\otimes I)$ . Here, ${\mathcal{F}}_{0}^{\bullet ,+}$ is the complexes of sheaves defined by ${\mathcal{F}}_{0}^{i,+}=0$ for $i\neq 0,1$ , ${\mathcal{F}}_{0}^{0,+}:={\mathcal{F}}_{0}^{0}$ ,

$$\begin{eqnarray}{\mathcal{F}}_{0}^{1,+}:=\left\{s\in {\mathcal{E}}nd(E\otimes A/m)\otimes \unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F4}}^{1}(D)\,\left|\,\begin{array}{@{}l@{}}\text{Tr}(s)=0,\\ \text{Res}_{p_{i}(s)\otimes A/m}(l_{j}^{(i)})_{A/m}\\ \subset (l_{j}^{(i)})_{A/m}\text{ for any}\\ i,j\text{ and the element of }\\ \text{End}((l_{j}^{(i)})_{A/m}/(l_{j+1}^{(i)})_{A/m})~\\ \text{induced by }\text{Res}_{p_{i}(s)\otimes A/m}~\\ \text{is a scalar}.\end{array}\right.\right\},\end{eqnarray}$$

and $d^{+}:{\mathcal{F}}_{0}^{0,+}\rightarrow {\mathcal{F}}_{0}^{1,+}$ maps $s$ to $\unicode[STIX]{x1D6FB}\circ s-s\circ \unicode[STIX]{x1D6FB}$ . We put ${\mathcal{T}}_{0}^{1}={\mathcal{F}}_{0}^{1.+}/{\mathcal{F}}_{0}^{1}$ and ${\mathcal{T}}_{0}^{\bullet }=[0\rightarrow {\mathcal{T}}_{0}^{1}]$ . Then, we have the following exact sequence of the complex on $\unicode[STIX]{x1D6F4}$ :

Note that ${\mathcal{T}}_{0}^{1}$ is a skyscraper sheaf. We consider the long exact sequence. Since $\mathbb{H}^{2}({\mathcal{F}}_{0}^{\bullet })=\mathbb{H}^{2}({\mathcal{T}}_{0}^{\bullet })=0$ , we obtain $\mathbb{H}^{2}({\mathcal{F}}_{0}^{\bullet ,+})=0$ .◻

The following proposition is shown by the same way as in the proofs [Reference Inaba8, Theorem 2.1] and [Reference Inaba and Saito9].

Proof. Since the fiber ${\mathcal{M}}_{\text{Hod}}^{\boldsymbol{\unicode[STIX]{x1D707}}}(\unicode[STIX]{x1D706},\boldsymbol{\unicode[STIX]{x1D709}})$ is smooth, we show that the tangent space $\unicode[STIX]{x1D6E9}_{{\mathcal{M}}_{\text{Hod}}^{\boldsymbol{\unicode[STIX]{x1D707}}}(\unicode[STIX]{x1D706},\boldsymbol{\unicode[STIX]{x1D709}})}(y)$ at $y=(\unicode[STIX]{x1D706},E,\unicode[STIX]{x1D6FB},\{l_{\ast }^{(i)}\}_{1\leqslant i\leqslant k})\in {\mathcal{M}}_{\text{Hod}}^{\boldsymbol{\unicode[STIX]{x1D707}}}(\unicode[STIX]{x1D706},\boldsymbol{\unicode[STIX]{x1D709}})$ is of dimension $n^{2}(2g-2+k)-\sum _{i,j}(\unicode[STIX]{x1D707}_{j}^{i})^{2}+2$ . Put

$$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{F}}^{0}:=\left\{s\in {\mathcal{E}}nd(E)\,\left|\,s_{|_{p_{i}}}(l_{j}^{i})\subset (l_{j}^{i})~\text{for any}~i,j\right.\right\} & \displaystyle \nonumber\\ \displaystyle & \displaystyle {\mathcal{F}}^{1}:=\left\{s\in {\mathcal{E}}nd(E)\otimes \unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F4}}^{1}(D)\,\left|\,\text{Res}_{p_{i}}(s)(l_{j}^{i})\subset (l_{j+1}^{i})~\text{for any}~i,j\right.\right\} & \displaystyle \nonumber\\ \displaystyle & \displaystyle \unicode[STIX]{x1D6FB}^{\dagger }:{\mathcal{F}}^{0}\ni s\mapsto \unicode[STIX]{x1D6FB}\circ s-s\circ \unicode[STIX]{x1D6FB}\in {\mathcal{F}}^{1}. & \displaystyle \nonumber\end{eqnarray}$$

Then we have an isomorphism

$$\begin{eqnarray}\unicode[STIX]{x1D6E9}_{{\mathcal{M}}_{\text{Hod}}^{\boldsymbol{\unicode[STIX]{x1D707}}}(\unicode[STIX]{x1D706},\boldsymbol{\unicode[STIX]{x1D709}})}(y)\cong \mathbb{H}^{1}({\mathcal{F}}^{\bullet }).\end{eqnarray}$$

By the same computation as in the proof of [Reference Inaba8, Theorem 2.1], we have

$$\begin{eqnarray}\dim \mathbb{H}^{1}({\mathcal{F}}^{\bullet })=n^{2}(2g-2+k)-\mathop{\sum }_{i,j}(\unicode[STIX]{x1D707}_{j}^{i})^{2}+2.\square\end{eqnarray}$$

3.3 Mixed Hodge structures of the moduli space ${\mathcal{M}}_{\text{Hod}}^{\boldsymbol{\unicode[STIX]{x1D707}}}(\unicode[STIX]{x1D706},\boldsymbol{\unicode[STIX]{x1D709}})$

We consider the natural $\mathbb{C}^{\times }$ -action on ${\mathcal{M}}_{\text{Hod}}^{\boldsymbol{\unicode[STIX]{x1D707}}}$

$$\begin{eqnarray}\displaystyle \mathbb{C}^{\times }\times {\mathcal{M}}_{\text{Hod}}^{\boldsymbol{\unicode[STIX]{x1D707}}} & \longrightarrow & \displaystyle {\mathcal{M}}_{Hod}^{\boldsymbol{\unicode[STIX]{x1D707}}}\nonumber\\ \displaystyle (t,(\unicode[STIX]{x1D706},E,\unicode[STIX]{x1D6FB},\{l_{\ast }^{(i)}\})) & \longmapsto & \displaystyle (t\unicode[STIX]{x1D706},E,t\unicode[STIX]{x1D6FB},\{l_{\ast }^{(i)}\}).\nonumber\end{eqnarray}$$

Since the relation between $\unicode[STIX]{x1D706}$ and $\boldsymbol{\unicode[STIX]{x1D709}}$ is $\unicode[STIX]{x1D706}d+\sum \unicode[STIX]{x1D707}_{j}^{i}\unicode[STIX]{x1D709}_{j}^{i}=0$ , the following $\mathbb{C}^{\times }$ action on $\unicode[STIX]{x1D6EF}_{n}^{\boldsymbol{\unicode[STIX]{x1D707}},d}$ is well defined,

$$\begin{eqnarray}\displaystyle \mathbb{C}^{\times }\times \unicode[STIX]{x1D6EF}_{n}^{\boldsymbol{\unicode[STIX]{x1D707}},d} & \longrightarrow & \displaystyle \unicode[STIX]{x1D6EF}_{n}^{\boldsymbol{\unicode[STIX]{x1D707}},d}\nonumber\\ \displaystyle (t,(\unicode[STIX]{x1D706},\boldsymbol{\unicode[STIX]{x1D709}})) & \longmapsto & \displaystyle (t\unicode[STIX]{x1D706},t\boldsymbol{\unicode[STIX]{x1D709}}).\nonumber\end{eqnarray}$$

Clearly, $\unicode[STIX]{x1D70B}:{\mathcal{M}}_{\text{Hod}}^{\boldsymbol{\unicode[STIX]{x1D707}}}\rightarrow \unicode[STIX]{x1D6EF}_{n}^{\boldsymbol{\unicode[STIX]{x1D707}},d}$ is a $\mathbb{C}^{\times }$ -equivariant morphism.

For any pairs $(\unicode[STIX]{x1D706},\boldsymbol{\unicode[STIX]{x1D709}})\in \unicode[STIX]{x1D6EF}_{n}^{\boldsymbol{\unicode[STIX]{x1D707}},d}$ where the fiber ${\mathcal{M}}_{Hod}^{\boldsymbol{\unicode[STIX]{x1D707}}}(\unicode[STIX]{x1D706},\boldsymbol{\unicode[STIX]{x1D709}})$ is nonempty, we consider the following subset of $\unicode[STIX]{x1D6EF}_{n}^{\boldsymbol{\unicode[STIX]{x1D707}},d}$

$$\begin{eqnarray}\unicode[STIX]{x1D6EF}_{(\unicode[STIX]{x1D706},\boldsymbol{\unicode[STIX]{x1D709}})}:=\{(t\unicode[STIX]{x1D706},t\boldsymbol{\unicode[STIX]{x1D709}})\mid t\in \mathbb{C}\}\cong \mathbb{C}.\end{eqnarray}$$

Let

(3.3.1) $$\begin{eqnarray}\unicode[STIX]{x1D70B}_{(\unicode[STIX]{x1D706},\boldsymbol{\unicode[STIX]{x1D709}})}:{\mathcal{M}}_{\unicode[STIX]{x1D6EF}_{(\unicode[STIX]{x1D706},\boldsymbol{\unicode[STIX]{x1D709}})}}\longrightarrow \unicode[STIX]{x1D6EF}_{(\unicode[STIX]{x1D706},\boldsymbol{\unicode[STIX]{x1D709}})}\end{eqnarray}$$

be the base change of ${\mathcal{M}}_{Hod}^{\boldsymbol{\unicode[STIX]{x1D707}}}\rightarrow \unicode[STIX]{x1D6EF}_{n}^{\boldsymbol{\unicode[STIX]{x1D707}},d}$ via $\unicode[STIX]{x1D6EF}_{(\unicode[STIX]{x1D706},\boldsymbol{\unicode[STIX]{x1D709}})}{\hookrightarrow}\unicode[STIX]{x1D6EF}_{n}^{\boldsymbol{\unicode[STIX]{x1D707}},d}$ . By the smoothness of the map $\unicode[STIX]{x1D70B}_{(\unicode[STIX]{x1D706},\boldsymbol{\unicode[STIX]{x1D709}})}$ , Lemma 3.12, and [Reference Hausel, Letellier and Rodriguez-Villegas5, Theorem B.1], there are isomorphisms

$$\begin{eqnarray}H^{\bullet }({\mathcal{M}}_{\text{Hod}}^{\boldsymbol{\unicode[STIX]{x1D707}}}(\unicode[STIX]{x1D706},\boldsymbol{\unicode[STIX]{x1D709}}),\mathbb{Q})\cong H^{\bullet }({\mathcal{M}}_{\text{Dol}}^{\boldsymbol{\unicode[STIX]{x1D707}}}(\mathbf{0}),\mathbb{Q})\end{eqnarray}$$

which preserve the mixed Hodge structures and the mixed Hodge structures of $H^{\bullet }({\mathcal{M}}_{\text{Hod}}^{\boldsymbol{\unicode[STIX]{x1D707}}}(\unicode[STIX]{x1D706},\boldsymbol{\unicode[STIX]{x1D709}}),\mathbb{Q})$ are pure. Therefore, we obtain the following theorem:

Here, note that the Poincaré duality is applied for each connected components of fibers of $\unicode[STIX]{x1D70B}:{\mathcal{M}}_{Hod}^{\boldsymbol{\unicode[STIX]{x1D707}}}\rightarrow \unicode[STIX]{x1D6EF}_{n}^{\boldsymbol{\unicode[STIX]{x1D707}},d}$ , which have same dimension (Proposition 3.11).

4.1 Character varieties

We fix integers $g\geqslant 0,k>0$ and $n>0$ . We also fix a $k$ -tuple of partition of $n$ , denoted by $\boldsymbol{\unicode[STIX]{x1D707}}=(\unicode[STIX]{x1D707}^{1},\ldots ,\unicode[STIX]{x1D707}^{k})$ , that is, $\unicode[STIX]{x1D707}^{i}=(\unicode[STIX]{x1D707}_{1}^{i},\ldots ,\unicode[STIX]{x1D707}_{r_{i}}^{i})$ such that $\unicode[STIX]{x1D707}_{1}^{i}\geqslant \unicode[STIX]{x1D707}_{2}^{i}\geqslant \cdots \,$ are nonnegative integers and $\sum _{j}\unicode[STIX]{x1D707}_{j}^{i}=n$ . Let $\unicode[STIX]{x1D6F4}$ be a smooth complex projective curve of genus $g$ . We fix $k$ -distinct points $p_{1},\ldots ,p_{k}$ in $\unicode[STIX]{x1D6F4}$ and we define a divisor by $D:=p_{1}+\cdots +p_{k}$ . We put $\unicode[STIX]{x1D6F4}_{0}=\unicode[STIX]{x1D6F4}\setminus D$ .

We now construct a variety, called a character variety, whose points parametrize representation of the fundamental group of $\unicode[STIX]{x1D6F4}_{0}$ into $\text{GL}(n,\mathbb{C})$ where the images of simple loops at the punctures associated to $p_{1},\ldots ,p_{k}$ are contained in semisimple conjugacy classes ${\mathcal{C}}_{1},\ldots ,{\mathcal{C}}_{k}$ , respectively. Assume that

(4.1.1) $$\begin{eqnarray}\mathop{\prod }_{i=1}^{k}\det {\mathcal{C}}_{i}=1\end{eqnarray}$$

and that $({\mathcal{C}}_{1},\ldots ,{\mathcal{C}}_{k})$ has type $\boldsymbol{\unicode[STIX]{x1D707}}=(\unicode[STIX]{x1D707}^{1},\ldots ,\unicode[STIX]{x1D707}^{k})$ ; that is, ${\mathcal{C}}_{i}$ has type $\unicode[STIX]{x1D707}^{i}$ for each $i=1,\ldots ,k$ , where the type of the semisimple conjugacy class ${\mathcal{C}}_{i}\subset \text{GL}(n,\mathbb{C})$ is defined as the partition $\unicode[STIX]{x1D707}^{i}=(\unicode[STIX]{x1D707}_{1}^{i},\ldots ,\unicode[STIX]{x1D707}_{r_{i}}^{i})$ describing the multiplicities of the eigenvalues of any matrix in ${\mathcal{C}}_{i}$ . Let $\unicode[STIX]{x1D708}^{i}=(\unicode[STIX]{x1D708}_{1}^{i},\ldots ,\unicode[STIX]{x1D708}_{r_{i}}^{i})\in (\mathbb{C}^{\times })^{r_{i}}$ be the eigenvalues of ${\mathcal{C}}_{i}$ . We denote the $k$ -tuple $(\unicode[STIX]{x1D708}^{1},\ldots ,\unicode[STIX]{x1D708}^{k})$ by $\boldsymbol{\unicode[STIX]{x1D708}}$ .

Definition 4.1. The $k$ -tuple $({\mathcal{C}}_{1},\ldots ,{\mathcal{C}}_{k})$ is generic if the following holds. If $V\subset \mathbb{C}^{n}$ is a subset which is stable by some $X_{i}\in {\mathcal{C}}_{i}$ for each $i$ such that

$$\begin{eqnarray}\mathop{\prod }_{i=1}^{k}\det (X_{i}|_{V})=1,\end{eqnarray}$$

then either $V=0$ or $V=\mathbb{C}^{n}$ .

Definition 4.3. For a $k$ -tuple of generic semisimple conjugacy classes $({\mathcal{C}}_{1},\ldots ,{\mathcal{C}}_{k})$ of type $\boldsymbol{\unicode[STIX]{x1D707}}$ , we define a subvariety of $\text{GL}(n,\mathbb{C})^{2g+n}$ by

$$\begin{eqnarray}\displaystyle {\mathcal{U}}^{\boldsymbol{\unicode[STIX]{x1D707}}}(\boldsymbol{\unicode[STIX]{x1D708}}) & := & \displaystyle \big\{(A_{1},B_{1},\ldots ,A_{g},B_{g};X_{1},\ldots ,X_{k})\in \text{GL}(n,\mathbb{C})^{2g}\times {\mathcal{C}}_{1}\times \cdots \times {\mathcal{C}}_{k}\nonumber\\ \displaystyle & & \displaystyle \quad \mid (A_{1},B_{1})\cdots (A_{g},B_{g})X_{1}\cdots X_{k}=I_{n}\big\},\nonumber\end{eqnarray}$$

where $(A,B):=ABA^{-1}B^{-1}$ . The group $\text{GL}(n,\mathbb{C})$ acts by conjugation on $\text{GL}(n,\mathbb{C})^{2g+n}$ . As the center acts trivially, the action induces that of $\text{PGL}(n,\mathbb{C})$ . The action induces that of $\text{PGL}(n,\mathbb{C})$ on ${\mathcal{U}}^{\boldsymbol{\unicode[STIX]{x1D707}}}(\boldsymbol{\unicode[STIX]{x1D708}})$ . We call the affine GIT quotient

$$\begin{eqnarray}{\mathcal{M}}_{\text{B}}^{\boldsymbol{\unicode[STIX]{x1D707}}}(\boldsymbol{\unicode[STIX]{x1D708}}):={\mathcal{U}}^{\boldsymbol{\unicode[STIX]{x1D707}}}(\boldsymbol{\unicode[STIX]{x1D708}})/\!/\text{PGL}(n,\mathbb{C})\end{eqnarray}$$

a generic character variety of type $\boldsymbol{\unicode[STIX]{x1D707}}$ . We denote by $\unicode[STIX]{x1D70B}_{\boldsymbol{\unicode[STIX]{x1D707}}}$ the quotient morphism

(4.1.2) $$\begin{eqnarray}\unicode[STIX]{x1D70B}_{\boldsymbol{\unicode[STIX]{x1D707}}}:{\mathcal{U}}^{\boldsymbol{\unicode[STIX]{x1D707}}}(\boldsymbol{\unicode[STIX]{x1D708}})\longrightarrow {\mathcal{M}}_{\text{B}}^{\boldsymbol{\unicode[STIX]{x1D707}}}(\boldsymbol{\unicode[STIX]{x1D708}}).\end{eqnarray}$$

Theorem 4.5. [Reference Hausel, Letellier and Rodriguez-Villegas5, Theorem 2.1.5]

If $({\mathcal{C}}_{1},\ldots ,{\mathcal{C}}_{k})$ is a generic type $\boldsymbol{\unicode[STIX]{x1D707}}$ , then the quotient

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{\boldsymbol{\unicode[STIX]{x1D707}}}:{\mathcal{U}}^{\boldsymbol{\unicode[STIX]{x1D707}}}(\boldsymbol{\unicode[STIX]{x1D708}})\longrightarrow {\mathcal{M}}_{\text{B}}^{\boldsymbol{\unicode[STIX]{x1D707}}}(\boldsymbol{\unicode[STIX]{x1D708}})\end{eqnarray}$$

is a geometric quotient and a principal $\text{PGL}(n,\mathbb{C})$ -bundle.

Theorem 4.6. [Reference Hausel, Letellier and Rodriguez-Villegas6, Theorem 1.1.1]

If nonempty, the generic character variety ${\mathcal{M}}_{B}^{\boldsymbol{\unicode[STIX]{x1D707}}}(\boldsymbol{\unicode[STIX]{x1D708}})$ is a connected nonsingular variety of dimension

$$\begin{eqnarray}n^{2}(2g-2+k)-\mathop{\sum }_{i,j}(\unicode[STIX]{x1D707}_{j}^{i})^{2}+2.\end{eqnarray}$$

4.2 Riemann–Hilbert correspondence

First, we define a generic tuple of the eigenvalues of the residue matrixes of a parabolic connection corresponding to Definition 4.1 for character varieties cases. We put

(4.2.1) $$\begin{eqnarray}\text{g.c.d.}(\boldsymbol{\unicode[STIX]{x1D707}}):=\text{g.c.d.}(\unicode[STIX]{x1D707}_{1}^{1},\ldots ,\unicode[STIX]{x1D707}_{j}^{i},\ldots ,\unicode[STIX]{x1D707}_{r_{k}}^{k}),\end{eqnarray}$$

where $r_{i}$ implies the number of the distinct eigenvalues of the residue matrix at $p_{i}$ of a parabolic connection. For an integer $d$ , we put

$$\begin{eqnarray}\unicode[STIX]{x1D6EF}_{n,\unicode[STIX]{x1D706}=1}^{\boldsymbol{\unicode[STIX]{x1D707}},d}:=\left\{\boldsymbol{\unicode[STIX]{x1D709}}=(\unicode[STIX]{x1D709}_{j}^{i})_{1\leqslant j\leqslant r_{i}}^{1\leqslant i\leqslant k}\in \mathbb{C}^{r}\,\left|\,d+\mathop{\sum }_{i,j}\unicode[STIX]{x1D707}_{j}^{i}\unicode[STIX]{x1D709}_{j}^{i}=0\right.\right\}\end{eqnarray}$$

where $r:=\sum r_{i}$ .

Let $\unicode[STIX]{x1D6EF}_{n,\unicode[STIX]{x1D706}=1}^{\boldsymbol{\unicode[STIX]{x1D707}},d,irr}$ be the locus of generic elements in $\unicode[STIX]{x1D6EF}_{n,\unicode[STIX]{x1D706}=1}^{\boldsymbol{\unicode[STIX]{x1D707}},d}$ , and let ${\mathcal{M}}_{DR}^{\boldsymbol{\unicode[STIX]{x1D707}},irr}$ be the inverse image of $\unicode[STIX]{x1D6EF}_{n,\unicode[STIX]{x1D706}=1}^{\boldsymbol{\unicode[STIX]{x1D707}},d,irr}$ via ${\mathcal{M}}_{DR}^{\boldsymbol{\unicode[STIX]{x1D707}}}\rightarrow \unicode[STIX]{x1D6EF}_{n,\unicode[STIX]{x1D706}=1}^{\boldsymbol{\unicode[STIX]{x1D707}},d}$ .

Remark 4.8. If $d$ and $\text{g.c.d.}(\boldsymbol{\unicode[STIX]{x1D707}})$ have the greatest common divisor $r^{\prime }\neq 1$ , then $\unicode[STIX]{x1D6EF}_{n,\unicode[STIX]{x1D706}=1}^{\boldsymbol{\unicode[STIX]{x1D707}},d,irr}=\emptyset$ , since

$$\begin{eqnarray}\mathop{\sum }_{i,j}\frac{\unicode[STIX]{x1D707}_{j}^{i}}{r^{\prime }}\unicode[STIX]{x1D709}_{j}^{i}=-\frac{d}{r^{\prime }}\in \mathbb{Z}\end{eqnarray}$$

for any $\boldsymbol{\unicode[STIX]{x1D709}}\in \unicode[STIX]{x1D6EF}_{n,\unicode[STIX]{x1D706}=1}^{\boldsymbol{\unicode[STIX]{x1D707}},d}$ .

Conversely, if $d$ and $\text{g.c.d.}(\boldsymbol{\unicode[STIX]{x1D707}})$ are coprime, then $\unicode[STIX]{x1D6EF}_{n,\unicode[STIX]{x1D706}=1}^{\boldsymbol{\unicode[STIX]{x1D707}},d,irr}$ is nonempty. (See Remark 4.10 as below and the proof of [Reference Hausel, Letellier and Rodriguez-Villegas5, Lemma 2.1.2].)

Next, we construct a family of all generic character varieties of type $\boldsymbol{\unicode[STIX]{x1D707}}$ . We put $r:=\sum r_{i}$ and

$$\begin{eqnarray}N_{n}^{\boldsymbol{\unicode[STIX]{x1D707}}}:=\left\{\boldsymbol{\unicode[STIX]{x1D708}}=(\unicode[STIX]{x1D708}_{j}^{i})_{1\leqslant j\leqslant r_{i}}^{1\leqslant i\leqslant k}\in \mathbb{C}^{r}\,\left|\,\mathop{\prod }_{i,j}{\unicode[STIX]{x1D708}_{j}^{i}}^{\unicode[STIX]{x1D707}_{j}^{i}}=1\right.\right\},\end{eqnarray}$$

which is the set of eigenvalues of $k$ -tuple of semisimple conjugacy classes $({\mathcal{C}}_{1},\ldots ,{\mathcal{C}}_{k})$ . We denote by ${\mathcal{U}}^{\boldsymbol{\unicode[STIX]{x1D707}}}$ the following subvariety of $N_{n}^{\boldsymbol{\unicode[STIX]{x1D707}}}\times \text{GL}(n,\mathbb{C})^{2g+n}$

$$\begin{eqnarray}\left\{(\boldsymbol{\unicode[STIX]{x1D708}},A_{1},B_{1},\ldots ,A_{g},B_{g},X_{1},\ldots ,X_{k})\,\left|\,\begin{array}{@{}l@{}}(1)~(A_{1},B_{1})\cdots (A_{g},B_{g})X_{1}\\ \qquad \cdots X_{k}=I_{n},\\ (2)~\text{For each}~i,\text{there is a filtration}\\ \quad ~~\mathbb{C}^{n}=W_{1}^{i}\supset W_{2}^{i}\supset \cdots \supset W_{r_{i}+1}^{i}\\ \quad ~~=0~\text{such that }\\ \quad ~~\dim W_{j}^{i}/W_{j+1}^{i}=\unicode[STIX]{x1D707}_{j}^{i}\text{ and }\\ \quad ~~(X_{i}-\unicode[STIX]{x1D708}_{j}^{i}\text{id})(W_{j}^{i})\subset W_{j+1}^{i}\\ \quad ~~\text{for any }i,j\end{array}\right.\right\}\end{eqnarray}$$

where $(\boldsymbol{\unicode[STIX]{x1D708}},A_{1},B_{1},\ldots ,A_{g},B_{g},X_{1},\ldots ,X_{k})\in N_{n}^{\boldsymbol{\unicode[STIX]{x1D707}}}\times \text{GL}(n,\mathbb{C})^{2g+n}$ . The group $\text{PGL}(n,\mathbb{C})$ acts on $N_{n}^{\boldsymbol{\unicode[STIX]{x1D707}}}\times \text{GL}(n,\mathbb{C})^{2g+n}$ which is trivial on $N_{n}^{\boldsymbol{\unicode[STIX]{x1D707}}}$ and conjugation on $\text{GL}(n,\mathbb{C})^{2g+n}$ . We take the categorical quotient of ${\mathcal{U}}^{\boldsymbol{\unicode[STIX]{x1D707}}}$ by the $\text{PGL}(n,\mathbb{C})$ -action;

$$\begin{eqnarray}\displaystyle {\mathcal{M}}_{B}^{\boldsymbol{\unicode[STIX]{x1D707}}} & := & \displaystyle {\mathcal{U}}^{\boldsymbol{\unicode[STIX]{x1D707}}}/\!/\text{PGL}(n,\mathbb{C})\nonumber\\ \displaystyle & = & \displaystyle \text{Spec}(\mathbb{C}[{\mathcal{U}}^{\boldsymbol{\unicode[STIX]{x1D707}}}]^{\text{PGL}(n,\mathbb{C})}).\nonumber\end{eqnarray}$$

The map

$$\begin{eqnarray}\displaystyle {\mathcal{M}}_{B}^{\boldsymbol{\unicode[STIX]{x1D707}}} & \longrightarrow & \displaystyle N_{n}^{\boldsymbol{\unicode[STIX]{x1D707}}}\nonumber\\ \displaystyle [(\boldsymbol{\unicode[STIX]{x1D708}},A_{1},B_{1},\ldots ,A_{g},B_{g},X_{1},\ldots ,X_{k})] & \longmapsto & \displaystyle \boldsymbol{\unicode[STIX]{x1D708}}\nonumber\end{eqnarray}$$

is well defined. Let $N_{n}^{\boldsymbol{\unicode[STIX]{x1D707}},irr}\subset N_{n}^{\boldsymbol{\unicode[STIX]{x1D707}}}$ be the set of generic eigenvalues in the sense of Definition 4.1. Then we take the base change of ${\mathcal{M}}_{B}^{\boldsymbol{\unicode[STIX]{x1D707}}}\rightarrow N_{n}^{\boldsymbol{\unicode[STIX]{x1D707}}}$ via the inclusion map $N_{n}^{\boldsymbol{\unicode[STIX]{x1D707}},irr}{\hookrightarrow}N_{n}^{\boldsymbol{\unicode[STIX]{x1D707}}}$ , denoted by

$$\begin{eqnarray}{\mathcal{M}}_{B}^{\boldsymbol{\unicode[STIX]{x1D707}},irr}\longrightarrow N_{n}^{\boldsymbol{\unicode[STIX]{x1D707}},irr},\end{eqnarray}$$

which is a family of any generic character varieties of type $\boldsymbol{\unicode[STIX]{x1D707}}$ . Observe that the fiber over $\boldsymbol{\unicode[STIX]{x1D708}}$ is precisely the space ${\mathcal{M}}_{B}^{\boldsymbol{\unicode[STIX]{x1D707}}}(\boldsymbol{\unicode[STIX]{x1D708}})$ introduced in Definition 4.3.

We define the morphism

$$\begin{eqnarray}rh_{d}:\unicode[STIX]{x1D6EF}_{n,\unicode[STIX]{x1D706}=1}^{\boldsymbol{\unicode[STIX]{x1D707}},d}\ni \boldsymbol{\unicode[STIX]{x1D709}}\longmapsto \boldsymbol{\unicode[STIX]{x1D708}}\in N_{n}^{\boldsymbol{\unicode[STIX]{x1D707}}}\end{eqnarray}$$

by $\unicode[STIX]{x1D708}_{j}^{i}=\text{exp}(-2\unicode[STIX]{x1D70B}\sqrt{-1}\unicode[STIX]{x1D709}_{j}^{i})$ for any $i,j$ .

Remark 4.11. For any $\boldsymbol{\unicode[STIX]{x1D708}}\in N_{n}^{\boldsymbol{\unicode[STIX]{x1D707}},irr}$ , there exist integers $d$ with $0\leqslant d<\text{g.c.d.}(\boldsymbol{\unicode[STIX]{x1D707}})$ such that $\boldsymbol{\unicode[STIX]{x1D708}}$ is contained in the images of the morphisms

$$\begin{eqnarray}rh_{d}:\unicode[STIX]{x1D6EF}_{n,\unicode[STIX]{x1D706}=1}^{\boldsymbol{\unicode[STIX]{x1D707}},d,irr}\longrightarrow N_{n}^{\boldsymbol{\unicode[STIX]{x1D707}},irr},\end{eqnarray}$$

that is, $\boldsymbol{\unicode[STIX]{x1D708}}\in \text{Im}(rh_{d})$ for some $d$ ( $0\leqslant d<\text{g.c.d.}(\boldsymbol{\unicode[STIX]{x1D707}})$ ).

For each member $(E,\unicode[STIX]{x1D6FB},\{l_{j}^{i}\})\in {\mathcal{M}}_{DR}^{\boldsymbol{\unicode[STIX]{x1D707}},irr}$ , $\text{Ker}(\unicode[STIX]{x1D6FB}^{an}|_{\unicode[STIX]{x1D6F4}_{0}})$ becomes a local system on $\unicode[STIX]{x1D6F4}_{0}$ , where $\unicode[STIX]{x1D6FB}^{an}$ means the analytic connection corresponding to $\unicode[STIX]{x1D6FB}$ . The local system $\text{Ker}(\unicode[STIX]{x1D6FB}^{an}|_{\unicode[STIX]{x1D6F4}_{0}})$ corresponds to a representation of $\unicode[STIX]{x1D70B}_{1}(\unicode[STIX]{x1D6F4}_{0})$ . Let $\unicode[STIX]{x1D6FE}_{i}$ be a loop around $p_{i}$ . The representation of $\unicode[STIX]{x1D6FE}_{i}$ is semisimple for $i=1,\ldots ,k$ , and the eigenvalues of the representation of $\unicode[STIX]{x1D6FE}_{i}$ are

$$\begin{eqnarray}\text{exp}(-2\unicode[STIX]{x1D70B}\sqrt{-1}\unicode[STIX]{x1D709}_{1}^{i}),\ldots ,\text{exp}(-2\unicode[STIX]{x1D70B}\sqrt{-1}\unicode[STIX]{x1D709}_{r_{i}}^{i})\end{eqnarray}$$

where the multiplicities are $\unicode[STIX]{x1D707}_{1}^{i},\ldots ,\unicode[STIX]{x1D707}_{r_{i}}^{i}$ , respectively. Then we can define the morphism

$$\begin{eqnarray}\mathbf{RH}_{\boldsymbol{\unicode[STIX]{x1D709}}}:{\mathcal{M}}_{\text{DR}}^{\boldsymbol{\unicode[STIX]{x1D707}}}(\boldsymbol{\unicode[STIX]{x1D709}})\longrightarrow {\mathcal{M}}_{\text{B}}^{\boldsymbol{\unicode[STIX]{x1D707}}}(\boldsymbol{\unicode[STIX]{x1D708}})\end{eqnarray}$$

where $\boldsymbol{\unicode[STIX]{x1D708}}=rh_{d}(\boldsymbol{\unicode[STIX]{x1D709}})$ . Then $\{\mathbf{RH}_{\boldsymbol{\unicode[STIX]{x1D709}}}\}$ induces the morphism

(4.2.2) $$\begin{eqnarray}\mathbf{RH}:{\mathcal{M}}_{\text{DR}}^{\boldsymbol{\unicode[STIX]{x1D707}},irr}\longrightarrow {\mathcal{M}}_{\text{B}}^{\boldsymbol{\unicode[STIX]{x1D707}},irr},\end{eqnarray}$$

which gives the commutative diagram

The following theorem follows from the result of Inaba [Reference Inaba8, Theorem 2.2] and Inaba–Saito [Reference Inaba and Saito9].

Theorem 4.12. (See [Reference Inaba8, Theorem 2.2] and [Reference Inaba and Saito9])

The morphism

$$\begin{eqnarray}\mathbf{RH}_{\boldsymbol{\unicode[STIX]{x1D709}}}:{\mathcal{M}}_{\text{DR}}^{\boldsymbol{\unicode[STIX]{x1D707}}}(\boldsymbol{\unicode[STIX]{x1D709}})\longrightarrow {\mathcal{M}}_{\text{B}}^{\boldsymbol{\unicode[STIX]{x1D707}}}(rh_{d}(\boldsymbol{\unicode[STIX]{x1D709}}))\end{eqnarray}$$

is an analytic isomorphism for any $\boldsymbol{\unicode[STIX]{x1D709}}\in \unicode[STIX]{x1D6EF}_{n,\unicode[STIX]{x1D706}=1}^{\boldsymbol{\unicode[STIX]{x1D707}},d,irr}$ .

Proof. We take any point $\unicode[STIX]{x1D70C}\in {\mathcal{M}}_{\text{B}}^{\boldsymbol{\unicode[STIX]{x1D707}}}(rh_{d}(\boldsymbol{\unicode[STIX]{x1D709}}))$ where $\boldsymbol{\unicode[STIX]{x1D709}}$ is generic. By [Reference Inaba8, Proposition 3.1], we obtain the following isomorphism,

$$\begin{eqnarray}{\mathcal{M}}_{\text{DR}}^{\boldsymbol{\unicode[STIX]{x1D707}}}(\boldsymbol{\unicode[STIX]{x1D709}})\cong {\mathcal{M}}_{\text{DR}}^{\boldsymbol{\unicode[STIX]{x1D707}}}(\boldsymbol{\unicode[STIX]{x1D709}}^{\prime })\end{eqnarray}$$

where $0\leqslant \text{Re}(\unicode[STIX]{x1D709}_{j}^{\prime i})<1$ for any $i,j$ . Hence, we assume that $\boldsymbol{\unicode[STIX]{x1D709}}$ satisfy $0\leqslant \text{Re}(\unicode[STIX]{x1D709}_{j}^{i})<1$ for any $i,j$ . By [Reference Deligne2, II, Proposition 5.4], there is a unique pair $(E,\unicode[STIX]{x1D6FB}_{E})$ where $E$ is a vector bundle on $\unicode[STIX]{x1D6F4}$ and $\unicode[STIX]{x1D6FB}_{E}:E\rightarrow E\otimes \unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6F4}}^{1}(D)$ is a logarithmic connection, such that the local system $\text{Ker}(\unicode[STIX]{x1D6FB}_{E}^{an})|_{\unicode[STIX]{x1D6F4}\setminus \{p_{1},\ldots ,p_{k}\}}$ corresponds to the representation $\unicode[STIX]{x1D70C}$ and all the eigenvalue of $\text{Res}_{p_{i}}(\unicode[STIX]{x1D6FB}_{E})$ lie in $\{z\in \mathbb{C}\mid 0\leqslant \text{Re}(z)<1\}$ . Since $\boldsymbol{\unicode[STIX]{x1D709}}$ is generic, we can define a parabolic structure of $(E,\unicode[STIX]{x1D6FB}_{E})$ , uniquely. Therefore, $\mathbf{RH}_{\boldsymbol{\unicode[STIX]{x1D709}}}$ gives a one to one correspondence between the points of ${\mathcal{M}}_{DR}^{\boldsymbol{\unicode[STIX]{x1D707}},irr}(\boldsymbol{\unicode[STIX]{x1D709}})$ and the points of ${\mathcal{M}}_{B}^{\boldsymbol{\unicode[STIX]{x1D707}},irr}(rh_{d}(\boldsymbol{\unicode[STIX]{x1D709}}))$ . We can define this correspondence between flat families. Hence,

$$\begin{eqnarray}\mathbf{RH}_{\boldsymbol{\unicode[STIX]{x1D709}}}:{\mathcal{M}}_{\text{DR}}^{\boldsymbol{\unicode[STIX]{x1D707}}}(\boldsymbol{\unicode[STIX]{x1D709}})\longrightarrow {\mathcal{M}}_{\text{B}}^{\boldsymbol{\unicode[STIX]{x1D707}}}(rh_{d}(\boldsymbol{\unicode[STIX]{x1D709}}))\end{eqnarray}$$

is an analytic isomorphism. ◻

Proof. We put

$$\begin{eqnarray}N_{d}:=\text{Im}(\unicode[STIX]{x1D6EF}_{n,\unicode[STIX]{x1D706}=1}^{\boldsymbol{\unicode[STIX]{x1D707}},d,irr})\subset N_{n}^{\boldsymbol{\unicode[STIX]{x1D707}},irr}.\end{eqnarray}$$

We can describe $N_{n}^{\boldsymbol{\unicode[STIX]{x1D707}},irr}$ as follows:

$$\begin{eqnarray}N_{n}^{\boldsymbol{\unicode[STIX]{x1D707}},irr}=N_{0}\cup N_{1}\cup \cdots \cup N_{\text{g.c.d.}(\boldsymbol{\unicode[STIX]{x1D707}})-1}\end{eqnarray}$$

(see Remark 4.11).

By Theorems 3.13 and 4.12, for any $\boldsymbol{\unicode[STIX]{x1D708}}_{1},\boldsymbol{\unicode[STIX]{x1D708}}_{2}\in N_{d}\;(d=1,\ldots ,\text{g.c.d.}(\boldsymbol{\unicode[STIX]{x1D707}})-1)$ the Poincaré polynomials of the character varieties ${\mathcal{M}}_{B}^{\boldsymbol{\unicode[STIX]{x1D707}}}(\boldsymbol{\unicode[STIX]{x1D708}}_{\mathbf{1}})$ and ${\mathcal{M}}_{B}^{\boldsymbol{\unicode[STIX]{x1D707}}}(\boldsymbol{\unicode[STIX]{x1D708}}_{\mathbf{2}})$ are same. On the other hand, there is a dense subset in the analytic topology of $N_{n}^{\boldsymbol{\unicode[STIX]{x1D707}},irr}$ such that the Poincaré polynomial is constant [Reference Hausel, Letellier and Rodriguez-Villegas5, Theorem 5.1.1]. Then the Poincaré polynomials is constant in $N_{n}^{\boldsymbol{\unicode[STIX]{x1D707}},irr}$ .◻