2.1.1 Decoupled harmonic bundles

Let $Y$ be a Riemann surface. Let $(V,\theta )$ be a Higgs bundle on $Y$ .

Note that the conditions (A1) and (A2) imply that $\theta$ and $\theta ^{\dagger }_h$ are commuting.

2.1.2 Symmetric Higgs bundles

Let $C$ be a non-degenerate symmetric product of $V$ . It is called a non-degenerate symmetric product of the Higgs bundle $(V,\theta )$ if $\theta$ is self-adjoint with respect to $C$ . Such a tuple $(V,\theta, C)$ is called a symmetric Higgs bundle. Let $V^{\lor }$ denote the dual bundle of $V$ . Let $\Psi _C:V\to V^{\lor }$ be the isomorphism induced by $C$ . Let $\theta ^{\lor }$ be the induced Higgs field of $V^{\lor }$ . The condition is equivalent to $\Psi _C$ inducing an isomorphism of the Higgs bundles $(V,\theta )\simeq (V^{\lor },\theta ^{\lor })$ .

A Hermitian metric $h$ of $V$ is said to be compatible with $C$ if $\Psi _C$ is isometric with respect to $h$ and its dual Hermitian metric $h^{\lor }$ of $V^{\lor }$ .

2.1.3 Generically regular semisimple Higgs bundles

Let $\Sigma _{V,\theta }\subset T^{\ast }Y$ denote the spectral curve of $(V,\theta )$ . We say that $(V,\theta )$ is regular semisimple if the projection $\Sigma _{V,\theta }\to Y$ is a covering map. We say that $(V,\theta )$ is generically regular semisimple if there exists a discrete subset $D\subset Y$ such that $(V,\theta )_{|Y\setminus D}$ is regular semisimple.

Let $\pi :\Sigma _{V,\theta }\to Y$ denote the projection. If $(V,\theta )$ is regular semisimple, there exists a line bundle $L_V$ on $\Sigma _{V,\theta }$ with an isomorphism $\pi _{\ast }L_V\simeq V$ such that $\theta$ is induced by $\mathcal{O}_{T^{\ast }Y}$ -action on $L_V$ .

2.2.1 Decoupled harmonic metrics

Suppose that $(V,\theta )$ is regular semisimple. We consider the following condition for a Hermitian metric $h$ of $V$ .

1. (A3) For any $P\in Y$ , the eigen decomposition of $\theta$ at $P$ is orthogonal with respect to $h$ .

Note that (A3) holds if and only if $\theta$ and $\theta ^{\dagger }_h$ are commuting. The following lemma is easy to see.

A flat metric $h_{L_V}$ of $L_V$ induces a Hermitian metric $\pi _{\ast }(h_{L_V})$ of $V$ . It is easy to check that $\pi _{\ast }(h_{L_V})$ is a decoupled harmonic metric of $(V,\theta )$ .

2.2.2 Symmetric products

The multiplication of $\mathcal{O}_{\Sigma _{V,\theta }}$ induces a multiplication

\begin{align*} \pi _{\ast }\mathcal {O}_{\Sigma _{V,\theta }} \otimes \pi _{\ast }\mathcal {O}_{\Sigma _{V,\theta }} \longrightarrow \pi _{\ast }\mathcal {O}_{\Sigma _{V,\theta }}. \end{align*}

Any local section $f$ of $\pi _{\ast }\mathcal{O}_{\Sigma _{V,\theta }}$ induces an endomorphism $F_f$ of the locally free $\mathcal{O}_Y$ -module $\pi _{\ast }\mathcal{O}_{\Sigma _{V,\theta }}$ . We obtain the local section ${\mathop{\textrm{tr}}\nolimits }(f):={\mathop{\textrm{tr}}\nolimits }(F_f)$ of $\mathcal{O}_Y$ .

Let $C_{L_V}$ be a non-degenerate symmetric pairing of $L_V$ . We obtain the non-degenerate pairing $C$ of $V=\pi _{\ast }L_V$ :

(3) \begin{align} V \otimes _{\mathcal{O}_Y} V \stackrel{\pi _{\ast }C_{L_V}}{\longrightarrow } \pi _{\ast }\mathcal{O}_{\Sigma _{V,\theta }} \stackrel{{\mathop{\textrm{tr}}\nolimits }}{\longrightarrow } \mathcal{O}_Y. \end{align}

We recall the following proposition.

Indeed, let $C_{L_V}$ be the non-degenerate symmetric pairing of $L_V$ corresponding to $C$ . Let $h_{L_V}$ be the unique Hermitian metric of $L_V$ satisfying $h_{L_V}(s,s)=|C_{L_V}(s,s)|$ . We obtain the Hermitian metric $h^C$ corresponding to $h_{L_V}$ . Then, $h^C$ is the decoupled harmonic metric compatible with  $C$ .

As for the converse, the following holds.

3.1.1 Hermitian metrics compatible with a non-degenerate symmetric pairing

Let $V$ be an $r$ -dimensional $\mathbb{C}$ -vector space. The dual space is denoted by $V^{\lor }$ . An $\mathbb{R}$ -structure of $V$ is an $r$ -dimensional $\mathbb{R}$ -subspace $V_{\mathbb{R}}$ such that the natural morphism $\mathbb{C}\otimes _{\mathbb{R}}V_{\mathbb{R}}\longrightarrow V$ is an isomorphism. A positive definite symmetric bilinear form $C_{\mathbb{R}}$ of $V_{\mathbb{R}}$ induces a Hermitian metric $h$ and a non-degenerate symmetric bilinear form $C$ of $V$ by $h(\alpha \otimes u,\beta \otimes v) =\alpha \overline{\beta } C_{\mathbb{R}}(u,v)$ and $C(\alpha \otimes u,\beta \otimes v) =\alpha \beta C_{\mathbb{R}}(u,v)$ for any $\alpha, \beta \in \mathbb{C}$ and $u,v\in V_{\mathbb{R}}$ . An orthogonal decomposition $V_{\mathbb{R}}=\bigoplus V_{\mathbb{R},i}$ with respect to $C_{\mathbb{R}}$ induces a decomposition $V=\bigoplus V_{\mathbb{R},i}\otimes \mathbb{C}$ which is clearly orthogonal with respect to both $h$ and $C$ .

Let $C$ be a non-degenerate symmetric pairing of $V$ . It induces a $\mathbb{C}$ -linear morphism $\Psi _C:V\to V^{\lor }$ . A Hermitian metric $h$ of $V$ is called compatible with $C$ if $\Psi _C$ is isometry between $(V,h)$ and $(V^{\lor },h^{\lor })$ , where $h^{\lor }$ denotes the Hermitian metric of $V^{\lor }$ induced by $h$ . If $h$ is compatible with $C$ , there uniquely exists an $\mathbb{R}$ -structure $V_{\mathbb{R}}$ of $V$ equipped with a positive definite symmetric bilinear form $C_{\mathbb{R}}$ such that (i) $V_{\mathbb{R}}\otimes \mathbb{C}=V$ and (ii) $h$ and $C$ are induced by $C_{\mathbb{R}}$ .

3.1.2 An estimate

Let $C$ be a non-degenerate symmetric form of $V$ . Let $V=\bigoplus _{i=1}^r V_i$ be an orthogonal decomposition with respect to $C$ such that $\dim V_i=1$ . The following lemma is obvious.

For any Hermitian metric $h$ of $V$ compatible with $C$ , let $s(h_0,h)$ be the automorphism of $V$ determined by the condition $h(u,v)=h_0(s(h_0,h)u,v)$ for any $u,v\in V$ . Note that $\det (s(h_0,h))=1$ . Let $\mathcal{H}(C;\epsilon )$ be the set of Hermitian metrics $h$ of $V$ compatible with $C$ such that the following holds for any $u\in V_i$ , $v\in V_j$ $(i\neq j)$ :

(4) \begin{align} |h(u,v)|\leqslant \epsilon |u|_h\cdot |v|_h. \end{align}

Lemma 3.2. There exists $C\gt 0$ , depending only on $r$ , such that the following holds for any $0\leqslant \epsilon \leqslant (2r)^{-1}$ and any $h\in \mathcal{H}(C;\epsilon )$ :

\begin{align*} \bigl |s(h_0,h)-{\mathop {\textrm {id}}\nolimits }_V\bigr |_{h_0} +\bigl |s(h_0,h)^{-1}-{\mathop {\textrm {id}}\nolimits }_V\bigr |_{h_0} \leqslant C\epsilon . \end{align*}

Proof. Let $e_i$ be a base of $V_i$ such that $C(e_i,e_i)=1$ . Note that the tuple $(e_1,\ldots, e_r)$ is an orthonormal base with respect to $h_0$ . Let $H$ be the matrix determined by $H_{i,j}=h(e_i,e_j)$ . Then, the linear map $s(h_0,h)$ is represented by the matrix ${}^{t}\!H$ with respect to the base $(e_1,\ldots, e_r)$ . Because $h$ is compatible with $C$ , ${}^{t}\!H\cdot H$ is the identity matrix. We obtain

(5) \begin{align} H_{i,i}^2-1 =-\sum _{\substack{1\leqslant j\leqslant r \\[5pt] j\neq i}}H_{i,j}H_{j,i}. \end{align}

By condition (4), we have $|H_{i,j}| \leqslant \epsilon H_{i,i}^{1/2}H_{j,j}^{1/2}$ for $i\neq j$ . We obtain

\begin{align*} H_{i,i}^2-1 \leqslant \epsilon \sum _{\substack {1\leqslant j\leqslant r \\[5pt] j\neq i}}H_{i,i}H_{j,j}. \end{align*}

We set $A=\sum _{j=1}^r H_{j,j}$ . We obtain

(6) \begin{align} H_{i,i}^2-1-\epsilon H_{i,i}A\leqslant 0. \end{align}

Lemma 3.3. The following holds: $H_{i,i}\leqslant \epsilon A+1$ .

Proof. Let $a\gt 0$ . Let us consider the $\mathbb{R}$ -valued function $f(s)=s^2-as-1$ $(s\in \mathbb{R})$ . We set $s_{\pm }=2^{-1}(a\pm \sqrt{a^2+4})$ , and then we have $f(s_{\pm })=0$ and $s_-\lt s_+$ . We obtain $f(s)\gt 0$ for any $s\gt s_+$ . Hence, if $f(s)\leqslant 0$ , we obtain

(7) \begin{align} s\leqslant s_+\leqslant 2^{-1}(a+a+2)=a+1. \end{align}

By setting $a=\epsilon A$ , we obtain the claim of the lemma from (6) and (7).

We obtain $A\leqslant \epsilon r A+r$ , and hence $A\leqslant (1-\epsilon r)^{-1}r\leqslant 2r$ . By (4) and (5), we obtain

\begin{align*} |H_{i,i}^2-1| \leqslant \sum _{\substack {1\leqslant j\leqslant r\\[5pt] j\neq i}} |H_{i,j}|\cdot |H_{j,i}| \leqslant \epsilon \sum _{1\leqslant j\leqslant r} H_{i,i}\cdot H_{j,j} \leqslant \epsilon A^2\leqslant 4r^2\epsilon . \end{align*}

Because $H_{i,i}$ are positive numbers, we obtain $\bigl |H_{i,i}-1\bigr |\leqslant 4r^2\epsilon$ . We also obtain $|H_{i,j}|\leqslant \epsilon (1+4r^2\epsilon )$ .

4.2.1 The case $\ell =0$

To simplify the notation we set $s_i=s(h_{0,t(i)},h_{t(i)})$ . By (19), there exist $C'(0),\epsilon '(0)\gt 0$ , depending only on $r$ , $C_1$ and $\epsilon _1$ , such that the following holds on $B(R_0)\setminus B(R_1)$ :

(20) \begin{align} {\mathop{\textrm{Tr}}\nolimits }(s_i-{\mathop{\textrm{id}}\nolimits }_{E}) \leqslant C'(0)\exp (-\epsilon '(0)t(i)). \end{align}

By [Reference SimpsonSim88, Lemma 3.1], we have

(21) \begin{align} -{\partial }_z{\partial }_{{\overline{z}}}{\mathop{\textrm{Tr}}\nolimits }(s_i-{\mathop{\textrm{id}}\nolimits }_E) =-\bigl | \overline{\partial }(s_i) s_i^{-1/2} \bigr |^2_{h_{0,t(i)}} -\bigl | [t(i)\theta, s_i]s_i^{-1/2} \bigr |^2_{h_{0,t(i)}}. \end{align}

In particular, ${\mathop{\textrm{Tr}}\nolimits }\bigl (s_i-{\mathop{\textrm{id}}\nolimits }_E\bigr )$ is a subharmonic function on $B(R_0)$ . By the maximum principle of subharmonic functions, (20) holds on $B(R_0)$ . Because $\det (s_i)=1$ , we obtain the claim in the case $\ell =0$ .

4.2.2 Estimates for $L^2$ -norms

We set $R_2=(R_0+R_1)/2$ and $R_3=(R_0+R_2)/2$ . Let $\chi :\mathbb{C}\to \mathbb{R}_{\geqslant 0}$ be a $C^{\infty }$ -function such that $\chi (z)=1$ $(|z|\leqslant R_2)$ and $\chi (z)=0$ $(|z|\geqslant R_3)$ . Let $g_z=dz\,d{\overline{z}}$ be the standard Euclidean metric. By using [Reference SimpsonSim88, Lemma 3.1], we obtain

(22) \begin{align} &\int _{B(R_2)} \Bigl ( \bigl | \overline{\partial }(s_i) s_i^{-1/2} \bigr |^2_{h_{0,t(i)},g_z} +\bigl | [t(i)\theta, s_i]s_i^{-1/2} \bigr |^2_{h_{0,t(i)},g_z} \Bigr )\mathop{\textrm{dvol}}\nolimits _{g_z} \leqslant \nonumber \\[5pt] &\qquad \qquad \qquad \qquad \int _{B(R_3)\setminus B(R_2)} \bigl |{\partial }_z{\partial }_{{\overline{z}}}(\chi ) \bigr |\cdot \bigl ({\mathop{\textrm{Tr}}\nolimits }(s_i-{\mathop{\textrm{id}}\nolimits }_E) \bigr )\mathop{\textrm{dvol}}\nolimits _{g_z}. \end{align}

Hence, there exist $C_5\gt 0,\epsilon _5\gt 0$ such that

(23) \begin{align} \int _{B(R_2)} \Bigl ( \bigl | \overline{\partial }(s_i) s_i^{-1} \bigr |^2_{h_{0,t(i)},g_z} +\bigl | s_i^{-1}[t(i)\theta, s_i] \bigr |^2_{h_{0,t(i)},g_z} \Bigr )\mathop{\textrm{dvol}}\nolimits _{g_z} \leqslant C_5\exp (-\epsilon _5t(i)). \end{align}

4.2.3 Rescaling

To study the derivatives, for any $t\gt t(1)$ , we define the map $\rho _t:\mathbb{C}_w\to \mathbb{C}_z$ by $\rho _t(w)=t^{-1}w$ . We have $\rho _t^{-1}(B(R))=B(tR)$ . We use the standard Euclidean metric $g_w=dw\,d\overline{w}$ on $\mathbb{C}_w$ .

We set $\widetilde{E}_{t}=\rho _t^{\ast }(E)$ on $B(tR_0)$ . It is equipped with the Higgs field $\widetilde{\theta }_t=\rho _t^{\ast }(t\theta )$ . We have $\widetilde{\theta }_t=\rho _t^{\ast }(f)\,dw$ . We have the harmonic metrics $\widetilde{h}_{0,t}=\varphi _t^{\ast }(h_{0,t})$ of the Higgs bundles $(\widetilde{E}_t,\overline{\partial }_{\widetilde{E}_t},\widetilde{\theta }_t)$ . Let $\widetilde{\nabla }^{0,t}$ denote the Chern connection of $(\widetilde{E}_t,\overline{\partial }_{\widetilde{E}_t},\widetilde{h}_{0,t})$ .

By Simpson’s main estimate, there exists $C_{10}\gt 0$ , depending only on $r$ and $C_0$ , such that the following holds on $B(tR_0-1)$ :

(24) \begin{align} \bigl | \widetilde{\theta }_{t} \bigr |_{\widetilde{h}_{0,t},g_w} \leqslant C_{10}. \end{align}

Let $R(\widetilde{h}_{0,t})$ denote the curvature of the Chern connection of $(\widetilde{E}_{t},\overline{\partial }_{\widetilde{E}_{t}}, \widetilde{h}_{0,t})$ . We have the following equality:

(25) \begin{align} R(\widetilde{h}_{0,t}) +\bigl [ \widetilde{\theta }_{t}, (\widetilde{\theta }_{t})^{\dagger }_{\widetilde{h}_{0,t}} \bigr ] =0. \end{align}

By (24) and (25), we have the following on $B(tR_0-1)$ :

(26) \begin{align} \bigl | R(\widetilde{h}_{0,t}) \bigr |_{\widetilde{h}_{0,t},g_w} \leqslant 2C_{10}^2. \end{align}

We also have the universal estimates for higher derivatives of $\widetilde{\theta }$ and $R(\widetilde{h}_{0,t})$ as in Proposition 4.1.

4.2.4 Estimates for higher derivatives

We also have the harmonic metrics $\widetilde{h}_{t(i)}=\varphi _{t(i)}^{\ast }(h_{t(i)})$ of $(\widetilde{E}_{t(i)},\overline{\partial }_{\widetilde{E}_{t(i)}},\widetilde{\theta }_{t(i)})$ . Let $\widetilde{s}_i=\varphi _{t(i)}^{\ast }(s_i)$ . We have $\widetilde{h}_{t(i)}=\widetilde{h}_{0,t(i)}\widetilde{s}_i$ . By (23), we have

(27) \begin{align} \int _{B(t(i)R_2)} \Bigl ( \bigl | \overline{\partial }(\widetilde{s}_i) \widetilde{s}_i^{-1} \bigr |^2_{\widetilde{h}_{0,t(i)},g_w} +\bigl | \widetilde{s}_i^{-1} [\widetilde{\theta }_{t(i)},\widetilde{s}_i] \bigr |^2_{\widetilde{h}_{0,t(i)},g_w} \Bigr )\mathop{\textrm{dvol}}\nolimits _{g_w} \leqslant C_5\exp (-\epsilon _5t(i)). \end{align}

This implies

(28) \begin{align} \int _{B(t(i)R_2)} \bigl | \widetilde{s}_i^{-1}{\partial }_{\widetilde{h}_{0,t(i)}}(\widetilde{s}_i) \bigr |^2_{\widetilde{h}_{0,t(i)},g_w}\mathop{\textrm{dvol}}\nolimits _{g_w} \leqslant C_5\exp (-\epsilon _5t(i)). \end{align}

Let $R(\widetilde{h}_{t(i)})$ denote the curvature of the Chern connection of $(\widetilde{E}_{t(i)},\overline{\partial }_{\widetilde{E}_{t(i)}},\widetilde{h}_{t(i)})$ . We have

\begin{align*} R(\widetilde {h}_{t(i)}) +\bigl [ \widetilde {\theta }_{t(i)}, (\widetilde {\theta }_{t(i)})^{\dagger }_{\widetilde {h}_{t(i)}} \bigr ] =0. \end{align*}

Note that

\begin{align*} (\widetilde {\theta }_{t(i)})^{\dagger }_{\widetilde {h}_{t(i)}} =\widetilde {s}_{t(i)}^{-1} (\widetilde {\theta }_{t(i)})^{\dagger }_{\widetilde {h}_{0,t(i)}} \widetilde {s}_{t(i)}. \end{align*}

We obtain

(29) \begin{align} \overline{\partial }\bigl ( \widetilde{s}_i^{-1}{\partial }_{\widetilde{h}_{0,t(i)}} \widetilde{s}_i \bigr ) &=R(\widetilde{h}_{t(i)})- R(\widetilde{h}_{0,t(i)}) =-\Bigl [ \widetilde{\theta }_{t(i)}, \widetilde{s}_i^{-1} (\widetilde{\theta }_{t(i)})^{\dagger }_{\widetilde{h}_{0,t(i)}} \widetilde{s}_i -(\widetilde{\theta }_{t(i)})^{\dagger }_{\widetilde{h}_{0,t(i)}} \Bigr ] \nonumber \\[5pt] & = -\Bigl [ \widetilde{\theta }_{t(i)}, \widetilde{s}_i^{-1} \bigl [ (\widetilde{\theta }_{t(i)})^{\dagger }_{\widetilde{h}_{0,t(i)}}, \widetilde{s}_i-{\mathop{\textrm{id}}\nolimits } \bigr ] \Bigr ]. \end{align}

Hence, there exist $C_{11}\gt 0$ and $\epsilon _{11}\gt 0$ such that the following holds on $B(t(i)R_0-1)$ :

(30) \begin{align} \bigl | \overline{\partial }\bigl ( \widetilde{s}_i^{-1}{\partial }_{\widetilde{h}_{0,t(i)}}\widetilde{s}_i \bigr ) \bigr |_{\widetilde{h}_{0,t(i)},g_w} \leqslant C_{11}\exp (-\epsilon _{11}t(i)). \end{align}

For any $w_0\in \mathbb{C}_w$ , we set $D(w_0,T)=\{|w-w_0|\lt T\}$ . By (26), (28) and (30), for any $p\geqslant 2$ , there exist $C_{12}(p)\gt 0,\epsilon _{12}(p)\gt 0$ such that the following holds for any $w_0\in B(t(i)R_2-1)$ :

(31) \begin{align} \bigl | \widetilde{s}_i^{-1}{\partial }_{\widetilde{h}_{0,t(i)}} \widetilde{s}_i \bigr |_{L_1^p(D(w_0,2/3)),\widetilde{h}_{0,t(i)},g_w} \leqslant C_{12}(p)\exp \bigl (-\epsilon _{12}(p)t(i)\bigr ). \end{align}

By (31) and the estimate in the case $\ell =0$ , for any $p\gt 1$ , there exist $C_{13}(p)\gt 0,\epsilon _{13}(p)\gt 0$ such that the following holds for any $w_0\in B(t(i)R_2-1)$ :

(32) \begin{align} \bigl | \widetilde{s}_i-{\mathop{\textrm{id}}\nolimits } \bigr |_{L_2^p(D(w_0,2/3)),\widetilde{h}_{0,t(i)},g_w} \leqslant C_{13}(p)\exp \bigl (-\epsilon _{13}(p)t(i)\bigr ). \end{align}

By a standard bootstrapping argument, for any $p\gt 1$ and $\ell \in \mathbb{Z}_{\geqslant 2}$ , there exist $C_{14}(\ell, p)\gt 0,\epsilon _{14}(\ell, p)\gt 0$ such that the following holds for any $w_0\in B(t(i)R_2-1)$ :

(33) \begin{align} \bigl | \widetilde{s}_i-{\mathop{\textrm{id}}\nolimits } \bigr |_{L_{\ell }^p(D(w_0,1/2)),\widetilde{h}_{0,t(i)},g_w} \leqslant C_{14}(\ell, p)\exp \bigl (-\epsilon _{14}(\ell, p)t(i)\bigr ). \end{align}

Then, we obtain the claim of Theorem4.3.

5.1.1 Vector bundles

Let $U\subset \mathbb{C}$ be a simply connected open neighbourhood of $0$ . We set $U^{\ast }=U\setminus \{0\}$ . Let $\iota :U^{\ast }\to U$ denote the inclusion. Let $V$ be a locally free $\mathcal{O}_{U^{\ast }}$ -module of rank $r$ . We obtain a locally free $\iota _{\ast }\mathcal{O}_{U^{\ast }}$ -module $\iota _{\ast }(V)$ . A meromorphic (respectively smooth) extension of $V$ is defined to be a locally free $\mathcal{O}_{U}(\ast 0)$ -submodule (respectively $\mathcal{O}_U$ -submodule) $\mathcal{V}\subset \iota _{\ast }(V)$ such that $\mathcal{V}_{|U^{\ast }}=V$ . A filtered extension of $V$ is defined to be a meromorphic extension $\mathcal{V}$ equipped with a filtered bundle $\mathcal{P}_{\ast }(\mathcal{V})$ over $\mathcal{V}$ .

For a positive integer $\ell$ , let $\varphi _{\ell }:\mathbb{C}\to \mathbb{C}$ be defined by $\varphi _{\ell }(\zeta )=\zeta ^{\ell }$ . We set $U^{(\ell )}=\varphi _{\ell }^{-1}(U)$ and $U^{(\ell )\ast }=U^{(\ell )}\setminus \{0\}$ . The induced morphisms $U^{(\ell )}\to U$ and $U^{(\ell )\ast }\to U^{\ast }$ are also denoted by $\varphi _{\ell }$ . Let $\mathop{\textrm{Gal}}\nolimits (\ell )$ denote the Galois group of the ramified covering $\varphi _{\ell }$ . Namely, we put $\mathop{\textrm{Gal}}\nolimits (\ell )=\{a\in \mathbb{C}^{\ast }\,|\,a^{\ell }=1\}$ , and we consider the action of $\mathop{\textrm{Gal}}\nolimits (\ell )$ on $U^{(\ell )}$ by the multiplication on the coordinate $\zeta$ . Let $\iota ^{(\ell )}:U^{(\ell )\ast }\to U^{(\ell )}$ denote the inclusion. We set $V^{(\ell )}:=\varphi _{\ell }^{\ast }(V)$ , which is naturally $\mathop{\textrm{Gal}}\nolimits (\ell )$ -equivariant. The $(\iota ^{(\ell )})_{\ast }\mathcal{O}_{U^{(\ell )\ast }}$ -module $(\iota ^{(\ell )})_{\ast }(V^{(\ell )})$ is also $\mathop{\textrm{Gal}}\nolimits (\ell )$ -equivariant. A $\mathop{\textrm{Gal}}\nolimits (\ell )$ -equivariant meromorphic extension of $V^{(\ell )}$ is defined to be a locally free $\mathcal{O}_{U^{(\ell )}}(\ast 0)$ -submodule $\mathcal{V}^{(\ell )}\subset \iota ^{(\ell )}_{\ast }(V^{(\ell )})$ which is preserved by the $\mathop{\textrm{Gal}}\nolimits (\ell )$ -action. A $\mathop{\textrm{Gal}}\nolimits (\ell )$ -equivariant filtered extension of $V^{(\ell )}$ is defined to be a filtered bundle $\mathcal{P}_{\ast }(\mathcal{V}^{(\ell )})$ over a $\mathop{\textrm{Gal}}\nolimits (\ell )$ -equivariant meromorphic extension $\mathcal{V}^{(\ell )}$ of $V^{(\ell )}$ such that each $\mathcal{P}_{a}\mathcal{V}^{(\ell )}$ is preserved by the $\mathop{\textrm{Gal}}\nolimits (\ell )$ -action.

A meromorphic extension $\mathcal{V}$ of $V$ induces a $\mathop{\textrm{Gal}}\nolimits (\ell )$ -equivariant meromorphic extension $\varphi _{\ell }^{\ast }(\mathcal{V})$ of $V^{(\ell )}$ . Conversely, for any $\mathop{\textrm{Gal}}\nolimits (\ell )$ -equivariant meromorphic extension $\mathcal{V}^{(\ell )}$ of $V^{(\ell )}$ , we obtain the $\mathcal{O}_{U}(\ast 0)$ -module $\varphi _{\ell \ast }(\mathcal{V}^{(\ell )})$ equipped with the $\mathop{\textrm{Gal}}\nolimits (\ell )$ -action. The $\mathop{\textrm{Gal}}\nolimits (\ell )$ -invariant part $\varphi _{\ell \ast }(\mathcal{V}^{(\ell )})^{\mathop{\textrm{Gal}}\nolimits (\ell )}$ is called the descent of $\mathcal{V}^{(\ell )}$ which is a meromorphic extension of $V$ .

For a filtered extension $\mathcal{P}_{\ast }\mathcal{V}$ of $V$ , we obtain a $\mathop{\textrm{Gal}}\nolimits (\ell )$ -equivariant filtered extension $\mathcal{P}_{\ast }(\varphi _{\ell }^{\ast }(\mathcal{V}))$ over $\mathcal{V}^{(\ell )}$ as follows:

\begin{align*} \mathcal {P}_a(\varphi _{\ell }^{\ast }\mathcal {V}) =\sum _{\substack {b\in \mathbb {R}, k\in \mathbb {Z}\\[5pt] \ell b+k\leqslant a}} \zeta ^{-k} \varphi _{\ell }^{\ast }(\mathcal {P}_{b}\mathcal {V}) \subset \varphi _{\ell }^{\ast }(\mathcal {V}). \end{align*}

The filtered bundle $\mathcal{P}_{\ast }(\varphi _{\ell }^{\ast }(\mathcal{V}))$ is denoted by $\varphi _{\ell }^{\ast }(\mathcal{P}_{\ast }\mathcal{V})$ .

For a $\mathop{\textrm{Gal}}\nolimits (\ell )$ -equivariant filtered extension $\mathcal{P}_{\ast }(\mathcal{V}^{(\ell )})$ of $V^{(\ell )}$ , we obtain a filtered extension $\mathcal{P}_{\ast }(\varphi _{\ell \ast }(\mathcal{V}^{(\ell )})^{\mathop{\textrm{Gal}}\nolimits (\ell )})$ as follows:

\begin{align*} \mathcal {P}_{a}(\varphi _{\ell \ast }(\mathcal {V}^{(\ell )})^{\mathop {\textrm {Gal}}\nolimits (\ell )}) =\varphi _{\ell \ast }(\mathcal {P}_{\ell a}\mathcal {V}^{(\ell )})^{\mathop {\textrm {Gal}}\nolimits (\ell )}. \end{align*}

It is called the descent of $\mathcal{P}_{\ast }(\mathcal{V}^{(\ell )})$ .

5.1.2 Non-degenerate symmetric pairing

For any $b\in \mathbb{R}$ , let $\mathcal{P}^{(b)}_{\ast }(\mathcal{O}_U(\ast 0))$ denote the filtered bundle over $\mathcal{O}_U(\ast 0)$ defined by

\begin{align*} \mathcal {P}^{(b)}_a(\mathcal {O}_U(\ast 0)) =z^{-[a-b]}\mathcal {O}_U. \end{align*}

Here, we set $[c]:=\max \{n\in \mathbb{Z}\,|\,n\leqslant c\}$ for any $c\in \mathbb{R}$ .

Let $C:V\otimes V\to \mathcal{O}_{U^{\ast }}$ be a holomorphic non-degenerate symmetric pairing. We say that a meromorphic extension $\mathcal{V}$ is compatible with $C$ if $C$ extends to a pairing $\mathcal{V}\otimes \mathcal{V}\to \mathcal{O}_{U}(\ast 0)$ . We say that a filtered extension $\mathcal{P}_{\ast }\mathcal{V}$ is compatible with $C$ if $C$ induces $\mathcal{P}_{\ast }\mathcal{V}\otimes \mathcal{P}_{\ast }\mathcal{V} \to \mathcal{P}^{(0)}_{\ast }(\mathcal{O}_U(\ast 0))$ . We say that $C$ is perfect with respect to $\mathcal{P}_{\ast }\mathcal{V}$ if $C$ induces an isomorphism $\mathcal{P}_{\ast }(\mathcal{V})\simeq \mathcal{P}_{\ast }(\mathcal{V}^{\lor })$ .

We have the induced symmetric pairing $\det (C)$ of $\det (V)$ . If $\mathcal{V}$ (respectively $\mathcal{P}_{\ast }\mathcal{V}$ ) is compatible with $C$ , then $\det (\mathcal{V})$ (respectively $\det (\mathcal{P}_{\ast }\mathcal{V})$ ) is compatible with $\det (C)$ .

Proof. We may assume that $U$ is a disc. Let $v_0$ be a frame of $\det (V)$ on $U^{\ast }$ . We obtain a holomorphic function $(\det C)(v_0,v_0)$ on $U^{\ast }$ . There exist an integer $k$ and a holomorphic function $g_1$ such that $(\det C)(v_0,v_0)=z^{-k}\exp (g_1)$ . We obtain a frame $v_1=\exp (-g_1/2)v_0$ of $\det (V)$ on $U^{\ast }$ . We set $\mathcal{L}=\mathcal{O}_{U}(\ast 0)v_1 \subset \iota _{\ast }(\det V)$ . Then, $\mathcal{L}$ is compatible with $\det (C)$ .

We have $\det (C)(v_1,v_1)=z^{-k}$ . We define

\begin{align*} \mathcal {P}^C_{a}(\mathcal {L}) =z^{-[a-k/2]}\mathcal {O}_U\cdot v_1. \end{align*}

Then, $\mathcal{P}^C_{\ast }\mathcal{L}$ satisfies the desired condition. The uniqueness is clear.

We set $C^{(\ell )}:=\varphi _{\ell }^{\ast }C$ which is a non-degenerate symmetric pairing of $V^{(\ell )}$ .

5.1.3 Higgs bundles

Let $\theta$ be a Higgs field of $V$ , i.e. $\theta :V\to V\otimes \Omega ^1_{U^{\ast }}$ . We obtain $\iota _{\ast }(\theta ):\iota _{\ast }(V)\to \iota _{\ast }(V)\otimes \Omega ^1_{U}$ . A meromorphic (respectively smooth) extension of $(V,\theta )$ is defined to be a meromorphic (respectively smooth) extension $\mathcal{V}$ of $V$ such that $\iota _{\ast }(\theta )(\mathcal{V})\subset \mathcal{V}\otimes \Omega ^1_U$ . The induced Higgs field of $\mathcal{V}$ is denoted by $\theta$ . A filtered extension of $(V,\theta )$ is defined to be a filtered extension $\mathcal{P}_{\ast }(\mathcal{V})$ over a meromorphic extension $\mathcal{V}$ of $(V,\theta )$ . A filtered extension $(\mathcal{P}_{\ast }\mathcal{V},\theta )$ is said to be regular (respectively good, unramifiedly good) if $(\mathcal{P}_{\ast }\mathcal{V},\theta )$ is a regular (respectively good, unramifiedly good) filtered Higgs bundle. (See [Reference MochizukiMoc21, §2.4] for the notion of good filtered Higgs bundles and unramifiedly good filtered Higgs bundles.)

We obtain the Higgs field $\theta ^{(\ell )}$ of $V^{(\ell )}$ . The following lemmas are clear.

5.2.1 Decomposable filtered extensions

We continue to use the notation in §5.1.1. Let $(V,\theta )$ be a regular semisimple Higgs bundle on $U^{\ast }$ . Assume that $\theta$ is wild. There exist $\ell \in \mathbb{Z}_{\gt 0}$ and the decomposition

(34) \begin{align} \varphi _{\ell }^{\ast }(V,\theta ) =\bigoplus _{i=1}^r (V_i,\theta _i), \end{align}

where $\mathop{\textrm{rank}}\nolimits V_i=1$ , and $\theta _i-\theta _j$ $(i\neq j)$ are nowhere vanishing on $U^{(\ell )\ast }$ . Let $\mathcal{V}$ be a meromorphic extension of $(V,\theta )$ . The decomposition (34) extends to

(35) \begin{align} \varphi _{\ell }^{\ast }(\mathcal{V},\theta ) =\bigoplus _{i=1}^r (\mathcal{V}_i,\theta _i), \end{align}

where each $\mathcal{V}_i$ is a meromorphic extension of $V_i$ .

Definition 5.10. A filtered bundle $\mathcal{P}_{\ast }\mathcal{V}$ over $\mathcal{V}$ is called a decomposable filtered extension of $(V,\theta )$ if the filtered bundle $\varphi _{\ell }^{\ast }(\mathcal{P}_{\ast }\mathcal{V})$ is compatible with the decomposition (35), i.e. the following holds for any $a\in \mathbb{R}$ :

\begin{align*} \mathcal {P}_{a}(\varphi _{\ell }^{\ast }\mathcal {V}) =\bigoplus _{i=1}^r \mathcal {P}_a(\varphi _{\ell }^{\ast }\mathcal {V}) \cap \mathcal {V}_{i}. \end{align*}

Such a $(\mathcal{P}_{\ast }\mathcal{V},\theta )$ is called a decomposable filtered Higgs bundle.

The following lemma is obvious by definition.

5.2.2 Filtered line bundles and decomposable filtered Higgs bundles

There exists the decomposition

(36) \begin{align} (V,\theta )=\bigoplus _{k\in S}(V^{[k]},\theta ^{[k]}), \end{align}

such that $\Sigma _{V^{[k]},\theta ^{[k]}}$ are connected. We set $r_k=\mathop{\textrm{rank}}\nolimits V^{[k]}$ . For each $k$ , there exists the decomposition of the Higgs bundle

(37) \begin{align} \varphi _{r_k}^{\ast }(V^{[k]},\theta ^{[k]}) =\bigoplus _{i=1}^{r_k} (V^{[k]}_i,\theta ^{[k]}_i), \end{align}

where $\mathop{\textrm{rank}}\nolimits V^{[k]}_i=1$ , and $\theta ^{[k]}_i$ are $1$ -forms such that $\theta ^{[k]}_i-\theta ^{[k]}_j$ $(i\neq j)$ are nowhere vanishing on $U^{(r_k)\ast }$ . A decomposable filtered extension $\mathcal{P}_{\ast }\mathcal{V}$ of $(V,\theta )$ induces filtered extensions $\mathcal{P}_{\ast }(\mathcal{V}^{[k]}_i)$ of $(V^{[k]}_i,\theta ^{[k]}_i)$ . Note that $\mathcal{P}_{\ast }(\mathcal{V}^{[k]}_i) =\sigma ^{\ast }\mathcal{P}_{\ast }(\mathcal{V}^{[k]}_1)$ for $\sigma \in \mathop{\textrm{Gal}}\nolimits (r_k)$ such that $\sigma ^{\ast }\theta ^{[k]}_1=\theta ^{[k]}_i$ . Conversely, a filtered extension $\mathcal{P}_{\ast }\mathcal{V}^{[k]}_1$ of $V^{[k]}_1$ induces a $\mathop{\textrm{Gal}}\nolimits (r_k)$ -equivariant filtered extension $\bigoplus _{\sigma \in \mathop{\textrm{Gal}}\nolimits (r_k)} \sigma ^{\ast }\mathcal{P}_{\ast }\mathcal{V}^{[k]}_1$ of $\varphi _{r_k}^{\ast }(V^{[k]})=\bigoplus _{i=1}^{r_k} V^{[k]}_i$ , and hence a decomposable filtered extension $\mathcal{P}_{\ast }\mathcal{V}^{[k]}$ of $(V^{[k]},\theta ^{[k]})$ . Thus, we obtain a decomposable filtered extension $\bigoplus _{k\in S}\mathcal{P}_{\ast }\mathcal{V}^{[k]}$ of $(V,\theta )$ . Note that $\mathcal{P}_{\ast }\mathcal{V}^{[k]}$ is also obtained as $(\varphi _{r_k})_{\ast }(\mathcal{P}_{\ast }\mathcal{V}^{[k]}_1)$ by the natural identification $(\varphi _{r_k})_{\ast }(V^{[k]}_1)=V^{[k]}$ . The following proposition is easy to see.

5.2.3 Decomposable filtered extension determined by determinant bundles

Let $\mathcal{V}$ be a meromorphic extension of $(V,\theta )$ . The decomposition (36) extends to a decomposition

(38) \begin{align} (\mathcal{V},\theta ) =\bigoplus _{k\in S} (\mathcal{V}^{[k]},\theta ^{[k]}). \end{align}

The decomposition (35) extends to a decomposition

(39) \begin{align} \varphi _{r_k}^{\ast }(\mathcal{V}^{[k]},\theta ^{[k]}) =\bigoplus _{i=1}^{r_k} (\mathcal{V}^{[k]}_i,\theta ^{[k]}_i). \end{align}