${\mathcal{E}}^{v}_{U}\otimes{\mathcal{O}}_{U[v]}/(\epsilon)=\tilde{E}|_{U}$.
Define the ${\mathcal{O}}_{U[v]}$-homomorphism

$\tilde{N}\colon\ {\mathcal{E}}^{v}_{U}\longrightarrow{\mathcal{E}}^{v}_{U}$

by the same matrix (10.3) as $N$. Then
$\big{(}{\mathcal{E}}^{v}_{U},\tilde{N}\big{)}$ becomes a lift of
$\big{(}\tilde{E}|_{U},N\big{)}$. Define matrices $\tilde{A}(z)$, $B(z)$,
$C(z)$ by setting

$\displaystyle\tilde{A}(z):=\sum_{k=0}^{r-1}\sum_{l=0}^{m-1}a_{k,l}z^{l}\tilde{N}^{k}+z^{m-1}\tilde{R}_{r}+z^{2m-1}\tilde{A}^{\prime}(z),$
$\displaystyle
B(z):=\sum_{k=0}^{r-1}\sum_{l=0}^{m-2}\frac{rb_{k,l}}{(-mr+lr+r+k)z^{m-l-1}}\tilde{N}^{k},$
$\displaystyle C(z)\frac{{\rm d}z}{z^{m}}:={\rm d}B(z)+[A(z),B(z)]\frac{{\rm
d}z}{z^{m}},$

where $\tilde{R}_{r}$ is the endomorphisms of ${\mathcal{E}}^{v}_{U}$ whose
representation matrix with respect to the basis $e_{0},\dots,e_{r-1}$ is the
same as that of $R_{r}$ in (10.5) and $\tilde{A}^{\prime}(z)$ is a lift of
$A^{\prime}(z)$ such that
$\frac{\partial\tilde{A}^{\prime}(z)}{\partial\epsilon}=0$. Using the
calculations

$\tilde{N}^{k}=\begin{pmatrix}0&\cdots&0&z&\cdots&0\\\
\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\ 0&\cdots&0&0&\cdots&z\\\
1&\cdots&0&0&\cdots&0\\\ \vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\
0&\cdots&1&0&\cdots&0\end{pmatrix},\qquad\big{[}\tilde{R}_{r},\tilde{N}^{k}\big{]}=\begin{pmatrix}0&\cdots&0&-\frac{r-k}{r}z&\cdots&0\\\
\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\
0&\cdots&0&0&\cdots&-\frac{r-k}{r}z\\\ \frac{k}{r}&\cdots&0&0&\cdots&0\\\
\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\
0&\cdots&\frac{k}{r}&0&\cdots&0\end{pmatrix},$

we can check the equality

${\rm d}\tilde{N}^{k}+\left[\tilde{R}_{r}\frac{{\rm
d}z}{z},\tilde{N}^{k}\right]=\frac{k}{r}\tilde{N}^{k}\frac{{\rm d}z}{z}.$
(10.6)

Then we can see

$\displaystyle\left({\rm d}B(z)+[A(z),B(z)]\frac{{\rm
d}z}{z^{m}}\right)\Big{|}_{(2m-1)\tilde{x}}$
$\displaystyle=\sum_{k=0}^{r-1}\sum_{l=0}^{m-2}\frac{rb_{k,l}}{-mr+lr+r+k}\left({\rm
d}\left(\frac{1}{z^{m-l-1}}\tilde{N}^{k}\right)+\left[\tilde{R}_{r}\frac{{\rm
d}z}{z},\frac{1}{z^{m-l-1}}\tilde{N}^{k}\right]\right)\Big{|}_{(2m-1)\tilde{x}}$
$\displaystyle\qquad{}=\sum_{k=0}^{r-1}\sum_{l=0}^{m_{i}-2}\frac{rb_{k,l}}{-mr+lr+r+k}\left(\frac{-m+l+1}{z^{m-l}}\tilde{N}^{k}{\rm
d}z+\frac{1}{z^{m-l-1}}\frac{k}{r}\tilde{N}^{k}\frac{{\rm
d}z}{z}\right)\Big{|}_{(2m-1)\tilde{x}}$
$\displaystyle\qquad{}=\sum_{k=0}^{r-1}\sum_{l=0}^{m-2}\frac{r(-m+l+1)+k}{-mr+lr+r+k}\frac{b_{k,l}}{z^{m-l}}\tilde{N}^{k}{\rm
d}z\bigg{|}_{(2m-1)\tilde{x}}=\nu_{v}(\tilde{N})\big{|}_{(2m-1)\tilde{x}}.$

So the matrix

$\big{(}\tilde{A}(z)+\epsilon C(z)\big{)}\frac{{\rm d}z}{z^{m}}+B(z){\rm
d}\epsilon$

determines an integrable connection
$\nabla^{\mathrm{flat}}_{U[v]}\colon{\mathcal{O}}_{U[v]}^{\oplus
r}\longrightarrow{\mathcal{O}}_{U[v]}^{\oplus
r}\otimes_{{\mathcal{O}}_{U[v]}}\tilde{\Omega}_{v}$ such that the induced
relative connection
$\overline{\nabla^{\mathrm{flat}}_{U[v]}}\colon{\mathcal{O}}_{U[v]}^{\oplus
r}\longrightarrow{\mathcal{O}}_{U[v]}^{\oplus
r}\otimes\Omega^{1}_{U[v]/{\mathcal{T}^{\prime}}[v]}({\mathcal{D}}_{\mathcal{T}^{\prime}}\cap
U)$ satisfies

$\overline{\nabla^{\mathrm{flat}}_{U[v]}}\big{|}_{(2m-1)\tilde{x}}=I_{v}^{*}\nu\big{(}\tilde{N}\big{)}\big{|}_{(2m-1)\tilde{x}}.$

We can give a filtration ${\mathcal{O}}_{U[v]}^{\oplus
r}\big{|}_{{\mathcal{D}}_{\tilde{M}^{\prime}[v]}}=V^{v}_{U,0}\supset
V^{v}_{U,1}\supset\cdots\supset V^{v}_{U,r-1}\supset V^{v}_{U,r}=zV^{v}_{U,0}$
by setting $V^{v}_{U,k}:=\mathop{\rm
Im}\nolimits\big{(}\tilde{N}^{k}\big{|}_{{\mathcal{D}}_{\tilde{M}^{\prime}[v]}}\big{)}$
for $k=0,1,\dots,r$. So we can see that $\\{V^{v}_{U,k}\\}$ induces
$\overline{V}^{v}_{U,k}$, $\overline{W}^{v}_{U,k}$ and that the homomorphism
$\overline{\tilde{N}|_{V^{v}_{U,k}}}\colon\overline{V}^{v}_{U,k}\longrightarrow\overline{V}^{v}_{U,k}$
induced by the restriction $\tilde{N}|_{V^{v}_{U,k}}$ has a factorization

$\overline{\tilde{N}|_{V^{v}_{U,k}}}\colon\
\overline{V}^{v}_{U,k}\xrightarrow{\kappa^{v}_{U,k}}\overline{W}^{v}_{U,k}\xrightarrow[\sim]{\theta^{v}_{U,k}}\overline{V}^{v}_{U,k}.$

Then $\big{(}V^{v}_{U,k},\theta^{v}_{U,k},\kappa^{v}_{U,k}\big{)}$ induces a
factorized ramified structure
$\big{(}V^{v}_{U,k},\vartheta^{v}_{U,k},\varkappa^{v}_{U,k}\big{)}$ on
$\big{(}{\mathcal{E}}^{v}_{U},\overline{\nabla^{\mathrm{flat}}_{U[v]}}\big{)}$,
where $\overline{\nabla^{\mathrm{flat}}_{U[v]}}$ is the relative connection
induced by $\nabla^{\mathrm{flat}}_{U[v]}$. Thus
$\big{(}{\mathcal{E}}^{v}_{U},,\nabla^{\mathrm{flat}}_{U[v]},\big{\\{}V^{v}_{U,k},\vartheta^{v}_{U,k},\varkappa^{v}_{U,k}\big{\\}}\big{)}$
becomes a local horizontal lift of
$\big{(}\tilde{E},\tilde{\nabla},\\{\tilde{V}_{k},\tilde{\vartheta}_{k},\tilde{\varkappa}_{k}\\}\big{)}\big{|}_{U}$.
∎

######  Lemma 10.6 (uniqueness of ramified irregular singular local horizontal
lift).

Under the same assumption as Lemma 10.5, a local horizontal lift
$\big{(}{\mathcal{E}}^{v}_{U},\nabla^{v}_{U},{\mathcal{V}}^{v}_{U}\big{)}$ of
$\big{(}\tilde{E}|_{U},\tilde{\nabla}|_{U},{\mathcal{V}}|_{U}\big{)}$ with
respect to $v$ is unique up to an isomorphism.

###### Proof.

Let
$\big{(}{\mathcal{E}}^{v}_{U},\nabla^{\mathrm{flat}}_{U[v]},\big{\\{}V^{v}_{U,k},\vartheta^{v}_{U,k},\varkappa^{v}_{U,k}\big{\\}}\big{)}$
be the local horizontal lift constructed in Lemma 10.5. Take another local
horizontal lift $\big{(}{\mathcal{O}}_{U_{\alpha}[v]}^{\oplus
r},\nabla^{\prime},\big{\\{}V^{\prime}_{k},\vartheta^{\prime}_{k},\varkappa^{\prime}_{k}\big{\\}}\big{)}$
of
$\big{(}\tilde{E},\tilde{\nabla},\big{\\{}\tilde{V}_{k},\tilde{\vartheta}_{k},\tilde{\varkappa}_{k}\big{\\}}\big{)}\big{|}_{U}$.
The connection $\nabla^{\prime}\colon{\mathcal{O}}_{U[v]}^{\oplus
r}\longrightarrow{\mathcal{O}}_{U[v]}^{\oplus
r}\otimes\Omega^{1}_{U[v]/M^{\prime}}(m(\tilde{x})_{M^{\prime}})$ can be given
by

$\nabla^{\prime}\begin{pmatrix}f_{1}\\\ \vdots\\\
f_{r}\end{pmatrix}=\begin{pmatrix}{\rm d}f_{1}\\\ \vdots\\\ {\rm
d}f_{r}\end{pmatrix}+\left(\big{(}\tilde{A}(z)+\epsilon
C^{\prime}(z)\big{)}\frac{{\rm d}z}{z^{m}}+B^{\prime}(z){\rm
d}\epsilon\right)\begin{pmatrix}f_{1}\\\ \vdots\\\ f_{r}\end{pmatrix}$

with $B^{\prime}(z)$ a rational function on $U$ admitting a pole at $z=0$ of
order at most $m-1$. Note that $\nabla^{\prime}$ satisfies the integrability
condition

$C^{\prime}(z)\frac{{\rm d}z}{z^{m}}={\rm
d}B^{\prime}(z)+[A(z),B^{\prime}(z)]\frac{{\rm d}z}{z^{m}}.$ (10.7)

Now we apply Proposition 4.1 in the case $q=2m-1$ to the relative connection
$\overline{\nabla^{\prime}}$ on ${\mathcal{O}}_{U[v]}^{\oplus r}$ induced by
$\nabla^{\prime}$. Then, after applying an automorphism of
${\mathcal{O}}_{U[v]}^{\oplus r}$ of the form $I_{r}+\epsilon h$, we may
assume that

$C^{\prime}(z)\frac{{\rm
d}z}{z^{m}}\Big{|}_{(2m-1)\tilde{x}}=\nu_{v}\big{(}\tilde{N}\big{)}\big{|}_{(2m-1)\tilde{x}},$
(10.8)

$V^{\prime}_{k}=\mathop{\rm
Im}\nolimits\big{(}\tilde{N}^{k}|_{m\tilde{x}}\big{)}$ and that
$\theta^{\prime}_{k}\circ\kappa^{\prime}_{k}$ is induced by the restriction
$\tilde{N}|_{V^{\prime}_{k}}$ for $0\leq k\leq r-1$.

By the equality (10.7), we have $\big{[}A(z),z^{m-1}B^{\prime}(z)\big{]}\equiv
0\pmod{z^{m-1}}$. Note that $A(z)$ satisfies the equality (10.4) with
$a_{1,0}\in{\mathcal{O}}_{\mathcal{T}}^{\times}$ and $B^{\prime}(z)\in
M_{r}({\mathcal{O}}_{U}({\mathcal{D}}^{\prime}_{\tilde{M}^{\prime}}\cap U))$
by the condition (ii) of Definition 10.1. So we can find
$c_{0}(z),\dots,c_{r-1}(z)\in{\mathcal{O}}_{U[v]}$ satisfying

$z^{m-1}B^{\prime}(z)\equiv\sum_{k=0}^{r-1}c_{k}(z)\tilde{N}^{k}\quad\big{(}{\rm
mod}\ z^{m-1}\mathop{\rm
End}\nolimits\big{(}{\mathcal{O}}_{U_{\alpha}[v]}^{\oplus r}\big{)}\big{)}$

since the equality
$\ker\big{(}\mathrm{ad}\big{(}\tilde{N}|_{z=0}\big{)}\big{)}={\mathcal{O}}_{U[v]}\big{[}\tilde{N}|_{z=0}\big{]}$
holds. Then we can write

$B^{\prime}(z)=\sum_{k=0}^{r-1}\frac{c_{k}(z)}{z^{m-1}}\tilde{N}^{k}+B_{m}(z).$

with $B_{m}(z)$ a matrix of regular functions. Furthermore, we can see that
$B_{m}(0)$ is a lower triangular matrix, since $\nabla^{v}={\rm
d}+\big{(}\tilde{A}(z)+\epsilon C(z)\big{)}\frac{{\rm
d}z}{z^{m}}+B^{\prime}(z){\rm d}\epsilon$ preserves the filtration
$(V^{v}_{k})$. Looking at the equality (10.7) again, we can see that

$\displaystyle C^{\prime}(z){\rm d}z-z^{m}\left({\rm
d}B^{\prime}(z)+[R_{r},B^{\prime}(z)]\frac{{\rm d}z}{z}\right)$
$\displaystyle\qquad{}=\big{[}\tilde{A}(z)-z^{m-1}R_{r},B^{\prime}(z)\big{]}{\rm
d}z$
$\displaystyle\qquad{}=\left[\sum_{k=0}^{r-1}\sum_{l=0}^{m-1}a_{k,l}z^{l}\tilde{N}^{k}+z^{2m-1}\tilde{A}^{\prime},\sum_{k=0}^{r-1}\frac{c_{k}(z)}{z^{m-1}}\tilde{N}^{k}+B_{m}(z)\right]{\rm
d}z$
$\displaystyle\qquad{}\equiv\sum_{k=0}^{r-1}\left[z^{m}c_{k}(z)\tilde{A}^{\prime}(z)-\sum_{l=0}^{m-1}a_{k,l}z^{l}B_{m}(z),\tilde{N}^{k}\right]{\rm
d}z\in\mathrm{ad}\big{(}\tilde{N}\big{)}{\rm d}z\quad\big{(}{\rm mod}\
z^{2m-1}{\rm d}z\big{)}.$

In particular, we have

$\mathop{\rm Tr}\nolimits\left(\tilde{N}^{l}\left(C^{\prime}(z){\rm
d}z-z^{m}\left({\rm d}B^{\prime}(z)+[R_{r},B^{\prime}(z)]\frac{{\rm
d}z}{z}\right)\right)\right)\equiv 0\quad\big{(}{\rm mod}\ z^{2m-1}{\rm
d}z\big{)}$ (10.9)

for $0\leq l\leq r-1$. Since

$\displaystyle z^{m}\left({\rm d}B^{\prime}(z)+[R_{r},B^{\prime}(z)]\frac{{\rm
d}z}{z}\right)$ $\displaystyle\qquad{}\equiv\sum_{k=0}^{r-1}z^{m}\left({\rm
d}\left(\frac{c_{k}(z)}{z^{m-1}}\right)+\frac{kc_{k}(z)}{rz^{m}}{\rm
d}z\right)\tilde{N}^{k}+z^{m-1}[R_{r},B_{m}(z)]{\rm d}z\quad\big{(}{\rm mod}\
z^{m}{\rm d}z\big{)}$

and since $[R_{r},B_{m}]|_{z=0}$ is lower triangular nilpotent matrix, we can
see that the condition (10.9) implies

$\displaystyle z^{m}\left({\rm
d}\left(\frac{c_{0}(z)}{z^{m-1}}\right)-\nu_{0,v}(z)\right)\equiv
0\pmod{z^{m}},$ $\displaystyle z^{m+1}\left({\rm
d}\left(\frac{c_{r-l}(z)}{z^{m-1}}\right)+\frac{(r-l)c_{r-l}(z)}{rz^{m}}{\rm
d}z-\nu_{r-l,v}(z)\right)\equiv 0\pmod{z^{m}}\quad(1\leq l\leq r-1).$

In other words, we have

${\rm
d}\left(\frac{c_{0}(z)}{z^{m-1}}\right)\Big{|}_{m\tilde{x}}=\nu_{0,v}(z)\big{|}_{m\tilde{x}},\qquad\left({\rm
d}\left(\frac{c_{k}(z)}{z^{m-1}}\right)+\frac{kc_{k}(z)}{rz^{m}}{\rm
d}z\right)\Big{|}_{(m-1)\tilde{x}}=\nu_{k,v}(z)\big{|}_{(m-1)\tilde{x}}$

for $1\leq k\leq r-1$, which implies that

$c_{k}(z)\equiv\sum_{l=0}^{m-2}\frac{rb_{k,l}}{-mr+lr+r+k}z^{l}\quad\big{(}{\rm
mod}\ z^{m-1}\big{)}.$

Thus

$Q(z):=B(z)-B^{\prime}(z)$

becomes a matrix of regular functions. Furthermore, (10.7) and (10.8) implies
the equality

${\rm d}Q(z)+[A(z),Q(z)]\frac{{\rm d}z}{z^{m}}\equiv 0\quad\big{(}{\rm mod}\
z^{2m-1}{\rm d}z/z^{m}\big{)},$

from which we can see
$Q(z)|_{m\tilde{x}}\in{\mathcal{O}}_{m\tilde{x}}\big{[}\tilde{N}\big{]}$. If
we apply the transform $I_{r}+\epsilon Q(z)$ to the connection
$\nabla^{\prime}$, then the consequent connection has the matrix form

$\displaystyle(I_{r}\\!+\epsilon Q(z))^{-1}{\rm d}(I_{r}+\epsilon
Q(z))+(I_{r}\\!+\epsilon Q(z))^{-1}\\!\left(\\!\big{(}\tilde{A}(z)+\epsilon
C^{\prime}(z)\big{)}\frac{{\rm d}z}{z^{m}}\\!+B^{\prime}(z){\rm
d}\epsilon\\!\right)\\!(I_{r}\\!+\epsilon Q(z))$
$\displaystyle\qquad{}=(I_{r}-\epsilon Q(z))(\epsilon{\rm d}Q(z)+Q(z){\rm
d}\epsilon)+\tilde{A}(z)\frac{{\rm
d}z}{z^{m}}+\epsilon([A(z),Q(z)]+C^{\prime}(z))\frac{{\rm
d}z}{z^{m}}+B^{\prime}(z){\rm d}\epsilon$
$\displaystyle\qquad{}=\tilde{A}(z)\frac{{\rm d}z}{z^{m}}+\epsilon\left({\rm
d}B(z)-{\rm
d}B^{\prime}(z)+\big{(}[A(z),B(z)-B^{\prime}(z)]+C^{\prime}(z)\big{)}\frac{{\rm
d}z}{z^{m}}\right)$ $\displaystyle\qquad\quad{}+(Q(z)+B^{\prime}(z)){\rm
d}\epsilon$ $\displaystyle\qquad{}=\tilde{A}(z)\frac{{\rm
d}z}{z^{m}}+\epsilon\left({\rm d}B(z)+[A(z),B(z)]\frac{{\rm d}z}{z^{m}}-{\rm
d}B^{\prime}(z)-[A(z),B^{\prime}(z)]\frac{{\rm
d}z}{z^{m}}+C^{\prime}(z)\frac{{\rm d}z}{z^{m}}\right)$
$\displaystyle\qquad\quad{}+B(z)){\rm d}\epsilon$
$\displaystyle\qquad{}=\big{(}\tilde{A}(z)+\epsilon C(z)\big{)}\frac{{\rm
d}z}{z^{m}}+B(z){\rm d}\epsilon,$

which means that $\big{(}{\mathcal{O}}_{U_{\alpha}[v]}^{\oplus
r},\nabla^{\prime}\big{)}$ is isomorphic to
$\big{(}{\mathcal{O}}_{U_{\alpha}[v]}^{\oplus
r},\nabla^{\mathrm{flat}}_{U_{\alpha}[v]}\big{)}$ via $I_{r}+\epsilon Q(z)$.
Since $Q(z)|_{m\tilde{x}}$ belongs to
${\mathcal{O}}_{m\tilde{x}}[\tilde{N}|_{m\tilde{x}}]$, we can see that
$I_{r}+\epsilon Q(z)$ induces an isomorphism which transforms
$\big{(}{\mathcal{O}}_{U[v]}^{\oplus
r},\nabla^{\prime},\\{V^{\prime}_{k},\vartheta^{\prime}_{k},\varkappa^{\prime}_{k}\\}\big{)}$
to $\big{(}{\mathcal{O}}_{U[v]}^{\oplus
r},\nabla^{\mathrm{flat}}_{U[v]},\big{\\{}V^{v}_{k},\vartheta^{v}_{k},\varkappa^{v}_{k}\big{\\}}\big{)}$.
We can see that such an isomorphism is unique, because it is determined by the
coefficient of $d\epsilon$. ∎

######  Proposition 10.7.

For any vector field $v\in
H^{0}({\mathcal{T}^{\prime}},T_{\mathcal{T}^{\prime}})$, there is a unique
global horizontal lift
$\big{(}{\mathcal{E}}^{v},\nabla^{v},l^{v},\ell^{v},{\mathcal{V}}^{v}\big{)}$
of
$\big{(}\tilde{E},\tilde{\nabla},\tilde{l},\tilde{\ell},\tilde{\mathcal{V}}\big{)}_{\tilde{M}^{\prime}}$.

###### Proof.

We take an affine open covering
${\mathcal{C}}_{\tilde{M}^{\prime}}=\bigcup_{\alpha}U_{\alpha}$ such that
$\tilde{E}|_{U_{\alpha}}\cong{\mathcal{O}}_{U_{\alpha}}^{\oplus r}$ for each
$\alpha$. We may assume that $\\#\big{\\{}\alpha\mid
U_{\alpha}\supset\tilde{x}\big{\\}}=1$ for each
$\tilde{x}=\big{(}\tilde{x}^{\mathrm{log}}_{i}\big{)}_{\tilde{M}^{\prime}}$,
$\tilde{x}=\big{(}\tilde{x}^{\mathrm{un}}_{i}\big{)}_{\tilde{M}^{\prime}}$ and
$\tilde{x}=\big{(}\tilde{x}^{\mathrm{ram}}_{i}\big{)}_{\tilde{M}^{\prime}}$.
We may further assume that, for each $\alpha$,
$U_{\alpha}\cap{\mathcal{D}}_{\tilde{M}^{\prime}}=\varnothing$ holds or
$U_{\alpha}\cap{\mathcal{D}}_{\tilde{M}^{\prime}}=\tilde{x}$ holds for some
$\tilde{x}=\big{(}\tilde{x}^{\mathrm{log}}_{i}\big{)}_{\tilde{M}^{\prime}}$,
$\tilde{x}=\big{(}\tilde{x}^{\mathrm{un}}_{i}\big{)}_{\tilde{M}^{\prime}}$ or
$\tilde{x}=\big{(}\tilde{x}^{\mathrm{ram}}_{i}\big{)}_{\tilde{M}^{\prime}}$.

Let $U_{\alpha}[v]$ be the open subscheme of
${\mathcal{C}}_{\tilde{M}^{\prime}[v]}$ whose underlying set is $U_{\alpha}$.
If $U_{\alpha}\cap{\mathcal{D}}_{\tilde{M}^{\prime}}=\varnothing$, then we can
write $\tilde{\nabla}|_{U_{\alpha}}={\rm d}+A_{\alpha}(z){\rm d}z$ for a
matrix $A_{\alpha}$ with values in ${\mathcal{O}}_{U_{\alpha}}$. We can take a
matrix $\tilde{A}_{\alpha}$ with values in ${\mathcal{O}}_{U_{\alpha}}[v]$
which is a lift of $A_{\alpha}$. After adding an element of $\epsilon
M_{r}({\mathcal{O}}_{U_{\alpha}})$, we can assume that
$\partial\tilde{A}_{\alpha}/\partial\epsilon=0$. Then $\nabla_{\alpha}={\rm
d}+\tilde{A}_{\alpha}{\rm d}z$ is an integrable connection and
$\big{(}{\mathcal{O}}_{U_{\alpha}[v]}^{\oplus r},\nabla_{\alpha}\big{)}$ is a
local horizontal lift of
$\big{(}\tilde{E}|_{U_{\alpha}},\tilde{\nabla}|_{U_{\alpha}}\big{)}$.
Furthermore, we can prove the uniqueness of the local horizontal lift by the
same proof as Lemma 10.3.

If $\alpha$ satisfies
$U_{\alpha}\cap{\mathcal{D}}_{\tilde{M}^{\prime}}=\tilde{x}$ for some
$\tilde{x}=\big{(}\tilde{x}^{\mathrm{log}}_{i}\big{)}_{\tilde{M}^{\prime}}$,
$\tilde{x}=\big{(}\tilde{x}^{\mathrm{un}}_{i}\big{)}_{\tilde{M}^{\prime}}$ or
$\tilde{x}=\big{(}\tilde{x}^{\mathrm{ram}}_{i}\big{)}_{\tilde{M}^{\prime}}$,
we can take a local horizontal lift
$\big{(}{\mathcal{E}}^{v}_{U_{\alpha}},\nabla^{v}_{U_{\alpha}},l^{v}_{U_{\alpha}},\ell^{v}_{U_{\alpha}},{\mathcal{V}}^{v}_{U_{\alpha}}\big{)}$
of
$\big{(}\tilde{E}|_{U_{\alpha}},\tilde{\nabla}|_{U_{\alpha}},\tilde{l}|_{U_{\alpha}},\tilde{\ell}|_{U_{\alpha}},\tilde{\mathcal{V}}|_{U_{\alpha}}\big{)}$
by Lemmas 10.3, 10.4 and 10.5. Since the local horizontal lifts are unique up
to unique isomorphisms, we can patch them and get a global horizontal lift
$\big{(}{\mathcal{E}}^{v},\nabla^{v},l^{v},\ell^{v},{\mathcal{V}}^{v}\big{)}$
of
$\big{(}\tilde{E},\tilde{\nabla},\tilde{l},\tilde{\ell},\tilde{\mathcal{V}}\big{)}_{\tilde{M}^{\prime}}$,
which is unique up to an isomorphism. ∎

For a Zariski open subset ${\mathcal{T}^{\prime}}\subset{\mathcal{T}}$,
consider a morphism

$u\colon\ \mathop{\rm
Spec}\nolimits{\mathcal{O}}_{\mathcal{T}^{\prime}}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{2}^{2}\big{)}\longrightarrow{\mathcal{T}^{\prime}}$

such that $u|_{\mathcal{T}^{\prime}}=\mathrm{id}_{\mathcal{T}^{\prime}}$. Let

$\bar{u}\colon\ \mathop{\rm
Spec}\nolimits{\mathcal{O}}_{\mathcal{T}^{\prime}}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{1}\epsilon_{2},\epsilon_{2}^{2}\big{)}\longrightarrow{\mathcal{T}^{\prime}}$

be the induced morphism which corresponds to a pair $(u_{1},u_{2})$ of vector
fields. We write

${\mathcal{T}^{\prime}}[\bar{u}]:={\mathcal{T}}^{\prime}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{1}\epsilon_{2},\epsilon_{2}^{2}\big{)},\qquad{\mathcal{T}^{\prime}}[u]:={\mathcal{T}}^{\prime}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{2}^{2}\big{)}$

with the structure morphisms
${\mathcal{T}^{\prime}}[\bar{u}]\xrightarrow{\bar{u}}{\mathcal{T}^{\prime}}$
and ${\mathcal{T}^{\prime}}[u]\xrightarrow{u}{\mathcal{T}^{\prime}}$,
respectively. We further set

$\displaystyle\tilde{M^{\prime}}[\bar{u}]:=\tilde{M}^{\prime}\times_{\mathcal{T}^{\prime}}{\mathcal{T}^{\prime}}[\bar{u}],\qquad$
$\displaystyle{\mathcal{C}}_{{\mathcal{M}^{\prime}}[\bar{u}]}:={\mathcal{C}}\times_{\mathcal{T}^{\prime}}{\mathcal{M}^{\prime}}[\bar{u}],$
$\displaystyle\tilde{M^{\prime}}[u]:=\tilde{M}^{\prime}\times_{\mathcal{T}^{\prime}}{\mathcal{T}^{\prime}}[u],\qquad$
$\displaystyle{\mathcal{C}}_{{\mathcal{M}^{\prime}}[u]}:={\mathcal{C}}\times_{\mathcal{T}^{\prime}}{\mathcal{M}^{\prime}}[u].$

We define a coherent subsheaf $\tilde{\Omega}_{u}$ of
$\Omega^{1}_{{\mathcal{C}}_{\tilde{M}^{\prime}[u]}/\tilde{M}^{\prime}}\big{(}{\mathcal{D}}_{\tilde{M^{\prime}}[u]}\big{)}$
in the same way as in (10.1) and define a coherent subsheaf
$\tilde{\Omega}_{\bar{u}}$ of
$\Omega^{1}_{{\mathcal{C}}_{\tilde{M}^{\prime}[\bar{u}]}/\tilde{M}^{\prime}}\big{(}{\mathcal{D}}_{\tilde{M^{\prime}}[\bar{u}]}\big{)}$
similarly.

######  Definition 10.8.

We say that
$\big{(}{\mathcal{E}}^{u},\nabla^{u},l^{u},\ell^{u},{\mathcal{V}}^{u}\big{)}$
(resp.
$\big{(}{\mathcal{E}}^{\bar{u}},\nabla^{\bar{u}},l^{\bar{u}},\ell^{\bar{u}},{\mathcal{V}}^{\bar{u}}\big{)}$)
is a horizontal lift of
$\big{(}\tilde{E},\tilde{\nabla},\tilde{l},\tilde{\ell},\tilde{\mathcal{V}}\big{)}_{\tilde{M}^{\prime}}$
with respect to $u$ (resp. $\bar{u}$) if the conditions (i), (ii), (iii), (iv)
and (v) of Definition 10.1 are satisfied after replacing
$\tilde{M}^{\prime}[v]$ with $\tilde{M^{\prime}}[u]$ (resp.
$\tilde{M^{\prime}}[\bar{u}]$), replacing $\tilde{\Omega}_{v}$ with
$\tilde{\Omega}_{u}$ (resp. $\tilde{\Omega}_{\bar{u}}$), replacing
$(\lambda,I_{v}^{*}\tilde{\mu},I_{v}^{*}\tilde{\nu})$-structure in (vi) with
$(\lambda,I_{u}^{*}\tilde{\mu},I_{u}^{*}\tilde{\nu})$-structure (resp.
$(\lambda,I_{\bar{u}}^{*}\tilde{\mu},I_{\bar{u}}^{*}\tilde{\nu})$-structure)
and replacing the equality of integrability condition in (iii) with

$\displaystyle\frac{\partial A}{\partial\epsilon_{1}}{\rm d}z\wedge{\rm
d}\epsilon_{1}+\frac{\partial A}{\partial\epsilon_{2}}{\rm d}z\wedge{\rm
d}\epsilon_{2}+\frac{\partial B_{1}}{\partial\epsilon_{2}}{\rm
d}\epsilon_{2}\wedge{\rm d}\epsilon_{1}+\frac{\partial
B_{2}}{\partial\epsilon_{1}}{\rm d}\epsilon_{1}\wedge{\rm d}\epsilon_{2}$
$\displaystyle\qquad{}={\rm d}B_{1}\wedge{\rm d}\epsilon_{1}+[A,B_{1}]{\rm
d}z\wedge{\rm d}\epsilon_{1}+{\rm d}B_{2}\wedge{\rm
d}\epsilon_{1}+[A,B_{2}]{\rm d}z\wedge d\epsilon_{2}+[B_{1},B_{2}]{\rm
d}\epsilon_{1}\wedge{\rm d}\epsilon_{2}$

for $\Gamma^{u}=A{\rm d}z+B_{1}{\rm d}\epsilon_{1}+B_{2}{\rm d}\epsilon_{2}$
(resp. replacing with

$\displaystyle\frac{\partial A}{\partial\epsilon_{1}}{\rm d}z\wedge{\rm
d}\epsilon_{1}+\frac{\partial A}{\partial\epsilon_{2}}{\rm d}z\wedge{\rm
d}\epsilon_{2}={\rm d}B_{1}\wedge d\epsilon_{1}+[A,B_{1}]{\rm d}z\wedge{\rm
d}\epsilon_{1}+{\rm d}B_{2}\wedge{\rm d}\epsilon_{1}+[A,B_{2}]{\rm
d}z\wedge{\rm d}\epsilon_{2}$

for $\Gamma^{\bar{u}}=Adz+B_{1}{\rm d}\epsilon_{1}+B_{2}{\rm d}\epsilon_{2}$).

The following proposition can be proved in the same way as Proposition 10.7.
So we omit its proof.

######  Proposition 10.9.

There exists a unique horizontal lift
$\big{(}{\mathcal{E}}^{\bar{u}},\nabla^{\bar{u}},l^{\bar{u}},\ell^{\bar{u}},{\mathcal{V}}^{\bar{u}}\big{)}$
of
$\big{(}\tilde{E},\tilde{\nabla},\tilde{l},\tilde{\ell},\tilde{\mathcal{V}}\big{)}_{\tilde{M}^{\prime}}$
with respect to $\bar{u}\colon{\mathcal{T}^{\prime}}[\bar{u}]=\mathop{\rm
Spec}\nolimits{\mathcal{O}}_{\mathcal{T}^{\prime}}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{1}\epsilon_{2},\epsilon_{2}^{2}\big{)}\longrightarrow{\mathcal{T}^{\prime}}$.

If a horizontal lift
$\big{(}{\mathcal{E}}^{u},\nabla^{u},l^{u},\ell^{u},{\mathcal{V}}^{u}\big{)}$
of
$\big{(}\tilde{E},\tilde{\nabla},\tilde{l},\tilde{\ell},\tilde{\mathcal{V}}\big{)}_{\tilde{M}^{\prime}}$
with respect to $u$ exists, it can be obtained as a lift of
$\big{(}{\mathcal{E}}^{\bar{u}},\nabla^{\bar{u}},l^{\bar{u}},\ell^{\bar{u}},{\mathcal{V}}^{\bar{u}}\big{)}$
whose existence is ensured by Proposition 10.9.

######  Proposition 10.10.

There exists a unique horizontal lift
$\big{(}{\mathcal{E}}^{u},\nabla^{u},l^{u},\ell^{u},{\mathcal{V}}^{u}\big{)}$
of
$\big{(}\tilde{E},\tilde{\nabla},\tilde{l},\tilde{\ell},\tilde{\mathcal{V}}\big{)}_{\tilde{M}^{\prime}}$
with respect to $u\colon{\mathcal{T}^{\prime}}[u]=\mathop{\rm
Spec}\nolimits{\mathcal{O}}_{\mathcal{T}^{\prime}}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{2}^{2}\big{)}\longrightarrow{\mathcal{T}^{\prime}}$.

###### Proof.

By Proposition 10.9, there is a unique horizontal lift
$\big{(}{\mathcal{E}}^{\bar{u}}\\!,\nabla^{\bar{u}}\\!,l^{\bar{u}}\\!,\ell^{\bar{u}}\\!,{\mathcal{V}}^{\bar{u}}\big{)}$
of
$\big{(}\tilde{E},\tilde{\nabla},\tilde{l},\tilde{\ell},\tilde{\mathcal{V}}\big{)}_{\tilde{M}^{\prime}}$
with respect to $\bar{u}\colon\mathop{\rm
Spec}\nolimits{\mathcal{O}}_{\mathcal{T}^{\prime}}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{1}\epsilon_{2},\epsilon_{2}^{2}\big{)}\rightarrow{\mathcal{T}^{\prime}}$.
So we only have to show the existence and the uniqueness of a lift of
$\big{(}{\mathcal{E}}^{\bar{u}},\nabla^{\bar{u}},l^{\bar{u}},\ell^{\bar{u}},{\mathcal{V}}^{\bar{u}}\big{)}$,
which is a horizontal lift of
$\big{(}\tilde{E},\tilde{\nabla},\tilde{l},\tilde{\ell},\tilde{\mathcal{V}}\big{)}_{\tilde{M}^{\prime}}$
with respect to the morphism $u\colon{\mathcal{T}^{\prime}}[u]=\mathop{\rm
Spec}\nolimits{\mathcal{O}}_{\mathcal{T}^{\prime}}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{2}^{2}\big{)}\longrightarrow{\mathcal{T}^{\prime}}$.
The method of the proof is similar to that of Proposition 10.7.

We take an affine open covering
${\mathcal{C}}\times_{\mathcal{T}}\tilde{M}^{\prime}=\bigcup U_{\alpha}$ as in
the proof of Proposition 10.7. If $U_{\alpha}$ is an open neighborhood of
$\big{(}\tilde{x}_{i}^{\mathrm{un}}\big{)}_{\tilde{M}^{\prime}}$, then the
existence and the uniqueness of the local horizontal lift with respect to $u$
is given in the proof of [12, Lemma 5.5]. If $U_{\alpha}$ is an open
neighborhood of
$\big{(}\tilde{x}_{i}^{\mathrm{log}}\big{)}_{\tilde{M}^{\prime}}$, then it is
much easier to prove the existence and the uniqueness of a logarithmic local
horizontal lift.

So assume that
$\tilde{x}:=\big{(}\tilde{x}_{i}^{\mathrm{ram}}\big{)}_{\tilde{M}^{\prime}}$
is contained in $U_{\alpha}$. If $u$ is given by

$\displaystyle u^{*}(\nu(w))$
$\displaystyle=\nu(w)+\epsilon_{1}\nu_{u_{1}}(w)+\epsilon_{2}\nu_{u_{2}}(w)+\epsilon_{1}\epsilon_{2}\nu_{u_{12}}(w)$
$\displaystyle=\sum_{k=0}^{r-1}\left(a_{k,m-1}z^{m-1}+\sum_{l=0}^{m-2}(a_{k,l}+\epsilon_{1}b_{1,k,l}+\epsilon_{2}b_{2,k,l}+\epsilon_{1}\epsilon_{2}b_{1,2,k,l})z^{l}\right)w^{k},$

then, by the proof of Proposition 10.7, the restriction of $\nabla^{\bar{u}}$
to
$U_{\alpha}[\bar{u}]=U_{\alpha}[u]\otimes{\mathcal{O}}_{{\mathcal{T}^{\prime}}[u]/(\epsilon_{1}\epsilon_{2})}$
can be given by

$A(z)\frac{{\rm d}z}{z^{m}}+\epsilon_{1}C_{1}(z)\frac{{\rm
d}z}{z^{m}}+\epsilon_{2}C_{2}(z)\frac{{\rm d}z}{z^{m}}+B_{1}(z){\rm
d}\epsilon_{1}+B_{2}(z){\rm d}\epsilon_{2}$

where $\frac{\partial A(z)}{\partial\epsilon_{1}}=\frac{\partial
A(z)}{\partial\epsilon_{2}}=0$ and

$\displaystyle
A(z)=\sum_{k=0}^{r-1}\sum_{l=0}^{m-1}a_{k,l}z^{l}\tilde{N}^{k}+z^{m-1}R_{r}+z^{3m-1}A^{\prime}(z),$
(10.10) $\displaystyle
B_{1}(z)=\sum_{k=0}^{r-1}\sum_{l=0}^{m-2}\frac{rb_{1,k,l}}{(-mr+lr+r+k)z^{m-l-1}}\tilde{N}^{k},$
$\displaystyle
B_{2}(z)=\sum_{k=0}^{r-1}\sum_{l=0}^{m-2}\frac{rb^{(i)}_{2,k,l}}{(-mr+lr+r+k)z^{m-l-1}}\tilde{N}^{k},$
$\displaystyle C_{1}(z)\frac{{\rm d}z}{z^{m}}={\rm
d}B_{1}(z)+[A(z),B_{1}(z)]\frac{{\rm d}z}{z^{m}},\qquad C_{2}(z)\frac{{\rm
d}z}{z^{m}}={\rm d}B_{2}(z)+[A(z),B_{2}(z)]\frac{{\rm d}z}{z^{m}}.$

Then we can see by the above equality that

$C_{j}(z)|_{2m\tilde{x}}=\sum_{k=0}^{r-1}\sum_{l=0}^{m-2}b_{j,k,l}z^{l}\tilde{N}^{k}|_{2m\tilde{x}}$

for $j=1,2$. So we have $[C_{1}(z),B_{2}(z)],[C_{2}(z),B_{1}(z)]\in
z^{m+1}\mathop{\rm End}\nolimits(\tilde{E}|_{U_{\alpha}})$.

######  Claim 10.11.

$[C_{1}(z),B_{2}(z)]=[C_{2}(z),B_{1}(z)]$.

###### Proof.

First notice that we can check the equality

${\rm
d}B_{1}(z)=\sum_{k=0}^{r-1}\sum_{l=0}^{m-2}\frac{r(-m+l+1)+k}{-mr+lr+r+k}\frac{b_{1,k,l}}{z^{m-l}}\tilde{N}^{k}{\rm
d}z+\left[B_{1}(z),R_{r}\right]\frac{{\rm d}z}{z}$

using (10.6). So we have

$\displaystyle[{\rm d}B_{1}(z),B_{2}(z)]$
$\displaystyle=\left[\left[B_{1}(z),R_{r}\right],B_{2}(z)\right]\frac{{\rm
d}z}{z}$
$\displaystyle=\left[\left[B_{2}(z),R_{r}\right],B_{1}(z)\right]\frac{{\rm
d}z}{z}+\left[[B_{1}(z),B_{2}(z)],R_{r}\right]\frac{{\rm d}z}{z}$
$\displaystyle=[{\rm d}B_{2}(z),B_{1}(z)],$

because $[B_{1}(z),B_{2}(z)]=0$. Thus we have

$\displaystyle\left[C_{1}(z)\frac{{\rm d}z}{z^{m}},B_{2}(z)\right]$
$\displaystyle=\left[{\rm d}B_{1}(z)+[A(z),B_{1}(z)]\frac{{\rm
d}z}{z^{m}},B_{2}(z)\right]$ $\displaystyle=[{\rm
d}B_{1}(z),B_{2}(z)]+[[A(z),B_{1}(z)],B_{2}(z)]\frac{{\rm d}z}{z^{m}}$
$\displaystyle=[{\rm d}B_{2}(z),B_{1}(z)]+[[A(z),B_{2}(z)],B_{1}(z)]\frac{{\rm
d}z}{z^{m}}=\left[C_{2}(z)\frac{{\rm d}z}{z^{m}},B_{1}(z)\right].\\!\\!\\!$ ∎

We put

$\displaystyle\tilde{A}(z):=\sum_{k=0}^{r-1}\sum_{l=0}^{m-1}a_{k,l}z^{l}\tilde{N}^{k}+z^{m-1}R_{r}+z^{3m-1}\tilde{A}^{\prime}(z),$
$\displaystyle
B_{1,2}(z):=\sum_{k=0}^{r-1}\sum_{l=0}^{m-2}\frac{rb_{1,2,k,l}}{(-mr+lr+r+k)z^{m-l-1}}\tilde{N}^{k},$
$\displaystyle C_{1,2}(z)\frac{{\rm d}z}{z^{m}}:={\rm
d}B_{1,2}(z)+\left([A(z),B_{1,2}(z)]+[C_{1}(z),B_{2}(z)]\right)\frac{{\rm
d}z}{z^{m}},$

where $\tilde{A}^{\prime}(z)$ is a lift of $A^{\prime}(z)$ as a matrix with
coefficients in ${\mathcal{O}}_{U_{\alpha}[u]}$ such that
$\frac{\partial\tilde{A}(z)}{\partial\epsilon_{1}}=\frac{\partial\tilde{A}(z)}{\partial\epsilon_{2}}=0$.
Define a connection
$\nabla^{u}_{\alpha}\colon{\mathcal{O}}_{U_{\alpha}[u]}^{\oplus
r}\longrightarrow{\mathcal{O}}_{U_{\alpha}[u]}^{\oplus
r}\otimes\Omega^{1}_{{\mathcal{C}}_{\tilde{M}^{\prime}[u]/\tilde{M}^{\prime}}}({\mathcal{D}}_{\tilde{M}^{\prime}[u]})$
by setting

$\nabla^{u}_{\alpha}={\rm
d}+\big{(}\tilde{A}+\epsilon_{1}C_{1}+\epsilon_{2}C_{2}+\epsilon_{1}\epsilon_{2}C_{1,2}\big{)}\frac{{\rm
d}z}{z^{m}}+B_{1}{\rm d}\epsilon_{1}+B_{2}{\rm
d}\epsilon_{2}+B_{1,2}(\epsilon_{1}{\rm d}\epsilon_{2}+\epsilon_{2}{\rm
d}\epsilon_{1}).$

Then $\nabla^{u}_{\alpha}$ is an integrable connection because its curvature
form becomes

$\displaystyle(C_{1}+\epsilon_{2}C_{1,2}){\rm d}\epsilon_{1}\wedge\frac{{\rm
d}z}{z^{m}}+(C_{2}+\epsilon_{1}C_{1,2}){\rm d}\epsilon_{2}\wedge\frac{{\rm
d}z}{z^{m}}+({\rm d}B_{1}+\epsilon_{2}{\rm d}B_{1,2})\wedge{\rm
d}\epsilon_{1}$ $\displaystyle\qquad\quad{}+({\rm d}B_{2}+\epsilon_{1}{\rm
d}B_{1,2})\wedge{\rm d}\epsilon_{2}+B_{1,2}{\rm d}\epsilon_{1}\wedge{\rm
d}\epsilon_{2}+B_{1,2}{\rm d}\epsilon_{2}\wedge{\rm d}\epsilon_{1}$
$\displaystyle\qquad\quad{}+[A,B_{1}+\epsilon_{2}B_{1,2}]\frac{{\rm
d}z}{z^{m_{i}}}\wedge{\rm
d}\epsilon_{1}+[A,B_{2}+\epsilon_{1}B_{1,2}]\frac{{\rm
d}z}{z^{m_{i}}}\wedge{\rm d}\epsilon_{2}+\epsilon_{2}[C_{2},B_{1}]\frac{{\rm
d}z}{z^{m}}\wedge{\rm d}\epsilon_{1}$
$\displaystyle\qquad\quad{}+\epsilon_{1}[C_{1},B_{2}]\frac{{\rm
d}z}{z^{m}}\wedge{\rm d}\epsilon_{2}+[B_{1},B_{2}]{\rm
d}\epsilon_{1}\wedge{\rm d}\epsilon_{2}$ $\displaystyle\qquad{}=\left({\rm
d}B_{1}+(-C_{1}+[A,B_{1}])\frac{{\rm d}z}{z^{m}}\right)\wedge{\rm
d}\epsilon_{1}+\left({\rm d}B_{2}+(-C_{2}+[A,B_{2}])\frac{{\rm
d}z}{z^{m}}\right)\wedge{\rm d}\epsilon_{2}$
$\displaystyle\qquad\quad{}+\epsilon_{2}\left({\rm
d}B_{1,2}+(-C_{1,2}+[A,B_{1,2}]+[C_{2},B_{1}])\frac{{\rm
d}z}{z^{m}}\right)\wedge{\rm d}\epsilon_{1}$
$\displaystyle\qquad\quad{}+\epsilon_{1}\left({\rm
d}B_{1,2}+(-C_{1,2}+[A,B_{1,2}]+[C_{1},B_{2}])\frac{{\rm
d}z}{z^{m}}\right)\wedge{\rm d}\epsilon_{2}=0.$

We can define $V^{u}_{k,\alpha}$, $\vartheta^{u}_{k,\alpha}$,
$\varkappa^{u}_{k,\alpha}$ on ${\mathcal{O}}_{{\mathcal{D}}[u]}^{\oplus r}$ in
the same way as in the proof of Lemma 10.5. So we can get a local horizontal
lift $\big{(}{\mathcal{O}}_{U_{\alpha}[u]}^{\oplus
r},\nabla^{u}_{\alpha},(V^{u}_{k,\alpha},\vartheta^{u}_{k,\alpha},\varkappa^{u}_{k,\alpha})_{0\leq
k\leq r-1}\big{)}$, which is a lift of the restriction
$\big{(}{\mathcal{E}}^{\bar{u}},\nabla^{\bar{u}},l^{\bar{u}},\ell^{\bar{u}},{\mathcal{V}}^{\bar{u}}\big{)}|_{U_{\alpha}[\bar{u}]}$.

Let $\big{(}{\mathcal{O}}_{U_{\alpha}[u]}^{\oplus
r},\nabla^{\prime}_{\alpha},(V^{\prime}_{k,\alpha},\vartheta^{\prime}_{k,\alpha},\varkappa^{\prime}_{k,\alpha})_{0\leq
k\leq r-1}\big{)}$ be another local horizontal lift with respect to $u$, which
is a lift of
$\big{(}{\mathcal{E}}^{\bar{u}},\nabla^{\bar{u}},l^{\bar{u}},\ell^{\bar{u}},{\mathcal{V}}^{\bar{u}}\big{)}|_{U_{\alpha}[\bar{u}]}$.
Then we can write

$\nabla^{\prime}_{\alpha}={\rm
d}+\big{(}\tilde{A}+\epsilon_{1}C_{1}+\epsilon_{2}C_{2}+\epsilon_{1}\epsilon_{2}C^{\prime}_{1,2}\big{)}\frac{{\rm
d}z}{z^{m}}+B_{1}{\rm d}\epsilon_{1}+B_{2}{\rm
d}\epsilon_{2}+B^{\prime}_{1,2}\epsilon_{1}{\rm
d}\epsilon_{2}+B^{\prime}_{2,1}\epsilon_{2}{\rm d}\epsilon_{1}.$

The integrability condition of $\nabla^{\prime}$ implies the equalities

$\displaystyle C^{\prime}_{1,2}(z)\frac{{\rm d}z}{z^{m}}$ $\displaystyle={\rm
d}B^{\prime}_{1,2}(z)+\big{(}[A(z),B^{\prime}_{1,2}(z)]+[C_{1}(z),B_{2}(z)]\big{)}\frac{{\rm
d}z}{z^{m}}$ $\displaystyle={\rm
d}B^{\prime}_{2,1}(z)+\big{(}[A(z),B^{\prime}_{2,1}(z)]+[C_{1}(z),B_{2}(z)]\big{)}\frac{{\rm
d}z}{z^{m}}$ (10.11)

and $B^{\prime}_{1,2}=B^{\prime}_{2,1}$. Since $\nabla^{\prime}$ has the
property of local horizontal lift, we have

$C^{\prime}_{1,2}(z)\big{|}_{m\tilde{x}}=\sum_{k=0}^{r-1}\sum_{l=0}^{m-2}b_{1,2,k,l}z^{l}\big{(}\tilde{N}\big{)}^{k}\big{|}_{m\tilde{x}}.$

We can see that $[A(z),B^{\prime}_{1,2}(z)]$ is regular from the equality
(10.11). Since $A(z)$ satisfies (10.10), we can first verify
$z^{m-1}B^{\prime}_{1,2}(z)\in{\mathcal{O}}_{(m-1)\tilde{x}}\big{[}\tilde{N}|_{(m-1)\tilde{x}}\big{]}$.
Combining with the condition (c) of Definition 10.1 (iv), we can take
$\beta_{k,l}$ such that

$B^{\prime}_{1,2}(z)-\sum_{k=0}^{r-1}\sum_{l=0}^{m-1}\beta_{k,l}z^{l-m-1}\tilde{N}^{k}\in\mathop{\rm
End}\nolimits\big{(}\tilde{E}|_{U_{\alpha}}\big{)}.$

is a matrix of regular functions whose constant term is a lower triangular
matrix. Using the same argument as in the proof of Lemma 10.6, we can see

$\beta_{k,l}=\frac{rb_{1,2,k,l}}{-mr+l+r+k}.$

So $B_{12}(z)-B^{\prime}_{12}(z)$ becomes a matrix of regular functions and
$I_{r}+\epsilon_{1}\epsilon_{2}(B_{12}(z)-B^{\prime}_{12}(z))$ gives an
automorphism of ${\mathcal{O}}_{U_{\alpha}[u]}^{\oplus r}$ which transform
$\nabla^{\prime}_{\alpha}$ to $\nabla^{u}_{\alpha}$ and which sends
$V^{\prime}_{k,\alpha}$ to $V^{u}_{k,\alpha}$. Furthermore, we can see that
such a transform is uniquely determined by the coefficient of
$\epsilon_{2}\,d\epsilon_{1}$. Thus the existence and the uniqueness of a
ramified local horizontal lift with respect to $u$ is proved.

Patching the local horizontal lifts together, we get a unique horizontal lift
$\big{(}{\mathcal{E}}^{\bar{u}},\nabla^{\bar{u}},l^{\bar{u}},\ell^{\bar{u}},{\mathcal{V}}^{\bar{u}}\big{)}$
of
$\big{(}\tilde{E},\tilde{\nabla},\tilde{l},\tilde{\ell},\tilde{\mathcal{V}}\big{)}_{\tilde{M}^{\prime}}$
on ${\mathcal{C}}\times_{\mathcal{T}}\mathop{\rm
Spec}\nolimits{\mathcal{O}}_{\tilde{M}^{\prime}}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{2}^{2}\big{)}$
with respect to $u$. ∎

##  11 Global generalized isomonodromic deformation

######  Definition 11.1.

For each vector field $v\in T_{\mathcal{T}^{\prime}}$, the relative connection
$\big{(}{\mathcal{E}}^{v},\overline{\nabla^{v}},l^{v},\ell^{v},{\mathcal{V}}^{v}\big{)}$
induced by the global horizontal lift
$\big{(}{\mathcal{E}}^{v},\nabla^{v},l^{v},\ell^{v},{\mathcal{V}}^{v}\big{)}$
(which exists by Proposition 10.7) defines a morphism

$I_{\Phi(v)}\colon\ \tilde{M}^{\prime}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon]/\big{(}\epsilon^{2}\big{)}\longrightarrow\tilde{M}^{\prime}$

which makes the diagram

$\begin{CD}\tilde{M}^{\prime}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon]/\big{(}\epsilon^{2}\big{)}@>{I_{\Phi(v)}}>{}>\tilde{M}^{\prime}\\\
@V{\pi_{\mathcal{T}^{\prime}}\times\mathrm{id}}V{}V@V{}V{\pi_{\mathcal{T}^{\prime}}}V\\\
{\mathcal{T}}^{\prime}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon]@>{I_{v}}>{}>{\mathcal{T}}^{\prime}\end{CD}$
(11.1)

commutative. We can see by the uniqueness of the horizontal lift that the
morphism $I_{\Phi(v)}$ descends to a morphism
$M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})_{\mathcal{T}^{\prime}}\times\mathbb{C}[\epsilon]\longrightarrow
M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})_{\mathcal{T}^{\prime}}$
which corresponds to a vector field

$\Phi(v)\in
H^{0}\big{(}M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})_{\mathcal{T}^{\prime}},T_{M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})_{\mathcal{T}^{\prime}}}\big{)}.$

We call this vector field $\Phi(v)$ a generalized isomonodromic vector field.

######  Proposition 11.2.

The map

$\Phi\colon\ H^{0}({\mathcal{T}^{\prime}},T_{\mathcal{T}^{\prime}})\ni
v\mapsto\Phi(v)\in
H^{0}\big{(}M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})_{\mathcal{T}^{\prime}},T_{M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})_{\mathcal{T}^{\prime}}}\big{)}$

is a homomorphism of
$H^{0}({\mathcal{T}^{\prime}},{\mathcal{O}}_{\mathcal{T}^{\prime}})$-modules.

###### Proof.

Take vector fields $v_{1},v_{2}\in
H^{0}({\mathcal{T}^{\prime}},T_{\mathcal{T}^{\prime}})$. Then $(v_{1},v_{2})$
corresponds to a morphism

$\bar{u}\colon\ {\mathcal{T}^{\prime}}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{1}\epsilon_{2},\epsilon_{2}^{2}\big{)}\longrightarrow{\mathcal{T}^{\prime}}$

such that the composition ${\mathcal{T}^{\prime}}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon_{i}]/\big{(}\epsilon_{i}^{2}\big{)}\hookrightarrow{\mathcal{T}^{\prime}}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{1}\epsilon_{2},\epsilon_{2}^{2}\big{)}\xrightarrow{\bar{u}}{\mathcal{T}^{\prime}}$
coincides with the morphism $I_{v_{i}}$ for $i=1,2$. Let

$\Delta_{\mathcal{T}^{\prime}}\colon\ {\mathcal{T}^{\prime}}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon]/\big{(}\epsilon^{2}\big{)}\longrightarrow{\mathcal{T}^{\prime}}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{1}\epsilon_{2},\epsilon_{2}^{2}\big{)}$

be the morphism corresponding to the ring homomorphism

${\mathcal{O}}_{\mathcal{T}^{\prime}}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{1}\epsilon_{2},\epsilon_{2}^{2}\big{)}\ni
a+b_{1}\epsilon_{1}+b_{2}\epsilon_{2}\mapsto
a+b_{1}\epsilon+b_{2}\epsilon\in{\mathcal{O}}_{\mathcal{T}^{\prime}}[\epsilon]/\big{(}\epsilon^{2}\big{)}.$

Then the composition

$\bar{u}\circ\Delta_{\mathcal{T}^{\prime}}\colon\
{\mathcal{T}^{\prime}}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon]/\big{(}\epsilon^{2}\big{)}\xrightarrow{\Delta_{\mathcal{T}^{\prime}}}{\mathcal{T}^{\prime}}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{1}\epsilon_{2},\epsilon_{2}^{2}\big{)}\xrightarrow{\bar{u}}{\mathcal{T}^{\prime}}$

coincides with the morphism $I_{v_{1}+v_{2}}$ corresponding to the vector
field $v_{1}+v_{2}$. By virtue of Proposition 10.9, there exists a horizontal
lift
$\big{(}{\mathcal{E}}^{\bar{u}},\nabla^{\bar{u}},l^{\bar{u}},\ell^{\bar{u}},{\mathcal{V}}^{\bar{u}}\big{)}$
of
$\big{(}\tilde{E},\tilde{\nabla},\tilde{l},\tilde{\ell},\tilde{\mathcal{V}}\big{)}_{\tilde{M}^{\prime}}$
with respect to $\bar{u}$. By the same procedure as Definition 11.1, the flat
family of connections induced by the horizontal lift
$\big{(}{\mathcal{E}}^{\bar{u}},\nabla^{\bar{u}},l^{\bar{u}},\ell^{\bar{u}},{\mathcal{V}}^{\bar{u}}\big{)}$
provides a morphism
$I_{\Phi(\bar{u})}\colon\tilde{M^{\prime}}\times\mathbb{C}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{1}\epsilon_{2},\epsilon_{2}^{2}\big{)}\longrightarrow\tilde{M^{\prime}}$
such that the right square of the diagram

$\begin{CD}\tilde{M}^{\prime}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon]/\big{(}\epsilon^{2}\big{)}@>{\Delta_{\tilde{M^{\prime}}}}>{}>\tilde{M}^{\prime}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{1}\epsilon_{2},\epsilon_{2}^{2}\big{)}@>{I_{\Phi(\bar{u})}}>{}>\tilde{M^{\prime}}\\\
@V{}V{}V@V{}V{}V@V{}V{}V\\\ {\mathcal{T}^{\prime}}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon]/\big{(}\epsilon^{2}\big{)}@>{{\Delta_{\mathcal{T}^{\prime}}}}>{}>{\mathcal{T}^{\prime}}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{1}\epsilon_{2},\epsilon_{2}^{2}\big{)}@>{\bar{u}}>{}>{\mathcal{T}^{\prime}}\end{CD}$

is commutative. The left square of the above diagram is defined as a Cartesian
diagram. By the definition of horizontal lift, the pullback
$\Delta_{\tilde{M}^{\prime}}^{*}\big{(}{\mathcal{E}}^{\bar{u}},\nabla^{\bar{u}},l^{\bar{u}},\ell^{\bar{u}},{\mathcal{V}}^{\bar{u}}\big{)}$
is a horizontal lift of
$\big{(}\tilde{E},\tilde{\nabla},\tilde{l},\tilde{\ell},\tilde{\mathcal{V}}\big{)}_{\tilde{M}^{\prime}}$
with respect to $I_{v_{1}+v_{2}}$. So the composition
$I_{\Phi(\bar{u})}\circ\Delta_{\tilde{M^{\prime}}}$ coincides with the
morphism $I_{\Phi(v_{1}+v_{2})}$ determined by the vector field
$\Phi(v_{1}+v_{2})$. On the other hand, the morphism $I_{\Phi(\bar{u})}$
corresponds to the pair $(\Phi(v_{1}),\Phi(v_{2}))$ of vector fields and the
composition $I_{\Phi(\bar{u})}\circ\Delta_{\tilde{M^{\prime}}}$ corresponds to
the vector field $\Phi(v_{1})+\Phi(v_{2})$. So we have the equality

$I_{\Phi(v_{1}+v_{2})}=I_{\Phi(v_{1})+\Phi(v_{2})}$

which means the equality $\Phi(v_{1}+v_{2})=\Phi(v_{1})+\Phi(v_{2})$.

Take a vector field $v\in
H^{0}({\mathcal{T}^{\prime}},T_{\mathcal{T}^{\prime}})$ and a regular function
$f\in H^{0}({\mathcal{T}^{\prime}},{\mathcal{O}}_{\mathcal{T}^{\prime}})$.
Consider the morphism

$\alpha_{f}\colon\ {\mathcal{T}^{\prime}}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon]/\big{(}\epsilon^{2}\big{)}\longrightarrow{\mathcal{T}^{\prime}}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon]/\big{(}\epsilon^{2}\big{)}$

corresponding to the ring homomorphism

${\mathcal{O}}_{\mathcal{T}^{\prime}}[\epsilon]/\big{(}\epsilon^{2}\big{)}\ni
a+\epsilon b\mapsto a+\epsilon
fb\in{\mathcal{O}}_{\mathcal{T}^{\prime}}[\epsilon]/\big{(}\epsilon^{2}\big{)}.$

Then the composition

${\mathcal{T}^{\prime}}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon]/\big{(}\epsilon^{2}\big{)}\xrightarrow{\alpha_{f}}{\mathcal{T}^{\prime}}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon]/\big{(}\epsilon^{2}\big{)}\xrightarrow{I_{v}}{\mathcal{T}^{\prime}}$

coincides with the morphism $I_{fv}$ corresponding to the vector field $fv$.
As in Definition 11.1, the horizontal lift
$\big{(}{\mathcal{E}}^{v},\nabla^{v},l^{v},\ell^{v},{\mathcal{V}}^{v}\big{)}$
induces a morphism $I_{\Phi(v)}\colon\tilde{M}^{\prime}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon]/\big{(}\epsilon^{2}\big{)}\longrightarrow\tilde{M^{\prime}}$
which makes the diagram

$\begin{CD}\tilde{M^{\prime}}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon]/\big{(}\epsilon^{2}\big{)}@>{(\alpha_{f})_{\tilde{M^{\prime}}}}>{}>\tilde{M^{\prime}}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon]/\big{(}\epsilon^{2}\big{)}@>{I_{\Phi(v)}}>{}>\tilde{M^{\prime}}\\\
@V{}V{}V@V{}V{}V@V{}V{}V\\\ {\mathcal{T}^{\prime}}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon]/\big{(}\epsilon^{2}\big{)}@>{\alpha_{f}}>{}>{\mathcal{T}^{\prime}}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon]/\big{(}\epsilon^{2}\big{)}@>{I_{v}}>{}>{\mathcal{T}^{\prime}}\end{CD}$

commutative, where the right square is Cartesian. By the definition of
horizontal lift, the pullback
$(\alpha_{f})_{\tilde{M^{\prime}}}^{*}\big{(}{\mathcal{E}}^{v},\nabla^{v},l^{v},\ell^{v},{\mathcal{V}}^{v}\big{)}$
is a horizontal lift of
$\big{(}\tilde{E},\tilde{\nabla},\tilde{l},\tilde{\ell},\tilde{\mathcal{V}}\big{)}_{\tilde{M}^{\prime}}$
with respect to $fv$. So the composition
$I_{\Phi(v)}\circ(\alpha_{f})_{\tilde{M^{\prime}}}$ coincides with the
morphism $I_{\Phi(fv)}$ corresponding to $\Phi(fv)$. On the other hand, the
composition $I_{\Phi(v)}\circ(\alpha_{f})_{\tilde{M^{\prime}}}$ coincides with
the morphism $I_{f\Phi(v)}$ corresponding to the vector field $f\Phi(v)$. So
we have $\Phi(fv)=f\Phi(v)$. ∎

By Proposition 11.2, $\Phi$ defines a homomorphism

$\Phi\colon\
T_{\mathcal{T}}\longrightarrow(\pi_{\mathcal{T}})_{*}T_{M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})}$

of sheaves of ${\mathcal{O}}_{\mathcal{T}}$-modules. By the adjoint property,
$\Phi$ corresponds to a homomorphism

$\Psi\colon\ (\pi_{\mathcal{T}})^{*}T_{\mathcal{T}}\longrightarrow
T_{M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})}$
(11.2)

Since the diagram (11.1) in Definition 11.1 is commutative, we can see ${\rm
d}\pi_{\mathcal{T}}\circ\Psi=\mathrm{id}_{T_{\mathcal{T}}}$ for the canonical
surjection ${\rm d}\pi_{\mathcal{T}}\colon
T_{M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})}\longrightarrow(\pi_{\mathcal{T}})^{*}T_{\mathcal{T}}$.
In particular, the image $\mathop{\rm Im}\nolimits\Psi$ is a subbundle of
$T_{M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})}$.

######  Definition 11.3.

We call $\mathop{\rm Im}\nolimits\Psi$ the generalized isomonodromic subbundle
of
$T_{M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})}$.

By using the generalized isomonodromic subbundle $\mathop{\rm
Im}\nolimits\Psi$, we can extend the relative symplectic form
$\omega_{M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})}$
constructed in Theorem 8.1 to a total $2$-form on the moduli space
$M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})$
in the following.

######  Definition 11.4.

We define a $2$-form
$\omega_{M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})}^{\mathrm{GIM}}$
on
$M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})$
by setting

$\omega_{M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})}^{\mathrm{GIM}}(v_{1},v_{2})=\omega_{M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})}\big{(}v_{1}-\Psi({\rm
d}\pi_{\mathcal{T}}(v_{1})),v_{2}-\Psi({\rm
d}\pi_{\mathcal{T}}(v_{2}))\big{)}$

for $v_{1},v_{2}\in
T_{M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})}$
and call it the generalized isomonodromic $2$-form.

######  Remark 11.5.

In the logarithmic case, the above formulation of isomonodromic $2$-form is
given by A. Komyo in [18]. For a vector field $v\in
T_{M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})}$
we can immediately see the equivalence

$v\in\mathop{\rm
Im}\nolimits\Psi\Leftrightarrow\omega_{M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})}^{\mathrm{GIM}}(v,w)=0\qquad\text{for
any $w\in
T_{M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})}$}$

from the definition of the generalized isomonodromic $2$-form. So the
generalized isomonodromic $2$-form recovers the generalized isomonodromic
subbundle.

######  Theorem 11.6.

For any vector fields $v_{1},v_{2}\in T_{\mathcal{T}}$, the equality

$\Phi([v_{1},v_{2}])=[\Phi(v_{1}),\Phi(v_{2})]$

holds, where $[v_{1},v_{2}]=v_{1}v_{2}-v_{2}v_{1}$ is the commutator of the
vector fields $v_{1}$, $v_{2}$. In particular, the generalized isomonodromic
subbundle $\mathop{\rm Im}\nolimits\Psi$ of
$T_{M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})}$
satisfies the integrability condition

$[\mathop{\rm Im}\nolimits\Psi,\mathop{\rm Im}\nolimits\Psi]\subset\mathop{\rm
Im}\nolimits\Psi.$

###### Proof.

Take vector fields $v_{1},v_{2}\in
H^{0}({\mathcal{T}^{\prime}},T_{\mathcal{T}}|_{\mathcal{T}^{\prime}})$ over a
Zariski open subset ${\mathcal{T}^{\prime}}$ of ${\mathcal{T}}$. Let

$\tilde{I}_{v_{1}}\colon\ {\mathcal{T}^{\prime}}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{2}^{2}\big{)}\longrightarrow{\mathcal{T}^{\prime}}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{2}^{2}\big{)}$

be the automorphism corresponding to the ring automorphism
$\tilde{I}_{v_{1}}^{*}$ of
${\mathcal{O}}_{\mathcal{T}^{\prime}}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{2}^{2}\big{)}$
defined by

$\tilde{I}_{v_{1}}^{*}\left(a+b_{1}\epsilon_{1}+b_{2}\epsilon_{2}+c\,\epsilon_{1}\epsilon_{2}\right)=a+(v_{1}(a)+b_{1})\epsilon_{1}+b_{2}\epsilon_{2}+(v_{1}(b_{2})+c)\epsilon_{1}\epsilon_{2}.$

Similarly, we can define an automorphism $\tilde{I}_{v_{2}}$ of
${\mathcal{T}^{\prime}}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{2}^{2}\big{)}$
corresponding to $v_{2}$. By construction, we can see that
$\tilde{I}_{-v_{1}}=\tilde{I}_{v_{1}}^{-1}$ and
$\tilde{I}_{-v_{2}}=\tilde{I}_{v_{2}}^{-1}$. The composition
$\tilde{I}_{v_{2}}\circ\tilde{I}_{v_{1}}\circ\tilde{I}_{-v_{2}}\circ\tilde{I}_{-v_{1}}$
corresponds to the ring automorphism of
${\mathcal{O}}_{\mathcal{T}^{\prime}}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{2}^{2}\big{)}$
determined by

$\displaystyle\tilde{I}_{-v_{1}}^{*}\circ\tilde{I}_{-v_{2}}^{*}\circ\tilde{I}_{v_{1}}^{*}\circ\tilde{I}_{v_{2}}^{*}(a+b_{1}\epsilon_{1}+b_{2}\epsilon_{2}+c\,\epsilon_{1}\epsilon_{2})$
$\displaystyle\quad{}=\tilde{I}_{-v_{1}}^{*}\circ\tilde{I}_{-v_{2}}^{*}\circ\tilde{I}_{v_{1}}^{*}(a+b_{1}\epsilon_{1}+(v_{2}(a)+b_{2})\epsilon_{2}+(c+v_{2}(b_{1}))\epsilon_{1}\epsilon_{2})$
$\displaystyle\quad{}=\tilde{I}_{-v_{1}}^{*}\circ\tilde{I}_{-v_{2}}^{*}(a+(v_{1}(a)+b_{1})\epsilon_{1}+(v_{2}(a)+b_{2})\epsilon_{2}+(v_{1}v_{2}(a)+v_{1}(b_{2})+c+v_{2}(b_{1}))\epsilon_{1}\epsilon_{2})$
$\displaystyle\quad{}=\tilde{I}_{-v_{1}}^{*}(a+(v_{1}(a)+b_{1})\epsilon_{1}+b_{2}\epsilon_{2}+(-v_{2}v_{1}(a)+v_{1}v_{2}(a)+v_{1}(b_{2})+c)\epsilon_{1}\epsilon_{2})$
$\displaystyle\quad{}=a+b_{1}\epsilon_{1}+b_{2}\epsilon_{2}+((v_{1}v_{2}-v_{2}v_{1})(a)+c)\epsilon_{1}\epsilon_{2}.$

Let

$\rho\colon\ {\mathcal{T}^{\prime}}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{2}^{2}\big{)}\longrightarrow{\mathcal{T}^{\prime}}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon]/\big{(}\epsilon^{2}\big{)}$

be the morphism corresponding to the ring homomorphism
$\rho^{*}\colon{\mathcal{O}}_{\mathcal{T}^{\prime}}[\epsilon]/\big{(}\epsilon^{2}\big{)}\rightarrow{\mathcal{O}}_{\mathcal{T}^{\prime}}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{2}^{2}\big{)}$
determined by $\rho^{*}(a+c\epsilon)=a+c\epsilon_{1}\epsilon_{2}$. Then the
composition

${\mathcal{T}^{\prime}}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{2}^{2}\big{)}\stackrel{{\scriptstyle\rho}}{{\longrightarrow}}{\mathcal{T}^{\prime}}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon]/\big{(}\epsilon^{2}\big{)}\xrightarrow{I_{v_{1}v_{2}-v_{2}v_{1}}}{\mathcal{T}^{\prime}}$
(11.3)

coincides with the composition

$\displaystyle{\mathcal{T}^{\prime}}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{2}^{2}\big{)}\xrightarrow{\tilde{I}_{v_{2}}\circ\tilde{I}_{v_{1}}\circ\tilde{I}_{v_{2}}^{-1}\circ\tilde{I}_{v_{1}}^{-1}}{\mathcal{T}^{\prime}}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{2}^{2}\big{)}$
$\displaystyle\hphantom{{\mathcal{T}^{\prime}}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{2}^{2}\big{)}\xrightarrow{\tilde{I}_{v_{2}}\circ\tilde{I}_{v_{1}}\circ\tilde{I}_{v_{2}}^{-1}\circ\tilde{I}_{v_{1}}^{-1}}}{}\xrightarrow{\text{trivial
projection}}{\mathcal{T}^{\prime}}.$ (11.4)

By Proposition 10.9, there exists a horizontal lift
$\big{(}{\mathcal{E}}^{\tilde{v}_{i}},\nabla^{\tilde{v}_{i}},l^{\tilde{v}_{i}},\ell^{\tilde{v}_{i}},{\mathcal{V}}^{\tilde{v}_{i}}\big{)}$
of
$\big{(}\tilde{E},\tilde{\nabla},\tilde{l},\tilde{\ell},\tilde{\mathcal{V}}\big{)}_{\tilde{M}^{\prime}}$
with respect to the morphism

${\mathcal{T}^{\prime}}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{2}^{2}\big{)}\xrightarrow{\tilde{I}_{v_{i}}}{\mathcal{T}^{\prime}}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{2}^{2}\big{)}\xrightarrow{\text{trivial
projection}}{\mathcal{T}^{\prime}}.$

For the relative connection $\overline{\nabla^{\tilde{v}_{i}}}$ induced by
$\nabla^{\tilde{v}_{i}}$, the flat family
$\big{(}{\mathcal{E}}^{\tilde{v}_{i}},\overline{\nabla^{\tilde{v}_{i}}},l^{\tilde{v}_{i}},\ell^{\tilde{v}_{i}},{\mathcal{V}}^{\tilde{v}_{i}}\big{)}$
determines a morphism
$I_{\Phi(\tilde{v}_{i})}\colon\tilde{M}^{\prime}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{2}^{2}\big{)}\longrightarrow\tilde{M}^{\prime}$
which is canonically extended to a morphism

$\tilde{I}_{\Phi(\tilde{v}_{i})}\colon\ \tilde{M}^{\prime}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{2}^{2}\big{)}\longrightarrow\tilde{M}^{\prime}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{2}^{2}\big{)}$

over $\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{2}^{2}\big{)}$.
Furthermore, the diagram

$\begin{CD}\tilde{M}^{\prime}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{2}^{2}\big{)}@>{\tilde{I}_{\Phi(\tilde{v}_{2})}\circ\tilde{I}_{\Phi(\tilde{v}_{1})}\circ\tilde{I}_{\Phi(\tilde{v}_{2})}^{-1}\circ\tilde{I}_{\Phi(\tilde{v}_{1})}^{-1}}>{}>\tilde{M}^{\prime}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{2}^{2}\big{)}\\\
@V{}V{}V@V{}V{}V\\\ {\mathcal{T}^{\prime}}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{2}^{2}\big{)}@>{\tilde{I}_{v_{2}}\circ\tilde{I}_{v_{1}}\circ\tilde{I}_{v_{2}}^{-1}\circ\tilde{I}_{v_{1}}^{-1}}>{}>{\mathcal{T}^{\prime}}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{2}^{2}\big{)}\end{CD}$

is commutative.

By the definition of horizontal lift, we can see that the pullback

$\big{(}\tilde{I}_{\Phi(\tilde{v}_{1})}^{-1}\big{)}^{*}\big{(}\tilde{I}_{\Phi(\tilde{v}_{2})}^{-1}\big{)}^{*}\tilde{I}_{\Phi(\tilde{v}_{1})}^{*}\big{(}{\mathcal{E}}^{\tilde{v}_{2}},\nabla^{\tilde{v}_{2}},l^{\tilde{v}_{2}},\ell^{\tilde{v}_{2}},{\mathcal{V}}^{\tilde{v}_{2}}\big{)}$

becomes a horizontal lift of
$\big{(}\tilde{E},\tilde{\nabla},\tilde{l},\tilde{\ell},\tilde{\mathcal{V}}\big{)}_{\tilde{M}^{\prime}}$
with respect to the morphism (11.4). On the other hand, there is a canonical
commutative diagram

$\begin{CD}\tilde{M^{\prime}}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{2}^{2}\big{)}@>{\rho_{\tilde{M^{\prime}}}}>{}>\tilde{M^{\prime}}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon]/\big{(}\epsilon^{2}\big{)}@>{I_{\Phi(v_{1}v_{2}-v_{2}v_{1})}}>{}>\tilde{M^{\prime}}\\\
@V{}V{}V@V{}V{}V@V{}V{}V\\\ {\mathcal{T}^{\prime}}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon_{1},\epsilon_{2}]/\big{(}\epsilon_{1}^{2},\epsilon_{2}^{2}\big{)}@>{\rho}>{}>{\mathcal{T}^{\prime}}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon]/\big{(}\epsilon^{2}\big{)}@>{I_{v_{1}v_{2}-v_{2}v_{1}}}>{}>{\mathcal{T}^{\prime}},\end{CD}$

whose left square is Cartesian. So we can see that the pullback

$\rho_{\tilde{M^{\prime}}}^{*}\big{(}{\mathcal{E}}^{v_{1}v_{2}-v_{2}v_{1}},\nabla^{v_{1}v_{2}-v_{2}v_{1}},l^{v_{1}v_{2}-v_{2}v_{1}},\ell^{v_{1}v_{2}-v_{2}v_{1}},{\mathcal{V}}^{v_{1}v_{2}-v_{2}v_{1}}\big{)}$

becomes a horizontal lift of
$\big{(}\tilde{E},\tilde{\nabla},\tilde{l},\tilde{\ell},\tilde{\mathcal{V}}\big{)}_{\tilde{M}^{\prime}}$
with respect to the morphism (11.3). Since the morphism (11.4) coincides with
the morphism (11.3), we can deduce an isomorphism

$\displaystyle\big{(}\tilde{I}_{\Phi(\tilde{v}_{1})}^{-1}\big{)}^{*}\big{(}\tilde{I}_{\Phi(\tilde{v}_{2})}^{-1}\big{)}^{*}\tilde{I}_{\Phi(\tilde{v}_{1})}^{*}\big{(}{\mathcal{E}}^{\tilde{v}_{2}},\nabla^{\tilde{v}_{2}},l^{\tilde{v}_{2}},\ell^{\tilde{v}_{2}},{\mathcal{V}}^{\tilde{v}_{2}}\big{)}$
$\displaystyle\qquad{}\cong\rho_{\tilde{M^{\prime}}}^{*}\big{(}{\mathcal{E}}^{v_{1}v_{2}-v_{2}v_{1}},\nabla^{v_{1}v_{2}-v_{2}v_{1}},l^{v_{1}v_{2}-v_{2}v_{1}},\ell^{v_{1}v_{2}-v_{2}v_{1}},{\mathcal{V}}^{v_{1}v_{2}-v_{2}v_{1}}\big{)}$

by the uniqueness of horizontal lift proved in Proposition 10.9. Considering
the induced morphism, we have

$(\text{trivial
projection})\circ\tilde{I}_{\Phi(\tilde{v}_{2})\circ\tilde{I}_{\Phi(\tilde{v}_{1})}\circ\tilde{I}_{\Phi(\tilde{v}_{2})}^{-1}\circ\tilde{I}_{\Phi(\tilde{v}_{1})}^{-1}}=I_{\Phi(v_{1}v_{2}-v_{2}v_{1})}\circ\rho_{\tilde{M^{\prime}}},$

from which we get
$\Phi(v_{1}v_{2}-v_{2}v_{1})=\Phi(v_{1})\Phi(v_{2})-\Phi(v_{2})\Phi(v_{1})$. ∎

######  Definition 11.7.

Since the subbundle $\mathop{\rm Im}\nolimits\Psi\subset
T_{M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})}$
satisfies the integrability condition by Theorem 11.6, it determines a
foliation ${\mathcal{F}}^{\rm
GIM}_{T_{M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})}}$
on the moduli space
$M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})$.
We call it the generalized isomonodromic foliation.

Take a point $t_{0}\in{\mathcal{T}}$ and a point $y$ of the fiber
$M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})_{t_{0}}$
over $t_{0}$. Then we can take an analytic open neighborhood
${\mathcal{M}^{\prime}}$ of $y$ in
$M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})$
and an analytic open neighborhood ${\mathcal{T}^{\prime}}$ of $t_{0}$ in
${\mathcal{T}}$ together with an analytic isomorphism

${\mathcal{M}^{\prime}}\cong{\mathcal{M}}^{\prime}_{t_{0}}\times{\mathcal{T}^{\prime}}$
(11.5)

such that the restriction $\pi_{\mathcal{T}}|_{\mathcal{M}^{\prime}}$ of
$\pi_{\mathcal{T}}\colon
M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})\longrightarrow{\mathcal{T}}$
coincides with the second projection and that the fibers
$\\{\\{y^{\prime}\\}\times{\mathcal{T}^{\prime}}\\}_{y^{\prime}\in{\mathcal{M}^{\prime}_{t_{0}}}}$
over ${\mathcal{M}^{\prime}_{t_{0}}}$ are leaves in ${\mathcal{F}}^{\rm
GIM}_{M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})}$.

Take a holomorphic system of coordinates
$\theta=(\theta_{1},\dots,\theta_{N})$ of ${\mathcal{T}^{\prime}}$. If we set

${\mathcal{T}^{\prime}}[\partial\theta]:={\mathcal{T}^{\prime}}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon_{1},\dots,\epsilon_{N}]\big{/}\big{(}\epsilon_{i}\epsilon_{j}\mid
1\leq i,j\leq N\big{)},$

then the tuple
$\partial\theta=(\partial/\partial\theta_{1},\dots,\partial/\partial\theta_{N})$
of vector fields on ${\mathcal{T}^{\prime}}$ corresponds to a morphism

$I_{\partial\theta}\colon\
{\mathcal{T}^{\prime}}[\partial\theta]\longrightarrow{\mathcal{T}^{\prime}},$

whose restriction to
${\mathcal{T}^{\prime}}\subset{\mathcal{T}^{\prime}}[\partial\theta]$ is the
identity morphism.

By the same proof as Proposition 10.9, we can construct a horizontal lift
$\big{(}{\mathcal{E}}^{\partial\theta},\nabla^{\partial\theta},l^{\partial\theta},\ell^{\partial\theta},\allowbreak{\mathcal{V}}^{\partial\theta}\big{)}$
of the universal family $(E,\nabla,l,\ell,{\mathcal{V}})$ on
${\mathcal{C}}\times_{\mathcal{T}}{\mathcal{M}^{\prime}}$ with respect to the
morphism $I_{\partial\theta}$. On a small open subset
$U\subset{\mathcal{C}}\times_{\mathcal{T}}{\mathcal{M}^{\prime}}$, we may
assume $E|_{U}\cong{\mathcal{O}}_{U}^{\oplus r}$. Then we can write
$\nabla|_{U}={\rm d}+A{\rm d}z$ where $z$ is a holomorphic coordinate on
${\mathcal{C}}\times_{\mathcal{T}}{\mathcal{M}^{\prime}}$ over
${\mathcal{M}^{\prime}}$ and $A$ is a matrix of meromorphic functions in $z$.
Let
$U[\partial\theta]\subset{\mathcal{C}}\times_{\mathcal{T}}{\mathcal{M}^{\prime}}[\partial\theta]$
be the open subscheme whose underlying set is $U$. Then we have
${\mathcal{E}}|_{U[\partial\theta]}\cong{\mathcal{O}}_{U[\partial\theta]}^{\oplus
r}$ and we can write $\nabla^{\partial\theta}={\rm d}+A(\epsilon){\rm
d}z+\sum_{j=1}^{N}B_{j}{\rm d}\epsilon_{j}$, where $A(\epsilon)$ is a lift of
$A$. By the integrability condition of $\nabla^{\partial\theta}$, we have the
equality

$\sum_{j=1}^{N}\frac{\partial A(\epsilon)}{\partial\epsilon_{j}}{\rm
d}\epsilon_{j}\wedge{\rm d}z+\sum_{j=1}^{N}\frac{\partial B_{j}}{\partial
z}{\rm d}z\wedge{\rm d}\epsilon_{j}+\sum_{j=1}^{N}[A,B_{j}]{\rm d}z\wedge{\rm
d}\epsilon_{j}=0.$

Take a holomorphic coordinate system $x_{1},\dots,x_{\delta}$ of
${\mathcal{M}}^{\prime}_{t_{0}}$. With respect to the coordinate system
$z,x_{1},\dots,x_{\delta},\theta_{1},\dots,\theta_{N}$, the partial derivative
$\partial/\partial\theta_{j}$ coincides with the vector field
$\Phi(\partial/\partial\theta_{j})$ and the partial derivative $\partial
A/\partial\theta_{j}$ coincides with $\partial
A(\epsilon)/\partial\epsilon_{j}$. So the above integrability condition of
$\nabla^{\partial\theta}$ is the same as the integrability condition

$\sum_{j=1}^{N}\frac{\partial A}{\partial\theta_{j}}{\rm
d}\theta_{j}\wedge{\rm d}z+\sum_{j=1}^{N}\frac{\partial B_{j}}{\partial z}{\rm
d}z\wedge{\rm d}\theta_{j}+\sum_{j=1}^{N}[A,B_{j}]{\rm d}z\wedge{\rm
d}\theta_{j}=0$

of the connection

$\nabla_{U}^{\rm flat}={\rm d}+A{\rm d}z+\sum_{j=1}^{N}B_{j}{\rm d}\theta_{j}$

on $E|_{U}$ relative to the composition

$U\hookrightarrow{\mathcal{C}}\times_{\mathcal{T}}{\mathcal{M}^{\prime}}\longrightarrow{\mathcal{M}^{\prime}}\cong{\mathcal{M}}^{\prime}_{t_{0}}\times{\mathcal{T}^{\prime}}\xrightarrow{\pi_{{\mathcal{M}}^{\prime}_{t_{0}}}}{\mathcal{M}}^{\prime}_{t_{0}},$

where
$\pi_{{\mathcal{M}}^{\prime}_{t_{0}}}\colon{\mathcal{M}^{\prime}}\cong{\mathcal{M}}^{\prime}_{t_{0}}\times{\mathcal{T}^{\prime}}\longrightarrow{\mathcal{M}}^{\prime}_{t_{0}}$
is the first projection. So we can see from Theorem 9.7 and Corollary 9.11
that $\nabla|_{U}$ is a local generalized isomonodromic deformation in the
sense of Definition 9.4 or Definition 9.10.

We can patch $\nabla_{U}^{\rm flat}$ together to get a global connection on
$E$. Indeed, take another open subset
$U^{\prime}\subset{\mathcal{C}}\times_{\mathcal{T}}{\mathcal{M}^{\prime}}$ and
write $\nabla|_{U^{\prime}}={\rm d}+A^{\prime}{\rm d}z$. Then we have
$P^{-1}{\rm d}P+P^{-1}A{\rm d}zP=A^{\prime}{\rm d}z$ for a transition matrix
$P$. There is a local horizontal lift ${\rm d}+A^{\prime}(\epsilon){\rm
d}z+\sum_{j=1}^{N}B^{\prime}_{j}{\rm d}\epsilon_{j}$ of $\nabla|_{U^{\prime}}$
and by the uniqueness of the local horizontal lift, we have a uniquely lift
$P(\epsilon)$ of $P$ satisfying

$\displaystyle P(\epsilon)^{-1}\left(\frac{\partial P(\epsilon)}{\partial
z}{\rm d}z+\sum_{j=1}^{N}\frac{\partial P(\epsilon)}{\partial\epsilon_{j}}{\rm
d}\epsilon_{j}\right)+P(\epsilon)^{-1}\left(A(\epsilon){\rm
d}z+\sum_{j=1}^{N}B_{j}{\rm d}\epsilon_{j}\right)P(\epsilon)$
$\displaystyle\qquad{}=A^{\prime}(\epsilon){\rm
d}z+\sum_{j=1}^{N}B^{\prime}_{j}{\rm d}\epsilon_{j}.$

Since $\partial P(\epsilon)/\partial\epsilon_{j}=\partial
P/\partial\theta_{j}$, the above equality yields the equality

$P^{-1}\left(\frac{\partial P}{\partial z}{\rm
d}z+\sum_{j=1}^{N}\frac{\partial P}{\partial\theta_{j}}{\rm
d}\theta_{j}\right)+P^{-1}\left(A{\rm d}z+\sum_{j=1}^{N}B_{j}{\rm
d}\theta_{j}\right)P=A^{\prime}{\rm d}z+\sum_{j=1}^{N}B^{\prime}_{j}{\rm
d}\theta_{j}.$

So we can patch the local connections $\nabla^{\rm flat}_{U}$ together to get
an integrable connection

$\nabla^{\rm flat}\colon\ E\longrightarrow
E\otimes\Omega_{{\mathcal{C}}\times_{\mathcal{T}}{\mathcal{M}^{\prime}}/{\mathcal{M}}^{\prime}_{t_{0}}}(D_{\mathcal{M}^{\prime}})$
(11.6)

relative to the composition
${\mathcal{C}}\times_{\mathcal{T}}{\mathcal{M}^{\prime}}\longrightarrow{\mathcal{M}^{\prime}}\cong{\mathcal{M}}^{\prime}_{t_{0}}\times{\mathcal{T}^{\prime}}\longrightarrow{\mathcal{M}}^{\prime}_{t_{0}}$.

######  Corollary 11.8.

The generalized isomonodromic $2$-form
$\omega_{M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})}^{\mathrm{GIM}}$
constructed in Definition 11.4 is ${\rm d}$-closed.

###### Proof.

Under the above notations, we will prove the equality

$\omega_{M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})}^{\mathrm{GIM}}\big{|}_{{\mathcal{M}}^{\prime}}=\pi_{{\mathcal{M}}^{\prime}_{t_{0}}}^{*}\big{(}\omega_{M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})}\big{|}_{{\mathcal{M}}^{\prime}_{t_{0}}}\big{)}$
(11.7)

where
$\pi_{{\mathcal{M}}^{\prime}_{t_{0}}}\colon{\mathcal{M}}^{\prime}\cong{\mathcal{M}}^{\prime}_{t_{0}}\times{\mathcal{T}^{\prime}}\longrightarrow{\mathcal{M}}^{\prime}_{t_{0}}$
corresponds to the first projection with respect to the isomorphism (11.5).
The corollary follows from this equality, since
$\omega_{M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})}|_{{\mathcal{M}}^{\prime}_{t_{0}}}$
is $d$-closed by Theorem 8.1.

Take two tangent vectors $v,v^{\prime}\in T_{{\mathcal{M}}^{\prime}}(y,t_{0})$
at $(y,t)\in{\mathcal{M}}^{\prime}_{t_{0}}\times{\mathcal{T}^{\prime}}$. We
have the equalities

$(\pi_{{\mathcal{M}}^{\prime}_{t_{0}}})_{*}(v)=(\pi_{{\mathcal{M}}^{\prime}_{t_{0}}})_{*}(v-\Phi(\pi_{\mathcal{T}^{\prime}})_{*}(v)),\qquad(\pi_{{\mathcal{M}}^{\prime}_{t_{0}}})_{*}(v^{\prime})=(\pi_{{\mathcal{M}}^{\prime}_{t_{0}}})_{*}(v-\Phi(\pi_{\mathcal{T}^{\prime}})_{*}(v^{\prime}))$

because
$\\{y\times{\mathcal{T}^{\prime}}\\}_{y\in{\mathcal{M}}^{\prime}_{t_{0}}}$ are
leaves of the foliation ${\mathcal{F}}^{\rm
GIM}_{M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})}$,
which is determined by the subbundle $\mathop{\rm Im}\nolimits\Phi$ of
$T_{M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})}$.
The tangent vector
$(\pi_{{\mathcal{M}}^{\prime}_{t_{0}}})_{*}(v-\Phi(\pi_{\mathcal{T}^{\prime}})_{*}(v))$
corresponds to a morphism $\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon]/\big{(}\epsilon^{2}\big{)}\longrightarrow{\mathcal{M}}^{\prime}_{t_{0}}$.
Let

$\tilde{I}_{v}\colon\ \mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon]/\big{(}\epsilon^{2}\big{)}\times{\mathcal{T}^{\prime}}\longrightarrow{\mathcal{M}}^{\prime}_{t_{0}}\times{\mathcal{T}^{\prime}}\cong{\mathcal{M}}^{\prime}$

be its base change.

We can construct a complex ${\mathcal{F}}^{\bullet}$ of sheaves on
${\mathcal{C}}\times_{\mathcal{T}}{\mathcal{M}^{\prime}}$ from the universal
family $(E,\nabla,l,\ell,{\mathcal{V}})$ in the same way as (6.14) in Section
6. Since $(\mathrm{id}\times\tilde{I}_{v})^{*}(E,\nabla,l,\ell,{\mathcal{V}})$
is a lift of
$(E,\nabla,l,\ell,{\mathcal{V}})|_{{\mathcal{C}}_{y\times{\mathcal{T}^{\prime}}}}$,
it induces a gluing data $\\{u_{\alpha\beta},v_{\alpha},\eta_{\alpha}\\}$ with
respect to an open covering $\\{U_{\alpha}\\}$ of
${\mathcal{C}}_{y\times{\mathcal{T}^{\prime}}}:={\mathcal{C}}\times_{\mathcal{T}}(y\times{\mathcal{T}^{\prime}})$.
as in Proposition 6.3. Set

$\tilde{v}:=[\\{u_{\alpha\beta},v_{\alpha},\eta_{\alpha}\\}]\in\mathbb{R}^{1}(p_{y\times{\mathcal{T}^{\prime}}})_{*}\big{(}{\mathcal{F}}^{\bullet}|_{{\mathcal{C}}_{y\times{\mathcal{T}^{\prime}}}}\big{)}.$

Then we can see from the construction of $\tilde{v}$ that the equalities

$\displaystyle\tilde{v}|_{(y,t_{0})}=(\pi_{{\mathcal{M}^{\prime}}_{t_{0}}})_{*}(v)\in\mathbb{H}^{1}\big{(}{\mathcal{F}}^{\bullet}|_{{\mathcal{C}}_{(y,t_{0})}}\big{)},$
$\displaystyle\tilde{v}|_{(y,t)}=v-\Phi(\pi_{\mathcal{T}^{\prime}})_{*}(v)\in\mathbb{H}^{1}\big{(}{\mathcal{F}}^{\bullet}|_{{\mathcal{C}}_{(y,t)}}\big{)}.$

hold. We can similarly construct an element
$\tilde{v^{\prime}}=[\\{u^{\prime}_{\alpha\beta},v^{\prime}_{\alpha},\eta^{\prime}_{\alpha}\\}]$
of
$\mathbb{R}^{1}(p_{y\times{\mathcal{T}^{\prime}}})_{*}\big{(}{\mathcal{F}}^{\bullet}|_{{\mathcal{C}}_{y\times{\mathcal{T}^{\prime}}}}\big{)}$
from the tangent vector $v^{\prime}$. Recall the construction of the complex
${\mathcal{F}}^{\bullet}$ in (6.14). Since the map
${\mathcal{G}}^{1}\longrightarrow G^{1}$ is surjective and the map
${\mathcal{G}}^{0}\longrightarrow{\mathcal{S}}^{1}_{\rm ram}$ is a surjection
to the kernel of the surjection ${\mathcal{S}}^{1}_{\rm ram}\longrightarrow
A^{1}$, we can replace $u_{\alpha\beta}$, $v_{\alpha}$ so that
$\eta_{\alpha}=0$ holds. Similarly we may assume $\eta^{\prime}_{\alpha}=0$.

Consider the pairing

$\omega\big{(}\tilde{v},\tilde{v^{\prime}}\big{)}:=\big{[}\big{\\{}\mathop{\rm
Tr}\nolimits(u_{\alpha\beta}u^{\prime}_{\beta\gamma}),-\mathop{\rm
Tr}\nolimits(u_{\alpha\beta}v_{\beta}-v_{\alpha}u^{\prime}_{\alpha\beta})\big{\\}}\big{]}\in\mathbb{R}^{2}(p_{y\times{\mathcal{T}^{\prime}}})_{*}\big{(}\Omega^{\bullet}_{{\mathcal{C}}_{y\times{\mathcal{T}^{\prime}}}/{\mathcal{T}^{\prime}}}\big{)}\cong{\mathcal{O}}_{\mathcal{T}^{\prime}}$

of $\tilde{v}$ and $\tilde{v^{\prime}}$. Then we have the equalities

$\displaystyle\pi_{{\mathcal{M}^{\prime}}_{t_{0}}}^{*}\big{(}\omega_{M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})}|_{{\mathcal{M}^{\prime}}_{t_{0}}}\big{)}(v,v^{\prime})=\omega_{M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})_{t_{0}}}\big{(}(\pi_{{\mathcal{M}^{\prime}}_{t_{0}}})_{*}(v),(\pi_{{\mathcal{M}^{\prime}}_{t_{0}}})_{*}(v^{\prime})\big{)}=\omega\big{(}\tilde{v},\tilde{v^{\prime}}\big{)}|_{(y,t_{0})},$
$\displaystyle\omega^{\rm
GIM}_{M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})}(v,v^{\prime})=\omega_{M^{\boldsymbol{\alpha}}_{{\mathcal{C}},{\mathcal{D}}}(\lambda,\tilde{\mu},\tilde{\nu})_{t}}\big{(}v-\Phi(\pi_{\mathcal{T}^{\prime}})_{*}(v),v^{\prime}-\Phi(\pi_{\mathcal{T}^{\prime}})_{*}(v^{\prime})\big{)}=\omega\big{(}\tilde{v},\tilde{v^{\prime}}\big{)}|_{(y,t)}.$

So, in order to prove (11.7), we only have to prove that
$\omega\big{(}\tilde{v},\tilde{v^{\prime}}\big{)}\in{\mathcal{O}}_{\mathcal{T}^{\prime}}$
is constant on ${\mathcal{T}^{\prime}}$. We may assume that
${\mathcal{T}}^{\prime}$ is isomorphic to a polydisk. Then it is sufficient to
show that $\omega\big{(}\tilde{v},\tilde{v^{\prime}}\big{)}$ belongs to the
image of the canonical map

$\displaystyle\mathbb{C}\cong\mathbb{H}^{2}\big{(}{\mathcal{O}}_{{\mathcal{C}}_{y\times{\mathcal{T}}^{\prime}}}\xrightarrow{{\rm
d}}\Omega^{1}_{{\mathcal{C}}_{y\times{\mathcal{T}}^{\prime}}}\xrightarrow{{\rm
d}}\cdots\xrightarrow{{\rm
d}}\Omega^{N+1}_{{\mathcal{C}}_{y\times{\mathcal{T}}^{\prime}}}\big{)}$
$\displaystyle\hphantom{\mathbb{C}}{}\hookrightarrow\mathbb{R}^{2}(p_{y\times{\mathcal{T}^{\prime}}})_{*}\big{(}{\mathcal{O}}_{{\mathcal{C}}_{y\times{\mathcal{T}}^{\prime}}}\xrightarrow{{\rm
d}}\Omega^{1}_{{\mathcal{C}}_{y\times{\mathcal{T}}^{\prime}}/{\mathcal{T}^{\prime}}}\big{)}\cong{\mathcal{O}}_{\mathcal{T}^{\prime}}.$
(11.8)

Recall that $(E,\nabla)$ can be extended to the family of integrable
connections $\big{(}E,\nabla^{\rm flat}\big{)}$ in (11.6). Then the pullback
$\big{(}\mathrm{id}\times\tilde{I}_{v}\big{)}^{*}\big{(}E,\nabla^{\rm
flat}\big{)}$ is a family of integrable connections relative to $\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon]$ whose induced relative connection is
$\big{(}\mathrm{id}\times\tilde{I}_{v}\big{)}^{*}(E,\nabla)$. So we can extend
the relative meromorphic differential $v_{\alpha}=B_{\alpha}{\rm d}z$ to a
total differential $v^{\rm flat}_{\alpha}=B_{\alpha}{\rm
d}z+\sum_{i=1}^{N}C^{i}_{\alpha}{\rm d}\theta_{i}$ which satisfies the
patching condition

$(\mathrm{id}+\epsilon u_{\alpha\beta})\circ\big{(}\nabla^{\rm flat}+\epsilon
v^{\rm flat}_{\beta}\big{)}=\big{(}\nabla^{\rm flat}+\epsilon v^{\rm
flat}_{\alpha}\big{)}\circ(\mathrm{id}+\epsilon u_{\alpha\beta})$

on $E|_{U_{\alpha\beta}}\otimes\mathbb{C}[\epsilon]$ and the integrability
condition

$\big{(}\nabla^{\rm flat}+\epsilon v^{\rm
flat}_{\alpha}\big{)}\circ\big{(}\nabla^{\rm flat}+\epsilon v^{\rm
flat}_{\alpha}\big{)}=0.$

Let

$\displaystyle\nabla^{\rm flat}_{{\dagger}}\colon\ {\mathcal{E}}{\rm
nd}(E_{y\times{\mathcal{T}^{\prime}}})\ni u\mapsto\nabla^{\rm flat}\circ
u-(u\otimes\mathrm{id})\circ\nabla^{\rm flat}\in{\mathcal{E}}{\rm
nd}(E_{y\times{\mathcal{T}^{\prime}}})\otimes\Omega^{1}_{{\mathcal{C}}\times_{\mathcal{T}}(y\times{\mathcal{T}}^{\prime})}(D_{y\times{\mathcal{T}}^{\prime}})$

be the induced connection on ${\mathcal{E}}{\rm
nd}(E|_{{\mathcal{C}}_{y\times{\mathcal{T}^{\prime}}}})$. Then the above two
equalities become

$\displaystyle\nabla^{\rm flat}_{{\dagger}}(u_{\alpha\beta})=v_{\beta}^{\rm
flat}-v_{\alpha}^{\rm flat},\qquad\nabla^{\rm flat}_{{\dagger}}\big{(}v^{\rm
flat}_{\alpha}\big{)}=0.$

We can check the equalities

$\displaystyle{\rm d}\mathop{\rm
Tr}\nolimits\big{(}u_{\alpha\beta}u^{\prime}_{\beta\gamma}\big{)}=\mathop{\rm
Tr}\nolimits\big{(}\nabla^{\rm
flat}_{{\dagger}}(u_{\alpha\beta}u^{\prime}_{\beta\gamma})\big{)}=\mathop{\rm
Tr}\nolimits\big{(}\nabla^{\rm
flat}_{{\dagger}}(u_{\alpha\beta})u^{\prime}_{\beta\gamma}+u_{\alpha\beta}\nabla^{\rm
flat}_{{\dagger}}(u^{\prime}_{\beta\gamma})\big{)}$
$\displaystyle\hphantom{{\rm d}\mathop{\rm
Tr}\nolimits\big{(}u_{\alpha\beta}u^{\prime}_{\beta\gamma}\big{)}}{}=\mathop{\rm
Tr}\nolimits\big{(}(v_{\beta}^{\rm flat}-v_{\alpha}^{\rm
flat})u^{\prime}_{\beta\gamma}+u_{\alpha\beta}(v^{\prime\rm
flat}_{\gamma}-v^{\prime\rm flat}_{\beta})\big{)}$
$\displaystyle\hphantom{{\rm d}\mathop{\rm
Tr}\nolimits\big{(}u_{\alpha\beta}u^{\prime}_{\beta\gamma}\big{)}}{}=\mathop{\rm
Tr}\nolimits\big{(}v_{\beta}^{\rm
flat}u^{\prime}_{\beta\gamma}-v_{\alpha}^{\rm
flat}(u^{\prime}_{\alpha\gamma}-u^{\prime}_{\alpha\beta})+(u_{\alpha\gamma}-u_{\beta\gamma})v^{\prime\rm
flat}_{\gamma}-u_{\alpha\beta}v^{\prime\rm flat}_{\beta}\big{)}$
$\displaystyle\hphantom{{\rm d}\mathop{\rm
Tr}\nolimits\big{(}u_{\alpha\beta}u^{\prime}_{\beta\gamma}\big{)}}{}=\mathop{\rm
Tr}\nolimits\big{(}{-}u_{\beta\gamma}v^{\prime\rm flat}_{\gamma}+v^{\rm
flat}_{\beta}u^{\prime}_{\beta\gamma}\big{)}-\mathop{\rm
Tr}\nolimits\big{(}{-}u_{\alpha\gamma}v^{\prime\rm flat}_{\gamma}+v^{\rm
flat}_{\alpha}u^{\prime}_{\alpha\gamma}\big{)}$ $\displaystyle\hphantom{{\rm
d}\mathop{\rm
Tr}\nolimits\big{(}u_{\alpha\beta}u^{\prime}_{\beta\gamma}\big{)}=}{}+\mathop{\rm
Tr}\nolimits\big{(}{-}u_{\alpha\beta}v^{\prime\rm flat}_{\beta}+v^{\rm
flat}_{\alpha}u^{\prime}_{\alpha\beta}\big{)},$ $\displaystyle{\rm
d}\mathop{\rm Tr}\nolimits\big{(}{-}u_{\alpha\beta}v^{\prime\rm
flat}_{\beta}+v^{\rm flat}_{\alpha}u^{\prime}_{\alpha\beta}\big{)}=\mathop{\rm
Tr}\nolimits\big{(}\nabla^{\rm
flat}_{{\dagger}}\big{(}{-}u_{\alpha\beta}v^{\prime\rm flat}_{\beta}+v^{\rm
flat}_{\alpha}u^{\prime}_{\alpha\beta}\big{)}\big{)}$
$\displaystyle\hphantom{{\rm d}\mathop{\rm
Tr}\nolimits\big{(}{-}u_{\alpha\beta}v^{\prime\rm flat}_{\beta}+v^{\rm
flat}_{\alpha}u^{\prime}_{\alpha\beta}\big{)}}{}=\mathop{\rm
Tr}\nolimits\big{(}{-}\nabla^{\rm flat}_{{\dagger}}(u_{\alpha\beta})\wedge
v^{\prime\rm flat}_{\beta}-v^{\rm flat}_{\alpha}\wedge\nabla^{\rm
flat}_{{\dagger}}(u^{\prime}_{\alpha\beta})\big{)}$
$\displaystyle\hphantom{{\rm d}\mathop{\rm
Tr}\nolimits\big{(}{-}u_{\alpha\beta}v^{\prime\rm flat}_{\beta}+v^{\rm
flat}_{\alpha}u^{\prime}_{\alpha\beta}\big{)}}{}=\mathop{\rm
Tr}\nolimits\big{(}\big{(}v^{\rm flat}_{\alpha}-v^{\rm
flat}_{\beta}\big{)}\wedge v^{\prime\rm flat}_{\beta}\big{)}-\mathop{\rm
Tr}\nolimits\big{(}v^{\rm flat}_{\alpha}\wedge\big{(}v^{\prime\rm
flat}_{\beta}-v^{\prime\rm flat}_{\alpha}\big{)}\big{)}$
$\displaystyle\hphantom{{\rm d}\mathop{\rm
Tr}\nolimits\big{(}{-}u_{\alpha\beta}v^{\prime\rm flat}_{\beta}+v^{\rm
flat}_{\alpha}u^{\prime}_{\alpha\beta}\big{)}}{}=-\mathop{\rm
Tr}\nolimits\big{(}v^{\rm flat}_{\beta}\wedge v^{\prime\rm
flat}_{\beta}\big{)}+\mathop{\rm Tr}\nolimits\big{(}v^{\rm
flat}_{\alpha}\wedge v^{\prime\rm flat}_{\alpha}\big{)}$

and

$\displaystyle{\rm d}\mathop{\rm Tr}\nolimits\big{(}v_{\alpha}^{\rm
flat}\wedge v^{\prime\rm flat}_{\alpha}\big{)}=\mathop{\rm
Tr}\nolimits\big{(}\nabla^{\rm flat}_{{\dagger}}\big{(}v^{\rm
flat}_{\alpha}\wedge v^{\prime\rm flat}_{\alpha}\big{)}\big{)}=0.$

Therefore, $\big{[}\big{\\{}\mathop{\rm
Tr}\nolimits(u_{\alpha\beta}u^{\prime}_{\beta\gamma}),-\mathop{\rm
Tr}\nolimits\big{(}u_{\alpha\beta}v^{\prime\rm flat}_{\beta}+v^{\rm
flat}_{\alpha}u^{\prime}_{\alpha\beta}\big{)},\mathop{\rm
Tr}\nolimits\big{(}v^{\rm flat}_{\alpha}\wedge v^{\prime\rm
flat}_{\alpha}\big{)}\big{\\}}\big{]}$ defines an element of
$\mathbb{H}^{2}\big{(}\Omega^{\bullet}_{{\mathcal{C}}_{y\times{\mathcal{T}}^{\prime}}}\big{)}\cong\mathbb{C}$
and its image by the map (11.8) coincides with

$\omega\big{(}\tilde{v},\tilde{v^{\prime}}\big{)}=\big{[}\big{\\{}\mathop{\rm
Tr}\nolimits(u_{\alpha\beta}u^{\prime}_{\beta\gamma}),-\mathop{\rm
Tr}\nolimits(u_{\alpha\beta}v^{\prime}_{\beta}+v_{\alpha}u^{\prime}_{\alpha\beta})\big{\\}}\big{]}.$

Thus we have proved the corollary. ∎

### Acknowledgements

The author would like to thank Professor Takuro Mochizuki for having a
discussion and giving a useful advice. The author also would like to thank
Professor Arata Komyo for having useful discussions frequently. The author
would like to thank the referee for valuable comments to improve of the paper.
This work is partially supported by JSPS Grant-in-Aid for Scientific Research
(C) 19K03422.

## References

  * [1] Babbitt D.G., Varadarajan V.S., Local moduli for meromorphic differential equations, Astérisque 169–170 (1989), 217 pages.
  * [2] Biquard O., Boalch P., Wild non-abelian Hodge theory on curves, Compos. Math. 140 (2004), 179–204, arXiv:math.DG/0111098.
  * [3] Biswas I., Heu V., Hurtubise J., Isomonodromic deformations of logarithmic connections and stability, Math. Ann. 366 (2016), 121–140, arXiv:1505.05327.
  * [4] Biswas I., Heu V., Hurtubise J., Isomonodromic deformations and very stable vector bundles of rank two, Comm. Math. Phys. 356 (2017), 627–640, arXiv:1703.07203.
  * [5] Boalch P., Symplectic manifolds and isomonodromic deformations, Adv. Math. 163 (2001), 137–205, arXiv:2002.00052.
  * [6] Boalch P., Geometry and braiding of Stokes data; fission and wild character varieties, Ann. of Math. 179 (2014), 301–365, arXiv:1111.6228.
  * [7] Boalch P., Yamakawa D., Twisted wild character varieties, arXiv:1512.08091.
  * [8] Bremer C.L., Sage D.S., Isomonodromic deformations of connections with singularities of parahoric formal type, Comm. Math. Phys. 313 (2012), 175–208, arXiv:1010.2292.
  * [9] Bremer C.L., Sage D.S., Moduli spaces of irregular singular connections, Int. Math. Res. Not. 2013 (2013), 1800–1872, arXiv:1004.4411.
  * [10] Diarra K., Loray F., Normal forms for rank two linear irregular differential equations and moduli spaces, Period. Math. Hungar. 84 (2022), 303–320, arXiv:1907.07678.
  * [11] Inaba M.-a., Moduli of parabolic connections on curves and the Riemann–Hilbert correspondence, J. Algebraic Geom. 22 (2013), 407–480, arXiv:math.AG/0602004.
  * [12] Inaba M.-a., Unfolding of the unramified irregular singular generalized isomonodromic deformation, Bull. Sci. Math. 157 (2019), 102795, 121 pages, arXiv:1903.08396.
  * [13] Inaba M.-a., Moduli space of irregular singular parabolic connections of generic ramified type on a smooth projective curve, Asian J. Math. 26 (2022), 1–36, arXiv:1606.02369.
  * [14] Inaba M.-a., Iwasaki K., Saito M.-H., Moduli of stable parabolic connections, Riemann–Hilbert correspondence and geometry of Painlevé equation of type VI. I, Publ. Res. Inst. Math. Sci. 42 (2006), 987–1089, arXiv:math.AG/0309342.
  * [15] Inaba M.-a., Saito M.-H., Moduli of unramified irregular singular parabolic connections on a smooth projective curve, Kyoto J. Math. 53 (2013), 433–482, arXiv:1203.0084.
  * [16] Jimbo M., Miwa T., Ueno K., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and $\tau$-function, Phys. D 2 (1981), 306–352.
  * [17] Komyo A., Description of generalized isomonodromic deformations of rank two linear differential equationsusing apparent singularities, arXiv:2003.08045.
  * [18] Komyo A., Hamiltonian structures of isomonodromic deformations on moduli spaces of parabolic connections, J. Math. Soc. Japan 74 (2022), 473–519, arXiv:1611.03601.
  * [19] Krichever I., Isomonodromy equations on algebraic curves, canonical transformations and Whitham equations, Mosc. Math. J. 2 (2002), 717–752, arXiv:hep-th/0112096.
  * [20] Malgrange B., Sur les déformations isomonodromiques. II. Singularités irrégulières, in Mathematics and Physics (Paris, 1979/1982), Progr. Math., Vol. 37, Birkhäuser Boston, Boston, MA, 1983, 427–438.
  * [21] Mochizuki T., Wild harmonic bundles and wild pure twistor $D$-modules, Astérisque 340 (2011), x+607 pages, arXiv:0803.1344.
  * [22] Pantev T., Toën B., Moduli of flat connections on smooth varieties, Algebr. Geom. 9 (2022), 266–310, arXiv:1905.12124.
  * [23] Sibuya Y., Linear differential equations in the complex domain: problems of analytic continuation, Transl. Math. Monogr., Vol. 82, Amer. Math. Soc., Providence, RI, 1990.
  * [24] Simpson C.T., Harmonic bundles on noncompact curves, J. Amer. Math. Soc. 3 (1990), 713–770.
  * [25] Simpson C.T., Moduli of representations of the fundamental group of a smooth projective variety. I, Inst. Hautes Études Sci. Publ. Math. 79 (1994), 47–129.
  * [26] Simpson C.T., Moduli of representations of the fundamental group of a smooth projective variety. II, Inst. Hautes Études Sci. Publ. Math. 80 (1994), 5–79.
  * [27] van der Put M., Saito M.-H., Moduli spaces for linear differential equations and the Painlevé equations, Ann. Inst. Fourier (Grenoble) 59 (2009), 2611–2667, arXiv:0902.1702.
  * [28] Wasow W., Asymptotic expansions for ordinary differential equations, Pure Appl. Math., Vol. 14, Interscience Publishers John Wiley & Sons, Inc., New York, 1965.


* [7] J. A. Barmak, E. G. Minian, _Strong homotopy types, nerves and collapses_ , Discrete Comput. Geom. 47 (2012), pp. 301–328.
  * [8] A. Björner, _Topological methods_ , Handbook of combinatorics 2 (1995), pp. 1819–1872.
  * [9] B. Braun, J. Browder, S. Klee, _Cellular resolutions of ideals defined by nondegenerate simplicial homomorphisms_ , Israel J. Math. 196 (2013), pp. 321 – 344.
  * [10] B. Braun, W. K. Hough, _Homomorphism complexes and maximal chains in graded posets_ , European J. Combin. 81 (2019), pp. 178–194.
  * [11] G. R. Brightwell, P. Winkler, _Gibbs Measures and Dismantlable Graphs_ , J. Comb. Theory Ser. B 78 (2000), no. 1, pp. 141–166.
  * [12] G. R. Brightwell, P. Winkler, _Graph homomorphisms and long range action_ , Graphs, morphisms and statistical physics, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 63, Amer. Math. Soc., Providence, RI, 2004, pp. 29–47.
  * [13] P. Bubenik, N. Milićević, _Applied topology using Čech’s closure spaces_ , preprint (2021), arxiv.org/abs/2104.10206.
  * [14] L. Cereceda, J. van den Heuvel, M. Johnson, _Connectedness of the graph of vertex-colourings_ , Discrete Math. 308 (2008), pp. 913–919.
  * [15] L. Cereceda, J. van den Heuvel, M. Johnson, _Mixing 3-colourings in bipartite graphs_ , European J. Combin. 30 (2009), no. 7, pp. 1593–1606.
  * [16] T. Chih, L. Scull, _A homotopy category for graphs_ , J. Algebraic Comb. 53 (2021), pp. 1231–1251.
  * [17] P. Csorba, _Fold and Mycielskian on homomorphism complexes_ , Contrib. Discrete Math. 3 (2008), pp. 1–8.
  * [18] P. Csorba, _Homotopy types of box complexes_ , Combinatorica 27 (2007), no. 6, pp. 669–682.
  * [19] P. Csorba, _Non-Tidy Spaces and Graph Colorings_. PhD thesis, ETH Zürich, 2005.
  * [20] P. Csorba, C. Lange, I. Schurr, A. Wassmer, _Box complexes, neighborhood complexes, and the chromatic number_ , J. Combin. Theory Ser. A 108 (2004) pp. 159–168.
  * [21] P. Csorba, F. Lutz, _Graph coloring manifolds_ , in: Algebraic and Geometric Combinatorics, Contemporary Mathematics 423, American Mathematical Society, Providence, RI, 2007.
  * [22] S. Čukíc, D. N. Kozlov, _Higher connectivity of graph coloring complexes_ , Int. Math. Res. Not. 2005, no. 25, pp. 1543–1562.
  * [23] S. Lj. Čukíc, D. N. Kozlov, _The homotopy type of the complexes of graph homomorphisms between cycles_ , Discrete Comput. Geom. 36 (2006), no. 2, pp. 313–329.
  * [24] J. De Loera, J. Rambau, F. Santos, _Triangulations, Structures for Algorithms and Applications_ , Algorithms and Computation in Mathematics, vol. 25, Springer-Verlag Berlin Heidelberg, 2010.
  * [25] M. Develin, B. Sturmfels, _Tropical convexity_ , Documenta Math. 9 (2004), pp. 1–27.
  * [26] A. Dochtermann, _Hom complexes and homotopy in the category of graphs_ , European J. Combin. 30 Issue 2 (2009), pp. 490–509.
  * [27] A. Dochtermann, _Homotopy groups of Hom complexes of graphs_ , J. Combin. Theory Ser. A 116, (2009), pp. 180–194.
  * [28] A. Dochtermann, _The universality of Hom complexes of graphs_ , Combinatorica 29 (2009), pp. 433–448.
  * [29] A. Dochtermann, A. Engström, _Cellular resolutions of cointerval ideals_ , Mathematische Zeitschrift 270 (2012), pp. 145–163.
  * [30] A. Dochtermann, R. Freij-Hollanti, _Warmth and edge spaces of graphs_ , Adv. Appl. Math. 96 (2018), pp. 176–194.
  * [31] A. Dochtermann, C. Schultz, _Topology of Hom complexes and test graphs for bounding chromatic number_ , Israel J. Math. 187 (2012), pp. 371–417.
  * [32] C. H. Dowker, _Homology Groups of Relations_ , Ann. of Math. (2) 56 (1952), pp. 84–95.
  * [33] J. M. Droz, _Quillen model structures on the category of graphs_ , Homology Homotopy Appl. 14 (2012), no. 2, pp. 265–284.
  * [34] A. Engström, _A short proof of a conjecture on the higher connectivity of graph coloring complexes_ , Proc. Amer. Math. Soc. 134 (2006), no. 12, pp. 3703–3705.
  * [35] L. Fajstrup, E. Goubault, M. Raussen, _Algebraic Topology and Concurrency_ , Theoretical Computer Science 357, (2006), pp. 241–278.
  * [36] R. Forman, _Morse theory for cell complexes_ , Adv. Math. 134 (1998), no. 1, pp. 90–145.
  * [37] P. Gaucher, _Six model categories for directed homotopy_ , Categories Gen. Algebraic Struct. 15 (2021), no. 1, pp. 145–181.
  * [38] C. Giusti, E. Pastalkova, C. Curto, V. Itskovm, _Clique topology reveals intrinsic geometric structure in neural correlations_ , PNAS 112 (2015), no. 44, pp. 13455–13460.
  * [39] S. Goyal, R. Santhanam, _(Lack of) model structures on the category of graphs_ , Appl. Categ. Structures 29 (2021), no. 4, pp. 671–683.
  * [40] A. Grigor’yan, Y. Lin, Y. V. Muranov, S. T. Yau, _Homotopy theory of digraphs_ , Pure and App. Math. Quat. 10 (2014), pp. 619–674.
  * [41] P. Hell, J. Nešetřil, Graphs and Homomorphisms, Oxford Lecture Series in Mathematics and Its Applications 28, Oxford University Press, 2004.
  * [42] K. Iriye, D. Kishimoto, _Hom complexes and hypergraph colorings_ , Topology and Appl. 160 (2013), pp. 1333–1344.
  * [43] M. Kahle, _The neighborhood complex of a random graph_ , J. Combin. Theory Ser. A 114 (2007), pp. 380–387.
  * [44] A. V. Kostochka, E. Sopena, X. Zhu, _Acyclic and oriented chromatic numbers of graphs_ , J. Graph Theory 24 (1997), no. 4, pp. 331–340.
  * [45] D. N. Kozlov, _A simple proof for folds on both sides in complexes of graph homomorphisms_ , Proc. Amer. Math. Soc. 134 (2006), no. 5, pp. 1265–1270.
  * [46] D. N. Kozlov, _Combinatorial Algebraic Topology,_ Algorithms and Computation in Mathematics 21, Springer, Berlin, 2008.
  * [47] H. G. Landau, _On dominance relations and the structure of animal societies. III. The condition for a score structure_ , Bulletin of Mathematical Biophysics 15 (1953), no. 2, pp. 143–148.
  * [48] C. Lange, G. Ziegler, _On generalized Kneser hypergraph colorings_ , J. Combin. Theory Ser. A 114 (2007), no. 1, pp. 159–166.
  * [49] L. Lovász, _Kneser’s conjecture, chromatic number, and homotopy_ , J. Combin. Theory Ser. A 25 (1978), no. 3, pp. 319–324.
  * [50] G. Malen, _Homomorphism complexes and $k$-cores_, Discrete Math. 341 (2018), no. 9, pp. 2567–2574.
  * [51] P. Masulli, A. E. P. Villam, _The topology of the directed clique complex as a network invariant_ , SpringerPlus, 5 (2016), no. 1.
  * [52] J. Matoušek, _Using the Borsuk-Ulam theorem. Lectures on topological methods in combinatorics and geometry_ , with A. Björner and G.M. Ziegler, Universitext, Springer-Verlag, Berlin, 2003.
  * [53] J. Matoušek, G. M. Ziegler, _Topological lower bounds for the chromatic number: A hierarchy_ , Jahresber. Dtsch. Math.-Ver. 106 (2004), pp. 71–90.
  * [54] T. Matsushita, _Morphism complexes of sets with relations_ , Osaka J. Math. 53 (2016), pp. 267–283.
  * [55] T. Matsushita, _Box complexes and homotopy theory of graphs_ , Homology Homotopy Appl. 19 (2017), no. 2, pp. 175–197.
  * [56] T. Matsushita, _Homotopy types of the Hom complexes of graphs_ , European J. Combin., 63 (2017), pp. 216–226.
  * [57] J. Mycielski, _Sur le coloriage des graphes_ , Colloq. Math. 3 (1955), pp. 161–162.
  * [58] C. M. Mynhardt, S. Nasserasr, _Reconfiguration of Colourings and Dominating Sets in Graphs_ , in 50 Years of Combinatorics, Graph Theory, and Computing, Chapman and Hall/CRC, 2019.
  * [59] U. Nagel, V. Reiner, _Betti numbers of monomial ideals and shifted skew shapes_ , Electron. J. Combin. 16 (2009), no. 2, Special volume in honor of Anders Björner, Research Paper 3, 59 pp.
  * [60] W. A. Stein et al. Sage Mathematics Software (Version 9.0), The Sage Development Team, 2020, http://www.sagemath.org.
  * [61] C. Schultz, _Small models of graph colouring manifolds and the Stiefel manifolds ${\text{H}om}(C_{5},K_{n})$_, J. Combin. Theory Ser. A 115 (2008), pp. 84–104.
  * [62] C. Schultz, _Graph colorings, spaces of edges and spaces of circuits_ , Adv. Math. 221 (2009), no. 6, pp. 1733–1756.
  * [63] G. Simonyi, G. Tardos, _Local chromatic number, Ky Fan’s theorem, and circular colorings_ , Combinatorica 26 (2006), no. 5, pp. 587–626.
  * [64] G. Simonyi, G. Tardos, S. Vrećica, _Local chromatic number and distinguishing the strength of topological obstructions_ , Trans. Amer. Math. Soc. 361 (2009), no. 2, pp. 889–908.
  * [65] B. Sturmfels, _Grobner Bases and Convex Polytopes_ , American Mathematical Society, Univ. Lectures Series, No 8, Providence, 1996.
  * [66] M. Wrochna, _Homomorphism reconfiguration via homotopy_ , SIAM J. Discrete Math. 34 (2014), no. 1, pp. 328–350.
  * [67] G. M. Ziegler, _Generalized Kneser coloring theorems with combinatorial proofs_ , Invent. Math. 147 (2002), pp. 671–691.
  * [68] R. T. Živaljević, _WI-posets, graph complexes and ${\mathbb{Z}}_{2}$-equivalences_, J. Combin. Theory Ser. A 111 (2005), pp. 204–223.