text:
\begin{document}

\frontmatter

\begin{abstract}
A.\ Girand has constructed an explicit two-parameter family of flat connections
over the complex projective plane $\mathbb{P}^2$.
These connections have dihedral monodromy
and their polar locus is a prescribed quintic composed of a conic and three tangent lines.
In this paper, we give a generalization of this construction.
That is, we construct an explicit $n$-parameter family of flat connections
over the complex projective space $\mathbb{P}^n$.
Moreover, we discuss the relation between these connections and the Garnier system.
\end{abstract}

\begin{altabstract}
A.\ Girand a construit une famille explicite de connexions \`{a} deux param\`{e}tres
sur le plan projectif complexe $\mathbb{P}^2$.
Ces connexions ont une monodromie di\'{e}drale
et leur lieu polaire est une quintique prescrite compos\'{e}e
d'une conique et de trois droites tangentes.
Dans cet article, nous donnons une g\'{e}n\'{e}ralisation de cette construction.
Autrement dit, nous construisons une famille de connexions plates explicite
\`a $n$ param\`etres
sur l'espace projectif complexe $\mathbb{P}^n$.
De plus, nous discutons de la relation entre ces connexions et le syst\`{e}me Garnier.
\end{altabstract}

\subjclass{14H05, 14F35, 34M55}
\keywords{algebraic function, Garnier system, isomonodromic deformation.}

\maketitle

\section{Introduction}

A \textit{meromorphic rank $2$ connection} $(E,\nabla)$ on a projective manifold $X$
is tha datum of a rank $2$ vector bundle $E$ equipped with a $\mathbb{C}$-linear
morphism $\nabla \colon E \ra E \otimes \Omega^1_X (D)$ satisfying
the Leibniz rule
\begin{equation*}
\nabla (f \cdot s) = f \cdot \nabla(s) + df \otimes s
\end{equation*}
for any section $s$ and function $f$.
Here $D$ is the polar divisor of the connection $\nabla$.
The connection $\nabla$ is \textit{flat} when the curvature vanishes,
that is $\nabla \cdot \nabla=0$.
For a flat meromorphic rank $2$ connection, we can define its monodromy representation.
When $\det(E) = \mathcal{O}_X$ and the trace connection $\mathrm{tr} (\nabla)$
is the trivial connection on $\mathcal{O}_X$,
we say that $(E,\nabla)$ is an \textit{$\mathfrak{sl}_2$-connection}.
A connection $(E,\nabla)$ is called \textit{regular}
if local $\nabla$-horizontal sections
have moderate growth near the polar divisor $D$
(for details, see \cite[Chap.\ II, Definition 4.2]{Deligne}).

In this paper, we introduce
a family, parametrized by $\boldsymbol{\lambda} \in \mathbb{C}^n$,
of meromorphic $\mathfrak{sl}_2$-connections
$\nabla_{\boldsymbol{\lambda}} = d + A_{\boldsymbol{\lambda}}$ on the trivial bundle
$\mathcal{O}_{\mathbb{P}^n}\oplus\mathcal{O}_{\mathbb{P}^n}$
over $\mathbb{P}^n$ with $n \ge 2$,
with an explicit connection matrix $A_{\boldsymbol{\lambda}}$.

\subsection{The explicit expression of $\nabla_{\boldsymbol{\lambda}}$}\label{2019.7.10.22.19}

The explicit connection matrix $A_{\boldsymbol{\lambda}}$ is described as follows.
Let $[x:y:z_1:\ldots:z_{n-2}:t]$ be the homogeneous coordinates of $\mathbb{P}^n$.
Set $f(x,y,t):=x^2 + y^2 +t^2  -2 (x y +yt   +  t x)$.
For $\boldsymbol{\lambda}= (\lambda_0 , \ldots, \lambda_{n-1}) \in \mathbb{C}^n$,
we define rational $1$-forms on $\mathbb{P}^n$ as follows:
\begin{equation*}
\begin{aligned}
\alpha_0(x,y) &:= -
\frac{(2 \lambda_0 + \lambda_1)dx - (2 \lambda_1 + \lambda_0)dy}{2} -
\frac{ \lambda_1 ( y-1)}{2}\frac{ dx}{x} +
\frac{ \lambda_0 ( x-1)}{2}\frac{ dy}{y}  ,\\
\alpha_1(x,y) &:= - \frac{1}{4}\frac{ d f(x,y,1)}{f(x,y,1)}, \quad
\alpha_2(x,y):= -\frac{\alpha_0(x,y)}{f(x,y,1)},
\end{aligned}
\end{equation*}
and
\begin{equation*}
\begin{aligned}
\alpha^i_0(x,y, z_i) &:=\lambda_{i+1} \left(  d z_i-
\frac{ z_id (f(x,y,1)-z_i^2) }{2(f(x,y,1)-z_i^2)} \right), \quad
\alpha^i_2(x,y, z_i):= -\frac{\alpha^i_0(x,y, z_i)}{f(x,y,1)}  ,
\end{aligned}
\end{equation*}
which are described by the affine coordinates $[x:y:z_1:\ldots:z_{n-2}:1]$.
We define a connection matrix $A_{\boldsymbol{\lambda}}$ as
\begin{equation*}
\begin{aligned}
A_{\boldsymbol{\lambda}}=
\begin{pmatrix}
\cA_{11}  & \cA_{12}  \\
-\cA_{21}  &- \cA_{11}
\end{pmatrix} + \sum_{i=1}^{n-2}
\begin{pmatrix}
\cA_{11}^i  & \cA_{12}^i  \\
-\cA_{21}^i &- \cA_{11}^i
\end{pmatrix},
\end{aligned}
\end{equation*}
\textit{where}
\begin{equation*}
\begin{aligned}
\cA_{11} &:=
 (x-1)  \alpha_2 (x,y)
 +  \alpha_1(x,y) +\frac{1}{2}\frac{d y}{y} ,\\
\cA_{11}^i &:=  (x-1)  \alpha^i_2 (x,y, z_i), \\
\cA_{12}  &:= \frac{dx    + (x-1)^2\alpha_2 (x,y)
 + 2 (x -1)\alpha_1(x,y)  + \alpha_0(x,y)  }{y} , \\
\cA_{12}^i &:=
 \frac{(x-1)^2\alpha^i_2 (x,y, z_i) + \alpha^{i}_0(x,y, z_i)  }{y},\\
\cA_{21}&:=  y  \alpha_2 (x,y),  \\
\cA_{21}^i &:= y \alpha^i_2 (x,y, z_i)  ,
\end{aligned}
\end{equation*}
in the affine coordinates $[x:y:z_1:\ldots:z_{n-2}:1]$.

\subsection{Main results}

Let $\mathcal{Q}_0$ and $\mathcal{Q}_i$ be the divisors on $\mathbb{P}^n$ defined by
$\mathcal{Q}_0:= ( f (x, y ,t) =0 )$ and
$\mathcal{Q}_i:= ( f (x, y ,t)  - z_i^2  =0 )$, $i=1,\ldots,n-2$,
respectively.
Let $D_n$ be the divisor on $\mathbb{P}^n$ defined by
\begin{equation*}
D_n := ( x=0 ) + ( y=0 ) + ( t=0 ) +
\mathcal{Q}_0 + \mathcal{Q}_1 + \cdots + \mathcal{Q}_{n-2}.
\end{equation*}
From the explicit expression of $\nabla_{\boldsymbol{\lambda}}$, it follows that
all $\nabla_{\boldsymbol{\lambda}}$ share the same polar divisor $D_n$.
Note that the conic $\mathcal{Q}_0$ plays a special role:
it is tangent to the conic $\mathcal{Q}_i$ inside the coordinate hyperplane $(z_i=0)$
for $i=1,\ldots,n-2$,
and it is tangent to the three coordinate hyperplanes $(x=0)$, $(y=0)$ and $(t=0)$.

\begin{theo}\label{2019.7.10.21.43}
\textit{
For each $\boldsymbol{\lambda}$, the connection $\nabla_{\boldsymbol{\lambda}}$
is flat and has at worst regular singularities.
}
\end{theo}

We say that two connections $(E,\nabla)$ and $(E',\nabla')$
are \textit{birationally equivalent} when there is a birational bundle transformation
$\phi \colon E \dashrightarrow E'$ that conjugates the two operators $\nabla$
and $\nabla'$.
We say that two connections $(E,\nabla)$ and $(E',\nabla')$
are \textit{projectively equivalent} if the induced $\mathbb{P}^1$-bundles
coincide $\mathbb{P}(E)=\mathbb{P}(E')$, and if moreover $\nabla$ and $\nabla'$
induce the same projective connection $\mathbb{P}(\nabla)= \mathbb{P}(\nabla')$.

\begin{theo}\label{2019.7.10.15.33}
\textit{Via a generically finite Galois morphism
$f \colon \mathbb{P}^n \rightarrow \mathbb{P}^n$, for each $\boldsymbol{\lambda}$,
the pull-back connection $f^*\nabla_{\boldsymbol{\lambda}}$ on the trivial bundle
is projectively birationally equivalent to a split connection of the form
\begin{equation*}
d+ \begin{pmatrix}
\omega & 0 \\
0 &-\omega
\end{pmatrix}
\end{equation*}
with $\omega$ a rational closed $1$-form on $X$.
}
\end{theo}

In this case, the generically finite Galois morphism is a genetically finite morphism of degree two.
Loray, Pereira, and Touzet have proved
the structure theorem of flat meromorphic $\mathfrak{sl}_2$-connections
on projective manifolds in \cite{LPT}
(see also \cite{CS}).
By Theorem \ref{2019.7.10.15.33}, each $\nabla_{\boldsymbol{\lambda}}$ is the first type of
the three possible types of flat meromorphic $\mathfrak{sl}_2$-connections
over projective manifolds in the sense of Loray, Pereira, and Touzet \cite[Theorem E]{LPT}.

Since the connection $\nabla_{\boldsymbol{\lambda}}$ is flat for each $\boldsymbol{\lambda}$,
we can define its monodromy representation
$\pi_1 (\mathbb{P}^n \setminus D_n)
\rightarrow
\mathrm{SL}_2(\mathbb{C})$
of $\nabla_{\boldsymbol{\lambda}}$ for each $\boldsymbol{\lambda}$.
Let $\boldsymbol{D}_{\infty}$ be the infinite dihedral group:
\begin{equation*}
\boldsymbol{D}_{\infty}:= \left\langle
\begin{pmatrix}
0 & \alpha \\
-\alpha^{-1} & 0
\end{pmatrix},
\begin{pmatrix}
\beta & 0 \\
0 & \beta^{-1}
\end{pmatrix}\
\middle| \
\alpha , \beta \in \mathbb{C}^*
 \right\rangle \le \mathrm{SL}_2 (\mathbb{C}).
\end{equation*}
For the monodromy representation of $\nabla_{\boldsymbol{\lambda}}$,
we have the following.

\begin{theo}\label{2019.7.10.22.03}
\textit{
For generic $\boldsymbol{\lambda}$,
the monodromy representation of $\nabla_{\boldsymbol{\lambda}}$ is conjugated to an explicit
representation
\begin{equation*}
\rho_{\boldsymbol{\lambda}} \colon \pi_1 (\mathbb{P}^n \setminus D_n)
\longrightarrow
\mathrm{SL}_2(\mathbb{C}),
\end{equation*}
which is virtually abelian, i.e. abelian after a finite cover of $\mathbb{P}^n \setminus D_n$,
and takes values in the infinite dihedral group $\boldsymbol{D}_{\infty}$.
}
\end{theo}

\subsection{Algebraic Garnier solution}

The $(2n-2)$-variable \textit{Garnier system} $\mathcal{G}_{2n-2}$
is the completely integrable Hamiltonian system
\begin{equation*}
\mathcal{G}_{2n-2} \colon
\left\{
\begin{aligned}
\frac{\partial \rho_j}{\partial t_i} &= -\frac{\partial K_i}{\partial \nu_j} &
i,j= 1,\ldots, 2n-2 \\
\frac{\partial \nu_j}{\partial t_i} &= \frac{\partial K_i}{\partial \rho_j} &
i,j= 1,\ldots, 2n-2,
\end{aligned}
\right.
\end{equation*}
where
\begin{equation*}
K_i= -\frac{\Lambda(t_i)}{T'(t_i)}
\left[ \sum^{2n-2}_{k=1} \frac{T(\nu_k)}{(\nu_k-t_i) \Lambda'(\nu_k)}
\left\{ \rho_k^2 -\sum^{2n}_{m=1} \frac{\theta_m-\delta_{im}}{\nu_k - t_m} \rho_k
+ \frac{\kappa}{\nu_k(\nu_k -1)} \right\}
\right]
\end{equation*}
with $t_{2n-1}=0$, $t_{2n}=1$,
$\kappa:= \frac{1}{4}\left\{ (\sum^{2n}_{m=1} \theta_m -1)^2 - (\theta_{\infty}^2 +1) \right\}$,
$\Lambda(t):= \prod^{2n-2}_{k=1} (t-\nu_k)$ and
$T(t):= \prod^{2n}_{k=1} (t-t_k)$
(see \cite{G1}, \cite{G2}, and \cite{Okamoto}).
Here $\theta_m$ ($m=1,\ldots,2n,\infty$) is the constant parameters defined by
\begin{equation*}
\begin{aligned}
\theta_1&= \frac{1}{2}, & \theta_2&= \frac{1}{2}, &
\theta_{2i+1}&= \lambda_{i+1},&
\theta_{2i+2}&= \lambda_{i+1}\ (i=1,\ldots,n-2) \\
\theta_{2n-1}&= \lambda_{1},&
\theta_{2n}&= \lambda_{0}-1,&
\theta_{\infty}&= \lambda_{0}+\lambda_{1}.
\end{aligned}
\end{equation*}

To give a solution of the Garnier system $\mathcal{G}_{2n-2}$,
we consider the Fuchsian system with $2n + 1$ regular singularities
at $0,1,t_1 , \ldots , t_{2n-2}, \infty$:
\begin{equation}\label{2018.5.22.12.11}
 d+
\tilde{H}_{2n-1}\frac{d\tilde{x}}{\tilde{x}} +
\tilde{H}_{2n}\frac{d\tilde{x}}{\tilde{x}-1} +
\sum_{i=1}^{2n-2} \tilde{H}_{i}\frac{d\tilde{x}}{\tilde{x}-t_i} ,
\end{equation}
where $\tilde{H}_{i}$ ($i=1, \ldots , 2n$) are
$2\times 2$ matrices independent of $\tilde{x}$ and
$t_i \neq t_j$ ($i \neq j$).
We assume that  $\tilde{H}_{2n+1}:=-\sum_{i=1}^{2n} \tilde{H}_{i}$ is a diagonal matrix and
the eigenvalues of $\tilde{H}_{i}$ ($i=1, \ldots , 2n+1$) are
as in Table \ref{2018.5.17.17.04}.

\begin{table}[htb]
\caption{The eigenvalues of the residue matrices
($i=1,\ldots , n-2$).}
  \begin{tabular}{c|ccccccc}\label{2018.5.17.17.04}
  Reside matrices  & $\tilde{H}_{1}$ &
    $\tilde{H}_{2}$ & $\tilde{H}_{2i+1}$ & $\tilde{H}_{2i+2}$
    &  $\tilde{H}_{2n-1}$ &  $\tilde{H}_{2n}$ & $\tilde{H}_{2n+1}$ \\\hline
  Eigenvalues
     & $\pm \frac{1}{4}$
      & $\pm \frac{1}{4}$
      & $\pm \frac{\lambda_{i+1}}{2}$
      & $\pm \frac{\lambda_{i+1}}{2}$
       & $\pm \frac{\lambda_1}{2}$
    & $\pm \frac{\lambda_0-1}{2}$
      &  $\pm\frac{\lambda_0+\lambda_1}{2}$
  \end{tabular}
\end{table}

We fix generators $\gamma_{\tilde{x}}$
($\tilde{x}=0,1, t_1 \ldots ,t_{2n-2}, \infty$)
 of the fundamental group $\pi_1(\mathbb{P}^1 \setminus \{ 0,1,t_1,\ldots,t_{2n-2}, \infty \},*)$.
Here the loop $\gamma_{\tilde{x}}$ on $\mathbb{P}^1$
is oriented counter-clockwise,
$\tilde{x}$ lies inside, while the other singular points lie outside.
Let $\rho'_{\boldsymbol{\lambda}}\colon
\pi_1(\mathbb{P}^1 \setminus\{ t_1,\ldots,t_{2n}, \infty \},*) \ra \mathrm{SL}_2(\mathbb{C})$
be the representation of the fundamental group defined by Table \ref{2018.5.14.12.41}.
If we have the isomonodromic deformation of the Fuchsian system (\ref{2018.5.22.12.11})
whose preserved monodromy representation is conjugated to
 $\rho'_{\boldsymbol{\lambda}}$,
 then we obtain a solution of the Garnier system $\mathcal{G}_{2n-2}$
 (see \cite[Section 2]{Mazz}).

\begin{table}[htb]
\caption{The representation of the fundamental group;
here $a_j = \exp(-\pi \sqrt{-1} \lambda_j)$ $j=0,1,\ldots,n-1$.}
\begin{center}
  \begin{tabular}{c|c|c|c}\label{2018.5.14.12.41}
     $\gamma_0$ &  $\gamma_1$ & $\gamma_{t_1}$ &
    $\gamma_{t_2}$ \\\hline
    $\begin{pmatrix} a_1 & 0 \\ 0 & a_1^{-1} \end{pmatrix}$
    & $\begin{pmatrix} -a_0 & 0 \\ 0 & -a_0^{-1} \end{pmatrix}$
     & $\begin{pmatrix} 0 & 1 \\ -1 &0 \end{pmatrix}$
      & $\begin{pmatrix} 0 & a_0^{2} \\ -a_0^{-2} &0 \end{pmatrix}$
  \end{tabular}\\
  \begin{tabular}{c|c|c}
     $\gamma_{t_{2i+1}}$  ($i=1,\ldots,n-2$) &  $\gamma_{t_{2i+2}}$  ($i=1,\ldots,n-2$)
      & $\gamma_{\infty}$\\\hline
    $\begin{pmatrix} a_{i+1} & 0 \\ 0 & a_{i+1}^{-1} \end{pmatrix}$
    & $\begin{pmatrix} a^{-1}_{i+1} & 0 \\ 0 & a_{i+1} \end{pmatrix}$
     & $\begin{pmatrix} a_0 a_1^{-1} & 0 \\ 0 &a_0^{-1} a_1 \end{pmatrix}$
  \end{tabular}
  \end{center}
\end{table}

We say $(\rho_j(t_1,\ldots,t_{2n-2}), \nu_j(t_1,\ldots,t_{2n-2}))_{j=1,...,2n-2} $
is an \textit{algebraic solution of $\mathcal{G}_{2n-2}$}
if $(\rho_j(t_1,\ldots,t_{2n-2}), \nu_j(t_1,\ldots,t_{2n-2}))_{j=1,...,2n-2} $
satisfies the Garnier system $\mathcal{G}_{2n-2}$
and the graph of the solution has Zariski closure of dimension $2n-2$.

\begin{theo}\label{2019.7.10.22.13}
Let $T$ be a certain Zariski open subset of $\mathbb{A}^{2n-2}$
parametrizing generic lines in $\mathbb{P}^n$.
From the natural morphism $\mathbb{P}^1 \times T \rightarrow \mathbb{P}^n$,
one obtains a relative connection $(\nabla_{\mathbb{P}^1 \times T/T})_{\boldsymbol{\lambda}}$
with $2n+1$ simple poles by the pull-back of $\nabla_{\boldsymbol{\lambda}}$.
\begin{itemize}
\item[(i)] Up to an \'etale base change $\tilde{T} \rightarrow T$, an isomorphism of the
relative trivial bundle, and up to relative M\"obius transformations in the base,
we can consider
the relative connection $(\nabla_{\mathbb{P}^1 \times T/T})_{\boldsymbol{\lambda}}$
as a family of the Fuchsian system (\ref{2018.5.22.12.11})
parametrized by $T$.
\item[(ii)] The family $(\nabla_{\mathbb{P}^1 \times T/T})_{\boldsymbol{\lambda}}$
is isomonodromic.
The preserved monodromy representation of the fundamental group
$\pi_1(\mathbb{P}^1 \setminus \{ 0,1,t_1,\ldots,t_{2n-2}, \infty \},*)$
of this isomonodromic family
 is conjugated to the representation given by Table \ref{2018.5.14.12.41}
 \item[(iii)] Since $\dim \tilde{T}=2n-2$, the connection matrices
 of the isomonodromic family $(\nabla_{\mathbb{P}^1 \times T/T})_{\boldsymbol{\lambda}}$
defines an algebraic solution of the Garnier system $\mathcal{G}_{2n-2}$.
\end{itemize}
\end{theo}

In the case of $n = 2$,
the family of connections $\nabla_{\boldsymbol{\lambda}}$
have been established by Girand in \cite{Girand}.
Moreover Girand have discussed
an explicit relation to certain algebraic solutions of the sixth Painlev\'e equation in \cite{Girand}.
Our argument is generalization of
Girand's idea of explicit construction of $\nabla_{\boldsymbol{\lambda}}$,
and of the proof of the main results, to the case $n \ge 2$.

The organization of this paper is as follows.
In Section \ref{2018.5.18.17.10}, we introduce
a family, parametrized by $\boldsymbol{\lambda} \in \mathbb{C}^n$,
of meromorphic $\mathfrak{sl}_2$-connections
$\nabla_{\boldsymbol{\lambda}} = d + A_{\boldsymbol{\lambda}}$ on the trivial bundle
$\mathcal{O}_{\mathbb{P}^n}\oplus\mathcal{O}_{\mathbb{P}^n}$
over $\mathbb{P}^n$ with $n \ge 2$,
with an explicit connection matrix $A_{\boldsymbol{\lambda}}$.
In Section \ref{2019.7.10.22.00},
we show Theorem \ref{2019.7.10.21.43} and Theorem \ref{2019.7.10.15.33}.
In Section \ref{2018.5.18.17.14}, we compute the monodromy representation of
$\nabla_{\boldsymbol{\lambda}}$ for generic $\boldsymbol{\lambda}$.
In Section \ref{2019.7.10.22.02},
we show Theorem \ref{2019.7.10.22.03}.
In Section \ref{2018.5.18.17.17},
we consider the natural morphism $\mathbb{P}^1 \times T \rightarrow \mathbb{P}^n$,
where $T$ is a certain Zariski open subset of $\mathbb{A}^{2n-2}$
parametrizing generic lines in $\mathbb{P}^n$.
Let $(\nabla_{\mathbb{P}^1 \times T/T})_{\boldsymbol{\lambda}}$ be
the relative connection with $2n+1$ simple poles
given by the pull-back of $\nabla_{\boldsymbol{\lambda}}$.
In Section \ref{2019.7.10.22.11},
we introduce an \'etale base change $\tilde{T} \rightarrow T$
to prove the assertion (i) of Theorem \ref{2019.7.10.22.13}.
In Section \ref{2019.7.12.11.27},
after the \'etale base change $\tilde{T} \rightarrow T$,
we compute the residue matrix of
$(\nabla_{\mathbb{P}^1 \times \tilde{T}/\tilde{T}})_{\boldsymbol{\lambda}}$
for each simple pole.
In Section \ref{2019.7.10.22.15},
we recall the relation between isomonodromic deformations and the Garnier system
following \cite{Mazz}.
In Section \ref{2019.7.12.11.28},
we show Theorem \ref{2019.7.10.22.13}.

\section{Construction of flat connections on projective spaces}\label{2018.5.18.17.10}

In this section,
we introduce
a family, parametrized by $\boldsymbol{\lambda} \in \mathbb{C}^n$,
of meromorphic $\mathfrak{sl}_2$-connections
$\nabla_{\boldsymbol{\lambda}} = d + A_{\boldsymbol{\lambda}}$ on the trivial bundle
$\mathcal{O}_{\mathbb{P}^n}\oplus\mathcal{O}_{\mathbb{P}^n}$
over $\mathbb{P}^n$ with $n \ge 2$,
with the explicit connection matrix $A_{\boldsymbol{\lambda}}$
described in Section \ref{2019.7.10.22.19}.
For this introduction, we start from
a family, parametrized by $\boldsymbol{\lambda} \in \mathbb{C}^n$,
of flat meromorphic $\mathfrak{sl}_2$-connections $(\nabla_0)_{\boldsymbol{\lambda}}$
on the trivial bundle
$\mathcal{O}_{\mathbb{C}^n}\oplus\mathcal{O}_{\mathbb{C}^n}$
over $\mathbb{C}^n$
whose connection matrix splits.
Next, we consider a birational transformation of the projective connection
$\mathbb{P}((\nabla_0)_{\boldsymbol{\lambda}})$.
We define a generically finite Galois morphism
$f \colon \mathbb{C}^n \rightarrow \mathbb{C}^n$.
We show that this birational transformation descend to
 a projective connection over $\mathbb{C}^n$.
We denote by $\mathbb{P}((\nabla_1)_{\boldsymbol{\lambda}})$ this projective connection.
The connection corresponding to $\mathbb{P}((\nabla_1)_{\boldsymbol{\lambda}})$
does not split.
If we extend the projective connection $\mathbb{P}((\nabla_1)_{\boldsymbol{\lambda}})$
over $\mathbb{C}^n$
to a projective connection over $\mathbb{P}^n$ naively,
then the extended projective connection over $\mathbb{P}^n$ has poles of oder $2$ along
the divisor $\mathbb{P}^n \setminus \mathbb{C}^n$.
Then we consider a birational transformation
of $\mathbb{P}((\nabla_1)_{\boldsymbol{\lambda}})$.
By this birational transformation, we obtain
the meromorphic $\mathfrak{sl}_2$-connections
$\nabla_{\boldsymbol{\lambda}} = d + A_{\boldsymbol{\lambda}}$
with the explicit connection matrix $A_{\boldsymbol{\lambda}}$
described in Section \ref{2019.7.10.22.19}.
Finally Theorem \ref{2019.7.10.21.43} and Theorem \ref{2019.7.10.15.33} follow from
this construction of $\nabla_{\boldsymbol{\lambda}}$.

\subsection{Flat connections $(\nabla_0)_{\boldsymbol{\lambda}}$ defined by
rational closed 1-forms}

Let $\lambda_0, \ldots, \lambda_{n-1} $ be complex numbers.
Set $Y:=\Spec \mathbb{C} [u_0,u_1,z_1,\ldots,z_{n-2}]$.
Let $\omega_0$ and $\psi_n$ be the closed rational 1-forms on $Y$ defined by
\begin{equation*}
\begin{aligned}
\omega_0 :=&\ \lambda_0 \left( \frac{du_0}{u_0} - \frac{du_1}{u_1}\right)
+ \lambda_1 \left( \frac{du_0}{u_0-1} - \frac{du_1}{u_1-1}\right)\\
\psi_n :=&\
\begin{cases}
\sum_{i=1}^{n-2} \lambda_{i+1} \left(
\frac{  d(u_0-u_1 + z_i)}{u_0-u_1 + z_i}
-\frac{d (u_0-u_1 - z_i)}{u_0-u_1 - z_i} \right) & n>2\\
0 & n=2.
\end{cases}
\end{aligned}
\end{equation*}
We have a family of flat connections
\begin{equation*}
(\nabla_0)_{\boldsymbol{\lambda}} :=d
+\frac{1}{2}
\begin{pmatrix}
\omega_0 + \psi_n & 0 \\
0 & -\omega_0 - \psi_n\\
\end{pmatrix}
\end{equation*}
on the trivial rank 2 vector bundle $E_0 \ra Y$.
The family $(\nabla_0)_{\boldsymbol{\lambda}}$
is parametrized by $\boldsymbol{\lambda}=(\lambda_0,\ldots,\lambda_{n-1})$.
On the associated projective bundle $\mathbb{P}(E_0)$,
we have the associated projective connection
$\mathbb{P}((\nabla_0)_{\boldsymbol{\lambda}})=d w_0 + (\omega_0 + \psi_n ) w_0$,
where $w_0$ is a projective coordinate on the fibers.

\subsection{Descent of the connection $(\nabla_0)_{\boldsymbol{\lambda}}$}\label{2019.7.10.23.01}

We consider the birational transformation of the projective connection
$\mathbb{P}((\nabla_0)_{\boldsymbol{\lambda}})$ defined by
 $\Phi \colon \mathbb{P}(E_0) \dashrightarrow \mathbb{P}(E_0)$;
\begin{equation*}
\begin{aligned}
(u_0,u_1,z_1,\ldots,z_{n-2}, [w_0^0:w_0^1])
\longmapsto (u_0,u_1,z_1,\ldots,z_{n-2},[\tilde{w}_0^0:\tilde{w}_0^1] ),
\end{aligned}
\end{equation*}
where
\begin{equation}\label{2018.5.4.11.20}
\begin{aligned}
\frac{\tilde{w}_0^1}{\tilde{w}_0^0}=
(u_0 - u_1) \frac{w^1_0+w^0_0}{w^1_0-w_0^0} .
\end{aligned}
\end{equation}
The rational function (\ref{2018.5.4.11.20}) is an invariant
of the involution $\iota\colon \mathbb{P}(E_0) \ra \mathbb{P}(E_0)$;
\begin{equation*}
\begin{aligned}
\iota\colon
(u_0,u_1,z_1,\ldots,z_{n-2}, [w_0^0:w_0^1])
\longmapsto (u_1,u_0,z_1,\ldots,z_{n-2},[w_0^1:w_0^0] ),
\end{aligned}
\end{equation*}
that is $(\tilde{w}_0^1/\tilde{w}_0^0) \circ \iota=\tilde{w}_0^1/\tilde{w}_0^0$ as
functions on $\mathbb{P}(E_0)$.
Put $w_0=w_0^1/w_0^0$ and $\tilde{w}_0=\tilde{w}_0^1/\tilde{w}_0^0$.
We can check the following proposition by direct computation.

\begin{prop}
We define a map $f \colon Y \ra \mathbb{P}^n$ by
\begin{equation*}
\begin{aligned}
(u_0,u_1,z_1, \dots,z_{n-2})
&\longmapsto [s_1:s_2 : z_1 : \ldots :z_{n-2}:1 ],
\end{aligned}
\end{equation*}
where $s_1=u_0+u_1$ and $s_2=u_0 u_1$.
The birational transformation $(\Phi^{-1})^* \mathbb{P}((\nabla_0)_{\boldsymbol{\lambda}})$
on $\mathbb{P}(E_0)$ descends to a projective connection on
$f(Y) \times \mathbb{P}^1 \ra f(Y)$:
\begin{equation}\label{2018.4.27.14.06}
\begin{aligned}
(\Phi^{-1})^* \mathbb{P}((\nabla_0)_{\boldsymbol{\lambda}})
&=\frac{d \tilde{w}_0}{d w_0} \left( dw_0 + (\omega_0 + \psi_n) w_0 \right)  \\
&=
d \tilde{w}_0 + \left( \alpha_2 (s_1,s_2) + \sum^{n-2}_{i=1}
\alpha^i_2 (s_1,s_2, z_i) \right) \tilde{w}_0^2 \\
&\quad + 2 \alpha_1(s_1,s_2) \tilde{w}_0
+ \left(\alpha_0 ( s_1,s_2) + \sum^{n-2}_{i=1}
\alpha^i_0 (s_1,s_2, z_i)\right),
\end{aligned}
\end{equation}
where
\begin{equation}\label{2018.4.27.14.04}
\begin{aligned}
\alpha_0 ( s_1,s_2) &:=
\frac{2 \lambda_0 (1  - s_1 + s_2 ) +  \lambda_1 (-s_1 + 2 s_2)}{2( 1-  s_1 +  s_2)} d s_1
 -
\frac{ \lambda_0 s_1 ( 1 - s_1  + s_2) +\lambda_1 s_2 (s_1-2)}{2 s_2 (1  - s_1 + s_2)}  ds_2, \\
\alpha^i_0 ( s_1,s_2,z_i)&:= \lambda_{i+1} \left(  d z_i
-
\frac{ z_id (s_1^2-4 s_2 - z_i^2) }{2(s_1^2-4 s_2 - z_i^2)} \right) ,\\
\alpha_1(s_1,s_2)
 &:=-\frac{1}{4}\frac{d(s_1^2 - 4 s_2)}{s_1^2 - 4 s_2}, \\
\alpha_2(s_1,s_2)
& := - \frac{\alpha_0(s_1,s_2)}{s_1^2 -4 s_2} , \\
\alpha^i_2(s_1,s_2, z_i) &:=-    \frac{\alpha^i_0(s_1,s_2, z_i)}{s_1^2-4 s_2}.
\end{aligned}
\end{equation}
\end{prop}

The corresponding connection $(\nabla_1)_{\boldsymbol{\lambda}}$
on $f(Y) \times \mathbb{C}^2 \ra f(Y)$
 is
\begin{equation*}
(\nabla_1)_{\boldsymbol{\lambda}}=
d+ \begin{pmatrix}
\alpha_1 (s_1,s_2) & \alpha_0 (s_1,s_2)\\
-\alpha_2 (s_1,s_2) & -\alpha_1 (s_1,s_2)
\end{pmatrix}+\sum^{n-1}_{i=1}
\begin{pmatrix}
0 & \alpha_0^i (s_1,s_2,z_i)\\
-\alpha_2^i (s_1,s_2,z_i) & 0
\end{pmatrix}.
\end{equation*}
We consider a relation between
this connection and the connection $(\nabla_0)_{\boldsymbol{\lambda}}$.
Let $\nabla_0'$ be the meromorphic connection on $Y \times \mathbb{C}\ra Y$
defined by
$\nabla_0':=d-\frac{1}{2}
\frac{d(u_0-u_1)}{u_0-u_1}$.
We define a matrix $M_1(u_0,u_1)$ on $Y$ by
\begin{equation*}
M_1(u_0,u_1):=
\begin{pmatrix}
-1 & -u_0+ u_1\\
-1 & u_0- u_1
\end{pmatrix}.
\end{equation*}
Let $\nabla_0''$ be the meromorphic connection on $Y \times \mathbb{C}^2\ra Y$
defined by
\begin{equation*}
\begin{aligned}
\nabla_0''&:=d+M_1(u_0,u_1)^{-1} dM_1(u_0,u_1) \\
&\qquad +
M_1(u_0,u_1)^{-1}
\frac{1}{2}
\begin{pmatrix}
\omega_0 + \psi_n & 0 \\
0 & -\omega_0 - \psi_n\\
\end{pmatrix}
M_1(u_0,u_1).
\end{aligned}
\end{equation*}
Then we have
\begin{equation}\label{2018.5.8.23.53}
f^* (\nabla_1)_{\boldsymbol{\lambda}} = \nabla_0'' \otimes \nabla_0'.
\end{equation}

Moreover, we consider the map $\mathbb{P}^n \ra \mathbb{P}^n$;
$[s_1:s_2:z_1:\ldots:z_{n-2}:t] \mapsto [x:y:z_1:\ldots :z_{n-2}:t]$,
where $x:=t - s_1 +s_2$ and $y:=s_2$.
Set
\begin{equation*}
\begin{aligned}
f(x,y) :=&\ x^2 + y^2 +1  -2 (x y + x + y)  .
\end{aligned}
\end{equation*}
Then the rational 1-forms (\ref{2018.4.27.14.04}) are transformed into
\begin{equation}\label{2018.4.27.13.15}
\begin{aligned}
\alpha_0(x,y) &= -
\frac{(2 \lambda_0 + \lambda_1)dx - (2 \lambda_1 + \lambda_0)dy}{2} -
\frac{ \lambda_1 ( y-1)}{2}\frac{ dx}{x} +
\frac{ \lambda_0 ( x-1)}{2}\frac{ dy}{y} , \\
\alpha^i_0(x,y, z_i) &=\lambda_{i+1} \left(  d z_i
-
\frac{ z_id (f(x,y)-z_i^2) }{2(f(x,y)-z_i^2)} \right), \\
\alpha_1(x,y) &= - \frac{1}{4}\frac{ d f(x,y)}{f(x,y)},\\
\alpha_2(x,y)&= -\frac{\alpha_0(x,y)}{f(x,y)}, \\
\alpha^i_2(x,y, z_i)&= -\frac{\alpha^i_0(x,y, z_i)}{f(x,y)}  ,
\end{aligned}
\end{equation}
which are described by the affine coordinates $[x:y:z_1:\ldots:z_{n-2}:1]$.

\subsection{Birational transformations of the connection
$(\nabla_1)_{\boldsymbol{\lambda}}$}\label{2019.7.10.22.00}

From the connection $(\nabla_1)_{\boldsymbol{\lambda}}$ on $ f(Y) \times \mathbb{C}^2 \ra f(Y)$,
we construct a connection on the trivial bundle $\mathbb{P}^n \times \mathbb{C}^2 \ra \mathbb{P}^n$
whose pole divisor is $D_n$.
If we extend the rational 1-forms (\ref{2018.4.27.13.15}) to
rational 1-forms on $\mathbb{P}^n$,
then
$\alpha_0(x,y)$,
$\alpha^i_0(x,y, z_i)$ and $\alpha_1(x,y)$ have
 poles of order $2$, $2$ and $1$ along the divisor $(t=0)$, respectively.
 On the other hand,
 the rational 1-forms $\alpha_2(x,y)$ and $\alpha_2^i(x,y, z_i)$ have no pole along
the divisor $(t=0)$.
So we consider a birational transformation of the projective connection (\ref{2018.4.27.14.06})
as follows.
The $dy/y$ part of the projective connection (\ref{2018.4.27.14.06}) is
\begin{equation*}
\begin{aligned}
& d \tilde{w}_0 - \frac{\lambda_0 (\tilde{w}_0 - x+1)(\tilde{w}_0 + x-1)}{x+1} \frac{dy}{y}  \\
&\quad + [\text{ terms whose pole divisors do not contain the divisor $(y=0)$ }] .
\end{aligned}
\end{equation*}
Then we consider the following birational map
\begin{equation}\label{2018.4.13.16.49}
\begin{aligned}
\mathbb{P}^n \times \mathbb{P}^1
&\dashrightarrow \mathbb{P}^n \times \mathbb{P}^1 \\
([x:y: z_1: \ldots: z_{n-2}:1],[1: \tilde{w}_0]) \
&\longmapsto ( [x: y: z_1: \ldots  :z_{n-2}:1], [1:w]),
\end{aligned}
\end{equation}
where $\tilde{w}_0 -x+1 = w y$.
By this birational transformation (\ref{2018.4.13.16.49}),
the projective connection (\ref{2018.4.27.14.06}) is transformed into
\begin{equation*}
\begin{aligned}
& d w + \left( \cA_{21} (x,y)
+ \sum^{n-2}_{i=1}  \cA_{21}^i (x,y, z_i)\right) w^2 \\
& \  + 2
 \left(\cA_{11}^i(x,y)
 + \sum_{i=1}^{n-2}   \cA_{11}^i (x,y, z_i)
 \right)
  w  +\cA_{12}^i(x,y)
 + \sum_{i=1}^{n-2}   \cA_{12}^i (x,y, z_i),
\end{aligned}
\end{equation*}
where
\begin{equation}\label{2018.5.21.12.20}
\begin{aligned}
\cA_{21} (x,y)&:=  y  \alpha_2 (x,y), \\
\cA_{21}^i (x,y,z_i)&:= y \alpha^i_2 (x,y, z_i) , \\
\cA_{11} (x,y)&:=
 (x-1)  \alpha_2 (x,y)
 +  \alpha_1(x,y) +\frac{1}{2}\frac{d y}{y} ,\\
\cA_{11}^i (x,y,z_i)&:=  (x-1)  \alpha^i_2 (x,y, z_i) ,\\
\cA_{12} (x,y) &:= \frac{dx    + (x-1)^2\alpha_2 (x,y)
 + 2 (x -1)\alpha_1(x,y)  + \alpha_0(x,y)  }{y} , \\
\cA_{12}^i (x,y,z_i)&:=
 \frac{(x-1)^2\alpha^i_2 (x,y, z_i) + \alpha^{i}_0(x,y, z_i)  }{y} .
\end{aligned}
\end{equation}
The corresponding connection $\nabla_{\boldsymbol{\lambda}}$ on
$ \mathbb{P}^n \times \mathbb{C}^2 \ra \mathbb{P}^n$ is
\begin{equation*}
\begin{aligned}
\nabla_{\boldsymbol{\lambda}}
=d+
\begin{pmatrix}
\cA_{11} (x,y) & \cA_{12} (x,y) \\
-\cA_{21} (x,y) &- \cA_{11} (x,y)
\end{pmatrix} + \sum_{i=1}^{n-2}
\begin{pmatrix}
\cA_{11}^i (x,y,z_i) & \cA_{12}^i (x,y,z_i) \\
-\cA_{21}^i (x,y,z_i) &- \cA_{11}^i (x,y,z_i)
\end{pmatrix},
\end{aligned}
\end{equation*}
whose polar divisor is $D_n$.
This connection $\nabla_{\boldsymbol{\lambda}}$
is the connection described in Section \ref{2019.7.10.22.19}.
We consider a relation between $\nabla_{\boldsymbol{\lambda}}$ and
$(\nabla_1)_{\boldsymbol{\lambda}}$.
Let $\nabla_1'$ be the meromorphic connection
on $\mathbb{P}^n \times \mathbb{C} \ra \mathbb{P}^n$ defined by
$\nabla_1':=d-\frac{1}{2}\frac{dy}{y}$.
We define a matrix $M_2(x,y)$ on $Y$ by
\begin{equation*}
M_2(x,y):=
\begin{pmatrix}
y & x-1\\
0 &1
\end{pmatrix} .
\end{equation*}
Let $\nabla_1''$ be the meromorphic connection
on $\mathbb{P}^n \times \mathbb{C}^2 \ra \mathbb{P}^n$ defined by
\begin{equation*}
\begin{aligned}
\nabla_1''=d+& M_2(x,y)^{-1} dM_2(x,y)
+
M_2(x,y)^{-1}
\begin{pmatrix}
\alpha_1(x,y)  & \alpha_0(x,y)  \\
-\alpha_2(x,y)  &- \alpha_1(x,y) \\
\end{pmatrix}
M_2(x,y)\\
&\quad +
\sum_{i=1}^{n-2}
M_2(x,y)^{-1}
\begin{pmatrix}
0  &
 \alpha^i_0(x,y, z_i) \\
-\alpha^i_2(x,y, z_i) & 0 \\
\end{pmatrix}
M_2(x,y).
\end{aligned}
\end{equation*}
We can check that
\begin{equation}\label{2018.5.8.23.54}
 \nabla_{\boldsymbol{\lambda}} =
\nabla''_1 \otimes \nabla'_1.
\end{equation}
By a combination of the equalities (\ref{2018.5.8.23.53}) and (\ref{2018.5.8.23.54}),
we have the following proposition:

\begin{prop}\label{2019.7.12.12.28}
The pull-back $ f^* \nabla_{\boldsymbol{\lambda}}$
is birationally equivalent to
$(\nabla_0)_{\boldsymbol{\lambda}}\otimes \nabla_0'\otimes f^* \nabla_1'$.
\end{prop}

\begin{proof}[Proof of Theorem \ref{2019.7.10.21.43} and Theorem \ref{2019.7.10.15.33}]
First, since $(\nabla_0)_{\boldsymbol{\lambda}}$,
$\nabla_0'$ and $\nabla_1'$ are flat and
$f$ is a generically finite Galois morphism, we have
the flatness of $\nabla_{\boldsymbol{\lambda}}$ for each $\boldsymbol{\lambda}$
by Proposition \ref{2019.7.12.12.28}.
Second, we have that $\nabla_{\boldsymbol{\lambda}}$
has at worst regular singularities for each $\boldsymbol{\lambda}$
by the explicit expression of $\nabla_{\boldsymbol{\lambda}}$ and
\cite[Chap. II, Theorem 4.1 (ii)]{Deligne}.
Finally, the assertion of Theorem \ref{2019.7.10.15.33}
is deduced by Proposition \ref{2019.7.12.12.28}.
\end{proof}

\section{Monodromy representation}\label{2018.5.18.17.14}

In this section, we consider the monodromy representation
$\pi_1(\mathbb{P}^n \setminus D_n,*) \ra \mathrm{SL}_{2}(\mathbb{C})$
of $\nabla_{\boldsymbol{\lambda}}$ for generic $\boldsymbol{\lambda}$.
In Section \ref{2019.7.10.23.32} and Section \ref{2019.7.10.23.33},
we discuss structure of the fundamental group $\pi_1(\mathbb{P}^n \setminus D_n,*)$
by using the Zariski's hyperplane section theorem
and the Zariski--Van-Kampen method.
In Section \ref{2019.7.10.22.02}, we show Theorem \ref{2019.7.10.22.03}
by using the results in Section \ref{2019.7.10.23.32} and Section \ref{2019.7.10.23.33}.

\subsection{Zariski's hyperplane section theorem}\label{2019.7.10.23.32}

Let $H_i$ ($i=1,\ldots,n-2$) be the hyperplanes in $\mathbb{P}^n$ defined by
\begin{equation*}
H_i := ( z_i - a_i x - b_i y -c_i t =0 ) \quad i=1,\ldots , n-2.
\end{equation*}
Here $a_i, b_i$, and $c_i $ $(i=1,\dots,n-2)$ are generic complex numbers.
For simplicity, we assume that $0<|a_i|\ll 1$ and $ 0 < |b_i | \ll 1$.
Let $f(x,y,t)$ be the following quadratic polynomial
\begin{equation*}
\begin{aligned}
f(x,y,t) :=&\ x^2 + y^2 +t^2  -2 ( x y +y t +  t x) \\
=&\ (y-x-t)^2 - 4 xt  .
\end{aligned}
\end{equation*}
Let $\tilde{\mathcal{Q}}_0$, $\tilde{\mathcal{Q}}_i$ and $\tilde{D}_n$
be the divisors on $\mathbb{P}^2=\mathbb{P}^n \cap (\cap_{i=1}^{n-2}H_i)$ defined by
\begin{equation*}
\begin{aligned}
\tilde{\mathcal{Q}}_0&:= (f(x,y,t)=0 ),   \\
\tilde{\mathcal{Q}}_i&:= (f(x,y,t) - ( a_i x + b_i y +c_i t)^2 =0)
\quad  (i=1,\ldots,n-2), \text{ and } \\
\tilde{D}_n &:= (x=0 ) + (y=0) + ( t=0) +
\tilde{\mathcal{Q}}_0 + \tilde{\mathcal{Q}}_1 + \cdots + \tilde{\mathcal{Q}}_{n-2},
\end{aligned}
\end{equation*}
respectively.
By Zariski's hyperplane section theorem (for example see \cite{HL}), we have the natural isomorphism
\begin{equation*}
\pi_1(\mathbb{P}^n \setminus D_n,*) \cong \pi_1(\mathbb{P}^2 \setminus \tilde{D}_n,*).
\end{equation*}

\subsection{Zariski--Van-Kampen method}\label{2019.7.10.23.33}

We derive some equalities in $\pi_1(\mathbb{P}^2 \setminus \tilde{D}_n,*)$
 by the Zariski--Van-Kampen method (see for example \cite{Degtyarev}).

Let $\pi\colon \mathbb{P}^{2} \setminus \tilde{D}_n \ra \mathbb{P}^1$ be
the projection defined by
\begin{equation*}
\begin{aligned}
\pi\colon \mathbb{P}^{2} \setminus \tilde{D}_n &\lra \mathbb{P}^1 \\
[x:y:t]&\longmapsto [x:t].
\end{aligned}
\end{equation*}
Let $\{[x^+_i:1] ,[ x_i^-:1 ]\} \subset \mathbb{P}^1$
be the roots of the discriminant of $f(x,y,t) - ( a_i x + b_i y +c_i t)^2$ with respect to $y$.
We denote $[x^+_i:1]$ and $[x^-_i:1]$ by $x^+_i$ and $x^-_i$, respectively.
Since $0<|a_i|\ll 1$ and $ 0 < |b_i | \ll 1$,
there exists an element of $\{x^+_i , x_i^-\}$ in a neighborhood of $\infty=[0:1]$.
We assume that $x_i^-$ is a point in a neighborhood of $\infty$.
Set $a=[ a:1 ]$ where $0<|a|\ll 0$.
For $i=0,1,\dots,n-2$,
let $y^+_{i}$ and $y^-_{i}$ be the intersection of
$\tilde{\mathcal{Q}}_i$ and $\pi^{-1}(a)$:
$\tilde{\mathcal{Q}}_i\cap \pi^{-1}(a) = \{ y^+_{i} , y^-_{i} \} $.
Here we assume that
\begin{equation*}
0<\mathrm{Arg} \left( \frac{y^+_1-(a+1)}{y^+_0-(a+1)} \right)
<\cdots<\mathrm{Arg} \left( \frac{y^+_{n-2}-(a+1)}{y^+_0-(a+1)} \right)
< \pi.
\end{equation*}
\begin{figure}[h]
\caption{Fibers of $\pi$.}
\begin{center}
\includegraphics[clip,width=9cm]{fig1.pdf}
\end{center}
\end{figure}

We define natural numbers $i_k$ and $j_k$ ($k=1,\dots,n-2$)
so that
$\{i_1 ,\ldots,i_{n-2} \}=\{1 ,\ldots,n-2 \}$,
$\{j_1 ,\ldots,j_{n-2} \}=\{1 ,\ldots,n-2 \}$,
\begin{equation*}
0<\mathrm{Arg}\,  (x^+_{i_1})
<\cdots<\mathrm{Arg}\,  (x^+_{i_{n-2}})
< 2\pi \text{ and }
0<\mathrm{Arg}\,  \frac{1}{x^-_{j_1}}
<\cdots<\mathrm{Arg}\,  \frac{1}{x^-_{j_{n-2}}}
< 2\pi.
\end{equation*}
Here we define the range of the principal value of arguments $\mathrm{Arg}$
by the closed-open interval $[0,2\pi)$.
Let $\Gamma$

be the group defined by
\begin{equation*}
\begin{aligned}
\Gamma &:=
\left\langle
\begin{array}{l}
\alpha_0, \alpha_{y^+_0} ,\ldots , \alpha_{y^+_{n-2}} \\
\alpha_{y^-_0} ,\ldots , \alpha_{y^-_{n-2}},\alpha_{\infty}
\end{array}
\middle| \
\alpha_0\alpha_{y^+_{1}}    \cdots \alpha_{y^+_{n-2}} \alpha_{y^-_{0}}
\alpha_{y^-_{1}}    \ldots \alpha_{y^-_{n-2}}
\alpha_{y^+_0}
 \alpha_{\infty}=1 \right\rangle .

\end{aligned}
\end{equation*}
Then we have
$\pi_1 (\pi^{-1} (a)  \setminus (\tilde{D}_n\cap \pi^{-1}(a) ) , *  ) \cong \Gamma$
and have an exact sequence
\begin{equation*}
 1  \longrightarrow \Gamma
 \longrightarrow \pi_1(\mathbb{P}^2 \setminus \tilde{D}_n,*)
 \longrightarrow \pi_1(\mathbb{P}^1 \setminus \{ 0,\infty \},a) \longrightarrow 1.
\end{equation*}
Let
\begin{equation}\label{1.21.16.34}
\gamma_0, \gamma_1 ,  \gamma_{x^+_1}, \ldots , \gamma_{x^+_{n-2}} ,
\gamma_{x^-_1}, \ldots , \gamma_{x^-_{n-2}}, \gamma_{\infty}
\end{equation}
be loops with base point $a$
on $\mathbb{P}^1\setminus \{ 0, 1 , x^{\pm}_1 ,\ldots ,x^{\pm}_{n-1}, \infty \}$
such that for $x\in \{ 0, 1 , x^{\pm}_1 ,\ldots ,x^{\pm}_{n-1}, \infty \}$,
the loop
$\gamma_x$ is oriented counter-clockwise,
$x$ lies inside, while the other points
$\{ 0, 1 , x^{\pm}_1 ,\ldots ,x^{\pm}_{n-1}, \infty \}\setminus \{x\}$
lie outside as in Figure 2.
\begin{figure}[h]
\caption{Loops on $\pi^{-1}(a)$ and $\mathbb{P}^1=(\mathbb{C}_x)_0 \cup (\mathbb{C}_x)_{\infty}.$}
\begin{center}
\includegraphics[clip,width=15cm]{fig2.pdf}
\end{center}
\end{figure}

Let $s \colon \mathbb{P}^1 \setminus \{ 0,\infty \} \rightarrow \mathbb{P}^2 \setminus \tilde{D}_n $
be a continuous section of $\pi$ such that
$s(a)=* \in \mathbb{P}^2 \setminus \tilde{D}_n$.
For the loops (\ref{1.21.16.34}), we define the monodromy actions of the
loops (\ref{1.21.16.34}) on $\Gamma$ as in \cite[Theorem 2.2.1]{Cousin}.
Namely, the action $(\gamma_x,\alpha) \mapsto \gamma_x(\alpha)$
for loops $\gamma_x$ and $\alpha \in \Gamma$  is characterized by the equality
$\gamma_x(\alpha) = \gamma_x^{-1} \alpha \gamma_x$
in $\pi_1(\mathbb{P}^2 \setminus \tilde{D}_n,*) $.
Here we denote by $\gamma_x$
the loop $s_*(\gamma_x) \in \pi_1(\mathbb{P}^2 \setminus \tilde{D}_n,*) $
for simplicity.
For explicit computation of this action, we consider the motion of the points
$y^{\pm}_i$ ($i=0,1,\ldots,n-2$) when $a$ varies along the loop $\gamma_x$
and a continuous deformation of
$\alpha\in \pi_1 (\pi^{-1} (a)  \setminus (\tilde{D}_n\cap \pi^{-1}(a) ) , *  )$
according to the motion of these points.
Note that the assumptions
$0<|a_i|\ll 1$, $ 0 < |b_i | \ll 1$ and $0<|a| \ll 1$
make the computation of the motion of the points $y_i^{\pm}$ simple.
By explicit computation of the action on some loops, we can check the following equalities:
\begin{align}
\gamma_0(\alpha_{y^+_0}) &= \alpha_{y^-_{0}}; &
\gamma_{x^+_i}(\alpha_{y^+_i}) &= \alpha_{y^+_i} \alpha_{y^-_{i}} \alpha_{y^+_i}^{-1}
\ (i=1,\ldots , n-2); \label{2018.5.12.13.30}\\
\gamma_0(\alpha_{0}) &= \alpha_{0};   &
\gamma_0(\alpha_{y^+_i})& = \alpha_{y^+_{i}}
\ (i=1,\dots,n-2) ; \label{2019.1.21.16.48} \\
\tilde{\gamma}_{\infty}(\alpha_{y^+_0}) &=  \alpha_{0}
\alpha_{y^-_0} \alpha_{0}^{-1}; \label{2018.5.12.13.28}
\end{align}
and
\begin{equation}\label{2019.1.12.16.51}
\begin{aligned}
\gamma_1(\alpha_{y^+_0}) &= ( \alpha_0 \alpha_{y^+_0}) \alpha_0
\alpha_{y^+_0}\alpha_0^{-1} (\alpha_{0} \alpha_{y^+_0})^{-1}  .
\end{aligned}
\end{equation}
Here we put
 $\tilde{\gamma}_{\infty}:= \gamma_{x^-_{j_1}} \cdots \gamma_{x^-_{j_{n-2}}} \gamma_{\infty}$.
In fact, if $a$ varies along the loop $\gamma_0$,
then $y_0^+$ moves to the location of $y_0^-$,
and $0$ and $y_i^+$ ($i=1,\ldots,n-2$) go back to the
prior locations, respectively.
If $a$ varies along the loop $\gamma_{x^{+}_i}$ for $i=1,\ldots,n-2$,
then $y_i^+$ moves to the location of $y_i^-$.
Here we assume that $-1 \ll \mathrm{Arg}\, (a)<0$
and $y_0^+$ closes to $0$ when $a$ approach $1$ along the real axis.
If $a$ varies along the loop $\tilde{\gamma}_{\infty}$,
then $y_0^+$ moves to the location of $y_0^-$ round by $0$.
If $a$ varies along the loop $\gamma_1$,
then $y_0^+$ go back to the prior locations round by $0$ twice.
If we consider continuous deformations of the corresponding loops
according to the motion of these points,
then we have the equalities (\ref{2018.5.12.13.30}), (\ref{2019.1.21.16.48}),
(\ref{2018.5.12.13.28}) and (\ref{2019.1.12.16.51}).
Here note that the images
of the intersection of $\mathcal{Q}_i$ and $\mathcal{Q}_j$ under $\pi$
are close to $\infty$ for $i,j =0,1, \ldots, n-2$
since $0<|a_i|\ll 1$ and $ 0 < |b_i | \ll 1$.

In $\pi_1(\mathbb{P}^2 \setminus \tilde{D}_n,*) $,
we have the equality
$\gamma_x(\alpha) = \gamma_x^{-1} \alpha \gamma_x$ for $\alpha \in \Gamma$.
By the equality (\ref{2018.5.12.13.30}), we can show that
 $\alpha_{y_i^-}$ ($i=0,\ldots,n-2$) are generated by
$\alpha_{y^+_0} ,\ldots , \alpha_{y^+_{n-2}}$, and $\gamma_0$
in $\pi_1(\mathbb{P}^2 \setminus \tilde{D}_n,*)$.
Then we obtain the following proposition.

\begin{prop}
The group $\pi_1(\mathbb{P}^2 \setminus \tilde{D}_n,*)$ is generated by
$\alpha_0, \alpha_{y^+_0} ,\ldots , \alpha_{y^+_{n-2}}$, and
$\gamma_0$.
\end{prop}

\begin{prop}
Set $\tilde{\alpha}= \alpha_{y^+_1} \cdots \alpha_{y^+_{n-2}}$.
For the elements $\alpha_{y^+_0}, \alpha_0,\gamma_{0}$, and $\tilde{\alpha}$
of $\pi_1(\mathbb{P}^2 \setminus \tilde{D}_n,*)$,
we have the following equalities:
\begin{align}
\ [ \alpha_0 , \gamma_{0} ]&= [ \tilde{\alpha}, \gamma_0 ]=1 \label{2018.5.10.22.52}, \\
((\gamma_{0}\tilde{\alpha} ) \alpha_{y^+_0})^2&=
(\alpha_{y^+_0}(\gamma_{0}
 \tilde{\alpha} ))^2, \label{2018.5.10.22.54} \\
(\alpha_{y^+_0} \alpha_0)^2 &= ( \alpha_0 \alpha_{y^+_0})^2 \label{2018.5.12.13.38}.
\end{align}
\end{prop}

\begin{proof}\label{2018.5.12.16.43}
By the equality (\ref{2019.1.21.16.48}),
we have the equality (\ref{2018.5.10.22.52}).
Second, we show the equality (\ref{2018.5.10.22.54}).
By the equalities
(\ref{2018.5.12.13.30}), (\ref{2018.5.12.13.28}),
$a_{\infty}=\gamma_{\infty}\gamma_0$, and (\ref{2018.5.10.22.52}),
we have
\begin{equation*}
\begin{aligned}
a_{y^+_0}&=  \gamma_{\infty} \alpha_{0}
\gamma_0^{-1}\alpha_{y^+_0}\gamma_0
\alpha_{0}^{-1} \gamma_{\infty}^{-1} \\

&= \alpha_{\infty} \alpha_{0}
\gamma^{-1}_{0}\gamma_0^{-1}\alpha_{y^+_0}\gamma_0\gamma_{0}
\alpha_{0}^{-1}\alpha_{\infty}^{-1} \\
&=
(\alpha_{y^+_1} \cdots \alpha_{y^+_{n-2}} \alpha_{y^-_0}
 \alpha_{y^-_1} \cdots \alpha_{y^-_{n-2}}  \alpha_{y^+_0})^{-1}
\gamma_{0}^{-1} \gamma_0^{-1}\alpha_{y^+_0}\gamma_0\gamma_0
(\alpha_{y^+_1} \cdots \alpha_{y^+_{n-2}} \alpha_{y^-_0}
 \alpha_{y^-_1} \cdots \alpha_{y^-_{n-2}} \alpha_{y^+_0})
 \\
 &=
( \tilde{\alpha} \gamma_{0}^{-1} \alpha_{y^+_0} \gamma_0
 \tilde{\alpha} \alpha_{y^+_0})^{-1}
\gamma_{0}^{-1}\gamma_0^{-1}\alpha_{y^+_0}\gamma_0\gamma_{0}
( \tilde{\alpha} \gamma_{0}^{-1} \alpha_{y^+_0} \gamma_0
 \tilde{\alpha} \alpha_{y^+_0})\\
  &=
( \tilde{\alpha}  \alpha_{y^+_0} \gamma_0
 \tilde{\alpha} \alpha_{y^+_0})^{-1}
\gamma_{0}^{-1}\alpha_{y^+_0}\gamma_0
( \tilde{\alpha}  \alpha_{y^+_0} \gamma_0
 \tilde{\alpha} \alpha_{y^+_0}).
\end{aligned}
\end{equation*}
Then we have the equality (\ref{2018.5.10.22.54}).
By the equality (\ref{2019.1.12.16.51}), we have the equality (\ref{2018.5.12.13.38}).
\end{proof}

\subsection{Monodromy representation of $\nabla_{\boldsymbol{\lambda}}$}\label{2019.7.10.22.02}

Let $\boldsymbol{D}_{\infty}$ be the infinite dihedral group:
\begin{equation*}
\boldsymbol{D}_{\infty}:= \left\langle
\begin{pmatrix}
0 & \alpha \\
-\alpha^{-1} & 0
\end{pmatrix},
\begin{pmatrix}
\beta & 0 \\
0 & \beta^{-1}
\end{pmatrix}\
\middle| \
\alpha , \beta \in \mathbb{C}^*
 \right\rangle \le \mathrm{SL}_2 (\mathbb{C}).
\end{equation*}

\begin{prop}\label{2018.5.8.15.13}
For generic $\boldsymbol{\lambda}$, the monodromy representation of
$\nabla_{\boldsymbol{\lambda}}$ is conjugated to
the dihedral representation $\rho_{\boldsymbol{\lambda}} \colon
\pi_1(\mathbb{P}^n \setminus D_n,*) \ra \boldsymbol{D}_{\infty}$ of
the fundamental group $\pi_1(\mathbb{P}^n \setminus D_n,*) $
defined by
\begin{equation*}
\begin{aligned}
\rho_{\boldsymbol{\lambda}}(\alpha_0)&=
\begin{pmatrix}
-\exp (- \pi \lambda_{0} ) & 0 \\
0 & -\exp ( \pi \lambda_{0} )
\end{pmatrix}, &
\rho_{\boldsymbol{\lambda}}(\gamma_0)&=\begin{pmatrix}
\exp (- \pi \lambda_1 ) & 0 \\
0 & \exp ( \pi \lambda_1 )
\end{pmatrix}, \\
\rho_{\boldsymbol{\lambda}}(\alpha_{y^+_0})&=
\begin{pmatrix}
0 & 1 \\
-1 & 0
\end{pmatrix}, &
\rho_{\boldsymbol{\lambda}}(\alpha_{y^+_i})&=
\begin{pmatrix}
\exp (- \pi \lambda_{i+1} ) & 0 \\
0 & \exp ( \pi \lambda_{i+1} )
\end{pmatrix},
\end{aligned}
\end{equation*}
where $i=1,\ldots , n-2$.
\end{prop}

\begin{proof}
Let $\rho_{\nabla_{\boldsymbol{\lambda}}}\colon
\pi_1(\mathbb{P}^n \setminus D_n,*) \ra \mathrm{SL}_{2}(\mathbb{C})$ be
a monodromy representation of $\nabla_{\boldsymbol{\lambda}}$.
Put $A_0 := \rho_{\nabla_{\boldsymbol{\lambda}}}(\alpha_0)$,
$A_{y^+_i} := \rho_{\nabla_{\boldsymbol{\lambda}}}(\alpha_{y^+_i})$
($i=0,\ldots,n-1$) and $C_0 := \rho_{\nabla_{\boldsymbol{\lambda}}}(\gamma_0)$.
Let $U$ be some analytic open subset of
$\mathbb{P}^2 \setminus (\tilde{\cQ}_0 \cup (y=0)\cup (t=0))$
such that $U$ is simply connected and $U$ contains the loops $\alpha_{y_i^+}$ ($i=1,\ldots,n-1$)
and $\gamma_0$.
On the open subset $U$, the connection $\nabla_{\boldsymbol{\lambda}}$
is isomorphic to $(\nabla_0)_{\boldsymbol{\lambda}}$.
Then by some conjugation, we may put
\begin{equation*}
C_0=
\begin{pmatrix}
\exp (- \pi \lambda_1 ) & 0 \\
0 & \exp ( \pi \lambda_1 )
\end{pmatrix}, \quad
A_{y^+_i}=
\begin{pmatrix}
\exp (- \pi \lambda_{i+1} ) & 0 \\
0 & \exp ( \pi \lambda_{i+1} )
\end{pmatrix}\  i=1,\ldots , n-2.
\end{equation*}
Assume that $\exp (- \pi \lambda_1 ) \neq \exp ( \pi \lambda_1 )$.
By Proposition \ref{2018.5.12.16.43}, we have the equality $A_0 C_0= C_0A_0$.
Then we have
\begin{equation*}
A_{0}=
\begin{pmatrix}
-\exp (- \pi \lambda_{0} ) & 0 \\
0 & -\exp ( \pi \lambda_{0} )
\end{pmatrix}.
\end{equation*}

Note that the image $\mathrm{Im}(\rho_{\nabla_{\boldsymbol{\lambda}}})$ is non-abelian.
Since $C_0$, $A_0$, and $A_{y_i^+}$ ($i=1,\ldots, n-2$) are diagonal matrices,
we may put
\begin{equation*}
A_{y^+_0}=
\begin{pmatrix}
a_{11} & a_{12} \\
-1 & a_{22}
\end{pmatrix}.
\end{equation*}
Put $\tilde{A}:=A_{y^+_1} \cdots A_{y^+_{n-2}}$.
By Proposition \ref{2018.5.12.16.43}, we have the equalities
$(A_{y^+_0} (C_0 \tilde{A}) )^2=
((C_0 \tilde{A} )A_{y^+_0})^{2}$
and
$(A_{y^+_0} A_0 )^2=(A_0A_{y^+_0})^{2}$.
Assume that $(\exp (-\pi \lambda_1 - \pi \sum_{i=1}^{n-2} \lambda_{i+1}  ) )^2 \neq 1$
and $ (- \exp (-\pi \lambda_0 ) )^2 \neq 1 $.
Since $A_{y^+_0} (C_0 \tilde{A})\neq  (C_0 \tilde{A})A_{y^+_0}$
and $A_{y^+_0} A_0 \neq  A_0A_{y^+_0}$, we have the equalities
$(A_{y^+_0}(C_0 \tilde{A} ))^{2}= - I_2$
and $(A_{y^+_0}A_0)^{2} = - I_2$.
Then we have the following equalities:
\begin{equation*}
\begin{cases}
a_{11}a_{22}+a_{12}=1 \\
a_{11} (\exp (-\pi \lambda_1 - \pi \sum_{i=1}^{n-2} \lambda_{i+1}  ) )^2 = a_{22} \\
a_{11} (- \exp (-\pi \lambda_0 ) )^2 = a_{22}.
\end{cases}
\end{equation*}
We assume that
$(\exp (-\pi \lambda_1 - \pi \sum_{i=1}^{n-2} \lambda_{i+1}  ) )^2 \neq (- \exp (-\pi \lambda_0 ) )^2$.
Then we have $a_{11}=0$, $a_{12}=1$, and $a_{22}=0$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{2019.7.10.22.03}]
By Theorem \ref{2019.7.10.15.33},
we have that the monodromy representation of
$\nabla_{\boldsymbol{\lambda}}$
is  virtually abelian.
By Proposition \ref{2018.5.8.15.13},
we have that the monodromy representation of
$\nabla_{\boldsymbol{\lambda}}$
is conjugated to the explicit
representation
\begin{equation*}
\rho_{\boldsymbol{\lambda}} \colon \pi_1 (\mathbb{P}^n \setminus D_n)
\longrightarrow
\mathrm{SL}_2(\mathbb{C}),
\end{equation*}
which takes values in the infinite dihedral group $\boldsymbol{D}_{\infty}$.
\end{proof}

\section{Algebraic Garnier solution}\label{2018.5.18.17.17}

Assume that an $n$-tuple of
complex numbers $\boldsymbol{\lambda}=(\lambda_0, \ldots, \lambda_{n-1} )$
is sufficiently generic.
In this section, we restrict the flat connection $\nabla_{\boldsymbol{\lambda}}$ to
a generic line $\mathbb{P}^n\cap (\cap_{i=0}^{n-2} H'_i)$, where
\begin{equation}\label{2018.5.3.19.42}
\begin{cases}
H'_0 = (y-ax -b t=0) &\\
H'_i =(z_i - c_{i} x - d_i t=0) & (i=1,2,\dots,n-2).
\end{cases}
\end{equation}
Here $a, b, c_i $, and $d_i$ $(i=1,2,\dots,n-2)$ are generic complex numbers.
We consider the transformation $\tilde{x}= -\frac{a}{b}x$.
Let $T$ be a Zariski open subset of $\Spec \mathbb{C} [a,b, c_i ,d_i ]_{i=1,\ldots,n-2}$.
We consider the map $\mathbb{P}^1 \times T \ra \mathbb{P}^n$ defined by (\ref{2018.5.3.19.42}).
Let $(\nabla_{\mathbb{P}^1 \times T})_{\boldsymbol{\lambda}}$ be the flat connection
on the trivial rank $2$ vector bundle $F_0$ over $\mathbb{P}^1 \times T$
induced by the flat connection $\nabla_{\boldsymbol{\lambda}}$ over $\mathbb{P}^n$.
Let
\begin{equation*}
(\nabla_{\mathbb{P}^1 \times T/T})_{\boldsymbol{\lambda}}
\colon F_0 \lra F_0 \otimes \Omega^1_{\mathbb{P}^1 \times T/T}(D_n)
\end{equation*}
 be the relative connection
on $F_0$ over $\mathbb{P}^1 \times T$
associated to $(\nabla_{\mathbb{P}^1 \times T})_{\boldsymbol{\lambda}}$.
In Section \ref{2019.7.10.22.11},
we introduce an \'etale base change $\tilde{T} \rightarrow T$
to prove the assertion (i) of Theorem \ref{2019.7.10.22.13}.
In Section \ref{2019.7.12.11.27},
after the \'etale base change $\tilde{T} \rightarrow T$,
we compute the residue matrix of
$(\nabla_{\mathbb{P}^1 \times \tilde{T}/\tilde{T}})_{\boldsymbol{\lambda}}$
for each simple pole.
In Section \ref{2019.7.10.22.15},
we recall the relation between isomonodromic deformations and the Garnier system
following \cite{Mazz}.
In Section \ref{2019.7.12.11.28},
we show Theorem \ref{2019.7.10.22.13}.

\subsection{Regular singular points of
$(\nabla_{\mathbb{P}^1 \times T/T})_{\boldsymbol{\lambda}}$}\label{2019.7.10.22.11}

By the pull-back of $x^2 + y^2 +t^2  -2 (x y + y t +t x )   $
and $x^2 + y^2 +t^2  -2 (x y + y t +t x ) -z_i^2  $
under $\mathbb{P}^1 \times T \ra \mathbb{P}^n$, we have the following polynomials over $T$:
\begin{equation*}
\begin{aligned}
f(a,b,\tilde{x}) &:=
\frac{(a-1)^2b^2}{a^2} \tilde{x}^2 +
\frac{2b(1+a+b-ab)}{a} \tilde{x} + (b-1)^2, \\
f_i(a,b,c_i,d_i,\tilde{x}) &:=f(a,b,\tilde{x})
-\frac{(a d_i -b c_i \tilde{x})^2}{a^2},
\end{aligned}
\end{equation*}
which are described on the affine coordinate $[\tilde{x}:1]$.
Let $I$ be the ideal of $\mathbb{C} [a,b,\tilde{b}, c_i ,d_i , \tilde{d}_i]_{i=1,\ldots,n-2}$ defined by
$I:=
(\tilde{b}^2-4 (a+b-ab),
\tilde{d}_i^2-\Delta^i_{\tilde{x}} )_{i=1,\ldots,n-2}$,
where $\Delta^i_{\tilde{x}} $ is the discriminant of $f_i(a,b,c_i,d_i,\tilde{x})$ with respect to $\tilde{x}$.
We have the natural morphism
\begin{equation*}
\Spec \mathbb{C} [a,b,\tilde{b}, c_i ,d_i , \tilde{d}_i]_{i=1,\ldots,n-2}
/ I \lra \Spec \mathbb{C} [a,b, c_i ,d_i ]_{i=1,\ldots,n-2}.
\end{equation*}
Let $\tilde{T}$ be the inverse image of $T$ under this morphism: $\tilde{T} \ra T$.
Let $t_{1}$ and $t_{2}$ be the rational functions on $\tilde{T}$ defined by
\begin{equation*}
t_{1}:=
\frac{a( \tilde{b}-2 )^2}{4(a-1)(\tilde{b}^2-a)} \text{  and  }
t_{2}:=
\frac{a(\tilde{b}+2 )^2}{4(a-1)(\tilde{b}^2-a)}.
\end{equation*}
Then $f(a,b,t_1)=f(a,b,t_2)=0$.
Moreover,
let $t_{2i+1}$ and $t_{2i+2}$ be the rational functions on $\tilde{T}$ defined by
\begin{equation*}
t_{2i+1}:=
\frac{2ab(1+a+b-ab- c_i d_i) -a^2  \tilde{d}_i}{2b^2((a-1)^2-c_i^2)} \text{  and  }
t_{2i+2}:=
\frac{2ab(1+a+b-ab- c_i d_i) +a^2  \tilde{d}_i}{2b^2((a-1)^2-c_i^2)}.
\end{equation*}
Then $ f_i (a,b,c_i,d_i,t_{2i+1})= f_i (a,b,c_i,d_i,t_{2i+2})=0$.
By these rational functions, we have a generically finite morphism
\begin{equation}\label{2018.5.12.23.51}
\tilde{T} \lra \Spec \mathbb{C}[t_1,t_2,\ldots ,t_{2n-2}],
\end{equation}
if the Zariski open subset $T$ shrinks.
We take the pull-back $(\nabla_{\mathbb{P}^1 \times \tilde{T}/\tilde{T}})_{\boldsymbol{\lambda}}$
of $(\nabla_{\mathbb{P}^1 \times T/T})_{\boldsymbol{\lambda}}$
under the morphism $\mathbb{P}^1 \times \tilde{T} \ra \mathbb{P}^1 \times T$.
Then $(\nabla_{\mathbb{P}^1 \times \tilde{T}/\tilde{T}})_{\boldsymbol{\lambda}}$
is a family of the
 Fuchsian systems with $2n + 1$ regular singularities
 at $\tilde{x}=0,1,t_1,\ldots , t_{2n-2}, \infty$ parametrized by $\tilde{T}$.

\subsection{Residue matrices of
$(\nabla_{\mathbb{P}^1 \times \tilde{T}/\tilde{T}})_{\boldsymbol{\lambda}}$ }\label{2019.7.12.11.27}

We describe the residue matrices of
$(\nabla_{\mathbb{P}^1 \times \tilde{T}/\tilde{T}})_{\boldsymbol{\lambda}}$
at the regular singular points.
Put
\begin{equation*}
\begin{aligned}
M_2(\tilde{x})&:=
\begin{pmatrix}
-ab (\tilde{x} -1) & - b \tilde{x} -a\\
0 &a
\end{pmatrix} \text{ and}\\
\alpha_0^i(\tilde{x})&:= \lambda_{i+1} \left(
- \frac{ b c_i}{a} -
\frac{ a d_i -b c_i  \tilde{x} }{2a(\tilde{x} -t_{2i+1})}
-\frac{ a d_i -b c_i \tilde{x} }{2a(\tilde{x} -t_{2i+2} ) } \right).
\end{aligned}
\end{equation*}
Let $H^{\tilde{T}}_{2n-1}$ be
the residue matrix at $\tilde{x}=0$.
We have the following equality
\begin{equation*}
\begin{aligned}
H^{\tilde{T}}_{2n-1}&=M_2(0)^{-1}
\begin{pmatrix}
0 & \frac{\lambda_1(\tilde{b}^2 -4)}{8(a-1)} \\
 \frac{2\lambda_1(a-1)}{\tilde{b}^2 -4} & 0
\end{pmatrix}
M_2(0).
\end{aligned}
\end{equation*}
Let $H^{\tilde{T}}_{2n}$ be
the residue matrix at $\tilde{x}=1$.
We have the following equality
\begin{equation*}
\begin{aligned}
H_{2n}^{\tilde{T}}
&=
\frac{1-\lambda_0}{2}
\begin{pmatrix}
1 & \frac{2 (\tilde{b}^2+4a^2-8a)}{\tilde{b}^2-4a^2} \\
0 & -1
\end{pmatrix}
+
\sum^{n-2}_{i=1}
 \alpha_0^i(1)
\begin{pmatrix}
0 &1  +  \frac{  ( a+b)^2}{b^2(a-1)^2(1 -t_1)(1 -t_2)}\\
0 & 0
\end{pmatrix}.
\end{aligned}
\end{equation*}
Let $H^{\tilde{T}}_{1}$ and $H^{\tilde{T}}_{2}$ be
the residue matrices  at $\tilde{x}=t_{1}$
and $\tilde{x}=t_{2}$, respectively.
We have the following equalities
\begin{equation*}
\begin{aligned}
H^{\tilde{T}}_{1}
&=M_2(t_{1})^{-1}
 \begin{pmatrix}
-\frac{1}{4} & 0 \\
-\frac{\lambda_0(a-1)}{2(\tilde{b} -2 a)}
-\frac{\lambda_1(a-1)}{2(\tilde{b}-2)}
+ \sum^{n-2}_{i=1}
\frac{ a^2 \alpha^i_0(t_1)}{b^2(a-1)^2(t_1 -t_2)}
& \frac{1}{4}
\end{pmatrix}
 M_2(t_{1}) \\
H^{\tilde{T}}_{2}
&=M_2(t_{2})^{-1}
\begin{pmatrix}
-\frac{1}{4} & 0 \\
\frac{\lambda_0(a-1)}{2(\tilde{b} +2 a)}
+\frac{\lambda_1(a-1)}{2(\tilde{b}+2)}
+ \sum^{n-2}_{i=1}
\frac{ a^2 \alpha^i_0(t_2)}{b^2(a-1)^2(t_2 -t_1)}
&\frac{1}{4}
\end{pmatrix}
M_2(t_{2}).
\end{aligned}
\end{equation*}
Let $H^{\tilde{T}}_{2i+1}$ and $H^{\tilde{T}}_{2i+2}$ be
the residue matrices at $\tilde{x}=t_{2i+1}$ and $\tilde{x}=t_{2i+2}$, respectively.
We have the following equalities
\begin{equation*}
\begin{aligned}
H^{\tilde{T}}_{2i+1}&=
M_2(t_{2i+1})^{-1}
\begin{pmatrix}
0 & \frac{ \lambda_{i+1} (a d_i -b c_i t_{2i+1}) }{{2a}} \\
\frac{ \lambda_{i+1} a(-a d_i +b c_i t_{2i+1})}{{2b^2(a-1)^2 (t_{2i+1} -t_1)(t_{2i+1} -t_2)}} &0
\end{pmatrix}
M_2(t_{2i+1})\\
H^{\tilde{T}}_{2i+2}&=
M_2(t_{2i+2})^{-1}
\begin{pmatrix}
0 & \frac{ \lambda_{i+1}   (a d_i -b c_i t_{2i+2}) }{{2a }} \\
\frac{ \lambda_{i+1}  a (-a d_i +b c_i t_{2i+2})}{{2b^2(a-1)^2 (t_{2i+2} -t_1)(t_{2i+2} -t_2)}} &0
\end{pmatrix}
M_2(t_{2i+2}).
\end{aligned}
\end{equation*}
Let $H^{\tilde{T}}_{2n+1}$ be
the residue matrix at $\tilde{x}= \infty$.
Let $\cA_{jk}^i (\tilde{x})$ be
the relative rational $1$-forms over $\tilde{T}$ which are the relativization of
the pull-backs of the rational $1$-forms (\ref{2018.5.21.12.20}) under
the composition $\mathbb{P}^1 \times \tilde{T} \ra\mathbb{P}^1 \times T \ra \mathbb{P}^n$.
Since $\lim_{\tilde{x}\ra 0 } \alpha_0^i(\frac{1}{\tilde{x}})=0$, we have
\begin{equation*}
\res_{\tilde{x}=\infty}
\begin{pmatrix}
\cA_{11}^i (\tilde{x}) & \cA_{12}^i (\tilde{x}) \\
-\cA_{21}^i (\tilde{x}) &- \cA_{11}^i (\tilde{x})
\end{pmatrix}=0 \quad (i=1,\ldots,n-2).
\end{equation*}
Then we have
\begin{equation*}
H_{2n+1}^{\tilde{T}}=
\begin{pmatrix}
-1 & a -2 \\
1&a
\end{pmatrix}
\begin{pmatrix}
\frac{\lambda_0 + \lambda_1}{2} &0 \\
0&-\frac{\lambda_0 + \lambda_1}{2}
\end{pmatrix}
\begin{pmatrix}
-1 & a -2 \\
1&a
\end{pmatrix}^{-1}.
\end{equation*}

\subsection{Garnier system}\label{2019.7.10.22.15}

Let $\cA(\tilde{x})$ be the Fuchsian system with $2n + 1$ regular singularities
at $t_1 , \ldots , t_{2n}, \infty$:
\begin{equation*}
\cA(\tilde{x})= d+
\sum_{i=1}^{2n} \tilde{H}_{i}\frac{d\tilde{x}}{\tilde{x}-t_i} ,
\end{equation*}
where $\tilde{H}_{i}$ ($i=1, \ldots , 2n$) are
$2\times 2$ matrices independent of $\tilde{x}$ and
$t_i \neq t_j$ ($i \neq j$).
We assume that  $\tilde{H}_{2n+1}:=-\sum_{i=1}^{2n} \tilde{H}_{i}$ is a diagonal matrix and
the eigenvalues of $\tilde{H}_{i}$ ($i=1, \ldots ,2n+1$) are
as in Table \ref{2018.5.20.22.24}.
\begin{table}[htb]
\caption{The eigenvalues of the residue matrices
($i=1,\ldots , n-2$).}
  \begin{tabular}{c|ccccccc}\label{2018.5.20.22.24}
  Reside matrices  & $\tilde{H}_{1}$ &
    $\tilde{H}_{2}$ & $\tilde{H}_{2i+1}$ & $\tilde{H}_{2i+2}$
    &  $\tilde{H}_{2n-1}$ &  $\tilde{H}_{2n}$ & $\tilde{H}_{2n+1}$ \\\hline
  Eigenvalues
     & $\pm \frac{1}{4}$
      & $\pm \frac{1}{4}$
      & $\pm \frac{\lambda_{i+1}}{2}$
      & $\pm \frac{\lambda_{i+1}}{2}$
       & $\pm \frac{\lambda_1}{2}$
    & $\pm \frac{\lambda_0-1}{2}$
      &  $\pm\frac{\lambda_0+\lambda_1}{2}$
  \end{tabular}
\end{table}

We fix generators $\gamma_{\tilde{x}}$
($\tilde{x}= t_1 \ldots ,t_{2n}, \infty$)
of the fundamental group $\pi_1(\mathbb{P}^1 \setminus \{ t_1,\ldots,t_{2n}, \infty \},*)$.
Here the loop $\gamma_{\tilde{x}}$ on $\mathbb{P}^1$
is oriented counter-clockwise,
$\tilde{x}$ lies inside, while the other singular points lie outside.
Let $\rho'_{\boldsymbol{\lambda}}\colon
\pi_1(\mathbb{P}^1 \setminus\{ t_1,\ldots,t_{2n}, \infty \},*) \ra \mathrm{SL}_2(\mathbb{C})$
be the representation of the fundamental group defined by Table \ref{2018.5.12.23.13}.
We consider the isomonodromic deformation of the Fuchsian system $\cA(\tilde{x})$
whose preserved monodromy representation is conjugated to
 $\rho'_{\boldsymbol{\lambda}}$.
Let
$d+
\sum_{i=1}^{2n} \tilde{H}^0_{i}\frac{d\tilde{x}}{\tilde{x}-t^0_i} $
be the Fuchsian system with $2n + 1$ regular singularities
at $t^0_1 , \ldots , t^0_{2n}, \infty$ whose monodromy representation is
conjugated to $\rho'_{\boldsymbol{\lambda}}$.
There exists an open neighbourhood $U_{t^0}\subset \mathbb{C}^{2n}$
of the point $t^0=(t^0_1 , \ldots , t^0_{2n})$
such that for any $t \in U_{t^0}$, there exists a unique tuple
$(\tilde{H}_{i}(t))_{i=1,\ldots,2n}$ of analytic matrix valued functions
such that $\tilde{H}_{i}(t^0) = \tilde{H}^0_{i}$, $i=1,\ldots,2n$,
and the monodromy representation of
 $d+
\sum_{i=1}^{2n} \tilde{H}_{i}(t)\frac{d\tilde{x}}{\tilde{x}-t_i} $
 is
conjugated to $\rho'_{\boldsymbol{\lambda}}$.
The matrices $\tilde{H}_{i}(t)$ $i=1,\ldots,2n$ are
the solutions of the Cauchy problem
with the initial data $(\tilde{H}_{i}(t^0))_{i=1,\ldots,2n}$
for the Schlesinger equations (see \cite[Theorem 2.7]{Mazz}).

\begin{table}[htb]
\caption{The representation $\rho'_{\boldsymbol{\lambda}}$ of the fundamental group;
here $a_j = \exp(-\pi \sqrt{-1} \lambda_j)$ $j=0,1,\ldots,n-1$.}
  \begin{tabular}{c|c|c|c}\label{2018.5.12.23.13}
     $\tilde{x}=t_{2n-1}$ &  $\tilde{x}=t_{2n}$ & $\tilde{x}=t_1$ &
    $\tilde{x}=t_2$ \\\hline
$ \rho'_{\boldsymbol{\lambda}}(\gamma_{t_{2n-1}})
 = \begin{pmatrix} a_1 & 0 \\ 0 & a_1^{-1} \end{pmatrix}$
&  $\rho'_{\boldsymbol{\lambda}}(\gamma_{t_{2n}})
=\begin{pmatrix} -a_0 & 0 \\ 0 & -a_0^{-1} \end{pmatrix}$
& $\rho'_{\boldsymbol{\lambda}}(\gamma_{t_1})=\begin{pmatrix} 0 & 1 \\ -1 &0 \end{pmatrix}$
& $\rho'_{\boldsymbol{\lambda}}(\gamma_{t_2})
=\begin{pmatrix} 0 &  a_0^{2} \\ - a_0^{-2} & 0 \end{pmatrix}$
  \end{tabular}\\
  \begin{tabular}{c|c|c}
     $\tilde{x}=t_{2i+1}$ ($i=1,\ldots,n-2$)
     &  $\tilde{x}=t_{2i+2}$ ($i=1,\ldots,n-2$)
      & $\tilde{x}=\infty$\\\hline
 $\rho'_{\boldsymbol{\lambda}}(\gamma_{t_{2i+1}})=
 \begin{pmatrix} a_{i+1} & 0 \\ 0 & a_{i+1}^{-1} \end{pmatrix}$
 & $\rho'_{\boldsymbol{\lambda} }(\gamma_{t_{2i+2}})
 = \begin{pmatrix} a^{-1}_{i+1} & 0 \\ 0 & a_{i+1} \end{pmatrix}$
 & $\rho'_{\boldsymbol{\lambda}}(\gamma_{\infty})=
 \begin{pmatrix} a_0 a_1^{-1} & 0 \\ 0 &a_0^{-1} a_1 \end{pmatrix}$
  \end{tabular}
\end{table}

Let $\cA(\tilde{x})$ be the Fuchsian system with $2n + 1$ regular singularities
at $t_1 , \ldots , t_{2n}, \infty$ as above.
We fix the poles $t_{2n-1}$ and $t_{2n}$ at $0$ and $1$, respectively.
Let $\{\nu_1,\ldots,\nu_{2n-2}\}$ be the roots of the following equation of degree $2n-2$:
\begin{equation}\label{2018.5.12.23.46}
\sum_{k=1}^{2n} \frac{(\tilde{H}_{k})_{12}}{\tilde{x}-t_k} =0.
\end{equation}
For each $\nu_i$, we define $\rho_i$ by
\begin{equation}\label{2018.5.12.23.47}
\begin{aligned}
\rho_i :=
\sum_{k=1}^{2n}  \frac{(\tilde{H}_{k})_{11}+\frac{\theta_{k}}{2}}{\nu_i-t_{k}}.
\end{aligned}
\end{equation}
If a tuple $(\tilde{H}_{i}(t))_{i=1,\ldots,2n}$ is
a solution of the Schlesinger equations,
then the corresponding functions $\nu_{j}(t_1,\ldots,t_{2n-2})$ and
$\rho_{j}(t_1,\ldots,t_{2n-2})$ ($j=1,\ldots,2n-2$)
satisfy the Garnier system $\mathcal{G}_{2n-2}$ (see \cite[Theorem 2.1]{Mazz}).

\subsection{Algebraic solution}\label{2019.7.12.11.28}

By the morphism (\ref{2018.5.12.23.51}), we have a generically finite morphism
\begin{equation*}
\Spec \mathbb{C}[\rho_i,\nu_i]_{1\le  i  \le2n-2} \times \tilde{T}\lra
\Spec \mathbb{C}[\rho_i,\nu_i]_{1\le  i  \le2n-2} \times \Spec \mathbb{C}[t_1,\ldots,t_{2n-2}].
\end{equation*}

We consider the algebraic solution of $\mathcal{G}_{2n-2}$ associated to
the representation $\rho'_{\boldsymbol{\lambda}}$.

\begin{proof}[Proof of Theorem \ref{2019.7.10.22.13}]
For the residue matrices $H^{\tilde{T}}_{i} $ of
$(\nabla_{\mathbb{P}^1 \times \tilde{T}/\tilde{T}})_{\boldsymbol{\lambda}}$,
we put
\begin{equation*}
\tilde{H}^{\tilde{T}}_{i}:=
\begin{pmatrix}
-1 & a -2 \\
1&a
\end{pmatrix}^{-1}
H^{\tilde{T}}_{i}
\begin{pmatrix}
-1 & a -2 \\
1&a
\end{pmatrix}
\end{equation*}
for $i = 1, \ldots, 2n$.
Let $\cA^{\tilde{T}}(\tilde{x})$ be the family of the Fuchsian systems
with $2n + 1$ regular singularities at $0,1,t_1 , \ldots , t_{2n-2}, \infty$
parametrized by $\tilde{T}$
defined by
\begin{equation}\label{2018.5.12.23.49}
\cA^{\tilde{T}}(\tilde{x}):=d+
\tilde{H}^{\tilde{T}}_{2n-1}\frac{d\tilde{x}}{\tilde{x}} +
\tilde{H}^{\tilde{T}}_{2n}\frac{d\tilde{x}}{\tilde{x}-1} +
\sum_{i=1}^{2n-2} \tilde{H}^{\tilde{T}}_{i}\frac{d\tilde{x}}{\tilde{x}-t_i} .
\end{equation}
Since $\tilde{H}^{\tilde{T}}_{2n+1}:=
-\sum_{i=1}^{2n} \tilde{H}^{\tilde{T}}_{i}$
is a diagonal matrix,
we have the assertion (i) of Theorem \ref{2019.7.10.22.13}.

By Proposition \ref{2018.5.8.15.13},
for each $\tilde{t} \in \tilde{T}$,
the Fuchsian system $\cA^{\tilde{T}}(\tilde{x})$
has the monodromy representation which is conjugated to $\rho'_{\boldsymbol{\lambda}}$,
which is independent of $\tilde{t} \in\tilde{T}$.
That is, the family $\cA^{\tilde{T}}(\tilde{x})$
of the Fuchsian systems parametrized by $\tilde{T}$
preserves
their monodromy representations.
Then we have the assertion (ii) of Theorem \ref{2019.7.10.22.13}.

By (\ref{2018.5.12.23.46}), (\ref{2018.5.12.23.47}), and (\ref{2018.5.12.23.49}),
we have algebraic functions $\nu_i$, $\rho_i$ ($i=1,\ldots,2n-2$) on $\tilde{T}$.
These algebraic functions give the solution of $\mathcal{G}_{2n-2}$ associated to
the representation $\rho'_{\boldsymbol{\lambda}}$.
Since $\dim \tilde{T} =2n-2$ and
$\tilde{T} \ra \Spec \mathbb{C}[t_1,t_2,\ldots ,t_{2n-2}]$
is a generically finite morphism,
we have the assertion (iii) of Theorem \ref{2019.7.10.22.13}.
\end{proof}

\noindent
{\bf Acknowledgments.}
The author thanks Frank Loray for many valuable discussions.
He also thanks Masa-Hiko Saito for warm encouragement.
He is supported by JSPS KAKENHI Grant Numbers 18J00245
and 19K14506.
He is grateful to the anonymous referee's suggestions which helped to improve the paper.


TITLE: second derivative of product of Jacobi with geodesic is ill
QUESTION [2 upvotes]: I have a Jacobi field $J$ along a geodesic $\gamma$ and a metric $g$. How do I show the following.
$$\frac{d^2}{dt^2}g(J_t,\dot{\gamma}(t)) = 0$$
Since the connexion is compatible, we have
$$\frac{d}{dt}g(J_t,\dot{\gamma}(t)) = g(\nabla_\dot{\gamma}J,\dot{\gamma}) + g(J,\nabla_\dot{\gamma}\dot{\gamma}) = g(\nabla_\dot{\gamma}J,\dot{\gamma})$$
so second derivative is $g(\nabla_\dot{\gamma}(\nabla_\dot{\gamma}J),\dot{\gamma})$
but after that I am always turning in circles to try and find something that is zero.

REPLY [2 votes]: Let $J=J(t)$ be a Jacobi vector field along $\gamma$.
Denote $J'=D_t (J)$ and $J''=D^{2}_{t}J$
Now
\begin{align}
D^{2}_{t} \langle J, \gamma' \rangle &= \langle J''(t), \gamma' \rangle \\
&= - \langle R(J, \gamma')\gamma',\gamma' \rangle \\
&= -R(J, \gamma', \gamma', \gamma') \\
&= 0
\end{align}