# Moduli spaces for the fifth Painlevé equation

Marius van der Put and Jaap Top  Bernoulli Institute, Nijenborgh 9, 9747 AG
Groningen, the Netherlands<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Date: September 2, 2024)

###### Abstract.

Isomonodromy for the fifth Painlevé equation $P_{5}$ is studied in detail in
the context of certain moduli spaces for connections, monodromy, the
Riemann–Hilbert morphism, and Okamoto-Painlevé spaces. This involves explicit
formulas for Stokes matrices and parabolic structures. The rank 4 Lax pair for
$P_{5}$, introduced by Noumi–Yamada et al., is shown to be induced by a
natural fine moduli space of connections of rank 4. As a by-product one
obtains a polynomial Hamiltonian for $P_{5}$, equivalent to the one of
Okamoto.

## Introduction and Summary

Historically, the list of classical Painlevé equations
$y^{\prime\prime}=F(y^{\prime},y,z)$ was deduced from the property that the
only moving singularities of the solutions are poles. The connection with
isomonodromy was quite early recognised. In the papers of Jimbo, Miwa and Ueno
[J-M-U, JM, Mi] et al. this theme is developed. In particular, isomonodromic
families of rank two matrix differential operators $\frac{d}{dz}+A$ are
provided for each of $P_{1}$-$P_{6}$. We refer to [C, C2, C3, G, O, O1, O2,
OO1] for some theory and detailed explicit equations. We note that this choice
does not do justice to the extensive literature related to Painlevé equations.

The family of matrix differential operators for $P_{3},P_{4}$ and for the
degenerate fifth Painlevé equation $degP_{5}$ has been refined, see [vdP-T3,
vdP-T4, A-P-T], to fine moduli spaces of connections on the projective line,
which are identified with Okamoto-Painlevé spaces. The detailed construction
of the moduli spaces supplements and continues some sections of [vdP-Sa].

The present paper applies a comparable technique for $P_{5}$. This supplements
and improves the sections [vdP-Sa, §3.2, 4.3] involving $P_{5}$. The family of
matrix differential operators will be refined to “natural” fine moduli spaces
$\mathcal{M}(\theta_{0},\theta_{1},\theta_{\infty})$ of connections of rank
$2$ on the projective line. A moduli space
$\mathcal{R}^{geom}(s_{1},s_{2},s_{3})$ for the monodromy is constructed. It
is shown that the extended Riemann–Hilbert morphism

$RH^{+}:\mathcal{M}(\theta_{0},\theta_{1},\theta_{\infty})\rightarrow\mathcal{R}^{geom}(s_{1},s_{2},s_{3})\times\mathbb{C}^{*}\mbox{
is an isomorphism.}$

This implies the Painlevé property for the solutions of $P_{5}$ and provides
the identification of $\mathcal{M}(\theta_{0},\theta_{1},\theta_{\infty})$
with the Okamoto-Painlevé spaces.
Some moduli spaces of connections of rank 4 are studied in relation with the
Lax pairs of Noumi–Yamada and Bäcklund transformations for $P_{5}$.

Now we discuss some details of the paper. The construction of moduli spaces
starts in §1 by defining a set ${\bf
S}(\theta_{0},\theta_{1},\theta_{\infty})$ of differential modules $M$ of
dimension 2 over $\mathbb{C}(z)$. This set consists (in short) of the $M$ with
regular singular points $z=0$ and $z=1$ with local eigenvalues
$\pm\frac{\theta_{0}}{2}$ and $\pm\frac{\theta_{1}}{2}$. Further, the point
$z=\infty$ is an irregular singularity with generalized eigenvalues
$\pm\frac{tz+\theta_{\infty}}{2}$. The definition of ${\bf
S}(\theta_{0},\theta_{1},\theta_{\infty})$ moreover involves ‘parabolic
structures’.

The monodromy data for objects of ${\bf
S}(\theta_{0},\theta_{1},\theta_{\infty})$ are given by local monodromy
matrices, Stokes matrices and parabolic structure. In §2 a fine moduli space
$\mathcal{R}^{geom}(s_{1},s_{2},s_{3})$ is constructed, for eigenvalues
$s_{1}=e^{\pi i\theta_{0}}+e^{-\pi i\theta_{0}}$, $s_{2}=e^{\pi
i\theta_{1}}+e^{-\pi i\theta_{1}}$, $s_{3}=e^{\pi i\theta_{\infty}}$. It is
shown that $\mathcal{R}^{geom}(s_{1},s_{2},s_{3})$ is simply connected and can
be identified with the minimal resolution of the monodromy space obtained in
[vdP-Sa, §3.2, §4.3]. The natural map

${\bf
S}(\theta_{0},\theta_{1},\theta_{\infty})\rightarrow\mathcal{R}^{geom}(s_{1},s_{2},s_{3})(\mathbb{C})\times\mathbb{C}^{*}\mbox{
is a bijection}.$

In §3, one considers a vector bundle $\mathcal{V}$, of rank $2$ and degree
$-1$ on $\mathbb{P}^{1}$. The data for every object $M\in{\bf
S}(\theta_{0},\theta_{1},\theta_{\infty})$ produce a unique connection
$\nabla(M):\mathcal{V}\rightarrow\Omega([0]+[1]+2[\infty])\otimes\mathcal{V}$
with prescribed data at the singular points $0,1,\infty$. The generic fibre of
$\nabla(M)$ is the differential module $M$ (including data). This leads to a
moduli functor and a fine moduli space
$\mathcal{M}(\theta_{0},\theta_{1},\theta_{\infty})$. It is shown that the
extended Riemann–Hilbert morphism $RH^{+}$ is an isomorphism if
$\theta_{0},\theta_{1}\not\in\mathbb{Z}$.

The reducible locus, corresponding to the reducible $M\in{\bf
S}(\theta_{0},\theta_{1},\theta_{\infty})$ is computed in §4. Isomonodromy in
this sublocus of $\mathcal{M}(\theta_{0},\theta_{1},\theta_{\infty})$ produces
the Riccati solutions of $P_{5}$. The parabolic structure is again studied in
§5 in order to prove the main result Theorem 5.1, namely, the extended
Riemann–Hilbert morphism is an isomorphism for all values of
$\theta_{0},\theta_{1},\theta_{\infty}$.

In §6, a standard computation with Lax Pairs produces the usual differential
equation for $P_{5}$.

From the extended Riemann–Hilbert map $RH^{+}$ one can read off and find the
formulas for some Bäcklund transformations. However, some are missing. Noumi
and Yamada constructed a Lax pair for $P_{5}$ from which one can read off all
Bäcklund transformation.

Our contribution in §7 to this is to provide a natural moduli space of
connections $\mathcal{M}_{4}$ and a corresponding monodromy space
$\mathcal{R}_{4}$, which produce the Lax pair of Noumi–Yamada. The moduli
space $\mathcal{M}_{4}$ is associated to the set of differential modules $M$
of dimension 4 over $\mathbb{C}(z)$ such that:

$\begin{array}[]{l}(\text{a})\quad\Lambda^{4}M\;\;\text{is trivial,}\\\
(\text{b})\quad z=0\;\;\text{is regular singular, }\\\ (\text{c})\quad
z=\infty\;\;\text{is irregular singular, and}\\\ \mbox{ }\hskip
25.60747pt\text{its generalized eigenvalues
are}\;\;z^{1/2}+tz^{1/4}\;\;\text{and its conjugates.}\end{array}$

A direct computation of a matrix differential operator for $\mathcal{M}_{4}$
seems rather difficult. Instead, one gives a module $M\in\mathcal{M}_{4}$ the
interpretation of a differential module $N$ over $\mathbb{C}(z^{1/4})$
provided with a symmetry $\sigma$ of order 4. Explicit formulas for $N$ and
$M:=N^{<\sigma>}$ are inspired by [vdP-S, §12.5]. The resulting Lax pair is
identical to the one in [S-H-C]. A calculation of the Lax pair for $N$ instead
of $M$ produces at once a polynomial Hamiltonian for $P_{5}$ which, up to a
linear change of coordinates, is also present in Okamoto’s work.

##  1\. The set ${\mathbf{S}}(\theta_{0},\theta_{1},\theta_{\infty})$ of
differential modules

We fix elements $\theta_{0},\theta_{1},\theta_{\infty}\in\mathbb{C}$ which are
called parameters. One considers the set
$\mathbf{S}(\theta_{0},\theta_{1},\theta_{\infty})$ of pairs $(M,t)$ (up to
isomorphism) with $t\in\mathbb{C}^{*}$ and $M$ a differential module over
$\mathbb{C}(z)$ with the properties:
(1). $M$ has dimension two; $\det M=\Lambda^{2}M$ is the trivial 1-dimensional
differential module; $M$ has at most singularities at $z=0,1,\infty$.
(2). The point $z=0$ is regular singular and $\mathbb{C}(\\!(z)\\!)\otimes M$
can be represented by the differential operator
$\frac{d}{dz}+\frac{1}{z}A_{0}$, where $A_{0}$ is a $2\times 2$-matrix with
entries in $\mathbb{C}[\\![z]\\!]$ and $A_{0}\mod(z)$ has eigenvalues
$\pm\frac{\theta_{0}}{2}$.
(3). The point $z=1$ is regular singular and $\mathbb{C}(\\!(z-1)\\!)\otimes
M$ can be represented by the operator $\frac{d}{dz}+\frac{1}{z-1}A_{1}$, where
$A_{1}$ is a $2\times 2$-matrix with entries in $\mathbb{C}[\\![z-1]\\!]$ and
$A_{1}\mod(z-1)$ has eigenvalues $\pm\frac{\theta_{1}}{2}$.
(4). The point $z=\infty$ is irregular singular and
$\mathbb{C}(\\!(z^{-1})\\!)\otimes M$ can be represented by the operator
$\frac{d}{dz}+\frac{1}{z}\cdot\left(\begin{array}[]{cc}\frac{tz+\theta_{\infty}}{2}&0\\\
0&-\frac{tz+\theta_{\infty}}{2}\end{array}\right)$.
(5). $M$ is not split, i.e., $M$ is not the direct sum of 1-dimensional
differential submodules.

The datum “$t$” is attached to the module $M$ in order to distinguish the two
generalized local exponents $\pm\frac{tz+\theta_{\infty}}{2}$ at $z=\infty$
(especially for the cases $\theta_{\infty}\in\\{-1,0\\}$).

We will construct a (fine) moduli space
$\mathcal{M}(\theta_{0},\theta_{1},\theta_{\infty})$ of connections on the
projective line such that the map
$\mathcal{M}(\theta_{0},\theta_{1},\theta_{\infty})(\mathbb{C})\rightarrow{\bf
S}(\theta_{0},\theta_{1},\theta_{\infty})$, which associates to a connection
its general fibre, is a bijection.

In order to obtain this one has to remove split modules (see (5)) and add
additional structure in some special cases (see (6) and (7)).

For $\theta_{0}\in\mathbb{Z}$ and/or $\theta_{1}\in\mathbb{Z}$ a refinement of
the definition of ${\bf S}(\theta_{0},\theta_{1},\theta_{\infty})$ is needed.
This consists of the additional data of one or two lines, namely:
(6). For $\theta_{0}\in\mathbb{Z}$, it is a line
$\mathbb{C}v\subseteq\mathbb{C}(\\!(z)\\!)\otimes M$ such that
$D(v)=-\frac{|\theta_{0}|}{2}v$.
(7). For $\theta_{1}\in\mathbb{Z}$, it is a line
$\mathbb{C}v\subseteq\mathbb{C}(\\!(z-1)\\!)\otimes M$ such that
$D(v)=-\frac{|\theta_{1}|}{2}v$.
The lines are called eigenlines and the additional data is called a parabolic
structure or a level structure.

Explanation of the conditions (6) and (7).
Consider a differential module $M$ over $\mathbb{C}(z)$ satisfying (1)–(5).
Write $D\colon M\rightarrow M$ for its differential operator with $D(f\cdot
m)=\frac{df}{dz}\cdot m+f\cdot D(m)$. We want to attach to $M$ an “invariant
lattice at $z=0$ with data $\theta_{0}$”. This is a
$\mathbb{C}[\\![z]\\!]$-lattice $\Lambda\subseteq\mathbb{C}(\\!(z)\\!)\otimes
M$ such that $zD(\Lambda)\subseteq\Lambda$ and $zD$ induces on
$\Lambda/z\Lambda$ a linear operator with eigenvalues
$\pm\frac{\theta_{0}}{2}$. In general $\Lambda$ is unique, except possibly for
$\theta_{0}\in\mathbb{Z}_{\neq 0}$. In the latter case, the parabolic
structure, namely the eigenline $\mathbb{C}v$, leads to a unique possibility
for $\Lambda$. Without the parabolic structure the (to be constructed) moduli
space of connections would have singularities or its set of closed points
would differ from ${\bf S}(\theta_{0},\theta_{1},\theta_{\infty})$. The
parabolic structure produces a natural resolution of the singularities that
one would have otherwise. The technical Lemma 1.1 provides the verification of
the above statements. In §1.5 there is another formulation of this Lemma.

######  Lemma 1.1.

Consider differential modules $M=(M,D)$ over $\mathbb{C}(\\!(z)\\!)$ with
$D(f\cdot m)=z\frac{df}{dz}\cdot m+f\cdot D(m)$. Let
$L=\mathbb{C}(\\!(z)\\!)e$ be defined by $De=\frac{1}{2}e$. Let $N$ be a
differential module with $\dim N=2$, $\det N$ is trivial and $N$ has a
$D$-invariant lattice $\Lambda$ such that $D$ has eigenvalues
$\pm\frac{\theta}{2}$ for its action on $\Lambda/z\Lambda$.
(1). The $D$-invariant lattice $\Lambda$ is unique except in the cases:
(a) $\theta\in\mathbb{Z}_{\neq 0}$ is even and $N$ is trivial,
(b) $\theta\in\mathbb{Z}_{\neq 0}$ is odd and $L\otimes N$ is trivial.
(2). Let $\theta=m\in\mathbb{Z}_{>0}$ and suppose that (1a) or (1b) holds. Let
$\mathcal{LA}$ denote the set of $D$-invariant lattices $\Lambda$ such that
$D$ has eigenvalues $\pm\frac{m}{2}$ on $\Lambda/z\Lambda$. Let $\mathcal{LI}$
denote the set of lines $\mathbb{C}v\neq 0$ in $N$ such that
$D(v)=-\frac{m}{2}v$. Every lattice $\Lambda\in\mathcal{LA}$ contains a unique
$\mathbb{C}v\in\mathcal{LI}$. This defines a bijection
$\mathcal{LA}\rightarrow\mathcal{LI}$.

Fix a basis $b_{1},b_{2}$ of $N$ such that $D(b_{1})=-\frac{m}{2}b_{1},\
D(b_{2})=\frac{m}{2}b_{2}$. The set $\mathcal{LI}$ identifies with the
projective complex line
$\mathbb{P}(\mathbb{C}\,b_{1}+\mathbb{C}\,z^{-m}b_{2})$.
(3). Suppose $\theta\in\mathbb{Z}_{\geq 0}$. There is a unique line
$\mathbb{C}v\neq 0$ in $N$ with $D(v)=-\frac{\theta}{2}v$, expect for cases
(1a), (1b) and the case where $N$ is trivial and $\theta=0$. In the last case
the set of those lines forms the projective line $\mathbb{P}(\\{n\in
N|D(n)=0\\})$.

###### Proof.

Suppose $\theta\neq 0$. The choice of a $D$-invariant lattice for $N$ produces
a differential operator $z\frac{d}{dz}+A$ where $A$ has entries in
$\mathbb{C}[\\![z]\\!]$ and
$A\equiv\genfrac{(}{)}{0.0pt}{1}{\lambda\;\;\;0}{0\ -\lambda}\bmod(z)$ and
$\lambda=-\frac{\theta}{2}$. By conjugation with a matrix
$U=1+U_{1}z+U_{2}z^{2}+\cdots$ (all $U_{i}$ are $2\times 2$-matrices,
$U_{0}:=1$) we try to obtain
$z\frac{d}{dz}+\genfrac{(}{)}{0.0pt}{1}{\lambda\;\;0}{0\ -\lambda}$. This
leads to
$U(z\frac{d}{dz}+A)=(z\frac{d}{dz}+\genfrac{(}{)}{0.0pt}{1}{\lambda\;\;0}{0\
-\lambda})U$. Write $A=A_{0}+A_{1}z+A_{2}z^{2}+\cdots$ with
$A_{0}=\genfrac{(}{)}{0.0pt}{1}{\lambda\;\;0}{0\ -\lambda}$ and all $A_{j}$
are $2\times 2$-matrices. Then one obtains equations

$\sum_{n\geq 1}\sum_{i=1}^{n}(U_{n-i}A_{i})z^{n}=\sum_{n\geq
1}(nU_{n}+A_{0}U_{n}-U_{n}A_{0})z^{n}\mbox{ for the matrices }U_{i}.$

$U_{n}\mapsto nU_{n}+A_{0}U_{n}-U_{n}A_{0}$ sends
$\genfrac{(}{)}{0.0pt}{1}{x_{1}\;\;x_{2}}{x_{3}\;\;x_{4}}$ to
$\left(\begin{array}[]{cc}nx_{1}&(n+2\lambda)x_{2}\\\
(n-2\lambda)x_{3}&nx_{4}\end{array}\right)$.

If $\theta=-2\lambda\not\in\mathbb{Z}$, there is a unique solution
$U=1+U_{1}z+U_{2}z^{2}+\cdots$. The given lattice has a basis $b_{1},b_{2}$
with $D(b_{1})=\lambda b_{1},\ D(b_{2})=-\lambda b_{2}$.

Any $D$-invariant lattice $\widetilde{\Lambda}$ such that $D$ has eigenvalues
$\pm\lambda$ on $\widetilde{\Lambda}/z\widetilde{\Lambda}$ has a basis
$\widetilde{b}_{1},\widetilde{b}_{2}$ with
$D(\widetilde{b}_{1})=\lambda\widetilde{b}_{1},\
D(\widetilde{b}_{2})=-\lambda\widetilde{b}_{2}$. The matrix
$\genfrac{(}{)}{0.0pt}{1}{x_{1}\;\;x_{2}}{x_{3}\;\;x_{4}}$ relating the two
bases has the property

$\genfrac{(}{)}{0.0pt}{1}{x_{1}\;\;x_{2}}{x_{3}\;\;x_{4}}(z\frac{d}{dz}+\genfrac{(}{)}{0.0pt}{1}{\lambda\;\;\;0}{0\
-\lambda})=(z\frac{d}{dz}+\genfrac{(}{)}{0.0pt}{1}{\lambda\;\;\;0}{0\
-\lambda})\genfrac{(}{)}{0.0pt}{1}{x_{1}\;\;x_{2}}{x_{3}\;\;x_{4}}.$

Using $\genfrac{(}{)}{0.0pt}{1}{x_{1}\;\;x_{2}}{x_{3}\;\;x_{4}}\in{\rm
GL}_{2}(\,\mathbb{C}(\\!(z)\\!)\,)$ one obtains $x_{1},x_{4}\in\mathbb{C}^{*}$
and $x_{2}=x_{3}=0$. Thus $\widetilde{\Lambda}$ coincides with the given
$D$-invariant lattice.

Case $\theta=-2\lambda\in\mathbb{Z}_{\neq 0}$. We suppose for convenience that
$\theta=m\in\mathbb{Z}_{>0}$. The above system of equations shows that the
equations for the $U_{n}$ have unique solutions for $n\neq m$. There might not
be a solution for $U_{m}$. This implies that the given lattice has a basis
$b_{1},b_{2}$ such that $D(b_{1})=\lambda b_{1}$ and $D(b_{2})=-\lambda
b_{2}+az^{m}b_{1}$ where $a\in\mathbb{C}$.

The subcase $a\neq 0$. We may normalize to $a=1$. A straightforward
calculation shows that there is no element $v\in N,\ v\neq 0$ with
$D(v)=-\lambda v$. Any $D$-invariant lattice $\widetilde{\Lambda}$ in $N$ such
that $D$ has eigenvalues $\pm\lambda$ on
$\widetilde{\Lambda}/z\widetilde{\Lambda}$ has a basis
$\widetilde{b}_{1},\widetilde{b}_{2}$ with similar properties. Comparing the
two lattices produces (as before) an equality
$\genfrac{(}{)}{0.0pt}{1}{x_{1}\;\;x_{2}}{x_{3}\;\;x_{4}}(z\frac{d}{dz}+\genfrac{(}{)}{0.0pt}{1}{\lambda\;\;z^{m}}{0\
-\lambda})=(z\frac{d}{dz}+\genfrac{(}{)}{0.0pt}{1}{\lambda\;\;z^{m}}{0\
-\lambda})\genfrac{(}{)}{0.0pt}{1}{x_{1}\;\;x_{2}}{x_{3}\;\;x_{4}}$ with
$\genfrac{(}{)}{0.0pt}{1}{x_{1}\;\;x_{2}}{x_{3}\;\;x_{4}}\in{\rm
GL}_{2}(\,\mathbb{C}(\\!(z)\\!)\,)$. This leads to the matrix equation

$z\frac{d}{dz}\genfrac{(}{)}{0.0pt}{1}{x_{1}\;\;x_{2}}{x_{3}\;\;x_{4}}=-m\cdot\left[\genfrac{(}{)}{0.0pt}{1}{x_{1}\;\;x_{2}}{x_{3}\;\;x_{4}},\genfrac{(}{)}{0.0pt}{1}{\frac{1}{2}\;\;\;0}{0\;\;\frac{-1}{2}}\right]+z^{m}\cdot\left[\genfrac{(}{)}{0.0pt}{1}{x_{1}\;\;x_{2}}{x_{3}\;\;x_{4}},\genfrac{(}{)}{0.0pt}{1}{0\;\;1}{0\;\;0}\right].$

The solutions of this system of equations are $x_{1}=x_{4}\in\mathbb{C}^{*}$,
$x_{3}=0$ and $x_{2}=c\cdot z^{m}$ with $c\in\mathbb{C}$. Hence
$\widetilde{\Lambda}$ coincides with the given $D$-invariant lattice.

The subcase $a=0$. Two $D$-invariant lattices such that $D\bmod(z)$ has the
eigenvalues $\pm\lambda$, are related by a matrix
$\genfrac{(}{)}{0.0pt}{1}{x_{1}\;\;x_{2}}{x_{3}\;\;x_{4}}\in{\rm
GL}_{2}(\,\mathbb{C}(\\!(z)\\!)\,)$ satisfying
$\genfrac{(}{)}{0.0pt}{1}{x_{1}\;\;x_{2}}{x_{3}\;\;x_{4}}(z\frac{d}{dz}+\genfrac{(}{)}{0.0pt}{1}{\lambda\;\;0}{0\;\;-\lambda})=(z\frac{d}{dz}+\genfrac{(}{)}{0.0pt}{1}{\lambda\;\;0}{0\;\;-\lambda})\genfrac{(}{)}{0.0pt}{1}{x_{1}\;\;x_{2}}{x_{3}\;\;x_{4}}$.
The solutions are
$\genfrac{(}{)}{0.0pt}{1}{x_{1}\;\;x_{2}}{x_{3}\;\;x_{4}}=\genfrac{(}{)}{0.0pt}{1}{c_{1}\;\;\;\;\;\;c_{2}z^{m}}{c_{3}z^{-m}\;\;c_{4}}$
with $c_{1},c_{2},c_{3},c_{4}\in\mathbb{C}$ and $c_{1}c_{4}-c_{2}c_{3}\neq 0$.
The two $D$-invariant lattices are equal if and only if $c_{3}=0$. The
conclusion is that there are many $D$-invariant lattices such that $D$ has,
modulo $(z)$, eigenvalues $\pm\lambda$. This proves statement (1) of the
Lemma.

Using the form of the matrix
$\genfrac{(}{)}{0.0pt}{1}{c_{1}\;\;\;\;\;\;c_{2}z^{m}}{c_{3}z^{-m}\;\;c_{4}}$
one deduces statement (2) of the Lemma. Furthermore, the case $a=0$
corresponds to $N$ is trivial if $\theta$ is even and $N\otimes L$ is trivial
if $\theta$ is odd.

The case $\theta=0$. Let $\Lambda$ be a $D$-invariant lattice such that $D$
has on $\Lambda/z\Lambda$ only $0$ as eigenvalue. Then $\Lambda$ has a basis
$b_{1},b_{2}$ such that $D$ becomes the operator
$z\frac{d}{dz}+\genfrac{(}{)}{0.0pt}{1}{0\;\;a}{0\;\;0}$ with
$a\in\mathbb{C}$. The module $N$ is trivial if and only if $a=0$. As above one
verifies that there is only one $D$-invariant lattice with this property.

Finally, if $a=0$ then a basis $b_{1},b_{2}$ exists with
$D(b_{1})=D(b_{2})=0$. Hence $D(v)=0$ if and only if
$v\in\mathbb{C}b_{1}+\mathbb{C}b_{2}$. ∎

##  2\. Definition of the monodromy space $\mathcal{R}^{\text{\it
geom}}(s_{1},s_{2},s_{3})$

The second step is to convert the analytic data for the modules in ${\bf
S}(\theta_{0},\theta_{1},\theta_{\infty})$ into a moduli space.
As in §1, additional data are added in order to have a good “geometric
quotient” without singularities. The analytic data added to any $M\in{\bf
S}(\theta_{0},\theta_{1},\theta_{\infty})$ is represented by the five matrices
defined below.
(i). The monodromy matrix
$\genfrac{(}{)}{0.0pt}{1}{a_{1}\;\;b_{1}}{c_{1}\;\;d_{1}}$ at $z=0$ with
eigenvalues $e^{\pm\pi i\theta_{0}}$ and with $s_{1}=e^{\pi
i\theta_{0}}+e^{-\pi i\theta_{0}}$,
(ii). The monodromy matrix
$\genfrac{(}{)}{0.0pt}{1}{a_{2}\;\;b_{2}}{c_{2}\;\;d_{2}}$ at $z=1$ with
eigenvalues $e^{\pm\pi i\theta_{1}}$ and with $s_{2}=e^{\pi
i\theta_{1}}+e^{-\pi i\theta_{1}}$,
(iii). The formal monodromy matrix
$\genfrac{(}{)}{0.0pt}{1}{\alpha\;\;0}{0\;\;\frac{1}{\alpha}}$ at $z=\infty$
with eigenvalue $s_{3}:=\alpha=e^{\pi i\theta_{\infty}}$ and the two Stokes
matrices $\genfrac{(}{)}{0.0pt}{1}{1\;\;0}{f_{2}\;\;1}$,
$\genfrac{(}{)}{0.0pt}{1}{1\;\;f_{1}}{0\;\;1}$.
(iv). The relation between these matrices, see [vdP-Sa, §3.2.1], is

$\genfrac{(}{)}{0.0pt}{1}{a_{1}\;\;b_{1}}{c_{1}\;\;d_{1}}\cdot\genfrac{(}{)}{0.0pt}{1}{a_{2}\;\;b_{2}}{c_{2}\;\;d_{2}}=\genfrac{(}{)}{0.0pt}{1}{\alpha\;\;0}{0\;\;\frac{1}{\alpha}}\cdot\genfrac{(}{)}{0.0pt}{1}{1\;\;0}{f_{2}\;\;1}\cdot\genfrac{(}{)}{0.0pt}{1}{1\;\;f_{1}}{0\;\;1}.$

(v). The analytic data is not split, i.e., not diagonalizable, which means
that at least one of the four items $b_{1},c_{1},b_{2},c_{2}$ is not zero.
(vi). If $\theta_{0}\in\mathbb{Z}$ or equivalently $s_{1}=\pm 2$, then there
is the additional datum of an eigenline $\mathbb{C}v\neq 0$ for the monodromy
matrix $\genfrac{(}{)}{0.0pt}{1}{a_{1}\;\;b_{1}}{c_{1}\;\;d_{1}}$.
(vii). If $\theta_{1}\in\mathbb{Z}$ or equivalently $s_{2}=\pm 2$, then there
is the additional datum of an eigenline $\mathbb{C}v\neq 0$ for the monodromy
matrix $\genfrac{(}{)}{0.0pt}{1}{a_{2}\;\;b_{2}}{c_{2}\;\;d_{2}}$.

For the explanation of (i)–(vii), we recall from [vdP-Sa, §3.2] the following.
At $z=\infty$ one chooses a basis $B_{1},B_{2}$ of
$\mathbb{C}(\\!(z^{-1})\\!)\otimes M$ such that the differential operator has
matrix
$\frac{1}{z}\cdot\genfrac{(}{)}{0.0pt}{1}{\frac{tz+\theta_{\infty}}{2}\;\;\;\;\;\;0}{0\;\;\;-\frac{tz+\theta_{\infty}}{2}}$
on this basis. Then $B_{1}$ and $B_{2}$ are unique up to multiplication by
elements of $\mathbb{C}^{*}$. Furthermore one chooses paths from $0$ to $1$
and from $1$ to $\infty$. The latter should have an ordinary direction at
$\infty$ such that (multi)summation gives a connection between the formal
solutions at $z=\infty$ and the local solutions at $z=1$ and $z=0$. All five
matrices are considered with respect to this basis $B_{1},B_{2}$. The identity
(iv) follows from the “monodromy identity” [vdP-S, Proposition 8.12]. The
above five matrices are considered modulo simultaneous conjugation by
$\mathbb{G}_{m}=\\{\genfrac{(}{)}{0.0pt}{1}{c\;\;0}{0\;\;1}\;|\;c\in\mathbb{C}^{*}\\}$,
since the basis $B_{1},B_{2}$ is unique up to multiplication by scalars.

Let $\text{Spec}(R)$ denote the affine variety describing the above five
matrices and their relations (for fixed $s_{1}:=e^{\pi i\theta_{0}}+e^{-\pi
i\theta_{0}}$, $s_{2}:=e^{\pi i\theta_{1}}+e^{-\pi i\theta_{1}}$,
$s_{3}:=\alpha$). We note that
$R=\mathbb{C}[a_{1},b_{1},c_{1},d_{1},a_{2},b_{2},c_{2},d_{2}]/\mbox{
(relations)}$. The relations are generated by

$a_{1}d_{1}-b_{1}c_{1}=1,a_{2}d_{2}-b_{2}c_{2}=1,a_{1}+d_{1}=s_{1},a_{2}+d_{2}=s_{2},a_{1}a_{2}+b_{1}c_{2}=s_{3}.$

One needs a quotient of $\text{Spec}(R)$ by the action of $\mathbb{G}_{m}$. A
categorical quotient $\mathcal{R}:=\text{Spec}(R_{0})$ with
$R_{0}=R^{\mathbb{G}_{m}}$ is computed in [vdP-Sa, §3.2]. It is an affine
cubic surface with three lines at infinity. This is given an interpretation in
[C-M-R]. The disadvantage is that it is not a geometric quotient and moreover
there are many singularities (depending on $s_{1},s_{2}$ and $s_{3}$). These
singularities are caused by resonance, reducibility and fixed points for the
action of $\mathbb{G}_{m}$ (see the table [vdP-Sa, §3.2.2]). Here we will
define a smooth “geometric quotient” $\mathcal{R}^{geom}(s_{1},s_{2},s_{3})$
by assuming (v) and adding additional data for $s_{1}=\pm 2$ and/or $s_{2}=\pm
2$ (see (vi) and (vii)).

Suppose that $\theta_{0},\theta_{1}\not\in\mathbb{Z}$. Then we consider the
open subspace $X$ of $\text{Spec}(R)$, defined by condition (v). A geometric
quotient $X/\mathbb{G}_{m}$ exists and is smooth as can be seen as follows.

The space $X$ is the union of the four Zariski open affine subspaces given by,
respectively, $b_{1}\neq 0$, $c_{1}\neq 0$, $b_{2}\neq 0$ and $c_{2}\neq 0$.
Let $X(b_{1}\neq 0)\subseteq X\subseteq\text{Spec}(R)$ denote the affine
subspace given by $b_{1}\neq 0$ and let $X(b_{1}\\!=\\!1)$ be the closed
subset given by $b_{1}=1$. The action of $\mathbb{G}_{m}$ on $X$ yields an
isomorphism $\mathbb{G}_{m}\times X(b_{1}\\!=\\!1)\rightarrow X(b_{1}\neq 0)$.
Therefore we can identify $X(b_{1}\neq 0)/\mathbb{G}_{m}$ with
$X(b_{1}\\!=\\!1)$. The affine space $X(b_{1}\\!=\\!1)$ is given by variables
$a_{1},c_{1},d_{1},a_{2},b_{2},c_{2},d_{2}$ and relations:

$d_{j}=s_{j}-a_{j}(j=1,2),\ a_{1}d_{1}-c_{1}=1,\ a_{2}d_{2}-b_{2}c_{2}=1,\
a_{1}a_{2}+c_{2}=s_{3}\neq 0.$

After elimination $d_{1},d_{2},c_{1},c_{2}$ the coordinate ring of
$X(b_{1}\\!=\\!1)$ is equal to
$\mathbb{C}[a_{1},a_{2},b_{2}]/(a_{2}(s_{2}-a_{2})-b_{2}(s_{3}-a_{1}a_{2})-1)$.
Hence $X(b_{1}\\!=\\!1)$ is connected, has dimension 2 and is smooth if
$s_{2}\neq\pm 2$.

We note that the condition $s_{2}\neq\pm 2$ is used above. Indeed,
$X(b_{1}\\!=\\!1)$ has for $s_{2}=\pm 2$ singularities, namely the line
$(a_{1},a_{2},b_{2})=(a_{1},\pm 1,0)$.

A similar calculation shows that the other three Zariski open subsets have
dimension 2 and are connected and smooth for $s_{1}\neq\pm 2$ and
$s_{2}\neq\pm 2$. The space $\mathcal{R}^{geom}(s_{1},s_{2},s_{3})$ is
obtained by glueing the four affine spaces
$X(b_{1}\\!=\\!1),X(c_{1}\\!=\\!1),X(b_{2}\\!=\\!1),X(c_{2}\\!=\\!1)$ and is
therefore smooth, connected and has dimension 2.

The cases $\theta_{0}\in\mathbb{Z}$ or $\theta_{1}\in\mathbb{Z}$. We explain
the case $\theta_{0}\in 2\mathbb{Z}$ in some detail. The other cases
$\theta_{0}\in 1+2\mathbb{Z}$ or $\theta_{1}\in\mathbb{Z}$ can be treated in a
similar way.
The space $\mathcal{R}^{geom}(2,s_{2},s_{3})$ is given by the tuples (modulo
$\mathbb{G}_{m}$-action)

$\left(\genfrac{(}{)}{0.0pt}{1}{a_{1}\;\;b_{1}}{c_{1}\;\;d_{1}},\genfrac{(}{)}{0.0pt}{1}{a_{2}\;\;b_{2}}{c_{2}\;\;d_{2}},[y_{1}:y_{2}]\right)\in\mathbb{A}^{4}\times\mathbb{A}^{4}\times\mathbb{P}^{1}\mbox{
with the following relations:}$

$a_{1}+d_{1}=2,a_{1}d_{1}-b_{1}c_{1}=1,a_{2}+d_{2}=s_{2},a_{2}d_{2}-b_{2}c_{2}=1,a_{1}a_{2}+b_{1}c_{2}=s_{3}$,
$\genfrac{(}{)}{0.0pt}{1}{a_{1}\;\;b_{1}}{c_{1}\;\;d_{1}}\genfrac{(}{)}{0.0pt}{1}{y_{1}}{y_{2}}=\genfrac{(}{)}{0.0pt}{1}{y_{1}}{y_{2}}$
and one of the elements $b_{1},c_{1},b_{2},c_{2}$ is $\neq 0$. We consider
again the geometric quotient $\mathcal{R}^{geom}(2,s_{2},s_{3})$ of the above
space by the action of $\mathbb{G}_{m}$. This quotient is obtained by gluing
some affine open subspaces. Each subspace is obtained by normalizing one of
the $b_{1},c_{1},b_{2},c_{2}$ to $1$ and also one of the elements
$y_{1},y_{2}$ is normalized to 1. For each of these open affine subsets one
can make the coordinate ring explicit and verify that there are no
singularities.

As an example we make the case $b_{2}=1,y_{2}=1$ explicit. Then
$d_{1}=2-a_{1}$, $d_{2}=s_{2}-a_{2}$, $c_{2}=-1+a_{2}(s_{2}-a_{2})$ and the
coordinate ring of this open affine subset is
$\mathbb{C}[a_{1},b_{1},c_{1},a_{2},y_{1}]/\mbox{(relations)}$. The relations
are generated by $a_{1}a_{2}+b_{1}(-1+a_{2}(s_{2}-a_{2}))-s_{3}$,
$a_{1}(2-a_{1})-b_{1}c_{1}-1$, $(a_{1}-1)y_{1}+b_{1}$, and
$c_{1}y_{1}+1-a_{1}$. Elimination of $a_{1}$ and $b_{1}$ leads to the
nonsingular algebra
$\mathbb{C}[c_{1},y_{1},a_{2}]/\left((1\\!+\\!c_{1}y_{1})a_{2}\\!+\\!c_{1}y_{1}^{2}(1\\!+\\!a_{2}(a_{2}\\!-\\!s_{2}))\\!-\\!s_{3}\right)$.

######  Proposition 2.1.

Let $s_{1}=e^{\pi i\theta_{0}}+e^{-\pi i\theta_{0}},s_{2}=e^{\pi
i\theta_{1}}+e^{-\pi i\theta_{1}},s_{3}=e^{\pi i\theta_{\infty}}$.
(1). $\mathcal{R}^{geom}(s_{1},s_{2},s_{3})$ is a simply connected, smooth
variety of dimension 2. It is a geometric quotient of the analytic data
including the eigenlines for the cases $s_{1}=\pm 2$ and/or $s_{2}=\pm 2$.
(2). The map ${\bf
S}(\theta_{0},\theta_{1},\theta_{\infty})\rightarrow\mathcal{R}^{geom}(s_{1},s_{2},s_{3})(\mathbb{C})\times\mathbb{C}^{*}$,
which associates to any $(M,t)$ the analytic data of $M$ (including additional
lines for $\theta_{0}\in\mathbb{Z}$ and/or $\theta_{1}\in\mathbb{Z}$) and
$t\in\mathbb{C}^{*}$, is a bijection.

###### Proof.

Most of the proof of (1) is sketched above. For the proof that the complex
algebraic variety $\mathcal{R}^{geom}(s_{1},s_{2},s_{3})$ is simply connected
one considers the morphism
$pr:\mathcal{R}^{geom}(s_{1},s_{2},s_{3})\rightarrow\mathbb{C}^{2}$ which
sends the data
$\genfrac{(}{)}{0.0pt}{1}{a_{1}\;\;b_{1}}{c_{1}\;\;d_{1}},\genfrac{(}{)}{0.0pt}{1}{a_{2}\;\;b_{2}}{c_{2}\;\;d_{2}}$
and $[y_{1}:y_{2}]$ (for special $s_{1},s_{2}$) to
$(a_{1},a_{2})\in\mathbb{C}$.

A computation shows that the fibre of $pr$ over a point $(a_{1},a_{2})$ with
$a_{1}a_{2}-s_{3}\neq 0$ and $a_{1}(s_{1}-a_{1})\neq 1$,
$a_{2}(s_{2}-a_{2})\neq 1$ consists of one point. For the case
$a_{1}a_{2}-s_{3}=0$ and $a_{1}(s_{1}-a_{1})\neq 1$, $a_{2}(s_{2}-a_{2})\neq
1$, the fibre is either empty, one point or $\mathbb{C}$. For the remaining
cases, including the special situations $s_{1}=\pm 2$ and/or $s_{2}=\pm 2$, a
computation of the fibres leads to the conclusion that
$\mathcal{R}^{geom}(s_{1},s_{2},s_{3})$ is simply connected.

The proof of (2) follows by combining the following observations:
(a). given formal and analytic data determine a unique differential module
over $\mathbb{C}(z)$, see [vdP-Sa, Theorem 1.7],
(b). $\mathcal{R}^{geom}(s_{1},s_{2},s_{3})$ is a geometric quotient (by
construction),
(c). ${\bf S}(\theta_{0},\theta_{1},\theta_{\infty})$ and
$\mathcal{R}^{geom}(s_{1},s_{2},s_{3})$ are provided with the same additional
data for the cases $\theta_{0}\in\mathbb{Z}$ and/or
$\theta_{1}\in\mathbb{Z}$.∎

##  3\. Construction of $\mathcal{M}(\theta_{0},\theta_{1},\theta_{\infty})$

The next step is a subtle construction of moduli spaces of connections.

First we give a construction of this moduli space of connections without
paying attention to a parabolic structure of eigenlines. For
$\theta_{0},\theta_{1}\not\in\mathbb{Z}$ this is sufficient. For
$\theta_{0}\in\mathbb{Z}$ and/or $\theta_{1}\in\mathbb{Z}$ we will refine this
construction since the space
$\mathcal{M}(\theta_{0},\theta_{1},\theta_{\infty})$ will, in general, have
singularities and
$\mathcal{M}(\theta_{0},\theta_{1},\theta_{\infty})(\mathbb{C})$ does not
coincide with ${\bf S}(\theta_{0},\theta_{1},\theta_{\infty})$.

The data for the moduli space of connections
$\mathcal{M}(\theta_{0},\theta_{1},\theta_{\infty})$ are:
(i). The sub-bundle $\mathcal{V}=Oe_{1}\oplus O(-[\infty])e_{2}$ of the free
bundle $Oe_{1}\oplus Oe_{2}$.
(ii). Local connections $L_{0},L_{1},L_{\infty}$, written as differential
operators, namely
(a). $L_{0}=\frac{d}{dz}+\frac{1}{z}A_{0}$ with $A_{0}$ a $2\times 2$-matrix
with entries in $\mathbb{C}[\\![z]\\!]$ and such that $A_{0}\\!\\!\mod(z)$ has
eigenvalues $\pm\frac{\theta_{0}}{2}$.
(b). $L_{1}=\frac{d}{dz}+\frac{1}{z-1}A_{1}$ with $A_{1}$ a $2\times 2$-matrix
with entries in $\mathbb{C}[\\![z-1]\\!]$ and such that $A_{1}\\!\\!\mod(z-1)$
has eigenvalues $\pm\frac{\theta_{1}}{2}$.
(c).
$L_{\infty}=\frac{d}{dz}+\frac{1}{z}\genfrac{(}{)}{0.0pt}{1}{x-1/2\;\;\;\;\;\;0}{0\;\;\;\;-x-1/2}$
at $z=\infty$ with $x=\frac{tz+\theta}{2}$, $\theta=\theta_{\infty}+1$ and
$t\in\mathbb{C}^{*}$.

The moduli functor $\mathcal{MF}$ (from $\mathbb{C}$-algebras to sets)
associates to each $\mathbb{C}$-algebra $R$ the set of equivalence classes of
pairs $(\nabla,t)$ consisting of an element $t\in R^{*}$ and a connection

$\nabla:R\otimes\mathcal{V}\rightarrow\Omega_{\mathbb{P}^{1}}([0]+[1]+2\cdot[\infty])\otimes(R\otimes\mathcal{V})$

satisfying the conditions stated below. We observe that
$D=\nabla_{\frac{d}{dz}}$ has, with respect to the basis $e_{1},e_{2}$, the
matrix

$\frac{1}{z(z-1)}\cdot\left(\begin{array}[]{cc}a_{0}+a_{1}z+a_{2}z^{2}&c_{0}+c_{1}z+\cdots+c_{3}z^{3}\\\
b_{0}+b_{1}z&d_{0}+d_{1}z+d_{2}z^{2}\end{array}\right)$

with $a_{*},b_{*},c_{*},d_{*}\in R$. The conditions are: $Rb_{0}+Rb_{1}=R$ and
the action of $D$ on the completions of $R\otimes\mathcal{V}$ at the points
$z=0,1,\infty$ is isomorphic to $R\otimes L_{0},R\otimes L_{1},R\otimes
L_{\infty}$, respectively.

Two pairs $(\nabla_{1},t_{1}),(\nabla_{2},t_{2})$ are called equivalent if
$t_{1}=t_{2}$ and there is an $R$-linear automorphism $g$ of
$R\otimes\mathcal{V}$ which transforms $\nabla_{1}$ into $\nabla_{2}$.
The $R$-linear automorphisms $g$ of $R\otimes\mathcal{V}$ are given by the
formulas

$g(e_{1})=\lambda_{1}e_{1},\ g(e_{2})=\lambda_{2}e_{2}+(\alpha+\beta
z)e_{1}\mbox{ with }\lambda_{1},\lambda_{2}\in R^{*}\mbox{ and
}\alpha,\beta\in R.$

Our aim is to show that the above functor $\mathcal{MF}$ is representable. The
resulting fine moduli space will be denoted by
$\mathcal{M}(\theta_{0},\theta_{1},\theta_{\infty})$. Moreover, the map
$\mathcal{M}(\theta_{0},\theta_{1},\theta_{\infty})(\mathbb{C})\rightarrow{\bf
S}(\theta_{0},\theta_{1},\theta_{\infty})$, which associates to any pair
$(\nabla,t)$ the pair $(M,t)$ with $M$ the generic fibre of $\nabla$, will be
shown to be bijective under the assumptions
$\theta_{0},\theta_{1}\not\in\mathbb{Z}$.

The above data produce an affine space
$M(\theta_{0},\theta_{1},\theta_{\infty})$ given by the variables
$t,a_{0},a_{1},a_{2},b_{0},b_{1},c_{0},c_{1},c_{2},c_{3},d_{0},d_{1},d_{2}$
and the relations:
$d_{0}=-a_{0},\ d_{1}=-a_{1},\ d_{2}=-a_{2}$ derived from the traces of
$L_{0},L_{1},L_{\infty}$, and
$a_{0}^{2}+b_{0}c_{0}=\frac{\theta_{0}^{2}}{4}$,
$(a_{0}+a_{1}+a_{2})^{2}+(b_{0}+b_{1})(c_{0}+c_{1}+c_{2}+c_{3})=\frac{\theta_{1}^{2}}{4}$,
$a_{2}^{2}+b_{1}c_{3}=\frac{t^{2}}{4}$,
$2a_{2}^{2}+2a_{1}a_{2}+a_{2}+b_{0}c_{3}+b_{1}c_{2}+2b_{1}c_{3}=\frac{t\theta}{2}$,
derived from the determinants of $L_{0},L_{1},L_{\infty}$.

We note that in case $\theta:=\theta_{\infty}+1\neq 0$, the variable $t$ can
be eliminated by one of the relations. For $\theta=0$, we will need the
variable $t$ for the construction of a good moduli space
$\mathcal{M}(\theta_{0},\theta_{1},\theta_{\infty})$.

From the open subspace of $M(\theta_{0},\theta_{1},\theta_{\infty})$, defined
by $b_{0}\neq 0$ or by $b_{1}\neq 0$, one has to take the geometric quotient
by the action of the automorphism group $G$ of the vector bundle
$\mathcal{V}$. Let $M_{1}(\theta_{0},\theta_{1},\theta_{\infty})$ denote the
open subset given by $b_{1}\neq 0$. Consider the closed subset
$\mathcal{M}_{1}(\theta_{0},\theta_{1},\theta_{\infty})$ of
$M(\theta_{0},\theta_{1},\theta_{\infty})$ defined by $b_{1}=1,a_{1}=a_{2}=0$.
The action of $G$ defines a morphism

$G\times\mathcal{M}_{1}(\theta_{0},\theta_{1},\theta_{\infty})\rightarrow
M_{1}(\theta_{0},\theta_{1},\theta_{\infty}).$

This is an isomorphism and therefore we identify
$M_{1}(\theta_{0},\theta_{1},\theta_{\infty})/G$ with
$\mathcal{M}_{1}(\theta_{0},\theta_{1},\theta_{\infty})$. In a similar way,
let $M_{2}(\theta_{0},\theta_{1},\theta_{\infty})$ be the open subset of
$M(\theta_{0},\theta_{1},\theta_{\infty})$ defined by $b_{0}\neq 0$. Then
$M_{2}(\theta_{0},\theta_{1},\theta_{\infty})/G$ can be identified with
$\mathcal{M}_{2}(\theta_{0},\theta_{1},\theta_{\infty})$ defined as the the
closed subset of $M(\theta_{0},\theta_{1},\theta_{\infty})$, given by
$b_{0}=1,a_{0}=a_{1}=0$. The above construction leads to

######  Proposition 3.1.

The space $\mathcal{M}(\theta_{0},\theta_{1},\theta_{\infty})$ obtained by
gluing the spaces $\mathcal{M}_{1}(\theta_{0},\theta_{1},\theta_{\infty})$ and
$\mathcal{M}_{2}(\theta_{0},\theta_{1},\theta_{\infty})$ represents the
functor $\mathcal{MF}$.

More details for the affine space
$\mathcal{M}_{1}=\mathcal{M}_{1}(\theta_{0},\theta_{1},\theta_{\infty})$.
Using an automorphism of $\mathcal{V}$, the matrix of
$D=\nabla_{\frac{d}{dz}}$ with respect to basis $e_{1},e_{2}$ can uniquely be
written as

$\frac{d}{dz}+\frac{1}{z(z-1)}\cdot\left(\begin{array}[]{cc}a_{0}&c_{0}+c_{1}z+c_{2}z^{2}+c_{3}z^{3}\\\
z+b_{0}&-a_{0}\end{array}\right),\mbox{ with equations}$
$a_{0}^{2}+b_{0}c_{0}=\frac{\theta_{0}^{2}}{4},\
a_{0}^{2}+(1+b_{0})(c_{0}+c_{1}+c_{2}+c_{3})=\frac{\theta_{1}^{2}}{4},$
$c_{3}=\frac{t^{2}}{4},\ (b_{0}+2)c_{3}+c_{2}=\frac{\theta t}{2}.$

Consecutively eliminating $c_{3}=\frac{t^{2}}{4}$ and then $c_{2}=\frac{\theta
t}{2}-(b_{0}+2)\frac{t^{2}}{4}$ and finally
$c_{0}=\frac{\theta_{1}^{2}}{4}-\frac{\theta_{0}^{2}}{4}-(b_{0}+1)(c_{1}-(b_{0}+1)\frac{t^{2}}{4}+\frac{\theta
t}{2})$ results in the equation

$a_{0}^{2}=-\frac{t^{2}}{4}b_{0}^{3}+(-\frac{t^{2}}{2}+\frac{\theta
t}{2}+c_{1})b_{0}^{2}+(-\frac{\theta_{1}^{2}}{4}+\frac{\theta_{0}^{2}}{4}+\frac{\theta
t}{2}-\frac{t^{2}}{4}+c_{1})b_{0}+\frac{\theta_{0}^{2}}{4}$

for the affine space $\mathcal{M}_{1}$. The singular locus of this
3-dimensional affine variety over $\mathbb{C}$ is the following:
(a). For $\theta_{0}=0$, the line $a_{0}=b_{0}=0$ and
$c_{1}=\frac{\theta_{1}^{2}}{4}-\frac{\theta t}{2}+\frac{t^{2}}{4}$.
After substitution of these values in $D$ the singularity at $z=0$ disappears
because $c_{0}=0$. On the other hand, if the module $M$ corresponding to a
point of $\mathcal{M}_{1}$ is locally trivial at $z=0$ then one concludes
$\theta_{0}=0$ and $a_{0}=b_{0}=c_{0}=0$.

The singularity at the surface $\mathcal{M}_{1}$ for $\theta_{0}=0$ can be
resolved by adding as extra data for the family an invariant line locally at
$z=0$. Then the singular point is replaced by a $\mathbb{P}^{1}$ of directions
for the local invariant line.
(b). For $\theta_{1}=0$, the line $a_{0}=0$, $b_{0}=-1$ and
$c_{1}=\frac{\theta_{0}^{2}}{4}-\frac{\theta t}{2}$.
It has the same interpretation as in (a). These singular loci coincide with
the loci where the map
$\mathcal{M}_{1}(\theta_{0},\theta_{1},\theta_{\infty})\rightarrow\text{Spec}(\mathbb{C}[t,t^{-1}])$
is not smooth.

More details for
$\mathcal{M}_{2}=\mathcal{M}_{2}(\theta_{0},\theta_{1},\theta_{\infty})$.
This Zariski open subset of
$\mathcal{M}(\theta_{0},\theta_{1},\theta_{\infty})$ has (apart from $t$) the
form

$D=\frac{d}{dz}+\frac{1}{z(z-1)}\cdot\left(\begin{array}[]{cc}A_{2}z^{2}&C_{0}+\dots+C_{3}z^{3}\\\
1+B_{1}z&-A_{2}z^{2}\end{array}\right),\mbox{ where}$
$C_{0}=\frac{\theta_{0}^{2}}{4},\
A_{2}^{2}+(1+B_{1})(C_{0}+C_{1}+C_{2}+C_{3})=\frac{\theta_{1}^{2}}{4},$
$A_{2}^{2}+B_{1}C_{3}=\frac{t^{2}}{4},\
2A_{2}^{2}+A_{2}+B_{1}C_{2}+2B_{1}C_{3}+C_{3}=\frac{\theta t}{2},\
\theta=\theta_{\infty}+1.$

Now $\mathcal{M}\setminus\mathcal{M}_{1}$ is the closed subset $B_{1}=0$ of
$\mathcal{M}_{2}$ and has equations:

$A_{2}^{2}=\frac{t^{2}}{4},\ C_{0}=\frac{\theta_{0}^{2}}{4},\
C_{1}+C_{2}=\frac{\theta_{1}^{2}}{4}-\frac{\theta_{0}^{2}}{4}-\frac{\theta
t}{2}+\frac{t^{2}}{4},C_{3}=\frac{\theta t}{2}-\frac{t^{2}}{2}.$

Seen as a variety over $\mathbb{C}$ it has 2 components which are affine
planes.

######  Theorem 3.2.

The extended Riemann-Hilbert map

$RH^{ext}:\mathcal{M}(\theta_{0},\theta_{1},\theta_{\infty})(\mathbb{C})\rightarrow\mathcal{R}^{geom}(s_{1},s_{2},s_{3})\times\mathbb{C}^{*}$

is an analytic isomorphism if $\theta_{0},\theta_{1}\not\in\mathbb{Z}$.

###### Proof.

It is well known that the monodromy matrices and the Stokes matrices depend in
an analytic way on the data of the connection [vdP-S, Proposition 12.20]).
Thus $RH^{ext}$ is an analytic map. Both spaces are smooth complex algebraic
varieties, since $\theta_{0},\theta_{1}\not\in\mathbb{Z}$. It is known that a
bijective analytic map between smooth complex varieties is an analytic
isomorphism. Thus we have only to show that $RH^{ext}$ is bijective.

In fact, we will show that
$\mathcal{M}(\theta_{0},\theta_{1},\theta_{\infty})(\mathbb{C})\rightarrow{\bf
S}(\theta_{0},\theta_{1},\theta_{\infty})$ is bijective. This suffices because
of Proposition 2.1. First we show surjectivity.

Consider a tuple $(M,t)\in{\bf S}(\theta_{0},\theta_{1},\theta_{\infty})$. One
can choose lattices $\Lambda_{0},\Lambda_{1},\Lambda_{\infty}$ at the points
$z=0,1,\infty$ with the required properties. These choices induce a connection
$\nabla$ on a vector bundle $\mathcal{W}$ of degree $-1$ with generic fibre
$M$.

If $\mathcal{W}\cong\mathcal{V}=Oe_{1}+O(-[\infty])e_{2}$, then the matrix of
$D:=\nabla_{\frac{d}{dz}}$ with respect to the basis $e_{1},e_{2}$ has the
required form (see the beginning of §1.3) except for the possibility that
$b_{0}+b_{1}z$ could be identical zero. Consider this case:

$D=\frac{d}{dz}+\frac{1}{z(z-1)}\cdot\left(\begin{array}[]{cc}a_{0}+a_{1}z+a_{2}z^{2}&c_{0}+c_{1}z+c_{2}z^{2}+c_{3}z^{3}\\\
0&d_{0}+d_{1}z+d_{2}z^{2}\end{array}\right).$

A computation at $z=\infty$ shows that $a_{2}=-d_{2}=\pm\frac{t}{2}\neq 0$.
Then there exist unique elements $\alpha_{0},\alpha_{1}\in\mathbb{C}$ such
that, after an automorphism of $\mathcal{V}$ which maps $e_{2}$ to
$e_{2}+(\alpha_{0}+\alpha_{1}z)e_{1}$, one obtains a case with
$c_{2}=c_{3}=0$. Let $D^{*}$ be the operator obtained by transposing the
matrix of $-D$. Since $M$ is not split, $c_{0}+c_{1}z+c_{2}z^{2}\neq 0$ and
one finds that $D^{*}$ belongs to
$\mathcal{M}(\theta_{0},\theta_{1},\theta_{\infty})(\mathbb{C})$. Furthermore,
$D^{*}$ has as generic fibre the dual of $M$. Since $\det M={\bf 1}$, the dual
of $M$ is isomorphic to $M$ and we are done in this case.

If $\mathcal{W}\not\cong\mathcal{V}$, then $\mathcal{W}\cong
O(m[\infty])e_{1}\oplus O((-1-m)[\infty])e_{2}$ for some $m\geq 1$ and the
matrix of $\nabla_{\frac{d}{dz}}$ with respect to the basis $e_{1},e_{2}$ has
the form
$\frac{1}{z(z-1)}\left(\begin{array}[]{cc}a_{0}+a_{1}z+a_{2}z^{2}&c_{0}+c_{1}z+\dots\\\
0&d_{0}+d_{1}z+d_{2}z^{2}\end{array}\right)$. In particular, $M$ is a
reducible differential module and is not split. As before one can normalize
the polynomial $c_{0}+c_{1}z+c_{2}z^{2}+\cdots$ such that it becomes a
polynomial of degree $\leq 1$. The dual differential operator defines an
element of $\mathcal{M}(\theta_{0},\theta_{1},\theta_{\infty})(\mathbb{C})$
with generic fibre isomorphic to $M$. This proves surjectivity.

For the smoothness of
$\mathcal{M}(\theta_{0},\theta_{1},\theta_{\infty})(\mathbb{C})$ one needs
$\theta_{0}\neq 0$ and $\theta_{1}\neq 0$. If moreover
$\theta_{0},\theta_{1}\not\in\mathbb{Z}$, then according to Lemma 1.1 the
required lattices $z=0,1$ are unique. The lattice for $z=\infty$ is also seen
to be unique. Thus for a pair $(M,t)$ is the image of at most one point of
$\mathcal{M}(\theta_{0},\theta_{1},\theta_{\infty})(\mathbb{C})$. This proves
injectivity. ∎

Theorem 3.2 implies that the solutions of the fifth Painlevé equation (at
least for $\theta_{0},\theta_{1}\not\in\mathbb{Z}$) are multivalued
meromorphic functions in $t$ or, equivalently, meromorphic functions on the
universal covering of $\mathbb{C}^{*}$ (compare [vdP-T4]). This was already
known from the paper [Mi].

More interesting is that $\mathcal{M}(\theta_{0},\theta_{1},\theta_{\infty})$
can be identified with the Okamoto–Painlevé variety for the fifth Painlevé
equation (at least for $\theta_{0},\theta_{1}\not\in\mathbb{Z}$). For this,
Proposition 2.1 shows that $\mathcal{R}^{geom}(s_{1},s_{2},s_{3})$ is simply
connected.

##  4\. The reducible locus in
$\mathcal{M}(\theta_{0},\theta_{1},\theta_{\infty})$

For $(\nabla,t)$ reducible, there is a line bundle
$\mathcal{W}\subseteq\mathcal{V}=Oe_{1}\oplus O(-[\infty])e_{2}$, “invariant”
under $\nabla$ and such that $\mathcal{V}/\mathcal{W}$ is again a line bundle.
By construction $\mathcal{W}\neq Oe_{1}$ and therefore one can choose $e_{2}$
such that $\mathcal{W}=O(-[\infty])e_{2}$. Furthermore, this $e_{2}$ is unique
up to multiplication by elements in $\mathbb{C}^{*}$. Therefore, the set of
the closed points of $\mathcal{M}(\theta_{0},\theta_{1},\theta_{\infty})$,
corresponding to reducible $(\nabla,t)$, consists of the differential
operators of the form

$\frac{d}{dz}+\frac{1}{z(z-1)}\cdot\left(\begin{array}[]{cc}a_{0}+a_{1}z+a_{2}z^{2}&0\\\
b_{0}+b_{1}z&-a_{0}-a_{1}z-a_{2}z^{2}\end{array}\right)$

with $b_{0}+b_{1}z\neq 0$, considered modulo the action of $\mathbb{G}_{m}$.
The equations are

$a_{0}^{2}=\frac{\theta_{0}^{2}}{4},\
(a_{0}+a_{1}+a_{2})^{2}=\frac{\theta_{1}^{2}}{4},\ a_{2}=\frac{t^{2}}{4},\
2a_{2}^{2}+a_{1}a_{2}+a_{2}=\frac{t\theta}{2}.$

The group $\mathbb{G}_{m}$ acts trivially on $a_{0},a_{1},a_{2}$. One has

$a_{0}=\epsilon_{0}\frac{\theta_{0}}{2},\
a_{1}=\epsilon_{1}\frac{\theta_{1}}{2}-\epsilon_{0}\frac{\theta_{0}}{2}-\epsilon_{2}\frac{t}{2},\
a_{2}=\epsilon_{2}\frac{t}{2}\mbox{ with
}\epsilon_{0},\epsilon_{1},\epsilon_{2}\in\\{1,-1\\}$

and $\theta=\epsilon_{2}(\epsilon_{1}\theta_{1}-\epsilon_{0}\theta_{0})$.
There are no conditions on $b_{0},b_{1}$ except for the action of
$\mathbb{G}_{m}$. Therefore the reducible locus consists of a number of at
most 8 projective lines (over $\mathbb{C}[t,t^{-1}]$), parametrized by
$[b_{0}:b_{1}]$ or $\frac{b_{1}}{b_{0}}$.

We fix a choice of the $\epsilon_{j},\ j=0,1,2$. Isomonodromy is obtained by
considering $b_{0},b_{1}$ as functions of $t$ and completing the Lax pair with
a suitable differential operator
$\frac{d}{dt}+\genfrac{(}{)}{0.0pt}{1}{\alpha\;\;\;0}{\beta\;\;-\alpha}$. This
results in a Riccati equation for the function $\frac{b_{1}}{b_{0}}$. All
Riccati solutions for $P_{5}$ are obtained in this way.

##  5\. The parabolic structure

Parabolic structure for connections and another way to formulate Lemma 1.1.
Consider a regular singular differential module $N=(N,D)$ over
$\mathbb{C}(\\!(z)\\!)$ with $D(f\cdot n)=z\frac{df}{dz}\cdot n+f\cdot D(n)$,
$\dim N=2$ and $\det N={\bf 1}$. An element $\eta\in\mathbb{C}$ will be called
an eigenvalue for $N$ if there is a $D$-invariant lattice $\Lambda$ for $N$
and $\eta$ is an eigenvalue of $D$ on $\Lambda/z\Lambda$.

Suppose that $\eta$ is an eigenvalue. Then $N$ is represented by a
differential operator of the form
$z\frac{d}{dz}+\genfrac{(}{)}{0.0pt}{1}{a\;\;b}{c\;\;d}$ with
$a,b,c,d\in\mathbb{C}[\\![z]\\!]$,
$\genfrac{(}{)}{0.0pt}{1}{a\;\;b}{c\;\;d}\bmod(z)$ has eigenvalue $\eta$ and
eigenvalue $-\eta+m$ for some integer $m$. One easily concludes that the set
of all eigenvalues is $(\eta+\mathbb{Z})\cup(-\eta+\mathbb{Z})$.

A parabolic structure for $(N,\eta)$ is an eigenline $\mathbb{C}e$ with $e\in
N,\ e\neq 0$ and $D(e)=\eta e$. If $(N,\eta)$ has a parabolic structure, then
$\eta$ is an eigenvalue for $N$. Now there are the following possibilities.

(i). $N$ has eigenvalue $\eta$ and $2\eta\not\in\mathbb{Z}$. Then
$(\eta+\mathbb{Z})\cap(-\eta+\mathbb{Z})=\emptyset$ and $(N,\alpha)$ has a
unique parabolic structure for every
$\alpha\in(\eta+\mathbb{Z})\cup(-\eta+\mathbb{Z})$.
Indeed, the condition $2\eta\not\in\mathbb{Z}$ implies that $N$ is a direct
sum, i.e., corresponds to an operator of the form
$z\frac{d}{dz}+\genfrac{(}{)}{0.0pt}{1}{\eta\;\;\;0}{0\;\;-\eta}$.

(ii). $\eta=0$ is eigenvalue of $N$. Then one can represent $N$ by the
operator $z\frac{d}{dz}+\genfrac{(}{)}{0.0pt}{1}{0\;\;a}{0\;\;0}$ with
$a\in\\{0,1\\}$.
(iia). If $a=0$, then $\ker D$ is a 2-dimensional complex vector space and the
parabolic structures for $\eta=0$ are the lines in $\ker D$. Thus the level
structures are parametrized by $\mathbb{P}^{1}$ (over $\mathbb{C}$).
(iib). If $a=1$, then $\ker D$ has dimension 1 over $\mathbb{C}$. This is the
only parabolic structure for $\eta=0$.

(iii). $\eta=\frac{m}{2}$ and $m\in\mathbb{Z}_{>0}$. It can be shown that $N$
is represented by an operator
$z\frac{d}{dz}+\genfrac{(}{)}{0.0pt}{1}{-\frac{m}{2}\;\;az^{m}}{0\;\;\;\;\frac{m}{2}}$
with $a\in\\{0,1\\}$, say on a basis $e_{1},e_{2}$ for $N$. For
$(N,\frac{m}{2})$ there is a unique parabolic structure.
(iiia). If $a=0$, then the parabolic structures for $-\frac{m}{2}$ are the
eigenlines $\mathbb{C}(c_{1}e_{1}+c_{2}z^{-m}e_{2})$. The eigenlines are
parametrized by $\mathbb{P}^{1}$.
(iiib). If $a=1$, then the only parabolic structure for $-\frac{m}{2}$ is
$\mathbb{C}e_{1}$.

Recall that ${\bf S}(\theta_{0},\theta_{1},\theta_{\infty})$ denotes the set
of equivalence classes of differential modules with prescribed singularities
and provided with a parabolic structure.

The definition of $\mathcal{M}(\theta_{0},\theta_{1},\theta_{\infty})$ for
$\theta_{0}\in\mathbb{Z}$ and/or $\theta_{1}\in\mathbb{Z}$ with parabolic
structure. Example: the case $\theta_{0}\in\mathbb{Z}$ and
$\theta_{1}\not\in\mathbb{Z}$.
Consider the affine space $M(\theta_{0},\theta_{1},\theta_{\infty})$ defined
on p. 3 by variables $a_{*},b_{*},c_{*},d_{*}$ and many relations. Take the
subspace of $\mathbb{P}^{1}\times M(\theta_{0},\theta_{1},\theta_{\infty})$
consisting of the pairs

$([y_{1}:y_{2}],\frac{d}{dz}+\frac{1}{z(z-1)}\cdot\left(\begin{array}[]{cc}a_{0}+a_{1}z+a_{2}z^{2}&c_{0}+c_{1}z+c_{2}z^{2}+c_{3}z^{3}\\\
b_{0}+b_{1}z&d_{0}+d_{1}z+d_{2}z^{2}\end{array}\right))$

with
$\genfrac{(}{)}{0.0pt}{1}{-a_{0}\;\;-c_{0}}{-b_{0}\;\;-d_{0}}\genfrac{(}{)}{0.0pt}{1}{y_{1}}{y_{2}}\equiv\frac{-|\theta_{0}|}{2}\genfrac{(}{)}{0.0pt}{1}{y_{1}}{y_{2}}\bmod(z)\mbox{
and }b_{0}+b_{1}z\neq 0$. We observe that there is a unique eigenline for
$-\frac{|\theta_{0}|}{2}$ which reduces modulo $(z)$ to the above element
$[y_{1}:y_{2}]\in\mathbb{P}^{1}$. Therefore the quotient of this space by the
action of the automorphism group $G$ of the vector bundle $\mathcal{V}$ is the
moduli space we are looking for.

The above space can be cut into four open affine subsets given by inequalities
($y_{1}\neq 0$ or $y_{2}\neq 0$) and ($b_{0}\neq 0$ or $b_{1}\neq 0$). On each
of these affine parts, taking the geometric quotient by $G$ is obtained by
normalizing ($y_{1}=1$ or $y_{2}=1$) and ($b_{0}=1$ or $b_{1}=1$).

The above definition and construction holds in all cases with
$\theta_{0}\in\mathbb{Z}$ and/or $\theta_{1}\in\mathbb{Z}$. One easily
verifies that by gluing one obtains an irreducible smooth variety of dimension
3, denoted by $\mathcal{M}(\theta_{0},\theta_{1},\theta_{\infty})$, such that
the natural map
$\mathcal{M}(\theta_{0},\theta_{1},\theta_{\infty})(\mathbb{C})\rightarrow{\bf
S}(\theta_{0},\theta_{1},\theta_{\infty})$ is a bijection.

For, say, $\theta_{0}\in 2\mathbb{Z}$, an eigenline for the connection at
$z=0$ is also an eigenline for the local monodromy at $z=0$ for $s_{1}=2$.
This holds for all cases with $\theta_{0}\in\mathbb{Z}$ and/or
$\theta_{1}\in\mathbb{Z}$. Thus we have, for all
$\theta_{0},\theta_{1},\theta_{\infty}$, an analytic morphism
$RH:\mathcal{M}(\theta_{0},\theta_{1},\theta_{\infty})(\mathbb{C})\rightarrow\mathcal{R}^{geom}(s_{1},s_{2},s_{3})$.
Using Proposition 2.1 one easily verifies that the proof of Theorem 3.2
extends to a proof of Theorem 5.1.

######  Theorem 5.1.

The extended Riemann-Hilbert map

$RH^{+}\colon\mathcal{M}(\theta_{0},\theta_{1},\theta_{\infty})(\mathbb{C})\rightarrow\mathcal{R}^{geom}(s_{1},s_{2},s_{3})\times\mathbb{C}^{*}$

is an analytic isomorphism for all $\theta_{0},\theta_{1},\theta_{\infty}$.

##  6\. The Lax pair and the fifth Painlevé equation

The explicit connection $\frac{d}{dz}+A$ with fixed
$\theta_{0},\theta_{1},\theta$ given on p. 3 (below Prop. 3.1), can be
completed to a Lax pair by using $\frac{d}{dt}+B$, where the entries of $B$
are polynomials in $z$ of degrees $\leq 2$ over a field of meromorphic
functions in $t$. The relation $[\frac{d}{dz}+A,\frac{d}{dt}+B]=0$ determines
$A$ and $B$ completely. The zero $q=-b_{0}$ of the entry $A[2,1]$ satisfies
the fifth Painlevé equation corresponding to the parameters
$\theta_{0},\theta_{1},\theta_{\infty}+1$ (the $+1$ comes from the shift we
made in the local data at $z=\infty$). This equation reads

$q^{\prime\prime}\\!=\\!\frac{1}{2}(\frac{1}{q}+\frac{1}{q\\!-\\!1})(q^{\prime})^{2}-\frac{q^{\prime}}{t}+\frac{q(q\\!-\\!1)}{t}\theta-\frac{q\\!-\\!1}{2qt^{2}}\theta_{0}^{2}+\frac{q}{2t^{2}(q\\!-\\!1)}\theta_{1}^{2}-\frac{(2q\\!-\\!1)(q\\!-\\!1)q}{2}.$

Substituting $q=\frac{y}{y-1}$ produces for $y$ the standard Painlevé equation

$P_{5}(\alpha=\frac{\theta_{1}^{2}}{2},\beta=-\frac{\theta_{0}^{2}}{2},\gamma=-\theta,\delta=-\frac{1}{2})\mbox{
and }\theta=1+\theta_{\infty}.$

The formula here seems to correct the one for $P_{5}$ in [vdP-Sa, (4.3)].

##  7\. Connections for the Lax pair of Noumi–Yamada

For certain tuples $(\theta_{0},\theta_{1},\theta_{\infty})$ and
$(\theta^{*}_{0},\theta^{*}_{1},\theta^{*}_{\infty})$, there are birational
maps
$\mathcal{M}(\theta_{0},\theta_{1},\theta_{\infty})\;\cdots\longrightarrow\mathcal{M}(\theta^{*}_{0},\theta^{*}_{1},\theta^{*}_{\infty})$.
These birational maps are the so-called Bäcklund transformations. Their action
on solutions can be made explicit. The Bäcklund transformations form the
extended affine Weyl group of $A_{3}^{(1)}$, see [O2, N-Y].

From Theorem 5.1 one can deduce that for instance shifts of
$\theta_{0},\theta_{1},\theta_{\infty}$ over integers are Bäcklund
transformations. However in our construction of
$\mathcal{M}(\theta_{0},\theta_{1},\theta_{\infty})$ some Bäcklund
transformations are not easily visible. Noumi and Yamada (and others) [N-Y]
studied a different Lax pair for $P_{5}$ from which one can read off all
Bäcklund transformations. Here we introduce a natural moduli space of
connections which leads to this Lax pair.

###  7.1. Moduli spaces for $q_{0}=z^{1/2}+tz^{1/4}$ with $t\in\mathbb{C}^{*}$

Consider the set $\mathbf{S_{4}}$ of isomorphism classes of differential
modules $M$ over $\mathbb{C}(z)$ given by:
(i) $\dim M=4$ and $\Lambda^{4}M$ is the trivial differential module and the
only singularities are $0,\infty$,
(ii) $z=0$ is a regular singularity,
(iii) $z=\infty$ is irregular and has eigenvalues $q_{0}$ and its conjugates.
We want to convert the set $\mathbf{S_{4}}$ into a family of connections
$(\nabla,\mathcal{V})$ on a vector bundle $\mathcal{V}$ on $\mathbb{P}^{1}$ of
rank 4 with singularities at $z=0$ and $z=\infty$, such that
$\nabla:\mathcal{V}\rightarrow\Omega_{\mathbb{P}^{1}}([0]+2[\infty])\otimes\mathcal{V}$,
or equivalently
$\delta=\nabla_{z\frac{d}{dz}}:\mathcal{V}\rightarrow\mathcal{V}([\infty])$,
has a (chosen) prescribed singularity at $z=\infty$.

A possible prescribed singularity at $z=\infty$ is obtained as follows.
For the fields $\mathbb{C}(z^{1/4})$, $\mathbb{C}(\\!(z^{-1/4})\\!)$, we write
$\sigma$ for the automorphism given by $\sigma(z^{1/4})=iz^{1/4}$. Let $N$
denote the free $\mathbb{C}[\\![z^{-1/4}]\\!]$ module with basis
$r_{0},r_{1},r_{2},r_{3}$ provided with a semi-linear action of $\sigma$ such
that $\sigma$ satisfies $r_{0}\mapsto r_{1}\mapsto r_{2}\mapsto r_{3}\mapsto
r_{0}$. Let $N_{0}\subseteq N^{<\sigma>}$ be the
$\mathbb{C}[\\![z^{-1}]\\!]$-module with free basis
$s_{0}=r_{0}+r_{1}+r_{2}+r_{3}$, $s_{1}=z^{1/4}(r_{0}+ir_{1}-r_{2}-ir_{3})$,
$s_{2}=z^{1/2}(r_{0}-r_{1}+r_{2}-r_{3})$,
$s_{3}=z^{3/4}(r_{0}-ir_{1}-r_{2}+ir_{3})$. We note that
$\mathbb{C}(\\!(z^{-1})\\!)\otimes N_{0}=\mathbb{C}(\\!(z^{-1})\\!)\otimes
N^{<\sigma>}$.

One defines $\delta_{0}\colon N\rightarrow
N\otimes\mathbb{C}(\\!(z^{-1/4})\\!)$ by
$\delta_{0}(r_{j})=(q_{j}-\frac{3}{8})r_{j}$ for $j=0,1,2,3$ and
$\delta_{0}(fn)=z\frac{df}{dz}n+f\delta_{0}(n)$ for
$f\in\mathbb{C}[\\![z^{-1/4}]\\!],\ n\in N$. The induced operator
$N_{0}\rightarrow N_{0}\otimes\mathbb{C}(\\!(z^{-1})\\!)$ is also denoted by
$\delta_{0}$. The term $-\frac{3}{8}$ is introduced in order that the operator
on $N_{0}$ has trace zero. We note that $(N_{0},\delta_{0})$ is irreducible
and therefore its group of automorphisms is $\mathbb{C}^{*}$. The chosen
prescribed singularity at $z=\infty$ is the formal local connection
$(N_{0},\delta_{0})$. On the basis $\\{s_{0},s_{1},s_{2},s_{3}\\}$ of $N_{0}$,
this is represented by
$L_{\infty}:=z\frac{d}{dz}+\scriptsize{\left(\begin{array}[]{cccc}-\frac{3}{8}&0&z&\frac{t}{4}z\\\
\frac{t}{4}&-\frac{1}{8}&0&z\\\ 1&\frac{t}{4}&\frac{1}{8}&0\\\
0&1&\frac{t}{4}&\frac{3}{8}\end{array}\right)}$.

######  Remark 7.1.

The choice $(N_{0},\delta_{0})$ and corresponding $L_{\infty}$ used here is
not unique. A canonical choice will be described in Remark 7.2.

We consider the moduli functor for the data:
“The isomorphy classes of the connections
$\delta:\mathcal{V}\rightarrow\mathcal{V}([\infty])$ on the free bundle
$\mathcal{V}$ such that the induced
$\delta:\widehat{\mathcal{V}}_{\infty}\rightarrow\widehat{\mathcal{V}}_{\infty}\otimes\mathbb{C}(\\!(z^{-1})\\!)$
is isomorphic to $(N_{0},\delta_{0})$”.

This isomorphism is unique up to $\mathbb{C}^{*}$. The connections $\delta$
(modulo isomorphisms) can be represented by the operators
$L:=z\frac{d}{dz}+A_{0}+A_{1}z$ (modulo equivalence) for which there exists a
$U\in{\rm GL}_{4}(\mathbb{C}[\\![z^{-1}]\\!])$ with $ULU^{-1}=L_{\infty}$.

The above is an example for the theory developed in Section 12 (especially
§12.5) of [vdP-S]. It is shown there that a fine moduli space exists by giving
an explicit construction for the general ramified case. For the present case,
this moduli space is denoted by $\mathcal{M}_{4}$ and we will produce explicit
formulas.

Consider the map
$\eta\colon\mathcal{M}_{4}(\mathbb{C})\rightarrow{\mathbf{S_{4}}}$ associating
to a connection its generic fibre (which is a differential module over
$\mathbb{C}(z)$ belonging to $\mathbf{S_{4}}$). From the observation that
$(N_{0},\delta_{0})$ has automorphism group $\mathbb{C}^{*}$, it follows that
$\eta$ is injective.

For $M\in{\mathbf{S_{4}}}$ an invariant lattice $\Lambda_{\infty}$ at
$z=\infty$ is defined by: $\Lambda_{\infty}$ is a free
$\mathbb{C}[\\![z^{-1}]\\!]$-submodule of $\mathbb{C}(\\!(z^{-1})\\!)\otimes
M$ of rank 4, such that $\delta(\Lambda_{\infty})\subseteq z\Lambda_{\infty}$.
Similarly, an invariant lattice $\Lambda_{0}$ for $M$ at $z=0$ is by
definition a free $\mathbb{C}[\\![z]\\!]$-submodule of
$\mathbb{C}(\\!(z)\\!)\otimes M$ of rank 4 such that
$\delta(\Lambda_{0})\subseteq\Lambda_{0}$.

From the definition of $\mathbf{S_{4}}$ it follows that any
$M\in\mathbf{S_{4}}$ has a unique invariant lattice
$\Lambda_{\infty}\subseteq\mathbb{C}(\\!(z^{-1})\\!)\otimes M$ at $z=\infty$
which is isomorphic to $(N_{0},\delta_{0})$. Further one chooses an invariant
lattice $\Lambda_{0}\subseteq\mathbb{C}(\\!(z)\\!)\otimes M$. These lattices
determine a connection $(\nabla,\mathcal{V})$ with generic fibre $M$. If
$\mathcal{V}$ happens to be the free vector bundle, then $M$ lies in the image
of $\eta$.

The regular singular differential module $\mathbb{C}(\\!(z)\\!)\otimes M$ can
be represented by a matrix differential operator $z\frac{d}{dz}+A$, such that
the eigenvalues of the constant matrix $A$ are distinct modulo integers. If
$A$ is diagonalizible, the $\mathbb{C}(\\!(z)\\!)\otimes M$ contains many
invariant lattices and, for a suitable choice of the lattice, the
corresponding vector bundle $\mathcal{V}$ is free (compare the method of
[vdP-S, §6.5]). This shows that the image of $\eta$ contains what could be
called a “Zariski open, dense subset of $\mathbf{S_{4}}$”. However, the map
$\eta$ is probably not surjective.

The parameter space $\mathcal{P}^{+}_{4}\cong\mathbb{C}^{3}$ is the space of
the polynomials of the form $T^{4}+c_{2}T^{2}+c_{1}T+c_{0}$. The morphism
$\mathcal{M}_{4}\rightarrow\mathcal{P}^{+}$ maps a connection
$z\frac{d}{dz}+A_{0}+zA_{1}\in\mathcal{M}_{4}(\mathbb{C})$ to the
characteristic polynomial of $A_{0}$.

Let $\mathcal{R}_{4}$ denote the moduli space for the analytic data (i.e., the
topological monodromy and the Stokes matrices) of the set ${\mathbf{S_{4}}}$.
The Riemann–Hilbert morphism is the obvious map
$RH:\mathcal{M}_{4}\rightarrow\mathcal{R}_{4}$. The map $RH$ forgets $t$ and
the choice of a logarithm of the topological monodromy at $z=0$. Thus the
fibres of $RH$ have dimension 1 and so
$\dim\mathcal{M}_{4}=1+\dim\mathcal{R}_{4}$. The connected components of the
fibres are parametrized by $t$.

We note that for every integer $n\geq 3$, one can define, in a similar way, a
family of differential modules $\bf S_{n}$, given by: $\dim M=n$;
$\Lambda^{n}M$ is the trivial differential module; $z=0$ is regular singular
and the (generalized) eigenvalues at $z=\infty$ are $z^{2/n}+tz^{1/n}$ and its
conjugates. The corresponding moduli spaces $\mathcal{M}_{n},\mathcal{R}_{n}$
produce the Lax pairs introduced by Noumi, Yamada et al.. For $n=3$ this Lax
pair yields $P_{4}$. This is exploited in [vdP-T4]. For $n=4$, the Lax pair
produces $P_{5}$ and its Bäcklund transformations. We will give the details
for $n=4$ and refer to a future paper for the general case.

####  7.1.1. $\mathcal{R}_{4}$ and the fibres of
$\mathcal{R}_{4}\rightarrow\mathcal{P}_{4}$

The monodromy of a differential module $M\in{\mathbf{S_{4}}}$ consists of the
topological monodromies $mon_{0}$ and $mon_{\infty}$ at the points $0$ and
$\infty$. Since there are no other singularities one has $mon_{0}\cdot
mon_{\infty}=1$. The monodromy at $\infty$ is given by the Stokes matrices and
the formal monodromy $\gamma$. The monodromy identity, [vdP-S], states that
$mon_{\infty}$ is equal to their product.

Now we compute the Stokes data at $z=\infty$ and refer to [vdP-S] for more
details. The (generalized) eigenvalues at $z=\infty$ are:
$q_{0}\\!=\\!z^{1/2}+tz^{1/4},\ q_{1}\\!=\\!-z^{1/2}+itz^{1/4},\
q_{2}\\!=\\!z^{1/2}-tz^{1/4},\ q_{3}\\!=\\!-z^{1/2}-itz^{1/4}$.
The differences $q_{0}-q_{1},\ q_{0}-q_{3},\ q_{2}-q_{1},\ q_{2}-q_{3}$ have
the form $2z^{1/2}+\dots$, and $q_{0}-q_{2}=2tz^{1/4}$ and
$q_{1}-q_{3}=2itz^{1/4}$. There is one singular direction in $[0,1)$ for the
terms $\pm 2z^{1/2}$ as well as for each of $\pm 2tz^{1/4},\pm 2iz^{1/4}$.
This leads to the monodromy identity

$mon_{\infty}=\scriptsize{\left(\begin{array}[]{cccc}&&&-1\\\ 1&&&\\\ &1&&\\\
&&1&\end{array}\right)}\scriptsize{\left(\begin{array}[]{cccc}1&&&\\\ &1&&y\\\
&&1&\\\ &&&1\end{array}\right)}\scriptsize{\left(\begin{array}[]{cccc}1&&&\\\
x_{1}&1&x_{2}&\\\ &&1&\\\ x_{3}&&x_{4}&1\end{array}\right)}.$

The basis for the above matrices is unique up to multiplying every basis
vector by the same scalar. Therefore $\mathcal{R}_{4}\cong\mathbb{C}^{5}$.

The parameter space $\mathcal{P}_{4}$ for $\mathcal{R}_{4}$ is given by the
characteristic polynomial of $mon_{\infty}$, which is written as
$T^{4}+p_{3}T^{3}+p_{2}T^{2}+p_{1}T+1$. Thus
$\mathcal{P}_{4}\cong\mathbb{C}^{3}$. One considers the commutative diagram

$\begin{array}[]{ccc}\mathcal{M}_{4}&\stackrel{{\scriptstyle
RH}}{{\rightarrow}}&\mathcal{R}_{4}\\\ \downarrow&&\downarrow\\\
\mathcal{P}_{4}^{+}&\stackrel{{\scriptstyle
rh}}{{\rightarrow}}&\mathcal{P}_{4}\end{array}$

where $rh$ maps the characteristic polynomial of a local differential operator
$z\frac{d}{dz}+A$ at $z=0$ to the characteristic polynomial of
$mon_{\infty}=mon_{0}^{-1}$.

One easily computes the fibre of $\mathcal{R}_{4}\rightarrow\mathcal{P}_{4}$
above $(p_{1},p_{2},p_{3})$. For a suitable choice of elimination of two
variables (e.g., $x_{1},x_{2}$), the fibres are described by a cubic equation
in three variables (say, $y,x_{3},x_{4}$) and parameters $p_{1},p_{2},p_{3}$.
The equation has the form

$v_{1}v_{2}v_{3}+*v_{1}^{2}+*v_{2}^{2}+*v_{1}+*v_{2}+*v_{3}+*=0$

for suitable affine expressions $*$’s in the parameters $p_{1},p_{2},p_{3}$.
The cubic equation for the monodromy of $P_{5}$ has the same features.
Comparing with the table of cubic surfaces associated to Painlevé equations in
[vdP-Sa, p. 2635-2636], [C-M-R], this is an indication that the (yet to be
determined) Lax pair produces $P_{5}$.

####  7.1.2. A matrix differential operator for $\mathcal{M}_{4}$, Lax pairs
and $P_{5}$

We want to represent a ‘general’ element $M\in\mathcal{M}_{4}$ by a matrix
differential operator $z\frac{d}{dz}+A_{0}+zA_{1}$. It turns out that a direct
computation is rather difficult. Instead, we adapt the computation of the
formal local connection $(N_{0},\delta_{0})$ and $L_{\infty}$, to the global
case. Then $M$ is replaced by $N:=\mathbb{C}(z^{1/4})\otimes M$ and we use
“equivariant formules” on $N$. Let $\gamma$ denote the automorphism of
$\mathbb{C}(z^{1/4})$ over $\mathbb{C}(z)$, defined by
$\gamma(z^{1/4})=iz^{1/4}$. Let $\sigma:N\rightarrow N$ be given by
$\sigma(f\otimes m)=\gamma(f)\otimes m$ for $f\in\mathbb{C}(z^{1/4})$ and
$m\in M$. This new differential module $N$ has no ramification at
$z^{1/4}=\infty$ and has Katz invariant 2. Let $D$ denote the differential
operator on $N$. One considers a basis $e_{0},e_{1},e_{2},e_{3}$ of $N$ such
that $\sigma$ acts by $e_{0}\mapsto e_{1}\mapsto e_{2}\mapsto e_{3}\mapsto
e_{0}$. We try to construct $D$ by a formula

$D(e_{0})=(z^{1/2}+\frac{t}{4}z^{1/4}-3/8)\,e_{0}+(a_{1}+b_{1}z^{1/4})\,e_{1}+a_{2}\,e_{2}+(a_{3}+b_{3}z^{1/4})\,e_{3}.$

This determines $D$ because $\sigma\circ D=D\circ\sigma$. The term $-3/8$ is
introduced (as before) in order to obtain a matrix for $D$ with trace zero.

The formula for $D$ follows from the construction of the moduli space for the
ramified case [vdP-S, §12.5]. The matrix of $D$ on the basis of invariants

$\begin{array}[]{l}B_{0}=e_{0}+e_{1}+e_{2}+e_{3},\\\
B_{1}=z^{1/4}(e_{0}+ie_{1}+i^{2}e_{2}+i^{3}e_{3}),\\\
B_{2}=z^{1/2}(e_{0}-e_{1}+e_{2}-e_{3}),\\\
B_{3}=z^{3/4}(e_{0}-ie_{1}-e_{2}+ie_{3})\end{array}$

is

$\left(\begin{array}[]{cccc}-\frac{3}{8}+a_{1}+a_{2}+a_{3}&0&z&z(\frac{t}{4}+b_{1}+b_{3})\\\
\frac{t}{4}-ib_{1}+ib_{3}&-\frac{1}{8}-ia_{1}-a_{2}+ia_{3}&0&z\\\
1&\frac{t}{4}-b_{1}-b_{3}&\frac{1}{8}-a_{1}+a_{2}-a_{3}&0\\\
0&1&\frac{t}{4}+ib_{1}-ib_{3}&\frac{3}{8}+ia_{1}-a_{2}-ia_{3}\\\
\end{array}\right)$

and $D$ is equal to the differential operator
$z\frac{d}{dz}+\left(\begin{array}[]{cccc}\epsilon_{0}&0&z&zf_{0}\\\
f_{1}&\epsilon_{1}&0&z\\\ 1&f_{2}&\epsilon_{2}&0\\\
0&1&f_{3}&\epsilon_{3}\end{array}\right)$ with $\sum\epsilon_{j}=0$,
$f_{0}+f_{2}=f_{1}+f_{3}=\frac{t}{2}$. The $\epsilon_{0},\dots,\epsilon_{3}$
are parameters.

The operator $D$ is completed to a Lax pair by the differential operator $E$
w.r.t. $\frac{d}{dt}$. This operator, written on the basis
$e_{0},e_{1},e_{2},e_{3}$ is $\sigma$-invariant and has the form
$E(e_{0})=z^{1/4}e_{0}+\sum_{j=1}^{3}h_{j}e_{j}$ for suitable functions
$h_{1},h_{2},h_{3}$ of $t$. On the basis $B_{0},B_{1},B_{2},B_{3}$ one obtains
$E:=\frac{d}{dt}+\left(\begin{array}[]{cccc}g_{0}&0&0&z\\\ 1&g_{1}&0&0\\\
0&1&g_{2}&0\\\ 0&0&1&g_{3}\end{array}\right)$ with $\sum g_{j}=0$. The
assumption that $E$ commutes with $D$ produces a nonlinear system of first
order differential equations for $f_{0},f_{1},f_{2},f_{3}$ as functions of
$t$.

These formulas are almost identical to those derived by Noumi–Yamada. After
eliminating $f_{2},f_{3}$ a combination of the resulting differential
equations for $f_{0}$ and $f_{1}$ leads to the standard $P_{5}$ equation, see
[N-Y, S-H-C] for details. The differential equations for $f_{0},f_{1}$ are:

$2tf^{\prime}_{0}=8f_{0}^{2}f_{1}-2f_{0}^{2}t-4f_{0}f_{1}t+f_{0}t^{2}-4\epsilon_{0}f_{0}+2\epsilon_{0}t+4\epsilon_{1}f_{0}-4\epsilon_{2}f_{0}+4\epsilon_{3}f_{0}-2\epsilon_{3}t-2f_{0}+2t,$
$2tf^{\prime}_{1}=-8f_{0}f_{1}^{2}+4f_{0}f_{1}t+2f_{1}^{2}t-f_{1}t^{2}+4\epsilon_{0}f_{1}-2\epsilon_{0}t-4\epsilon_{1}f_{1}+2\epsilon_{1}t+4\epsilon_{2}f_{1}-4\epsilon_{3}f_{1}+2f_{1}$

In fact, the equations coincide with the ones in [S-H-C] for
$f_{0},f_{1},f_{2},f_{3}$. The symmetry of these equations with respect to the
cyclic permutation $f_{0}\mapsto f_{1}\mapsto f_{2}\mapsto f_{3}\mapsto f_{0}$
produces an element $\pi$ of order 4 in the group of Bäcklund transformations.
This is the “missing element” needed to obtain the extended affine Weyl group
of $A_{3}^{(1)}$.

An alternative computation for the Lax pair is based on observing that
isomonodromy is given by $DE(e_{0})=ED(e_{0})$ and that $a_{1},a_{2},a_{3}$
are independent of $t$. A straightforward computation then yields

$4t\cdot\frac{db_{1}}{dt}=-16ib_{1}^{2}b_{3}+t^{2}b_{1}i+16ib_{3}^{3}-4ia_{1}t+4a_{1}t-32a_{2}b_{3},$
$4t\cdot\frac{db_{3}}{dt}=-16ib_{1}^{3}+16ib_{1}b_{3}^{2}-t^{2}b_{3}i+4ia_{3}t-32a_{2}b_{1}+4a_{3}t,$
$h_{1}=-b_{1}i/2+b_{1}/2,\ t\cdot h_{2}=2b_{1}^{2}i-2b_{3}^{2}i+4a_{2},\
h_{3}=b_{3}i/2+b_{3}/2.$

These equations for $b_{1},b_{3}$ form a Hamiltonian system

$t\frac{db_{1}}{dt}=\frac{\partial H}{\partial b_{3}},\
t\frac{db_{3}}{dt}=-\frac{\partial H}{\partial b_{1}},$
$H=\frac{i(b_{1}^{2}-b_{3}^{2})^{2}}{t}+\frac{itb_{1}b_{3}}{4}+\frac{4a_{2}(b_{1}^{2}-b_{3}^{2})}{t}-(1+i)a_{3}b_{1}+(1-i)a_{1}b_{3}.$

The above Hamiltonian $H$ is, up to a change of variables, the same as
Okamoto’s polynomial Hamiltonian for $P_{5}$, see [O1, p.265].

######  Remark 7.2.

Here we describe a more canonical choice for the invariant lattice (see p.
7.1). Let $N$ be the free $\mathbb{C}[\\![z^{-1/4}]\\!]$-module with basis
$e_{0},e_{1},e_{2},e_{3}$. The action of $\sigma$ is given by $e_{0}\mapsto
e_{1}\mapsto e_{2}\mapsto e_{3}\mapsto e_{0}$ and $\sigma(z^{1/4})=iz^{1/4}$.
The operator $\delta_{0}$ on $N$ commutes with $\sigma$ and is determined by
$\delta_{0}(e_{0})=(z^{1/2}+\frac{t}{4}z^{1/4}+\frac{3}{8})e_{0}$. Choose for
the lattice $\widetilde{N}_{0}$ at $z=\infty$ the module $N^{\sigma}$ of the
$\sigma$-invariant elements of $N$. It has a basis

$\begin{array}[]{l}B_{0}=e_{0}+e_{1}+e_{2}+e_{3},\\\
\widetilde{B}_{1}=z^{-3/4}(e_{0}+ie_{1}-e_{2}-ie_{3}),\\\
\widetilde{B}_{2}=z^{-1/2}(e_{0}-e_{1}+e_{2}-e_{3}),\\\
\widetilde{B}_{3}=z^{-1/4}(e_{0}-ie_{1}-e_{2}+ie_{3}).\end{array}$

The global operator $D$, commuting with $\sigma$, is given by

$D(e_{0})=(z^{1/2}+\frac{t}{4}z^{1/4}+\frac{3}{8})e_{0}+(a_{1}+b_{1}z^{1/4})e_{1}+(a_{2})e_{2}+(a_{3}+b_{3}z^{1/4})e_{3}.$

The matrix of $D$ with respect to the basis
$B_{0},\widetilde{B}_{1},\widetilde{B}_{2},\widetilde{B}_{3}$ is

$\left(\begin{array}[]{cccc}\frac{3}{8}+a_{1}+a_{2}+a_{3}&0&1&b_{1}+b_{3}+\frac{t}{4}\\\
z(\frac{t}{4}-ib_{1}+ib_{3})&-\frac{3}{8}-ia_{1}-a_{2}+ia_{3}&0&z\\\
z&\frac{t}{4}-b_{1}-b_{3}&-\frac{1}{8}-a_{1}+a_{2}-a_{3}&0\\\
0&1&\frac{t}{4}+ib_{1}-ib_{3}&\frac{1}{8}+ia_{1}-a_{2}-ia_{3}\end{array}\right).$

This can be rewritten as
$\left(\begin{array}[]{cccc}\epsilon_{0}&0&1&g_{0}\\\
zf_{0}&\epsilon_{1}&0&z\\\ z&g_{1}&\epsilon_{2}&0\\\
0&1&f_{1}&\epsilon_{3}\end{array}\right)$ with entries satisfying
$f_{0}+f_{1}=g_{0}+g_{1}=\frac{t}{2}$ and
$\epsilon_{0}+\cdots+\epsilon_{3}=0$.

Using the Lax pair conditions, elimination of $f_{1},g_{1},\epsilon_{3}$ and
writing $f=f_{0},g=g_{0}$ a system of differential equations essentially the
same as the one in [S-H-C] is obtained:

$\left\\{\begin{array}[]{l}2tf^{\prime}\\!=\\!-8f^{2}g\\!+\\!2f^{2}t\\!+\\!4fgt\\!-\\!ft^{2}\\!+\\!8\epsilon_{0}f\\!-\\!2\epsilon_{0}t\\!+\\!2\epsilon_{1}t\\!+\\!8\epsilon_{2}f\\!-\\!2f\\!+\\!2t\\\
2tg^{\prime}\\!=\\!8fg^{2}\\!-\\!4fgt\\!-\\!2g^{2}t\\!+\\!gt^{2}\\!-\\!8\epsilon_{0}g\\!+\\!4\epsilon_{0}t\\!+\\!2\epsilon_{1}t\\!-\\!8\epsilon_{2}g\\!+\\!2\epsilon_{2}t\\!+\\!2g.\end{array}\right.$

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