Isomonodromic deformation of Lamé connections, Painlevé VI equation and Okamoto symmetry

The paper is self-contained in the sense that it contains several background/survey sections, namely:

$\bullet$ §8 (appendix) contains basic facts about bundles, connections, moduli spaces, parabolic structures, elementary transformations;

$\bullet$ §4 describes how the Painlevé VI equation arises as an isomonodromy condition for linear differential equations;

$\bullet$ §5 gives a geometric and modular description of the phase portrait of the Painlevé VI equation, including Okamoto's space of initial conditions.

The original part of the paper is concentrated in the following sections.

$\bullet$ §2: construction of Lamé connections by elliptic pullback of some systems on $\mathbb P^1$;

$\bullet$ §3: how to compute the Tu invariant of the bundle in terms of the parabolic structure of the downward system;

$\bullet$ §6: why most of the Lamé connections are elliptic pullbacks (a Riemann– Hilbert approach);

$\bullet$ §7: conclusion of the proof of the main theorem.

To compute the modular invariant $\lambda(\tilde E'')$ for the elliptic pullback, it is convenient to work with the associated projective bundle. Given any vector bundle $E$ over a curve $X$, we denote the associated $\mathbb P^1$-bundle by $\mathbb P(E)$. Another vector bundle $E'$ gives rise to the same $\mathbb P^1$-bundle if and only if $E'=L\otimes E$ for some line bundle $L$. When $E$ and $E'$ are both determinant-free, $L$ is a $2$-torsion point of the Jacobian. Hence there are at most $2^{2g}$ determinant-free vector bundles $E'$ giving rise to the same $\mathbb P^1$-bundle, where $g$ is the genus of $X$. Line bundles $L\subset E$ are in a one-to-one correspondence with sections $\sigma\colon X\to\mathbb P(E)$. The total space $S$ of the bundle $\mathbb P(E)$ is a ruled surface, and every section $\sigma$ determines a curve on $S$, to be denoted by $\sigma$ for simplicity. The normal bundle of $\sigma$ in $S$ is identified with the line bundle $\det(E)\otimes L^{-2}$, where $L\subset E$ is the corresponding line bundle, so that the self-intersection is given by

$$\begin{equation*} \sigma\cdot \sigma=\deg(E)-2\deg(L). \end{equation*}$$

Recall that $E$ is semistable if and only if $\deg(E)-2\deg(L)\ge 0$ for every line bundle $L\subset E$. If $L'$ is any line bundle distinct from $L$, then the composite

$$\begin{equation*} L'\to E\to E/L\simeq \det(E)\otimes L^{-1} \end{equation*}$$

is a non-trivial homomorphism of line bundles. It determines a non-trivial holomorphic section of $\det(E)\otimes L^{-1}\otimes L'^{-1}$ vanishing at those points where $L$ and $L'$ coincide. This yields a formula for the intersection number of the corresponding sections $\sigma$ and $\sigma'$:

$$\begin{equation*} \sigma\cdot \sigma'=\deg(E)-\deg(L)-\deg(L')=\frac{1}{2}(\sigma\cdot \sigma+\sigma'\cdot \sigma')\ge 0. { } \end{equation*}$$

A vector bundle $E$ is decomposable (that is, of the form $E=L\oplus L'$) if and only if $\mathbb P(E)$ admits two non-intersecting sections $\sigma$ and $\sigma'$ (they correspond to $L$ and $L'$) or, equivalently, two sections $\sigma$ and $\sigma'$ having opposite self-intersection numbers. In this case the $\mathbb P^1$-bundle $\mathbb P(E)$ may be viewed as the fibrewise compactification $\overline{L'\otimes L^{-1}}$ of the line bundle $L'\,{\otimes}\, L^{-1}$ obtained by adding a section at infinity. To be precise, $\sigma$ is the zero section and $\sigma'$ is the section at infinity in $\mathbb P(E)\simeq\overline{L'\otimes L^{-1}}$.

If $X$ is an elliptic curve (for example, $X\colon \{y^2=x(x-1)(x-t)\}$) and $\det(E)=\mathcal O_X$, then $E$ is semistable if and only if $\mathbb P(E)$ has a section with zero self-intersection. The four indecomposable bundles defined in (8) correspond to the same $\mathbb P^1$-bundle $\mathbb P(E_0)$, which is given by the unique non-trivial extension

$$\begin{equation*} 0\to \mathcal O_X\to E_0\to \mathcal O_X\to 0. { } \end{equation*}$$

The corresponding ruled surface $S_0$ is characterized by the existence of one and only one section with zero self-intersection. By the results of Atiyah [12] (see also [13]), all other semistable determinant-free vector bundle are decomposable: the corresponding $\mathbb P^1$-bundles are of the form

$$\begin{equation*} \mathbb P(E)=\overline{\mathcal O_X([\omega]-[-\omega])} , \end{equation*}$$

where $\pm\omega\in X$ are the two points of the fibre $\pi^{-1}(\lambda(E))$ (see the definition (7)). We note that the modular invariant $\lambda(E)$ is determined by $\mathbb P(E)$ up to the action of the $2$-torsion points of the elliptic curve $X$: the determinant-free vector bundles with $\mathbb P^1$-bundle $\mathbb P(E)$ are the four semistable bundles with modular invariants

$$\begin{equation*} \lambda, \quad \frac{t}{\lambda}, \quad\frac{\lambda-t}{\lambda-1}\quad \text{and}\quad t\frac{\lambda-1}{\lambda-t}. \end{equation*}$$

Let us now describe the construction of §2 in terms of ruled surfaces. We consider a rational ruled surface $p\colon S=\mathbb P(E)\to\mathbb P^1$ equipped with the parabolic structure $\boldsymbol{l}$ determined by a point $l_i$ on the fibre $S\vert_i=p^{-1}(i)$ for $i=0,1,t,\infty$.

Step 1. The elliptic ruled surface $\tilde p\colon \tilde S=\mathbb P(\tilde E)\to\mathbb P^1$ is obtained as the two-fold ramified covering $\Pi\colon \tilde S\to S$ branching along the four fibres $S\vert_i$. It makes the following diagram commutative:

We equip $\tilde S$ with a parabolic structure $\tilde{\boldsymbol{l}}$ by putting $\tilde l_i=\Pi^{-1}(l_i)\in \tilde S\vert_{\omega_i}=\tilde p^{-1}(\omega_i)$ for $i=0,1,t,\infty$.

Step 2 (and 3). The birational transformation

is obtained by blowing up the four points $\tilde l_i$ and then blowing down the strict transforms of the four fibres. Step 3 is irrelevant from the projective point of view since it suffices to multiply $\tilde E'$ by a line bundle in order to obtain $\tilde E''$. Therefore $\tilde S''= \tilde S'$.

By Lemma 6, the parabolic ruled surface $(S,\boldsymbol{l})$ we start with is of the following form:

$\bullet$ either $S=\mathbb P^1\times\mathbb P^1$ and no three points $l_i\in S$ lie on the same horizontal line;

$\bullet$ or $S=\mathbb F_2$ is the second Hirzebruch surface, that is, the total space of $\mathbb P(\mathcal O(-1)\oplus\mathcal O(1))$, and none of the points $l_i$ lies on the `negative' section $\sigma$ corresponding to $\mathcal O(1)$.

In either case we write $\tilde S'$ for the corresponding elliptic pullback constructed in §3.2.

Proposition 7.  When $S=\mathbb P^1\times\mathbb P^1$, choose a coordinate $w$ and consider the cross-ratio

$$\begin{equation*} c=\frac{w_t-w_0}{w_1-w_0}\,\frac{w_1-w_\infty}{w_t-w_\infty}\in\mathbb P^1 , { } \end{equation*}$$

where the points $w_i$ are defined by the formula $l_i=(i,w_i)$. Then the following three assertions hold.

$\bullet$ If $c\neq 0,1,t,\infty$, then $\tilde S'$ is a decomposable ruled surface

$$\begin{equation*} \tilde S'=\overline{\mathcal O_X([\omega]-[-\omega])} , \end{equation*}$$

where $\pm \omega=(x,\pm y)\in X$ are the two points over $x=t(c-1)/(c-t)$.

$\bullet$ If $c=t$ (all four points $l_i$ lie on the same irreducible curve of bidegree $(1,1)$), then $\tilde S'$ is the indecomposable ruled surface $S_0$.

$\bullet$ If $c=0,1,\infty$, then at least two points $l_i$ lie on the same horizontal line. Moreover, if the two other points lie on another horizontal line, then $\tilde S'$ is a trivial bundle. Otherwise $\tilde S'$ is the indecomposable ruled surface $S_0$.

When $S=\mathbb F_2$, the following two assertions hold.

$\bullet$ If all four points $l_i$ lie on a section with self-intersection $+2$ (that is, on a section induced by an arbitrary embedding $\mathcal O_{\mathbb P^1}(-1)\hookrightarrow\mathcal O_{\mathbb P^1}(-1)\oplus \mathcal O_{\mathbb P^1}(1)$), then $\tilde S'$ is a trivial bundle.

$\bullet$ Otherwise $\tilde S'$ is the indecomposable ruled surface $S_0$.

Proof.  We first consider the generic case when $S=\mathbb P^1\times\mathbb P^1$ and $c\neq 0,1,t,\infty$. Then one can choose the vertical coordinate $w$ in such a way that

$$\begin{equation*} l_0=(0,0), \quad l_1=(1,1), \quad l_t=(t,c) \quad \text{and}\quad l_\infty=(\infty,\infty). \end{equation*}$$

One easily checks by computation that there is a unique curve $C\subset\mathbb P^1\times\mathbb P^1$ of bidegree $(2,2)$ intersecting each fibre $x=i$ at the point $l_i$ with multiplicity $2$. The equation of $C$ is of the form $F= 0$, where

$$\begin{equation} F(x,w)=\bigl((c-t)x-t(c-1)\bigr)w^2+2(t-1)cxw-cx\bigl((c-1)x-(c-t)\bigr). \end{equation} \tag{ 12 }$$

The discriminant with respect to the variable $w$ is given by

$$\begin{equation*} \mathrm{disc}_w(F)=4c(c-1)(c-t)x(x-1)(x-t). \end{equation*}$$

Since it does not vanish identically and its roots are simple, the curve $C$ is reduced, irreducible and smooth. Its lifting $\tilde C$ to the trivial bundle $X\times\mathbb P^1$ splits into the union of two distinct sections, $\tilde\sigma_0$ and $\tilde\sigma_\infty$, which intersect each other at exactly the four points $\tilde l_i$ without multiplicity. After elementary transformations with centres $\tilde l_i$ we obtain disjoint sections $\sigma_0$ and $\sigma_\infty$ of $\tilde S'$. This already means that $\tilde S'$ is the compactification of a line bundle. To determine this line bundle, we consider the horizontal section $w=\infty$ of $\mathbb P^1\times\mathbb P^1$ passing through $l_\infty$. It intersects the $(2,2)$-curve $C$ at two points ($l_\infty$ and the point $s=(x,\infty)$ with coordinate $x=t(c-1)/(c-t)$) and lifts to a section $\tilde\sigma$ of $X\times\mathbb P^1$ intersecting $\tilde\sigma_0$ over $0$ and, say, $\omega$, and then intersecting $\tilde\sigma_\infty$ over $0$ and $-\omega$, where $\pm \omega\in X$ are the two points of $X$ over $x$. After elementary transformations with centres $\tilde l_i$ we obtain a section $\sigma$ of $\tilde S'$ intersecting $\sigma_0$ at $\omega$ and $\sigma_\infty$ at $-\omega$. Thus $\tilde S'=\overline{\mathcal O([\omega]-[-\omega])}$. This argument is summarized in Fig. 1.

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  When $c=t$, the curve $F= 0$ degenerates to twice the $(1,1)$-curve passing through all the points $l_i$, namely, the diagonal section $\sigma(x)=x$. Its lifting to $\tilde S$ is the graph $\tilde\sigma$ of the two-fold covering $X\to\mathbb P^1$ with self-intersection $+4$. After elementary transformations we obtain a section $\tilde\sigma'$ of $\tilde S'$ having zero self-intersection. More precisely, the normal bundle of $\tilde\sigma'$ is trivial. Indeed, the section $\tilde\sigma$ is induced in $\tilde S=\mathbb P(\tilde E)$, $\tilde E=\mathcal O_X\oplus\mathcal O_X$, by the line bundle $\tilde L\subset\tilde E$ generated by its meromorphic section $(x,y)\mapsto\begin{pmatrix}x\\1\end{pmatrix}$ whose divisor is $-2[\omega_\infty]$. The normal bundle of $\tilde\sigma$ in $\tilde S$ is therefore given by

$$\begin{equation*} \mathcal N_{\tilde\sigma}=\det(\tilde E)\otimes\tilde L^{-2}= \mathcal O_X\otimes\mathcal O_X(-2[\omega_\infty])^{-2}= \mathcal O_X(4[\omega_\infty]). \end{equation*}$$

After elementary transformations we obtain that

$$\begin{equation*} \begin{aligned} \, \mathcal N_{\tilde\sigma'} &=\det(\tilde E')\otimes L'^{-2} \\ &=\mathcal O_X\bigl([\omega_0]+[\omega_1]+[\omega_t]+[\omega_\infty]\bigr)\otimes\mathcal O_X\bigl([\omega_0]+[\omega_1]+[\omega_t]-[\omega_\infty]\bigr)^{-2} =\mathcal O_X. \end{aligned} \end{equation*}$$

If $\tilde S'$ were decomposable, it would be a trivial $\mathbb P^1$-bundle, and we must now exclude this possibility. Consider the centres $\tilde l_i'\in\tilde S'$ of the inverse elementary transformations. If $\tilde S'$ were the trivial $\mathbb P^1$-bundle, most of horizontal sections would avoid the four points $\tilde l_i'$ and would define, back on $\tilde S=X\times \mathbb P^1$, a section with self- intersection $-4$. This is impossible. Thus $\tilde S'$ is the indecomposable bundle $S_0$.

We now assume that $c=\infty$ (the other cases $c= 0$ or $1$ are similar). We are in one of the following three cases:

1) $w_0= 0$, $w_1= 1$ and $w_t=w_\infty=\infty$;

2) $w_0=w_1= 0$, $w_t= 1$ and $w_\infty=\infty$;

3) $w_0=w_1= 0$ and $w_t=w_\infty=\infty$.

The last case is easy since the three horizontal sections $\sigma_0$, $\sigma_1$ and $\sigma_\infty$ (which are defined by $w= 0$, $w= 1$ and $w=\infty$ respectively) are transformed on the elliptic pullback $\tilde S'$ into disjoint sections $\tilde\sigma_0'$ and $\tilde\sigma_\infty'$ and a third section $\tilde\sigma_1'$ that intersects $\tilde\sigma_0'$ at $\omega_t$ and $\omega_\infty$, and $\tilde\sigma_\infty'$ at $\omega_0$ and $\omega_1$. We promptly deduce that

$$\begin{equation*} \tilde S'=\overline{\mathcal O_X([\omega_t]+[\omega_\infty]-[\omega_0]-[\omega_1])}=X\times\mathbb P^1. \end{equation*}$$

We now study the first case, where only $w_t$ and $w_\infty$ coincide (this is similar to the diagonal case). The section $w=\infty$ of $S$ induces a section $\tilde \sigma_\infty'$ of $\tilde S'$ having a trivial normal bundle. If $\tilde S'$ were decomposable, it would be the trivial bundle, and a generic constant section $\tilde\sigma'$ would provide a section $\tilde\sigma$ of the trivial bundle $\tilde S$ having self-intersection $-4$, a contradiction. Thus $\tilde S'$ is the indecomposable bundle $S_0$.

We finally consider the case when $S=\mathbb F_2$. As above, we easily see that the exceptional section $\sigma_\infty$, which is induced by $\mathcal O_{\mathbb P^1}(1)$, yields a section $\tilde\sigma_\infty'$ of $\tilde S'$ having a trivial normal bundle. Again, as in the diagonal case, consider the centres $l_i'\in \tilde S'$ of the elementary transformations inverse to $\phi$; they are contained in $\tilde\sigma_\infty'$. If $\tilde S'$ is the trivial bundle, then its constant sections $\tilde\sigma'$ give rise to a pencil of sections $\tilde\sigma$ of $\tilde S$ having the four points $\tilde l_i$ as base points. A special member of this pencil is given by the union of $\tilde\sigma_\infty$ and the four fibres over the points $\omega_i$, and the pencil itself consists of all sections of $\tilde S$ passing through the four points $\tilde l_i$ and disjoint from $\tilde\sigma_\infty$ (plus the special one). The elliptic involution $\tau\colon (x,y)\mapsto(x,-y)$ permutes these sections, and so acts on the parameter space $\mathbb P^1$. This action has at least two fixed points, namely, $\tilde\sigma_\infty$ and another section $\tilde\sigma_0$ that can be pushed down as a section $\sigma_0$ of $S$. By construction, $\sigma_0$ passes through all the points $l_i$ and does not intersect $\sigma_\infty$; it is a $+2$-curve, as required. Conversely, when all $l_i$ lie on a $+2$-curve $\sigma_0$, we obtain a second section $\tilde\sigma_0'$ of $\tilde S'$, which yields a trivialization. We note that the pencil considered above comes from a pencil not of sections of $S$, but of curves that intersect each fibre twice. $\square$

Let $X_0$ be a complex projective curve and let $(E_0,\nabla_0)$ be a vector bundle of rank $2$ over $X_0$ equipped with a (flat) logarithmic $\mathrm{sl}(2,\mathbb C)$-connection with a reduced effective polar divisor $D_0$. For every topologically trivial analytic deformation $(X_t,D_t)$ of the punctured curve there is a unique deformation $(E_t,\nabla_t)$ of the vector bundle and the connection such that the monodromy data remain constant. This follows from the Riemann–Hilbert correspondence (Proposition 24 with parameters). The monodromy data consist of the monodromy representation complemented by the `parabolic structure' at apparent singular points. Equivalently, if we denote the total space of the deformation by $\boldsymbol X$ and the corresponding (smooth) divisor by $\boldsymbol D\subset \boldsymbol X$, then the deformation $(E_t,\nabla_t)$ is induced by the unique flat logarithmic $\mathrm{sl}(2,\mathbb C)$- connection $(E,\nabla)$ with polar divisor $\boldsymbol D$ inducing $(E_0,\nabla_0)$ on the slice $X_0$.

In this paper we consider the case of the 4-punctured sphere and the once- punctured torus. Their deformations are parametrized by the corresponding Teichmüller spaces that are both isomorphic to the Poincaré half-plane $\mathbb H$. To be precise, we start with the isomonodromic deformation of a logarithmic $\mathrm{sl}(2,\mathbb C)$- connection $(E_t,\nabla_t)$ over the Riemann sphere with poles at $0$, $1$, $t$ and $\infty$, where $t$ ranges over the universal covering $T\simeq\mathbb H\to \mathbb P^1\setminus\{0,1,\infty\}$. Consider the deformation $(\tilde E_t,\tilde\nabla_t)$ of the corresponding elliptic pullback constructed in §2. We easily see that $(\tilde E_t,\tilde\nabla_t)$ is still an isomonodromic deformation of some logarithmic connection over the Legendre family of elliptic curves $X_t$ with poles contained in the ramification locus $\{\omega_0,\omega_1,\omega_t,\omega_\infty\}$ of the elliptic curve. The parameter space $T$ is now understood as the Teichmüller space of the torus. For the special parameters $\boldsymbol{\theta}=({1}/{2},{1}/{2},{1}/{2},{1}/{2}+{\vartheta}/{2})$ we obtain the isomonodromic deformation of a Lamé connection.

Although we do not really need it, it is interesting to recall how the Painlevé VI equation was originally derived as an isomonodromic equation for Fuchsian projective structures on the 4-punctured sphere with one extra branch point. After normalizing the singular points as $0$, $1$, $t$ and $\infty$ by a Möbius transformation, the corresponding Fuchsian second-order differential equation $u_{xx}+f(x)u_x+g(x)u= 0$ takes the form

$$\begin{equation} \begin{cases} f(x)=\dfrac{1-\kappa_0}{x}+\dfrac{1-\kappa_1}{x-1} +\dfrac{1-\kappa_t}{x-t}-\dfrac{1}{x-q}, \\ g(x)=\dfrac{-\frac{t(t-1) H}{x-t}+\frac{q(q-1)p}{x-q}+\rho(\kappa_\infty+\rho)}{x(x-1)}. \end{cases} \end{equation} \tag{ 13 }$$

Here $q\not\in\{0,1,t,\infty\}$ is the branch point, $\kappa_i$ is the local exponent at $i=0,1,t,\infty$, and $\rho$ is fixed by the relation

$$\begin{equation} \kappa_0+\kappa_1+\kappa_t+\kappa_\infty+2\rho=1. { } \end{equation} \tag{ 14 }$$

Note that the parameters $p$ and $H$ are residues of $g$:

$$\begin{equation} H=-\operatorname{Res}_{x=t}g(x) \quad \text{and}\quad p=\operatorname{Res}_{x=q}g(x). \end{equation} \tag{ 15 }$$

The singular point $q$ has exponent $2$. This point is apparent (that is, a branch point of the projective chart) if and only if the parameter $H$ is given by

$$\begin{equation} H=\frac{q(q-1)(q-t)}{ t(t-1)}\biggl(p^2-\biggl(\frac{\kappa_0}{q}+\frac{\kappa_1}{q-1} +\frac{\kappa_t-1}{q-t}\biggr)p +\frac{\rho(\kappa_\infty+\rho)}{q(q-1)}\biggr). \end{equation} \tag{ 16 }$$

Under these assumptions, the local charts $\phi={u_1}/{u_2}$, where $u_1$ and $u_2$ range over independent solutions of (13), form a projective atlas on the complement of $0$, $1$, $t$, $q$ and $\infty$ in the Riemann sphere. At each of the singular points $i=0,1,t,\infty$ one of the projective charts takes the form $\phi=z^{\kappa_i}$ (or, possibly, $\phi={1}/{z^m}+\log(z)$ in the case when $\kappa_i=\pm m\in\mathbb Z_{>0}$) for a convenient local coordinate $z$ at $i$. At the point $q$ one of the projective charts takes the form $\phi=z^2$ (a simple branch point). Conversely, every projective structure on the Riemann sphere having five singularities with moderate growth, one of which is a simple branch point, is conjugate by a Möbius transformation to an element of the family above. Such projective structures are characterized by the following data:

$\bullet$ the positions $t$ and $q$ of the singular points,

$\bullet$ the exponents $\boldsymbol{\kappa}=(\kappa_0,\kappa_1,\kappa_t,\kappa_\infty)$,

$\bullet$ the monodromy representation of a projective chart $\phi$ up to conjugacy.

A small deformation of an equation (13) with moving singular points $t$ and $q$ is said to be isomonodromic if the projective charts have constant monodromy representation (up to conjugacy). Such deformations are characterized by the following classical theorem.

Theorem 8.  (Fuchs–Malmquist) A deformation of (13) parametrized by the position $t$ of the singular point is isomonodromic if and only if the exponents $\kappa_i$ remain fixed and the parameters $(p(t),q(t))$ satisfy the non-autonomous Hamiltonian system

$$\begin{equation} \frac{\partial q}{\partial t}=\frac{\partial H}{\partial p} \quad \text{and\/} \quad \frac{\partial p}{\partial t}=-\frac{\partial H}{\partial q}. \end{equation} \tag{ 17 }$$

The first Hamiltonian equation (17) is of the form

$$\begin{equation} p=\frac{1}{2}\biggl(\frac{t(t-1)}{q(q-1)(q-t)}\,\frac{\partial q}{\partial t} +\frac{\theta_0}{q}+\frac{\theta_1}{q-1}+\frac{\theta_t-1}{q-t}\biggr). \end{equation} \tag{ 18 }$$

Substituting (18) in the second equation (17), we obtain the Painlevé VI equation (9) with parameters $\boldsymbol{\kappa}$. From a chronological point of view, the Painlevé VI equation was first derived by Fuchs, and the Hamiltonian form was discovered later by Malmquist.

Let us now recall how Painlevé VI solutions correspond to isomonodromic deformations of logarithmic $\mathrm{sl}(2,\mathbb C)$-connections $(E_t,\nabla_t)$ with singular points $0$, $1$, $t$ and $\infty$ over the Riemann sphere. Let $(E_t,\nabla_t)$ be such a deformation. Assume that it is irreducible (this depends only on the monodromy and not on the value of $t$). By Lemma 6, the underlying bundle $E_t$ is either trivial or equal to $\mathcal O(-1)\oplus\mathcal O(1)$. It turns out that $E_t$ must be trivial for all values of $t$ outside a discrete subset of $T$: there are no non-trivial irreducible isomonodromic deformations of such connections on the bundle $\mathcal O(-1)\oplus\mathcal O(1)$ (see [15]). It is thus enough to consider isomonodromic deformations of $\text{sl}(2,\mathbb C)$-Fuchsian systems

$$\begin{equation} \frac{d Y}{dx}=\biggl(\frac{A_0}{x}+\frac{A_1}{x-1}+\frac{A_t}{x-t}\biggr)Y, \qquad A_i\in\mathrm{sl}(2,\mathbb C). \end{equation} \tag{ 19 }$$

The residual matrix at the singular point $x=\infty$ is given by

$$\begin{equation} A_0+A_1+A_t+A_\infty=0. \end{equation} \tag{ 20 }$$

We denote the eigenvalues of $A_i$ by $\pm {\theta_i}/2$. Then

$$\begin{equation} A_i=\biggl(\begin{matrix}a_i&b_i\\c_i&-a_i\end{matrix}\biggr), \quad\text{where } \ a_i^2+b_ic_i=\frac{\theta_i^2}{4}, \quad i=0,1,t,\infty. \end{equation} \tag{ 21 }$$

After a change of variables $Y:=MY$, where $M\in\mathrm{SL}(2,\mathbb C)$, we normalize

$$\begin{equation} A_\infty= \begin{pmatrix}\dfrac{\theta_\infty}{2}&0\\ \ast&-\dfrac{\theta_\infty}{2}\end{pmatrix}. \end{equation} \tag{ 22 }$$

We exclude the case when $A_\infty= 0$: the connection $\nabla$ must be singular at $\infty$. Then the following theorem holds.

Theorem 9.  A small deformation $A_i=A_i(t)$ of the system (10), normalized by (22), is isomonodromic if and only if the eigenvalues $\pm{\theta_i}/2$ are constant and the function $q:=tb_0/(tb_0 +(t-1)b_1)$ satisfies the differential equation

$$\begin{equation} \frac{dq}{dt}=-2a_0\frac{q-1}{t-1}-2a_1\frac{q}{t}+(1-\theta_\infty)\frac{q(q-1)}{t(t-1)} \end{equation} \tag{ 23 }$$

and the Painlevé VI equation (9) with parameters

$$\begin{equation*} \boldsymbol\kappa=(\kappa_0,\kappa_1,\kappa_t,\kappa_\infty) =(\theta_0,\theta_1,\theta_t,\theta_\infty-1). \end{equation*}$$

We first deduce Theorem 9 from the Fuchs–Malmquist theorem (Theorem 8).

Proof.  The vector $\begin{pmatrix}0\\ 1\end{pmatrix}$ is an eigenvector with eigenvalue $-{\theta_\infty}/{2}$ at $\infty$. Since the system (10) is irreducible, this vector is not invariant and can be chosen for a cyclic vector to derive a scalar Fuchsian equation. Namely, if $Y=\begin{pmatrix}y_1\\ y_2\end{pmatrix}$ is a solution of (10), then the function

$$\begin{equation*} u:=\sqrt{x^{\theta_0}(x-1)^{\theta_1}(x-t)^{\theta_t}}\,y_1 \end{equation*}$$

satisfies the scalar equation $(13)$ with exponents

$$\begin{equation} (\kappa_0,\kappa_1,\kappa_t,\kappa_\infty)=(\theta_0,\theta_1,\theta_t,\theta_\infty-1) \end{equation} \tag{ 24 }$$

and parameters

$$\begin{equation} q=\frac{tb_0}{tb_0 +(t-1)b_1}, \end{equation} \tag{ 25 }$$

$$\begin{equation} p=\frac{a_0+{\theta_0}/{2}}{q} +\frac{a_1+{\theta_1}/{2}}{q-1}+\frac{a_t+{\theta_t}/{2}}{q-t}. \end{equation} \tag{ 26 }$$

In fact, the Darboux coordinates have the following interpretation. The point $x= q$ is the unique other point at which $\begin{pmatrix}0\\ 1\end{pmatrix}$ is again an eigenvector of the system (10). The corresponding eigenvalue is

$$\begin{equation*} -p+\frac{\theta_0}{2q}+\frac{\theta_1}{2(q-1)}+\frac{\theta_t}{2(q-t)}. \end{equation*}$$

It follows from Theorem 8 that a deformation of the system (10) is isomonodromic if and only if the auxiliary variables $p$ and $q$ defined by (25) and (26) satisfy the Hamiltonian system (17) with the non-autonomous Hamiltonian $H(t,p,q)$ defined in (16). $\square$

Let us now explain how to reconstruct the system (10) uniquely (up to a gauge transformation) from a solution $q(t)$ of the Painlevé VI equation. We first introduce an auxiliary variable $p$ by the formula (18). This gives us a unique scalar equation (13) from which one can reconstruct a Fuchsian system by the standard method. The resulting system is defined (up to a gauge transformation) by the formula (10) with equations (21) and

$$\begin{equation} \begin{cases} a_0=\dfrac{\boldsymbol p}{t}-\dfrac{\kappa_0}{2}, \\ a_1=-\dfrac{\boldsymbol p}{t-1}-\dfrac{q-1}{t-1}(\rho+\kappa_\infty)-\dfrac{\kappa_1}{2}, \\ a_t=\dfrac{\boldsymbol p}{t(t-1)}+\dfrac{q-t}{t-1}(\rho+\kappa_\infty)-\dfrac{\kappa_t}{2}, \end{cases} \qquad \begin{cases} b_0=-\dfrac{q}{t}, \\ b_1=\dfrac{q-1}{t-1}, \\ b_t=-\dfrac{q-t}{t(t-1)}, \end{cases} \end{equation} \tag{ 27 }$$

where $\boldsymbol p=q(q-1)(q-t)p$ and $\rho$ is defined by (14). The coefficients $c_i$ of this system can immediately be deduced from the equations (21). The standard formulae given by Jimbo and Miwa (see [26], pp. 199, 200) assume that $\theta_\infty\neq 0$ so that the matrix $A_\infty$ can be further normalized as a diagonal matrix by an additional gauge transformation. The resulting formulae are much more complicated than those above. The way we obtain the formulae (27) is described in §§5.3–5.5.

Coming back to our initial problem and wishing to apply Theorem 5, we would like to parametrize the parabolic structure $\boldsymbol l$ induced by $\nabla_t$ (or, equivalently, by the system (10)) in terms of the Painlevé VI transcendent $q(t)$ that parametrizes the deformation. We deduce from (21) and (27) that the eigenline $l_i$ associated with the eigenvalue $-{\theta_i}/{2}$ over the pole $i=0,1,t$ is given by

$$\begin{equation*} l_i=\biggl(-b_i:a_i+\frac{\theta_i}{2}\biggr), \end{equation*}$$

whence

$$\begin{equation} \begin{cases} l_0=\biggl(1:\dfrac{\boldsymbol p}{q}\biggr), \\ l_1=\biggl(1:\dfrac{\boldsymbol p}{q-1}+\rho+\kappa_\infty\biggr), \\ l_t=\biggl(1:\dfrac{\boldsymbol p}{q-t}+(\rho+\kappa_\infty)t\biggr), \\ l_\infty=(0:1) \end{cases} \end{equation} \tag{ 28 }$$

(recall that $l_\infty$ was normalized by (22)). These expressions for $l_i$ no longer hold for a gauge-equivalent system (10) (for example, through the Jimbo–Miwa normalization), but the cross-ratio

$$\begin{equation} c=\frac{l_t-l_0}{l_1-l_0}\,\frac{l_1-l_\infty}{l_t-l_\infty} =t\frac{(q-1)p+\rho+\kappa_\infty}{(q-t)p+\rho+\kappa_\infty} \end{equation} \tag{ 29 }$$

depends only on the system (22) up to gauge transformations. We note that the formula (29) gives an elegant definition of the auxiliary variable $p$ in terms of the parabolic structure of the connection, $q$ and $t$.

Corollary 10.  Let $(E_t,\nabla_t)$ be the isomonodromic deformation defined by the Painlevé VI solution $q(t)$ as above, and let $(\tilde E_t'',\tilde\nabla_t'')$ be the elliptic pullback of this deformation. Then the bundle $\tilde E_t''$ is semistable and has invariant

$$\begin{equation*} \lambda(\tilde E_t'')=q(t)+\frac{\rho+\kappa_\infty}{p(t)}. \end{equation*}$$

Proof.  By construction, the bundle $E_t$ is trivial. By Theorem 5, $\tilde E_t''$ is semistable and has invariant

$$\begin{equation*} \lambda(\tilde E_t'')=t\frac{c-1}{c-t}=q+\frac{\rho+\kappa_\infty}{p}. {\qquad\square} \end{equation*}$$

All computations above hold only under the generic assumptions that $q\neq 0,1,t,\infty$. On the other hand, it is well known that constant solutions $q(t)\equiv0$, $1$, $t$ or $\infty$ correspond to isomonodromic deformations of reducible connections. Thus Corollary 10 is enough for proving Theorem 1. However, we can be more precise and check for special values of $q$ and $p$ whether $\tilde E''$ is indecomposable or not.

The Painlevé property, which characterizes the Painlevé equations among all differential equations of the form $\lambda''=F(t,\lambda,\lambda')$, says that all Painlevé VI solutions can be analytically continued as meromorphic solutions along any path avoiding $0$, $1$ and $\infty$. Painlevé VI solutions become meromorphic and global on the universal covering of the 3-punctured sphere.

The (naive) space of initial conditions $\mathbb C^2\ni(q(t_0),q'(t_0))$ for the Painlevé VI equation at some point $t_0\in\mathbb P^1\setminus\{0,1,\infty\}$ fails to describe all solutions in a neighbourhood of $t_0$. Painlevé VI solutions are meromorphic and some of them have a pole at $t_0$; we have to add them. The good space of initial conditions is

$$\begin{equation} \mathcal M^{\boldsymbol\theta}_{t}:=\{\text{germs of meromorphic $\mathrm{P}_{\mathrm{VI}}^{\boldsymbol\theta}$-solutions at }t\}. \end{equation} \tag{ 30 }$$

To construct it, Okamoto considers the phase portrait of the Painlevé VI equation in the variables

$$\begin{equation*} (t,q,q')\in\bigl(\mathbb P^1\setminus\{0,1,\infty\}\bigr) \times\mathbb C^2 \end{equation*}$$

(introducing an auxiliary equation ${dq}/{dt}=q'$): it is defined by a rational vector field that determines a singular holomorphic foliation on any rational compactification. For example, we can start with $\mathbb P^1\times\mathbb P^2$ and observe that the singularities of the foliation are located on the special fibres $t=0,1,\infty$ (these do not concern us) and at the infinity of the $\mathbb P^2$-factor. The latter are degenerate and lie along a one-dimensional section of the $t$-projection; we must blow up this section in order to reduce the degeneracy of the singular points. After nine successive blow-ups, Okamoto obtains a fibre bundle

$$\begin{equation} t\colon \overline{\mathcal M^{\boldsymbol\theta}}\to \mathbb P^1\setminus\{0,1,\infty\} \end{equation} \tag{ 31 }$$

(we just ignore what happens over $t=0,1,\infty$) which is not locally trivial as an analytic bundle (but is as a topological bundle). There is a non-vertical divisor $Z\subset \overline{\mathcal M^{\boldsymbol\theta}}$ consisting of vertical leaves (with respect to the $t$-projection) and singular points. On the complement $\mathcal M^{\boldsymbol\theta}:= \overline{\mathcal M^{\boldsymbol\theta}}\setminus Z$ of this divisor, the Painlevé foliation is transversal to $t$ and induces a local analytic trivialization of the bundle

$$\begin{equation} t\colon \mathcal M^{\boldsymbol\theta}\to \mathbb P^1\setminus\{0,1,\infty\}. \end{equation} \tag{ 32 }$$

By construction, the fibre $\mathcal M_{t}^{\boldsymbol\theta}$ over any point $t\neq 0,1,\infty$ may be interpreted as a set of germs of meromorphic $\mathrm{P}_{\mathrm{VI}}^{\boldsymbol\theta}$-solutions. Actually, for special parameters $\boldsymbol\theta$, there are leaves staying at the infinity of the affine chart $(q,q')$. These cannot be regarded as meromorphic solutions. They should rather be viewed as `constant solutions $q\equiv \infty$'. The divisor $Z$ actually coincides with the reduced polar divisor of the closed $2$-form defined in the affine chart as

$$\begin{equation*} dt\wedge dH+dp\wedge dq , \end{equation*}$$

where $H$ is defined by (16). The kernel of this $2$-form determines the Painlevé fibration.

We now describe the parameter space $\mathcal M_{t}^{\boldsymbol\theta}$ starting with the Hirzebruch ruled surface $\mathbb F_2$. Define a reduced divisor $Z_t\subset\mathbb F_2$ as the union of the section $\sigma\colon \mathbb P^1\to\mathbb F_2$ having self-intersection $-2$ and the four fibres over $0$, $1$, $t_0$ and $\infty$. Next, we fix two points on each vertical component of $Z_t$, none of which lies on the horizontal component. Blowing up these eight points, we obtain the compact space $\overline{\mathcal M_t^{\boldsymbol\theta}}$. Let us preserve the notation $Z_t$ for the strict transform of this divisor. The complement $\mathcal M_t^{\boldsymbol\theta}:= \overline{\mathcal M_t^{\boldsymbol\theta}}\setminus Z_t$ is the space of initial conditions. It remains to define the position of these eight points as a function of $\boldsymbol{\theta}$ and $t$ (see §5.6).

We go back to the approach of Fuchs, where the Painlevé transcendents parametrize isomonodromic deformations of Fuchsian projective structures with $4+1$ singular points (see §4.2). Such a structure can be defined by a Fuchsian second-order differential equation (13). One can also determine it by the data consisting of a logarithmic $\mathrm{sl}(2,\mathbb C)$-connection $(E,\nabla)$ together with a line subbundle $L\subset E$ which is not $\nabla$-invariant and plays the role of a cyclic vector (see §4.3). In [5] these data are referred to as `$\mathrm{sl}(2,\mathbb C)$-opers'. A more geometrical picture inspired by the works of Ehresman uses the notion of a `projective oper'. This is a triple $(\mathbb P(E),\mathbb P(\nabla),\sigma)$, where $\mathbb P(E)$ is the associated $\mathbb P^1$-bundle, $\mathbb P(\nabla)$ is the induced projective connection and $\sigma\colon \mathbb P^1\to \mathbb P(E)$ is the section corresponding to $L$. For example, the system (10) determines the Riccati equation

$$\begin{equation} \frac{dy}{dx}=-b(x)y^2-2a(x)y+c(x) \end{equation} \tag{ 33 }$$

on a trivial $\mathbb P^1$-bundle by the formula $(1:y)=(y_1:y_2)$, where

$$\begin{equation*} Y=\begin{pmatrix}y_1\\ y_2\end{pmatrix}, \qquad A=\begin{pmatrix}a&b\\ c&-a\end{pmatrix}. \end{equation*}$$

We prefer to consider the associated phase portrait, that is, the singular holomorphic foliation $\mathcal F_0$ induced by this equation on the ruled surface $\mathbb F_0=\mathbb P^1\times\mathbb P^1$ and also called a Riccati foliation (see [28] or §8.8). Its singular points are located at the poles of the Riccati equation. Precisely, in the notation (21), the singular points are

$$\begin{equation*} \begin{cases} l_i=\biggl(-b_i:a_i+\dfrac{\theta_i}{2}\biggr)=\biggl(a_i-\dfrac{\theta_i}{2}:c_i\biggr), \\ l_i'=\biggl(-b_i:a_i-\dfrac{\theta_i}{2}\biggr)=\biggl(a_i+\dfrac{\theta_i}{2}:c_i\biggr). \end{cases} \end{equation*}$$

From the point of view of foliations ¹ we say that $l_i$ and $l_i'$ have exponents $\theta_i$ and $-\theta_i$ respectively. They correspond to the eigenlines of the system with eigenvalues $-{\theta_i}/{2}$ and ${\theta_i}/{2}$ (pay attention to the sign). If $\theta_i= 0$, then either the singular point of the system is logarithmic and the two singular points of $\mathcal F_1$ coincide, or the singular point of the system is apparent and the Riccati foliation is non-singular. In the latter case we shall introduce an additional parabolic structure (see below). The section $\sigma$ defined by $y=\infty$ plays the role of a cyclic vector. It has two tangencies with the Riccati foliation, namely, at $x=q$ and $x=\infty$ (where $\sigma$ passes through a singular point of the foliation), see the bottom diagram in Fig. 2.

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In this picture the foliation $\mathcal F_0$ is regular, transversal to the $\mathbb P^1$-fibre over a generic point $x$ and therefore transversely projective (see [29]). Thus it induces a projective structure on the section $\sigma$ that projects to the base $\mathbb P^1$. Clearly, the resulting projective structure on (a Zariski- open subset of) $\mathbb P^1$ is preserved by birational bundle transformations, and it is natural to look for the simplest birational model. In the triple $(\mathbb F_0,\mathcal F_0,\sigma)$ above, the point $x=\infty$ artificially plays a special role since we require in the normalization (22) that the section $\sigma$ passes through the singular point $(x,y)=(\infty,\infty)$ of the foliation. It is more natural to apply an elementary transformation at this point and obtain the following more symmetric picture (see [30]): a Riccati foliation $\mathcal F_1$ on the Hirzebruch ruled surface $\mathbb F_1$ having singular points over $0$, $1$, $t$ and $\infty$, and the section $\sigma$ now has a simple tangency with the foliation at the point $x=q$. In fact, $\sigma\colon\mathbb P^1\to\mathbb F_1$ is the unique negative section (that is, with self-intersection $-1$). The exponents (eigenvalues) of the foliation are now given by

$$\begin{equation*} \boldsymbol{\kappa}=(\kappa_0,\kappa_1,\kappa_t,\kappa_\infty) :=(\theta_0,\theta_1,\theta_t,\theta_\infty-1). \end{equation*}$$

Over each pole $x=i$ of the Riccati equation, the foliation $\mathcal F_1$ has one or two singular points depending on the exponent $\kappa_i$; when $q=i$, one of the singular points accidentally lies on $\sigma$ (see the middle diagram in Fig. 2).

We now add a parabolic structure which will be convenient in our further application of elementary transformations. It is a necessary tool in order to get a smooth moduli space when one of the exponents $\kappa_i$ vanishes.

The Okamoto space $\mathcal M^{\boldsymbol\theta}_{t}$ of initial conditions can be viewed as the moduli space of such Riccati foliations $\mathcal F_1$. To be precise, we fix the exponents $\boldsymbol{\kappa}$ and the parameter $t$ and consider the data $(\mathbb F_1,\mathcal F_1,\sigma,\boldsymbol{l})$, where

$\bullet$ $\mathbb F_1$ is the Hirzebruch ruled surface equipped with the ruling structure $\mathbb F_1\to \mathbb P^1$,

$\bullet$ $\mathcal F_1$ is a regular Riccati foliation on $\mathbb F_1$ which is transversal to the ruling structure outside $x=0,1,t,\infty$,

$\bullet$ over each $i=0,1,t,\infty$, either the fibre $x=i$ is invariant and there is one singular point for each exponent $\pm\kappa_i$, or $\kappa_i= 0$ and the foliation is regular and transversal to the ruling,

$\bullet$ $\boldsymbol{l}=(l_0,l_1,l_t,l_\infty)$, where either $l_i$ is a singular point with exponent $\kappa_i$ over $x=i$, or $\kappa_i= 0$, $\mathcal F_i$ is regular over $x=i$ and $l_i$ is any point of the fibre.

Such data correspond exactly to projectivizations of semistable parabolic connections, which were used in [25] to construct the moduli space $\mathcal M^{\boldsymbol\theta}_{t}$.

Putting $y=x(x-1)(x-t)u'/u$ in the scalar equation (13), we obtain the Riccati equation

$$\begin{equation} \begin{aligned} \, \frac{dy}{dx} &=-\frac{y^2}{x(x-1)(x-t)}+ \biggl(\frac{\kappa_0}{x}+\frac{\kappa_1}{x-1}+\frac{\kappa_t}{x-t}+\frac{1}{x-q}\biggr) y \\ &\qquad -\frac{q(q-1)(q-t)p}{x-q}-q(q-1)p+t(t-1)H-\rho(\kappa_\infty+\rho)(x-t). \end{aligned} \end{equation} \tag{ 34 }$$

Recall that $\kappa_0+\kappa_1+\kappa_t+\kappa_\infty+2\rho= 1$ and $H$ is defined by the formula (16). The equation (34) defines a Riccati foliation $\mathcal F_2$, which naturally compactifies on the Hirzebruch surface $\mathbb F_2$ given by two charts

$$\begin{equation*} (x,y)\in(\mathbb P^1\setminus\{\infty\})\times\mathbb P^1 \quad\text{and}\quad (x,\tilde y)\in(\mathbb P^1\setminus\{0\})\times\mathbb P^1 \end{equation*}$$

with transition map $\tilde y=y/x^2$. The singular points of the foliation lie on the five fibres:

$$\begin{equation*} x=0,\ 1,\ t,\ q\ \text{and}\ \infty, \end{equation*}$$

with corresponding exponents (up to a sign)

$$\begin{equation*} \kappa_0,\quad\kappa_1,\quad\kappa_t,\quad \kappa_q=1\quad\text{and} \quad\kappa_\infty. \end{equation*}$$

To be precise, the singular points in the first chart $(x,y)$ are given by

$$\begin{equation} \begin{gathered} \, \begin{cases} s_0=(0,0), \\ s'_0=(0,t\kappa_0), \end{cases} \quad \begin{cases} s_1=(1,0), \\ s'_1=(1,(1-t)\kappa_1), \end{cases} \quad \begin{cases} s_t=(t,0), \\ s'_t=(t,t(t-1)\kappa_t), \end{cases} \\ \begin{cases} s_q=(q,\boldsymbol p), \\ s'_q=(q,\infty). \end{cases} \end{gathered} \end{equation} \tag{ 35 }$$

Here $s_i$ (resp. $s'_i$) is the singular point with exponent $\kappa_i$ (resp. $-\kappa_i$) and $\boldsymbol p=q(q-1)(q-t)p$. At $x=\infty$, the singular points are given in the chart $(x,\tilde y)$ by the equations

$$\begin{equation} \begin{cases} s_\infty=(\infty,-\rho), \\ s'_\infty=(\infty,-\rho-\kappa_\infty). \end{cases} \end{equation} \tag{ 36 }$$

The `cyclic vector' is given by the section $\sigma$ defined in these charts by $y=\infty$ and $\tilde y=\infty$ respectively.

After an elementary transformation centred at the nodal singular point $s_q=(q,\boldsymbol p)$ of the section $\mathcal F_0$ we obtain a Riccati foliation $\mathcal F_1$ on the Hirzebruch surface $\mathbb F_1$ with poles $x= 0$, $1$, $t$, $\infty$ and exponents $\kappa_0$, $\kappa_1$, $\kappa_t$, $\kappa_\infty$ respectively, as in §5.2. The apparent singular point has disappeared.

If we define $\mathbb F_1$ by the usual charts

$$\begin{equation*} (x,y)\in(\mathbb P^1\setminus\{\infty\})\times\mathbb P^1 \quad\text{and}\quad (x,\tilde y)\in(\mathbb P^1\setminus\{0\})\times\mathbb P^1 \end{equation*}$$

with transition map $\tilde y=y/x$, then the negative section $\sigma$ is given by the formulae $y=\infty$ and $\tilde y=\infty$ respectively and the Riccati foliation $\mathcal F_1$ is induced by the equation

$$\begin{equation} \begin{aligned} \, \frac{dy}{dx} &=\frac{(qy-\boldsymbol p)(qy-\boldsymbol p+t\kappa_0)}{tqx} \\ &\qquad -\frac{((q-1)y-\boldsymbol p)((q-1)y-\boldsymbol p+(1-t)\kappa_1)}{(t-1)(q-1)(x-1)} \\ &\qquad +\frac{((q-t)y-\boldsymbol p)((q-t)y-\boldsymbol p+t(t-1)\kappa_t)}{t(t-1)(q-t)(x-t)} -\rho(\kappa_\infty+\rho). \end{aligned} \end{equation} \tag{ 37 }$$

This equation is obtained from (34) by setting $y:=(x-q)y+\boldsymbol p$. The singular points in the first chart $(x,y)$ are now given by

$$\begin{equation} \begin{gathered} \, \begin{cases} s_0=\biggl(0,\dfrac{\boldsymbol p}{q}\biggr), \\ s'_0=\biggl(0,\dfrac{\boldsymbol p-t\kappa_0}{q}\biggr), \end{cases} \qquad \begin{cases} s_1=\biggl(1,\dfrac{\boldsymbol p}{q-1}\biggr), \\ s'_1=\biggl(1,\dfrac{\boldsymbol p-(1-t)\kappa_1}{q-1}\biggr), \end{cases} \\ \begin{cases} s_t=\biggl(t,\dfrac{\boldsymbol p}{q-t}\biggr), \\ s'_t=\biggl(t,\dfrac{\boldsymbol p-t(t-1)\kappa_t}{q-t}\biggr), \end{cases} \end{gathered} \end{equation} \tag{ 38 }$$

and the singular points at $x=\infty$ are given in the chart $(x,\tilde y)$ by

$$\begin{equation} \begin{cases} s_\infty=(\infty,-\rho), \\ s'_\infty=(\infty,-\rho-\kappa_\infty) \end{cases} \end{equation} \tag{ 39 }$$

(here again $s_i$ has exponent $\kappa_i$).

Finally, by making the last elementary transformation centred at the singular point $s_\infty'$ of $\mathcal F_1$, we obtain a Riccati foliation $\mathcal F_0$ with eigenvalues

$$\begin{equation*} \boldsymbol\theta=(\theta_0,\theta_1,\theta_t,\theta_\infty)=(\kappa_0,\kappa_1,\kappa_t,\kappa_\infty+1). \end{equation*}$$

The underlying ruled surface depends on the relative position of the singular point $s'_\infty$ of $\mathcal F_1$ with respect to the negative section $\sigma_{-1}$: the Riccati foliation $\mathcal F_0$ is defined on

$\bullet$ $\mathbb F_0$ when $s'_\infty\not\in\sigma_{-1}$ (generic case),

$\bullet$ $\mathbb F_2$ when $s'_\infty\in\sigma_{-1}$ (codimension $1$ case).

In particular, $q=\infty$ in the latter case.

If $q\neq \infty$, then the Riccati equation defining the foliation $\mathcal F_0$ on $\mathbb F_0$ can be obtained from (37) by setting $y:=y-(\kappa_\infty+\rho)$. We obtain the equation

$$\begin{equation} \begin{aligned} \, \frac{dy}{dx} &=\frac{(qy-\boldsymbol p)(qy-\boldsymbol p+t\kappa_0)}{tqx} \\ &\qquad -\frac{((q-1)(y-\rho-\kappa_\infty)-\boldsymbol p)((q-1)(y-\rho-\kappa_\infty)-\boldsymbol p+(1-t)\kappa_1)}{(t-1)(q-1)(x-1)} \\ &\qquad +\frac{((q-t)(y-t(\kappa_\infty+\rho))-\boldsymbol p)((q-t)(y-t(\kappa_\infty+\rho))-\boldsymbol p+t(t-1)\kappa_t)}{t(t-1)(q-t)(x-t)}. \end{aligned} \end{equation} \tag{ 40 }$$

This is precisely the projectivization of the Fuchsian system in (27). The singular points $s_i$ with eigenvalues $\kappa_i$ correspond to the parabolic structure $\boldsymbol{l}$ defined by the formulae (28):

$$\begin{equation} \begin{cases} s_0=\biggl(0,\dfrac{\boldsymbol p}{q}\biggr), \\ s_1=\biggl(1,\dfrac{\boldsymbol p}{q-1}+\rho+\kappa_\infty\biggr), \\ s_t=\biggl(t,\dfrac{\boldsymbol p}{q-t}+(\rho+\kappa_\infty)t\biggr), \\ s_\infty=(\infty,\infty), \end{cases} \qquad \begin{cases} s_0'=\biggl(0,\dfrac{\boldsymbol p-t\kappa_0}{q}\biggr), \\ s_1'=\biggl(1,\dfrac{\boldsymbol p-(1-t)\kappa_1}{q-1}+\rho+\kappa_\infty\biggr), \\ s_t'=\biggl(t,\dfrac{\boldsymbol p-t(t-1)\kappa_t}{q-t}+(\rho+\kappa_\infty)t\biggr) \end{cases} \end{equation} \tag{ 41 }$$

and

$$\begin{equation} \begin{aligned} \, s'_\infty &=-\frac{\boldsymbol p(\boldsymbol p-t\kappa_0)}{tq(\kappa_\infty+1)} +\frac{\boldsymbol p(\boldsymbol p-(1-t)\kappa_1)}{(t-1)(q-1)(\kappa_\infty+1)} -\frac{\boldsymbol p(\boldsymbol p-t(t-1)\kappa_t)}{t(t-1)(q-t)(\kappa_\infty+1)} \\ &\qquad\qquad\qquad\qquad\qquad\qquad -\frac{\kappa_\infty+\rho}{\kappa_\infty+1} \bigl((\kappa_\infty+\rho)(q-t-1)-\kappa_1-t\kappa_t\bigr). \end{aligned} \end{equation} \tag{ 42 }$$

With each pair $(p,q)\in\mathbb C\times\bigl(\mathbb P^1\setminus\{0,1,t,\infty\}\bigr)$ we have associated a Riccati foliation $\mathcal F_1$ on the Hirzebruch surface $\mathbb F_1$ with poles $0$, $1$, $t$, $\infty$ and exponents $\boldsymbol{\kappa}$.

Conversely, let $\mathcal F_1$ be such a Riccati foliation on $\mathbb F_1$ with poles $0$, $1$, $t$ and $\infty$. Then the unique tangency $x=q$ between the negative section $\sigma_{-1}\colon\mathbb P^1\to\mathbb F_1$ and the foliation $\mathcal F_1$ uniquely determines $q\in\mathbb P^1$. Now, if $q\neq 0,1,t,\infty$, then we perform an elementary transformation at this tangency point and thus define a Riccati foliation $\mathcal F_2$ on $\mathbb F_2$. There is a unique section $\sigma_2\colon\mathbb P^1\to\mathbb F_2$ with self-intersection $+2$ passing through the singular points $s_0$, $s_1$ and $s_t$ of $\mathcal F_2$. Choose a chart $(x,y)\in\mathbb C\times \mathbb P^1$ such that the section $\sigma_2$ and the negative section $\sigma_{-2}$ are given by the equations $y= 0$ and $y=\infty$ respectively. In fact, $y$ is uniquely determined if we normalize the $y^2$-coefficient of the Riccati equation as in (34). We then observe that all singular points $s_i$ and $s'_i$ for $i=0,1,t,\infty$ depend only on the parameters $\boldsymbol{\kappa}$ and $t$; the singular point $s_q$ given by $(x,y)=(q,\boldsymbol p)$ is the only one depending on the particular foliation $\mathcal F_1$. Define $p$ by the formula $\boldsymbol p=q(q-1)(q-t)p$. We note that the Riccati foliation $\mathcal F_1$ is characterized by the position of the nodal singular point of $\mathcal F_2$ on the Hirzebruch surface $\mathbb F_2$ (after the normalization above).

Theorem 11.  [25] Fix the parameters

$$\begin{equation*} \boldsymbol\kappa=(\kappa_0,\kappa_1,\kappa_t,\kappa_\infty) \end{equation*}$$

and consider Riccati foliations on the Hirzebruch surface $\mathbb F_1$ having at most simple poles at $0$, $1$, $t$ and $\infty$, with exponents $\pm\kappa_i$ over $x=i$ such that the negative section $\sigma_{-1}$ of $\mathbb F_1$ is not $\mathcal F$-invariant (the semistability condition). Add moreover a parabolic structure $\boldsymbol l=(l_0,l_1,l_t,l_\infty)$ which consists of the following data over each point $i=0,1,t,\infty$:

$\bullet$ either of the singular point $l_i\in\mathbb F_1$ with exponent $\kappa_i$;

$\bullet$ or, when $\kappa_i= 0$ and there is actually no singular point, of any point of the fibre $x= i$ (the only case when the parabolic structure is relevant, that is, not determined by the foliation itself).

The moduli space $\mathcal M^{\boldsymbol\kappa}_t$ of such pairs $(\mathcal F,\boldsymbol l)$ up to bundle automorphisms is a quasi- projective rational surface that can be described as follows. Start with the Hirzebruch surface $\mathbb F_2$ and write $Z_t$ for the reduced divisor consisting of the four fibres $x=i$ together with the negative section $\sigma_{-2}$. Define eight points $s_i$ and $s_i'$ by the formulae

$$\begin{equation} \begin{cases} s_0=(0,0), \\ s'_0=(0,t\kappa_0), \end{cases} \qquad \begin{cases} s_1=(1,0), \\ s'_1=(1,(1-t)\kappa_1), \end{cases} \qquad \begin{cases} s_t=(t,0), \\ s'_t=(t,t(t-1)\kappa_t) \end{cases} \end{equation} \tag{ 43 }$$

in the first chart $(x,y)$ and

$$\begin{equation} \begin{cases} s_\infty=(\infty,-\rho), \\ s'_\infty=(\infty,-\rho-\kappa_\infty) \end{cases} \end{equation} \tag{ 44 }$$

in the chart $(x,\tilde y={y}/{x^2})$. Carry out the following operations for each $i=0,1,t,\infty$.

$\bullet$ If $\kappa_i\neq 0$, then blow up the two points $s_i$ and $s_i'$.

$\bullet$ If $\kappa_i= 0$, then successively blow up the point $s_i=s_i'$ and the point where the exceptional divisor meets the strict transform of the fibre $x=i$.

Denote the resulting eight-point blow-up of $\mathbb F_2$ by $\overline{\mathcal M^{\boldsymbol\kappa}_t}$. Then the moduli space $\mathcal M_t^{\boldsymbol\kappa}$ is the complement in $\overline{\mathcal M_t^{\boldsymbol\kappa}}$ of the strict transform of the divisor $Z_t$.

The eight points $s_i$ and $s_i'$ in this theorem are nothing but the singular points of the foliation on $\mathbb F_2$ described in §5.3. When the nodal singular point $(x,y)=(q,\boldsymbol p)$ tends to $s_i$ (resp. $s_i'$), the singular point $s_i'$ (resp. $s_i$) of the corresponding foliation on $\mathbb F_1$ (see §5.4) tends to the negative section $\sigma_{-1}$. The exact limit depends on the way in which the nodal point tends to $s_i$ or $s_i'$. If $\kappa_i\neq 0$, then the fibre of $\mathcal M_t^{\boldsymbol\kappa}$ over $x=i$ consists of two disjoint copies $S_i$ and $S_i'$ of the affine line $\mathbb C$ which contain $s_i$ and $s_i'$ respectively: they stand for the moduli spaces of those foliations $\mathcal F$ whose singular points $s_i'$ (resp. $s_i$) lie on the negative section $\sigma_{-1}$. If $\kappa_i= 0$, then the fibre of $\mathcal M_t^{\boldsymbol\kappa}$ over $x=i$ is the union of an affine line $S_i\simeq\mathbb F$ and a projective line $S_i'\simeq\mathbb P^1$ that intersect each other transversally at one point and are projected to the point $s_i=s_i'$. The component $S_i$ consists of those foliations whose point $s_i$ lies on the negative section $\sigma_{-1}$. The compact component $S_i'$ consists of those parabolic foliations $(\mathcal F,\boldsymbol l)$ which actually have no pole over $x=i$. If we neglect the parabolic structure, then the rational curve $S_i'$ blows down to a quadratic singular point.

Sketch of the proof of Theorem 11. Let $\mathcal F$ be a Riccati foliation on $\mathbb F_1$ having at most simple poles over $x= 0$, $1$, $t$, $\infty$ such that the negative section $\sigma_{-1}$ is not invariant (semistability). In the standard chart $(x,y)$ the foliation is given by

$$\begin{equation*} \frac{dy}{dx}=\frac{(a_1x+a_0)y^2+(b_2x^2+b_1x+b_0)y +(c_3x^3+c_2x^2+c_1x+c_0)}{x(x-1)(x-t)}, \end{equation*}$$

where $a_k,b_k,c_k\in\mathbb C$ and the coefficients $a_0$ and $a_1$ do not vanish simultaneously. Bundle automorphisms are given by changes of coordinates of the form $y:=ay+bx+c$, $a,b,c\in\mathbb C$, $a\neq 0$. The single tangency between $\mathcal F$ and the negative section $\sigma_{-1}$ is given by $x=q:=-{a_0}/{a_1}$. By making the change of coordinates $y:=ay$, one can achieve the following normalization:

$$\begin{equation*} \text{either}\quad a_1x+a_0=x-q, \qquad\text{or}\quad a_1x+a_0=\tilde qx-1, \end{equation*}$$

where $\tilde q={1}/{q}$. We first assume that $q\neq \infty$, whence we can assume that $a_1x+a_0=x-q$. Using a change of coordinates of the form $y:=y+bx+c$, we can further assume that $b_1=b_2= 0$. Incidentally, assuming that $q\neq \infty$, we obtain a unique normal form

$$\begin{equation} \frac{dy}{dx}=\frac{(x-q)y^2+b_0y}{x(x-1)(x-t)}+\frac{c_0}{tx} +\frac{c_1}{(1-t)(x-1)}+\frac{c_t}{t(t-1)(x-t)}+c_\infty. \end{equation} \tag{ 45 }$$

It follows from (37) that

$$\begin{equation*} b_0=-2\boldsymbol p+(q-1)(q-t)\kappa_0+q(q-t)\kappa_1+q(q-1)\kappa_t. \end{equation*}$$

The residue at $x= 0$ is found from the formula

$$\begin{equation} {dy}=\frac{-qy^2+b_0y+c_0}{t}\frac{dx}{x}+(\text{holomorphic at } x=0), \end{equation} \tag{ 46 }$$

and the exponents $\pm\kappa_0$ are determined by the discriminant

$$\begin{equation*} \Delta_0:=\frac{b_0^2+4qc_0}{t^2}=\kappa_0^2. \end{equation*}$$

We similarly get

$$\begin{equation*} \begin{gathered} \, \Delta_1:=\frac{b_0^2+4(q-1)c_1}{(t-1)^2}=\kappa_1^2, \qquad \Delta_t:=\frac{b_0^2+4(q-t)c_t}{t^2(t-1)^2}=\kappa_t^2, \\ \Delta_\infty:=1-4c_\infty=\kappa_\infty^2. \end{gathered} \end{equation*}$$

Once the parameter $\boldsymbol{\kappa}$ is fixed, one can uniquely determine $c_0$, $c_1$, $c_t$ and $c_\infty$ as functions of $\boldsymbol\kappa$, $q$ and $b_0$ (that is, $\boldsymbol p$) provided that $q\neq 0,1,t,\infty$. In a neighbourhood of $q= 0$ one can still express $c_1$, $c_t$ and $c_\infty$ as functions of $\boldsymbol\kappa$, $q$ and $b_0$, and the moduli space of such foliations is locally isomorphic to the surface

$$\begin{equation*} \bigl\{(q,b_0,c_0)\in\mathbb C^3,\ b_0^2+2qc_0=(t\kappa_0)^2\bigr\}. \end{equation*}$$

We promptly see that the moduli space consists, over $q= 0$, of an affine line parametrized by $c_0$ over each of the points $b_0=\pm t\kappa_0$.

If $\kappa_0\neq 0$, then the graph

$$\begin{equation*} c_0=-\frac{(b_0-t\kappa_0)(b_0+t\kappa_0)}{4q} \end{equation*}$$

is clearly obtained by blowing up the two points and then deleting the level $c_0=\infty$, that is, the strict transform of $q= 0$. The formula (46) for the residue shows that the following conditions are equivalent over $q= 0$:

• $b_0=t\kappa_0$ (resp. $b_0=-t\kappa_0$),
• $\boldsymbol p= 0$ (resp. $\boldsymbol p=t\kappa_0)$,
• the singular point $s_0'$ (resp. $s_0$) of the corresponding foliation $\mathcal F_1$ lies on the negative section $y=\infty$.

If $\kappa_0= 0$, then the graph

$$\begin{equation*} c_0=-\frac{b_0^2}{4q} \end{equation*}$$

is obtained after two blow-ups, and the affine line parametrized by $c_0$ is the set of all foliations $\mathcal F_1$ whose singular point $s_0=s_0'$ lies on the negative section $y= \infty$. The surface with equation $b_0^2+4qc_0= 0$ has a singular point at $(q,b_0,c_0)=(0,0,0)$. This corresponds to the case when the residue (46) vanishes, that is, $\mathcal F_1$ actually has no singular point at $x= 0$. The parabolic data at $x= 0$ provide a desingularization of the surface. Indeed, the moduli space of pairs $(\mathcal F_1,\boldsymbol l)$ is locally parametrized in a neighbourhood of $q= 0$ by the set

$$\begin{equation*} \bigl\{(q,b_0,c_0,s_0)\in\mathbb C^3\times\mathbb P^1,\ b_0^2+2qc_0=0,\ 2q s_0=b_0\bigr\} \end{equation*}$$

(or it would be better to write $2q u_0=b_0 v_0$, where $(u_0:v_0)=s_0$). The parabolic data $s_0={b_0}/({2q})$ parametrize the exceptional divisor $S_0'$ (thus realizing the blow-up). The intersection $S_0\cap S_0'$ is given by $(q,b_0,c_0,s_0)=(0,0,0,\infty)$ and describes the foliation without singular points at $x= 0$ whose parabolic structure lies on the negative section $y=\infty$.

The study of $q=1,t,\infty$ is similar except that one must choose another normalization for $q=\infty$:

$$\begin{equation*} \frac{dy}{dx}=\frac{(\tilde qx-1)y^2+\tilde b_2x^2y}{x(x-1)(x-t)}+\frac{\tilde c_0}{x}+\frac{\tilde c_1}{x-1}+\frac{\tilde c_t}{x-t}+\tilde c_\infty, \end{equation*}$$

where $(\tilde q,\tilde b_2)=({1}/{q},{b_0}/{q^2})$ is the other chart of $\mathbb F_2$. $\square$

The deformation $t\mapsto \mathcal M^{\boldsymbol\kappa}_t$ is analytically (but not algebraically) trivial and the trivialization is given by the Painlevé flow (see [31], [25]). The good phase space for the Painlevé VI equation is the (locally analytically trivial) fibration

$$\begin{equation*} t\colon\mathcal M^{\boldsymbol\kappa}\to\mathbb P^1\setminus\{0,1,\infty\}. \end{equation*}$$

The map $q\colon \mathcal M^{\boldsymbol\kappa}_t\to\mathbb P^1$ is regular (when none of the $\kappa_i$ vanish) and endows $\mathcal M^{\boldsymbol\kappa}_t$ (for $t$ fixed) with the structure of an affine $\mathbb A^1$-bundle with double fibres over $0,1,t,\infty$. Finally, we note that the formula (37) defines an explicit section (universal Riccati foliation)

$$\begin{equation*} (t,q,\boldsymbol p)\mapsto (\mathbb F_1,\mathcal F_1) \end{equation*}$$

over $\mathcal M^{\boldsymbol\kappa}\setminus\{q=0,1,t,\infty\}$.

By §5.5 we have an isomorphism

$$\begin{equation*} \mathrm{elm}_{s_\infty'}\colon \mathcal M^{\boldsymbol\kappa}_t\stackrel{\sim}{\longrightarrow} \mathcal M^{\boldsymbol\theta}_t \end{equation*}$$

from the previous moduli space to the moduli space $\mathcal M^{\boldsymbol\theta}_t$ of $\mathrm{sl}(2,\mathbb C)$-connections $(E,\nabla)$ with exponents

$$\begin{equation*} \boldsymbol{\theta}=(\theta_0,\theta_1,\theta_t,\theta_\infty) :=(\kappa_0,\kappa_1,\kappa_t,\kappa_\infty+1). \end{equation*}$$

We recall that for every point of this moduli space the underlying vector bundle $E$ is either trivial or equal to $\mathcal O(-1)\oplus\mathcal O(1)$. It is actually trivial on a Zariski- open subset of $\mathcal M^{\boldsymbol\theta}$. The locus of the non-trivial bundles is Malgrange's theta divisor $\Theta\subset\mathcal M^{\boldsymbol\theta}$ defined for fixed $t$ by the exceptional divisor $S_\infty$ arising from the blow-up of $s_\infty$. Indeed, the points of the last divisor correspond to those foliations $\mathcal F_1$ for which $s_\infty'$ lies on the negative section $\sigma_{-1}$; applying $\mathrm{elm}_{s_\infty'}$ gives a foliation $\mathcal F_0$ on the Hirzebruch surface $\mathbb F_2=\mathbb P(\mathcal O(-1)\oplus\mathcal O(1))$. In fact we have $\mathcal F_0=\mathcal F_2$ in this case: we are back to the foliation of §5.3, where $s_q\to s_\infty$. To be precise, putting

$$\begin{equation*} p=\frac{\lambda-\rho q}{(q-1)(q-t)} \end{equation*}$$

in (13) and letting $q$ tend to $\infty$, we obtain the Heun equation

$$\begin{equation*} g(x)=\frac{\rho(\rho+\kappa_\infty+1)x-\lambda\kappa_\infty -\rho((t+1)\rho+\kappa_1+t\kappa_t)}{x(x-1)(x-t)}. { } \end{equation*}$$

For our special choice of the parameters $\boldsymbol{\theta}=({1}/{2},{1}/{2},{1}/{2},{1}/{2}+{\vartheta}/{2})$ we obtain the Lamé equation (1) with $n={\vartheta}/{2}$ and $c=2\lambda(\vartheta-1)+(t+1){\vartheta}/{2}({\vartheta}/{2}-1)$.

Finally, the open set $\mathcal M^{\boldsymbol\theta}\setminus\Theta$ (resp. the closed subset $\Theta$) may be viewed as the moduli space of Riccati foliations $\mathcal F_0$ on $\mathbb F_0$ (resp. $\mathbb F_2$) with simple poles $(0,1,t,\infty)$ and exponents $(\theta_0,\theta_1,\theta_t,\theta_\infty)$, excluding those foliations for which the section passing through $s_\infty$ (resp. the exceptional section) is totally $\mathcal F_0$-invariant.

There are many birational transformations

$$\begin{equation*} (\boldsymbol\kappa,t,p,q)\mapsto(\tilde{\boldsymbol\kappa}, \tilde{t},\tilde{p},\tilde{q}) \end{equation*}$$

that induce biregular diffeomorphisms between moduli spaces

$$\begin{equation*} \mathcal M^{\boldsymbol\kappa}\to\mathcal M^{\tilde{\boldsymbol\kappa}}, \end{equation*}$$

equivariant with respect to the projection $t$. They were studied in [32]. Some of them are the classical Schlesinger transformations arising from geometrical transformations of connections (resp. Riccati foliations); together with a strange extra symmetry, they generate the full group of Okamoto symmetries.

5.8.1. Change of signs

First, we can change the `spin structure', that is, the signs of the parameters $\pm\kappa_i$. This does not change the Riccati foliation or the coefficients $t$, $q$, $b_0$ of the normal form (45), but the variable $p$ is modified as follows:

$$\begin{equation*} \begin{gathered} \, (-,+,+,+)\colon \begin{cases} (\kappa_0,\kappa_1,\kappa_t,\kappa_\infty) \mapsto(-\kappa_0,\kappa_1,\kappa_t,\kappa_\infty), \\ t\mapsto t, \\ (q,p)\mapsto\biggl(q,p-\dfrac{\kappa_0}{q}\biggr), \end{cases} \\ (+,-,+,+)\colon \begin{cases} (\kappa_0,\kappa_1,\kappa_t,\kappa_\infty) \mapsto(\kappa_0,-\kappa_1,\kappa_t,\kappa_\infty), \\ t\mapsto t, \\ (q,p)\mapsto\biggl(q,p-\dfrac{\kappa_1}{q-1}\biggr), \end{cases} \\ (+,+,-,+)\colon \begin{cases} (\kappa_0,\kappa_1,\kappa_t,\kappa_\infty)\mapsto (\kappa_0,\kappa_1,-\kappa_t,\kappa_\infty), \\ t\mapsto t, \\ (q,p)\mapsto\biggl(q,p-\dfrac{\kappa_t}{q-t}\biggr), \end{cases} \\ (+,+,+,-)\colon \begin{cases} (\kappa_0,\kappa_1,\kappa_t,\kappa_\infty)\mapsto (\kappa_0,\kappa_1,\kappa_t,-\kappa_\infty), \\ t\mapsto t, \\ (q,p)\mapsto(q,p). \end{cases} \end{gathered} \end{equation*}$$

5.8.2. Permutation of poles

One can permute the four poles of the Riccati foliation. The resulting group of order $24$ is generated by the elements

$$\begin{equation*} \begin{gathered} \, (01)\colon \begin{cases} (\kappa_0,\kappa_1,\kappa_t,\kappa_\infty)\mapsto (\kappa_1,\kappa_0,\kappa_t,\kappa_\infty), \\ t\mapsto 1-t, \\ (q,p)\mapsto(1-q,-p), \end{cases} \\ (1t)\colon \begin{cases} (\kappa_0,\kappa_1,\kappa_t,\kappa_\infty)\mapsto (\kappa_0,\kappa_t,\kappa_1,\kappa_\infty), \\ t\mapsto \dfrac{1}{t}, \\ (q,p)\mapsto\biggl(\dfrac{q}{t},tp\biggr), \end{cases} \\ (0\infty)(1t)\colon \begin{cases} (\kappa_0,\kappa_1,\kappa_t,\kappa_\infty)\mapsto (\kappa_\infty,\kappa_t,\kappa_1,\kappa_0), \\ t\mapsto t, \\ (q,p)\mapsto\biggl(\dfrac{t}{q},-\dfrac{q(qp+\rho)}{t}\biggr) \end{cases} \end{gathered} \end{equation*}$$

(the change of variable is given by $\tilde x=1-x$, ${x}/{t}$ and ${t}/{x}$ respectively; we get $\tilde b_0=b_0$, ${b_0}/{t^2}$ and $(t(q-1)(q-t)-tb_0)/{q^2}$ respectively).

Together with sign changes, we already get a linear group of order $384$ acting on our moduli space.

5.8.3. Elementary transformations

Let $\mathcal F$ be a Riccati foliation on $\mathbb F_1$ representing a point of $\mathcal M^{\boldsymbol\kappa}_t$. An elementary transformation centred at one of the singular points (say, $s_0$) of $\mathcal F$ gives a new Riccati foliation with simple poles over $0$, $1$, $t$, $\infty$ and with the shifted parameters $\tilde{\boldsymbol\kappa}= (\kappa_0-1,\kappa_1,\kappa_t,\kappa_\infty)$. The resulting bundle is either the trivial bundle $\mathbb F_0$, or the Hirzebruch surface $\mathbb F_2$. However, after two (or, more generally, an even number of) such elementary transformations we are back to $\mathbb F_1$. Indeed, since the type $n$ of the Hirzebruch surface $\mathbb F_n$ shifts by $\pm 1$ under each elementary transformation, we must only exclude the possibility that, for example,

This would mean that each of the two successive elementary transformations has centre on the negative section. In this case, the negative section $\sigma_{-3}$ of $\mathbb F_3$ is the strict transform of $\sigma_{-1}$. Since $\sigma_{-1}$ is not $\mathcal F$-invariant, we obtain that $\sigma_{-3}$ is not $\mathcal F''$-invariant. But then Proposition 26 (see §8.8) gives a negative tangency, a contradiction. Thus we have defined a biregular transformation

$$\begin{equation*} \mathrm{elm}_{s_\infty}\circ\mathrm{elm}_{s_0} \colon \begin{cases} (\kappa_0,\kappa_1,\kappa_t,\kappa_\infty)\mapsto (\kappa_0-1,\kappa_1,\kappa_t,\kappa_\infty-1), \\ t\mapsto t, \\ (q,p)\mapsto(\tilde q,\tilde p). \end{cases} \end{equation*}$$

Omitting the huge formula, we note that $\tilde q$ is given by the unique tangency point between $\mathcal F$ and the only section $\sigma$ of $\mathbb F_1$ that has self-intersection $+1$ and passes through $s_0$ and $s_\infty$.

More generally, given any quadruple

$$\begin{equation*} \boldsymbol{n}=(n_0,n_1,n_t,n_\infty)\in\mathbb Z^4, \qquad n=n_0+n_1+n_t+n_\infty\in2\mathbb Z, \end{equation*}$$

we construct a biregular transformation

$$\begin{equation*} \mathrm{elm}_{\boldsymbol{l}}^{\boldsymbol{n}} \colon \begin{cases} (\kappa_0,\kappa_1,\kappa_t,\kappa_\infty)\mapsto (\kappa_0-n_0,\kappa_1-n_1,\kappa_t-n_t,\kappa_\infty-n_\infty), \\ t\mapsto t, \\ (q,p)\mapsto(\tilde q,\tilde p), \end{cases} \end{equation*}$$

where

$$\begin{equation*} \mathrm{elm}_{\boldsymbol{l}}^{\boldsymbol{n}}= \mathrm{elm}_{s_0}^{n_0}\circ\mathrm{elm}_{s_1}^{n_1}\circ \mathrm{elm}_{s_t}^{n_t}\circ\mathrm{elm}_{s_\infty}^{n_\infty} { } \end{equation*}$$

with the convention that $\mathrm{elm}_{s_i}^{n_i}:=\mathrm{elm}_{s_i'}^{-n_i}$ when $n_i< 0$. As above, there is a unique section $\sigma$ with self-intersection $\sigma\cdot\sigma=n-1$ and tangency of multiplicity $n_i$ with the foliation at each point $s_i$; the extra tangency between $\mathcal F$ and $\sigma$ is at $x=\tilde q$.

We now get an infinite affine group of transformations. We denote it by $H$.

5.8.4. The Okamoto symmetry

To generate the full group $G$ of biregular transformations described in [32], we need an extra symmetry,

$$\begin{equation*} \begin{cases} (\kappa_0,\kappa_1,\kappa_t,\kappa_\infty) \mapsto (\kappa_0+\rho,\kappa_1+\rho,\kappa_t+\rho,\kappa_\infty+\rho), \\ t\mapsto t, \\ (q,p)\mapsto\biggl(q+\dfrac{\rho}{p},p\biggr) \end{cases} \end{equation*}$$

(called $s_2$ in [19]) or any of its conjugates. So far there is no geometric interpretation of this symmetry as long as we interpret $\mathcal M^{\boldsymbol\kappa}$ as the moduli space of rank-$2$ connections (or Riccati foliations). To derive the full Okamoto group from natural transformations of connections, one has to deal with isomonodromic deformations of connections of rank $3$ (see [26]) or more (see [19]).

The conjugate of the Okamoto symmetry above by the change of signs $(+,+,+,-)$ is equal to $s_2s_1s_2$ in the notation of [19] and is given by

$$\begin{equation*} \begin{cases} (\kappa_0,\kappa_1,\kappa_t,\kappa_\infty)\mapsto (\kappa_0+\rho+\kappa_\infty,\kappa_1+\rho+\kappa_\infty,\kappa_t+\rho+\kappa_\infty,-\rho), \\ t\mapsto t, \\ (q,p)\mapsto\biggl(q+\dfrac{\rho+\kappa_\infty}{p},p\biggr). \end{cases} \end{equation*}$$

We recognize in $\tilde q=q+(\rho+\kappa_\infty)/{p}$ the Tu invariant of the underlying vector bundle of the elliptic pullback (see Corollary 10).

For special values of the parameter $\boldsymbol\kappa$ the moduli space $\mathcal M^{\boldsymbol\kappa}_t$ contains complete rational curves (independently of $t$). They arise from the curves on $\mathbb F_2$ avoiding the negative section and passing through $s_i$ or $s_i'$ for each value of $i=0,1,t,\infty$. They correspond either to the locus of reducible connections or to the locus of connections with an apparent singular point. It turns out that there are no other complete curves in $\mathcal M^{\boldsymbol\kappa}_t$ (see [25]). In particular, there are no such curves for generic values of $\boldsymbol\kappa$. Here are some examples.

Suppose that $\kappa_t= 1$ and the foliation $\mathcal F$ has an apparent singular point at $x=t$. Then the pole disappears after one elementary transformation centred at $s_t$. We thus obtain a Riccati foliation $\mathcal F_0$ of hypergeometric type on $\mathbb F_0$ (it cannot be $\mathbb F_1$ by Proposition 26), that is, a foliation with poles at $0$, $1$, $\infty$ and exponents $(\kappa_0,\kappa_1,\kappa_\infty)$. Conversely, a foliation $\mathcal F$ as above can be recovered from $\mathcal F_0$ by an elementary transformation at any point of the fibre $x=t$. This gives us a rational family of foliations $\mathcal F$ (parametrized by the fibre $x=t$). The corresponding rational curve $C$ in the moduli space $\mathcal M^{\boldsymbol\kappa}_t$ is given by the equation

$$\begin{equation*} q(q-1)p^2-((q-1)\kappa_0+q\kappa_1)p+\frac{(\kappa_0+\kappa_1)^2-\kappa_\infty^2}{4}. \end{equation*}$$

This is the only curve $C$ on $\mathbb F_2$ with the following properties:

$\bullet$ $q\colon C\to\mathbb P^1$ has degree $2$;

$\bullet$ $C$ does not intersect the negative section $\sigma_{-2}$;

$\bullet$ $C$ intersects the fibre $q=i$ at both $s_i$ and $s_i'$ for $i=0,1,\infty$;

$\bullet$ $C$ intersects the fibre $q=t$ twice at $s_t$;

$\bullet$ $C$ is singular at $s_t$: it has two smooth branches.

Proof.  To compute this family in the moduli space, we start with the Riccati foliation $\mathcal F_0$ that can be normalized to

$$\begin{equation*} \frac{dy}{dx}=\frac{-y^2-(\kappa_0+x\kappa_\infty)y+cx}{x(x-1)}, \qquad c=\frac{\kappa_1^2-(\kappa_0+\kappa_\infty)^2}{4} \end{equation*}$$

(we exclude some reducible cases here) and choose a parabolic structure $s_t=(t,\lambda)$ over $x=t$. The horizontal section $y=\lambda$ is sent by the elementary transformation $\mathrm{elm}_{s_t}\colon \mathbb F_0\dashrightarrow\mathbb F_1$ to the negative section: the corresponding value $q=(\lambda^2+\kappa_0\lambda)/(c-\kappa_\infty\lambda)$ corresponds to the unique tangency point between $\mathcal F_0$ and $y=\infty$. We have already noted that the map $\lambda\mapsto q(\lambda)$ has degree $2$. We now claim that the foliation $\mathcal F=\mathrm{elm}_{s_t}\mathcal F_0$ determines a point over $s_0$ (resp. $s_0'$) in the moduli space $\mathcal M^{\boldsymbol\kappa}_t$ if and only if the parabolic structure $s_t=(t,\lambda)$ and the singular point $s_0'$ (resp. $s_0$) of $\mathcal F_0$ lie on the same horizontal section, namely, on $y=\lambda$. Indeed, this section is transformed by $\mathrm{elm}_{s_t}$ into the negative section of $\mathbb F_1$. Thus the curve $C$ passes once through each of the points $s_0$, $s_0'$. The same holds over $q= 1$ and $\infty$. Finally, when $s_t$ is an apparent singular point for $\mathcal F_1$, it cannot lie on the negative section $\sigma_{-1}$ (otherwise an application of $\mathrm{elm}_{s_t}$ would give a hypergeometric foliation on $\mathbb F_2$, having by Proposition 26 a tangency of multiplicity $-1$ with the negative section, a contradiction). We can now compute the equation of the curve $C$. Since $C$ has degree $2$ and does not intersect the negative section of $\mathbb F_2$, the equation of $C$ takes the form $P^2+A(q)P+B(q)$, where $A$ and $B$ are polynomials of degrees $2$ and $4$ respectively. The condition that $C$ passes through all the points $s_i$ and $s_i'$, except $s_t'$, determines not only $C$, but a pencil of curves; they are all smooth and vertical at $s_t$ (and thus escape from $\mathcal M^{\boldsymbol\kappa}_t$ over this point), except one of them which has a singularity with normal crossing at $s_t$. $\square$

In the case when $\kappa_t=2$, the foliations $\mathcal F$ with apparent singular points at $x=t$ are obtained as follows. Take the hypergeometric foliation $\mathcal F_0$ on $\mathbb F_1$ with exponents $(\kappa_0,\kappa_1,\kappa_\infty)$, choose a parabolic structure $s_t$ at $x= 0$ and apply $\mathrm{elm}_{s_t}$ twice. We again get a rational curve in the moduli space which projects down to a curve $C\subset \mathbb F_2$ with the following properties:

$\bullet$ $q\colon C\to\mathbb P^1$ has degree $4$;

$\bullet$ $C$ does not intersect the negative section $\sigma_{-2}$;

$\bullet$ $C$ intersects the fibre $q=i$ at both $s_i$ and $s_i'$ for $i=0,1,\infty$;

$\bullet$ $C$ intersects the fibre $q=t$ three times at $s_t$ and once at $s_t'$.

A remarkable result of Bolibrukh ([9], §5.2, Proposition 5.6) asserts in our context that the isomonodromic deformation $t\mapsto(E_t,\nabla_t)\in\mathcal M^{\boldsymbol\theta}_t$ of an irreducible $\mathrm{sl}(2,\mathbb C)$-connection is `mostly' defined on the trivial bundle.

Theorem 12.  (Bolibrukh) Let $t\mapsto(E_t,\nabla_t)\in\mathcal M^{\boldsymbol\theta}_t$ be a local isomonodromic deformation. Then one of the following cases holds.

$\bullet$ The underlying bundle $E_t$ is trivial outside a discrete subset of the parameter space $T$.

$\bullet$ $E_t\equiv\mathcal O(-1)\oplus\mathcal O(1)$ and the destabilizing subsheaf $\mathcal O(1)$ is $\nabla_t$-invariant.

In particular, the former case holds when $(E_t,\nabla_t)$ is irreducible. Bolibrukh proved a more general result for certain logarithmic connections of arbitrary rank on the Riemann sphere. The rank $2$ case was considered by Heu [15] in full generality (for regular or irregular $\mathrm{sl}(2,\mathbb C)$-connections on an arbitrary Riemann surface). We will use the following corollary.

Proposition 13.  Let $t\mapsto(E_t,\nabla_t)\in\mathcal M^{\boldsymbol\theta}_t$ be a local isomonodromic deformation of an $\mathrm{sl}(2,\mathbb C)$-connection. Assume that $E_t$ is trivial and two eigenlines $l_i$ and $l_j$, $i,j\in \{0,1,t,\infty\}$, coincide along the whole deformation. Then the connection $(E_t,\nabla_t)$ is reducible: the constant line bundle $L_t\subset E_t$ determined by the lines $l_i=l_j$ is $\nabla_t$-invariant.

Proof.  Applying $\operatorname{elm}_{s_i}\circ\operatorname{elm}_{s_j}$ to the deformation, we obtain an isomonodromic deformation on the Hirzebruch surface $\mathbb F_2$. By Theorem 12, this is possible only when the connection is reducible. $\square$

Remark 14.  The following more general result holds. Consider an irreducible Riccati foliation $\mathcal F_0$ in $\mathcal M^{\boldsymbol\theta}_t$ defined on the trivial bundle $\mathbb P^1\times\mathbb P^1$. The sections $\sigma_d$ with self-intersection $d\ge0$ ($d$ even) form a $(d+1)$-dimensional family. The smooth curve $\sigma_d$ has exactly $d+2$ tangencies with $\mathcal F_0$ counting multiplicities. Then the tangency locus can be completely contained in the fibres over $\{0,1,t,\infty\}$ only for isolated points of the parameter space $T$. When $d= 0$, this yields the previous proposition. When $d=2$, we obtain, for example, that all four parabolics cannot lie on a curve of bidegree $(1,1)$ along the deformation.

For every exponent $\vartheta\in\mathbb C$ we have an analytic map

$$\begin{equation*} \begin{gathered} \, RH\colon \\ \biggl\{\begin{matrix}\text{Lam\'e connections over }X \\ \text{with exponent }\vartheta\end{matrix}\biggr\}/\!\sim \ \to \ \biggl\{\begin{matrix}\rho\colon\pi_1(X^*)\to\mathrm{SL}(2,\mathbb C), \\ \mathrm{tr}(\rho(\delta))=2\cos(\pi\vartheta)\end{matrix}\biggr\}/\!\sim, \\ (E,\nabla)\;\mapsto\;\rho, \end{gathered} \end{equation*}$$

which sends every Lamé connection (considered up to holomorphic bundle isomorphisms) to its monodromy representation (considered up to $\mathrm{SL}(2,\mathbb C)$-conjugacy). This map is almost one-to-one: it is surjective, and its restriction to the set of connections without apparent singular points (that is, connections with $\rho(\delta)\neq \pm I$) is injective.

If $(E,\nabla)$ has an apparent singular point at $\omega_\infty$, where $\vartheta=n\in\mathbb Z_{>0}$, then all horizontal sections have meromorphic extension to $\omega_\infty$. Holomorphic sections of this kind are contained in a line subbundle (say, $L$) of $E$ near $\omega_\infty$. In terms of the normal form (47), $L$ is the constant line bundle generated by $\begin{pmatrix}1\\0\end{pmatrix}$. Except in very special cases, $L$ cannot be extended to a line bundle $L\subset E$ on the whole of $X$: it is only defined in a neighbourhood of $\omega_\infty$. The fibre of $L$ at $\omega_\infty$ coincides with the eigenline of the residual matrix associated with the positive eigenvalue ${n}/{2}$. By the Riemann– Hilbert correspondence over the punctured curve $X^*$, any two Lamé connections with the same monodromy representation are conjugate to each other by a gauge transformation over $X^*$. This conjugacy extends to a global gauge transformation if and only if it conjugates the corresponding local line bundles as defined above. One can restore the injectivity of the Riemann– Hilbert map in the following way. Regard the monodromy representation $\rho$ as an action of the fundamental group $\pi_1(X^*,p)$ on the space $E_p\simeq\mathbb C^2$ of germs of solutions at the base point $p$. Then analytic continuation of local holomorphic solutions at $\omega_\infty$ to the base point $p$ along (half-)$\delta$ yields a one-dimensional subspace $L_p\subset E_p$. In other words, $L_p$ is obtained by analytic continuation of $L$ (as a $\nabla$-invariant line bundle) along $\delta$. The Lamé connection (with an apparent singular point) is characterized by the pair

$$\begin{equation*} (\rho,L_p)\in\mathrm{Hom}\bigl(\pi_1(X^*,p),\mathrm{SL}(E_p)\bigr)\times\mathbb P(E_p) { } \end{equation*}$$

up to conjugacy:

$$\begin{equation*} (\rho,L_p)\sim(M^{-1}\rho M,M^{-1}L_p), \qquad M\in\mathrm{SL}(2,\mathbb C). \end{equation*}$$

This is a kind of parabolic structure for the space of representations.

We recall how to describe the moduli space of representations following Fricke (see [2]). Fix standard generators $\alpha,\beta\in\pi_1(X^*)$ of the fundamental group in such a way that their commutator $\delta:=[\alpha,\beta]=\alpha\beta\alpha^{-1}\beta^{-1}$ represents a small loop turning once around the puncture, as before. We neglect the base point since it plays no role in our discussion. A representation $\rho$ is determined by the images of generators,

$$\begin{equation*} A:=\rho(\alpha)\quad\text{and}\quad B:=\rho(\beta). \end{equation*}$$

The ring of polynomial functions on $\mathrm{SL}(2,\mathbb C)\times\mathrm{SL}(2,\mathbb C)$ that are invariant under $\mathrm{SL}(2,\mathbb C)$-conjugacy, is generated by the functions

$$\begin{equation*} a:=\operatorname{tr}(A),\quad b:=\operatorname{tr}(B)\quad \text{and}\quad c:=\operatorname{tr}(AB). \end{equation*}$$

For example, the trace of the commutator $\delta=[\alpha,\beta]$ is given by

$$\begin{equation*} d:=\operatorname{tr}([A,B])=a^2+b^2+c^2-abc-2, \end{equation*}$$

whence we have

$$\begin{equation*} \rho\ \text{is reducible}\quad \Longleftrightarrow\quad a^2+b^2+c^2-abc-2=2. \end{equation*}$$

The geometric quotient of $\mathrm{Hom}(\pi_1(X_t^*),\mathrm{SL}(2,\mathbb C))$ by the action of $\mathrm{SL}(2,\mathbb C)$- conjugacy is identified with $\mathbb C^3$ by means of the composite

$$\begin{equation*} \begin{gathered} \, \mathrm{Hom}(\pi_1(X^*),\mathrm{SL}(2,\mathbb C)) \to \mathrm{SL}(2,\mathbb C)\times\mathrm{SL}(2,\mathbb C) \to\mathbb C^3, \\ \rho\mapsto(A,B)\mapsto(a,b,c). \end{gathered} \end{equation*}$$

To be precise (see [33], [2]), if $a^2+b^2+c^2-abc-2\neq 2$, then the fibre over $(a,b,c)$ consists of the single $\mathrm{SL}(2,\mathbb C)$-conjugacy class of the irreducible representation defined by the matrices

$$\begin{equation*} A=\begin{pmatrix} a&-1 \\ 1&0 \end{pmatrix} \quad\text{and}\quad B=\begin{pmatrix} 0&\gamma^{-1} \\ -\gamma&b\end{pmatrix}, \quad \text{where}\quad \gamma+\gamma^{-1}=c. \end{equation*}$$

This normal form is obtained in any basis of the form $(v,-\gamma B\cdot v)$, where $v$ is an eigenvector for the product $A\cdot B$ with eigenvalue $\gamma$. It depends only on the choice of the root $\gamma$. The commutator is therefore given by

$$\begin{equation*} [A,B]=\begin{pmatrix} -\dfrac{2\gamma^2+1}{\gamma^2}&\dfrac{a-b\gamma}{\gamma^2} \\ \dfrac{a\gamma-b}{\gamma}&\dfrac{1}{\gamma^2} \end{pmatrix}. \end{equation*}$$

One can check by a direct computation that the matrices $A$ and $B$ above share a common eigenvector if and only if $d=2$.

Corollary 15.  A Lamé connection is reducible if and only if $\vartheta\in\mathbb Z$.

The elliptic involution $\sigma\colon X\to X$, $(x,y)\mapsto(x,-y)$, acts on our moduli space by sending the $\mathrm{SL}(2,\mathbb C)$-class defined by $(A,B)$ to that defined by $(A^{-1},B^{-1})$. This may be seen by choosing a fixed point of $\sigma$ for the base point of the fundamental group. The following lemma is one of the key points in our construction.

Lemma 16.  An $\mathrm{SL}(2,\mathbb C)$-conjugacy class is stabilized by the elliptic involution $\sigma$ if and only if it consists of either irreducible or Abelian representations. In other words, for every pair $(A,B)$ generating an irreducible or Abelian subgroup of $\mathrm{SL}(2,\mathbb C)$ there is a matrix $M\in\mathrm{SL}(2,\mathbb C)$ such that

$$\begin{equation*} M^{-1}AM=A^{-1}\quad\textit{and\/}\quad M^{-1}BM=B^{-1}. \end{equation*}$$

In the irreducible case $M$ is unique up to a sign and $\operatorname{tr}(M)= 0$.

Proof.  In $\mathrm{SL}(2,\mathbb C)$ we have $\operatorname{tr}(A^{-1})=\operatorname{tr}(A)$ and

$$\begin{equation*} \operatorname{tr}(A^{-1}B^{-1})=\operatorname{tr}(B^{-1}A^{-1})=\operatorname{tr}((AB)^{-1})=\operatorname{tr}(AB), \end{equation*}$$

so that the involution acts trivially on the quotient, that is, on the triples $(a,b,c)$. Since irreducible $\mathrm{SL}(2,\mathbb C)$-classes are characterized by the corresponding triples $(a,b,c)$, they are $\sigma$-invariant. Another way to prove this is to note that the matrix $M$ in the statement of the lemma must transpose the two eigenvectors of each of the matrices $A$, $B$. We look for an element $\overline{M}$ of $\mathrm{PSL}(2,\mathbb C)$ transposing the corresponding points of $\mathbb P^1$, that is, sending a quadruple $(a_1,a_2,b_1,b_2)$ to the quadruple $(a_2,a_1,b_2,b_1)$. Since the cross-ratios of these quadruples coincide, such an element $\overline{M}$ exists. Moreover, $\overline{M}$ is an involution since its square fixes the four points and, therefore, $\operatorname{tr}(M)= 0$. One must study the degenerate cases when $a_1=a_2$ and/or $b_1=b_2$ separately; we omit this discussion. In the reducible case we have, for example, $a_1=b_1$ and $\overline{M}$ exists if and only if $a_2=b_2$, which characterizes the Abelian case. Finally, $\sigma$ interchanges the upper-triangular and lower-triangular $\mathrm{SL}(2,\mathbb C)$-representations but stabilizes the diagonal ones. $\square$

When the matrices $A$ and $B$ are in the normal form above, the matrix $M$ in the lemma is given, up to a sign, by the formula

$$\begin{equation} M=\begin{pmatrix} \dfrac{\gamma^2-1}{2\gamma}&\dfrac{a-b\gamma}{2\gamma} \\ \dfrac{a\gamma-b}{2}&-\dfrac{\gamma^2-1}{2\gamma} \end{pmatrix}. \end{equation} \tag{ 48 }$$

We resume our discussion of the non-resonant case.

Corollary 17.  Given an elliptic curve $X$ and $\vartheta\not\in\mathbb Z$, the Lamé connections on $X$ with exponent $\vartheta$ are in one-to-one correspondence with points of the smooth affine hypersurface

$$\begin{equation*} S_d:=\{(a,b,c)\in\mathbb C^3;\ a^2+b^2+c^2-abc-2=d\}, \qquad d=2\cos(\pi\vartheta). \end{equation*}$$

They are irreducible and $\sigma$-invariant.

Proof.  The connections are irreducible ($\vartheta\not\in\mathbb Z$) and have no apparent singular points. Therefore the Riemann–Hilbert correspondence is injective and the desired assertions follow from Lemma 16. $\square$

We now complete the picture by studying the case of resonant parameters $\vartheta\in\mathbb Z$.

6.3.1. $\vartheta\in\mathbb Z\setminus 2\mathbb Z$

The monodromy representation is irreducible and is characterized by the corresponding triple $(a,b,c)$. The local monodromy around $\omega_\infty$ is unipotent, with repeated eigenvalue $-1$, and is given by the commutator

$$\begin{equation*} [A,B]=\begin{pmatrix} a^2+b^2+\gamma^2-abc&\gamma^{-2}(a-b\gamma) \\ a-b\gamma^{-1}&\gamma^{-2}\end{pmatrix}, \qquad \gamma+\gamma^{-1}=c. \end{equation*}$$

We have $\operatorname{tr}([A,B])=a^2+b^2+c^2-abc-2=-2$ and $[A,B]=-I$ precisely when $(a,b,c)=(0,0,0)$, the unique singular point of the surface.

Proposition 18.  The Lamé connections with exponent $\vartheta\in\mathbb Z\setminus 2\mathbb Z$ having a logarithmic singular point are in one-to-one correspondence with the smooth points of the Markov affine hypersurface

$$\begin{equation*} S_{-2}=\{(a,b,c)\in\mathbb C^3;\ a^2+b^2+c^2-abc=0\}. \end{equation*}$$

They are irreducible and $\sigma$-invariant.

Proof.  The same as for Corollary 17. $\square$

When $(a,b,c)=(0,0,0)$, the image of the monodromy representation is the dihedral group of order $8$ (that is, quaternionic):

$$\begin{equation*} A=\begin{pmatrix}0&-1\\1&0\end{pmatrix} \quad\text{and}\quad B=\begin{pmatrix}0&i\\ i&0\end{pmatrix}, \end{equation*}$$

and the singular point $\omega_\infty$ of the connection is apparent: $[A,B]=-I$. In this case the Lamé connection is not characterized by its monodromy representation. Indeed, we have the following proposition.

Proposition 19.  The Lamé connections over the singular point $(a,b,c)=(0,0,0)$ are in one-to-one correspondence with $\mathbb P^1$. They have the same monodromy representation into the dihedral subgroup of order $8$ in $\mathrm{SL}(2,\mathbb C)$, and thus are irreducible. All of them are $\sigma$-invariant.

Proof.  The Lamé connection is determined by its monodromy representation $\rho$ (acting on the space of solutions $E_p\simeq\mathbb C^2$) and the line $L_p\subset E_p$ corresponding to the solutions holomorphic at $\omega_\infty$ after analytic continuation along $\delta$. In other words, the connection is determined by a triple

$$\begin{equation*} (A,B,L)\in\mathrm{SL}(2,\mathbb C)\times\mathrm{SL}(2,\mathbb C)\times\mathbb P^1. \end{equation*}$$

Another triple $(\tilde A,\tilde B,\tilde L)$ represents a connection gauge equivalent to the initial one if and only if

$$\begin{equation*} (\tilde A,\tilde B,\tilde L)=(M^{-1}AM,M^{-1}BM,M^{-1}L), \qquad M\in\mathrm{SL}(2,\mathbb C). \end{equation*}$$

The monodromy representation, being irreducible here, has centralizer $\pm I$ acting trivially on $\mathbb P^1$: once the monodromy representation $(A,B)$ is fixed, the gauge equivalence classes of connections are in one-to-one correspondence with $\mathbb P^1\supset L$.

One can easily check that the action of $\sigma$ on Lamé connections induces the following action on the corresponding triples:

$$\begin{equation*} \sigma\colon (A,B,L)\mapsto(A^{-1},B^{-1},(AB)^{-1}L). \end{equation*}$$

It turns out that the matrix $M$ in Lemma 16 (see (48)) takes the following form when $(a,b,c)=(0,0,0)$ (and, for example, $\gamma=i$):

$$\begin{equation*} M=A\cdot B=\begin{pmatrix}i&0\\ 0 &-i\end{pmatrix}. \end{equation*}$$

It conjugates $\sigma(A,B,L)$ to $(A,B,L)$, thus proving the $\sigma$-invariance of the corresponding connection, a kind of miracle. $\square$

Remark 20.  In fact, the moduli space of Lamé connections with a fixed exponent $\vartheta\in\mathbb Z\setminus 2\mathbb Z$ may be viewed, over the surface $S_{-2}\colon\{a^2+b^2+c^2-abc=0\}$, as the minimal resolution obtained by a single blow-up of the singular point $(0,0,0)$: the exceptional divisor stands for the set of connections with apparent singular point considered in Proposition 19. This will follow immediately from a similar result obtained in [25] for connections over $\mathbb P^1$ after our descent construction. Let us also give some direct arguments. For every point $p$ in the smooth part of the affine surface $S_{-2}$, we have a one-dimensional subspace $L_p\subset E_p$ of all solutions holomorphic at $\omega_\infty$ after analytic continuation along $\delta$. Using the local model (47) in the logarithmic case, one can check that $L_p$ coincides with an eigenspace of the local monodromy $[A,B]$, namely,

$$\begin{equation*} L=\mathbb C\cdot\begin{pmatrix}\gamma^2+1\\ -a\gamma^2+b\gamma \end{pmatrix}. \end{equation*}$$

The exceptional divisor of $S_{-2}$ is given by the equation $a^2+b^2+c^2= 0$ in the homogeneous coordinates $(a:b:c)$ and can be parametrized by

$$\begin{equation*} \mathbb P^1\hookrightarrow\mathbb P^2 ; \qquad s\mapsto (i(s^2+1):s^2-1:2s). \end{equation*}$$

One can easily verify that the line $L_p$ tends to $\mathbb C\cdot\begin{pmatrix}1\\ s\end{pmatrix}$ as the representation $\rho$ tends to the point $s$ in the parametrization above.