Real Slices of ${\rm SL}(r,\mathbb{C})$-Opers

Real Slices of $\boldsymbol{{\rm SL}(r,\mathbb{C})}$-Opers

Indranil BISWAS a, Sebastian HELLER b and Laura P. SCHAPOSNIK c

I. Biswas, S. Heller and L.P. Schaposnik

a) Department of Mathematics, Shiv Nadar University,
a) NH91, Tehsil Dadri, Greater Noida, Uttar Pradesh 201314, India
<EMAIL_ADDRESS>
b) Beijing Institute of Mathematical Sciences and Applications, Beijing
101408, P.R. China<EMAIL_ADDRESS>
c) Department of Mathematics, Statistics, and Computer Science,
c) University of Illinois at Chicago, 851 S Morgan St, Chicago, IL 60607, USA
<EMAIL_ADDRESS>
Received April 12, 2023, in final form September 05, 2023; Published online
September 16, 2023

Through the action of an anti-holomorphic involution $\sigma$ (a real
structure) on a Riemann surface $X$, we consider the induced actions on ${\rm
SL}(r,\mathbb{C})$-opers and study the real slices fixed by such actions. By
constructing this involution for different descriptions of the space of ${\rm
SL}(r,\mathbb{C})$-opers, we are able to give a natural parametrization of the
fixed point locus via differentials on the Riemann surface, which in turn
allows us to study their geometric properties.

opers; real structure; differential operator; anti-holomorphic involution;
real slice

14H60; 33C80; 53A55

##  1 Introduction

Anti-holomorphic involutions on Riemann surfaces have long been studied, and
their induced actions on associated moduli spaces (such as the moduli space of
vector bundles or Higgs bundles on them) have led to very fruitful results. In
particular, by considering the induced actions on those spaces, one can
consider the special subsets of fixed points, often leading to interesting
Lagrangians of the moduli spaces (in the case of Higgs bundles, see, for
instance, [3, 4]).

The space of ${\rm SL}(r,\mathbb{C})$-opers on $X$, which we denote by ${\rm
OSL}_{X}(r)$, admits several diverse, but equivalent, descriptions. In the
present paper, starting with an anti-holomorphic involution $\sigma$ (a real
structure) on a compact Riemann surface $X$, we construct the induced action
of ${\mathbb{Z}}/2\mathbb{Z}$ on various spaces associated to $X$:

  * •

an involution $F$ on the space of projective structures on $X$ (see Lemma
2.5),

  * •

an involution $\mathcal{A}_{1}$ induced by $F$ on the space of projective
structures on $X$ times the space of holomorphic differentials (5.14),

  * •

an involution $\beta$ on the space of holomorphic differential operators (see
Lemma 4.10),

  * •

an involution $\mathcal{B}$ on the space of vector bundles with connections
(see Lemma 4.13),

  * •

and an involution $\Gamma$ on the space of equivariant projective embeddings
of the universal cover (see Lemma 6.5).

The fixed point locus in ${\rm OSL}_{X}(r)$ for the action of
${\mathbb{Z}}/2\mathbb{Z}$ on it will be called the real slice of ${\rm
SL}(r,{\mathbb{C}})$-opers.

Along the paper, we first describe the induced involution from the point of
view of just one description of ${\rm OSL}_{X}(r)$. Then the space ${\rm
OSL}_{X}(2)$ of ${\rm SL}(2,{\mathbb{C}})$-opers on $X$ is identified with the
space of all projective structures on $X$. The anti-holomorphic involution
$\sigma$ of $X$ produces an involution of the space of projective structures
on $X$. Also $\sigma$ produces a conjugate linear involution of
$H^{0}\bigl{(}X,K^{\otimes i}_{X}\bigr{)}$ for every $i\geq 0$ by sending any
$\omega\in H^{0}\bigl{(}X,K^{\otimes i}_{X}\bigr{)}$ to
$\sigma^{*}\overline{\omega}$. On the other hand, we have an isomorphism

$\Psi\colon\ {\rm OSL}_{X}(r)\longrightarrow{\rm
OSL}_{X}(2)\times\left(\bigoplus_{i=3}^{r}H^{0}\bigl{(}X,K^{\otimes
i}_{X}\bigr{)}\right).$

######  Theorem A.

The above isomorphism $\Psi$ takes the involution of ${\rm OSL}_{X}(r)$ given
by $\sigma$ to the involution of ${\rm
OSL}_{X}(2)\times\bigl{(}\bigoplus_{i=3}^{r}H^{0}\bigl{(}X,K^{\otimes
i}_{X}\bigr{)}\bigr{)}$ given by the diagonal action of
${\mathbb{Z}}/2\mathbb{Z}$ on the product space. In particular, an element

$\left(\eta,\bigoplus_{i=3}^{r}\omega_{i}\right)\in{\rm
OSL}_{X}(2)\times\left(\bigoplus_{i=3}^{r}H^{0}\bigl{(}X,K^{\otimes
i}_{X}\bigr{)}\right)$

is in the real slice of ${\rm SL}(r,{\mathbb{C}})$-opers if and only if $\eta$
is in the real slice of ${\rm SL}(2,{\mathbb{C}})$-opers and
$\omega_{i}=\sigma^{*}\overline{\omega}_{i}$ for all $3\leq i\leq r$.

The above mentioned ${\mathbb{Z}}/2\mathbb{Z}$ actions produce involutions of
${\rm OSL}_{X}(r)$ from four different points of view, and we finish the paper
by showing that all these different involutions of ${\rm OSL}_{X}(r)$ actually
coincide, leading to our second main result.

######  Theorem B.

The four involutions $\Gamma$, $\beta$, $\mathcal{B}$ and
$\mathcal{A}=\Psi^{-1}\circ\mathcal{A}_{1}\circ\Psi$ of ${\rm OSL}_{X}(r)$
coincide.

##  2 Projective structures

###  2.1 Involution of the space of projective structures

The standard action of the group ${\rm GL}(2,{\mathbb{C}})$ on
${\mathbb{C}}^{2}$ produces an action of ${\rm PGL}(2,{\mathbb{C}})$ on the
complex projective line ${\mathbb{C}}{\mathbb{P}}^{1}$, giving an
identification of ${\rm PGL}(2,{\mathbb{C}})$ with the holomorphic
automorphism group ${\rm Aut}\bigl{(}{\mathbb{C}}{\mathbb{P}}^{1}\bigr{)}$ of
${\mathbb{C}}{\mathbb{P}}^{1}$. Through this identification, one can define a
projective structure on an oriented $C^{\infty}$ surface in the following
fashion.

Let $S$ be a compact oriented $C^{\infty}$ surface of genus $g$, with $g\geq
2$. Recall that a coordinate chart on $S$ is a pair $(U,\varphi)$, where
$U\subset S$ is an open subset and $\varphi\colon
U\longrightarrow{\mathbb{C}}{\mathbb{P}}^{1}$ is an orientation preserving
$C^{\infty}$ embedding, and a coordinate atlas is a collection of coordinate
charts $\\{(U_{i},\varphi_{i})\\}_{i\in I}$ such that $\bigcup_{i\in
I}U_{i}=S$.

######  Definition 2.1.

A projective structure on $S$ is an equivalence class of
$\\{(U_{i},\allowbreak\varphi_{i})\\}_{i\in I}$ such that for all $i,j\in I$,
and every nonempty connected component $U^{c}_{ij}\subset U_{i}\cap U_{j}$,
there is an element

$T^{c}_{ij}\in{\rm PGL}(2,{\mathbb{C}})$ (2.1)

such that the restriction of the maps $\varphi_{i}\circ\varphi^{-1}_{j}$ to
$\varphi_{j}(U^{c}_{ij})$ coincides with the restriction of the automorphism
$T^{c}_{ij}\in{\rm Aut}\bigl{(}{\mathbb{C}}{\mathbb{P}}^{1}\bigr{)}$. Two
coordinate atlases are called equivalent if their union also satisfies the
condition (2.1) on the transition functions.

Note that the condition (2.1) above uniquely determines the element
$T^{c}_{ij}$. Also a projective structure on $S$ defines a complex structure
on $S$ because the automorphisms of ${\mathbb{C}}{\mathbb{P}}^{1}$ given by
the elements of ${\rm PGL}(2,{\mathbb{C}})$ are holomorphic. In what follows
we shall denote by

$\displaystyle\widetilde{\mathcal{P}}(S)$ (2.2)

the space of all projective structures on $S$.

A projective structure on a Riemann surface $X$ is a projective structure
$P_{0}$ on the underlying $C^{\infty}$ surface $X_{0}$ such that the complex
structure on $X_{0}$ given by $P_{0}$ coincides with the complex structure
defining $X$. Consider the orientation reversing involution

$\displaystyle\sigma_{0}\colon\ {\mathbb{C}}{\mathbb{P}}^{1}$
$\displaystyle\longrightarrow{\mathbb{C}}{\mathbb{P}}^{1},$
$\displaystyle\quad(z_{1},z_{2})$
$\displaystyle\longmapsto(\overline{z}_{1},\overline{z}_{2}),$ (2.3)

for $z_{i}\in\mathbb{C}$, and let

$\sigma\colon\ S\longrightarrow S$ (2.4)

be an orientation reversing diffeomorphism of $S$ such that
$\sigma\circ\sigma={\rm Id}_{S}$.

######  Lemma 2.2.

Given a projective structure $P$ on $S$ defined by a coordinate atlas

$\\{(U_{i},\varphi_{i})\\}_{i\in I},$

the corresponding atlas

$\big{\\{}\bigl{(}\sigma^{-1}(U_{i}),\sigma_{0}\circ\varphi_{i}\circ\sigma\bigr{)}\big{\\}}_{i\in
I}$

also defines a projective structure on $S$, where $\sigma_{0}$ and $\sigma$
are the involutions in (2.3) and (2.4) respectively. This produces an
involution of the space $\widetilde{\mathcal{P}}(S)$ in (2.2).

###### Proof.

Given any $i,j\in I$ and any nonempty connected component $U^{c}_{ij}\subset
U_{i}\cap U_{j}$, the composition of maps on
$\sigma_{0}\bigl{(}\varphi_{j}(U^{c}_{ij})\bigr{)}$

$\sigma_{0}\circ\varphi_{i}\circ\sigma\circ(\sigma_{0}\circ\varphi_{j}\circ\sigma)^{-1}=\sigma_{0}\circ\bigl{(}\varphi_{i}\circ\varphi^{-1}_{j}\bigr{)}\circ\sigma^{-1}_{0}=\sigma_{0}\circ\bigl{(}\varphi_{i}\circ\varphi^{-1}_{j}\bigr{)}\circ\sigma_{0}$

is holomorphic, and it is the restriction of the global holomorphic
automorphism $\sigma_{0}\circ T^{c}_{ij}\circ\sigma_{0}$ of
${\mathbb{C}}{\mathbb{P}}^{1}$. If $\\{(U_{j},\varphi_{j})\\}_{j\in J}$ is
equivalent to $\\{(U_{i},\varphi_{i})\\}_{i\in I}$, then
$\\{(\sigma(U_{j}),\sigma_{0}\circ\varphi_{j}\circ\sigma)\\}_{j\in J}$ is also
equivalent to
$\\{(\sigma(U_{i}),\sigma_{0}\circ\varphi_{i}\circ\sigma)\\}_{i\in I}$.
Consequently, we get a map on the space of all projective structures on $S$

$\widetilde{\sigma}\colon\
\widetilde{\mathcal{P}}(S)\longrightarrow\widetilde{\mathcal{P}}(S)$ (2.5)

that sends any projective structure defined by a coordinate atlas
$\\{(U_{i},\varphi_{i})\\}_{i\in I}$ to the one defined by the coordinate
atlas $\\{(\sigma(U_{i}),\sigma_{0}\circ\varphi_{i}\circ\sigma)\\}_{i\in I}$.
From the construction of $\widetilde{\sigma}$ it follows immediately that
$\widetilde{\sigma}\circ\widetilde{\sigma}={\rm
Id}_{\widetilde{\mathcal{P}}(S)}$. ∎

######  Remark 2.3.

The map $\widetilde{\sigma}$ defined in (2.5) is independent of the choice of
the anti-holomorphic involution $\sigma_{0}$ of
${\mathbb{C}}{\mathbb{P}}^{1}$, but in general it depends on $\sigma$. Indeed,
in its construction, one may replace $\sigma_{0}$ (defined in (2.3)) by any
other anti-holomorphic involution $\sigma^{\prime}_{0}$ of
${\mathbb{C}}{\mathbb{P}}^{1}$ without changing the map $\widetilde{\sigma}$
because $\sigma^{\prime}_{0}$ and $\sigma_{0}$ differ by an element of ${\rm
PGL}(2,{\mathbb{C}})={\rm Aut}\bigl{(}{\mathbb{C}}{\mathbb{P}}^{1}\bigr{)}$.

We shall denote the group of orientation preserving diffeomorphisms of $S$ by
${\rm Diff}^{+}(S)$. Let

${\rm Diff}^{0}(S)\subset{\rm Diff}^{+}(S)$

be the connected component of it consisting of diffeomorphisms homotopic to
the identity map of $S$; it is a normal subgroup of ${\rm Diff}^{+}(S)$. The
mapping class group can then be seen as the quotient

${\rm MCG}(S):={\rm Diff}^{+}(S)/{\rm Diff}^{0}(S).$

Moreover, considering the space ${\mathcal{C}}(S)$ of complex structures on
$S$ compatible with its $C^{\infty}$ structure and orientation, the
Teichmüller space can be seen as the quotient

${\mathcal{T}}(S):={\mathcal{C}}(S)/{\rm Diff}^{0}(S),$ (2.6)

which is a complex manifold of dimension $3(g-1)$.

The group ${\rm Diff}^{+}(S)$ acts on $\widetilde{\mathcal{P}}(S)$ as follows.
Let $P$ be a projective structure on $S$ given by
$\\{(U_{i},\varphi_{i})\\}_{i\in I}$. Then the action of any $\psi\in{\rm
Diff}^{+}(S)$ sends it to the projective structure $\psi\cdot P$ defined by
$\big{\\{}\bigl{(}\psi(U_{i}),\varphi_{i}\circ\psi^{-1}\bigr{)}\big{\\}}_{i\in
I}$. Consider the corresponding quotient space

${\mathcal{P}}(S):=\widetilde{\mathcal{P}}(S)/{\rm Diff}^{0}(S),$ (2.7)

which is a complex manifold of dimension
$\dim{\mathcal{P}}(S)=6(g-1)=2\dim{\mathcal{T}}(S)$. Since a projective
structure on $S$ produces a complex structure on $S$, we have a map

$\Phi\colon\ {\mathcal{P}}(S)\longrightarrow{\mathcal{T}}(S),$ (2.8)

where ${\mathcal{P}}(S)$ and ${\mathcal{T}}(S)$ are constructed in (2.7) and
(2.6), respectively.

######  Remark 2.4.

Any Riemann surface admits a projective structure. For example, there is a
natural projective structure given by the uniformization theorem.
Consequently, the map $\Phi$ in (2.8) is surjective. In fact, $\Phi$ is a
holomorphic map, and ${\mathcal{P}}(S)$ is a holomorphic torsor over
${\mathcal{T}}(S)$ for the holomorphic cotangent bundle
$T^{*}{\mathcal{T}}(S)$ (see [15, 16, 19]).

######  Lemma 2.5.

The map $\widetilde{\sigma}$ in (2.5) produces an anti-holomorphic involution

$F\colon\ {\mathcal{P}}(S)\longrightarrow{\mathcal{P}}(S).$ (2.9)

###### Proof.

Since ${\rm Diff}^{0}(S)$ is the connected component of ${\rm Diff}(S)$
containing the identity element, it is a normal subgroup of ${\rm Diff}(S)$.
Moreover, ${\rm Diff}^{+}(S)$ is also a normal subgroup of ${\rm Diff}(S)$,
and ${\rm Diff}(S)/{\rm Diff}^{+}(S)={\mathbb{Z}}/2{\mathbb{Z}}$. Take any
$\psi\in{\rm Diff}^{0}(S)$ and any projective structure $P$ on $S$ given by
$\\{(U_{i},\varphi_{i})\\}_{i\in I}$. Then the projective structure
$\widetilde{\sigma}(\psi\cdot P)$ is given by
$\big{\\{}\bigl{(}\sigma\circ\psi(U_{i}),\sigma_{0}\circ\varphi_{i}\circ\psi^{-1}\circ\sigma\bigr{)}\big{\\}}_{i\in
I}$. Therefore, we have

$\widetilde{\sigma}(\psi\cdot
P)=(\sigma\circ\psi\circ\sigma)\cdot\widetilde{\sigma}(P).$

Since $\sigma\circ\psi\circ\sigma\in{\rm Diff}^{0}(S)$ (recall that ${\rm
Diff}^{0}(S)$ is a normal subgroup), we conclude $\widetilde{\sigma}$ produces
a self-map $F$ of ${\mathcal{P}}(S)$. This map $F$ is evidently an involution.
Also $F$ is clearly anti-holomorphic.∎

For any $C^{\infty}$ complex structure $J\colon TS\longrightarrow TS$ on $S$
compatible with its orientation, the pullback $-\sigma^{*}J$ is a $C^{\infty}$
complex structure on $S$ compatible with its orientation, and thus there is an
induced anti-holomorphic involution

$\displaystyle f\colon\ {\mathcal{T}}(S)$
$\displaystyle\longrightarrow{\mathcal{T}}(S),$ $\displaystyle J$
$\displaystyle\longmapsto-\sigma^{*}J.$ (2.10)

For $f$ and $F$ the involutions constructed in (2.10) and (2.9), respectively,
one can see that

$\Phi\circ F=f\circ\Phi,$ (2.11)

where $\Phi$ is the projection in (2.8). Given any Riemann surface
$X\\!\\!\in\\!{\mathcal{T}}(S)$, the fiber
${\Phi^{-1}(X)\\!\subset\\!{\mathcal{P}}(S)}$ is an affine space for
$H^{0}\bigl{(}X,K^{\otimes 2}_{X}\bigr{)}$, where $K_{X}$ is the holomorphic
cotangent bundle of $X$ [15, 19]. Moreover, using the isomorphism

$T^{*}_{X}{\mathcal{T}}(S)=H^{0}\bigl{(}X,K^{\otimes 2}_{X}\bigr{)},$

${\mathcal{P}}(S)$ is realized as a holomorphic affine bundle over
${\mathcal{T}}(S)$ for the holomorphic vector bundle $T^{*}{\mathcal{T}}(S)$
[15, 19]. The involution $F$ satisfies the equation

$F(P+\omega)=F(P)+\sigma^{*}\overline{\omega}$ (2.12)

for all $P\in\Phi^{-1}(X)$ and $\omega\in H^{0}\bigl{(}X,K^{\otimes
2}_{X}\bigr{)}$, where $\sigma^{*}\overline{\omega}$ is the corresponding
holomorphic quadratic differential on the Riemann surface
$f(X)\in{\mathcal{T}}(S)$.

##  3 Real slices of the character variety and Teichmüller space

In what follows, we shall study the induced real slices of the character
variety and Teichmüller space obtained through the anti-holomorphic
involutions introduced in the previous section.

###  3.1 The character variety and symplectic form

As before, $S$ is a compact oriented surface of genus at least two. Given a
base point $x_{0}\in S$, recall that a homomorphism
$\rho\colon\pi_{1}(S,x_{0})\longrightarrow\operatorname{PSL}(2,{\mathbb{C}})$
is called irreducible if the image of $\rho$ is not contained in any Borel
subgroup of $\operatorname{PSL}(2,{\mathbb{C}})$. Let

$\operatorname{Hom}(\pi_{1}(S,x_{0}),\operatorname{PSL}(2,{\mathbb{C}}))^{\rm
ir}\subset\operatorname{Hom}(\pi_{1}(S,x_{0}),\operatorname{PSL}(2,{\mathbb{C}}))$

be the space of all irreducible homomorphisms.

######  Definition 3.1.

The irreducible ${\rm PSL}(2,{\mathbb{C}})$-character variety of $S$ is the
quotient space

${\mathcal{R}}_{2}(S):=\operatorname{Hom}\left(\pi_{1}(S,x_{0}),\operatorname{PSL}(2,{\mathbb{C}})\right)^{\rm
ir}/\operatorname{PSL}(2,{\mathbb{C}})$ (3.1)

for the action of $\operatorname{PSL}(2,{\mathbb{C}})$ on
$\operatorname{Hom}\left(\pi_{1}(S,x_{0}),\operatorname{PSL}(2,{\mathbb{C}})\right)^{\rm
ir}$ given by the conjugation action of $\operatorname{PSL}(2,{\mathbb{C}})$
on itself.

It should be mentioned that ${\mathcal{R}}_{2}(S)$ is independent of the
choice of the base point $x_{0}$. In fact, ${\mathcal{R}}_{2}(S)$ parametrizes
the isomorphism classes of flat
$\operatorname{PSL}(2,{\mathbb{C}})$-connections on $S$.

The space ${\mathcal{R}}_{2}(S)$ has two connected components, which are
parametrized by the second Stiefel–Whitney class of the projective bundles
associated to the $\operatorname{PSL}(2,{\mathbb{C}})$-bundles on $S$. Also
each connected component of ${\mathcal{R}}_{2}(S)$ is an irreducible complex
affine variety of dimension $6(g-1)$. We note that its algebraic structure is
given by the algebraic structure of $\operatorname{PSL}(2,{\mathbb{C}})$ and
the fact that the group $\pi_{1}(S,x_{0})$ is finitely presented. Moreover,
the character variety ${\mathcal{R}}_{2}(S)$ is a smooth orbifold and is
equipped with an algebraic symplectic structure [2, 12], which we shall denote
by

$\Theta_{2}\in
H^{0}\bigl{(}{\mathcal{R}}_{2}(S),\Omega^{2}_{{\mathcal{R}}_{2}(S)}\bigr{)}.$
(3.2)

The involution $\sigma$ of $S$ in (2.4) induces an algebraic involution

$\displaystyle I_{2}\colon\ {\mathcal{R}}_{2}(S)$
$\displaystyle\longrightarrow{\mathcal{R}}_{2}(S),$ $\displaystyle(E,\nabla)$
$\displaystyle\longmapsto(\sigma^{*}E,\sigma^{*}{\nabla}),$

where $\nabla$ is a flat connection on a principal
$\operatorname{PSL}(2,{\mathbb{C}})$-bundle $E$, equivalently, $I_{2}$ sends
any homomorphism

$\rho\colon\
\pi_{1}(S,x_{0})\longrightarrow\operatorname{PSL}(2,{\mathbb{C}})$

to the homomorphism
$\pi_{1}(S,\sigma(x_{0}))\longrightarrow\operatorname{PSL}(2,{\mathbb{C}})$
that maps any $\gamma\in\pi_{1}(S,\sigma(x_{0}))$ to

$\rho(\sigma(\gamma))\in\operatorname{PSL}(2,{\mathbb{C}}).$

The anti-holomorphic involution $A\longmapsto\overline{A}$ of
$\operatorname{PSL}(2,{\mathbb{C}})$ defines an anti-holomorphic involution of
the representation space
$\operatorname{Hom}\left(\pi_{1}(S,x_{0}),\operatorname{PSL}(2,{\mathbb{C}})\right)^{\rm
ir}$; this anti-holomorphic involution

$\operatorname{Hom}(\pi_{1}(S,x_{0}),\operatorname{PSL}(2,{\mathbb{C}}))^{\rm
ir}\longrightarrow\operatorname{Hom}(\pi_{1}(S,x_{0}),\operatorname{PSL}(2,{\mathbb{C}}))^{\rm
ir}$

in turn produces an anti-holomorphic involution

$b\colon\ {\mathcal{R}}_{2}(S)\longrightarrow{\mathcal{R}}_{2}(S)$ (3.3)

of ${\mathcal{R}}_{2}(S)$. It is straightforward to check that $b\circ
I_{2}=I_{2}\circ b$, and hence the composition

$h:=b\circ I_{2}\colon\
{\mathcal{R}}_{2}(S)\longrightarrow{\mathcal{R}}_{2}(S)$ (3.4)

is an anti-holomorphic involution of ${\mathcal{R}}_{2}(S)$ (recall that the
map $I_{2}$ is holomorphic).

######  Lemma 3.2.

The symplectic form $\Theta_{2}$ in (3.2) satisfies the equation

$h^{*}\Theta_{2}=-\overline{\Theta}_{2},$

where $h$ is the involution in (3.4).

###### Proof.

We have $I^{*}_{2}\Theta_{2}=-\Theta_{2}$, because $\sigma$ is an orientation
reversing involution. We also have $b^{*}\Theta_{2}=\overline{\Theta}_{2}$. In
view of (3.4), these two together imply that
$h^{*}\Theta_{2}=-\overline{\Theta}_{2}$. ∎

Giving a projective structure on a Riemann surface $X$ is equivalent to giving
a holomorphic fiber bundle $p\colon Z\longrightarrow X$, a holomorphic
connection $\nabla$ on the fiber bundle and a holomorphic section $s\colon
X\longrightarrow Z$ of $p$, such that

  * •

the fibers of $p$ are isomorphic to ${\mathbb{C}}{\mathbb{P}}^{1}$, and

  * •

$s$ is transversal to the horizontal distribution for the connection $\nabla$

(see [15, 19] for details). Therefore the fiber bundle $Z$ produces a
holomorphic principal $\operatorname{PSL}(2,{\mathbb{C}})$-bundle
$\mathcal{P}$ over $X$; the fiber of $\mathcal{P}$ over any $x\in X$ is the
space of all holomorphic isomorphisms from ${\mathbb{C}}{\mathbb{P}}^{1}$ to
the fiber $p^{-1}(x)$. The connection $\nabla$ produces a holomorphic
connection on $\mathcal{P}$. The monodromy representation
$\pi_{1}(X,x_{0})\longrightarrow\operatorname{PSL}(2,{\mathbb{C}})$ for
$\nabla$ is irreducible.

######  Lemma 3.3.

The symplectic form $\Theta_{2}$ on the character variety
${\mathcal{R}}_{2}(S)$ defined in (3.2) induces a natural symplectic form
$\Theta_{\mathcal{P}}$ on the quotient space of projective structures
${\mathcal{P}}(S)$ introduced in (2.7).

###### Proof.

We have seen above that a projective structure on $S$ defines a flat
$\operatorname{PSL}(2,{\mathbb{C}})$-connection on $S$. Therefore, assigning
the monodromy representation to the flat connections we get a map

$\mu\colon\ {\mathcal{P}}(S)\longrightarrow{\mathcal{R}}_{2}(S),$ (3.5)

from the quotient space of projective structures ${\mathcal{P}}(S)$ introduced
in (2.7) to the $\operatorname{PSL}(2,{\mathbb{C}})$-character variety
${\mathcal{R}}_{2}(S)$ constructed in (3.1). It is known that $\mu$ is a local
biholomorphism (see [19, p. 272] and [16]) and thus the pullback

$\Theta_{\mathcal{P}}:=\mu^{*}\Theta_{2}\in
H^{0}\bigl{(}{\mathcal{P}}(S),\Omega^{2}_{{\mathcal{P}}(S)}\bigr{)}$ (3.6)

of the symplectic form in (3.2) is a holomorphic symplectic form on
${\mathcal{P}}(S)$.∎

######  Remark 3.4.

From the constructions of $F$ and $h$ in (2.9) and (3.4) respectively, it
follows that

$\mu\circ F=h\circ\mu,$ (3.7)

where $\mu$ is the monodromy map in (3.5).

######  Corollary 3.5.

The symplectic form $\Theta_{\mathcal{P}}$ in (3.6) satisfies the equation

$F^{*}\Theta_{\mathcal{P}}=-\overline{\Theta}_{\mathcal{P}}.$

###### Proof.

From (3.7) we have
$F^{*}\Theta_{\mathcal{P}}=F^{*}\mu^{*}\Theta_{2}=\mu^{*}h^{*}\Theta_{2}$.
Therefore, from Lemma 3.2 and (3.6) it follows that
$F^{*}\Theta_{\mathcal{P}}=-\mu^{*}\overline{\Theta}_{2}=-\overline{\Theta}_{\mathcal{P}}$.
∎

###  3.2 Real projective structures

A projective structure $P\in{\mathcal{P}}(S)$ is called real if it is a fixed
point of the anti-holomorphic involution

$F\colon\ {\mathcal{P}}(S)\longrightarrow{\mathcal{P}}(S)$

introduced in (2.9). Moreover, given a real projective structure
$P\in{\mathcal{P}}(S)$, from (2.11) one obtains an induced real point
$\Phi(P)\in{\mathcal{T}}(S)$ with respect to the anti-holomorphic involution
$f$, meaning

$f(\Phi(P))=\Phi(P).$

The following lemma is a sort of converse of it.

######  Lemma 3.6.

Given any $J\in{\mathcal{T}}(S)$ such that $f(J)=J$, there exists a real
projective structure $P$ such that $\Phi(P)=J$.

###### Proof.

Let $P$ be the projective structure given by the uniformization of the Riemann
surface $(S,J)$. Then it is evident that $P$ is real and satisfies the
equation $\Phi(P)=J$. ∎

In what follows we shall denote by
${\mathcal{T}}(S)^{f}\subset{\mathcal{T}}(S)$ and
${\mathcal{P}}(S)^{F}\subset{\mathcal{P}}(S)$ the fixed point loci for the
involutions $f$ and $F$ respectively. The fixed point locus
${\mathcal{P}}(S)^{F}$ (respectively, ${\mathcal{T}}(S)^{f}$) is a
$C^{\infty}$ real submanifold of ${\mathcal{P}}(S)$ (respectively,
${\mathcal{T}}(S)$) of dimension $6(g-1)$ (respectively, $3(g-1)$). Let

$\Phi_{0}\colon\ {\mathcal{P}}(S)^{F}\longrightarrow{\mathcal{T}}(S)^{f}$
(3.8)

be the restriction of the map $\Phi$ in (2.8); as noted above, from (2.11) it
follows immediately that $\Phi$ maps ${\mathcal{P}}(S)^{F}$ to
${\mathcal{T}}(S)^{f}$. From Lemma 3.6, it follows that the map $\Phi_{0}$ is
surjective.

The differential ${\rm d}f$ of the anti-holomorphic involution $f$ of
${\mathcal{T}}(S)$ produces an anti-holomorphic involution of the holomorphic
cotangent bundle $T^{*}{\mathcal{T}}(S)$. Let

$T^{*}{\mathcal{T}}(S)^{f}\subset T^{*}{\mathcal{T}}(S)$

be the fixed point locus of this involution. This $T^{*}{\mathcal{T}}(S)^{f}$
is a $C^{\infty}$ real vector bundle over ${\mathcal{T}}(S)^{f}$. We note that
$T^{*}{\mathcal{T}}(S)^{f}$ is identified with the (real) cotangent bundle
$T^{*}\big{(}{\mathcal{T}}(S)^{f}\big{)}$ of ${\mathcal{T}}(S)^{f}$.

######  Lemma 3.7.

The space ${\mathcal{P}}(S)^{F}$ is an affine bundle over
${\mathcal{T}}(S)^{f}$ for the real vector bundle
$T^{*}{\mathcal{T}}(S)^{f}=T^{*}\bigl{(}{\mathcal{T}}(S)^{f}\bigr{)}$.

###### Proof.

Given any $J\in{\mathcal{T}}(S)^{f}$, let $X$ be the corresponding Riemann
surface $(S,J)$, and denote the holomorphic cotangent bundle of $X$ by
$K_{X}$. Let

$H^{0}\bigl{(}X,K^{\otimes 2}_{X}\bigr{)}^{\sigma}\subset
H^{0}\bigl{(}X,K^{\otimes 2}_{X}\bigr{)}$

be the fixed point locus of the conjugate linear involution
$w\longmapsto\sigma^{*}\overline{w}$ of $H^{0}\bigl{(}X,K^{\otimes
2}_{X}\bigr{)}$. The fiber over $X$ of the map
$\Phi_{0}\colon\mathcal{P}(S)^{F}\longrightarrow{\mathcal{T}}(S)^{f}$ in (3.8)
is actually an affine space for the real vector space
$H^{0}\bigl{(}X,K^{\otimes 2}_{X}\bigr{)}^{\sigma}$. Indeed, this follows
immediately from the combination of the fact that the space of all projective
structures on the Riemann surface $X$ is an affine space, for the complex
vector space $H^{0}\bigl{(}X,K^{\otimes 2}_{X}\bigr{)}$, and (2.12). We note
that

$T^{*}_{X}\bigl{(}{\mathcal{T}}(S)^{f}\bigr{)}=H^{0}\bigl{(}X,K^{\otimes
2}_{X}\bigr{)}^{\sigma}.$

Consequently, ${\mathcal{P}}(S)^{F}$ is an affine bundle over
${\mathcal{T}}(S)^{f}$ for the real vector bundle
$T^{*}\bigl{(}{\mathcal{T}}(S)^{f}\bigr{)}$ as needed. ∎

######  Proposition 3.8.

The manifold ${\mathcal{P}}(S)^{F}$ is connected, and it is homeomorphic to
the Euclidean space.

###### Proof.

The manifold ${\mathcal{T}}(S)^{f}$ is connected, and it is homeomorphic to
the Euclidean space (see [10, p. 122] and [20]). On the other hand,
${\mathcal{P}}(S)^{F}$ is an affine bundle over ${\mathcal{T}}(S)^{f}$ for the
vector bundle $T^{*}{\mathcal{T}}(S)^{f}$ from Lemma 3.7, and hence the
proposition follows. ∎

Since $F$ is an anti-holomorphic involution, Corollary 3.5 gives the
following.

######  Corollary 3.9.

Let $\Theta^{F}_{\mathcal{P}}$ be the restriction of the holomorphic
symplectic form $\Theta_{\mathcal{P}}$ to ${\mathcal{P}}(S)^{F}$. Then the
real part $\operatorname{Re}\bigl{(}\Theta^{F}_{\mathcal{P}}\bigr{)}$
satisfies the equation

$\operatorname{Re}\bigl{(}\Theta^{F}_{\mathcal{P}}\bigr{)}=0,$

and the imaginary part
$\operatorname{Im}\bigl{(}\Theta^{F}_{\mathcal{P}}\bigr{)}$ is a real
symplectic form on ${\mathcal{P}}(S)^{F}$.

###  3.3 Projective structures with Fuchsian monodromy

It should be mentioned that the notion of real projective structure is used in
the literature also in a different context than ours above, see, for example,
[11] and also [17, 21, 22]. In these papers, a real projective structure is a
projective structure with Fuchsian monodromy. In our notation, projective
structures with Fuchsian monodromy are given by certain components of

$\\{P\in\mathcal{P}(S)\mid\mu(P)=b\circ\mu(P)\\},$

where $b$ is the involution in (3.3). Examples are given by the uniformization
of (any) Riemann surface, but there are others obtained by grafting. In fact,
it was shown by Goldman [13] that any projective structure with Fuchsian
monodromy is obtained by the process of $2\pi$-grafting of the projective
structure obtained from the uniformization of an appropriate Riemann surface.

##  4 Real and quaternionic $\boldsymbol{{\rm SL}(r,\mathbb{C})}$-opers

Having studied anti-holomorphic involutions on character varieties, projective
structures and Teichmüller space, we shall now consider the induced actions on
different moduli spaces related to them, and in particular, on ${\rm
SL}(r,\mathbb{C})$-opers.

###  4.1 Real and quaternionic bundles

As in the previous sections, consider a pair $(X,\sigma_{X})$ of a compact
connected Riemann surface $X$ of genus $g\geq 2$, and an anti-holomorphic
involution

$\sigma_{X}\colon\ X\longrightarrow X.$ (4.1)

For any complex vector bundle $E$ on $X$, let $\overline{E}$ denote the
complex vector bundle on $X$ whose underlying real vector bundle is identified
with $E$, while the multiplication by $\sqrt{-1}$ on $\overline{E}$ coincides
with the multiplication by $-\sqrt{-1}$ on $E$.

######  Lemma 4.1.

For any holomorphic vector bundle $V$ on $X$, there is a holomorphic structure
on the $C^{\infty}$ vector bundle $\sigma^{*}_{X}\overline{V}$ given by the
holomorphic structure on $V$.

###### Proof.

We shall construct the holomorphic structure on $\sigma^{*}_{X}\overline{V}$
in the following way. Consider a $C^{\infty}$ section $s$ of
$\sigma^{*}_{X}\overline{V}$ defined over an open subset $U\subset X$. So $s$
gives a $C^{\infty}$ section of $\overline{V}$ over $\sigma_{X}(U)$. As
$V=\overline{V}$ as real bundles, this section of $\overline{V}$ over
$\sigma_{X}(U)$ gives a $C^{\infty}$ section of $V$, which we denote by
$\widetilde{s}$, over $\sigma_{X}(U)$. The holomorphic structure on
$\sigma^{*}_{X}\overline{V}$ is defined by the following condition: Any
section $s$ of $\sigma^{*}_{X}\overline{V}$ as above is holomorphic if and
only if the corresponding section $\widetilde{s}$ of $V$ is holomorphic.∎

Recall that a theta characteristic on a compact connected Riemann surface $Y$
is a holomorphic line bundle $K^{1/2}_{Y}$ on $Y$ such that
$K^{1/2}_{Y}\otimes K^{1/2}_{Y}$ is holomorphically isomorphic to the
holomorphic cotangent bundle $K_{Y}$. There are exactly $2^{2k}$ theta
characteristics on $Y$, where $k$ is the genus of $Y$.

It is known that there are theta characteristics $K^{1/2}_{X}$ on $X$ such
that $\sigma^{*}_{X}\overline{K^{1/2}_{X}}$ is holomorphically isomorphic to
$K^{1/2}_{X}$ (see [1, p. 61, Remark 2]).

Take a holomorphic line bundle $\mathbf{L}$ on $X$ such that
$\sigma^{*}_{X}\overline{\mathbf{L}}$ is holomorphically isomorphic to
$\mathbf{L}$. For any holomorphic isomorphism
$h\colon{\mathbf{L}}\longrightarrow\sigma^{*}_{X}\overline{\mathbf{L}}$, the
composition of homomorphisms $\bigl{(}\sigma^{*}_{X}\overline{h}\bigr{)}\circ
h$ is a holomorphic automorphism of ${\mathbf{L}}$. Multiplying $h$ by a
suitable nonzero complex number, we may ensure that
$\bigl{(}\sigma^{*}_{X}\overline{h}\bigr{)}\circ h$ is either the identity map
${\rm Id}_{\mathbf{L}}$ of $\mathbf{L}$ or it is $-{\rm Id}_{\mathbf{L}}$. The
holomorphic line bundle $\mathbf{L}$ is called real (respectively,
quaternionic) if $\bigl{(}\sigma^{*}_{X}\overline{h}\bigr{)}\circ h$ can be
made to be ${\rm Id}_{\mathbf{L}}$ (respectively, $-{\rm Id}_{\mathbf{L}}$);
see [9, p. 206, Definition 3.3].

Given a theta characteristic $K^{1/2}_{X}$ on $X$ such that
$\sigma^{*}_{X}\overline{K^{1/2}_{X}}$ is holomorphically isomorphic to
$K^{1/2}_{X}$, since $\text{degree}\bigl{(}K^{1/2}_{X}\bigr{)}=g-1$, the
following two hold:

  * (1)

Assume that $\sigma_{X}$ does not have any fixed point. If $g$ is even, then
$K^{1/2}_{X}$ must be quaternionic because
$\text{degree}\bigl{(}K^{1/2}_{X}\bigr{)}$ is odd [9, p. 210, Proposition
4.2].

  * (2)

Assume that $\sigma_{X}$ has fixed points. Then $K^{1/2}_{X}$ must be real
(see [9, p. 210, Section 4.2]).

######  Definition 4.2.

For simplicity, we shall denote the dual line bundle
$\bigl{(}K^{1/2}_{X}\bigr{)}^{*}$ by $\mathcal{L}$, and thus one has that
$\mathcal{L}$ is isomorphic to $\sigma^{*}_{X}\overline{\mathcal{L}}$. We note
that $\mathcal{L}$ is real (respectively, quaternionic) if and only if
$K^{1/2}_{X}$ is real (respectively, quaternionic).

###  4.2 Definition of opers

We shall briefly recall here the construction of the jet bundles $J^{m}(W)$,
$m\geq 0$, of a holomorphic vector bundle $W$ on $X$. For $i=1,2$, let

$p_{i}\colon\ X\times X\longrightarrow X$ (4.2)

be the projection to the $i$-th factor, and let

$\Delta:=\\{(x,x)\in X\times X\mid x\in X\\}\subset X\times X$ (4.3)

be the reduced diagonal divisor. Then for any $m\geq 0$, one can define the
associated Jet bundle as

$J^{m}(W):=p_{1*}((p^{*}_{2}W)/(p^{*}_{2}W\otimes{\mathcal{O}}_{X\times
X}(-(m+1)\Delta)))\longrightarrow X.$ (4.4)

The vector bundle $J^{m}(W)$ fits in the short exact sequence of holomorphic
vector bundles

$0\longrightarrow K^{\otimes(m+1)}_{X}\otimes W\longrightarrow
J^{m+1}(W)\longrightarrow J^{m}(W)\longrightarrow 0.$ (4.5)

Using (4.4), it is straightforward to deduce that there is a natural
holomorphic isomorphism

$\sigma^{*}_{X}\overline{J^{m}(W)}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}J^{m}\bigl{(}\sigma^{*}_{X}\overline{W}\bigr{)}.$
(4.6)

######  Lemma 4.3.

Take a holomorphic line bundle $\mathcal{L}$ on $X$ such that
${\mathcal{L}}\otimes{\mathcal{L}}=TX$. The determinant line bundle $\det{\rm
Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}$ of the $(r-1)$-th symmetric
product is

$\det{\rm
Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}=\bigwedge\nolimits^{r}{\rm
Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}={\mathcal{O}}_{X}.$ (4.7)

###### Proof.

From (4.5), it follows that the line bundle $\det
J^{1}(\mathcal{L}):=\bigwedge^{2}J^{1}(\mathcal{L})$ is

$\det
J^{1}(\mathcal{L})=K_{X}\otimes{\mathcal{L}}\otimes{\mathcal{L}}=K_{X}\otimes
TX={\mathcal{O}}_{X}.$ (4.8)

Since $\det{\rm Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}=\bigl{(}\det
J^{1}(\mathcal{L})\bigr{)}^{\otimes r(r-1)/2}$, the isomorphism in (4.7)
follows from (4.8) for every $r\geq 2$. ∎

Via the notion of jet bundles, we can now recall from [5] the definition of an
${\rm SL}(r,{\mathbb{C}})$-oper on Riemann surfaces.

######  Definition 4.4.

An ${\rm SL}(r,{\mathbb{C}})$-oper on a Riemann surface $X$ of genus $g\geq 2$
is a holomorphic connection $D$ on ${\rm
Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}$ such that the connection on
$\det{\rm Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}$ induced by $D$ is the
trivial connection on ${\mathcal{O}}_{X}$.

######  Remark 4.5.

Let ${\mathcal{L}}^{\prime}$ be another holomorphic line bundle on $X$ such
that ${\mathcal{L}}^{\prime}\otimes{\mathcal{L}}^{\prime}=TX$. So the
holomorphic line bundle $\xi:={\mathcal{L}}^{\prime}\otimes{\mathcal{L}}^{*}$
satisfies the equation $\xi\otimes\xi={\mathcal{O}}_{X}$. Therefore,
$\xi\otimes\xi$ has a unique holomorphic connection $D_{0}$ with trivial
monodromy. In other words, $D_{0}$ is the trivial connection. There is a
unique holomorphic connection $D_{\xi}$ on $\xi$ such that the connection on
$\xi\otimes\xi$ induced by $D_{\xi}$ coincides with $D_{0}$. Using $D_{\xi}$,
there is a canonical isomorphism

$\phi_{\xi}\colon\
J^{1}({\mathcal{L}})\otimes\xi\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}J^{1}({\mathcal{L}}^{\prime}).$

To construct $\phi_{\xi}$, let $\underline{\xi}$ denote the locally constant
sheaf on $X$ defined by the sheaf of flat sections of $\xi$ for the connection
$D_{\xi}$. Consider the natural homomorphism

$J^{1}({\mathcal{L}})\otimes_{\mathbb{C}}\underline{\xi}\longrightarrow
J^{1}({\mathcal{L}}^{\prime}).$

This homomorphism extends uniquely to $\phi_{\xi}$. Note that $\phi_{\xi}$
produces a holomorphic isomorphism

$\displaystyle\phi^{r}_{\xi}\colon\ {\rm
Sym}^{r-1}\bigl{(}J^{1}({\mathcal{L}})\bigr{)}\otimes\xi^{\otimes
r}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\rm
Sym}^{r-1}\bigl{(}J^{1}({\mathcal{L}}^{\prime})\bigr{)}.$ (4.9)

The connection $D_{\xi}$ on $\xi$ induces a connection $D^{r}_{\xi}$ on
$\xi^{\otimes r}$. Therefore, the isomorphism $\phi^{r}_{\xi}$ in (4.9)
produces a bijection between the space of holomorphic connections on ${\rm
Sym}^{r-1}\bigl{(}J^{1}({\mathcal{L}})\bigr{)}$ and the space of holomorphic
connections on ${\rm Sym}^{r-1}\bigl{(}J^{1}({\mathcal{L}}^{\prime})\bigr{)}$.
Consequently, the space of holomorphic connections $D$ on ${\rm
Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}$ such that the connection on
$\det{\rm Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}$ induced by $D$ is the
trivial connection on ${\mathcal{O}}_{X}$ (see Definition 4.4) does not depend
on the choice of the line bundle $\mathcal{L}$.

Let ${\mathcal{C}}_{X}(r)$ denote the space of all holomorphic connections on
${\rm Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}$ such that their induced
connection on $\det{\rm Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}$ is the
trivial connection on ${\mathcal{O}}_{X}$. Let

$\text{Aut}\bigl{(}{\rm Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}\bigr{)}$

denote the group of all holomorphic automorphisms of ${\rm
Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}$. If $D$ is a holomorphic
connection on ${\rm Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}$ such that
the connection on the line bundle

$\det{\rm Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}$

induced by $D$ is the trivial connection, and

$A\in\text{Aut}\bigl{(}{\rm
Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}\bigr{)},$

then $(A\otimes{\rm Id}_{K_{X}})\circ D\circ A^{-1}$ also has the property
that the connection on $\det{\rm Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}$
induced by it is the trivial connection. Indeed, this follows immediately from
the fact that the holomorphic automorphisms of a line bundle $\xi$ act
trivially on the space of all holomorphic connections on $\xi$.

######  Definition 4.6.

The moduli space of ${\rm SL}(r,{\mathbb{C}})$-opers on $X$, which will be
denoted by ${\rm OSL}_{X}(r)$, is defined as follows

${\rm OSL}_{X}(r):={\mathcal{C}}_{X}(r)/\text{Aut}\bigl{(}{\rm
Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}\bigr{)}.$ (4.10)

Using the isomorphism $\phi^{r}_{\xi}$ in (4.9), we conclude that

$\text{Aut}\bigl{(}{\rm
Sym}^{r-1}\bigl{(}J^{1}({\mathcal{L}}^{\prime})\bigr{)}\bigr{)}=\text{Aut}\bigl{(}{\rm
Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}\bigr{)}.$

Therefore, from Remark 4.5 it follows that ${\rm OSL}_{X}(r)$ in Definition
4.6 is independent of the choice of the line bundle $\mathcal{L}$.

######  Remark 4.7.

We need to explain the reason for the assumption that the genus of $X$ is at
least two. On ${\mathbb{C}}{\mathbb{P}}^{1}$ there is no nontrivial ${\rm
SL}(r,{\mathbb{C}})$-oper. If $\text{genus}(X)=1$, then the underlying
holomorphic vector bundle for an ${\rm SL}(r,{\mathbb{C}})$-oper is not unique
up to tensoring with line bundles of order two (unlike in the case of higher
genus Riemann surfaces). For example, when ${\text{genus}(X)=1}$, both
${\mathcal{O}}_{X}\oplus{\mathcal{O}}_{X}$ and the unique nontrivial extension
of ${\mathcal{O}}_{X}$ by ${\mathcal{O}}_{X}$ admit an ${\rm
SL}(2,{\mathbb{C}})$-oper structure.

###  4.3 Differential operators and opers

For completion, we shall quickly recall the definition of holomorphic
differential operators. Given holomorphic vector bundles $V$ and $W$ on $X$,
and a nonnegative integer $d$, define the holomorphic vector bundle

$\operatorname{Diff}^{d}_{X}(W,V):=V\otimes
J^{d}(W)^{*}=\operatorname{Hom}\bigl{(}J^{d}(W),V\bigr{)}.$

A holomorphic differential operator of order $d$ from $W$ to $V$ is a
holomorphic section of the vector bundle $\operatorname{Diff}^{d}_{X}(W,V)$.
Consider the homomorphism $K^{\otimes d}_{X}\otimes W\hookrightarrow J^{d}(W)$
in (4.5). Using its dual

$\nu\colon\ J^{d}(W)^{*}\longrightarrow\bigl{(}K^{\otimes d}_{X}\otimes
W\bigr{)}^{*},$

we get a surjective homomorphism

$\operatorname{Diff}^{d}_{X}(W,V)=V\otimes J^{d}(W)^{*}\xrightarrow{{\rm
Id}\otimes\nu}V\otimes\bigl{(}K^{\otimes d}_{X}\otimes
W\bigr{)}^{*}=(TX)^{\otimes d}\otimes\operatorname{Hom}(W,V),$ (4.11)

which is known as the symbol map.

The following notation will be used: For a line bundle $\xi$ on $X$ and a
positive integer $d$, the line bundle $(\xi^{*})^{\otimes d}$ will be denoted
by $\xi^{\otimes-d}$. Also $\xi^{0}$ will denote the trivial line bundle
${\mathcal{O}}_{X}$.

When considered on ${\rm
Diff}^{r}_{X}\bigl{(}{\mathcal{L}}^{\otimes(r-1)},{\mathcal{L}}^{-r-1}\bigr{)}$,
the symbol map in (4.11) sends it to

$(TX)^{\otimes
r}\otimes\operatorname{Hom}\bigl{(}{\mathcal{L}}^{\otimes(r-1)},{\mathcal{L}}^{-r-1}\bigr{)}=(TX)^{\otimes
r}\otimes{\mathcal{L}}^{-2r}={\mathcal{O}}_{X},$

where we use for a holomorphic line bundle $A$ and a positive integer $d$ the
convention ${A^{-d}\\!\\!=\\!(\\!A^{*}\\!)^{\otimes d}}$.

The following proposition is well known.

######  Proposition 4.8.

The moduli space of ${\rm SL}(r,{\mathbb{C}})$-opers on $X$ in (4.10) is
isomorphic to the subset

$\displaystyle\mathcal{U}\subset H^{0}\bigl{(}X,{\rm
Diff}^{r}_{X}\bigl{(}{\mathcal{L}}^{\otimes(r-1)},{\mathcal{L}}^{-r-1}\bigr{)}\bigr{)}=H^{0}\bigl{(}X,{\rm
Diff}^{r}_{X}\bigl{(}{\mathcal{L}}^{\otimes(r-1)},\bigl{(}{\mathcal{L}}^{\otimes(r+1)}\bigr{)}^{*}\bigr{)}\bigr{)}$

consisting of all differential operators $\delta\in H^{0}\bigl{(}X,{\rm
Diff}^{r}_{X}\bigl{(}{\mathcal{L}}^{\otimes(r-1)},\bigl{(}{\mathcal{L}}^{\otimes(r+1)}\bigr{)}^{*}\bigr{)}\bigr{)}$
such that the symbol of $\delta$ is the constant function $1$ on $X$ and the
sub-leading term of $\delta$ is vanishing identically.

######  Remark 4.9.

One should note that the above condition in Proposition 4.8 that the sub-
leading term of $\delta$ vanishes identically makes sense intrinsically.
Indeed, this condition is equivalent to the condition that the local system on
$X$ given by the sheaf of solutions of $D$ is an ${\rm
SL}(r,{\mathbb{C}})$-system (meaning the $r$-th exterior product of it is the
constant sheaf with stalk $\mathbb{C}$).

We shall briefly describe here the isomorphism

${\rm
OSL}_{X}(r)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathcal{U}\subset
H^{0}\bigl{(}X,{\rm
Diff}^{r}_{X}\bigl{(}{\mathcal{L}}^{\otimes(r-1)},\bigl{(}{\mathcal{L}}^{\otimes(r+1)}\bigr{)}^{*}\bigr{)}\bigr{)}$

in Proposition 4.8 in more detail. Given a differential operator

$\delta\in H^{0}\bigl{(}X,{\rm
Diff}^{r}_{X}\bigl{(}{\mathcal{L}}^{\otimes(r-1)},\bigl{(}{\mathcal{L}}^{\otimes(r+1)}\bigr{)}^{*}\bigr{)}\bigr{)}$

such that the symbol of $\delta$ is the constant function $1$ on $X$, its
sheaf of solutions produces a flat vector bundle on $X$. The holomorphic
vector bundle underlying this flat bundle is isomorphic to
$J^{r-1}(\mathcal{L})$. In other words, the sheaf of solutions of $\delta$
produces a holomorphic connection $D$ on $J^{r-1}(\mathcal{L})$. The
holomorphic vector bundle $J^{r-1}(\mathcal{L})$ is holomorphically isomorphic
to ${\rm Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}$. After fixing a
holomorphic isomorphism between $J^{r-1}(\mathcal{L})$ and ${\rm
Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}$, the holomorphic connection on
$D$ on $J^{r-1}(\mathcal{L})$ gives a holomorphic connection on ${\rm
Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}$.

To describe the reverse map

${\rm OSL}_{X}(r)\longrightarrow H^{0}\bigl{(}X,{\rm
Diff}^{r}_{X}\bigl{(}{\mathcal{L}}^{\otimes(r-1)},\bigl{(}{\mathcal{L}}^{\otimes(r+1)}\bigr{)}^{*}\bigr{)}\bigr{)},$
(4.12)

take a holomorphic connection $D$ on ${\rm
Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}$ such that the connection on the
line bundle $\det{\rm Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}$ induced by
$D$ is the trivial connection on ${\mathcal{O}}_{X}$. Using $D$, we will
construct a holomorphic isomorphism between ${\rm
Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}$ and $J^{r-1}(\mathcal{L})$.
Consider the projection

$\psi\colon\ {\rm
Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}\longrightarrow{\rm
Sym}^{r-1}\bigl{(}J^{0}(\mathcal{L})\bigr{)}={\mathcal{L}}^{\otimes(r-1)},$

which is the $(r-1)$-th symmetric power of the natural projection
$J^{1}(\mathcal{L})\longrightarrow\mathcal{L}$ (see the exact sequence in
(4.5)). Take any $x\in X$ and $v\in{\rm
Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}_{x}$. Let $\widetilde{v}$ denote
the unique flat section of ${\rm
Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}\big{|}_{U}$, for the connection
$D$, defined on a simply connected open neighborhood $x\in U\subset X$, such
that $\widetilde{v}(x)=v$. Restricting $\psi(\widetilde{v})$ to the $(r-1)$-th
order infinitesimal neighborhood of $x$ we get an element
$\widetilde{v}^{\prime}\in
J^{r-1}\bigl{(}{\mathcal{L}}^{\otimes(r-1)}\bigr{)}_{x}$. Consequently, we
have a holomorphic map

$\displaystyle\widetilde{\psi}\colon\ {\rm
Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}$ $\displaystyle\longrightarrow
J^{r-1}\bigl{(}{\mathcal{L}}^{\otimes(r-1)}\bigr{)},$ $\displaystyle v$
$\displaystyle\longmapsto\widetilde{v}^{\prime},$ (4.13)

which is in fact an isomorphism. Let
$\bigl{(}\widetilde{\psi}^{-1}\bigr{)}^{*}D$ be the holomorphic connection on
the vector bundle $J^{r-1}\bigl{(}{\mathcal{L}}^{\otimes(r-1)}\bigr{)}$
obtained by pulling back the connection $D$ on ${\rm
Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}$ using $\widetilde{\psi}^{-1}$.
Then there is a unique

$\delta\in H^{0}\bigl{(}X,{\rm
Diff}^{r}_{X}\bigl{(}{\mathcal{L}}^{\otimes(r-1)},\bigl{(}{\mathcal{L}}^{\otimes(r+1)}\bigr{)}^{*}\bigr{)}\bigr{)}$
(4.14)

such that

  * •

the symbol of $\delta$ is the constant function $1$ on $X$, and

  * •

the flat vector bundle on $X$ given by the sheaf of solutions of $\delta$ is
isomorphic to $\bigl{(}\widetilde{\psi}^{-1}\bigr{)}^{*}D$ on
$J^{r-1}\bigl{(}{\mathcal{L}}^{\otimes(r-1)}\bigr{)}$ equipped with the above
flat connection.

Since $\bigl{(}\widetilde{\psi}^{-1}\bigr{)}^{*}D$ is a holomorphic
$\text{SL}(r,{\mathbb{C}})$ connection, the sub-leading term of $\delta$
vanishes identically as required.

###  4.4 Real and quaternionic slices of $\boldsymbol{{\rm
SL}(r,\mathbb{C})}$-opers

As before, $\mathcal{L}$ is a holomorphic line bundle on $X$ with
${\mathcal{L}}\otimes{\mathcal{L}}=TX$. The Riemann surface is now equipped
with an anti-holomorphic involution $\sigma_{X}$.

In order to define and study real and quaternionic slices of ${\rm
SL}(r,\mathbb{C})$-opers, we shall consider certain compatibility conditions
with the anti-holomorphic involutions studied in Section 4.1. We shall begin
by fixing a holomorphic isomorphism of line bundles

$\widehat{\sigma}\colon\
{\mathcal{L}}\longrightarrow\sigma^{*}_{X}\overline{\mathcal{L}}$ (4.15)

such that
$\bigl{(}\sigma^{*}_{X}\overline{\widehat{\sigma}}\bigr{)}\circ\widehat{\sigma}\in\pm{\rm
Id}_{\mathcal{L}}$, where $\sigma_{X}$ is the anti-holomorphic involution in
(4.1). Note that such pairs $\bigl{(}{\mathcal{K}},\widehat{\sigma}\bigr{)}$
exist. The isomorphism $\widehat{\sigma}$ in (4.15) above produces a
holomorphic isomorphism

$\displaystyle\widehat{\sigma}^{m}_{k}\colon\
J^{m}\bigl{(}{\mathcal{L}}^{k}\bigr{)}\longrightarrow
J^{m}\bigl{(}\sigma^{*}_{X}\overline{\mathcal{L}}^{k}\bigr{)}=\sigma^{*}_{X}\overline{J^{m}\bigl{(}{\mathcal{L}}^{k}\bigr{)}}$
(4.16)

for all $m\geq 0$ and $k\in\mathbb{Z}$, where the second isomorphism in (4.16)
is given by (4.6). Moreover, since
$\bigl{(}\sigma^{*}_{X}\overline{\widehat{\sigma}}\bigr{)}\circ\widehat{\sigma}\in\pm{\rm
Id}_{\mathcal{L}}$ (see (4.15)), it follows that

$\bigl{(}\sigma^{*}_{X}\overline{\widehat{\sigma}^{m}_{k}}\bigr{)}\circ\widehat{\sigma}^{m}_{k}\in\pm{\rm
Id}_{J^{m}({\mathcal{L}}^{k})}.$ (4.17)

The homomorphism $\widehat{\sigma}^{1}_{1}$ defined through (4.16) produces,
in turn, a holomorphic isomorphism of symmetric products

${\rm Sym}^{r-1}\bigl{(}\widehat{\sigma}^{1}_{1}\bigr{)}\colon\ {\rm
Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}\longrightarrow{\rm
Sym}^{r-1}\bigl{(}\sigma^{*}_{X}\overline{J^{1}(\mathcal{L})}\bigr{)}=\sigma^{*}_{X}\overline{{\rm
Sym}^{r-1}\bigl{(}J^{1}({\mathcal{L}})\bigr{)}}.$ (4.18)

From (4.17) for $m=1$, we deduce that

$\bigl{(}\sigma^{*}_{X}\overline{{\rm
Sym}^{r-1}\bigl{(}\widehat{\sigma}^{1}_{1}\bigr{)}}\bigr{)}\circ{\rm
Sym}^{r-1}\bigl{(}\widehat{\sigma}^{1}_{1}\bigr{)}\in\pm{\rm Id}_{{\rm
Sym}^{r-1}(J^{1}(\mathcal{L}))}.$

Observe that $\bigl{(}\sigma^{*}_{X}\overline{{\rm
Sym}^{r-1}\bigl{(}\widehat{\sigma}^{1}_{1}\bigr{)}}\bigr{)}\circ{\rm
Sym}^{r-1}\bigl{(}\widehat{\sigma}^{1}_{1}\bigr{)}={\rm Id}_{{\rm
Sym}^{r-1}(J^{1}(\mathcal{L}))}$ whenever $r$ is an odd integer.

Recall that ${\mathcal{C}}_{X}(r)$ denotes the space of all holomorphic
connections on ${\rm Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}$ such that
their induced connection on $\det{\rm
Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}$ is the trivial connection on
${\mathcal{O}}_{X}$. Let

$\gamma\colon\ {\mathcal{C}}_{X}(r)\longrightarrow{\mathcal{C}}_{X}(r)$ (4.19)

be the anti-holomorphic map that sends any holomorphic connection $D$ to the
holomorphic connection on ${\rm Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}$
given by the holomorphic connection ${\rm
Sym}^{r-1}\bigl{(}\widehat{\sigma}^{1}_{1}\bigr{)}^{*}\sigma^{*}_{X}\overline{D}$
through the isomorphism in (4.18), where $\sigma^{*}_{X}\overline{D}$ is the
connection on $\sigma^{*}_{X}\overline{J^{1}(\mathcal{L})}$ induced by $D$. It
can be shown that $\gamma$ in (4.19) is an involution. Indeed, the
automorphism of ${\rm Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}$ given by
multiplication by $-1$ preserves any connection on ${\rm
Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}$. So even if the holomorphic line
bundle $\mathcal{L}$ is quaternionic, the map $\gamma$ remains to be of order
two.

The group ${\mathbb{Z}}/2{\mathbb{Z}}$ acts on ${\mathcal{C}}_{X}(r)$ via the
involution $\gamma$ in (4.19), and the group of automorphisms
$\text{Aut}\bigl{(}{\rm Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}\bigr{)}$
also acts on ${\mathcal{C}}_{X}(r)$ through (4.10). On the other hand, there
is an anti-holomorphic group automorphism of order two

$\displaystyle\tau_{r}\colon\ \text{Aut}\bigl{(}{\rm
Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}\bigr{)}\longrightarrow\text{Aut}\bigl{(}{\rm
Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}\bigr{)}$ (4.20)

that sends any $\beta\in\text{Aut}\bigl{(}{\rm
Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}\bigr{)}$ to

$\bigl{(}{\rm
Sym}^{r-1}\bigl{(}\widehat{\sigma}^{1}_{1}\bigr{)}\bigr{)}^{-1}\circ\bigl{(}\sigma^{*}_{X}\overline{\beta}\bigr{)}\circ{\rm
Sym}^{r-1}\bigl{(}\widehat{\sigma}^{1}_{1}\bigr{)}\in\text{Aut}\bigl{(}{\rm
Sym}^{r-1}(J^{1}(\mathcal{L}))\bigr{)},$

where
$\sigma^{*}_{X}\overline{\beta}\in\text{Aut}\bigl{(}\sigma^{*}_{X}\overline{{\rm
Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}}\bigr{)}$ is the automorphism
given by $\beta$.

######  Lemma 4.10.

The action of ${\mathbb{Z}}/2{\mathbb{Z}}$ on ${\mathcal{C}}_{X}(r)$ given by
the involution $\gamma$ in (4.19) descends to an involution

$\beta\colon\ {\rm OSL}_{X}(r)\longrightarrow{\rm OSL}_{X}(r)$ (4.21)

of ${\mathbb{Z}}/2{\mathbb{Z}}$ on the quotient space of ${\rm
SL}(r,\mathbb{C})$-opers ${\rm OSL}_{X}(r)$ $($see Definition 4.6$)$.

###### Proof.

The lemma follows from the above construction of $\gamma$ and from noticing
that the actions of ${\mathbb{Z}}/2{\mathbb{Z}}$ and ${\rm Aut}\bigl{(}{\rm
Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}\bigr{)}$ on
${\mathcal{C}}_{X}(r)$ are related as follows:

$\gamma\circ\psi(D)=\tau_{r}(\psi)\circ\gamma(D)$

for all $\psi\in{\rm Aut}\bigl{(}{\rm
Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}\bigr{)}$ and
$D\in{\mathcal{C}}_{X}(r)$, where $\tau_{r}$ and $\gamma$ are constructed in
(4.20) and (4.19) respectively. ∎

######  Remark 4.11.

Since $\gamma$ in (4.19) is anti-holomorphic, it follows immediately that
$\beta$ in (4.21) is also anti-holomorphic.

######  Definition 4.12.

An ${\rm SL}(r,{\mathbb{C}})$-oper on $X$ is in the real slice of ${\rm
SL}(r,\mathbb{C})$-opers if it is a fixed point of the involution $\beta$ in
(4.21) of Lemma 4.10.

In order to study the slice of real ${\rm SL}(r,{\mathbb{C}})$-opers on $X$,
we shall construct a conjugate linear involution of $H^{0}\bigl{(}X,{\rm
Diff}^{r}_{X}\bigl{(}{\mathcal{L}}^{r-1},{\mathcal{L}}^{-r-1}\bigr{)}\bigr{)}$.

######  Lemma 4.13.

There is a natural conjugate linear involution

${\mathcal{B}}\colon\ H^{0}\bigl{(}X,{\rm
Diff}^{r}_{X}\bigl{(}{\mathcal{L}}^{r-1},{\mathcal{L}}^{-r-1}\bigr{)}\bigr{)}\longrightarrow
H^{0}\bigl{(}X,{\rm
Diff}^{r}_{X}\bigl{(}{\mathcal{L}}^{r-1},{\mathcal{L}}^{-r-1}\bigr{)}\bigr{)}$
(4.22)

induced by the anti-holomorphic involution $\sigma$ of $X$.

###### Proof.

Given any $\delta\in H^{0}\bigl{(}X,{\rm
Diff}^{r}_{X}\bigl{(}{\mathcal{L}}^{r-1},{\mathcal{L}}^{-r-1}\bigr{)}\bigr{)},$
the holomorphic differential operator on
$\sigma^{*}_{X}\overline{\mathcal{L}}^{r-1}$ defined by
$\sigma^{*}\overline{\delta}\colon
J^{r}\bigl{(}\sigma^{*}_{X}\overline{\mathcal{L}}^{r-1}\bigr{)}\longrightarrow\sigma^{*}_{X}\overline{\mathcal{L}}^{-r-1}$
sends any locally defined holomorphic section $s$ of ${\mathcal{L}}^{r-1}$ to
$\sigma^{*}\overline{\delta(s)}$; recall that the holomorphic sections of
$\sigma^{*}_{X}\overline{\mathcal{L}}^{r-1}$ over an open subset $U\subset X$
are identified with the holomorphic sections of ${\mathcal{L}}^{r-1}$ over
$\sigma_{X}(U)$. Consider the diagram

$\begin{matrix}J^{r}\bigl{(}{\mathcal{L}}^{r-1}\bigr{)}&\xrightarrow{\widehat{\sigma}^{r}_{r-1}}&J^{r}\bigl{(}\sigma^{*}_{X}\overline{\mathcal{L}}^{r-1}\bigr{)}\\\
\Big{\downarrow}\delta&&\Big{\downarrow}\sigma^{*}\overline{\delta}\\\
{\mathcal{L}}^{-r-1}&\xrightarrow{\widehat{\sigma}^{0}_{-r-1}}&\sigma^{*}_{X}\overline{\mathcal{L}}^{-r-1},\end{matrix}$

where $\widehat{\sigma}^{r}_{r-1}$ and $\widehat{\sigma}^{0}_{-r-1}$ are the
holomorphic isomorphisms in (4.16), which needs not be commutative. This
diagram shows that

$\bigl{(}\widehat{\sigma}^{0}_{-r-1}\bigr{)}^{-1}\circ\bigl{(}\sigma^{*}\overline{\delta}\bigr{)}\circ\widehat{\sigma}^{r}_{r-1}\in
H^{0}\bigl{(}X,{\rm
Diff}^{r}_{X}\bigl{(}{\mathcal{L}}^{r-1},{\mathcal{L}}^{-r-1}\bigr{)}\bigr{)}.$

The map ${\mathcal{B}}$ from (4.22) can then be defined as the map that sends
any $\delta$ to the differential operator
$\bigl{(}\widehat{\sigma}^{0}_{-r-1}\bigr{)}^{-1}\circ\bigl{(}\sigma^{*}\overline{\delta}\bigr{)}\circ\widehat{\sigma}^{r}_{r-1}$
constructed above from it. Finally, it should be noted that ${\mathcal{B}}$ is
an involution even if the holomorphic line bundle $\mathcal{L}$ is
quaternionic. ∎

######  Theorem 4.14.

The involution $\mathcal{B}$ in (4.22) preserves the subset ${\rm OSL}_{X}(r)$
in Proposition 4.8. The restriction of the map $\mathcal{B}$ to this subset
${\rm OSL}_{X}(r)$ coincides with the involution $\beta$ in (4.21).

###### Proof.

Recall from Proposition 4.8 that ${\rm OSL}_{X}(r)$ is the locus of all

$\delta\in H^{0}\bigl{(}X,{\rm
Diff}^{r}_{X}\bigl{(}{\mathcal{L}}^{r-1},{\mathcal{L}}^{-r-1}\bigr{)}\bigr{)}$

such that the symbol of $\delta$ is the constant function $1$ and the sub-
leading term of $\delta$ vanishes identically. Therefore, it is evident that
$\mathcal{B}$ preserves the subset ${\rm OSL}_{X}(r)$. Given a holomorphic
connection $D$ on ${\rm Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}$ such
that the induced connection on $\det{\rm
Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}$ is the trivial connection on
${\mathcal{O}}_{X}$, through the map $\widetilde{\psi}$ in (4.13) we have

$\widetilde{\psi}\circ{\rm
Sym}^{r-1}\bigl{(}\widehat{\sigma}^{1}_{1}\bigr{)}=\widehat{\sigma}^{r-1}_{r-1}\circ\widetilde{\psi}$
(4.23)

(see (4.18) and (4.16) for ${\rm
Sym}^{r-1}\bigl{(}\widehat{\sigma}^{1}_{1}\bigr{)}$ and
$\widehat{\sigma}^{r-1}_{r-1}$, respectively). Moreover, from (4.14) there is
a unique differential operator $\delta$ such that

  * •

the symbol of $\delta$ is the constant function $1$ on $X$, and

  * •

the flat vector bundle on $X$ given by the sheaf of solutions of $\delta$ is
isomorphic to
$J^{r-1}\\!\bigl{(}\\!{\mathcal{L}}^{\otimes(\\!r-1\\!)}\bigr{)}$ equipped
with the above connection $\bigl{(}\widetilde{\psi}^{-1}\bigr{)}^{*}D$.

Since $\delta$ is the image of $D$ under the map in (4.12), the second part of
the theorem follows from (4.23). ∎

Theorem 4.14 has the following immediate consequence, which characterizes the
real slice of ${\rm SL}(r,\mathbb{C})$-opers defined above.

######  Proposition 4.15.

An element of the moduli space of ${\rm SL}(r,\mathbb{C})$-opers

$\delta\in{\rm OSL}_{X}(r)\subset H^{0}\bigl{(}X,{\rm
Diff}^{r}_{X}\bigl{(}{\mathcal{L}}^{\otimes(r-1)},{\mathcal{L}}^{-r-1}\bigr{)}\bigr{)}$

is in the real slice of ${\rm SL}(r,{\mathbb{C}})$-opers if and only if
${\mathcal{B}}(\delta)=\delta$, where $\mathcal{B}$ is the involution defined
in (4.22).

##  5 Another description of the involution

In order to study the real slice of ${\rm SL}(r,\mathbb{C})$-opers defined
through Proposition 4.15 in the previous section, we shall dedicate this
section to describing the involution from a different perspective.

###  5.1 Anti-holomorphic involutions on differential operators

Consider as in previous sections (see (4.2)) the projections

$p_{i}\colon\ X\times X\longrightarrow X.$

Then, given two holomorphic vector bundles $A$ and $B$ on $X$, we shall denote
the holomorphic vector bundle $(p^{*}_{1}A)\otimes(p^{*}_{2}B)$ on $X\times X$
by $A\boxtimes B$. These holomorphic vector bundles produce the natural vector
bundles

$\displaystyle\mathcal{J}:=A\boxtimes(B^{*}\otimes K_{X})\qquad\text{ and
}\qquad\mathcal{J}_{d}:=A\boxtimes(B^{*}\otimes
K_{X})\otimes{\mathcal{O}}_{X\times X}((d+1){\Delta})$ (5.1)

on $X\times X$, where $\Delta$ is the diagonal divisor defined in (4.3). These
vector bundles fit in a short exact sequence of coherent sheaves on $X\times
X$ given by

$\displaystyle
0\longrightarrow\mathcal{J}\longrightarrow\mathcal{J}_{d}\longrightarrow{\mathcal{Q}}_{d}(A,B):=\frac{\mathcal{J}_{d}}{\mathcal{J}}\longrightarrow
0,$ (5.2)

where the support of ${\mathcal{Q}}_{d}(A,B)$ in (5.2) is the non-reduced
divisor $(d+1){\Delta}$. The direct image

${\mathcal{K}}_{d}(A,B):=p_{1*}{\mathcal{Q}}_{d}(A,B)$ (5.3)

is a holomorphic vector bundle on $X$, and from [6, Section 2.1], [8, p. 25,
equation (5.1)], [7, Section 3.1, p. 1314]) one has that

${\mathcal{K}}_{d}(A,B)=\operatorname{Hom}\bigl{(}J^{d}(B),A\bigr{)}=\operatorname{Diff}^{d}_{X}(B,A).$
(5.4)

For $d\geq 1$, the sheaf ${\mathcal{Q}}_{d}(A,B)$ in (5.2) fits in the
following short exact sequence of sheaves on $X\times X$, where the
abbreviations of (5.1) are used,

$\displaystyle
0\longrightarrow{\mathcal{Q}}_{d-1}(A,B)\longrightarrow{\mathcal{Q}}_{d}(A,B)\longrightarrow\frac{\mathcal{J}_{d}}{\mathcal{J}_{d-1}}\longrightarrow
0.$ (5.5)

Note that the above sheaf $\mathcal{J}_{d}/\mathcal{J}_{d-1}$ is supported on
the reduced divisor ${\Delta}$. Taking direct image of the short exact
sequence in (5.5) by the projection $p_{1}$, we get the following short exact
sequence of holomorphic vector bundles on $X$

$\displaystyle 0\longrightarrow p_{1*}{\mathcal{Q}}_{d-1}(A,B)\longrightarrow
p_{1*}{\mathcal{Q}}_{d}(A,B)={\mathcal{K}}_{d}(A,B)\stackrel{{\scriptstyle
p_{0}}}{{\longrightarrow}}p_{1*}\left(\frac{\mathcal{J}_{d}}{\mathcal{J}_{d-1}}\right)\longrightarrow
0.$ (5.6)

From Poincaré adjunction formula, one has that ${\mathcal{O}}_{X\times
X}({\Delta})\big{|}_{\Delta}$ is the normal bundle of ${\Delta}$ [14, p. 146],
and therefore ${\mathcal{O}}_{X\times X}({\Delta})\big{|}_{\Delta}=TX$, using
the identification of ${\Delta}$ with $X$ defined by $x\longmapsto(x,x)$.
Hence, we have

$p_{1*}\left(\frac{\mathcal{J}_{d}}{\mathcal{J}_{d-1}}\right)=\operatorname{Hom}(B,A)\otimes(TX)^{\otimes
d}.$

######  Remark 5.1.

The isomorphism in (5.4) and the projection $p_{0}$ in (5.6) together produce
a homomorphism

$\operatorname{Diff}^{d}_{X}(B,A)\longrightarrow\operatorname{Hom}(B,A)\otimes(TX)^{\otimes
d},$ (5.7)

which is the symbol map on differential operators constructed in (4.11).

Recall from Section 4.1 that $\mathcal{L}:=\bigl{(}K^{1/2}_{X}\bigr{)}^{*}$,
and consider now an element of the moduli space of ${\rm
SL}(r,{\mathbb{C}})$-opers on $X$ given by

$\delta\in{\rm OSL}_{X}(r)\subset H^{0}\bigl{(}X,{\rm
Diff}^{r}_{X}\bigl{(}{\mathcal{L}}^{\otimes(r-1)},{\mathcal{L}}^{-r-1}\bigr{)}\bigr{)}.$
(5.8)

From (5.3) and (5.4), we know that $\delta$ corresponds to a section

$\widetilde{S}_{\delta}\in
H^{0}((r+1)\Delta,\bigl{(}{\mathcal{L}}^{-r-1}\boxtimes{\mathcal{L}}^{-r-1}\bigr{)}{\mathcal{O}}_{X\times
X}((r+1)\Delta)).$ (5.9)

Note that the restriction of

$\bigl{(}{\mathcal{L}}^{-r-1}\boxtimes{\mathcal{L}}^{-r-1}\bigr{)}{\mathcal{O}}_{X\times
X}((r+1)\Delta)\longrightarrow X\times X$

to $\Delta\subset X\times X$ is ${\mathcal{O}}_{\Delta}$, and that
$\mathcal{J}_{r}=\bigl{(}{\mathcal{L}}^{-r-1}\boxtimes{\mathcal{L}}^{-r-1}\bigr{)}{\mathcal{O}}_{X\times
X}((r+1)\Delta)$ for the vector bundles in (5.1) defined as

$A={\mathcal{L}}^{-r-1}\qquad\text{ and }\qquad B={\mathcal{L}}^{r-1}.$

######  Lemma 5.2.

There is a natural section

$S_{\delta}\in
H^{0}(3\Delta,\mathcal{J}_{0})=H^{0}\bigl{(}3\Delta,\bigl{(}{\mathcal{L}}^{-1}\boxtimes{\mathcal{L}}^{-1}\bigr{)}{\mathcal{O}}_{X\times
X}(\Delta)\bigr{)}$

such that

  * •

$\widetilde{S}_{\delta}\big{|}_{3\Delta}=(S_{\delta})^{\otimes(r+1)}$,

  * •

the restriction of $S_{\delta}$ to $\Delta\subset X\times X$ coincides with
the section of ${\mathcal{O}}_{\Delta}$ given by the constant function $1$,
and

  * •

$S_{\delta}\big{|}_{2\Delta}$ is anti-invariant under the involution of
$2\Delta$ obtained by restricting the involution of $X\times X$ defined by
$(x_{1},x_{2})\longmapsto(x_{2},x_{1})$, in other words, this involution takes
$S_{\delta}\big{|}_{2\Delta}$ to $-S_{\delta}\big{|}_{2\Delta}$.

###### Proof.

This follows from the above analysis, by noting that the given condition that
the symbol of $\delta$ is the constant function $1$ is equivalent to the
statement that the restriction of the section $\widetilde{S}_{\delta}$
(constructed in (5.9)) to $\Delta\subset X\times X$ is the section of
${\mathcal{O}}_{\Delta}$ given by the constant function $1$ (see (5.7)). The
given condition that the sub-leading term of $\delta$ vanishes identically is
equivalent to the condition that the section
$\widetilde{S}_{\delta}\big{|}_{2\Delta}$ is anti-invariant under the
involution of $2\Delta$ obtained by restricting the involution of $X\times X$
defined by $(x_{1},x_{2})\longmapsto(x_{2},x_{1})$. ∎

We will show that the section of $\mathcal{J}_{r}$ for $r=2$ given by

$(S_{\delta})^{\otimes 3}\in H^{0}(3\Delta,\mathcal{J}_{2})$ (5.10)

naturally defines a projective structure on the Riemann surface $X$.

######  Proposition 5.3.

The section $(S_{\delta})^{\otimes 3}$ in (5.10) naturally defines a
projective structure on the Riemann surface $X$.

###### Proof.

From (5.3) and (5.4), we know that $(S_{\delta})^{\otimes 3}$ in (5.10) gives
a differential operator

$\delta_{0}\in H^{0}\bigl{(}X,{\rm
Diff}^{2}_{X}\bigl{(}{\mathcal{L}},{\mathcal{L}}^{-3}\bigr{)}\bigr{)}.$ (5.11)

Since the restriction of $S_{\delta}$ to $\Delta\subset X\times X$ coincides
with the section of ${\mathcal{O}}_{\Delta}$ given by the constant function
$1$, we conclude that the symbol of the differential operator $\delta_{0}$ is
the constant function $1$ (see (5.7)). Since $S_{\delta}\big{|}_{2\Delta}$ is
anti-invariant under the involution of $2\Delta$ obtained by restricting the
involution of $X\times X$ defined by $(x_{1},x_{2})\longmapsto(x_{2},x_{1})$,
we conclude that the sub-leading term of the differential operator
$\delta_{0}$ vanishes identically. Therefore, we have

$\delta_{0}\in{\rm OSL}_{X}(2)$

(see Proposition 4.8). Since the space ${\rm OSL}_{X}(2)$ is identified with
the space of projective structures on $X$, the differential operator
$\delta_{0}$ in (5.11) defines a projective structure on the Riemann surface
$X$. ∎

Consider the flat connection $D_{0}$ given by the sheaf of solutions of the
differential operator $\delta_{0}$ in (5.11). The underlying holomorphic
vector bundle for it is $J^{1}(\mathcal{L})$. The holomorphic connection on
$\text{Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}$ induced by $D_{0}$ will
be denoted by $\text{Sym}^{r-1}(D_{0})$, and from (4.13) it produces a
holomorphic isomorphism

$\text{Sym}^{r-1}\bigl{(}J^{1}(\mathcal{L})\bigr{)}\longrightarrow
J^{r-1}\bigl{(}{\mathcal{L}}^{r-1}\bigr{)}.$

The holomorphic connection on $J^{r-1}\bigl{(}{\mathcal{L}}^{r-1}\bigr{)}$
given by $\text{Sym}^{r-1}(D_{0})$ using this isomorphism will be denoted by
$\text{Sym}^{r-1}(D_{0})^{\prime}$.

For $D$ denote the holomorphic connection on
$J^{r-1}\bigl{(}{\mathcal{L}}^{r-1}\bigr{)}$ given by the sheaf of solutions
of the differential operator $\delta$ in (5.8), we shall define

$\theta_{\delta}:=D-\text{Sym}^{r-1}(D_{0})^{\prime}\in
H^{0}\bigl{(}X,\text{End}\bigl{(}J^{r-1}\bigl{(}{\mathcal{L}}^{r-1}\bigr{)}\bigr{)}\otimes
K_{X}\bigr{)}.$

Taking the trace for all $1\leq i\leq r$ given by

$\text{trace}\bigl{(}(\theta_{\delta})^{i}\bigr{)}\in
H^{0}\bigl{(}X,K^{\otimes i}_{X}\bigr{)},$ (5.12)

note that
$\text{trace}(\theta_{\delta})=0=\text{trace}\bigl{(}(\theta_{\delta})^{2}\bigr{)}$.

###  5.2 Real slices of fibrations

In search of an interesting parametrization of real slices of ${\rm
SL}(r,\mathbb{C})$-opers similar to those one can define for Higgs bundles
though Hitchin’s integrable system [18], consider the map

$\Psi\colon\ {\rm OSL}_{X}(r)\longrightarrow{\rm
OSL}_{X}(2)\times\left(\bigoplus_{i=3}^{r}H^{0}\bigl{(}X,K^{\otimes
i}_{X}\bigr{)}\right)$ (5.13)

that sends any differential operator $\delta$ as defined in (4.8) to
$\bigl{(}\delta_{0},\bigoplus_{i=3}^{r}\text{trace}\bigl{(}(\theta_{\delta})^{i}\bigr{)}\bigr{)}$,
where ${\delta_{0}\in{\rm OSL}_{X}(2)}$ is constructed in (5.11) and the
traces $\text{trace}\bigl{(}(\theta_{\delta})^{i}\bigr{)}$ are the sections in
(5.12).

Consider an anti-holomorphic involution $\sigma_{X}$ of $X$ as in Section 4,
and the induced involution $F$ from Lemma 2.5. From equation (2.11), it
follows that $F$ restricts to an anti-holomorphic involution

$F_{X}\colon\ {\rm OSL}_{X}(2)\longrightarrow{\rm OSL}_{X}(2).$

From Theorem 4.14, in the case when $r=2$ this $F_{X}$ coincides with
$\mathcal{B}$ constructed in (4.22), and $\beta$ constructed in (4.21). On the
other hand, the differential ${\rm d}\sigma\colon
T^{\mathbb{R}}X\longrightarrow\sigma^{*}T^{\mathbb{R}}X$ produces a
holomorphic involution $TX\longrightarrow\sigma^{*}\overline{TX}$. Through
these maps, one can define an anti-holomorphic linear automorphism of
$\omega\longmapsto\sigma^{*}_{X}\overline{\omega}$ of
$H^{0}\bigl{(}X,K^{\otimes i}_{X}\bigr{)}$. Then there is a natural involution

$\displaystyle{\mathcal{A}}_{1}\colon\ {\rm
OSL}_{X}(2)\times\left(\bigoplus_{i=3}^{r}H^{0}\bigl{(}X,K^{\otimes
i}_{X}\bigr{)}\right)$ $\displaystyle\longrightarrow{\rm
OSL}_{X}(2)\times\left(\bigoplus_{i=3}^{r}H^{0}\bigl{(}X,K^{\otimes
i}_{X}\bigr{)}\right),$
$\displaystyle\left(\eta,\bigoplus_{i=3}^{r}\omega_{i}\right)$
$\displaystyle\longmapsto\left(F_{X}(\eta),\bigoplus_{i=3}^{r}\sigma^{*}_{X}\overline{\omega}_{i}\right).$
(5.14)

Through the above involution, Theorem 4.14 leads to the following statement.

######  Proposition 5.4.

Let ${\mathcal{A}}=\Psi^{-1}\circ{\mathcal{A}}_{1}\circ\Psi$ be the involution
of ${\rm OSL}_{X}(r)$, where ${\mathcal{A}}_{1}$ and $\Psi$ are constructed in
(5.14) and (5.13) respectively. Then ${\mathcal{A}}$ coincides with the
involution $\beta$ in (4.21). Also ${\mathcal{A}}$ coincides with the
restriction of $\mathcal{B}$ $($in (4.22)$)$ to ${\rm OSL}_{X}(r)$.

Finally, Proposition 5.4 has the following immediate consequence.

######  Corollary 5.5.

An element

$\left(\eta,\bigoplus_{i=3}^{r}\omega_{i}\right)\in{\rm
OSL}_{X}(2)\times\left(\bigoplus_{i=3}^{r}H^{0}\bigl{(}X,K^{\otimes
i}_{X}\bigr{)}\right)$

is within the real slice of ${\rm SL}(r,{\mathbb{C}})$-opers if and only if
$F_{X}(\eta)=\eta$ and $\omega_{i}=\sigma^{*}_{X}\overline{\omega}_{i}$ for
all $3\leq i\leq r$.

##  6 Nondegenerate immersions into a projective space

Let $X$ be a connected Riemann surface and consider a holomorphic immersion

$\phi\colon\ X\longrightarrow{\mathbb{C}}{\mathbb{P}}^{d}.$

Given a point $x\in X$ and a hyperplane $H\subset{\mathbb{C}}{\mathbb{P}}^{d}$
such that $\phi(x)\in H$, we say that the order of contact of $X$ with $H$ at
$x$ is at least $n$ if the following condition holds: Every holomorphic
function $h$ defined on some neighborhood $U$ of
$\phi(x)\in{\mathbb{C}}{\mathbb{P}}^{d}$ such that $h\big{|}_{U\cap H}=0$, the
order of vanishing of $h\circ\phi$ at $x$ is at least $n$.

######  Definition 6.1.

The order of contact of $X$ with $H$ at $x$ is said to be $n$ if

  * •

the order of contact of $X$ with $H$ at $x$ is at least $n$, and

  * •

the order of contact of $X$ with $H$ at $x$ is not at least $n+1$.

In other words, the order of contact of $X$ with $H$ at $x$ is $n$ if

  * •

the order of contact of $X$ with $H$ at $x$ is at least $n$, and

  * •

there is a holomorphic function $h$ defined on some neighborhood $U$ of
$\phi(x)\in{\mathbb{C}}{\mathbb{P}}^{d}$ such that $h\big{|}_{U\cap H}=0$ and
the order of vanishing of $h\circ\phi$ at $x$ is exactly $n$.

For any point $x\in X$, there is a hyperplane
$H^{x}\subset{\mathbb{C}}{\mathbb{P}}^{d}$ such that the order of contact of
$X$ with $H^{x}$ at $x$ is at least $d-1$.

######  Definition 6.2.

We say that the map $\phi$ is nondegenerate at $x$ if the order of contact of
$X$, at $x$, with every hyperplane $H\subset{\mathbb{C}}{\mathbb{P}}^{d}$
containing $\phi(x)$ is at most $d-1$. Equivalently, $\phi$ is nondegenerate
at $x$ if there is no hyperplane $H\subset{\mathbb{C}}{\mathbb{P}}^{d}$ such
that the order of contact of $X$ with $H$ at $x$ is at least $d$. The map
$\phi$ is said to be nondegenerate if it is nondegenerate at every $x\in X$.

Consider now $X$ to be a compact connected Riemann surface and fix a point
$x_{0}\in X$. Given the universal cover
$\varpi\colon\widetilde{X}\longrightarrow X$ one has that $\pi_{1}(X,x_{0})$
acts on $\widetilde{X}$ as deck transformations. Consider all pairs of the
form $(\rho,\phi)$, where
$\rho\colon\pi_{1}(X,x_{0})\longrightarrow\text{SL}(r,{\mathbb{C}})$ is a
homomorphism and

$\phi\colon\ \widetilde{X}\longrightarrow{\mathbb{C}}{\mathbb{P}}^{r-1}$

is a holomorphic immersion such that

  * (1)

$\phi$ is nondegenerate, and

  * (2)

$\phi\circ\gamma=\rho(\gamma)\circ\phi$ for all $\gamma\in\pi_{1}(X,x_{0})$
(the standard action of $\text{SL}(r,{\mathbb{C}})$ on
${\mathbb{C}}{\mathbb{P}}^{r-1}$ is being used here).

######  Definition 6.3.

Two pairs $(\rho,\phi)$ and $(\rho_{1},\phi_{1})$ as above are called
equivalent if there is an element $G\in\text{SL}(r,{\mathbb{C}})$ such that

  * (1)

$\rho_{1}(\gamma)=G^{-1}\rho(\gamma)G$ for all $\gamma\in\pi_{1}(X,x_{0})$,
and

  * (2)

$\phi_{1}=G^{-1}\circ\phi$.

For any given $r\geq 2$, by ${\mathcal{M}}(r)$ we shall denote the space of
all equivalence classes of pairs $(\rho,\phi)$ of the above type. Recall that
given another base point $x^{\prime}_{0}\in X$, the fundamental group
$\pi_{1}(X,x^{\prime}_{0})$ is identified with $\pi_{1}(X,x_{0})$ uniquely up
to an inner automorphism. More precisely, a choice of a homotopy class of
paths between $x_{0}$ and $x^{\prime}_{0}$ identifies $\pi_{1}(X,x_{0})$ with
$\pi_{1}(X,x^{\prime}_{0})$ and it also identifies $\widetilde{X}$ with the
universal cover of $X$ corresponding to $x_{0}^{\prime}$. From these it
follows immediately that ${\mathcal{M}}(r)$ does not depend on the choice of
the base point $x_{0}$. Moreover, it is known that ${\mathcal{M}}(r)$ is
identified with the space of all ${\rm SL}(r,{\mathbb{C}})$-opers on $X$. This
identification is briefly described in the proof of Proposition 6.6.

###  6.1 Real structures

Consider as in previous sections an anti-holomorphic involution

$\sigma\colon\ X\longrightarrow X,$

and denote by

$\sigma_{*}\colon\ \pi_{1}(X,x_{0})\longrightarrow\pi_{1}(X,\sigma(x_{0}))$
(6.1)

the homomorphism induced by $\sigma$. Using the homomorphism
$(\sigma_{*})^{-1}$, the action of $\pi_{1}(X,x_{0})$ on $\widetilde{X}$
produces an action of $\pi_{1}(X,\sigma(x_{0}))$ on $\widetilde{X}$. Let

$\varpi^{\prime}\colon\ \widetilde{X}^{\prime}\longrightarrow X$

be the universal covering corresponding to the point $\sigma(x_{0})$. Note
whilst that there is no natural map between the universal covers
$\widetilde{X}^{\prime}$ and $\widetilde{X}$, the self-map $\sigma$ of $X$ has
a natural lift to the universal covering space

$\widetilde{\sigma}\colon\
\widetilde{X}\longrightarrow\widetilde{X}^{\prime}.$ (6.2)

######  Remark 6.4.

From the above constructions, one has the following:

  * (1)

$\varpi^{\prime}\circ\widetilde{\sigma}=\varpi$,

  * (2)

$\widetilde{\sigma}$ takes the base point in $\varpi^{-1}(x_{0})$ to the base
point in $(\varpi^{\prime})^{-1}(\sigma(x_{0}))$,

  * (3)

$\widetilde{\sigma}$ is an anti-holomorphic diffeomorphism, and

  * (4)

$\widetilde{\sigma}$ is $\pi_{1}(X,\sigma(x_{0}))$-equivariant (recall that
$\pi_{1}(X,\sigma(x_{0}))$ acts on $\widetilde{X}$).

######  Lemma 6.5.

The involution $\sigma$ on $X$ induces an involution $\Gamma$ on
${\mathcal{M}}(r)$ of all equivalence classes of pairs $(\rho,\phi)$.

###### Proof.

In what follows we shall describe an involution on the space
${\mathcal{M}}(r)$ of all equivalence classes of pairs $(\rho,\phi)$
constructed through Definition 6.3. Given a pair

$(\rho,\phi)\in{\mathcal{M}}(r),$

one can define a homomorphism

$\displaystyle\widehat{\rho}\colon\ \pi_{1}(X,\sigma(x_{0}))$
$\displaystyle\longrightarrow\text{SL}(r,{\mathbb{C}}),$ $\displaystyle\gamma$
$\displaystyle\mapsto\overline{\rho((\sigma_{*})^{-1}(\gamma))},$ (6.3)

where $\sigma_{*}$ is the homomorphism in (6.1). Moreover, one can also define
the map

$\displaystyle\widehat{\phi}\colon\ \widetilde{X}^{\prime}$
$\displaystyle\longrightarrow{\mathbb{C}}{\mathbb{P}}^{r-1},$ $\displaystyle
y$
$\displaystyle\mapsto\overline{\phi\bigl{(}\widetilde{\sigma}^{-1}(y)\bigr{)}},$
(6.4)

where $\widetilde{\sigma}$ is the map in (6.2). Now it is straightforward to
check that the pair $\bigl{(}\widehat{\rho},\widehat{\phi}\bigr{)}$
constructed in (6.3) and (6.4) defines an element of ${\mathcal{M}}(r)$. Let

$\Gamma\colon\ {\mathcal{M}}(r)\longrightarrow{\mathcal{M}}(r)$ (6.5)

be the map that sends any pair $(\rho,\phi)$ to
$\bigl{(}\widehat{\rho},\widehat{\phi}\bigr{)}$ constructed from it in (6.3)
and (6.4). It is evident that $\Gamma$ is an involution.∎

As mentioned before, ${\mathcal{M}}(r)$ is identified with the space of all
${\rm SL}(r,{\mathbb{C}})$-opers on $X$. Hence, the involution $\Gamma$
constructed in Lemma 6.5 via (6.5) is an involution of ${\rm OSL}_{X}(r)$ (see
Definition 4.6) and it can be understood in terms of the involutions studied
in previous sections.

######  Proposition 6.6.

The involution $\Gamma$ of ${\rm OSL}_{X}(r)$ coincides with the involution
$\beta$ constructed in Lemma 4.10.

###### Proof.

This is deduced by examining the identification between ${\rm OSL}_{X}(r)$ and
${\mathcal{M}}(r)$. The map ${\mathcal{M}}(r)\longrightarrow{\rm OSL}_{X}(r)$
is constructed as follows. Take any pair $(\rho,\phi)$ giving an element of
${\mathcal{M}}(r)$. Let

$E_{0}={\mathbb{C}}{\mathbb{P}}^{r-1}\times{\mathbb{C}}^{r}\longrightarrow{\mathbb{C}}{\mathbb{P}}^{r-1}$

be the trivial holomorphic vector bundle equipped with the unique holomorphic
connection $D_{0}$ on $E_{0}$ (it is the trivial connection). The vector
bundle $E_{0}$ has a tautological holomorphic line subbundle

${\mathcal{L}}={\mathcal{O}}_{{\mathbb{C}}{\mathbb{P}}^{r-1}}(-1)\subset
E_{0},$ (6.6)

whose fiber over any $z\in{\mathbb{C}}{\mathbb{P}}^{r-1}$ is the line in
${\mathbb{C}}^{r}$ represented by $z$.

The homomorphism
$\rho\colon\pi_{1}(X,x_{0})\longrightarrow\text{SL}(r,{\mathbb{C}})$ and the
standard action of $\text{SL}(r,{\mathbb{C}})$ on
${\mathbb{C}}{\mathbb{P}}^{r-1}$ together produce an action of
$\pi_{1}(X,x_{0})$ on ${\mathbb{C}}{\mathbb{P}}^{r-1}$. Consider the diagonal
action of $\pi_{1}(X,x_{0})$ on the pulled back vector bundle

$\phi^{*}E_{0}=\widetilde{X}\times{\mathbb{C}}^{r}\longrightarrow\widetilde{X}.$

This action of $\pi_{1}\\!(X,x_{0})$ on $\phi^{*}E_{0}$ preserves the
holomorphic line subbundle $\phi^{*}{\mathcal{L}}\\!\subset\\!\phi^{*}E_{0}$
(see (6.6)). Moreover, the action of $\pi_{1}(X,x_{0})$ on $\phi^{*}E_{0}$
preserves the pulled back connection $\phi^{*}D_{0}$. Therefore, one can
define the vector bundle

${\mathcal{V}}:=(\phi^{*}E_{0})/\pi_{1}(X,x_{0})\longrightarrow\widetilde{X}/\pi_{1}(X,x_{0})=X,$
(6.7)

which is a holomorphic vector bundle equipped with

  * •

a holomorphic connection $D$ induced by $\phi^{*}D_{0}$, and

  * •

a holomorphic line subbundle

$\widetilde{\mathcal{L}}:=(\phi^{*}{\mathcal{L}})/\pi_{1}(X,x_{0})\subset(\phi^{*}E_{0})/\pi_{1}(X,x_{0})={\mathcal{V}}.$

The holomorphic section of the line bundle
$\bigwedge^{r}(\phi^{*}E_{0})=\widetilde{X}\times{\mathbb{C}}\longrightarrow\widetilde{X}$
given by the constant function $1$ on $\widetilde{X}$ is preserved by the
action of $\pi_{1}(X,x_{0})$ on $\bigwedge^{r}(\phi^{*}E_{0})$ induced by the
action of $\pi_{1}(X,x_{0})$ on $\phi^{*}E_{0}$. Furthermore, this constant
section is preserved by the connection on $\bigwedge^{r}(\phi^{*}E_{0})$
induced by the connection $\phi^{*}D_{0}$ on $\phi^{*}E_{0}$. Consequently,
the vector bundle ${\mathcal{V}}$ in (6.7) defines a principal
$\text{SL}(r,{\mathbb{C}})$-bundle on $X$, and $D$ is a holomorphic connection
on this principal $\text{SL}(r,{\mathbb{C}})$-bundle. From the non-degeneracy
condition of the map $\phi$ it can be deduced that the line subbundle
$\widetilde{\mathcal{L}}$ produces an $\text{SL}(r,{\mathbb{C}})$-oper
structure on this flat principal $\text{SL}(r,{\mathbb{C}})$-bundle. This way
we obtain the map ${\mathcal{M}}(r)\longrightarrow{\rm OSL}_{X}(r)$.

The converse map ${\rm OSL}_{X}(r)\longrightarrow{\mathcal{M}}(r)$ is fairly
straightforward. Let $F$ be a holomorphic vector bundle of rank $r$ on $X$
such that the line bundle $\bigwedge^{r}F$ is holomorphically trivial, and let
$D$ be a holomorphic connection on $F$ such that the induced connection on
$\bigwedge^{r}F$ is the trivial one. Let ${\mathbb{L}}\subset F$ be a
holomorphic line subbundle such that the triple $(F,D,{\mathbb{L}})$ defines
an $\text{SL}(r,{\mathbb{C}})$-oper on $X$. Let

$\rho\colon\ \pi_{1}(X,x_{0})\longrightarrow\text{SL}(F_{x_{0}})$ (6.8)

be the monodromy representation for $D$.

Since $\widetilde{X}$ is simply connected, the pulled back connection
$(\varpi^{*}F,\varpi^{*}D)$ is trivializable, where $\varpi$ is the projection
of $\widetilde{X}$ to $X$. Let $\widetilde{x}_{0}\in\widetilde{X}$ be the base
point over the chosen base point $x_{0}\in X$. So $\varpi^{*}F$ is identified
with the trivial vector bundle ${\widetilde{X}}\times
F_{x_{0}}\longrightarrow{\widetilde{X}}$ and $\varpi^{*}D$ is the trivial
connection on it. The line subbundle

$\varpi^{*}{\mathbb{L}}\subset\varpi^{*}F={\widetilde{X}}\times F_{x_{0}}$

produces a map

$\psi\colon\ {\widetilde{X}}\longrightarrow{\mathbb{P}}(F_{x_{0}});$

the map $\psi$ sends any $z\in{\widetilde{X}}$ to the line
$(\varpi^{*}{\mathbb{L}})_{z}\subset(\varpi^{*}F)_{z}=F_{x_{0}}$. Now the pair
$(\rho,\psi)$, where $\rho$ is the homomorphism in (6.8), defines an element
of ${\mathcal{M}}(r)$. This construction gives the opposite map ${\rm
OSL}_{X}(r)\longrightarrow{\mathcal{M}}(r)$.

Using the above isomorphism ${\rm
OSL}_{X}(r)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{M}}(r)$
it is straightforward to deduce that the involution $\Gamma$ of ${\rm
OSL}_{X}(r)$ coincides with the involution $\beta$ constructed in Lemma 4.10.
∎

Proposition 6.6, Theorem 4.14 and Proposition 5.4 together give the following.

######  Theorem 6.7.

The four involutions $\Gamma$, $\beta$, $\mathcal{B}$ and ${\mathcal{A}}$ of
${\rm OSL}_{X}(r)$ coincide.

### Acknowledgements

We thank the referees for their valuable comments. The work of LPS is
partially supported by a Simons Fellowship, NSF CAREER Award DMS 1749013 and
NSF FRG Award 2152107. The material presented here is partially based upon
work supported by the National Science Foundation under Grant No. DMS-1928930
while LPS was in residence at the Mathematical Sciences Research Institute in
Berkeley, California, during the Fall 2022 semester.

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