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

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.


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 SL(r, C)-opers on X, which we denote by OSL X (r), admits several diverse, but equivalent, descriptions.In the present paper, starting with an anti-holomorphic involution σ (a real structure) on a compact Riemann surface X, we construct the induced action of Z/2Z on various spaces associated to X: an involution F on the space of projective structures on X (see Lemma 2.5), an involution A 1 induced by F on the space of projective structures on X times the space of holomorphic differentials (5.14), an involution β on the space of holomorphic differential operators (see Lemma 4.10), an involution B on the space of vector bundles with connections (see Lemma 4.13), and an involution Γ on the space of equivariant projective embeddings of the universal cover (see Lemma 6.5).
The fixed point locus in OSL X (r) for the action of Z/2Z on it will be called the real slice of SL(r, C)-opers.
Along the paper, we first describe the induced involution from the point of view of just one description of OSL X (r).Then the space OSL X (2) of SL(2, C)-opers on X is identified with the space of all projective structures on X.The anti-holomorphic involution σ of X produces an involution of the space of projective structures on X.Also σ produces a conjugate linear involution of H 0 X, K ⊗i X for every i ≥ 0 by sending any ω ∈ H 0 X, K ⊗i X to σ * ω.On the other hand, we have an isomorphism Theorem A. The above isomorphism Ψ takes the involution of OSL X (r) given by σ to the involution of OSL X (2)× r i=3 H 0 X, K ⊗i X given by the diagonal action of Z/2Z on the product space.In particular, an element η, is in the real slice of SL(r, C)-opers if and only if η is in the real slice of SL(2, C)-opers and ω i = σ * ω i for all 3 ≤ i ≤ r.
The above mentioned Z/2Z actions produce involutions of OSL X (r) from four different points of view, and we finish the paper by showing that all these different involutions of OSL X (r) actually coincide, leading to our second main result.
Theorem B. The four involutions Γ, β, B and 2 Projective structures

Involution of the space of projective structures
The standard action of the group GL(2, C) on C 2 produces an action of PGL(2, C) on the complex projective line CP 1 , giving an identification of PGL(2, C) with the holomorphic automorphism group Aut CP 1 of CP 1 .Through this identification, one can define a projective structure on an oriented C ∞ surface in the following fashion.
Let S be a compact oriented C ∞ surface of genus g, with g ≥ 2. Recall that a coordinate chart on S is a pair (U, φ), where U ⊂ S is an open subset and φ : U −→ CP 1 is an orientation preserving C ∞ embedding, and a coordinate atlas is a collection of coordinate charts {(U i , φ i )} i∈I such that i∈I U i = S. Definition 2.1.A projective structure on S is an equivalence class of {(U i , φ i )} i∈I such that for all i, j ∈ I, and every nonempty connected component U c ij ⊂ U i ∩ U j , there is an element such that the restriction of the maps φ i • φ −1 j to φ j (U c ij ) coincides with the restriction of the automorphism T c ij ∈ Aut CP 1 .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 CP 1 given by the elements of PGL(2, C) are holomorphic.In what follows we shall denote by 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 ∞ 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 for z i ∈ C, and let be an orientation reversing diffeomorphism of S such that σ • σ = Id S .
Lemma 2.2.Given a projective structure P on S defined by a coordinate atlas the corresponding atlas also defines a projective structure on S, where σ 0 and σ are the involutions in (2.3) and (2.4) respectively.This produces an involution of the space P(S) in (2.2).
Proof .Given any i, j ∈ I and any nonempty connected component is holomorphic, and it is the restriction of the global holomorphic automorphism Consequently, we get a map on the space of all projective structures on S σ : P(S) −→ P(S) (2.5) that sends any projective structure defined by a coordinate atlas {(U i , φ i )} i∈I to the one defined by the coordinate atlas {(σ From the construction of σ it follows immediately that σ • σ = Id P(S) .■ Remark 2.3.The map σ defined in (2.5) is independent of the choice of the anti-holomorphic involution σ 0 of CP 1 , but in general it depends on σ.Indeed, in its construction, one may replace σ 0 (defined in (2.3)) by any other anti-holomorphic involution σ ′ 0 of CP 1 without changing the map σ because σ ′ 0 and σ 0 differ by an element of PGL(2, C) = Aut CP 1 .
We shall denote the group of orientation preserving diffeomorphisms of S by Diff + (S).Let Diff 0 (S) ⊂ Diff + (S) be the connected component of it consisting of diffeomorphisms homotopic to the identity map of S; it is a normal subgroup of Diff + (S).The mapping class group can then be seen as the quotient MCG(S) := Diff + (S)/Diff 0 (S).
Moreover, considering the space C(S) of complex structures on S compatible with its C ∞ structure and orientation, the Teichmüller space can be seen as the quotient T (S) := C(S)/Diff 0 (S), (2.6) which is a complex manifold of dimension 3(g − 1).The group Diff + (S) acts on P(S) as follows.Let P be a projective structure on S given by {(U i , φ i )} i∈I .Then the action of any ψ ∈ Diff + (S) sends it to the projective structure ψ • P defined by ψ(U i ), φ i • ψ −1 i∈I .Consider the corresponding quotient space P(S) := P(S)/Diff 0 (S), (2.7) which is a complex manifold of dimension dim P(S) = 6(g − 1) = 2 dim T (S).Since a projective structure on S produces a complex structure on S, we have a map where P(S) and 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 Φ in (2.8) is surjective.In fact, Φ is a holomorphic map, and P(S) is a holomorphic torsor over T (S) for the holomorphic cotangent bundle T * T (S) (see [15,16,19]).
Lemma 2.5.The map σ in (2.5) produces an anti-holomorphic involution Proof .Since Diff 0 (S) is the connected component of Diff(S) containing the identity element, it is a normal subgroup of Diff(S).Moreover, Diff + (S) is also a normal subgroup of Diff(S), and Diff(S)/Diff + (S) = Z/2Z.Take any ψ ∈ Diff 0 (S) and any projective structure P on S given by {(U i , φ i )} i∈I .Then the projective structure σ(ψ Since σ • ψ • σ ∈ Diff 0 (S) (recall that Diff 0 (S) is a normal subgroup), we conclude σ produces a self-map F of P(S).This map F is evidently an involution.Also F is clearly antiholomorphic.■ For any C ∞ complex structure J : T S −→ T S on S compatible with its orientation, the pullback −σ * J is a C ∞ complex structure on S compatible with its orientation, and thus there is an induced anti-holomorphic involution (2.10) For f and F the involutions constructed in (2.10) and (2.9), respectively, one can see that where Φ is the projection in (2.8).Given any Riemann surface X∈ T (S), the fiber Φ −1 (X) ⊂ P(S) is an affine space for H 0 X, K ⊗2 X , where K X is the holomorphic cotangent bundle of X [15,19].Moreover, using the isomorphism P(S) is realized as a holomorphic affine bundle over T (S) for the holomorphic vector bundle T * T (S) [15,19].The involution F satisfies the equation for all P ∈ Φ −1 (X) and ω ∈ H 0 X, K ⊗2 X , where σ * ω is the corresponding holomorphic quadratic differential on the Riemann surface f (X) ∈ T (S).

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.

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 ∈ S, recall that a homomorphism ρ : be the space of all irreducible homomorphisms.Definition 3.1.The irreducible PSL(2, C)-character variety of S is the quotient space for the action of PSL(2, C) on Hom (π 1 (S, x 0 ), PSL(2, C)) ir given by the conjugation action of PSL(2, C) on itself.
It should be mentioned that R 2 (S) is independent of the choice of the base point x 0 .In fact, R 2 (S) parametrizes the isomorphism classes of flat PSL(2, C)-connections on S.
The space R 2 (S) has two connected components, which are parametrized by the second Stiefel-Whitney class of the projective bundles associated to the PSL(2, C)-bundles on S. Also each connected component of 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 PSL(2, C) and the fact that the group π 1 (S, x 0 ) is finitely presented.Moreover, the character variety R 2 (S) is a smooth orbifold and is equipped with an algebraic symplectic structure [2,12], which we shall denote by The involution σ of S in (2.4) induces an algebraic involution where ∇ is a flat connection on a principal PSL(2, C)-bundle E, equivalently, I 2 sends any homomorphism is an anti-holomorphic involution of R 2 (S) (recall that the map I 2 is holomorphic).
■ Giving a projective structure on a Riemann surface X is equivalent to giving a holomorphic fiber bundle p : Z −→ X, a holomorphic connection ∇ on the fiber bundle and a holomorphic section s : X −→ Z of p, such that the fibers of p are isomorphic to CP 1 , and s is transversal to the horizontal distribution for the connection ∇ (see [15,19] for details).Therefore the fiber bundle Z produces a holomorphic principal PSL(2, C)-bundle P over X; the fiber of P over any x ∈ X is the space of all holomorphic isomorphisms from CP 1 to the fiber p −1 (x).The connection ∇ produces a holomorphic connection on P. The monodromy representation π 1 (X, x 0 ) −→ PSL(2, C) for ∇ is irreducible.
Lemma 3.3.The symplectic form Θ 2 on the character variety R 2 (S) defined in (3.2) induces a natural symplectic form Θ P on the quotient space of projective structures P(S) introduced in (2.7).
Proof .We have seen above that a projective structure on S defines a flat PSL(2, C)-connection on S. Therefore, assigning the monodromy representation to the flat connections we get a map from the quotient space of projective structures P(S) introduced in (2.7) to the PSL(2, C)character variety R 2 (S) constructed in (3.1).It is known that µ is a local biholomorphism (see [19, p. 272] and [16]) and thus the pullback of the symplectic form in (3.2) is a holomorphic symplectic form on P(S).■ Remark 3.4.From the constructions of F and h in (2.9) and (3.4) respectively, it follows that where µ is the monodromy map in (3.5).

Real projective structures
A projective structure P ∈ P(S) is called real if it is a fixed point of the anti-holomorphic involution introduced in (2.9).Moreover, given a real projective structure P ∈ P(S), from (2.11) one obtains an induced real point Φ(P ) ∈ T (S) with respect to the anti-holomorphic involution f , meaning The following lemma is a sort of converse of it.
Lemma 3.6.Given any J ∈ T (S) such that f (J) = J, there exists a real projective structure P such that Φ(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 In what follows we shall denote by T (S) f ⊂ T (S) and P(S) F ⊂ P(S) the fixed point loci for the involutions f and F respectively.The fixed point locus be the restriction of the map Φ in (2.8); as noted above, from (2.11) it follows immediately that Φ maps P(S) F to T (S) f .From Lemma 3.6, it follows that the map Φ 0 is surjective.
The differential df of the anti-holomorphic involution f of T (S) produces an anti-holomorphic involution of the holomorphic cotangent bundle T * T (S).Let be the fixed point locus of this involution.This T * T (S) f is a C ∞ real vector bundle over T (S) f .We note that T * T (S) f is identified with the (real) cotangent bundle T * T (S) f of T (S) f .Lemma 3.7.The space P(S) F is an affine bundle over T (S) f for the real vector bundle Proof .Given any J ∈ T (S) f , let X be the corresponding Riemann surface (S, J), and denote the holomorphic cotangent bundle of X by K X .Let The fiber over X of the map Φ 0 : P(S) F −→ T (S) f in (3.8) is actually an affine space for the real vector space H 0 X, K ⊗2 X σ .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 X, K ⊗2 X , and (2.12).We note that Consequently, P(S) F is an affine bundle over T (S) f for the real vector bundle T * T (S) f as needed.■ Proposition 3.8.The manifold P(S) F is connected, and it is homeomorphic to the Euclidean space.
Proof .The manifold T (S) f is connected, and it is homeomorphic to the Euclidean space (see [10, p. 122] and [20]).On the other hand, P(S) F is an affine bundle over T (S) f for the vector bundle T * 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 Θ F P be the restriction of the holomorphic symplectic form Θ P to P(S) F .Then the real part Re Θ F P satisfies the equation Re Θ F P = 0, and the imaginary part Im Θ F P is a real symplectic form on P(S) F .

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 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 2π-grafting of the projective structure obtained from the uniformization of an appropriate Riemann surface.

Real and quaternionic SL(r, 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 SL(r, C)-opers.

Real and quaternionic bundles
As in the previous sections, consider a pair (X, σ X ) of a compact connected Riemann surface X of genus g ≥ 2, and an anti-holomorphic involution For any complex vector bundle E on X, let E denote the complex vector bundle on X whose underlying real vector bundle is identified with E, while the multiplication by √ −1 on E coincides with the multiplication by − √ −1 on E.
Lemma 4.1.For any holomorphic vector bundle V on X, there is a holomorphic structure on the C ∞ vector bundle σ * X V given by the holomorphic structure on V .Proof .We shall construct the holomorphic structure on σ * X V in the following way.Consider a C ∞ section s of σ * X V defined over an open subset U ⊂ X.So s gives a C ∞ section of V over σ X (U ).As V = V as real bundles, this section of V over σ X (U ) gives a C ∞ section of V , which we denote by s, over σ X (U ).The holomorphic structure on σ * X V is defined by the following condition: Any section s of σ * X V as above is holomorphic if and only if the corresponding section s of V is holomorphic. ■ Recall that a theta characteristic on a compact connected Riemann surface Y is a holomorphic line bundle 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 Remark 2]).Take a holomorphic line bundle L on X such that σ * X L is holomorphically isomorphic to L. For any holomorphic isomorphism h : L −→ σ * X L, the composition of homomorphisms σ * X h • h is a holomorphic automorphism of L. Multiplying h by a suitable nonzero complex number, we may ensure that σ the following two hold: (1) Assume that σ X does not have any fixed point.If g is even, then K (2) Assume that σ 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 K 1/2 X * by L, and thus one has that L is isomorphic to σ * X L. We note that L is real (respectively, quaternionic) if and only if K 1/2 X is real (respectively, quaternionic).

Definition of opers
We shall briefly recall here the construction of the jet bundles J m (W ), m ≥ 0, of a holomorphic vector bundle W on X.For i = 1, 2, let be the projection to the i-th factor, and let be the reduced diagonal divisor.Then for any m ≥ 0, one can define the associated Jet bundle as The vector bundle J m (W ) fits in the short exact sequence of holomorphic vector bundles Using (4.4), it is straightforward to deduce that there is a natural holomorphic isomorphism Lemma 4.3.Take a holomorphic line bundle L on X such that L ⊗ L = T X.The determinant line bundle det Sym r−1 J 1 (L) of the (r − 1)-th symmetric product is Proof .From (4.5), it follows that the line bundle det Since det Sym r−1 J 1 (L) = det J 1 (L) ⊗r(r−1)/2 , the isomorphism in (4.7) follows from (4.8) for every r ≥ 2. ■ Via the notion of jet bundles, we can now recall from [5] the definition of an SL(r, C)-oper on Riemann surfaces.Definition 4.4.An SL(r, C)-oper on a Riemann surface X of genus g ≥ 2 is a holomorphic connection D on Sym r−1 J 1 (L) such that the connection on det Sym r−1 J 1 (L) induced by D is the trivial connection on O X .Remark 4.5.Let L ′ be another holomorphic line bundle on X such that L ′ ⊗ L ′ = T X.So the holomorphic line bundle ξ := L ′ ⊗ L * satisfies the equation ξ ⊗ ξ = O X .Therefore, ξ ⊗ ξ 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 ξ on ξ such that the connection on ξ ⊗ ξ induced by D ξ coincides with D 0 .Using D ξ , there is a canonical isomorphism To construct ϕ ξ , let ξ denote the locally constant sheaf on X defined by the sheaf of flat sections of ξ for the connection D ξ .Consider the natural homomorphism This homomorphism extends uniquely to ϕ ξ .Note that ϕ ξ produces a holomorphic isomorphism The connection D ξ on ξ induces a connection D r ξ on ξ ⊗r .Therefore, the isomorphism ϕ r ξ in (4.9) produces a bijection between the space of holomorphic connections on Sym r−1 J 1 (L) and the space of holomorphic connections on Sym r−1 J 1 (L ′ ) .Consequently, the space of holomorphic connections D on Sym r−1 J 1 (L) such that the connection on det Sym r−1 J 1 (L) induced by D is the trivial connection on O X (see Definition 4.4) does not depend on the choice of the line bundle L.
Let C X (r) denote the space of all holomorphic connections on Sym r−1 J 1 (L) such that their induced connection on det Sym r−1 J 1 (L) is the trivial connection on O X .Let Aut Sym r−1 J 1 (L) denote the group of all holomorphic automorphisms of Sym r−1 J 1 (L) .If D is a holomorphic connection on Sym r−1 J 1 (L) such that the connection on the line bundle det Sym r−1 J 1 (L) induced by D is the trivial connection, and also has the property that the connection on det Sym r−1 J 1 (L) induced by it is the trivial connection.Indeed, this follows immediately from the fact that the holomorphic automorphisms of a line bundle ξ act trivially on the space of all holomorphic connections on ξ.
Definition 4.6.The moduli space of SL(r, C)-opers on X, which will be denoted by OSL X (r), is defined as follows OSL X (r) := C X (r)/Aut Sym r−1 J 1 (L) . (4.10) Using the isomorphism ϕ r ξ in (4.9), we conclude that Therefore, from Remark 4.5 it follows that OSL X (r) in Definition 4.6 is independent of the choice of the line bundle L.
Remark 4.7.We need to explain the reason for the assumption that the genus of X is at least two.On CP 1 there is no nontrivial SL(r, C)-oper.If genus(X) = 1, then the underlying holomorphic vector bundle for an SL(r, 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 genus(X) = 1, both O X ⊕ O X and the unique nontrivial extension of O X by O X admit an SL(2, C)-oper structure.

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 A holomorphic differential operator of order d from W to V is a holomorphic section of the vector bundle Diff d X (W, V ).Consider the homomorphism K ⊗d X ⊗ W → J d (W ) in (4.5).Using its dual we get a surjective homomorphism which is known as the symbol map.
The following notation will be used: For a line bundle ξ on X and a positive integer d, the line bundle (ξ * ) ⊗d will be denoted by ξ ⊗−d .Also ξ 0 will denote the trivial line bundle O X .
When considered on Diff r X L ⊗(r−1) , L −r−1 , the symbol map in (4.11) sends it to where we use for a holomorphic line bundle A and a positive integer d the convention The following proposition is well known.
Proposition 4.8.The moduli space of SL(r, C)-opers on X in (4.10) is isomorphic to the subset consisting of all differential operators δ ∈ H 0 X, Diff r X L ⊗(r−1) , L ⊗(r+1) * such that the symbol of δ is the constant function 1 on X and the sub-leading term of δ is vanishing identically.
Remark 4.9.One should note that the above condition in Proposition 4.8 that the sub-leading term of δ 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 SL(r, C)-system (meaning the r-th exterior product of it is the constant sheaf with stalk C).
We shall briefly describe here the isomorphism OSL X (r) in Proposition 4.8 in more detail.Given a differential operator such that the symbol of δ 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 (L).In other words, the sheaf of solutions of δ produces a holomorphic connection D on J r−1 (L).The holomorphic vector bundle J r−1 (L) is holomorphically isomorphic to Sym r−1 J 1 (L) .After fixing a holomorphic isomorphism between J r−1 (L) and Sym r−1 J 1 (L) , the holomorphic connection on D on J r−1 (L) gives a holomorphic connection on Sym r−1 J 1 (L) .
To describe the reverse map take a holomorphic connection D on Sym r−1 J 1 (L) such that the connection on the line bundle det Sym r−1 J 1 (L) induced by D is the trivial connection on O X .Using D, we will construct a holomorphic isomorphism between Sym r−1 J 1 (L) and J r−1 (L).Consider the projection , which is the (r − 1)-th symmetric power of the natural projection J 1 (L) −→ L (see the exact sequence in (4.5)).Take any x ∈ X and v ∈ Sym r−1 J 1 (L) x .Let v denote the unique flat section of Sym r−1 J 1 (L) U , for the connection D, defined on a simply connected open neighborhood x ∈ U ⊂ X, such that v(x) = v.Restricting ψ( v) to the (r − 1)-th order infinitesimal neighborhood of x we get an element v ′ ∈ J r−1 L ⊗(r−1) x .Consequently, we have a holomorphic map which is in fact an isomorphism.Let ψ −1 * D be the holomorphic connection on the vector bundle J r−1 L ⊗(r−1) obtained by pulling back the connection D on Sym r−1 J 1 (L) using ψ −1 .Then there is a unique such that the symbol of δ is the constant function 1 on X, and the flat vector bundle on X given by the sheaf of solutions of δ is isomorphic to ψ −1 * D on J r−1 L ⊗(r−1) equipped with the above flat connection.
Since ψ −1 * D is a holomorphic SL(r, C) connection, the sub-leading term of δ vanishes identically as required.

Real and quaternionic slices of SL(r, C)-opers
As before, L is a holomorphic line bundle on X with L ⊗ L = T X.The Riemann surface is now equipped with an anti-holomorphic involution σ X .
In order to define and study real and quaternionic slices of SL(r, 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 , where σ X is the anti-holomorphic involution in (4.1).Note that such pairs K, σ exist.The isomorphism σ in (4.15) above produces a holomorphic isomorphism for all m ≥ 0 and k ∈ Z, where the second isomorphism in (4.16) is given by (4.6).Moreover, since σ * X σ • σ ∈ ±Id L (see (4.15)), it follows that (4.17) The homomorphism σ 1 1 defined through (4.16) produces, in turn, a holomorphic isomorphism of symmetric products From (4.17) for m = 1, we deduce that Observe that σ * X Sym r−1 σ 1 1 • Sym r−1 σ 1 1 = Id Sym r−1 (J 1 (L)) whenever r is an odd integer.Recall that C X (r) denotes the space of all holomorphic connections on Sym r−1 J 1 (L) such that their induced connection on det Sym r−1 J 1 (L) is the trivial connection on O X .Let be the anti-holomorphic map that sends any holomorphic connection D to the holomorphic connection on Sym r−1 J 1 (L) given by the holomorphic connection Sym r−1 σ 1 1 * σ * X D through the isomorphism in (4.18),where σ * X D is the connection on σ * X J 1 (L) induced by D. It can be shown that γ in (4.19) is an involution.Indeed, the automorphism of Sym r−1 J 1 (L) given by multiplication by −1 preserves any connection on Sym r−1 J 1 (L) .So even if the holomorphic line bundle L is quaternionic, the map γ remains to be of order two.
The group Z/2Z acts on C X (r) via the involution γ in (4.19), and the group of automorphisms Aut Sym r−1 J 1 (L) also acts on C X (r) through (4.10).On the other hand, there is an antiholomorphic group automorphism of order two is the automorphism given by β.
Lemma 4.10.The action of Z/2Z on C X (r) given by the involution γ in (4.19) descends to an involution of Z/2Z on the quotient space of SL(r, C)-opers OSL X (r) (see Definition 4.6).
Proof .The lemma follows from the above construction of γ and from noticing that the actions of Z/2Z and Aut Sym r−1 J 1 (L) on C X (r) are related as follows: In order to study the slice of real SL(r, C)-opers on X, we shall construct a conjugate linear involution of H 0 X, Diff r X L r−1 , L −r−1 .

Lemma 4.13.
There is a natural conjugate linear involution induced by the anti-holomorphic involution σ of X.
Proof .Given any δ ∈ H 0 X, Diff r X L r−1 , L −r−1 , the holomorphic differential operator on σ * X L r−1 defined by σ * δ : J r σ * X L r−1 −→ σ * X L −r−1 sends any locally defined holomorphic section s of L r−1 to σ * δ(s); recall that the holomorphic sections of σ * X L r−1 over an open subset U ⊂ X are identified with the holomorphic sections of L r−1 over σ X (U ).Consider the diagram where σ r r−1 and σ 0 −r−1 are the holomorphic isomorphisms in (4.16), which needs not be commutative.This diagram shows that The map B from (4.22) can then be defined as the map that sends any δ to the differential operator σ 0 r−1 constructed above from it.Finally, it should be noted that B is an involution even if the holomorphic line bundle L is quaternionic.■ Theorem 4.14.The involution B in (4.22) preserves the subset OSL X (r) in Proposition 4.8.
The restriction of the map B to this subset OSL X (r) coincides with the involution β in (4.21).
Proof .Recall from Proposition 4.8 that OSL X (r) is the locus of all such that the symbol of δ is the constant function 1 and the sub-leading term of δ vanishes identically.Therefore, it is evident that B preserves the subset OSL X (r).Given a holomorphic connection D on Sym r−1 J 1 (L) such that the induced connection on det Sym r−1 J 1 (L) is the trivial connection on O X , through the map ψ in (4.13) we have (see (4.18) and (4.16) for Sym r−1 σ 1 1 and σ r−1 r−1 , respectively).Moreover, from (4.14) there is a unique differential operator δ such that the symbol of δ is the constant function 1 on X, and the flat vector bundle on X given by the sheaf of solutions of δ is isomorphic to J r−1 L ⊗(r−1) equipped with the above connection ψ −1 * D.
Since δ 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 SL(r, C)-opers defined above.Proposition 4.15.An element of the moduli space of SL(r, C)-opers is in the real slice of SL(r, C)-opers if and only if B(δ) = δ, where B is the involution defined in (4.22).

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

Anti-holomorphic involutions on differential operators
Consider as in previous sections (see (4.2)) the projections Then, given two holomorphic vector bundles A and B on X, we shall denote the holomorphic vector bundle (p * 1 A) ⊗ (p * 2 B) on X × X by A ⊠ B. These holomorphic vector bundles produce the natural vector bundles and on X × X, where ∆ is the diagonal divisor defined in (4.3).These vector bundles fit in a short exact sequence of coherent sheaves on X × X given by where the support of Q d (A, B) in (5.2) is the non-reduced divisor (d + 1)∆.The direct image 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 For d ≥ 1, the sheaf Q d (A, B) in (5.2) fits in the following short exact sequence of sheaves on X × X, where the abbreviations of (5.1) are used, Note that the above sheaf J d /J d−1 is supported on the reduced divisor ∆.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 From Poincaré adjunction formula, one has that O X×X (∆) ∆ is the normal bundle of ∆ [14, p. 146], and therefore O X×X (∆) ∆ = T X, using the identification of ∆ with X defined by x −→ (x, x).Hence, we have Remark 5.1.The isomorphism in (5.4) and the projection p 0 in (5.6) together produce a homomorphism which is the symbol map on differential operators constructed in (4.11).
Recall from Section 4.1 that L := K 1/2 X * , and consider now an element of the moduli space of SL(r, C)-opers on X given by (5.8) From (5.3) and (5.4), we know that δ corresponds to a section (5.9) Note that the restriction of and that J r = L −r−1 ⊠ L −r−1 O X×X ((r + 1)∆) for the vector bundles in (5.1) defined as Lemma 5.2.There is a natural section the restriction of S δ to ∆ ⊂ X × X coincides with the section of O ∆ given by the constant function 1, and S δ 2∆ is anti-invariant under the involution of 2∆ obtained by restricting the involution of X × X defined by (x 1 , x 2 ) −→ (x 2 , x 1 ), in other words, this involution takes S δ 2∆ to −S δ 2∆ .
Proof .This follows from the above analysis, by noting that the given condition that the symbol of δ is the constant function 1 is equivalent to the statement that the restriction of the section S δ (constructed in (5.9)) to ∆ ⊂ X × X is the section of O ∆ given by the constant function 1 (see (5.7)).The given condition that the sub-leading term of δ vanishes identically is equivalent to the condition that the section S δ 2∆ is anti-invariant under the involution of 2∆ obtained by restricting the involution of X × X defined by (x 1 , x 2 ) −→ (x 2 , x 1 ).■ We will show that the section of J r for r = 2 given by (S δ ) ⊗3 ∈ H 0 (3∆, J 2 ) (5.10) naturally defines a projective structure on the Riemann surface X.
Proof .From (5.3) and (5.4), we know that (S δ ) ⊗3 in (5.10) gives a differential operator (5.11) Since the restriction of S δ to ∆ ⊂ X × X coincides with the section of O ∆ given by the constant function 1, we conclude that the symbol of the differential operator δ 0 is the constant function 1 (see (5.7)).Since S δ 2∆ is anti-invariant under the involution of 2∆ obtained by restricting the involution of X × X defined by (x 1 , x 2 ) −→ (x 2 , x 1 ), we conclude that the sub-leading term of the differential operator δ 0 vanishes identically.Therefore, we have (see Proposition 4.8).Since the space OSL X (2) is identified with the space of projective structures on X, the differential operator δ 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 δ 0 in (5.11).The underlying holomorphic vector bundle for it is J 1 (L).The holomorphic connection on Sym r−1 J 1 (L) induced by D 0 will be denoted by Sym r−1 (D 0 ), and from (4.13) it produces a holomorphic isomorphism The holomorphic connection on J r−1 L r−1 given by Sym r−1 (D 0 ) using this isomorphism will be denoted by Sym r−1 (D 0 ) ′ .
For D denote the holomorphic connection on J r−1 L r−1 given by the sheaf of solutions of the differential operator δ in (5.8), we shall define Taking the trace for all 1 ≤ i ≤ r given by trace (θ δ ) i ∈ H 0 X, K ⊗i X , (5.12) note that trace(θ δ ) = 0 = trace (θ δ ) 2 .

Real slices of fibrations
In search of an interesting parametrization of real slices of SL(r, C)-opers similar to those one can define for Higgs bundles though Hitchin's integrable system [18], consider the map H 0 X, K ⊗i X that sends any differential operator δ as defined in (4.8) to δ 0 , r i=3 trace (θ δ ) i , where δ 0 ∈ OSL X (2) is constructed in (5.11) and the traces trace (θ δ ) i are the sections in (5.12).
Consider an anti-holomorphic involution σ 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 From Theorem 4.14, in the case when r = 2 this F X coincides with B constructed in (4.22), and β constructed in (4.21).On the other hand, the differential dσ : T R X −→ σ * T R X produces a holomorphic involution T X −→ σ * T X.Through these maps, one can define an anti-holomorphic linear automorphism of ω −→ σ * X ω of H 0 X, K ⊗i X .Then there is a natural involution 6 Nondegenerate immersions into a projective space Let X be a connected Riemann surface and consider a holomorphic immersion ϕ : X −→ CP d .
Given a point x ∈ X and a hyperplane H ⊂ CP d such that ϕ(x) ∈ 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 ϕ(x) CP d such that h U ∩H = 0, the order of vanishing of h • ϕ 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 ϕ(x) ∈ CP d such that h U ∩H = 0 and the order of vanishing of h • ϕ at x is exactly n.
For any point x ∈ X, there is a hyperplane H x ⊂ CP 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 ϕ is nondegenerate at x if the order of contact of X, at x, with every hyperplane H ⊂ CP d containing ϕ(x) is at most d − 1. Equivalently, ϕ is nondegenerate at x if there is no hyperplane H ⊂ CP d such that the order of contact of X with H at x is at least d.The map ϕ is said to be nondegenerate if it is nondegenerate at every x ∈ X.
Consider now X to be a compact connected Riemann surface and fix a point x 0 ∈ X.Given the universal cover ϖ : X −→ X one has that π 1 (X, x 0 ) acts on X as deck transformations.Consider all pairs of the form (ρ, ϕ), where ρ : π 1 (X, x 0 ) −→ SL(r, C) is a homomorphism and ϕ : X −→ CP r−1 is a holomorphic immersion such that (1) ϕ is nondegenerate, and (2) ϕ • γ = ρ(γ) • ϕ for all γ ∈ π 1 (X, x 0 ) (the standard action of SL(r, C) on CP r−1 is being used here).Definition 6.3.Two pairs (ρ, ϕ) and (ρ 1 , ϕ 1 ) as above are called equivalent if there is an element G ∈ SL(r, C) such that For any given r ≥ 2, by M(r) we shall denote the space of all equivalence classes of pairs (ρ, ϕ) of the above type.Recall that given another base point x ′ 0 ∈ X, the fundamental group π 1 (X, x ′ 0 ) is identified with π 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 ′ 0 identifies π 1 (X, x 0 ) with π 1 (X, x ′ 0 ) and it also identifies X with the universal cover of X corresponding to x ′ 0 .From these it follows immediately that M(r) does not depend on the choice of the base point x 0 .Moreover, it is known that M(r) is identified with the space of all SL(r, C)-opers on X.This identification is briefly described in the proof of Proposition 6.6.

Real structures
Consider as in previous sections an anti-holomorphic involution σ : X −→ X, and denote by the homomorphism induced by σ.Using the homomorphism (σ * ) −1 , the action of π 1 (X, x 0 ) on X produces an action of π 1 (X, σ(x 0 )) on X.Let ϖ ′ X ′ −→ X be the universal covering corresponding to the point σ(x 0 ).Note whilst that there is no natural map between the universal covers X ′ and X, the self-map σ of X has a natural lift to the universal covering space σ : X −→ X ′ .(6.2) Remark 6.4.From the above constructions, one has the following: (1) σ takes the base point in ϖ −1 (x 0 ) to the base point in (ϖ ′ ) −1 (σ(x 0 )), (3) σ is an anti-holomorphic diffeomorphism, and (4) σ is π 1 (X, σ(x 0 ))-equivariant recall that π 1 (X, σ(x 0 )) acts on X .where σ * is the homomorphism in (6.1).Moreover, one can also define the map ϕ : X ′ −→ CP r−1 , y → ϕ σ −1 (y) , (6.4) where σ is the map in (6.2).Now it is straightforward to check that the pair ρ, ϕ constructed in (6.3) and (6.4) defines an element of M(r).Let Γ : M(r) −→ M(r) (6.5) be the map that sends any pair (ρ, ϕ) to ρ, ϕ constructed from it in (6.3) and (6.4).It is evident that Γ is an involution.■ As mentioned before, M(r) is identified with the space of all SL(r, C)-opers on X.Hence, the involution Γ constructed in Lemma 6.5 via (6.5) is an involution of 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 Γ of OSL X (r) coincides with the involution β constructed in Lemma 4.10.
Proof .This is deduced by examining the identification between OSL X (r) and M(r).The map M(r) −→ OSL X (r) is constructed as follows.Take any pair (ρ, ϕ) giving an element of M(r).Let E 0 = CP r−1 × C r −→ CP 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 whose fiber over any z ∈ CP r−1 is the line in C r represented by z.
The holomorphic section of the line bundle r (ϕ * E 0 ) = X × C −→ X given by the constant function 1 on X is preserved by the action of π 1 (X, x 0 ) on r (ϕ * E 0 ) induced by the action of π 1 (X, x 0 ) on ϕ * E 0 .Furthermore, this constant section is preserved by the connection on r (ϕ * E 0 ) induced by the connection ϕ * D 0 on ϕ * E 0 .Consequently, the vector bundle V in (6.7) defines a principal SL(r, C)-bundle on X, and D is a holomorphic connection on this principal SL(r, C)-bundle.From the non-degeneracy condition of the map ϕ it can be deduced that the line subbundle L produces an SL(r, C)-oper structure on this flat principal SL(r, C)bundle.This way we obtain the map M(r) −→ OSL X (r).The converse map OSL X (r) −→ M(r) is fairly straightforward.Let F be a holomorphic vector bundle of rank r on X such that the line bundle r F is holomorphically trivial, and let D be a holomorphic connection on F such that the induced connection on r F is the trivial one.Let L ⊂ F be a holomorphic line subbundle such that the triple (F, D, L) defines an SL(r, C)-oper on X.Let ρ : π 1 (X, x 0 ) −→ SL(F x 0 ) (6.8) be the monodromy representation for D.

. 14 )Proposition 5 . 4 .
Through the above involution, Theorem 4.14 leads to the following statement.Let A = Ψ −1 • A 1 • Ψ be the involution of OSL X (r), where A 1 and Ψ are constructed in (5.14) and (5.13) respectively.Then A coincides with the involution β in (4.21).Also A coincides with the restriction of B (in (4.22)) to OSL X (r).Finally, Proposition 5.4 has the following immediate consequence.Corollary 5.5.An element η, r i=3
Definition 4.12.An SL(r, C)-oper on X is in the real slice of SL(r, C)-opers if it is a fixed point of the involution β in (4.21) of Lemma 4.10.