Moduli spaces of vector bundles on a curve and opers

Abstract

Let X be a compact connected Riemann surface of genus g, with $g\, \ge \,2$, and let $\xi$ be a holomorphic line bundle on X with $\xi ^{\otimes 2}\,=\, {\mathcal O}_X$. Fix a theta characteristic ${\mathbb {L}}$ on X. Let ${\mathcal M}_X(r,\xi )$ be the moduli space of stable vector bundles E on X of rank r such that $\bigwedge ^r E\,=\, \xi$ and $H^0(X,\, E\otimes {\mathbb L})\,=\, 0$. Consider the quotient of ${\mathcal M}_X(r,\xi )$ by the involution given by $E\, \longmapsto \, E^*$. We construct an algebraic morphism from this quotient to the moduli space of $\textrm{SL}(r,{\mathbb C})$ opers on X. Since $\dim {\mathcal M}_X(r,\xi )$ coincides with the dimension of the moduli space of $\textrm{SL}(r,{\mathbb C})$ opers, it is natural to ask about the injectivity and surjectivity of this map.

1 Introduction

Opers were introduced by Beilinson and Drinfeld [2, 3]. Our aim here is to construct $\textrm{SL}(n, {\mathbb C})$ opers from stable vector bundles of degree zero. While a stable vector bundle of degree zero has a unique unitary flat connection, unitary connections are never an oper.

Take a compact connected Riemann surface of genus g, with $g\, \ge \, 2$, and fix a theta characteristic $\mathbb L$ on X. Let ${\mathcal M}_X(r)$ be the moduli space of stable vector bundles E of rank r and degree zero on X such that $H^0(X,\, E\otimes {\mathbb L})\,=\, 0$. For $i\,=\, 1,\, 2$, the projection $X\times X\,\longrightarrow \, X$ to the i-th factor is denoted by $p_i$. The diagonal divisor in $X\times X$ is denoted by $\Delta$; it is identified with X by $p_i$. For any $E\, \in \, {\mathcal M}_X(r)$, there is a unique section

$$\begin{aligned} {\mathcal A}_E\, \in \, H^0(X\times X,\, p^*_1 (E\otimes {\mathbb L})\otimes p^*_2 (E^*\otimes {\mathbb L})\otimes {\mathcal O}_{X\times X}(\Delta )) \end{aligned}$$

whose restriction to $\Delta$ is $\textrm{Id}_E$ (using the identification of $\Delta$ with X).

Using ${\mathcal A}_E$, we construct an $\textrm{SL}(n,{\mathbb C})$ oper on X for every $n\, \ge \, 2$; see Theorem 2 and Proposition 3. Related construction of opers from vector bundles were carried out in [5].

Let $\textrm{Op}_X(n)$ denote the moduli space of $\textrm{SL}(n,{\mathbb C})$ opers on X. The above mentioned map

$$\begin{aligned} {\mathcal M}_X(r) \, \longrightarrow \, \textrm{Op}_X(n) \end{aligned}$$

factors through the quotient of ${\mathcal M}_X(r)$ by the involution $\mathcal I$ defined by $E\, \longmapsto \, E^*$.

Fix a holomorphic line bundle $\xi$ on X such that $\xi ^{\otimes 2}\,=\, {\mathcal O}_X$. Let

$$\begin{aligned} {\mathcal M}_X(r, \xi )\, \subset \, {\mathcal M}_X(r) \end{aligned}$$

be the subvariety defined by the locus of all E such that $\bigwedge ^r E\,=\, \xi$. We have

$$\begin{aligned} \dim {\mathcal M}_X(r,\xi )/{\mathcal I}\,=\, (r^2-1)(g-1)\,=\, \dim \textrm{Op}_X(r). \end{aligned}$$

We end with a question (see Question (6)).

2 Vector bundles with trivial cohomology

Let X be a compact connected Riemann surface of genus g, with $g\, \ge \, 2$. The holomorphic cotangent bundle of X will be denoted by $K_X$. Fix a theta characteristic $\mathbb L$ on X. So, $\mathbb L$ is a holomorphic line bundle on X of degree $g-1$, and ${\mathbb L}\otimes \mathbb L$ is holomorphically isomorphic to $K_X$.

For any $r\, \ge \, 1$, let $\tilde{\mathcal M}_X(r)$ denote the moduli space of stable vector bundles on X of rank r and degree zero. It is an irreducible smooth complex quasiprojective variety of dimension $r^2(g-1)+1$. Let

$$\begin{aligned} {\mathcal M}_X(r)\, \subset \, \tilde{\mathcal M}_X(r) \end{aligned}$$

(1)

be the locus of all vector bundles $E\, \in \, \tilde{\mathcal M}_X(r)$ such that $H^0(X,\, E\otimes {\mathbb L})\,=\, 0$. From the semicontinuity theorem, [10, p. 288, Theorem 12.8], we know that ${\mathcal M}_X(r)$ is a Zariski open subset of $\tilde{\mathcal M}_X(r)$. In fact, ${\mathcal M}_X(r)$ is known to be the complement of a theta divisor on $\tilde{\mathcal M}_X(r)$ [11]. For any $E\, \in \, \tilde{\mathcal M}_X(r)$, the Riemann–Roch theorem says

$$\begin{aligned} \dim H^0(X,\, E\otimes {\mathbb L}) - \dim H^1(X,\, E\otimes {\mathbb L})\,=\, 0; \end{aligned}$$

so $H^0(X,\, E\otimes {\mathbb L})\,=\,0$ if and only if we have $H^1(X,\, E\otimes {\mathbb L})\,=\,0$.

We will now recall a construction from [6, 7].

For $i\,=\, 1,\, 2$, let $p_i:\, X\times X\, \longrightarrow \, X$ be the projection to the i-th factor. Let

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

be the reduced diagonal divisor. We will identify $\Delta$ with X using the map $x\, \longmapsto \, (x,\, x)$. Using this identification, the restriction of the line bundle ${\mathcal O}_{X\times X}(\Delta )$ to $\Delta \, \subset \, X\times X$ gets identified with the holomorphic tangent bundle TX by the Poincaré adjunction formula [9, p. 146].

Take any $E\,\in \, {\mathcal M}_X(r)$. The restriction of the vector bundle

$$\begin{aligned} p^*_1 (E\otimes {\mathbb L})\otimes p^*_2 (E^*\otimes {\mathbb L})\otimes {\mathcal O}_{X \times X}(\Delta ) \end{aligned}$$

to $\Delta$ is identified with the vector bundle $\text {End}(E)$ on X. Indeed, this follows immediately from the following facts:

• the restriction of $(p^*_1 E)\otimes (p^*_2 E^*)$ to $\Delta$ is identified with the vector bundle $\text {End}(E)$ on X, and

• the above identification of ${\mathcal O}_{X\times X}(\Delta )\vert _\Delta$ with TX produces an identification of $(p^*_1 {\mathbb L})\otimes (p^*_2 {\mathbb L})\otimes {\mathcal O}_{X \times X}(\Delta )\vert _\Delta$ with $K_X\otimes TX\,=\, {\mathcal O}_X$.

Consequently, we have the following short exact sequence of sheaves on $X\times X$:

$$\begin{aligned}{} & {} 0\, \longrightarrow \, p^*_1 (E\otimes {\mathbb L})\otimes p^*_2 (E^*\otimes {\mathbb L}) \longrightarrow \, p^*_1 (E\otimes {\mathbb L})\nonumber \\{} & {} \quad \otimes p^*_2 (E^*\otimes {\mathbb L})\otimes {\mathcal O}_{X \times X}(\Delta ) \, \longrightarrow \, \text {End}(E) \, \longrightarrow \, 0, \end{aligned}$$

(2)

where $\text {End}(E)$ is supported on $\Delta$, using the identification of $\Delta$ with X. For $k\,=\, 0,\, 1$, since $H^k(X,\, E\otimes {\mathbb L})\,=\, 0$, the Serre duality implies that $H^{1-k}(X,\, E^*\otimes {\mathbb L})\,=\, 0$. Using Künneth formula,

$$\begin{aligned} H^j(X\times X,\, p^*_1 (E\otimes {\mathbb L})\otimes p^*_2 (E^*\otimes {\mathbb L}))\,=\, 0 \end{aligned}$$

for $j\,=\, 0,\, 1,\, 2$. Therefore, the long exact sequence of cohomologies for the short exact sequence of sheaves in (2) gives

$$\begin{aligned}{} & {} 0\,=\, H^0(X\times X,\, p^*_1 (E\otimes {\mathbb L})\otimes p^*_2 (E^*\otimes {\mathbb L}))\, \longrightarrow \, \nonumber \\{} & {} \quad H^0(X\times X,\, p^*_1 (E\otimes {\mathbb L})\otimes p^*_2 (E^*\otimes {\mathbb L})\otimes {\mathcal O}_{X \times X}(\Delta ))\nonumber \\{} & {} \quad {\mathop {\longrightarrow }\limits ^{\gamma }}\, H^0(X,\, \text {End}(E)) \, \longrightarrow \, \nonumber \\{} & {} \quad H^1(X\times X,\, p^*_1 (E\otimes {\mathbb L})\otimes p^*_2 (E^*\otimes {\mathbb L}))\,=\,0. \end{aligned}$$

(3)

So the homomorphism $\gamma$ in (3) is actually an isomorphism. For this isomorphism $\gamma$, let

$$\begin{aligned} {\mathcal A}_E\, :=\, \gamma ^{-1}(\textrm{Id}_E)\, \in \, H^0(X\times X,\, p^*_1 (E\otimes {\mathbb L})\otimes p^*_2 (E^*\otimes {\mathbb L})\otimes {\mathcal O}_{X\times X}(\Delta )) \end{aligned}$$

(4)

be the section corresponding to the identity automorphism of E.

3 A section around the diagonal

Using the section ${\mathcal A}_E$ in (4), we will construct a section of $(p^*_1 {\mathbb L})\otimes (p^*_2 {\mathbb L})\otimes {\mathcal O}_{X\times X}(\Delta )$ on an analytic neighborhood of the diagonal $\Delta$. For that, we first recall a description of the holomorphic differential operators on X.

3.1 Differential operators

Fix holomorphic vector bundles $V,\, W$ on X, and also fix an integer $d\, \ge \, 1$. The ranks of V and W are denoted by r and $r'$ respectively. Let $\text {Diff}^d_X(V,\, W)$ denote the holomorphic vector bundle on X of rank $rr'(d+1)$ corresponding to the sheaf of differential operators of degree d from V to W. We recall that $\text {Diff}^d_X(V,\, W)\,=\, W\otimes J^d(V)^*$, where

$$\begin{aligned} J^d(V):=\, p_{1*}\left( (p^*_2V)/((p^*_2V)\otimes {\mathcal O}_{X\times X}(-(d+1)\Delta ))\right) \, \longrightarrow \, X \end{aligned}$$

is the d-th order jet bundle for V.

We have a short exact sequence of coherent analytic sheaves on $X\times X$ as follows:

$$\begin{aligned}&0\, \longrightarrow \, (p^*_1 W)\otimes p^*_2(V^*\otimes K_X)\, \longrightarrow \,(p^*_1 W)\otimes p^*_2(V^*\otimes K_X) \otimes {\mathcal O}_{X\times X}((d+1){\Delta })\nonumber \\&\quad \longrightarrow \, {\mathcal Q}_d(V,\,W)\, :=\, \frac{(p^*_1 W)\otimes p^*_2(V^*\otimes K_X) \otimes {\mathcal O}_{X\times X}((d+1){\Delta })}{(p^*_1 W)\otimes p^*_2(V^*\otimes K_X)} \, \longrightarrow \, 0\, ; \end{aligned}$$

(5)

the support of the quotient sheaf ${\mathcal Q}_d(V,\,W)$ in (5) is $(d+1){\Delta }$. The direct image

$$\begin{aligned} {\mathcal K}_d(V,\,W)\, :=\, p_{1*} {\mathcal Q}_d(V,\,W) \, \longrightarrow \, X \end{aligned}$$

(6)

is a holomorphic vector bundle on X of rank $rr'(d+1)$. It is known that

$$\begin{aligned} {\mathcal K}_d(V,\,W)\,=\, \text {Hom}(J^d(V),\, W)\,=\, \text {Diff}^d_X(V,\, W)\, , \end{aligned}$$

(7)

where ${\mathcal K}_d(V,\,W)$ is the vector bundle constructed in (6) (see [4, Section 2.1]).

Note that $R^1p_{1*} {\mathcal Q}_d(V,\,W)\,=\, 0$, because ${\mathcal Q}_d(V,\,W)$ is supported on $(d+1){\Delta }$. We have $H^0(X,\, p_{1*} {\mathcal Q}_d(V,\,W))\,=\, H^0(X\times X,\, {\mathcal Q}_d(V,\,W))$. So from (6) and (7), it follows that

$$\begin{aligned} H^0(X,\, \text {Diff}^d_X(V,\, W))\, =\, H^0(X\times X,\, {\mathcal Q}_d(V,\,W))\, . \end{aligned}$$

(8)

The restriction of the vector bundle $(p^*_1 W)\otimes p^*_2(V^*\otimes K_X) \otimes {\mathcal O}_{X\times X}((d+1){\Delta })$ to $\Delta \, \subset \, X\times X$ is $\text {Hom}(V,\, W)\otimes (TX)^{\otimes d}$, because the restriction of ${\mathcal O}_{X\times X}({\Delta })$ to $\Delta$ is TX. Therefore, we get a surjective homomorphism

$$\begin{aligned} {\mathcal K}_d(V,\,W)\,&\longrightarrow \, p_{1*}\left( \frac{(p^*_1 W)\otimes p^*_2(V^*\otimes K_X) \otimes {\mathcal O}_{X\times X}((d+1){\Delta })}{(p^*_1 W)\otimes p^*_2(V^*\otimes K_X) \otimes {\mathcal O}_{X\times X}(d{\Delta })}\right) \\&=\, \text {Hom}(V,\, W)\otimes (TX)^{\otimes d}\, , \end{aligned}$$

where ${\mathcal K}_d(V,\,W)$ is constructed in (6). Using (7), this gives a surjective homomorphism

$$\begin{aligned} \text {Diff}^d_X(V,\, W)\, \longrightarrow \,\text {Hom}(V,\, W)\otimes (TX)^{\otimes d}\, . \end{aligned}$$

(9)

The homomorphism in (9) is known as the symbol map.

3.2 Construction of a connection

Consider the de Rham differential operator $d\,:\, {\mathcal O}_X\, \longrightarrow \, K_X$. Using the isomorphism in (8), this d produces a section

$$\begin{aligned} d_1\, \in \, H^0(X\times X,\, {\mathcal Q}_1({\mathcal O}_X,\,K_X)). \end{aligned}$$

From (5), we conclude that $d_1$ is a section of $(p^*_1 K_X)\otimes (p^*_2 K_X)\otimes {\mathcal O}_{X\times X}(2{\Delta })$ over $2{\Delta }$. The restriction of $d_1$ to $\Delta \, \subset \,2\Delta$ is the section of ${\mathcal O}_X$ given by the constant function 1 (see (9)); note that the symbol of the differential operator d is the constant function 1.

As before, ${\mathbb L}$ is a theta characteristic on X. There is a unique section

$$\begin{aligned} \delta \,\in \, H^0(2{\Delta },\,(p^*_1{\mathbb L})\otimes (p^*_2{\mathbb L})\otimes {\mathcal O}_{X\times X}({\Delta })) \end{aligned}$$

(10)

such that

• (1) $\delta \otimes \delta \,=\, d_1$, and

• (2) the restriction of $\delta$ to $\Delta \, \subset \,2\Delta$ is the constant function 1 (note that since the restriction of ${\mathcal O}_{X\times X}({\Delta })$ to $\Delta$ is TX, the restriction of $(p^*_1{\mathbb L})\otimes (p^*_2{\mathbb L})\otimes {\mathcal O}_{X\times X}({\Delta })$ to $\Delta$ is $K_X\otimes TX\,=\, {\mathcal O}_X$).

See [8, p. 754, Theorem 2.1(b)] for an alternative construction of $\delta$.

There is a unique section

$$\begin{aligned} \Phi _E\,\in \, H^0(2{\Delta },\, (p^*_1 E)\otimes (p^*_2 E^*)) \end{aligned}$$

(11)

such that $({\mathcal A}_E)\vert _{2\Delta }\, =\, \Phi _E\otimes \delta$, where ${\mathcal A}_E$ and $\delta$ are the sections in (4) and (10) respectively. Indeed, this follows immediately from the fact that the section $\delta$ is nowhere zero, so $\delta ^{-1}$ is a holomorphic section of $((p^*_1{\mathbb L}) \otimes (p^*_2{\mathbb L})\otimes {\mathcal O}_{X\times X}({\Delta }))^*\vert _{2\Delta }$. Now set

$$\begin{aligned} \Phi _E\,=\, (({\mathcal A}_E)\big \vert _{2\Delta })\otimes \delta ^{-1} \end{aligned}$$

and consider it as a section of $((p^*_1 E)\otimes (p^*_2 E^*))\big \vert _{2\Delta }$ using the natural duality pairing

$$\begin{aligned} ((p^*_1{\mathbb L}) \otimes (p^*_2{\mathbb L})\otimes {\mathcal O}_{X\times X}({\Delta }))\vert _{2\Delta }\otimes ((p^*_1{\mathbb L}) \otimes (p^*_2{\mathbb L})\otimes {\mathcal O}_{X\times X}({\Delta }))^*\vert _{2\Delta } \, \longrightarrow \, {\mathcal O}_{2\Delta }. \end{aligned}$$

Since the restriction of ${\mathcal A}_E$ to $\Delta$ is $\textrm{Id}_E$ (see (4)), and the restriction of $\delta$ to $\Delta$ is the constant function 1, it follows that the restriction of the section $\Phi _E$ in (11) to $\Delta$ is $\textrm{Id}_E$. Therefore, $\Phi _E$ defines a holomorphic connection on U, which will be denoted by $D^E$. To describe the connection $D^E$ explicitly, take an open subset $U\, \subset \, X$ and a holomorphic section $s\, \in \, H^0(U,\, E\big \vert _U)$. Consider the section $\Phi _E\otimes p^*_2 s$ over ${\mathcal U}\,:=\, (2\Delta )\bigcap (U\times U)$. Using the natural pairing $E^*\otimes E\, \longrightarrow \, {\mathcal O}_X$, it produces a section of $((p^*_1 E)\otimes (p^*_2{\mathcal O}_X))\vert _{\mathcal U} \,=\,(p^*_1 E)\vert _{\mathcal U}$; denote this section of $(p^*_1 E)\vert _{\mathcal U}$ by $\tilde{s}$. Since $\Phi _E\vert _{\Delta }\,=\, \textrm{Id}_E$, we know that $\tilde{s}$ and $p^*_1 s$ coincide on $\Delta \bigcap (U\times U)$. So

$$\begin{aligned} \tilde{s}- (p^*_1 s)\vert _{2\Delta }\, \in \, H^0(U,\, E\otimes K_X); \end{aligned}$$

the Poincaré adjunction formula identifies $K_X$ with the restriction of the line bundle ${\mathcal O}_{X\times X}(-\Delta )$ to $\Delta$. Then we have

$$\begin{aligned} D^E (s) \,=\, \tilde{s}- (p^*_1 s)\vert _{2\Delta } \, \in \, H^0(U,\, E\otimes K_X)\, . \end{aligned}$$

(12)

It is straightforward to check that $D^E$ satisfies the Leibniz identity thus making it a holomorphic connection on E.

4 Opers from vector bundles

We will construct an $\textrm{SL}(n,{\mathbb C})$-oper on X, for every $n\, \ge \, 2$, from the section ${\mathcal A}_E$ in (4).

We will use that any holomorphic connection on a holomorphic bundle over X is integrable (same as flat) because $\Omega ^2_X\,=\, 0$.

As before, take any $E\, \in \, {\mathcal M}_X(r)$. Consider the holomorphic connection $D^E$ on E in (12). Let $U\, \subset \, X$ be a simply connected open subset, and let $x_0\, \in \, U$ be a point. Since the connection $D^E$ is integrable, using parallel translations, for $D^E$, along paths emanating from $x_0$, we get a holomorphic isomorphism of $E\vert _U$ with the trivial vector bundle $U\times E_{x_0}\, \longrightarrow \, U$. This isomorphism takes the connection $D^E\vert _U$ on $E\vert _U$ to the trivial connection on the trivial bundle. Let

$$\begin{aligned} \Delta \, \subset \, {\mathcal U}\, \subset \, X\times X \end{aligned}$$

be an open neighborhood of $\Delta$ that admits a deformation retraction to $\Delta$. For $i\,=\,1,\, 2$, the restriction of the projection $p_i:\, X\times X\,\longrightarrow \, X$ to the open subset ${\mathcal U}\, \subset \, X\times X$ will be denoted by $q_i$. There is a unique holomorphic isomorphism over ${\mathcal U}$,

$$\begin{aligned} f \, :\, q^*_1 E\, \longrightarrow \, q^*_2 E \end{aligned}$$

(13)

that satisfies the following two conditions:

• (1) the restriction of f to $\Delta$ is the identity map of E, and

• (2) f takes the connection $q^*_1 D^E$ on $q^*_1 E$ to the connection $q^*_2 D^E$ on $q^*_2 E$.

Since the inclusion map $\Delta \, \hookrightarrow \, {\mathcal U}$ is a homotopy equivalence, the flat vector bundle $(E,\, D^E)$ on $X\,=\,\Delta$ has a unique extension to a flat vector bundle on ${\mathcal U}$. On the other hand, both $(q^*_1 E,\, q^*_1 D^E)$ and $(q^*_2 E,\, q^*_2 D^E)$ are extensions of $(E,\, D^E)$. Therefore, there is a unique holomorphic isomorphism f as in (13) satisfying the above two conditions.

Using the isomorphism f in (13), the section ${\mathcal A}_E$ in (4) produces a holomorphic section

$$\begin{aligned} {\mathcal A}'_E \, \in \, H^0({\mathcal U},\, p^*_1 (E\otimes E^*\otimes {\mathbb L})\otimes (p^*_2 {\mathbb L})\otimes {\mathcal O}_{X\times X}(\Delta ))\, . \end{aligned}$$

(14)

Consider the trace pairing

$$\begin{aligned} tr\,:\,\, E\otimes E^*\,=\, \text {End}(E) \, \longrightarrow \, {\mathcal O}_X, \ \ B\, \longmapsto \, \frac{1}{r}\text {trace}(B); \end{aligned}$$

recall that $r\,=\, \text {rank}(E)$. Note that $r\cdot tr$ is the natural pairing $E\otimes E^*\, \longrightarrow \, {\mathcal O}_X$. Using tr, the section ${\mathcal A}'_E$ in (14) produces a section

$$\begin{aligned} \widehat{\beta }_E\, \in \, H^0({\mathcal U},\, (p^*_1 {\mathbb L})\otimes (p^*_2 {\mathbb L})\otimes {\mathcal O}_{X\times X}(\Delta ))\, . \end{aligned}$$

(15)

The following lemma is straightforward to prove.

Lemma 1

The restriction of the section $\hat{\beta }_E$ (in (15)) to $2\Delta \, \subset \, {\mathcal U}$ coincides with the section $\delta$ in (10).

Proof

This follows from the constructions of $\Phi _E$ (in (11)) and $\hat{\beta }_E$. $\square$

For any integer $k\, \ge \, 1$, the holomorphic line bundles ${\mathbb L}^{\otimes k}$ and $({\mathbb L}^{\otimes k})^*$ will be denoted by ${\mathbb L}^k$ and ${\mathbb L}^{-k}$ respectively.

Theorem 2

Take any integer $n\, \ge \, 2$. The section $\hat{\beta }_E$ in (15) produces a holomorphic connection ${\mathcal D}(E)$ on the holomorphic vector bundle $J^{n-1}({\mathbb L}^{(1-n)})$.

Proof

Consider the $(n+1)$-th tensor power of $\hat{\beta }_E$:

$$\begin{aligned} (\hat{\beta }_E)^{\otimes (n+1)}\, \in \, H^0({\mathcal U},\, (p^*_1 {\mathbb L}^{(n+1)})\otimes (p^*_2 {\mathbb L}^{(n+1)})\otimes {\mathcal O}_{X\times X}((n+1)\Delta )), \end{aligned}$$

and restrict it to $(n+1)\Delta \, \subset \, {\mathcal U}$. From (8), we have

$$\begin{aligned} \beta ^n_E \, :=\,(\hat{\beta }_E)^{\otimes (n+1)}\vert _{(n+1)\Delta }\, \in \, H^0(X,\, \text {Diff}^n_X({\mathbb L}^{(1-n)},\, {\mathbb L}^{(n+1)}))\, . \end{aligned}$$

(16)

The symbol of the differential operator $\beta ^n_E$ in (16) is the section of ${\mathcal O}_X$ given by the constant function 1. Indeed, this follows immediately from the fact that the restriction of $\hat{\beta }_E$ to $\Delta$ is the constant function 1 (see Lemma 1).

We recall that there is a natural injective homomorphism $J^{m+n}(V)\, \longrightarrow \, J^m(J^n(V))$ for all $m,\, n\, \ge \, 0$ and every holomorphic vector bundle V. We have following commutative diagram of vector bundles on X:

(17)

where the rows are exact. From (7), we know that the differential operator $\beta ^n_E$ in (16) produces a homomorphism

$$\begin{aligned} \rho :\, J^n({\mathbb L}^{(1-n)})\, \longrightarrow \, {\mathbb L}^{(n+1)}. \end{aligned}$$

Since the symbol of $\beta ^{n}_{E}$ is the constant function 1, we have

$$\begin{aligned} \rho \circ \iota _1\,=\, \textrm{Id}_{{\mathbb L}^{(n+1)}}\,, \end{aligned}$$

(18)

where $\iota _1$ is the homomorphism in (17). From (18), it follows immediately that $\rho$ gives a holomorphic splitting of the top exact sequence in (17). Let

$$\begin{aligned} {\mathcal D}_1:\, J^{n-1}({\mathbb L}^{(1-n)})\, \longrightarrow \, J^n({\mathbb L}^{(1-n)}) \end{aligned}$$

be the unique holomorphic homomorphism such that

• $\rho \circ {\mathcal D}_1\,=\, 0$, and

• $\psi _1\circ {\mathcal D}_1\,=\, \textrm{Id}_{J^{n-1}({\mathbb L}^{(1-n)})}$, where $\psi _1$ is the projection in (17).

Now consider the homomorphism

$$\begin{aligned} {\mathcal D}_2\, :=\, {\textbf{h}}\circ {\mathcal D}_1\, :\, J^{n-1}({\mathbb L}^{(1-n)})\, \longrightarrow \, J^1(J^{n-1}({\mathbb L}^{(1-n)}))\, , \end{aligned}$$

(19)

where $\textbf{h}$ is the homomorphism in (17). Since $\psi _1\circ {\mathcal D}_1\,=\, \textrm{Id}_{J^{n-1}({\mathbb L}^{(1-n)})}$, from the commutativity of (17), it follows that

$$\begin{aligned} \psi _2\circ {\mathcal D}_2\,=\, \psi _2\circ {\textbf{h}}\circ {\mathcal D}_1\,=\, \textrm{Id}_{J^{n-1}({\mathbb L}^{(1-n)})} \circ \psi _1\circ {\mathcal D}_1\,=\, \textrm{Id}_{J^{n-1}({\mathbb L}^{(1-n)})}, \end{aligned}$$

where $\psi _2$ is the projection in (17). This implies that ${\mathcal D}_2$ in (19) gives a holomorphic splitting of the bottom exact sequence in (17). Let

$$\begin{aligned} {\mathcal D}(E):\, J^1(J^{n-1}({\mathbb L}^{(1-n)}))\, \longrightarrow \, J^{n-1}({\mathbb L}^{(1-n)})\otimes K_X \end{aligned}$$

be the unique holomorphic homomorphism such that

• ${\mathcal D}(E)\circ {\mathcal D}_2\,=\, 0$, and

• ${\mathcal D}(E)\circ \iota _2\,=\, \textrm{Id}_{J^{n-1}({\mathbb L}^{(1-n)})\otimes K_X}$, where $\iota _2$ is the homomorphism in (17).

Using (7), we know that

$$\begin{aligned} {\mathcal D}(E)\, \in \, H^0(X,\, \text {Diff}^1_X(J^{n-1}({\mathbb L}^{(1-n)}),\, J^{n-1}({\mathbb L}^{(1-n)})\otimes K_X))\, . \end{aligned}$$

(20)

From the homomorphism in (9) and the above equality ${\mathcal D}(E)\circ \iota _2\,=\, \textrm{Id}_{J^{n-1} ({\mathbb L}^{(1-n)})\otimes K_X}$, it follows that the symbol of the differential operator ${\mathcal D}(E)$ in (20) is $\textrm{Id}_{J^{n-1}({\mathbb L}^{(1-n)})}$. This implies that ${\mathcal D}(E)$ satisfies the Leibniz rule. Consequently, ${\mathcal D}(E)$ is a holomorphic connection on the holomorphic vector bundle $J^{n-1}({\mathbb L}^{(1-n)})$. $\square$

For $1\, \le \, i\, \le \, n-1$, consider the short exact sequence

$$\begin{aligned} 0 \, \longrightarrow \, {\mathbb L}^{(1-n)}\otimes K^{\otimes i}_X\, \longrightarrow \, J^{i}({\mathbb L}^{(1-n)}) \, \longrightarrow \, J^{i-1}({\mathbb L}^{(1-n)})\, \longrightarrow \, 0. \end{aligned}$$

Using these together with the fact that ${\mathbb L}\otimes {\mathbb L}\,=\, K_X$, it is deduced that

$$\begin{aligned} \det J^i({\mathbb L}^{(1-n)}):=\, \bigwedge \nolimits ^{i+1} J^i({\mathbb L}^{(1-n)})\,=\, {\mathbb L}^{(i+1)(i+1-n)}. \end{aligned}$$

In particular, we have

$$\begin{aligned} \det J^{n-1}({\mathbb L}^{(1-n)}):=\, \bigwedge \nolimits ^n J^{n-1}({\mathbb L}^{(1-n)})\,=\, {\mathcal O}_X. \end{aligned}$$

So $\det J^{n-1}({\mathbb L}^{(1-n)})$ has a unique holomorphic connection whose monodromy is trivial; it will be called the trivial connection on $\det J^{n-1}({\mathbb L}^{(1-n)})$.

PROPOSITION 3

The holomorphic connection on $\det J^{n-1}({\mathbb L}^{(1-n)})$ induced by the connection ${\mathcal D}(E)$ on $J^{n-1}({\mathbb L}^{(1-n)})$ (see Theorem 2) is the trivial connection.

Proof

Any holomorphic connection D on $\det J^{n-1}({\mathbb L}^{(1-n)})\,=\, {\mathcal O}_X$ can be uniquely expressed as

$$\begin{aligned} D\,=\, D_0+\omega , \end{aligned}$$

where $D_0$ is the trivial connection on $\det J^{n-1}({\mathbb L}^{(1-n)})$ and $\omega \, \in \, H^0(X,\, K_X)$. Let $D^1$ be the holomorphic connection on $\det J^{n-1}({\mathbb L}^{(1-n)})$ induced by the connection ${\mathcal D}(E)$ on $J^{n-1}({\mathbb L}^{(1-n)})$. Decompose it as

$$\begin{aligned} D^1\,=\, D_0+\omega ^1, \end{aligned}$$

where $\omega ^1\, \in \, H^0(X,\, K_X)$. Then

$$\begin{aligned} \omega ^1\,=\, (n-1)\cdot ((\hat{\beta }_E)\vert _{2\Delta }-\delta ), \end{aligned}$$

where $\hat{\beta }_E$ and $\delta$ are the sections in (15) and (10) respectively. Note that two sections of $((p^*_1{\mathbb L})\otimes (p^*_2{\mathbb L})\otimes {\mathcal O}_{X\times X}({\Delta }))\vert _{2\Delta }$ that coincide on $\Delta \, \subset \, 2\Delta$ differ by an element of $H^0(\Delta ,\, ((p^*_1{\mathbb L})\otimes (p^*_2{\mathbb L}))\vert _{\Delta }) \,=\,H^0(X,\, K_X)$. Now from Lemma 1, it follows that $\omega ^1\,=\, 0$. $\square$

Let $\textrm{Op}_X(n)$ denote the moduli space of $\textrm{SL}(n, {\mathbb C})$ opers on X [3]. It is a complex affine space of dimension $(n^2-1)(g-1)$. We recall a description of $\textrm{Op}_X(n)$. Let ${\mathcal C}_n(X)$ denote the space of all holomorphic connections $D'$ on $J^{n-1}({\mathbb L}^{(1-n)})$ such that the holomorphic connection on $\det J^{n-1}({\mathbb L}^{(1-n)})$ induced by $D'$ is the trivial connection on $\det J^{n-1}({\mathbb L}^{(1-n)})\,=\, {\mathcal O}_X$. Then

$$\begin{aligned} \textrm{Op}_X(n)\,=\, {\mathcal C}_n(X)/\textrm{Aut}(J^{n-1}({\mathbb L}^{(1-n)})), \end{aligned}$$

where $\textrm{Aut}(J^{n-1}({\mathbb L}^{(1-n)}))$ denotes the group of all holomorphic automorphisms of the vector bundle $J^{n-1}({\mathbb L}^{(1-n)})$; note that $\textrm{Aut}(J^{n-1}({\mathbb L}^{(1-n)}))$ has a natural action on ${\mathcal C}_n(X)$.

The moduli space $\textrm{Op}_X(n)$ also coincides with the space of all holomorphic differential operators

$$\begin{aligned} {\mathcal B}\, \in \, H^0(X,\, \text {Diff}^n_X({\mathbb L}^{(1-n)},\, {\mathbb L}^{(n+1)})) \end{aligned}$$

such that

• (1) the symbol of ${\mathcal B}$ is the section of ${\mathcal O}_X$ given by the constant function 1, and

• (2) the sub-leading term of ${\mathcal B}$ vanishes.

From Theorem 2 and and Proposition 3, we get an algebraic morphism

$$\begin{aligned} \tilde{\Psi }\, :\, {\mathcal M}_X(r)\,\longrightarrow \, \textrm{Op}_X(n) \end{aligned}$$

(21)

that sends any $E\, \in \, {\mathcal M}_X(r)$ to the image in $\textrm{Op}_X(n)$ of the holomorphic connection ${\mathcal D}(E)$ (see Theorem 2).

Since $H^i(X,\, E\otimes {\mathbb L})\,=\, H^{1-i}(X,\, E^*\otimes {\mathbb L})$ (Serre duality), we have an involution

$$\begin{aligned} {\mathcal I}\, :\,{\mathcal M}_X(r)\, \longrightarrow \, {\mathcal M}_X(r)\, ,\ \ F\, \longmapsto \, F^*\, . \end{aligned}$$

(22)

Let

$$\begin{aligned} \tau \, :\, X\times X\, \longrightarrow \,X\times X\, ,\ \ (x_1,\, x_2)\, \longmapsto \, (x_2,\, x_1) \end{aligned}$$

(23)

be the involution.

PROPOSITION 4

For any $E\, \in \, {\mathcal M}_X(r)$, the sections

$$\begin{aligned} {\mathcal A}_E\, \in \, H^0(X\times X,\, p^*_1 (E\otimes {\mathbb L})\otimes p^*_2 (E^*\otimes {\mathbb L})\otimes {\mathcal O}_{X\times X}(\Delta )) \end{aligned}$$

and

$$\begin{aligned} {\mathcal A}_{{\mathcal I}(E)}\, \in \, H^0(X\times X,\, p^*_1 (E^*\otimes {\mathbb L})\otimes p^*_2 (E\otimes {\mathbb L})\otimes {\mathcal O}_{X\times X}(\Delta )) \end{aligned}$$

(see (4) for ${\mathcal A}_E$ and (22) for ${\mathcal I}$) satisfy the equation

$$\begin{aligned} \tau ^*{\mathcal A}_E\,=\, {\mathcal A}_{{\mathcal I}(E)}, \end{aligned}$$

where $\tau$ is the involution in (23).

Proof

We recall that ${\mathcal A}_E$ is the unique section of $p^*_1 (E^*\otimes {\mathbb L})\otimes p^*_2 (E\otimes {\mathbb L})\otimes {\mathcal O}_{X\times X}(\Delta )$ over $X\times X$ whose restriction to $\Delta$ coincides with $\textrm{Id}_E$ using the identification of $\Delta$ with X. Now the restriction of $\tau ^*{\mathcal A}_{{\mathcal I}(E)}$ to $\Delta$ is also $\textrm{Id}_E$. So, $\tau ^*{\mathcal A}_{{\mathcal I}(E)}\,=\, {\mathcal A}_E$, which implies that $\tau ^*{\mathcal A}_E\,=\, {\mathcal A}_{{\mathcal I}(E)}$. $\square$

COROLLARY 5

The map $\tilde{\Psi }$ in (21) descends to a map

$$\begin{aligned} \tilde{\Psi }^0:\, {\mathcal M}_X(r)/{\mathcal I}\,\longrightarrow \,\textrm{Op}_X(n), \end{aligned}$$

where $\mathcal I$ is the involution in (22).

Let $\xi$ be a holomorphic line bundle on X such that $\xi \otimes \xi \,=\, {\mathcal O}_X$; for example, $\xi$ can be ${\mathcal O}_X$. Let

$$\begin{aligned} {\mathcal M}_X(r,\xi )\, \subset \, {\mathcal M}_X(r) \end{aligned}$$

be the sub-variety consisting of all $E\, \in \, {\mathcal M}_X(r)$ such that $\bigwedge ^r E\,=\, \xi$. Since $\xi ^{\otimes 2}\,=\, {\mathcal O}_X$, it follows that

$$\begin{aligned} {\mathcal I}({\mathcal M}_X(r,\xi ))\,=\, {\mathcal M}_X(r,\xi ), \end{aligned}$$

where $\mathcal I$ is defined in (22). So restricting the map $\tilde{\Psi }^0$ in Corollary 5 to ${\mathcal M}_X(r,\xi )/{\mathcal I}$, we get morphism

$$\begin{aligned} \Psi \, :\, {\mathcal M}_X(r,\xi )/{\mathcal I}\,\longrightarrow \, {\textrm{Op}}_X(n)\, . \end{aligned}$$

(24)

We note that when $n\,=\, r$,

$$\begin{aligned} \dim {\mathcal M}_X(r,\xi )/{\mathcal I}\,=\, (r^2-1)(g-1)\,=\, \dim {\textrm{Op}}_X(r). \end{aligned}$$

So it is natural to ask the following question.

Question 6

When $n\,=\, r$, how close is the map $\Psi$ (constructed in (24)) to being injective or surjective?