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.
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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
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
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
be the subvariety defined by the locus of all E such that \(\bigwedge ^r E\,=\, \xi \). We have
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
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
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
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
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 \(X\times X\):
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,
for \(j\,=\, 0,\, 1,\, 2\). Therefore, the long exact sequence of cohomologies for the short exact sequence of sheaves in (2) gives
So the homomorphism \(\gamma \) in (3) is actually an isomorphism. For this isomorphism \(\gamma \), let
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
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:
the support of the quotient sheaf \({\mathcal Q}_d(V,\,W)\) in (5) is \((d+1){\Delta }\). The direct image
is a holomorphic vector bundle on X of rank \(rr'(d+1)\). It is known that
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
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
where \({\mathcal K}_d(V,\,W)\) is constructed in (6). Using (7), this gives a surjective homomorphism
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
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
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
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
and consider it as a section of \(((p^*_1 E)\otimes (p^*_2 E^*))\big \vert _{2\Delta }\) using the natural duality pairing
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
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
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
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 \({\mathcal U}\),
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
Consider the trace pairing
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
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 \(\hat{\beta }_E\):
and restrict it to \((n+1)\Delta \, \subset \, {\mathcal U}\). From (8), we have
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:
where the rows are exact. From (7), we know that the differential operator \(\beta ^n_E\) in (16) produces a homomorphism
Since the symbol of \(\beta ^{n}_{E}\) is the constant function 1, we have
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
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
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
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
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
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
Using these together with the fact that \({\mathbb L}\otimes {\mathbb L}\,=\, K_X\), it is deduced that
In particular, we have
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
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
where \(\omega ^1\, \in \, H^0(X,\, K_X)\). Then
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
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
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
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
Let
be the involution.
PROPOSITION 4
For any \(E\, \in \, {\mathcal M}_X(r)\), the sections
and
(see (4) for \({\mathcal A}_E\) and (22) for \({\mathcal I}\)) satisfy the equation
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
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
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
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
We note that when \(n\,=\, r\),
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?
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Biswas, I., Hurtubise, J. & Roubtsov, V. Moduli spaces of vector bundles on a curve and opers. Proc Math Sci 133, 15 (2023). https://doi.org/10.1007/s12044-023-00738-2
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DOI: https://doi.org/10.1007/s12044-023-00738-2