Moduli spaces of vector bundles on a curve and opers (Q6175458)

Let \(X\) be a compact connected Riemann surface of genus \(g \ge 2\). Let \({\mathcal M}_X (r)\) be the moduli space of stable vector bundles of rank \(r \ge 1\) and degree zero over \(X\). Fix a theta characteristic on \(X\); that is, a line bundle \(\mathbb{L} \to X\) such that \(\mathbb{L}^{\otimes 2}\) is isomorphic to the canonical bundle \(K_X\). For any \(n \ge 1\), the authors show (in several steps) how to associate naturally to each \(E \in {\mathcal M}_X (r)\) a connection \({\mathcal D} (E)\) on the bundle \(J^{n-1} \left( \mathbb{L}^{\otimes (1-n)} \right)\) of jets of order \(n-1\) of \(\mathbb{L}^{\otimes (1 - n)}\). As \(\mathbb{L}^{\otimes 2} \cong K_X\), one has \(\det J^{n-1} \left( \mathbb{L}^{\otimes (1-n)} \right) \cong {\mathcal O}_X\). The authors show that \({\mathcal D} (E)\) induces the trivial connection on \(\det J^{n-1} \left( \mathbb{L}^{\otimes (1-n)} \right)\). Then by \textit{A. Beilinson} and \textit{V. Drinfeld} [``Opers'', Preprint, \url{arXiv:math/0501398}], the connection \({\mathcal D} (E)\) defines an \(\mathrm{SL}(n)\)-oper over \(X\). The above association defines an algebraic morphism \(\tilde{\Psi} \colon {\mathcal M}_X ( r ) \to \mathrm{Op}_X ( n )\), where \(\mathrm{Op}_X (n)\) is the moduli space of \(\mathrm{SL}(n)\)-opers on \(X\) (see [loc. cit.]). The authors observe moreover that \(\tilde{\Psi}\) factorises via the involution \({\mathcal I}\) of \({\mathcal M}_X ( r )\) given by \(F \mapsto F^*\). Now let \(\xi \to X\) be a line bundle such that \(\xi^{\otimes 2}\) is trivial. The involution \({\mathcal I}\) restricts to an involution of the sublocus \({\mathcal M}_X ( r, \xi )\) of \({\mathcal M}_X ( r )\) consisting of bundles of determinant \(\xi\), and so \(\tilde{\Psi}\) gives rise to an algebraic morphism \(\Psi \colon {\mathcal M}_X ( r , \xi ) / {\mathcal I} \to \mathrm{Op}_X ( n )\). The authors go on to observe that for \(n = r\), there is a coincidence of dimensions \[ \dim {\mathcal M}_X ( r , \xi ) / {\mathcal I} \ = \ ( r^2 - 1 ) ( g - 1 ) \ = \ \dim \mathrm{Op}_X ( r ) . \] They conclude by posing the question of how close \(\Psi\) is to being injective or surjective.