Meromorphic connections on vector bundles over curves (Q2018808)

Let \(E\) be a holomorphic vector bundle over a compact Riemann surface \(X\). Let \[ 0=E_0 \subset E_1 \subset E_2 \dots \subset E_l = E \] be a filtration of holomorphic subbundles of \(E\). It is well known that any holomorphic vector bundle can be endowed with a meromorphic connection. Let \(\Delta\) be an effective divisor on \(X\) and for each \(i \in [1, l]\), let \(D_i\) be a meromorphic connection on \(E_i/E_{i-1}\) with polar divisor (at most) \(\Delta\). Consider the maximal subbundle \(\text{ End}(E_{\bullet})\) of \(\text{ End}(E)\) such that \(\sigma(\text{ End}(E_{\bullet}) \otimes E_i) \subset E_i\) for all \(i \in [1, l]\), where \(\sigma: \text{ End}(E) \otimes E \longrightarrow E\) is the evaluation map. In this article authors have shown that if \(H^0(X, \text{End}(E)/\text{End}(E_{\bullet}) \otimes \mathcal{O}_X(-\Delta)) = 0\), then there is a meromorphic connection \(D\) on \(E\) with polar divisor (at most) \(\Delta\) such that \(D\) preserves each subbundle \(E_i, i \in [1, l]\), and the connection on \(E_i/E_{i-1}\) induced by \(D\) coincides with \(D_i\).