Weil's Theorem for Logarithmic Connections on Irreducible Nodal Curves

Abstract

We establish an analogue of Andr\'e Weil's classical theorem for irreducible nodal curves. Let \(X_0\) be an irreducible projective nodal curve. We prove that an indecomposable vector bundle or torsion-free coherent sheaf \(E\) on \(X_0\) admits a holomorphic logarithmic connection \(\nabla\colon E\to E\otimes\omega_{X_0}\) with respect to the dualizing sheaf if and only if \(\deg E=0\). Moreover, when \(\deg E=0\), such a connection can be chosen so that the induced logarithmic connection on the normalization has scalar residues \(\lambda\cdot I\) at one preimage of the node and \(-\lambda\cdot I\) at the other, for some \(\lambda\in\mathbb{C}\). Explicit one-parameter families of flat connections are constructed on the irreducible rational nodal cubic curve as an illustration.