Weil's Theorem for Logarithmic Connections on Irreducible Nodal Curves

Abstract

We establish an analogue of André 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 \(\nabla\colon E\to E\otimesω_{X_0}\) with respect to the dualizing sheaf if and only if \(°E=0\). Moreover, when \(°E=0\), such a connection can be chosen so that the induced logarithmic connection on the normalization has scalar residues \(λ\cdot I\) at one preimage of the node and \(-λ\cdot I\) at the other, for some \(λ\in\mathbb{C}\). Explicit one-parameter families of flat connections are constructed on the irreducible rational nodal cubic curve as an illustration.