Weil's Theorem for Logarithmic Connections on Irreducible Nodal Curves

Source 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.