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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 X0 be an irreducible projective nodal curve. We prove that an indecomposable vector bundle or torsion-free coherent sheaf E on X0 admits a holomorphic logarithmic connection ∇: E→ E⊗ωX0 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· I at one preimage of the node and −λ· I at the other, for some λ∈ℂ. Explicit one-parameter families of flat connections are constructed on the irreducible rational nodal cubic curve as an illustration.
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BibTeX
@misc{das2026weils,
title = {{Weil's Theorem for Logarithmic Connections on Irreducible Nodal Curves}},
author = {Sourav Das},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29827},
eprint = {2608.29827},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29827},
}
RIS
TY - UNPB
TI - Weil's Theorem for Logarithmic Connections on Irreducible Nodal Curves
AU - Sourav Das
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29827
UR - https://arxiv.org/abs/2608.29827
AB - 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 ∇: E→ E⊗ω_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· I at one preimage of the node and −λ· I at the other, for some λ∈ℂ. Explicit one-parameter families of flat connections are constructed on the irreducible rational nodal cubic curve as an illustration.
ER -