# Abelianisation of Logarithmic $\mathfrak{sl}_2$-Connections

@article{Nikolaev2019AbelianisationOL, title={Abelianisation of Logarithmic \$\mathfrak\{sl\}_2\$-Connections}, author={Nikita S. Nikolaev}, journal={arXiv: Algebraic Geometry}, year={2019} }

We prove a functorial correspondence between a category of logarithmic $\mathfrak{sl}_2$-connections on a curve $X$ with fixed generic residues and a category of abelian logarithmic connections on an appropriate spectral double cover $\pi : \Sigma \to X$. The proof is by constructing a pair of inverse functors $\pi^{\text{ab}}, \pi_{\text{ab}}$, and the key is the construction of a certain canonical cocycle valued in the automorphisms of the direct image functor $\pi_\ast$.

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