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Lie groupoids and logarithmic connections

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Abstract

Using tools from the theory of Lie groupoids, we study the category of logarithmic flat connections on principal G-bundles, where G is a complex reductive structure group. Flat connections on the affine line with a logarithmic singularity at the origin are equivalent to representations of a groupoid associated to the exponentiated action of \(\mathbb {C}\). We show that such representations admit a canonical Jordan–Chevalley decomposition and may be linearized by converting the \({\mathbb {C}}\)-action to a \({\mathbb {C}}^{*}\)-action. We then apply these results to give a functorial classification. Flat connections on a complex manifold with logarithmic singularities along a hypersurface are equivalent to representations of a twisted fundamental groupoid. Using a Morita equivalence, whose construction is inspired by Deligne’s notion of paths with tangential basepoints, we prove a van Kampen type theorem for this groupoid. This allows us to show that the category of representations of the twisted fundamental groupoid can be localized to the normal bundle of the hypersurface. As a result, we obtain a functorial Riemann–Hilbert correspondence for logarithmic connections in terms of generalized monodromy data.

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Acknowledgements

I began this project at the start of my Ph.D. studies in 2015 under the supervision of M. Gualtieri, and several ideas were worked out during a visit to Paris in 2016 where I had the opportunity to talk with P. Boalch. In the end, it took the confinement imposed by the coronavirus pandemic to force me to finalize the details. Many of the ideas of this project were developed in collaboration with M. Gualtieri, in particular the work in the final section. I would like to thank M. Gualtieri, P. Boalch, and B. Pym for many useful discussions. This work was supported by an NSERC postdoctoral fellowship.

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    Francis Bischoff

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Bischoff, F. Lie groupoids and logarithmic connections. Sel. Math. New Ser. 30, 44 (2024). https://doi.org/10.1007/s00029-024-00929-3

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