Lie groupoids and logarithmic connections

TL;DR

This paper employs Lie groupoid theory to classify logarithmic flat connections on principal bundles, establishing a functorial Riemann-Hilbert correspondence and analyzing their monodromy representations.

Contribution

It introduces a Lie groupoid framework for logarithmic connections, providing a functorial classification and a van Kampen type theorem for their monodromy representations.

Findings

Canonical Jordan-Chevalley decomposition for representations

Functorial classification of logarithmic connections

Riemann-Hilbert correspondence via generalized monodromy

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 use this 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.