Representations of Atiyah algebroids and logarithmic connections

TL;DR

This paper explores the relationship between Atiyah algebroid representations and logarithmic connections, providing new functorial links and applying Lie algebroid theory to classical problems like Deligne extensions and the Riemann-Hilbert correspondence.

Contribution

It introduces functors connecting Atiyah algebroid modules with logarithmic connections, enabling new approaches to classical problems using Lie algebroid theory.

Findings

Established functors relating Atiyah algebroids and logarithmic connections

Derived invariants such as monodromy for Atiyah algebroid representations

Reproduced classical results like Deligne extensions via pull-back from Atiyah algebroid modules

Abstract

In this paper, we investigate representations of $\operatorname{At}(N)$, the Atiyah algebroids of a holomorphic line bundles $N$ over a complex manifold $Y$. In particular, we relate $\operatorname{At}(N)$-modules with logarithmic connections through two functors. On the one hand, we use these functors to the define invariants (monodromy) for representations of Atiyah algebroids. On the other hand, this opens the way to use the theory of Lie algebroids to study problems about logarithmic connections; we will give an example of this by showing that the existence of Deligne's extensions of flat connections and the Riemann-Hilbert correspondence for regular flat meromorphic connections may be obtained as pull-back of similar results for $\operatorname{At}(N)$-modules, and, at this level, these results are a direct consequence of the second theorem of Lie.