Normal forms and moduli stacks for logarithmic flat connections

We establish normal form theorems for a large class of singular flat connections on complex manifolds, including connections with logarithmic poles along weighted homogeneous Saito free divisors. As a result, we show that the moduli spaces of such connections admit the structure of algebraic quotient stacks. In order to prove these results, we introduce homogeneous Lie groupoids and study their representation theory. In this direction, we prove two main results: a Jordan–Chevalley decomposition theorem and a linearization theorem. We give explicit normal forms for several examples of free divisors, such as homogeneous plane curves, reductive free divisors, and one of Sekiguchi’s free divisors.