Normal forms and moduli stacks for logarithmic flat connections

TL;DR

This paper develops normal form theorems for singular flat connections with logarithmic poles, showing their moduli spaces are algebraic quotient stacks, and introduces homogeneous Lie groupoids with new structural theorems.

Contribution

It introduces a framework for normal forms of logarithmic flat connections and establishes the structure of their moduli spaces as algebraic quotient stacks.

Findings

Normal form theorems for singular flat connections.

Moduli spaces are algebraic quotient stacks.

Explicit normal forms for specific free divisors.

Abstract

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.