Local Structure of Moduli Spaces

TL;DR

This paper explores the local structure of moduli spaces of connections using geometric invariant theory and Luna Slice Theorem, analyzing singularities in specific examples.

Contribution

It introduces a method to study the local structure and singularities of moduli spaces of connections via GIT and Luna Slice Theorem.

Findings

Representation of moduli space germ as quotient of Kuranishi space

Identification of singularities in specific connection moduli spaces

Application of Luna Slice Theorem to moduli space analysis

Abstract

We provide a sketch of the GIT construction of the moduli spaces for the three classes of connections: the class of meromorphic connections with fixed divisor of poles $D$ and its subclasses of integrable and integrable logarithmic connections. We use the Luna Slice Theorem to represent the germ of the moduli space as the quotient of the Kuranishi space by the automorphism group of the central fiber. This method is used to determine the singularities of the moduli space of connections in some examples.