Introduction to moduli spaces of connections: some explicit constructions

TL;DR

This paper provides explicit constructions of moduli spaces of rank 2 Fuchsian systems and logarithmic connections on the Riemann sphere with four poles, illustrating their geometric properties and differences.

Contribution

It offers concrete examples of moduli spaces, clarifying their structure and algebraic properties, especially distinguishing between affine cubic surfaces and non-algebraic affine surfaces.

Findings

Fuchsian systems form affine cubic surfaces

Logarithmic connections are analytically but not algebraically isomorphic to affine surfaces

Remarks on moduli spaces with arbitrary poles

Abstract

We give some concrete examples of moduli spaces of connections. Precisely, we explain how to explicitly construct the moduli spaces of rank 2 fuchsian systems and logarithmic connections on the Riemann sphere with 4 poles. The former ones are affine cubic surfaces; the latter ones are analytically isomorphic to affine surfaces but not algebraically: they do not carry non constant regular functions. We end with some remark on arbitrary number of poles.