Moduli space of $G$-connections on an elliptic curve

TL;DR

This paper studies the moduli space of algebraic $G$-connections on an elliptic curve, showing it has no nonconstant algebraic functions despite being biholomorphic to an affine variety.

Contribution

It proves that the moduli space of topologically trivial algebraic $G$-connections on an elliptic curve admits no nonconstant algebraic functions, revealing its complex structure.

Findings

The moduli space is biholomorphic to an affine variety.

It admits no nonconstant algebraic functions.

The space's structure contrasts with its algebraic function properties.

Abstract

Let $X$ be a smooth complex elliptic curve and $G$ a connected reductive affine algebraic group defined over $\mathbb C$. Let ${\mathcal M}_X(G)$ denote the moduli space of topologically trivial algebraic $G$--connections on $X$, that is, pairs of the form $(E_G\, , D)$, where $E_G$ is a topologically trivial algebraic principal $G$--bundle on $X$, and $D$ is an algebraic connection on $E_G$. We prove that ${\mathcal M}_X(G)$ does not admit any nonconstant algebraic function while being biholomorphic to an affine variety.