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

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.