Unipotent Schottky bundles on Riemann surfaces and complex tori (Q2874682)

Let \(X\) be a complex manifold, \(\mathcal{O}_X\) its sheaf of holomorphic functions, and \(\pi:=\pi_1(X)\) its fundamental group. There is a functor from the category of \(\mathbb{C}\pi\)-modules to the category of \(\mathcal{O}_X\)-modules by taking a \(\mathbb{C}\pi\)-module \(M\) and identifying it with a representation \(\rho:\pi\to \mathrm{Aut}(M)\), and then using \(\rho\) to construct a holomorphic vector bundle over \(X\), which in turn gives an \(\mathcal{O}_X\)-module on \(X\). Let \(\Sigma\) be a free group (abelian or non-abelian), together with a surjective homomorphism \(\alpha:\pi\to \Sigma\). A \(\mathbb{C}\pi\)-module \(M\) is called \(\Sigma\)-Schottky if it is induced by a \(\mathbb{C}\Sigma\)-module via \(\alpha\). The restriction of the aforementioned functor to the Schottky modules is called the \textit{Schottky functor}. The main theorems of this interesting paper show that the Schottky functor induces an equivalence of categories between unipotent \(\mathbb{C}\Sigma\)-modules and unipotent \(\mathcal{O}_X\)-modules on \(X\) when either:NEWLINENEWLINE(a) \(X\) is a complex torus of dimension \(g\), and \(\Sigma\) is free abelian of rank \(g\), orNEWLINENEWLINE(b) \(X\) is a compact Riemann surface of genus \(g\) and \(\Sigma\) is a free group of rank \(g\).NEWLINENEWLINE Note that in both cases the homomorphism \(\alpha\) is explicitly given. These results then imply:NEWLINENEWLINE (1) a holomorphic vector bundle \(E\) over a complex torus admits a flat holomorphic connection if and only if \(E\) is Schottky, andNEWLINENEWLINE (2) a principal \(G\)-bundle \(P\), for \(G\) a complex connected linear algebraic group, over a complex torus admits a flat holomorphic connection if and only if \(P\) is Schottky.