On holomorphic principal bundles over a compact Riemann surface admitting a flat connection
Let \(X\) be a compact connected Riemann surface, and \(E\) be a holomorphic vector bundle over \(X.\) A theorem of A. Weil says that \(E\) admits a flat connection, compatible with its holomorphic structure, if and only if every direct summand of \(E\) is of degree zero. The aim of this article is to establish an analog of the criterion for principal bundles. Let \(G\) be a connected reductive linear algebraic group over \({\mathbb C},\) and \(L \subset G\) be a Levi factor of some parabolic subgroup of \(G,\) with \(L_{0} = L/[L, L]\) its maximal abelian quotient. Theorem 4.1. A holomorphic \(G\)-bundle \(E_{G}\) over \(X\) admits a flat connection, compatible with its holomorphic structure, if and only if for every parabolic subgroup \(P\) containing the fixed Borel \(B\) and every holomorphic reduction \(E_{L} \subset E_{G}\) of its structure group to \(L,\) the principal \(L_{0}\)-bundle, obtained by extending the structure group of \(E_{G},\) is topologically trivial.
scientific article; zbMATH DE number 1747293