Flat holomorphic connections on principal bundles over a projective manifold

Abstract

Let G G be a connected complex linear algebraic group and R u ( G ) R_u(G) its unipotent radical. A principal G G –bundle E G E_G over a projective manifold M M will be called polystable if the associated principal G / R u ( G ) G/R_u(G) –bundle is so. A G G –bundle E G E_G over M M is polystable with vanishing characteristic classes of degrees one and two if and only if E G E_G admits a flat holomorphic connection with the property that the image in G / R u ( G ) G/R_u(G) of the monodromy of the connection is contained in a maximal compact subgroup of G / R u ( G ) G/R_u(G) .