On the finite principal bundles

Abstract

Let G be a connected linear algebraic group defined over \({\mathbb C}\). Fix a finite dimensional faithful G-module V0. A holomorphic principal G-bundle EG over a compact connected Kahler manifold X is called finite if for each subquotient W of the G-module V0, the holomorphic vector bundle EG(W) over X associated to EG for W is finite. Given a holomorphic principal G-bundle EG over X, we prove that the following four statements are equivalent: (1) The principal G-bundle EG admits a flat holomorphic connection whose monodromy group is finite. (2) There is a finite etale Galois covering \({f: Y \longrightarrow X}\) such that the pullback f*EG is a holomorphically trivializable principal G-bundle over Y. (3) For any finite dimensional complex G-module W, the holomorphic vector bundle EG(W) = E × GW over X, associated to the principal G-bundle EG for the G-module W, is finite. (4) The principal G-bundle EG is finite.