On the finite principal bundles

Abstract

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