Abstract
Let G be a connected linear algebraic group defined over \({\mathbb C}\). Fix a finite dimensional faithful G-module V 0. A holomorphic principal G-bundle E G over a compact connected Kähler manifold X is called finite if for each subquotient W of the G-module V 0, 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 étale Galois covering \({f: Y \longrightarrow X}\) 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 × G W 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.
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Biswas, I. On the finite principal bundles. Ann Glob Anal Geom 35, 181–190 (2009). https://doi.org/10.1007/s10455-008-9129-5
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DOI: https://doi.org/10.1007/s10455-008-9129-5