Étale triviality of finite vector bundles over compact complex manifolds

Abstract

A vector bundle E over a projective variety M is called finite if it satisfies a nontrivial polynomial equation with nonnegative integral coefficients. Introducing finite bundles, Nori proved that E is finite if and only if the pullback of E to some finite étale covering of M is trivializable [14]. The definition of finite bundles extends naturally to holomorphic vector bundles over compact complex manifolds. We prove that a holomorphic vector bundle over a compact complex manifold M is finite if and only if the pullback of E to some finite étale covering of M is holomorphically trivializable. Therefore, E is finite if and only if it admits a flat holomorphic connection with finite monodromy. In [4] this result was proved under the extra assumption that the compact complex manifold M admits a Gauduchon astheno-Kähler metric.