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We prove that a holomorphic vector bundle E over a compact connected Kdhler manifold admits a flat connection, with a finite group as its monodromy, if and only if there are two distinct polynomials f and g, with nonnegative integral coefficients, such that the vector bundle f (E) is isomorphic to g(E). An analogous result is proved for vector bundles over connected smooth quasi-projective varieties, of arbitrary dimension, admitting a flat connection with finite monodromy group. When the base space is a connected projective variety, or a connected smooth quasi-projective curve, the above characterization of vector bundles admitting a flat connection with finite monodromy group was established by M. V. Nori.

On a characterization of finite vector bundles as vector bundles admitting a flat connection with finite monodromy group