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

We prove that a holomorphic vector bundle $E$ over a compact connected Kähler 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.