On semi-finite vector bundles with connection over Kahler manifolds

Let X be a compact connected K¨ahler manifold. We consider the category C EC(X) of flat holomorphic connections (E, ∇E) over X satisfying the condition that the underlying holomorphic vector bundle E admits a filtration of holomorphic subbundles preserved by the connection ∇E such that the monodromy of the induced connection on each successive quotient has finite image. The category C EC(X), equipped with the neutral fiber functor that sends any object (E, ∇E) to the fiber Ex0 , where x0 ∈ X is a fixed point, defines a neutral Tannakian category over C. Let ϖEC(X, x0) denote the affine group scheme corresponding to this neutral Tannakian category C EC(X). Let π EN(X, x0) be an extension of the Nori fundamental group scheme over C [8]. We show that π EN(X, x0) is a closed subgroup scheme of ϖEC(X, x0). Finally, we discuss an example illustrating that if X is not K¨ahler, then the natural homomorphism π EN(X, x0) −→ ϖEC(X, x0) might fail to be an embedding.