On the semi-finite vector bundles with connection over Kähler manifolds

Let X be a compact connected Kähler manifold. We consider the category CEC(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 CEC(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 CEC(X). Let πEN(X,x0) be an extension of the Nori fundamental group scheme over C [8]. We show that πEN(X,x0) is a closed subgroup scheme of ϖEC(X,x0). Finally, we discuss an example illustrating that if X is not Kähler, then the natural homomorphism πEN(X,x0)⟶ϖEC(X,x0) might fail to be an embedding.