On semi-finite vector bundles with connection over Kahler manifolds

Abstract

Let X be a compact connected Kähler manifold. We consider the category \mathcal{C}^\mathrm{EC}(X) of flat holomorphic connections (E,\, \nabla^E) over X satisfying the condition that the underlying holomorphic vector bundle E admits a filtration of holomorphic subbundles preserved by the connection \nabla^E such that the monodromy of the induced connection on each successive quotient has finite image. The category \mathcal{C}^\mathrm{EC}(X), equipped with the neutral fiber functor that sends any object (E,\, \nabla^E) to the fiber E_{x_0}, where x_0\, \in\, X is a fixed point, defines a neutral Tannakian category over \mathbb{C}. Let \varpi^{\mathrm{EC}}(X,\, x_0) denote the affine group scheme corresponding to this neutral Tannakian category \mathcal{C}^\mathrm{EC}(X). Let \pi^{\mathrm{EN}}(X,\, x_0) be an extension of the Nori fundamental group scheme over \mathbb{C}. We show that \pi^{\mathrm{EN}}(X,\, x_0) is a closed subgroup scheme of \varpi^{\mathrm{EC}}(X,\, x_0). Finally, we discuss an example illustrating that if X is not Kähler, then the natural homomorphism \pi^{\mathrm{EN}}(X,\, x_0)\, \longrightarrow\, \varpi^{\mathrm{EC}}(X,\, x_0) might fail to be an embedding.