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 ${\pi }^{\mathrm{EN}}(X,x_0)⟶{\varpi }^{\mathrm{EC}}(X,x_0)$ might fail to be an embedding.

Final version