On semi-finite vector bundles with connection over Kahler manifolds

TL;DR

This paper introduces a new Tannakian category of semi-finite flat connections over compact Kähler manifolds, relating it to the Nori fundamental group scheme and exploring their subgroup relations.

Contribution

It defines a novel Tannakian category for semi-finite vector bundles with connection and establishes its relation to the Nori fundamental group scheme over Kähler manifolds.

Findings

The category $ ext{EC}(X)$ forms a neutral Tannakian category.

The affine group scheme $ ext{EN}(X, x_0)$ embeds as a subgroup of $ ext{EC}(X)$.

The embedding property may fail for non-Kähler manifolds.

Abstract

Let $X$ be a compact connected K\"ahler 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 K\"ahler, then the natural homomorphism $\pi^{\mathrm{EN}}(X,\, x_0)\, \longrightarrow\, \varpi^{\mathrm{EC}}(X,\, x_0)$ might fail to be an embedding.