On certain Tannakian categories of integrable connections over Kaehler manifolds

TL;DR

This paper explores a specific Tannakian category formed by trivial holomorphic bundles with integrable connections over compact Kaehler manifolds, revealing that for Riemann surfaces, the associated group scheme uniquely identifies the surface.

Contribution

It introduces a new Tannakian category based on trivial bundles with integrable connections and shows its pro-algebraic group scheme uniquely characterizes Riemann surfaces.

Findings

The category forms a neutral Tannakian category.

The pro-algebraic group scheme encodes geometric information.

For Riemann surfaces, the scheme determines the surface uniquely.

Abstract

Given a compact Kaehler manifold X, it is shown that pairs of the form (E, D), where E is a trivial holomorphic vector bundle on X, and D is an integrable holomorphic connection on $E$, produce a neutral Tannakian category. The corresponding pro-algebraic affine group scheme is studied. In particular, it is shown that this pro-algebraic affine group scheme for a compact Riemann surface determines uniquely the isomorphism class of the Riemann surface.