Higgs bundles and fundamental group schemes

TL;DR

This paper introduces the Higgs fundamental group scheme for smooth projective varieties, linking it to the properties of numerically flat Higgs bundles and a conjecture on Chern class vanishing.

Contribution

It defines the Higgs fundamental group scheme and establishes its relation to the category of numerically flat Higgs bundles and a key conjecture.

Findings

The category of numerically flat Higgs bundles is Tannakian.

The Higgs fundamental group scheme is introduced and characterized.

Connections to a conjecture on Chern class vanishing are explored.

Abstract

Relying on a notion of "numerical effectiveness" for Higgs bundles, we show that the category of "numerically flat" Higgs vector bundles on a smooth projective variety $X$ is a Tannakian category. We introduce the associated group scheme, that we call the "Higgs fundamental group scheme of $X$," and show that its properties are related to a conjecture about the vanishing of the Chern classes of numerically flat Higgs vector bundles.