Higgs bundles and fundamental group schemes

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.