Higgs bundles and fundamental group schemes (Q2322405)

Given a projective scheme \(X\) over a field \(k\), a line bundle \(L\) on \(X\) is said to be numerically effective (abbreviated as nef) if \(\mathrm{deg }f*L=0\) for every morphism \(f:\mathcal{C}\longrightarrow X\), where \(\mathcal{C}\) is an irreducible smooth projective curve. A notion of numerical effectiveness for a vector bundle \(E\) can be defined by asking that the relative hyperplane bundle \(\mathcal{O}_{\mathbb{P}(E)}(1)\) on the projective bundle \(\mathbb{P}(E)\) is nef (see [\textit{R. Hartshorne}, Publ. Math., Inst. Hautes Étud. Sci. 29, 63--94 (1966; Zbl 0173.49003); Positivity in algebraic geometry. I. Classical setting: line bundles and linear series. Berlin: Springer (2004; Zbl 1093.14501); Positivity in algebraic geometry. II. Positivity for vector bundles, and multiplier ideals. Berlin: Springer (2004; Zbl 1093.14500)]). Recall that Tannakian categories are abelian tensor categories that satisfy some additional properties and are equipped with a functor to the category of vector spaces. They all turn out to be equivalent to categories of representations of pro algebraic affine group schemes, so that there is natural duality between Tannakian categories and such group schemes. Relying on a notion of numerical effectiveness for Higgs bundles, the authors show that the category of numerically flat Higgs vector bundles on a smooth projective variety \(X\) is a Tannakian category. They introduce the associated group scheme, that they 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. This paper is organized as follows: the first section is an introduction to the subject. Section 2 deals with numerically effective Higgs bundles. In Section 3, the authors give some properties of \(H\)-nef Higgs bundles. In Section 4, the authors study categories of numerically flat bundles.