In this paper we study the extension of structure group of principal bundles with a reductive algebraic group as structure group on a smooth projective variety defined over an algebraically closed field. Our main result is to show that given a finitedimensional representation ρ of a reductive algebraic group G, there exists an integer N which is quantifiable in terms of G and ρ such that any semistable G-bundle whose first N Frobenius pullbacks are semistable induces a semistable bundle on extension of structure group via ρ. We do this by quantifying the fields of definition of the instability parabolics associated to various parabolic reductions of the induced bundle.
Let G be a reductive algebraic group over a field k of characteristic zero, let X → S be a smooth projective family of curves over k, and let E be a principal G bundle on X. The main result of this note is that for each Harder-Narasimhan type τ there exists a locally closed subscheme S τ (E) of S which satisfies the following universal property. If f : T → S is any base-change, then f factors via S τ (E) if and only if the pullback family f * E admits a relative canonical reduction of Harder-Narasimhan type τ . As a consequence, all principal bundles of a fixed Harder-Narasimhan type form an Artin stack. We also show the existence of a schematic Harder-Narasimhan stratification for flat families of pure sheaves of -modules (in the sense of Simpson) in arbitrary dimensions and in mixed characteristic, generalizing the result for sheaves of O-modules proved earlier by Nitsure. This again has the implication that -modules of a fixed Harder-Narasimhan type form an Artin stack.
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.
Let G be a connected split reductive group over a field k of arbitrary characteristic, chosen suitably. Let X → S be a smooth projective morphism of locally noetherian k-schemes, with geometrically connected fibers. We show that for each Harder-Narasimhan type τ for principal G-bundles, all pairs consisting of a principal G-bundle on a fiber of X → S together with a given canonical reduction of HN-type τ form an Artin algebraic stack Bun τ X/S (G) over S. Moreover, the forgetful 1-morphism Bun τ X/S (G) → Bun X/S (G) to the stack of all principal G-bundles on fibers of X → S is a schematic morphism, which is of finite type, separated and injective on points.The notion of a relative canonical reduction that we use was defined earlier in arXiv:1505.02236, where we showed that a stronger result holds in characteristic zero, namely, the 1-morphisms Bun τ X/S (G) → Bun X/S (G) are locally closed imbeddings which stratify Bun X/S (G) as τ varies.
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