Schematic Harder–Narasimhan stratification for families of principal bundles and Λ-modules

Abstract

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 ,a nd letE 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 univer- sal 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 strati- fication 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.