Schematic Harder–Narasimhan stratification for families of principal bundles in higher dimensions

Abstract

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.

A guide to moduli theory beyond GIT

“…A reduction to a parabolic subgroup is also defined in [AB83] by a different method, and it is proved in [AAB02] to coincide with the canonical filtration. Gurjar and Nitsure shows that this reduction also gives a schematic stratification (i.e., a stratification of the stack) [GN14,GN18]. For principal bundles in positive characteristics, Gurjar and Nitsure [GN16] show that there are algebraic stacks corresponding to each HNtype, and these are radicial over the algebraic stack of all principal bundles.…”

“…Gurjar and Nitsure shows that this reduction also gives a schematic stratification (i.e., a stratification of the stack) [GN14,GN18]. For principal bundles in positive characteristics, Gurjar and Nitsure [GN16] show that there are algebraic stacks corresponding to each HNtype, and these are radicial over the algebraic stack of all principal bundles. If the Behrend conjecture holds then these stacks are shown to define a stratification of the stack of all principal bundles (see also [Hei08a]).…”