Harder-Narasimhan stacks for principal bundles in higher dimensions and arbitrary characteristics

TL;DR

This paper constructs algebraic stacks parametrizing principal G-bundles with fixed Harder-Narasimhan type over higher-dimensional varieties in arbitrary characteristic, extending previous results and providing new proofs of stability properties.

Contribution

It develops a theory of Harder-Narasimhan stacks for principal bundles in higher dimensions and arbitrary characteristic, with new proofs and detailed geometric properties.

Findings

Constructed algebraic stacks for fixed HN types of principal G-bundles.

Proved the schematic nature and finiteness properties of the forgetful morphism.

Established openness of semistability and semicontinuity of canonical reductions.

Abstract

Let $G$ be a split reductive group over a field $k$ of arbitrary characteristic, chosen suitably. Let $X\to S$ be a smooth projective morphism of locally noetherian $k$-schemes, with geometrically connected fibers. We show that for each Harder-Narasimhan type $\tau$ for principal $G$-bundles, all pairs consisting of a principal $G$-bundle on a fiber of $X\to S$ together with a given canonical reduction of HN-type $\tau$ form an algebraic stack $Bun_{X/S}^{\tau}(G)$ over $S$. The forgetful $1$-morphism $Bun_{X/S}^{\tau}(G) \to Bun_{X/S}(G)$ to the algebraic stack of all principal $G$-bundles on fibers of $X\to S$ is a schematic morphism, which is of finite type, separated, radicial, and induces an isomorphism on residue fields of all points of $Bun_{X/S}^{\tau}(G)$. It factors via an open substack $Bun_{X/S}^{\ngtr \tau}(G)$ of $Bun_{X/S}(G)$, inducing a finite morphism $Bun_{X/S}^{\tau}(G) \to Bun_{X/S}^{\ngtr \tau}(G)$. This is a closed embedding if the Behrend conjecture is satisfied by $G$. The results of this paper hold in arbitrary characteristic, and in fact it gives better proofs of the results of our earlier papers which had assumed the characteristic to be zero. Along the way we give a new proof of the existence of a canonical reduction over any base field in all dimensions, and we also prove openness of semistability and semicontinuity of canonical type in a family.