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

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.