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Let $G$ be a split reductive group over a field $k$ of arbitrarycharacteristic, chosen suitably. Let $X\to S$ be a smooth projective morphismof locally noetherian $k$-schemes, with geometrically connected fibers. We showthat for each Harder-Narasimhan type $\tau$ for principal $G$-bundles, allpairs consisting of a principal $G$-bundle on a fiber of $X\to S$ together witha 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 fibersof $X\to S$ 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 Behrendconjecture is satisfied by $G$. The results of this paper hold in arbitrarycharacteristic, and in fact it gives better proofs of the results of ourearlier papers which had assumed the characteristic to be zero. Along the waywe give a new proof of the existence of a canonical reduction over any basefield in all dimensions, and we also prove openness of semistability andsemicontinuity of canonical type in a family.

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