Harder-Narasimhan stacks for principal bundles in higher dimensions

Sudarshan Gurjar, Nitin Nitsure

Published 2016-05-29Version 1

Let $G$ be a connected 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 Artin algebraic stack $Bun_{X/S}^{\tau}(G)$ over $S$. Moreover, the forgetful $1$-morphism $Bun_{X/S}^{\tau}(G) \to Bun_{X/S}(G)$ to the stack of all principal $G$-bundles on fibers of $X\to S$ 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 $Bun_{X/S}^{\tau}(G) \to Bun_{X/S}(G)$ are locally closed imbeddings which stratify $Bun_{X/S}(G)$ as $\tau$ varies.