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

Let G be a connected split reductive group over a field k of characteristic zero. Let be a smooth projective morphism of k-schemes, with geometrically connected fibers. We formulate a natural definition of a relative canonical reduction, under which principal G-bundles of any given Harder-Narasimhan type on fibers of X / S form an Artin algebraic stack over S, and as varies, these stacks define a stratification of the stack by locally closed substacks. This result extends to principal bundles in higher dimensions the earlier such result for principal bundles on families of curves. The result is new even for vector bundles, that is, for G = GL(n,k)