Abstract: Around 1975, Harder-Narasimhan defined a canonical filtration of a vector bundle which proved to be of central importance to the subject. A natural question is to understand the variation of this filtration in families. Associated to such a filtration, one can define a ''type'' which is a measure of its non-semistability. We show that this type varies upper-semicontinuously for families of vector bundles. Furthermore, for each type, we define a locally-closed subscheme structure on the subset of the parameter space, consisting of points corresponding to this type and show that it satisfies the universal property that a base-change admits a relative Harder-Narasimhan filtration if and only if the base-change factors through that stratum.

As a consequence we show that vector bundles of a fixed Harder-Narasimhan type form an Artin stack which is a locally-closed substack of the stack of all vector bundles.

These results hold more generally for Principal Bundles with a reductive structure group over higher dimensional varieties as well but I will mostly stick to vector bundles in the talk, making some comments about Principal Bundles in the end. Basic familiarity of algebraic geometry will be assumed.