TITLE: Unstable Vector Bundles
QUESTION [10 upvotes]: As a follow up to me other question, what can be said about unstable vector bundles? I know this is rather open ended, but what sorts of horrible things does having a subbundle of strictly greater slope imply?

REPLY [3 votes]: Another reason behind the notion of stability is the "jumping phenomenon."  Namely, you can construct a family of vector bundles parametrized by the disk where all the fibers apart from the origin are mutually isomorphic, but not isomorphic to the fiber at the origin.  Concretely, you can realize this by scaling an extension class.  In the Atiyah-Bott-Kempf-Ness picture, this corresponds to non-closed orbits of the complex gauge group. When the base manifold has dimension greater than one, the pieces in the Harder-Narasimahan filtration may not be subbundles, so the limiting object is in general a torsion-free sheaf, and not necessarily a bundle.