As a follow up to me other [question](https://mathoverflow.net/questions/1527/pushforwards-of-line-bundles-and-stability), 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?



Unstable Vector Bundles
Well, I don't know about horrible. There's a lot you can say that's good! I'll start rambling and see where I end up.


I'm going to pretend you said principal GL(n)-bundle instead of rank n vector bundle. Same thing, really, since we have the standard representation.


The collection Bun(n,C) of all principal GL(n) bundles P on a smooth curve C is a very nice geometric object: it's an Artin stack. It's not connected; the different components are labelled by topological data, like the Chern class. The tangent "space" (complex, really) to Bun(n,C) at a point P is naturally the derived global sections RGamma(C,ad(P)), where ad(P) is the associated bundle with fiber the adjoint representation of GL(n). The zero-th cohomology gives the infinitesimal automorphisms and the 1st cohomology gives the deformations. So the stabilizer group of any point V in Bun(n,C) is finite-dimensional, and the dimension of the stack is n(g-1) (by Riemann-Roch). Bun(n,C) is smooth, and unobstructed, thanks to the vanishing of H^2(C,ad(P)).


Bun(n,C) has a very nice stratification, too. It's an increasing union of quotient stacks [A/G] of projective varieties by finite-dimensional groups. Roughly, A is the stack of pairs (P,t), where t is a trivialization of P in an infinitesimal neighborhood of some point in C. Make the neighborhood large enough, i.e., r-th order, and you can kill off all the automorphisms of P. Unfortunately, except for n=1, there is no uniform bound on r that works for all bundles. So, Bun(n,C) isn't a finite type quotient stack. 


You can also realize Bun(n,C) (homotopically) as the infinite type quotient stack of U(n)- connections modulo complexified gauge transformations. That's what Atiyah & Bott do in their paper "The Yang-Mills Equations on Riemann Surfaces". (They also have a nice discussion of slope-stability and the stratification.)


The top component of the stratification (those bundles where the stabilizer group is as small as possible) is the stack of (semi-)stable vector bundles. If you take the coarse moduli space of this substack, you get the usual moduli space of stable bundles. 


In summary: If you drop the stability conditions, you get a lot more geometry with a similar flavor, and without the random bits of weirdness that crop up in the theory of moduli spaces. (e.g., the stack always carries a universal bundle, you don't need the rank and the chern class to be coprime.) 


OK, I'll stop evangelizing now. 


