Unstable Vector Bundles

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?

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A non-horrible thing: it has a "Harder-Narasimhan filtration" by smaller bundles, where each quotient is stable. Instead of making a moduli space of stable vector bundles, you could make a moduli space of vector bundles with fixed dimensions in the H-N filtration. See Atiyah-Bott's Yang-Mills paper for more. –  Allen Knutson Dec 3 '10 at 1:52

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.

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Well, I'm actually quite fine with Artin stacks and such. Here, I was more thinking about what happens when you take a specific unstable bundle, and what you can say about it, and also partly why (other than to get a GIT quotient) it might be bad. However, thanks for doing a few things explicitly that I'd been meaning to work through. –  Charles Siegel Oct 21 '09 at 3:05
Heh. I did say I'd ramble, not that I'd answer your question. But you might say that what's distinctive about unstable bundles is that their automorphism groups can get bigger. (Caution: misleading when n=1) –  userN Oct 21 '09 at 11:58
" It's an increasing union of quotient stacks [A/G] of projective varieties by finite-dimensional groups." Shouldn't it be "quasi-projective varieties"? (Or maybe something strange is going on here.) –  t3suji Dec 3 '10 at 2:43
Yes, thanks! In particular the statement I have in mind is the one gotten by considering trivializations of the bundles over formal neighborhoods of points. –  userN Dec 3 '10 at 4:17