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The slope of a vector bundle E is defined as mu(E) = deg(E)/rank(E). Then a vector bundle E is called semistable if mu(E') ≤ mu(E) for all proper sub-bundles E'. It is called stable if mu(E') < mu(E).

I've heard that moduli spaces of stable and semistable vector bundles are somehow well-behaved, but I don't know why this is, nor do I know exactly what well-behaved should mean in this context. What goes wrong if we try to consider moduli of more general vector bundles? Moreover the definitions of slope and (semi)stable seem a bit artificial -- where do they come from?

I've also only seen the above definitions made in the context of vector bundles over curves. Why just curves? Does something stop working in higher dimensions or in greater generality?

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Err, I should be "Kevin Lin", not "unknown (google)", what's going on? – Kevin H. Lin Oct 9 '09 at 22:12
@Kevin: I merged the two users and added the other OpenID as an alternate on your main account. For some reason, Google is producing multiple OpenIDs for the same person, which confuses this site. Out of curiosity, were you using a different computer from your usual one? If you have this kind of problem with your account again, flag the post for moderator attention (just like you did). – Anton Geraschenko Oct 9 '09 at 22:38
Yes, at the time I posted the question, I was using a different computer from my usual one. – Kevin H. Lin Oct 10 '09 at 21:10
up vote 7 down vote accepted

I believe "nice" here means "is a quasi-projective variety."

As for why, the reason is geometric invariant theory, which is roughly a way of looking at moduli problems, or actions of groups (which is roughly the same thing) and picking out a subset of the quotient (which itself is only nice as a stack) which is a quasi-projective variety. So there's a general definition of semi-stable points for any action of an affine algebraic group acting on a projective variety with choice of equivariant projective embedding (it depends on the choice of embedding) and the quotient of the semi-stable points is always a quasi-projective variety.

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Are there any standard references for this material? --- At the moment I don't necessarily want to learn all of the ins and outs of GIT, I'm mainly just curious about what this general definition of semistable is (probably something to do with automorphisms or stabilizers?), and how we get from it the definition of semistable vector bundle. – Kevin H. Lin Oct 10 '09 at 18:37
If you just want definitions, read Wikipedia: The standard reference is the book of Mumford, Fogarty and Kirwan called simply "Geometric Invariant Theory." – Ben Webster Oct 11 '09 at 13:33
A good survey with actual information is in this thesis here: – Urs Schreiber Jul 14 '14 at 16:54

From a topological viewpoint, I believe the idea is that one wants to have a Hausdorff quotient space. In other words, consider the space of all holomorphic structures on a fixed (topological) vector bundle on a curve. Holomorphic structures can be viewed as differential operators on sections of the bundle, such that a section is holomorphic if and only if this operator evaluates to zero on the section. (See, for example, sections 5 and 7 of Atiyah and Bott's "The Yang-Mills equations over Riemann surfaces.") This makes the space of holomorphic structures (i.e. the space of bundles with a fixed topological type) into an affine space. The group of complex automorphisms of the bundle acts on this space, and the quotient is the moduli space of holomorphic bundles. If you don't restrict to stable bundles, this quotient space fails to be Hausdorff. Atiyah and Bott reference this to Mumford's 1965 GIT book. Actually, they just say that the moduli space of stable bundles is Hausdorff, due to the fact that the orbits of stable bundles are closed. (Hmmm... that really just says points are closed in the quotient...) I don't know how much of this is spelled out in Mumford; in particular, I don't know whether there's a proof in the literature that the full quotient space fails to be Hausdorff.

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Since I can't comment yet, I'll have to recommend this in an answer: The book by Huybrechts and Lehn: "The Geometry of the Moduli Space of Sheaves" might be worth a look. I really liked the parts that I read.

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Another interpretation of "well-behaved" might be that the collection of semistable sheaves with a fixed Hilbert polynomial is a bounded family of sheaves. This is equivalent to saying that there is a coherent sheaf F with the property that all semistable sheaves with fixed Hilbert polynomial P can all be realized as a quotient of F. See 1.7 of Huybrechts and Lehn.

Compare with the Hilbert scheme: there, we only need to fix the Hilbert polynomial, but that is because all structure sheaves of subschemes come equipped with the structure of a quotient of OX. So boundedness of the family of structure sheaves is equivalent to fixing the Hilbert polynomial.

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Some reasons people study bundles on curves are:

  • It's easier, e.g. space of curves themselves is fairly simple.

  • You expect some correspondence to representations of fundamental group (indeed, by Narasimhan-Seshadri stable bundles correspond to irreducible reps of fundamental group). Incidentally, I just learned from wikipedia that it's been proven for surfaces as well.

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I'm only intressted in the case of bundles over curves. But as far as I know, there is also a lot of work in higher dimensions. A nice book about this is Kobayashis "DG of complex vector bundles."

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