We’ve been talking about moduli problems, and one notoriously hard type of moduli problem is to “classify” vector bundles on some variety. Even when you restrict yourself to some special case like a specific surface and try to classify only vector bundles of certain rank or Chern class you run into trouble. To this day, these types of problems are a very active area of research.
It is fairly well-known (due to Grothendieck, but a problem in Hartshorne as well) that any finite rank vector bundle over is just a finite direct sum . The next interesting case would be to move up to genus curves. It turns out that Atiyah in the 1957 paper, Vector Bundles over an Elliptic Curve, worked out a classification. I just learned about this a month or two ago, and it is pretty cool so I’d like to briefly describe the idea.
Fix an algebraically closed field of characteristic , and let be an elliptic curve. Define to be the set of indecomposable vector bundles (up to isomorphism) on of rank and degree , where degree just means the degree of the determinant. It is well known that can be identified with , because is the set of degree divisors. In particular, this says that the moduli space of degree divisors (line bundles) on is fine and representable by .
Let’s continue with this idea of classifying degree vector bundles. It turns out there is a unique vector bundle with the property that , i.e. there are non-trivial global sections. Now, recall how works. Let be the origin. Then our isomorphism is given by . Our is going to play the role of here. In complete analogy we get a bijection by .
This finishes off the case of degree vector bundles, because we get that (roughly speaking, of course there’s a lot more structure we need to know about to say something about the moduli spaces).
In more general situations we can use the same sort of trick. Note that we always have a bijection given by . Thus one reduction Atiyah makes right away is that (by the Euclidean algorithm) we only need to consider for .
Now suppose the rank and degree are non-zero, and . We can establish a bijection , so that the composed map is the map multiplication by . The idea again is that one can find a certain vector bundle that is unique up to isomorphism from which all the others can be produced. Most people call this the Atiyah bundle. As a corollary, we see that if , then the moduli space of indecomposable rank and degree vector bundles on is fine and again representable by .
In modern language, we could work out that stable or semi-stable sheaves are the appropriate things to classify if we want a hope for the moduli space to be fine. It turns out that this condition of certain numerical invariants being coprime often implies that the sheaves are stable. For example, on an elliptic curve, a K3 surface, or even when dealing with relative moduli of sheaves on a K3 fibration. We may get to this another day.