We’re going to talk about Chern classes, but first, a note on the last post. For any scheme , there’s a pairing , taken by restricting the line bundle to the curve and taking the degree (or doing the intersection as we described, and integrating). In the case of a surface, , and so we have the usual intersection pairing, as we mentioned by reproving Bezout’s Theorem in . So, at the least, our notion of cycles and intersections is recovering the basic intersection theory that we know from Hartshorne.
Now, let be a line bundle, and let a subvariety. Then is given by some Cartier divisor, which gives us a well defined element of . We’re going to denote this element by . Extending linearly, we have (after using inclusions, of course). This has the nice property that if is a divisor representing , then , so we get a lot of the same properties we had before. But we also get some other things, here’s a list of basic properties:
- Flat Pullback:
- Additivity: and .
So, why don’t we just define the Chern class itself, rather than the map it induces on ? Well, really, what we have is just a map. The Chern classes are somewhat cohomological objects, and so they pair with homological objects. We’ll find the Chern class naturally once we have a cohomological object to work from.
We’re going to stop after defining all the Chern classes of a vector bundle, and will start playing with them later. So first, we need to define the Segre clases. Let be a rank vector bundle on , and the projectivization, with the projection from . This bundle has a natural line bundle, .
We define by . Now, the specific properties here aren’t too important, because we won’t be using them very much. (And, if I’m wrong, they’re pretty much the standard properties)
So now, define as a formal power series with coefficients endomorphisms of . Then the Chern polynomial is , and the coefficient of is . These are the Chern classes. Now, we’re mostly going to forget this (it’s important to construct them) and use their nice properties when we’re doing math:
- Vanishing: For all , we have $c_i(E)=0$.
- Whitney Sum: For any exact sequence , we have .
- Normalization: when is a line bundle with .