We’re going to talk about Chern classes, but first, a note on the last post.  For any scheme X, there’s a pairing \mathrm{Pic}(X)\times A_1(X)\to \mathbb{Z}, 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, \mathrm{Pic}(X)\cong A_1(X), and so we have the usual intersection pairing, as we mentioned by reproving Bezout’s Theorem in \mathbb{P}^2.  So, at the least, our notion of cycles and intersections is recovering the basic intersection theory that we know from Hartshorne.

Now, let L be a line bundle, and let V\in A_k(X) a subvariety.  Then L|_V is given by some Cartier divisor, which gives us a well defined element of A_{k-1}(X).  We’re going to denote this element by c_1(L)\cap [V].  Extending linearly, we have c_1(L)\cap \cdot:A_k(X)\to A_{k-1}(X) (after using inclusions, of course).  This has the nice property that if D is a divisor representing L, then c_1(L)\cap \alpha=D\cdot \alpha, 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:

  1. Commutativity: c_1(L)\cap (c_1(L')\cap \alpha)=c_1(L')\cap (c_1(L)\cap \alpha)
  2. Projection: f_*(c_1(f^*L)\cap \alpha)=c_1(L)\cap f_*(\alpha)
  3. Flat Pullback: c_1(f^*L)\cap f^*\alpha=f^*(c_1(L)\cap\alpha)
  4. Additivity: c_1(L\otimes L')\cap \alpha=c_1(L)\cap\alpha+c_1(L')\cap\alpha and c_1(L^\vee)\cap\alpha=-c_1(L)\cap\alpha.

So, why don’t we just define the Chern class itself, rather than the map it induces on A_k(X)? 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 E be a rank e+1 vector bundle on X, and P=\mathbb{P}(E) the projectivization, with p the projection from P\to X.  This bundle has a natural line bundle, \mathscr{O}(1).

We define s_i(E)\cap \cdot:A_k(X)\to A_{k-i}(X) by s_i(E)\cap\alpha=p_*(c_1(\mathscr{O}(1))^{e+i}\cap p^*\alpha).  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 s_t(E)=\sum_{i=0}^\infty s_i(E)t^i as a formal power series with coefficients endomorphisms of A_*(X).  Then the Chern polynomial is c_t(E)=s_t(E)^{-1}, and the coefficient of t^n is c_n(E).  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:

  1. Vanishing: For all i>\mathrm{rank}(E), we have $c_i(E)=0$.
  2. Commutativity: c_i(E)\cap (c_j(F)\cap\alpha)=c_j(F)\cap (c_i(E)\cap\alpha)
  3. Projection: f_*(c_i(f^*E)\cap\alpha)=c_i(E)\cap f_*(\alpha)
  4. Pullback: c_i(f^*E)\cap f^*\alpha=f^*(c_i(E)\cap\alpha)
  5. Whitney Sum: For any exact sequence 0\to E'\to E\to E''\to 0, we have c_t(E)=c_t(E')c_t(E'').
  6. Normalization: c_1(E)\cap [X]=[D] when E is a line bundle with E=\mathscr{O}(D).
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