## Chern Classes: Part 1

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 $E=\mathscr{O}(D)$.