Chern class

Definition

Let $X$ be a smooth quasi-projective variety over a field, with Chow ring $CH^\bullet(X)$ built from its Chow groups. For an invertible sheaf $L$ on $X$, let $c_1(L) \in CH^1(X)$ be the class of a Cartier divisor $D$ with $L \cong \mathcal{O}_X(D)$. Let $E$ be a vector bundle of rank $r$ on $X$. Let $\pi \colon \mathbb{P}(E) \to X$ be the associated projective bundle, $\mathbb{P}(E) = \operatorname{Proj} \operatorname{Sym} E$, and let $\xi = c_1(\mathcal{O}_{\mathbb{P}(E)}(1)) \in CH^1(\mathbb{P}(E))$, where $\mathcal{O}_{\mathbb{P}(E)}(1)$ is the twisting sheaf. Then $CH^\bullet(\mathbb{P}(E))$ is a free $CH^\bullet(X)$-module, through $\pi^*$, with basis $1, \xi, \ldots, \xi^{r-1}$. The Chern classes $c_i(E) \in CH^i(X)$, for $0 \le i \le r$, are the unique classes with $c_0(E) = 1$ and $$\sum_{i=0}^{r} (-1)^i \, \pi^* c_i(E) \, \xi^{r-i} = 0.$$ One sets $c_i(E) = 0$ for $i > r$, and the total Chern class is $c(E) = \sum_i c_i(E)$.