Chern character

Remarks

Formally, if $x_1, \ldots, x_r$ are the Chern roots of $E$, then $\operatorname{ch}(E) = e^{x_1} + \cdots + e^{x_r}$. The first terms are $\operatorname{ch}(E) = r + c_1 + \tfrac{1}{2}(c_1^2 - 2c_2) + \tfrac{1}{6}(c_1^3 - 3c_1c_2 + 3c_3) + \cdots$. It satisfies $\operatorname{ch}(E \oplus F) = \operatorname{ch}(E) + \operatorname{ch}(F)$ and $\operatorname{ch}(E \otimes F) = \operatorname{ch}(E) \operatorname{ch}(F)$, so it gives a ring homomorphism from topological K-theory $K^0(X)$ to rational cohomology.

Example: on $\mathbb{CP}^1$ let $h \in H^2(\mathbb{CP}^1; \Q)$ be the positive generator, so $h^2 = 0$. The line bundle $\mathcal{O}(n)$ has $c_1 = nh$, so $\operatorname{ch}(\mathcal{O}(n)) = e^{nh} = 1 + nh$.

Non-example: the total Chern class $c(E) = 1 + c_1 + \cdots + c_r$ is not additive: $c(E \oplus F) = c(E) c(F)$. So $c$ does not give a homomorphism of additive groups from K-theory, while $\operatorname{ch}$ does.

In algebraic geometry the same formula, with Chern classes in the rational Chow ring $\operatorname{CH}^*(X) \otimes \Q$ of a smooth variety, defines $\operatorname{ch} \colon K_0(X) \to \operatorname{CH}^*(X) \otimes \Q$.