Chern character via Chern–Weil theory

Let $M$ be a smooth manifold and let $E\to M$ be a complex vector bundle equipped with a connection $\nabla$ with curvature $F_\nabla\in\Omega^2(M;\mathrm{End}(E))$.

Definition (Chern character form and Chern character)

The Chern character form of $\nabla$ is the even differential form

where the exponential is taken as a formal power series and $\mathrm{tr}$ is the fiberwise trace. Writing by degree,

Then:

1. Each $\mathrm{ch}_k(\nabla)$ is closed: $d\,\mathrm{ch}_k(\nabla)=0$, where $d$ is the exterior derivative .
2. The de Rham class $[\mathrm{ch}(\nabla)]\in H^{\mathrm{even}}_{\mathrm{dR}}(M)$ is independent of $\nabla$.
3. The Chern character $\mathrm{ch}(E)\in H^{\mathrm{even}}(M;\mathbb Q)$ is the rational cohomology class whose image in de Rham cohomology equals $[\mathrm{ch}(\nabla)]$.

The Chern character is natural under pullback: for any smooth map $f:N\to M$,

It is also additive under direct sum (already at the level of forms): if $\nabla=\nabla_1\oplus\nabla_2$ on $E_1\oplus E_2$, then

Examples

1. Rank. The degree-zero component is the rank:

   $$\mathrm{ch}_0(E)=\mathrm{rank}_{\mathbb C}(E)\in H^0(M;\mathbb Q).$$

2. Line bundles. For a complex line bundle $L\to M$ with first Chern class $c_1(L)\in H^2(M;\mathbb Z)$,

   $$\mathrm{ch}(L)=\exp(c_1(L))=1+c_1(L)+\tfrac12 c_1(L)^2+\cdots$$

   in rational cohomology.

3. Trivial / flat bundles. If $E$ admits a flat connection ($F_\nabla=0$), then $\mathrm{ch}(\nabla)=\mathrm{rank}(E)$, so $\mathrm{ch}_k(E)=0$ for all $k\ge 1$ in de Rham cohomology.