Integrality of Chern classes

Let $M$ be a smooth manifold and let $E\to M$ be a complex vector bundle of rank $r$, equipped with a connection on a vector bundle $\nabla$ with curvature $F_\nabla$.

Theorem (integrality)

Form the total Chern–Weil representative

where each $c_k(E,\nabla)$ is a closed differential form of degree $2k$ (closed because its exterior derivative vanishes by Chern–Weil theory and the Bianchi identity).

Then:

1. The de Rham cohomology class $[c_k(E,\nabla)]\in H^{2k}_{\mathrm{dR}}(M)$ is independent of the choice of $\nabla$, and
2. There exists a unique class $c_k(E)\in H^{2k}(M;\mathbb{Z})$ whose image under the change-of-coefficients map $H^{2k}(M;\mathbb{Z})\to H^{2k}(M;\mathbb{R})\cong H^{2k}_{\mathrm{dR}}(M)$ equals $[c_k(E,\nabla)]$.

Equivalently: for every smooth singular $2k$-cycle $\Sigma$ in $M$,

This integrality is the precise sense in which Chern classes are “integral” characteristic classes, even though the Chern–Weil representatives are differential forms.

Examples

1. Complex line bundles over the 2-sphere.
   For a complex line bundle $L\to S^2$ with any connection, the $2$-form $c_1(L,\nabla)=\frac{i}{2\pi}F_\nabla$ satisfies

   $$\int_{S^2} \frac{i}{2\pi}F_\nabla \in \mathbb{Z},$$

   and that integer is the degree (first Chern number) of $L$.

2. The tautological line bundle over complex projective space.
   For the tautological line bundle $\mathcal{O}(-1)\to \mathbb{CP}^n$, the class $c_1(\mathcal{O}(-1))$ generates $H^2(\mathbb{CP}^n;\mathbb{Z})\cong\mathbb{Z}$. Any Chern–Weil representative integrates to an integer over any embedded $\mathbb{CP}^1\subset\mathbb{CP}^n$.

3. Direct sums preserve integrality.
   If $E=L_1\oplus\cdots\oplus L_r$ is a sum of line bundles, then the total Chern class satisfies

   $$c(E)=\prod_{j=1}^r \bigl(1+c_1(L_j)\bigr),$$

   so each $c_k(E)$ lies in integral cohomology and is determined by integral products of the $c_1(L_j)$.