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 Hermitian connection $\nabla$ with curvature $F_\nabla$ .

Form the total Chern–Weil representative

c(E,\nabla)\;=\;\det\!\left(I+\frac{i}{2\pi}F_\nabla\right) \;=\;1+c_1(E,\nabla)+\cdots+c_r(E,\nabla),

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:

The de Rham cohomology class $[c_k(E,\nabla)]\in H^{2k}_{\mathrm{dR}}(M)$ is independent of the choice of $\nabla$ , and
The canonical integral Chern class $c_k(E)\in H^{2k}(M;\mathbb{Z})$ maps to $[c_k(E,\nabla)]$ under the change-of-coefficients map $H^{2k}(M;\mathbb{Z})\to H^{2k}(M;\mathbb{R})\cong H^{2k}_{\mathrm{dR}}(M)$

In particular, for every smooth singular $2k$ -cycle $\Sigma$ in $M$ ,

\int_\Sigma c_k(E,\nabla)\in\mathbb{Z}.

This integrality is the precise sense in which Chern classes are “integral” characteristic classes, even though the Chern–Weil representatives are differential forms. The de Rham class or its periods do not, however, determine an integral lift uniquely: torsion classes vanish under the change-of-coefficients map to $\mathbb R$ .

Examples

Complex line bundles over the 2-sphere. For a complex line bundle $L\to S^2$ with any Hermitian 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 $L$ .

The tautological line bundle over complex projective space. For the tautological line bundle $\mathcal{O}(-1)\to \mathbb{CP}^n$ with $n\ge1$ , the class $c_1(\mathcal{O}(-1))$ is the negative of the complex-orientation generator of $H^2(\mathbb{CP}^n;\mathbb{Z})\cong\mathbb{Z}$ . Any Chern–Weil representative integrates to an integer over any embedded $\mathbb{CP}^1\subset\mathbb{CP}^n$ .

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 $c_1(L_j)$ .