Chern class via Chern–Weil theory

Let $E\to M$ be a smooth complex vector bundle of rank $n$ , with a complex-linear connection $\nabla$ and curvature $F_\nabla$ . Its total Chern form is

c(\nabla)=\det\!\left(I+\frac{i}{2\pi}F_\nabla\right) =1+c_1(\nabla)+\cdots+c_n(\nabla), \qquad c_k(\nabla)\in\Omega^{2k}(M;\mathbb C).

Here the determinant uses its usual permutation formula, with multiplication replaced by the wedge product of forms. The even-degree entries commute, so this formula is well defined and invariant under changes of frame. Complex forms mean forms $\alpha+i\beta$ with real forms $\alpha,\beta$ , with exterior derivative extended complex-linearly.

The forms $c_k(\nabla)$ are closed and their de Rham classes are independent of $\nabla$ . The Chern–Weil Chern class is this class

c_k^{\mathrm{dR}}(E)=[c_k(\nabla)]\in H^{2k}_{\mathrm{dR}}(M;\mathbb C).

It lies in the image of real de Rham cohomology: choosing a Hermitian metric and a compatible connection gives real Chern forms representing the same class. For an arbitrary complex connection the forms themselves need not be real. Set $c_0^{\mathrm{dR}}=1$ and $c_k^{\mathrm{dR}}=0$ for $k>n$ .

Integral classes and naturality

The integral Chern classes $c_k(E)\in H^{2k}(M;\mathbb Z)$ are defined topologically by naturality, the Whitney sum axiom, rank normalization, and the tautological-line normalization. Their images in complex cohomology, identified with de Rham cohomology, are $c_k^{\mathrm{dR}}(E)$ . Curvature alone does not define the integral classes, since change of coefficients can lose integral information, including torsion.

For any smooth map $f:N\to M$ ,

c_k^{\mathrm{dR}}(f^*E)=f^*c_k^{\mathrm{dR}}(E).

The integral classes satisfy the corresponding integral naturality identity.

Examples

Trivial bundle. If $E\cong M\times\mathbb C^n$ admits the flat connection ( $F_\nabla=0$ ), then $c(\nabla)=1$ and hence $c_k(E)=0$ for all $k\ge 1$ .

Complex line bundle. If $\mathrm{rank}_{\mathbb C}E=1$ , then $c(\nabla)=1+\frac{i}{2\pi}F_\nabla,$ so $c_1(E)$ is represented in de Rham cohomology by the complex 2-form $\frac{i}{2\pi}F_\nabla$ , which is real when the connection is Hermitian.

Whitney sum behavior (curvature-level). If $E=E_1\oplus E_2$ with a block-diagonal connection $\nabla=\nabla_1\oplus\nabla_2$ , then $F_\nabla$ is block-diagonal and $c(\nabla)=c(\nabla_1)\wedge c(\nabla_2),$ recovering the usual multiplicativity of total Chern classes under direct sum.