So, now with the preliminaries on connections and curvature established, and the Chern classes summarized, it’s time to see how they connect with one another. Namely, we want to say that, given a complex vector bundle, we can compute the Chern classes in de Rham cohomology by picking a connection — any connection — on it,  computing the curvature, and then applying various polynomials.

We shall start by warming up with a special case, of a line bundle, where the algebra needed is easier. Let ${M}$ be a smooth manifold, ${L \rightarrow M}$ a complex line bundle. Let ${\nabla}$ be a connection on ${L}$, and let ${\Theta}$ be the curvature.

Thus, ${\Theta}$ is a global section of ${\mathcal{A}^2 \otimes \hom(L, L)}$; but since ${L}$ is a line bundle, this bundle is canonically identified with ${\mathcal{A}^2}$. (Recall the notation that $\mathcal{A}^k$ is the bundle (or sheaf) of smooth $k$-forms on the manifold $M$.)

Proposition 1 (Chern-Weil for line bundles) ${\Theta}$ is a closed form, and the image in ${ H^2(M; \mathbb{C})}$ is ${2\pi i}$ times the first Chern class of the line bundle ${L}$.

Proof: Let us suppose that we have an open cover ${\left\{U_\alpha\right\}}$ of ${M}$ such that each finite intersection of elements of this cover is either empty or contractible; we can do this by choosing a Riemannian metric on ${M}$, and then taking geodesically convex neighborhoods.

The line bundle ${L}$ is described by nonvanishing, complex-valued continuous functions ${g_{\alpha \beta}: U_\alpha \cap U_\beta \rightarrow \mathbb{C}^*}$, satisfying the usual cocycle condition. To compute the first Chern class, we take the family of functions ${\log g_{\alpha \beta}}$ (this can be done, since each intersection is contractible!), and take their Cech 2-coboundary. That is, for a triple ${\alpha, \beta, \gamma}$, we consider the integer

$\displaystyle \frac{1}{2\pi i } \left( \log g_{\alpha \beta} + \log g_{\beta \gamma} - \log g_{\alpha \gamma} \right);$

these integers, for ${\alpha, \beta, \gamma}$ varying, form a Cech 2-cocycle (integer-valued), which is the first Chern class, by definition of the connecting homomorphisms in sheaf cohomology.

Now let us try to understand where the curvature ${\Theta}$ lives in de Rham cohomology, and first that it is actually a closed 2-form. Over each ${U_\alpha}$, the bundle ${L}$ is trivial, and we have chosen an isomorphism of it with the trivial bundle (this is what choosing the ${g_{\alpha \beta}}$ amounted to), so we have a canonical frame ${\{e_\alpha\}}$ over ${U_\alpha}$ (so ${e_\alpha \in L(U_\alpha)}$). The transition from ${e_\alpha }$ to ${e_\beta}$ over ${U_\alpha \cap U_\beta}$ is given by

$\displaystyle e_\beta = g_{\alpha \beta} e_\alpha.$

This is the definition of the local trivializations.

Now, we know that the connection form is a simply a 1-form ${\theta_\alpha}$ for each ${\alpha}$ (as it’s a one-by-one matrix) such that ${\nabla e_\alpha = \theta_\alpha e_\alpha}$, and the transition rule is, by what we saw last time,

$\displaystyle \theta_\beta = g_{\alpha \beta} \theta_\alpha g_{\alpha \beta}^{-1} + (d g_{\alpha \beta}) g_{\alpha \beta}^{-1} = \theta_\alpha + d(\log g_{\alpha \beta}). \ \ \ \ \ (1)$

The curvature form is given by, locally,

$\displaystyle \Theta_\alpha = d \theta_\alpha - \theta_\alpha \wedge \theta_\alpha = d \theta_\alpha, \ \ \ \ \ (2)$

because ${\theta_\alpha}$ is a 1-form (and not a large matrix), so ${\theta_\alpha \wedge \theta_\alpha = 0}$.

With these preliminaries established, we can figure out what is happening. Note first that (2) implies that ${\Theta_\alpha}$ is a closed 1-form, and consequently ${\Theta}$ (which is obtained by gluing the ${\Theta_\alpha}$ together) is a closed 1-form itself. We next need to figure out where ${\Theta_\alpha}$ maps to in ${H^2(M; \mathbb{C})}$. To do this, we need to unwind the de Rham isomorphism, and use (1). Namely, the de Rham isomorphism came from the sheaf-theoretic resolution

$\displaystyle 0 \rightarrow \mathbb{C} \rightarrow \mathcal{A}^0 \rightarrow \mathcal{A}^1 \rightarrow \mathcal{A}^2 \rightarrow \dots.$

So, if we have a global closed 2-form ${\omega}$, and we want to figure out where it goes in ${H^2(M; \mathbb{C})}$ (as a Cech 2-cocycle), we need to start by lifting ${\omega}$ over each ${U_\alpha}$: that is, we need to find 1-forms ${\tau_\alpha \in \mathcal{A}^1(U_\alpha)}$ such that ${d \tau_\alpha = \omega|_{U_\alpha}}$.

Then, we need to form the associated 1-cocycle ${(\alpha, \beta) \mapsto \tau_\beta - \tau_\alpha}$, which is an element of ${H^1(d \mathcal{A}^0)}$. Then we have to unapply ${d}$, and take the 2-coboundary of this. This is how the de Rham isomorphism works.

So, let’s do it. Locally we can lift ${\Theta}$ to ${d \theta_\alpha}$. So we can take the ${\tau_\alpha = \theta_\alpha}$. The differences ${\tau_\beta - \tau_\alpha}$ are given by ${d(\log g_{\alpha \beta})}$, by the transition rules. Now we have to unapply ${d}$ and take the 2-coboundary of this.

But then we get precisely the differences ${\log g_{\alpha \beta} + \log g_{\beta \gamma} - \log g_{\alpha \gamma}}$ which were used to define the first Chern class. $\Box$