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 be a smooth manifold, a complex line bundle. Let be a connection on , and let be the curvature.
Thus, is a global section of ; but since is a line bundle, this bundle is canonically identified with . (Recall the notation that is the bundle (or sheaf) of smooth -forms on the manifold .)
Proposition 1 (Chern-Weil for line bundles) is a closed form, and the image in is times the first Chern class of the line bundle .
Proof: Let us suppose that we have an open cover of 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 , and then taking geodesically convex neighborhoods.
The line bundle is described by nonvanishing, complex-valued continuous functions , satisfying the usual cocycle condition. To compute the first Chern class, we take the family of functions (this can be done, since each intersection is contractible!), and take their Cech 2-coboundary. That is, for a triple , we consider the integer
these integers, for 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 lives in de Rham cohomology, and first that it is actually a closed 2-form. Over each , the bundle is trivial, and we have chosen an isomorphism of it with the trivial bundle (this is what choosing the amounted to), so we have a canonical frame over (so ). The transition from to over is given by
This is the definition of the local trivializations.
Now, we know that the connection form is a simply a 1-form for each (as it’s a one-by-one matrix) such that , and the transition rule is, by what we saw last time,
The curvature form is given by, locally,
because is a 1-form (and not a large matrix), so .
With these preliminaries established, we can figure out what is happening. Note first that (2) implies that is a closed 1-form, and consequently (which is obtained by gluing the together) is a closed 1-form itself. We next need to figure out where maps to in . 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
So, if we have a global closed 2-form , and we want to figure out where it goes in (as a Cech 2-cocycle), we need to start by lifting over each : that is, we need to find 1-forms such that .
Then, we need to form the associated 1-cocycle , which is an element of . Then we have to unapply , and take the 2-coboundary of this. This is how the de Rham isomorphism works.
So, let’s do it. Locally we can lift to . So we can take the . The differences are given by , by the transition rules. Now we have to unapply and take the 2-coboundary of this.
But then we get precisely the differences which were used to define the first Chern class.