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 ${L}$. (more…)

I’ve been away from this blog for too long–partially it’s because most of my expository energy has gone into preparing a collection of notes on algebraic geometry (to help me learn the subject). Someday I’ll post them.

Today, however, I’d like to talk about a clever proof I learned recently.

The following result is neat:

Theorem: Let ${M}$ be a compact smooth manifold. Then the de Rham cohomology groups ${H^i_{DR}(X, \mathbb{R})}$ are finite-dimensional.

I’m pretty sure it follows from Hodge theory and the finite-dimensionality of the harmonic forms. However, I learned a neat elementary proof that I’d like to discuss.

By de Rham’s theorem, we can compute the de Rham cohomology groups as the sheaf cohomology groups ${H^i(X, \mathbb{R})}$ for ${\mathbb{R}}$ denoting the constant sheaf associated to the group ${\mathbb{R}}$. Now, pick a Riemannian metric on ${M}$. Each point has a neighborhood ${U}$ such that any two points in ${U}$ are joined by a unique geodesic contained in ${U}$—such a neighborhood is called geodesically convex. It is clear that a geodesically convex neighborhood is homeomorphic (via the exponential map) to a convex set in ${\mathbb{R}^n}$, which has trivial de Rham cohomology, and also that the intersection of two geodesically convex sets is geodesically convex.

So pick a finite cover of ${M}$ by geodesically convex sets ${U_1, \dots, U_k}$. Then on every intersection ${U_{i_1} \cap \dots \cap U_{i_n}}$, the sheaf ${\mathbb{R}}$ has trivial cohomology because this intersection is geodesically convex, hence diffeomorphic to a convex set in ${\mathbb{R}^n}$. In particular, the cover ${\{U_i\}}$ satisfies the hypotheses of Leray’s theorem. We can apply Cech cohomology with this cover to compute ${H^i(X, \mathbb{R})}$, or equivalently the de Rham cohomology.

But there are finitely many sets in this cover, and the sections of the sheaf ${\mathbb{R}}$ over each of these sets is just the abelian group ${\mathbb{R}}$ by connectedness of anything geodesically convex. So the Cech complex consists of finite-dimensional vector spaces; its cohomology thus consists of finite-dimensional vector spaces. $\Box$

I learned this from Bott and Tu’s Differential Forms in Algebraic Topology, which appears to be a really fun read.