Chern classes

In this section we define the Chern class which is a (topological) invariant of a line bundle. Before doing this we collect some facts about the curvature.

Proposition 3.1   The curvature $F$ of a connection $\nabla$ satisfies:

(i)

$dF = 0$

(ii)

If $\nabla, \nabla'$ are two connections then $\nabla = \nabla' + \eta$ for $\eta$ a 1-form and $F_{\nabla} = F_{\nabla'} + d \eta$.

(iii)

If $\Sigma$ is a closed surface then ${\frac{1}{2 \pi i}} \int_{\Sigma} F_{\nabla}$ is an integer independent of $\nabla$.

Proof. (i) $dF\vert _{U \alpha} = d(d A_{\alpha}) = 0.$

(ii) Locally $A'_{\alpha} = A_{\alpha} + \eta \alpha$ as $\eta_{\alpha} = A'_{\alpha} - A_{\alpha}$. But $A_{\beta} = A_{\alpha} - g_{\alpha \b }^{-1}d g_{\alpha \beta}$ and $A'_{\beta} = A'_{\alpha} - g_{\alpha \b }^{-1}d g_{\alpha \beta}$ so that $\eta_{\beta} = \eta_{\alpha}$. Hence $\eta$ is a global 1-form and $F_ {\nabla} = d A_{\alpha}$ so $F'_{\nabla} = F_{\nabla} + d \eta$.

(iii) If $\Sigma$ is a closed surface then $\partial \Sigma =\emptyset$ so by Stokes' theorem $\int_{\Sigma} F_{\nabla} = \int_{\Sigma} F_{\nabla}'$. Now choose a family of disks $D_t$ in $\Sigma$ whose limit as $t \to 0$ is a point. For any $t$ the holonomy of the connection around the boundary of $D_t$ can be calculated by integrating the curvature over $D_t$ or over the complement of $D_t$ in $\Sigma$ and using Proposition 2.1. Taking into account orientation this gives us

$$\exp(\int_{\Sigma - D_t} F ) = \exp(- \int_{D_t} F )$$

and taking the limit as $t \to 0$ gives

$$\exp(\int_{\Sigma} F )= 1$$

which gives the required result. $\qedsymbol$

The Chern class, $c (L)$, of a line bundle $L \to \Sigma$ where $\Sigma$ is a surface is defined to be the integer ${\frac{1}{2 \pi i}} \int_{\Sigma} F_{\nabla}$ for any connection $\nabla$.

Example 3.1   For the case of the two sphere previous results showed that $F = - i$   vol$_{S^2}$. Hence

$$c (TS^{2}) = {\frac{- i}{2 \pi i}} \int_{S^{2}}$$   vol$$= {\frac{- i}{2 \pi i}} \ 4 \pi = - 2.$$

Some further insight into the Chern class can be obtained by considering a covering of $S^2$ by two open sets $U_{0}, U_{1}$ as in Figure 2. Let $L \to S^{2}$ be given by a transition for $g_{01} : U_{0} \cap U_{1} \to {\bf C}^\times$. Then a connection is a pair of 1-forms $A_{0}$, $A_{1}$, on $U_{0}, U_{1}$ respectively, such that

$$A_{1} = A_{0} + dg_{10}{g_{10}^{-1}} \ {\mbox {on}} \ U_{0} \cap U_{1}.$$

Take $A_{0} = 0$ and $A_{1}$ to be any extension of $dg_{10}{g_{10}^{-1}}$ to $U_1$. Such an extension can be made by shrinking $U_0$ and $U_1$ a little and using a cut-off function. Then $F = dA_{0} = 0$ on $U_0$ and $F = dA_{1}$ on $U_1$. To find $c (L)$ we note that by Stokes theorem:

$$\int_{S^2} F = \int_{U_1} F = \int_{\partial U{_1}} A_{1} = \int_{\partial U_{1}}{dg_{10}g_{10}^{-1}}.$$

But this is just $2 \pi i$ the winding number of $g_{10}$. Hence the Chern class of $L$ is the winding number of $g_{10}$. Note that we have already seen that for $TS^2$ the winding number and Chern class are both $-2$. It is not difficult to go further now and prove that isomorphism classes of line bundles on $S^2$ are in one to one correspondence with the integers via the Chern class but will not do this here.

Example 3.2   Another example is a surface $\Sigma_g$ of genus $g$ as in Figure 5.

Figure 5: A surface of genus $g$.

We cover it with $g$ open sets $U_1, \dots, U_g$ as indicated. Each of these open sets is diffeomorphic to either a torus with a disk removed or a torus with two disks removed. A torus has a non-vanishing vector field on it. If we imagine a rotating bicycle wheel then the inner tube of the tyre (ignoring the valve!) is a torus and the tangent vector field generated by the rotation defines a non-vanishing vector field. Hence the same is true of the open sets in Figure 5. There are corresponding transition functions $g_{12}, g_{23}, \dots, g_{g-1g}$ and we can define a connection in a manner analogous to the two-sphere case and we find that

$$c(T\Sigma_g) = \sum^{g - 1}_{i = 1}$$   winding number$$(g_{i, i + 1}).$$

All the transition functions have winding number $-2$ so that

$$c(T \Sigma_g) = 2 - 2g.$$

This is a form of the Gauss-Bonnet theorem. It would be a good exercise for the reader familiar with the classical Riemannian geometry of surfaces in $\mathbb{R}^3$ to relate this result to the Gauss-Bonnet theorem. In the classical Gauss-Bonnet theorem we integrate the Gaussian curvature which is the trace of the curvature of the Levi-Civita connection.

So far we have only defined the Chern class for a surface. To define it for manifolds of higher dimension we need to recall the definition of de Rham cohomology [4]. If $M$ is a manifold we have the de Rham complex

$$0 \to \Omega^{0} (M) \to \Omega^{1} (M) \to . . . \to \Omega^{m} (M) \to 0.$$

where $\Omega^{p}(M)$ is the space of all $p$ forms on $M$, the horizontal maps are $d$ the exterior derivative and $m = \dim (M)$. Then $d^{2} = 0$ and it makes sense to define:

$$H^{p} (M) = {\frac{\hbox{kernel} \ d : \Omega^{p} \ (M) \to \Omega^{p + 1} \ (M)}{\hbox{image} \ d : \Omega^{p - 1} \ (M) \to \Omega^{p} \ (M)}}$$

This is the pth de Rham cohomology group of $M$ - a finite dimensional vector space if $M$ is compact or otherwise well behaved.

The general definition of $c (L)$ is to take the cohomology class in $H^{2} (M)$ containing ${\frac{1}{2 \pi i}} F_{\nabla}$ for some connection.

It is a standard result [4] that if $M$ is oriented, compact, connected and two dimensional integrating representatives of degree two cohomology classes defines an isomorphism

$$H^{2}(M) \to \mathbb{R}$$

$$\left[\omega\right] \mapsto \int_M \omega$$

where $\left[\omega\right]$ is a cohomology class with representative form $\omega$. Hence we recover the definition for surfaces.