Connections, The Kähler Condition, and Curvature – V. Chern Forms and Chern Classes

This blog post is inspired by conversation from a while back with Zach Kirsche, the DANKEST MEMER I have met in a long time.

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Now we go back to the case of a real smooth manifold ${M}$ of real dimension ${m}$ and a smooth ${\mathbb{C}}$-vector bundle ${V}$ over ${M}$. We take a smooth connection ${\omega}$ of ${V}$ and get a curvature form ${\Omega}$. The curvature form ${\Omega}$ is an ${\text{End}(V)}$-valued ${2}$-form on ${M}$. With respect to a local trivialization of the bundle ${V}$, the curvature form is a matrix valued ${2}$-form. From a matrix, we can get numerical invariants like the trace, the determinant, and other symmetric functions of the eigenvalues of the matrix. By using the same method, from ${\Omega}$ we can to construct from ${2k}$-forms on ${M}$. Let us look at the construction of numerical invariants of an ${r\times r}$ matrix ${A = (A_{ij})}$. Let the eigenvalues of ${A}$ be ${\lambda_1, \dots, \lambda_r}$. Then ${\sum_{\nu = 1}^r \lambda_\nu^k}$ is simply the trace the trace of ${A^k}$, as one can easily see by reducing ${A}$ to its Jordan normal form. So the ${k}$th elementary symmetric function ${\sigma_k}$ of the eigenvalues of ${A}$ is simply a universal homogeneous polynomial ${P_k}$ of weight ${k}$ in ${\text{Tr}\,A^\nu}$ ${(1 \le \nu \le k)}$ when ${\text{Tr}\,A^\nu}$ is given the weight ${\nu}$. We now define the ${k}$th Chern form ${\gamma_k}$ to be ${P_k(\text{Tr}\,\Omega, \dots, \text{Tr}\,\Omega^k)}$. Thus, the Chern form ${\gamma_k}$ is a global form of degree ${2k}$ on ${M}$ and is a homogeneous polynomial of degree ${k}$ in the coefficients ${\Omega_{\alpha i j}^\beta}$ of ${\Omega}$. Recall that we have the Bianchi identity

$\displaystyle D_\wedge \Omega = 0.$

So we have

$\displaystyle D_\wedge P_k(\text{Tr}\,\Omega,\dots \text{Tr}\,\Omega^k) = 0,$

and we conclude that ${\gamma_k}$ is always a closed form.

The most important aspect of the theory of Chern classes is that the Chern form ${\gamma_k}$, up to a global exact form, is independent of the choice of the connection. Let us rename our connection ${\omega}$ and give it the new name ${\omega_0}$. Suppose we have another connection ${\omega_1}$. Our original curvature is now called ${\Omega_0}$, and our new new curvature is ${\Omega_1}$. We denote by ${\gamma_k^{(\nu)}}$ the ${k}$th Chern form from ${\Omega_\nu}$. Consider the new manifold

$\displaystyle \tilde{M} = M \times \mathbb{R}$

and the vector bundle ${\tilde{V}}$ over ${\tilde{M}}$ obtained by pulling back ${V}$ via the natural projection map ${\pi: \tilde{M} \rightarrow M}$ onto the first factor. Let ${t}$ be the variable of ${\mathbb{R}}$. We can define a connection

$\displaystyle \tilde{\omega} = (1 - t)\omega_0 + t\omega_1.$

More precisely, we should have used ${\pi^* \omega_\nu}$ instead of ${\omega_\nu}$ in the above definition of ${\tilde{\omega}}$, but for the sake of simpler notation, we allow this slight abuse of language. We have the curvature ${\tilde{\Omega}}$ of ${\tilde{\omega}}$ on ${\tilde{M}}$ and Chern forms ${\tilde{\gamma}_k}$ on ${\tilde{M}}$. We can write

$\displaystyle \tilde{\gamma}_k = \alpha_k + dt \wedge \beta_k,$

where ${\alpha_k}$ and ${\beta_k}$ are respectively a ${2k}$-form and a ${(2k-1)}$-form not containing ${dt}$ (i.e., expressible in terms of the differentials of the local coordinates of ${M}$ with coefficients depending on ${t}$ and the local coordinate of ${M}$). Since ${\tilde{\gamma}_k}$ is closed, we have

$\displaystyle d_M\alpha_k + dt \wedge {\partial\over{\partial t}}\alpha_k - dt \wedge d_M \beta_k = 0,$

where ${d_M}$ means (partial) exterior differentiation with the coordinate ${t}$ kept constant, and ${(\partial/\partial t)\alpha_k}$ means differentiating the coefficients of ${\alpha_k}$ with respect to the variable ${t}$. Hence,

$\displaystyle {\partial\over{\partial t}}\alpha_k = d_M \beta_k,$

and

$\displaystyle \alpha_k|_{t = 1} - \alpha_k|_{t = 0} = d_M \left(\int_{t=0}^1 \beta_k\,dt\right).$

Since

$\displaystyle \alpha_k|_{t = \nu} = \tilde{\gamma}_k |_{t = \nu} = \gamma_k^{(\nu)}$

for ${\nu = 0, 1}$, it follows that ${\gamma_k^{(1)} - \gamma_k^{(0)}}$, being equal to ${d_M(\int_{t = 0}^1 \beta_k\,dt)}$, is exact on ${M}$.

We define as the de Rham cohomology group of degree ${k}$ over ${M}$ the abelian group of all smooth closed ${k}$-forms on ${M}$ quotiented by the abelian subgroup of all smooth exact ${k}$-forms on ${M}$. An element of the de Rham cohomology group of degree ${k}$ is called a de Rham cohomology class of degree ${k}$. We will later discuss de Rham cohomology in the context of sheaf cohomology. The ${k}$th Chern form ${\gamma_k}$ defines a de Rham cohomology class of degree ${2k}$, and this class is independent of the choice of the connection and is called the ${k}$th Chern class of the ${\mathbb{C}}$-vector bundle ${V}$ over ${M}$. Chern classes are invariants of smooth vector bundles.

When ${M}$ is a complex manifold and ${V}$ is a holomorphic vector bundle, we can choose as our connection the complex metric connection of a Hermitian metric along the fibers of ${V}$. Then the curvature form is of type ${(1, 1)}$, and the ${k}$th Chern form is of type ${(k, k)}$. So we conclude that the ${k}$th Chern class of a holomorphic vector bundle over a complex manifold can be represented by a form of type ${(k, k)}$.