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)}.

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