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 of real dimension and a smooth -vector bundle over . We take a smooth connection of and get a curvature form . The curvature form is an -valued -form on . With respect to a local trivialization of the bundle , the curvature form is a matrix valued -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 we can to construct from -forms on . Let us look at the construction of numerical invariants of an matrix . Let the eigenvalues of be . Then is simply the trace the trace of , as one can easily see by reducing to its Jordan normal form. So the th elementary symmetric function of the eigenvalues of is simply a universal homogeneous polynomial of weight in when is given the weight . We now define the th Chern form to be . Thus, the Chern form is a global form of degree on and is a homogeneous polynomial of degree in the coefficients of . Recall that we have the Bianchi identity
So we have
and we conclude that is always a closed form.
The most important aspect of the theory of Chern classes is that the Chern form , up to a global exact form, is independent of the choice of the connection. Let us rename our connection and give it the new name . Suppose we have another connection . Our original curvature is now called , and our new new curvature is . We denote by the th Chern form from . Consider the new manifold
and the vector bundle over obtained by pulling back via the natural projection map onto the first factor. Let be the variable of . We can define a connection
More precisely, we should have used instead of in the above definition of , but for the sake of simpler notation, we allow this slight abuse of language. We have the curvature of on and Chern forms on . We can write
where and are respectively a -form and a -form not containing (i.e., expressible in terms of the differentials of the local coordinates of with coefficients depending on and the local coordinate of ). Since is closed, we have
where means (partial) exterior differentiation with the coordinate kept constant, and means differentiating the coefficients of with respect to the variable . Hence,
for , it follows that , being equal to , is exact on .
We define as the de Rham cohomology group of degree over the abelian group of all smooth closed -forms on quotiented by the abelian subgroup of all smooth exact -forms on . An element of the de Rham cohomology group of degree is called a de Rham cohomology class of degree . We will later discuss de Rham cohomology in the context of sheaf cohomology. The th Chern form defines a de Rham cohomology class of degree , and this class is independent of the choice of the connection and is called the th Chern class of the -vector bundle over . Chern classes are invariants of smooth vector bundles.
When is a complex manifold and is a holomorphic vector bundle, we can choose as our connection the complex metric connection of a Hermitian metric along the fibers of . Then the curvature form is of type , and the th Chern form is of type . So we conclude that the th Chern class of a holomorphic vector bundle over a complex manifold can be represented by a form of type .