Why is the constant relating the chern class and curvature form always $2\pi i$?

I'm reading Milnor's book on Characteristic Classes. In Appendix C, Milnor shows the invariant polynomial of the curvature form and the Chern class differ by powers of $2\pi i$. He first shows that the first Chern class and the trace of the curvature form are multiples of each other by a constant $a$. He calculates this constant for a bundle on a sphere, and finds it is $2\pi i$. Milnor says it's enough to calculate $a$ for one specific case, and that this constant is the same for all vector bundles.

My question is: why is it enough to find $a$ for one specific case? Is there any reason why the constant can't be different for different bundles?

The proof is done on page 306.

The point of the proof is to show that what Milnor calls $\Omega_{12}(M)$ is a characteristic class for complex line bundles $\zeta\to M^2$. It's not an arbitrary element of $H^*(M, \mathbb{C})$; it's a map from bundles to cohomology that's natural and defined in terms of a certain classifying space. On p. 298, Milnor proves that certain forms induce characteristic classes; he then proves on p 306 that $\Omega_{12}(M)$ is a form of that type and thus defines a characteristic class. The only characteristic classes for complex line bundles are of the form $\zeta \to \alpha c(\zeta)\in H^*(X, \mathbb{C}) = H^*(X)\otimes \mathbb{C}$ for some fixed $\alpha\in\mathbb{C}$ (see, for example, the axiomatic definition of the Chern class--- it's determined by its value on the tautological bundle of $\mathbb{CP}^\infty$), and going through the computation for the case of $M = S^2, \zeta = T^*S^2$ determines $\alpha$.

This is meant as a supplement to anomaly’s elegant answer above. In what follows, all manifolds are assumed to be closed and oriented.

For each smooth manifold $ M $, let

• $ \mathscr{L}(M;\mathbb{C}) $ denote the set of all isomorphism classes of smooth (complex) line bundles over $ M $;
• $ {H^{*}}(M;\mathbb{C}) $ denote the cohomology ring of $ M $.

Hence, we may view $ \mathscr{L}(\bullet;\mathbb{C}) $ and $ {H^{*}}(\bullet;\mathbb{C}) $ as contravariant functors from the category of smooth manifolds to the category of sets.

Milnor essentially establishes that $ \Omega_{12} $ is a natural transformation from $ \mathcal{L}(\bullet;\mathbb{C}) $ to $ {H^{*}}(\bullet;\mathbb{C}) $, i.e., for each morphism $ f: M \to N $ of smooth manifolds, we have the following commutative diagram: $$ \require{AMScd} \begin{CD} \mathscr{L}(N;\mathbb{C}) @>{f^{*}}>> \mathscr{L}(M;\mathbb{C}) \\ @V{{\Omega_{12}}(N)}VV @VV{{\Omega_{12}}(M)}V \\ {H^{*}}(N;\mathbb{C}) @>>{f^{*}}> {H^{*}}(M;\mathbb{C}) \end{CD} $$

It is a well-known result (see Theorem 14.5 of Milnor’s and Stasheff’s book) that any element of the cohomology ring $ {H^{*}}(\mathbb{C P}^{\infty};\mathbb{C}) $ is a polynomial in the first Chern class of the tautological line bundle $ \gamma^{1} $ over $ \mathbb{C P}^{\infty} $. Having said so, let $ P \in \mathbb{C}[X] $ be a polynomial such that $$ [{\Omega_{12}}(\mathbb{C P}^{\infty})](\gamma^{1}) = P({c_{1}}(\gamma^{1})). $$ Let $ M $ be any smooth $ 2 $-manifold and $ E $ any smooth line bundle over $ M $. As $ \gamma^{1} $ is a universal line bundle, there exists a smooth mapping $ f: M \to \mathbb{C P}^{\infty} $ such that $ E = {f^{*}}(\gamma^{1}) $. The naturality of the first Chern class $ c_{1} $ then yields \begin{align} [{\Omega_{12}}(M)](E) & = [{\Omega_{12}}(M)]({f^{*}}(\gamma^{1})) \\ & = {f^{*}}([{\Omega_{12}}(\mathbb{C P}^{\infty})](\gamma^{1})) \\ & = {f^{*}}(P({c_{1}}(\gamma^{1}))) \\ & = P({f^{*}}({c_{1}}(\gamma^{1}))) \\ & = P({c_{1}}({f^{*}}(\gamma^{1}))) \\ & = P({c_{1}}(E)). \end{align} Now, $ [{\Omega_{12}}(M)](E) \in {H^{2}}(M;\mathbb{C}) $, so $ P $ is a linear polynomial with no constant term, i.e., $ P = \alpha X $ for some $ \alpha \in \mathbb{C} $. Therefore, for all smooth $ 2 $-manifolds $ M $ and all smooth line bundles $ E $ over $ M $, $$ [{\Omega_{12}}(M)](E) = \alpha \cdot {c_{1}}(E). $$

Conclusion: We can determine the value of $ \alpha $ by considering the cotangent bundle $ T^{*} \mathbb{S}^{2} $ over $ \mathbb{S}^{2} $.