# If a complex vector bundle arises as a complexification, why is its first Chern number zero?

I'm attempting to understand the following example from Weinstein's "Lectures on Symplectic Manifolds":

If $$P\to X$$ is a symplectic vector bundle, it may admit half dimensional subbundles, but no Lagrangian subbundles. Concretely, consider $$E_1 \oplus E_2 \to S^2$$, where $$E_1$$ is the tangent bundle of $$S^2$$ and $$E_2$$ is the trivial bundle $$\mathbb R^2 \to S^2$$.

The idea is that $$c_1(E_1 \oplus E_2)= 2$$, but if $$E_1 \oplus E_2$$ admits a Lagrangian subbundle $$L$$, then $$E_1 \oplus E_2$$ is isomorphic to $$L \oplus L^*$$ and hence $$L_{\mathbb C}$$. But this requires $$c_1(E_1\oplus E_2) = 0$$. Apparently "$$c_1(L_{\mathbb C}) = 0$$" is a very well known result since I haven't found any proofs but I've seen it stated in various forms while trying to find a proof online!

Unfortunately I know nothing about Chern classes or Chern numbers (as I'm still just learning introductory symplectic geometry), but I would like to understand this to some extent.

In McDuff-Salamon, the first Chern number of a complex vector bundle over an oriented closed surface is defined to satisfy the following four properties. (It is proved that there exists a unique functor that satisfies these.) Currently this is exactly the extent of my understanding of the first Chern number.

1. Two complex vector bundles over $$\Sigma$$ are isomorphic if and only if they have the same rank and first Chern number.

2. Whenever $$\varphi: \Sigma_2 \to \Sigma_1$$ is a smooth map between surfaces and $$E \to \Sigma_1$$ is a complex vector bundle, $$c_1(\varphi^* E) = \deg(\varphi) c_1(E).$$

3. If $$E_1, E_2$$ are complex vector bundles over $$\Sigma$$, $$c_1(E_1\oplus E_2) = c_1(E_1) + c_1(E_2.)$$

4. The first Chern number of the tangent bundle of $$\Sigma$$ is the Euler characteristic of $$\Sigma$$.

Is there a proof that whenever $$E \to \Sigma$$ is a real vector bundle over a closed oriented surface, then the complexification $$E_{\mathbb C} \to \Sigma$$ has vanishing first Chern number using the above axioms?

$$c_k(\bar F)=(-1)^k c_k(F)$$ for a complex bundle $$F$$ and its complex conjugate $$\bar F$$. Now $$\bar E_{\mathbb{C}}=E_{\mathbb{C}}$$ for the complexification of a real bundle $$E$$.