TITLE: A complex line bundle is trivial if and only if the first Chern class is zero
QUESTION [5 upvotes]: Let $\xi$ be a complex line bundle over a CW-complex $B$. I want to prove that $\xi$ is trivial if and only if $c_1(\xi)=0$.
My attempt: Suppose $c_1(\xi)=0$. Then the Euler class $e(\xi)=0$. Since $e(\xi)=o_2(\xi)$,  there exists a nonvanishing cross section of $\xi|_{sk^2 (B)}$, denoted as $X\in \Gamma(\xi|_{sk^2 (B)})$. $X,JX$ are linearly independent cross sections of $\xi|_{sk^2 (B)}$. Thus $\xi|_{sk^2 (B)}$ is a trivial bundle. How to prove that $\xi$ is trivial?

REPLY [6 votes]: Let us approach the case of complex manifolds by means of sheaf cohomology.
On a complex manifold $M$, the homolorphic/$C^\infty$ line bundles may be identified with $H^1(M, \mathcal O^*)$ and $H^1(M, \mathcal A)$, respectively (the sheaf cohomology with coefficients in non-vanishing holomorphic/$C^\infty$ functions).
The holomorphic exponential exact sequence of sheaves
$$0 \to \mathbb Z \stackrel{2\pi\cdot}\to \mathcal O \stackrel{\exp}{\to} \mathcal O^* \to 0 \hspace{2em} (1)$$
induces boundary homomorphism
$$c_1: H^1(M, \mathcal O^*) \stackrel{\delta}{\to} H^2(M, \mathbb Z),$$
which may be taken as the definition of the first Chern class of a holomorphic line bundle.
Moreover, the inclusion of $(1)$ into the $C^\infty$ exponential exact sequence of sheaves
$$0 \to \mathbb Z \stackrel{2\pi\cdot}\to \mathcal A \stackrel{\exp}{\to} \mathcal A^* \to 0, \hspace{3em}$$
by the functoriality of sheaf cohomology, induces the following commutative diagram:
$$\begin{array}{ccl}
H^1(M, \mathcal O)&\to&H^1(M,\mathcal O^*)&\to&H^2(M,\mathbb Z)\\
\downarrow&&\hspace{2em}\downarrow&&\hspace{2em}\|\\
H^1(M, \mathcal A)&\to&H^1(M,\mathcal A^*)&\to&H^2(M,\mathbb Z),\\
\end{array}$$
where $H^1(M, \mathcal A)=0$ holds from the existence of a (real) partition of unity, so $$H^1(M, \mathcal A^*) \hookrightarrow H^2(M, \mathbb Z)$$ is an inclusion, and the map $H^1(M, \mathcal O^*) \to H^1(M, \mathcal A^*)$ is the tautological forgetting map from holomorphic line bundles to $C^\infty$ line bundles.
Therefore, the first Chern class of a holomophic line bundle completely  determines it as a $C^\infty$ bundle. For more details, see Griffiths, Harris, "Principles of Algebraic Geometry", section 1.1 or the following answer on ResearchGate.net.
By the way, the kernel 
$$\mathfrak P = \ker c_1: H^1(M, \mathcal O^*) \to H^2(M, \mathbb Z)$$
is called the Picard variety of a complex manifold $M$, or the Jacobi variety if $M$ is a curve, which is a complex torus of the dimension equal to the genus of the curve.
Moreover, the Torelli theorem states that a smooth complex projective curve is completely determined by its Jacobi variety with some additional structure, therefore there is a wide gap between holomorphic line bundles which are holomorhically and $C^\infty$ trivial.