{line bundles} $\neq$ {divisor line bundles} Let $X$ be a compact complex manifold, and with the following sheaves


*

*$\mathscr O$, the sheaf of holomorphic function,

*$\mathscr O^*$, the sheaf of nonvanishing holomorphic function

*$\mathscr K^*$ the sheaf of nonidentically zero meromorphic function.


A divisor is an element in $\Gamma(X, \mathscr K^*/\mathscr O^*)$ (Cartier divisor). The short exact sequence of sheaves 
$$ 0 \to \mathscr O^* \to \mathscr K^* \to \mathscr K^*/\mathscr O^* \to 0$$
induces a long exact sequence in cohomology
$$\cdots \to \Gamma(X, \mathscr K^*) \to \Gamma(X, \mathscr K^*/\mathscr O^*) \to H^1(X, \mathscr O^*) \to \cdots$$
and $\Gamma(X, \mathscr K^*/\mathscr O^*) \to H^1(X, \mathscr O^*)$ identify a divisor to a line bundle. 
I suppose that the map is not in general surjective. Can anyone give an example? 
 A: The short paper "The sheaf of nonvanishing meromorphic functions in the projective algebraic case is not acyclic" (www.math.wustl.edu/~matkerr/MerR.pdf) explains that even for a projective algebraic variety $X/\mathbb{C}$, the sheaf $\mathcal{K}^*$ in the analytic topology is not acyclic in general, and in particular that as soon as $\dim(X)\geq 2$ and $H^1(X,\mathbb{Z})\neq 0$, one has $H^1(X,\mathcal{K}^*)\neq 0$ !
The proof is short and clear, so I will not reproduce it here. This provides plenty of examples, for instance any abelian variety of dimension $\geq 2$. 
This does not contradict GAGA because $\mathcal{K}^*$ is not coherent.
There may be simpler, complex analytic counter-examples, and I hope others will provide some.
Edit: This does not answer the question, because in the projective case the map $H^1(X,\mathcal{O}^*_X)\rightarrow H^1(X,\mathcal{K}^*_X)$ is $0$. Hence one really need a non-projective example.
Edit 2: In fact this question has already been answered in math.SE : When is $CaCl(X) \to Pic(X)$ surjective? It turns out that a very general, non-algebraic complex torus of dimension 2 answers your question.
