How do I prove that gauge-equivalence classes of $U(1)$ connections on a line bundle $L\to M$ are determined uniquely by pairs $(\alpha,F)$, where 
$$\alpha\in\text{Hom}(\pi\_1(M),U(1)),~~~~F\in \Omega^2(M)?$$



Is a non-flat hermitian connection determined uniquely by its holonomy and curvature?
The 2-form $F$ has to be closed. Then you choose an open covering $U\_i$ on which $F$ has a primitive $\theta\_i$, i.e. $d \theta\_i = F$. Now you try to glue together the trivial bundles $U\_i \times U(1)$ with connection $\theta\_i$. In order to do this, the curvature form has to be integral. 


Finally, equivalence classes of flat connections are parametrized by $Hom(\pi\_1(M), U(1)$. Every such flat bundle arises from a homomorphism $\lambda: \pi\_1(M) \to U(1)$ as the associated bundle $\tilde{M} \times\_\lambda U(1)$.


A classical reference for this material is Woddhouse 1997 "Geometric Quantization", Proposition 8.3.1


