Let $X$ be a complex compact manifold, let $\mathcal{E^*}$ denotes the sheaf of smooth complex valued nowhere vanishing functions from $X$ to $\mathbb{C}$.
Consider the map $H^1(X,\mathcal{E^*})\rightarrow H^2(X,\mathbb{Z})\rightarrow H^2(X,\mathbb{R})$ (the first one is from the $0\rightarrow\mathbb{Z}\rightarrow \mathcal{O}_X\rightarrow \mathcal{O}_X ^*\rightarrow 0$, the second one is tensoring by $\mathbb{R}$). It associates to each complex line bundle a real 2-form.
On the other hand we have a map $H^1(X,\mathcal{E}^*)\rightarrow H^2(X,\mathbb{R})$ $L\mapsto i\theta (L)/2\pi$, where $\theta (L)$ is defined to be $D\circ D$ for an arbitrary connection $D$.
This two maps shall differ be sign. Please correct if there are mistakes.
1) Why the second map is a group homomorphism? I haven't really understood why the form is real, but I can believe that.
2) Is the second map what we call the Chern class homomorphism? I am not sure about the coefficients.
3) Why the maps differ be sign?