I have been reading a rigorous treatment of Riemann surfaces described in terms of a given complete analytic function and a sheaf of germs. My question is the following: Suppose we are considering $\Omega \subset \mathbb{C}$ and say we assume it's simply connected. If we have a single valued function holomorphic defined on all of $\Omega$ does this imply that the sheaf of germs is in some sense trivial? My thinking is that each stalk will consist of exactly one germ and then it seems the Riemann surface will just be the graph of the function sitting in $\mathbb{C}^2$.
I definitely could be missing something here.
Thanks for your input.