If I remember by Riemann surfaces course correctly, then the following should be true:
Let $X$ be a Riemann surface, $U\subset X$ be open, and $x\in U$. Then the map $\Gamma(U,\mathcal{O}_X)\to \mathcal{O}_{X,x}$ is injective.
This should follow from the uniqueness of analytic continuation. Of course the map is not surjective, as can be seen by considering $X=\mathbb{C}^*$, $x=1$, and taking the germ of a logarithm.
My question is if this theorem is still true on higher dimensional complex manifolds? So:
Let $X$ be a complex manifold, $U\subset X$ be open, and $x\in U$. Then the map $\Gamma(U,\mathcal{O}_X)\to \mathcal{O}_{X,x}$ is injective.
Also, a related question:
Let $X$ be a complex manifold, $U\subset X$ be open and simply connected, and $x\in U$. Then the map $\Gamma(U,\mathcal{O}_X)\to \mathcal{O}_{X,x}$ is an isomorphism.
Are these statements true?