This question consists of $3$ points, and each of them deals with the relationship between the "algebraic" structure on a projective curve and its "analytic" structure. I know that this argument has been made explicit by Serre's GAGA, but here I'd like to understand some elementary concepts.
Let $(X,\mathscr O_X)$ an irreducible, smooth projective algebraic curve over $\mathbb C$. Then $X$ is also a Riemann surface equipped with the sheaf of holomorphic functions $\mathscr O_X^{\operatorname{an}}$. Every Zariski open set $U$ is also an open set in the strong topology and we have an embedding: $$\mathscr O_X(U)\subset\mathscr O^{\operatorname{an}}_X(U)$$
First of all I'd like to visualize each section $s\in\mathscr O_X(U)$ as an holomorphic map on $U$, so please tell me if the following argument is correct: for each point $x\in U $ consider the natural map $x\mapsto s_x+\mathfrak m_x\in k(x)=\mathbb C$, which sends a point to the image of $s$ in the residue field at $x$. This map should be holomorphic, one can check it by working on charts.
We have also $\mathscr O_{X,x}\subseteq \mathscr O^{\operatorname{an}}_{X,x}$ so every element $s_x$ can be seen as a germ of holomorphic functions at $x$. Now consider the completion $\widehat{\mathscr O_{X,x}}$ with respect the valuation $v_x$ associated to a closed point in $x$. What is the relationship between $\widehat{\mathscr O_{X,x}}$ and $\mathscr O^{\operatorname{an}}_{X,x}$? Roughly speaking I have the following idea: note that $\widehat{\mathscr O_{X,x}}\cong \mathbb C[[t]]$ so here it seems that we are considering the holomorphic functions in $\mathscr O_{X,x}$ plus the functions with eliminable discontinuity at $x$.
- One can repeat the same reasoning with the field of rational functions $K(X)$ (i.e. the meromorphic functions on $X$). What is its completion $K(X)_x=\operatorname{Frac }(\widehat{\mathscr O_{X,x}})\cong \mathbb C((t))$ in the framework of meromorphic functions? It seems that we are considering germs of meromorphic functions with a fixed pole in $x$.