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: $$\mathscr O_X(U)\subset\mathscr O^{\operatorname{an}}_X(U)$$

  1. 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.

  2. 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 $x$.

  3. 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 $x$.
  • $\begingroup$ May I, very humbly, suggest that you change $\mathcal{O}_{X,x}^h$ to $\mathcal{O}_{X^\text{an},x}$ or something? Namely, the superscript '$h$' already has a meaning, which makes total/interesting sense in the context of question 2. I wrote a long answer before realizing you meant something else. I'll come back and answer this question later if no one else has. $\endgroup$ – Alex Youcis Sep 25 '16 at 0:27
  • $\begingroup$ Yes, I will edit. Sorry for the misunderstanding $\endgroup$ – Dubious Sep 25 '16 at 0:32
  • $\begingroup$ 1. is correct--that's how you think of the functions as being honest holomorphic functions. That said, for this to be faithful in any sense you should assume that $X$ is reduced (maybe curve already means that for you). For 2. you should note that the equality $\widehat{\mathcal{O}_{X,x}}=\mathbb{C}[[T]]$ really presupposes that $X$ is normal=non-singular=smooth. Before answering the rest of 2., can I clarify precisely what you mean by the relationship between the two rings? Namely, again assuming that $X$ is normal, we have that $\widehat{\mathcal{O}_{X,x}}=\mathbb{C}[[T]]$ $\endgroup$ – Alex Youcis Sep 25 '16 at 1:59
  • $\begingroup$ and $\mathcal{O}_{X^\text{an},x}$ is $\mathbb{C}\{T\}$ the ring of convergent power series (i.e. those elements of $\mathbb{C}[[T]]$ with positive radius of convergence). Does that answer your question, or are you trying to get a deeper analogy? A similar question I have for 3. What precisely are you looking for in the comparision between the two objects? $\endgroup$ – Alex Youcis Sep 25 '16 at 2:01
  • $\begingroup$ @Alex Youcis, X is integral so reduced; moreover you are right, I have to assume $X$ smooth (I'll edit). So, from your second message it seems that $\widehat{\mathscr O_{X,x}}\cong\mathscr O_{X,x}^{\operatorname{an}}$. Is this true? $\endgroup$ – Dubious Sep 25 '16 at 9:27

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