Question

I'm in the situation to have a smooth proper curve $X$ over $Spec(\mathbb C)$, from which I consider the analytification $X^{an}$, which I consider as a compact Riemann surface.

Furthermore I have given a vector bundle $F$ on $X$ with analytification $F^{an}$.

Let $p$ denote a closed point of the curve.

Now I am given an analytic section of $F^{an}$ on the complement of the point $s \in H^0(X^{an}-p, F^{an})$. The question is if there is an algebraic section $\in H^0(X-p,F)$, which goes to $s$ under analytification.

Answer 1

No, because the section $s$ may have essential singularities. This is really the only issue. In general, suppose that $X$ is a compact Riemann surface with a vector bundle $F$ and $\lbrace p_1,p_2,\ldots\rbrace$ a finite subset. Then a section of $H^0(X^{an}-\lbrace p_i\rbrace,F^{an})$ with poles of finite order at these points would be algebraic. To see this, apply GAGA to $F(\sum n_ip_i)= F\otimes\mathcal{O}_X(\sum n_ip_i)$ for suitable coefficients $n_i$.