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.



Can one apply GAGA in this special case?
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 $n\_i$.


