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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.

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What if you take the $q$-coordinate on the complex upper half plane? Its differential gives a differential form which is defined on the modular curve $X(1)$. I don't think this is algebraic, but I might be wrong. More generally, if you uniformize your Riemann surface $X=E-0$, you can construct coordinates $z$ around the cusps by using exponential maps. These aren't algebraic but they give differential forms $dz$. –  Harry Mar 20 '12 at 10:09
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up vote 12 down vote accepted

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 $n_i$.

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Thanks to Donu! I also have generalized my question in the spirit of his answer. –  Veen Mar 20 '12 at 13:48
    
@Donu: what is the precise definition of $F(\sum n_ip_i)$? –  Veen Mar 22 '12 at 16:50
    
I added a definition. –  Donu Arapura Mar 22 '12 at 17:22
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