I have something elementary to ask. Let $E\rightarrow X$ be a holomorphic line bundle over a Riemann surface. Then in general a section of $E$ is a meromorphic function on $X$, since $O_{div(s)}\cong E$. But we also know that the first Dolbeault cohomology group be the group of holomorphic sections on $X$. The space of meromorphic functions on $X$ with poles in $div(s)$ (over $\mathbb{C}$) should be the same as the first Dolbeault cohomology group. But the two objects are different. What am I missing at here? Any ideas?
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3$\begingroup$ To identify a section of $E$ with a meromorphic function, you must first choose a divisor $D$ and write $E$ as $\mathcal{O}_X(D)$. The question would be more appropriate on MSE. $\endgroup$– abxCommented Jul 8, 2018 at 4:53
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2$\begingroup$ I do not really follow. I thought one need to pick a local chart and check the degree of the section over the chart. $\endgroup$– GuestCommented Jul 8, 2018 at 5:19
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I think you make some confusion, and also that this question is perhaps more appropriate for MSE.
Anyway, if you are given a holomorphic line bundle $\pi\colon E\to X$, a holomorphic section is a holomorphic map $s\colon X\to E$ such that $\pi\circ s=\operatorname{Id}_X$.
More concretely, if your line bundle is described by some cocycle $\{g_{\alpha\beta}\}$ relative to some open cover $\{U_\alpha\}$ of $X$ which trivializes $E$, then a holomorphic section of $E$ is given by a collection of holomorphic functions $\sigma_\alpha\colon U_\alpha\to\mathbb C$ such that on the overlapping $U_\alpha\cap U_\beta$ you have $$ \sigma_\alpha=g_{\alpha\beta}\cdot\sigma_\beta. $$ Now, fix any holomorphic section $s$ of $E$, given locally on $U_\alpha$ by holomorphic functions $\sigma_\alpha$. Then, you can identify holomorphic sections of $E$ with meromorphic functions $f$ on $X$ such that $f\cdot s$ (which is a priori a meromorphic section of $E$) is again holomorphic (in the language of divisors this means than $\operatorname{div}(f)+\operatorname{div}(s)\ge 0$).
One direction of this correspondence is clear: if $f$ is such that $f\cdot s$ is a holomorphic section of $E$, well you have your holomorphic section. For the other direction, if $t$ is a holomorphic section of $E$, given locally on $U_\alpha$ by holomorphic functions $\tau_\alpha$, then you set $f|_{U_\alpha}:=\tau_\alpha/\sigma_\alpha$. Begin a quotient of holomorphic function, $f$ is meromorphic. It actually defines a globally defined function since on the overlappings you have $$ \tau_\alpha/\sigma_\alpha=(g_{\alpha\beta}\cdot\sigma_\beta)/(g_{\alpha\beta}\cdot\tau_\beta)=\tau_\beta/\sigma_\beta. $$
