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?



holomorphic sections of line bundles on Riemann surfaces
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
$$
\tau\_\alpha/\sigma\_\alpha=(g\_{\alpha\beta}\cdot\sigma\_\beta)/(g\_{\alpha\beta}\cdot\tau\_\beta)=\tau\_\beta/\sigma\_\beta.
$$ 


