Let $X$ be a compact Riemann surface, $p\in X$, and $\varphi$ be a smooth 2-form on a $X-\{p\}$, and hence exact. I'm wondering if it is possible to find a form of type (1,0) whose differential is $\varphi$. I know that for example if $\varphi$ is smooth and non-exact on $X$, this is always possible using Green functions.
I know this is very basic and not really research level, but I need it for something to work, and unfortunately cannot find a reference for it. Any reference would be greatly appreciated.