This is the question from Lectures on Riemann surfaces (Otto forster), exercise 10.3:

Suppose $X$ is a Riemann surface and $\omega$ is a meromorphic 1-form on $X$ which has residue zero at every pole. Show that there is a covering $p: Y \to X$ and a meromorphic function $F$ on $X$ such that $dF=p^*w$.



The condition you have means $\omega$ is closed on $X$. Now consider $f:Y\rightarrow X$ where $Y$ is the universal cover of $X$, then since $\pi_{1}(Y)=0$ and the period map factorizes through the first cohomology group $$ Hom(\pi_{1}(X),\mathbb{C})\rightarrow Hom(H_{1}(X),\mathbb{C})\cong H^{1}(X,\mathbb{C})\cong \mathbb{C}^{2g} $$ we have the pull-back form $f^{*}\omega$ to be exact. In other words there exist some $F$ such that $dF=\omega$.

  • $\begingroup$ Why does the condition mean that $\omega$ is closed on $X$? Is it obvious? $\endgroup$ – Acton Jan 20 '18 at 8:25
  • $\begingroup$ Use Stokes formula. $\endgroup$ – Bombyx mori Jan 22 '18 at 11:09

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