2
$\begingroup$

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 $dF=p^*w$.

Thanks.

$\endgroup$
0
$\begingroup$

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 $dF=\omega$.

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.