3
$\begingroup$

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

1 Answer 1

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 $dF=\omega$.

$\endgroup$
2
  • $\begingroup$ Why does the condition mean that $\omega$ is closed on $X$? Is it obvious? $\endgroup$
    – Acton
    Jan 20, 2018 at 8:25
  • $\begingroup$ Use Stokes formula. $\endgroup$ Jan 22, 2018 at 11:09

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .