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Can anyone give me a hint on how to start solving this problem?

Let $\Gamma \subset \mathbb{C}$ be a lattice. Show that the form $dz$ on $\mathbb{C}$ defines a holomorphic $1$-form on $\mathbb{C}/\Gamma$.

Any help appreciated!

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    @Gregor: Thanks for the help! The def. of hol. $1$-forms $\omega$ on a RS $X$: $\omega$ is hol. if, w.r.t. every chart $(U,z)$, $\omega$ may be written $\omega = f dz$ on $U \cap Y$, where $f \in \mathcal{O}(U \cap Y)$, $Y \subseteq X$, $Y$ open. Charts: Let $V \subset \mathbb{C}$ be open s.t. no two points are equiv. under $\Gamma$. Then $U:= \pi(V)$ open in $\mathbb{C}/\Gamma$, and $\pi| V \rightarrow U$ is a homeom., its inverse $\varphi :U \rightarrow V$ is a chart. But I'm not sure what I have to prove. Should I let $\varphi : U \rightarrow V$ be any chart, and find an appropriate $f$?2012-11-20

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Thank you for the information provided. Here are some little hints to get you started:

  • Your idea is the right one. Choose a chart, find the appropriate function, and prove it.
  • Your holomorphic $f$ will be in fact $1$.
  • Your $Y$ is in fact the whole torus $\mathbb{C}/\Gamma$.

The solution, if you write it down, should look almost trivial.

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    Yes, seems right to me. Thanks a lot for the help :) I might have said the converse in the first comment, I'm a bit new to these questions, and thus easily confused. Btw, I don't know if this is appropriate, but could you have a look at [another of my questions?](http://math.stackexchange.com/questions/241419/how-to-prove-that-the-weierstrass-wp-function-is-a-well-defined-meromorphic-f)2012-11-21