I have a big problem understanding the meaning of holomorphic coordinates on Riemann surfaces, especially in relation to 1-forms.

Holomorphic coordinates on a Riemann surface $X$ is an open set $U \subset X$ and a complex chart

$\varphi:U \rightarrow \varphi(U)=:V \subset \mathbb{C}$

As far as clear. Now you can define a differentiable function $f:U \rightarrow \mathbb{C}$ by: for every open $W \subset U$ and each chart $\psi:W \rightarrow \psi(W)$ you have $f \circ \psi^{-1}: \psi(W) \rightarrow \mathbb{C}$ is differentiable.

Now we defined Residues on Riemann surfaces in the following way:

Let $X$ be a Riemann surface, $U \subset X$ open, $a \in X$ and $\alpha$ a holomorphic 1-form on $U \setminus \{a\}$. Let $\alpha=f \ dz$ be the local expression for $\alpha$ in some hol. coordinate $z = \varphi:V \rightarrow \varphi(V) \subset \mathbb{C}$, where $V$ is a neighborhood of $a$ and $z(a)=0$. Then $Res_0(f \circ \varphi^{-1})$ is coordinate independent. As a consequence we can define $Res_a(\alpha):= Res_0(f \circ \varphi^{-1})$. \ Proof: The right-hand side is precisely $c_{-1}$ in the Laurent expansion $f(z) = \sum\limits_{\nu= - \infty}^{\infty} c_{\nu} z^{\nu}$.

I'm not even able to understand the following: If $z:U \rightarrow V$ is a local coordinate, look at $f(z)$, but if $f(z)$ is meant to be the composition, it doesn't make any sense for me since $z(U) \subset \mathbb{C}$ and $f$ is defined on $U$ ...

I am really hoping tha someone can lead me the way to understand this...

  • 2
    $\begingroup$ I don't know where you saw "look at $f(z)$", but whoever said it was being careless. Most likely by $f(z)$, they actually mean the map $f\circ z^{-1}$. More verbosely, when they called $z$ a local coordinate, they actually mean that there is some chart $\phi$ that they will not explicitly mention, and by means of this chart, they are identifying $f$ with $f\circ \phi^{-1}$. $z$ in then just a variable over the complex numbers, and by $f(z)$ they mean $f(\phi^{-1}(z))$. $\endgroup$ – Paul Sinclair Feb 20 at 1:17
  • $\begingroup$ And what about the differential form $f \ dz$, where $z$ is a local coordinate, how to understand this? $\endgroup$ – User1 Feb 20 at 9:38
  • $\begingroup$ Thanks btw, tha already helped me, it seems like no big deal but since it was never properly defined I was confused. $\endgroup$ – User1 Feb 20 at 9:39
  • $\begingroup$ $f\,dz$ is $(f\circ \phi^{-1})dz$ on $\phi(U)$. $\endgroup$ – Paul Sinclair Feb 20 at 17:01
  • $\begingroup$ but what does $dz$ mean here? $\endgroup$ – User1 Feb 20 at 18:47

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