In Miranda's Algebraic Curves and Riemann Surfaces he defines the multiplicity of a non-constant holomorphic map $F\colon X \rightarrow Y$ between Riemann surfaces as the unique integer $m$ such that there are local coordinates $\phi$ and $\psi$, near $p$ and $F(p)$ respectively, with $\psi\circ F\circ\phi^{-1}$ having the form $z\mapsto z^m$.
On page 45 he then tries to give a way to determine the multiplicity of a non-constant holomorphic map without determining its local normal form.
He writes:
Take any local coordinates $z$ near $p$ and $w$ near $F(p)$; say that $p$ corresponds to $z_0$ and $F(p)$ to $w_0$. In terms of these coordinates, the map $F$ may be written as $w=h(z)$ where $h$ is holomorphic.
What does the expression $w=h(z)$ actually mean? It seems that the domains and codomains do not really match up (i.e. $z\colon U\rightarrow V\subseteq \mathbb{C}$ and $w\colon \tilde{U}\rightarrow \tilde{V}\subseteq \mathbb{C}$, where $U\subseteq X$ and $\tilde{U}\subseteq Y$ are open, and $h\colon ?\rightarrow ?$). Is $h=w\circ F \circ z^{-1}$? Why does Miranda use this (to my eye) careless and weird notation?
Best Answer
Here is a more precise formulation:
Take a chart $\phi : U \to U' \subset \mathbb C$ with $p \in U$ and a chart $\psi : V \to V' \subset \mathbb C$ with $F(p) \in V$. Set $z_0 = \phi(p)$ and $w_0 = \psi(F(p))$. Now consider the holomorphic map $$h : \phi(F^{-1}(V) \cap U) \stackrel{\phi^{-1}}{\to} F^{-1}(V) \cap U \stackrel{F}{\to} V \stackrel{\psi}{\to} V' .$$ We can write $h = \psi \circ F \circ \phi^{-1}$ if we agree that the domain of this composition is the maximal subset $M \subset U'$ such that $\psi(F(\phi^{-1}(q)))$ is defined for all $q \in M$.
Alternatively we can start by first choosing $\psi$ and then choosing $\phi$ such that $U \subset F^{-1}(V)$. This gives the nicer representation $$h : U' \stackrel{\phi^{-1}}{\to} U \stackrel{F}{\to} V \stackrel{\psi}{\to} V' .$$
"In terms of these (local) coordinates" the map $F$ is therefore uniqueley described by the holomorphic map $h$ whose input is a complex variable $z$ and whose output is a complex variable $w$, i.e.has the form $w = h(z)$.
In my opinion Miranda should better use a phrase like "local representation of $F$ via charts" in such cases. It would also be okay to say "using local coordinates, we may assume w.lo.g.that $F$ is a holomorphic map between open subsets of $\mathbb C$".