Multiplicity of an holomorphic map between Riemann surfaces

Question

I need help understanding the meaning of multiplicity in a point of an holomorphic map between Riemann sufaces. So $F\colon X \to Y$ be an holomorphic, not constant map between Riemann surfaces and fix $p\in X$.

We know that it is possible to find local charts $(U,\phi), (V,\psi)$ centered in $p$ and in $F(p)$ respectively such that the local expression $f:=\psi\circ F\circ \phi^{-1}$ is $z\mapsto z^m$ for an integer $m$. It is possible to prove that this $m$ does not depend on the choice of the charts and it is called multiplicity of $F$ in $p$.

Now, what is the meaning of this $m$? I think that $m$ indicates the number of preimages in $X$ near $p$. More precisely, if $(U,\phi), (V,\psi)$ are local charts centered in $p$ and in $F(p)$ respectively, if $m=mult_p(F)$, then for each $y\in V\cap \text{Im} F$ there are exactly $m$ preimages in $U$. Is it correct or am I missing something? So it is intuitively true that the definition of multeplicity does not depend on the charts, but only on $p$ and $F$.

Answer 1

To simplify, let's assume X and Y compacts. Then, $\, f : X \longrightarrow Y$ is a branched covering. Which means that expect over a finite number of points $A = \{y_0, \ldots, y_n \} \subset Y$, $f : X \backslash f^{-1}(A) \longrightarrow Y \backslash A$ is a covering map. Then the multiplicity is exactly the degree of this covering map: the number of sheets over a point.

Answer 2

You need some extra assumptions:$f$ not constant on any component of $X$, $Y$ connected, ($X$ compact, or some behavior of $f$ at infinity if $X$ not compact). The degree of the map $f\colon X \to Y$ is a positive integer $m$ so that:

for every $y$ in $Y$ the equation $f(x)=y$ has exactly $m$ solutions $x \in X$ ( counting multiplicities). The fact that this number is constant is a theorem. There will be just finitely many points $y$ in $Y$ for which the multiplicities of some roots are $>1$.

A basic example is the map $z \mapsto z^m$ from $\mathbb{C}$ to $\mathbb{C}$ or even better, from $\hat{ \mathbb{C}}$ to $\hat{ \mathbb{C}}$. Another important example are polynomial maps of degree $m\ $ $$z \mapsto a_m z^m + \ldots + a_0$$

Answer 3

You are right: m indicates the number of pre-images of $f$ in X near p, counted with multiplicity. As you write, locally $f$ equals the m-th power $z\mapsto z^m$. Hence each point from $V$ different from zero has exactly $m$ different roots in $U$. All roots come together in the singleton $\{0\}$ containing the unique root of $0$ with multiplicity $m$.