Holomorphic maps and meromorphic functions on Riemann surfaces

Let $X$ and $Y$ be Riemann surfaces with atlases ${\mathcal{A}}_X$ and ${\mathcal{A}}_Y$ (Riemann surfaces and holomorphic atlases), and let $\widehat{\mathbb{C}}$ be the Riemann sphere with its standard holomorphic charts ${\varphi }_0,{\varphi }_{\mathrm{\infty }}$ (The standard holomorphic charts on the Riemann sphere, with holomorphy and poles at infinity).

Holomorphic maps. A map $f:X\to Y$ is holomorphic at $x\in X$ when there are charts $\phi \in {\mathcal{A}}_X$ with $x\in \mathrm{dom}\phi$ and $\psi \in {\mathcal{A}}_Y$ with $f(x)\in \mathrm{dom}\psi$ such that the chart expression is defined on $\phi (W)$ for some open neighbourhood $W\subseteq \mathrm{dom}\phi$ of $x$ with $f(W)\subseteq \mathrm{dom}\psi$ and is holomorphic at $\phi (x)$: $\psi \circ f\circ {\phi }^{-1}:\phi (W)\to \mathbb{C}$ Since $\phi$ is a chart, $\phi (W)$ is open in $\mathbb{C}$. The map $f$ is holomorphic when it is holomorphic at every point of $X$.

Independence of the charts. The condition does not depend on the charts chosen, and it is enough to test it on one atlas on each side. If ${\phi }^{\mathrm{\prime }}\in {\mathcal{A}}_X$ and ${\psi }^{\mathrm{\prime }}\in {\mathcal{A}}_Y$ are further charts with $x\in \mathrm{dom}{\phi }^{\mathrm{\prime }}$ and $f(x)\in \mathrm{dom}{\psi }^{\mathrm{\prime }}$, the given local chart expression makes $f$ continuous on a neighbourhood of $x$. Shrink that neighbourhood to an open set $W$ on which both source charts are defined and $f(W)\subseteq \mathrm{dom}\psi \cap \mathrm{dom}{\psi }^{\mathrm{\prime }}$. Then near ${\phi }^{\mathrm{\prime }}(x)$, ${\psi }^{\mathrm{\prime }}\circ f\circ {\phi }^{\mathrm{\prime }-1}=({\psi }^{\mathrm{\prime }}\circ {\psi }^{-1})\circ (\psi \circ f\circ {\phi }^{-1})\circ (\phi \circ {\phi }^{\mathrm{\prime }-1}),$ and all three factors are holomorphic on neighbourhoods of the relevant points: the outer two are transitions between compatible charts, and the middle one is holomorphic by hypothesis. The chain rule gives holomorphy of the composite. A map tested with one pair of charts at $x$ is therefore holomorphic at $x$ whatever charts an alternative atlas supplies. In particular, replacing either atlas by its maximal one does not change the class of holomorphic maps, so the notion depends only on the two complex structures.

A holomorphic map is continuous, since in the charts above it is locally the composite of continuous maps; in particular a holomorphic map is determined by its values on a nonempty open set when the target is Hausdorff and the source connected, a fact used later but not proved here. Constant maps, the identity $X\to X$, restrictions $f{\mathrm{\mid }}_U$ to open $U\subseteq X$ (with the restricted atlas and the subspace topology), and composites of holomorphic maps are holomorphic.

Meromorphic functions. A meromorphic function on $X$ is a holomorphic map $f:X\to \widehat{\mathbb{C}}$ that is not the constant map with value $\mathrm{\infty }$. Equivalently, writing $f$ in the charts of $\widehat{\mathbb{C}}$: $f$ is meromorphic when for every $x\in X$ there is a chart $\phi$ of $X$ with $x\in \mathrm{dom}\phi$ such that the local expression $f\circ {\phi }^{-1}$ is either holomorphic at $\phi (x)$ (when $f(x)\mathrm{\ne }\mathrm{\infty }$) or has a pole at $\phi (x)$ in the sense of Isolated singularities: removable, poles, and essential singularities (when $f(x)=\mathrm{\infty }$), and the set $f^{-1}(\mathrm{\infty })$ is not all of $X$. The points of $f^{-1}(\mathrm{\infty })$ are the poles of $f$; at such a point the reciprocal chart expression $w\mapsto 1\mathrm{/}F(w)$ near $w=\phi (x)$, where $F=f\circ {\phi }^{-1}$, is holomorphic with value $0$, so poles are isolated and coincide with the usual plane-domain notion in a chart.

Agreement with plane domains. If $X=\mathrm{\Omega }$ is a plane domain with its identity atlas and $g:\mathrm{\Omega }\setminus P\to \mathbb{C}$ is meromorphic in the sense of Meromorphic functions on a plane domain, with pole set $P$, then the map $f:\mathrm{\Omega }\to \widehat{\mathbb{C}}$ with $f(z)=g(z)$ for $z\mathrm{\notin }P$ and $f(z)=\mathrm{\infty }$ for $z\in P$ is holomorphic: away from $P$ this is the ordinary statement that $g$ is holomorphic, and at $p\in P$ the chart expression in ${\varphi }_{\mathrm{\infty }}$ is $z\mapsto 1\mathrm{/}g(z)$, which is holomorphic near $p$ with value $0$, because $g$ has a pole at $p$. Conversely, if $f:\mathrm{\Omega }\to \widehat{\mathbb{C}}$ is holomorphic and not constant $\mathrm{\infty }$, then $g:=f$ restricted to the open set $f^{-1}(\mathbb{C})$ is holomorphic there and every point of $f^{-1}(\mathrm{\infty })$ is a pole of $g$, so $g$ is meromorphic on $\mathrm{\Omega }$ in the plane-domain sense. The two notions therefore agree, and this is the sense in which a meromorphic function on a Riemann surface is locally a usual meromorphic function.

Conventions.

• The map $X\to \widehat{\mathbb{C}}$ constant at $\mathrm{\infty }$ is holomorphic but is excluded from being meromorphic; it is the analogue of the zero function being excluded from having a well-defined finite order. Every nonconstant holomorphic map $X\to \widehat{\mathbb{C}}$ is meromorphic.
• Constants, the identity, and $z\mapsto 1\mathrm{/}z$ on ${\mathbb{C}}^{\times }$ (and the standard chart transition of $\widehat{\mathbb{C}}$) are examples of meromorphic functions; the definition of holomorphic map is used here for maps between surfaces of possibly different complex dimension conventions only in dimension one, so all chart expressions are functions of one complex variable.
• No choice principle is used. Charts are quantified over an atlas, which is a set; the independence argument is a chain-rule computation. No chart is selected.