Meromorphic differentials, orders and residues

Let $X$ be a Riemann surface with its maximal holomorphic atlas $\mathcal{A}$ (Riemann surfaces and holomorphic atlases). Let ${\mathcal{A}}_c$ be the subatlas of all charts in $\mathcal{A}$ with nonempty connected domain. It covers $X$: a chart restricts to each connected component of its domain, and these components are open because its image is open in $\mathbb{C}$. Thus every image of a chart in ${\mathcal{A}}_c$ is a plane domain (Meromorphic functions on a plane domain).

A meromorphic differential $\omega$ on $X$ is a family $(h_{\phi }{)}_{\phi \in {\mathcal{A}}_c}$, where each $h_{\phi }$ is a meromorphic function on the domain $\phi (U_{\phi })$ (Meromorphic functions on a plane domain), subject to the transition law: for charts $\phi ,\psi$ with coordinates $z=\phi (x)$ and $w=\psi (x)$ and transition $z=z(w)=\phi \circ {\psi }^{-1}(w)$, on each connected component of $\psi (U_{\phi }\cap U_{\psi })$,

$h_{\psi }(w)=h_{\phi }(z(w))z^{\mathrm{\prime }}(w)\mathrm{.}$

One writes $\omega =h_{\phi }dz$ in the chart $\phi$, so that the law is the familiar $h_{\psi }dw=h_{\phi }dz$. The differential is zero when every $h_{\phi }$ is the zero function, and nonzero otherwise; a nonzero differential has a local expression that is not identically zero near every point, as the well-definedness argument below shows. The differential is holomorphic at $p\in X$ when some, equivalently every, local expression $h_{\phi }$ is holomorphic at $\phi (p)$, and holomorphic when it is holomorphic at every point.

The pole set of $\omega$ is the set of points $p$ at which some, equivalently every, local expression of a nonzero $\omega$ has a pole. Since poles of a meromorphic function on a plane domain are isolated (Isolated singularities: removable, poles, and essential singularities), the pole set of a nonzero $\omega$ is a discrete subset of $X$.

Order and residue. Let $\omega \mathrm{\ne }0$ and $p\in X$, and choose a chart $\phi$ with $\phi (p)=0$; such centred charts exist, since translations of charts are again compatible with the maximal atlas. Writing $h=h_{\phi }$, the order of $\omega$ at $p$ is

${\mathrm{ord}}_p(\omega ):=\{\begin{array}{ll}k,& h\text{ has a zero of order }k\text{ at }0,\\ -k,& h\text{ has a pole of order }k\text{ at }0,\\ 0,& h(0)\mathrm{\ne }0,\end{array}$

and the residue of $\omega$ at $p$ is the coefficient ${\mathrm{Res}}_p(\omega ):=c_{-1}$ of $z^{-1}$ in the Laurent expansion of $h$ at $0$ (The residue of an isolated singularity, Laurent expansion on an annulus); in particular ${\mathrm{Res}}_p(\omega )=0$ when $\omega$ is holomorphic at $p$. The point $p$ is a zero, respectively a pole, of $\omega$ of order $k\ge 1$ when ${\mathrm{ord}}_p(\omega )=k$, respectively ${\mathrm{ord}}_p(\omega )=-k$.

Conventions.

• On a disconnected chart of $\mathcal{A}$, its coefficient is determined component by component by the coefficients of its restrictions in ${\mathcal{A}}_c$; meromorphic on such an open set means meromorphic on each nonempty connected component. No plane-domain definition is applied to a disconnected set.
• Each local expression $h_{\phi }$ is meromorphic on a possibly different domain, and the transition law is required only on connected components of overlaps; the two charts may be taken from any atlas contained in $\mathcal{A}$, because compatibility is a local condition on overlaps.
• On a plane domain $\mathrm{\Omega }\subseteq \mathbb{C}$ with its identity atlas, a meromorphic differential is exactly an expression $h(z)dz$ with $h$ meromorphic on $\mathrm{\Omega }$, and the order and residue above are the usual Laurent order and residue of $h$ (Isolated singularities: removable, poles, and essential singularities, The residue of an isolated singularity).
• No choice principle is used: the data are functions indexed by the charts of a fixed atlas, and the well-definedness argument below uses only the identity theorem and local Laurent expansions.

The transition law $h_{\psi }(w)=h_{\phi }(z(w))z^{\mathrm{\prime }}(w)$ is the statement that $hdz$ transforms as a differential, and it is exactly what makes the order invariant: the Jacobian factor $z^{\mathrm{\prime }}(w)$ is a unit in the local ring at $w=0$ because a change of coordinates is injective. The residue is invariant for the same reason, and the computation in step 4.1 isolates the one term $k=-1$; a ramified map such as $w\mapsto w^2$ is not a change of coordinates, which is consistent with the pullback formula ${\mathrm{ord}}_x(f^{\ast }\eta )=e_x{\mathrm{ord}}_{f(x)}(\eta )+e_x-1$ proved later for branched maps.