COMPLEX GEOMETRY Course notes

2.3 Meromorphic functions and differentials
Definition 2.3.1. A meromorphic function on a Riemann surface Z is a holomorphic function on an
open subset U ⊆ Z where Z − U is a discrete set consisting of at most poles of the function.
Recall that a pole p ∈ Z − U is defined by one of the following equivalent conditions:
(a) lim z→p f(z) = ∞.
(b) f can be written locally as a Laurent series
∞∑
f(z) = a i z i
−∞
with a i = 0 for every i < n ∈ Z.
(c) f = g/h, where g, h ∈ O(p), g(p) ≠ 0 and h(p) = 0.
The set of such functions is denoted M(Z). We have
f ∈ M(Z) ⇐⇒ f : Z hol
−→ CP 1
and that
poles of f = f −1 (∞)
Example 2.3.1.
(1) A non-constant polynomial defines a meromorphic function from CP 1 with pole order at ∞ equal to
deg(f) ≥ 1.
(2) A rational function p(z)/g(z) defines a meromorphic function with pole order at ∞ equal to deg(p) −
deg(g). If this difference is negative then f has a zero at ∞.
Fact 2.3.1. M(Z) is a field.
A finite map of Riemann surfaces f : Z 1 −→ Z 2 corresponds to a finite field extension
f ∗ : M(Z 2 ) ↩→ M(Z 1 ).
Definition 2.3.2. A meromorphic differential on a Riemann surface Z is a holomorphic differential ω
on an open U ⊆ Z whose complement Z − U is discrete and consist of poles of ω. Locally, ω = fdz even at
a pole. The pole order of ω is defined by that of f (locally) and its residue at p is the same as that of fdz
(p = 0), denoted Res p (ω).
Theorem 2.3.1 (Residue). V ⊂⊂ Z with rectifiable boundary ∂V and ω differentiable on Z. Then
∫
ω = ∑ Res p (ω)
p∈V
∂V
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