Compact Riemann surfaces
Definition. A compact Riemann surface is a compact connected one-dimensional complex manifold. Local coordinates make holomorphic and meromorphic functions meaningful exactly as in plane domains.
Meromorphic functions are maps to the sphere
Proposition. A meromorphic function on a compact Riemann surface is the same as a holomorphic map to \(\widehat{\mathbb C}\). Poles are points mapped to \(\infty\).
No nonconstant holomorphic functions
Theorem. Every holomorphic function on a compact Riemann surface is constant. Its absolute value attains a maximum, so the maximum principle forces constancy.
Divisors of meromorphic functions
Definition. The divisor of a nonzero meromorphic function records zeros with positive multiplicity and poles with negative multiplicity. On a compact surface, the total degree of this divisor is zero.
Residue theorem on compact surfaces
Theorem. The sum of residues of a meromorphic one-form on a compact Riemann surface is zero. There is no boundary; local residue contributions must cancel globally.
Sphere example
Example. On \(\widehat{\mathbb C}\), meromorphic functions are rational functions. The equality between total zero order and total pole order is the usual degree count for rational functions.
Holomorphic maps to the sphere
A meromorphic function on a Riemann surface \(X\) is equivalently a holomorphic map \[ f:X\to\widehat{\mathbb C}. \] At a pole, the coordinate \(1/f\) is holomorphic and vanishes. This definition is independent of local charts and treats zeros and poles symmetrically.
Compactness forces constancy or surjectivity
Theorem. Every holomorphic \(f:X\to\mathbb C\) on a compact connected Riemann surface is constant. Every nonconstant meromorphic \(f:X\to\widehat{\mathbb C}\) is surjective and has finite fibers.
Proof. \(|f|\) attains a maximum on compact \(X\), so the maximum principle makes a complex-valued holomorphic function constant. A nonconstant map to the sphere is open, while its compact image is closed; connectedness of the sphere makes the image all of it. Fibers are discrete by local factorization and compact, hence finite.
Divisors and degree
For nonzero meromorphic \(f\), its divisor is \[ \operatorname{div}(f) =\sum_{p\in X}\operatorname{ord}_p(f)\,[p]. \] Only finitely many terms are nonzero on compact \(X\).
Theorem. \[ \deg\operatorname{div}(f) =\sum_{p\in X}\operatorname{ord}_p(f)=0. \] Thus a meromorphic function has equally many zeros and poles, counted with multiplicity.
Proof. The logarithmic differential \(\mathrm{d}f/f\) is meromorphic with residue \(\operatorname{ord}_p(f)\) at \(p\). The sum of residues of a meromorphic one-form on a compact Riemann surface is zero, proving the identity.
Global residue theorem
Theorem. For every meromorphic one-form \(\omega\) on compact \(X\), \[ \sum_{p\in X}\operatorname{Res}_p\omega=0. \]
Proof. Remove disjoint coordinate discs around the poles and triangulate the remaining compact surface. Integrals over internal edges cancel pairwise. Cauchy's theorem makes the total boundary integral zero; each negatively oriented removed boundary tends to \(-2\pi i\operatorname{Res}_p\omega\).
Degree and Riemann--Hurwitz
A nonconstant meromorphic map has a degree \(d\), the number of preimages of a regular value counted with multiplicity. Its ramification orders satisfy \[ 2g_X-2=d(2g_Y-2)+\sum_{p\in X}(e_p-1) \] for a holomorphic map \(X\to Y\). For \(Y=\widehat{\mathbb C}\), branching is therefore constrained by the genus of \(X\). This is the bridge from elementary complex analysis to the topology of compact Riemann surfaces.