Riemann Surfaces 1.2: Analysis on Compact Riemann Surfaces
The Differential Forms and Analysis.
1 Differential Forms
In the context of standard calculus, differential forms are not a novel concept; indeed, they serve as the primary tool for the rigorous formulation of Stokes’ Theorem. Within the theory of differential manifolds, they further allow for the construction of De Rham cohomology, proving to be an indispensable methodology.
A differential form on an open subset \(U \subseteq \mathbb{C}\) is an expression of the form \(f(z)dz\), where \(f\) is a meromorphic function on \(U\). For any meromorphic function \(F\) on \(U\), we associate a corresponding differential form defined by \(dF := \frac{dF}{dz} dz\).
Suppose \(w: U \to V\) is a holomorphic mapping between two open subsets of \(\mathbb{C}\). If \(\alpha = f(z)dz\) is a differential form on \(V\), its pullback under \(w\), denoted by \(w^*(\alpha)\), is defined as: \[w^*(\alpha) = f(w(z)) \frac{dw(z)}{dz} dz\]
Utilizing these local definitions on \(\mathbb{C}\), we can extend the notion of differential forms to a general Riemann surface \(X\).
A differential form \(\omega\) on a Riemann surface \(X\) is defined by a collection of local differential forms \(\{\alpha_i = f_i(z_i)dz_i\}\) associated with a coordinate atlas \(\{(U_i, \phi_i)\}\), such that they are compatible on intersections \(U_i \cap U_j\). Specifically, let \(h_{ij} = \phi_i \circ \phi_j^{-1}\) be the transition map such that \(z_i = h_{ij}(z_j)\). We require the local forms to satisfy: \[\alpha_j = h_{ij}^*(\alpha_i) = f_i(h_{ij}(z_j)) \frac{dh_{ij}}{dz_j} dz_j\]
In the language of meromorphic functions, specifying a meromorphic function \(F\) on \(X\) is equivalent to providing a collection of local meromorphic functions \(\{F_i(z_i)\}\) that are compatible under the transition maps, satisfying: \[F_j(z_j) = F_i(h_{ij}(z_j))\] In contrast to differential forms, the transformation law for meromorphic functions does not involve the derivative of the transition map, reflecting their nature as scalar fields.
We further classify differential forms based on their singularity properties:
A differential form \(\omega\) is categorized as follows:
First Kind (Holomorphic): If \(\omega\) has no poles on \(X\).
Second Kind: If \(\omega\) is meromorphic and all its residues at its poles are zero.
Third Kind: If it is a meromorphic differential form that is not of the second kind (i.e., at least one residue is non-zero).
- Logarithmic Differentials: If \(f\) is a meromorphic function on \(X\), then \(\omega = \frac{df}{f}\) is a differential form of the third kind. Its poles correspond to the zeros and poles of \(f\), and the residues are always integers.
- On the Riemann Sphere: On \(\mathbb{P}^1\), consider the form \(\omega = dz\) in the standard affine coordinate \(z\). To examine its behavior at infinity, we transition to the coordinate \(w = 1/z\). The pullback yields \(d(1/w) = -w^{-2}dw\), revealing a pole of order 2 at \(w=0\). Thus, \(\omega\) is not of the first kind.
- On Complex Tori: For \(X = \mathbb{C}/\Lambda\), the differential form \(dz\) is invariant under the translation action of the lattice \(\Lambda\). It descends to a well-defined, non-vanishing holomorphic differential (First Kind) on the torus.
2 Analysis on Compact Riemann Surfaces
Our objective is to generalize the analytic properties of the extended complex plane to compact Riemann surfaces. While these generalizations are often intuitive, they involve subtle technical details.
For any meromorphic differential form \(\omega\) on a compact Riemann surface \(X\), the sum of the residues at all its poles is zero: \[\sum_{p \in X} \text{Res}_p(\omega) = 0\]1
1 In modern algebraic geometry, the Residue Theorem is the manifestation of Serre Duality. The sum of residues defines a canonical isomorphism \(H^1(X, \Omega^1_X) \xrightarrow{\sim} \mathbb{C}\), often referred to as the trace map. The fact that the sum is zero for a global meromorphic form reflects the fact that principal divisors represent the zero class in cohomology.
To provide a sketch of the proof, recall that if \(\alpha = f(z)dz\) is a differential form on an open subset of \(\mathbb{C}\) and \(L\) is a closed path not passing through any poles, then by the classical Residue Theorem: \[\int_L \alpha = 2\pi i \sum \text{Res}(\alpha)\] On a compact Riemann surface \(X\), we fix a finite coordinate atlas and choose a triangulation such that each triangle is entirely contained within some coordinate neighborhood \(U_i\). The sum \(2\pi i \sum_{p \in X} \text{Res}_p(\omega)\) then corresponds to the sum of the integrals of \(\omega\) along the boundaries of all triangles in the triangulation. Since each edge of the triangulation is traversed exactly twice in opposite directions, the line integrals along the interior edges cancel out perfectly, leaving a total sum of zero2.
2 Note: This proof is a conceptual sketch. A fully rigorous treatment requires a formal discussion of the orientation of the triangulation and the consistency of the differential form across coordinate transitions, which is analogous to the proof of Stokes’ Theorem on manifolds.
Any non-constant meromorphic function \(f\) on a compact Riemann surface \(X\) has an equal number of zeros and poles (counted with multiplicity).
This follows immediately by considering the logarithmic differential \(\omega = \frac{df}{f}\). Applying the previous proposition to \(\omega\), we note that the residues of \(\frac{df}{f}\) at the zeros and poles of \(f\) are precisely their respective multiplicities (with poles having negative residues). Thus, the sum of residues being zero implies that the number of zeros equals the number of poles.
A meromorphic function \(f\) on a compact Riemann surface takes every value in the extended complex plane \(\mathbb{C} \cup \{\infty\}\) the same number of times.
Let \(f\) be a meromorphic function and \(c \in \mathbb{C}\) be any complex value. By applying the preceding results to the function \(f - c\), we observe that the number of points where \(f(p) = c\) (the zeros of \(f-c\)) must equal the number of poles of \(f-c\). Since the poles of \(f-c\) are identical to the poles of \(f\), the number of times \(f\) attains any value \(c\) is constant and equal to the number of poles of \(f\).
This constant number of occurrences is referred to as the valence of \(f\). By convention, the valence of a constant function is defined to be zero. Consequently, a meromorphic function of valence \(n\) naturally induces a map \(f: X \to S^2\) which can be viewed as an \(n\)-sheeted branched covering of the Riemann sphere.
A meromorphic function on the Riemann sphere \(S^2\) is a rational function. In the language of sheaf theory, the function field \(\mathcal{M}(S^2)\) is isomorphic to the field of rational functions \(\mathbb{C}(z)\).
By applying a translation if necessary, we may assume without loss of generality that a given meromorphic function \(g(z)\) has no poles or zeros at the point at infinity (the North Pole \(N\)). Let \(g(z)\) possess poles of order \(m_i\) at points \(p_i\) and zeros of order \(n_j\) at points \(q_j\). We consider the auxiliary function \[\Phi(z) = g(z) \frac{\prod (z - p_i)^{m_i}}{\prod (z - q_j)^{n_j}}\] By construction, \(\Phi(z)\) is a meromorphic function on \(S^2\) with neither zeros nor poles. On a compact Riemann surface, such a function must be constant. Consequently, \(g(z)\) can be expressed as a quotient of polynomials, proving it is a rational function3.
3 Note: Rational functions on \(S^2\) are precisely the quotients of polynomials, and \(\mathcal{M}(S^2)\) coincides with the function field of the projective line \(\mathbb{P}^1\) in algebraic geometry.
If \(f\) is a meromorphic function of valence \(n\) on a compact Riemann surface \(X\), then any other meromorphic function \(g \in \mathcal{M}(X)\) is a root of a polynomial with coefficients in \(\mathbb{C}(f)\). Furthermore, the degree of this polynomial is at most \(n\).
We view \(f\) as a branched covering \(X \to S^2\). For a generic value \(c \in S^2\), the fiber \(f^{-1}(c)\) consists of \(n\) points \(\{z_1, z_2, \dots, z_n\}\). We define a polynomial in \(T\) whose roots are the values of \(g\) on the fiber: \[\prod_{i=1}^n (T - g(z_i)) = T^n + \sigma_1(z) T^{n-1} + \dots + \sigma_n(z) = 0\] where \(\sigma_i(z)\) are the elementary symmetric polynomials in \(g(z_1), \dots, g(z_n)\). Although the points \(z_i\) are permuted as \(z\) moves, the coefficients \(\sigma_i\) are well-defined meromorphic functions on \(S^2\), and thus by our previous proposition, \(\sigma_i \in \mathbb{C}(f)\). Therefore, \(g\) satisfies an algebraic equation over \(\mathbb{C}(f)\) of degree \(n\).
On any compact Riemann surface \(X\), there exist non-constant meromorphic functions. Furthermore, these functions form a finitely generated field \(\mathcal{M}(X)\) over \(\mathbb{C}\) with transcendence degree 1.
The proof concerning the algebraic structure follows from the previous results. The set of meromorphic functions on \(X\) clearly forms a field. Let \(f\) be a meromorphic function of valence \(n\). As demonstrated, every \(g \in \mathcal{M}(X)\) is algebraic over \(\mathbb{C}(f)\) with a degree not exceeding \(n\). If the degree of the extension \([\mathcal{M}(X) : \mathbb{C}(f)]\) were greater than \(n\), there would exist a subfield \(L\) such that \(n < [L : \mathbb{C}(f)] = n_0 < \infty\). By the Primitive Element Theorem, \(L = \mathbb{C}(f, g)\) for some \(g\) whose minimal polynomial has degree \(n_0\), which contradicts the bound \(n_0 \le n\). Thus, \(\mathcal{M}(X)\) is a finite extension of \(\mathbb{C}(f)\) of degree at most \(n\)4.
4 Note: We omit the existence proof of non-constant meromorphic functions (the Riemann-Roch theorem or the use of subharmonic functions), as it is significantly more involved.
- The Riemann Sphere: The function field \(\mathcal{M}(S^2) = \mathbb{C}(f)\), where \(f\) is any meromorphic function of valence 1 (a Möbius transformation).
- Higher Dimensions: For compact complex manifolds of dimension greater than 1, the meromorphic functions still form a field, but its transcendence degree may be strictly less than the dimension of the manifold. For instance, there exist compact complex surfaces (dimension 2) that possess no non-constant meromorphic functions5.
5 Note: Such examples often appear in the study of non-Kähler manifolds, like certain Hopf surfaces.