Riemann Surfaces 1.3: Riemann-Roch Theorem and Genus
A famous theorem and some corollaries
1 The Riemann-Roch Theorem
The Riemann-Roch Theorem stands as a monumental result in complex geometry, algebraic geometry, and number theory. It provides a precise link between the analytic properties of a compact Riemann surface and the dimension of the spaces of meromorphic functions with prescribed zeros and poles.
Let \(X\) be a compact Riemann surface. We define the divisor group, denoted by \(\text{Div}(X)\), as the free abelian group generated by the points of \(X\). A divisor \(D \in \text{Div}(X)\) is a formal sum \(\sum_{P \in X} n_P P\), where \(n_P \in \mathbb{Z}\) and only finitely many coefficients are non-zero. A divisor is said to be effective if \(n_P \ge 0\) for all \(P\).
For a meromorphic function \(f\), we associate a divisor \(\text{div}(f) = \sum \text{ord}_P(f) P\), where \(\text{ord}_P(f)\) denotes the order of the zero or pole at \(P\). Divisors of this form are called principal divisors. Two divisors \(D_1, D_2\) are linearly equivalent, denoted \(D_1 \sim D_2\), if their difference is a principal divisor.
The degree of a divisor, defined by the homomorphism \(\text{deg}: \text{Div}(X) \to \mathbb{Z}\) where \(\text{deg}(\sum n_P P) = \sum n_P\), is a well-defined topological invariant of the linear equivalence class. This is ensured by the fact that any principal divisor on a compact surface has degree zero, a consequence of the equality between the number of zeros and poles.
The construction extends naturally to meromorphic differential forms. For a form \(\omega\) represented locally by \(f_i(z_i) dz_i\), we define \(\text{ord}_P(\omega) = \text{ord}_P(f_i)\). The resulting divisor \(\text{div}(\omega) = \sum \text{ord}_P(\omega) P\) is independent of the coordinate system1 because the transition maps in a complex structure are required to be holomorphic and non-vanishing.
Since any two non-zero meromorphic differential forms differ only by a meromorphic function, all such divisors belong to a unique linear equivalence class \(k\), known as the canonical class. Any divisor \(K \in k\) is referred to as a canonical divisor.
1 Note: The independence of the coordinate choice is crucial; in the transformation \(\alpha_j = f_i(h(z_j)) \frac{dh}{dz_j} dz_j\), the term \(\frac{dh}{dz_j}\) is a non-vanishing holomorphic function, thus its order at any point \(P\) is zero.
As we have seen, the relation \(\text{div}(f \omega) = \text{div}(f) + \text{div}(\omega)\) mirrors the group structure of the divisors. This algebraic framework allows us to translate analytic constraints into the language of linear systems, which ultimately leads to the statement of the Riemann-Roch formula.
For a given divisor \(D\), we define the following set: \[L(D) = \{f \in \mathcal{M}(X)^\times \mid \text{div}(f) + D \ge 0\} \cup \{0\}\] This set constitutes a vector space over \(\mathbb{C}\). Furthermore, if \(D_0 = D + \text{div}(g)\) is a divisor linearly equivalent to \(D\), then the assignment \(f \mapsto fg^{-1}\) defines a linear isomorphism between \(L(D)\) and \(L(D_0)\). Consequently, the dimension \(\ell(D) = \dim_{\mathbb{C}} L(D)\) depends solely on the linear equivalence class of \(D\).
Let \(X\) be a compact Riemann surface. There exists a non-negative integer \(g\), called the genus of \(X\), such that for every divisor \(D \in \text{Div}(X)\), the following equality holds: \[\ell(D) = \deg(D) + 1 - g + \ell(K - D)\]
The proof of the Riemann-Roch theorem in any of its formulations is non-trivial and fundamentally deep. For a comprehensive treatment, one may refer to classical texts such as Fulton (1969, Chapter 8) or Neukirch (1999, Chapter 3).
The degree of a canonical divisor \(K\) is given by \(\deg(K) = 2g - 2\), and the dimension of its associated space is \(\ell(K) = g\).
By setting \(D = 0\) in the Riemann-Roch formula, we observe that the only meromorphic functions satisfying \(\text{div}(f) \ge 0\) on a compact surface are the constant functions, whence \(\ell(0) = 1\). The formula then yields \(1 = 0 + 1 - g + \ell(K)\), which simplifies to \(\ell(K) = g\).
Next, substituting \(D = K\) into the formula, we obtain \(\ell(K) = \deg(K) + 1 - g + \ell(0)\). Substituting the previously determined values \(\ell(K) = g\) and \(\ell(0) = 1\), we arrive at \(g = \deg(K) + 1 - g + 1\), which solves to \(\deg(K) = 2g - 2\).
Furthermore, if we fix a non-zero meromorphic differential \(\omega\) such that \(K = \text{div}(\omega)\), then the map \(f \mapsto f\omega\) provides an isomorphism between \(L(K)\) and the space of holomorphic differentials on \(X\). Thus, the space of holomorphic differentials has dimension \(g\).
If \(\deg(D) > 2g - 2\), then for any \(f \in \mathcal{M}(X)^\times\), we have \(\deg(\text{div}(f) + K - D) = 2g - 2 - \deg(D) < 0\). Since no effective divisor can have a negative degree, it follows that \(L(K - D) = \{0\}\).
If \(\deg(D) > 2g - 2\), then the Riemann-Roch formula simplifies to: \[\ell(D) = \deg(D) + 1 - g\]2
2 Note: The term \(\ell(K-D)\) in the Riemann-Roch theorem is often referred to as the “index of speciality,” and it is generally difficult to estimate without further geometric information.
On the Riemann sphere \(S^2\), the genus is \(g = 0\). Consider the divisor \(D = m P_\infty\) for \(m \ge 0\), where \(P_\infty\) is the point at infinity. The degree is \(\deg(D) = m\). According to the simplified formula (since \(m > 2(0) - 2 = -2\)), we have: \[\ell(m P_\infty) = m + 1 - 0 = m + 1\] This is consistent with our knowledge of rational functions: the space \(L(m P_\infty)\) consists of all polynomials in \(z\) of degree at most \(m\), which indeed forms a vector space of dimension \(m + 1\) with basis \(\{1, z, z^2, \dots, z^m\}\). This confirms the Riemann-Roch theorem for \(S^2\) via the isomorphism of linear equivalence classes.
2 Genus and the Riemann-Hurwitz Formula
For a compact Riemann surface \(X\), we may consider its structure as a topological space and compute its homology groups. A fundamental fact in algebraic topology is that for homology groups with rational coefficients \(\mathbb{Q}\), the dimensions of \(H_0(X, \mathbb{Q})\) and \(H_2(X, \mathbb{Q})\) are both 1, while the dimension of \(H_1(X, \mathbb{Q})\) is \(2g\)3.
3 Note: The \(g\) appearing here is indeed identical to the \(g\) in the Riemann-Roch theorem. This equivalence is a deep result relating the analytic and topological properties of the surface.
Consequently, the value \(g\) depends solely on the topological properties of \(X\) rather than its specific complex structure. The Euler-Poincaré characteristic is defined as: \[\chi(X) := \dim H_0 - \dim H_1 + \dim H_2 = 2 - 2g\] Since \(X\) is orientable, it admits a triangulation, which naturally yields the relation: \[2 - 2g = V - E + F\] where \(V, E, \text{ and } F\) denote the number of vertices, edges, and faces, respectively.
- The Sphere (\(g=0\)): As studied in simplicial homology, the sphere can be triangulated using a tetrahedron (\(V=4, E=6, F=4\)), an octahedron (\(V=6, E=12, F=8\)), or an icosahedron (\(V=12, E=30, F=20\)). In every case, the alternating sum yields \(\chi(X) = 2\), confirming \(g=0\).
- Local Branching: Consider the map \(f: \mathbb{D} \to \mathbb{D}\) defined by \(z \mapsto z^e\) for \(e \ge 1\), where \(\mathbb{D}\) is the unit disk. This is an \(e\)-to-\(1\) mapping away from the origin. The origin is a ramification point with branching index \(e\). Under the transformation \(w = z^e\), the pullback of the differential form \(dw\) is given by \(d(z^e) = ez^{e-1}dz\). Thus, the pullback \(f^*(dw)\) possesses a zero of order \(e-1\) at the origin.
Let \(f: Y \to X\) be a holomorphic map of degree \(m\) between compact Riemann surfaces. For each point \(Q \in Y\), let \(e_Q\) be the ramification index (multiplicity) of \(f\) at \(Q\). Then the genera of \(X\) and \(Y\) are related by: \[2g(Y) - 2 = m(2g(X) - 2) + \sum_{Q \in Y} (e_Q - 1)\]
To establish this result, let \(\alpha\) be a non-zero meromorphic differential form on \(X\). We may assume \(\alpha\) is chosen such that none of its zeros or poles coincide with the branch points in \(X\). The pullback \(f^*\alpha\) is a meromorphic differential form on \(Y\). For any point \(P \in X\) that is not a branch point, the \(m\) points in the fiber \(f^{-1}(P)\) contribute to the degree of the divisor of \(f^*\alpha\) exactly as they did for \(\alpha\), multiplied by the degree \(m\). However, at each ramification point \(Q \in Y\), the local behavior \(w = z^{e_Q}\) introduces an additional zero of order \(e_Q - 1\). Therefore, we have the following relation between the degrees of the canonical divisors: \[\deg(\text{div}(f^*\alpha)) = m \cdot \deg(\text{div}(\alpha)) + \sum_{Q \in Y} (e_Q - 1)\] By substituting the previously proven result that the degree of a canonical divisor is \(2g - 2\), we arrive at the Riemann-Hurwitz formula. 4
4 Note: An alternative proof can be constructed using topological methods by lifting a triangulation from \(X\) to \(Y\) and accounting for the defect in the vertex count caused by ramification.