The Riemann-Roch Theorem

The Riemann-Roch theorem is a fundamental result relating the topology of a compact Riemann surface (its genus) to the algebra of meromorphic functions and differential forms.

Setup and Statement

Definition10.9Linear system of a divisor

For a divisor $D = \sum n_p \cdot p$ on a compact Riemann surface $S$ , the linear system $L(D)$ is the vector space

$L(D) = \{f \in \mathcal{M}(S)^* : \text{div}(f) + D \geq 0\} \cup \{0\}$

where $\text{div}(f) + D \geq 0$ means $\text{ord}_p(f) + n_p \geq 0$ for all $p$ . Informally, $L(D)$ consists of meromorphic functions whose poles are "no worse than" $D$ . We write $\ell(D) = \dim_\mathbb{C} L(D)$ .

Theorem10.8Riemann-Roch theorem

Let $S$ be a compact Riemann surface of genus $g$ , $K$ a canonical divisor (the divisor of any non-zero meromorphic $1$ -form), and $D$ any divisor. Then

$\ell(D) - \ell(K - D) = \deg D - g + 1.$

Special Cases

ExampleImportant special cases

Case $D = 0$ : $\ell(0) - \ell(K) = 0 - g + 1$ , so $\ell(K) = g$ (since $\ell(0) = 1$ , only constants). This confirms that the space of holomorphic $1$ -forms has dimension equal to the genus.

Case $\deg D > 2g - 2$ : Since $\deg(K-D) < 0$ , we have $\ell(K-D) = 0$ , so $\ell(D) = \deg D - g + 1$ . For large degree, the dimension grows linearly.

Case $D = K$ : $\ell(K) - \ell(0) = \deg K - g + 1$ , giving $g - 1 = \deg K - g + 1$ , so $\deg K = 2g - 2$ . The canonical divisor always has degree $2g - 2$ .

Case $g = 0$ : $\ell(D) = \deg D + 1$ for $\deg D \geq 0$ (and $0$ for $\deg D < 0$ ). On $\hat{\mathbb{C}}$ , $L(n \cdot \infty)$ is the space of polynomials of degree $\leq n$ , which has dimension $n + 1$ .

Applications

Theorem10.9Embedding theorem

Every compact Riemann surface of genus $g$ can be holomorphically embedded in $\mathbb{CP}^{3g-3}$ (for $g \geq 2$ ). For $g = 1$ , the embedding is in $\mathbb{CP}^2$ as a cubic curve. For $g = 0$ , the surface is $\mathbb{CP}^1$ itself.

ExampleWeierstrass gap theorem

At a general point $p$ on a surface of genus $g$ , the "gap sequence" — the values $n$ for which there is no meromorphic function with a pole of exact order $n$ at $p$ and no other poles — consists of exactly $g$ values, all in $\{1, 2, \ldots, 2g-1\}$ . At a general point, these are $1, 2, \ldots, g$ . The exceptional points where the gap sequence differs are called Weierstrass points, and a surface of genus $g \geq 2$ has finitely many of them.

The Serre Duality Perspective

RemarkSerre duality

The term $\ell(K - D)$ in the Riemann-Roch formula has a cohomological interpretation. By Serre duality, $\ell(K - D) = \dim H^1(S, \mathcal{O}(D))$ , so the Riemann-Roch theorem becomes

$\chi(\mathcal{O}(D)) = \ell(D) - h^1(\mathcal{O}(D)) = \deg D - g + 1.$

This is the Euler characteristic of the line bundle $\mathcal{O}(D)$ . In this form, the Riemann-Roch theorem generalizes to higher dimensions as the Hirzebruch-Riemann-Roch theorem and ultimately the Atiyah-Singer index theorem.

RemarkGAGA principle

The Riemann-Roch theorem illustrates the GAGA principle (Serre, 1956): on compact complex manifolds, analytic and algebraic geometry coincide. Every meromorphic function on a compact Riemann surface is algebraic, and the Riemann-Roch theorem is simultaneously a theorem in complex analysis and algebraic geometry.