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The Riemann-Roch theorem on a compact Riemann surface

Statement

Assume the full Axiom of Choice (The Axiom of Choice). Let $X$ be a compact Riemann surface of topological genus $g$ , let $D$ be a divisor on $X$ , and put $E_{D} : = O_{X} (D)$ and $F_{D} : = K_{X} \otimes E_{D}^{*}$ , where $K_{X} : = Λ^{1, 0} T^{*} X$ is the canonical holomorphic line bundle (Divisors, principal divisors and canonical divisors on a Riemann surface, The holomorphic line bundle associated to a divisor, Serre duality on a compact Riemann surface). Write $h^{q} (X, G) : = \dim_{C} H^{q} (X, G)$ when finite, $ℓ (D) : = \dim H^{0} (X, E_{D})$ , $i (D) : = \dim H^{1} (X, E_{D})$ , and $χ (E_{D}) : = ℓ (D) - i (D)$ (Finite-dimensionality of the cohomology of a divisor on a compact Riemann surface). Then:

The spaces $H^{0} (X, E_{D})$ , $H^{1} (X, E_{D})$ , and $H^{0} (X, F_{D})$ are finite-dimensional, and $i (D) = h^{0} (X, F_{D})$ (Finite-dimensionality of the cohomology of a divisor on a compact Riemann surface, Serre duality on a compact Riemann surface).
The intrinsic Riemann–Roch formula is $ℓ (D) - h^{0} (X, F_{D}) = χ (E_{D}) = \deg D + 1 - g .$
There exists a nonzero meromorphic differential on $X$ . For any such differential $η$ , let $K_{η} : = (η)$ be its canonical divisor. The isomorphism $O_{X} (K_{η}) ≅ K_{X}$ identifies $H^{0} (X, F_{D})$ with $L (K_{η} - D)$ , so $i (D) = ℓ (K_{η} - D)$ and the formula becomes $ℓ (D) - ℓ (K_{η} - D) = \deg D + 1 - g .$ This expression is independent of the differential, and the identity is invariant under replacing $D$ by a linearly equivalent divisor. For $D = 0$ , it gives $ℓ (0) = 1$ , $i (0) = g$ , and $χ (O_{X}) = 1 - g$ .

Facts & Assumptions

Given: Full AC, a compact Riemann surface $X$ of topological genus $g$ , and a divisor $D$ .

[F1]

Full AC implies countable choice by restricting a choice function to any countable family of nonempty sets (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F2]

A divisor on compact $X$ has finite support and degree the sum of its finitely many coefficients. Every principal divisor has degree zero, and any two canonical divisors are linearly equivalent (Divisors, principal divisors and canonical divisors on a Riemann surface).

[F3]

For a canonical divisor $K_{η} = (η)$ , the divisor bundle satisfies $O_{X} (K_{η}) ≅ K_{X}$ , and tensoring with $O_{X} (- D) ≅ O_{X} (D)^{*}$ identifies $O_{X} (K_{η} - D)$ with $F_{D}$ (The holomorphic line bundle associated to a divisor).

[F4]

Under countable choice, a compact Riemann surface admits a compatible Riemannian metric (Hermitian metric and $L^{2}$ pairing on a compact Riemann surface).

[F5]

Under countable choice, every holomorphic line bundle admits a Hermitian metric (Hermitian metric and $L^{2}$ pairing on a compact Riemann surface).

[F6]

With compatible metrics supplied, the groups $H^{0} (X, O_{X} (D))$ and $H^{1} (X, O_{X} (D))$ are finite-dimensional (Finite-dimensionality of the cohomology of a divisor on a compact Riemann surface).

[F7]

The notation is $ℓ (D) = \dim H^{0} (X, O_{X} (D)) = \dim L (D)$ , $i (D) = \dim H^{1} (X, O_{X} (D))$ , and $χ (O_{X} (D)) = ℓ (D) - i (D)$ (Finite-dimensionality of the cohomology of a divisor on a compact Riemann surface).

[F8]

For every divisor $A$ and point $p$ , $χ (O_{X} (A + [p])) = χ (O_{X} (A)) + 1$ (The point-divisor exact sequence and the Euler-characteristic step).

[F9]

The adjacent cohomology spaces in that point-divisor sequence are finite-dimensional (The point-divisor exact sequence and the Euler-characteristic step).

[F10]

For topological genus $g$ , $ℓ (0) = 1$ , $i (0) = h^{1} (X, O_{X}) = g$ , and $χ (O_{X}) = 1 - g$ (The Euler characteristic of the structure sheaf is one minus the genus).

[F11]

For $E_{D} = O_{X} (D)$ , Serre duality identifies $H^{1} (X, E_{D})$ with the complex-linear dual of $H^{0} (X, F_{D})$ , and both spaces are finite-dimensional with equal dimensions (Serre duality on a compact Riemann surface).

[F12]

Every principal divisor has degree zero (Divisors, principal divisors and canonical divisors on a Riemann surface).

[F13]

Any two canonical divisors are linearly equivalent (Divisors, principal divisors and canonical divisors on a Riemann surface).

[F14]

If $D^{'} \sim D$ , multiplication by the inverse of a meromorphic function with divisor $D^{'} - D$ gives the line-bundle isomorphism $O_{X} (D) ≅ O_{X} (D^{'})$ (The holomorphic line bundle associated to a divisor).

[F15]

The zero-divisor bundle is canonically trivial, $O_{X} (0) ≅ X \times C$ (The holomorphic line bundle associated to a divisor).

[F16]

For every divisor $A$ , the global holomorphic sections of $O_{X} (A)$ identify with $L (A)$ (The holomorphic line bundle associated to a divisor).

[F17]

A Riemann surface is nonempty and connected and has holomorphic coordinate charts (Riemann surfaces and holomorphic atlases).

[F18]

If $X$ is compact and $\deg A < 0$ , then $L (A) = 0$ (Divisors, principal divisors and canonical divisors on a Riemann surface).

[F19]

The divisor bundle $O_{X} (A)$ has a canonical meromorphic section $s_{A} = f_{i} e_{i}$ with $(s_{A}) = A$ , where $e_{j} = (f_{i} / f_{j}) e_{i}$ ; also $O_{X} (- A) ≅ O_{X} (A)^{*}$ (The holomorphic line bundle associated to a divisor).

[F20]

Holomorphic sections have holomorphic coefficients in holomorphic frames, and meromorphic sections of $K_{X}$ are exactly meromorphic differentials. A nonzero meromorphic differential has no identically zero germ on connected $X$ (Holomorphic line bundles and meromorphic sections on a Riemann surface, Meromorphic differentials, orders and residues).

Proof

The proof computes the Euler characteristic by finite point-divisor increments, then applies Serre duality to the canonical line bundle. Applying the intrinsic formula to a sufficiently negative point divisor produces a nonzero meromorphic differential, and hence the unconditional divisor form.

1.1F1F4F5F6F7given

Full AC gives countable choice by [F1]. Choose compatible metrics on $X$ and $E_{D}$ using [F4, F5], solely to invoke the finiteness result [F6]; the formula below does not depend on these auxiliary metrics. Thus $ℓ (D)$ and $i (D)$ are finite and $χ (E_{D}) = ℓ (D) - i (D)$ by [F7].

2.1F1F2F4F5F6F8F9F10step 1.1algebra

Write $D = \sum_{p \in S} n_{p} [p]$ , where $S$ is finite by [F2]. The finite sequence from $0$ to $D$ uses only finitely many intermediate divisors. By [F1], [F4], and [F5], choose compatible metrics on $X$ and each associated line bundle; [F6] then makes every intermediate Euler characteristic finite and defined. Starting at the zero divisor, apply [F8] $n_{p}$ times to add $[p]$ when $n_{p} > 0$ . When $n_{p} < 0$ , apply [F8] to $A - [p]$ to obtain $χ (O_{X} (A - [p])) = χ (O_{X} (A)) - 1$ , and repeat $- n_{p}$ times; [F9] supplies finiteness for each adjacent pair. This finite sequence reaches $D$ and changes $χ$ by $\sum_{p \in S} n_{p} = \deg D$ . By [F10], its initial value is $χ (O_{X}) = 1 - g$ , so $χ (E_{D}) = 1 - g + \deg D$ .

3.1F7F11step 1.1step 2.1algebra

Set $F_{D} = K_{X} \otimes E_{D}^{*}$ . By [F11], $i (D) = h^{0} (X, F_{D})$ . Substituting this equality into the definition of $χ (E_{D})$ and using step 2.1 gives $ℓ (D) - h^{0} (X, F_{D}) = 1 - g + \deg D$ . Finiteness of $H^{0} (X, F_{D})$ follows from the same perfect duality and finiteness of $H^{1} (X, E_{D})$ .

4.1F2F17F18F19F20step 3.1chooseconstructalgebra

By [F17], choose $p \in X$ and put $n = g + 2 > 0$ and $A = - n [p]$ . Since $\deg A = - n < 0$ , [F18] gives $ℓ (A) = 0$ . Applying step 3.1 to $A$ and using the dual-bundle identity in [F19] gives $h^{0} (X, K_{X} \otimes O_{X} (n [p])) = n + g - 1 = 2 g + 1 > 0$ . Choose a nonzero holomorphic section $σ$ of this bundle. In local coordinates and holomorphic divisor-bundle frames, write $σ = a_{i} d z_{i} \otimes e_{i}$ and $s_{n [p]} = f_{i} e_{i}$ as in [F19], and define $η ∣_{U_{i}} = (a_{i} / f_{i}) d z_{i}$ . Each quotient is meromorphic. If $e_{j} = g_{i j} e_{i}$ and $d z_{j} = k_{i j} d z_{i}$ on an overlap, then $a_{i} = k_{i j} g_{i j} a_{j}$ and $f_{i} = g_{i j} f_{j}$ , so $a_{i} / f_{i} = k_{i j} (a_{j} / f_{j})$ ; the differentials therefore glue by [F20]. Near $p$ , the local equation $f_{i}$ has order $n$ , so $η$ has pole order at most $n$ ; away from $p$ , $f_{i}$ is a holomorphic unit, so $η$ is holomorphic. Since $σ \neq 0$ , division by $s_{n [p]}$ gives $η \neq 0$ . By [F20], its germs are not identically zero, and [F2] gives the finite-support canonical divisor $(η)$ .

5.1F3F12F13F14F16step 3.1step 4.1algebra

A nonzero meromorphic differential exists by step 4.1. For any such differential $η$ , [F3] identifies $F_{D} ≅ O_{X} (K_{η} - D)$ , and [F16] identifies its global sections with $L (K_{η} - D)$ . Together with step 3.1, this gives $i (D) = ℓ (K_{η} - D)$ and the divisor form. If $η^{'}$ is another nonzero meromorphic differential, [F13] gives $K_{η^{'}} \sim K_{η}$ , so [F3] yields the same dimension. If $D^{'} \sim D$ , then [F14] identifies $O_{X} (D^{'})$ with $O_{X} (D)$ and their dual twists with the same $K_{X}$ ; [F12] gives $\deg D^{'} = \deg D$ . Hence both sides of the formula are invariant under this replacement.

6.1F10F15step 2.1algebra∎

For $D = 0$ , [F15] identifies $E_{0}$ with $O_{X}$ . By [F10], $ℓ (0) = 1$ and $i (0) = g$ , so the formula reads $1 - g = 1 - g$ .

Depends on

Used by

Dependency tree · two levels

117 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Karl Otto Forster, Lectures on Riemann Surfaces (GTM 81, Springer 1981), translated by Bruce Gilligan (standard reference, not scraped)
Eduard Looijenga, Riemann Surfaces (2007 author lecture notes) (standard reference, not scraped)
Curtis T. McMullen, Riemann Surfaces, Harvard Math 213b course notes (2026) (standard reference, not scraped)
Anand Deopurkar, Riemann-Roch (MATH 8320/2017 algebraic curves course notes, University of California Davis) (standard reference, not scraped)