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The space of holomorphic differentials and the degree of the canonical divisor

Statement

Assume the full Axiom of Choice (The Axiom of Choice). Let $X$ be a compact connected Riemann surface of topological genus $g$ (Genus and Euler characteristic of a compact Riemann surface) and let $Ω (X)$ be its complex vector space of holomorphic differentials (Meromorphic differentials, orders and residues). The Riemann–Roch theorem supplies a nonzero meromorphic differential; for any such $η$ , put $K : = (η)$ and define $ℓ (K) : = \dim_{C} H^{0} (X, O_{X} (K))$ . Then:

$\dim_{C} Ω (X) = ℓ (K) = g$ ; hence $Ω (X) = 0$ for $g = 0$ and $Ω (X) \neq 0$ for $g \geq 1$ (The Riemann-Roch theorem on a compact Riemann surface, The holomorphic line bundle associated to a divisor).
Every nonzero $ω \in Ω (X)$ has effective divisor $(ω)$ of degree $2 g - 2$ , so it has exactly $2 g - 2$ zeros counted with multiplicity. In particular, for $g = 1$ a nonzero holomorphic differential has no zeros, and for $g = 0$ there is no nonzero holomorphic differential (The Riemann-Roch theorem on a compact Riemann surface, Divisors, principal divisors and canonical divisors on a Riemann surface).
If $ω \neq 0$ , its zeros are isolated and its zero set is finite (Zeros of a nonzero holomorphic function are isolated, Meromorphic differentials, orders and residues).
Every holomorphic differential is closed: $d ω = 0$ (The d, partial and dbar identities, Bigraded complex forms and the Dolbeault operators, The Wirtinger derivatives $\partial_{z} f$ and $\partial_{\overset{ˉ}{z}} f$ , and antiholomorphic functions, Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with $\partial_{\overset{ˉ}{z}} f = 0$ , or with the Cauchy–Riemann equations).

Full AC enters through the genus and Riemann–Roch interfaces used for the dimension and canonical-degree claims; the local zero and closedness arguments use no choice.

Facts & Assumptions

Given: Full AC, a compact connected Riemann surface $X$ of topological genus $g$ , and its holomorphic differentials.

[F1]

For every divisor $D$ , Riemann–Roch gives $ℓ (D) - h^{0} (X, K_{X} \otimes O_{X} (D)^{*}) = \deg D + 1 - g$ , with the relevant spaces finite-dimensional (The Riemann-Roch theorem on a compact Riemann surface).

[F2]

For a canonical divisor $K_{η} = (η)$ , $O_{X} (K_{η}) ≅ K_{X}$ (The holomorphic line bundle associated to a divisor).

[F3]

The holomorphic sections of $K_{X}$ are exactly the holomorphic differentials (The holomorphic line bundle associated to a divisor).

[F4]

The divisor-bundle construction gives $O_{X} (- D) ≅ O_{X} (D)^{*}$ and $O_{X} (0) ≅ X \times C$ (The holomorphic line bundle associated to a divisor).

[F5]

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

[F6]

Principal divisors have degree zero, so degree is constant on linear-equivalence classes (Divisors, principal divisors and canonical divisors on a Riemann surface).

[F7]

A nonzero meromorphic differential has no local coefficient that vanishes identically near a point; its local order defines its divisor (Meromorphic differentials, orders and residues).

[F8]

Every divisor on compact $X$ has finite support (Divisors, principal divisors and canonical divisors on a Riemann surface).

[F9]

A holomorphic function on a complex domain that is not identically zero has only isolated zeros (Zeros of a nonzero holomorphic function are isolated).

[F10]

The exterior derivative decomposes as $d = \partial + \overset{ˉ}{\partial}$ on complex forms (The d, partial and dbar identities).

[F11]

On a complex curve, a local $(1, 0)$ -form is $h d z$ and $d z \land d z = 0$ (Bigraded complex forms and the Dolbeault operators).

[F12]

For a smooth coefficient, $\partial h = (\partial_{z} h) d z$ (The Wirtinger derivatives $\partial_{z} f$ and $\partial_{\overset{ˉ}{z}} f$ , and antiholomorphic functions).

[F13]

A holomorphic function is real-totally differentiable and satisfies $\partial_{\overset{ˉ}{z}} h = 0$ (Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with $\partial_{\overset{ˉ}{z}} f = 0$ , or with the Cauchy–Riemann equations).

[F14]

The genus $g$ is the nonnegative integer determined by the topological type of $X$ (Genus and Euler characteristic of a compact Riemann surface).

[F15]

Riemann–Roch on compact $X$ supplies a nonzero meromorphic differential $η$ (The Riemann-Roch theorem on a compact Riemann surface).

[F16]

Riemann–Roch states $ℓ (0) = 1$ (The Riemann-Roch theorem on a compact Riemann surface).

[F17]

Full AC is assumed by the genus and Riemann–Roch interfaces used for the dimension and canonical-degree claims (The Axiom of Choice).

Proof

technique · Riemann–Roch at the zero and canonical divisors, followed by local differential calculations

1.1F2F3F4F5F6F15F16F17given

By [F15] choose a nonzero meromorphic differential $η$ and put $K : = (η)$ . By [F2], $O_{X} (K) ≅ K_{X}$ , so $ℓ (K) = \dim_{C} Ω (X)$ ; also $O_{X} (0)$ is trivial and $ℓ (0) = 1$ . The degree of $K$ is independent of this choice by [F5, F6].

1.2F7F8F9given

At any point choose a holomorphic coordinate and write $ω = h d z$ . For nonzero $ω$ , [F7] ensures $h$ is not identically zero near that point; [F9] makes its zeros isolated. The zero set is contained in the finite support of $(ω)$ by [F8], so it is finite.

2.1F1F2F3F4F14step 1.1algebra

Apply [F1] with $D = 0$ . Since $K_{X} \otimes O_{X} (0)^{*} ≅ K_{X}$ , this gives $ℓ (0) - \dim Ω (X) = 1 - g$ . Using step 1.1 yields $\dim Ω (X) = g$ . Therefore $Ω (X) = 0$ for $g = 0$ and is nonzero for $g \geq 1$ .

3.1F1F2F4F6step 1.1step 2.1algebra

Apply [F1] with $D = K$ . By [F2], $K_{X} \otimes O_{X} (K)^{*} ≅ X \times C$ , so the right-hand cohomology term has dimension $ℓ (0) = 1$ . Step 2.1 gives $ℓ (K) = g$ , and hence $g - 1 = \deg K + 1 - g$ . Thus $\deg K = 2 g - 2$ .

4.1F5F6F7F8step 2.1step 3.1given

If $ω \neq 0$ is holomorphic, it has no poles, so $(ω)$ is effective. It is a canonical divisor, hence linearly equivalent to $K$ by [F5]; therefore $\deg (ω) = \deg K$ by [F6] and equals $2 g - 2$ by step 3.1. This degree is the number of zeros counted with multiplicity. For $g = 1$ the effective divisor has degree zero and is empty; for $g = 0$ no such $ω$ exists by step 2.1.

5.1F10F11F12F13given∎

In a holomorphic chart write $ω = h d z$ . By [F13], $\overset{ˉ}{\partial} h = 0$ . Using [F10], [F11] and [F12], $d ω = d h \land d z = (\partial h + \overset{ˉ}{\partial} h) \land d z = (\partial_{z} h) d z \land d z = 0$ . This holds in every chart, so $d ω = 0$ globally.

Depends on

Used by

The Jacobian of a compact Riemann surface Definition
The period pairing and the period subgroup Definition
Period matrix and Jacobian of the pentagon curve Example
Periods of a complex torus Example
Principal divisor tests via the Abel-Jacobi map Example
Holomorphic differentials separate generic points Lemma
The period pairing is well defined and computed by integration Lemma
Jacobi inversion Theorem
The Riemann bilinear relations and the period lattice Theorem

Dependency tree · two levels

71 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)
Curtis T. McMullen, Riemann Surfaces, Harvard Math 213b course notes (2026) (standard reference, not scraped)
Eduard Looijenga, Riemann Surfaces (2007 author lecture notes) (standard reference, not scraped)