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Nondegeneracy of the residue pairing

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $X$ be a compact Riemann surface, $D$ a divisor, and $E = O_{X} (D)$ . Put $F : = K \otimes E^{*}$ , where $K = Λ^{1, 0} T^{*} X$ is the canonical bundle (The residue pairing for line-bundle cohomology); Choose a compatible Riemannian metric $g$ on $X$ and a Hermitian metric $h$ on $E$ , which exist under full AC (Hermitian metric and $L^{2}$ pairing on a compact Riemann surface), and equip $F$ with the tensor metric induced by $g$ and the dual of $h$ on $E^{*}$ . Equivalently, the customary $K - D$ twist means $K \otimes O_{X} (- D) ≅ K \otimes E^{*}$ (The holomorphic line bundle associated to a divisor). Let $B_{D} : H^{1} (X, O_{X} (D)) \times H^{0} (X, F) ⟶ C$ be the residue pairing of The residue pairing for line-bundle cohomology. Then $B_{D}$ is perfect:

For every nonzero $ξ \in H^{1} (X, O_{X} (D))$ , there is $ω \in H^{0} (X, F)$ such that $B_{D} (ξ, ω) \neq 0$ .
For every nonzero $ω \in H^{0} (X, F)$ , there is $ξ \in H^{1} (X, O_{X} (D))$ such that $B_{D} (ξ, ω) \neq 0$ .

Consequently the induced maps $H^{1} (X, O_{X} (D)) ⟶ H^{0} (X, F)^{*}, H^{0} (X, F) ⟶ H^{1} (X, O_{X} (D))^{*}$ are complex-linear isomorphisms between finite-dimensional vector spaces (Finite-dimensionality of the cohomology of a divisor on a compact Riemann surface, Dolbeault cohomology of a compact riemann surface is finite dimensional).

Facts & Assumptions

Given: Full AC, a compact Riemann surface $X$ , a divisor $D$ , $E = O_{X} (D)$ , compatible supplied metrics on $X$ and $E$ , the induced metrics on $E^{*}$ and $F = K \otimes E^{*}$ , and the intrinsic residue pairing of the preceding item.

[F1]

The intrinsic residue pairing is well defined and complex-bilinear, with $B_{D} (ξ, ω) = (2 π i)^{- 1} \int_{X} θ \land ω$ for a smooth Dolbeault representative $θ$ of $ξ$ (The residue pairing for line-bundle cohomology).

[F2]

The canonical global comparison identifies $H^{1} (X, O_{X} (D))$ with the smooth Dolbeault quotient $H^{0, 1} (X, E)$ , naturally in the bundle and without a finite-cover hypothesis (Cech--Dolbeault comparison for holomorphic line bundles on a compact Riemann surface).

[F3]

The space $H^{1} (X, O_{X} (D))$ is finite-dimensional (Finite-dimensionality of the cohomology of a divisor on a compact Riemann surface).

[F4]

The divisor construction gives $E^{*} ≅ O_{X} (- D)$ , so $F = K \otimes E^{*}$ is the canonical twist denoted by $K - D$ (The holomorphic line bundle associated to a divisor).

[F5]

The supplied metrics define the conjugate-linear bundle map $# = ⋆_{E}$ and the positive identity $u \land # u = ∣ u ∣^{2} d V_{g}$ (Hermitian metric and $L^{2}$ pairing on a compact Riemann surface).

[F6]

The maximal Dolbeault operator defines the Hilbert harmonic space $H^{0, 1} (X, E) = \ker {\overset{ˉ}{D}}^{*}$ and the Dolbeault Laplacian, with their stated domains (The maximal Dolbeault operator and its Hilbert adjoint on a compact Riemann surface).

[F7]

For the supplied metrics, $#$ is a conjugate-linear isomorphism from $H^{0, 1} (X, E)$ onto $H^{0} (X, F)$ for $F = K \otimes E^{*}$ , and $u \land # u = ∣ u ∣^{2} d V_{g}$ (Harmonic star duality for line bundle valued dolbeault cohomology).

[F8]

The Dolbeault group $H^{0, 1} (X, E)$ is finite-dimensional and every class has a unique smooth harmonic representative. Together with [F7], this also makes $H^{0} (X, F)$ finite-dimensional (Dolbeault cohomology of a compact riemann surface is finite dimensional).

[F9]

The scalar Hodge star on an oriented Riemannian manifold is characterized by $α \land * β = ⟨ α, β ⟩_{g} {vol}_{g}$ (Riemannian hodge star).

[F10]

Full AC is assumed by the sheaf-cohomology, finiteness, and Hodge inputs. No further choice is made in the pairing or the kernel arguments (The Axiom of Choice).

Proof

Transfer the pairing to the smooth Dolbeault model. The Hodge-star map identifies harmonic representatives with the dual holomorphic space; its positive norm identity proves both nondegeneracy directions.

1.1F2F3F4F6F7F8F10given

Put $V : = H^{0, 1} (X, E)$ and $W : = H^{0} (X, F)$ . By [F10], the cohomology and Hodge inputs below inherit the stated full Axiom of Choice. By [F2], the comparison isomorphism identifies $H^{1} (X, O_{X} (D))$ with $V$ . The finite-dimensionality of $H^{1} (X, O_{X} (D))$ follows from [F3], while [F8] gives a unique harmonic representative in $H^{0, 1} (X, E)$ for each class; [F7] carries that finite-dimensional harmonic space onto $W$ .

2.1F1F5F6F7F8F9step 1.1algebra

Let $0 \neq ξ \in H^{1} (X, O_{X} (D))$ , and let $0 \neq u \in H^{0, 1} (X, E)$ be its harmonic representative under [F8]. By [F7], $ω : = # u$ lies in $W$ . The Hodge identity [F5, F7, F9] gives $\int_{X} u \land ω = \int_{X} ∣ u ∣^{2} d V_{g} = ∥ u ∥_{L^{2}}^{2} > 0$ . Using $u$ as the Dolbeault representative in [F1], $B_{D} (ξ, ω) = (2 π i)^{- 1} ∥ u ∥_{L^{2}}^{2} \neq 0$ . Thus the induced map $H^{1} (X, O_{X} (D)) \to W^{*}$ is injective.

2.2F1F2F5F6F7F8F9step 1.1algebra

Let $0 \neq ω \in W$ . By the isomorphism in [F7], there is a unique harmonic $0 \neq u \in H^{0, 1} (X, E)$ with $# u = ω$ . Let $ξ$ be its class under the inverse comparison [F2]. Then [F1] and the same positive norm identity give $B_{D} (ξ, ω) = (2 π i)^{- 1} ∥ u ∥_{L^{2}}^{2} \neq 0$ . Thus the induced map $W \to H^{1} (X, O_{X} (D))^{*}$ is injective.

3.1F1F3F6F7F8step 2.1step 2.2algebra∎

The conjugate-linear isomorphism in [F7] and the harmonic-representative identification in [F8] give $\dim_{C} V = \dim_{C} W$ . Both are finite-dimensional by [F3, F7, F8]. The two induced maps are complex-linear because $B_{D}$ is bilinear by [F1]; steps 2.1 and 2.2 show each is injective. An injective linear map between finite-dimensional spaces of equal dimension is surjective, so both maps are isomorphisms and $B_{D}$ is perfect.

Depends on

Used by

Serre duality on a compact Riemann surface Theorem

Dependency tree · two levels

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Sources

Karl Otto Forster, Lectures on Riemann Surfaces (GTM 81, Springer 1981), translated by Bruce Gilligan (standard reference, not scraped)
Jean-Pierre Demailly, Complex Analytic and Differential Geometry (author manuscript, Universite Grenoble Alpes) (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)