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The period pairing and the period subgroup

Definition

Assume full AC (The Axiom of Choice), used for the symplectic basis and $\dim_{C} Ω (X) = g$ interfaces below. Let $X$ be a compact connected Riemann surface of genus $g$ , with a symplectic basis $a_{1}, b_{1}, \dots, a_{g}, b_{g}$ of $H_{1} (X; Z)$ supplied by the paired side loops of a one-polygon normal form (A symplectic homology basis of a compact Riemann surface). Write $A_{1}, B_{1}, \dots, A_{g}, B_{g}$ for those fixed continuous closed side-loop representatives on $X$ . The path integral of a holomorphic differential along each such path is defined by Path integral of a holomorphic differential on a Riemann surface.

For $ω \in Ω (X)$ , the period pairing on this fixed basis and its chosen representatives is $P ⁣ (\sum_{i = 1}^{g} (m_{i} a_{i} + n_{i} b_{i}), ω) : = \sum_{i = 1}^{g} (m_{i} \int_{A_{i}} ω + n_{i} \int_{B_{i}} ω), m_{i}, n_{i} \in Z .$ The symplectic basis makes the coordinates $m_{i}, n_{i}$ unique. Thus $P$ is additive in its homology-class argument and $C$ -linear in $ω$ . This definition uses the fixed polygon representatives; representative- and basis-independence are separate claims.

Define $e : H_{1} (X; Z) \to Ω (X)^{*}$ by $e (γ) (ω) : = P (γ, ω)$ , and let $Λ : = e (H_{1} (X; Z)) \subseteq Ω (X)^{*} .$ This is the period subgroup, generated by the $2 g$ period vectors $e (a_{1}), e (b_{1}), \dots, e (a_{g}), e (b_{g})$ . The later bilinear-relations result establishes when this subgroup is a full lattice; discreteness and fullness are not asserted here. Since $\dim_{C} Ω (X) = g$ , after choosing a $C$ -basis $ω_{1}, \dots, ω_{g}$ of $Ω (X)$ the period matrix is the $g \times 2 g$ matrix whose columns are the coordinates of those period vectors. For $g = 0$ , both bases are empty, $P = 0$ , $Λ = {0}$ , and the period matrix is the empty $0 \times 0$ matrix.

Facts & Assumptions

Given: Full AC, a compact connected Riemann surface $X$ of genus $g$ , a chosen symplectic basis with its polygon side-loop representatives, and $ω \in Ω (X)$ .

[F1]

Under full AC, the one-polygon side-loop classes form a symplectic basis of $H_{1} (X; Z)$ ; every class has unique integer coordinates in that basis, and the genus-zero group is zero (A symplectic homology basis of a compact Riemann surface, The Axiom of Choice).

[F2]

Under full AC, $Ω (X)$ is a finite-dimensional complex vector space of dimension $g$ (The space of holomorphic differentials and the degree of the canonical divisor, The Axiom of Choice).

[F3]

The path integral of a holomorphic differential along a continuous path exists independently of its chart subdivision and local primitives; it is additive under concatenation and $C$ -linear in the differential (Path integral of a holomorphic differential on a Riemann surface).

[F4]

$Ω (X)$ denotes the complex vector space of holomorphic differentials on $X$ (Meromorphic differentials, orders and residues).

Verification

Given: The objects and hypotheses in the Definition.

Proof technique: direct.

1.1F1F3F4given

Each side loop $A_{i}$ or $B_{i}$ is a continuous path, so [F3] defines its period integral for every $ω \in Ω (X)$ . When $g = 0$ there are no side loops and the finite sum is empty.

2.1F1step 1.1given

Every $γ \in H_{1} (X; Z)$ has unique integer coordinates in the chosen basis by [F1]. Substituting those coordinates and the already-defined side-loop integrals from step 1.1 therefore gives one value of $P (γ, ω)$ for the fixed basis data.

3.1F3step 2.1algebra

The finite coordinate formula and [F3] show that $P$ is additive in $γ$ and complex-linear in $ω$ . Hence for each $γ$ , $e (γ)$ is a complex-linear functional on $Ω (X)$ , and $e$ is a homomorphism of abelian groups.

4.1F1F2step 3.1algebra∎

Because the basis generates $H_{1} (X; Z)$ , the image $Λ$ is the subgroup generated by the $2 g$ values $e (a_{i}), e (b_{i})$ ; for $g = 0$ it is the zero subgroup. By [F2], the algebraic dual has complex dimension $g$ , so choosing a basis of $Ω (X)$ gives exactly $g$ coordinates for each of the $2 g$ period vectors and hence the stated $g \times 2 g$ period matrix. This verifies the definition without asserting that $Λ$ is discrete or full.

Remarks

The side loops are topological representatives from the polygonal normal form; the local-primitive path integral defines their periods even if those fixed representatives are only continuous. The later well-definedness result proves that the resulting pairing agrees with integration over arbitrary smooth cycles and is independent of representative and symplectic basis. The later bilinear relations prove that the period subgroup is a full lattice.

Depends on

Used by

The Abel-Jacobi map Definition
The Jacobian of a compact Riemann surface Definition
Period matrix and Jacobian of the pentagon curve Example
Periods of a complex torus Example
The Abel-Jacobi map is well defined and its degree-zero extension is base-point independent Lemma
The cut surface, primitives of closed forms, and their boundary jumps Lemma
The period pairing is well defined and computed by integration Lemma
Abel's theorem for divisors Theorem
The Riemann bilinear relations and the period lattice Theorem
The symplectic period formula for integrals of wedge products Theorem

Dependency tree · two levels

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Sources

Eduard Looijenga, Riemann Surfaces (2007 author lecture notes) (standard reference, not scraped)
Karl Otto Forster, Lectures on Riemann Surfaces (GTM 81) (standard reference, not scraped)