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A failed principal-parts problem detected by residues on a complex torus

Example

Assume full AC (The Axiom of Choice). Let $X = C / Λ$ for a full complex lattice $Λ$ , with origin $o = [0]$ and local quotient coordinate $z$ centred at $o$ . Prescribe the function-valued principal parts $η_{o} = [1 / z] \in M_{o} / O_{o}, η_{p} = 0 (p \neq o) .$ There is no global meromorphic function with these principal parts. A putative solution would have divisor $[q] - [o]$ for a point $q \neq o$ , and would give a degree-one proper holomorphic map to the sphere, contradicting the torus genus $1$ .

The obstruction is the residue of a differential, rather than an invariant residue of a function germ: the nowhere-vanishing holomorphic differential $ω = d z$ gives $\sum_{p \in X} {Res}_{p} (η_{p} ω) = {Res}_{o} (d z / z) = 1 \neq 0.$ All holomorphic differentials are constant multiples of $d z$ , so the corresponding residue functional is $c d z \mapsto c$ . For a finite good cover and compatible comparison data as constructed below, the necessary-and-sufficient criterion of Prescribed principal parts on a compact Riemann surface detects exactly this obstruction.

Facts & Assumptions

Given: Full AC, a full lattice $Λ$ , its torus $X$ , and the principal part $1 / z$ at $o$ with zero principal parts elsewhere.

[F1]

Full AC is inherited through RR, duality and the good-cover comparison chain (The Axiom of Choice).

[F2]

The quotient torus is compact with local lift charts and translation transitions. Its proof gives $δ > 0$ such that distinct lattice points are separated by at least $δ$ (Complex lattice and quotient torus, The quotient $C / Λ$ is a compact Riemann surface).

[F3]

A principal part is a finite negative Laurent polynomial. A differential's residue is its coefficient of $z^{- 1} d z$ , independent of coordinates (The principal part at an isolated singularity, Meromorphic differentials, orders and residues).

[F4]

Analytic RR gives $ℓ (0) = 1$ , $i (0) = g$ and the canonical-divisor formula; Serre duality gives $i (A) = ℓ (K - A)$ (The Riemann-Roch theorem on a compact Riemann surface, Serre duality on a compact Riemann surface).

[F5]

Principal divisors have degree zero; meromorphic functions on compact $X$ are proper maps to the sphere when nonconstant, with pole order equal to fibre multiplicity (Divisors, principal divisors and canonical divisors on a Riemann surface).

[F6]

Proper nonconstant holomorphic maps have positive weighted fibre degree, and multiplicity one gives a holomorphic local inverse; genus is invariant under biholomorphism and the sphere has genus zero (Degree of a proper holomorphic map of Riemann surfaces, Local power-map normal form on Riemann surfaces, Genus and Euler characteristic of a compact Riemann surface).

[F7]

The residues of a global meromorphic differential on compact $X$ sum to zero (Residue theorem on a compact Riemann surface).

[F8]

The bundle $O_{X} (0)$ has a global frame $1$ . A finite good cover has disc chart members and disc-biholomorphic nonempty finite intersections. Under full AC a contractible proper plane domain is biholomorphic to a disc by the grand simple-connectivity equivalence (The holomorphic line bundle associated to a divisor, Cech cohomology of holomorphic sections of a line bundle on finite good covers, For a plane domain, the complement, homology, primitive, logarithm, conjugate, conformal, homotopy, and contractibility conditions are equivalent).

[F9]

Compatible metrics exist under countable choice, hence full AC (Hermitian metric and $L^{2}$ pairing on a compact Riemann surface).

[F10]

A supplied finite good cover subordinate to holomorphic frame domains has the canonical sheaf/Čech/Dolbeault comparison (Cech--Dolbeault comparison for holomorphic line bundles on a compact Riemann surface). With this comparison and supplied compatible metrics, principal parts are realizable if and only if their residue sum paired against every holomorphic differential is zero; the pairing is independent of representatives and cover (Prescribed principal parts on a compact Riemann surface).

Verification

1.1F1F2F3F4givenalgebra

By [F2], local lift coordinates differ by translations, so their differentials glue to a nowhere-zero holomorphic differential $ω = d z$ with $K = (ω) = 0$ . By [F4], $i (0) = ℓ (K) = ℓ (0) = 1$ , so $g = 1$ and $h^{0} (X, K_{X}) = 1$ . Thus every holomorphic differential is $c d z$ . The prescribed principal part is nonzero by [F3] and would force a simple pole at $o$ with no other poles.

2.1F3F5F6F7step 1.1algebra

If a meromorphic function $f$ realized the data, [F5] would give $(f) = E - [o]$ with $E$ effective of degree one. Hence $E = [q]$ and $q \neq o$ . The weighted fibre over infinity has one simple point, so [F6] gives degree one; every fibre is then one point of multiplicity one. The local holomorphic inverses in [F6] glue to a global inverse, making $X$ biholomorphic to the sphere, contrary to $g = 1$ in step 1.1. Independently, $f ω$ would have residue $1$ at $o$ and zero elsewhere by [F3], contradicting [F7]. This directly proves nonsolvability without a good-cover hypothesis.

3.1F2F3F8F9F10step 1.1chooseconstructalgebra∎

Choose $0 < r < δ / 4$ using [F2]. The quotient images of radius- $r$ plane balls cover $X$ and are disc chart domains; compactness gives a finite subcover. In the lift of any one member, another member that intersects it has at most one relevant translated radius- $r$ ball: two such centres would be within $4 r < δ$ of each other, contrary to [F2]. Thus every nonempty finite intersection lifts injectively to an intersection of finitely many plane balls. It is bounded, open and convex; straight-line contraction to an interior point makes it contractible, and [F8] makes it disc-biholomorphic. This is a finite good cover subordinate to the global holomorphic frame $1$ of $O_{X} (0)$ from [F8]. Supply compatible metrics by [F9]; the canonical comparison in [F10] supplies the comparison map used by its residue criterion. By [F3], pairing the data with $c d z$ gives ${Res}_{o} (c d z / z) = c$ , since all other terms vanish; replacing the representative $1 / z$ by a holomorphic perturbation leaves this residue unchanged. Step 1.1 identifies the entire differential space, so this is the complete residue functional, and [F10] is the exact necessary-and-sufficient obstruction criterion. Its value $1$ at $d z$ proves the claimed failure.

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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)