Cousin problems

Ask whether compatible local meromorphic data on a complex manifold glue to a global meromorphic function, with additive and multiplicative versions carrying distinct cohomological obstructions.

Version

v2 · 2026-09-19 · History

Domain-specific #

1572

Domain group

Formal Sciences

Origin domain

Mathematics

Subdomains

Several Complex Variables, Complex Analytic Geometry → Mathematics

Core Idea

The Cousin problems are the paired local-to-global existence questions in several complex variables: the first asks for a global meromorphic function matching prescribed local principal parts up to holomorphic differences, and the second asks for one matching local zero-and-pole divisors up to nowhere-zero holomorphic ratios. Local representatives define an additive cocycle in the holomorphic-function sheaf for the first problem or a multiplicative cocycle in its units for the second; a global meromorphic representative exists exactly when the corresponding gluing obstruction vanishes under the stated hypotheses. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

How would you explain it like I'm…

 

No faithful explanation at this level. All three generators agree a child picture collapses into generic patchwork gluing, erasing the paired additive (principal parts) versus multiplicative (divisor) problems and their different obstructions, and implying matching pieces always glue.

The Two Blow-Up Matching Puzzles

In an advanced part of math, people study functions of several complex numbers that can blow up to infinity at some places. The Cousin problems ask: if you know what such a function should look like near every spot, and nearby descriptions agree where they overlap, can you find one single function that works everywhere? The first Cousin problem is about matching the 'blow-up part' of the function near each spot. The second is about matching where the function is zero or blows up, and how strongly. Whether the answer is yes depends on the shape of the space.

Additive and Multiplicative Gluing Problems

The Cousin problems are two related 'local-to-global' questions about meromorphic functions (functions that are complex-differentiable except at poles) in several complex variables. In the first Cousin problem, you are given, on each piece of an open cover, a local meromorphic function describing the principal part (the part that blows up), and on overlaps any two local choices differ by a holomorphic function; the question is whether one global meromorphic function has exactly these principal parts. In the second Cousin problem, you are given local zero-and-pole data (a divisor), and on overlaps the local choices differ by a nowhere-zero holomorphic factor; the question is whether one global meromorphic function has exactly these zeros and poles. The first involves differences (additive), the second ratios (multiplicative), so they have different obstructions. A solution exists when the relevant gluing obstruction vanishes under suitable hypotheses on the space.

 

The Cousin problems are the paired local-to-global existence problems for meromorphic data in several complex variables. In the first (additive) Cousin problem, local meromorphic functions on an open cover are prescribed whose pairwise differences on overlaps are holomorphic; one seeks a global meromorphic function whose difference from each local datum is holomorphic, i.e. with the prescribed principal parts. The differences define a 1-cocycle in the sheaf of holomorphic functions, and the problem is solvable exactly when its cohomology class vanishes. In the second (multiplicative) Cousin problem, local data are prescribed whose ratios on overlaps are nowhere-vanishing holomorphic functions, i.e. a divisor of zeros and poles; the ratios form a 1-cocycle in the sheaf of holomorphic units, and a global meromorphic function realizing the divisor exists exactly when that class vanishes. Because the additive and multiplicative cocycles live in different sheaves, their obstruction groups differ: on a Stein space the additive obstruction vanishes, while the multiplicative one can still be blocked by a topological obstruction. Correct use requires stating which problem is meant, checking overlap compatibility, forming the right cocycle, and applying vanishing theorems only under their hypotheses, not by transferring one-variable results.

Scope of Application

Cousin problems belongs to several complex variables and is useful where the analyst can specify an open cover of a complex manifold together with local meromorphic functions or local divisor data on its members, then evaluate local meromorphic data agree on overlaps in the typed additive or multiplicative sense and the question is whether one global meromorphic object realizes those data. The scope is broad within that domain but bounded by the need for local meromorphic data agree on overlaps in the typed additive or multiplicative sense and the question is whether one global meromorphic object realizes those data.

Clarity

The abstraction clarifies a crowded vocabulary by making local meromorphic data agree on overlaps in the typed additive or multiplicative sense and the question is whether one global meromorphic object realizes those data the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because Cousin also names unrelated lemmas, complexes, and family relationships, so the two meromorphic gluing problems must be named explicitly.

Manages Complexity

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived consequences, boundary cases, and validation obligations specific to Cousin problems. Cousin problems compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning

1. Identify the carrier. State what the elements, states, objects, or observations are: an open cover of a complex manifold together with local meromorphic functions or local divisor data on its members. Reject examples whose alleged carrier belongs to a different problem.
2. Lock the constitutive rule. Express local meromorphic data agree on overlaps in the typed additive or multiplicative sense and the question is whether one global meromorphic object realizes those data independently of one notation or implementation.

Knowledge Transfer

Knowledge transfers strongly among subfields of several complex variables because they reuse an open cover of a complex manifold together with local meromorphic functions or local divisor data on its members, Local representatives define an additive cocycle in the holomorphic-function sheaf for the first problem or a multiplicative cocycle in its units for the second; a global meromorphic representative exists exactly when the corresponding gluing obstruction vanishes under the stated hypotheses., and state first or second Cousin type, verify overlap compatibility, form the correct sheaf cocycle, identify its cohomology class, and invoke a vanishing or topological theorem only with its manifold hypotheses.

Relationships to Other Abstractions

Current abstraction Cousin problems Domain-specific

Parents (1) — outgoing typed relations

• Cousin problems is a kind of Local-to-Global Aggregation Prime

  Each Cousin problem literally tests whether locally valid, overlap-compatible data promote to one global object; the additive or multiplicative cocycle and obstruction theory provide the several-complex-variable residual.

Hierarchy path (1) — routes to 1 parentless root

• Cousin problems → is a kind of Local-to-Global Aggregation

Neighborhood in Abstraction Space

Cousin problems sits in a crowded region of the domain-specific corpus (25^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Manifolds, Stacks & Classifying Spaces (18 abstractions)

Nearest neighbors

• Twisted sheaf — 0.90
• Haefliger structure — 0.90
• Gerbe — 0.89
• Coherent sheaf — 0.89
• Meromorphic function — 0.89

Computed from structural-signature embeddings · 2026-10-25