Cousin problems

Cousin I asks whether compatible local meromorphic principal parts come from a global meromorphic function; Cousin II asks whether a prescribed divisor is globally principal.

Sheaf cohomology: A theory that measures how local sections of a sheaf fail to fit together globally. Cousin I and II become vanishing questions for first cohomology groups.
Meromorphic function: A function holomorphic except for poles, locally expressible as a quotient of holomorphic functions. Both problems seek global meromorphic functions from local or divisor data.
Pierre Cousin: A French mathematician whose work introduced the two classical problems of complex analysis. His name is attached to the paired questions about globalizing meromorphic data.
Mittag-Leffler theorem: A theorem giving conditions for a meromorphic function with prescribed principal parts on a complex domain. It is the one-variable counterpart most closely associated with Cousin I.
Čech cohomology: A cohomology theory computed from overlaps in an open cover. Cousin I records discrepancies between local meromorphic functions on overlaps.
Principal part: The negative-power terms in a function's Laurent expansion at a pole. Cousin I prescribes these local singular terms and asks for one global function.
Weierstrass theorem on analytic continuation: A nineteenth-century result establishing broad existence principles for analytic functions on complex domains. Cousin's questions extend the period's search for global existence theorems.
Weierstrass factorization theorem: A theorem constructing entire functions with prescribed zeros in the complex plane. It realizes zero data in one variable, unlike the general divisor problem on complex spaces.
Coherent analytic sheaf: A sheaf of modules locally presented by finitely many generators and relations over holomorphic functions. The analytic coherence theorem supplies finiteness needed for Cousin solvability.
Divisor: A formal integer-weighted sum of codimension-one subvarieties recording zeros and poles. Cousin II prescribes this global pattern of zeros and poles.

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