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In mathematics, the Cousin problems are two questions in several complex variables, concerning the existence of meromorphic functions that are specified in terms of local data. They were introduced in special cases by Pierre Cousin in 1895. They are now posed, and solved, for any complex manifold ''M'', in terms of conditions on ''M''. For both problems, an open cover of ''M'' by sets ''Ui'' is given, along with a meromorphic function ''fi'' on each ''Ui''. ==First Cousin problem== The first Cousin problem or additive Cousin problem assumes that each difference :''fi'' − ''fj'' is a holomorphic function, where it is defined. It asks for a meromorphic function ''f'' on ''M'' such that :''f'' − ''fi'' is ''holomorphic'' on ''Ui''; in other words, that ''f'' shares the singular behaviour of the given local function. The given condition on the ''fi'' − ''fj'' is evidently ''necessary'' for this; so the problem amounts to asking if it is sufficient. The case of one variable is the Mittag-Leffler theorem on prescribing poles, when ''M'' is an open subset of the complex plane. Riemann surface theory shows that some restriction on ''M'' will be required. The problem can always be solved on a Stein manifold. The first Cousin problem may be understood in terms of sheaf cohomology as follows. Let K be the sheaf of meromorphic functions and O the sheaf of holomorphic functions on ''M''. A global section ''ƒ'' of K passes to a global section φ(''ƒ'') of the quotient sheaf K/O. The converse question is the first Cousin problem: given a global section of K/O, is there a global section of K from which it arises? The problem is thus to characterize the image of the map : By the long exact cohomology sequence, : is exact, and so the first Cousin problem is always solvable provided that the first cohomology group ''H''1(''M'',O) vanishes. In particular, by Cartan's theorem B, the Cousin problem is always solvable if ''M'' is a Stein manifold. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cousin problems」の詳細全文を読む スポンサード リンク
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