Cousin problems

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. The first Cousin problem or additive Cousin problem assumes that each difference is a holomorphic function, where it is defined. It asks for a meromorphic function f on M such that is holomorphic on Ui; in other words, that f shares the singular behaviour of the given local function. The given condition on the 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 of K passes to a global section 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 H1(M,O) vanishes. In particular, by Cartan's theorem B, the Cousin problem is always solvable if M is a Stein manifold. The second Cousin problem or multiplicative Cousin problem assumes that each ratio is a non-vanishing holomorphic function, where it is defined. It asks for a meromorphic function f on M such that is holomorphic and non-vanishing.