Cousin problems

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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 P. Cousin in 1895. They are now posed, and solved, for any complex manifold M, in terms of conditions on M. Assume there is an open cover of M by sets U_i and a meromorphic function f_i is given on each U_i. The first Cousin problem or additive Cousin problem assumes that each difference

f_i − f_j

is a holomorphic function, where it is defined. It asks for a meromorphic function f on M such that

f − f_i

is holomorphic on U_i; in other words, that f shares the singular behaviour of the given local function. The given condition on the f_i − f_j 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 set of complex numbers. Riemann surface theory shows that some restriction on M will be required. The problem can always be solved on a Stein manifold. The second Cousin problem or multiplicative Cousin problem assumes that each ratio

f_i/f_j

is a non-vanishing holomorphic function, where it is defined. It asks for a meromorphic function f on M such that

f/f_i

is holomorphic and non-vanishing. The attack on this problem by means of taking logarithms, to reduce it to the additive problem, meets an obstruction in the form of the first Chern class; and this implies that it cannot always be solved on a Stein manifold M unless the cohomology group

H²(M,Z) = {0}.

In terms of sheaf theory, these problems can be expressed via quotient sheaves: of the sheaf of meromorphic functions modulo holomorphic functions, for the first problem, and for the sheaf of non-vanishing meromorphic functions modulo non-vanishing holomorphic functions, in the second case.