Cousin Problems - First Cousin Problem

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 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 H1(M,O) vanishes. In particular, by Cartan's theorem B, the Cousin problem is always solvable if M is a Stein manifold.