Cousin Problems - Second Cousin Problem

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 second Cousin problem is a multi-dimensional generalization of the Weierstrass theorem on the existence of a holomorphic function of one variable with prescribed zeros.

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. In terms of sheaf theory, let O∗ be the sheaf of holomorphic functions that vanish nowhere, and K∗ the sheaf of meromorphic functions that are not identically zero. These are both then sheaves of abelian groups, and the quotient sheaf K∗/O∗ is well-defined. The multiplicative Cousin problem then seeks to identify the image of quotient map φ

The long exact sheaf cohomology sequence associated to the quotient is

so the second Cousin problem is solvable in all cases provided that H1(M,O∗) = 0. The quotient sheaf K∗/O∗ is the sheaf of germs of Cartier divisors on M. The question of whether every global section is generated by a meromorphic function is thus equivalent to determining whether every line bundle on M is trivial.

The cohomology group H1(M,O∗), for the multiplicative structure on O∗, can be compared with the cohomology group H1(M,O) with its additive structure by taking a logarithm. That is, there is an exact sequence of sheaves

where the leftmost sheaf is the locally constant sheaf with fiber . The obstruction to defining a logarithm at the level of H1 is in, from the long exact cohomology sequence

When M is a Stein manifold, the middle arrow is an isomorphism because Hq(M,O) = 0, for so that a necessary and sufficient condition in that case for the second Cousin problem to be always solvable is that .