Editing Cousin problems (section)

==Second Cousin problem==
The '''second Cousin problem''' or '''multiplicative Cousin problem''' assumes that each ratio

:<math>f_i/f_j</math>

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

:<math>f/f_i</math>

is holomorphic and non-vanishing. The second Cousin problem is a multi-dimensional generalization of the [[Weierstrass factorization theorem|Weierstrass theorem]] on the existence of a holomorphic function of one variable with prescribed zeros.

The attack on this problem by means of taking [[logarithm]]s, to reduce it to the additive problem, meets an obstruction in the form of the first [[Chern class]] (see also [[exponential sheaf sequence]]). In terms of sheaf theory, let <math>\mathbf{O}^*</math> be the sheaf of holomorphic functions that vanish nowhere, and <math>\mathbf{K}^*</math> the sheaf of meromorphic functions that are not identically zero. These are both then sheaves of [[abelian group]]s, and the quotient sheaf <math>\mathbf{K}^*/\mathbf{O}^*</math> is well-defined. The multiplicative Cousin problem then seeks to identify the image of quotient map <math>\phi</math>

:<math>H^0(M,\mathbf{K}^*)\xrightarrow{\phi} H^0(M,\mathbf{K}^*/\mathbf{O}^*).</math>

The long exact sheaf cohomology sequence associated to the quotient is

:<math>H^0(M,\mathbf{K}^*)\xrightarrow{\phi} H^0(M,\mathbf{K}^*/\mathbf{O}^*)\to H^1(M,\mathbf{O}^*)</math>

so the second Cousin problem is solvable in all cases provided that <math>H^1(M,\mathbf{O}^*)=0.</math> The quotient sheaf <math>\mathbf{K}^*/\mathbf{O}^*</math> is the sheaf of germs of [[Cartier divisor]]s 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 bundle|trivial]].

The cohomology group <math>H^1(M,\mathbf{O}^*),</math> for the multiplicative structure on <math>\mathbf{O}^*</math> can be compared with the cohomology group <math>H^1(M,\mathbf{O})</math> with its additive structure by taking a logarithm. That is, there is an exact sequence of sheaves

:<math>0\to 2\pi i\Z\to \mathbf{O} \xrightarrow{\exp} \mathbf{O}^* \to  0</math>

where the leftmost sheaf is the locally constant sheaf with fiber <math>2\pi i\Z</math>.  The obstruction to defining a logarithm at the level of ''H''<sup>1</sup> is in <math>H^2(M,\Z)</math>, from the long exact cohomology sequence

:<math>H^1(M,\mathbf{O})\to H^1(M,\mathbf{O}^*)\to 2\pi i H^2(M,\Z) \to H^2(M, \mathbf{O}).</math>

When ''M'' is a Stein manifold, the middle arrow is an isomorphism because <math>H^q(M,\mathbf{O}) = 0</math> for <math>q > 0</math> so that a necessary and sufficient condition in that case for the second Cousin problem to be always solvable is that <math>H^2(M,\Z)=0.</math>