Second Cousin problem

Remarks

The data prescribe the zeros and poles of $f$ with multiplicities, that is, a divisor on $X$. The problem asks whether this divisor is the divisor of a global meromorphic function. The quotients $g_{ij} = f_i / f_j$ form a Čech $1$-cocycle with values in the sheaf $\mathcal{O}^*$ of nowhere-zero holomorphic functions, and they define a holomorphic line bundle. The problem is solvable for these data if and only if this line bundle is holomorphically trivial. So if $H^1(X, \mathcal{O}^*) = 0$, every second Cousin problem on $X$ is solvable. On a Stein manifold, Cartan's theorem B and the exponential sequence $0 \to \Z \to \mathcal{O} \to \mathcal{O}^* \to 0$ give an isomorphism $H^1(X, \mathcal{O}^*) \cong H^2(X, \Z)$. Hence on a Stein manifold the problem is solvable if and only if the line bundle is topologically trivial, a result of Oka. In particular it is always solvable on a Stein manifold with $H^2(X, \Z) = 0$, for example on $\C^n$ or on a convex domain. For a domain in $\C$ the problem is solved by the Weierstrass factorization theorem. There are domains of holomorphy on which some second Cousin problem has no solution. An example is $X = \set{(z, w) \in \C^2 : (\log \abs{z})^2 + (\log \abs{w})^2 < 1}$, a logarithmically convex Reinhardt domain that retracts onto the torus $\abs{z} = \abs{w} = 1$, so $H^2(X, \Z) \cong \Z$. A holomorphic line bundle on $X$ with nonzero Chern class has a meromorphic section that is not identically zero, and its local expressions are second Cousin data without solution.