First Cousin problem

Remarks

The data $(U_i, f_i)$ are called Cousin I data; they prescribe the principal parts of $f$. The differences $g_{ij} = f_i - f_j$ form a Čech $1$-cocycle with values in the sheaf $\mathcal{O}$ of holomorphic functions. The problem is solvable for these data if and only if this cocycle is a coboundary, that is, $g_{ij} = h_j - h_i$ for holomorphic $h_i$ on $U_i$. Then $f = f_i + h_i$ on $U_i$ is a solution. So if $H^1(X, \mathcal{O}) = 0$, every first Cousin problem on $X$ is solvable. By Cartan's theorem B this holds on every Stein manifold, in particular on every domain of holomorphy in $\C^n$. For $n = 1$ and $X$ a domain in $\C$, the problem is solved by the Mittag-Leffler theorem. It can fail on other domains: on $X = \C^2 \setminus \set{0}$, with $U_1 = \set{z_1 \ne 0}$, $U_2 = \set{z_2 \ne 0}$, $f_1 = 1/(z_1 z_2)$ and $f_2 = 0$, there is no solution. Pierre Cousin studied the problem in 1895 for product domains in $\C^n$.