---
tag: 537J79VYQR
kind: problem
name: First Cousin problem
msc: [32A20, "32E10", 32C35]
uses: [JSGWAN2RJV, W6FTSF57T1, 507C34W0ZT]
about: [W6FTSF57T1]
sources:
  - url: https://en.wikipedia.org/wiki/Cousin_problems
history:
  posed_by: "Pierre Cousin"
  first_stated: "1895"
---

## Problem

Let $X$ be a [[complex manifold|JSGWAN2RJV]] and let $(U_i)_{i \in I}$ be an open cover of $X$.
For each $i$, let $f_i$ be a [[meromorphic function|W6FTSF57T1]] on $U_i$, such that $f_i - f_j$ is [[holomorphic|507C34W0ZT]] on $U_i \cap U_j$ for all $i, j$.
Find a meromorphic function $f$ on $X$ such that $f - f_i$ is holomorphic on $U_i$ for every $i$.

## 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 $\C^n$.

## Explanations

### L1 human

In one variable, the Mittag-Leffler theorem builds a meromorphic function with prescribed poles and principal parts.
The first Cousin problem asks the same in several variables.
The local singular behaviour is given on pieces of a manifold, with the pieces agreeing up to holomorphic terms.
One wants one global meromorphic function with exactly this behaviour.
Whether this is possible depends on the shape of the manifold.
On Stein manifolds, for example on $\C^n$ and on convex domains, it always is.