$n$-th root of meromorphic functions of several complex variables

Question

Let $f:\Omega\rightarrow\mathbb{C}$ be a "nice" meromorphic function of several complex variables on some domain $\Omega$. I wonder if the following claim is true.

Claim. $f$ admits a global $n$-th root over $\Omega$ if and only if the divisor of $f$ is an $n$-tuple of some divisor. Namely, there exists a meromorphic function $g:\Omega\rightarrow\mathbb{C}$ such that $f=g^{n}$ if and only if there is a divisor $D$ on $\Omega$ such that $\mathrm{div}(f)=nD$.

It seems like this is true for the single complex variable case, but I am a bit unsure of what happens in the higher dimensional case.

Answer 1

If $\Omega$ is simply connected, this is true. Let $\tilde \Omega$ be the space of pairs $(z, g)$ where $z \in \Omega$ and $g$ is the germ of an $n$th root of $f$ at $z$. If $\mathrm{div}(f) = nD$, then $f$ has an $n$th root in a small neighborhood around each point in $\mathrm{div}(f)$ (where we can express $f = h^nk$ for nonvanishing $k$ and meromorphic $h$). Further $\tilde \Omega \to \Omega$ is an $n$-fold covering space (in fact, it is a torsor for the group of $n$th roots of unity). Since $\Omega$ is simply connected, this cover has a section, which gives an $n$th root for $f$.