Meromorphic function of several variables

Remarks

On the dense open set where some $h_i$ is nonzero, a meromorphic function $f$ is the holomorphic function $g_i / h_i$. Its polar set and its set of indeterminacy points are analytic subsets; for $n \ge 2$ the indeterminacy set has codimension at least $2$.

Example: on $D = \C^2$ take the single set $U_1 = \C^2$ with $g_1(z) = z_1$ and $h_1(z) = z_2$. Clause 1 holds since $z_2$ is not the zero function, and clause 2 is empty, so $z_1 / z_2$ is meromorphic on $\C^2$.

Non-example: $e^{1/z_1}$, defined where $z_1 \ne 0$, is not the restriction of a meromorphic function on $\C^2$. Suppose it were $g/h$ near a point $(0, c)$. Then, for a suitable fixed $z_2$ close to $c$, the one-variable function $z_1 \mapsto e^{1/z_1}$ would be meromorphic at $0$. But it has an essential singularity there.

For $n = 1$ this definition gives the usual meromorphic functions on an open set of $\C$. In the language of sheaves, a meromorphic function is a global section of the sheaf of quotient fields of the sheaf of holomorphic functions.