Algebraic dimension

Remarks

By Siegel's theorem, $a(X)$ is an integer with $0 \le a(X) \le \dim X$.

Example: for $X = \mathbb{CP}^n$, the functions $z_1/z_0, \ldots, z_n/z_0$ are meromorphic. They are algebraically independent over $\C$, because on the chart $\set{z_0 \ne 0} \cong \C^n$ they are the coordinate functions, and a nonzero polynomial does not vanish on all of $\C^n$. So $a(\mathbb{CP}^n) \ge n$, and by Siegel's bound $a(\mathbb{CP}^n) = n$.

Example: the diagonal Hopf surface has algebraic dimension $1$, and an Inoue surface has algebraic dimension $0$.

Non-example: if $X$ is the disjoint union of two copies of $\mathbb{CP}^1$, the connectedness hypothesis fails. Then $\mathcal{M}(X)$ is not a field: the function equal to $1$ on one copy and $0$ on the other is a nonzero zero divisor.

The notion extends to irreducible compact complex spaces. A compact connected complex manifold $X$ with $a(X) = \dim X$ is a Moishezon manifold.