Moishezon manifold

Remarks

Example: $\mathbb{CP}^n$ is Moishezon, since $z_1/z_0, \ldots, z_n/z_0$ are $n$ algebraically independent meromorphic functions. More generally, every smooth projective variety of dimension $n$ is Moishezon, since its field of rational functions has transcendence degree $n$.

Non-example: the diagonal Hopf surface has dimension $2$ but algebraic dimension $1$, so it is not Moishezon.

Since $a(X) \le \dim X$ by Siegel's theorem, Moishezon manifolds are those with the largest possible algebraic dimension. Equivalently, $X$ is Moishezon if and only if it is bimeromorphic to a projective manifold. A Moishezon manifold is projective if and only if it admits a Kähler metric, so non-projective Moishezon manifolds are never Kähler.