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  "body": "\n## Definition\n\nLet $X$ be a compact, connected [[complex manifold|JSGWAN2RJV]] of dimension $n$, and let $a(X)$ be its [[algebraic dimension|F0GF0JEZTG]].\nThen $X$ is a *Moishezon manifold* if $a(X) = n$, that is, if $X$ has $n$ algebraically independent meromorphic functions.\n\n## Remarks\n\nExample: $\\mathbb{CP}^n$ is Moishezon, since $z_1/z_0, \\ldots, z_n/z_0$ are $n$ algebraically independent meromorphic functions.\nMore generally, every smooth projective variety of dimension $n$ is Moishezon, since its field of rational functions has transcendence degree $n$.\n\nNon-example: the diagonal [[Hopf surface|PNFNGBXRM3]] has dimension $2$ but algebraic dimension $1$, so it is not Moishezon.\n\nSince $a(X) \\le \\dim X$ by [[Siegel's theorem|ENXCWPTG92]], Moishezon manifolds are those with the largest possible algebraic dimension.\nEquivalently, $X$ is Moishezon if and only if it is bimeromorphic to a projective manifold.\nA Moishezon manifold is projective if and only if it admits a Kähler metric, so non-projective Moishezon manifolds are never [[Kähler|X60JG9AEQJ]].\n",
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