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  "body": "\n## Definition\n\nLet $X$ be a [[compact|0TVJZYB6SH]], [[connected|0V032VWTHD]] [[complex manifold|JSGWAN2RJV]].\nLet $\\mathcal{M}(X)$ be the set of [[meromorphic functions|W6FTSF57T1]] on $X$; with pointwise operations it is a field that contains $\\C$ as the constant functions.\nThe *algebraic dimension* of $X$ is\n$$a(X) = \\operatorname{trdeg}_\\C \\mathcal{M}(X),$$\nthe [[transcendence degree|D7TR0YWR9H]] of $\\mathcal{M}(X)$ over $\\C$.\n\n## Remarks\n\nBy [[Siegel's theorem|ENXCWPTG92]], $a(X)$ is an integer with $0 \\le a(X) \\le \\dim X$.\n\nExample: for $X = \\mathbb{CP}^n$, the functions $z_1/z_0, \\ldots, z_n/z_0$ are meromorphic.\nThey 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 $\\C^n$.\nSo $a(\\mathbb{CP}^n) \\ge n$, and by Siegel's bound $a(\\mathbb{CP}^n) = n$.\n\nExample: the diagonal [[Hopf surface|PNFNGBXRM3]] has algebraic dimension $1$, and an [[Inoue surface|GPSVJK3R4X]] has algebraic dimension $0$.\n\nNon-example: if $X$ is the disjoint union of two copies of $\\mathbb{CP}^1$, the connectedness hypothesis fails.\nThen $\\mathcal{M}(X)$ is not a field: the function equal to $1$ on one copy and $0$ on the other is a nonzero zero divisor.\n\nThe notion extends to irreducible compact complex spaces.\nA compact connected complex manifold $X$ with $a(X) = \\dim X$ is a [[Moishezon manifold|8ZW9P9JRQD]].\n",
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