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  "body": "\n## Summary\n\nThe algebraic dimension $a(X)$ of a compact connected [[complex manifold|JSGWAN2RJV]] $X$ is the [[transcendence degree|D7TR0YWR9H]] over $\\C$ of its field of meromorphic functions.\nIt measures how many independent meromorphic functions $X$ carries.\nBy [[Siegel's theorem|ENXCWPTG92]], $0 \\le a(X) \\le \\dim X$.\nThe extreme case $a(X) = \\dim X$ defines [[Moishezon manifolds|8ZW9P9JRQD]], which are close to projective varieties, while $a(X) = 0$ occurs for generic complex tori.\n\n## Explanations\n\n### L1 human\n\nOn a compact complex manifold the holomorphic functions are all constant, but meromorphic functions, which may have poles, can be interesting.\nThe algebraic dimension counts how many of them are independent, in the sense that no polynomial relation links them.\nOn complex projective space of dimension $n$ there are $n$ independent ones, such as $z_1/z_0, \\ldots, z_n/z_0$.\nOn some complex tori there are none besides the constants.\nSo the algebraic dimension tells how far a compact complex manifold is from being algebraic.\n",
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