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  "body": "\n## Statement\n\nLet $X$ be a [[compact|0TVJZYB6SH]], [[connected|0V032VWTHD]] [[complex manifold|JSGWAN2RJV]], and let $f \\colon X \\to \\C$ be a holomorphic function.\nThen $f$ is constant.\n\n## Remarks\n\nProof idea: since $X$ is compact, $\\abs{f}$ attains its maximum at some point $p$.\nIn a connected chart around $p$, the [[maximum modulus principle in several variables|91DW8V0PET]] shows that $f$ equals $f(p)$ near $p$.\nThe set of points near which $f \\equiv f(p)$ is open, and by the [[identity theorem|7GZ908QZ04]] it is also closed, so by connectedness it is all of $X$.\n\nBoth hypotheses are needed.\nOn the non-compact manifold $\\C^n$, the coordinate function $z_1$ is holomorphic and not constant.\nOn a disjoint union of two compact complex manifolds, take the function equal to $0$ on one piece and $1$ on the other.\nIt is holomorphic and not constant; in general $f$ is only locally constant.\n\nConsequence: a compact connected complex manifold of positive dimension is not a complex submanifold of any $\\C^N$, since the coordinate functions restricted to it would be constant.\nFor example, [[complex projective space|7GJ4STCNNM]] does not embed holomorphically in $\\C^N$.\n\n## Explanations\n\n### L1 human\n\nOn a compact complex manifold, such as the Riemann sphere or a torus, every holomorphic function is constant.\nThe reason is the maximum principle: the absolute value of a holomorphic function cannot have a peak inside its domain.\nOn a compact space there is no boundary to escape to.\nThis is a strong contrast with smooth functions, which exist in great supply on every manifold.\n\n### L2 human\n\nThis result is the several-variable version of the fact that a compact Riemann surface has no nonconstant holomorphic functions, and it is a cousin of [[Liouville's theorem|0EHKVRF4J6]].\nBecause global holomorphic functions are useless on a compact complex manifold, one studies instead meromorphic functions and holomorphic sections of line bundles.\nThe opposite extreme is a [[Stein manifold|HMCD0JBFST]], which has enough holomorphic functions to embed in some $\\C^N$.\n",
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