Suppose $X$ is a proper variety over $\mathbb{C}$, is every meromorphic function rational? In the case of projective variety, can this be derived from Chow lemma? How does the GAGA principal illustrate on these ?
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You have Chow's theorem in mind, not Chow's lemma , which is about obtaining complete varieties from projective ones – Georges Elencwajg Apr 21 '14 at 13:29
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Thanks for your clarification! (The chow lemma says the existence of a proper birational morphism from a projective variety to any complete variety, while the chow theorem says an analytic subvariety is an algebraic variety.)And thanks for your references below. – Apr 21 '14 at 14:35
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You are welcome, dear mqx. – Georges Elencwajg Apr 21 '14 at 14:38
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Every meromorphic function on a complete complex variety $X$ over $\mathbb C$ is indeed rational.
The proof is in Shafarevich's Basic Algebraic Geometry 2, Second Edition, Chap.VIII, §3, Theorem 1, pages 179-180.
The generalization of GAGA from projective varieties to complete ones is due to Grothendieck and can be found in SGA 1, exposé XII.
There is however no explicit mention of meromorphic functions in that exposé.
Georges Elencwajg
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