meromorphic functions on proper varieties are rational

Question

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 ?

Answer 1

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.