Cartan's theorem A says that on for a coherent sheaf ${\mathcal{F}}$ on a Stein manifold X, the fibres ${\mathcal{F}}\_x$ over each point x in X are generated by global sections.


I'm wondering if there are compact analogues of these theorem. Here I consider holomorphic line bundles over a compact complex manifold X. Consider the  fibre stalk ${\mathcal{O}}\_{X,x}$ of holomorphic germs over some point x. Is this  fibre stalk generated by quotients of global holomorphic line bundle sections? That is, given two global sections $s$ and $t$ of the same line bundle $L\to X$. The quotient $s/t$ is a global meromorphic function. Given a holomoprhic function germ $f\_x\in {\mathcal{O}}\_{X,x}$, we can always find such $s$ and $t$ such that $(s/t)\_x=f\_x$? 


An analogue for Cartan's theorem B would be nice too. But I can't phrase this precisely.



Are there compact analogues of Cartan's theorems A and B?
Colin, I think you have answered your own question in your response to Brian Conrad. The fraction field of $\mathcal O\_{X,x}$ has infinite transcendence degree over $\mathbb C$, while $\mathcal M(X)$, in Elencwajg's notation, has finite transcendence degree. For any global sections $s,t$ of a holomorphic line bundle, the ratio $s/t$ lies in $\mathcal M(X)$. So most $f$ cannot be written as $s/t$.


