2 added 48 characters in body

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.

1

# Are there compact analogues of Cartan's theorems A and B?

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 ${\mathcal{O}}_{X,x}$ of holomorphic germs over some point x. Is this fibre 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.