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.

algebraicholomorphic germ is generated by quotients of holomorphic line bundle sections? (assuming X compact) $\endgroup$equalto dim($X$). $\endgroup$2more comments