cv.complex variables - Are there compact analogues of Cartan's theorems A and B?

Dear Colin , for $X$ a holomorphic connected manifold, denote by $mathcal M (X)$ its field of meromorphic functions.

A) It is not true that a germ of holomorphic function $f_xin mathcal O_{X,x}$ is induced by a global meromorphic function : many compact complex manifolds only have $mathbb C$ as meromorphic functions:
$mathcal M (X)=mathbb C$. There is an example with $X$ a surface in Shafarevich's Basic Algebraic Geometry, volume 2, page 164.

B) The best analogon to Theorem B is probably Cartan-Serre's result that for any coherent sheaf $mathcal F$ on the compact manifold $X$, the cohomology vector spaces $H^q(X,mathcal F), qgeq 1$ are finite-dimensional over $mathbb C$.

(Original article: Cartan-Serre, C.R.Acad.Sci. Paris 237 (1953), 128-130)