I was reading [link text](https://mathoverflow.net/questions/10724/relationship-between-algebraic-and-holomorphic-differential-forms)
and these two much simpler questions occurred to me: 
(i) What type of algebraic functions on complex *projective* varieties do the holomorphic functions correspond to? The rational functions?
(ii) What type of algebraic functions on complex *algebraic* varieties do the holomorphic functions correspond to? The affine coordinate ring?



Algebraic and Holomorphic Functions
Well, here's a few theorems that might help:


1: On a complex projective variety, a function that is meromorphic on the whole variety is a rational function. You can get this out of the embedding into projective space.


2: On a compact complex manifold, the only globally holomorphic functions are constant. This follows from the maximum principle.


Additionally, if you're looking locally, then on an affine variety, there are a LOT more holomorphic functions than algebraic functions. On $\mathbb{C}$, you have $\mathbb{C}[z]$ for the algebraic functions, and convergent power series for holomorphic, so $e^z$ is holo but not algebraic. But every algebraic function is holomorphic.


