ag.algebraic geometry - 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.