Algebraic and Holomorphic Functions I was reading link text
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?
 A: 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.
A: On a complex (connected, reduced...) projective variety, the only holomorphic functions are constants, by the maximum principle. The interesting comparison theorem is the following :(algebraic) rational functions are the same as (analytic) meromorphic functions. It is a consequence of GAGA, but it is a much simpler statement that was known long before Serre.
If the variety is not supposed projective anymore, the holomorphic functions have no reason at all to be algebraic : the exponential function on $\mathbb{C}$ is not !
A: We have the following theorem due to S.T.Yau and Uhlenbeck which play an important rule in regularity theory of solutions of Hermitian-Einstein metric on stable vector bundles

Any weakly holomorphic map into a projectively algebraic manifold is
  rational.

Definition:For a weakly holomorphic map from the ball $B \subset \mathbb C^n$ into an algebraic manifold $M$, We
assume that $M$ is isometrically embedded in $\mathbb CP^k$ which is embedded in some
Euclidian space $\mathbb R^N$. The map $F : B\to M \subset \mathbb  R^N$ is then a vector-valued map and it makes sense to say that it is $L_1^2$.  When $n > 1$, we call $F$ weakly holomorphic if for any linear holomorphic
coordinate system $\{ z_l,...,z_n\}$ on $B$ and for almost every point $\{ z_2;..., z_n\}$ the
restrictions of $F$ to each complex line in the $z_1$ variable is weakly holomorphic.
