# Holomorphic line bundles are algebraic on compact Riemann surfaces

By looking at Griffiths-Harris book one can see that for a compact Riemann surface $X$ we have that: $$\operatorname{Div }(X) /\mathcal M(X)^\times\cong \operatorname{hPic }(X)$$

where $\mathcal M$ is the sheaf of meromorphic functions, and $\operatorname{hPic }(X)$ is the group of holomorphic line bundles on $X$ up to isomorphism.

But since $M(X)=K(X)$, the above isomorphism implies that $\operatorname{hPic }(X)\cong \operatorname{Pic }(X)$. In other words any holomorphic line bundle on $X$ is algebraic! Isn't this a bit counterintuitive? Where is my mistake?

1) Every compact Riemann surface can be holomorphically embedded into some projective space $\mathbb P^N(\mathbb C)$: this is a difficult theorem, using Analysis in an essential way !
2) Every holomorphic submanifold of $\mathbb P^N(\mathbb C)$ is in fact algebraic, i.e. is the zero locus of finitely many homogeneous polynomials in $N+1$ variables. This is Chow's theorem.
In particular it is true that $\operatorname{hPic }(X)\cong \operatorname{Pic }(X)$ and you were perfectly right all along. Bravo!