I am reading a proof of Chow's theorem from Mumford's complex projective varieties. The theorem establishes that every analytic subvariety of $\mathbb{P}^N$ is in fact, algebraic.
I have seen people cite this as proof that every compact Riemann surface $X$ is an algebraic curve. The way I understand it is that by Riemann existence, one constructs many meromorphic functions on $X$. So we pick a very ample divisor and embed $X$ into $\mathbb{P}^{g-1}$ by picking a basis for $L(D)$ and sending $x$ to $[f_1(x): \dots :f_g(x)]$.
I am not too sure why this embedding gives us an analytic subvariety though. Could someone elaborate this, or let me know if I'm understanding it incorrectly?