If $i : M \to \mathbb{C}^n$ is a holomorphic map, then each coordinate function on $\mathbb{C}^n$ restricts to a global holomorphic function on the image. In particular, there is no holomorphic analogue of the Whitney embedding theorem; the only connected, compact holomorphic manifold that embeds holomorphically in $\mathbb{C}^n$ is a point.
Could someone here explain this passage to me? I know the maximum principle that states that every holomorphic function on a connected, compact holomorphic manifold is constant, but I'm not sure how it applies here? What is meant by each coordinate function on $\mathbb{C}^n$ restricting to a global holomorphic function on the image?
If $M\subset \Bbb C^n$ is a compact submanifold, then the injection map $\iota\colon M \to \Bbb C^n$ is holomorphic by definition. For $j \in \{1,\ldots,n\}$, the coordinate map $z^j \colon \Bbb C^n \to \Bbb C$ is holomorphic. Therefore, the composition map $f^j = z^j\circ \iota \colon M \to \Bbb C^n$ is holomorphic.
The maximum principle implies that $f^j$ is constant whenever $M$ is compact. In this case, the image of $\iota \colon M \to \Bbb C^n$ is constant, that is, a point. Since $\iota$ is an embedding, it is in particular a bijection, and $M$ is a point.