I've actually a problem understanding this assertion:
There no compact complex submanifolds of $\mathbb{C}^n$ of positive dimension.
This assertion is given as a corollary of the following theorem:
Let $X$ be a connected compact complex manifold and let $f \in \mathcal{O}(X)$. Then $f$ is constant.
I don't see how this theorem should imply the first assertion I wrote. The book I am reading sketches the proof of the first assertion I wrote by saying: Otherwise at least one of the coordinate functions $z_1,...,z_n$ would be a nonconstant function when restricted to such a submanifold.
So far, because of the theorem to me, it makes sense to have $z_1, ..., z_n$ nonconstant since such a submanifold may not be connected. But, how this should imply their nonexistence?
Suppose that such a connected compact manifold $X$, exists. Since $dimX>0$, it has two distinct points $u=(u_1,..,u_n); v=(v_1,...,v_n)$. There exists $i$ such that $u_i\neq v_i$, we deduce that $z_i(u)\neq z_i(v)$ contradiction since the restriction of $z_i$ to $X$ must be constant.