Let $\mathbb P^n$ denote the $n$-dimensional complex projective space.
From Forster's book Lectures on Riemann surfaces p.11, Theorem 2.7, we know any non-constant holomorphic map $f:\mathbb P^1\to \mathbb P^1$ is surjective.
So, similarly, do we have any non-constant holomorphic map $f:\mathbb P^n\to \mathbb P^n$ is surjective?
One way to see this is to use the GAGA principle to establish that $f$ must be algebraic, since it is a map between projective varieties over $\Bbb{C}$. Then, an algebraic map from $\Bbb{P}^n$ to itself is finite or constant. If it is finite, it maps $\Bbb{P}^n$ to a (Zariski) closed subset of $\Bbb{P}^n$ of dimension $n$. Since $\Bbb{P}^n$ is irreducible, this implies that $f(\Bbb{P}^n) = \Bbb{P}^n$ and we have surjectivity.