Let $X$ be a compact connected complex manifold.
Let $f:X \to X$ be a surjective holomorphic map then can we say that $f$ is a finite map (i.e., every point has finitely many preimages)?
If $X$ is one dimensional then the answer to the above question is yes. If $X$ is a complex projective space then also answer is yes.
If we don’t assume surjectivity then we can easily construct non constant non finite map (just consider product manifold and $f$ to be projection on one of the coordinates).
I am unable to produce counterexample in the general case. Thanks for any help!