How to prove that any $\mathbb{P}^n$ is finitely covered by an Abelian variety of the same dimension? Maybe it is elementary, but I can not see it.

@nfdc23 Thank you for your answer. In my case, there could be some multiple fibers.

@KarlSchwede Thank you very much for your reference. I will read that.

Thanks Jason. That is just what I want to know. I have revised my question according to your suggestion.

Thanks, Christian. For the fibered surface case, the degree of $f_* \omega_{X/C}$ is non-negative in semistable cases. But is there any result related to the higher dimensional case which involves the higher direct images?