I am finding some nontrivial examples of surjective holomorphic selfmap of compact Kähler manifold of $\operatorname{dim} \geq 3$. That is, find $X$ a compact Kähler manifold of $\operatorname{dim} X \geq 3$, $f:X\to X$ a surjective holomorphic selfmap so that:

1. $X$ is not a projective space.
2. $f$ is not a submersion (or étale?)
3. $f$ has positive entropy.
4. $f$ is not a product map.

References or nonexistence results are also welcome.

Thank you very much.