Non-trivial surjective holomorphic selfmap of compact Kähler manifold of $\operatorname{dim} \geq 3$? 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:


*

*$X$ is not a projective space.

*$f$ is not a submersion (or étale?) 

*$f$ has positive entropy. 

*$f$ is not a product map.


References or nonexistence results are also welcome.
Thank you very much. 
 A: If you want some non-existence results, you might be interested in the work of Beauville who proves the following theorem:
Theorem. A smooth complex projective hypersurface of dimension $\ge 2$ and degree $\ge 3$ admits no endomorphism of degree $> 1$. http://math.unice.fr/~beauvill/pubs/endo.pdf
Note, that this theorem holds as well for quadrics of dimensions $\ge 3$.
You can easily conclude from this theorem that any hypersurface $X$ that is not Calabi-Yau (i.e. $X\subset \mathbb P^n$, $deg(X)\ne n+1$) don't have self-maps of positive entropy (of course, we impose $deg(X)\ge 2$, $X$ is not a quadric in $\mathbb CP^3$). 
On the other hand Calabi-Yau manifolds (in particular Tori) have self-maps of positive entropy quite often. This is especially well studied in dimension two (for K3 surfaces): http://www.math.harvard.edu/~ctm/papers/home/text/papers/glue/glue.pdf
Apart from this, I would like to give a reference (that you probably know) on one preprint of Gromov on this topic, that you can find on his webpage: ON THE ENTROPY OF HOLOMORPHIC MAPS:  http://www.ihes.fr/~gromov/PDF/10[24].pdf
A: The article "Quelques aspects des systemes dynamiques polynomiaux" (it is not just about polynomials!) by S. Cantat available from
http://perso.univ-rennes1.fr/serge.cantat/Articles/etats_cantat.pdf
contains a very nice survey.
