# 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:

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.

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Thank you Dmitri and Ulrich. The references are very helpful. I am studying the references (I already knew the papers of McMullen and Ulrich). In fact, I need examples that have more constraints than in the question I posted before. I edited the answer correspondingly. –  anonymousInd Jul 12 '11 at 20:23
@Dmitry: Thank you for adding more tags to my question. –  anonymousInd Jul 12 '11 at 20:25
I mean I knew the papers of McMullen and Gromov. –  anonymousInd Jul 12 '11 at 21:50
You can let $X$ be any smooth projective toric variety; most of these will not be products and any selfmap of degree $>1$ will not be a submersion since $X$ is simply connected. For any $n \in \mathbb{Z}$, the $n$'th power map on the torus extends to a selfmap of the toric variety which has positive entropy if $|n|> 1$. –  ulrich Jul 13 '11 at 5:53

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

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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.

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Very nice indeed! –  Dmitri Jul 13 '11 at 11:32