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This questions if related to a cute article of Beauville where he proves in particular the following theorem:

http://math1.unice.fr/~beauvill/pubs/endo.pdf

Theorem.− A smooth complex projective hypersurface of dimension $\ge 2$ and degree $\ge 3$ admits no endomorphism of degree $>1$.

Here "endomorphism" means a holomorphic self-map and the proof of the theorem uses algebraic geometry bien sure:) . But the following question is not mentioned in the article:

Question. What are smooth complex projective hypersufaces in $\mathbb CP^n$ that admit topological self-maps of degree $>1$?

I wonder if this question is trivial and completely answered for $n>2$?

Added. The comment of algori settles this question (apperently). But still I wonder for example how one can construct a self-map of a quintic in $\mathbb CP^3$ of degree $>1$.

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    $\begingroup$ aglearner -- as pointed out by Donu Arapura in this thread: mathoverflow.net/questions/112572/…, by a formality argument the answer is "all except curves of genus $\leq 1$"; the argument is applicable since smooth projective hypersurfaces and, more generally, complete intersections are simply-connected unless they are curves by Lefschetz theorem. $\endgroup$
    – algori
    Commented Nov 16, 2012 at 16:41
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    $\begingroup$ .. erm.. that should have been "all except curves of genus $>1$". $\endgroup$
    – algori
    Commented Nov 16, 2012 at 16:44
  • $\begingroup$ aglearner -- welcome! $\endgroup$
    – algori
    Commented Nov 16, 2012 at 21:56
  • $\begingroup$ Hi aglearner, I don't understand your "added"... Proposition 2 of the paper you linked states that if $X$ is a compact manifold, with an endomorphism $f$ of degree $>1$ then the Kodaira dimension $\kappa(X)$ is $<\dim(X)$. But smooth a quintic in $\mathbb P^3$ has ample canonical bundle by adjunction, thus it is of maximal Kodaira dimension. $\endgroup$
    – diverietti
    Commented Nov 17, 2012 at 12:11
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    $\begingroup$ diverietti -- that's precisely the point: holomorphic endomorphisms of positive degree don't exist (that's what Beauville proves) but continuous do. The way I understand it, aglearner is asking for an explicit construction of such an endomorphism. $\endgroup$
    – algori
    Commented Nov 17, 2012 at 15:37

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