# Topological self-maps of smooth complex hypersufaces in complex projective spaces

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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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. –  algori Nov 16 '12 at 16:41
.. erm.. that should have been "all except curves of genus $>1$". –  algori Nov 16 '12 at 16:44
Thank you algori! –  aglearner Nov 16 '12 at 17:59
aglearner -- welcome! –  algori Nov 16 '12 at 21:56
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. –  algori Nov 17 '12 at 15:37