This questions if related to a cute article of Beauville where he proves in particular the following theorem:
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 $>1$.