The Amerik–Kollár–Peternell conjecture on endomorphisms of Fano manifolds

Let $X$ be a Fano manifold of Picard number $1$, and let $f:X\to X$ be a surjective endomorphism. Here, $\deg f$ denotes the degree of $f$, and $\mathbb{P}^n$ denotes projective $n$-space. Amerik–Kollár–Peternell conjecture. If

$$\deg f>1,$$

then

$$X\cong\mathbb{P}^n.$$

The conjecture asserts that, among Fano manifolds of Picard number $1$, only projective spaces admit surjective endomorphisms of degree greater than one. It generalizes the corresponding result for hypersurfaces of projective space; its general status is not established here.