The Amerik–Rovinsky–Van de Ven conjecture on endomorphisms of Fano varieties

Let $X$ be a smooth Fano variety of Picard number $1$ over an algebraically closed field of characteristic zero. Suppose that $X$ admits a non-invertible surjective endomorphism.

Amerik–Rovinsky–Van de Ven conjecture. Then $X$ is isomorphic to projective space.

This is a long-standing conjecture concerning the classification of smooth Fano varieties of Picard number one that admit non-invertible surjective endomorphisms. The paper studies related classification results and proves extensions in arbitrary characteristic, but the stated conjecture is presented without a resolution here.