Endomorphisms of Fano threefolds and hypersurfaces in arbitrary characteristic

Tatsuro Kawakami

Published 2023-02-23Version 1

Let $k$ be an algebraically closed field of characteristic $p\geq 0$. Let $X$ be a smooth Fano threefold over $k$ of Picard number one or a smooth hypersurface over $k$ of dimension bigger than two that admits a non-invertible endomorphism. We assume that the degree of the endomorphism is coprime to $p$ when $p>0$. When $p=0$, it is known that $X$ is isomorphic to the projective space by the work of Amerik-Rovinsky-Van de Ven, Hwang-Mok, Paranjape-Srinivas, and Beauville. In this paper, we generalize this result to arbitrary characteristic. In the proof, we prove that $X$ satisfies the Bott vanishing theorem. This is a new approach even when $p=0$.