Non-isomorphic endomorphisms of Fano threefolds

TL;DR

This paper classifies smooth Fano threefolds that admit non-isomorphic surjective endomorphisms, showing they are either toric or products of P^1 with a del Pezzo surface, and identifies conditions for polarized endomorphisms.

Contribution

It provides a complete characterization of Fano threefolds with non-isomorphic and polarized endomorphisms, linking geometric properties to endomorphism types.

Findings

Fano threefolds with non-isomorphic surjective endomorphisms are toric or products of P^1 and a del Pezzo surface.

Such threefolds are rational varieties.

Only toric Fano threefolds admit polarized endomorphisms.

Abstract

Let $X$ be a smooth Fano threefold. We show that $X$ admits a non-isomorphic surjective endomorphism if and only if $X$ is either a toric variety or a product of $\mathbb{P}^1$ and a del Pezzo surface; in this case, $X$ is a rational variety. We further show that $X$ admits a polarized (or amplified) endomorphism if and only if $X$ is a toric variety.