Amplified endomorphisms of Fano fourfolds

TL;DR

This paper characterizes Fano fourfolds with conic bundle structures that admit amplified endomorphisms, showing they are toric and rational, thus linking geometric structure with endomorphism properties.

Contribution

It establishes a precise equivalence between being toric and admitting an amplified endomorphism for certain Fano fourfolds with conic bundle structures.

Findings

Fano fourfolds with conic bundle structures are toric if and only if they admit an amplified endomorphism.

Such fourfolds are necessarily rational varieties.

The paper provides a characterization linking endomorphisms to the toric property.

Abstract

Let $X$ be a smooth Fano fourfold admitting a conic bundle structure. We show that $X$ is toric if and only if $X$ admits an amplified endomorphism; in this case, $X$ is a rational variety.