Polarized endomorphisms of uniruled varieties (with Appendix by Y. Fujimoto and N. Nakayama)

TL;DR

This paper classifies polarized endomorphisms on certain uniruled varieties, showing they originate from simpler building blocks, and proves that smooth Fano threefolds with such endomorphisms are rational.

Contribution

It provides a structural description of polarized endomorphisms on uniruled threefolds and fourfolds, linking them to known varieties and establishing rationality of certain Fano threefolds.

Findings

Polarized endomorphisms are constructed from those on Q-Fano threefolds, Gorenstein log del Pezzo surfaces, and P^1.

Every smooth Fano threefold with a polarized endomorphism of degree > 1 is rational.

Results extend to uniruled threefolds and fourfolds.

Abstract

We show that polarized endomorphisms of rationally connected threefolds with at worst terminal singularities are equivariantly built up from those on Q-Fano threefolds, Gorenstein log del Pezzo surfaces and P^1. Similar results are obtained for polarized endomorphisms of uniruled threefolds and fourfolds. As a consequence, we show that every smooth Fano threefold with a polarized endomorphism of degree > 1, is rational.