Endomorphisms of varieties and Bott vanishing

TL;DR

This paper demonstrates that projective varieties with certain endomorphisms satisfy Bott vanishing, extending classification results across characteristics and linking endomorphisms to global F-regularity.

Contribution

It introduces a new approach to analyze varieties with nontrivial endomorphisms and extends classification results to arbitrary characteristic.

Findings

Varieties with int-amplified endomorphisms satisfy Bott vanishing.

Extended classification of endomorphism-admitting varieties to any characteristic.

Bounded degrees of morphisms into such varieties.

Abstract

We show that a projective variety with an int-amplified endomorphism of degree invertible in the base field satisfies Bott vanishing. This is a new way to analyze which varieties have nontrivial endomorphisms. In particular, we extend some classification results on varieties admitting endomorphisms (for Fano threefolds of Picard number one and several other cases) to any characteristic. The classification results in characteristic zero are due to Amerik-Rovinsky-Van de Ven, Hwang-Mok, Paranjape-Srinivas, Beauville, and Shao-Zhong. Our method also bounds the degree of morphisms into a given variety. Finally, we relate endomorphisms to global $F$-regularity.