Polarized endomorphisms of log Calabi-Yau pairs

TL;DR

The paper proves that dlt log Calabi-Yau pairs with polarized endomorphisms are essentially finite quotients of toric fibrations over abelian varieties, revealing structural constraints under these conditions.

Contribution

It establishes a structural classification of dlt log Calabi-Yau pairs with polarized endomorphisms as finite quotients of toric fibrations over abelian varieties, and explores the necessity of the dlt condition.

Findings

Dlt log Calabi-Yau pairs with polarized endomorphisms are finite quotients of toric fibrations.

The dlt condition is crucial; dropping it can invalidate the classification.

A birational modification can produce a similar structure for klt type varieties.

Abstract

Let $(X,\Delta)$ be a dlt log Calabi-Yau pair admitting a polarized endomorphism. We show that $(X,\Delta)$ is a finite quotient of a toric log Calabi-Yau fibration over an abelian variety. We provide an example which shows that the previous statement does not hold if we drop the dlt condition of $(X,\Delta)$ even if $X$ is a smooth variety. Given a klt type variety $X$ and a log Calabi-Yau pair $(X,\Delta)$ admitting a polarized endomorphism, we show that a suitable birational modification of $(X,\Delta)$ is a finite quotient of a toric log Calabi-Yau fibration over an abelian variety.