Log Calabi-Yau structure of projective threefolds admitting polarized endomorphisms

TL;DR

This paper proves that smooth projective threefolds with polarized endomorphisms are of Calabi-Yau type, confirming a conjecture by Broustet and Gongyo using the equivariant minimal model program and canonical bundle formula.

Contribution

It establishes the conjecture for smooth projective threefolds admitting polarized endomorphisms, advancing understanding of their structure and Calabi-Yau properties.

Findings

Proved the conjecture for smooth projective threefolds with polarized endomorphisms.

Established a general guideline using equivariant minimal model program.

Applied the canonical bundle formula to confirm Calabi-Yau type.

Abstract

Let $X$ be a normal projective variety admitting a polarized endomorphism $f$, i.e., $f^*H\sim qH$ for some ample divisor $H$ and integer $q>1$. It was conjectured by Broustet and Gongyo that $X$ is of Calabi-Yau type, i.e., $(X,\Delta)$ is lc for some effective $\mathbb{Q}$-divisor such that $K_X+\Delta\sim_{\mathbb{Q}} 0$. In this paper, we establish a general guideline based on the equivariant minimal model program and the canonical bundle formula. In this way, we prove the conjecture when $X$ is a smooth projective threefold.