Log Calabi–Yau Structure of Projective Threefolds Admitting Polarized Endomorphisms

Abstract

Abstract Let $X$ be a normal projective variety admitting a polarized endomorphism $f$, that is, $f^*H\sim qH$ for some ample divisor $H$ and integer $q>1$. It was conjectured by Broustet and Gongyo that $X$ is of Calabi–Yau type, that is, $(X,\Delta )$ 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.