Log Calabi-Yau structure of projective threefolds admitting polarized endomorphisms

Abstract

Let $X$ be a normal projective variety admitting a polarized endomorphism $f$, i.e., $f^{\ast }H\sim qH$ 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.

arXiv commentThe proof of Theorem 2.6 has been revised to fix the incomplete reasoning in its earlier version