Log Calabi-Yau structure of algebaic varieties admitting a polarized endomorphism

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 $q>1$. Then Broustet and Gongyo proposed the conjecture that $X$ is of Calabi-Yau type (CY for short), i.e., $(X,\Delta )$ is lc for some effective $\mathbb{Q}$-divisor $\Delta$ and $K_X+\Delta {\sim }_{\mathbb{Q}}0$. We prove the conjecture when $X$ is a Gorenstein terminal 3-fold, extending the result of Sheng Meng for smooth threefolds. We then study the singularity type and CY property for $(X,\Delta +\frac{R_{\Delta }}{q-1})$ when $(X,\Delta )$ is an $f$-pair, i.e., $K_X+\Delta =f^{\ast }(K_X+\Delta )+R_{\Delta }$ with $\Delta ,R_{\Delta }$ being effective. In particular, we show: (1) $K_X+\frac{R_f}{q-1}$ is $\mathbb{Q}$-Cartier and numerically trivial when $X$ is a $\mathbb{Q}$-factorial (or of klt type) $3$-fold; (2) $(X,\frac{R_f}{q-1})$ is log Calabi-Yau when $X$ is a surface with the Picard number $\rho (X)>1$ or $f^{-s}(P)=P$ for some prime divisor $P$ and $s>0$.

28 pages; comments are welcome!