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

Let X be a normal projective variety admitting a polarized endomorphism f, i.e., f^*H~ 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., $(X,Δ)$ is lc for some effective Q-divisor $Δ$ and KX+Δ~Q0. 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,Δ+R_Δ/q-1) when $(X,Δ)$ is an f-pair, i.e., KX+Δ=f^*(KX+Δ)+R_Δ with Δ, R_Δ being effective. In particular, we show: (1) KX + Rf/q-1 is Q-Cartier and numerically trivial when X is a Q-factorial (or of klt type) $3$-fold; (2) (X, Rfq-1) is log Calabi-Yau when X is a surface with the Picard number $ρ(X)>1$ or f⁻ˢ(P)=P for some prime divisor P and $s>0$.