Log Calabi-Yau Structure of Projective Threefolds Admitting Polarized Endomorphisms

Let X be a normal projective variety admitting a polarized endomorphism f, that is, f *H ∼ 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, Δ) is lc for some effective Q-divisor such that K_X + Δ ∼ _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.