Varieties Admitting Polarized Galois Endomorphisms

Let $X$ be an $n$-dimensional smooth complex projective variety admitting an int-amplified Galois endomorphism. We prove that its maximal rationally connected fibration is represented by a smooth toric fibration over a smooth $Q$-abelian variety. Consequently, if $X$ is rationally connected, then it is toric. If it also has Picard number one, then $X\cong {\mathbf{P}}^n$. As another application, we prove that such $X$ is of log Calabi-Yau type as conjectured by Gongyo.