Totally invariant divisors of int-amplified endomorphisms of normal projective varieties

Abstract

We consider an arbitrary int-amplified surjective endomorphism $f$ of a normal projective variety $X$ over $\mathbb{C}$ and its $f^{-1}$-stable prime divisors. We extend the early result for the case of polarized endomorphisms to the case of int-amplified endomorphisms. Assume further that $X$ has at worst Kawamata log terminal singularities. We prove that the total number of $f^{-1}$-stable prime divisors has an optimal upper bound $\dim X+\rho(X)$, where $\rho(X)$ is the Picard number. Also, we give a sufficient condition for $X$ to be rationally connected and simply connected. Finally, by running the minimal model program (MMP), we prove that, under some extra conditions, the end product of the MMP can only be an elliptic curve or a single point.