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

## TL;DR

This paper studies the structure of invariant divisors under int-amplified endomorphisms of normal projective varieties, establishing bounds, conditions for rational connectivity, and implications of the minimal model program.

## Contribution

It extends results on invariant divisors from polarized to int-amplified endomorphisms and analyzes their geometric properties and classifications.

## Key findings

- Number of invariant prime divisors is bounded by dimension plus Picard number.
- Provides conditions for the variety to be rationally connected and simply connected.
- Shows the minimal model program leads to an elliptic curve or a point under certain conditions.

## 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.