# Characterizations of Toric Varieties via Polarized Endomorphisms

## TL;DR

This paper provides new characterizations of toric varieties using polarized endomorphisms, focusing on geometric and group action conditions, and shows under certain conditions, such varieties admit covers that are toric.

## Contribution

It introduces two novel criteria for identifying toric varieties based on polarized endomorphisms and geometric properties, extending previous characterizations.

## Key findings

- If $X$ is $Q$-factorial and $G$-almost homogeneous with $f$ $G$-equivariant, then $X$ is toric.
- Under certain geometric conditions, $X$ admits a cover that is toric, and $f$ lifts to this cover.
- Smoothness assumptions lead to $X$ itself being toric.

## Abstract

Let $X$ be a normal projective variety and $f:X\to X$ a non-isomorphic polarized endomorphism. We give two characterizations for $X$ to be a toric variety. First we show that if $X$ is $\mathbb{Q}$-factorial and $G$-almost homogeneous for some linear algebraic group $G$ such that $f$ is $G$-equivariant, then $X$ is a toric variety. Next we give a geometric characterization: if $X$ is of Fano type and smooth in codimension 2 and if there is an $f^{-1}$-invariant reduced divisor $D$ such that $f|_{X\backslash D}$ is quasi-\'etale and $K_X+D$ is $\mathbb{Q}$-Cartier, then $X$ admits a quasi-\'etale cover $\widetilde{X}$ such that $\widetilde{X}$ is a toric variety and $f$ lifts to $\widetilde{X}$. In particular, if $X$ is further assumed to be smooth, then $X$ is a toric variety.