Characterizations of toric varieties via polarized endomorphisms

Abstract

Let X be a normal projective variety and $f:X\rightarrow X$ 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-étale and $K_X+D$ is $\mathbb {Q}$-Cartier, then X admits a quasi-étale 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.