Characterizations of toric varieties via polarized endomorphisms

Let X be a normal projective variety and f: X→ X a non-isomorphic polarized endomorphism. We give two characterizations for X to be a toric variety. First we show that if X is 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 \ D} is quasi-étale and K_X+ D is Q-Cartier, then X admits a quasi-étale cover X~ such that X~ is a toric variety and f lifts to X~. In particular, if X is further assumed to be smooth, then X is a toric variety.