Characterizations of toric varieties via polarized endomorphisms
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Abstract
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 KX+ 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.
Keywords
Science & Technology, Physical Sciences, Mathematics, Polarized endomorphism, Toric pair, Complexity
Source Title
MATHEMATISCHE ZEITSCHRIFT
Publisher
SPRINGER HEIDELBERG
Series/Report No.
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Date
2019-08-01
DOI
10.1007/s00209-018-2160-8
Type
Article