Characterizations of toric varieties via polarized endomorphisms

  • Sheng MengEmail author
  • De-Qi Zhang


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.


Polarized endomorphism Toric pair Complexity 

Mathematics Subject Classification

14M25 32H50 20K30 08A35 



The second author thanks Mircea Mustata for valuable discussions and warm hospitality during his visit to Univ. of Michigan in December 2016; he is also supported by an ARF of National University of Singapore. The authors thank the referee for suggestions to improve the paper.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsNational University of SingaporeSingaporeSingapore

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