Examples of Kähler–Einstein toric Fano manifolds associated to non-symmetric reflexive polytopes

Original Paper


In this note we report on examples of 7- and 8-dimensional toric Fano manifolds whose associated reflexive polytopes are not symmetric, but they still admit a Kähler–Einstein metric. This answers a question first posed by Batyrev and Selivanova. The examples were found in the classification of ≤8-dimensional toric Fano manifolds obtained by Øbro. We also discuss related open questions and conjectures. In particular, we notice that the alpha-invariants of these examples do not satisfy the assumptions of Tian’s theorem.


Fano varieties Kähler–Einstein manifolds Lattice polytopes Toric varieties 

Mathematics Subject Classification (2000)

Primary 14M25 32Q20 Secondary 14J45 52B20 53C25 


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© The Managing Editors 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of GeorgiaAthensUSA
  2. 2.Fachbereich Mathematik, TU DarmstadtDarmstadtGermany

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