Let X be a complex toric Fano n-fold and N (T ) the normalizer of a maximal torus T in the group of biholomorphic authomorphisms Aut(X). We call X symmetric if the trivial character is a single N (T )-invariant algebraic character of T . Using an invariant α G (X) introduced by Tian, we show that all symmetric toric Fano n-folds admit an Einstein-Kähler metric. We remark that so far one doesn't know any example of a toric Fano n-fold X such that Aut(X) is reductive, the Futaki character of X vanishes, but X is not symmetric.
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