The Α-invariant on Toric Fano Manifolds

The global holomorphic invariant αG(X) introduced by Tian [?], Tian and Yau [?] is closely related to the existence of Kähler-Einstein metrics. In his solution to the Calabi conjecture, Yau [?] proved the existence of a Kähler-Einstein metric on compact Kähler manifolds with nonpositive first Chern class. Kähler-Einstein metrics do not always exist in the case when the first Chern class is positive, for there exist known obstructions such as the Futaki invariant. For a compact Kähler manifold X with positive first Chern class, Tian [?] proved that X admits a Kähler-Einstein metric if αG(X) > n n+1 , where n = dimX . In the case of compact complex surfaces, he proved that any compact complex surface with positive first Chern class admits a Kähler-Einstein metric except CP#1CP(CP blown up at one point) and CP#2CP(CP blown up at two points) [?].