The Α-invariants on Toric Fano Manifolds

The global holomorphic invariant αG(M) introduced by Tian[14], Tian and Yau[13] is closely related to the existence of Kähler-Einstein metrics. In his solution to the Calabi conjecture, Yau[19] 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 M with positive first Chern class, Tian[14] proved that M admits a Kähler-Einstein metric if αG(M) > n n+1 , where n = dimM . 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 2 and CP #2CP 2[16].