The Α-invariant on Cp2#2cp2

The global holomorphic invariant αG(M) introduced by Tian[6], Tian and Yau[5] is closely related to the existence of Kähler-Einstein metrics. In his solution of the Calabi conjecture, Yau[11] proved the existence of a KählerEinstein 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 are known obstructions such as the Futaki invariant. For a compact Kähler manifold M with positive Chern class, Tian[6] 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[8]. It would be also interesting to find the estimate of the α invariant for CP #1CP 2 and CP #2CP 2. In this paper, we apply the Tian-Yau-Zelditch expansion of the Bergman kernel on polarized Kähler metrics to approximate plurisubharmonic functions and compute the α-invariant of CP #2CP 2. This gives an improvement of Abdesselem’s result[1]. More precisely, we shall show that: