Ricci Curvature on the Blow-up of Cp2 at Two Points

It is well-known that the αG(M)-invariant introduced by Tian plays an important role in the study of the existence of Kähler-Einstein metrics on complex manifolds with positive first Chern class ([T1], [T2], [TY]). Based on the estimate of αG(M)-invariant, Tian in 1990 proved that any complex surface with c1(M) > 0 always admits a Kähler-Einstein metric except in two cases CP2#1CP2 and CP2#2CP2, i.e., the blow-ups of CP2 at one point and two points respectively ([T2]). Instead of Kähler-Einstein metric, Koiso constructed a Kähler-Ricci soliton on CP2#1CP2 ([Ko]). But it is still unknown that there is a Kähler-Ricci soliton on CP2#2CP2 or not. Recently, the author studied a sufficient condition for the existence of Kähler-Ricci soliton on a complex manifold with c1(M) > 0 in the sense of Tian’s αG(M)invariant ([Zh]). In this note, we compute the Tian’s αG(M)-invariant on CP2#2CP2 and wish that our estimate was an important step towards finding the Kähler-Ricci soliton on CP2#2CP2. Kähler-Ricci soliton can be regarded as a good replacement when a Kähler manifold with c1(M) > 0 doesn’t admit a Kähler-Einstein metric ([Ca], [Ha]). The uniqueness problem of such metrics was solved by Tian and the author recently ([TZ1], [TZ2], [TZ3]). Our result is also an improvement of Abdesselem’s result ([Ab]). As a consequence, we obtain a good estimate of Ricci curvature on CP2#2CP2 by studying certain complex Monge-Ampère equation.