On the α-invariants of cubic surfaces with Eckardt points

In this paper, we show that the αm,2-invariant (introduced by Tian (1991) [27], and (1997) [29]) of a smooth cubic surface with Eckardt points is strictly bigger than . This can be used to simplify Tian's original proof of the existence of Kähler–Einstein metrics on such manifolds. We also sketch the computations on cubic surfaces with one ordinary double points, and outline the analytic difficulties to prove the existence of orbifold Kähler–Einstein metrics.