On the \(\alpha \)-invariants of cubic surfaces with Eckardt points (Q2637911)

An effective way to prove the existence of Kähler-Einstein metrics is to use Tian's \(\alpha\)-invariants. From the definitions of \(\alpha\), \(\alpha_m\) and \(\alpha_{m,2}\) one can see that these invariants are not easy to be computed. The main theorem here states that whenever \(X\) is a smooth cubic surface with Eckardt points then for any integer \(m>0\) it holds \(\alpha_{m,2}(X)>\frac23\). In this work new and simplified proofs of some known results are given. The author discusses basic properties of Tian's invariants and he computes the \(\alpha\)-invariant for cubic surfaces with Eckardt points.