On the \alpha-Invariants of Cubic Surfaces with Eckardt Points

TL;DR

This paper proves that smooth cubic surfaces with Eckardt points have an -invariant greater than 2/3, simplifying the proof of Ke4hler-Einstein metrics existence, and discusses related computations and challenges.

Contribution

It establishes a lower bound for the -invariant of cubic surfaces with Eckardt points, aiding in the proof of Ke4hler-Einstein metrics.

Findings

-invariant > 2/3 for cubic surfaces with Eckardt points

Simplifies Tian's proof of Ke4hler-Einstein metrics

Discusses computations and difficulties for orbifold cases

Abstract

In this paper, we show that the \alpha_{m,2}-invariant of a smooth cubic surface with Eckardt points is strictly bigger than 2/3. This can be used to simplify Tian's original proof of the existence of Kaehler-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 Kaehler-Einstein metrics.