Sasaki-Einstein metrics on simply connected rational homology 5-spheres

By developing the method introduced by Kobayashi in 1960's, Boyer, Galicki and Kollár found many examples of simply connected Sasaki- Einstein 5-manifolds. For such examples they veried existence of orbifold Kähler-Einstein metrics on various log del Pezzo surfaces. Their verification is essentially based on the $\alpha$-invariant method developed by Tian, Nadel, Demailly and Kollár. Using the $\delta$-invariant instead of the $\alpha$-invariant on the leaf spaces of the Reeb foliations of Sasakian 5-manifolds, I complete the classication of simply connected rational homology 5-spheres that admits Sasaki-Einstein metrics. This is a joint work with Joonyeong Won.