Abstract:

Theory on quasi-regular Sasaki-Einstein metrics has progressed  along with research works on Kähler-Einstein metrics on Fano varieties. Indeed, since the alpha-invariant method was adapted for Fano orbifolds by Demailly and Kollär, numerous Kähler-Einstein Fano orbifolds have been detected, in particular, in dimension 2. Such Kähler-Einstein Fano orbifolds  could yield many examples of Sasaki-Einstein manifolds using the method introduced by Kobayashi and fully matured by Boyer, Galicki and Kollär. Since the seminal works of Chen-Donaldson-Sun and Tian, we have been strongly reinforced by new technologies for detecting Kähler-Einstein Fano orbifolds, in particular, the delta-invariant method, so it would be natural to expect that many hidden Sasaki-Einstein manifolds can be detected by the new methods. In this talk we apply the delta-invariant method to certain hypersurfaces in 3-dimensional weighted projective spaces and we complete the classification of simply connected Sasaki-Einstein rational homology 5-spheres, which was partially done by Boyer, Galicki and Kollär before.