Regularity for singular Kahler-Einstein metrics

Singular Kahler-Einstein metrics arise naturally when studying limits of sequence of smooth Kahler-Einstein manifolds. Through the work of Donaldson-Sun and Li-Wang-Xu we know that the tangent cones of such singular KE metrics are determined by the underlying complex variety, however it is important to have more refined geometric information. For certain classes of isolated singularities, such as ordinary double points, Hein-Sun provided a precise asymptotic description. I will discuss results extending this work to singularities with more general tangent cones, including those with non-isolated singular sets, as well as ”unstable” examples where the tangent cone is not locally biholomorphic to the original complex variety. This is joint work with Shih-Kai Chiu.