Uniqueness of tangent cone of Kähler-Einstein metrics on singular varieties with crepant singularities

Let ( X , L ) be a polarized Calabi-Yau variety (or canonical polarized variety) with crepant singularities. Suppose ω _KE∈ c_1(L) (or ω _KE∈ c_1(K_X)) is the unique Ricci flat current (or Käher-Einstein current with negative scalar curvature) with local bounded potential constructed in (Eyssidieux in J Am Math Soc 22: 607-639, 2009), we show that the local tangent at any point p∈ X of metric ω _KE is unique.