Abstract: Let (X,L) be a polarized Calabi Yau variety (or canonical polarized variety) with crepant singularity. Suppose omega_KE in c_1(L) (or omega_KE in c_1(KX)) is the unique Ricci flat current (or Kähler Einstein current with negative scalar curvature) with local bounded potential constructed in [18], we show that the local tangent at any point p in X of metric omega_KE is unique.