We study singular Kähler-Einstein metrics that are obtained as non-collapsed limits of polarized Kähler-Einstein manifolds. Our main result is that if the metric tangent cone at a point is locally isomorphic to the germ of the singularity, then the metric converges to the metric on its tangent cone at a polynomial rate on the level of Kähler potentials. When the tangent cone at the point has a smooth cross section, then the result implies polynomial convergence of the metric in the usual sense, generalizing a result due to Hein-Sun. We show that a similar result holds even in certain cases where the tangent cone is not locally isomorphic to the germ of the singularity. Finally we prove a rigidity result for complete ∂ ∂-exact Calabi-Yau metrics with maximal volume growth. This generalizes a result of Conlon-Hein, which applies to the case of asymptotically conical manifolds.Theorem 1.2. Suppose that, as above, (Z, p) is the pointed Gromov-Hausdorff limit of a non-collapsing sequence of polarized Kähler-Einstein manifolds, with singular Kähler-Einstein metric ω Z . Suppose that the germ (Z, p) is isomorphic to the germ (A p−1 , 0) at the origin. Then for some r 0 > 0 there is a biholomorphism φ : B(0, r 0 ) → U ⊂ Z, with φ(0) = p, and constants Λ, C, α > 0, such thatfor some u r defined on B(0, r), and sup B(0,r) |u r | ≤ Cr 2+α
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