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Ric(ω)= λω

for λ 1, 0, 1 ; if λ = 0, we further assume KX is holomorphically trivial;∈ { − } (B) Uniform non-collapsing condition:

Vol(B(q, r)) κr2n (1.3) ≥ for all q X and r (0, 1]. ∈ ∈ (C) Uniform volume bound: Vol(X,ω) V. (1.4) ≤ By the Bishop-Gromov volume comparison theorem, (B) and (C) together are equivalent to a uniform diameter bound on X, and the latter is indeed a consequence of the Einstein condition when λ = 1. It is proved in [16] that the (polarized) Gromov-Hausdorff limit of a sequence of spaces in 1(n,κ,V ) is naturally a normal projective variety. Theorem 1.1 is an extensionK of this result. Our main interest in this paper is on rescaled limits. For this purpose we let (n,κ,V ) be the set of polarized K¨ahler manifolds of the form (X,La,aω,p) K for some (X,L,ω,p) 1(n,κ,V ) and a 1. Clearly (n,κ,V ) is a sub- set of (n,κ) so Theorem∈ K 1.1 applies to Gromov-Hausdorff≥ K limits of spaces in (n,κ,VK ). Let (Z,p) be such a Gromov-Hausdorff limit. We consider the fam- ilyK of spaces given by rescaling (Z,p) by a factor √a for a positive integer a. Let a , by passing to a subsequence we obtain limit spaces, called the tan- gent cones→ ∞at p. These can themselves be viewed as Gromov-Hausdorff limits of elements in (n,κ,V ), so by Theorem 1.1 they are naturally