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There has been increasing interest lately in compactness theorems of Riemannian manifolds under various geometric assumptions (see, among others, [3], [10], [1], [7], and [19]). More recently, it has been found that the boundedness condition on the curvature as in [3] and [10] can be replaced by some integral norms of the curvature tensor. One of those often used is the L^-norm on the curvature tensor, where n is the real dimension of the underlying manifold. For instance, in [1] and [19], the authors show that if {(Mi, gt)} is a sequence of Einstein manifolds of real dimension 2/i satisfying: (i) d i a m ^ . , g() < μ; (ii) / M WRm^g^dV^ < μ; and (iii) Vol(M., g.) > ~ , where μ is a uniform constant, then the subsequence of {(Mi, gz)} converges to an Einstein orbifold with finitely many isolated singular points. Also see [20] for the case of Kahler-Einstein surfaces. The case that the limit is an orbifold does occur in dimension four (cf. [15], [20]). However, in this paper, we show that it cannot occur for Kahler-Einstein manifolds of higher dimension and nonzero scalar curvature. In order to give our main theorem precisely, we need to introduce some notation first. For any fixed constant μ > 0 and positive integer n > 0, denote by K(μ, n) the set of all Kahler-Einstein manifolds (M, g) of complex dimension n satisfying:

Compactness theorems for Kähler-Einstein manifolds of dimension 3 and up