A compactness result for Kähler Ricci solitons

In this paper we prove a compactness result for compact Kahler Ricci gradient shrinking solitons. If ( M i , g i ) is a sequence of Kahler Ricci solitons of real dimension n ⩾ 4 , whose curvatures have uniformly bounded L n / 2 norms, whose Ricci curvatures are uniformly bounded from below and μ ( g i , 1 / 2 ) ⩾ A (where μ is Perelman's functional), there is a subsequence ( M i , g i ) converging to a compact orbifold ( M ∞ , g ∞ ) with finitely many isolated singularities, where g ∞ is a Kahler Ricci soliton metric in an orbifold sense (satisfies a soliton equation away from singular points and smoothly extends in some gauge to a metric satisfying Kahler Ricci soliton equation in a lifting around singular points).