The compactness result for K\"ahler Ricci solitons

In this paper we prove the compactness result for compact K"ahler Ricci gradient shrinking solitons. If $(M_i,g_i)$ is a sequence of K"ahler Ricci solitons of real dimension $n\ge 4$, whose curvatures have uniformly bounded $L^{n\mathrm{/}2}$ norms, whose Ricci curvatures are uniformly bounded from below and $\mu (g_i,1\mathrm{/}2)\ge A$ (where $\mu$ is Perelman’s functional), there is a subsequence $(M_i,g_i)$ converging to a compact orbifold $(M_{\mathrm{\infty }},g_{\mathrm{\infty }})$ with finitely many isolated singularitites, where $g_{\mathrm{\infty }}$ is a K"ahler 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 K"ahler Ricci soliton equation in a lifting around singular points).