The compactness result for K\"ahler Ricci solitons

Abstract

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 $L^{n/2}$ norms, whose Ricci curvatures are uniformly bounded from below and $\mu(g_i,1/2) \ge A$ (where $\mu$ is Perelman's functional), there is a subsequence $(M_i,g_i)$ converging to a compact orbifold $(M_{\infty},g_{\infty})$ with finitely many isolated singularitites, where $g_{\infty}$ 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).