The compactness result for Kähler Ricci solitons

Abstract

In this paper we prove the compactness result for compact Kähler Ricci gradient shrinking solitons. If $(M_i,g_i)$ is a sequence of Kähler 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 Kähler 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 Kähler Ricci soliton equation in a lifting around singular points).