Degeneration of K\"ahler-Ricci solitons

Abstract

Let $(Y, d)$ be a Gromov-Hausdorff limit of $n$-dimensional closed shrinking K\"ahler-Ricci solitons with uniformly bounded volumes and Futaki invariants. We prove that off a closed subset of codimension at least 4, Y is a smooth manifold satisfying a shrinking K\"ahler-Ricci soliton equation. A similar convergence result for K\"ahler-Ricci flow of positive first Chern class is also obtained.