Degeneration of Kähler-Ricci solitons

Let $(Y, d)$ be a Gromov-Hausdorff limit of $n$-dimensional closed shrinking Kähler-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 Kähler-Ricci soliton equation. A similar convergence result for Kähler-Ricci flow of positive first Chern class is also obtained.

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