D G ] 8 M ay 2 01 8 Compactness of Kähler-Ricci solitons on Fano manifolds 1

In this short paper, we improve the result of Phong-Song-Sturm on degeneration of Fano Kähler-Ricci solitons by removing the assumption on the uniform bound of the Futaki invariant. Let KR(n) be the space of Kähler-Ricci solitons on n-dimensional Fano manifolds. We show that after passing to a subsequence, any sequence in KR(n) converge in the Gromov-Hausdorff topology to a Kähler-Ricci soliton on an n-dimensional Q-Fano variety with log terminal singularities.